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 sample’s 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

5. 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 ?

6. 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 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) …….. n Inan ??? Temperature at nanoscale • [Hartmann et al. PRL 93 80402 (2004)]

7. [Hartmann et al. PRL 93 80402 (2004)] • [Hartmann et al. EPL 65 613 (2004)] RESULTS : • Inter-group interaction Ina • 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,j+1 = harmonic potentialnmin = constant for T >qD a(T/qD)3 for T <qD nmindepends on T(quantum effect) lmin = nmin a0 Carbon : lmin = 10 µm at 300K Silicium : lmin = 10 cm at 1K !!! (1D chain) (~100nm for 3D)

8. Hill’s nanothermodynamics • 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…) • 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…)

9. [Hill NanoLett. 1 273 (2001)] 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= systemcontaining Nequivalent and non-interacting small systems Sis 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, soE= 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 ! )

11. [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) • [Hill & Chamberlain NanoLett. 2 609 (2002)] 1/N for macrosystem 1 for small system

12. 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)

13. [Rajagopal, Pande & Abe, Proceedings of Indo-US Workshop (2004)] • c²-distribution ofb=1/kT ; width = q-1 • theory of large deviations Tsallis Entropy (pi= microstate probability) If q=1 (no temerature fluctuations) : one recovers Gibbs entropy

14. 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)]

15. 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: lT=0.1 µm • 100 mK: lT=1 µm • Bulk Diamond has the higher k reported ; what about carbon nanotubes ? • Analogy with Landauer description of transport : one thermal conductance quantum per channel

16. Thermal Conductance of CNTs CNTs vs Silicium nanowires • SWNT : d~1nm real 1D behavior • C-C = strongest chemical bond in nature (Diamond : k=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 (k~6600 W/mK) [Berber et al. PRL 84 4613 (2000)] Possible Waveguide for heat transfer ??

17. Thermal Conductance of CNTs • [Hone et al. PRB 59 R2514 (1999)] • Macroscopic Bundles of SWCNTs (d~1.4 nm) K(T) measured by a comparative method Measure of selec(T)(non metallic for T<150K) Room T : ksingleCNT = 1750-5800W/m.K • Wiedman Franz ratio : • k/(selec T) > 100 L0 • Transport is dominated by phonons at low T • Low T : kaT for T<30K Energy-independent mean free path ~ 0.5-1.5 µm , due to surface scattering

18. 10 µm • [Pop et al. NanoLett 6 96 (2005)] Single SWCNT : k~3500 W/m.K • [Kim et al. PRL 87 215502 (2001)] • First measure of kof a single MWCNT • (d~14 nm, L~2.5µm) • Suspended SiN device ; T = 8-370 K • Room T • k > 3000 W/m.K ; mfp ~ 500 nm • T > 320 K : Umklapp phonon scattering • T<320 K : nearly ballistic transport Ballistic or diffusive transport ? remains unclear !

19. Need for separating e- and phonons : • n+GaAs/iGaAs : heterostructure with separated transducers and conductor • still an electronic pathway ! • [Tighe et al. APL 70 2687 (1997)] thermometer conductors 200nm*300nm cavity heater bath 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

20. Isolation of phonon contribution[Fon et al. PRB 66 45302 (2002)] • Better understanding of phonon scattering mechanisms : • kbeam<< 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)

21. Resistive film [ R(T) ] 1D conductor (section S) substrate X=0 X=L 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

22. Application of 3w method • 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 toldom (T)> 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 Heron’s poster for latest measurements]

23. a = modes • v = group velocity • = transmission probability hi= Bose distribution of phonons in reservoir i at Ti Quantized Thermal Conductance [Maynard PRB 32 5440 (1985)] disordered systems prediction of universal regime of phonon thermal conductance [Rego et al. PRL 81 232 (1998)] Landauer formalism ; heat flow between two phonon reservoirs : L TL R TR va(k)=dwa/dk is canceled by the 1D DOS= dk/dwa

24. 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 • 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)

25. Measurementof 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…

26. 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

27. Linked to Debye temperature Linked to Density of States N(0) • Great deal of infos about lattice and electronic properties • (ex : Einstein’s model invalidated in 1911 leading to Debye ‘scalculation in 1912) • Useful for studying every phase transition (e.g. magnetic, superconducting, structural) Low temperature Specific heat (LTSH) Isolated system dQ introduced dT measured Adiabatic method

28. 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 • In both cases : C = Csyst + Caddenda • need for high resolution DC/C • need for highly sensitive thermometry

29. 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

30. ac calorimetry method [ F. Sullivan and G. Seidel,Phys. Rev.679173 (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

31. Simplifications : • Structuration of calorimeter : Kb << 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

32. Recent achievements

33. 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 DC/C=1x10-4 at 2K • sensitivity = 36000 kB/object • [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/C=5x10-5 at O.5 K • sensitivity ~ 500 kB/object

34. 1 µm In a nanostructure, one cannot speak of specific heat, extensivity is lost 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

35. L=0 L=1 L=2 L=3 L=4 L=5 L=6 D = 2.10 µm Thickness = 160 nm Mass ~ 1.5 pg 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

36. 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

37. H=3.8 mT 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)

38. 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 !