- Email: [email protected]
Neutron scattering linewidth for cerium Kondo impurities
Journal of Magnetism and Magnetic Materials 63 & 64 (1987) 219-221 North-Holland, Amsterdam
219
N E U T R O N SCATTERING LINEWIDTH FOR CERIUM K O N D O IMPURITIES T H O H N and J K E L L E R
Fakultat fur Physlk, UnwersltatRegensburg, D-8400 Regensburg, Fed Rep Germany The neutron scattenng lmewidth of Ce lmpurmes m a cubic crystalhne electric field is investigated by different methods Starting from the Anderson Hamlltonmn we calculate the dynamic suscepUbdlty with help of a self-consistent perturbation theory and compare tt with results obtained by a mean-field approxlmauon
1. Introduction In most Ce alloys and c o m p o u n d s the mixing of the Ce f-electrons with the conduction electrons leading to Intermediate valence or K o n d o behavlour cannot be ignored In the magnetic properties it manifests itself in a finite magnetic susceptIbihty at T - - 0 for single Ce ions and a large hnewldth in the neutron scattering spectra In this report we present calculations of the dynamical susceptlblhty for Ce impurities m a metal in the presence of a cubic crystalhne electric field including the K o n d o effect We assume that the physical properties of the Ce ions can be described by an Anderson Hamiltonmn of the form H--~.
+ + ~ Efmf ,.,f,,, + ¢k,,,Ck,,,Ck,,,
kra
2. Perturbation theory
m
+ ~ (Vc~,.f,,, + h c ) + ( U / 2 ) ~ /+/+'/,.,'fro klan
For the dynamical properties one has to use approximations Recently Gunnarson and S c h o n h a m m e r [1] have calculated the dynamical susceptibility including crystal field effects using a variational approach Cox et al [2] made a systematic investigation of the quasielastic hnewldth using a perturbation theory developed by Kelter and Grewe [3], K u r a m o t o [4] and C o l e m a n [5] Maekawa et al [6] used a similar perturbation theory starting from a C o q b h n Schrieffer Hamlltonian In our contribution we want to c o m p a r e results for the dynamic susceptIblhty obtained by the perturbation theory with calculations using a generalization of a meanfield theory developed by C o l e m a n [7] and Newns et al [8]
m?rl'
(1) containing the hybridization of the f-electrons in the Ce / = 5/2 groundstate multiplet with conductlon electrons of the same symmetry In the presence of a cubic crystalline electric field this multtplet is split into a F7-doublet and a Faquartet We assume that all the f-state energies Err. are far below the fermi energy and the local C o u l o m b repulsion U is sufficiently large that only the e m p t y and single occupied f-states have to be considered T h e static susceptibility for this model can be calculated exactly with Bethe ansatz methods
Within the so-called non-crossing approximation the imaginary part of the dynamical susceptibility which is closely related to the neutron scattering cross section can be written as [4] )(,,(to) _- z~_l( 1 _ e_t3o,) ~.. _1 f dE e -tu
x I(miJzim')i2R~(e)R~(e + to),
(2)
with Zf=-~l
I d ~ R " ( e ) e -t3~
H e r e Rm = ( z - E . . - ~ m (z)) -1 are propagators for the different f-states Their self-energies caused by the mixing with conduction electrons
0304-8853/87/$03 50 © Elsevier Science Pubhshers B V (North-Holland Physics Pubhshlng Dwlslon)
T Hohn, J Keller I Cerium Kondo irnpunues
220
are calculated selfconsistently from a set of integral equations containing also the propagator for the empty f-state [4] We have solved these equations for a level scheme with E r, = - 1200 K and Er. = - 1 1 0 0 K, a hybridization energy of zXo= ~ V 2 N o = 120 K (No is the conduction electron density of states at the fermi energy), and a cut-off of the conduction electron bandwidth at D = 104 K Then we obtain an occupation of the f-level of nf = 0 95 for low temperatures Results for the dynamic susceptlblhty are shown in fig 1 These may be compared with calculations by Maekawa et al [6] based on an equivalent Coqbhn-Schrleffer Hamlltonlan with an exchange interaction given by 1 , - , - = V 2 / 2 (llEf,.+llE~,..) For the parameters given above the value of the exchange interaction is l~,.,No ~ - 0 03 for all m, which Is also the value used m [6] The results of the two calculations are rather similar for this set of parameters, where nf is close to 1 and the Coqbhn-Schrmffer model is applicable One finds broad quasielastlc and inelastic peaks If one calculates instead the hnewidth of the elastic peak in the C o q b h n Schrmffer model using second order perturbation theory one obtains a K o r r m g a law of the form 3' = 4"rr(INo)2T at low temperatures The resulting llnewidth is much smaller (factor of 20) than the one shown m fig 1 Because of numerical dlfficultms we were not able to con-
0168K-1
7~ 0 I0
Ai--20 K H"d ~50K
0 08
I1_\
3 006
tmue our calculations to lower temperatures, but already at these high temperatures the influence of the K o n d o effect is clearly seen We have also performed calculations for a large hybridization (V2No = 70 K) In that case an inelastic peak can still be distinguished, but at a much larger energy
3. Mean-field theory In the mean-field approximation [7,8] the original Anderson Hamdtonlan is replaced by a simple hybridization Hamdtonlan where the local Coulomb repulsion is neglected, the fstates are shifted up to the fermi energy Ef,.---* El,. = El,. + A and the mixing energy is reduced Ao---~A At T = 0 the quantities A and A are determined by the following set of equations
1
nf = ~ nf,~ = ~ - arctan(A/~.,) = 1 - A/Ao, m
(3)
m
(Ao/-rr) ln((~,. + A2)I/2/D) =
-- A
(4)
rtl
The dynamic susceptlblhty can then be calculated from
x Gim'(lW. + IWs),
(5)
where G,.(lW.) = 0w. - el,. + iA sign(w.)) , are the Green's functions of the f-states correspondlng to the hybridization Hamlltoman From (5) It is straightforward to calculate the susceptlbdlty We do not write down the lengthy formula for the dynamic susceptlbdlty, but will give only the numerical results below For the static susceptlblhty (o)s = 0), however, one obtains a very simple result at T = 0
x = Y. IJ~-.,Iza/('rr(~2,- + A2))
3
m
0
I
50
i
100
I
150 200 WIK]
Fig 1 Dynamical susceptlblhty calculated by self-consistent p e r t u r b a t i o n theory with El 7 = - 1200 K, Ei~ = - 1100 K, V2No = 38 K and D = 104 K for different t e m p e r a t u r e s
+
Z ]JZrn'la(/'lfm -- n f m ) / ( E f m -- l~f,n ) , mT~rn
(6)
where nf,. is defined m (3) It shows a separation m a van Vleck term (rn 4: rn') and a Curie term ( m = m') where, however, the singular tem-
T Hohn, J Keller/Cerium Kondo :mpunaes p e r a t u r e d e p e n d e n c e of the f o r m 1 / T has b e e n r e p l a c e d by a finite v a l u e at T = 0 w h i c h c a n be identified with the a d d m o n a l d e n s i t y of states at the fermi e n e r g y d u e to the l o c a h z e d states U s i n g the same p a r a m e t e r s as a b o v e we h a v e solved the set of e q u a t i o n s (3),(4) F o r the e n e r g i e s a n d o c c u p a t i o n of the q u a s l p a r t l c l e fstates we o b t a i n ~r7 = 4 K a n d ~r8 = 104 K, A = 11 K, nf = 0 9 Results for the d y n a m i c s u s c e p tlblhty c a l c u l a t e d for T = 0 are s h o w n m fig 2 T h e y s h o u l d serve as a n e x t r a p o l a t i o n to T = 0 of the results from fig 1 b u t the a g r e e m e n t is n o t
000~
0003 7 0002 3
-X 0 001
~
//r-\
100
t.,O[ K ]
200
Fig 2 Dynamical susceptlblhty calculated by mean-field theory with the same parameters as m fig 1 The dashed curve are the results from fig 1 for T = 20 K
221
v e r y g o o d T h i s m a y be d u e to the a b s e n c e of q u a s l p a r t l c l e i n t e r a c t i o n s in the m e a n - f i e l d t h e o r y T h e s e m a y be sizeable tf the K o n d o t e m p e r a t u r e ts small c o m p a r e d to the crystalfield s p h t t m g a n d the g r o u n d s t a t e m u l t l p l e t has a low d e g e n e r a c y as in the p r e s e n t case
References [1] O Gunnarson and K Schonhammer, in Theory of Heavy Ferrmons and Valence Fluctuauons eds T Kasuya and T Saso (Springer, Berhn, 1985) p 100, J Magn Magn Mat 52 (1985)227 [2] D L Cox, N E Bickers and J W Wllklns, J Appl Phys 57 (1985) 3166 N E Bickers, DL Cox and JW Wdktns, Phys Rev Lett 54 (1985) 230 [3] N Grewe and H Kelter, Phys Rev B 24 (1981) 4420 N Grewe, Z Physlk B 53 (1983) 271 For a review see H Kelter and G Morandl, Phys Rep 109 (1984) 227 [4] Y Kuramoto, Z Physlk B 53 (1983) 37 H KOllma, Y Kuramoto and M Tachlkl, Z Physlk B 54 (1984) 293 [5] P Coleman, Phys Rev B 29 (1984) 3035 [6] S Maekawa, S Takahashl, S Kashlba and M Tachlkl, J Phys Soc Japan 54 (1985)1955, J Magn Magn Mat 52 (1985) 149 [7] P Coleman, J Magn Magn Mat 47&48 (1985)323, 52 (1985) 223 P Coleman, m Theory of Heavy Fermlons and Valence Fluctuations, eds T Kasuya and T Saso (Springer, Berhn, 1985) p 163 [8] D Newns and N Read, J Phys C 16 (1983) 3273 D W Newns, N Read and A C Hewson, m Moment Formarion m Sohds, ed W J K Buyers (Plenum, New York, 1984) p 257
219
N E U T R O N SCATTERING LINEWIDTH FOR CERIUM K O N D O IMPURITIES T H O H N and J K E L L E R
Fakultat fur Physlk, UnwersltatRegensburg, D-8400 Regensburg, Fed Rep Germany The neutron scattenng lmewidth of Ce lmpurmes m a cubic crystalhne electric field is investigated by different methods Starting from the Anderson Hamlltonmn we calculate the dynamic suscepUbdlty with help of a self-consistent perturbation theory and compare tt with results obtained by a mean-field approxlmauon
1. Introduction In most Ce alloys and c o m p o u n d s the mixing of the Ce f-electrons with the conduction electrons leading to Intermediate valence or K o n d o behavlour cannot be ignored In the magnetic properties it manifests itself in a finite magnetic susceptIbihty at T - - 0 for single Ce ions and a large hnewldth in the neutron scattering spectra In this report we present calculations of the dynamical susceptlblhty for Ce impurities m a metal in the presence of a cubic crystalhne electric field including the K o n d o effect We assume that the physical properties of the Ce ions can be described by an Anderson Hamiltonmn of the form H--~.
+ + ~ Efmf ,.,f,,, + ¢k,,,Ck,,,Ck,,,
kra
2. Perturbation theory
m
+ ~ (Vc~,.f,,, + h c ) + ( U / 2 ) ~ /+/+'/,.,'fro klan
For the dynamical properties one has to use approximations Recently Gunnarson and S c h o n h a m m e r [1] have calculated the dynamical susceptibility including crystal field effects using a variational approach Cox et al [2] made a systematic investigation of the quasielastic hnewldth using a perturbation theory developed by Kelter and Grewe [3], K u r a m o t o [4] and C o l e m a n [5] Maekawa et al [6] used a similar perturbation theory starting from a C o q b h n Schrieffer Hamlltonian In our contribution we want to c o m p a r e results for the dynamic susceptIblhty obtained by the perturbation theory with calculations using a generalization of a meanfield theory developed by C o l e m a n [7] and Newns et al [8]
m?rl'
(1) containing the hybridization of the f-electrons in the Ce / = 5/2 groundstate multiplet with conductlon electrons of the same symmetry In the presence of a cubic crystalline electric field this multtplet is split into a F7-doublet and a Faquartet We assume that all the f-state energies Err. are far below the fermi energy and the local C o u l o m b repulsion U is sufficiently large that only the e m p t y and single occupied f-states have to be considered T h e static susceptibility for this model can be calculated exactly with Bethe ansatz methods
Within the so-called non-crossing approximation the imaginary part of the dynamical susceptibility which is closely related to the neutron scattering cross section can be written as [4] )(,,(to) _- z~_l( 1 _ e_t3o,) ~.. _1 f dE e -tu
x I(miJzim')i2R~(e)R~(e + to),
(2)
with Zf=-~l
I d ~ R " ( e ) e -t3~
H e r e Rm = ( z - E . . - ~ m (z)) -1 are propagators for the different f-states Their self-energies caused by the mixing with conduction electrons
0304-8853/87/$03 50 © Elsevier Science Pubhshers B V (North-Holland Physics Pubhshlng Dwlslon)
T Hohn, J Keller I Cerium Kondo irnpunues
220
are calculated selfconsistently from a set of integral equations containing also the propagator for the empty f-state [4] We have solved these equations for a level scheme with E r, = - 1200 K and Er. = - 1 1 0 0 K, a hybridization energy of zXo= ~ V 2 N o = 120 K (No is the conduction electron density of states at the fermi energy), and a cut-off of the conduction electron bandwidth at D = 104 K Then we obtain an occupation of the f-level of nf = 0 95 for low temperatures Results for the dynamic susceptlblhty are shown in fig 1 These may be compared with calculations by Maekawa et al [6] based on an equivalent Coqbhn-Schrleffer Hamlltonlan with an exchange interaction given by 1 , - , - = V 2 / 2 (llEf,.+llE~,..) For the parameters given above the value of the exchange interaction is l~,.,No ~ - 0 03 for all m, which Is also the value used m [6] The results of the two calculations are rather similar for this set of parameters, where nf is close to 1 and the Coqbhn-Schrmffer model is applicable One finds broad quasielastlc and inelastic peaks If one calculates instead the hnewidth of the elastic peak in the C o q b h n Schrmffer model using second order perturbation theory one obtains a K o r r m g a law of the form 3' = 4"rr(INo)2T at low temperatures The resulting llnewidth is much smaller (factor of 20) than the one shown m fig 1 Because of numerical dlfficultms we were not able to con-
0168K-1
7~ 0 I0
Ai--20 K H"d ~50K
0 08
I1_\
3 006
tmue our calculations to lower temperatures, but already at these high temperatures the influence of the K o n d o effect is clearly seen We have also performed calculations for a large hybridization (V2No = 70 K) In that case an inelastic peak can still be distinguished, but at a much larger energy
3. Mean-field theory In the mean-field approximation [7,8] the original Anderson Hamdtonlan is replaced by a simple hybridization Hamdtonlan where the local Coulomb repulsion is neglected, the fstates are shifted up to the fermi energy Ef,.---* El,. = El,. + A and the mixing energy is reduced Ao---~A At T = 0 the quantities A and A are determined by the following set of equations
1
nf = ~ nf,~ = ~ - arctan(A/~.,) = 1 - A/Ao, m
(3)
m
(Ao/-rr) ln((~,. + A2)I/2/D) =
-- A
(4)
rtl
The dynamic susceptlblhty can then be calculated from
x Gim'(lW. + IWs),
(5)
where G,.(lW.) = 0w. - el,. + iA sign(w.)) , are the Green's functions of the f-states correspondlng to the hybridization Hamlltoman From (5) It is straightforward to calculate the susceptlbdlty We do not write down the lengthy formula for the dynamic susceptlbdlty, but will give only the numerical results below For the static susceptlblhty (o)s = 0), however, one obtains a very simple result at T = 0
x = Y. IJ~-.,Iza/('rr(~2,- + A2))
3
m
0
I
50
i
100
I
150 200 WIK]
Fig 1 Dynamical susceptlblhty calculated by self-consistent p e r t u r b a t i o n theory with El 7 = - 1200 K, Ei~ = - 1100 K, V2No = 38 K and D = 104 K for different t e m p e r a t u r e s
+
Z ]JZrn'la(/'lfm -- n f m ) / ( E f m -- l~f,n ) , mT~rn
(6)
where nf,. is defined m (3) It shows a separation m a van Vleck term (rn 4: rn') and a Curie term ( m = m') where, however, the singular tem-
T Hohn, J Keller/Cerium Kondo :mpunaes p e r a t u r e d e p e n d e n c e of the f o r m 1 / T has b e e n r e p l a c e d by a finite v a l u e at T = 0 w h i c h c a n be identified with the a d d m o n a l d e n s i t y of states at the fermi e n e r g y d u e to the l o c a h z e d states U s i n g the same p a r a m e t e r s as a b o v e we h a v e solved the set of e q u a t i o n s (3),(4) F o r the e n e r g i e s a n d o c c u p a t i o n of the q u a s l p a r t l c l e fstates we o b t a i n ~r7 = 4 K a n d ~r8 = 104 K, A = 11 K, nf = 0 9 Results for the d y n a m i c s u s c e p tlblhty c a l c u l a t e d for T = 0 are s h o w n m fig 2 T h e y s h o u l d serve as a n e x t r a p o l a t i o n to T = 0 of the results from fig 1 b u t the a g r e e m e n t is n o t
000~
0003 7 0002 3
-X 0 001
~
//r-\
100
t.,O[ K ]
200
Fig 2 Dynamical susceptlblhty calculated by mean-field theory with the same parameters as m fig 1 The dashed curve are the results from fig 1 for T = 20 K
221
v e r y g o o d T h i s m a y be d u e to the a b s e n c e of q u a s l p a r t l c l e i n t e r a c t i o n s in the m e a n - f i e l d t h e o r y T h e s e m a y be sizeable tf the K o n d o t e m p e r a t u r e ts small c o m p a r e d to the crystalfield s p h t t m g a n d the g r o u n d s t a t e m u l t l p l e t has a low d e g e n e r a c y as in the p r e s e n t case
References [1] O Gunnarson and K Schonhammer, in Theory of Heavy Ferrmons and Valence Fluctuauons eds T Kasuya and T Saso (Springer, Berhn, 1985) p 100, J Magn Magn Mat 52 (1985)227 [2] D L Cox, N E Bickers and J W Wllklns, J Appl Phys 57 (1985) 3166 N E Bickers, DL Cox and JW Wdktns, Phys Rev Lett 54 (1985) 230 [3] N Grewe and H Kelter, Phys Rev B 24 (1981) 4420 N Grewe, Z Physlk B 53 (1983) 271 For a review see H Kelter and G Morandl, Phys Rep 109 (1984) 227 [4] Y Kuramoto, Z Physlk B 53 (1983) 37 H KOllma, Y Kuramoto and M Tachlkl, Z Physlk B 54 (1984) 293 [5] P Coleman, Phys Rev B 29 (1984) 3035 [6] S Maekawa, S Takahashl, S Kashlba and M Tachlkl, J Phys Soc Japan 54 (1985)1955, J Magn Magn Mat 52 (1985) 149 [7] P Coleman, J Magn Magn Mat 47&48 (1985)323, 52 (1985) 223 P Coleman, m Theory of Heavy Fermlons and Valence Fluctuations, eds T Kasuya and T Saso (Springer, Berhn, 1985) p 163 [8] D Newns and N Read, J Phys C 16 (1983) 3273 D W Newns, N Read and A C Hewson, m Moment Formarion m Sohds, ed W J K Buyers (Plenum, New York, 1984) p 257