Thermoelectricité comme sonde de l orgnisation électronique dans les solides

Transcription

1 Thermoelectricité comme sonde de l orgnisation électronique dans les solides Kamran Behnia Ecole Supérieure de Physique et de Chimie Industrielles

2 Collaborators, ESPCI Benoit Fauqué CNRS researcher Zengwei Zhu Postdoc Xiao Lin, PhD Aurélie Callaudin PhD German Bridoux postdoc Cyril Proust and his group, LNCMI- Toulouse and Grenoble Jacques Flouquet and his group, SPSMS-CEA, Grenoble

3 Outline I. Introduction : electric vs. thermoelectric conductance II. From Kondo effect to heavy-electron metals III. The Nernst effect IV. Quantum limit and dilute metals J. M. Ziman, Electrons and phonons D. K. C. Macdonald, Thermoelectricity

4 Flow of heat and charge J J e Q E T T E ' T Electric conductivity Thermal conductivity Thermo-electric conductivity In general,, and are tensors! Off-diagonal components emerge in presence of magnetic field: Hall effect Nernst effect Righi-Leduc effect

5 Experimental access to thermoelectric coefficients Impose a thermal gradient! Impede charge flow (J e =0 )! E T B E x Seebeck Nernst S N S xy E x x T E x y T hot T E y J Q [ Ey ] Bz xt cold

6 The linearized Boltzmann equation 1 1 )/ ( 0 T k B e f ).( ) ( E f T T f v f r f k k k Scattering time velocity

7 Electric and thermoelectric conductivity Always positive Dominant contributors: Can be positive or negative Dominant contributors: < and >

8 Electrons which carry charge and those which carry heat! Quanta of conductance electric thermoelectric thermal Transport distribution function

9 Transport distribution function (Mahan & Sofo, 1996) It depends on both the electronic structure and scattering mechanism. In the material. In a bulk solid, it replaces the transmission probability introduced by Landauer in the case of 1D systems! ) ( ) ( ) ( ) ( ) ( ) ( ) ( k v k N h N h

10 Electric and thermoelectric conductivity Parabolic dispersion with constant mean-free-path d f h e ) ( ) ( High-energy electrons are faster and denser d T k f h ek B B ) ( ) ( ) (

11 The free electron gas k db m k * B T S k e B < k < k db F > > 3 k e B T T F de Broglie wavelength measures the thermal fuzziness of the Fermi surface! The Seebeck coefficient becomes scattering-independent, if the mean-free-path is independent of energy.

12 In a degenerate Fermi liquid (k B T<< F ) The Mott formula: T e k B 3 ln 3 T e k S B Thermoelectricity probes electronic states near,but not exactly at, the chemical potential,! 3 e k T B The Wiedemann-Franz law:

13 The Kondo effect Mobile e k k k Localised moment 1936: Mysterious upturn in resistivity of gold seen by de Haas 1950s: Role of MAGNETIC impurities pinned down 1960s: Giant Seebeck effect in thermopower in impure noble metals 1964: Kondo offers a solution (and defines a new problem)! The isolated magnetic moment couples to the Fermi sea Incoming electrons are spin-flipped and visit states of opposite spin!

14 The Kondo effect 1ppb to 10ppb! Resistivity Seebeck coefficient Iron impurities in gold, Kopp 1975

15 Magnitude of Seebeck peak in Au-Fe at 4. K Still moves! The system becomes a spin glass! The instrinsic diffusive Seebeck coefficient of gold is only AuFe (0.0.3 %): the most sensitive thermocouple at low T! 14V/K peak at ppm!

16 Why the effect is so drastic? A resonance peak in one side of the dividing line!

17 Kondo lattices Intermetallics with atoms hosting f electrons (CeAl 3, YbAl 3, UPt 3 ) At high temperature, isolated magnetic moments in a Fermi sea At low temperature, no dichotomy between isolated spins and mobile charges. Both merge in to a sea of heavy electrons! C e n kbtn( F ) kbt kbt 3 3 T F * 1/3 m n An electronic specific heat several orders of magnitude larger than copper!

18 Large Seebeck coefficient in heavy-electron metals Jaccard & Sierro 198

19 Thermopower and specific heat In a free electron gas : Thermopower is specific heat per carrier [Macdonald 196, Ziman, 1961, ] The dimensionless ratio: q en Av S C is equal to 1 (+1) for free electrons (holes).

20 In many heavy-fermion metals q is close to unity! Behnia, Jaccard & Flouquet, JPCM 004 See also Sakurai & Isikawa JSPJ 005 Theory: Miyake & Kohno JPSJ 005 Zlatic et al., PRB 007

21 Enhanced specific heat Heavy Fermi liquids 3 k N B ( F ) Enhanced Pauli Susceptibility ( B N F ) Enhanced inelastic resistivity 0 AT 1 A N( F ) Enhanced Seebeck coefficient S T 3 k B N( n F )

22 Fermi liquid ratios The Kadowaki-Woods ratio The Wilson ratio R W KW A k B 3 B

23 Current research on thermoelectricity of heavy electrons Main themes Quantum criticality Metamagnetic transitions Hidden electronic orders Fermi surface reconstruction Ferromagnetic superconductors Kohler et al., 008 Dresden Principal actors Grenoble (Flouquet ) Dresden (Steglich ) Tokyo (Izawa) Ames (Canfield) Geneva (Jaccard)

24 Ferromagnetic superconductors UGe URhGe UCoGe Saxena S.S. et al. Nature 406, 587 (000) Aoki D. et al. Nature 413, 613 (001) Huy N.T. et al. PRL 99, (007) T c = 5 K, T SC =0.75 K T c = 9.5 K, T sc = 0.5 K T c =.5 K, T sc = 0.6 K For a review, see: D. Aoki & J. Flouquet, JPSJ 01 : Reentrant superconductivity: Ferromagnetic fluctuations generate a SC dome!

25 S-shape H c (T) in UCoGe Both T sc and S/T peak at H *! Logarithmic divergence of S/T at H *! Theory for QCP Paul, Kotliar, 03 Pourret et al., 013

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27 The effective Fermi temperature Resistivity (WcmK - ) A h e a 1 T F Specfic heat (JK -1 mol -1 ) 3 k B n 1 T F Material-dependent length scale Seebeck (VK -1 ) S kb 1 T 3 e T F Carrier density All would diverge if T F vanishes! Seebeck coefficient is the only intensive coefficient (no geometric factor)!

28 T T E J T E J Q e The Nernst effect E T 0 T T E E T T E E y yy x xy y yy x xy y xy x xx y xy x xx 0 T y T E x xy xx xx xy y yy xx xy ] [ ] [ xy xy xx xx xy yy xx x y T E N is now a tensor

29 Nernst effect and Hall mobility N xy xx xx xx xy xy H xy xx H xy xx xx xx xy xy Extended Mott formula: xx k B T k B T xy xx xy 3 e 3 e Nernst effect measures the change in the Hall mobility induced by shifting the Fermi level!

30 The Hall mobility F B H B e k T eb k T F 3 3 F H H k e m e * ) H (

31 Quantum oscillations and the quantum limit

32 Landau quantization E k E m * ( k x k y ) Electron energy spectrum is no more a continuum.

33 Fermi surface truncated by Landau tubes Shoenberg 1984

34

35 The magnetic field required to attain the quantum limit When the magnetic length becomes shorter than the Fermi wave length! l F 1/3 n l B 1/ B Particle (radius of curved trajectory) Wave (wave-length) The lower the carrier density, the lower B QL! Copper bismuth Carrier density (cm -3 ) l F (nm) B QL (T)

36 Giant quantum oscillations of the Nernst response The last expected 9 T B// trigonal T= 30 mk

37 S xx,s xy (VK -1 ) S xy (VK -1 ) Giant Nernst quantum oscillations in graphite Zhu et al., Nature Physics K T=0.49 K 0 xx HOPG sample 1 8 K 4. K xx (mw cm) xy (mwcm) K xy HOPG sample 1.6 K 1.65 K T=0.55 K B(T) 0.98K 0.77 K 0.55 K 0.34 K 0.9 K Graphite B//c S xy S xx B(T)

38 Nernst quantum oscillations in Graphene Resistivity Hall effect Zuev et al. (Philip Kim s group) PRL 009 Magnetic field is kept constant and the gate volage is scanned. Seebeck effect Nernst effect

39 S xy (V/K) When a Landau level intersects the Fermi level, the Nernst response 100 peaks in graphite! 0.343K 0.549K 0.678K 0.771K 0.805K 0.853K 0.976K 1.651K vanishes in graphene! B -1 (T -1 )

40 S xy (µv/k) Experiment: graphene Theory :D (Girvin, Jonson PRB 1984) 30 Experiment:graphite Theory :3D (Bergman&Oganesyan PRL 010) 1 L T=0.34 K 0 L 3 L S 1 S 4 L 3 S 5 L 6 L 5 S 4 S 10 0,0 0,3 0,6 0,9 1, B -1 (T -1 )

41 Recent theory on Nernst quantum oscillations

42 Angle-resolved Nernst effect in bismuth Binary Nersnt peaks shift as the magnetic field tilts! Trigonal T= 0.49K T=1.5K Zhu et al., PNAS (01)

43 Angle-resolved Landau spectrum in bismuth (experiment and theory) Symbols: experimental data; Lines: theory Agreement is very good for holes, but only fair for electrons Zhu et al., PNAS (01)

44 The band picture of metals and insulators! Interesting thermoelectricity J.-P. Issi, Aust. J. Phys. (1979) Chemical potential

45 How does an isulator become a metal? Rosenbaum et al. PRL 1980 doping X. Blase et al., Nature Materials (009) phosphorus-doped silicon

46 Fermiology of doped semi-conductors Self-doped Bi Se 3 believed to be a bulk topological insulator with nontrivial surface states. SrTiO 3-d discovered in 1964 as the first case of a semiconducting superconductor Many others to be explored: PbTe, InSb, )

47 Which side of the metal-insulator transition? Toplogical? Sure! Insulator? Hum

48 The effective Bohr radius: Critical density for metal-insulator transition Edwards and Sienko, PRB 1978 a H r * m e The Mott criterion: Metallicity emerges when two length scales become comparable Metal a H n 1/3 0.6 Non-metal The interdopant distance: n 1/3

49 Quantum oscillations of the Nernst effect in Bi Se 3 n = cm -3 Fauqué et al., arxiv: n = cm -3 A sharp bulk Fermi surface down to cm -3

50 Two routes towards Fermi temperature Thermopower Quantum 19 cm -3 S KB 1 T 3 e T F Fauqué et al., PRB (013) k B T F k m F 17 cm -3 S/T= VK - F=170 T S/T= -6.1 VK - F=15 T m* =0.18m e m*=0.m e T F =900 K T F =87 K

51 An old superconducting dome! n- doped SrTiO 3

52 Superconductivity in bulk n-doped SrTiO 3 Lin et al., PRX 013 n-doping: removing oxygen or replacing Ti with Nb Carrier density window: between to cm -3

53 Emergence of Nernst oscillations with underdoping Lin et al., PRX 013

54 Emergence of Nernst oscillations with underdoping Lin et al., PRX 013

55 Emergence of Nernst oscillations with underdoping Lin et al., PRX 013

56 Emergence of Nernst oscillations with underdoping Lin et al., PRX 013

57 Emergence of Nernst oscillations with underdoping Lin et al., PRX 013

58 Emergence of Nernst oscillations with underdoping Lin et al., PRX 013

59 The lowest concentration has a single frequency Band calculations: van der Marel, van Mechelen and Mazin PRB (011) Lin et al., PRX cm -3

60 Two ways to estimate the Fermi energy ev F =1.1meV S/T =( /3)(k B /e) 1/T F T F = 1.9 k Lin et al., PRX 013

61 Dilute superconductors K Ueno et. al., Nature Nanotechnology 6, SrTiO 3-d [ With d 10-5 ] is the most dilute superconductor!

62 SUMMARY Thermoelectricity is poorly understood and barely explored in many solids! Often, it opens another window to electronic properties. At the boundary between metals and insulators, a large entropy is shared by few carriers with a remarkable thermoelectric response. In dilute metals, when few landau tubes survive, the exit of each of them generates a large Nernst oscillation. Many interesting thermoelectric materials are on the metalic side of metal-insulator transition and have a Fermi surface.

63 Neville Mott on metals and non-metals Edwards, Phil. Trans. R. Soc. A 368, (010)

64 Everyone knows what a metal is and can describe many of its characteristics. It is safe to say, however, that few people would define a metal as a solid with a Fermi surface. This may nevertheless be the most meaningful definition of a metal and provides a precise explanation of the main physical properties A. R. Mackintosh Scientific American 1963

65 First-principle calculations for Bi Te 3 Scheidemental et al., PRB 68, 1510 (003) gap ZT peaks on the metallic side of the metal-insulator transition!