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Itinerant electron correlation and the ideal Lorenz number of transition metals
Volume 26A.
number 2
PHYSICS
ITINERANT ELECTRON LORENZ NUMBER
LETTERS
18 December
CORRELATION OF TRANSITION
AND THE METALS
1967
IDEAL
M. J. RICE
SolidState Theory Group, Departmentof Physics, Imperial College, London, England Received
14 November
1967
A simple discussion of the effect of itinerant electron correlation on the ideal Lorenz tion metals is given and applied to reconcile data recently taken on Pd.
It is well known in the theory of transition metals that interband electron-electron scatter ing yields a characteristic Lorenz number L,_, which is temperature independent *. In this letter we calculate this quantity when the interband scattering takes the form of an s-d exchange interaction and point out that this contribution can give an indirect measure of the magnitude of itinerant d-band electron correlation: the calculation, straight forward extensions of which may be useful in characterizing in a simple way the low-temperature transport properties of a whole class of strongly correlated transition metal systems, accounts well for data recently taken on Pd [2,3]. We consider a simple itinerant model [4] in which the d-electrons interact via a semi-phenomenological short range repulsion Zb(r) and are coupled to a non-interacting s-band (assumed to make the principal contribution to the electrical currents) through an interband exchange interaction Hsd = w,d Jd3,S(r)
* s(f)
'*l/l-~
(1)
per unit cell, where S(r)and s(r) denote respectively the fermion spin density operators for the d and s electrons. (1) can be analysed into terms describing the scattering of s electrons from either transverse or longitudinal d-band spin density fluctuations (d-SDF). The effect on the resistivities of the former type of scattering, in which the s-electron spin is flipped, has been discussed in the literature recently [5,6], while the latter form of scattering is of the same type * That is, yielding an electrical and thermal resistivity varying as T2 and T, respectively, e.g. [ 11. 86
I
Fig.
I. he_,
number of transi-
I
2o
3
as a function of the RPA d-band electron correlation.
discussed some years ago by Baber [‘7]. Following closely the theory of the electrical and thermal resistivities developed in [6] it is not difficult to write down the expression for low-temperature (L. T.) Lorenz number L,_, resulting from both types of scattering when the d-SDF correlations are treated in the RPA manner of Izuyama et al. 141. Denoting by fthe dimensionless parameter IN, (No = density of d-band states at the fermi surface) and assuming spherical d
Volume 26A, number 2
PHYSICS
and s bands, we obtain Le-e = !&/{I
+4[Zrt(O) +ZTr(O)j/[Z,,(V +Zt$(2)]] (2)
with Z?,(N) = J2dq &[I
+&(q)12;
0
2 Z?+(N)= 3 I dq
(3)
&[l
- i;c(q)12
0
where Lo denotes the classical Lorenz number and u (ix) the Lindhard function (u(0) = 1) [8]. Eq. (2) expresses L,_, as a function of the RPA d-band electron correlation. We note that asr1, from below, L,_, - 0, showing that the effect of increasing electron correlation in the d-band is to decrease the Lorentz number. This arises because the thermal resistivity (associated with the moments Z(0) increases more rapidly with increasing F than does the electrical resistivity (associated with the moments Z(2)). This variation is shown in fig. 1 where the integrals (3) have been evaluated numerically. From susceptibility measurements Foner [3]* has recently estimated r to be 2 0.875 for pure Pd. With this value (2) gives Le_e -' 0.56x lo-%V which is in good accord with the recent low temperature ideal Lorenz ratio L.i = 0.68-+0.05 x x 10-6~ (observed in Pd by Schriempf [2]. Schriempf’s value is actually obtained from (2) if TN<0.80. * A. P. W. band-structure
calculations ly indicated that I M 0.92.
[9] have previous-
LETTERS
18 December 1967
In conclusion we point out that for i= 0, Le_e = AL, = 1.02 x 10-3~. It is seen from eqs. (l-3) that this value coincides with that for ordinary “Baber-scattering” for a strongly screened Coulomb potential. We note that the low temperature ideal Lorenz ratio of pure Ni is 1.0 x lo-6w [lo]. The author wishes to thank Drs. A. I. Schindler and J. T. Schriempf for several stimulating discussions at N. R. L., Washington D. C., and the latter for communicating experimental data prior to publication. 1. J. M. Ziman, Electrons and Phonons, (Clarendon Press, Oxford, England, 1960), Chap. 9. 2. J. T. Schriempf, Phys. Rev. Letters 19 (1967) 1131. 3. S. Foner, R. Doclo and E. J. McNiff Jr., Proc. Intern. Congr. on Magnetism. Boston. Sept. 1967. to be published. 4. T.Izuyama, D.J.Kim and R.Kubo, J. Phys. Sot. Japan 18 (1963) 1025. 5. D. L. Mills and P. Lederer, J. Phys. Chem. Solids 27 (1966) 1805, and to be published. and M. J.Rice, Phys. Rev. to be 6. A.I.Schindler published; M. J.Rice, Phys. Rev. 159 (1967) 155. Possible numerical errors in a variational calculation of the transport coefficients, mentioned in the former reference, have since been shown by the author to be negligibly small for the present scattering mechanism. Proc. Roy. Sot. (London) Al58 (1937) 7. W.G.Baber, 133. 8. J.Lindhard, Dan. Vid. Selsk. Mat. fys. Mdd., 28 No. 8 (1954). J.O.DimmockandA.M.Furdyna, 9. A.J.Freeman, J. Appl. Phys. 37 (1966) 1256. 10. G. K.White and R. J. Tainsh, Phys. Rev. Letters 19 (1967) 165.
*****
87
number 2
PHYSICS
ITINERANT ELECTRON LORENZ NUMBER
LETTERS
18 December
CORRELATION OF TRANSITION
AND THE METALS
1967
IDEAL
M. J. RICE
SolidState Theory Group, Departmentof Physics, Imperial College, London, England Received
14 November
1967
A simple discussion of the effect of itinerant electron correlation on the ideal Lorenz tion metals is given and applied to reconcile data recently taken on Pd.
It is well known in the theory of transition metals that interband electron-electron scatter ing yields a characteristic Lorenz number L,_, which is temperature independent *. In this letter we calculate this quantity when the interband scattering takes the form of an s-d exchange interaction and point out that this contribution can give an indirect measure of the magnitude of itinerant d-band electron correlation: the calculation, straight forward extensions of which may be useful in characterizing in a simple way the low-temperature transport properties of a whole class of strongly correlated transition metal systems, accounts well for data recently taken on Pd [2,3]. We consider a simple itinerant model [4] in which the d-electrons interact via a semi-phenomenological short range repulsion Zb(r) and are coupled to a non-interacting s-band (assumed to make the principal contribution to the electrical currents) through an interband exchange interaction Hsd = w,d Jd3,S(r)
* s(f)
'*l/l-~
(1)
per unit cell, where S(r)and s(r) denote respectively the fermion spin density operators for the d and s electrons. (1) can be analysed into terms describing the scattering of s electrons from either transverse or longitudinal d-band spin density fluctuations (d-SDF). The effect on the resistivities of the former type of scattering, in which the s-electron spin is flipped, has been discussed in the literature recently [5,6], while the latter form of scattering is of the same type * That is, yielding an electrical and thermal resistivity varying as T2 and T, respectively, e.g. [ 11. 86
I
Fig.
I. he_,
number of transi-
I
2o
3
as a function of the RPA d-band electron correlation.
discussed some years ago by Baber [‘7]. Following closely the theory of the electrical and thermal resistivities developed in [6] it is not difficult to write down the expression for low-temperature (L. T.) Lorenz number L,_, resulting from both types of scattering when the d-SDF correlations are treated in the RPA manner of Izuyama et al. 141. Denoting by fthe dimensionless parameter IN, (No = density of d-band states at the fermi surface) and assuming spherical d
Volume 26A, number 2
PHYSICS
and s bands, we obtain Le-e = !&/{I
+4[Zrt(O) +ZTr(O)j/[Z,,(V +Zt$(2)]] (2)
with Z?,(N) = J2dq &[I
+&(q)12;
0
2 Z?+(N)= 3 I dq
(3)
&[l
- i;c(q)12
0
where Lo denotes the classical Lorenz number and u (ix) the Lindhard function (u(0) = 1) [8]. Eq. (2) expresses L,_, as a function of the RPA d-band electron correlation. We note that asr1, from below, L,_, - 0, showing that the effect of increasing electron correlation in the d-band is to decrease the Lorentz number. This arises because the thermal resistivity (associated with the moments Z(0) increases more rapidly with increasing F than does the electrical resistivity (associated with the moments Z(2)). This variation is shown in fig. 1 where the integrals (3) have been evaluated numerically. From susceptibility measurements Foner [3]* has recently estimated r to be 2 0.875 for pure Pd. With this value (2) gives Le_e -' 0.56x lo-%V which is in good accord with the recent low temperature ideal Lorenz ratio L.i = 0.68-+0.05 x x 10-6~ (observed in Pd by Schriempf [2]. Schriempf’s value is actually obtained from (2) if TN<0.80. * A. P. W. band-structure
calculations ly indicated that I M 0.92.
[9] have previous-
LETTERS
18 December 1967
In conclusion we point out that for i= 0, Le_e = AL, = 1.02 x 10-3~. It is seen from eqs. (l-3) that this value coincides with that for ordinary “Baber-scattering” for a strongly screened Coulomb potential. We note that the low temperature ideal Lorenz ratio of pure Ni is 1.0 x lo-6w [lo]. The author wishes to thank Drs. A. I. Schindler and J. T. Schriempf for several stimulating discussions at N. R. L., Washington D. C., and the latter for communicating experimental data prior to publication. 1. J. M. Ziman, Electrons and Phonons, (Clarendon Press, Oxford, England, 1960), Chap. 9. 2. J. T. Schriempf, Phys. Rev. Letters 19 (1967) 1131. 3. S. Foner, R. Doclo and E. J. McNiff Jr., Proc. Intern. Congr. on Magnetism. Boston. Sept. 1967. to be published. 4. T.Izuyama, D.J.Kim and R.Kubo, J. Phys. Sot. Japan 18 (1963) 1025. 5. D. L. Mills and P. Lederer, J. Phys. Chem. Solids 27 (1966) 1805, and to be published. and M. J.Rice, Phys. Rev. to be 6. A.I.Schindler published; M. J.Rice, Phys. Rev. 159 (1967) 155. Possible numerical errors in a variational calculation of the transport coefficients, mentioned in the former reference, have since been shown by the author to be negligibly small for the present scattering mechanism. Proc. Roy. Sot. (London) Al58 (1937) 7. W.G.Baber, 133. 8. J.Lindhard, Dan. Vid. Selsk. Mat. fys. Mdd., 28 No. 8 (1954). J.O.DimmockandA.M.Furdyna, 9. A.J.Freeman, J. Appl. Phys. 37 (1966) 1256. 10. G. K.White and R. J. Tainsh, Phys. Rev. Letters 19 (1967) 165.
*****
87