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Anisotropic mixing in Kondo systems
Journal of Magnetism and Magnetic Materials 76 & 77 (1988) 73-74 North-Holland, Amsterdam
73
ANISOTROPIC MIXING IN KONDO SYSTEMS Peter M. L E V Y and Shufeng Z H A N G
Department of Pl~vsics, New York Uniuersi(v, 4 Washington Place, New York, NY 10003. USA We take account of the anisotropy of the mixing interaction in the Anderson Hamiltonian to lift the degeneracy of the local state. Without introducing new parameters we are able to calculate the crystal field splittings and linewidths of the local f states in Kondo-type cerium compounds. Our theoretical understanding of the behavior of K o n d o - t y p e systems is based on the A n d e r s o n model of local m o m e n t s in metals. The central feature of this model is the mixing interaction between electrons in local and conduction states [1],
ttmi~ =
Y'. Vk .... ct,+of,,, + h.c. ko,
(1)
m
where
V
-
if
~/7,,(r)V(r)ECk'R,ao(r--Ri)
dr
We assume the local state to be an f electron in a s p i n - o r b i t coupled j state, as we are interested in, interalia, cerium compounds, and the conduction electron state is written in terms of Wannier functions. In the conventional treatment of the K o n d o problem only the isotropic part of the mixing interaction is considered, i.e., the dependence on the orbital index m in eq. (1) is suppressed [2]. When one accounts for the dependence of the mixing on the orbital index m the local state is split. We find the magnitude of the splitting due to the anisotropic c o m p o n e n t of the mixing interaction accounts for the "crystal field" splittings observed in K o n d o - t y p e systems containing cerium. Anisotropic mixing has been previously used to account for magnetocrystalline anisotropy in the cerium pnictides [3]. The new point we are making is that for K o n d o systems, at least those containing cerium, it is unnecessary to introduce
additional crystal field terms in the Anderson model when one properly accounts for the anisotropy present in the mixing interaction itself. We have taken into account the anisotropy in mixing interaction, and have found a solution for the ground state properties of the single-ion A n d e r s o n model in the K o n d o regime (in the limit of infinite U) [4]. We used the variational approach [5] to obtain the g r o u n d state to first order in I / N , where N = 2 j + 1 is the degeneracy of the
local state, N = 6 for cerium j = 5 / 2 . We find the splittings of the local f state are A ....
l(iv = ~-
-F~)ln
ef-D ~f
,
(2)
where F,, - vN(O) [ Vm [ z, and c f is the position of the f state relative to the Fermi level, D is the half-width of the conduction b a n d which is assumed to be flat, N(0) is the single particle density of states, and V is the anisotropic mixing parameter eq. (1), which is assumed to be independent of k and o. The index m denotes linear combinations of states I j m j ) which transform according to the irreducible representations of the point group symmetry at the site of the local state. Also, we find the widths of the f levels are approximately given by [4],
F* = N.----~ 1 +
E
] - ~ ~
,
(3)
m* ~ m
where [6] T,, =
(4)
mVI,m A"m'-D-T'N°''Fm'/N''c'' is the low temperature energy scale for the ruth sublevel, and N,, is the multiplicity of the sublevel. The splittings (2) and linewidths (3) can be written in terms of the isotropic mixing parameter
1 E/m__. ,~U(0) F = -~ U Y~ [ V,, [ 2 m
(5)
m
and the conventional low temperature energy scale of the K o n d o problem [2] r 0 = D e -'~I~fl/NF
(6)
We note that the overall energy scale for the local f level splittings is that of the isotropic mixing parameter F which is typically of the order of 15 to 20 meV for K o n d o - t y p e alloys containing cerium. On the contrary, the scale for the linewidths is given by To which is of the order of 1 to 10 K.
0 3 0 4 - 8 8 5 3 / 8 8 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V.
74
P.M. Let'v, S. Zhany; / Anisotropic mixing in Kondo systems
To determine the anisotropy of the mixing parameters entering the expressions for the splittings and linewidths of the local f levels, we evaluated the two-center intergals entering the mixing parameters V,, in terms of radial (Slater Koster) integrals and related them to a comparable set of integrals entering the isotropic mixing parameter F. In this manner we relate the F,, to F. To evaluate the integrals and sum over neighboring anion sites we used the Slater-Koster tables for f electrons developed by Takegahara et al. [7]. To keep the number of parameters to a minimum we made the following assumptions. 1) The Wannier function in eq. (1) is replaced by an atomic orbital with definite angular momentum. 2) We take the mixing interaction V(r) in eq. (1) to be a function only of its radial variable. With these two assumptions the two-center integrals in the mixing interaction separate into a product of radial and angular integrals. The radial integrals are unknown parameters, which are used to relate F,,, and F. 3) When the atomic orbital on the anion site has p or higher angular momentum, we consider only the o configuration in which the anion orbital points towards the central ion(cerium). 4) When there are more than one set of equivalent neighbors we fix their relative contributions, so as to reproduce the observed ratio of the f level splittings. As an example of the above procedure we considered CeA12 which has a cubic structure [8]. The six states of the ground manifold of cerium ]jm/) split into a doublet F 7 and quartet I"s. From Eqs. (2) and (5) we find the F~-I7 splitting is a..~ 87 =
0.32,~ F In - - c~ffD .
(7)
By using a set of parameters suggested in Bickers, Cox and Wilkins [2], for N = 6 , ~ f / D = - 0 . 6 7 , F / D = 0.05, and D = 3 eV, i.e., parameter set no. 1 in table 1 of ref. [2] we find for CeA12 As7 = 162 K. This value compares quite favorably with the data found from neutron scattering data for CeA12 [8,9], i.e., a F7 ground doublet with As7 = 100 K. The widths of the levels are estimated from eqs. (3) and (4) by using the observed splitting As7, and the T~ found from eq. (6) when the above parameters are used. We find /'7* = 2 K and Fs* = 21 K. These values are close to those observed [9,10] Fv*=3 5 K and Fs* = 2 0 K. We have also calculated the splittings and linewidths for CeA13 and find comparable results [4].
We have found that the estimated splittings for both CeA12 and CeAI 3 are larger than those observed. This is understandable when one realizes that the orbitals on the anion(aluminum) sites are more diffuse than the atomic orbitals used in our calculations. On the other hand, the linewidth is relatively unaffected by this redistribution of charge, and our estimates should be close to those observed. In conclusion, we can account for the "crystal field" splittings of local states in Kondo-type systems without introducing new interaction or parameters. From our examples we find anisotropic mixing is the dominant mechanism for the splittings of the local 4f states observed in Kondo-type cerium compounds. Indubitably, there are other contributions to the "'crystal field" splittings, but they are small compared to those coming from the mixing interaction. From eqs. (2) and (7) we note that the overall scah, of the splittings is governed by the isotropic mixing parameter I', while the linewidths of the levels eqs. (3) and (4) are determined by the temperature T,. The actual size of the splittings and the variation of the linewidths from one level to another are governed by the anisotropy of the mixing parameter. We would like to thank Professor S. Horn for very fruitful discussions and comments, and to Prof. E. Mtiller-Hartmann and Dr. H.J. Schmidt for pointing out the need to include higher order in 1 / N corrections to determine the linewidth.
References [1] P.W. Anderson, Phys. Rev. 124 (1961) 41. [2] N.E. Bickers, D . L Cox and J.W. Wilkins, Phys. Rev. B36 (1987) 2036. [3] K. Takegahara, H. Takahashi, A. Yanase and T. Kasuya, J. Phys. C14 (1981) 737. [4] P.M. Levy and S. Zhang, to be published. [5] O. Gunnarsson and K. SchOnhammer, Phys. Rev. B 28 (1983) 4315. N.E. Bickers, Rev. Mod. Phys. 59 (1987) 845. See also N. Grewe and H. Keiter, Phys. Rev. B 24 (1981) 4420. [6] For a two level system this result reduces to the Kondo temperature found by K. Yamada, K. Yosida and K. Hanzawa, Prog. Theor. Phys. 71 (1984) 450. Similar results have been recently reported by H.J. Schmidt, K.-A. Ulrich and E. Miiller-Hartmann, presented at this conference (A9). [7] K. Takegahara, Y. Aoki and A. Yanase, J. Phys. C13 (1980) 583. [8] J.A. White, H.J. Williams, J.H. Wernick and R.C. Sherwood. Phys. Rev. 131 (1963) 1039. [9] B.D. Rainford, in: Crystal Field Effects in Metals and Alloys, ed. A. Furrer (Plenum Press, New York 1977) p. 243.
73
ANISOTROPIC MIXING IN KONDO SYSTEMS Peter M. L E V Y and Shufeng Z H A N G
Department of Pl~vsics, New York Uniuersi(v, 4 Washington Place, New York, NY 10003. USA We take account of the anisotropy of the mixing interaction in the Anderson Hamiltonian to lift the degeneracy of the local state. Without introducing new parameters we are able to calculate the crystal field splittings and linewidths of the local f states in Kondo-type cerium compounds. Our theoretical understanding of the behavior of K o n d o - t y p e systems is based on the A n d e r s o n model of local m o m e n t s in metals. The central feature of this model is the mixing interaction between electrons in local and conduction states [1],
ttmi~ =
Y'. Vk .... ct,+of,,, + h.c. ko,
(1)
m
where
V
-
if
~/7,,(r)V(r)ECk'R,ao(r--Ri)
dr
We assume the local state to be an f electron in a s p i n - o r b i t coupled j state, as we are interested in, interalia, cerium compounds, and the conduction electron state is written in terms of Wannier functions. In the conventional treatment of the K o n d o problem only the isotropic part of the mixing interaction is considered, i.e., the dependence on the orbital index m in eq. (1) is suppressed [2]. When one accounts for the dependence of the mixing on the orbital index m the local state is split. We find the magnitude of the splitting due to the anisotropic c o m p o n e n t of the mixing interaction accounts for the "crystal field" splittings observed in K o n d o - t y p e systems containing cerium. Anisotropic mixing has been previously used to account for magnetocrystalline anisotropy in the cerium pnictides [3]. The new point we are making is that for K o n d o systems, at least those containing cerium, it is unnecessary to introduce
additional crystal field terms in the Anderson model when one properly accounts for the anisotropy present in the mixing interaction itself. We have taken into account the anisotropy in mixing interaction, and have found a solution for the ground state properties of the single-ion A n d e r s o n model in the K o n d o regime (in the limit of infinite U) [4]. We used the variational approach [5] to obtain the g r o u n d state to first order in I / N , where N = 2 j + 1 is the degeneracy of the
local state, N = 6 for cerium j = 5 / 2 . We find the splittings of the local f state are A ....
l(iv = ~-
-F~)ln
ef-D ~f
,
(2)
where F,, - vN(O) [ Vm [ z, and c f is the position of the f state relative to the Fermi level, D is the half-width of the conduction b a n d which is assumed to be flat, N(0) is the single particle density of states, and V is the anisotropic mixing parameter eq. (1), which is assumed to be independent of k and o. The index m denotes linear combinations of states I j m j ) which transform according to the irreducible representations of the point group symmetry at the site of the local state. Also, we find the widths of the f levels are approximately given by [4],
F* = N.----~ 1 +
E
] - ~ ~
,
(3)
m* ~ m
where [6] T,, =
(4)
mVI,m A"m'-D-T'N°''Fm'/N''c'' is the low temperature energy scale for the ruth sublevel, and N,, is the multiplicity of the sublevel. The splittings (2) and linewidths (3) can be written in terms of the isotropic mixing parameter
1 E/m__. ,~U(0) F = -~ U Y~ [ V,, [ 2 m
(5)
m
and the conventional low temperature energy scale of the K o n d o problem [2] r 0 = D e -'~I~fl/NF
(6)
We note that the overall energy scale for the local f level splittings is that of the isotropic mixing parameter F which is typically of the order of 15 to 20 meV for K o n d o - t y p e alloys containing cerium. On the contrary, the scale for the linewidths is given by To which is of the order of 1 to 10 K.
0 3 0 4 - 8 8 5 3 / 8 8 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V.
74
P.M. Let'v, S. Zhany; / Anisotropic mixing in Kondo systems
To determine the anisotropy of the mixing parameters entering the expressions for the splittings and linewidths of the local f levels, we evaluated the two-center intergals entering the mixing parameters V,, in terms of radial (Slater Koster) integrals and related them to a comparable set of integrals entering the isotropic mixing parameter F. In this manner we relate the F,, to F. To evaluate the integrals and sum over neighboring anion sites we used the Slater-Koster tables for f electrons developed by Takegahara et al. [7]. To keep the number of parameters to a minimum we made the following assumptions. 1) The Wannier function in eq. (1) is replaced by an atomic orbital with definite angular momentum. 2) We take the mixing interaction V(r) in eq. (1) to be a function only of its radial variable. With these two assumptions the two-center integrals in the mixing interaction separate into a product of radial and angular integrals. The radial integrals are unknown parameters, which are used to relate F,,, and F. 3) When the atomic orbital on the anion site has p or higher angular momentum, we consider only the o configuration in which the anion orbital points towards the central ion(cerium). 4) When there are more than one set of equivalent neighbors we fix their relative contributions, so as to reproduce the observed ratio of the f level splittings. As an example of the above procedure we considered CeA12 which has a cubic structure [8]. The six states of the ground manifold of cerium ]jm/) split into a doublet F 7 and quartet I"s. From Eqs. (2) and (5) we find the F~-I7 splitting is a..~ 87 =
0.32,~ F In - - c~ffD .
(7)
By using a set of parameters suggested in Bickers, Cox and Wilkins [2], for N = 6 , ~ f / D = - 0 . 6 7 , F / D = 0.05, and D = 3 eV, i.e., parameter set no. 1 in table 1 of ref. [2] we find for CeA12 As7 = 162 K. This value compares quite favorably with the data found from neutron scattering data for CeA12 [8,9], i.e., a F7 ground doublet with As7 = 100 K. The widths of the levels are estimated from eqs. (3) and (4) by using the observed splitting As7, and the T~ found from eq. (6) when the above parameters are used. We find /'7* = 2 K and Fs* = 21 K. These values are close to those observed [9,10] Fv*=3 5 K and Fs* = 2 0 K. We have also calculated the splittings and linewidths for CeA13 and find comparable results [4].
We have found that the estimated splittings for both CeA12 and CeAI 3 are larger than those observed. This is understandable when one realizes that the orbitals on the anion(aluminum) sites are more diffuse than the atomic orbitals used in our calculations. On the other hand, the linewidth is relatively unaffected by this redistribution of charge, and our estimates should be close to those observed. In conclusion, we can account for the "crystal field" splittings of local states in Kondo-type systems without introducing new interaction or parameters. From our examples we find anisotropic mixing is the dominant mechanism for the splittings of the local 4f states observed in Kondo-type cerium compounds. Indubitably, there are other contributions to the "'crystal field" splittings, but they are small compared to those coming from the mixing interaction. From eqs. (2) and (7) we note that the overall scah, of the splittings is governed by the isotropic mixing parameter I', while the linewidths of the levels eqs. (3) and (4) are determined by the temperature T,. The actual size of the splittings and the variation of the linewidths from one level to another are governed by the anisotropy of the mixing parameter. We would like to thank Professor S. Horn for very fruitful discussions and comments, and to Prof. E. Mtiller-Hartmann and Dr. H.J. Schmidt for pointing out the need to include higher order in 1 / N corrections to determine the linewidth.
References [1] P.W. Anderson, Phys. Rev. 124 (1961) 41. [2] N.E. Bickers, D . L Cox and J.W. Wilkins, Phys. Rev. B36 (1987) 2036. [3] K. Takegahara, H. Takahashi, A. Yanase and T. Kasuya, J. Phys. C14 (1981) 737. [4] P.M. Levy and S. Zhang, to be published. [5] O. Gunnarsson and K. SchOnhammer, Phys. Rev. B 28 (1983) 4315. N.E. Bickers, Rev. Mod. Phys. 59 (1987) 845. See also N. Grewe and H. Keiter, Phys. Rev. B 24 (1981) 4420. [6] For a two level system this result reduces to the Kondo temperature found by K. Yamada, K. Yosida and K. Hanzawa, Prog. Theor. Phys. 71 (1984) 450. Similar results have been recently reported by H.J. Schmidt, K.-A. Ulrich and E. Miiller-Hartmann, presented at this conference (A9). [7] K. Takegahara, Y. Aoki and A. Yanase, J. Phys. C13 (1980) 583. [8] J.A. White, H.J. Williams, J.H. Wernick and R.C. Sherwood. Phys. Rev. 131 (1963) 1039. [9] B.D. Rainford, in: Crystal Field Effects in Metals and Alloys, ed. A. Furrer (Plenum Press, New York 1977) p. 243.