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The travel of heat and charge g in solids K Kamran Behnia B h i Ecole Supérieure de Physique et de Chimie Industrielles Paris, FRANCE Contents I. Introduction • • • Kinetic picture of heat travel in metals and insulators Phonons and electrons Superconductors (conventional and unconventional) II Transport Equations II. • • • The three conductivity tensors The Wiedemann-Franz law The Bridgman relation III. Electronic correlations • • • T-square Resistivity Enhanced linear specific heat Seebeck coefficient • • • correlated electrons superconducting vortices short-lived Cooper pairs IV. Nernst effect ff Kinetic theory y of gases g = 1/3 C v l mean-free-path Thermal conductivity Heat capacity velocity Jq (W cm-2) hot ∇T (K cm-1) cold = Jq /(-∇T) [WK-1cm-1] Heat conduction in insulators Phonondefect scattering Phononphonon h scattering Only phonons carry heat! Heat conduction in insulators Phonons are both carriers and scatterers As T increases, increases • More carriers ! • More scattering centers! Heat conduction in metals Electrondefect scattering Electronphonon h scattering Carriers:electrons & phonons Scatterers: phonons, deffects (and electrons) Electrons are dominant carriers of heat! The best conductor at room temperature! Graphene at room T Non-interacting phonons ! In the zero temperature limit • Mean-free-path p attains its maximum value and then: ph ∝T3 (phonons ( h are b bosons)) ∝T ((electrons are fermions)) e In principle, one can separate the two contributions! Example Taillefer et al., 1997 Thermal conductivity of superconductors • Above Tc a superconductor is a metal (mobile electrons carry heat!) • Below Tc, mobile electrons condensate in a macroscopic quantum state: electronic heat carriers vanish! • A superconductor can be assimilated to a thermal insulator In conventional superconductors p • Electron thermal conductivity decreases exponentially ti ll • Phonon thermal conductivity increases due to a diminished electron scattering At finite temperature, the temperature dependence of thermal conductivityy in a superconductor p is complex p ! Example: niobium A phonon peak, whenever the phonon h meanfree-path can increase below Tc! Kes et al. J. Low Temp. Phys. (1974) normal superconducting e- component Unconventional superconductors • The order parameter of the unconventional superconductors is less symmetric than the Fermi surface of the mother metal. • Th The gap function f i can vanish i h along l particular orientations (nodes). • Nodal quasi-particles quasi particles can carry heat! Effect of an unconventional superconducting d ti ttransition iti on th thermall transport • The electronic thermal transport does NOT decrease exponentially • It can even increase below Tc (due to an increase in the electronic mean-free-path = 1/3 C v l Scattering g events are restricted in an unconventional superconductor s-wave d-wave Heat conduction in YBCO Aubin 1997 The increase in thermal conductivity below Tc is due to electrons! Enhanced electronic thermal conductivity in other unconventional superconductors CeCoIn5 (Seyfarth et al., 2008) Thermal transport in the zero-temperature limit • A finite linear term in thermal conductivity • A residual normal fluid of nodal quasi-particles = 00T +...+ + + bT3 Example: heavy-fermion superconductor CePt3Si Izawa et al.,, PRL ‘05 A superconductor with no inversion symmetry and yet with nodal quasiparticles! Universal thermal conductivity • In a d-wave superconductor, in a first approximation 0 is independent of impurity concentration Decrease the mean-free-path Impurities: Enhance the specific heat These two cancel out in a subset of unconventional superconductors superconductors. A TALE OF TWO VELOCITIES! ( (Durst , Lee ’99)) E it ti spectrum Excitation t in i the vicinity of a node: E(k) = (k2 + k2)1/2 = (vF2k12 + v22k22)1/2 Fermi velocity :vF = dk/dk1 Gap velocity: v2= dk/dk2 00/T =(nkB2/3ħ) (vF / v2) Experimental observation of universal thermal conductivity A 30-fold 30 f ld d decrease iin mean-free-path f th iin Z Zn-doped d d YBCO leaves 0 unchanged (Taillefer et al., PRL 1997) Universal conductivity in other unconventional superconductors 0 independent of impurity concentration checked in: • • Bi2Sr2CaCu2O8 (Nakamae et al. , 2001) Sr2RuO4 ((Suzuki et al. , 2004)) 0 of the right order of magnitude found in: • • • • κ-(BEDT-TTF)2Cu(NCS)2 (Belin et al., al 1998) CePt3Si (Izawa et al., 2005) URu2Si2 (Kasahara et al., 2007) CeCoIn5 (Seyfarth et al., 2008) Mesauring thermal conductivity Temperature captors: Resistive thermometers or thermocouples! Part II- transport equations Heat and charge current in a solid Je E J Q T J e E T J Q E T T T Kelvin relation, (1860) Onsager relation (1930) Four vectors Je : charge current density JQ : heat h t currentt density d it E : electric field DT : thermal gradient Three tensors electric conductivity thermal conductivity thermoelectric conductivity Definition of thermoelectric coefficients • In p presence of a thermal g gradient,, electrons produce an electric field. • Seebeck and Nernst effect refer to the longitudinal and the transverse components of this field. Ex B hot Ey JQ cold ld T E x S xT N S xy Ey xT [ Ey ] BzxT Link to the Peltier tensor: Seebeck coefficient Nernst coefficient S xx xx N/B xy xx xx 2 2 xx xy Experimentally, p y what is measured is S and xy Seebeck and Peltier coefficients Peltier effect: effect A thermal gradient created by b an electric ccurrent rrent JQ Je Je JQ Seebeck and Peltier coefficients Seebeck effect: effect An electric field created b by a thermal ccurrent rrent Ex S xT E T The Kelvin relation : ST Nernst and Ettingshausen coefficients B N Nernst t coefficient ffi i t S xy Ey xT T Ey Nernst and Ettingshausen g coefficients B Ettingshausen Etti h coefficient ffi i t = yT/Je T Je Nernst and Ettingshausen g coefficients Sxy= / T Bridgman g relationship, p Thermal conductivity Ettingshausen Etti h coefficient ffi i t [ = yT/Je] B T Je Nernst and Ettingshausen g coefficients Sxy= / T Bridgman g relationship, p Sommerfeld and Frank, Rev. Mod. Phys., 1931 B Thermal conductivity Ettingshausen Etti h coefficient ffi i t [ = yT/Je] T Qy = y T = Je Je Qy Nernst and Ettingshausen g coefficients Sxy= / T Bridgman g relationship, p Sommerfeld and Frank, Rev. Mod. Phys., 1931 Thermal conductivity B Ettingshausen Etti h coefficient ffi i t [ = yT/Je] Qy = y T = Je T Je Qy Energy cost : (Qy /T) y T = Je y T / T Can b C be provided id d by b an Electric El t i field fi ld Ex JeEx = k Je y T / T Ex / y T = / T The Semi-classic Semi classic picture J e E T J Q T E T k 2 3 T e 2 B k 2 2 3 e B 2 T F The Wiedemann-Franz law! The Wiedemann Wiedemann-Franz Franz law • Strictly valid in absence of inelastic scattering tt i • A robust signature of a Fermi liquid (expected to break down in case of spin-charge separation) • Even in Fermi liquids a finite downward d i ti is deviation i expected t d in i presence off inelastic scattering Carriers of charge are carriers of heat The ratio of two conductivities is li k d to linked t the th ratio ti off the th ttwo quanta t of charge and entropy! 2 k 2B T 3 e2 Can be derived using the kinetic equation! C 2 3 k B2TN ( F ) 1 2 k 2B Cel vF T 2 3 3 e 1 3 e 2 N ( F )vF e The effect of an electric field and a thermal gradient on a Fermi surface T. Ziman El Electrons andd phonons h The electric field displaces the Fermi surface, but a thermal gradient makes it fuzzier in the hotter end! The Wiedemann-Franz law is recovered only at T 0 and T= d att high hi h temperatures! t t ! At high temperatures vertical scattering beomes marginal because of the thermal broadening of the Fermi surface III Correlated III. C l d electrons l Landau theory of Fermi liquids • Why band theory is successful in spite of its neglect of electronic interactions? • Interaction electrons can be mapped to noninteracting quasi-particles quasi particles with the same spin and charge • Normalized mass of quasi-particles contains all information on the magnitude g of the interactions Heavy Fermi liquids • Intermetalics containing a 4f (Ce, Yb…) or 5f (U, Pu,…) element. • High temperature: A Fermi sea and localised spins • Low L t temperature: t H Heavy quasi-particles i ti l as a result of hybridization of f electrons and conduction electtrons Heavy Fermi liquids • Enhanced specific heat • Enhanced Pauli Susceptibility p y • Enhanced T2-resistivity 2 3 k N ( F ) 2 B N ( F ) 2 B =0 +A A T2 A N ( F ) 2 Fermi liquid ratios The Wilson ratio 2 k 2B RW 3 B2 Fermi liquid ratios The Kadowaki Kadowaki-Woods Woods ratio KW A 2 Thermoelectric response hot cold hot cold 1.0 O Occupation F 0.5 A finite thermal gradient! A finite electric field! But no charge current! Response to a gradient in chemical potential! 0.0 Energy Thermoelectric response hot cold Carriers with a charge, q, and an entropy, Sex will suffer two forces: •Electric force: F= E q •Thermal force: F= Sex ∇T hot cold 1.0 F Thermopower measures entropy per [charged] carrier. S = (kB T/ EF)/ e Occupatiion S = E/ ∇T = Sex /q 0.5 0.0 Energy Seebeck coefficient of the free electron gas In the Boltzmann picture thermopower is linked to electric conductivity: This yields: t transport t Th Thermodynamic d i For a free electron gas, with =0, this becomes: Thermopower and specific heat In a free electron gas (with =0): Thermopower is a measure of specific heat per carrier The dimensionless ratio: is equal to –1 (+1) for free electrons (holes) [if one assumes a constant mean-free-path , then =1/2 and q=2/3] Thermoelectricity in real metals • Even in simplest metals, the free-electron-gas free electron gas picture does not work at finite temperature ! Structure in the thermopower of AlKali metals (MacDonald 1961) ! Phonon Dragg h t hot cold ld Flow of heat-carrying phonons •If electrons and phonons exchange energy then electrons would be dragged by phonons Phonon dragg thermopower Carrier density Sg L tti specific Lattice ifi heat h t CL Ne e- - ph coupling (0<<1) Order of magnitude of phonon dragg (T/D)3 S g / S diff T/TF CL Ce Frequency of ph-e- scattering events e- per atom MacDonald 1961 • • Expected to vanish in the zero-temperature limit Becomes negligible when Ce>> CL Heavy electrons in the T=0 limit Two-decades-old data replotted! Extrapolated to T=0, data yields a q close to unity! Another plot linking two distinct signatures of electron correlation KB, D. Jaccard, J Flouquet J. Fl t 2004 Data since 2004 S/T q UPt3 430 2.5 0.6 NpPd5Al2 200 -1.3 -0.65 CeNi2Al3 30 0.9 PrFe4As12 340 -1 0.3 YbRh2Si2 750 -6.7 -0.8 FeTe 34 -0.3 -0.9 MgB2 3 0.04 -1.3 1.3 SrRu03 30 0.24 0.8 SrRh2O4 10 0.22 2.2 0.27 In semi-metals q is large! •URu URu2Si2 (q=-11) (q= 11) •PrFe4P12 (q=-58) •PrRu4P12 (q=-43) •CeNiSn (q=107) ( 107) •Bi0.96Sb0.04 (q=10 ) 4 In these systems the FS occupies a small fraction (~1/2q) of the BZ. Theory on correlation between S and Miyake & Kohno, JPSJ (2005) In both unitary and Born limits, q ~ 1 Zlatic, Monnier, Freericks & Becker, PRB 2007 (Single-impurity Anderson model) Paul & Kotliar, PRB 2001 Near a QCP both expected to diverge logarithmically Haule and Kotliar, CorrelatedThermoelectricity workshop (2008) DMFT Measuring the Fermi energy of an electronic system • Resistivity: • Specfific heat • Thermopower h 1 A 2 a 2 e F 2 3 2 B k n 1 F S kB 1 T 3 e F 2 Electronic correlations… correlations • Do not enhance or diminish the absolute value of or components in the T T=00 limit • But, they do enhance the magnitude of , the thermoelectric response IV Nernst IV. N t effect ff t Thermoelectric coefficients • • In presence of a thermal gradient, electrons produce an electric field. Seebeck and Nernst effect refer to the longitudinal and the transverse components of this field. Ex B hot Ey JQ cold ld T E x S xT N e y S xy Ey xT [ Ey B z xT ] Nernst effect in a single-band metal Absence of charge current leads to a counterflow of hot and cold electrons: JQ 0 ; Je= 0 ; Ey= 0 B e- JQ e- T In an ideally simple metal metal, the Nernst effect vanishes! (« Sondheimer cancellation », 1948) Ey Nernst coefficient in remarkable metals! A large diffusive component in the zero-temperature limit! Close-up on Sondheimer cancellation J e E T JQ T E T Je=0 0 Boltzmann picture: If the Hall angle angle, H, does not depend on the position of the Fermi level level, then the Nernst signal vanishes! Roughly, the Nernst coefficient tracks cF… N ~ 2/3 k2BT/e c / F Recipe for a large diffusive Nernst response: •High mobility •Small Fermi energy gy •Ambipolarity In graphite at 20 K and 1T ~3mcm Sxy ~3mVK‐1,, Sxy2/ = 3000 W K‐2cm‐1 Nernst effect as a probe of quantum criticality The case of CeCoIn5 Izawa et al. 2007 Paglione et al. ,2003, Bianchi et al. 2003 logarithmic color plot of /T Nernst effect directly reveals the quantum critical point! Nernst effect in the vortex state B Ey T A superconducting vortex is: • A quantum of magnetic flux • An entropy py reservoir • A topological defect • Thermal force on the vortex : • • The vortex moves Th movementt leads The l d tto a transverse voltage: Ey=vx Bz F=-S T (S : vortex entropy) Nernst effect in optimally-doped YBCO The Nernst coefficient is finite only in the vortex liquid state! (Ri, Huebner et al. 1994) Vortex-like excitations in the normal state of the underdoped cuprates? Ong et collaborators 2006 A finite Nernst signal in a wide temperature range above Tc A small Fermi surface Highly mobile electrons The smallll electron Th l t pocket generates a sizeable negative Nernst signal! Within a factor of 2 of what h t is i expected t d for f the normal state! Nernst effect due to Gaussian fluctuations of the amplitude of the superconducting order parameter (Usshishkin, Sondhi & Huse, 2002) In 2D: Magnetic length Quantum of thermo-electric conductance (21 nA/K) In two dimensions, the coherence length is the unique p parameter! The coherence length g above Tc How do the fluctuating Cooper pairs generate a Nernst signal? cold hot •Above Tc, the lifetime of the Cooper pairs d decrease with ith iincreasing i temperature cold h t hot dT/dx Ey B •Therefore, those pairs which travel from the hot side along the cold side live longer! Superconductivity in Nb0.15Si 0.85 thin films 1500 d=125 A 1200 900 800 d=250 A R() Rsquare() 1200 600 d12.5 nm Tc=220 mK mcm 400 d=500 A 300 Nb0.15Si0.85 d=1000 A 0 00 0.0 05 0.5 10 1.0 15 1.5 20 2.0 T(K) 25 2.5 30 3.0 35 3.5 40 4.0 0 0 50 100 150 T(K) 200 The normal state is a simple dirty metal: le~a~ 1/kF ! 250 300 Nernst effect across the resistive transition • A large vortex signal below Tc Pourret 2006 •A long g tail above Tc A signal distinct from the vortex signal Deep into the normal state! Pourret 2006 Can this signal come from normal electrons? The Nernst signal of the normal electrons is negligible! ~10 10 -55 T F ~104 K Even at 6K : The expected normal state contribution is three orders of magnitude smaller than /T ! Comparison with theory Experiment: Rsquare SC XY B Theory: 0.01 xy/B(A A/TK) 1E-3 xy/B (Sample1) (B=0) 1E-4 xy/B (Sample2) (B=0) xy/B (USH) 1E-5 xy B USH k Be 2 2 .ξ 2 d 6π 0.02 0.1 1 -2 ln(T/TC) d Satisfactory agreement close to Tc 4 Pourret 2008 The g ghost critical field Sample 2 Contour plot of N= -Ey /(dT/dx) A unique correlation length Contour plot of the Nernst coefficient =N/B N/B Pourret 2008 1 3 v F d 0.36 2 k BTc