Surface Vibrational Surface Vibrational Surface Vibrational Surface Vibrational PropertiesPropertiesPropertiesPropertiesPropertiesPropertiesPropertiesProperties

Unlike surface crystallography, which deals with average positions of atoms in a crystal, lattice dynamics (LD) extends the concept of crystal lattice to an array of atoms with finite masses that are capable of motion. This motion is not random but a coherent superposition of vibrations of atoms around their equilibrium sites due to the interaction with neighbor atoms. A collective vibration of atoms in the crystal forms a wave with given wavelength and amplitude.

Just as light is a wave motion that is considered as composed of particles called photons, we can think of the normal modes of vibration in a solid as being particle-like. The quantum of lattice vibration is called the phonon. The problem of lattice dynamics is to find the normal modes of vibration of a crystal and to calculate their energies (or frequencies, ω) as a function of their wavevector k . The relationship ω(k) is called phonon dispersion .

LD offers two different ways of finding the dispersion relation: Quantum-mechanical approachSemiclassical treatment of lattice vibrations

There are two possible polarizations for the vibrations of atoms in a crystal: longitudinal and transverse

In case of longitudinal modes the displacement of the atoms from their equilibrium position coincides with the propagation direction of the wave, whereas for transverse modes, atoms move perpendicular to the propagation of the wave.

For one atom per unit cell the phonon dispersion curves are represented only by acoustical branches. However, if we have more than one atom in the unit cell optical branches will appear than one atom in the unit cell optical branches will appear additionally.

The difference between acoustical and optical branches arises because of the options for the vibration of the atoms in the unit cell. For example, atoms A and B of diatomic cell can move together in phase (acoustical branch) or out of phase (optical branch).

Generally, for N atoms per unit cell there will be 3 acoustical branches (1 longitudinal and 2 transverse) and 3N-3 optical branches (N-1 longitudinal and 2N-2 transverse)

Lets consider a linear chain of identical atoms of mass M spaced at a distance a,

the lattice constant, connected by Hook's law springs.

For simplicity we will consider longitudinal deformations - that is, displacements of

atoms are parallel to the chain.

Let Un=displacement of atom n from its equilibrium position

Un-1=displacement of atom n-1 from its equilibrium position

Un+1=displacement of atom n+1 from its equilibrium position

The force on atom n will be given by its displacement and the displacement of its

nearest neighbors :

Acoustical phononsAcoustical phononsAcoustical phononsAcoustical phonons

The equation of motion is:

where β is a spring constant.

The right hand side is a second derivative with respect to space, so that we have

a differential equation of solution

with Uno = Uo amplitude of the wave.

If we substitute our trial wave solution into the equation of motion we find a phonon's dispersion relation for linear monatomic chain as follows:

One important feature of the dispersion curve is the periodicity of the function. For unit cell length a ,

the repeat period is equal to the unit cell length in the reciprocal lattice. Therefore the useful information is contained in the waves with wave vectors lying between the limits

Optical phononsOptical phononsOptical phononsOptical phonons

Consider a lattice with two kinds of atoms - that is a lattice with a basis of two atoms in the primitive cell. Now we have to write two solutions for the displacement corresponding to the two masses m and M. The equations of motion are:

We try solution like:

where A and B are the amplitudes of vibration of atom of mass m and M, respectively. The diatomic case has two solutions of the dispersion relation:

The methods to measure the dispersion relation of lattice vibrations are:

Bulk: Neutron Inelastic Scattering and Raman scattering of X-Rays

Surface: Electron Energy Loss Spectroscopy (EELS) and Inelastic He

Atom Scattering

BULK phonons

Neutron Inelastic Scattering

Thermal neutrons have an energy in the same range as the lattice Thermal neutrons have an energy in the same range as the lattice

vibrations and a wavelength of the same order as the crystal lattice spacing.

By analyzing the energy and direction of the scattered neutrons, the phonon

dispersion relation is obtained. Neutrons are well suited for bulk phonon

dispersion studies because of their very small cross section with matter

which allows them to penetrate deeply into the crystal.

More recently, inelastic X ray scattering has become available thanks to the

high flux obtained at synchrotron radiation sources. The photon energy has

to be high to match the wavevector transfer. The interaction mechanism is

Raman scattering

HREELS and He Atom ScatteringHREELS and He Atom ScatteringHREELS and He Atom ScatteringHREELS and He Atom Scattering

HREELS and He atom inelastic scattering are currently employed to study the dispersion of surface phonons.

The He atom measurements cover the region of energy transfers up to about 30-50meV with a resolution down to 0.5-1meV.

HREELS measurements have no limit in the energy loss and can easily access the high frequency range of light adsorbate modes (e.g O-H access the high frequency range of light adsorbate modes (e.g O-H approx 500meV), albeit with the lower resolution of about 2-3 meV. Due to the high cross section of the electrons with matter, ~5A2, their penetration depth in HREELS experiments (kinetic energy from 20 to 200 eV) is about 2-3 surface layers. Multiple scattering phenomena can be used to tune of particular modes.

He atoms, however, have lower kinetic energy (~0.05eV) and reflect at relatively large distances, the order of 3-4 A2, from the surface plane. Thus in general multiple interactions of the He atoms with the surface are negligible.

Phonons Phonons Phonons Phonons are collective lattice vibration modes, and surface phononssurface phononssurface phononssurface phonons are those particular modes associated with surfaces; they are an artifact of periodicity, symmetrysymmetrysymmetrysymmetry, and the termination of bulk crystal structure associated with the surface layer of a solid [1]. The study of surface phonons provides valuable insight into the surface structure and other properties specific to the surface region, which often differ from bulk. The figure below gives a pictoral representation of the atomic motion in a phonon mode.

Surface phonons are represented by a wave vectorwave vectorwave vectorwave vector along the surface, qqqq, and an energy corresponding to a particular vibrational mode frequency, ω [1]. The surfacesurfacesurfacesurface Brillouin zone (SBZBrillouin zone (SBZBrillouin zone (SBZBrillouin zone (SBZ) is two dimensional. For example, the face centered cubic (100) surface is described by the directions ΓX and ΓM, referring to the [110] direction and [100] direction, respectively [2].

Surface phononsSurface phononsSurface phononsSurface phononsSurface phonon branches may occur in specific parts of the SBZ or encompass it entirely across. These modes can show up both in the bulk phonon dispersion bands, resonance, or run outside of these bands as a pure surface phonon mode. Thus, surface phonons can be purely surface existing vibrations, or simply the expression of bulk vibrations in the presence of a surface, known as a surface-excess property.A particular mode, the Rayleigh phonon mode, exists across the entire BZ and is known by special characteristics, including a linear frequency versus wave number relation near the SBZ center.

due to the very different masses, the motion of electrons and nuclei takes

place on very different time scales (femto- vs pico- seconds) so that

electrons follow the motion of the nuclei adjusting adiabatically to their

positions

The potential Φ{r(n)} felt by an atom in the solid, depends therefore , to a

very good approximation only on the atomic coordinates {r(n)} .

To write the equation of motion, let’s expand Φ{r(n)} into a Taylor series

around the equilibrium positions {r0(n)}.

Born and Oppenheimer approximationBorn and Oppenheimer approximationBorn and Oppenheimer approximationBorn and Oppenheimer approximation

around the equilibrium positions {r0(n)}.

The first derivative vanishes around the minimum and the other terms may

be neglected since the displacements uα(n) are small with respect to the

lattice spacing. Given M(n), the mass of the nth atom, we can write:

...)()(),;,(

)})(({)}(({,,,

2

00 +∂∂

Φ∂+Φ=Φ ∑

βαβα

βα

βα

mn

munuuu

mnnrnr

rr

∑ ∂∂Φ∂

−=β

ββα

αβα

,

2

)(),;,(

)()(m

muuu

mnnunM &&

The sum extends over all atoms of the solid, but the coupling decreases

rapidly with distance. Considering nearest and possibly next nearest

neighbours contributions is in general sufficient to describe a system.

Angular forces are in addition needed to describe rigid lattices like those of

semiconductors.

For a periodic structure the system of equations separates into subsets of

3s equations, with s the number of unit cells, using Bloch’s theorem which

allows for plane wave solutions with frequency ω and wavevector q

))()(( nrqtqirrr

⋅−−= ω

∑ ∂∂Φ∂

−=β

ββα

α

βα

,

2

)(),;,(

)()(m

muuu

mnnunM &&

))()((

,00)(nrqtqi

eunurrr

⋅−−= ωαα

which are the solutions for phonons of the 3D solid.At the surface the 3D symmetry is broken and we have to consider the

coordinates in the surface plane separately from the vertical direction.

It is useful to introduce mass normalised amplitudes ξ. Denoting the unit

cells by l|| and lz and the positions of atoms within the unit cell with κ and

the spatial directions with α,β,A we get:

)()()( |||| κκκξ αα zzz llulMll =

looking for solutions with a time dependence of the form e-iωt

we get from the equation of motion

);( ''' κκαβ llllΦ

0)();()( '''

||

'''

||||||

2

'''||

=− ∑ κξκκκξω βκβ

αβα z

ll

zzz llllllDll

z

With Dαβ the dynamical matrix of the system given by:

)()(

);();(

''

'''

||||'''

||||

κκ

κκκκ αβ

αβ

zz

zz

zz

lMlM

llllllllD

Φ=

Dαβ is symmetric with respect to the interchange of primed and of non primed

symbols and depends only on l // -l//’ because of translational symmetry in the

surface plane. Given the eigenvectors eα(q//;lzκ) we can rewrite:

);(

||||||0||);()(

καα κκξ zlqrqi

zz elqellrrr

r ⋅−=

and inserting ξ into the equation of motion we obtain the secular equation

in which

is the Fourier transformed dynamical matrix

0)'';()'';;;();()( ||

)(

''

||||||

)(

||

2 =− ∑ κκκκω ακβ

αβα z

s

l

zzz

s

s lqelqlqdlqeqz

rrr

)'';;( || κκαβ zz llqd

Whose solutions, the eigenfrequencies, are given by the zeros of the

determinant

)'''()((

||||

'

||||0||0||

||

)'',';,()'';;(κκ

αβαβ κκκκ zz llrllrqi

zz

l

zz ellllDllqdrrr

−⋅−∑=

( ) 0)'';;()(det ||||

2 =− κκω αβ zzs llqdqr

Note that the solutions are for ω2 not for ω

Projection of the 3D Brillouin Zone on the surface

3D fcc Wigner Seitz cell

2D fcc (111) Brillouin Zone

Projection to 2 D

Modes of an infinite and of a semi-infinite one dimensional

diatomic chain

VibrationalVibrationalVibrationalVibrational dynamicsdynamicsdynamicsdynamics

k force constant between

nearest neighbors

Equations of motion:

( ) ( ) ( )( ) ( ) ( )

−−=−−−=

−−=−−−=

++

−−(1)

1n

(1)

n

(2)

n

(1)

1n

(2)

n

(2)

n

(1)

n

(2)

n

(2)

1n

(2)

n

(1)

n

(2)

1n

(1)

n

(1)

n

(2)

n

(1)

n

ss-s2ksskssksm

ss-s2ksskssksM

&&

&&

We look for a plane wave solution

with wavevector q

=

=−+

−−

ωt))4

1i(qa(n

2

(2)

n

ωt))4

1i(qa(n

1

(1)

n

ec s

ec Ms

m

( ) ( ) ( ) −−=−−−= −−(2)(2)(1)(2)(1)(1)(2)(1) ss-s2ksskssksM &&

( )

( )

=−+

+

=

++−

⇒

−

0c 2kmωc eeM

mk

0c eem

Mkc 2kMω

2

2

12

iqa

2

iqa-

22

iqa

2

iqa

1

2

2

qacos

( ) ( ) ( )( ) ( ) ( )

−−=−−−=

−−=−−−=

++

−−(1)

1n

(1)

n

(2)

n

(1)

1n

(2)

n

(2)

n

(1)

n

(2)

n

1nnn1nnnnn

ss-s2ksskssksm

ss-s2ksskssksM

&&

&&

+−

+−

2

2

mω2k2

qacos

M

m2k

2

qacos

m

M2kMω2k

DYNAMICAL MATRIXDYNAMICAL MATRIXDYNAMICAL MATRIXDYNAMICAL MATRIX

Solutions are given by the zeros

of the determinant

i.e.:

( )( ) 02

qacos2k m ω2kM ω2k

2

22 =

−+−+−

( ) ( ) ( )( )

−−+±+=⇒ ± qacos12MmmMmM

mM

kω

22

Bulk modes

modes Optical (q)ω

modes Acoustical (q)ω

+

−

q

LA

TA

Three polarizations

TO

Transverse optical

modes (TO)

generate a dipole

moment (centroid of

diatomic chain

Longitudinal acoustical mode at q=0

longitudinal optical mode at q=0

Transverse optical mode at q=0

Eigenvectors

moment (centroid of

positive charge

moves with respect to

the centroid of the

negative charges).

Can be excited

optically by absorbing

Infrared Radiation or

by electron energy

loss in dipole

scattering.

Equilibrium configuration

IR inactive phonon

IR active phonon

mode at q=0

Surface modesSurface modesSurface modesSurface modesIR iqqq~ +=

ωωωω REALREALREALREAL

a)a)sinh(qisin(qa)a)cosh(qcos(qa)q~cos( IRIR −=⇒

{ } 0a)a)sinh(qsin(q0a)q~cos(Im ===⇒

We repeat the calculation with

( ) ( ) ( )( )

−−+±+=± aq~cos12MmmMmM

mM

kω

22

{ } 0a)a)sinh(qsin(q0a)q~cos(Im IR ===⇒

0q I = VOLUME BRANCHESVOLUME BRANCHESVOLUME BRANCHESVOLUME BRANCHES

K21,0,nwith

nπaq e 0q RI

±±=

=≠SURFACE SOLUTIONSSURFACE SOLUTIONSSURFACE SOLUTIONSSURFACE SOLUTIONS

( ) ( )( ) ( )zisinhizsin

zcoshizcos

=

= ( ) ωt)rqi(zq

qqq////

//eeeArs

−⋅− ⊥

⊥=

rr

r

r

( ) a)cosh(q1)(a)cosh(q ncosa)q~cos( I

n

I −== πWe thus get:

in the first Brillouin zone

( )

( ) ( ) ( )( )

BULKs

0

I

22

IIR

ωω

aqcosh12MmmMmMmM

kω

aq 0a)cosh(q-1 0n0qΓ

>⇒

−−+±+=

∀<=⇒=⇒

>

±44444 344444 21

in the first Brillouin zone

Surface modes can exist only above

the bulk highest frequency

for n=1 the solutions are located at the 3D zone boundary

MAXI

22

I

R

q2Mm

mMarccosh

a

1q

a

πq

=

+<

=

Solutions exist only inside the band-gap i.e. between upper and lower bulk

bands

( ) ( ) 00qω 2km)2f(M

0qω II ===+

==⇒ −+ µ( ) ( ) 00qω

Mm0qω II =====⇒ −+ µ

( )

( )

( )µ

kqqω

M

2k0qω

m

2k0qω

MAXII

I

I

==

==

==⇒

±

−

+

qIIII

qRRRR

( ) )1

M

1k(qqω max II

m+==⇒Surface bands

Boundary conditions:

)~((i)

ns tzqi

i

ineC

ω−≈)

4

1-a(n =i

nz )4

1a(n +=i

nzwith for atom (1)=(i) and for atom (2)=(i)

GivenIIR iq

a

πiqqq~ +±=+= we obtain solutions of the form (qI>0)

qIIII

qRRRR

tizqee

inI ω−−≈

~(i)

ns

a

whose vibrational amplitude

decays towards the bulk

For a three dimensional crystal with surfaces

there are solutions of the formtirqizq

qqq eeeAe ω−⋅− ⊥

⊥⊥= )(~

,,q||||

||||s

With an additional index in case there is more than one atom

per unit cell.

A surface phonon is thus characterized by its frequency, its

wavevector and by the way it decays towards the bulk. wavevector and by the way it decays towards the bulk.

These quantities are related by the boundary condition that no

forces act on the topmost layer of atoms.

We obtain a set of: ),( || ⊥qqω

One mode, the Rayleigh wave, characterized by an acoustic

dispersion, survives up to the continuum limit.

In earth quakes it is responsible for the largest damages, at

surfaces it influences the rate of catalytic reactions.

Surface Modes

Γ ΓM K

Penetration of surface modes into the bulk

The character of the surface modes may change over the 2D zoneCase of fcc (100)

Along Г-X S1 is SH, along Г-M it is sagittal

Elastic continuum limitThe Rayleigh wave persists

in the elastic continuum limit.

Its velocity is smaller than

those of all bulk waves

moving in the same

direction.

The motion takes place in

the sagittal plane, i.e. The

atoms move both vertical Rayleigh wave

atoms move both vertical

and parallel to the surface.

fcc (001) anisotropy of the Rayleigh wave and pseudo wave

Pseudo surface wave decoupled by symmetry from the bulk modes

Application: high frequency filtersApplication: high frequency filtersApplication: high frequency filtersApplication: high frequency filters

Transmitter: The high voltage

electric field induces strain in the

piezoelectric crystal with q=2π/λ.

When such frequency coincides

with the dispersion of the Rayleigh

wave ω =ωR(q)=vRq

Rayleigh waves are excited in a

narrow spectral range

∑=jk

jkijki dP ε

q

q

ijkij Ed∑=ε

Receiver: An electric polarization

P is induced by the strain field

narrow spectral range

Effect of surface stress on the Rayleigh wave

In points of high symmetry such as the point of the 2D BZ of

fcc(100) the frequency of the Rayleigh wave can be calculated

easily without solving the secular equation and reads:

12

2

12

2 245cos4)(1

kkMM S =°=ω

)(245cos)(4)( 12

2

12

2

2 bbS kkkkMM +=°+=ω

1st layer mode

2nd layer mode

M

k12 force constant

between first and

second layer

kb bulk force constant

Effect of surface stress on the Rayleigh wave

In points of high symmetry such as the point of the 2D BZ of

fcc(100) the frequency of the Rayleigh wave can be calculated

easily without solving the secular equation and reads:

12

2

12

2 245cos4)(1

kkMM S =°=ω

)(245cos)(4)( 12

2

12

2

2 bbS kkkkMM +=°+=ω

b kMM

4)(2

=ω

while the uppermost bulk frequency is

1st layer mode

2nd layer mode

M

bb kMM

42

)(=

ωwhile the uppermost bulk frequency is

In presence of a stress field the force )(

11

'

11

saσϕ =acts on the surface atoms changing the equation of motion.

For the Nearest Neighbour central force model we obtain:

akM S /'2)M( 12

2

1ϕω +=

Clean metallic surfaces have a tensile stress which may stiffen the

Rayleigh wave

Rayleigh wave anomaly

on metal surfaces

The frequency of the Rayleigh

wave at the 2D Zone Border

is higher than expected for

most metal surfaces.

The effect may be either due

to an increase of the force

constant between first and

second layer or to surface

Tensile (compressive)

stress at metal surfaces

causes a stiffening

(softening) of the Rayleigh

wave frequency

second layer or to surface

stress. The latter explanation

is the currently accepted one.

tensileno stress

compressive

• Slab method

• Green function method

Theoretical methods of Surface Dynamics

• Green function method

• Molecular dynamics

Slab method

Analogous to the electronic state calculations. The equation of motion is solved for a slab and one gets a spaghetti diagram

Green Function method

The surface is considered as a perturbation affecting the bulk phonon spectrum. The secular equation is a differential equation and it is solved mathematically by the Green function by the Green function method. The surface term is introduced as a perturbation of the bulk solution.

Molecular dynamics

The molecular dynamics method consists in calculating the atomic positions vs time. The Fourier transform vs space gives the k vector of the phonons, while vs time it allows to recover the phonon frequency spectrum to recover the phonon frequency spectrum

Ab initio methodsAb initio methodsAb initio methodsAb initio methods

By virtue of the Born Oppenheimer approximation, phonons are a property of the electronic ground state. At the high symmetry points of the 3D Brillouin Zone the eigenvectors correspond to particularly simple motions (e.g. at fcc X to the sliding of the (100) planes against each other) and the potential energy associated to such motion can be calculated relatively easily by doubling of the unit cell (frozen phonon method). At the surface things are not as easy, but also in that case one can calculate the energy of the distorted lattice and then compute the second derivative of the E(x) curve.

Adsorbate modesAdsorbate modesAdsorbate modesAdsorbate modes

Dynamics in presence of an adsorbed layer

Adsorbate modes contain information on the

lateral interaction between the adsorbates

Phonon anomalies

• Soft phonons induced by adsorbates which modify the surface stress

• Soft phonons induced by electron –phonon interaction (Kohn anomaly)

Sulfur induces a softening of the RW because of loading

while oxygen induces a much larger effect because of the

compressive stress it introduces into the surface layer

The Rayleigh wave goes soft for 2/''12

)(

11 ϕσ −≤s

corresponding to compressive stress (soft phonon)

aM S /'''2)M( 12

2

1ϕϕω +=

Before this happens the A2 mode at X goes however soft, too, for:

4/''12

)(

11 ϕσ −≤s

1211

so that the surface reconstructs with p4g symmetry.

This situation happens e.g. for the c(2x2) overlayers of C and N on Ni(100).

For O/Ni(100), on the contrary, the stress is large but still insufficient to

cause the freezing in of the A2 mode.

Effect of surface stressEffect of surface stressEffect of surface stressEffect of surface stress

Phase transition and soft modes

Reconstruction of W(001)

Atoms are pairing in rows

Artificially stressed surfaces

Kohn anomalyKohn anomalyKohn anomalyKohn anomaly

The electron phonon interaction causes a rigid shift of the Fermi sphereand a failure of the Born – oppenheimer approximation since the electrons do not follow any more the ionic motion adiabatically.

Fermi sphere

Kohn anomaly

The Kohn anomaly occurs e.g. for bulk Pt. The phenomenon is expected to be particularly strong for materials which present parallel planes in the Fermi surface, the so called surface nesting.

direct lattice

reciprocal lattice

For 2D materials the Kohn anomaly is observed only for

W(110)-H and Mo(110)-H. In this case the Fermi surface

reduces to a Fermi –line.

The upper branch is of phononic character and is observed

both in HREELS and in inelastic He atom scattering (IHAS).

The lower branch is due to (e-h) pairs and is excited only by

.

anomaly

line

Filled symbols HREELS: phonon only

Open symbols IHAS: phonon and e-h

pairs

The anomaly is observed all along

the line running parallel to Г-N.

The effect is thus due to one-

dimensional nesting

2qF

Explanation of the Kohn anomaly. Since the effect is observed only in presence of the (1x1) H phase, nesting is due to an adsorbate induced electronic state