thermodynamics and thermal measurements at the nanoscale florian ong, olivier bourgeois institut...
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Thermodynamics and thermal measurements
at the nanoscale
Florian ONG, Olivier BOURGEOIS
Institut Néel, Grenoble
GDR Physique Mésoscopique, Aussois Mars 2007
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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
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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
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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
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)]
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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)]
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–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)
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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)]
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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)]
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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)
²
²²
X
XX 1/N for macrosystem
1 for small system
[Hill NanoLett. 1 273 (2001)]
[Hill & Chamberlain NanoLett. 2 609 (2002)]
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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)
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• 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)]
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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)]
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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
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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 ??
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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
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[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 !
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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
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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
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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 :
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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) ~
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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
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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
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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…
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 !
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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
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•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
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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
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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
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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
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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
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[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
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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
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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
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•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
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•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
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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 : – 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 !