Electronic refrigeration and

thermometry in nanostructures at

low temperatures

Jukka Pekola

Low Temperature Laboratory

Aalto University, Finland

Nanostructures

Temperature

Energy relaxation

Thermometry

Refrigeration

Present topics of interest

1. New physical phenomena and versatile fabrication

techniques lead to new electronic device concepts.

- quantum mechanical effects

- single-electron effects

- superconductivity

- non-equilibrium effects

2. Thermal properties are very different from those of

macroscopic systems, especially at low temperature. Local

refrigeration and thermometry become possible.

What new can micro- and nanostructures

provide at low temperatures?

Technological basis: micro- and

nanofabrication

1. Lithography and thin film

deposition (structures in 2D)

A. Kemppinen et al.,

Single-electron transistor

2. Etching (structures in 3D)

A. Luukanen et al.,

Antenna Coupled Micro-

bolometer

Temperature in an electronic device

0.0

0.2

0.4

0.6

0.8

1.0

hot

intermediate

cold

f(E

)

E

E = m

Generic thermal model for an

electronic ”thermometer”

The energy distribution of electrons

in a small metal conductor

Equilibrium –Thermometer measures the temperature of the ”bath”

Quasi-equilibrium –Thermometer measures the temperature of the electron system which

can be different from that of the ”bath”

Non-equilibrium –There is no well defined temperature measured by the ”thermometer”

Illustration: diffusive

normal metal wire

H. Pothier et al. 1997

The distribution is determined by energy relaxation:

Electron-phonon relaxation in

metals at low T

Tunnelling and thermometry by

tunnel junctions

METAL METAL

G+

G-

Symbols:

Normal junction Superconducting junction

Tunnel barrier

Examples of aluminium-oxide tunnel barriers

Basics of tunnel junctions

Tunneling from occupied states to empty states

Energy is conserved

E

V

1 2

Metal – Insulator – Metal (MIM or

NIN) tunnel junction

E

V

Now, density of states (DOS) is almost constant over the

small energy interval:

If f1 = f2 = f, we may use

Ohmic, no temperature dependence NOT A THERMOMETER!

Noise?

Average current through a basic tunnel junction is not sensing T. How about

fluctuations around the mean current?

Equilibrium noise: Iave = 0

Tunnel junction behaves like a resistor

SI SV

Noise thermometry is possible, but it is seldom used as such in nanostructures.

Non-equilibrium: Shot

noise thermometry (SNT)

L. Spietz et al., Science 300, 1929 (2003); PhD thesis

2006; APL 89, 183123 (2006).

SNT – quantitative analysis

Idea: measure the

crossover voltage between

thermal noise and shot

noise; not necessary to

know the absolute

calibration for noise

measurement.

Single-electron tunneling and

Coulomb blockade thermometry

Single-electron transistor (SET)

V/2

RT1 , C1

V/2

RT2 , C2

Cg

Vg

neUnit of charging

energy:

-2 0 2

-2

0

2

I (e

/2R

TC

)

V (e/C)

Coulomb Blockade Thermometry

– principal idea

With increasing temperature the IV

curve of a SET transistor gets

increasingly smeared.

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

4

6

8

10

CU

RR

EN

T

VOLTAGE

-10 -8 -6 -4 -2 0 2 4 6 8 10

0.985

0.990

0.995

1.000

VOLTAGE

CO

ND

UC

TA

NC

E d

I/d

V

IV curve

conductance

Basic properties of a Coulomb

blockade thermometer (CBT)

primary thermometer

secondary thermometer

An array of N tunnel junctions in weak Coulomb blockade,

EC << kBT:

J.P. et al., PRL 73, 2903 (1994)

CBT performance at

low T

3He MCT (mK)

CB

T (

mK

)

Solution: Larger islands and make a (known) correction.

CBTs produced by VTT optical

lithography process (2007)

M. Meschke et al., J. Low Temp. Phys. 134, 1119 (2004); arXiv:1006.1609 (2010).

Thermal

decoupling

Experiments

against MCT at

CNRS Grenoble

and at PTB

CBT in magnetic field

Magnetic field has no

observable influence on CBT

J. Appl. Phys. 83, 5582 (1998)

J. Low Temp. Phys. 128, 263 (2002)

B (T)

T = 1.46 K

T = 1.46 K

Single junction thermometer

(SJT)

CBT

SJT I

V

e

TNkV B439.521

In both cases:I

V

PRL 101, 206801 (2008)

Single Junction Thermometer for T > 77 K

FABRICATED BY Yu. Pashkin, NEC

Single Junction Thermometer

-100 -75 -50 -25 0 25 50 75 1000.88

0.90

0.92

0.94

0.96

0.98

1.00

V1/2

T=77 K

T=42 K

G

/GT

U BIAS

(mV)

V1/2

•Error of ~1% mainly

due to background

NIS-tunnelling and thermometry

0 1 2 30

1

2

3

eR

T I/

eV/

NIS-thermometry

0 100 200 300 400 500 600 700 8000.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

V (

mV

)

T (mK)

I = 3 pA

Probes electron temperature of N island (and not of S!)

-0.6 -0.4 -0.2 0.0 0.2 0.4-6

-4

-2

0

2

4

6

215mK

250mK

285mK

320mK

360mK

395mK

430mK

I (n

A)

V (mV)

38mK

48mK

78mK

110mK

145mK

180mK

Outlook

Electronic primary thermometers, SNT, CBT and SJT

presented as prime examples, show great promise

1. as practical calibration-free on-chip thermometers over a

wide temperature range

2. in realising the temperature scale, based on a fixed value

of the Boltzmann constant, in the sub - 10 K regime

Problems:

Measurement of SNTs is not straightforward

CBTs suffer from self-heating at the lowest temperatures

NANO-REFRIGERATION

Refrigeration on-chip

Thermoelectric refrigeration

Peltier refrigerators,

Peltier 1834

Thermionic refrigeration,

Mahan, 1994

Korotkov and Likharev, 1999

Quantum-dot refrigerator,

Edwards et. al., 1993

Experiment: Prance et al.,

PRL 102, 146602 (2009).

Refrigeration by tunnelling

Energy current in a tunnel junction

In order to find energy current (from conductor 1) one replaces e by E – eV in the

corresponding expression:

Compare to:

For a NIN junction, we obtain

This means that the Joule power is divided evenly between 1 and 2.

NIS junction as a refrigerator

Optimum cooling power is

reached at V /e:

Cooling power of a

NIS junction:

Optimum cooling power of a NIS junction at TS ,TN<< TC

Efficiency (coefficient of performance) of a NIS

junction refrigerator:

SINIS structure

S

probe junctions

N SVc/2

S

V/2

-Vc/2

cooler junctions

Basic idea:

- bias voltage Vc modifies electron distribution f(E) on N island

- probe junctions measure modified f(E)

S

-V/2

N

Early experiments

M. Leivo et al., 1996

Cooling of a superconductor

(SIS’IS cooler)

Ti – Al

sample

[TC(Ti) = 0.5 K,

TC(Al) = 1.3 K]

A. J. Manninen et al., Appl. Phys. Lett. 74, 3020 (1999).

COOLING FROM NORMAL TO

SUPERCONDUCTING STATE

Experimental status

A. Clark et al., Appl. Phys. Lett. 86, 173508 (2005).

Refrigeration of a ”bulk” object

Nahum, Eiles, Martinis 1994 Demonstration of NIS cooling

Leivo, Pekola, Averin 1996, Kuzmin 2003, Rajauria et al. 2007 Cooling electrons 300 mK

-> 100 mK by SINIS

Manninen et al. 1999 Cooling by SIS’IS see also Chi and Clarke 1979 and Blamire et al.

1991, Tirelli et al. 2008

Manninen et al. 1997, Luukanen et al. 2000 Lattice refrigeration by SINIS

Savin et al. 2001 S – Schottky – Semic – Schottky – S cooling

Clark et al. 2005, Miller et al. 2008 x-ray detector refrigerated by SINIS

Prance et al. 2009 Electronic refrigeration of a 2DEG

Kafanov et al. 2009 RF-refrigeration

For a review, see Rev. Mod. Phys. 78, 217 (2006).

RF-refrigerator

Question: can one cool the island of a single-electron box by gate?

Typical cooling cycle

J. P. , F. Giazotto, O.-P. Saira, PRL 98, 037201 (2007)

Influence of photon assisted tunneling: N. Kopnin et al., PRB 77, 104517 (2008)

RF-refrigerator - experiment

2

S. Kafanov et al., PRL 2009

Hybrid single-electron turnstile

(SINIS)

DC properties of the coolerSource-drain voltage cools the island, measured current yields temperature

Results of the RF experiments

eV

A

G- G+

G- / G+ = e -eV/kT

DC current provides

thermometry for RF

cooling

[e] [e]

[K]

Curiosity experiments and (till now)

Gedanken experiments

Quantum of thermal conductance

Brownian refrigerator

Temperature fluctuations

Quantized conductance

Electrical

conductance in a

ballistic contact:

Thermal

conductance:

GQ and sQ related by Wiedemann-Franz law

More generally:

Expression of GQ is expected to hold for carriers obeying arbitrary

statistics, in particular for electrons, phonons, photons (Pendry 1983,

Greiner et al. 1997, Rego and Kirczenow 1999, Blencowe and Vitelli

1999).

Electrons:

Example of quantized thermal

conductance: phonons in a nanobridge

K. Schwab et al., Nature 404, 974

(2000)C. Yung, D. Schmidt and A. Cleland,

Appl. Phys. Lett. 81 31 (2002)

Electromagnetic transfer of heat

(photons)

Lattice

Electrical

environment

Electron

system

Schmidt et al., PRL 93, 045901 (2004)

Meschke et al., Nature 444, 187 (2006)

Ojanen et al., PRB 76, 073414 (2007), PRL 100, 155902 (2008)D. Segal, PRL 100, 105901 (2008)

Gn

Heat transported between two resistors

Impedance matching:

Radiative contribution to net heat flow

between electrons of 1 and 2:

Linearized expression

for small temperature

difference

T = Te1 – Te2:

Classical or quantum heat transport?

w

wC

w

wC

”Classical” ”Quantum”

Classical or quantum heat transport?

Classical:

Quantum limited:

Johnson,

Nyquist 1928

Demonstration of photonic heat

conduction

10 µm

R2 R1

x

x

x

x

LJ

CJ

Tunable

impedance

matching using

DC-SQUIDs

F

M. Meschke, W. Guichard and J.P., Nature 444, 187 (2006)

enGn

90

95

100

105

110

115

120

125

155

160

165

170

105mK

118mK

T0 = 167mK

60mK

75mK

157mK

T (m

K)

F (a.u.)

e1

Thermal

model

2nd experiment

SAMPLE A in a loop (”matched”)

[SAMPLE B without loop (”not

matched”)]

Heat transport in different set-ups

r << 1T1

R

T2

R

C

~ ~

r ~ 1T1

R

T2

R

C

~ ~

Linear

geometry

(Sample B)

Loop

geometry

(Sample A)

for small temperature difference

in the present experiment

Results in the two

sample geometries

Heat transported by

residual quasiparticles

at T > 0.3 K and by

photons (in the loop

sample) at T < 0.3 K

QuasiparticlesA. Timofeev et al., PRL 2009

Brownian refrigerator

Heat flows from hot to cold by

photon radiation

This happens between

two resistors

The situation is identical if

we replace one resistor by

an ordinary tunnel junction

N N

R, TR

RT, TN

N N

R, TR

RT, TN

Harmonic vs stochastic drive in

refrigeration

N S

R, TR

RT, TN,S

N S

R, TR

RT, TN,S

N S

R, TR

RT, TN,S

N S

R, TR

RT, TN,S

N S

R, TR

RT, TN,S

N S

R, TR

RT, TN,SSinusoidal bias –

Refrigerates N if

frequency and

amplitude are not

too high

Stochastic drive –

Refrigerates N if

spectrum is

”suitable”

Brownian

refrigerator?

Brownian refrigerator

CO

OL

ING

PO

WE

R O

F N

(fW

)

N S

R, TR

RT, TN,S

N S

R, TR

RT, TN,S

J.P. and F. Hekking, PRL 98, 210604 (2007)

Preliminary experiment on the

Brownian refrigerator

A. Timofeev et al., unpublished

Temperature fluctuations

Temperature fluctuations

Classical temperature fluctuationsAssume that all Ti are equal, and equal to the

average of T (equilibrium fluctuations)

(fluctuation-dissipation theorem)

(balance equation)

(Fourier transform into noise spectra)

Classical temperature fluctuations

Example system: electrons in the

phonon bath

In this grain at T = 100 mK, = (10 mK)2.

0.01 0.1 1 10

ST

f / fC

Cut-off frequency fcdetermined by electron-

phonon relaxation rate, it

varies in the range 10 kHz –

10 MHz: suitable for a

measurement.

electrons

phonon bath

Preliminary measurements

Experimental setup

Magnitude response: Phase response:

Resonance:

Quality factor:

Optimal bias point:

-1 0 1

-2

0

2

Vp [mV]

I p[¹

A]

I p

CBT results for an array

V (mV)G

/GT

Uniform array, EC << kBT:

CBT sensors

1 mm

Long arrays suppress efficiently the errors

due to co-tunnelling and finite impedance of

the electromagnetic environment.

Parallel arrays lower the sensor impedance

to a convenient value.

Another configuration is a 2D array,

employed by T. Bergsten et al., APL 78, 1264

(2001).

”Low T”

”High T”

NIS-thermometry

This method is self-calibrating and can be considered

(almost) primary

On-Chip Cooling/small junctions

Cooling of electrons from 0.3 K to 0.1 K (M.M. Leivo, J.P. Pekola, and D.V:

Averin, Appl. Phys. Lett. 68, 1996 (1996).)

Cooling of lattice from 0.2 K to 0.1 K (A. Luukanen, M.M. Leivo, J.K. Suoknuuti,

A.J. Manninen, and J.P. Pekola, JLTP, 120, 281 (2000).)

Schottky barrier coolers with super-

conductor and heavily doped silicon

60mm

30mm

I

Ith

I

Uth

A. M. Savin et al., Appl. Phys. Lett. 79, 1471 (2001).

Schottky barrier coolers – results

-0.4 0.0 0.4

100

200

300

400

30.04.2001, X3F2, cooler 20x30

T /m

K

V / mVI0

I

V

-0.5 0.0 0.5

-0.1

0.0

0.1

812 mK

940 mK

1060 mK

1340 mK

100 mK

163 mK

255 mK

424 mK

631 mK

I / mA

V / mV

Thermometry by S-Sm-S Schottky barrier Cooling results

2nd experiment

SAMPLE B (”mismatched”)

N S N

Results in the two sample geometries

BLUE LINE: Directly cooled resistor

RED LINE: Remote resistor