Thermodynamics and thermal measurements

at the nanoscale

Florian ONG, Olivier BOURGEOIS

Institut Néel, Grenoble

GDR Physique Mésoscopique, Aussois Mars 2007

OverviewHow is macroscopic thermodynamical description affected as one reduces system sizes ?

– thermodynamical limit is not reached

importance of fluctuations– High Surface/Volume ratio : Surface Energy term in Uint

Loss of extensivity of U and S– Are local variables well defined ? – What are the effects of confinement ?– What changes in heat transfer when phonon mean free

path and/or wavelength exceeds sample’s dimensions ?

N

Motivations

• To bring a different and innovative point

of view on mesoscopic physics

(complementary to electrical transport,

magnetization, spectroscopy…)

• To predict heat transfers in

nanodevices, to control heating

processes

Outline

• Temperature at the nanoscale• Some thermodynamic descriptions

of small systems• Thermal transport in

nanoconductors• Specific heat : nanocalorimetry

Existence of temperature at nanoscale

Thermodynamical limit fundation : interaction I between parts of a system becomes

negligible, and so extensivity can hold

•How does I scale when N is finite ?•What is the minimum size of a system to define T ?

Temperature at nanoscale

MODEL :• 1D macroscopic chain of N identical particles at

temperature T (ie described by a canonical state

at T)• First neighbour interaction Vj,j+1

• Division into NG groups of n particlesQUESTIONS :• How does In scale with n ?• What is the minimal groupe size nmin so as Tloc is defined (ie so

as reduced density matrix may be approximated by a canonical one at Tloc)

……..

n In n ???

[Hartmann et al. PRL 93 80402 (2004)]

EXAMPLE :

Vj,j+1 = harmonic potential nmin = constant for T > D

(T/D )3 for T < D

nmin depends on T (quantum effect)lmin = nmin a0 Carbon : lmin = 10 µm at 300K

Silicium : lmin = 10 cm at 1K !!! (1D chain)

(~100nm for 3D)

RESULTS :

• Inter-group interaction In

• Condition on n so as a group can be described by thermodynamics :

• If Tloc exists, Tloc = T

n

1

kT

En

Width of the energy

distribution of the total system

[Hartmann et al. PRL 93 80402 (2004)][Hartmann et al. EPL 65 613 (2004)]

–Late XIXth : Gibbs generalized by allowing variations of the number of molecules : dE = TdS – pdV + µdN

introduction of Free Energy functions

treatment of various equilibria (chemical reactions, phase transitions…)

• Motivations :– Early 1960’s : study of macromolecular solutions– 2000’s : growing interest due to nanofabrication progress– Growing interest in completely open systems (µ,p,T)

(open aggregates in biology, metastable droplets in vapor…)

Hill’s nanothermodynamics

•Philosophy :–Before Gibbs : dE = TdS – pdV at equilibrium (1st principle)

Hill’s contribution : Gibbs’ description cannot hold for small systems, because a surface energy term ~ N2/3 cannot be neglected

there should be another term added in the right-hand side of the 1st principle

Modification of Gibbs equation at the ensemble level :

S = system containing N equivalent and non-interacting small systems

S is a macroscopic system obeying {Eq Gibbs + new term}

dEt = TdSt - pdVt + µdNt + EdN

E = subdivision potential ~ system chemical potential

[Hill NanoLett. 1 273 (2001)]

Consequences:– In macrosystems : surface/edges effectsare negligible, so E = 0

-SdT + Vdp – Ndµ = 0 Gibbs-Duhem relation : intensive variables (µ,T,p) are not

independent ! (usual choice of (T,p) couple to describe systems)

– Back to small systems :

Integration gives : Et = TSt – pVt + µNt + EN

dE = -SdT + Vdp – Ndµ

In contrast to macrosystems, (µ,T,p ) are independent( A macrosystem has one less degree of

freedom ! )

[Hill NanoLett. 1 273 (2001)]

Consequences– Influence of environnement

• Energy, entropy depend on the choice of environnemental variables

– Fluctuations• completely open systems (µ,p,T) : large fluctuations

of extensive parameters (N,V,S)

²

²²

X

XX 1/N for macrosystem

1 for small system

[Hill NanoLett. 1 273 (2001)]

[Hill & Chamberlain NanoLett. 2 609 (2002)]

Abe’s Nanothermodynamics

• Hill : Modification of thermodynamical relation by adding a term. Consequence :

large fluctuations of variables

• Abe : Incorporation of fluctuations at the beginning ( by averaging the Boltzmann-Gibbs distribution over a T distribution)

• ²-distribution of =1/kT ; width = q-1• theory of large deviations

1

1

q

pS i

qi

q

Tsallis Entropy(pi = microstate probability)

i

ii ppS logIf q=1 (no temerature fluctuations) :

one recovers Gibbs entropy

[Rajagopal, Pande & Abe, Proceedings of Indo-US Workshop (2004)]

Thermodynamics with Tsallis entropy

• relevant for systems with long range interactions, and for systems with T fluctuations and/or dissipation of energy :– Hydrodynamic tubulence– Scattering processes in particle physics

– Self-gravitating systems in astrophysics

• Non additivity of Tsallis Entropy :

• Thermodynamics principles :– 1st law : OK (conservation of Energy)– 3rd law : OK (defines the ordered state)– 2nd law : OK if [Abe et al. PRL 91 120601

(2003)]

1q

IIq

Iq

IIq

Iq

IIIq SSqSSS )1(

2,0q

[Beck EPL 57 329 (2002)]

Thermal transport in 1D conductors

-- Study of thermal conductivity in monocrystaline conductors whose size is smaller than the dominant phonon wavelength.For silicium [D(Si)=625K] : – 1K: T=0.1 µm– 100 mK: T=1 µm

– Bulk Diamond has the higher reported ; what about carbon nanotubes ?

– Analogy with Landauer description of transport : one thermal conductance quantum per channel

T

aDebyeT

h

Tkk B

Q 3

22

Thermal Conductance of CNTs

CNTs vs Silicium nanowires– SWNT : d~1nm real 1D behavior– C-C = strongest chemical bond in nature

(Diamond : =2300-3320 W/m.K)– ph-ph scattering limited by interfaces with vacuum (restricted

number of final states)– other scattering processes limited by high structural perfection

Exceptionally high thermal conductivity is predicted (~6600 W/mK)

[Berber et al. PRL 84 4613 (2000)]

Possible Waveguide for heat transfer ??

Low T : T for T<30K

Energy-independent mean free path ~ 0.5-1.5 µm , due to surface scattering

[Hone et al. PRB 59 R2514 (1999)]

Macroscopic Bundles of SWCNTs (d~1.4 nm)

(T) measured by a comparative method

Measure of elec(T) (non metallic for T<150K)

Room T : singleCNT = 1750-5800

W/m.KWiedman Franz ratio :

/(elec T) > 100 L0

Transport is dominated by phonons at low T

Thermal Conductance of CNTs

[Kim et al. PRL 87 215502 (2001)]

Room T > 3000 W/m.K ; mfp ~ 500 nmT > 320 K : Umklapp phonon scatteringT<320 K : nearly ballistic transport

10 µm

First measure of of a single MWCNT

(d~14 nm, L~2.5µm)Suspended SiN device ; T = 8-

370 K

[Pop et al. NanoLett 6 96 (2005)] Single SWCNT : ~3500 W/m.K

Ballistic or diffusive transport ? remains unclear !

Thermal conductance of crystaline nanostructures

Conductive wires : metals, n+GaAs… (electron heating technique, 1985-1995)

• poor e-ph scattering at low T

• e- short-circuit the thermal transport

Phonon contribution hard to isolate

[Tighe et al. APL 70 2687 (1997)]

Need for separating e- and phonons : • n+GaAs/iGaAs :

heterostructure with separated transducers and conductor

• still an electronic pathway !

cavitybath heater

thermometerconductors

200nm*300nm

bathcavity TT

Q

Isolation of phonon contribution [Fon et al. PRB 66 45302 (2002)]

Comparative measurement (4-40K)Better understanding of phonon scattering mechanisms :

beam<< bulk : reduction of mfp due to– enhanced surface scattering– reduction of group velocity– reduction of DOS

4-10 K : diffuse surface scattering( do (4K) ~ 10 nm ; 3D model )

20-40 K : Umklapp processes turn on

Thermal conductance : 3 method

[Lu et al. RevSciInst 72 2996 (2001)]

X=0 X=L

Resistive film [ R(T) ]1D conductor (section S)substrate

- V1(T) gives access to R(T) and R’(T)

- V3(T) carries thermal information :

4 point probe resistance

measurement : transducer is ac-

biased by a current I and V is measured

with a lock-in amplifier

= characteristic time for axial thermal processes

Limiting cases :

Application of 3 method

[Bourgeois et al. JAP 101 16104 (2007)]

• T>1.3K K(T) = 2,6.10-11 T3 W/K With fitting param = mfp set to 620 nm : scattering by specular reflexions on surfaces

• Low T deviation : increased mfp due to dom

(T)> roughness

Roughness effect : experimental study of conductors with a modulated width

[see Cleland et al. PRB 64 172301 (2001) for predictions]

[see Jean-Savin Heron’s poster for latest measurements]

dom(T) ~

Quantized Thermal Conductance

[Rego et al. PRL 81 232 (1998)]

Landauer formalism ; heat flow between two phonon reservoirs :

LTL

RTR

Q

)())(()(20

kkvkdk

Q LR

v(k)=d/dk is canceled by the 1D DOS= dk/d

= modesv = group velocity = transmission probability

iBose distribution of phonons

in reservoir i at Ti

[Maynard PRB 32 5440 (1985)] disordered systemsprediction of universal regime of phonon thermal conductance

Now 2 hypotheses :

1) Adibaticity of contacts :

2) Only acoustic phonons contribute to thermal transport at low temperature :

In this limit, the conductance of one 1D ballistic channelhas the upper bound :

1)(),,(

0)0( k

h

Tk

TT

Qg B

RL 3

22

0

-Depends only on T and fundamental csts

-Independent of material and of disorder

-And also independent of the statistics of heat carriers ! Universal Thermal Conductance Quantum

(Another derivation [Blencowe et al. PRB 59 4992 (1998)] is based on quantization of classical mechanics describing the lattice)

g0 ~ 1 pW/K x T

)()(20

LR

dQ

Measurement of g0[Schwab et al. Nature 404 974 (2000)]

- SiN suspended membrane (60nm thick)

- 2 Cr/Au transducers- Noise thermometry

- Adiabaticity achieved through catenoidal contacts (cf Rego PRL 1998)

4 modes per conductor (1 longitudinal, 2 transverse, 1 torsional)4 conductors

A plateau at 16g0 is expected at low T

Limits of this (beautiful) experiment :- never reproduced- parasitic thermal conductance of superconducting Nb leads : unclear that it can be neglected…

Why are there no conductance steps ?

Quantization of electronic transport : sharp steps each time a conductance channel opens up

Quantization of thermal transport : we observe only a plateau at low T…

Electron case :

- states are full or empty : discontinuous steps characterize change of occupation

- F tuned by gate voltage Width of thermal broadening tuned by T : two independent parameters

Phonon case :

- occupation tuned by T : when T increases more states are occupied

- Range of effective modes and thermal broadening are both tuned by T : the width of the distribution masks the quantum signature of transport !

Low temperature Specific heat (LTSH)

Qintroduced

Tmeasured

Isolated system

• Great deal of infos about lattice and electronic properties (ex : Einstein’s model invalidated in 1911 leading to Debye ‘s calculation in 1912)

• Useful for studying every phase transition (e.g. magnetic, superconducting, structural)

3TTC Linked to Debye temperature

Linked to Density of States N(0)

Adiabatic method

•In both cases : C = Csyst + Caddenda

need for high resolution C/Cneed for highly sensitive

thermometry

LTSH techniques for small systems

• Adiabatic method : impossible to isolate system from thermal bath !

• Two methods adapted to T<1K and small systems :

- Relaxation method (time constant method)- ac method

Relaxation Method[Bachmann et al. RevSciInst 43 205 (1972)]

•Heating power P0

Sample heated at T0 + T •Heater turned of :

)/exp( 10 tTTT

= relaxation time

= C/K = C(T/P0)

Advantages :- accuracy ~ 1%- easy to average numerous decays- can be used with sample of poor thermal conductivity

Drawbacks :

- small C : need for fast electronics

- difficulty to determine accurately

ac calorimetry method

Oscillating power P0 injected at frequency f

Oscillations of temperature Tac

at same frequency f

[ F. Sullivan and G. Seidel, Phys. Rev. 679 173 (1968) ]

sb

acKKC

PT

3/2/11 22

221

2

0

1 = relaxation time to the bath2 = internal diffusion timeKb = thermal conductance to the bathKs = internal thermal conductance

C = P0/(2fTac)

Simplifications :

- Structuration of calorimeter : Kb << Ks - Choice of frequency (experimental) :

Conditions of Quasi-adiabaticity :1

21

1 f

Advantages :

- detect very small changes of C- stationary method; averaging

Drawbacks :

- accuracy ~ 5%- restriction of frequencies- high internal heat conduction required

sb

acKKC

PT

3/2/11 22

221

2

0

Recent achievementsSource Metho

dEnvironne

mentCaddenda

Best resolution/system size

[Denlinger et al. RevSciInst 65 946 (1994)]

relaxation

T=1.5–800 K

2 nJ/K at 4K(180nm SiN)

thin films of a few µg

[Riou et al. RevSciInst 68 1501 (1997)]

ac method

T=40-160K 1.5 µK/K at 100 K(polymer mb)

C/C = 10-4

(bulk samples ~10µg)

[Fominaya et al. RevSciInst 68 4191 (1997)]

ac method

T=1.5-20K 0.5 nJ/K at 1.5 K(2-10 µm Si)

C/C = 10-4

(samples~1µg)

[Zink et al. RevSciInst 73 1841 (2001)]

relaxation

T=2–300 KH= 0-8 T

1 nJ/K at 2K(180nm SiN)

thin films of a few µg

[Bourgeois et al. PRL 94 57007 (2005)]

ac method

T=0.5-15KH= 0-1 T

50 pJ/K at 0.5K

C/C = 5.10-

5

(samples~50 ng)

[Fon et al. Nanolett 5 1968 (2005)]

relaxation

T=0.5-8K 0.4 fJ/K at 0.6K

C/C = 10-4 at 2 K

[Bourgeois et al. PRL 94 57007 (2005)]

* Suspended Silicium membrane (5-10 µm thick)

• assembly of ~106 non interacting objects

•addenda = 50 pJ/K at 0.5 K

•ac method

* Copper heater and NbN thermometer (metal-insulator transition at tunable T)

•Best Resolution C/C=5x10-5 at O.5 K•sensitivity ~ 500 kB/object

4 mm

[Fon et al. Nanolett 5 1968 (2005)]

•Suspended SIN (120 nm thick)

•Single object

•addenda = 0.4 fJ/K at 0.6 K

•relaxation method

* Au heater and AuGe thermometer (resistive)

•Best resolution C/C=1x10-4 at 2K•sensitivity = 36000 kB/object

Thermal signature of Little-Parks effect

0-periodic Modulation of phase

diagram :

first free-contact measure0-periodic

modulation of the height of the C jump

at the transition

In a nanostructure, one cannot speak of specific heat, extensivity is lost

[F.R. Ong et al. PRB 74 140503(R) (2006) ]

1 µm

Vortex matter in superconducting disks

Modulation by external magnetic field H of Tc and of

C :

-more pronounced than in the ring geometry

-no periodicity ! (fluxoid is quantized in a non-rigid contour)

D = 2.10 µmThickness = 160 nm

Mass ~ 1.5 pg

Giant vortex states : (r,f(r)exp(2L) vorticity L = number of vorticies threading a single disk

L=0 L=1 L=2

L=3

L=4

L=5

L=6

•phase transitions between successive giant vortex states•strong hysteresis and metastability•Hn

up = penetration field of the nth vortex Hn

dwn = expulsion field of the nth vortex

Vortex matter in superconducting disks

•good agreement and complementary to [Baelus et al., PRB 58 140502] near

Tc

•different behaviors are expected between FC and zero field cooled (ZFC) scans of CH(T)

Vortex matter in superconducting disks

H=3.8 mT

Summary• Theoritical descriptions of thermodynamics of small

systems do exist– their experimental demonstration is still challenging– only non-extensivity has been demonstrated

(modulation of heat capacity by external parameter, geometry dependence)

• Thermal conductance of 1D conductors : – CNT’s subject to large uncertainties– quantum of thermal conductance : still has to be

demonstrated– better knowledge needed to improve heat capacity

nanosensors

• Heat capacity sensors : – towards the measurement of a single nano-object– behaviour at low T (<100 mK) is problematic (e-ph coupling,

internal conduction…) : better knowledge through experiments !