University of Colorado, BoulderCU Scholar

Physics Graduate Theses & Dissertations Physics

Spring 1-1-2017

Nanoscale and Ultrafast Imaging and Spectroscopyto Probe Heterogeneity of Novel Materials andCoherence of Thermal Near-FieldsBrian Thomas O'CallahanUniversity of Colorado at Boulder, btocallahan@gmail.com

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Recommended CitationO'Callahan, Brian Thomas, "Nanoscale and Ultrafast Imaging and Spectroscopy to Probe Heterogeneity of Novel Materials andCoherence of Thermal Near-Fields" (2017). Physics Graduate Theses & Dissertations. 209.https://scholar.colorado.edu/phys_gradetds/209

Nanoscale and ultrafast imaging and spectroscopy to probe

heterogeneity of novel materials and coherence of thermal

near-fields

by

Brian T. O’Callahan

B.S., University of Washington, 2011

M.S., University of Colorado, Boulder, 2015

A thesis submitted to the

Faculty of the Graduate School of the

University of Colorado in partial fulfillment

of the requirements for the degree of

Doctor of Philosophy

Department of Physics

2017

This thesis entitled:Nanoscale and ultrafast imaging and spectroscopy to probe heterogeneity of novel

materials and coherence of thermal near-fieldswritten by Brian T. O’Callahan

has been approved for the Department of Physics

Prof. Markus Raschke

Prof. Dmitry Reznik

Date

The final copy of this thesis has been examined by the signatories, and we find that boththe content and the form meet acceptable presentation standards of scholarly work in the

above mentioned discipline.

O’Callahan, Brian T. (Ph.D., Physics)

Nanoscale and ultrafast imaging and spectroscopy to probe heterogeneity of novel materials

and coherence of thermal near-fields

Thesis directed by Prof. Markus Raschke

Novel optical phenomena emerge on nanometer length scales which determine the

macroscopic material response. By bringing a sharp AFM tip close to a surface and il-

luminating with either a laser, a broadband light source, or the intrinsic thermal fields of the

material itself, we can probe near-field optical properties with spatial resolution only limited

by the apex radius of the tip. These properties include the spectral, spatial, and coherence

properties of the thermal near-fields that emerge at sub-wavelength distances from any mat-

ter at non-zero temperature that affect thermal emission and nanoscale heat transfer. I also

apply near-field imaging to correlated electron and 2D materials where nanoscale properties

strongly influence the bulk properties. The nanoscale heterogeneity of the metal-insulator

transition of vanadium dioxide and the strong light confinement provided by polaritons of

boron nitride and graphene may lead to novel electronic or photonic devices, however appli-

cation of these materials requires additional insight and understanding. Nano-imaging and

-spectroscopy can provide the needed spatial resolution and specificity to interrogate these

phenomenon on their natural length scales.

Acknowledgements

This work would not have been possible without the tremendous support from my

family, friends, and coworkers. I would like to thank all who helped me through the setbacks

and celebrated the successes with me.

I am very thankful to have been a part of a collaborate and supportive research group.

I am indebted those past group members who taught me so much of what I know now,

including Joanna Atkin, Andy Jones, Sam Berweger, and Honghua Yang. Their seemingly

endless patience toward my constant questioning played no small part toward my success as

a graduate student. I would also like to thank the support of fellow graduate students and

postdocs Benjamin Pollard, Vasily Kravtsov, Omar Khatib, and Eric Muller, whom have

help me along the research process every step of the way. In particular, I’d like to thank Jun

Yan for his enthusiasm and motivation he brought into the lab. Finally, I am thankful for

the guidance provided by my advisor, Markus, to ensure the success of our research and my

education.

I am also thankful to have worked with many great collaborators through the PhD

process. Graphene samples were provided by the labs of Thomas Schibli (CU Boulder) and

Jun Zhu (Pennsylvania State University), VO2 microrods were grown by the Cobden group

(University of Washington), and the graphene-hBN heterostructure was provided by Ingrid

Barcelos and Alisson Cadore (Federal University of Minas Gerais).

This process toward my PhD would not have nearly as enjoyable if it weren’t for the

incredible group of friends that I met in the department. From commiserating over endless

v

Jackson assignments to lunch crew, the sense of community was amazing and kept me going

through the good and bad times.

Finally, I would like to thank my family for their love and support through this process.

I can’t imagine completing this with out them.

Contents

Chapter

1 Introduction 1

2 An introduction to optical properties of materials 5

2.1 Dielectric function models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Lorentzian model for phonon resonances . . . . . . . . . . . . . . . . 7

2.1.2 Drude model for conductivity . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Surface polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 scattering Scanning Near-field Optical Microscopy 11

3.1 Diffraction limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 scattering Scanning Near-field Optical Microscopy . . . . . . . . . . . . . . . 13

3.2.1 s-SNOM models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2.2 Sources for s-SNOM . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2.3 Noise sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Thermal near-field spectroscopy and optical forces 38

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Historical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vii

4.3 Thermal infrared near-field spectroscopy . . . . . . . . . . . . . . . . . . . . 43

4.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4.1 Laser heating of scanning probe tips for thermal near-field spectroscopy

and imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.2 Spectral frustration in TINS . . . . . . . . . . . . . . . . . . . . . . . 53

4.5 Spectral frustration Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.7 Optical forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.7.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.7.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.7.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.8 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Inhomogeneity of the insulator to metal transition of vanadium dioxide studied by

ultrafast microscopy and infrared nano-imaging 84

5.0.1 Ultrafast IMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.0.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.0.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.0.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.0.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.0.6 Control of IMT nano-domains . . . . . . . . . . . . . . . . . . . . . . 109

5.1 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Nano-spectroscopy and imaging of polariton propagation in 2D materials 112

6.1 Hexagonal boron nitride . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.1.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

viii

6.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.1.3 Dispersion calculations . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.1.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.2 Imaging of Graphene SPPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.2.1 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.2.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.2.3 Conclusion and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

7 Conclusion 127

8 Appendix A: Crystallinity in metal-porphyrines 129

Bibliography 133

Tables

Table

5.1 Fit parameters and Raman peak positions of all crystals shown in Fig. 5.4 . 102

5.2 Table of fit parameters shown in Figure 5.3c. . . . . . . . . . . . . . . . . . . 102

6.1 Parameters used for Lorentizan model for hBN dielectric function . . . . . . 117

Figures

Figure

1.1 Diffraction limit of conventional optical spectroscopy . . . . . . . . . . . . . 3

2.1 Interface between two semi-infinite half-spaces . . . . . . . . . . . . . . . . . 8

3.1 Diffraction limit concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 s-SNOM concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 s-SNOM concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 s-SNOM concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 s-SNOM using a blackbody source . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Blackbody source approach curves . . . . . . . . . . . . . . . . . . . . . . . . 24

3.7 hBN spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.8 hBN dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.9 Blackbody source beamspot measurement . . . . . . . . . . . . . . . . . . . 28

3.10 SiO2 and PTFE spectra using blackbody radiation . . . . . . . . . . . . . . . 31

3.11 Signal-to-noise ratio comparison of different sources and samples . . . . . . . 32

3.12 Signal-to-noise dependence on irradiance and acquisition time . . . . . . . . 36

4.1 EM-LDOS phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 TINS setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 TINS setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.4 Laser heating TINS schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 52

xi

4.5 Heating characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.6 hBN TINS spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.7 TINS SiC spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.8 TINS distance dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.9 Effective medium theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.10 TINS spectra of PTFE and SiC using laser heating . . . . . . . . . . . . . . 66

4.11 Experimental setup of photo-induced force measurement . . . . . . . . . . . 70

4.12 Photo-induced force spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.13 Force dipole model schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.14 Spatial variation of force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.15 Force dependence on PMMA thickness . . . . . . . . . . . . . . . . . . . . . 77

4.16 Approach curves of optical signal and force strength . . . . . . . . . . . . . . 77

4.17 Force and s-SNOM image of PMMA coated antenna . . . . . . . . . . . . . . 79

4.18 Fine details in PiFM map on PS-b-PMMA . . . . . . . . . . . . . . . . . . . 81

5.1 Degenerate far-field pump-probe setup . . . . . . . . . . . . . . . . . . . . . 86

5.2 Raman characterization of microrods . . . . . . . . . . . . . . . . . . . . . . 88

5.3 Fluence dependence of dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.4 Transition time variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5.5 Thermal IMT inhomogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.6 Band structure schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.7 Thermal IMT hyteresis data on crystal #30 . . . . . . . . . . . . . . . . . . 104

5.8 Nanoscale heterogeneity of ultrafast IMT measured with s-SNOM . . . . . . 105

5.9 Variable domain pattern in the thermal IMT . . . . . . . . . . . . . . . . . . 107

5.10 Thermal IMT behavior of VO2 rod under complex strain . . . . . . . . . . . 108

5.11 Thermal IMT behavior controlled with laser-heating . . . . . . . . . . . . . . 110

5.12 Thermal IMT behavior upon heating . . . . . . . . . . . . . . . . . . . . . . 111

xii

6.1 Dielectric function of hBN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.2 SINS setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.3 Influence of substrate and graphene on the near-field spectra of hBN . . . . . 118

6.4 Spectrally resolved linescan of hBN across substrate interfaces . . . . . . . . 120

6.5 Graphene SPP interferometry . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.6 Graphene SPP interferometry of grain boundaries in CVD graphene . . . . . 124

6.7 Graphene SPP grain boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.1 Far-field FTIR on RuOEP/P3HT . . . . . . . . . . . . . . . . . . . . . . . . 129

8.2 Topography and near-field response of RuOEP crystallites . . . . . . . . . . 130

8.3 Nanoscale heterogeneity of RuOEP vibrational response . . . . . . . . . . . . 131

Chapter 1

Introduction

The macroscopic properties of everyday objects arise from the motions and mutual

interactions of their microscopic constituents such as electrons, ion cores, and photons.

However, these microscopic constituents are “blurred out” and are invisible to the naked

eye. Bulk material properties can all be traced back to this microscopic origin, and a full

understanding of macroscopic phenomenon requires a description of its microscopic physical

origin.

The importance of nanoscale phenomenon is evidence in Si-based electronics, where

nanoscale defects or dopants can trap or introduce carriers which dramatically alter the bulk

conductivity. The high sensitivity of Si to doping concentrations of just a few parts in a billion

of single atom defects demonstrates the dramatic role that nanoscale phenomenon play in

determination of bulk properties. Similar sensitivity in more complex, correlated-electron

materials is increasingly evident and in many cases limits their technological applicability.

Additionally, the recent rise in interest of low dimensional materials such as graphene and

layered heterostructures has lead to renewed focus on the quantum-mechanical phenomena

that dominate in these nanoscale confined materials. As electronic devices such as cell

phones and computers reach smaller sizes and faster performance, the ability to engineer

compact electronics depends on our ability control matter on short length scales. Even

thermal emission, which is ubiquitous in the natural world from sunlight to warmth from a

kitchen stove burner, originates from microscopic current densities within a material. Clearly,

2

understanding of many everyday phenomena requires knowledge of material properties on

nanometer length scales.

However, the precision and sensitivity required to probe matter on its natural length,

time, and frequency scales prohibit the use of many conventional microscopy techniques. The

natural scales for an electron in a typical material are determined by its mean free path of 10’s

of nm, its scattering times on the order of 10 fs and typical frequency response in the visible

spectrum (Fig. 1.1). Clearly, for a complete understanding, we need to simultaneous probe

the electronic and vibrational properties of matter on the femtosecond timescale, nanometer

length scale and THz to PHz optical frequency range.

A wide collection of techniques exist to probe these scales individually. Optical spec-

troscopy provides access to the frequency response of matter by the measuring the emission,

absorption, reflection, scattering, or transmission of light by a material. It provides sen-

sitivity to the fundamental excitations of matter with energies matched to electronic and

vibrational degrees of freedom in the 10 meV-1eV range. Additionally, ultrafast optical

pulses provide temporal resolution down to 10’s of fs, with the ability to probe slow ∼ps

lattice thermalization down to fast carrier relaxation. However, conventional optical tech-

niques have a fundamental limit on spatial resolution, which is given by the minimum spot

size of focused beam. In Gaussian optics, this minimum spot size w of a beam focused by

a half angle θ is w = λ/2n sin(θ), where n is the index of refraction of the medium. First

identified by Ernst Abbe in the late 19th century [1], this is known as the “diffraction limit”,

it is limited by the “numerical aperture” of the focusing optics NA = n sin(θ) (dashed black

line in Fig. 1.1). The beam waist may be reduced and the resolution improved by increasing

the focusing angle θ or by imaging through a medium with a higher index of refraction.

However, even the most powerful imaging techniques are limited to only a few times bet-

ter than the wavelength, and with infrared wavelengths on the order of λ ∼ 10µm, spatial

resolution of conventional optical techniques is very limited in the infrared. With typical

phonon polariton wavelength on the order of 1µm or less, this prohibits their direct spatial

3

imaging with diffraction-limited studies. Additionally, while a typical local vibrational mode

has energy in the infrared range, they are localized to a single bond length of < 1 nm. By

reducing the wavelength to the X-ray regime, structures may be imaged with high resolution.

However, with high photon energies may damage samples, and sensitivity to many material

properties is lost since the energies are no longer matched to the natural energy scales of

the material. Electron microscopy such as transmission electron microscopy and scanning

electron microscopy also achieves atomic-scale spatial resolution, but these techniques are

limited to vacuum environments and again provide little chemically specific information.

1 10 100 1000 10000

1014

1015

1016

Freq

uenc

y (H

z)

Spatial resolution (nm)

Diffraction limit

Optical spectroscopy

Phonons

Electrons

Molecularresonances

Figure 1.1: Outline of the relevant frequency and length scales of electrons, phonons, molec-ular resonances. Electonic excitations have typically higher energy and shorter mean freepaths and wavelengths than phonons. Localized molecular resonances span a broad fre-quency range and are localized down to single bond lengths. Optical spectroscopy in generalcannot access these length scales due to the diffraction limit. Figure after [79].

Scanning probe microscopy (SPM) can achieve high spatial resolution of a sample sur-

face. In SPM, a sharp tip is raster-scanned across the surface using various feedback signals

4

to record the height (e.g. the tunneling current for scanning tunneling microscopy). Various

SPM implementations are sensitive to material properties. Scanning tunneling spectroscopy

measures the local electronic density of states, magnetic force microscopy measures the mag-

netic polarization of a surface, and conductive atomic force microscopy images variations of

the conductivity. However, none of the above SPM techniques are directly sensitive to the

optical properties.

Other techniques exist to improve the spatial resolution of optical microscopy, such as

super-resolution fluorescence microscopy which relies on switchable fluorescence of molecules

[38]. Though this technique has seen success mapping biological structures with ∼ 10 nm res-

olution, it requires the use of fluorescent tracer molecules and is not specific to the materials’

intrinsic optical response.

Simultaneous realization of high spatial resolution, temporal resolution, and vibrational

specificity, can be achieved by combining SPM and optical spectroscopy techniques. In

scattering-scanning near-field microscopy (s-SNOM), light illuminates a sharp AFM tip in

close proximity to a sample. The tip then acts like a lightning rod, whereby the sharp apex

of the tip localized a strong electric field that probes the sample surface with resolution equal

to the apex radius of curvature. The resulting scattered light reflects the local properties

of samples surface and provides a spatial map as the tip scans the surface. s-SNOM is a

inherently broadband technique, is compatible with ultrafast pulses, and can be performed

in ambient conditions with few sample constraints.

Chapter 2

An introduction to optical properties of materials

Here, I present an introduction to several key optical properties which are discussed in

this thesis. First, I describe surface polaritons, which are surface confined electromagnetic

waves that propagate at the interface of a material. I also present two dielectric function

models: the Lorentzian model for vibrational material responses and the Drude model for

materials with free carriers.

Description of optical properties begin with Maxwell’s equations that describe electro-

magnetic fields and how they interact with matter. In the presence of a charge density ρ

and a current density ~J , the electric ~E and magnetic induction ~B fields are given by:

∇ · ~E = ρ/ε0 (2.1)

∇× ~E = −∂~B

∂t(2.2)

∇ · ~B = 0 (2.3)

∇× ~B = µ0

(~J + ε0

∂ ~E

∂t

). (2.4)

In vacuum, there is no charge density or current density (ρ, ~J = 0), we can separate the

magnetic induction field from the equation describing the electric field and obtain c2∇2 ~E =

∂2 ~E/∂t2. This equation is known as the electromagnetic wave equation and describes the

evolution of a wave traveling at the speed of light c = 1/√ε0µ0. Its solution is given by

E(t) = E0 exp(i(ωt− kx)), (2.5)

6

where E0 is the electric field amplitude, ω is the angular frequency, and k is the wavevector.

The wavevector of propagating light

k = kxx+ kyy + kz z (2.6)

has a magnitude confined to∣∣∣~k∣∣∣2 =

√k2x + k2

y + k2z = ω/c, where c is the speed of light.

In a linear dielectric, we introduce the dielectric functions (or relative dielectric func-

tions) ε and µ to describe the materials response. We also introduce the displacement field

~D = εε0 ~E and the magnetic field ~B = µµ0~H. The electromagnetic wave equation then

takes the form: c2∇2 ~E = ε∂2 ~E/∂t2, where µ is set to 1 for typical, non-magnetic materials.

This equation then describes the light-matter interaction for linear dielectrics through the

dielectric function ε.

2.1 Dielectric function models

Here, I will describe how we model the dielectric function of materials based on their

electronic or vibrational properties. The dielectric function describes the bulk, frequency-

dependent light-matter interaction within a material. In general, when we apply a DC

electric field on a dielectric, we expect the bound electrons to displace slightly from their

associated ion core to create a small dipole oriented parallel to the applied field, and the

internal field will increase. This describes a positive dielectric function. In contrast, we

know that a metal will screen any applied electric field to ensure its internal electric field is

zero. This describes a negative dielectric function, since the material creates a polarization

to oppose the applied field.

As the electric field is switched at faster and faster frequencies, the electron displace-

ment will start to lag behind the applied electric field, and the material polarization reduces

from the DC value. When the electric field begins oscillating, it of course can be consid-

ered an electromagnetic wave with a defined frequency. When this frequency approaches

a natural frequency such as a transverse optical phonon mode in a polar material, it can

7

start driving the phonon mode of the lattice. Just like a driven harmonic oscillator, when

the electromagnetic frequency sweeps over the phonon frequency, the electric polarization of

the lattice goes from in-phase to out-of-phase with the driving field. Thus the electric field

should be strengthened just below resonance, but the field should be reduced slightly above

resonance. The description of the dielectric function of a material near a phonon resonance

is thus well described by a damped, driven harmonic oscillator.

2.1.1 Lorentzian model for phonon resonances

The functional form for the response of a damped, driven harmonic oscillator is a

complex-valued Lorentzian. It takes the form

I(ω) = I01

ω20 − ω2 − iωΓ

. (2.7)

The lorentizan model for the dielectric function follows the same form, except it is offset

from zero by a constant ε∞ by residual electron conductivity or by nearby phonon modes.

Thus, in the vicinity of a phonon frequency ωTO with damping Γ, the dielectric function is

ε(ω) = ε∞

(1 +

ω2LO − ω2

TO

ω2TO − ω2 − iωΓ

). (2.8)

Here, the resonance occurs at ωTO, which is the transverse optical phonon mode of the

material. The other characteristic frequency is ωLO, which is the longitudinal optical phonon

mode. The effect of other phonon modes on the dielectric function is additive, so to account

for multiple phonon modes, additional Lorentzians may be easily added. For strong phonon

resonances, the dielectric function can dip below zero at frequencies between the LO and

the TO optical phonon modes. In this spectral region, known as the Reststrahlen band, the

material acts like a metal with high reflectivity due to the presence of SPhPs.

2.1.2 Drude model for conductivity

The Drude model for conductivity is surprisingly effective, given its simplistic formu-

lation. It describes the motion of electrons through a metal driven by an electric field E(t)

8

with classical kinetic theory:

m∗x+m∗Γx = −eE(t), (2.9)

where m∗ is the effective mass of the electron, and Γ is the damping rate. This is the

description of damped, driven motion with no restoring force, or equivalently a harmonic

oscillator with resonance frequency at zero. The dielectric function is then of a similar form

to above, except with ω0 = 0:

ε0(ω) = 1−ω2p

ω2 + IωΓ(2.10)

where ωp = Ne2/m∗ε0 is the plasma frequency with N , e, m∗ the electron number density,

charge, and effective mass. The Drude model well describes the dielectric function and

optical conductivity of most metals across the infrared [190]. From this analysis, we see that

an oscillating electric field can drive electrons, resulting in a fluctuating electron density.

This electron density fluctuation is called a plasmon.

e1, m

1

e2, m

2

z

E1, B

1

E2, B

2

Figure 2.1: Illustration of geometry and definition of variables used in Sec. 2.2.

2.2 Surface polaritons

Surface polaritons (SPs) are electromagnetic waves that travel at the interface between

two materials. By definition, they are surface confined and have imaginary out of plane

wavevectors. For simplicity, I narrow the discussion to a planar interface between two semi-

infinite half-spaces. The upper half space (z > 0) has dielectric function ε1 and the lower

half-space (z < 0) has dielectric function ε2. They are described by Maxwell’s equations and

9

the following boundary conditions:

~n12 × ( ~E2 − ~E1) = ~0 (2.11)

( ~D2 − ~D1) · ~n12 = ρs (2.12)

( ~B2 − ~B1) · ~n12 = 0 (2.13)

~n12 × ( ~H2 − ~H1) = ~js, (2.14)

where ~n12 is the vector normal to the interface pointing in the direction from z < 0 to z > 0,

and ρs,~js are the surface charge and surface current, respectively. I will also narrow the

discussion to transverse magnetic (TM) modes, since there is no transverse electric field for

nonmagnetic materials. The electric field can then be expressed as

~E1(~r, ω) = ~E1 exp(−i~k‖ · ~R + kz,1z) exp(iωt) (2.15)

~E2(~r, ω) = ~E2 exp(−i~k‖ · ~R− kz,2z) exp(iωt) (2.16)

where i = 1, 2 correspond to the upper and lower half spaces, respectively, ~Ei is the electric

field amplitude, R = xx + yy is the in-plane distance, and the wavevector ~k is split into its

component in the xy plane ~k‖ and its surface normal component kz so that |~k‖|2 +k2z,i = |~ki|2.

Using the electromagnetic boundary conditions, we can determine the vector pre-factors

~Ei = (0, Ey,i, Ez,i).

Continuity of the in-plane component of the electric field gives Ey,1 = Ey,2, and with no

free charges, the out-of-plane components of the displacement fields Di are also continuous,

yielding ε1Ez,1 = ε2Ez,2. Finally Gauss’ law requires ∇ · ~E = 0, and thus

k‖Ey,2 − kz,2Ez,2 = 0 (2.17)

k‖Ey,1 − kz,1Ez,1 = 0. (2.18)

From these expressions, the dispersion relation for surface polaritons can be derived:

k‖(ω) =ω

c

√ε1(ω)ε2(ω)

ε1(ω) + ε2(ω). (2.19)

10

From this expression, there is a notable divergence when ε1(ω) = −ε2(ω), at the surface

polariton resonance. Also, we see that for frequencies where ε1(ω)ε2(ω)ε1(ω)+ε2(ω)

< 0, the wavevector

is imaginary and surface polaritons are not permitted. So far, I have not specified the origin

of the surface polaritons. If they arise from coupling of photons to plasmons, they are termed

surface plasmon polaritons (SPPs). If the underlying material excitation is due to phonons,

they are called surface phonon polaritons (SPhPs).

2.3 Conclusion

In this chapter, I describe a few fundamental properties of the light-matter interaction.

I present the field structure and dispersion relation of surface polaritons. Additionally, I

outline some relevant models for the dielectric function of a material, which I use in this

thesis to model several key material responses.

Chapter 3

scattering Scanning Near-field Optical Microscopy

With photon energies matched to vibrational and electronic energies in materials, op-

tical spectroscopy provides sensitivity to many chemical properties. However, the minimum

focal spot of light using conventional optics is limited by diffraction. When a point emitter

is imaged through a collection optic of diameter d, the image forms an Airy disk diffraction

pattern. Akin to single slit Fraunhofer diffraction, spatially dispersed pattern is due to the

diffraction that occurs when light is spatially confined through the aperture of the focusing

optic. The radius of the Airy disk to its first minimum is given by rmin ' 0.61λ/NA, where

NA = n sin θ is the numerical aperture of the collection optics with focus half angle θ imaged

through a medium of index of refraction n. As a result, when two emitters are closer than

rmin, the resulting intensity pattern is not distinguishable from a single point emitter and

the two emitters are unresolved (Fig. 3.1a)). According to the Raleigh criteron, two point

emitters are “barely resolved” when the minimum of the Airy disk of one emitter overlaps the

the maximum of the other emitter’s Airy disk. This occurs when their separation d = rmin,

and thus rmin is the smallest length scale with which spatial features that can be resolved

in standard optical microscopy techniques. To improve this resolution limit, the numerical

aperture can be increased by using, e.g., oil-immersion techniques to increase the index of

refraction n, or by using higher collection angles θ. However, in order to probe many mate-

rials on their natural vibrational energy scales, infrared light with wavelengths on the order

of λ1− 10 µm must be used, which limits the spatial resolution to similar length scales. In

12

order to overcome the diffraction limit, we use a fundamentally different approach to optical

spectroscopy that accesses optical near-fields.

d < rmin

d = rmin

a) b)

Figure 3.1: When imaged through a conventional optic microscope, a point emitter appearsas an Airy disk of diameter determined by the focal conditions. a) When two emittersare separated by distances d smaller than the minimum resolution rmin determined by theRayleigh criterion, they cannot be distinquished. b) When the minimum of the Airy disk ofone emitter overlaps with the maximum of the other, the Raleigh criterion is satisfied. Wesay that the two emitters are “barely resolved”.

3.1 Diffraction limit

The wavevector determines the spatial variation in the optical field, so propagating

radiation is inherently limited to spatial variations on the order of ω/c = 2πλ. As the

spatial variation of light fundamentally determines the spatial resolution of conventional

optical microscopy, by this analysis, the smallest separation that two object can be resolved

is limited to the order of the wavelength.

In order to overcome the diffraction limit, we can appeal to the fundamental limit on

the wavenumber described above. If one of the wavevector components in Eq. 2.6 is allowed

to be imaginary, then the other components can exceed the far-field limit. The resulting

solutions are called evanescent waves, and can increase the spatial variation of the optical

fields, and therefore the improve spatial resolution. However, the imaginary component of

13

the wavevector results in a decaying exponential in the corresponding spatial dimension,

which is surface confined to sub-wavelength distances. In order to use evanescent waves for

high resolution microscopy, we need to gain optical access to sub-wavelength distances to

samples.

3.2 scattering Scanning Near-field Optical Microscopy

The first implementations of near-field microscopy used a small nanoscale aperture

brought within the evanescent field of a sample to capture these high-wavevector virtual

photons (see Fig. 3.2a)) [104, 157, 17]. Tapered optical fibers are metal-coated, leaving only

an aperture a few 10’s of nm in diameter at the end. Piezoelectric motors as used in scanning

tunneling microscopy bring these tips in close proximity of a sample surface and the sample

is illuminated by light sent through the fiber or through external illumination (Fig. 3.2b)).

Near field light that enters the fiber through the aperture is measured as the tip scans the

surface to create a spatial image.

This technique, known now as near-field scanning optical microscopy (NSOM), was used

to obtain < 20nm spatial resolution of nano-pattered surfaces [17]. While this technique

provided high spatial resolution determined by the aperture size, it has a inherent flaw.

To increase spatial resolution, the aperture size must decrease, which in turn decreases

the optical throughput. This trade-off limits the minimum spatial resolution achievable to

aperture diameters that allow sufficient near-field signal. Additionally, the transparency of

the optical fiber limits the bandwidth of this technique and precludes the use of ultrafast

pulses.

The next major advance in near-field optical imaging occurred when a sharp metallic tip

replaced the aperture probes in what is termed scattering-Scanning Near-field Microscopy

(s-SNOM) [71, 43, 194, 171]. Acting like a lightning rod, when illuminated with light,

the sharp apex of the tip enhances and confines the electromagnetic fields to length scales

comparable to the radius of curvature of the tip apex (typically 1-10 nm). s-SNOM does

14

not suffer from the sample trade-off between signal strength and resolution that exists for

NSNOM; in fact as the tip apex radius decreases, the local electric field strength increases

instead. Considering the metallic tip as an equipotential surface, the electric field near the

surface goes as the spatial derivative of the potential, so for a smaller radius of curvature,

the electric field increases. Thus s-SNOM overcomes the sensitivity limitation of NSNOM

and other far-field techniques as a decreasing tip apex radius increases signal strength and

improves spatial resolution simultaneously.

p

p'

Ein

Enf

Ein

Enf

dd

d

a) b) c)

Ein

Enf

Figure 3.2: a) Illuminating a small aperture brought close to a sample surface with incidentlight Einc can locally probe the sample. Back-scattered light Enf collected through theaperture contains local optical information of the sample with spatial resolution on theorder of the aperture size d. b) NSOM schematic using a tapered optical fiber coated withmetal except for an aperture at the end of diameter d. Light sent through the fiber canprovide sample illumination. c) Schematic of s-SNOM using a sharp metal tip close to asample surface. The enhanced electric fields near the tip apex locally probe the sample withspatial resolution on the order of the tip apex size d.

s-SNOM is based on atomic force microscopy (AFM). AFM has several modes of op-

eration. In most modes, a sharp tip on mounted on a cantilever and the tip position is

monitored by a laser reflected off the backside of the cantilever which is measured by a posi-

tion sensitive photodiode. In contact-mode, a tip on a soft cantilever is brought into physical

contact with the sample surface. As the tip presses into the sample, the cantilever defects

and the reflected laser light moves along the photodiode. Constant tip-sample force is then

maintained by keeping the laser position on the photodiode fixed at the tip scans over the

surface by moving the sample z-position. In tapping-mode, or dynamic-mode, the cantilever

15

oscillates at its resonance frequency at amplitudes ranging from 30− 100 nm. When the tip

begins to touch the surface, the amplitude of its motion, or tapping amplitude is damped.

The tapping amplitude is used as the feedback signal to keep a constant height above the

surface. Alternatively, the phase of the tip motion with respect to the drive signal can be

used as the feedback signal.

When the tip is brought close to a sample, the nanoconfined light locally polarizes the

sample on length scales equal to the tip apex radius. The resulting polarization radiates

and emits scattered radiation that reflects the local optical properties of the sample. The

light is collected and collimated with a collection optic, typically a high numerical aperture

off-axis parabolic mirror, and detected with an infrared detector. Typical IR detectors in-

clude mercury-cadmium-telluride detectors (MCTs) or bolometers which convert IR photons

into current which can be read through conventional electronics. In s-SNOM, the external

illumination is typically split into two paths by a 50:50 beamsplitter: one path goes to the

tip and the other goes to a movable mirror in a reference arm.

Typically, s-SNOM suffers from a large background due to the large focal size com-

pared to the nanometer scale tip apex. Conventional IR s-SNOM uses the tip motion to

efficiently subtract this background. Since the near-field signal is highly surface confined,

the scattered signal strength has a highly non-linear dependence on tip-sample separation.

When the tip moves over the surface sinusoidally at frequency νd, the scattered field will

vary anharmonically. The degree of anharmonicity is determined directly by the near-field

scattering strength. Since the tip only oscillates 50-100 nm and the far-field focal spot is on

the order of 10’s of µm, any variation of intensity within that motion will be nearly linear

and its contribution to the background will only appear at f = νd after lock-in filtration.

Thus sending the detector output into a lock-in amplifier and demodulating at harmonics of

the tip frequency nνd for n > 1, the background can be largely removed and the near-field

scattered signal may be preferentially selected. Increasing harmonic order improves spatial

resolution and background suppression, although at the cost of a factor of 5-10 lower SNR

16

per harmonic order [94]. Typically, n = 3 provides a good compromise between signal levels

and background suppression for laser-based experiments, although for low irradiance sources,

only n = 1 provides usable signal levels.

Light scattered by the tip is weak, Pnf = |Enf |2 ∼ 1 − 10 fW. We typically combine

the scattered light with the light from a reference arm Eref with powers in the range of

a few mW, which serves to interferometrically amplify the light to detectable powers. By

controlling the delay of the reference arm, we can also extract spectroscopic information of

the scattered light. However, there is always a background field Ebkg of uncontrolled phase

and magnitude that reaches the detector. The resulting intensity at the detector is given by:

I = |Enf + Ebkg + Eref |2 (3.1)

= |Enf |2 + |Ebkg|2 + |Eref |2 + 2 · Re(E∗bkgEref

)+ 2 · Re (E∗nfEbkg) + 2 · Re (E∗nfEref) .

(3.2)

The background term can come from reflections and scattering from the portion of sample

and tip within the far-field focus. Thus demodulation at higher harmonics removes the

second, third and fourth terms from the expression. As stated above, Enf Ebkg, Eref and

thus the first term is negligible compared to the last two terms that are interferometrically

amplified. The detected intensity is then

I ∝ Re (E∗nfEbkg) + Re (E∗nfEref) . (3.3)

We see that any variation in the magnitude of Enf will provide a change in intensity at the

detector, which can provide high spatial resolution as the tip scans across an inhomogenous

sample. However, a change in phase of Enf relative to the uncontrolled background can also

change the intensity of the first term. The first term is known as the “self-homodyne” term,

and creates ambiguous contrast, where either a change in the amplitude or phase can modify

the scattered intensity. To unambiguously resolve the near-field amplitude and phase, we

must find a way to completely remove the first term in the above expression.

17

3.2.0.1 Interferometric s-SNOM measurements

The reference arm can be use in several different modalities to measure the phase and

amplitude of the scattered near-field light. Two phase homodyne is performed by acquiring

two images of the same sample location but with the reference arm shifted by φ = π/2 = λ/4

[44]. This is akin to measuring the real and the imaginary part of a signal. The phase is then

calculated Φ = arctan(I(φref − pi/2)/I(φref)) and summing the two images in quadrature

gives the amplitude. Typically, the two images can be acquired by synchronizing the AFM

scan with the piezo mirror to shift the phase between the trace and retrace of the raster scan.

This is a relatively straightforward and efficient way of measurement, however it typically

requires post processing to visualize the data, and relies on minimal sample drift during

the scan and little trace/retrace disagreement to provide an accurate image. Since it relies

only on algebraic manipulation of the two images and requires no additional actively moving

parts, no additional noise is added to obtain the phase and amplitude.

One disadvantage of two-phase homodyne is that it does not eliminate the self-homodyne

contribution to the detected intensity. This can create an error particularly in the phase sig-

nal. The use of a strong reference field compared to the background field Eref Ebkg

reduces the effect of the self-homodyne term and minimizes error in the phase calculation.

Alternatively, by acquiring a self-homodyne image with the reference arm blocked and sub-

tracting that from the two phase-shifted images, the self-homodyne term can be completely

removed. This typically requires the acquisition of another image immediately before or af-

ter the phase-shifted images which is usually cumbersome and increases the total acquisition

time by a factor of 2.

Another technique known as psuedoheterodyne for background-free measurement of

near-field amplitude and phase is performed by sinusoidially oscillating the reference arm

mirror a fraction of the wavelength.By oscillating the reference arm mirror at frequency Ω,

the optical signal is double modulated at by the tip motion and the change in interference

18

condition. The optical signal is described by I = exp(iA cos(Ωt)) exp(iωt + φref), and in

frequency-space signal, In,m, appears at the mixing frequencies of ν = nνd±mΩ for integers

n,m. The near-field signal can be expressed as Enf,n ∝ In,2 + iIn,1, and the amplitude and

phase can be then extracted.

In addition to providing phase and amplitude data, the reference arm also allow near-

field spectroscopic measurements when a broadband source is used. Here, a spectrum is

acquired by scanning the reference arm across the zero path delay to create an interferogram

as in FTIR. The spectrum obtained by this technique is also background-free, since the

self-homodyne term EnfEbkg is delay-independent and only creates an intensity offset in the

interferogram that can be subtracted out when performing the Fourier transform.

3.2.1 s-SNOM models

To describe the tip-sample interaction and its spectral and spatial variation, we can

appeal to various simple geometric approximations to the tip-sample geometry.

3.2.1.1 Point dipole model

As a first approximation to the tip-sample interaction, we model the tip and sample as a

spheres of radius r with polarizability αt,s = 4πr3ε0εt,s−1

εt,s+2, where εt,s are the dielectric function

of the tip and sample, respectively [94]. This is known as the point dipole model (PDM)

for s-SNOM. When the tip is close to a sample surface, the tip dipole p = αtE0 induces an

image dipole in the sample of strength p′ = pβ, where β = ε2(ω)−1ε2(ω)+1

and E0 is the incident

electric field. The electric field on the tip dipole is the sum of the incident electric field and

the field due to the image dipole E = E0 + βp8πε0(r+h)3 where h is the separation between the

bottom of the sphere and the surface. The field due to the image dipole acts back on to the

tip dipole and repolarizes it, giving a modified tip polarization of p = αt

(E0 + p′

2πε0(r+h)3

),

where the second term is the radiation reaction term.

We can describe the tip-sample mutual response by assigning the system an effective

19

polarizability αeff given by αeff = p/E0 = αt(1−βαt/16πε0(r+h)3)

. The effective polarizability

then determines the scattering field emitted from the tip-sample region through the usual

Rayleigh scattering result:

Iscat =k4

2|αeff |2 (1 + cos2(θ))I0 (3.4)

where θ is the angle between the polarization vector and the vector to the observation

point. For typical s-SNOM collecting at near-normal incidence, θ ≈ 0. Thus, αeff relates the

scattered intensity to the nanoscale spatio-spectral properties of a sample.

The magnitude of αeff roughly follows a 1/z3 distance dependence, and its spectral

dependence arises from spectral variations in β(ω) 3.3. Typically, non-resonant tip materials

are used to minimize spectral variations in αt.

r = 5 nm

r = 50 nm

1700 1710 1720 1730 1740 1750 1760 1770

0

2.×10- 36

4.×10- 36

6.×10- 36

8.×10- 36

1.×10- 35

Frequency (cm- 1)

C2

r = 5 nm

r = 50 nm

900 910 920 930 940 950 960 970

0

1.×10- 34

2.×10- 34

3.×10- 34

4.×10- 34

Frequency (cm- 1)

C2

a) PTFE

b) SiC

Figure 3.3: Scattered intensity at the second harmonic calculated for a) PMMA and b) SiCusing the point dipole model for varying tip radii. For SiC, its strong oscillator strengthleads to red-shifts of the scattered intensity with large tip radius. In contrast, the spectralresponse of PMMA is minimally perturbed by tip-sample coupling even with large tip radius.

20

3.2.1.2 Finite dipole model

arb

. u

.)

z = 1 nm

z = 2 nm

z = 10 nm

5.×10- 34

1.×10- 33

1.5×10- 33

2.×10- 33

2.5×10- 33

αe

ff

z = 1 nm

z = 2 nm

z = 10 nm

900 920 940 960 980

2

4

6

8

Frequency (cm-1)

αe

ff(

b) FDM

a) PDM

Figure 3.4: Effective polarizability αeff over a SiC surface calculated for a) point dipolemodel (PDM) using a 20 nm tip and b) the finite dipole model (FDM) with R = 20 nm,and L = 200 nm. The values are plotted at varying distances from the surface and show ared-shift with decreasing distance.

While the point dipole describes the s-SNOM spectra of many materials, it often fails to

match the spectra obtained on strongly resonant materials which can be highly red-shifted

due to perturbation by the tip. Additionally, the point-dipole model fails to predict the

distance dependence of the s-SNOM signal. The finite dipole model (FDM) models the tip

as a more extended object with the addition of several other point charges distributed along

the tip length to simulate a more extended charge separation [40]. The resulting effective

polarizability at tip-sample separation z is then given by

αeff ∝ 2 +β(g − R+z

L

)log(

4L4z+3R

)log(

4LR

)− β

(g − 3R+4z

4L

)log(

2L2z+R

) , (3.5)

where R and L are fit parameters. R can roughly be attributed to the tip radius determined

through SEM or topographic resolution, but L cannot be attributed to a particular observable

21

tip dimension. The parameter g characterizes the ratio of near apex to total charge induced

along the tip. The FDM was shown to model SPhP near-field laser s-SNOM spectra of SiC

[40, 174] and SiO2 [3] to good agreement.

3.2.2 Sources for s-SNOM

Even with conventional IR optical studies, stringent infrared light source requirements

make simultaneously high-sensitivity and broadband techniques difficult. These requirements

have so far prevented IR s-SNOM from becoming a routine technique. Continuous wave lasers

have found use for single frequency nano-imaging, however spectrally resolved studies require

laser tunability and a spectrally resolved measurement typically requires acquiring many

images at different wavelengths. While femtosecond lasers provide higher bandwidth and

temporal resolution [156, 130, 9], they typically rely on a series of nonlinear processes which

reduce stability, increase cost, and limit bandwidths to typically spanning ∼ 1 vibrational

resonance at a time. For chemical nano-FTIR spectroscopy, broadband IR radiation that

spans the full mid-IR spectral range is desired from the few 100’s of cm−1 finger print region

up to ∼ 4000 cm−1 for the highest frequency normal modes (spanning 3 octaves). Such

broadband IR s-SNOM spectroscopy has been enabled using the high spectral irradiance

of synchrotron IR radiation [13, 66], but this technique relies on using shared beamtime at

large-scale facilities.

3.2.2.1 Nano-spectroscopy using thermal blackbody radiation

A simple blackbody thermal source in the form of tungsten or ceramic filaments can

provide IR bandwidth comparable to that at a synchrotron. They are routinely used in

commercial far-field FTIR microscopes and its use has been demonstrated for s-SNOM [70,

72]. However these demonstrations were over a limited spectral range, with low signal-to-

noise ratio, and only for the strong responses of collective phonon polaritons of SiO2 or free

carriers of doped Si. Despite the compelling simplicity, low cost, and low irradiance noise,

22

the use of a thermal emitter as a light source for s-SNOM has so far not been explored

further.

Continuing the work of undergraduate researcher Jared Stanley and in collaboration

with Silke Mobius and William Lewis, I demonstrate IR s-SNOM nano-spectroscopy using a

blackbody emitter with high sensitivity based on improved source and tip illumination and

an optimized short-path asymmetric Michelson interferometer [144]. Together with William

Lewis, I designed an improved optical setup to reduce the beam path length in order to

minimize propagation losses of the source and reduced the reference arm length down to

hardware limitations. These changes dramatically improved stability and the irradiance on

the tip.

I implement optical interferometric heterodyne near-field amplification and detection to

obtain the complex-valued nanoscale vibrational material response. In addition to the nano-

spectroscopy and -imaging of the surface phonon polariton of SiO2, I resolve the phonon

polariton response and its dispersion in hexagonal boron nitride (hBN), with thicknesses

as low as 8 nm (25 layers), and demonstrate the extension to the two orders-of-magnitude

weaker molecular resonances of C-F vibrational modes in polytetrafluoroethylene (PTFE) as

an example. Despite the incoherent nature of the radiation and the associated low spectral

irradiance, this work shows the general feasibility of the use of a thermal emitter as a light

source for nano-spectroscopy for a broader range of s-SNOM applications than previously

considered.

3.2.2.2 Experiment

The s-SNOM setup as shown in Fig. 3.5(a) is based on a modified atomic force mi-

croscope (AFM) (Vesta AFM-SP, Anasys Instruments) operated in dynamic mode, with

asymmetric heterodyne interferometric signal detection [8]. Dry air purge enclosure reduces

atmospheric absorption by more than a factor of 2 and allows us to investigate vibrational

modes in the water absorption region, e.g., the in-plane mode of hBN. A heated ceramic

23

globar (EverGlo, Thermo-Fisher, IR emissivity ε ≈ 0.9) operating at T ' 1400 K provides

IR thermal blackbody radiation, which is collected with an off-axis parabolic mirror (f = 50

mm, φ = 25 mm, NA = 0.24). Light passes through the KBr beamsplitter (BS) with a

transmission coefficient of 50% and is focused onto the AFM tip by another high-NA off-axis

parabolic mirror (f = 20.8 mm, NA = 0.48). Commercial dynamic mode metallic AFM tips

(Arrow NCPt, NanoWorld) provide local field enhancement of the incident IR radiation as

described above. The mirror position in the reference arm is controlled with a closed loop

Ω

MCT Det.

Lock-in

Amp.

Delay arm

BSCP

IR source

Figure 3.5: (a) Schematic of infrared nano-spectroscopy using broadband infrared thermalradiation from a ceramic globar, utilizing heterodyne detection by asymmetric Michelsoninterferometry with beamsplitter (BS), compensation plate (CP) and mercury-cadmium-telluride (MCT) detector. (b) Reference interferogram of blackbody source. (c) Correspond-ing broadband Fourier transform of (b).

delay stage with 50 mm maximum travel distance corresponding to a spectral resolution

limit of 0.1 cm−1 (ANT95-50-L, Aerotech). The compensation plate (CP) in the reference

arm accounting for the beamsplitter substrate thickness facilitates the extraction of near-

field phase spectra which aids in the identification of vibrational modes. Short beam paths

in both signal and reference arm of length ≈ 9 cm provide high stability (sub-250 nm mean

drift in 30 min) and ease of alignment. The short beam path further helps to increase the

infrared fluence at the tip due to the limited beam collimation of the blackbody source.

After interference with the reference beam, the tip-scattered light is focused by an off-axis

parabolic mirror (f = 10 cm, φ = 25 mm, NA = 0.12) onto a mercury-cadmium-telluride

24

(MCT) detector (J15D14-M204-S250U-30, EG&G, peak specific detectivity D∗ = 4 · 1010

√Hz/cm−1 W−1).

0 20 40 60 80

0.0

0.2

0.4

0.6

0.8

1.0

Am

plit

ud

e (

no

rma

lize

d)

(arb

. u

.)

Distance (nm)

1Ω

2Ω

Figure 3.6: Approach curves of the s-SNOM signal taken above a gold surface with referencearm at zero path difference. Demodulation at both the first (Ω) and second harmonic (2Ω)show strong near-field content with an order-of-magnitude increase within the last 50 nm.

Lock-in demodulation on the first and second harmonics of the cantilever tapping

frequency Ω provides for effective far-field background suppression at tapping amplitudes

below ∼ 80 nm. Approach curves (see Fig. 3.6) verify the near-field content of the first

harmonic, with an order of magnitude signal increase when in contact with the surface with

a 1/e decay length of ∼ 60 nm. All data in this section are reported for first harmonic

demodulation, which is found to be sufficient for far-field background suppression while

providing high quality spectra. Typically, demodulation at higher orders of the cantilever

tapping frequency (nΩ for n ≥ 2) is desirable for high near-field contrast and improved spatial

resolution [57], however for smooth samples with low scattering the first harmonic can suffice

for near-field detection with tapping amplitudes . 80 nm. With proper alignment and high

quality beam focus we also observe sufficient background suppression at first harmonic in

approach curves and imaging on even highly scattering samples. We acquire interferograms

by collecting the real component of the lock-in signal while sweeping the delay by moving the

reference arm. The delay stage is operated in a step-scan modality, with a step size of 500

25

nm and step dwell time ranging from 50− 200 ms, depending on signal strength. Typically

5-60 consecutive interferograms are averaged to improve signal-to-noise ratio. To reduce

acquisition times and improve signal-to-noise at positive delays, we acquire interferograms

asymmetrically around zero path delay. Corresponding asymmetric apodization is applied

using the product of a Gaussian and an error function centered around zero path delay

before performing the Fourier transform to extract the complex valued near-field spectrum.

We normalize the amplitude spectrum by dividing by the gold reference amplitude spectrum.

The gold phase spectrum is subtracted from the sample phase spectrum.

Figure 3.5(b) shows an interferogram of the source radiation with corresponding Fourier

transform spectrum (c). Broadband optics (KBr beamsplitter transmission window 400-

30000 cm−1), high source temperatures, and dry air purging extend the available bandwidth

of previous studies [70, 72] with our bandwidth primarily limited by the sensitivity range

of the MCT detector used. A high temperature thermal source (T ≈ 1400 K) increases

irradiance and further improves usable bandwidth.

3.2.2.3 Results

Typical near-field interferograms are shown in Fig. 3.7(a) obtained on gold (red) and

hBN (blue). Due to the non-resonant response of gold in the mid-IR, interferograms are

symmetric about zero path delay. In contrast, hBN yields an asymmetric interferogram with

distinct oscillations persisting for long optical path delays arising from the phonon polariton

resonance. Figures 3.7(b) and 3.7(c) show the resulting broadband Fourier-transform near-

field amplitude (blue) and phase (red) spectra normalized to the Au reference spectrum for

two hBN flakes on a SiO2 substrate with flake thicknesses 60 nm (b) and 8 nm (c), respec-

tively. The distinct peaks in the amplitude spectrum correspond to the out-of-plane and

in-plane transverse optical (TO) phonon modes at νTO⊥ ' 800 cm−1 and νTO

‖ ' 1370 cm−1,

respectively [52]. The broad dispersive feature in the amplitude spectra between 1000 cm−1

to 1200 cm−1 is the phonon polariton response of the SiO2 substrate. It is more pronounced

26

relative to the in-plane TO phonon mode of hBN for the thinner flake, as expected. Despite

the longer integration time for the 8 nm thick flake (56 min) than the 60 nm thick flake (15

min), the signal-to-noise ratio of the phonon modes is smaller for the thinner flake due to

the decreased material volume in the near-field probe region. In addition, we see an increase

of the linewidth of the in-plane phonon polariton response with increasing thickness. This

linewidth increase is not due to defects as we verify by the thickness independent linewidth

of the TO phonon mode measured by Raman spectroscopy of both samples (not shown).

Instead, we attribute the increase in linewidth to increasingly bulk-like polariton properties

for thicker flakes, and the formation of the associated Reststrahlen band[21].

Figure 3.9 shows the focal spot size of the thermal beam profile at the tip measured us-

ing a knife edge. Error function analysis shows that the focused beam profile is approximately

Gaussian, with a FWHM of ≈ 70 µm. This is larger than the theoretical diffraction-limited

spot size of ∼ 10 µm for a numerical aperture of NA = 0.48. We attribute this difference to

the lack of spatial coherence across the phase front which is limited by the conservation of

ν

250 nm

270 nm

0 nm

65 pW

25 pW

hBN Au

d)

e)

f)

0.0

0.2

0.4

0.6

0.8

0

2

4

6

8

10

600 800 1000 1200 1400

0.1

0.2

0.3

0.4

Wavenumber ν (cm-1)

-1

0

1

2

3

Au

hBN (x5)

a) b) hBN

60 nm

c) hBN

8 nm

SiO2

TO TO

|A(ν

)| (

arb

. u

.) Φ(ν) (ra

d.)

ν^

ν||

-100 0 100 200 300

Path delay (µm)

0.0 0.5 1.0 1.5

0

100

200

Distance (µm)

He

igh

t (n

m)

30

40

50

60

Int. (p

W)

25 nm

-0.5

0.0

0.5

1.0

1.5

Inte

nsity (

mV

)

Figure 3.7: Thermal source asymmetric s-SNOM interferograms of hexagonal boron nitrideflake on SiO2 (hBN, blue) and Au reference sample (red, offset for clarity) obtained at firstharmonic demodulation (a). Normalized near-field amplitude and phase spectra acquired ona 60 nm ((b), integration time t = 15 minutes and spectral resolution δν = 26 cm−1) and a8 nm thick flake ((c), t = 56 minutes, δν = 31 cm−1). AFM topography (d), and spectrallyintegrated optical image (e) across an hBN/Au step edge. Line traces shown in panel (f)correspond to the dashed line in panel (e) and show ∼ 25 nm spatial resolution (5-95% signallevel).

27

0 2 4 6 8 101350

1400

1450

1500

Phonon wavevector k = 8!/d (104 cm-1)P

ho

no

n fre

qu

en

cy (

cm

-1) 21.5

17.2

12.9

8.6

4.3

0.0

500 nm

71.5 nm

0 nm

1200 1300 1400 15000.0

0.5

1.0

1.5

400 nm

Am

plit

ud

e (

se

pa

rate

d fo

r cla

rity

)

Wavenumber (cm-1)

Bulk

500 nm

750 nm

1 µm Im(r

p ) (arb

. u.)

da) b)hBN

c)

Figure 3.8: (a) Spatially resolved spectra near an hBN edge with emergence of blue-shiftedresonance near the in-plane TO phonon peak at 1370 cm−1 due to the constructive inter-ference of phonon polaritons excited by the tip and reflected by the edge. (b) Topographyof the ∼ 60 nm thick flake with the colored points representing the acquisition positionsof the spectra in (a). Distance d from the edge is indicated. (c) Dispersion relation of thephonon polaritons with red points extracted from the spectra in (a). Grayscale plot is Im(rp)calculated using results from [41]. Blue line is the dispersion relation based on the modelpresented in [170].

etendue of our optical system.

Figure 3.7(d) and 3.7(e) show AFM topography and near-field signal acquired simulta-

neously across an hBN/Au step edge. Spectrally integrated images were obtained with the

reference arm at zero path difference for maximum heterodyne amplification. Line traces

(dashed line in Fig. 3.7(e)) show ' 25 nm spatial resolution limited only by the apex radius

of the tip. Strong near-field contrast is observed with a large signal on Au due to its high

Drude response. The spatial image also shows a strong contrast across the ∼ 140 nm step

between the 100 nm and 240 nm thick sections of hBN (left) and the absence of discernible

contrast for the ∼ 25 nm step between the 75 nm and the 100 nm regions (center).

The correlation of contrast with step height results from the finite IR s-SNOM probe

28

-100 -50 0 50 100

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Po

we

r (m

W)

Distance (µm)

Knife-edge beam profile

70 µm FWHM Gaussian

Figure 3.9: Measurement of the focal spot at the tip using a knife edge technique. Data(black) is approximately described by a Gaussian intensity profile with FHWM of 70 µm(red).

depth within hBN and thus a variable residual IR contribution from the Au substrate de-

pending on flake thickness. The amount of the contrast is then determined by the relative

change in height across the hBN flake, i.e. larger for the greater step height compared to

the smaller one. As the s-SNOM probe depth decreases for increasing harmonic orders, the

observed contrast between the 100 nm and 240 nm thick hBN layers is larger than would

otherwise be expected for second or higher order harmonic demodulation [57].

The heterodyne amplified signal can contain different spectral information depending

on the fixed delay length of the reference arm [16]. By imaging at zero path difference, the

nonresonant Drude response from gold dominates compared to the resonant phonon response

of the hBN. In the case of a longer reference arm delay the contrast can be dominated by the

long-lived phonon modes of hBN and contrast may be reversed based on the relative phase

between the reference arm light.

In addition, due to the low absorption of hBN at infrared frequencies outside of the

Reststrahlen bands and the associated finite IR penetration depth through the flake, incident

IR light may propagate through the hBN flake and reflect off the Au substrate. This re-

29

flection would vary depending on flake thickness and may cause variations in the homodyne

background that may amplify the near field contrast between the 100 nm and the 240 nm

thick flakes.

Figure 3.8(a) shows spatially resolved s-SNOM spectra acquired at different distances

from the edge of a ∼ 60 nm thick hBN flake. At < 1 µm proximity to the edge (positions

indicated in Fig. 3.8(b)), a second peak appears at frequencies higher than the in-plane

TO phonon peak of 1370 cm−1. This second peak, which continuously blue-shifts with

decreasing tip-edge distance, is due to tip-excited phonon polaritons that reflect at the

edge and constructively interfere with the incoming field at the tip position, as established

previously [41, 170]. Spatial resolution of better than 100 nm is demonstrated through the

resolution of the ∼ 10 cm−1 blue shift of this peak from 500 to 400 nm positions. Figure

3.8(c) shows the resulting dispersion of the phonon polaritons (red points) extracted from

the peak positions in Fig. 3.8(a). The wavevectors are given by the cavity condition of

constructive interference d = λ/4 = 8π/k, where d is the distance from the hBN flake edge.

Error in the wavevectors is determined by the uncertainty in d during the scan due to sample

drift, and the error in the phonon frequencies is determined by the spectral resolution. The

results are in good agreement with the phase space dispersion relation calculated from the

imaginary part of the reflection coefficient rp of the air/hBN/SiO2 system [41] (grayscale

plot), and the dispersion relation of 2D polaritons in the long wavelength limit [170] (blue

line).

Figure 3.10(a) shows near-field spectra of a 300 nm thick amorphous SiO2 layer on

a Si substrate. The amplitude (blue) and phase (red) spectra are in good agreement with

predictions from the point dipole model describing the tip-sample interaction [94]. The

dielectric function for SiO2 is modeled using a Lorentzian oscillator model with three phonon

modes at frequencies ν = 447 cm−1, 811 cm−1, and 1050 cm−1, with FWHM linewidths

γ = 49 cm−1, 69 cm−1, and 70 cm−1, and relative oscillator strengths 0.923, 0.082, and

0.663, respectively [61].

30

Figure 3.10(b) shows near-field amplitude and phase spectra of a ∼ 5 µm thick section

of PTFE. Two distinct vibrational resonances around 1150 cm−1 and 1220 cm−1 are resolved

in the phase spectrum, corresponding to the symmetric νs and antisymmetric νas C-F stretch

modes, respectively (dashed vertical lines). Given the lateral spatial resolution of 100 nm

demonstrated in Fig. 3 with corresponding probe volume of ∼ (100 nm)3 and a molar

volume of 0.02 mol/cm3, this corresponds to 20 attomol sensitivity of the C-F vibrational

mode of PTFE. The near-field amplitude and phase spectra (solid lines) approximate the

real and imaginary parts of the index of refraction [188], with their corresponding dispersive

and absorptive features, respectively [105, 97].

Although the appearance of symmetric and antisymmetric modes were reproducible

between consecutive scans, we observed spectral variations at different positions across the

PTFE sample. Sample drift across spectrally distinct nanoscale regions during measurements

limits integration time to t ≤ 5 min and prevents summation of many acquisitions. Though

there can be spectral shifts due to sample thickness observed in s-SNOM, in the limit of

film thickness much larger than tip radius these effects become negligible [123]. This spatial

heterogeneity in the spectral features may be due to local variations in the polymer chain

order and crystallinity [122].

3.2.2.4 Discussion

In the following, we analyze the normalized signal-to-noise ratios (NSNR) normalized

by spectral resolution and acquisition time for the different samples (Fig. 3.11). The NSNR

is defined by NSNR = signal powerrms noise

/(δν√t) where δν is the spectral resolution and t is the

integration time of the spectrum. The calculated NSNRs allow us to compare published

results using different s-SNOM sources with spectra collected at different resolutions and

acquisition times.

NSNRs determined from amplitude data in Figs. 2b),c) and 3a),b) range from 0.001√

Hz/cm−1 up to 0.02√

Hz/cm−1. We compare these thermal source near-field NSNRs to

31

PD

M Φ

(ν) (

arb

. u.)

Wavenumber ν (cm-1)ν

0.0

0.2

0.4

0.6

0.0

0.2

0.4

0.6

0.0

0.1

0.2

0.3

0.90

0.95

1.00

n

800 1000 1200 1400

0.0

0.5

1.0

1.5

0.0

0.5

1.0

1.5

1100 1150 1200 1250

0.0

0.5

1.0

1.5

0.0

0.2

0.4

0.6

k

a) SiO2

b) PTFE

νWavenumber ν (cm-1)

νΦ

(ν)

(ra

d.)

|A(ν)| (a

rb

. u

.)

νΦ

(ν)

(ra

d.)

|A(ν)| (a

rb

. u

.)

νs

νas

PD

M |A

(ν)| (

arb

. u.)

Figure 3.10: Near-field magnitude (blue) and phase (red) spectra for (a) a 300 nm thickamorphous SiO2 layer and (b) PTFE. Spectra were obtained with integration times of 30 minand 5 min, and spectral resolution of 26 cm−1 and 21 cm−1 for SiO2 and PTFE, respectively.The SiO2 spectrum matches the predictions from the point dipole model (PDM) over a broadspectral range. The PTFE near-field spectrum is plotted in comparison with the complexindex of refraction n(ν) and k(ν) [97].

the range of signal levels and their NSNRs from s-SNOM data based on other light sources

as displayed in Fig. 5. The extracted NSNRs from synchrotron infrared nano-spectroscopy

(SINS) [13] with demodulation at the second harmonic of the tip dither frequency (2Ω) is

consistently almost two orders-of-magnitude higher than that from our blackbody source

demodulated at the first harmonic of the tip (Ω). Furthermore, several NSNRs for laser

based near-field spectroscopy obtained at 3Ω [158, 41, 3, 42, 9] are included for reference. In

general, higher NSNRs are possible with laser sources but they vary widely, often limited by

laser stability, noise, and irradiance fluctuations across the mid-infrared.

While the NSNRs are normalized by spectral resolution and acquisition time, we do

not account for the typically weaker signal strengths with increasing harmonic order [94].

32

10-3

0.01

0.1

1

10

BN, in

plane

BN, out

of plane

SiO2

PTFE PMMA

Sample

SINS (2Ω)

Blackbody source (1Ω)

Laser (3Ω)

[32]

[13][34]

[10]

[20]

[33]

NS

NR

(√H

z/c

m-1)

Figure 3.11: Signal-to-noise ratio analysis for s-SNOM spectra of different samples using thethermal blackbody source (red, first harmonic), synchrotron infrared near-field spectroscopy(SINS) (blue, second harmonic) and different laser sources (black, third harmonic). NSNRanalysis for published work are referenced [X] with corresponding citation number.

Note that our NSNRs are extracted from the amplitude spectra normalized by dividing

by a reference spectrum, which may add additional noise that would not be present in

unnormalized spectra.

The estimated irradiance incident on the tip using the blackbody source is I ∼ 25

mW/(cm2 cm−1), estimated by a knife-edge beam profile measurement (Fig. 3.9). This is

in general 1-2 orders of magnitude less than that available with SINS, which ranges from

0.1−1 W/(cm2 cm−1). The large difference in irradiance is in part due to the large focal spot

of the spatially incoherent thermal radiation compared to coherent synchrotron radiation.

Although the tip scattered near-field can be treated as Rayleigh point-dipole emission from

the apex vicinity exhibiting approximately planar wavefronts, the large incident beam spot

reduces the driving field strength at the tip. We estimate the peak field strength at the tip

to be 280 V/cm, and the spectrally integrated intensity across the cross-sectional area of the

25 nm radius tip apex to be 0.5 nW.

33

Tunable continuous wave laser sources can achieve the highest spectral irradiance with

NSNRs greater than 10√

Hz/cm−1, e.g., for PMMA [158], or hBN [41, 42], albeit at the

expense of wavelength tuning range. Note that the carbonyl vibrational resonance, though

relatively strong, is weaker than the C-F vibrational mode of PTFE, and could not be

observed in our s-SNOM measurements using the blackbody source. Ultrafast laser sources

allow for broader spectral bandwidth, with typical irradiances of over 10 W/(cm2 cm−1). In

particular, the 3 orders of magnitude higher spectral irradiance compared to the blackbody

source yields correspondingly higher NSNRs (e.g., PTFE, with a NSNR of ∼ 2.5√

Hz/cm−1

attained at 3Ω) [9].

In the following, we estimate the signal strength and degree of heterodyne amplification.

The intensity at the detector includes contributions from the tip scattered near-field Enf , the

uncontrolled background scattered from the tip-shaft and the sample Ebkg, and the reference

field Eref , given by Eq. 3.2. The second, third, and fourth terms in the expanded equation

are not modulated at the tip frequency and are thus suppressed by the lock-in detection.

The first term is negligible compared to the remaining terms which are interferometrically

amplified (described below). The last two terms describe the self-homodyne amplification

by Ebkg which is delay-independent yielding a small offset in the interferograms (dashed line

in Fig. 3.7(a)), and the delay-dependent heterodyne amplification of the near-field scattered

light by Eref , respectively.

We now use the characteristics of a typical interferogram (see Fig. 3.5(a)) to estimate

the relative contributions from the remaining relevant terms in Eq. 2. The sensitivity of the

MCT detector and amplifier system is ∼ 107 V/W, determined through external far-field

calibration with a known power incident on the detector and using an optical chopper for

demodulation. At large negative delays, in the absence of interference with the reference

arm, the small signal (dotted line) is from the term 2 ·Re (E∗nfEbkg) only, which we estimate

to be 20− 100 pW. Near zero path delay, the intensity is dominated by the 2 · Re (E∗nfEref)

term, with a peak power of 2−10 nW. Based on the heterodyne amplification with a typical

34

measured reference arm power of Pref ∼ 3 mW, this corresponds to a tip scattered power of

Pnf ∼ 2 − 30 fW. This value agrees well with estimates of the scattered power of a 25 nm

radius metallic sphere near the sample surface from the point dipole model based on our

incident fluence [94]. Given a peak incident fluence of 25 mW/cm2/cm−1 at the tip apex, we

calculate the scattered power to be 0.1 pW, corresponding to Enf ∼ 1 mV/cm electric field

strength. We can also estimate the background power to be Pbkg ∼ 10 µW.

3.2.3 Noise sources

The fundamental limitation in opto-electronic systems is shot noise, and is prevalent

when the signal is small enough that the discrete nature of light and current influences the

measurement statistics. This occurs when light intensities are small enough that only a few

photons are recorded during one acquisition period. These low-intensity measurements then

follow Poisson statistics, since they are uncorrelated random events. The standard deviation

of a Poisson distribution scales as√N for N events, and the signal scales as N , so the

SNR is given by SNR = N/√N =

√N . Shot noise is especially prominent in spectroscopy

techniques that use gratings to spectrally disperse onto detector arrays. Then, the signal

is spread over many detectors which receive only a small fraction of the original number

of photons. FTIR spectroscopy minimizes this noise source by measuring all frequencies at

once with one detector, which is known at Jacquinot’s advantage.

However, in this study, the lower bound of the unamplified tip-scattered power Pnf ∼ 2

fW corresponds to ∼ 100 photons/s within 1 cm−1 bandwidth, which is approaching the

shot noise limit for typical step-scan dwell times of ∼ 100 ms. The self-homodyne and

heterodyne amplification are thus crucial for the detection of such weak near-field signals.

From the power estimates of Pnf , Pbkg, and Pref , we estimate a self-homodyne intensity

amplification factor of Re (E∗nfEbkg) /|Enf |2 ∼ 103 and a heterodyne intensity amplification

factor of Re (E∗nfEref) /|Enf |2 ∼ 105. These amplification steps result in more practical signal

intensities well above the shot noise limit.

35

To further explore the limitation of low spectral irradiance on detectable s-SNOM signal

levels, we acquired spectra at variable source temperature with a corresponding variation

in intensity (Fig. 3.12). The NSNR versus irradiance is ideally described by NSNR(I) =

Signal/Noise = A·I/(√

I +B)

where A,B are fit parameters (dashed lines).This expression

describes the roughly√I dependence for large irradiance, with a small offset given by B

for the detector noise at zero irradiance, suggesting that we are shot noise limited at low

irradiances. Near-field spectra of a 60 nm thick hBN flake could be obtained using an

estimated peak spectral irradiance as low as I ∼ 2 mW/cm2/cm−1 at the tip apex and ≈ 40

pW spectrally integrated power incident on the cross-sectional area of the 25 nm radius tip,

and a reference arm power ≈ 200 µW. The corresponding NSNR around the in-plane mode

is found to be ∼ 1.2 · 10−3√

Hz/cm−1, which for a typical acquisition time of 103 s and a

spectral resolution of 20 cm−1 leads to a signal-to-noise ratio of only 0.75:1 for the in-plane

mode and 2:1 for the out-of-plane mode.

The NSNR increases rapidly with temperature for the in-plane mode due to the change

in color temperature, and leads to a signal-to-noise ratio of, e.g., 2.3:1 for an irradiance of

∼ 6 mW/cm2/cm−1 measured under otherwise identical acquisition parameters as above.

In comparison, the signal-to-noise ratio for the out-of-plane mode increases only by a factor

of 2, to 4:1. Note that a unique feature of the use of a blackbody emitter as a source for

s-SNOM in contrast to most laser sources is the observation that the NSNRs are in general

higher for spectroscopy at low frequencies. This results from the combination of the antenna

enhancement of the radiative s-SNOM signal [149] and the higher emission efficiency of a

blackbody at lower energies.

Intensity noise from the thermal source is small (typical rms power fluctuations are ≈

0.1%), and interferometer instability is minimized with the short beam path. The dominant

noise source at higher irradiance is detector noise from the photoconductive MCT detector,

in the form of Johnson noise from thermally excited carriers. By cooling the detectors

to cryogenic temperatures, we reduce this noise, but it cannot be completely eliminated. It

36

0 5 10 15 20 25 300

2

4

6

8

10

12

14

Peak spectral irradiance I (mW/cm2/cm-1)

Out-of-plane (700-800 cm-1)In-plane (1320-1420 cm-1)

NS

NR

(√H

z/c

m-1)

x10-3

0 200 400 600 800 1000 120040

60

80

100

120

140

160

180

SN

R (

arb

. u

.)

Acquisition time t (s)

Data

t1/2

a) b)

Figure 3.12: a) NSNRs of the in-plane (red) and out-of-plane (blue) TO phonon responseof hBN as a function of spectral irradiance I incident on the tip. The results were obtainedby decreasing the temperature of the thermal source primarily resulting in a decrease in IR

intensity. Dashed blue and red lines are fits to a theoretical NSNR(I) = A · I/(√

I +B)

irradiance dependence. b) SNR dependence on acquisition time t. Data match a√t depen-

dence.

typically has a white noise spectral response with a 1/f corner around 100 kHz. We eliminate

the 1/f noise contribution by lock-in filtration at the cantilever frequency νf . The detector

noise in our spectral region of interest can therefore be approximated as white noise, and we

see a√t acquisition time dependence on the SNR. The 1/f noise can be further reduced by

demodulating at higher harmonics of the cantilever frequency. Detectors of smaller active

area have correspondingly higher detectivity D* and can also increase NSNRs.

Due to the sub-wavelength size of the apex, the tip scatters near-field light approx-

imating a point emitter, with spatially coherent wavefronts primarily polarized along the

tip axis. However, the tip-scattered light is combined with incoherent and unpolarized light

from the reference arm resulting in incomplete interference at the detector. A mode matched

detector size can serve to mode filter the reference arm field by preferentially selecting the

spatially coherent mode of smaller focus size, and thus can improve the fringe visibility of

37

the interferograms. The poor beam quality arises from imperfect collimation of the spatially

extended and incoherent blackbody source. The fraction of collimated, spatially coherent

light is limited by the conservation of etendue of the optical system, primarily determined

by the source collection optics as well as the source geometry This can be improved by

collecting light from the source with high NA optics. The spatial coherence of the source

phase fronts is fundamentally limited by the finite coherence length of the emitter within the

focal volume of the collection optics. Due to the finite decay length lc of the first-order field

correlation functions of a blackbody on the order of lc ∼ ~c/kBT ∼ 2 µm at T = 1400 K

[119], collecting a spatially coherent mode from the blackbody source would require confocal

and mode filtered collection and imaging of the source.

Despite these challenges, high quality spectra from an 8 nm thin hBN flake demon-

strates the high sensitivity of this technique. The spectrum from PTFE demonstrates the

applicability of the technique to molecularly resonant materials and could allow its use for

broadband chemical identification with nanoscale resolution.

3.3 Conclusion

s-SNOM overcomes the diffraction limit by accessing the high wavevectors of evanes-

cent, surface-confined waves. As a combination of AFM and optical spectroscopy, it provides

chemical sensitivity with 10’s of nm spatial resolution. With no specific bandwidth limita-

tions, s-SNOM can be used with broadband or single frequency, and continuous wave or

ultrafast pulsed light sources. Interferometric detection provides signal amplification and

allows spectroscopic characterization of the scattered near-field using FTIR techniques. As

one application, I discussed my work implementing the weak, but broadband emission from

a thermal source. My work demonstrated an advance on the sensitivity of the technique

with the observation of the spectral signatures of a few-layer hBN flake and the molecularly-

resonant PTFE.

Chapter 4

Thermal near-field spectroscopy and optical forces

4.1 Introduction

One property that all matter at non-zero temperature shares is the emission of thermal

radiation originating from stochastically excited current densities within the medium. While

the far-field properties of thermal emission are well-understood through Plank’s law and its

modification to incorporate the emissivity of specific materials, its evanescent properties at

distances very close to the emitter are less well-explored. In particular, while far-field thermal

emission is largely incoherent, thermal radiation can be coherent in the near-field [24, 84].

The evanescent near-fields, characterized by large wavevectors, exhibit near-monochromatic

emission arising from material resonances. Understanding and ultimately control of thermal

near-field radiation may aid the engineering of thermal transport and emission properties.

However, experimental access to the thermal near-field has only recently been achieved

[185, 82]. In order to advance the nascent study of the thermal near-field, I use a variation

of s-SNOM to explore the spatial, spectral and coherence properties of thermal infrared

near-field radiation.

4.2 Historical background

The study of thermal radiation dates back to 1800, when William Herschel discovered

the infrared contributions of dispersing solar light with a prism and placing a thermometer

below the red end of the spectrum[67, 131]. While no visible radiation was incident on the

39

thermometer, its temperature reading rose due to what Herschel termed “infrared” light,

or “below red” light. Further, more precise measurements of solar radiation performed

by Joseph Fraunhofer led to the observation of narrow absorption lines [49], which where

soon related to the emission lines of gasses heated to incandescence [90]. This relationship

between the absorptivity of the emissivity of a material is characterized by “Kirchoff’s law”,

which states for a material in thermal equilibrium with its surrounding environment, its

absorptivity α(ω) equals its emissivity ε(ω) [91]. The development of Kirchoff’s law lead to

the concept of a “perfect blackbody”, a material that perfectly absorbs all incident light.

The description and characterization of perfect blackbody emission became an experimental

and theoretical challenge for researchers in the late 1800s [125].

Pioneering experiments performed by Otto Lummer, Ernst Pringsheim, and Ferdinand

Kurlbaum in the 1890’s advanced the understanding of thermal blackbody emission [114, 184,

112, 115, 113]. A blackbody was simulated by a cavity with walls made of highly-absorptive

blackened platinum, with only one small aperture in a wall at the end. Any light that enters

the aperture has to reflecs off the wall many times before it could exit the cavity back through

the aperture. If the walls are highly absorptive, the probability of re-emission is low and thus

the absorption of the aperture is high and closely simulates a blackbody. When the walls

are heated, the emission from the aperture will closely resemble blackbody emission. This

cavity structure allowed the first spectrally resolved measurement of blackbody radiation.

However, the measured spectrum had no theoretical explanation.

Early attempts to theoretically describe the emission were performed by Wilhelm Wien,

Lord Rayleigh, and James Jean [183, 78], however none of the derived distributions accu-

rately described the experimental results. In 1900, Max Planck presented a distribution he

empirically fit to measured spectrum. This function, known as “Planck’s law of blackbody

radiation” is given by

ξbb(ν, T ) =8πν2

c3

hν

exp( hνkBT

)− 1, (4.1)

40

where Plank’s constant h, was an empirical factor used to fit the distribution.

Plank’s law can now be derived using modern quantum optics, by modelling the black-

body as a enclosure formed by perfectly reflective walls a uniform temperature T . Modes

must take discrete modes with wavevectors given by nπ/L for enclosure size L and inte-

gers n. For a three dimensional rectangular cavity, there are three wavenumbers (kx,ky,kz)

that determine the cavity mode given by kx,y,z = nx,y,zπ/L. These wavevector components

must satisfy the relation k2x + k2

y + k2z = (ω/c)2, which defines a sphere in k-space of ra-

dius kmax = ω/c. By counting the number of modes within this sphere, we may obtain the

spectral electromagnetic mode density in free space,

D(ω) =ω2

π2c3dω. (4.2)

As photons obey Bose-Einstein statistics, their mode occupation number is

n(ω, T ) =1

exp ~ωkBT− 1

. (4.3)

This contribution from modern quantum mechanics resolved the so-called “ultraviolet catas-

trophe” that plagued early attempts to theoretically describe blackbody radiation by re-

ducing the occupation probability of states above a characteristic energy given by kBT .

Combined with the energy of a photon E(ω) = ~ω, the spectral energy density of blackbody

radiation is

uff(ω, T ) = E(ω)n(ω, T )D(ω) (4.4)

=~ω3

π2c3

dω

exp(

~ωkBT

)− 1

. (4.5)

This is known as Plank’s law of blackbody radiation. For real materials or “grey bod-

ies”, the emission is modified by the materials emissivity ε(ω), to give emission E(ω, T ) =

ε(ω)uff(ω, T ). We can identify Θ(ω, T ) = E(ω)n(ω, T ) = ~ω/(

exp ~ωkBT− 1)

as the mean

energy of a quantum harmonic oscillator at temperature T , whereby Plank’s law can be put

in a more general form

u(ω,~r, T ) = Θ(ω, T )ρ(ω,~r). (4.6)

41

Here, the energy density is written in terms of a generalized, spatially dependent density of

states ρ(ω,~r, T ) which is known as the electromagnetic local density of states (EM-LDOS).

The EM-LDOS contains information of the system geometry and dielectric properties. It

is derived by through combination of the Green’s dyadic function, and the fluctuation-

dissipation theorem.

The electromagnetic energy density is also given by Poynting’s theorem, which yields

u(ω,~r, T ) =ε

2〈|E(~r, ω)|2〉+

µ

2〈|H(~r, ω)|2〉. (4.7)

within a linear dielectric with permittivity ε and permeability µ. We can relate the electric

and magnetic field to the thermally excited electric ~j(~r) and magnetic ~m(~r) currents in a

medium using Green’s dyadic functions:

~E(~r, ω) = iµ0ω

∫ ←→G E(~r, ~rb, ω) ·~j(~rb, ω)d3~rb (4.8)

~H(~r, ω) =

∫ ←→G H(~r, ~rb, ω) · ~m(~rb, ω)d3~rb (4.9)

where the volume integrals are computed across the entire thermal body. Here,←→G E and

←→G H are the electric and magnetic Green’s dyadic tensors that describe the system geometry

and have integral expressions given in Ref Ref. [24]. The fluctuation-dissipation theorem

relates the statistics of the thermally-excited currents to the material’s losses, given by the

imaginary part of its dielectric function:

〈ji(ω, ~r1)jj(ω′, ~r2)〉 = 4πε0ωIm(ε(ω)) ·Θ(ω, T )δijδ(~r1 − ~r2)δ(ω − ω′). (4.10)

Following the derivation in Ref. [83], I introduce the correlation functions Eij and Hij for the

electric and magnetic fields, respectively, for a stationary system:

Eij(~r, ~r′, ω, ω′) = 〈Ei(~r, ω)E∗j (~r′, ω′)〉 (4.11)

Hij(~r, ~r′, ω, ω′) = 〈Hi(~r, ω)H∗j (~r′, ω′)〉. (4.12)

42

Using the results from Eq. 4.9 and Eq. 4.12, these correlation functions can be expressed as

Eij(~r, ~r′, ω) = Θ(ω, T ) · µ0ω

2πImGEij(~r, ~r

′, ω)

(4.13)

Hij(~r, ~r′, ω) = Θ(ω, T ) · ε0ω

2πImGHij (~r, ~r

′, ω). (4.14)

Using the expression for the energy density u(ω,~r, T ) = 2∑3

i=1 (ε0Eii(~r, ~r, ω) + µHii(~r, ~r, ω)),

we can then write

u(ω,~r, T ) = Θ(ω, T ) · ωπc2

Im

Tr(←→G E(~r, ~r, ω) +

←→G H(~r, ~r, ω)

), (4.15)

identifying the EM-LDOS ρ(ω,~r) as

ρ(ω,~r) =ω

πc2Im[Tr(←→G E(~r, ~r, ω) +

←→G H(~r, ~r, ω)

)]. (4.16)

We consider now a specific geometry consisting of the interface between two semi-

infinite media of dielectric functions ε1, ε2 of the upper (z > 0) and lower (z < 0) half

spaces, respectively. The Green’s function for this system relates a microscopic current

density in the lower half space to the fields generated in the upper half space through the

momentum dependent Fresnel coefficients rp,s(ω, q) for a planar interface. Using the known

forms of the Green’s functions, we can express ρ(ω,~r) as an integral over k-space:

ρ(z, ω) =ρv(ω)

2

∫ ω/c

0

k‖k0 |k1,⊥|

2− |rs12|2 − |rp12|

2

2dk‖︸ ︷︷ ︸

propagating

+

∫ ∞ω/c

4k3‖

k30 |k1,⊥|

Im(rs12) + Im(rp12)

2e−2Im(k1,⊥)·zdk‖︸ ︷︷ ︸

evanescent

,

(4.17)

where k‖ and k⊥ are the wavevectors parallel and perpendicular to the interface. This

integral is split into a contribution from propagating wavevectors k‖ < ω/c and evanescent

wavevectors k‖ > ω/c. In the quasistatic limit (k‖ k0), the contribution from propagating

wavevectors may be neglected, the Fresnel coefficients reduce to rs1,2 = 0 and rp1,2 = (ε2 −

ε1)/(ε2 − ε1. The EM-LDOS then takes the form

ρnf(z, ω) ≈ 1

4π2ωz3

Im(ε2)

|ε2 + ε1|2. (4.18)

43

From this quasistatic form, we see that the EM-LDOS is peaked at frequencies when either

Im(ε2(ω)) is peaked as for a vibrational resonance, or when ε2(ω) = −ε1(ω), as is the case for

a surface polariton resonance. Also, the distance dependence scales as 1/z3 in the extreme

near-field, although there may be shallower distance dependencies for intermediate distances.

The spatio-spectral dependence of ρ(z, ω) is shown in Fig. 4.1 for a) PTFE, b) SiC,

and c) SiO2. The pink dashed lines are the calculated dispersion relations for each material,

and the white dashed line is the light line. Far from the surfaces of these materials, the EM-

LDOS is dominated by contributions with momenta below the light line. However, at close

distances (ii,iii), high wavevectors begin to appear at the corresponding material resonance

frequencies. This occur at the vibrational resonance frequency for PTFE, where Im(ε(ω)) is

peaked, or when ε(ω) = −1 for the polariton resonances of SiC and SiO2.

4.3 Thermal infrared near-field spectroscopy

In thermal infrared near-field spectroscopy (TINS), a sharp tip is brought within the

sub-wavelength distances of a surface to probe the spectral spatial and coherence properties

of the thermal near-field. Fig. 4.2 shows the experimental concept of TINS. A sample surface

is heated resulting in an evanescent EM-LDOS ρevan(z) that decays quickly above the sample

surface. The confined electric near-field (relate to rho) polarizes a small sphere when it is in

close proximity with the surface. The polarization results in scattered propagation photons.

Simlarily, a tip with a sharp apex radius a acts like a local scatterer of the thermal near-field.

TINS is thus similar to s-SNOM, where the thermal evanescent near-field instead of a laser

or external source illuminates the tip-sample region.

As the first implementation of TINS, Ref. [185] revealed variations of the spatial dis-

tribution of the EM-LDOS above Au stripes on SiC and SiO2substrates were imaged with

∼ 100 nm resolution. Due to their high IR conductivity, the Au stripes showed strong

IR contrast from their dielectric substrates. Additionally, spectrally filtering the scattered

near-field intensity at the SPhP resonance of SiC caused the contrast to reverse, and the

44

700 900 1100 900 1100 13001000 1200 1400

1

2

0

2

0

4

6

2

0

4

6

8

10z=0.1µmz=0.1µmz=0.1µm

z=1µmz=1µmz=1µm

z=10µmz=10µmz=10µm

Scaled 10x

Scaled 10x

[1/H

z∙m

3]

[1/H

z∙m

3]

[1/H

z∙m

3]

1∙10-3

1∙10-2

5∙10-2

0

0

0

(a) PTFE (b) SiC (c) SiO2

k|| /k

0 [ω=

10

00

cm

-1]

Wavenumber [cm-1]

(i)

(ii)

(iii)

Figure 4.1: Momentum-frequency phase space plot of the EM-LDOS ρ(~r, ω) for a) PTFE, b)SiC, and c) SiO2at distances i) z = 10µm, ii) z = 1µm, and iii) z = 100 nm. The white andpink dashed lines are the dispersion relation for free space light and the material, respectively.At z = 10µm, the behavior reproduces the far-field behavior, where only contributions belowthe light line are present. At closer distances (ii, iii), high momentum contributions appearat the vibrational resonance frequencies and begin to dominate the EM-LDOS. Figure anddata courtesy of Ref. [81].

45

a) b) c)

aρ

evan(z)

z z z

aP P

Figure 4.2: a) Surface confined evanescent electromagnetic local density of states (EM-LDOS) ρevan(z) at the interface. b) A small polarizable sphere with radius a, can scatter the

near-field into detectable far-field radiation due to the induced radiating polarization ~P . c)Similarly, a sharp scanning probe tip with apex radius a, can act as the near-field scatterer.From Ref. [81]

SPhP resonance of SiC dominates the TINS signal. This was a signature of the predicted

resonantly enhanced EM-LDOS, however no direct spectrum was recorded. Follow up work

from our group developed and performed by Dr. Andrew Jones was the first spectroscopic

measurement of the resonantly enhanced EM-LDOS [82]. Spectra were obtained of the SPhP

resonance of SiC and SiO2, as well as the C-F vibrational response of PTFE. This work con-

firmed theoretical predictions that random, stochastic excitations in a medium can couple

to nonlocalized coherent material excitations.

Scanning probe microscopy techniques have also enabled measurement of nano-scale

temperature variations through scanning thermal microscopy [54]. This is performed by em-

bedding a thermocouple thermometer into the apex of the tip and scanning the tip across the

sample in contact mode AFM. These probes are also used to measure nanoscale heat transfer

[101, 88]. By studying the material dependence of the tip coating and the sample, Ref. [88]

studied the effect of mutual resonant coupling on the thermal transport properties. Strong

thermal transport was observed between SiO2surfaces, due to the shared SPhP resonances,

while much weaker transport was observed between non-resonant SiN and Au surfaces.

46

Lock-in

BS

Delay

arm

MCT

AFM Tipν

d

CP

Δx(a)

OAP

Retracted

In contact

Window

CB removed

(b)

SiC −400 −200 0 200 400−2

0

2

4

6

8

10

12

Optical delay [µm]

Inte

nsity [a

rb. u

.]0 200 400 600

0

100

200

x [nm]

He

igh

t [n

m]

200 nm

Resistor

10 µm 1 µm 200 nm

(c) (d) (e) (f)

Ttip

=700K

50 nm

Figure 4.3: (a) Thermal infrared near-field spectroscopy using heated AFM tips to provideboth local excitation and scattering of the thermal near-field of the sample, as detected byFTIR spectroscopy. (b) Interferograms with tip retracted (blue), tip in contact above SiC(red), and with the centerburst removed (green) by a window function (magenta). Long rangeinterference in contact (red) indicates coherent contribution due to the SPhP resonance. (c,d) SEM images of thermal cantilever tips. (e) SEM image of test structure with correspondingAFM line trace (f) with apex radius of ∼ 50 nm. Figure from Ref. [143].

4.4 Experiment

The experimental setup (Fig. 4.3(a)) is based on an atomic force microscope (AFM,

Anasys Instruments) using either customized silicon tips with a built-in resistive heater or

conventional AFM tips heated with focused laser light [82, 143, 145]. Fig. 4.3(c,d) show

SEM images of Joule-heated cantilever assembly (c) and tip apex region (d). Briefly, these

cantilevers consist of two separate legs that join at the end where the sharp AFM resides.

The cantilever legs are made from highly-doped conductive Si, while the tip region is made

of low-doped Si with lower conductivity. The tip region then acts like a resistor, and when

a current flow between the cantilever legs the tip region experiences Joule heating while the

cantilevers remain relatively cool [134].

47

AFM line trace (f) of a ∼ 200 nm high test structure (e) allows us to verify a regular

tip shape and derive an apex radius of ∼ 50 nm. The use of these tips allows for local

sample heating up to ∼ 550 K while maintaining stable non-contact AFM feedback. The

enhanced EM-LDOS of the thermal evanescent near-field scatters when interacting with a

nanoscopic tip providing a perturbation for k-vectors ∝ 1/r, with r the tip apex radius.

The scattered IR light is collected with an off-axis parabolic mirror (OAP), sent through a

Michelson-type interferometer with a beam splitter (BS) and a compensation plate (CP), and

detected with a mercury-cadmium-telluride detector (MCT) (J15D14, EG&G). We use either

a BaF2 beamsplitter (with a IR frequency cut-off at 900 cm−1) or a KBr beamsplitter (with

transparency down to 500 cm−1, however with slightly lower IR transmission) depending

on the desired IR spectral region. Interferograms obtained through scanning the delay arm

provide spectral information through Fourier transform infrared spectroscopy (FTIR).

Lock-in demodulation at the tip-dither frequency νd discriminates the near-field signal

from the far-field background. As noted above, in conventional s-SNOM experiments, higher

order harmonics are used to discriminate the near-field light from the large background.

However, in TINS the weak near-field strength yields weak scattered field with only the

first harmonic showing appreciable signal. Since the tip cantilever is heated directly and

is thus the main source of thermal radiation, there is typically a background signal as the

tip oscillates in the focus of the parabolic mirror. However, for tapping amplitudes < 80 −

100 nm, this background can be minimized. In contrast, in conventional IR s-SNOM the

illumination focal spot is much larger than the tip shaft, and interference from multiple

reflections gives rise to a large and distance dependent background. Our finding is consistent

with previous results of first-harmonic demodulation in TINS, showing ∼ 50 nm spatial

resolution for spectrally integrated studies [82, 143, 156].

48

4.4.1 Laser heating of scanning probe tips for thermal near-field spectroscopy

and imaging

TINS and other nanoscale thermal measurements based on scanning probe microscopy

typically require independent control of tip and sample temperature. Tip temperature con-

trol has often been provided through Joule heating by flowing current through highly-doped,

conductive cantilever legs across a low-doped, resistive tip region as discussed above. Such

electrically-heated tips mounted at the end of cantilevers have been successfully used for

chemical nano-identification based on melting point [133], for nano-machining [93], and

to thermally excite a sample to perform thermal infrared near-field spectroscopy (TINS)

[82, 143].

However, the micron-scale precision needed to fabricate electrodes and doping varia-

tions in cantilevers creates difficulties and prevents reproducibility. Joule-heated tips typi-

cally have sub-optimal radii of curvature limiting spatial resolution, require complex fabrica-

tion, and yield low near-field optical signal as the electrodes often preclude metallic coating

and limit the type of possible tip materials. Additionally, electrically heated tips are limited

to maximum temperatures of up to ∼ 900 K by “thermal runaway”, whereby an increas-

ing thermally-excited carrier density reduces the resistance with increasing temperature and

creates a positive feedback mechanism. This effect makes precise control of the temperature

difficult. In addition, the heating is no longer localized to the tip as cantilever legs also start

heating significantly [35].

As an alternative, AFM probe heating through focused laser irradiation has been used

to heat tips for thermal conductivity measurements [137, 164], for photothermal actuation

of cantilevers [2], and for nano-machining [34]. I integrate and demonstrate laser heating

and thermal control of simple cantilever tips to perform TINS to perform spectroscopy on

vibrational resonances of hexagonal boron nitride (hBN), silicon carbide (SiC), and polyte-

trafluoroethylene (PTFE). I characterize heating and cooling rates as well as achievable tip

49

temperatures. Good agreement between theory and experiment is found with the vibrational

frequencies of PTFE and the in-plane and out-of-plane phonon polariton (PhP) modes of

hBN. The surface phonon polariton (SPhP) resonance of SiC is strongly red-shifted since it

is highly sensitive to extrinsic perturbation.

4.4.1.1 Laser heating experiment

I use a continuous-wave laser (Verdi-G, Coherent, λ = 532 nm) for photothermal

heating of Si cantilevers (Access-NC, AppNano) while operating in dynamic-mode AFM

feedback. As shown in Fig. 4.4a), the laser output is fiber-coupled using a multimode fiber

(core diameter φ = 62.5 µm) into an overhead imaging arm and focused onto the backside

of the cantilever by a microscope objective (NA = 0.3, f = 16 mm), which in turn allows for

monitoring the φ ∼ 10 µm laser spot with a CCD imaging system. A long-pass filter blocks

the scattered and reflected light from the AFM feedback laser (λ = 670 nm) to prevent

interference with the AFM operation. I designed custom-coated cantilevers with platinum-

iridium on the tip-side for enhanced the interaction between the tip and thermal near-field yet

uncoated on the backside to allow strong laser absorption. The heating is largely localized to

the end of the cantilever. The cantilever resonance frequency at f ∼ 300 kHz shifts by 1-2%

at moderate temperatures, which is small enough to not disturb dynamic-mode feedback.

Far-field tip emission spectra indicate that heating up to the melting point of Si can be

achieved (∼ 1800 K), with correspondingly broad IR emission spectrum (Fig. 4.4b)).

The AFM tip is brought into sample contact and used with the standard TINS configu-

ration as described above. Again, lock-in demodulation of the optical signal at the tip-dither

frequency (Ω) discriminates the near-field light from the far-field background [94]. Despite

the higher tip cantilever temperature, the low emissivity of the metal tip coating, and the

localized heating generally confined to the tip shaft and its immediate surroundings, the

relative far-field background is small and we find first harmonic demodulation sufficient for

near-field extraction, in contrast to the use of higher harmonic demodulation as used in

50

s-SNOM under external illumination.

Although higher order harmonics provide higher spatial resolution and improved back-

ground subtraction, they have generally weaker signal-to-noise ratios [94]. We perform

Fourier transform infrared spectroscopy (FTIR) by scanning the delay arm controlled with a

closed-loop delay stage (ANT95-50-L, Aerotech). A compact design (delay arm length ∼ 12

cm) minimizes propagation losses and facilitates alignment.

4.4.1.2 Laser heating results

Fig. 4.5 shows the IR emission characteristics of the laser-heated tips far from a sur-

face. Spectra acquired with increasing laser power P (Fig. 4.5a)) show an increase in overall

intensity and a shift to higher frequencies with higher temperature. The spectra are fit to a

blackbody spectral energy dependence u(ν, T ) = 2hc2ν3 1ehcν/kBT−1

multiplied by a power law

D(ν) ∝ ν1.2 to incorporate the spectral dependence in MCT detectivity (dashed lines). This

power law dependence was determined by far-field calibration using a blackbody emitter at

a known temperature. With increasing laser intensity, we see an increase in temperature up

to 610 K. The highest usable temperatures for near-field imaging and spectroscopy is limited

by AFM feedback stability to a more moderate temperature range of 600-700 K. At higher

temperatures the cantilever spring constant is reduced and its resonance frequency simul-

taneously red-shifts and broadens. Additionally, the resonance becomes less well-defined,

possibly due to temperature inhomogeneities across the cantilever.

Using an optical chopper to periodically modulate the heating laser intensity (P = 450

mW) and measuring the spectrally integrated IR emission at the chopper frequency, we can

estimate the cooling rate of the tip. When the laser is on, the tip heats up rapidly until

an equilibrium is reached between the constant heating power and the cooling pathways

(described below). When the laser is off, the temperature decreases exponentially and the

emitted radiation intensity decays. For sufficiently long modulation times, the tip relaxes

to ambient temperature and the temperature difference, and therefore the change in optical

51

signal, between heating cycles saturates. In Fig. 4.5b) this saturation occurs below 200 Hz,

indicating a cooling time of less than 5 ms.

Fig. 4.6a) shows interferograms of tip-scattered IR light of a hBN/SiO2 structure (hBN

thickness = 200 nm) (blue), and with the tip far from the sample (red) as a reference

measurement. The green curve is a fit to the long-lived oscillations away from zero path

delay, which captures the resonant, temporally-coherent contribution to the signal due to the

PhP resonance of hBN. Fig. 4.6b) shows the corresponding spectra from the interferograms in

Fig. 4.6a). The near-field spectrum of hBN shows peaked emission at the in-plane transverse-

optical (TO) phonon mode frequency at νTO‖ = 1350 cm−1.

The blue curves in Figs. 4.6c) and d) show the interferogram and spectrum acquired on

a hBN/Au structure (hBN thickness ∼ 2 µm). In this case, the out-of-plane mode at νTO⊥ =

780 cm−1 dominates the spectrum, while the in-plane mode is absent (discussed below). The

spectral positions of the in-plane mode and the out-of-plane mode from Figs. 4.6b),d) agree

with literature values [20, 21]. The near-field spectrum of hBN/SiO2 shows peaked emission

at νTO‖ = 1350 cm−1, matching the in-plane phonon mode resonance frequency (Fig. 4.6b)

[20, 21]. The out-of-plane mode response at νTO⊥ = 780 cm−1 is not observed due to the use of

a BaF2 beamsplitter with a IR bandwidth cut-off at ∼ 900 cm−1. In contrast, the near-field

spectrum using a KBr beamsplitter of hBN on a Au substrate shows peaked signal at the

out-of-plane phonon mode resonance frequency νTO⊥ = 780 cm−1, while the in-plane mode is

not observed. The absence of the in-plane phonon response may be due to screening from

the Au substrate which reduces the in-plane mode strength while enhancing the out-of-plane

phonon response.

The TINS spectrum of SiC shows a strong peak at ∼ 885 cm−1 (Fig.4.10c),d)). The red

curve shows calculations of uden(ω, z, T ) at temperature T = 600 K and distance z = 50 nm

from the surface using literature values of the dielectric function, which peaks at ω = 948

cm−1, due to the surface phonon polariton of SiC [152]. The observed near-field peak is

significantly red-shifted due to tip-sample coupling, as discussed below. In contrast, the

52

near-field spectrum of PTFE shown in Fig.4.10b) shows peaks that are not shifted with

respect to uden(ω, z, T ) calculated using dielectric values from literature [122].

The peaks closely match the symmetric νs = 1158 cm−1 and the unresolved superpo-

sition of the two antisymmetric νas = 1210 and 1240 cm−1 C-F vibration modes.

Delay

arm

MCT

det. CP

Δx

λ = 532 nm

Objective

AFM tip

CCD

WL LED

BS

2000 4000 6000 80000

1

2

3

4

5

Inte

nsity (

arb

. u

.)

Wavenumber ν (cm-1)

TINS Fit, 1800 K

b)a)

ENF

z

xy

Figure 4.4: Laser radiation (λ = 532 nm) is combined with white light from an LED witha dichroic beamsplitter, and focused onto the backside of the cantilever with a microscopeobjective. A CCD allows in-situ imaging of the tip, the heating laser, and the samplesurface. Inset image shows overhead view of tip cantilever, with depiction of heating laserspot. The sample is indirectly heated locally through conduction across the tip-samplegap. The evanescent thermal near-field is scattered by the apex of the tip into far-fieldradiation, collected by an off-axis parabolic mirror and detected interferometrically. b) Far-field emission spectrum from tip indicates heating level of ∼ 1800 K

The high laser powers necessary for significant heating is due to high thermal dissipation

of the cantilevers, which occurs through several mechanisms. The heat flow through the

cantilever (Φcant = − κ∆xA∆T ) is roughly 3·10−4 W/K, using a cantilever cross-section of A '

300 µm2 and Si thermal conductivity of κSi = 1.3 W/cm/K. Radiated heat flow calculated

with the Stefan-Boltzmann law is negligible (Φrad ∼ 10−13 W). Heat conduction from the

cantilever through air is more difficult to quantify and depends on numerous parameters such

53

1000 1500 2000 2500 30000

100

200

300

400

500

600

800 mW, 610 K

300 mW, 420 K500 mW, 530 K

Wavenumber (cm-1)

Inte

nsity (

arb

. u

.)a) b) 1100

Φrad.

Φair

Φcant.

0 100 200 300 400 500

800

900

1000

Te

mp

era

ture

ch

an

ge

(a

rb. u

.)

Laser modulation frequency (Hz)

Figure 4.5: a) Far-field tip emission spectra as a function of laser heating power P . Spectraare fit to a blackbody model multiplied by a power law to correct for MCT detectivity. b)Heating dynamics measured by modulating heating laser power P = 450 mW and measuringthe spectrally integrated infrared emission.

as tip height, however it has been estimated to be on the order of Φair ∼ 10−5 W/K[89]. These

thermal dissipation pathways determine the temperature dynamics shown in Fig. 4.5b) which

may prove critical for frequency-resolved thermal measurements [99], thermal actuation of

cantilevers [2], improving signal-to-noise by reducing 1/f measurement noise [102] or for

high-speed fabrication techniques using laser-heated tips [118, 36]. Careful control of these

thermal pathways could allow optimization of the thermal time constant for the desired

implementation.

4.4.2 Spectral frustration in TINS

TINS fundamentally relies on the frustration of the evanescent field through scattering

by a nano-particle or apex of a scanning probe tip. This raises the question of the degree

of perturbation and thus relationship of the measured signal to the unperturbed EM-LDOS.

In Ref. [82], a generally good agreement has been found between TINS experiment and

54

-100 -50 0 50 100

1000

2000

3000

4000

5000

6000

1000 1200 1400 1600 1800 2000

100

200

300

400

500

-100 -50 0 50 100

500

600

700

800

900

1000

1100

600 800 1000 1200 1400

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity (

arb

. u

.)

Path delay (µm) Wavenumber ν (cm-1)

Inte

nsity (

arb

. u

.)

Path delay (µm) Wavenumber ν (cm-1)

Inte

nsity (

arb

. u

.)In

ten

sity (

arb

. u

.)

TOν||TINS

Fit

Retracted

a) hBN/SiO2

b)

TOν^

c) hBN/Au d)

Figure 4.6: a) TINS interferogram and b) spectrum of hBN on a SiO2 substrate. Peakedthermal emission occurs at the in-plane phonon mode frequency νTO

‖ = 1350 cm−1. Red

curves are reference tip emission measurements acquired far from the sample surface. c)Interferogram and d) spectrum of hBN on a Au substrate dominated by out-of-plane phononresponse at νTO

⊥ ' 780 cm−1.

theoretical resonant EM-LDOS peak positions and linewidths of both intrinsic molecular

resonances, as well as for weak extrinsic surface phonon polariton (SPhP) resonances of

SiO2 [82]. However, follow-up work from Ref. [10] on SiC found a large ∼ 40 cm−1 redshift

and significant broadening of the SPhP resonance. This effect was attributed to tip-sample

55

coupling, but modeling the tip as a spherical scatterer required the assumption of a 1.6 µm

tip apex radius which is 1− 2 orders of magnitude larger than the actual tip radius [10].

I performed TINS on the strong SPhP response of SiC to investigate the role of the tip

and its degree of perturbation of the intrinsic response. As a strongly dispersive resonance,

SiC is a good candidate for the study of perturbation of the tip. My spectra using tips with

different apex radii of curvature revealed a large and variable redshift of the SPhP resonance

frequency ranging from ∼ 5 cm−1 up to ∼ 50 cm−1, indicating that the SPhP resonance

is highly sensitive to external perturbation as introduced by the AFM tip. Modeling per-

formed by William Lewis using conventional tip-sample interactions through dipole-dipole

coupling explains some of the shift, it seems insufficient to explain the most extreme spec-

tral red-shift. This suggests a modification of the SPhP condition through a previously not

considered effective medium change due to the tip proximity to the surface as a possible ad-

ditional contribution. In addition I observe an exponential distance dependence as compared

to a much steeper scaling as expected and previously observed for the thermal evanescent

near-field of uncorrelated local emitter [82]. This exponential distance dependence can be

explained by spatial coherence arising from the correlation of thermal polarization via the

non-local nature of the SPhP, despite the excitation through stochastic fluctuations. How-

ever, a breakdown of macroscopic electrodynamics for distances shorter than the phonon

mean-free path is also discussed as a possible alternative [106]. The results highlight scat-

tering and frustration as a sensitive probe of, as well as means to control, thermal near-field

properties.

4.5 Spectral frustration Results

Fig. 4.3(b) shows typical TINS interferograms. With the tip retracted (blue), the nar-

row centerburst reflects the large spectral bandwidth and low coherence of the heated tip

blackbody spectrum. When near SiC, the long-lived interference fringes in the tip-scattered

interferogram (red) indicates the increased temporal coherence of the quasi-monochromatic

56

800 1000 1200 1400 1600 1800 20000.0

0.2

0.4

0.6

0.8

1.0

Wavenumber [cm−1]

TIN

S s

ign

al [a

rb. u

.]

800 900 1000 11000.0

0.2

0.4

0.6

0.8

1.0

Wavenumber [cm−1]

TIN

S s

ign

al [a

rb. u

.]

800 900 1000 11000.0

0.2

0.4

0.6

0.8

1.0

Wavenumber [cm−1]

TIN

S s

ign

al [a

rb. u

.]

TINS spectrum

Model

unf(z,ω,T)

TINS SiC Spectra

[cm−1]

De

lay [µ

m]

800 1000 1200

−400

−200

0

200

400−5

−4

−3

−2

−1

0

log

(In

t.)

(a)

(b) (c)

Figure 4.7: Fourier transformed TINS spectra of SiC (blue, with corresponding spectro-gram). Fits (black dashed) using finite dipole model (a,b) and effective medium model (c).Calculated spectral energy density u(100 nm, ω, 300 K) (red, arb. units). Spectra are red-shifted by about 5 cm−1(b), 25 cm−1(a), and 50 cm−1(c), indicating the high sensitivity ofthermal SPhP response to extrinsic effects.

SPhP resonantly-enhanced thermal near-field. Fig. 4.7(a) shows the corresponding Fourier

transform and spectrogram (inset) of the in-contact interferogram in Fig. 4.3(b). The en-

hanced emission peak is induced by the SPhP resonance with flat spectral response otherwise.

The narrow linewidth ∼ 15 cm−1 is evidence of the strong temporal coherence of the SPhP

response, corresponding to a coherence time of several ps. We note that the technique is

inherently broadband with no specific resonances due to tip material or geometry of the Si

tips in the spectral range investigated.

For subsequent data, we first apply a window function (Fig. 4.3(b), magenta) to sup-

57

press the centerburst of the interferogram (see Fig. 4.3(b), green), which is dominated by the

incoherent both near- and far-field off-resonant background. Two additional representative

Fourier transformed (pre-filtered) interferograms are shown in Fig. 4.7(b) and (c). Each

spectrum in Fig. 4.7(a-c), taken using different tips, show the peaked SPhP near-field signal.

It is associated with the spectral energy density of SiC u(z, ω, T ) (red, calculated for z = 100

nm and T = 300 K), yet redshifted by variable amounts ranging from 5 cm−1 (b), 25 cm−1

(a), and to 50 cm−1 (c).

I performed TINS distance dependence measurements by acquiring spectra while fix-

ing the tip-sample distances as measured through approach curves. Each spectra shows

peaked emission at the SPhP resonance frequency, but at variable strength relative to the

background. Integration between 875 cm−1 and 945 cm−1 extracts the on-resonance signal,

while integrating between 1275 cm−1 and 1345 cm−1 gives the background emission, primar-

ily due to far-field emission from the tip-shaft. Fig. 4.8 shows the resulting TINS distance

dependencies with integration on- (a), and off-resonance (b), for my measurements using

long acquisition times and averaging over several interferograms (red circles), and measure-

ments from Ref. [82] with shorter acquisition times (blue circles). For comparison, laser

s-SNOM (λext = 10.8 µm) distance dependencies using the same tips are shown for dif-

ferent harmonic demodulations (νd and 2νd,) exhibiting much stronger near-surface spatial

confinement compared to TINS.

4.6 Discussion

Spectral shifts I chose SiC as a system with a strongly dispersive SPhP resonance ex-

pected to peak at 948 cm−1 at the SiC/air interface to investigate the degree of perturbation

imposed by the AFM tip. In contrast to earlier experiments [82, 10], the variable and large

spectral shifts of the TINS SPhP peak observed here suggest more complex mechanisms with

regards to optical tip-sample coupling and frustration of the evanescent field than previously

assumed.

58

0.01

0.1

1

Inte

nsity [a

rb. u

.]

100 1000

0.1

1

Tip−sample distance [nm]

(b) Off-resonance

(a) On-resonance

Int. [a

rb. u

.]

TINSExp. fit

Laser s−SNOM

1st Harmonic2nd Harmonicu(z,ω,T)

20

Figure 4.8: Distance dependence of spectrally integrated TINS measurements of SiC (blueand red circles). On-resonance data (a) show exponential dependence, with a decay constantof (5.5± 1.0) · 103 cm−1 (black curve, grey region shows uncertainty of fit). Approach curvesfrom laser s-SNOM at the first (green) and second (blue) cantilever harmonic for comparison.The energy density u(z, 948 cm−1, 300 K) shows a 1/z3 divergence at short distances.

In principle a few wavenumber red-shift can arise from phonon softening when heating

the sample [148]. The effect on the dispersion relation for the SPhP condition gives rise to a

at most 6-7 cm−1 spectral shift and a 2-3 cm−1 increase in linewidth in TINS when heating

from 300 to 550 K. This may account for small shifts such as the 5 cm−1 shift in Fig. 4.7(b)

but the larger shifts observed would require higher temperatures than were experimentally

accessible (T > 800 K).

s-SNOM spectra, in general, have been modeled via a modified frequency and tip-

sample distance dependent tip polarizability [10, 40, 94, 124]. In the simplest implementation

we model TINS as Rayleigh scattering by the tip with scattered power Pscat(z, ω, T ) =

Ceff · c · u(z, ω, T ). Ceff = k4|αeff |4/6π is the effective scattering cross section and αeff the

effective tip polarizability which depends on optical properties of the tip and sample, as well

59

as tip geometry and position above the surface. The spectral energy density is given by

u(z, ω, T ) = Θ(ω, T ) · ρ(z, ω) with EM-LDOS ρ(z, ω) and mean thermal photon energy at

frequency ω, Θ(ω, T ) [84, 81].

While TINS spectra of local mode molecular resonances can accurately be described

with at most few cm−1 shifts using the point dipole model (PDM) for αeff [82], it fails for

the strongly dispersive and extrinsic SPhP modes [10]. In contrast, the finite dipole model

(FDM) [40] simulates αeff via an extended induced charge distribution along the tip-shaft

over a characteristic length L, with tip radius R, and using a complex parameter g to describe

the effective tip-material response as described in Sec. 3.2.1.2.

Fig. 4.7(a) and (b) show our FDM fits (black dashed) using R = 150 nm and L = 3 µm

(a), and R = 50 nm and L = 300 nm (b), both using g = 0.8 exp(iπ/25) and convoluted with

experimental spectral resolution of 11 cm−1 (a) and 49 cm−1 (b). While the smallest 5 cm−1

spectral shift in (b) is readily accommodated using physically meaningful fit parameters, the

25 cm−1 shift (a) is already pushing the validity of the quasistatic nature of the model.

A spectral shift as large as 50 cm−1 (c) pushes the limits of the FDM. However, follow

up work has found parameters that can fit the red shift in (c)[77]. This suggests that either

the FDM does not capture the essential physics of the tip-sample coupling or additional

processes contribute to the spectral shift [124].

Note that larger values of R and L are required to fit in particular the 25 cm−1 shifted

spectrum (Fig. 4.7(a)) compared to those used to fit spectra obtained using laser-based s-

SNOM. For example, in a laser s-SNOM study of SiC with Pt tips a ∼ 25 cm−1 shift was

modeled with fit parameters R ≈ 35 nm and L = 300 nm [40]. The larger values of R

and L could reflect the differences in the local field distribution induced by the incident

laser versus those induced by the thermal near-field. Compared to the s-SNOM value of

g = 0.7 exp(i0.06) [40], the larger phase value of g reflects a smaller conductivity of our

uncoated Si tips compared to Pt-coated tips.

The thermal near-field interaction may alternatively be described as the tip modifying

60

the SPhP dispersion relation through the near-field interaction when brought close to the

surface. This model was used to describe the interaction of the probe with a laser-excited

surface plasmon field in a total internal reflection geometry in a photon scanning tunneling

microscopy (PSTM) study [155].

To describe the thermal near-field tip-sample interaction resulting in the 50 cm−1

redshift in Fig. 4.7(c), we therefore propose the possibility that the presence of the tip

near the surface gives rise to an effective change in dielectric environment ε1 = εeff 6= 1,

thus altering the SPhP condition in Eq. 4.18 (Re(εeff) = −Re(εSiC)) from that of vacuum

(εeff = 1), a method that has been applied to s-SNOM studies of nanoantennas [167], and

is a valid consideration when an inhomogenous medium is brought in the near-field region

(z . λ/2π ≈ 1− 2 µm) of the surface.

We take the local surrounding within the near-field region of the tip to first approxima-

tion as a uniform effective medium of tip material and vacuum given by a linear combination

as εeff = (1 − δ) + δεtip with 0 ≤ δ ≤ 1. Due to strong dispersion of SiC (Fig. 4.9(a)),

an increasing fraction of Si tip material (b) (dielectric function from Ref. [151]) results in a

continuous redshift of the SPhP spectral energy density (c). A modest ∼ 10% Si fraction can

already result in an up to 20 cm−1 redshift of the spectral energy density (c). With δ = 0.35

we can readily fit the experimental 50 cm−1 spectral shift in Fig. 4.7(c) (black dashed, convo-

luted with 39 cm−1 spectral resolution), even without consideration of additional geometric

factors that affect field-enhancement, retardation, and scattering.

Using metal coated tips in the laser-heating modality, the TINS spectrum peak position

of SiC is even more strongly red-shifted with respect to the peak frequency of uden(ω, z, T ).The

∼ 60 cm−1 red-shift is larger than previously observed and cannot be adequately explained

by conventional tip-sample coupling models or phonon softening [40]. Additionally, the ef-

fective medium model is not applicable to metallic tip materials with negative dielectric

functions [143]. Instead, we attribute this large red-shift to stronger tip-sample coupling and

modification of the local field distribution due to the metallic tip. The origin and description

61

900 910 920 930 940 950 960 970 980 990 10000.0

0.2

0.4

0.6

0.8

1.0

Wavenumber [cm−1]

900 920 940 960 980 1000

−2.0

−1.5

−1.0

Wavenumber [cm−1]Wavenumber [cm−1]

ee

ffe SiC

Vacuum

2% Si

4% Si

6% Si

8% Si

10% Si

Re

(eS

iC)

u(ω

) [a

rb. u

.]

eSiC

Re(eSiC

)

Im(eSiC

)

eeff

2.0

1.5

1.0

@

(b)

(c)

700 800 900 1000

−200

0

200

400(a)

Figure 4.9: Effective medium theory modifying SPhP resonance condition. (a) ε(ω) of SiCshowing strong dispersion due to the bulk transverse-optical phonon mode at 790 cm−1. (b)Effective dielectric function εeff(ω) for various fractions of Si in the spectral region marked(dashed box in a). The near-field spectral energy density u(100 nm, ω, 300 K) peaks forRe(εeff) = −Re(εSiC) (black dashed circles), with calculated spectral energy density (c) cor-responding to the effective medium of (b).

of the spectral shift requires additional follow up work.

In contrast to the large and variable shifts observed for strongly dispersive SiC, the

TINS peak positions of PTFE are not shifted compared to calculations of the energy density

Fig. 4.10b). This is because its localized vibrational modes are minimally sensitive to external

perturbation by tip-sample coupling. Additionally, the peak positions of hBN closely match

the phonon mode frequencies. Since it is not a truly surface confined polariton, it is expected

to be less susceptible to extrinsic perturbation.

While the physical mechanisms of the models of near-field coupling and effective

medium change appear different, both show how the tip locally tunes the EM-LDOS, in

62

particular for the case of extrinsic resonances such as SPPs and SPhPs. This is in contrast

to weakly dispersive local molecular vibrational resonances as shown, e.g., for PTFE where

the peak positions of the C-F molecular resonances are essentially unaffected by the presence

of the tip [82]. These results are also consistent with s-SNOM spectra where extrinsic or

strongly dispersive [39] resonances show significant spectral sensitivity to tip material and

geometry [51], while intrinsic resonance peak widths and positions are essentially unaffected

[56]. Irrespectively, the results show that the use of TINS for measuring the EM-LDOS of

materials with strongly dispersive resonances requires a minimally perturbing probe.

Distance dependence Along with the temporal coherence that results in the sharp

spectral features in TINS, the thermal near-field is also predicted to exhibit spatial coherence

when coupled to non-local resonances. In particular, for propagating surface polaritons, the

field will retain a phase correlation for distances on the order of their propagation length,

which can be up to 0.4 mm for SPhPs of SiC [24]. This dramatic effect has seen application

in the development of directional thermal emitter[5] However, this spatial coherence only

appears when the EM-LDOS is dominated by the evanescent contribution of the polariton.

Thus, the coherence is lost during propagation from the surface, and the coherence length

converges to the free space value for a blackbody of ∼ λ/2. Not only is the spatial coherence

near-field specific, it vanishes at very short distances from the surface, where the thermal

near-field is dominated by the 1/z3 contributions from local excitations [103].

Direct evidence of spatial coherence is thus far limited. Ref. [185] claimed the obser-

vation of spatial coherence of theramlly excited SPPs. In particular, fringes in intensity

appeared on gold stripes parallel to their edges when the signal was spectrally filtered (cen-

ter wavelength = 10µm, bandwidth = 1µm). These fringes were attributed to interference

of thermally excited plasmons in the Au stripe which acts like a cavity along its width.

However, a recent imaging study with ∼ 60 nm resolution on similar metal nano-structures

observed a spatially uniform infrared signal with no intensity fringes observed [95]. Addi-

tionally, calculations and approach curves in Ref. [95] indicate that the thermal near-field

63

intensity on Au is dominated by uncorrelated, localized dipolar excitations with short spatial

correlation length, and not SPPs in the MIR.

My results on the distance dependence of the TINS signal of the SPhP of SiC provides

direct evidence of spatial coherence in the thermal near-field. We now consider the spectrally-

integrated on- and off-resonance contributions to our TINS signal as a function of distance.

In agreement with theory, the experimentally observed (Fig. 4.8(b)) off-resonance near-field

signal is small, and only weakly distance dependent; except an experimentally not resolved

moderate increase expected at very close proximity to the surface [169]. On-resonance (a),

over the almost two orders of magnitude distance variation we find an exponential scaling

down to the shortest tip-sample distance of ∼ 20 nm. The behavior can be fit (black curve)

by |Ez|2 ∝ e−2kzz, with kz = (5.5± 1.0) · 103 cm−1. While a pronounced increase in spectral

energy density near the surface is expected, for the SPhP resonance, an exponential scaling

of u(z, 948 cm−1, 300 K) starting from the far-field regime (z > 4 µm) should transition to a

much steeper 1/z3 distance dependence below ∼ 1 µm (see Fig. 4.8(a), magenta) [24, 65].

We propose that the observed exponential behavior reflects the spatial coherence and

polarization properties of the SiC SPhP thermal near-field. It should be noted, however,

that the underlying mechanism is different compared to the exponential distance behavior

observed in photon tunneling microscopy of laser-excited surface plasmon polaritons, where

spatial coherence is expected due to the coherent excitation process [147]. In general, the

thermally induced random fluctuations of the transient optical polarization density in the

material lead to a spatially incoherent field [24], with a low degree of polarization [168]. It

is this incoherent field that underlies the 1/z3 dependent increase in spectral energy density.

In the case of SiC, this source polarization couples to the delocalized SPhP excitation, that

is spatially coherent in the sample plane, with coherence length of 10’s of microns [24]. This

gives rise to an exponentially decaying coherent SPhP field confined to within about ∼ 5 µm

of the surface with a decay constant ktheory = 2.2 · 103 cm−1 as derived from the SPhP

dispersion relation for SiC in air.

64

This spatially coherent and polarized thermal SPhP near-field is more efficient to induce

a superradiant polarization in the tip due to the finite spatial extent of the apex compared

to the incoherent term. The coherent thermal SPhP field can thus can be scattered more

efficiently compared to a spatially incoherent field from a randomly fluctuating optical po-

larization, thus giving rise to the experimentally observed exponential distance dependence.

This interpretation is supported by the distinct TINS behavior for thermally excited local

molecular vibrational resonances. Here, the lack of spatial coherence gives rise to a steeper

super-exponential distance dependence as experimentally demonstrated for PTFE previously

[82].

Note that tip-heating of the sample is less efficient at larger distances. This explains

the shorter experimental decay length with kz = (5.5 ± 1.0) · 103 cm−1, compared to the

theoretical value of ktheory = 2150 cm−1. Nevertheless, the long-range exponential scaling of

the TINS data is consistent with the coherent excitation of SPhPs, indicating that the hot tip

is effective at heating the sample at least within a distance of ∼ 1 µm 1 . Alternatively, the

effective medium model from above would also predict a shorter decay length with increasing

Si content in εeff which follows from the dispersion relation.

While SPhP coherence is a compelling explanation for the exponential distance behav-

ior, an alternative explanation is a possible breakdown of the macroscopic theory used to

derive the spectral energy density. The macroscopic theory assumes that the polarization

source currents j(r, ω) are delta correlated, 〈jα(r, ω)j∗β(r′, ω′)〉 ∝ δαβδ (ω−ω′)δ(r−r′) [106]. A

possible breakdown of the macroscopic theory was previously invoked to describe deviations

from the theoretically expected distance dependence of heat transfer in scanning thermal

microscopy [92]. Here the mean free path of the electrons (`e ≈ 10 nm) sets the scale for the

optical current correlations, that need to be compared to the tip length scale which describes

the tip-sample interaction. In our experiments, the relevant mean free phonon path, mea-

1 Thermal conduction between macroscopic cantilever with a large aspect ratio and a short tip providesefficient heating through ballistic heat thermal air conduction. Surface melting of polycarbonate sampleshave verified local heating up to ∼ 550 K.

65

sured to be on the order of 3−5 µm [50], is longer than apex radius or tip dipole extent that

follows from FDM to be in the 0.1− 1 µm range. A different explanation for the absence of

the divergent behavior was proposed in Ref. [31] whereby the heat transfer between metallic

plates does not exhibit the divergent distance dependence as attributed to the generation of

eddy currents in the medium.

While we believe that these latter two processes are less likely to be the dominant

mechanisms responsible for the exponential distance dependence observed, they highlight the

unique properties of the thermal near-field and the insight it can provide into the different

microscopic processes underlying the optical physics of thermal radiation.

In summary, the spectral and distance dependent response of TINS performed on

SiC as an example of an extrinsic and strongly momentum-dependent resonance highlights

the ability of thermal near-field scattering using TINS for gaining insight into microscopic

and coherent processes underlying thermal radiation. In addition, the high spatio-spectral

sensitivity demonstrates the ability of TINS for testing the validity of established models

used to describe s-SNOM in general which are otherwise difficult to discern. Furthermore, the

results indicate how coupled nano-structures or devices can locally modify the EM-LDOS

and tune the surface polariton resonance, offering nanoscale control of extrinsic resonant

material properties.

In summary, we have demonstrated laser tip heating for TINS with temperature range

up to the melting point of Si and providing near-field spectra across a wide range of sam-

ples. Further improvements to improve TINS sensitivity include utilizing higher-sensitivity

photodetectors, and modifying cantilevers to improve AFM stability at higher tempera-

tures. Heating through photoexcitation can provide a platform for nanoscale heat transfer

experiments by controlling input heat while monitoring tip temperature. These heat trans-

fer and thermal conductivity measurements can be performed simultaneously with near-

field spectroscopy to identify the underlying optical processes. Also, this technique may

replace electrically heated tips for thermal-desorption-based chemical analysis using mass

66

Wavenumber ν (cm-1)

-150 -100 -50 0 50 100 150

400

500

600

700

800

-200 -100 0 100 200

200

300

400

500

600

800 900 1000

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Inte

nsity (

arb

. u

.)

Path delay (µm)

1050 1100 1150 1200 1250 1300 1350

0.0

0.2

0.4

0.6

0.8

1.0

Wavenumber ν (cm-1)

Inte

nsity (

arb

. u

.)

0.0

0.2

0.4

0.6

0.8

1.0

ud

en(ω

,z,T

) (a

rb. u

.)

Inte

nsity (

arb

. u

.)

Path delay (µm)

Inte

nsity (

arb

. u

.)

ud

en(ω

,z,T

) (a

rb. u

.)

νs

νas

TINS

Fit

Envelope

a)

c)

b)

d)

Figure 4.10: a) Interferogram and b) spectrum of PTFE showing peaks at the symmetricand antisymmetric C-F vibrational mode frequencies. c) Interferogram and d) spectrum ofSiC showing a strongly red-shifted response of the SPhP.

spectroscopy [150]. The high temperatures achievable through photothermal excitation also

allows tip-based nano-fabrication of a wider range of materials with higher melting tempera-

tures than previously accessible. Due to the simplicity of the approach, cantilever materials

are not limited to doped-Si and have more design versatility to improve heating dynamics

and localization, extending the application space of tip-based thermal measurements and

nano-machining.

67

4.7 Optical forces

Transient excitations of charge densities through either thermal excitation or quantum

fluctuations not only results in novel electromagnetic field properties as probed by TINS, but

are also the origin of optical forces such as the Van der Waals, Casimir, and Casimir-Polder

forces. Optical forces are used for optical tweezers for nanoparticle manipulation [6], atom

trapping [18], and are the focus of research of the fundamental interactions between light

and matter that include Casimir and Casimir-Polder force [25, 141].

Even at zero temperature, the Casimir effect results in an attractive force between

closely-spaced objects due to zero-point energy arising from the Heisenberg uncertainty prin-

ciple [25]. It is typically described as originating from the quantization of the electric field

described through quantum optics and the density of states picture described above. As a

simple geometry I consider a cavity formed by two parallel reflective plates. At zero tem-

perature, the presence of the plates reduces the allowed photon states that can exist in the

cavity to photons of wavelengths λ = nL/2 for cavity length L and integers n. The exclusion

of certain photon states reduces the energy density inside the cavity. Since the energy of the

open system of the cavity and its surroundings increases when L decreases, there will be an

attractive force between the plates to decrease the cavity length.

An alternative description of the Casimir force appeals to the transfer of momentum

of virtual photons and the cavity walls. Since the density of states is lower inside the cavity

than the free space density of states, there is a difference in vacuum photon states across

the mirrors, with more virtual photons hitting and imparting momentum on the mirror from

the outside the cavity than from the inside. This results in a force on the mirror toward the

other end mirror, and a net attraction force.

For non-zero temperatures, thermal excitations give rise to additional optical forces

including the van der Waals (or dispersion) and the Casimir-Polder force. These optical

forces fundamentally originate from the Lorentz force, which gives the force ~F on a charge

68

q in spatially-varying electric ~E and magnetic ~B fields [139]:

~F = q ~E + q~v × ~B, (4.19)

where ~v is the velocity of the charge. The force on a dipole ~p can be derived by treating the

dipole as two point charges separated by a vanishingly small distance. The result is [75]

~F = (~p · ∇) ~E +d~p

dt× ~B. (4.20)

The force on a dipole is thus determined by the spatial and temporal variations of the fields.

As discussed above, a thermally-excited object will generate associated electric fields ~E which

reflect the geometry and material properties of the object. In the near-field region, this field

will be highly spatially varying, and can result in strong optical forces. When a material with

polarizability α = α′ + iα′′ is brought within this field, it will experience a time-averaged

Lorentz force given by

〈~F 〉 = 〈(~p · ∇) ~E +d~p

dt× ~B〉 (4.21)

=α′

2〈(~E · ∇

)~E〉+ ωα′′〈 ~E × ~B〉 (4.22)

= α1

2∇〈E2〉+ ωα′′〈~S〉 (4.23)

due to its induced polarization ~p = α~E, where the vector equality(~E · ∇

)~E = 1

2∇E2− ~E×

(∇ × ~E) and the assumption that ∇ × ~E = 0 were used to simplify the above expression

and ~S = ~E × ~B is the Poynting vector. The first term is known as the gradient force, since

it is proportional to the gradient of the field intensity. Since the polarizability is generally a

positive quantity, this results in an attractive force toward the point of maximum intensity

such as a beam focal spot. The second term is known as the scattering force, and arises when

a material absorbs or scatters an incident photon. Its strength is related to the imaginary

part of the polarizability of the material, which is closely related to the absorption coefficient

of the material.

The gradient force has been successfully implemented in a wide variety of applications

including crystallizing collections of dielectric particles [19] as well as numerous biological

69

applications [172] including viscoelastic studies of cell membranes [7]. The scattering force

can provide forces along the direction of the light propagation and has even been proposed

as a mechanism for space travel by capturing the force from solar photons [177].

A proposed method for measuring optical forces uses an AFM tip in feedback with a

sample surface. In this implementation, known as photo-induced force microscopy (PiFM),

light illumination of the tip induces an electric dipole p at the tip apex to first approximation.

This dipole in the tip induces an image dipole in the sample of strength p′ = pβ, where β =

εs(ω)−1εs(ω)+1

for sample dielectric function εs(ω) as determined by the electromagnetic boundary

conditions. The resulting mutual optical gradient force is given by ~F = ~p · ∇ ~E, where ~E is

the resulting electric field due to the two dipoles.

Recent implementations of PiFM demonstrate a powerful new imaging and nano-

spectroscopy modality complementing s-SNOM and AFM-IR [76, 69, 176, 140, 162, 161,

110, 100, 111]. However, these studies paint a confusing picture regarding the actual con-

trast mechanism with some studies assigning the observed force exerted onto the tip to the

optical gradient force [76, 69, 176, 140, 162, 161], while others assigning it to material ab-

sorption and thermal expansion [110, 100, 111]. The observed force magnitudes vary from

0.85− 2.7 pN, and force gradient range from 2.1 · 10−5 to 3.7 · 10−5 N/m [162, 161]. Recent

theory [191, 4, 136] and calculations presented here predict optical gradient force magnitudes

of 1-10 fN for electric field strengths of 106 V/m. Measured spectra of the photo-induced

force also show an absorptive lineshape [140, 162, 161], while theory in Ref. [191] and results

presented here predict a dispersive spectral lineshape for the optical gradient force.

By spectrally and spatially resolving the photoinduced force, I identify thermal expan-

sion as the dominant contribution to the measured force signal in our implementation [146].

I observe symmetric force spectra that match the sample absorption spectrum, and estimate

the force magnitude to range from from 1-10 pN induced by local thermal expansion, corre-

sponding to ∼ 2 K temperature rise of the sample. In contrast, calculations of the optical

gradient force based on dipole models yield a dispersive spectral lineshape. I also perform

70

νd

νr

Fopt

Fexp

nνd

νd

Lock-in Amp.

ref. sig. out

QC

L

νr

νd

2νd

3νd

s-S

NO

M

νd

ν1

νd+ν

rν

d-ν

r

Ca

ntile

ve

r

mo

tio

n

c)b)a)

d)p

p'

h(t)

Figure 4.11: a) Experimental concept: tip-sample gap is illuminated with mid-infrared lightpulsed at repetition rate νr. An optical dipole force Fdip results between induced dipole ~pin the tip and its image dipole in the sample ~p′. Thermal expansion due to IR absorptionresults in an impulsive force Fexp on the tip. Scattered near-field light can also be monitoredby collecting scattered light and demodulating at harmonics of the tip motion nνd. b)Lock-inamplifier receives tip dither frequency νd and tip motion h(t) as measured by a laser diodeand a four-quadrant photodiode. Lock-in output sets QCL repetition rate νr. c) Frequencydependence of optical signal measured by s-SNOM, as described previously. d) Frequencydependence of the cantilever motion. Sidebands at frequencies νd ± νr appear due to theperiodic sample thermal expansion. Repetition rate is tuned to the difference of the cantileverfrequencies νr = ν1 − ν0 or the second cantilever frequency νr = ν1.

thickness dependence measurements of the force on PMMA, and see increase of signal up to

thickness of ∼ 250 nm, whereas the calculated optical gradient force is expected to saturate

below 30 nm thicknesses. Approach curves also indicate that the force signal is confined to

direct tip-sample contact.

4.7.1 Experiment

Figure 4.11a) shows the experimental concept of this study. An AFM (Vesta AFM-SP,

Anasys Instruments) operates in dynamic mode feedback by driving and monitoring the

cantilever motion at its fundamental resonance νd = ν0. Light illuminates the tip-sample

region which results in polarization density in the tip apex ~p and image polarization density

in the sample ~p′ resulting in a mutual electromagnetic force Fopt. Additionally, as the

sample absorbs the incident light, it expands and provides an impulsive force Fexp = kdh

71

on the cantilever, where dh is the thermal expansion distance and k is the cantilever spring

constant. A four-quadrant photodiode measures the tip height as a function of time h(t),

which is sent to a lock-in amplifier to spectrally filter the tip motion due to the photo-

induced force (Fig. 4.11b)). A TTL pulse output triggers a pulsed quantum cascade laser

(QCL, Daylight solutions) at a frequency νr set by an internal oscillator in the lock-in

amplifier. A high numerical aperture (NA = 0.48) off-axis parabolic mirror focuses the IR

radiation onto the apex of the tip. The periodic expansion of the sample yields tip motion

at the sideband mixing frequencies ν0 ± νr (Fig. 4.11d)). In sideband mode, we set the

repetition rate to the difference of the first two vibrational frequencies of the cantilever ν0, ν1

so that νr = ν1 − ν0. The thermal expansion resonantly drives the second cantilever mode,

dramatically improving sensitivity. Alternatively, in direct-drive mode we set the laser

repetition rate to the second cantilever vibrational mode frequency νr = ν1. Here, the second

cantilever mode still resonantly enhances the localized force signal, however a background

offset exists when the tip is far from the sample due to, e.g., thermal expansion of the

cantilever itself. In both sideband and direct-drive mode, we measure the optically induced

force by recording the tip motion at the second cantilever resonance and we obtain spectra

of the photothermal response by sweeping the laser wavelength. Additionally, the sample

near-field optical response can be monitored by collecting scattered light and demodulating

at higher harmonics of the tip motion nνd for integers n, as typically done is s-SNOM (Fig.

4.11c)).

4.7.2 Model

In the dipole approximation, we treat the sample as a semi-infinite half-space with

a planar interface and we model the tip as a sphere of radius r with polarizability αt =

4πr3ε0εt−1εt+2

, where εt is the dielectric function of the tip (Fig. 4.13) [94]. When the tip is

close to a sample surface, the tip dipole p induces an image dipole in the sample of strength

p′ = pβ, where β = εs(ω)−1εs(ω)+1

and εs(ω) is the dielectric function of the sample. The electric

72

0.5

1

1.5

2

7.5

8

8.5

Dip

ole

fo

rce

(fN

)

1

1.5

2

2.5

3

Re

()

Coupled dipoles

Uncoupled dipoles

Re( )

ASNOM

1710 1720 1730 1740 1750

Frequency (cm-1

)

1

2

3

Fo

rce

(a

rb. u

.)

0

0.2

0.4

0.6

0.8

1

/ r

ad

.

Sideband mode

Direct drive

Fit

Abs.

SNOM

Figure 4.12: a) Calculated dipole force spectra, s-SNOM amplitude, and the real part ofPMMA dielectric all show dispersive features. b) Measured force spectra match IR absorp-tion, and s-SNOM phase ΦSNOM

field at the tip position is the sum of the incident electric field and the field due to the image

dipole E = E0 + βp16πε0(r+h)3 where h is the separation between the bottom of the sphere and

the sample surface (Fig. 4.13). The field due to the image dipole acts back on to the tip dipole

and repolarizes it, giving a modified tip polarization of p = αt

(E0 + βp

16πε0(r+h)3

), where the

second term is the radiation reaction term. We can describe the tip-sample mutual response

by solving the recursive relation for p, and assigning the system an effective polarizability

αeff given by αeff = p/E0 = αt(1−βαt/16πε0(r+h)3)

. The field acting on the tip dipole is given by

E = E0 + βαeffE0

16πε0(r+h)3 and the force experienced by the tip is then given by:

~F = ~p · ∇ ~E = Re

(α2

eff

3E20

16πε0(r + h)4

). (4.24)

73

r

h

p

p' = βp

h

E0

Figure 4.13: Schematic of dipole model of optical gradient force between tip and sample.The tip is approximated by a sphere of radius r at height h above the surface with dipolepolarization p. The sample response is modeled by an image dipole of strength p′ = βp.

Past predictions of the optical force have differed from the above treatment by considering

the sample as a sphere with polarizability αs = 4πr3ε0εs−1εs+2

instead of as a semi-infinite half-

space [140]. Additionally, the radiation reaction term between the two spheres was neglected.

Instead, while the electric field on the tip was given by E = E0 + αsE0

2πε0(2r+h)3 , the strength of

the tip dipole was given by pt = αtE0 and neglected the additional electric field contribution

due to the sample dipole. In this case, the optical gradient force is given by

F = ~pt · ∇ ~E = −1

2Re

(1

4πε0

6αtα∗sE

2

z4

)∝ Re(αtα

∗s)E2

z4∝ (α′tα

′s + α′′tα

′′s)E2

z4. (4.25)

74

Calculations using Eq. 4.24 and Eq. 4.25 both show dispersive spectral lineshapes, however

the inconsistent model presented previously which results in Eq. 4.25 underestimates the

force magnitude by close to an order of magnitude. Eq.4.24 is based on the more accurate

sample geometry, and is based on a self-consistent model which should result in more accurate

predictions.

Our samples consist of a PMMA layer of thickness t on a Si substrate creating a

air/PMMA/Si structure. To describe this layered system and the PMMA thickness depen-

dence, we use an effective dielectric function derived from the boundary conditions across

the two interfaces [195]

ε∗(ω, q) = εP (ω)εS(ω)kzP (ω, q)− εP (ω)kzP (ω, q) tanh(ikzP (ω, q)t)

εP (ω)kzS(ω, q)− εS(ω)kzP (ω) tanh(ikzP (ω)t). (4.26)

Here, εP,S(ω) refer to the dielectric functions of PMMA and Si, respectively, and kzP,S(ω, q) =√εP,S(ω)ω2/c2 − q2 is the z-component of the wavevector in PMMA and Si, respectively.

4.7.3 Results

Figure 4.12 compares calculated and experimental spectra of several optical properties

of PMMA around the carbonyl vibrational resonance (k0 = 1735 cm−1). Optical force

calculations shown in Fig. 4.12a) use dipole approximations to represent the polarizabilities

of the tip and sample using the above equations (blue curves). When the radiation reaction

term is neglected, the force magnitude between the dipoles varies from ∼ 0.2− 0.5 fN, and

is proportional to αtα∗s (Eq. 4.25). Whereas, when the radiation reaction term is included,

the force is stronger by a factor of ∼ 5 and is proportional to α2eff (Eq. 4.24). The force

is stronger since the induced dipole in the sample and the dipole in the tip back-act on

each other recursively and increase the induced polarization. These calculations show a

dispersive shape, agreeing with recent electromagnetic simulations [191]. For comparison,

the s-SNOM amplitude spectrum (ASNOM(ω)) and the real part measured dielectric function

ε(ω) are also plotted.

75

Figure 4.12b) shows the measured PiFM spectra obtained in either sideband mode

(circles) and direct drive mode (squares), with their respective Lorentizan fits. These spec-

tra were acquired by sweeping the laser wavelength at a fixed sample location, and were

normalized by dividing by the corresponding force spectra acquired on a Si reference loca-

tion. Also included in the plot are the absorption coefficient α(ω) obtained using far-field

FTIR, and the measured s-SNOM phase spectrum (ΦSNOM(ω)) obtained using SINS at the

Advanced Light Source [129]. The peak frequency of the measured force spectra matches

the absorption coefficient obtained using far-field FTIR.

Figure 4.15 shows the thickness dependence of the force signal measured on-resonance

with the carbonyl resonance on a tapered film of PMMA with thickness ranging from ∼

7 − 250 nm. Both sideband mode and direct drive mode show a smooth increase in signal

up to 200 nm. Calculations of the film thickness dependence of the thermal expansion were

performed by calculating the electric field intensity within the PMMA film approximating

the electric field of the tip as an electric dipole and including the exponential decay of the

electric field of the far-field illumination:

E(z) =αeffE0

2πε0ε2(ω)(z + h)3+ E0e

−z/2δ, (4.27)

where δ ≈ 0.8 µm is the intensity absorption depth of PMMA and z is the distance from the

PMMA sample. The thermal expansion is calculated at different film thicknesses t using

∆z(t) ∝ αT,PMMA

∫ t

0

|E(z)|2dz + αT,Si

∫ ∞t

|E(z)|2dz (4.28)

where αT,x is the thermal expansion coefficient of x = Si,PMMA. The calculated thermal

expansion shows an increase with t up to the absorption depth of PMMA δ. In contrast,

calculations show a slight increase in optical force below a PMMA thickness of 30 nm.

Figure 4.16 shows the tip-sample distance dependence of the tip oscillation amplitude,

the s-SNOM intensity, and the force signal, measured by approaching the sample at a con-

stant rate. The sharp corner in the tip oscillation amplitude indicates the point of tip-sample

76

c) F (1735 cm-1) d) F (1680 cm-1)

a) Height b) Tapping phase

300 nm0 nm

10 nm

0 V

1.2 V

0 V

1.2 V

0°

10°

0 50 100 150 200 250

Distance (nm)

-2

0

2

4

6

He

igh

t (n

m)

0

0.2

0.4

0.6

Fo

rce

(a

rb. u

.)

HeightPhaseForce

45 nme)

Figure 4.14: a) Height, b) tapping phase, c) on-resonance force signal, and d) off-resonanceforce signal of a PS-PMMA block copolymer sample. The force channel shows high contraston the carbonyl resonance at ν = 1735 cm−1 with high signal on PMMA and low signal onPS. e) Line cut along dashed line in c) showing ∼ 45 nm spatial resolution.

contact, below which the tip oscillation amplitude decreases linearly. While s-SNOM inten-

sity at the third harmonic of the tip oscillation frequency extends 10’s of nm from the point

of contact, the force signal is more closely confined.

The dependence of the force magnitude was also studied as a function of IR intensity us-

ing a Au nanowire as a plasmonic antenna. Plasmonic antennae are commonly used to confine

electromagnetic light down to nanometer length scales [149]. These structures have also been

used to enhance chemical vibrational sensitivity by matching its resonant wavelength to the

77

10 100

PMMA thickness (nm)

0

1

2

3

4

Fo

rce

(a

rb. u

.)

7.75

8

8.25

8.5

Fo

rce

(fN

)

Force (sideband)

Force (direct)

Thermal expansion (arb. u.)

Optical force

Figure 4.15: Dependence on PMMA thickness on measured force values, coupled dipole forcecalculations (green solid) and relative thermal expansion (green dashed).

30 40 50 60 70Distance (nm)

65

70

75

Tap

ping

am

plitu

de (

nm)

0

0.2

0.4

0.6

0.8

1

Nor

mal

ized

sig

nal (

arb.

u.)

Tapping amplitudeSideband modeDirect drive mode3rd Harm.

Figure 4.16: Tapping amplitude, optical, and force signals as a function of tip-sample dis-tance. Black dashed indicates point of contact between the tip and sample surface. Forcesignals show confinement to region of physical contact. s-SNOM signal at the third harmonicextends 10’s of nm from the sample surface.

78

vibrational resonance frequency [186]. When excited on-resonance with external radiation,

plasmonic antennae focus and enhance incident electric fields which improves the interaction

with nanoscale sample structures. Using s-SNOM and PiFM, I measure the interaction of

a thin PMMA film with a resonant antenna. Fig. 4.17a) shows the topography of such an

antenna coated with a ∼ 10 nm thick layer of PMMA. The s-SNOM intensity (Fig. 4.17d))

shows the field structure of the antenna when excited on-resonance at λ = 1735 cm−1. This

shows an asymmetric structure along the antenna length with intense fields on the right

end and weaker fields on the left end. The on-resonance force map (Fig. 4.17b)) shows a

similar structure with the peak force signal appearing at the point of high s-SNOM intensity.

Off-resonance, the force map shows no discernible contrast with low signal throughout the

scan (Fig. 4.17c)). These results demonstrate that the force signal scales with the near-field

intensity on the antenna structure. This measurement does not differentiate between the op-

tical gradient force and the force due to thermal expansion since both of these contributions

scale with the intensity. However, this does demonstrate the ability of plasmonic antennae

to improve sensitivity of tip-based force microscopy with a factor of 20 increase of signal on

the bright end of the antenna compared to the background signal off the antenna.

4.7.4 Discussion

The measured absorptive lineshapes, the PMMA film thickness dependence, and ap-

proach curves indicate that the measured force is dominated by material heating from reso-

nant infrared absorption, and subsequent thermal expansion. The spectrum of the force is

symmetric and matches the IR absorption spectrum. This spectrum agrees with the past

spectra of the photoinduced force between a tip and a PMMA surface acquired using side-

band mode [140], though these results had been assigned to the optical gradient force. This

conclusion was supported by an incorrect analysis of the expression of the optical force.

First, the uncoupled dipole expression was used (Eq. 4.25), which neglects the radiation

reaction term that results in a mutual repolarization of the tip and sample. Additionally,

79

500 nm

160 nm

0 nm

1.3 V

0 V

1.3 V

0 V

1 V

0 V

b) F (1735 cm-1)

c) F (1760 cm-1)

d) ASNOM

(1735 cm-1)

a) Height

Figure 4.17: An infrared antenna structure coated with PMMA provides a platform forincreased resonant force sensitivity. a) Topography of Au nanoantenna structure coatedwith a ∼ 10 nm layer of PMMA. b) Photoinduced force signal on PMMA resonance showsenhanced signal on the region of the antenna with enhanced IR field intensity (d). c) Offthe PMMA resonance, the force signal is low.

Ref. [140] predicted a symmetric lineshape dominated by α′′tα′′s in Eq. 4.25, however for

metallic tips as used in these experiments, ε′t ε′′t at infrared frequencies (in particular

gold: ε ≈ −1600 + 360i at k = 1735 cm−1), and thus the product of the real parts of the

polarizability dominates as shown in Fig. 4.12a).

The force magnitudes observed here and in recent studies [76, 162, 161] also disagree

with predictions of the optical gradient force. Our predictions of the strength of the optical

gradient force between gold and PMMA spheres calculated using Eq. 4.24 is ∼ 8 fN. In

contrast, the thermal noise ∆Hrms of an AFM cantilever at temperature T is expressed by

80

∆Hrms =√

(4kBTQ/ωk) where kB is Boltzmann’s constant and k and Q are the cantilever

spring constant and the quality factor of the cantilever vibrational mode, respectively. In

typical room temperature AFM operation, this is typically Frms ≈ 0.3 pN. Thus the optical

gradient force is not detectable unless the AFM is operated in special conditions, e.g., at

low-T, or using high Q cantilevers.

Our estimated force magnitude from a 10 nm thick layer PMMA on-resonance is 6

pN. This is estimated by by assigning the noise floor of the force signal as the minimum

detectable force using our cantilevers of Frms ≈ 0.3 pN and using the signal-to-noise ratio

of ∼ 20. This agrees with previously observed values [76, 162, 161] and corresponds to a

thermal expansion of ∆H ≈ 3 pm, or a local heating of the PMMA of ∼ 2K.

The magnitude of optical forces in optical trapping potentials has been measured on

Au nano-spheres by measuring the minimum external force needed to remove the particle

from the trap [64]. The forces on particles with radii ranging from r = 50 nm to r = 130 nm

were 0.56 pN to 2.2 pN in a field strength of ∼ 2× 106 V/m. While the field strength in this

measurement is comparable to those considered here, the particle sizes are about an order

of magnitude larger than the values considered in the above calculations. Since the optical

gradient force scales as r3, in our application with tip radius values of r = 10− 20 nm, the

expected force based on these measurements would be only ∼ 0.5− 1 fN. This is consistent

with the force values from calculations presented here.

The measured force dependence on the PMMA film thickness acquired using in side-

band mode matches calculations of thermal expansion dependence on thickness given in

Eq. 4.28. While data acquired using direct drive mode has a larger offset at small thickness

due to thermal expansion of the cantilever itself, it also shows a monotonic increase in signal

up to thickness > 100 nm. The calculated force due to thermal expansion increases with

film thickness up to approximately the absorption depth of δ ≈ 0.8µm and is roughly linear

for thickness < 300 nm. The calculated optical gradient force shows a qualitatively different

trend. It is peaked at zero PMMA thickness, and decreases to a minimum value within about

81

0 V

1.2 V

PS

PMMA

0 50 100 150 200 250

Distance (nm)

0.2

0.4

0.6

0.8

1

1.2

Fo

rc

e (

arb

. u

.)

Min

Max

PM

MA

co

nc

. (a

rb

. u

.)

a) b) c)

300 nm

Figure 4.18: a) s-SNOM image on-resonance with the carbonyl resonances provides a chem-ical map of a block-copolymer blend of polystyrene and PMMA. The high intensity regionsshow where the PMMA concentration is peaked. b) PiFM image across the same region andlaser wavelength as in a). There is additional nanoscale structure in the force image that isnot completely correlated with the chemical map. c) A line cut along the white line in b)of the s-SNOM and the PiFM signal. PiFM signal appears peaked at the interfaces of thePS-PMMA copolymer, possibly due to their nanomechanical interactions.

30 nm. The peaked value at zero thickness is due to the larger force magnitude between the

tip and the Si substrate since Si has a larger IR polarizability. The thickness dependence of

the optical force is qualitatively similar to the s-SNOM probing depth, since the thickness

dependence is incorporated through αeff , which has the same form as in s-SNOM.

Approach curves on PMMA shown in Fig. 4.16 demonstrate the spatial confinement of

PiFM signal to direct sample contact. Since the expected thermal expansion is only on the

order of a few pm, the force due to thermal expansion is expected to be confined to similar

length scales. This is in contrast to results in Ref. [140] that the force is long range, acting

10’s of nm above the sample surface. While we see an onset of the force in the last ∼ 1 nm

from the surface, this is likely due to the water layer or electrostatic forces. These effects are

evident in the tapping amplitude, which shows a small variation in the same spatial region.

In contrast, the s-SNOM signal at the third harmonic of the tip oscillation frequency extends

∼ 10’s of nm away from the sample surface. Calculations of the force distance dependence

are calculated using the dipole model, and show a decay on the 10 nm length scale.

Figure 4.18 shows a) s-SNOM and b) PiFM signal of a PS-b-PMMA block co-polymer

82

blend. The s-SNOM map shown in Fig. 4.18a) acquired on-resonance with the PMMA

carbonyl resonance provides a chemical map of the region, with high signal corresponding to

regions with high PMMA concentration. The PiFM image shown in Fig. 4.18b) was acquired

across the same region and wavelength but shows slightly different spatial features than the

chemical map. This is highlighted in the linecut along the white line in Fig. 4.18b) shown in

Fig. 4.18c), where the PiFM signal is peaked close to the interfaces between the PS-PMMA

regions. The discrepancy between the chemical map and the force map could possibly due

to the thermal and nanomechanical properties in the heterogeneous film, where the IR light

is absorbed by the PMMA, but heat is transferred throughout the film heterogeneously with

the varying thermal conductivity κ. This effect, combined with the variation in coefficient of

thermal expansion α and the elastic modulus E of the material blend can mix the nanoscale

thermal and mechanical properties during the thermal excitation and expansion process

probed by PiFM.

4.7.5 Conclusion

The results presented above demonstrate that thermal expansion dominates the signal

in our implementation of photo-induced force spectroscopy and microscopy. The measured

force is distinguished from the optical gradient force through the symmetric spectral line-

shape which approximates the IR absorption, through the thickness dependence of a PMMA

film, through the force magnitude on the order of 6 pN, and through spatial localization

of the force to the direct contact region seen with approach curves. While our results re-

fute the claim of direct observation of the optical gradient force between tip and samples as

previously claimed, they also highlight the high spatio-spectral sensitivity of this technique.

4.8 Conclusion and outlook

The resonantly-enhanced thermal near-field exhibits unique spectral, spatial, and co-

herence properties not present in far-field thermal emission. When coupled to resonant

83

vibrational or electronic modes, stochastic thermal current densities gain a surprising de-

gree of temporal coherence. Additionally, the thermal excitation of propagating polariton

modes creates spatial coherence over several polariton cycles. However, these unique prop-

erties are confined close to a sample surface and do not propagate to the far-field. Using

a sharp tip close to the surface, TINS can interrogate and scatter the thermal near-field,

and transmit this coherence to the far-field. This is evident with the narrow linewidth and

spatially-extended field of the SPhP of SiC. Further investigation of the novel properties of

the thermal near-field will yield the ability to tailor thermal emission and transport proper-

ties of materials, and expand our understanding of thermal emission.

Optical forces have found many applications, however, their direct measurement is

difficult due to their steep distance dependence and generally weak magnitude. By incorpo-

rating an extension of AFM with new sensitivity to optically induced forces, including the

optical gradient force, I study their spectral and spatial variation, with a particular focus on

the differentiation between the optical gradient force and the force due to material thermal

expansion. Approach curves, spectral lineshapes, force magnitude estimates, and material

thickness dependence suggest that the force is dominated by thermal expansion for a molecu-

larly resonant sample. Further work includes imaging plasmonic systems with small thermal

expansion coefficients to isolate the pure optical gradient force, as well as careful study of

the mechanical interaction of the tip and sample with tapping amplitude and set-point vari-

ation and comparison between attractive and repulsive mode AFM operation. This work

will expand our understanding of optical forces and allow better control of the light-matter

interaction.

Chapter 5

Inhomogeneity of the insulator to metal transition of vanadium dioxide studied

by ultrafast microscopy and infrared nano-imaging

As a canonical correlated electron system, vanadium dioxide (VO2) exemplifies many

interesting phenomenon due to the interplay between its lattice, electronic, and spin degrees

of freedom. Its most interesting property is its insulator-metal transition (IMT) just above

room temperature at T = 340 K with a change in resistivity of several orders of magnitude

first discovered in the late 1950’s [126]. Yet despite almost 60 years of intense research, the

mechanism of the insulator-metal transition (IMT) of VO2 remains unclear and the center

of much scientific debate. There is clear evidence for electron correlations in VO2 [196, 128]

evidenced by the doping dependence of magnetic properties[159] and predicted by band

models of its d-orbitals [55] which suggests that the low-temperature state VO2 could be a

Mott-Hubbard insulator due to Coulomb repulsion [128]. However, simultaneous with its

electronic transition VO2 undergoes a structural transition from a low-symmetry monoclinic

to a high-symmetry rutile structure. This is primarily characterized by a paring and tilting

of vanandium atoms along the c-axis. This suggests that the insulating state may be caused

by a Peierls mechanism where the dimerization opens a bandgap. While its switchable

conductivity and optical properties has many technological applications including electric

transistors, thermochromatic films[59], or for switchable THz devices [15], this confusing

picture has precluded reliable device fabrication using VO2.

85

5.0.1 Ultrafast IMT

In addition to its thermally driven transition, VO2 undergoes a photoinduced IMT

when optically excited with ultrafast laser pulses of sufficient fluence [29]. The photoinduced

IMT occurs on sub-picosecond timescales, at rates too fast for complete thermalization,

and therefore has a non-thermal basis. This ultrafast transition has been studied using a

wide variety of spectroscopies, with short-pulse optical [14, 29, 27, 179], X-ray diffraction

[28, 26, 63], terahertz [98, 132, 189], and electron diffraction [58, 109, 12] techniques. These

studies have addressed the ultrafast electron dynamics and lattice structural processes which

occur during the transition, in addition to slower behavior over multi-picosecond to nanosec-

ond time scales. However, the mechanism underlying both the thermal and photoinduced

insulator-to-metal transition remains unclear, and results conflict with both Mott or Peierls

explanations [128, 197, 160, 98, 173, 108]. For example, degenerate pump-probe studies

[26] revealed a limiting transition timescale of 75 fs, suggesting a phonon bottleneck and

therefore a structurally-limited transition. In contrast, the observation of coherent phonon

oscillations above the apparent threshold for triggering the photoinduced phase transition

[98] indicates that the metallic phase appears after one V-V phonon oscillation cycle, even

though the lattice is still presumably far from equilibrium. Ultrafast electron diffraction re-

veal that the structural dynamics occur on longer timescales than the conductivity change,

indicating a metallic monoclinic state [127]. These results suggests that the photoinduced

IMT is decoupled from the structural transition.

Much of the work on VO2 has focused on polycrystalline thin films, grown by a variety

of techniques [182]. Recently, differences observed in the ultrafast and thermal properties

due to anisotropy and grain size in thin film samples [116, 107, 189] suggest that growth

conditions can substantially modify the measured response. These results contributed to

the confusion in the interpretation of previous measurements, which is further amplified

by the use of ultrafast techniques that average over multiple crystallites, subject to large

86

Delay τ

PD

PolarizerSample

95:5 B.S.

Probe arm

Pump arm

Optical

chopper

λ = 800 nm

τ = 45 fs

Rep. rate = 125 kHz

Ep

uls

e

0

Ph

ase

(rad

)

0 50-50-100Time (fs)

1000 50-50-100Time (fs)

100

800

820

Wa

ve

len

gth

(n

m) b) FROG trace

a

c b) FROG trace

780

b

150

0 nm1 µm

d

e

25 µm

π

z

xy

τ

Figure 5.1: Schematic of optical layout for degenerate 800 nm pump-probe (a). TypicalFROG spectrogram of pulse (b), with full phase and amplitude reconstruction of pulse intime domain (c), showing an approximately 45 fs pulse with small chirp. Optical microscopeimage of VO2 microcrystals on Si/SiO2 substrate (d). AFM topography (e) showing thehomogeneous nature of the microcrystals.

inhomogeneous strain, which could, for example, create mixtures of the different insulating

phases.

In order to access the homogeneous response, I investigate individual VO2 single

micro-crystals. I perform both degenerate pump-probe microscopy (Fig. 5.1a) to moni-

tor the femtosecond dynamics following the ultrafast photoinduced excitation, and infrared

scattering-scanning near-field optical microscopy (s-SNOM) to probe the details of the evo-

lution of the spatial phase competition process in the thermal IMT [142].

5.0.2 Experiment

The VO2 micro-crystals, grown by vapor transport by the Cobden group at University

of Washington [60], show distinct behavior from thin film samples, with an abrupt, first

87

order transition in the absence of strain or doping. With their well defined, controllable

strain state, our group was able to establish the strain-temperature phase diagrams for the

insulating polymorphs below the transition temperature through Raman microscopy (see

Fig. 5.2b) [8]. This work built on previous work that had mapped the phase diagram close to

the transition temperature, but had not observed the triclinic phase [23, 175]. Ref. [153] has

done additional work to identify a triple point between the M1,M2, and R phase. Peltier and

Joule heating effects at metal-insulating domain interfaces have also been observed, probing

the complex domain wall structure between the phases [46]. Microcrystals have been used

for nano-mechanical devices [68] and their domain structure and transition behavior can be

readily tuned by the application of external mechanical stress [22].

The crystals have a well-defined size (Fig. 5.1d), crystallographic orientation, and ther-

mal transition temperature, which I characterize with atomic force microscopy (Fig. 5.2e),

micro-Raman spectroscopy (Fig. 5.2a), and scattering-scanning near-field optical microscopy

(s-SNOM.) [80]. I use Raman spectroscopy to measure the phonon spectra of each microcrys-

tal in order to identify the strain state prior to performing pump-probe microscopy. Raman

spectroscopy is performed by focusing narrowband laser light on a sample and measuring

the back-scattered light. When an incident photon inelastically scatters off a phonon in the

sample, it loses energy by an amount equal to the phonon energy. This slight red-shift can

be measured and gives a phonon energy spectrum.

The very low luminescence background and narrow linewidths in Raman spectroscopy

indicate a low defect density, and the atomic force microscopy image show that the crystals

are highly structurally homogeneous with a flat surface (rms roughness < 1.8 nm). The

microcrystals sharp thermal phase transition behavior, without percolation [160]. The shift

of the ωV−O Raman mode allows us to estimate the amount of strain within the crystal due

to substrate interactions or doping and distinguished Three possible insulating structures:

monoclinic 1 or 2, or intermediate triclinic (M1, M2, T) [8]. The combination of Raman with

pump-probe microscopy allows us to confirm the high structural quality of the individual VO2

88

Te

nsile

str

ain

M1

M2

T

0

1

2

3

4

5

Inte

nsity (

arb

. u

.)

200 300 400 500 600 700

Raman shift (cm-1)

a

Perpendicular

Parallel

6 ωV-O

-∆

R/R

6543210 Pump-probe delay τ (ps)

0

45

90

135

180

225

270

315

-∆R/R(θ) for τ = 3 ps

cR

cR

10-4

10-5

10-3

c

e

Temp.

T

M1

M2

Wavenumber (cm-1)

CP

Sp

ec. (a

rb. u

.)

180 200 220 240

0

1

2

3

4

b

d

θ

ωV2

ωV1

ωV2

ωV1

R0

338 K

Figure 5.2: Raman spectra of the three insulating structures, M1 (green), M2 (dark blue),and triclinic T (light blue), allowing full characterization of the initial phase of individualsingle crystals (a). Perpendicular (black) and parallel (purple) polarization with respect tocR for microcrystal in triclinic phase. Strain - temperature diagram showing relation of threeinsulating phases to metallic rutile phase (gray) (b). Transient reflectivity traces, −∆R/R,for low fluences where coherent phonon excitation is visible, with probe polarization parallel(purple) and perpendicular (gray) to the cR axis (c). The initial pulse excitation is indicatedby the red line. Fourier transform spectra of the reflectivity traces (d), showing phononmodes at 200 cm−1 and 220 cm−1. Reflectivity at 3 ps for different probe polarizations (e).

micro-crystals and to systematically study the relationship of crystallographic orientation,

insulating phase, and temperature with the photoinduced response dynamics. IR nano-

imaging using s-SNOM probes the Drude response and allows us to monitor the growth

and evolution of metallic and insulating domains through the thermal transition with 10 nm

spatial resolution.

The single crystal vanadium dioxide microcrystals studied were grown by vapor phase

89

transport on an oxidized silicon substrate [60]. This produces rectangular microcrystals of

varying sizes and orientations, as shown in Fig. 5.1d). Typically the rods have width 100

nm - 15 µm, and length up to 1 mm. The rutile c-axis (cR) is along the length of the rod.

The photoinduced IMT in individual microcrystals is studied using a degenerate pump

and probe transient reflectivity measurement designed and built by Andrew Jones, with

experimental configuration shown in Fig. 5.1a. Excitation is provided by a regenerative

amplifier Ti:S system (K&M Labs, Wyvern), which produces < 50 fs pulses at 800 nm, with

a variable repetition rate from 10 − 350 kHz. The regenerative amplifier is pumped by 18

W of a solid-state continuous wave laser, of which approximately 4 W is split off and pumps

a Ti:S oscillator which emits pulses at 80 MHz with pulse energies on the order of ∼ nJ.

These pulses are sent into the regenerative amplifier cavity, which pulse-picks an oscillator

pulse by rotating its polarization and retaining it in the cavity. The pulse is sent through

another Ti:S crystal pumped with the remaining of the 18 W pump power. The pulses are

amplified up to approximately 10 µJ, before being sent out of the cavity.

At 1.5 eV, the 800 nm pump pulses provide above band gap excitation for VO2 (optical

gap 0.6-0.7 eV [178]), while the probe at this wavelength is dominated by the response of

electrons in the d‖ bands close to the Fermi level [179]. A 5:95% beamsplitter separates the

incident light into the probe and pump arms, with a delay of up to 100 ps introduced, with

resolution 0.3 fs. The two beams are then recombined with a small spatial offset, passed

through a dual frequency optical chopper, and focused onto the sample using an off-axis

parabolic mirror with probe focus diameter ∼ 15 µm. The pump focal spot is larger, with

diameter ∼ 30 µm to ensure the probe region is uniformly pumped. This produces pump

fluences of up to 10’s of mJ/cm2. The reflected light is collected again by the parabolic

mirror, and the probe light is detected by a photodiode (New Focus, Nirvana 2007) with

lock-in amplification in order to improve the signal-to-noise ratio. In addition, the pump

and probe polarizations are orthogonal, and a polarizer is used to select the probe signal

from the pump background. Pulses are fully characterized with frequency resolved optical

90

gating (FROG) (Fig. 5.1b,c). In FROG, we perform a spectrally resolved autocorrelation of

the pulse. This is done by splitting the pulse by a 50:50 beamsplitter and sending one half

into a delay arm. The pulses are recombined non-colinearly into an optimally phase matched

beta-barium borate (BBO) crystal which generates SHG of each of the 800 nm pulses. When

the pulses are temporally overlapped, additional SHG from their mixing is generated, and

measured with a spectrometer. The full amplitude and phase profile of the pulse can be

reconstructed from this spectrally resolved autocorrelation.

The VO2 samples are mounted on a resistive heater with a thermocouple to enable

temperature control within ± 0.5 K. Atomic force microscopy (AFM) measurements are

used to characterize the heights of the rods chosen for measurements. From this, we find

the rods have rectangular cross-sections with thicknesses of 25 - 200 nm (Fig. 5.1e). Si-

multaneous with the AFM measurements, we perform scattering-scanning near-field optical

microscopy (s-SNOM) with a CO2 laser source to probe the changes in reflectivity at 10.6

µm as crystallites are heated through the IMT. We see stripe domain formation due to the

strain between the microcrystal and the substrate, shown in Fig. 5.5, as previously observed

[80, 108]. Since the rods are chemically bonded to the Si substrate, their crystal structural

transition occurs differently than if they were free standing. The IMT is characterized by a

1-2% structural contraction along the cR axis, however when the crystal is clamped to the

substrate, it cannot achieve this full structural change. Instead of exhibiting a first order

transition from the monoclinic to the rutile structure, the substrate-bound crystals form

alternating regions of monoclinic and rutile structure to minimize the free energy due to

strain, creating the characteristic striped structure.

I characterize the crystallography of individual microcrystals through Raman spec-

troscopy, using a home-built microscope with HeNe laser excitation (λ = 632.8 nm) and a

0.8 NA objective (Olympus). The Raman scattered light is detected by a spectrometer with

600 groove/mm grating and liquid nitrogen-cooled CCD after passing through a cut-off fil-

ter, enabling 2 cm−1 spectral resolution. The three insulating structural phases, monoclinic

91

M1, monoclinic M2, and triclinic T, can be distinguished principally through the position

of the 610-650 cm−1 Raman mode, denoted ωV−O (Fig. 5.2a) [8]. In addition to this char-

acterization of insulting phase, I observe a polarization dependence of the Raman modes.

Microcrystals with ωV−O close to 650 cm−1 are in the M2 phase, which can be produced

by substrate strain or ∼ 2% Cr doping [80, 8] . While the rutile phase is characterized

by equally spaced vanadium atoms along the cR axis, the M2 phase is distinguished by a

dimerization of this vanadium chain [8]. It is this dimerization that suggests a Peierls dis-

tortion mechanism could be behind the conductivity change in the IMT. For microcrystals

in the M1 phase, ωV−O ∼ 620 cm−1, the crystals are relatively unstrained, and the amount

of doping or impurity is less than 1% [121]. The M1 phase is differentiated from the M2

phase by a rotation of the V-V dimers away from the cR axis. The triclinic phase is viewed

as a continuous distortion of the M1 phase through a smooth rotation of the dimers from

their M1 locations toward the cR axis [8]. The transition from M1-T can be induced by

applying tensile strain and with large enough strain, the T phase abruptly transitions to

the M2 phase. The Raman signature of the M1-T-M2 transition is a splitting of the ωV−O

into a blue shifted and weaker red-shifted peak. The peak separation continuously increases

until the red-shifted peak suddenly vanishes and the blue-shifted peak aburptly shifts to the

M2 phase peak location. During this transition, the peak position of the blue shifted peak

begins at the M1 value of ωV−O ∼ 620 cm−1, continuously shifts from 620cm−1to 645cm−1,

and finally jumps up to the M2 position at ωV−O ∼ 650 cm−1. The ωV−O mode then provides

a good proxy for the strain state of the VO2 crystal.

5.0.3 Results

For low pump fluences, the transient reflectivity response shows an initial electronic

excitation, due to the above-gap excitation of the pump, followed by relaxation on a picosec-

ond time scale, as seen in Fig. 5.2c. Probing at 800 nm (1.5 eV), the response is expected to

be dominated by electrons in the d‖ bands close to the Fermi level [179]. The observed mod-

92

ulations in the reflectivity signal indicate the excitation of coherent phonons in the insulating

phase [193].

An orientational anisotropy is evident in the relaxation and coherent phonon behavior,

with faster decay and more prominent oscillations for pump polarization perpendicular to

the crystallographic c-axis (gray line), compared to parallel polarization (purple line). The

Fourier transform phonon spectrum, shown in Fig. 5.2d, reveals an even more pronounced

anisotropy. For polarization parallel to the cR axis, only one low energy phonon peak is

resolved, at ∼ 200 cm−1 (6 THz). For perpendicular polarization, both phonon modes

emerge, with the second at approximately 225 cm−1 (6.7 THz). Similarly, the reflectivity

change at 3 ps also shows an angular anisotropy, with a cos2θ dependence with angle θ of

pump polarization with respect to the cR axis (Fig. 5.2e).

At higher fluences, the increasing carrier density screens the coherent phonon response,

which eventually vanishs for fluences sufficiently high to drive the microcrystal through the

photoinduced insulator-metal transition (Fig. 5.3). The persistence of the reflectivity change

(up to microseconds) indicates that we have induced a quasi-stable metal-like state. We

cannot simultaneously measure the structural change, and therefore cannot determine if this

state corresponds to the postulated monoclinic metallic state [87, 173]. The inset shows

the reflectivity −∆R/R at 1 ps as a function of fluence, in order to derive the threshold

fluence Fth. Fth for different microcrystals varies between 2 mJ/cm2 and 6 mJ/cm2, which

we attribute to variable coupling to the substrate. These values are generally lower than

those observed in thin films, which range from approximately 5.5 mJ/cm2 up to > 15 mJ/cm2

[179, 26, 37], due to stronger substrate coupling.

In order to quantify the transition dynamics and relate to the physical characteristics

of the different microcrystals, I fit the above-threshold transient reflectivity behavior to a

double exponential, with

−∆R/R = Afe−t/τf + Ase

−t/τs . (5.1)

93

Here τf describes the initial, ultrashort timescale transition behavior, and is constrained to

sub-1 ps. τs captures the long timescale behavior, which incorporates thermalization of the

microcrystal, and is constrained to be in the range 2 ps < τs < 10 ps. In the fit process

I convolute the fit function with the laser pulse amplitude measured through FROG, to

deconvolute the transient reflectivity trace from the pulse intensity trace in order to more

accurately resolve the sub-50 fs dynamics.

Representative fits of the transient reflectivity response for a single microcrystal for a

range of above-threshold fluences are shown in Fig. 5.3b (black dashed lines), with −∆R/R

normalized for clarity. For the < 500 fs range shown here, the response is dominated by the

τf term. We observe three distinct characteristics in the ultrafast initial response dynamics.

First, for the lowest fluence investigated we see a transition time of τf = 40 ± 8 fs, shorter

than previously observed dynamics in any VO2 sample, as discussed further below. Second,

we observe a dramatic, up to three-fold increase in transition time with increasing fluence

(inset), in contrast to a decrease observed in previous work on polycrystalline films [179].

Third, we see the transition time τf decrease with increasing sample temperature (Fig. 5.3c).

Previous observations note a decrease in threshold fluence when the initial temperature is

increased [98, 154]. Through the combination with Raman spectroscopy we can correlate this

behavior with the structural changes of the insulating phase. The pump-probe measurement

at T = 352 K shows only a very small change in transient reflectivity with very fast dynamics,

due to the presence of metallic stripe domains in the crystal in the initial state.

The data shown in Fig. 5.3 are for three different microcrystals. The trends of an

increase in τf with fluence and a decrease with temperature are seen for all measured mi-

crocrystals, though extracted transition times vary between crystals. The transition times

τf obtained from all measurements on 20 microcrystals of different sizes, and for selected

different initial temperatures and range of fluences are summarized in Fig. 5.4. The values

are shown as a function of the measured ωV−O phonon frequency from Raman spectroscopy,

as a proxy for the different insulating phases of the crystals (indicated by green-blue color

94

0 500 1000 1500

10−4

10−3

10−2

Pump-probe delay τ (fs)

-ΔR

/R

0 100 200 300 400 500

2

3

4

5

678

10

Delay τ (fs)

-ΔR

/R (

no

rma

lize

d)

300 320 340 3600

20

40

60

80

100

Tra

nsitio

n tim

e τ

f (fs

)

625

630

635

640

645

650

Temperature (K)

ωV

-O p

ea

k p

ositio

n (c

m−

1)a b

c

2.0 2.4 2.8 3.2

40

80

120

Fluence (mJ cm-2)

τf (

fs)

2 3 4 5 6 70

1

2

Fluence (mJ cm-2)

-ΔR

/R (

x1

0-4

)

Threshold

fluence

-ΔR/R at 1 ps

1.6

2.1

3.43.2

4.04.3 mJ/cm2

Incr.

Fluence

Figure 5.3: Fluence dependence of transient reflectivity −∆R/R for an individual micro-crystal (a). Change in short timescale dynamics with fluence for F > Fth (b). Black linesshow fits to exponential recovery behavior, with extracted transition time τf shown in theinset. Variation of transition time with temperature, showing a decrease in τf with increas-ing temperature (c). This is correlated with a change in insulating structure, as monitoredthrough the ωV−O Raman mode. Blue and red dashed lines are a guide to the eye.

bar for the M1, T, and M2 phases). The error bars are based on the uncertainty of the

fit of Eq. 5.1 in the y-direction, and a Lorentzian fit to the V-O phonon Raman line in

the x-direction. Different microcrystal widths are indicated by the size of the data symbols

(see legend), from less than 5 µm to greater than 15 µm. The fluences used are all above

the threshold fluence for the specific microcrystal, with values indicated using false color.

The transition times τf vary from 40 fs to 200 fs. Notably, the average value of τf over all

microcrystals is found to be τf = 80 ± 25 with τf−1 =

∑i τ−1i (blue circle). This value is

in striking agreement with transition times τTF ∼ 80 fs from thin film studies [27].

95

M1 T

615 620 625 630 635 640 645 6500

50

100

150

200

250

ωV-O

peak position (cm−1)

Tra

nsitio

n T

ime

τf (

fs)

Fluence (mJ/cm2)

2

Fig. 3

τTF

τi

τi

ba

Wall et al.,

[6]

Cavalleri

et al., [5]

τf

>7

τiτ

i

F

TF

M2

>15 µm

< 5 µm

5-10 µm

10-15 µm

Rod width

Fig. 3

Figure 5.4: Transition time for ultrafast photoinduced transition, τf , plotted against initialambient insulating phase, as given by the position of the Raman mode ωV−O. Colorbar attop shows insulating phases. Size of the data points reflects the microcrystal width, fromless than 5 µm, to more than 15 µm. Color indicates the fluence at which the transition timeτf was measured, from 2 mJ/cm2 (orange) to > 6 mJ/cm2 (black). The measured transitiontimes vary from 40 fs to 200 fs, with no clear correlation with insulating phase. Dashed ovalsindicate data sets also shown in Fig. 5.3. The blue dashed line and circle show the averagetransition time based on the ensemble of measurements, of 80 ± 25 fs. The black dashedline and square show 75 fs, the limiting timescale observed in thin film samples by Cavalleriet al.[27]. The red-black triangles show fast time constants extracted from data by Wall etal. [179], where a strong decrease in time constant is observed with increasing fluence, incontrast the behavior observed here for single crystals. Schematic of thin film (top, τTF) vssingle crystal (bottom, τi) measurements of the transition time, where the average transitionrate over all crystals provides a value close to the thin film value.

The ultrafast pump-probe results indicate that the photoinduced IMT is highly inho-

mogeneous among the single crystallites. This inhomogeneity appears to be uncorrelated

with structure, strain, temperature, or microcrystal size, with no consistent pattern of be-

havior observed even between crystals attached to the substrate (i.e., strained) or free. This

suggests a sensitivity to moderate variations in the doping, stoichiometry, or defects. These

96

343.7 K

340.3 K68.0

2 μm

343.2 K

MCT Det.

VO2

λinc

=10.8 μm

OAP

AFM tip

SiO2

a b 344.2 K

341.2 K

353.2 K

1 μm 1 μm 340.2 K 343.2 K

343.2 K

340.2 K

340.2 K

c

Figure 5.5: Schematic of set-up for scattering scanning near-field optical microscopy (s-SNOM) imaging (a). 10.6 µm is focused onto an AFM tip, and the scattered light is collectedand detected using an MCT detector. When heating through the metal-insulator transition,the microcrystals form alternating metallic (gray) and insulating (dark blue) domains inorder to minimize substrate strain [80] (b). On cooling, the metallic domains similarly breakup along the c-axis direction, narrow, and continue to break up, as shown in the inset to (b)and (c). The domain walls also roughen, rather than maintaining a straight line, and formnanoscale metallic puddles ((c), insets). The scale bars in the insets of (b) and (c) are 500nm and 200 nm, respectively. Over repeated cooling cycles, the domains break in differentlocations and at varying temperatures (see supplement).

variations are small enough that they are not reflected in the lattice structure at the level

detectable by few-wavenumber Raman shift, nor in the overall strain or temperature depen-

dence. These variations between different crystals in turn may be spatially inhomogeneous

and lead to spatial variations in the IMT on the intra-crystalline level, within the individual

single crystals. We take advantage of the phase behavior of microcrystals attached to the

substrate, where the minimization of strain to accommodate the different thermal expansions

of VO2 and the silicon substrate leads to the formation of metallic stripe domains along the

c-axis [80] (Fig.5.5). The presence of these mesoscopic domains with the nanoscale spatial

phase coexistence of both metallic and insulating regions provides us with the sensitivity to

97

resolve the effect of microscopic inhomogeneities on the thermal IMT, from the interplay of

extrinsic strain and localized intrinsic defects and impurities at the domain walls.

Fig. 5.5a shows a schematic of s-SNOM nano-IR probing of the Drude dielectric re-

sponse of VO2 microcrystals on heating and cooling through the phase transition, with

∼ 10 nm spatial resolution. Fig. 5.5b shows the formation of mesoscopic metallic stripe

domains perpendicular to the c-axis on cooling from the metallic state, with straight domain

walls and homogeneous behavior as the insulating states begin to form in the center of the

microcrystal. The insets show the gradual narrowing of the metallic domains upon cooling,

associated with the emergence of complex, meandering domain walls with details very sen-

sitive to temperature. Once the metallic domains have narrowed sufficiently, they begin to

break up in the direction perpendicular to the c-axis (insets, Fig. 5.5b and c). This inho-

mogeneous domain wall roughening and disappearance of the metallic domains on cooling is

highly variable with repeated temperature cycling (see Fig. 5.9) This thermal behavior, with

both reproducible and non-reproducible spatial features, supports a hypothesis of an elec-

tronically driven transition with both static and dynamic variations in the local properties,

as discussed further below.

5.0.4 Discussion

In the following we discuss the implications of the above observations of the ultrafast

photoinduced response dynamics and thermal spatial behavior of single VO2 micro-crystals,

especially with regard to the interpretation of previous experiments on polycrystalline thin

films.

Below threshold, we resolve the dynamical response of the two low-frequency modes in

the coherent phonon spectrum, at approximately 200 and 225 cm−1. These modes are of A1g

symmetry, and are attributed to twisting of vanadium dimers, with their relative strength

depending on the type of insulating phase and crystal orientation with respect to pump and

probe polarization [166, 193]. Frequencies and line-widths are consistent with corresponding

98

incoherent Raman scattering and track the different insulating phases, of M1, M2, and

triclinic with their characteristic frequencies. Previous coherent phonon measurements have

disagreed in whether one or two phonon modes are observed in the 200 cm−1 range (6 THz

vicinity), and in their precise frequencies [87, 180]. These findings can now be reconciled

given possible different insulating phases of the crystallites in the thin films. The modes

that are observed can thus depend on the number and relative orientation of the ensemble

of crystallites probed in polycrystalline films.

Above the fluence threshold for the transition, we observe ultrafast dynamics on the

tens of fs timescale, with a collapse in the gap in the insulating phase. In thin film measure-

ments, the fluence-dependent behavior has been divided into three regimes: below threshold,

where coherent phonons are resolved; above threshold, where the system is driven into the

metallic state but thermal effects are visible over longer timescales; and a saturation regime,

where the magnitude of the transient reflectivity signal saturates and long timescale thermal

behavior is no longer observed [179]. In contrast, our fluence dependent measurements on

single micro-crystals show only two distinct regimes (Fig. 5.3): the below threshold regime

where coherent phonons are observed and the reflectivity relaxes over picoseconds, and the

saturation regime with emergence of the metallic phase. In the saturation regime, in general

the rapid initial change and persistence of the change in reflectivity indicates that the entire

probed volume experiences an ultrafast photoinduced transition to a metal-like state. The

micro-crystal thickness does not appear to affect the transition time. Since the crystals are

thinner than the penetration depth of 800 nm light of approximately 180 nm, a homogeneous

excitation of the microcrystal can be assumed. In contrast, a polycrystalline ensemble con-

sists of crystallites with different threshold fluences and different dynamics. The ensemble

measurement will then appear to be a superposition of multiple timescales, requiring a larger

number of fitting parameters [179].

Most notably we find that the initial insulating phase (M1, M2, or T) has no influence

on the dynamics of the ultrafast transition. Furthermore, crystals with apparently identical

99

lattice structure as concluded from identical Raman spectra reveal different photoinduced

transition time scales. This suggests that the emergence of the metallic phase in the pho-

toinduced IMT is not a lattice related effect and the variations in the IMT dynamics point

to an electronic delocalization transition.

With the shortest timescales observed of 40± 8 fs we deduce that the 150 fs timescale

for breaking the V-V dimer bonds is not relevant as a rate limiting step for the the formation

of the metallic state of the photoinduced transition, as originally proposed by Cavalleri et

al. [27, 12]. Our range of timescale values is similar to those reported in Wall et al. [179],

but their results are not directly comparable to ours since they are based on a model with

additional time constants and fitting parameters.

The electronic origin of the dynamics is supported by the decrease in transition time

observed on heating the microcrystals. While the free energy change with the increase in

temperature is small [159] compared to the energy of the pump pulse, the change in dynamics

we observe is substantial. The insulating phase also changes with increasing temperature,

following the progression M1-T-M2 (as shown in in Fig. 5.3, but the structural change in itself

has no effect on the transition time, as discussed for the comparison between microcrystals.

The temperature-dependent decrease in transition time is an interesting counterpoint

to the increase in transition time with increasing fluence. This is in contradiction to the

observations of Wall et al., where the time constants in their model all decrease with in-

creasing fluence [179]. Our results suggest a possible artefact of thin-film polycrystalline

studies which probe an ensemble averaged response of variable numbers of crystallites, each

with varying transition times and threshold fluences.

Fluence behavior similar to that observed here has been previously observed in graphite

[73] and Cr-doped V2O3 [120]. For VO2, this could suggest that the higher fluences drive the

system further out of equilibrium and lead to slower transition times to the metallic state. A

possible mechanism would be non-equilibrium interband excitations, with variable fractional

pump-induced occupation depending on the density of states and its variation with doping

100

and impurities, to states supporting or opposing band structure collapse and saturation of

states favorable to fast transitions at higher fluences (Fig. 5.6b).

We conclude that in thin films the intrinsic dynamics can be masked by the inhomoge-

neous distribution and complicating extrinsic interactions among crystallites, with different

sizes, orientations, and strain (Fig. 5.4). However, even on the individual crystal level,

for nominally homogeneous single crystals, we observe inhomogeneous behavior. s-SNOM

imaging of the thermal transition also shows inter- and intra-crystal inhomogeneity: on

heating, the nucleation and growth of the metallic domains is highly reproducible, but the

break-up on these metallic domains on cooling can vary over repeated thermal cycles (see

Fig. 5.9). The formation of metallic and insulating domains during the transition arises due

to strain reduction, but minimization of free energy in a homogeneous system, considering

bulk thermodynamic energy, strain energy, domain wall energy, twinning, and insulating

lattice structure, should favor straight domain walls rather than the meandering structure

we resolve in Fig. 5.6b and c. Furthermore, the metallic puddles remaining after break-up of

the stripe domains appear at varying positions and are extremely sensitive to temperature,

indicating that the VO2 microcrystals are in a highly dynamic state close to the critical

temperature TC .

Crystal

No.

Fluence

(mJ/cm−2)

ωV−O (cm−1) τf (fs) τs (fs) a Crystal size

(µm)

1 3.3 617.5 168 3580 0.02 33.6

2 5.3 631.4 70 2000 0.03 12.3

3 4.5 629.3 80 2050 0.1 5.5

4 3.3 630.2 92 2500 0.3 7.3

5 4.5 632.1 66 2000 0.1 10.4

6 4.5 632.5 71 2500 0.01 11.3

7 3.3 621.3 147 2000 0.3 4.9

101

8 3.0 627.2 125 2860 0.4 4.8

3.7 627.2 128 2010 0.2 4.8

9 3.3 624.9 119 2000 0.3 6

10 3.3 618.7 118 2100 0.3 2.9

11 2.5 616.9 68 2000 0.1 3

3.3 616.9 111 2590 0.2 3

12 2.4 626.2 55 2020 0.2 10

13 2.2 616.9 42 8620 3E-06 3

2.4 616.9 77 5000 0.01 3

2.5 616.9 92 2000 0.01 3

2.8 616.9 92 2000 0.2 3

14 2.4 625.5 71 10000 0.01 9

15 2.3 626.3 76 9720 3E-05 9

2.5 626.3 80 2010 0.05 9

3.3 626.3 130 2010 0.2 9

3.3 626.3 121 2760 0.4 9

16 2.4 617.1 97 9950 0.009 2.3

2.6 617.1 84 9990 0.01 2.3

17 2.6 619.8 108 2000 0.2 4.8

18 3.6 625.5 97 2000 0.1 9

3.6 625.5 83 3600 0.3 9

19 7.0 614.4 157 2000 0.2 6

20 7.0 648.7 156 8250 0.002 6

21 5.3 632.2 42 9990 5E-07 11.5

5.3 632.2 52 6840 2E-07 11.5

21(p) 5.3 632.2 107 2000 0.05 11.5

22 8.2 640.2 71 10000 0.04 13

102

23 4.1 628.9 199 5860 0.01 2.5

24 4.1 641.5 119 2000 0.03 4.5

25 8.2 643.0 76 6710 5E-06 10.4

26 8.2 640.1 115 2010 0.2 5.6

27 8.2 642.7 42 6290 5E-07 7.5

28 8.2 642.7 107 2000 0.05 10

29 0.8 636 7.2

Table 5.1: Fit parameters shown in Figure 5.4 of main text. Measurement

21(p) was taken on crystal #21, with pump polarization perpendicular

to the cR-axis, while the other measurements were performed with pump

parallel to the cR-axis. Coherent phonon measurements were performed

on crystal #29, at the specified fluence.

Crystal

No.

Fluence

(mJ cm−2)

ωV−O (cm−1) τf (fs) τs (fs) a Temperature (K)

30 3.3 628.3 84.2 2000 0.2 297

3.3 630.0 79.5 3150 0.45 313

3.3 630.9 75.3 2010 0.15 323

3.3 632.5 73.7 9090 1.4E-5 334

3.3 634.9 62.1 9920 0.012 343

3.3 640.0 7.6 5000 0.01 353

Table 5.2: Table of fit parameters shown in Figure 5.3c.

103

E -

EF (

eV

)

-1

0

1

Density of States

0.5

-0.5 E -

EF (

eV

)

-1

0

1

Density of States

0.5

-0.5Monoclinic insulating

σ σ

Defect statesE

g

e- ω

c cI

M

Eg

Monoclinic insulating

Rutile metallic

s-SNOM s-SNOM

a b

dc

Figure 5.6: Schematic representation showing the electronic band structure of VO2 (a) andpossible modifications from defects or impurities (yellow lines, b), which could alter the tran-sition time of the ultrafast IMT and produce the variable dynamics observed. Defects couldalso change the spatial arrangement of metallic and insulating domains in single crystals,producing a shift from straight domain walls, as predicted by theory (c), to more complicatedstructures (d).

5.0.5 Conclusion

Both the ultrafast photoinduced studies and thermal IMT s-SNOM therefore show a

high degree of sensitivity to the effect of dopants and defects. These could alter the density

of available states and therefore the redistribution of holes and electrons due to the exciting

pulse, and thereby change the rate of electron delocalization and bandgap collapse (see

Fig. 5.6a,b). Similarly, defects and complex strain could disrupt the free energy uniformity

and produce complex domain topology in the thermal transition, as we resolve at domain

walls for attached crystals through s-SNOM (Fig. 5.6c,d).

We reveal a wide variation in static and dynamic properties of apparently homoge-

neous, well-characterized single crystal sub-systems. These results raise the question of how

104

0 200 400 600 800 1000

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Raman shift (cm-1)

302.2 K313.2 K323.2 K333.2 K343.2 K348.2 K350.7 K353.2 K363.2 K373.2 K383.2 K

Inte

nsity (

se

pa

rate

d fo

r cla

irty

)

0 200 400 600 800 1000

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

Raman shift (cm-1)

Inte

nsity (

se

pa

rate

d fo

r cla

irty

)

302.2 K313.2 K323.2 K333.2 K343.2 K353.2 K363.2 K373.2 K383.2 K

290 300 310 320 330 340 350 360 370 380 390625

630

635

640

645

650

655

Temperature (K)

HeatingCooling

ωV

-O (

cm

-1)

a b c

HeatingCooling

Figure 5.7: Raman spectra of crystal #30 on heating (a) and cooling (b) through the thermaltransition. Metallic stripes appear at 348 K and the microcrystal is fully metallic above 385K. Extracted ωV−O peak position showing the evolution of the insulating phase through theT phase upon heating. At 363 K, the remaining insulating stripes transition to the M2phase.

to access the intrinsic response of VO2, and strongly correlated electron materials more gen-

erally. The rich and diverse properties of these materials that can be induced and controlled

through doping, strain, external fields, etc. may be more sensitive to disorder and impurities

than previously expected.

5.0.5.1 Near-field pump-probe microscopy

My far-field pump-probe microscopy suggested inhomogeneity in the dynamics in nom-

inally homogeneous single-crystals results from nanoscale variations in strain, doping, or

stoichiometry. In order to further explore these effects, follow-up experimental work within

our group has been performed by Sven Donges, Omar Khatib, and Joanna Atkin. By com-

bining s-SNOM and pump-probe microscopy, they were able to directly measure the spatial

variations in the dynamics across single crystals with 10’s of nm spatial resolution [45].

Pump light at λpump = 1032 nm drives the ultrafast IMT and its evolution is monitored at

λprobe = 4.7µm, probing the tail of the Drude conductivity change. The probe pulse is gen-

erated with an optical parametric amplifier (OPA)/difference frequency generation (DFG)

system pumped at 1032 nm at a 1 MHz repetition rate by an amplified laser. This creates

105

Figure 5.8: a) Red solid lines show near-field pump-probe traces on VO2 rod at differentdistances from crystal edges (locations i-iii indicated in b)). Dashed lines are fits to the traces.Blue points are extracted from spatial images at locations A-D indicated in b). b) Spatialimages acquired at different pump-probe delays. Lower 3D plot shows the topography. c)Colored lines show spatial linecuts perpendicular to the cR axis of the probe signal at constantdelays. Black line is the topography. Figure and data courtesy of Ref. [45].

200 fs pulses tunable from 2-16 µm. The pump arm is sent to a delay arm and the pump

and probe pulses are combined on a Ge beam combiner and sent to an AFM tip. Scattered

signal is detected with an MCT detector and demodulated at the second harmonic of the

tip motion with a lock-in detector to extract the near-field response.

By scanning single-crystal microrods, intra-crystallite variation was probed. Figure 5.8

summarizes the results from this study. Fluence-dependent measurements indicate the exci-

tation to a non-thermal metastable metallic state with fluences above ∼ 2 mJ/cm2. Charac-

106

teristic pump-probe traces are shown in Fig. 5.8a). Spatial scans as displayed in Fig. 5.8b)

at different delays shows heterogeneity in directions perpendicular to the c-axis. The spa-

tial linecuts of probe signal at constant pump-probe delay (Fig. 5.8c)) shows a pronounced

spatial variation perpendicular to the cR axis from a quick relaxation due to excitations of

carriers, to full excitation of the metastable state at constant fluence. The inhomogeneity

at the edges could be caused by variations in doping or stoichiometry that can occur during

crystal growth.

5.0.5.2 Irreproducibility and the role of polymorphs and crystallite edges in

thermal IMT

To further investigate the possible role of nanoscale phenomenon of the ultrafast IMT,

I perform further nanoscale images showing the irreproducible break up pattern of metallic

puddles upon cooling through the thermal transition. Figure 5.9 shows several thin metallic

domain patterns during two repeated cooling cycles imaged with s-SNOM. Images taken at

the same temperature are stacked vertically. The domains on the left break at roughly the

same location at T = 344.0 K, but they show different domain patterns upon cooling to

T = 343.2 K, as highlighted in the red circles. Since the domain patterns begin at 347.2 K

and end at 342.8 K in qualitatively the same spatial arrangement, the different intermediate

behavior cannot be explained by hysteresis effects. The large difference in the repeated scans

at 343.2 K also cannot arise due to uncertainties in the temperature. Additionally, these

metastable nanoscale metallic domains are observed to relax to a completely insulating state

over temperature steps as small as ∆T = 0.1 K, indicating their dynamic nature.

The irreproducibility illustrate the highly dynamical state of the microcrystals near

the critical temperature TC which is extremely sensitive to spatial variations in the chemical

potential and perturbations in the electronic structure. This highly dynamical state is in

part a result of the free energy degeneracy of the rutile metallic and the two monoclinic

insulating structures at TC [153]. As the crystal is cooled from TC , the degeneracy is broken

107

347.2 K 344.8 K 344.0 K 343.2 K 342.8 K

Figure 5.9: s-SNOM imaging during repeated cooling cycles showing irreproducible break-uppattern of metallic domains on a VO2 microcrystal. Scale bars are 500 nm. Scans from thefirst cooling sequence (top row) and the second cooling cycle (bottom row, images at thesame temperatures are stacked vertically) show a meandering pair of domains on the leftthat break around T = 344.0 K at nearly the same locations in the different cycles, butexhibit different domain patterns at T = 343.2 K (red circles) that cannot be explained byhysteresis effects since they begin and end with roughly the same domain pattern. Thisalso may not be explained by difference in temperature since they are qualitatively differentdomain patterns that do not match a corresponding temperature in the other cycle.

and free energies begin to deviate, favoring the insulating phases. The insulating areas that

first nucleate are observed to be in the M2 phase [80] due to the substrate-induced strain,

which is predicted to have a structural free energy contribution closer to the rutile structure

∆HM1→R > ∆HM2→R [159] and is therefore more sensitive to fine spatial variations in the

free energy landscape.

The above results demonstrate the rich phenomenon that can occur in VO2 resulting

from the interplay of nanoscale variations of strain, doping, and stoichiometry. To further

108

0 nm

110 RT 370 K 344 K

Min

Maxa) b) c) d)

M1 M2

Figure 5.10: Spatial images of a single-crystal VO2 section with complex strain. Althoughthe topography shows a homogenous surface (a), s-SNOM imaging at λ = 10.8 µm showsalternating sections of M1 and M2 (b). M2 is identified by the small stripes in the opticalimage parallel to the c-axis due to twinning. c) When heating, metallic domains first appearat the regions that are M1 at room temperature, and grow outward into the M2 regionswith increasing temperature. d) Cooling behavior is qualitatively different than the heatingbehavior. Metallic bands break and persist on the boundaries with the M2 phase. Addi-tionally, small puddles can persist on the edges of the crystallites below the bulk transitiontemperature. Scale bars are 1 µm.

explore these nanoscale phenomenon in the IMT of VO2, I studied in-depth the role of

strain and temperature through the thermal transition using infrared s-SNOM. By probing

the Drude conductivity at λ = 10.8µm, I study the evolution of the thermally driven IMT

with 10’s of nm resolution. A ceramic heating element provides sample temperature control

from RT to ∼ 400 K. By focusing on crystal edges and boundaries of regions of different

polymorphs, I probe the high sensitivity of the IMT to nanoscale strain and structural

inhomogeneities.

In Fig. 5.10, I show a region of a crystal with co-existing M1 and M2 polymorphs

due to complex strain along the c-axis. The M2 phase can be distinguished from the M1

phase through characteristic ripples in the s-SNOM contrast. These variations occur due to

crystal twinning in the M2 phase, and allows for identification of the polymorph through s-

SNOM [82].As the crystal is heated through the bulk transition temperature, metallic regions

form in the M1 regions and gradually grow into the M1 regions with increasing temperature.

The phase competition during the transition follows the progression: M1/M2 → R/M2 →

R. This is consistent with the strain minimization since the M1 phase cR axis is 1% longer

109

than the R phase while that for the M2 phase is 1.6% longer.

Fine details of the sensitivity of the thermal IMT to nanoscale strain emerge during

cooling from above the transition temperature. When cooling, the domains begin to shrink

along the c-axis within the M2 regions. As the temperature continues to decrease, super-

cooled metallic regions persist in the sections that are M1 at room temperature, until they

abruptly break and leave thin metallic stripes at the interface between the M1/M2 regions.

Additionally, small metallic “puddles” can persist on the edges of the crystallites. The

behavior is qualitatively similar to the behavior seen in the ultrafast nano-imaging, where

inhomogeneities exist at the edges. These puddles have previously be ascribed to the metallic

monoclinic phase [108].

5.0.6 Control of IMT nano-domains

Control of IMT nano-domains has been achieved through nano-mechanical stress on

VO2 nanorods previously, whereby the domain shape was readily manipulated [22]. This

was performed by clamping nanorods on one end and pressing on the free end. This creates

compressive strain on one side of the rod, while the other end experiences tensile strain. This

was done at constant temperature, but by heating locally I may also manipulate the domain

structure. With our independent control of tip and sample temperature, I investigate the

effect of local heating on the IMT.

With our newly developed laser-heating technique, I could heat tips to high tempera-

ture and simultaneously obtain s-SNOM images. This tip acts as a local heat source with

which I could study the material thermal near-fields. However, I can also manipulate the lo-

cal structure of the IMT of VO2, and monitor their dynamic evolution. Figure 5.11a) and b)

shows the topography and metallic domain nano-structure of VO2 measured at λ = 10.8 µm

at high temperature. The IR contrast shows the presence of many dielectric impurities on

the VO2 surface due to sample contamination.

Upon cooling, the metallic domains start to thin and break (Fig. 5.11c)). However,

110

0 nm

130

Max

Min

1 µm

b) 80°C

c) 70°C e) 65°C

d) 70°C + 0.5W f) 65°C + 0.5W

a)

Figure 5.11: a) Height of VO2 single crystal structure. b) s-SNOM image well above thetransition temperature showing metallic stripes orthogonal to the c-axis. c) s-SNOM imageat 70C after cooling from 80C. Metallic stripes begin to thin and split. d) Applying localheating through laser-heating of cantilever with 0.5 W of power creates metallic domainstructure qualitatively similar to b). e) s-SNOM image after cooling from 70C to 65C. f)Applying laser heating with 0.5 W of power enlarges puddles at the edges.

the metallic domains regrow during laser-heating the tip with a laser power of P ≈ 0.5 W

(Fig. 5.11d)). Similarly, applying laser-heating to super-cooled metallic puddles (Fig. 5.11)

creates larger puddles. This behavior is qualitatively similar to heating up from a tempera-

ture where the rod exhibits puddles (as shown in Fig. 5.12).

5.1 Conclusion and outlook

My work on VO2 reveals a complex picture with many variables determining the dy-

namics and spatial properties of the IMT. While the observed inhomogeneity in the dy-

namics of the ultrafast IMT cannot be conclusively attributed to an underlying physical

mechanism, its lack of correlation with crystal strain or size suggests a high sensitivity to

other growth-specific parameters such as variations in doping, defects, or variations in stoi-

chiometry. Additionally, the ultrafast imaging of the IMT reveals strong heterogeneity even

111

343 K 344 K

345 K 346 K 348 K

1 µm

a)

d)

b) c)

e) f)

Figure 5.12: a) Height of the same rod shown in Fig. 5.10. b-f) s-SNOM images takenheating from 343 K, where the VO2 rod has small metallic puddles persisting on the edges,up to 348 K in 1-2 K increments. The puddles generally grow uniformly outwards until fullstripes can form across the VO2 crystal width.

within a nominally homogenous single crystal. Results from s-SNOM imaging of the Drude

conductivity during the thermally induced IMT show pronounced heterogeneity, with metal-

lic puddles persisting both within otherwise homogenous regions of crystallites, as well as

at the edges where crystal strain and stoichiometry is expected to vary. This work suggests

that future studies of VO2 must account for this heterogeneity in their interpretations of

spatially-averaged data. Additional work to identify the origin of this heterogeneity may

be required for a complete understanding of the IMT, and for its successful technological

application.

Chapter 6

Nano-spectroscopy and imaging of polariton propagation in 2D materials

Polaritons in 2D materials provide a novel platform for photonics by supplying strong

confinement and control of light down to nm length scales [21, 187, 11]. However, the

successful application of the polaritons in these materials requires a good understanding of

their propagation and resonant properties. With their high-wavevectors and resulting high

confinement compared to the free space photon wavelengths, they cannot be directly imaged

through diffraction-limited studies. Using s-SNOM, I can directly measure these polariton

spatial propagation parameters such as their spatial wavelength and damping.

6.1 Hexagonal boron nitride

Photonics is dominated by the use of plasmonics, which limits the spectral range to

visible frequencies and generally broad resonances due to the ohmic losses in metals. Phonon-

based photonics applications have many promising advantages, including dramatically lower

losses and longer propagation lengths and extends the applicability to the MIR. However,

phonon resonances in the MIR are in general narrowband and have limited mechanisms for

tunability. As a result, phonon polaritons have not seen much application in photonics.

Hexagonal boron nitride (hBN) has shown promise and generated much recent interest

due to its unique dispersion properties and high degree of electromagnetic field confinement.

hBN has a graphite-like structure, consisting of stacked honeycomb layers. As a result of

its highly anisotropic structure, it has two distinct vibrational modes, one at ν ' 1370

113

cm−1 corresponding to in-plane vibrations and one at ν ' 790 cm−1 which corresponds to

out-of-plane vibration of the lattice. This results in very different dielectric functions in

the in-plane and out-of-plane directions which yield two Reststrahlen bands where one of

the dielectric function components is negative while the other component is positive. The

dispersion relation is given by

ω2

c2=∣∣∣~k∣∣∣2 =

k2x + k2

y

ε‖+k2z

ε⊥. (6.1)

For typical materials, this defines elliptical isofrequency surfaces. However, in Reststrahlen

bands of hBN, the iso-frequency surfaces of the dispersion relation form hyperboloids which

extend to very high wavevector components. In theory, the hyperboloids extend indefinitely,

however for hBN the highest wavevectors are limited by the dimension of the unit cell and

cannot exceed kmax = 1/a for bond length a.

The hyperbolic dispersion results in numerous interesting properties. In the Rest-

strahlen bands, hBN supports phonon polaritons (PhPs) with high wavevectors with phase

velocities that travel through the crystal bulk at angles defined by θph = arctan( √

ε‖i√ε⊥

)with

respect to the c-axis. The PhP group velocity is orthogonal to the phase velocity, and makes

an angle θg = π/2 − θph, which defines the propagation of PhP energy through the crystal

bulk.

Furthermore, when coupled to graphene as in a graphene/hBN heterostructure, the

PhPs may interact with graphene surface plasmon polaritons to create hyperbolic phonon-

plasmon polaritons (HP3’s). Graphene SPPs have generated much recent interest due to their

ability to confine IR light down to nanometer-scale gate-tunable wavelengths[53, 48, 47, 33].

Additionally, when on a metal substrate, the PhPs may also couple similarly to SPPs excited

on the surface of the metal. By performing broadband nano-imaging and -spectroscopy on

such a heterostructure, collaborators at the Brazilian Synchrotron Light Laboratory (LNLS)

have studied the unique properties of these HP3s [117].

This coupling is evidenced by an enhancement of the out-of-plane mode of hBN through

114

a) b)

1300 1400 1500 1600

Frequency (cm-1)

700 750 800 850 900

Frequency (cm-1)

400

200

0

-200

100

50

0

-50

Re(e )

Re(e )

Im(e )

Im(e )

Figure 6.1: Real and imaginary part of the in-plane εx and out-of-plane εz dielectric functionsin the spectral range near the a) out-of-plane and b) in-plane phonon resonance. The redshaded region indicate spectral regions with hyperbolic dispersion.

the interaction with the SPPs of graphene and a metallic substrate. The primarily c-axis

polarization of this mode can excite SPPs which have a similar field structure. In contrast,

the in-plane mode is quenched by SPPs. Since the atomic motion of the in-plane PhP is

primarily in the ab plane, the generated electric fields can induced currents in graphene

and a metal substrate which screen the electric fields and reduce their strength. Another

consequence of this phenomenon is that when PhPs encounter a change between a metallic

and dielectric substrate, they can be either reflected or transmitted, depending on their

electric field symmetry.

6.1.1 Experiment

The following experiment is performed by Francisco Maia and Raul Freitas at the

synchrotron at LNLS. Infrared light from a 1.67 T bending magnet of the synchrotron is

collimated and then sent to a commercial s-SNOM (NeaSnom, Neaspec GmbH) and used for

nano-spectroscopy. This technique, named synchrotron infrared nano-spectroscopy (SINS),

has been developed by our group at the LBNL synchrotron at Berkeley [13]. This provides

bandwidth spanning much of the MIR, with good irradiance. The system is a typical s-

115

Figure 6.2: a) Experimental schematic. G/hBN is stacked on Au stripes to create sub-strate interfaces. AFM tip is illuminated by broadband synchrotron light to perform nano-spectroscopy. Backgate is used to tune the graphene Fermi level. b) AFM topography of theheterostructure. c) Spectrally-integrated s-SNOM image showing contrast dominated by thechange in conductivity of the underlying substrates. Figure and data courtesy of Ref. [117].

SNOM, with MCT detector and reference arm for interferometric detection.

The sample is assembled by fabricating Au stripes on a SiO2/Si substrate using stan-

dard thermal metal deposition and electron-beam lithography, and was provided by Ingrid

Barcelos and Alisson Cadore (Federal University of Minas Gerais). 20 nm thick hBN and

single layer graphene flakes are created using mechanical exfoliation with adhesive tape. The

hBN and graphene are stacked across the Au stripes to create the G/hBN/Au, G/hBN/air,

and G/hBN/SiO2 heterostructures. Electronic contacts are added on the graphene for gate

dependent measurements.

Spectra are acquired at 5 cm−1 resolution, and linescans were performed by acquiring

spectra at 10 nm steps along the substrate interfaces.

116

6.1.2 Results

The near-field spectra of hBN on different substrate shown in fig. 6.2 show several

distinct effects. For the hBN/SiO2 structure (Fig. 6.3a)), three prominent peaks are present.

The in-plane PhP response appears at ' 1365 cm−1, and the out-of-plane PhP response is at

' 780 cm−1. The broad response around ∼ 1100 cm−1is due to the SiO2 phonon polariton

resonance and is visible due to the partial IR transparency of the hBN flake. The spectrum

of suspended hBN (Fig. 6.3b)) shows the same three peaks, although the SiO2 response is

weaker due to the increased distance from the SiO2 surface. In contrast, the spectrum from

the hBN/Au shows a weaker in-plane PhP response (Fig. 6.3c)). The addition of a graphene

top layer (solid lines) creates hybidized HP3 modes. The show a dramatically narrower and

stronger peak at the out-of-plane HP3 frequency compared to the bare hBN response, while

the in-plane HP3 response is slightly quenched.

A spectrally resolved linescan across the interface between G-hBN/SiO2-G-hBN/Au-

suspended G-hBN structures reveals the change in strength of the in- and out-of-plane HP3

mode responses as they are enhanced or quenched by the substrate. Additionally, within

a few hundred nanometers of the interfaces, there are areas of higher signals at frequencies

blue of the in-plane mode and red of the out-of-plane modes. Similar features are observed

near edges of hBN crystals, caused by interference of tip-excited PhPs that are reflected by

the edge. When illuminated by resonant light, the tip can couple the incident photons into

propagating HP3s, thus acting as a polariton source. The HP3s propagate and reflect off of

crystal edges or other boundaries in the flake, and can interfere with counter-propagating

HP3s[41, 42, 170]. As the tip scans close to an edge, a standing wave is created between the

tip and the edge. As the tip-edge distance changes, the interference condition changes and

the field strength at the tip changes, resulting in periodic spatial patterns as the tip scans

closer to a HP3s scatterer. This can be used to measure the HP3s propagation parameters,

such as the spatial wavelength and the damping.

117

ωTO (cm−1) ωLO (cm−1) Γ (cm−1) ε∞‖ 1367 1610 5 4.1⊥ 783 828 4 4.95

Table 6.1: Parameters used for Lorentizan model for hBN dielectric function

6.1.3 Dispersion calculations

To calculate the dispersion properties of hBN, I use Lorentian models for the hBN

dielectric function:

ε‖ = ε∞,‖

(1 +

ω2LO,‖ − ω2

TO,‖

ω2TO,‖ − ω2 − iωΓ‖

)(6.2)

ε⊥ = ε∞,⊥

(1 +

ω2LO,⊥ − ω2

TO,⊥

ω2TO,⊥ − ω2 − iωΓ⊥

)(6.3)

using literature values[52] (parameters displayed in Table 6.1). The dielectric functions are

plotted in Fig. 6.1. The red shaded regions indicate the regions of hyperbolic dispersion,

where the real parts of the ε‖ and ε⊥ are of opposite sign.

The polariton resonances correspond to peaks in the imaginary part of the reflection

coefficient[139]. The reflection coefficient is calculated through the Fresnel coefficients of the

layered structure air/G/hBN/substrate for each substrate. They are given by

ra =ε⊥k

za − ε1kze + 4πσkzak

ze/ω

ε⊥kza + ε1kze + 4πσkzakze/ω

(6.4)

rs =ε2k

ze − ε⊥kzs

ε2kze + ε⊥kzs(6.5)

rp =ra + rse

i2kzed

1 + rarsei2kzed

(6.6)

where ra, rs, and rp are the Fresnel coefficients for the air-G/hBN, hBN/SiO2, and com-

bined system, respectively. The wavevectors kzi for i = air, SiO2 are given by kzi (ω, q) =√εiω2/c2 − q2 for angular frequency ω and wavevector q. The wavevector kz⊥ for hBN is

given by kz⊥(ω, q) =√ε⊥ω2/c2 − q2. The conductivity of graphene σ is given by the Drude

118

Figure 6.3: Near-field spectra of a) hBN on SiO2, b) suspended hBN, and c) hBN on Auwith graphene top layer (solid lines) and without graphene (dashed lines). Figure and datacourtesy of Ref. [117].

model for intraband excitations

σ =e2EF

π~2

i

ω + i/τ(6.7)

for Fermi level EF, and relaxation time τ . I use a EF = 1 eV and τ = 100 fs.

My calculations show that the substrate dielectric function strongly influences the

dispersion relation of hBN. For the in-plane mode the HP3 momentum is higher by a factor

of ∼ 5 for a Au substrate than for a dielectric function, while the out-of-plane mode shows

the opposite trend.

119

6.1.4 Discussion

This effect has been observed previously on hBN[41, 42, 170, 144] and for SPPs of

graphene[53, 48, 47, 33], however only close to material edges. We observe this effect at

substrate boundaries, on sections where the hBN is continuous and has no physical boundary.

Additionally, we see a pronounced asymmetry in this feature particularly at the SiO2/Au

interface. For the in-plane mode, we see the interference only on the G-hBN/SiO2, with no

corresponding feature on the G-hBN/Au side. The opposite is true for the out-of-plane mode,

where the interference only appears on the G-hBN/Au side. A similar trend occurs for the

suspended G-hBN/G-hBN/Au interface. This suggests that some unidirectionality in terms

of HP3s transmission across the interface, or that the interface acts as a “diode” for HP3s.

When in-plane HP3s are excited on the G-hBN/SiO2 structure, they are reflected when they

encounter the G-hBN/Au structure, while in-plane HP3s excited on the G-hBN/Au structure

transmit across the interface with the SiO2 interface. For out-of-plane HP3s, the situation

reverses. Out of plane HP3s excited on the G-hBN/Au structure are reflected by the interface

with G-hBN/SiO2, while counter-propagating HP3s transmit across the interface.

This interpretation is supported by the above observation of enhancement and quench-

ing of the two HP3 modes. As observed in Fig. 6.3, the Au substrate screens the in-plane

mode and reduces its strength. When in-plane HP3s traveling from the G-hBN/SiO2 or

suspended G-hBN structures encounter the interface with the G-hBN/Au structure, this

screening acts like a barrier for the HP3s and they are reflected. However, HP3s that are

excited on the G-hBN/Au structure do not see this barrier and can transmit across the

interface. Similarly, out-of-plane HP3s that are excited on the G-hBN/Au structure reflect

when they encounter interfaces with a substrate that would reduce their intensity.

The calculated HP3s momentum dispersion relation provides additional support for

this interpretation. As seen in Fig. 6.4f) the in-plane HP3s momentum is significantly higher

for a Au substrate than for either suspended hBN or hBN on a SiO2 substrate, while the

120

Figure 6.4: a) Spectrally integrated s-SNOM image of interface between G-hBN/SiO2/G-hBN/Au/suspended G-hBN structures. High contrast due to the Au substrate is visiblethrough the 20 nm thick hBN flake. b) Spectrally resolved linescan across interface. c)Schematic of the tip-mediated excitation and propagation of hBN HP3s. Incident resonantlight is coupled to propagating HP3s through the spatial momentum imparted by the tip.They then reflect when the encounter barriers. Diode-like rectification of the d) in-plane ande) out-of-plane HP3s occur when traveling on and off the G-hBN/Au structure, respectively.Momentum calculations for the three different substrates for the f) in-plane and g) out-of-plane modes. Figure and data courtesy of Ref. [117].

out-of-plane HP3s momentum is lower for the Au substrate. This means that in order to

transmit from G-hBN/SiO2 or suspended G-hBN to G-hBN/Au, the HP3 must absorb mo-

mentum from some source. A HP3 may absorb momentum from a defect, sample roughness,

a thermally excited HP3, or through merging multiple HP3s together, although is in gen-

eral unlikely at room temperature with a low density of thermally excited HP3s. However,

HP3s transmission to a region of lower momentum can occur by transferring momentum to

multiple HP3s, electron, or photons much more readily[192, 30, 163].

121

6.2 Imaging of Graphene SPPs

With its linear dispersion and readily tunable Fermi level, graphene exhibits many

interesting plasmonic and photonic properties [138]. As a single atomic layer carbon atoms,

graphene promises extreme electromagnetic confinement. Its electronic dispersion can be

solved within the tight-binding approximation [135], which at the K and K′ point in reciprocal

space is approximately given by

E±(~q) ≈ ±vF |q| (6.8)

for wavevector q and Fermi velocity vF ' 1× 106 m/s [181].

For intraband excitations below the Fermi level EF (~ω < EF ), the conductivity of

graphene is described well with a Drude model σ(ω) = σ0i

ω+iτ−1 , where σ0 = e2EFπ~2 and τ is

the electron scattering time [96]. Graphene supports surface plasmon polaritons in the MIR,

with dispersion relation given by [74]

kSPP =2πh2c2ε0κ

e2EFλ20

=iωκ

2πσ(ω)(6.9)

where λ is the light wavelength, and κ = (ε1 + ε2)/2 is the effective dielectric function

described by the average of the dielectric functions of the upper and lower half-spaces. While

graphene promises many novel applications in the field of plasmonics, the sensitivity of its

plasmonic damping on many external parameters has precluded its widespread use.

The damping parameter is approximately given by γp ≈ σ1/σ2 + κ2/κ1, which gives

the separate contributions of the intrinsic conductivity of graphene and of its substrate [47].

For a typical SiO2 substrate, κ2/κ1 ≈ 0.05, however typical plasmon damping parameters

(derived through spatial imaging) are several times higher than Drude conductivity values

extrapolated from DC mobility measurements [62, 86, 165, 32, 74]. This could originate from

nanoscale impurities or charge pooling occurring due to charged impurities of the substrate.

To further investigate this discrepancy, I studied the influence of substrate and fabrication

process on graphene SPP propagation.

122

kin

kr

kt

GB

k0

AFM tip

kin k

r

6 nm 0 nm

30 V 6.5 V

0.9 rad. 0 rad.

a) b)

c)

d)

G SiO2

Graphene edge

Figure 6.5: a) SPP interferometry set-up. A sharp tip is brought close to a graphene sheet,and couples far-field radiation with wavevector k0 into propagating SPPs which reflect whenthey encounter grain boundaries or the edge of the graphene sheet. Wavevectors kin cor-respond to the outward propagation of SPPs excited by tip, while kr and kt correspond topropagation of SPPs reflected by and transmitted through by boundaries. b) Topography ofa thin graphene ribbon on a SiO2 substrate. c) Near-field amplitude and d) phase acquiredusing s-SNOM, showing interference of SPPs due to edge reflections as fringes parallel to thegraphene edges. Scalebars are 200 nm.

6.2.1 Experiment

As described previously, a sharp tip is brought close to the graphene sheet and is illu-

minated with MIR light (Fig. 6.5a)) [53, 48, 47, 33]. The tip couples incident photons into

propagating SPPs, which travel along the graphene sheet and are reflected by grain bound-

aries, graphene edges, or other defects. The reflected SPPs interfere with newly launched

counter-propagating SPPs, creating a spatial interference pattern as the tip scans the surface.

The interference pattern can then be directly related to SPP propagation properties such as

the SPP wavelength and damping parameter. To extract these values, I fit the oscillations

to a cavity model developed in our group [53].

123

6.2.2 Results and discussion

SPP interference is visible near flake edges, grain boundaries, and other linear defects

in a graphene section. Fig. 6.5b) shows the topographic image of a bilayer graphene (BLG)

ribbon with pronounced SPP propagation. The amplitude value is lower on the graphene

due to the low ambient doping, which reduces the IR conductivity, whereas by probing at

λ = 10.8µm is close to a phonon polariton mode of SiO2. The plasmon wavelength of 110±20

nm is extracted from a linecut across the width of the graphene ribbon. This is shorter than

plasmon wavelengths observed previously in Ref. [53] for bilayer graphene, due to the low

doping of around EF = 0.2 eV.

Fig. 6.6 shows SPP interferometry images of a single layer graphene (SLG) sheet on a

Ta2O5 substrate fabricated by the Schibli group at CU-Boulder. The graphene was grown

using chemical vapor deposition, which generally results in lower mobility samples than

mechanical exfoliation due to, e.g., higher defect density. This lower mobility is evident in

the much shorter propagation length of the plasmons, where only a decrease in amplitude at

the grain boundaries in Fig. 6.6b), and no significant interference fringes. In contrast, the

phase displayed in Fig. 6.6b) shows one peak and one minimum parallel to the grain boundary

due to SPP interference, but this decays before a second maximum is visible. Since a grain

boundary is theoretically atomically thin, they do not appear in the topography (Fig. 6.6a)).

Other types of linear defects include folds or wrinkles in the graphene sheet. Figure 6.6d-f)

show the topography, optical phase and amplitude of such a defect. The interference pattern

in the phase correlates with a small linear inhomogeneity in the topography, which acts as a

partial reflector of SPPs.

In general, my SPP interferometry images of CVD graphene show a much shorter

propagation length than those of exfoliated graphene. This is due to the generally lower

mobility of CVD graphene compared to exfoliated graphene due to a higher density of defects

and impurities. However, the substrate has a significant influence as described previously.

124

2 µm

0 nm

30 nm

0 rad.

2.2 rad.

0.1 V

7.9 Va) b) c)

0 nm

15.7 nm

0.2 rad.

1.6 rad.

0.3 V

8.1 Vd) e) f)

500 nm

Figure 6.6: SPP interferometry images of CVD graphene on a high κ dielectric substrate(Ta2O5). a) Topography of grain boundaries show a mostly smooth surface, with no lineardefect visible. However, b) phase and c) amplitude show signs of plasmon reflection off ofgrain boundaries. d) Topography of a fold or wrinkle in the graphene sheet that showsplasmon reflection in both the near-field e) phase and f) amplitude.

For Ta2O5, the substrate contribution to the damping constant is κ2/κ1 ≈ 0.46, where for

SiO2 the same contribution is only κ2/κ1 ≈ 0.05. Thus the substrate-induced damping

is much higher for a Ta2O5 substrate, and only a weak plasmonic effect can be observed.

Also visible in the topography (Fig. 6.6d)) is comparably large roughness compared to the

graphene shown in Fig. 6.5. This also contributes to damping as the SPPs can scatter off of

this roughness.

Through a collaboration with the Zhu group at Penn State, we identified grain bound-

aries in bilayer graphene. These linear defects are known to exhibit ballistic electron valley

transport along their length due to topological protection of K and K′ electrons [85]. Addi-

tionally, they can reflect SPPs due to the change in wavevector right at the boundary where

the bilayer graphene band structure changes. Fig. 6.7 shows my results mapping these grain

boundaries. While the topography shows no visible defect (Fig. 6.7a)) and a smooth struc-

125

ture, the near-field b) phase and c) amplitude shows a small linear feature. Here, due to

a very small incidental doping level, the plasmon propagation length is very short, and the

near-field amplitude of the graphene is lower than the SiO2 substrate. Additionally, the SPP

reflection is also weaker than that observed for SLG since the bilayer SPPs have a higher

transmission coefficient and are more likely to traverse the boundary.

6.2.3 Conclusion and outlook

Using SPP interferometry, I studied the propagation parameters of graphene SPPs.

Using the tip to launch graphene SPPs, I used s-SNOM to image the resulting interference

patterns which directly relate to the SPP wavelength and damping. I performed a comparison

between CVD graphene on a high κ dielectric substrate and exfoliated graphene on SiO2, and

saw significantly smaller damping on the exfoliated sample. This shorter propagation length

can be attributed to stronger substrate damping due to a higher imaginary component of the

dielectric function, the generally lower mobility of CVD graphene over exfoliated samples,

and the higher substrate roughness of the Ta2O5 surface. I was also able to image atomic-

scale grain boundaries in both SLG and BLG, despite their complete lack of topographic

contrast. Since these boundaries prohibit electron transport, their presence would debilitate

an electronic device based on graphene. This work is therefore relevant to nano-engineering

of graphene devices that rely on electron transport, where the ability to identify these defects

on large scale is particularly useful.

6.3 Conclusion

While the polaritons of hBN and graphene show promise for many photonics applica-

tions, it is still necessary to explore and further characterize their propagation properties.

Precise knowledge of their polariton dispersion relation is needed to design efficient coupling

with far-field light, and their propagation length and damping properties set limitation on

the spatial dimensions of possible photonics devices that implement polaritons. Using polari-

126

0.4 nm

2.8 nm

0.4 rad.

1.2 rad.

0.7 V

1.5 V

2 µm

a) b) c)

Figure 6.7: a) Topography of bilayer graphene near the sheet edge. Near-field b) phase andc) amplitude shows plasmon reflection due to a grain boundary.

ton interferometry, I directly measure these spatial properties for both hBN and graphene.

Phonon polariton rectification was observed in hBN, and the effects of substrate and defects

on polariton damping were explored graphene. These results reveal new optical phenomenon

in materials, as well as measure the limitations of polaritons, both of which must be under-

stood for efficient technological applications.

Chapter 7

Conclusion

This thesis presents fundamental work advancing the technique of s-SNOM, as well

as new discoveries of nanoscale material properties. I first describe my work to incorporate

a blackbody emitter as a source for s-SNOM. I achieve sensitivity to a broader range of

samples that past implementations, including weaker molecular resonances. My work also

outlines the general irradiance requirements for s-SNOM and provides a comparison of source

performance for s-SNOM.

My work on VO2 reveals inhomogeneity of the ultrafast dynamics and nanoscale mor-

phology of the IMT in single crystal structures. Uncorrelated with strain, size or orientation,

this inhomogeneity suggests that the IMT of VO2 is highly sensitive to nanoscale defects or

variations in stoichiometry.

I also performed polariton interferometry on the 2D material graphene and hBN which

provides a direct measurement of the polariton propagation parameters. Using this tech-

nique, I can map defects in graphene and study the role of substrate on the damping con-

stant. Additionally, I discuss PhP rectification in hBN, which may occur due to the strong

role of the substrate on the dispersion relation of PhPs.

Using TINS, I show the high sensitivity of the SPhP of SiC to perturbation as nec-

essarily imposed by the tip. My work also provides evidence of spatial coherence of the

thermal near-fields due to contributions from non-local SPhP excitations. By incorporating

laser heating as an alternative to specialized Joule-heated tips, I broaden the applicability

128

of TINS to arbitrary tip design, including metal coated tips.

I also describe my work to measure optically induced forces. I discuss the possible

contributions to optically-induced force spectroscopy and microscopy using an AFM tip.

My work suggests that the force due to heating and subsequent thermal expansion of the

sample dominates the measured signal.

These results advance the field of nano-optics by improving the s-SNOM technique,

while also revealing new optical phenomena on nanoscale lengths.

Chapter 8

Appendix A: Crystallinity in metal-porphyrines

As a continuation of work initiated by Benjamin Pollard and Eric Muller from our

group, I have studied the crystallization process of metal-porphyrines. These materials

consist of a transition metal atom centered in an aromatic ring. In particular, I study

ruthenium-porphyrine (ReOEP), which demonstrates strong coupling between the d-orbitals

of ruthenium and a carbonyl bond. This hybridization results in strong, long-lived vibrational

resonance around 1930 cm−1, which can serve as a marker resonance for changes in the local

Figure 8.1: Far-field FTIR spectroscopy of RuOEP/P3HT samples with different annealingtimes. As the annealing time increases, the resonance red-shifts (20 min spectrum), andthen splits (60 min spectrum). There is evidence in inhomogeneity within the sample dueto the non-monotonic trend with annealing time.

130

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

Delay (µm)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 nm

60 nm

0 nm

220 nm

0 nm

220 nm

1 µm 1 µm 1 µm

a) b) c)

d) RuOEP

Reference

Figure 8.2: a-c) Topography of various crystals after varying annealing times. With increas-ing annealing time, crystals grow larger and more regular pyramidal shapes. d) Interferogramof a non-resonant reference (red) and of a well-formed crystallite (blue), showing a long-livedvibrational resonance.

structure or dielectric environment.

Changes to the structure and local ordering of the metal-porphyrine molecules can be

induced by vapor-annealing. To do this, I mix pure RuOEP with poly(3-hexylthiophene)

(P3HT) in a 1:5 mass ratio and dissolve in a chloroform solution and spin coated on Si sub-

strates. Then, the samples are placed in a chloroform vapor environment for 2-60 min for the

annealing process. During the annealing process, the RuOEP congregate and begin to crys-

tallize. As they crystallize, the molecules mutually orient so that the carbonyl bonds align.

As this occurs, coupling between neighboring carbonyl bonds arises and strengthens contin-

uously. As with coupled oscillators, this results in mode hybridization and the creation of

131

0 0.1 0.2 0.3 0.4 0.5

Distance (µm)

0

0.1

0.2

0.3

0.4

0.5

0.6

Dis

tan

ce

(µ

m)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 nm

220 nm

500 nm

a)

1880 1900 1920 1940 1960 1980

Wavenumber (cm-1)

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

b)

c)A

1 /A2

A1

A2

Figure 8.3: a) Crystal topography. b) Spectra at different locations of the crystal. Thespectra show variable strength of vibrational mode at 1930 cm−1. c) Spatial variation of themode strength in the region indicated in a).

symmetric and antisymmetric modes with distinct energies. The energy gap ∆ = h(νs−νas)

between these modes is directly proportional to the coupling strength. Infrared spectroscopy

can be used to probe this energy gap.

Far-field FTIR microscopy was performed on the samples to measure the resonant

response as a function of annealing time (Fig. 8.1). With increasing annealing time, we

see a red-shift and a splitting of the peak at 1930 cm−1, indicating mode hybridization.

The samples become highly heterogeneous during the annealing time, becoming increasingly

rough with the formation and growth of microscopic crystal. Our FTIR spectroscopy shows

some evidence of this inhomogeneity, with the non-monotonic trend of the spectra with

annealing time.

We then performed near-field spectroscopy on these samples. As expected, samples

with longer annealing time show more regular and larger crystals than those annealed for

132

brief times.

Bibliography

[1] E. Abbe. On the estimation of aperture in the microscope. Journal of the RoyalMicroscopical Society, 1(3):388–423, 1881.

[2] Holger Adam, Sebastian Rode, Martin Schreiber, Kei Kobayashi, Hirofumi Yamada,and Angelika Kuhnle. Photothermal excitation setup for a modified commercial atomicforce microscope. Review of Scientific Instruments, 85(2), 2014.

[3] S. Amarie and F. Keilmann. Broadband-infrared assessment of phonon resonance inscattering-type near-field microscopy. Phys. Rev. B, 83:045404, Jan 2011.

[4] J. R. Arias-Gonzalez, M. Nieto-Vesperinas, and M. Lester. Modeling photonic forcemicroscopy with metallic particles under plasmon eigenmode excitation. Phys. Rev.B, 65:115402, Feb 2002.

[5] C. Arnold, F. Marquier, M. Garin, F. Pardo, S. Collin, N. Bardou, J.-L. Pelouard, andJ.-J. Greffet. Coherent thermal infrared emission by two-dimensional silicon carbidegratings. Physical Review B, 86(3):035316, 2012.

[6] A. Ashkin and J. M. Dziedzic. Optical levitation in high vacuum. Applied PhysicsLetters, 28(6):333–335, 1976.

[7] A Ashkin and J M Dziedzic. Internal cell manipulation using infrared laser traps.Proceedings of the National Academy of Sciences, 86(20):7914–7918, 1989.

[8] Joanna M. Atkin, Samuel Berweger, Emily K. Chavez, Markus B. Raschke, JinboCao, Wen Fan, and Junqiao Wu. Strain and temperature dependence of the insulatingphases of VO2 near the metal-insulator transition. Phys. Rev. B, 85:020101, Jan 2012.

[9] Joanna M. Atkin, Paul M. Sass, Paul E. Teichen, Joel D. Eaves, and Markus B.Raschke. Nanoscale probing of dynamics in local molecular environments. J. Phys.Chem. Lett., 2015. In press.

[10] Arthur Babuty, Karl Joulain, Pierre-Olivier Chapuis, Jean-Jacques Greffet, and Yan-nick De Wilde. Blackbody spectrum revisited in the near field. Phys. Rev. Lett.,110:146103, Apr 2013.

134

[11] D. N. Basov, M. M. Fogler, and F. J. Garcıa de Abajo. Polaritons in van der waalsmaterials. Science, 354(6309):aag1992, 2016.

[12] Peter Baum, Ding-Shyue Yang, and Ahmed H. Zewail. 4d visualization of transitionalstructures in phase transformations by electron diffraction. Science, 318(5851):788–792, 2007.

[13] Hans A. Bechtel, Eric A. Muller, Robert L. Olmon, Michael C. Martin, and Markus B.Raschke. Ultrabroadband infrared nanospectroscopic imaging. Proceedings of theNational Academy of Sciences, 111(20):7191–7196, 2014.

[14] Michael F. Becker, A. Bruce Buckman, Rodger M. Walser, Thierry Lepine, PatrickGeorges, and Alain Brun. Femtosecond laser excitation of the semiconductor-metalphase transition in VO2. Applied Physics Letters, 65(12):1507–1509, 1994.

[15] T. Ben-Messaoud, G. Landry, J.P. Gariepy, B. Ramamoorthy, P.V. Ashrit, andA. Hache. High contrast optical switching in vanadium dioxide thin films. OpticsCommunications, 281(24):6024 – 6027, 2008.

[16] Stefanie Bensmann, Fabian Gaußmann, Martin Lewin, Jochen Wuppen, SebastianNyga, Christoph Janzen, Bernd Jungbluth, and Thomas Taubner. Near-field imagingand spectroscopy of locally strained gan using an ir broadband laser. Opt. Express,22(19):22369–22381, Sep 2014.

[17] E. BETZIG, J. K. TRAUTMAN, T. D. HARRIS, J. S. WEINER, and R. L. KOSTE-LAK. Breaking the diffraction barrier: Optical microscopy on a nanometric scale.Science, 251(5000):1468–1470, 1991.

[18] Immanuel Bloch. Ultracold quantum gases in optical lattices. Nat Phys, 1(1):23–30,October 2005.

[19] Michael M. Burns, Jean-Marc Fournier, and Jene A. Golovchenko. Optical matter:Crystallization and binding in intense optical fields. Science, 249(4970):749–754, 1990.

[20] Yongqing Cai, Litong Zhang, Qingfeng Zeng, Laifei Cheng, and Yongdong Xu. In-frared reflectance spectrum of BN calculated from first principles. Solid StateCommunications, 141(5):262 – 266, 2007.

[21] Joshua D. Caldwell, Andrey V. Kretinin, Yiguo Chen, Vincenzo Giannini, Michael M.Fogler, Yan Francescato, Chase T. Ellis, Joseph G. Tischler, Colin R. Woods, Alexan-der J. Giles, Minghui Hong, Kenji Watanabe, Takashi Taniguchi, Stefan A. Maier,and Kostya S. Novoselov. Sub-diffractional volume-confined polaritons in the naturalhyperbolic material hexagonal boron nitride. Nat Commun, 5:5221, October 2014.

[22] J. Cao, E. Ertekin, V. Srinivasan, W. Fan, S. Huang, H. Zheng, J. W. L. Yim, D. R.Khanal, D. F. Ogletree, J. C. Grossman, and J. Wu. Strain engineering and one-dimensional organization of metal-insulator domains in single-crystal vanadium dioxidebeams. Nature Nanotech., 4(11):732–737, September 2009.

135

[23] J Cao, Y Gu, W Fan, L Q Chen, D F Ogletree, K Chen, N Tamura, M Kunz, C Barrett,J Seidel, and J Wu. Extended mapping and exploration of the vanadium dioxide stress-temperature phase diagram. Nano Lett., 10(7):2667–73, 2010.

[24] Remi Carminati and Jean-Jacques Greffet. Near-field effects in spatial coherence ofthermal sources. Phys. Rev. Lett., 82:1660–1663, Feb 1999.

[25] H. B. G. Casimir and D. Polder. The influence of retardation on the london-van derwaals forces. Phys. Rev., 73:360–372, Feb 1948.

[26] A. Cavalleri, H. H. W. Chong, S. Fourmaux, T. E. Glover, P. A. Heimann, J. C. Kieffer,B. S. Mun, H. A. Padmore, and R. W. Schoenlein. Picosecond soft x-ray absorptionmeasurement of the photoinduced insulator-to-metal transition in VO2. Phys. Rev. B,69:153106, Apr 2004.

[27] A. Cavalleri, Th. Dekorsy, H. H. W. Chong, J. C. Kieffer, and R. W. Schoenlein.Evidence for a structurally-driven insulator-to-metal transition in VO2: A view fromthe ultrafast timescale. Phys. Rev. B, 70:161102, Oct 2004.

[28] A. Cavalleri, M. Rini, H. H. W. Chong, S. Fourmaux, T. E. Glover, P. A. Heimann,J. C. Kieffer, and R. W. Schoenlein. Band-selective measurements of electron dynamicsin VO2 using femtosecond near-edge x-ray absorption. Phys. Rev. Lett., 95:067405,Aug 2005.

[29] A. Cavalleri, Cs. Toth, C. W. Siders, J. A. Squier, F. Raksi, P. Forget, and J. C.Kieffer. Femtosecond structural dynamics in VO2 during an ultrafast solid-solid phasetransition. Phys. Rev. Lett., 87:237401, Nov 2001.

[30] C. W. Chang, D. Okawa, A. Majumdar, and A. Zettl. Solid-state thermal rectifier.Science, 314(5802):1121–1124, 2006.

[31] Pierre-Olivier Chapuis, Sebastian Volz, Carsten Henkel, Karl Joulain, and Jean-Jacques Greffet. Effects of spatial dispersion in near-field radiative heat transfer be-tween two parallel metallic surfaces. Phys. Rev. B, 77:035431, Jan 2008.

[32] Jian-Hao Chen, Chaun Jang, Shudong Xiao, Masa Ishigami, and Michael S. Fuhrer.Intrinsic and extrinsic performance limits of graphene devices on sio2. Nat Nano,3(4):206–209, April 2008.

[33] Jianing Chen, Michela Badioli, Pablo Alonso-Gonzalez, Sukosin Thongrattanasiri,Florian Huth, Johann Osmond, Marko Spasenovic, Alba Centeno, Amaia Pesquera,Philippe Godignon, Amaia Zurutuza Elorza, Nicolas Camara, F. Javier Garciade Abajo, Rainer Hillenbrand, and Frank H. L. Koppens. Optical nano-imaging ofgate-tunable graphene plasmons. Nature, 487(7405):77–81, July 2012.

[34] A. Chimmalgi, T. Y. Choi, C. P. Grigoropoulos, and K. Komvopoulos. Femtosec-ond laser aperturless near-field nanomachining of metals assisted by scanning probemicroscopy. Applied Physics Letters, 82(8):1146–1148, 2003.

136

[35] B. W. Chui, M. Asheghi, Y. S. Ju, K. E. Goodson, T. W. Kenny, and H. J.Mamin. Intrinsic-carrier thermal runaway in silicon microcantilevers. MicroscaleThermophysical Engineering, 3(3):217–228, 1999.

[36] B. W. Chui, T. D. Stowe, Yongho Sungtaek Ju, K. E. Goodson, T. W. Kenny, H. J.Mamin, B. D. Terris, R. P. Ried, and D. Rugar. Low-stiffness silicon cantilevers withintegrated heaters and piezoresistive sensors for high-density afm thermomechanicaldata storage. Journal of Microelectromechanical Systems, 7(1):69–78, Mar 1998.

[37] T. L. Cocker, L. V. Titova, S. Fourmaux, G. Holloway, H.-C. Bandulet, D. Brassard,J.-C. Kieffer, M. A. El Khakani, and F. A. Hegmann. Phase diagram of the ultrafastphotoinduced insulator-metal transition in vanadium dioxide. Phys. Rev. B, 85:155120,Apr 2012.

[38] Nobel Prize Committee. Super-resolved fluorescence microscopy. The Nobel Prize inChemistry 2014, 2014.

[39] Ian M. Craig, Matthew S. Taubman, A. Scott Lea, Mark C. Phillips, Erik E. Josberger,and Markus B. Raschke. Infrared near-field spectroscopy of trace explosives using anexternal cavity quantum cascade laser. Opt. Express, 21(25):30401–30414, Dec 2013.

[40] A. Cvitkovic, N. Ocelic, and R. Hillenbrand. Analytical model for quantitative pre-diction of material contrasts in scattering-type near-field optical microscopy. Opt.Express, 15(14):8550–8565, Jul 2007.

[41] S. Dai, Z. Fei, Q. Ma, A. S. Rodin, M. Wagner, A. S. McLeod, M. K. Liu, W. Gannett,W. Regan, K. Watanabe, T. Taniguchi, M. Thiemens, G. Dominguez, A. H. Cas-tro Neto, A. Zettl, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N. Basov.Tunable phonon polaritons in atomically thin van der waals crystals of boron nitride.Science, 343(6175):1125–1129, 2014.

[42] S. Dai, Q. Ma, T. Andersen, A. S. Mcleod, Z. Fei, M. K. Liu, M. Wagner, K. Watanabe,T. Taniguchi, M. Thiemens, F. Keilmann, P. Jarillo-Herrero, M. M. Fogler, and D. N.Basov. Subdiffractional focusing and guiding of polaritonic rays in a natural hyperbolicmaterial. Nat. Commun., 6:6963, April 2015.

[43] R.B.G. de Hollander, N.F. van Hulst, and R.P.H. Kooyman. Near field plasmon andforce microscopy. Ultramicroscopy, 57(2):263 – 269, 1995.

[44] B. Deutsch, R. Hillenbrand, and L. Novotny. Near-field amplitude and phase recoveryusing phase-shifting interferometry. Opt. Express, 16(2):494–501, Jan 2008.

[45] Sven A. Donges, Omar Khatib, Brian T. O’Callahan, Joanna M. Atkin, Jae HyungPark, David Cobden, and Markus B. Raschke. Ultrafast nanoimaging of the photoin-duced phase transition dynamics in VO2. Nano Letters, 16(5):3029–3035, 2016. PMID:27096877.

137

[46] Tela Favaloro, Joonki Suh, Bjorn Vermeersch, Kai Liu, Yijia Gu, Long-Qing Chen,Kevin X. Wang, Junqiao Wu, and Ali Shakouri. Direct observation of nanoscale Peltierand Joule effects at metal-insulator domain walls in vanadium dioxide nanobeams.Nano Letters, 14(5):2394–2400, 2014.

[47] Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang,Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau,F. Keilmann, and D. N. Basov. Gate-tuning of graphene plasmons revealed by infrarednano-imaging. Nature, 487(7405):82–85, July 2012.

[48] FeiZ., RodinA. S., GannettW., DaiS., ReganW., WagnerM., LiuM. K., McLeodA.S., DominguezG., ThiemensM., Castro NetoAntonio H., KeilmannF., ZettlA., Hil-lenbrandR., FoglerM. M., and BasovD. N. Electronic and plasmonic phenomena atgraphene grain boundaries. Nat Nano, 8(11):821–825, November 2013.

[49] Joseph Fraunhofer. Bestimmung des brechungs- und des farben-zerstreuungs -vermogens verschiedener glasarten, in bezug auf die vervollkommnung achromatischerfernrohre. Denkschriften der Koniglichen Akademie der Wissenschaften zu Munchen,5:193–226, 1872.

[50] J.P. Freedman, J.H. Leach, E.A. Preble, Z. Sitar, R.S. Davis, and J.A. Malen. Universalphonon mean free path spectra in crystalline semiconductors at high temperature. Sci.Rep., 3:2963, Oct 2013.

[51] Aitzol Garcia-Etxarri, Isabel Romero, F. Javier Garcia de Abajo, Rainer Hillenbrand,and Javier Aizpurua. Influence of the tip in near-field imaging of nanoparticle plas-monic modes: Weak and strong coupling regimes. Phys. Rev. B, 79:125439, Mar 2009.

[52] R. Geick, C. H. Perry, and G. Rupprecht. Normal modes in hexagonal boron nitride.Phys. Rev., 146:543–547, Jun 1966.

[53] Justin A. Gerber, Samuel Berweger, Brian T. O’Callahan, and Markus B. Raschke.Phase-resolved surface plasmon interferometry of graphene. Phys. Rev. Lett.,113:055502, Jul 2014.

[54] Severine Gomes, Ali Assy, and Pierre-Olivier Chapuis. Scanning thermal microscopy:A review. physica status solidi (a), 212(3):477–494, 2015.

[55] John B. Goodenough. The two components of the crystallographic transition in vo2.Solid State Chemistry, 3:490, 1971.

[56] Alexander A. Govyadinov, Iban Amenabar, Florian Huth, P. Scott Carney, and RainerHillenbrand. Quantitative measurement of local infrared absorption and dielectricfunction with tip-enhanced near-field microscopy. The Journal of Physical ChemistryLetters, 4(9):1526–1531, 2013.

138

[57] Alexander A. Govyadinov, Stefan Mastel, Federico Golmar, Andrey Chuvilin, P. ScottCarney, and Rainer Hillenbrand. Recovery of permittivity and depth from near-fielddata as a step toward infrared nanotomography. ACS Nano, 8(7):6911–6921, 2014.PMID: 24897380.

[58] Michael S. Grinolds, Vladimir A. Lobastov, Jonas Weissenrieder, and Ahmed H. Ze-wail. Four-dimensional ultrafast electron microscopy of phase transitions. Proceedingsof the National Academy of Sciences, 103(49):18427–18431, 2006.

[59] Frederic Guinneton, Laurent Sauques, Jean-Christophe Valmalette, Frederic Cros, andJean-Raymond Gavarri. Optimized infrared switching properties in thermochromicvanadium dioxide thin films: role of deposition process and microstructure. Thin SolidFilms, 446(2):287 – 295, 2004.

[60] Beth S Guiton, Qian Gu, Amy L Prieto, Mark S Gudiksen, and Hongkun Park. Single-crystalline vanadium dioxide nanowires with rectangular cross sections. J. Am. Chem.Soc., 127(2):498–9, January 2005.

[61] Marta Klanjsek Gunde. Vibrational modes in amorphous silicon dioxide. Physica B:Condensed Matter, 292(34):286 – 295, 2000.

[62] V P Gusynin, S G Sharapov, and J P Carbotte. On the universal ac optical backgroundin graphene. New Journal of Physics, 11(9):095013, 2009.

[63] Masaki Hada, Kunio Okimura, and Jiro Matsuo. Characterization of structural dy-namics of VO2 thin film on c-Al2O3 using in-air time-resolved x-ray diffraction. Phys.Rev. B, 82:153401, Oct 2010.

[64] Poul Martin Hansen, Vikram Kjøller Bhatia, Niels Harrit, and Lene Oddershede. Ex-panding the optical trapping range of gold nanoparticles. Nano Letters, 5(10):1937–1942, 2005. PMID: 16218713.

[65] C. Henkel, K. Joulain, R. Carminati, and J.-J. Greffet. Spatial coherence of thermalnear fields. Optics Communications, 186(13):57 – 67, 2000.

[66] Peter Hermann, Arne Hoehl, Georg Ulrich, Claudia Fleischmann, Antje Hermelink,Bernd Kastner, Piotr Patoka, Andrea Hornemann, Burkhard Beckhoff, Eckart Ruhl,and Gerhard Ulm. Characterization of semiconductor materials using synchrotronradiation-based near-field infrared microscopy and nano-ftir spectroscopy. Opt.Express, 22(15):17948–17958, Jul 2014.

[67] F.W. Herschel. Phil. T. R. Soc. Lond., 90:437, 1800.

[68] Aaron Holsteen, In Soo Kim, and Lincoln J. Lauhon. Extraordinary dynamic me-chanical response of vanadium dioxide nanowires around the insulator to metal phasetransition. Nano Letters, 14(4):1898–1902, 2014.

139

[69] Fei Huang, Venkata Ananth Tamma, Zahra Mardy, Jonathan Burdett, and H Kumar.Imaging Nanoscale Electromagnetic Near-Field Distributions Using Optical Forces.Scientific reports, 5:10610, 2015.

[70] F. Huth, M. Schnell, J. Wittborn, N. Ocelic, and R. Hillenbrand. Infrared-spectroscopicnanoimaging with a thermal source. Nat. Mater., 10(5):352–356, May 2011.

[71] Yasushi Inouye and Satoshi Kawata. Near-field scanning optical microscope with ametallic probe tip. Opt. Lett., 19(3):159–161, Feb 1994.

[72] Michio Ishikawa, Makoto Katsura, Satoru Nakashima, Yuka Ikemoto, and HidekazuOkamura. Broadband near-field mid-infrared spectroscopy and application to phononresonances in quartz. Opt. Express, 20(10):11064–11072, May 2012.

[73] Kunie Ishioka, Muneaki Hase, Masahiro Kitajima, Ludger Wirtz, Angel Rubio, andHrvoje Petek. Ultrafast electron-phonon decoupling in graphite. Phys. Rev. B,77:121402, Mar 2008.

[74] Marinko Jablan, Hrvoje Buljan, and Marin Soljacic. Plasmonics in graphene at infraredfrequencies. Physical review B, 80(24):245435, 2009.

[75] Junghoon Jahng, Jordan Brocious, Dmitry A Fishman, Fei Huang, Xiaowei Li,Venkata Ananth Tamma, H Kumar Wickramasinghe, Eric Olaf Potma, and A Time-averaged Lorentz. Gradient and scattering forces in photoinduced force microscopy.Physical Review B, 90:155417, 2014.

[76] Junghoon Jahng, Jordan Brocious, Dmitry A Fishman, Steven Yampolsky, DerekNowak, Fei Huang, Junghoon Jahng, Jordan Brocious, Dmitry A Fishman, StevenYampolsky, Derek Nowak, Fei Huang, Vartkess A Apkarian, and H Kumar Wickra-masinghe. Ultrafast pump-probe force microscopy with nanoscale resolution Ultra-fast pump-probe force microscopy with nanoscale resolution. Applied Physics Letters,106:083113, 2015.

[77] Amun Jarzembski and Keunhan Park. Finite dipole model for extreme near-fieldthermal radiation between a tip and planar sic substrate. Journal of QuantitativeSpectroscopy and Radiative Transfer, 191:67 – 74, 2017.

[78] J. H. Jeans and J. W. S. Rayleigh. The dynamical theory of gases and of radiation.Nature, 72:54, 1905.

[79] Andrew C. Jones. Nano-scale imaging and spectroscopy of plasmonic systems, thermalnear-fields, and phase separation in complex oxides. PhD thesis, University of Wash-ington, 2012.

[80] Andrew C Jones, Samuel Berweger, Jiang Wei, David Cobden, and Markus B Raschke.Nano-optical investigations of the metal-insulator phase behavior of individual VO2

microcrystals. Nano Lett., 10:1574–1581, 2010.

140

[81] Andrew C. Jones, Brian T. O’Callahan, Honghua U. Yang, and Markus B. Raschke.The thermal near-field: Coherence, spectroscopy, heat-transfer, and optical forces.Progress in Surface Science, 88(4):349 – 392, 2013.

[82] Andrew C. Jones and Markus B. Raschke. Thermal infrared near-field spectroscopy.Nano Letters, 12(3):1475–1481, 2012. PMID: 22280474.

[83] Karl Joulain, Remi Carminati, Jean-Philippe Mulet, and Jean-Jacques Greffet. Def-inition and measurement of the local density of electromagnetic states close to aninterface. Phys. Rev. B, 68:245405, Dec 2003.

[84] Karl Joulain, Jean-Philippe Mulet, Franois Marquier, Rmi Carminati, and Jean-Jacques Greffet. Surface electromagnetic waves thermally excited: Radiative heattransfer, coherence properties and casimir forces revisited in the near field. SurfaceScience Reports, 57(34):59 – 112, 2005.

[85] Long Ju, Zhiwen Shi, Nityan Nair, Yinchuan Lv, Chenhao Jin, Jairo Velasco Jr, Clau-dia Ojeda-Aristizabal, Hans A. Bechtel, Michael C. Martin, Alex Zettl, James Analytis,and Feng Wang. Topological valley transport at bilayer graphene domain walls. Nature,520(7549):650–655, April 2015.

[86] K. Kechedzhi and S. Das Sarma. Plasmon anomaly in the dynamical optical conduc-tivity of graphene. Phys. Rev. B, 88:085403, Aug 2013.

[87] Hyun-Tak Kim, Yong Wook Lee, Bong-Jun Kim, Byung-Gyu Chae, Sun Jin Yun,Kwang-Yong Kang, Kang-Jeon Han, Ki-Ju Yee, and Yong-Sik Lim. Monoclinic andcorrelated metal phase in VO2 as evidence of the Mott transition: Coherent phononanalysis. Phys. Rev. Lett., 97:266401, Dec 2006.

[88] Kyeongtae Kim, Bai Song, Vıctor Fernandez-Hurtado, Woochul Lee, Wonho Jeong,Longji Cui, Dakotah Thompson, Johannes Feist, M. T. Homer Reid, Francisco J.Garcıa-Vidal, Juan Carlos Cuevas, Edgar Meyhofer, and Pramod Reddy. Radiativeheat transfer in the extreme near field. Nature, 528(7582):387–391, December 2015.

[89] William P King, Bikramjit Bhatia, Jonathan R Felts, Hoe Joon Kim, Beomjin Kwon,Byeonghee Lee, Suhas Somnath, and Matthew Rosenberger. Heated atomic forcemicroscope cantilevers and their applications. Annual Review of Heat Transfer,16(16):287–326, 2013.

[90] G. Kirchhoff. Uber den Zusammenhang zwischen Emission und Absorption von Lichtund Warme. Monatsber. Preuss Akad., pages 783–87, 1859.

[91] G. Kirchhoff. Uber das Verhaltniss zwischen dem Emissionsvermogen und dem Ab-sorptionsvermogen der Korper fur Warme und Licht. Annalen der Physik und Chemie,109:275–301, 1860.

141

[92] Achim Kittel, Wolfgang Muller-Hirsch, Jurgen Parisi, Svend-Age Biehs, Daniel Reddig,and Martin Holthaus. Near-field heat transfer in a scanning thermal microscope. Phys.Rev. Lett., 95:224301, Nov 2005.

[93] A. Knoll, P. Bachtold, J. Bonan, G. Cherubini, M. Despont, U. Drechsler, U. Durig,B. Gotsmann, W. Haberle, C. Hagleitner, D. Jubin, M.A. Lantz, A. Pantazi, H. Pozidis,H. Rothuizen, A. Sebastian, R. Stutz, P. Vettiger, D. Wiesmann, and E.S. Eleftheriou.Integrating nanotechnology into a working storage device. Microelectronic Engineering,83(49):1692 – 1697, 2006. Micro- and Nano-Engineering MNE 2005Proceedings ofthe 31st International Conference on Micro- and Nano-Engineering.

[94] Bernhard Knoll and Fritz Keilmann. Enhanced dielectric contrast in scattering-typescanning near-field optical microscopy. Optics Communications, 182(46):321 – 328,2000.

[95] S. Komiyama, Y. Kajihara, K. Kosaka, T. Ueda, , and Zhenghua An. Near-fieldnanoscopy of thermal evanescent waves on metals. arXiv:1601.00368, 2016.

[96] Frank H. L. Koppens, Darrick E. Chang, and F. Javier Garcıa de Abajo. Grapheneplasmonics: A platform for strong light-matter interactions. Nano Letters, 11(8):3370–3377, 2011. PMID: 21766812.

[97] Ernst Heiner Korte and Arnulf Roseler. Infrared reststrahlen revisited: commonlydisregarded optical details related to n¡1. Analytical and Bioanalytical Chemistry,382(8):1987–1992, 2005.

[98] C. Kubler, H Ehrke, R Huber, R Lopez, A Halabica, R F Haglund, and A Leitenstorfer.Coherent structural dynamics and electronic correlations during an ultrafast insulator-to-metal phase transition in VO2. Phys. Rev. Lett., 99:116401, 2007.

[99] Ohmyoung Kwon, Li Shi, and Arun Majumdar. Scanning thermal wave microscopy(stwm). Journal of Heat Transfer, 125(1):156–163, January 2003.

[100] Basudev Lahiri, Glenn Holland, and Andrea Centrone. Chemical Imaging Beyond theDiffraction Limit : Experimental Validation of the PTIR Technique. Small, 9:439–445,2013.

[101] Woochul Lee, Kyeongtae Kim, Wonho Jeong, Linda Angela Zotti, Fabian Pauly,Juan Carlos Cuevas, and Pramod Reddy. Heat dissipation in atomic-scale junctions.Nature, 498(7453):209–212, June 2013.

[102] Stephane Lefevre and Sebastian Volz. 3ω-scanning thermal microscope. Review ofScientific Instruments, 76(3):033701, 2005.

[103] Lasse-Petteri Leppanen, Ari T. Friberg, and Tero Setala. Partial polarization of opticalbeams and near fields probed with a nanoscatterer. J. Opt. Soc. Am. A, 31(7):1627–1635, Jul 2014.

142

[104] A. Lewis, M. Isaacson, A. Harootunian, and A. Muray. Development of a 500 spatialresolution light microscope. Ultramicroscopy, 13(3):227 – 231, 1984.

[105] C. Y. Liang and S. Krimm. Infrared spectra of high polymers. iii. polytetrafluoroethy-lene and polychlorotrifluoroethylene. The Journal of Chemical Physics, 25(3):563–571,1956.

[106] E.M. Lifshitz and L.P. Pitaevskii. Course of Theoretical Physics, volume 9.Butterworth-Heinemann, Oxford, 2002.

[107] Haihua Liu, Oh-Hoon Kwon, Jau Tang, and Ahmed H. Zewail. 4d imaging and diffrac-tion dynamics of single-particle phase transition in heterogeneous ensembles. NanoLetters, 14(2):946–954, 2014.

[108] M. K. Liu, M. Wagner, E. Abreu, S. Kittiwatanakul, A. McLeod, Z. Fei, M. Goldflam,S. Dai, M. M. Fogler, J. Lu, S. A. Wolf, R. D. Averitt, and D. N. Basov. Anisotropicelectronic state via spontaneous phase separation in strained vanadium dioxide films.Phys. Rev. Lett., 111:096602, Aug 2013.

[109] Vladimir A. Lobastov, Jonas Weissenrieder, Jau Tang, and Ahmed H. Zewail. Ultrafastelectron microscopy (UEM): Four-dimensional imaging and diffraction of nanostruc-tures during phase transitions. Nano Letters, 7(9):2552–2558, SEP 2007.

[110] Feng Lu and Mikhail A Belkin. Infrared absorption nano-spectroscopy using sam-ple photoexpansion induced by tunable quantum cascade lasers. Optics Express,19(21):19942, 2011.

[111] Feng Lu, Mingzhou Jin, and Mikhail A Belkin. Tip-enhanced infrared nanospec-troscopy via molecular expansion force detection. Nature Photonics, 8(4):307–312,2014.

[112] O. Lummer and E. Pringsheim. Die strahlung eines “schwarzen” korpers zwischen 100und 1300C. Ann. Phys., 299:395–410, 1897.

[113] O. Lummer and E. Pringsheim. Die verteilung der im spektrum des schwarzen korpersund des blanken platins. Verh. Dt. Phys. Ges., 1:215–235, 1899.

[114] O. Lummer and E. Pringsheim. Die vertheilung der energie im spektrum des schwartzenkorpers. Verh. phys. Ges., 1:23–41, 1899.

[115] O. Lummer and E. Pringsheim. Uber die strahlung des schwarzen korpers fur langewellen. Verh. Dt. Phys. Ges., 2:163–180, 1900.

[116] Sergiy Lysenko, Felix Fernandez, Armando Rua, and Huimin Liu. Ultrafast lightscattering imaging of multi-scale transition dynamics in vanadium dioxide. Journal ofApplied Physics, 114(15):–, 2013.

143

[117] Francisco C. B. Maia, Raul de Oliveira Freitas, Brian T. O’Callahan, and M. B.Raschke. Novel dispersion and phonon rectification in hexagonal boron nitride.Submitted, 2017.

[118] H. J. Mamin. Thermal writing using a heated atomic force microscope tip. AppliedPhysics Letters, 69(3):433–435, 1996.

[119] L. Mandel and E. Wolf. Optical Coherence and Quantum Optics. Cambridge Univer-sity, 1995. Chapter 13.

[120] B. Mansart, D. Boschetto, S. Sauvage, A. Rousse, and M. Marsi. Mott transition inCr-doped V2O3 studied by ultrafast reflectivity: Electron correlation effects on thetransient response. EPL (Europhysics Letters), 92(3):37007, 2010.

[121] C. Marini, E. Arcangeletti, D. Di Castro, L. Baldassare, A. Perucchi, S. Lupi,L. Malavasi, L Boeri, E. Pomjakushina, K. Conder, and P. Postorino. Optical prop-erties of V1−xCrxO2 compounds under high pressure. Phys. Rev. B, 77:235111, Jun2008.

[122] G. Masetti, F. Cabassi, G. Morelli, and G. Zerbi. Conformational order and disorder inpoly(tetrafluoroethylene) from the infrared spectrum. Macromolecules, 6(5):700–707,September 1973.

[123] Stefan Mastel, Alexander A. Govyadinov, Thales V. A. G. de Oliveira, Iban Amenabar,and Rainer Hillenbrand. Nanoscale-resolved chemical identification of thin organicfilms using infrared near-field spectroscopy and standard fourier transform infraredreferences. Applied Physics Letters, 106(2):023113, 2015.

[124] A. S. McLeod, P. Kelly, M.D Goldflam, Z. Gainsforth, A. J. Westphal, G. Domingues,M. H. Thiemens, M. M. Gogler, and D. N. Basov. Model for quantitativenear-field spectroscopy and the extraction of nanoscale-resolved optical constants.arXiv:1308.1784, 2013.

[125] J Mehra and H Rechenberg. The Historical Development of Quantum Theory.Springer, 2000.

[126] F. J. Morin. Oxides which show a metal-to-insulator transition at the Neel temperature.Phys. Rev. Lett., 3(1):34, 1959.

[127] Vance R. Morrison, Robert. P. Chatelain, Kunal L. Tiwari, Ali Hendaoui, AndrewBruhcs, Mohamed Chaker, and Bradley J. Siwick. A photoinduced metal-like phase ofmonoclinic vo2 revealed by ultrafast electron diffraction. Science, 346(6208):445–448,2014.

[128] N. F. Mott. Metal-Insulator Transition. Rev. Mod. Phys., 40(4):677, 1968.

[129] Eric A. Muller.

144

[130] Eric A. Muller, Benjamin Pollard, and Markus B. Raschke. Infrared chemical nano-imaging: Accessing structure, coupling, and dynamics on molecular length scales. TheJournal of Physical Chemistry Letters, 6(7):1275–1284, 2015.

[131] Ingo Muller. A History of Thermodynamics. Springer, 2007.

[132] M. Nakajima, N. Takubo, Z. Hiroi, Y. Ueda, and T. Suemoto. Photoinduced metal-lic state in VO2 proved by the terahertz pump-probe spectroscopy. Applied PhysicsLetters, 92(1):011907, 2008.

[133] B.A. Nelson and W.P. King. Temperature calibration of heated silicon atomic forcemicroscope cantilevers. Sensors and Actuators A: Physical, 140(1):51 – 59, 2007.

[134] Brent A. Nelson and William P. King. Applications of heated atomic force microscopecantilevers. In Bharat Bhushan and Harald Fuchs, editors, Applied Scanning ProbeMethods IV, NanoScience and Technology, pages 251–275. Springer Berlin Heidelberg,2006.

[135] AH Castro Neto, F Guinea, Nuno MR Peres, Kostya S Novoselov, and Andre K Geim.The electronic properties of graphene. Reviews of modern physics, 81(1):109, 2009.

[136] M Nieto-Vesperinas, P C Chaumet, and A Rahmani. Near-field photonic forces.Philosophical transactions. Series A, Mathematical, physical, and engineering sciences,362(1817):719–737, 2004.

[137] M. Nonnenmacher and H. K. Wickramasinghe. Scanning probe microscopy of thermalconductivity and subsurface properties. Applied Physics Letters, 61(2):168–170, 1992.

[138] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grig-orieva, S. V. Dubonos, and A. A. Firsov. Two-dimensional gas of massless diracfermions in graphene. Nature, 438(7065):197–200, November 2005.

[139] L. Novotny and B. Hecht. Principles of Nano-Optics. Cambridge University Press,2006.

[140] Derek Nowak, William Morrison, H Kumar Wickramasinghe, Junghoon Jahng, EricPotma, Lei Wan, Ricardo Ruiz, Thomas R Albrecht, Kristin Schmidt, Jane Frommer,Daniel P Sanders, and Sung Park. Nanoscale chemical imaging by photoinduced forcemicroscopy. Science Advances, 2(March), 2016.

[141] J. M. Obrecht, R. J. Wild, M. Antezza, L. P. Pitaevskii, S. Stringari, and E. A. Cornell.Measurement of the temperature dependence of the casimir-polder force. Phys. Rev.Lett., 98:063201, Feb 2007.

[142] Brian T. O’Callahan, Andrew C. Jones, Jae Hyung Park, David H. Cobden, Joanna M.Atkin, and Markus B. Raschke. Inhomogeneity of the ultrafast insulator-to-metaltransition dynamics of VO2. Nat Commun, 6:–, April 2015.

145

[143] Brian T. O’Callahan, William E. Lewis, Andrew C. Jones, and Markus B. Raschke.Spectral frustration and spatial coherence in thermal near-field spectroscopy. Phys.Rev. B, 89:245446, Jun 2014.

[144] Brian T. O’Callahan, William E. Lewis, Silke Mobius, Jared C. Stanley, Eric A. Muller,and Markus B. Raschke. Broadband infrared vibrational nano-spectroscopy using ther-mal blackbody radiation. Opt. Express, 23(25):32063–32074, Dec 2015.

[145] Brian T. O’Callahan and Markus B. Raschke. Laser heating of scanning probe tips forthermal near-field spectroscopy and imaging. APL Photonics, 2(2):021301, 2017.

[146] Brian T. O’Callahan and Markus B. Raschke. Optically-induced tip-sample forces inthe infrared. In preparation, 2017.

[147] Motoichi Ohtsu. Progress of high-resolution photon scanning tunneling microscopydue to a nanometric fiber probe. Lightwave Technology, Journal of, 13(7):1200–1221,Jul 1995.

[148] Diego Olego and Manuel Cardona. Temperature dependence of the op-tical phonons and transverse effective charge in ¡span class=”aps-inline-formula”¿¡math¿¡mn¿3¡/mn¿¡mi¿c¡/mi¿¡/math¿¡/span¿-sic. Phys. Rev. B, 25:3889–3896, Mar 1982.

[149] R L Olmon and M B Raschke. Antennaload interactions at optical frequencies:impedance matching to quantum systems. Nanotechnology, 23(44):444001, 2012.

[150] Olga S. Ovchinnikova, Kevin Kjoller, Gregory B. Hurst, Dale A. Pelletier, and GaryJ. Van Berkel. Atomic force microscope controlled topographical imaging and prox-imal probe thermal desorption/ionization mass spectrometry imaging. AnalyticalChemistry, 86(2):1083–1090, 2014. PMID: 24377265.

[151] E.D. Palik, editor. Handbook of Optical Constants of Solids, volume 1. AcademicPress, 1985.

[152] E.D. Palik. Handbook of Optical Constants of Solids, Five-Volume Set: Handbook ofThermo-Optic Coefficients of Optical Materials with Applications. Elsevier Science,1997.

[153] Jae Hyung Park, Jim M. Coy, T. Serkan Kasirga, Chunming Huang, Zaiyao Fei, ScottHunter, and David H. Cobden. Measurement of a solid-state triple point at the metal-insulator transition in VO2. Nature, 500:431–434, 2013.

[154] A. Pashkin, C. Kubler, H. Ehrke, R. Lopez, A. Halabica, R. F. Haglund, R. Huber,and A. Leitenstorfer. Ultrafast insulator-metal phase transition in VO2 studied bymultiterahertz spectroscopy. Phys. Rev. B, 83:195120, May 2011.

[155] A. Passian, R. H. Ritchie, A. L. Lereu, T. Thundat, and T. L. Ferrell. Curvature effectsin surface plasmon dispersion and coupling. Phys. Rev. B, 71:115425, Mar 2005.

146

[156] Florian Peragut, Jean-Blaise Brubach, Pascale Roy, and Yannick De Wilde. In-frared near-field imaging and spectroscopy based on thermal or synchrotron radiation.Applied Physics Letters, 104(25), 2014.

[157] D. W. Pohl, W. Denk, and M. Lanz. Optical stethoscopy: Image recording withresolution λ/20. Applied Physics Letters, 44(7):651–653, 1984.

[158] Benjamin Pollard, Eric A. Muller, Karsten Hinrichs, and Markus B. Raschke. Vibra-tional nano-spectroscopic imaging correlating structure with intermolecular couplingand dynamics. Nat. Commun., 5, 04 2014.

[159] J. P. Pouget, H. Launois, T.M. Rice, P. D. Dernier, A. Gossard, G. Villeneuve, andP. Hagenmuller. Dimerization of a linear Heisenberg chain in the insulating phases ofV1−xCrxO2. Phys. Rev. B, 10(5):1801, 1974.

[160] M M Qazilbash, M Brehm, Byung-Gyu Chae, P-C Ho, G O Andreev, Bong-Jun Kim,Sun Jin Yun, A V Balatsky, M B Maple, F Keilmann, Hyun-Tak Kim, and D N Basov.Mott transition in VO2 revealed by infrared spectroscopy and nano-imaging. Science,318(5857):1750–3, 2007.

[161] I Rajapaksa, K Uenal, H Kumar Wickramasinghe, I Rajapaksa, K Uenal, and H Ku-mar Wickramasinghe. Image force microscopy of molecular resonance : A microscopeprinciple Image force microscopy of molecular resonance : A microscope principle.Applied Physics Letters, 97:073121, 2010.

[162] I Rajapaksa and H Kumar Wickramasinghe. Raman spectroscopy and microscopybased on mechanical force detection. Applied Physics Letters, 99:161103, 2011.

[163] N. A. Roberts and D. G. Walker. Phonon wave-packet simulations of ar/kr interfacesfor thermal rectification. Journal of Applied Physics, 108(12):123515, 2010.

[164] Dror Sarid, Brendan McCarthy, and Ranjan Grover. Scanning thermal-conductivitymicroscope. Review of Scientific Instruments, 77(2), 2006.

[165] Benedikt Scharf, Vasili Perebeinos, Jaroslav Fabian, and Phaedon Avouris. Effectsof optical and surface polar phonons on the optical conductivity of doped graphene.Phys. Rev. B, 87:035414, Jan 2013.

[166] P Schilbe. Raman scattering in VO2. Physica B, 316(1):600–602, 2002.

[167] M. Schnell, A. Garcia-Etxarri, A. J. Huber, K. Crozier, J. Aizpurua, and R. Hillen-brand. Controlling the near-field oscillations of loaded plasmonic nanoantennas. NaturePhotonics, 3(5):287–291, May 2009.

[168] Tero Setala, Matti Kaivola, and Ari T. Friberg. Degree of polarization in near fieldsof thermal sources: Effects of surface waves. Phys. Rev. Lett., 88:123902, Mar 2002.

147

[169] Andrei V. Shchegrov, Karl Joulain, Remi Carminati, and Jean-Jacques Greffet. Near-field spectral effects due to electromagnetic surface excitations. Phys. Rev. Lett.,85:1548–1551, Aug 2000.

[170] Zhiwen Shi, Hans A. Bechtel, Samuel Berweger, Yinghui Sun, Bo Zeng, Chenhao Jin,Henry Chang, Michael C. Martin, Markus B. Raschke, and Feng Wang. Amplitude-and phase-resolved nanospectral imaging of phonon polaritons in hexagonal boronnitride. ACS Photonics, 2(7):790–796, 2015.

[171] M. Specht, J. D. Pedarnig, W. M. Heckl, and T. W. Hansch. Scanning plasmon near-field microscope. Phys. Rev. Lett., 68:476–479, Jan 1992.

[172] K Svoboda and S M Block. Biological applications of optical forces. Annual Reviewof Biophysics and Biomolecular Structure, 23(1):247–285, 1994. PMID: 7919782.

[173] Zhensheng Tao, Tzong-Ru T. Han, Subhendra D. Mahanti, Phillip M. Duxbury, FeiYuan, Chong-Yu Ruan, Kevin Wang, and Junqiao Wu. Decoupling of structural andelectronic phase transitions in VO2. Phys. Rev. Lett., 109:166406, Oct 2012.

[174] T. Taubner, F. Keilmann, and R. Hillenbrand. Nanomechanical resonance tuning andphase effects in optical near-field interaction. Nano Letters, 4(9):1669–1672, 2004.

[175] A Tselev, I A Luk’yanchuk, I N Ivanov, J D Budai, J Z Tischler, E Strelcov, A Kol-makov, and S V Kalinin. Symmetry relationship and strain-induced transitions be-tween insulating M1 and M2 and metallic R phases of vanadium dioxide. Nano Lett.,10(11):4409–4416, 2010.

[176] Thejaswi U Tumkur, Xiao Yang, Benjamin Cerjan, Naomi J Halas, Peter Nordlander,and Isabell Thomann. Photoinduced Force Mapping of Plasmonic Nanostructures.Nano Letters, 16:7942–7949, 2016.

[177] M. Urbanczyk and Redstone Scientific Information Center. Translation Branch. SolarSails: A Realistic Propulsion for Spacecraft. Translation Branch, Redstone ScientificInformation Center, Research and Development Directorate, U.S. Army Missile Com-mand, 1967.

[178] Hans W. Verleur, A. S. Barker, and C. N. Berglund. Optical properties of vo2 between0.25 and 5 ev. Phys. Rev., 172:788–798, Aug 1968.

[179] S. Wall, L. Foglia, D. Wegkamp, K. Appavoo, J. Nag, R. F. Haglund, J. Stahler, andM. Wolf. Tracking the evolution of electronic and structural properties of VO2 duringthe ultrafast photoinduced insulator-metal transition. Phys. Rev. B, 87:115126, Mar2013.

[180] S. Wall, D. Wegkamp, L. Foglia, K. Appavoo, J. Nag, R.F. Haglund, J. Stahler, andM. Wolf. Ultrafast changes in lattice symmetry probed by coherent phonons. NatCommun, 3:721, 2012.

148

[181] P. R. Wallace. The band theory of graphite. Phys. Rev., 71:622–634, May 1947.

[182] Michael E. A. Warwick and Russell Binions. Advances in thermochromic vanadiumdioxide films. J. Mater. Chem. A, 2:3275–3292, 2014.

[183] W. Wien. Uber die energieverteilung im emissionsspektrum eines schwarzen korpers.Ann. Phys., 299:662–669, 1896.

[184] W. Wien and O. Lummer. Methode zur Prufung des Strahlungsgesetzes absolutschwarzer Korper. Ann. Phys., 292:451–456, 1897.

[185] Y. De Wilde, F. Formanek, R. Carminati, B. Gralek, P.A. Lemoine, K. Joulain, J.P.Mulet, Y. Chen, and J.J. Greffet. Thermal radiation scanning tunneling microscopy.Nature, 444:740–743, 2006.

[186] Chihhui Wu, Alexander B Khanikaev, Ronen Adato, Nihal Arju, Ahmet Ali Yanik,Hatice Altug, and Gennady Shvets. Fano-resonant asymmetric metamaterials for ul-trasensitive spectroscopy and identification of molecular monolayers. Nature materials,11(1):69–75, 2012.

[187] Fengnian Xia, Han Wang, Di Xiao, Madan Dubey, and Ashwin Ramasubramaniam.Two-dimensional material nanophotonics. Nat. Photon., 8(12):899–907, December2014.

[188] Xiaoji G. Xu and Markus B. Raschke. Near-field infrared vibrational dynamics andtip-enhanced decoherence. Nano Letters, 13(4):1588–1595, 2013. PMID: 23387347.

[189] Xin Xue, Meng Jiang, Gaofang Li, Xian Lin, Guohong Ma, and Ping Jin. Photoin-duced insulator-metal phase transition and the metallic phase propagation in VO2

films investigated by time-resolved terahertz spectroscopy. Journal of Applied Physics,114(19):193506, 2013.

[190] Honghua U. Yang, Jeffrey D’Archangel, Michael L. Sundheimer, Eric Tucker, Glenn D.Boreman, and Markus B. Raschke. Optical dielectric function of silver. Phys. Rev. B,91:235137, Jun 2015.

[191] Honghua U. Yang and Markus B. Raschke. Resonant optical gradient force interactionfor nano-imaging and -spectroscopy. New Journal of Physics, 18(5):053042, 2016.

[192] Nuo Yang, Nianbei Li, Lei Wang, and Baowen Li. Thermal rectification and negativedifferential thermal resistance in lattices with mass gradient. Phys. Rev. B, 76:020301,Jul 2007.

[193] H. J. Zeiger, J. Vidal, T. K. Cheng, E. P. Ippen, G. Dresselhaus, and M. S. Dresselhaus.Theory for displacive excitation of coherent phonons. Phys. Rev. B, 45:768–778, Jan1992.

149

[194] F. Zenhausern, Y. Martin, and H. K. Wickramasinghe. Scanning interferomet-ric apertureless microscopy: Optical imaging at 10 angstrom resolution. Science,269(5227):1083–1085, 1995.

[195] L. M. Zhang, G. O. Andreev, Z. Fei, A. S. McLeod, G. Dominguez, M. Thiemens,A. H. Castro-Neto, D. N. Basov, and M. M. Fogler. Near-field spectroscopy of silicondioxide thin films. Phys. Rev. B, 85:075419, Feb 2012.

[196] A. Zylbersztejn and N. F. Mott. Metal-insulator transition in vanadium dioxide. Phys.Rev. B, 11:4383–4395, Jun 1975.

[197] A. Zylbersztejn and N. F. Mott. Metal-insulator transition in vanadium dioxide. Phys.Rev. B, 11(11):4383, 1975.