- Email: [email protected]
Conductivity and thermopower of heavy fermion systems with disorder
ARTICLE IN PRESS
Physica B 359–361 (2005) 732–734 www.elsevier.com/locate/physb
Conductivity and thermopower of heavy fermion systems with disorder Claas Grenzebach, Gerd Czycholl Institut fu¨r Theoretische Physik, Universita¨t Bremen Otto-Hahn-Allee 1, 28359 Bremen, Germany
Abstract We investigate the influence of disorder (impurities) on the electronic (transport) properties of heavy fermion systems, which are described by the periodic Anderson model (PAM). In this paper we consider disorder within the conduction band. The PAM is mapped onto an effective single impurity Anderson model (SIAM) by dynamical mean-field theory (DMFT), and the modified perturbation theory (MPT)—an approximation that is exact up to second order in the Coulomb correlation U and reproduces the atomic limit—is used for the effective SIAM. To include the scattering by the impurities we use the coherent potential approximation (CPA) which is consistent with the DMFT. For various concentrations c of the impurities we calculate the resistivity as well as the thermoelectrical power in a situation describing a Kondo insulator in the pure case (c ¼ 0). r 2005 Elsevier B.V. All rights reserved. PACS: 71.27.+a; 71.10.Fd; 71.28.+d; 72.15.Qm Keywords: Heavy fermions; Disorder; DMFT; MPT; CPA
The typical behavior of heavy fermion systems depends on the occurrence of disorder, see for example [1]. The periodic Anderson model (PAM) is a simple model for heavy fermions. Its standard form with only a two-fold degeneracy of the flevels is given by (ncks :¼ cyks cks and nfRs :¼ f yRs f Rs
as usual): H¼
k ncks
X
þ
þ
V ðcyRs f Rs
þ ðf yRs ; cyRs Þ
" f nfRs þ
Rs
ks
Corresponding author. Tel.: +1 421 218 4465;
fax: +1 421 218 4869. E-mail address: [email protected] (C. Grenzebach).
X
þ c:c:Þ
V^ R
f Rs cRs
U f f n n 2 Rs Rs
!# :
ð1Þ
Here k is the band electron dispersion, f is the bare f-level energy, U is the Coulomb correlation
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.209
ARTICLE IN PRESS C. Grenzebach, G. Czycholl / Physica B 359– 361 (2005) 732–734
In this paper we consider only impurities in the conduction band, therefore we have V^ ¼ ð00 ~0c Þ: The Green functions of the PAM can be obtained by matrix inversion ! hhf j f y iiz hhf j cy iiz ^ GðzÞ ¼ hhc j f y iiz hhc j cy iiz " ! z f Scorr ðzÞ V 1 X ¼ N k V z k #1 ^ imp ðzÞ : ð3Þ S Here Scorr ðzÞ is the correlation self-energy, which contains all contributions of the interaction U, and ^ imp ðzÞ is the impurity self-energy (matrix), S which—for configurationally averaged quantities—replaces the random potential V^ R by a translationally invariant, ‘‘coherent potential’’. For c ¼ 0 and c ¼ 1 the trivial exact limits are ^ imp ¼ 0 and S ^ imp ¼ V^ ; respectively. S We calculate the correlation self-energy using dynamical mean-field theory (DMFT, [2,3]) to map the PAM onto an effective SIAM and solve the effective SIAM within modified perturbation theory (MPT, [4]), see also [5]. For the impurity self-energy we use the coherent potential approximation (CPA, [6]), which is consistent with the DMFT and leads to the self-consistency equation ^ 1 ðV^ S ^ imp Þ cð1 ðV^ S^ imp ÞGÞ 1 ^ S ^ imp : ¼ ð1 cÞð1 þ S^ imp GÞ
ð4Þ
The complete self-consistency scheme consists in: Start with an initial guess for both self-energies. Then compute an effective hybridization function DðzÞ ¼ z f Scorr ðzÞ 1=G f ðzÞ for the DMFTmapping. The hereby obtained effective SIAM is solved within MPT and a new correlation selfenergy is obtained. Finally the CPA-equation is
solved. Having now new values for both selfenergies we start over with the DMFT. All quantities to be calculated can be obtained from the one-particle (band and f-electron) Green functions and thus from the self-energies. The resistivity and the thermopower are given by (using the Kubo formula and the fact that for high dimensions vertex corrections vanish, see [7,2]) Z 1 df ðEÞ LðEÞ dE X0; (5) RðTÞ dE R SðTÞ ¼
ðdf dE ðEÞÞ ðE mÞ LðEÞ dE ; R eT ðdf dE ðEÞÞ LðEÞ dE
(6)
with LðEÞ ¼
2 X ðIm Gcks ðE þ i0ÞIm Gcks ði0ÞÞ: N ks
(7)
We use the parameters V ¼ 0:2; U ¼ 1; f ¼ 0:5 and ~c ¼ 0:25: Then the Coulomb correlation U is of the same magnitude as the bandwidth. Furthermore we take ntotal ¼ 2 c=5; making the total occupation c-dependent, and we assume a semielliptical unperturbed conduction band, i.e. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X 1 (8) ¼ 8zð1 1 ð4zÞ2 Þ; N k z k
104
Resistivity R
between two f-electrons and V the hybridization between f- and conduction electrons at the same lattice site. The additional last term introduces impurities with a concentration c, by the matrix ( 0 with probability 1 c; V^ R ¼ (2) V^ with probability c:
733
c = 0.00 c = 0.10 c = 0.20 c = 0.50 c = 0.90
102
100
10 2
0
0.005
0.01 0.015 0.02 Temperature T
0.025
0.03
Fig. 1. Static resistivity for the parameters V ¼ 0:2; U ¼ 1:0; f ¼ 0:5; ntotal ¼ 2 c=5; ~c ¼ 0:25:
ARTICLE IN PRESS C. Grenzebach, G. Czycholl / Physica B 359– 361 (2005) 732–734
734
Results for the temperature dependence of the resistivity RðTÞ and the thermopower SðTÞ are shown in Figs. 1 and 2. The trends observed are in rough qualitative agreement with the experimental results of Ref. [1]. We conclude that the combined DMFT and CPA treatment of the PAM extended by a disorder term is able to qualitatively describe the effects of substitutional doping (with arbitrary concentration c) in heavy fermion systems.
50
Thermopower S [µV/K]
0 –50 –100 –150 c = 0.10 c = 0.20 c = 0.50 c = 0.90
–200 –250 –300 0
0.01
0.02 0.03 Temperature T
0.04
0.05
Fig. 2. Thermoelectric power, parameters as in Fig. 1.
thereby introducing the conduction bandwidth as energy unit. For this choice of the parameters one obtains the Fermi energy within a hybridization gap in the pure case c ¼ 0: Therefore, this is a model for Kondo insulators like Ce3 Pt3 Sb4 ; and disorder in the conduction band may simulate the substitution of Pt-atoms by Cu-atoms, as investigated experimentally in Ref. [1].
References [1] C.D.W. Jones, K.A. Regan, F.J. DiSalvo, Phys. Rev. B 60 (1999) 5282. [2] A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [3] W. Metzner, D. Vollhardt, Phys. Rev. Lett. 62 (1989) 324. [4] M. Potthoff, T. Wegner, W. Nolting, Phys. Rev. B 55 (1997) 16132. [5] C. Grenzebach, G. Czycholl, Acta Phys. Pol. B 34 (2003) 971. [6] R.J. Elliott, J.A. Krumhansl, P.L. Leath, Rev. Mod. Phys. 46 (1974) 465. [7] H. Schweitzer, G. Czycholl, Phys. Rev. Lett. 67 (1991) 3724.
Physica B 359–361 (2005) 732–734 www.elsevier.com/locate/physb
Conductivity and thermopower of heavy fermion systems with disorder Claas Grenzebach, Gerd Czycholl Institut fu¨r Theoretische Physik, Universita¨t Bremen Otto-Hahn-Allee 1, 28359 Bremen, Germany
Abstract We investigate the influence of disorder (impurities) on the electronic (transport) properties of heavy fermion systems, which are described by the periodic Anderson model (PAM). In this paper we consider disorder within the conduction band. The PAM is mapped onto an effective single impurity Anderson model (SIAM) by dynamical mean-field theory (DMFT), and the modified perturbation theory (MPT)—an approximation that is exact up to second order in the Coulomb correlation U and reproduces the atomic limit—is used for the effective SIAM. To include the scattering by the impurities we use the coherent potential approximation (CPA) which is consistent with the DMFT. For various concentrations c of the impurities we calculate the resistivity as well as the thermoelectrical power in a situation describing a Kondo insulator in the pure case (c ¼ 0). r 2005 Elsevier B.V. All rights reserved. PACS: 71.27.+a; 71.10.Fd; 71.28.+d; 72.15.Qm Keywords: Heavy fermions; Disorder; DMFT; MPT; CPA
The typical behavior of heavy fermion systems depends on the occurrence of disorder, see for example [1]. The periodic Anderson model (PAM) is a simple model for heavy fermions. Its standard form with only a two-fold degeneracy of the flevels is given by (ncks :¼ cyks cks and nfRs :¼ f yRs f Rs
as usual): H¼
k ncks
X
þ
þ
V ðcyRs f Rs
þ ðf yRs ; cyRs Þ
" f nfRs þ
Rs
ks
Corresponding author. Tel.: +1 421 218 4465;
fax: +1 421 218 4869. E-mail address: [email protected] (C. Grenzebach).
X
þ c:c:Þ
V^ R
f Rs cRs
U f f n n 2 Rs Rs
!# :
ð1Þ
Here k is the band electron dispersion, f is the bare f-level energy, U is the Coulomb correlation
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.01.209
ARTICLE IN PRESS C. Grenzebach, G. Czycholl / Physica B 359– 361 (2005) 732–734
In this paper we consider only impurities in the conduction band, therefore we have V^ ¼ ð00 ~0c Þ: The Green functions of the PAM can be obtained by matrix inversion ! hhf j f y iiz hhf j cy iiz ^ GðzÞ ¼ hhc j f y iiz hhc j cy iiz " ! z f Scorr ðzÞ V 1 X ¼ N k V z k #1 ^ imp ðzÞ : ð3Þ S Here Scorr ðzÞ is the correlation self-energy, which contains all contributions of the interaction U, and ^ imp ðzÞ is the impurity self-energy (matrix), S which—for configurationally averaged quantities—replaces the random potential V^ R by a translationally invariant, ‘‘coherent potential’’. For c ¼ 0 and c ¼ 1 the trivial exact limits are ^ imp ¼ 0 and S ^ imp ¼ V^ ; respectively. S We calculate the correlation self-energy using dynamical mean-field theory (DMFT, [2,3]) to map the PAM onto an effective SIAM and solve the effective SIAM within modified perturbation theory (MPT, [4]), see also [5]. For the impurity self-energy we use the coherent potential approximation (CPA, [6]), which is consistent with the DMFT and leads to the self-consistency equation ^ 1 ðV^ S ^ imp Þ cð1 ðV^ S^ imp ÞGÞ 1 ^ S ^ imp : ¼ ð1 cÞð1 þ S^ imp GÞ
ð4Þ
The complete self-consistency scheme consists in: Start with an initial guess for both self-energies. Then compute an effective hybridization function DðzÞ ¼ z f Scorr ðzÞ 1=G f ðzÞ for the DMFTmapping. The hereby obtained effective SIAM is solved within MPT and a new correlation selfenergy is obtained. Finally the CPA-equation is
solved. Having now new values for both selfenergies we start over with the DMFT. All quantities to be calculated can be obtained from the one-particle (band and f-electron) Green functions and thus from the self-energies. The resistivity and the thermopower are given by (using the Kubo formula and the fact that for high dimensions vertex corrections vanish, see [7,2]) Z 1 df ðEÞ LðEÞ dE X0; (5) RðTÞ dE R SðTÞ ¼
ðdf dE ðEÞÞ ðE mÞ LðEÞ dE ; R eT ðdf dE ðEÞÞ LðEÞ dE
(6)
with LðEÞ ¼
2 X ðIm Gcks ðE þ i0ÞIm Gcks ði0ÞÞ: N ks
(7)
We use the parameters V ¼ 0:2; U ¼ 1; f ¼ 0:5 and ~c ¼ 0:25: Then the Coulomb correlation U is of the same magnitude as the bandwidth. Furthermore we take ntotal ¼ 2 c=5; making the total occupation c-dependent, and we assume a semielliptical unperturbed conduction band, i.e. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 X 1 (8) ¼ 8zð1 1 ð4zÞ2 Þ; N k z k
104
Resistivity R
between two f-electrons and V the hybridization between f- and conduction electrons at the same lattice site. The additional last term introduces impurities with a concentration c, by the matrix ( 0 with probability 1 c; V^ R ¼ (2) V^ with probability c:
733
c = 0.00 c = 0.10 c = 0.20 c = 0.50 c = 0.90
102
100
10 2
0
0.005
0.01 0.015 0.02 Temperature T
0.025
0.03
Fig. 1. Static resistivity for the parameters V ¼ 0:2; U ¼ 1:0; f ¼ 0:5; ntotal ¼ 2 c=5; ~c ¼ 0:25:
ARTICLE IN PRESS C. Grenzebach, G. Czycholl / Physica B 359– 361 (2005) 732–734
734
Results for the temperature dependence of the resistivity RðTÞ and the thermopower SðTÞ are shown in Figs. 1 and 2. The trends observed are in rough qualitative agreement with the experimental results of Ref. [1]. We conclude that the combined DMFT and CPA treatment of the PAM extended by a disorder term is able to qualitatively describe the effects of substitutional doping (with arbitrary concentration c) in heavy fermion systems.
50
Thermopower S [µV/K]
0 –50 –100 –150 c = 0.10 c = 0.20 c = 0.50 c = 0.90
–200 –250 –300 0
0.01
0.02 0.03 Temperature T
0.04
0.05
Fig. 2. Thermoelectric power, parameters as in Fig. 1.
thereby introducing the conduction bandwidth as energy unit. For this choice of the parameters one obtains the Fermi energy within a hybridization gap in the pure case c ¼ 0: Therefore, this is a model for Kondo insulators like Ce3 Pt3 Sb4 ; and disorder in the conduction band may simulate the substitution of Pt-atoms by Cu-atoms, as investigated experimentally in Ref. [1].
References [1] C.D.W. Jones, K.A. Regan, F.J. DiSalvo, Phys. Rev. B 60 (1999) 5282. [2] A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13. [3] W. Metzner, D. Vollhardt, Phys. Rev. Lett. 62 (1989) 324. [4] M. Potthoff, T. Wegner, W. Nolting, Phys. Rev. B 55 (1997) 16132. [5] C. Grenzebach, G. Czycholl, Acta Phys. Pol. B 34 (2003) 971. [6] R.J. Elliott, J.A. Krumhansl, P.L. Leath, Rev. Mod. Phys. 46 (1974) 465. [7] H. Schweitzer, G. Czycholl, Phys. Rev. Lett. 67 (1991) 3724.