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Lattice distortion in mixed-valent materials
003%1098/81/290671-03$02.00/O
Solid State Communications, Vol. 39, pp. 67 l-673. Pergamon Press Ltd. 1981. Printed in Great Britain.
LATTICE DISTORTION
IN MIXED-VALENT
MATERIALS
K. Baba and Y. Kuroda Physics Department,
Nagoya University,
Nagoya, Japan
(Received 23 February 198 1 by J. Kanamori)
Using a simple model, we calculate rigorously the distribution function of displacements of anions in mixed-valent rare-earth compounds, and find that the anions always adopt dynamically distorted positions corresponding to the valence fluctuations in the rare-earth ions contrary to the previous study.
THE VALENCE FLUCTUATIONS in the rare-earth compounds have been receiving considerable attention in the last decade [l-3]. In these systems the valence fluctuations are supposed to couple strongly to the lattice vibrations, because changes in valences of the rareearth ions mean changes in the ionic-sizes and hence may induce local lattice distortions. Such problems have been rather intensively investigated by many authors [4-61. More recently, Martin et al. [7] have done EXAFS-measurements on the mixed-valent phase of Sm0.,sYo.2sS, and tried to determine directly both the effective valence and near neighbour environments of the Sm ions. They claimed that the results show the S atoms nearest to each Sm atom adopt single-average positions rather than dynamically distorted environments with two different stable positions corresponding to the two different valence states in the Sm ions and concluded that the characteristic width of the Sm 4f-band is not greatly modified by the polaron effects. In the present note, using a simple model, we calculate rigorously distribution function of the displacements of the S ions in such systems. The result shows that the S atoms always adopt dynamically distorted positions correponding to the valence fluctuations in the Sm ions, but the degree of the distortion is crucially dependent upon the ratio of the characteristic frequencies of the lattice vibrations and the valence fluctuations in the system, and hence by doing a more quantitative study than the previous study [7] we are able to determine the strength of the polaron effect from the EXAFS data. For such purpose, we take the same model as that used in [4]. For the study of the metallic phase, the Hamiltonian is given by
where f:,df and al are the creation operators of spinless localized f-electrons, itinerant d-electrons and Einstein phonons of S-ions, respectively. The suffices i and j denote the Sm-ion sites and S-ion sites, respectively. In this model we neglect the oscillations of Sm-ions. The electron-phonon coupling constant gij is assumed to be non-zero only for the nearest-neighbor pairs. si, is the average number of f-electrons per Sm-site. As regards the detailed meaning of the model we refer to
141. Carrying out the so-called polaron transformation on this Hamiltonian in order to remove the simple electron-phonon couplings, we obtain
a =C
(
Ef~+fi-_OC C g”(fitfi-ii~)
i
j
i
OO
I 2
•k 1 Ekdkt,dko + c Wo(U~fZj + a) •k v c i,k.u k.0 i
The displacement operator of the S-ion, tij = ao(af + a,) (a0 = (h2/2uoMs)1’2), is also transformed as follows:
Then the distribution is given by
function
P(u)
= (6(u -ii)>2
= (6(U_li)>H
of the displacement
P(u)
ODdq = _oo s g
x eiPu( e-i4h )h, H = c
i
erfi+fi + c
ku
e&k&o
+ C wO(afaj + 4) + 1
j
Lj
+ I’
1
i.k,o
(fit&o
+ h.c.)
gij(f;tf, -fitXal + aj)* (l) 671
where ( k )fi z Tr (2 esPrr)/Tr (e-@). Evaluating this average exactly up to order of V2 in the limit of T = 0, we obtain
672
LATTICE DISTORTION
u.
IN MIXED-VALENT
q =
1 _
Af(2
_
eiq*Aa
_
Fig. 2. The inverse of the effective polaron factor, e+niff , and the maximum value, P;lax, in the corrections to P(U) due to the f-d mixing effect: as functions of wo/(Zf -cl). The left scale is for e+qeff (solid line) and the right scale is for Pi” (dashed line). The thick arrows denote the limiting values of e+n:rf and Pf”= for ~O/Gf - P)+ m, respectively.
e-iq.A,a)
i
x fq 2
+pp2
c I=0
21
x’Fln
l+,-
.I
Ef-j-l
I
-(-l)” 2 ?&m=r n!m!
2 (n +m)oe
(2;-/.L>+(l+n)wo (Ff -/J)
+ 100
+l(l+n)wo)e
q2 E F (gii/We)2,
v2 E V2 e-n*,
- 1)q200, ~1is the chemical potential, Ef=ff+(2Bf and we have assumed that the density of the d-states is given by a constant p. We note that Aa denotes essentially the difference between the ionic radii of Sm*+ ion and of Sm3+ ion. Af is given by [4] jjf”2VZP
*I
_p)
+ the terms with (q + - q)] ,
(n + owe &-iJ
x 2 %ln ,=e 11
)2nlzo$(,
where Aa - 2oeg/we,
IWO
Vol. 39, No. 5
H
Fig. 1. The distribution functions of the displacement, P(U), for the various values of w,,/(Ef -P). Solid lines are the distribution functions P(U), short-dashed line is the unperturbed distribution function PO(u) (i.e. in the case of V = 0), and dashed lines are PI(u) -P(U) P,(U), the corrections to P(U) due to the f-d mixing effect, where the parameters used are iif = 0.3, Aa = 0.18A and o. = 6.8 x 10”Hz. ( e-iqfi jH x e&i
MATERIALS
1 f 92” n! noo+Ef-& n=o
If 2; - n 9 wo, the result is simplified as . {_
1 +
eiq*Ao}
2 r/z (1 -&-($[I P(U) = (2ncuo)-
-5))
Note that the previous study [4] suggests that the phonon frequencies are reduced by including the electron-phonon interaction when oe/(zf -CC) Q 1. Therefore, we have normally been inclined to expect
LATTICE DISTORTION
Vol. 39, No. 5
IN MIXED-VALENT
that the restoring force for Einstein oscillation of a S-ion must be reduced correspondingly, and hence the width of Gaussian distribution of the S-ion’s displacement must be broadened. However, the present results suggest that it isn’t the case. The distribution function of the displacement is given simply by the superpose of the original Gaussian and the correction including the second higher harmonics of the original Einstein oscillator, the latter of which has peaks at u = *fi (Y,, and hence denotes a distorted configuration of the S-ion.
(1 -22Af)exp
P(U) = (2n(u$)-“2 (
-2 (
1
MATERIALS
673
function of wo/(Ef - cc) together with the maximum values, PyBx, in the corrections to P(u) due to the f-d mixing effect. We note that both Ppax and eO:ff are given by rather smooth and similar functions of OO/Gf
-d.
In conclusion, we note that in order to extract the strength of the polaron effects from the EXAFS data, we have to do more quantitative analysis of the data than that done in [7]. Then we are able to obtain important information about the ground state of the mixed-valent phase. Since the present results give the exact values of the initial slope of P(u) as a function of Af within the present model, we may use these results to compare with the experiments at least in the semiquantitative ways. The details of such study will be communicated in a forthcoming paper.
+~~,exp~-(~~~~‘1+exp~-~~~‘I REFERENCES
As for the intermediate-energy region, we can calculate P(U) only numerically. Some results are shown in Fig. 1 for various values of oo/(Ef -/.L). In order to see the strength of the polaron effect we calculate the effective polaron factor, e-n:ff , given by Af(Ef -p)/2pV2, which is equal to 1 in the absence of the polaron effect (i.e. Ef -p % oo) or to e-n in the strong limit of the polaron effect (i.e. o. % Zf - g). Some numerical results for e”iff are shown in Fig. 2 as a
1. 2.
3. 4. 5. 6. 7.
C.M. Varma, Rev. Mod. Phys. 48,219 (1976). Valence Instabilities and Related NarrowBand Phenomena (Edited by R.D. Parks). Plenum, New York (1977). \-J.M. Robinson, Physics Reports 51,l (1979). Y. Kuroda & K.H. Benneman, To be published in Phys. Rev. B15, (1981). K. Baba, M. Kobayashi, H. Kaga & I. Yokota, Solid State Commun. 35,175 (1980). References quoted in [4]. R.M. Martin, J.B. Boyce, J.W. Allen & F. Holtzberg, Phys. Rev. Lett. 44, 1275 (1980).
Solid State Communications, Vol. 39, pp. 67 l-673. Pergamon Press Ltd. 1981. Printed in Great Britain.
LATTICE DISTORTION
IN MIXED-VALENT
MATERIALS
K. Baba and Y. Kuroda Physics Department,
Nagoya University,
Nagoya, Japan
(Received 23 February 198 1 by J. Kanamori)
Using a simple model, we calculate rigorously the distribution function of displacements of anions in mixed-valent rare-earth compounds, and find that the anions always adopt dynamically distorted positions corresponding to the valence fluctuations in the rare-earth ions contrary to the previous study.
THE VALENCE FLUCTUATIONS in the rare-earth compounds have been receiving considerable attention in the last decade [l-3]. In these systems the valence fluctuations are supposed to couple strongly to the lattice vibrations, because changes in valences of the rareearth ions mean changes in the ionic-sizes and hence may induce local lattice distortions. Such problems have been rather intensively investigated by many authors [4-61. More recently, Martin et al. [7] have done EXAFS-measurements on the mixed-valent phase of Sm0.,sYo.2sS, and tried to determine directly both the effective valence and near neighbour environments of the Sm ions. They claimed that the results show the S atoms nearest to each Sm atom adopt single-average positions rather than dynamically distorted environments with two different stable positions corresponding to the two different valence states in the Sm ions and concluded that the characteristic width of the Sm 4f-band is not greatly modified by the polaron effects. In the present note, using a simple model, we calculate rigorously distribution function of the displacements of the S ions in such systems. The result shows that the S atoms always adopt dynamically distorted positions correponding to the valence fluctuations in the Sm ions, but the degree of the distortion is crucially dependent upon the ratio of the characteristic frequencies of the lattice vibrations and the valence fluctuations in the system, and hence by doing a more quantitative study than the previous study [7] we are able to determine the strength of the polaron effect from the EXAFS data. For such purpose, we take the same model as that used in [4]. For the study of the metallic phase, the Hamiltonian is given by
where f:,df and al are the creation operators of spinless localized f-electrons, itinerant d-electrons and Einstein phonons of S-ions, respectively. The suffices i and j denote the Sm-ion sites and S-ion sites, respectively. In this model we neglect the oscillations of Sm-ions. The electron-phonon coupling constant gij is assumed to be non-zero only for the nearest-neighbor pairs. si, is the average number of f-electrons per Sm-site. As regards the detailed meaning of the model we refer to
141. Carrying out the so-called polaron transformation on this Hamiltonian in order to remove the simple electron-phonon couplings, we obtain
a =C
(
Ef~+fi-_OC C g”(fitfi-ii~)
i
j
i
OO
I 2
•k 1 Ekdkt,dko + c Wo(U~fZj + a) •k v c i,k.u k.0 i
The displacement operator of the S-ion, tij = ao(af + a,) (a0 = (h2/2uoMs)1’2), is also transformed as follows:
Then the distribution is given by
function
P(u)
= (6(u -ii)>2
= (6(U_li)>H
of the displacement
P(u)
ODdq = _oo s g
x eiPu( e-i4h )h, H = c
i
erfi+fi + c
ku
e&k&o
+ C wO(afaj + 4) + 1
j
Lj
+ I’
1
i.k,o
(fit&o
+ h.c.)
gij(f;tf, -fitXal + aj)* (l) 671
where ( k )fi z Tr (2 esPrr)/Tr (e-@). Evaluating this average exactly up to order of V2 in the limit of T = 0, we obtain
672
LATTICE DISTORTION
u.
IN MIXED-VALENT
q =
1 _
Af(2
_
eiq*Aa
_
Fig. 2. The inverse of the effective polaron factor, e+niff , and the maximum value, P;lax, in the corrections to P(U) due to the f-d mixing effect: as functions of wo/(Zf -cl). The left scale is for e+qeff (solid line) and the right scale is for Pi” (dashed line). The thick arrows denote the limiting values of e+n:rf and Pf”= for ~O/Gf - P)+ m, respectively.
e-iq.A,a)
i
x fq 2
+pp2
c I=0
21
x’Fln
l+,-
.I
Ef-j-l
I
-(-l)” 2 ?&m=r n!m!
2 (n +m)oe
(2;-/.L>+(l+n)wo (Ff -/J)
+ 100
+l(l+n)wo)e
q2 E F (gii/We)2,
v2 E V2 e-n*,
- 1)q200, ~1is the chemical potential, Ef=ff+(2Bf and we have assumed that the density of the d-states is given by a constant p. We note that Aa denotes essentially the difference between the ionic radii of Sm*+ ion and of Sm3+ ion. Af is given by [4] jjf”2VZP
*I
_p)
+ the terms with (q + - q)] ,
(n + owe &-iJ
x 2 %ln ,=e 11
)2nlzo$(,
where Aa - 2oeg/we,
IWO
Vol. 39, No. 5
H
Fig. 1. The distribution functions of the displacement, P(U), for the various values of w,,/(Ef -P). Solid lines are the distribution functions P(U), short-dashed line is the unperturbed distribution function PO(u) (i.e. in the case of V = 0), and dashed lines are PI(u) -P(U) P,(U), the corrections to P(U) due to the f-d mixing effect, where the parameters used are iif = 0.3, Aa = 0.18A and o. = 6.8 x 10”Hz. ( e-iqfi jH x e&i
MATERIALS
1 f 92” n! noo+Ef-& n=o
If 2; - n 9 wo, the result is simplified as . {_
1 +
eiq*Ao}
2 r/z (1 -&-($[I P(U) = (2ncuo)-
-5))
Note that the previous study [4] suggests that the phonon frequencies are reduced by including the electron-phonon interaction when oe/(zf -CC) Q 1. Therefore, we have normally been inclined to expect
LATTICE DISTORTION
Vol. 39, No. 5
IN MIXED-VALENT
that the restoring force for Einstein oscillation of a S-ion must be reduced correspondingly, and hence the width of Gaussian distribution of the S-ion’s displacement must be broadened. However, the present results suggest that it isn’t the case. The distribution function of the displacement is given simply by the superpose of the original Gaussian and the correction including the second higher harmonics of the original Einstein oscillator, the latter of which has peaks at u = *fi (Y,, and hence denotes a distorted configuration of the S-ion.
(1 -22Af)exp
P(U) = (2n(u$)-“2 (
-2 (
1
MATERIALS
673
function of wo/(Ef - cc) together with the maximum values, PyBx, in the corrections to P(u) due to the f-d mixing effect. We note that both Ppax and eO:ff are given by rather smooth and similar functions of OO/Gf
-d.
In conclusion, we note that in order to extract the strength of the polaron effects from the EXAFS data, we have to do more quantitative analysis of the data than that done in [7]. Then we are able to obtain important information about the ground state of the mixed-valent phase. Since the present results give the exact values of the initial slope of P(u) as a function of Af within the present model, we may use these results to compare with the experiments at least in the semiquantitative ways. The details of such study will be communicated in a forthcoming paper.
+~~,exp~-(~~~~‘1+exp~-~~~‘I REFERENCES
As for the intermediate-energy region, we can calculate P(U) only numerically. Some results are shown in Fig. 1 for various values of oo/(Ef -/.L). In order to see the strength of the polaron effect we calculate the effective polaron factor, e-n:ff , given by Af(Ef -p)/2pV2, which is equal to 1 in the absence of the polaron effect (i.e. Ef -p % oo) or to e-n in the strong limit of the polaron effect (i.e. o. % Zf - g). Some numerical results for e”iff are shown in Fig. 2 as a
1. 2.
3. 4. 5. 6. 7.
C.M. Varma, Rev. Mod. Phys. 48,219 (1976). Valence Instabilities and Related NarrowBand Phenomena (Edited by R.D. Parks). Plenum, New York (1977). \-J.M. Robinson, Physics Reports 51,l (1979). Y. Kuroda & K.H. Benneman, To be published in Phys. Rev. B15, (1981). K. Baba, M. Kobayashi, H. Kaga & I. Yokota, Solid State Commun. 35,175 (1980). References quoted in [4]. R.M. Martin, J.B. Boyce, J.W. Allen & F. Holtzberg, Phys. Rev. Lett. 44, 1275 (1980).