Thermodynamics and thermal measurements at the nanoscale

1
Thermodynamics and thermal measurements at the
nanoscale

• Florian ONG, Olivier BOURGEOIS
• Institut Néel, Grenoble
• GDR Physique Mésoscopique, Aussois Mars 2007

2
Overview

• How is macroscopic thermodynamical description
  affected as one reduces system sizes ?
• a thermodynamical limit is not reached
• a importance of fluctuations
• High Surface/Volume ratio
• a Surface Energy term in Uint
• a 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 samples
  dimensions ?

3
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

4
Outline

• Temperature at the nanoscale
• Some thermodynamic descriptions of small systems
• Thermal transport in nanoconductors
• Specific heat nanocalorimetry

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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 ?

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Temperature at nanoscale
Hartmann et al. PRL 93 80402 (2004)

• MODEL
• 1D macroscopic chain of N identical particles at
  temperature T
• (ie described by a canonical state at T)
• First neighbour interaction Vj,j1
• Division into NG groups of n particles

• QUESTIONS
• 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)

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

• RESULTS
• Inter-group interaction In a
• Condition on n so as a group can be described by
  thermodynamics
• If Tloc exists, Tloc T

Width of the energy distribution of the total
system
EXAMPLE Vj,j1 harmonic potential
nmin constant for T gt qD a (T/ qD )3
for T lt qD 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)
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Hills nanothermodynamics

• Motivations
• Early 1960s study of macromolecular solutions
• 2000s growing interest due to nanofabrication
  progress
• Growing interest in completely open systems
  (µ,p,T)
• (open aggregates in biology, metastable
  droplets in vapor)

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

• 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)

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Hill NanoLett. 1 273 (2001)

• Hills 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

10
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
  ! )

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• 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)

Hill NanoLett. 1 273 (2001)
Hill Chamberlain NanoLett. 2 609 (2002)
1/N for macrosystem 1 for small system
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Abes 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)

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

• c²-distribution of b1/kT width q-1
• theory of large deviations

Tsallis Entropy (pi microstate probability)
If q1 (no temerature fluctuations) one
recovers Gibbs entropy
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Thermodynamics with Tsallis entropy
Beck EPL 57 329 (2002)

• 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)

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Thermal transport in 1D conductors

• -- Study of thermal conductivity k in
  monocrystaline conductors whose size is smaller
  than the dominant phonon wavelength.
• For silicium qD(Si)625K
• 1K lT0.1 µm
• 100 mK lT1 µm
• Bulk Diamond has the higher k reported what
  about carbon nanotubes ?
• Analogy with Landauer description of transport
  one thermal conductance quantum per channel

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Thermal Conductance of CNTs

• CNTs vs Silicium nanowires
• SWNT d1nm real 1D behavior
• C-C strongest chemical bond in nature
• (Diamond k2300-3320 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 (k6600 W/mK)
• Berber et al. PRL 84 4613 (2000)
• Possible Waveguide for heat transfer ??

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Thermal Conductance of CNTs
Hone et al. PRB 59 R2514 (1999)

• Macroscopic Bundles of SWCNTs (d1.4 nm)
• K(T) measured by a comparative method
• Measure of selec(T) (non metallic for
  Tlt150K)
• Room T ksingleCNT 1750-5800 W/m.K

Wiedman Franz ratio k/(selec T) gt 100
L0 Transport is dominated by phonons at low T

• Low T
• ka T for Tlt30K
• Energy-independent mean free path
• 0.5-1.5 µm , due to surface scattering

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Kim et al. PRL 87 215502 (2001)
First measure of k of a single MWCNT (d14 nm,
L2.5µm) Suspended SiN device T 8-370 K
Room T k gt 3000 W/m.K mfp 500 nm T gt 320 K
Umklapp phonon scattering Tlt320 K nearly
ballistic transport
Ballistic or diffusive transport ? remains
unclear !
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Thermal conductance of crystaline nanostructures

• Conductive wires metals, nGaAs
  (electron heating technique, 1985-1995)
• poor e-ph scattering at low T
• e- short-circuit the thermal transport
• Phonon contribution hard to isolate

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

• Better understanding of phonon scattering
  mechanisms
• kbeamltlt kbulk reduction of mfp due to
• enhanced surface scattering
• reduction of group velocity
• reduction of DOS
• 4-10 K diffuse surface scattering
• ( ldo (4K) 10 nm 3D model )
• 20-40 K Umklapp processes turn on

Comparative measurement (4-40K)
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Thermal conductance 3w method
Lu et al. RevSciInst 72 2996 (2001)
4 point probe resistance measurement
transducer is ac-biased by a current I and V is
measured with a lock-in amplifier
- V1w(T) gives access to R(T) and R(T) - V3w(T)
carries thermal information
Limiting cases
g characteristic time for axial thermal
processes
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Application of 3w method

• Tgt1.3K
• 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 ldom (T)gt
  roughness

Bourgeois et al. JAP 101 16104 (2007)

• Roughness effect experimental study of
  conductors with a modulated width
• see Cleland et al. PRB 64 172301 (2001) for
  predictions

ldom(T) L
L
see Jean-Savin Herons poster for latest
measurements
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Quantized Thermal Conductance
Maynard PRB 32 5440 (1985) disordered
systems prediction of universal regime of
phonon thermal conductance
L TL

• Rego et al. PRL 81 232 (1998)
• Landauer formalism heat flow between two phonon
  reservoirs

R TR
va(k)dwa/dk is canceled by the 1D DOS dk/dwa
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• Now 2 hypotheses
• Adibaticity of contacts
• Only acoustic phonons contribute to thermal
  transport at low temperature
• In this limit, the conductance of one 1D
  ballistic channel
• has the upper bound

g0 1 pW/K x T
(Another derivation Blencowe et al. PRB 59 4992
(1998) is based on quantization of classical
mechanics describing the lattice)
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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
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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
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 !
Electron case - states are full or empty
discontinuous steps characterize change of
occupation - eF tuned by gate voltage Width of
thermal broadening tuned by T two independent
parameters
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Low temperature Specific heat (LTSH)
Isolated system
dQ introduced
dT measured
Adiabatic method
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LTSH techniques for small systems

• Adiabatic method impossible to isolate system
  from thermal bath !

• Two methods adapted to Tlt1K and small systems
• - Relaxation method (time constant method)
• - ac method

• In both cases C Csyst Caddenda
• need for high resolution DC/C
• need for highly sensitive thermometry

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

• Heating power P0
• Sample heated at T0 DT
• Heater turned of

t1 relaxation time t1 C/K C(DT/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 t1 accurately

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ac calorimetry method
F. Sullivan and G. Seidel, Phys. Rev. 679 173
(1968)
Oscillating power P0 injected at frequency f
Oscillations of temperature dTac at same
frequency f
t1 relaxation time to the bath t2 internal
diffusion time Kb thermal conductance to the
bath Ks internal thermal conductance
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• Simplifications
• Structuration of calorimeter Kb ltlt Ks
• Choice of frequency (experimental)
• Conditions of Quasi-adiabaticity

C P0/(2pfdTac)

• Drawbacks
• accuracy 5
• restriction of frequencies
• high internal heat conduction required

• Advantages
• - detect very small changes of C
• stationary method averaging

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Recent achievements
33

• 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 DC/C5x10-5 at O.5 K
• sensitivity 500 kB/object

• 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 DC/C1x10-4 at 2K
• sensitivity 36000 kB/object

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Thermal signature of Little-Parks effect
F.R. Ong et al. PRB 74 140503(R) (2006)
f0-periodic Modulation of phase diagram first
free-contact measure
f0-periodic modulation of the height of the C
jump at the transition
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Vortex matter in superconducting disks

• Modulation by external magnetic field H of Tc and
  of DC
• more pronounced than in the ring geometry
• no periodicity !
• (fluxoid is quantized in a non-rigid contour)

Giant vortex states Y(r,q)f(r)exp(2pLq)
vorticity L number of vorticies threading a
single disk
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Vortex matter in superconducting disks

• phase transitions between successive giant vortex
  states
• strong hysteresis and metastability
• Hnup penetration field of the nth vortex
• Hndwn expulsion field of the nth vortex

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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)

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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
• CNTs 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 (lt100 mK) is problematic (e-ph
  coupling, internal conduction) better
  knowledge through experiments !