Low temperature transport in correlated systems and its evolution - - PowerPoint PPT Presentation

Low temperature transport in correlated systems and its evolution under pressure

• V. Zlatic, R. M., J. K. Freericks

(Phys. Rev. B 78, 045113 (2008))

2 Hvar 2008

Thermoelectric device efficiency in the temperature range of interest given by the Figure of Merit (ZT) : S = Seebeck coefficient ( V/K) ρ = electrical resistivity (Ωcm) κ = thermal conductivity (W/cm K) Normal Fermi liquid (FL) : T = 3e2 2kB

2 (Wiedemann-Franz)

ZT = T (T)(T)S2(T) ZT > 1 S > 155 µV/K

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Experimentally, FL behavior is often observed: As well as the (quasi-) universal ratios:

K.Behnia, D. Jaccard and J. Flouquet, J. Phys.: Condens. Matter 16, 5187 (2004)

• K. Kadowaki and S. B. Woods, Solid State Commun. 71, 1149 (1987)

(T) 0 + AT 2 S(T) T cV(T) T

• Kadowaki- Woods ratio:

(T)- 0

• cV(T)
• 2

A 2 = cst.

• thermopower - entropy ratio:

e S(T) cV(T) cst.

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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

1 2 N(N 1)

=

• 1

2 N(N 1)

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Thermopower- entropy ratio:

data from K.Behnia et al.,

• p.cit.
• K. Myiake and H. Kohno, J.Phys. Soc. Japan 74, 254 (2005): due to scattering off impurities

6 Hvar 2008

Modelization by simplified, SU(N), version of orbitally degenerate periodic Anderson model (N “flavors“ for both f and conduction states) in the Ulimit: Hybridization occurs only between states of the same flavor and is taken as local (k-independent). Assume a FL ground state with energy scale kBT0 2kB

2

3 Vcell 1

• (γ given per unit volume)

ˆ H = ˆ HBand + ˆ Hf + ˆ HHyb µ ˆ Nel

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Transport coefficients can all be expressed in terms of usual transport integrals (G. D. Mahan, in Solid State Physics 51, 81 (1997) ) : (T) = e2L11(T) Electrical conductivity Thermoelectric power Thermal conductivity Lij(T) = d df() d

• +
• i+ j 2(,T)

S(T) = 1 e T L12(T) L11(T) (T) = 1 T L22(T) L12

2 (T)

L11(T)

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Need the transport function:

(,T) = (,T)

flavors

• Neglect vertex corrections:

Renormalized c-DOS: Transport relaxation time: vkF

= velocity of unhybridized, independent conduction electrons

Nc

( ) =

1 Vcell ImGc

( +,T = 0) , Gc (,T) =

1 Nsites Gc

(k,,T) k

• ( 0,T 0) 1

3 vkF

• (

)

2

Nc

( ) ,T ( )

,T

( ) =

• Im

c ( +,T)

9 Hvar 2008

c- and f- Green‘s functions (GF) from equations of motion: DMFT I : f-electron self-energy local (k-independent)

• Gc

(k,z,T) =

1 z k

+ µ c (z,T)

with c

(z,T) =

V 2 z Ef

f (z,T)

Gc

(k,z,T) =

z Ef

f (k,z,T) + µ

z k

+ µ

• z Ef

f (k,z,T) + µ)

• V 2

Gf

(k,z,T) =

z k

+ µ

z k

+ µ

• z Ef

f (k,z,T) + µ)

• V 2

(only f-levels are correlated)

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DMFT II : local f-electron GF must be equal to GF for an effective impurity with the same self-energy

Gf

(z,T) =

1 Nsites Gf

(k,z,T) k

• = Gimp
• 1

z Ef + µ (z) f

(z,T)

Numerical treatments for SU(2) PAM:

• 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

David E. Logan and N. S. Vidhyardhiraja, J. Phys.: Condens. Matter 17, 2935 (2005): Solve iteratively for Σf

σ(z,T) by the local moment approach.

• r

11 Hvar 2008

The quasiparticle description of the FL state: Linearize f

around the pole:

Renormalized QP weight Renormalized f-level Ef

+ Re f () µ

• f

( )Zf

1 + O( 2) ,

QP excitations: k

+ µ

( )

f

( )

Vf

2 = 0

• f = Ef

+ Re f (0) µ

• Zf
• f

µ

• (

)

With effective hybridization Vf =V Zf Zf

1 = 1 f / =0

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Quasiparticle excitation energy relative to chemical potential: k

± =

k

± µ

( ) = 1

2

k

µ +

f

( ) ±

k

µ

f

( )

2 + 4

Vf

2

• Figure from:
• H. Okamura et al., J. Phys. Soc.

Japan, 76, 023703 (2007) Homogeneous paramagnet same for all flavors σ.

µ = f „Large“ Fermi surface defined by k

± = 0 and k = µ +

• Vf

2

• f

13 Hvar 2008

QP density of states per channel: Relation of the FL scale T0 to the renormalized DOS : N

QP

( ) =

1 NsitesVcell k

±

( )

k

• = Nc

( ) + Zf

1 Nf

( )

Next step: relate to (known) bare c-electron DOS, Nc z

( )

= 2kB

2

6 N

QP 0

( )

• =

2kB

2

3Vcell

• 1

kBT0

• 1

kBT0 = Vcell 2 Nc 0

( ) + Zf

1 Nf 0

( )

• Vcell

2 Zf

1 Nf 0

( )

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1 kBT0 = Vcell 2 1+ µ

• f
• Nc

µ + µ

( )

• , µ =
• Vf

2

• f

Combining these two results yields From Dyson equation for Gc

:

Nc

( ) = Nc

+ µ

• Vf

2

f

• , Nc

z

( ) =

1 NsitesVcell z k

• (

)

k

• From spectral decomposition of Gf

and defining eq. for QP:

Nf

( ) = Zf

Vf

2

f

( )

2 Nc

( )

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1 kBT0 = NVcell 2 1+ µ

• f
• Nc

µ + µ

( )

• , µ =
• Vf

2

• f

Homogeneous paramagnet (all σ equivalent) from now on: relation between f and kBT0 :

• f NVcell

2 µNc µ + µ

( )kBT0

µ fixed by particle number conservation, µ follows from Luttinger's theorem. Remember : T0 is an equilibrium property, which can be obtained from experiment or computed accurately by numerical methods (eg. DMFT + NRG).

16 Hvar 2008

Wanted:

• 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)

Imf ,T

( )

2 2 + kBT

( )

2

• N1

( ) VcellNf 0 ( )

• 3 f

2

f = irred. four-point scattering vertex for e's with different flavors. f = Zf

1

N 1

( )VcellNf(0)

Ward identity for U (charge fluctuations suppressed):

( )(,T) 1

3 vkF

( )

( )

2

Nc

( ) ( ) ,T ( )

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leads to: Imf ,T

( ) 2

2 + kBT

( )

2

• N1

( )N2VcellNf (0) kBT0 ( )

2

insert into c ,T

( ), from which, to leading order in T and :

(,T)

• Imc( +,T)

0(T) 1

• f
• 2

1 2 2kB

2T 2

• 0(T) = (N 1)N2VcellNc

0 (µ + µ)

2 3 T0 T

• 2

with f = Zf

1

N 1

( )VcellNf(0)

together with 1 kBT0 Vcell 2 N Zf

1 Nf 0

( )

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Lij(T) = N d df() d

• +
• i+ j 2
• (,T)

Compute by Sommerfeld expansion: Lij(T) i+ j 2 (,T)

• =0 + 2kB

2T 2

6 2 2 i+ j2 (,T)

• =0

(,T) 1

3 vkF

• (

)

2

Nc

( ) ,T ( )

with

19 Hvar 2008

Fermi liquid power laws: Thermopower: S(T) = 4 2kB e NVcellNc

0 (µ + µ)µ

• T

T0 Thermal conductivity: (T) = T 2kB

2

2e2 (T) 1 32 2 NVcellNc

0 (µ + µ)µ

• 2

T T0

• 2
• Resistivity:

(T) = 9 3Vcellµ2 e2 (vkF

)2 N N 1

( ) NVcellNc

0 (µ + µ)µ

• 2

T T0

• 2

20 Hvar 2008

Universal ratios follow from = 2kB

2

3Vcell

• 1

kBT0 : thermopower - entropy ratio: NAvo e S(T) MolT

• 12

N VcellNc

0 (µ + µ)µ

• 12

±nf for N large Kadowaki- Woods ratio: A 2 = 81 3Vcell

3µ2

e2 (vkF

)2 N N -1

( ) N VcellNc

0 (µ + µ)µ

• 2 kB

2

where for large N : N VcellNc

0 (µ + µ)µ

• nf

21 Hvar 2008

Pressure dependence of A coefficient:

Example 1: CePd2Ge2 p≈0: AF, TN ≅ 5.1 K, µneutron ≅ .8 4 µB

f el. essentially localized, with

CF-doublet ground state light QP‘s, small FS, T0 large Pressure increases hybridization. crossover to N=2 heavy QP bands large FS, low velocity, small T0 At a critical pressure, hybridization

• vercomes CF-splitting.

formation of N=6 heavy QP bands

• ut of complete j=5/2 multiplet.
• red. of A by a factor 15 at least!

A

A 1 N(N 1) 1 T0

2

22 Hvar 2008

Example 2: CeCu2Ge2 p≈0 GPa: incomm.(spiral) magnetic order, TN ≅ 4.1 K, µneutron ≅ .7 4 µB

CF-doublet ground state ;

8.5≤p≤11 GPa: f el. delocalized, N=2 heavy QP bands A high. 11≤p≤15 Gpa: crossover to N=4 heavy QP bands A(4)=A(2) / 6 15≤p≤16.5 Gpa: crossover to N=6 heavy QP bands A(6)=A(2) /15 16.5 Gpa<p: crossover to intermediate-valent régime A drops. Also explains behavior of residual resistivity! i.e., the more channels are heavy (slow), the larger is ρ0 , if the scattering rate by impurities stays the same. @1 K( 200 mJ/K2)

23 Hvar 2008

Example 3: YbCu2Si2 Intermediate valence state at ambient pressure 1≤p≤12 GPa: crossover to N=8 heavy QP bands; AF order at p~ 8 GPa 12 GPa≤p : crossover to N=4,2 heavy QP bands; ( 145 mJ/K2)

A &

24 Hvar 2008

Conclusions:

• Periodic SU(N) model captures the low temperature

physics of HF‘s

• Transport coefficients are given by power laws
• The low-T Fermi volume differs from the high-T one
• Properties are „nearly“ universal on a T/T0 scale
• Dependence on explicit band structure cannot be

lumped into T0

• Pressure-dependence of A and ρ0 can be explained

by different degeneracies N of CF states Pressure changes degeneracy by quenching (Ce) or enhancing (Yb) the CF splitting. It also shifts the chemical potential with respect to the band center.

25 Hvar 2008

High temperature régime:

c-f scattering incoherent Anderson impurity model adequate Solution via the Non-Crossing Approximation

(V. Zlatic, R. M., Phys. Rev. B 71, 165109 (2005))

26 Hvar 2008

Thermoelectric power displays a complex behavior as a function of pressure, e.g. CeRu2Ge2

(H. Wilhelm and D. Jaccard,Phys. Rev. B 69, 214408 (2004)):

Typical shapes: (a) For p< 2GPa (low pressure) (b) above 2 Gpa (intermediate p) (c) above 4 Gpa (high pressure) (d) above 8 Gpa (very high p)

T

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Yb-compounds: chemical pressure

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YbCu2Si2: physical pressure

30 Hvar 2008

f-level shifts under pressure in Yb-compounds