Low temperature transport in correlated systems and its evolution under pressure V. Zlatic, R. M., J. K. Freericks (Phys. Rev. B 78 , 045113 (2008))
Thermoelectric device efficiency in the temperature range of interest given by the Figure of Merit ( ZT ) : T ZT = � ( T ) � ( T ) S 2 ( T ) S = Seebeck coefficient ( V/K) ρ = electrical resistivity ( Ω cm) κ = thermal conductivity (W/cm K) �� = 3 e 2 Normal Fermi liquid (FL) : T 2 (Wiedemann-Franz) � 2 k B � ZT > 1 � S > 155 µ V/K Hvar 2008 2
Experimentally, FL behavior is often observed: � ( T ) � � 0 + A � T 2 S ( T ) � T c V ( T ) � � � T As well as the (quasi-) universal ratios: � � � ( T )- � 0 � A � � • Kadowaki- Woods ratio : � 2 = cst. 2 � � c V ( T ) � � K. Kadowaki and S. B. Woods, Solid State Commun. 71 , 1149 (1987) e S ( T ) • thermopower - entropy ratio : c V ( T ) � cst. K.Behnia, D. Jaccard and J. Flouquet, J. Phys.: Condens. Matter 16 , 5187 (2004) Hvar 2008 3
Kadowaki-Woods ratio: H. Kontani, J. Phys. Soc. Jap. 73 , 515 (2004) N. Tsujii, H. Kontani and K. Yoshimura, Phys. Rev. Lett. 94 , 057201 (2005) Importance of the low temperature degeneracy, N, of the f-levels (can be changed by pressure!) A A � � A = 2 N ( N � 1) 1 � � � � � = 2 N ( N � 1) 1 Hvar 2008 4
Thermopower- entropy ratio: data from K.Behnia et al., op.cit. K. Myiake and H. Kohno, J.Phys. Soc. Japan 74 , 254 (2005): due to scattering off impurities Hvar 2008 5
Modelization by simplified, SU(N), version of orbitally degenerate periodic Anderson model ( N “flavors“ for both f and c onduction states) in the U ��� limit: H = ˆ ˆ H Band + ˆ H f + ˆ H Hyb � µ ˆ N el Hybridization occurs only between states of the same flavor and is taken as local ( k -independent). Assume a FL ground state with energy scale k B T 0 � � 2 k B 2 � 1 � 3 V cell ( γ given per unit volume) Hvar 2008 6
Transport coefficients can all be expressed in terms of usual transport integrals ( G. D. Mahan, in Solid State Physics 51 , 81 (1997) ) : + � � � d � � df ( � ) � � i + j � 2 � ( � , T ) L ij ( T ) = � � � d � � � �� Electrical conductivity � ( T ) = e 2 L 11 ( T ) L 12 ( T ) 1 S ( T ) = � Thermoelectric power e � T L 11 ( T ) 2 ( T ) � � L 22 ( T ) � L 12 � ( T ) = 1 � � Thermal conductivity T L 11 ( T ) � � � � Hvar 2008 7
� � ( � , T ) � ( � , T ) = � Need the transport function: flavors � Neglect vertex corrections: ( ) � � ( � � 0, T � 0) � 1 ( ) � � � , T ( ) 2 � N c � � v k F 3 � = velocity of unhybridized, independent conduction electrons v k F Renormalized c-DOS: 1 1 ( ) = � N c � � � ( � + , T = 0) , G c � ( � , T ) = � ( k , � , T ) � Im G c G c � V cell N sites k Transport relaxation time: ( ) = � � � � � , T � ( � + , T ) Im � c Hvar 2008 8
c - and f- Green‘s functions (GF) from equations of motion: � � � f � ( k , z , T ) + µ z � E f � ( k , z , T ) = G c � � V 2 � + µ � � � f � � � � � ( k , z , T ) + µ ) z � � k � z � E f � � � + µ z � � k � ( k , z , T ) = G f � + µ � � � f � � � � � ( k , z , T ) + µ ) z � � k � z � E f � � V 2 � � DMFT I : f -electron self-energy local ( k- independent) � V 2 1 � ( k , z , T ) = with � c � ( z , T ) = G c � + µ � � c � � � f � ( z , T ) � ( z , T ) z � � k z � E f (only f -levels are correlated) Hvar 2008 9
Numerical treatments for SU(2) PAM: David E. Logan and N. S. Vidhyardhiraja, J. Phys.: Condens. Matter 17 , 2935 (2005): Solve iteratively for Σ f σ ( z ,T) by the local moment approach . or DMFT II : local f -electron GF must be equal to GF for an effective impurity with the same self-energy � � 1 1 G f � � ( z , T ) = � ( k , z , T ) � = G imp � � G f � z � E f + µ � � � ( z ) � � f � ( z , T ) N sites � � k C. Grenzebach et al., Phys. Rev. B 74 , 195119 (2006): NRG for impurity problem C. Grenzebach et al., Phys. Rev. B 77 , 115125 (2008): include disorder via CPA Hvar 2008 10
The quasiparticle description of the FL state: � around the pole: Linearize � f ( ) Z f � � + Re � f � 1 + O ( � 2 ) , � � � � E f � ( � ) � µ � � � � � � f � � Renormalized QP weight � 1 = 1 � �� f � / � � � � Z f � � � � = 0 Renormalized f -level ( ) � + Re � f � � µ � � � � � f � = E f � (0) � µ � Z f � � � f � � � � � QP excitations: ( ) � � � ( ) � � � + µ 2 = 0 � � � k � f � V f � With effective hybridization � V f � = V Z f � Hvar 2008 11
Quasiparticle excitation energy relative to chemical potential: 2 + 4 � ( ) = 1 � ( ) ± ( ) � ± = � � ± � µ � � µ + � � � µ � � � k � � k � k � f � � k � f � 2 V f � � � 2 � � = � � f Figure from: H. Okamura et al., J. Phys. Soc. µ � Japan, 76 , 023703 (2007) Homogeneous paramagnet �� same for all flavors σ . � 2 V f � ± = 0 and � k � = µ + � k � „Large“ Fermi surface defined by � f � � Hvar 2008 12
QP density of states per channel: ( ) 1 ( ) = ( ) + Z f � ( ) QP � � � � � � k � ± = N c � � � 1 N f � � N � N sites V cell k �� Relation of the FL scale T 0 to the renormalized DOS : � � = � 2 k B 2 ( ) = V cell 1 ( ) + Z f � ( ) QP 0 � � � N � � 1 N f � 0 � � N c � 0 � � 6 � k B T 0 2 � � � � � � � 2 k B 2 1 � V cell � ( ) � = � � � � � 1 N f � 0 � � Z f � � � � 3 V cell k B T 0 � � 2 � � ( ) 0 N c � z Next step: relate to (known) bare c -electron DOS, Hvar 2008 13
� : From Dyson equation for G c � � � � � 2 ( ) V f � ( ) = N c � ( ) = 1 � N c � � � + µ � � z � � k � 0 0 � , N c � z � � � � � � � f � N sites V cell � � � � k � and defining eq. for QP: From spectral decomposition of G f ( ) = Z f � � 2 V f � ( ) N f � � 2 N c � � ( ) � � � � f � Combining these two results yields � � � � � 2 1+ � µ = V cell V f � 1 ( ) � µ + � µ , � µ = 0 � � N c � � � � f � � � f � � k B T 0 2 � � � � � � � Hvar 2008 14
Homogeneous paramagnet (all σ equivalent) from now on: � � � � � 1+ � µ 2 = NV cell V f 1 ( ) µ + � µ , � µ = 0 � � N c � � � � f � � f � k B T 0 2 � � � � � � � relation between � � f and k B T 0 : � f � NV cell ( ) k B T 0 � µ N c � µ + � µ � 0 2 Remember : T 0 is an equilibrium property, which can be obtained from experiment or computed accurately by numerical methods (eg. DMFT + NRG). µ fixed by particle number conservation, � µ follows from Luttinger's theorem. Hvar 2008 15
� � ( ) � ( � ) ( � , T ) � 1 ( ) � ( � ) � , T ( ) 2 ( � ) N c � � Wanted: v k F 3 Need imaginary part of f -electron self energy (local in DMFT!): K. Yamada and K. Yosida, Prog. Theor. Phys. 76 , 681 (1986) H. Kontani, J. Phys. Soc. Jap. 73 , 515 (2004) 3 � f ) � � � ( ( ) ( ) V cell N f � 0 ( ) 2 � 2 + � k B T � � 2 � � Im � f � , T � N � 1 2 � � � � � � f = irred. four-point scattering vertex for e 's with different flavors. Ward identity for U � � (charge fluctuations suppressed): � 1 Z f � f = ( ) V cell N f � (0) N � 1 Hvar 2008 16
� 1 Z f � V cell 1 ( ) � 1 N f � 0 � f = together with 2 N Z f ( ) V cell N f � (0) N � 1 k B T 0 leads to: ( ) � 2 + � k B T � � 2 � � ( ) � � 2 � � � Im � f � , T ( ) N 2 V cell N f � (0) k B T 0 ( ) 2 N � 1 ( ) , from which, to leading order in T and � : insert into � c � , T 2 � � � � � � 0 ( T ) 1 � � � 2 � � ( � , T ) � � 1 � � � � � � f Im � c ( � + , T ) � � 2 k B 2 T 2 � � � � 0 ( µ + � µ ) 2 � � � 0 ( T ) = � ( N � 1) N 2 V cell N c � � T 0 with � � 2 � 3 � T � Hvar 2008 17
Compute + � � � � d � � df ( � ) � � i + j � 2 � L ij ( T ) = N ( � , T ) � � � d � � � �� by Sommerfeld expansion : � � � � =0 + � 2 k B � 2 2 T 2 L ij ( T ) � � i + j � 2 � � ( � , T ) � � � � 2 � i + j � 2 � � ( � , T ) � � � � � � � � � � � 6 � � � =0 with ( ) � � ( � , T ) � 1 ( ) � � � , T ( ) 2 � N c � � v k F 3 Hvar 2008 18
Recommend
More recommend
