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Spatial correlations of rare-earth ions in mixed-valence systems
Solid State Communications, Vol. 27, pp. 991-993. © Pergamon Press Ltd. 1978. Printed in Great Britain.
0038-1098/78/0908-0991 $02.00/0
SPATIAL CORRELATIONS OF RARE-EARTH IONS IN MIXED-VALENCE SYSTEMS A. Sakurai* and P. Schlottmann Institut fiir Theoretische Physik, Freie Universit~it Berlin, 1000 Berlin 33, Arnimallee 3, West Germany
(Received 28 February 1978 by M. Cardona) An effective interaction between 4f-occupied rare-earth ions arises from the scattering of the conduction electrons with the localized 4f-electrons. Due to the space dependence of this interaction the ground state energy depends on the distribution of the localized 4f-electrons. We discuss the possibility of clustering and ordering in simple cases. IN A GROUP of rare-earth (RE) metals and compounds known as mixed valence systems the average number of localized 4f-electrons is not an integer. This average value may change with temperature and pressure continuously as, for example, for SmSe and SmTe or discontinuously as for Ce and SmS [ 1 ]. These systems are characterized by two configurations of the RE ions, i.e. 4/"" and 4 f n-I 5d, having similar energies and which may coexist in the crystal. The electronic properties of such a system are appropriately described by a two-band Hubbard model with band degeneracy and hybridization. One band is a narrow 4f-band with a large intraband Coulomb exchange Uff, the other one consists of extended 5d-states. Uad can be neglected for the present purposes, while the interband interaction Ufd plays an important role. The electron-phonon interaction can be eliminated through a canonical transformation and included into effective Coulomb interactions. We are interested in the groundstate energy of the system, the average occupation number of the f-levels and possible correlations between ions with occupied f-level. If the hybridization (or f-level width) is large enough the system will have a homogeneous groundstate with nonintegral 4f-occupancy number. In order to discuss an inhomogeneous groundstate the 4f-level width should be small compared to the energy gain by the spatial correlations caused by Urd. For this purpose we use a simplified version of the two-band Hubbard model postulated by Falicov et al. [2, 3]. The hybridization between the bands and the hopping integral within the 4f-band are neglected. The f-levels have now a completely atomic character. Moreover Uff is assumed to be very large, so that a double occupancy of the filevels is excluded. The Hamiltonian is given by
* On leave of absence from Institute for Solid State Physics, University of Tokyo, Japan. 991
a = kZ, o
-
× ei(k-k')'Itia~ak'o],
N
L
k~ k'o"
(1)
where a~a(ako) is the creation (annihilation) operator of a 5d-electron with~vave vector k, spin o and band energy ek, while nj (= 0 or 1) is the occupation number of the f-electrons at the site Rj. N is the total number of" lattice sites and E the energy of the f-level. Due to the simplifying assumptions in the model Hamiltonian we are limited to a qualitative and suggestive discussion of the problem. However, it is worth to mention that this model is sufficiently general to describe metal insulator transitions [2, 4] and with only small modifications the phase transitions of the magnetite (Fea04) [5]. In the molecular field approximation [3] at zero temperature the equilibrium condition is given by the shifted f-level equated to the shifted Fermi level of the d-band:
E + ncUtd = eF + ntUta,
(2)
where nc = No[N; nf = Nf[N and Nf(Ne) is the total number o f f (conduction)-electrons. For a system with one electron per site, N c + N t = N, and a square density of states the Fermi energy ev is given by 6F "~ ( E + U f d ) / [ 1 - - 4 U f d P F ] ,
(3)
where OF is the density of states at the Fermi level. If PFUfd < 1/4 the f-level occupation number varies continuously with E, while if PFUfa >i 1/4 nf shows a jump from 0 to 1 at EPF = --J~ i f E is lowered. This is easily extended to a general density of states p(e): if UfdP(eF) >1~ there is a first order transition as a function of E, while if UfdP(eF) < ~ the variation is continuous. The system can be considered as an alloy of ions with occupied and empty f-level. Alloy theories like the
SPATIAL CORRELATIONS OF RARE-EARTH IONS
992
coherent potential approximation (CPA) [4] yield a similar picture: a metallic phase (n t = 0), an insulating phase (n t = 1) and first order, as well as, smooth transitions among them. Excitonic phases,have been found within the split-band regime of CPA, i.e. for large values of Uta. Within these effective scattering theories the energy of the system is only given by the average f-level occupancy, but independent of the configuration of the scatterers. However, for a given number off-electrons, N t, the energy of the system at low temperatures should depend on the spatial distribution of the f-electrons [6]. There will be one or more spatial configurations which are energetically favorable at T = 0. The object of this communication is to discuss the possibility of a short range order and clustering of ions with f-electron at low T. We expand the ground-state energy perturbationally in terms of the scattering strength U:d for given N e and N t. The second order correction to the mean field theory is I
--U~a ~ Ar ~ x ( q ) e x p [ - - i q . ( R i - - Rj)],
(4)
where the summation over i and / is taken over the f-occupied lattice sites and
I ~ f(ek):f_..(ek__+q) x(q) = ~
(5)
ek+q--ek
is the susceptibility of the conduction electrons. This result is well-known from the RKKY-theory. For a parabolic band we have
N: Ne 2 -- U~a "~ qZ x(q) + 3npF ~ - U/a ~, F ( 2 k F I R i - R i l )
(6) with kF being the Fermi momentum and
F ( x ) = (x cos x -- sin x)[x 4. As in the RKKY-theory the interaction between scatterers is mediated by the conduction electrons. The oscillating function F(2 kFR) has its first and deepest minimum at Ro ~ 4.5[kF. This favors the f-occupied sites to arrange themselves such that the distance between them is nearly Ro, if the lattice fits this condition. Hence, the f-sites are likely to form clusters by adjusting the bonding distances nearly Ro, in order to minimize the energy. A difference with respect to the usual RKKYinteraction should be pointed out. While the usual RKKYqnteraction between magnetic moments is originated by spin density oscillations in the electron gas, the interaction given by equation (6) is caused by
(7)
Vol. 27, No. 10
charge density oscillations, which are screened by the Coulomb interaction between d-electrons. In order to retain the interaction, equation (6), we must assume that kFRo < (UaaPF) -I/2. It should be mentioned that hybridization between f- and d-electrons also tends to suppress the interaction; on the other hand, the interaction itself, equation (6), reduces the hopping of the f-electrons and hence the width of the f-level. A criterion for the stability of a state is difficult to give, unless eF >> Urn ~ Via, Vta being the hybridization. The size of the clusters sensitively depends on N r and the crystal symmetry. At concentrations where N r is of the order of No we should also consider the formation of order in a more extended region. We discuss below the extreme case of formation of a superlattice for N t = ~N. For small N e we have a small kF and a large Ro. In this case almost all the ions have a f-electron and the problem can be regarded as interactions between f-holes, which arrange themselves in order to minimize the energy. Formation of excitons between f-holes and d-electrons is also expected in this case. The above discussion is valid for the groundstate. The temperature range for which the spatial correlations are important is determined by the excited configurations. Except for special cases, like the superlattice formation presented below, the excitations essentially form a continuum above the groundstate. A small hybridization may allow transitions between different configurations and the new groundstate becomes a combination of energetically favorable inhomogeneous states. At high temperatures a large number of configurations play a role and effective field theories are applicable. Let us discuss now the case of formation of a superlattice. We consider the exaggerated case of a triple periodicity in a one-dimensional lattice with N t = ~N. The new Brillouin zone is reduced to one-third of the original one. The original band is divided into three bands within the reduced zone scheme. Energy gaps of 2Urn/3 are opened at k = + rr/3a and + 21r13a (a = lattice constant), i.e. at the center of the zone and its boundary. The available conduction electrons, i.e. N e = I N can just be accommodated in the first band, if we consider the spin degeneracy. This is clearly the groundstate configuration of the system for given N r = 1[3N. Low energy excited configurations for fixed N t can be obtained by creating "dislocations" in the ordered state. It remains to analyze the stability condition of the ordered configuration with respect to changes in N r. If we localize an additional f-electron or remove one from the superlattice, the problem can be regarded as the ordered superlattice with I N -T-1 conduction electrons scattered by the additional f-electron or hole. An additional f-electron or hole introduces three simple
Vol. 27, No. 10
SPATIAL CORRELATIONS OF RARE-EARTH IONS
"dislocations" in the superlattice, which can be distri. buted in several possible spatial arrangements. Since the detailed calculation of the energy change Ae is tedious, we limit ourselves to a perturbational estimation:
Ae = [e(nc = ~ + x ) - - e ( n c = ~)]/IxlN = - - ( E - - e F + ~Ura) sign x + ~Ura -
v
aor + . . . .
(8)
where ev is the Fermi-energy for the periodic n c = case, x N is the excess of conduction electrons over ~N and a is a positive number of the order of one. The system is then stable in the superlattice configuration if Ae > 0, i.e. if (9)
I f -- e F q- ~Ufdl < ~Ufd --olU~d~ F.
In the one-electron picture, neglecting the second order term, equation (9) is equivalent to the condition that the shifted f-level lies within the gap of the shifted d-band. This provides the range for E in which the system will be locked at Nf = N/3 within the superlattice configuration. A similar situation appears in a two-dimensional triangular lattice, where the superlattice forms triangles x/3 times larger in size than the original ones. In this case, however, the Fermi surface does not match
993
exactly the boundaries of the first Brillouin zone of the reduced zone scheme and a larger Ufd is required to produce the above effect. We do not find such idealized cases in simple three dimensional lattices. However, the formation of a superlattice in a certain plane (charge density waves) may reduce the energy of the system even if the Fermi surface does not strictly coincide with the Brillouin zone boundary, according to the H u m e Rothery rule for alloys. The density of states at the Fermi level and the scattering of the electrons determine the low temperature resistivity. In the presence of order in an extended region the density of states will be reduced at certain energies. If the Fermi level happens to lie in one of these possible minima, the available number of electrons for conduction is reduced. In this case, on the other hand, the electrons are scattered coherently by the potential favoring the conductivity. For a quantitative discussion of the magnetic and transport properties the lifetime of the f-levels, not considered here, must be taken into account. Although no experimental evidence for spatial correlations has yet been reported, it is interesting to notice that the average number off-occupied ions is Kept around 1/3 in some of the valence-fluctuating systems, like ~-Ce, Sml_xY~,S (x >~ 0.15) and SmB6 [7].
REFERENCES 1.
VARMA C.M., Rev. Mod. Phys. 48, 219 (1976).
2.
FALICOV L.M. & KIMBALL J.C., Phys. Rev. Lett. 22, 997 (1969).
3.
RAMIREZ R. & FALICOV L.M.,Phys. Rev. B3, 2425 (1971).
4.
GON~ALESDASILVA C.E.T.&FALICOVL.M.,J. Phys. CS,906(1972);BRINGERA.,Z. Phys. B21,21 (1975).
5.
RAJAN V.T., AVIGNON M. & FALICOV L.M., Solid State Commun. 14, 149 (1974).
6.
The spatial correlation in mixed valence systems is contrasted to the correlation in alloys, since the scattering centers are not fixed but can choose cooperatively the most favorable configuration.
7.
FRANCESCHI E. & OLCESE G.L.,Phys. Rev. Lett. 22, 1299 (1969); PENNY T. & HOLTZBERG F.,Phys. Rev. Lett. 34, 322 (1975); NICKERSON J.C., WHITE R.M., LEE K.N., BACHMANN R., GEBALLEL T.H. & HULL G.W., JR.,Phys. Rev. 133, 2030 (1971).
0038-1098/78/0908-0991 $02.00/0
SPATIAL CORRELATIONS OF RARE-EARTH IONS IN MIXED-VALENCE SYSTEMS A. Sakurai* and P. Schlottmann Institut fiir Theoretische Physik, Freie Universit~it Berlin, 1000 Berlin 33, Arnimallee 3, West Germany
(Received 28 February 1978 by M. Cardona) An effective interaction between 4f-occupied rare-earth ions arises from the scattering of the conduction electrons with the localized 4f-electrons. Due to the space dependence of this interaction the ground state energy depends on the distribution of the localized 4f-electrons. We discuss the possibility of clustering and ordering in simple cases. IN A GROUP of rare-earth (RE) metals and compounds known as mixed valence systems the average number of localized 4f-electrons is not an integer. This average value may change with temperature and pressure continuously as, for example, for SmSe and SmTe or discontinuously as for Ce and SmS [ 1 ]. These systems are characterized by two configurations of the RE ions, i.e. 4/"" and 4 f n-I 5d, having similar energies and which may coexist in the crystal. The electronic properties of such a system are appropriately described by a two-band Hubbard model with band degeneracy and hybridization. One band is a narrow 4f-band with a large intraband Coulomb exchange Uff, the other one consists of extended 5d-states. Uad can be neglected for the present purposes, while the interband interaction Ufd plays an important role. The electron-phonon interaction can be eliminated through a canonical transformation and included into effective Coulomb interactions. We are interested in the groundstate energy of the system, the average occupation number of the f-levels and possible correlations between ions with occupied f-level. If the hybridization (or f-level width) is large enough the system will have a homogeneous groundstate with nonintegral 4f-occupancy number. In order to discuss an inhomogeneous groundstate the 4f-level width should be small compared to the energy gain by the spatial correlations caused by Urd. For this purpose we use a simplified version of the two-band Hubbard model postulated by Falicov et al. [2, 3]. The hybridization between the bands and the hopping integral within the 4f-band are neglected. The f-levels have now a completely atomic character. Moreover Uff is assumed to be very large, so that a double occupancy of the filevels is excluded. The Hamiltonian is given by
* On leave of absence from Institute for Solid State Physics, University of Tokyo, Japan. 991
a = kZ, o
-
× ei(k-k')'Itia~ak'o],
N
L
k~ k'o"
(1)
where a~a(ako) is the creation (annihilation) operator of a 5d-electron with~vave vector k, spin o and band energy ek, while nj (= 0 or 1) is the occupation number of the f-electrons at the site Rj. N is the total number of" lattice sites and E the energy of the f-level. Due to the simplifying assumptions in the model Hamiltonian we are limited to a qualitative and suggestive discussion of the problem. However, it is worth to mention that this model is sufficiently general to describe metal insulator transitions [2, 4] and with only small modifications the phase transitions of the magnetite (Fea04) [5]. In the molecular field approximation [3] at zero temperature the equilibrium condition is given by the shifted f-level equated to the shifted Fermi level of the d-band:
E + ncUtd = eF + ntUta,
(2)
where nc = No[N; nf = Nf[N and Nf(Ne) is the total number o f f (conduction)-electrons. For a system with one electron per site, N c + N t = N, and a square density of states the Fermi energy ev is given by 6F "~ ( E + U f d ) / [ 1 - - 4 U f d P F ] ,
(3)
where OF is the density of states at the Fermi level. If PFUfd < 1/4 the f-level occupation number varies continuously with E, while if PFUfa >i 1/4 nf shows a jump from 0 to 1 at EPF = --J~ i f E is lowered. This is easily extended to a general density of states p(e): if UfdP(eF) >1~ there is a first order transition as a function of E, while if UfdP(eF) < ~ the variation is continuous. The system can be considered as an alloy of ions with occupied and empty f-level. Alloy theories like the
SPATIAL CORRELATIONS OF RARE-EARTH IONS
992
coherent potential approximation (CPA) [4] yield a similar picture: a metallic phase (n t = 0), an insulating phase (n t = 1) and first order, as well as, smooth transitions among them. Excitonic phases,have been found within the split-band regime of CPA, i.e. for large values of Uta. Within these effective scattering theories the energy of the system is only given by the average f-level occupancy, but independent of the configuration of the scatterers. However, for a given number off-electrons, N t, the energy of the system at low temperatures should depend on the spatial distribution of the f-electrons [6]. There will be one or more spatial configurations which are energetically favorable at T = 0. The object of this communication is to discuss the possibility of a short range order and clustering of ions with f-electron at low T. We expand the ground-state energy perturbationally in terms of the scattering strength U:d for given N e and N t. The second order correction to the mean field theory is I
--U~a ~ Ar ~ x ( q ) e x p [ - - i q . ( R i - - Rj)],
(4)
where the summation over i and / is taken over the f-occupied lattice sites and
I ~ f(ek):f_..(ek__+q) x(q) = ~
(5)
ek+q--ek
is the susceptibility of the conduction electrons. This result is well-known from the RKKY-theory. For a parabolic band we have
N: Ne 2 -- U~a "~ qZ x(q) + 3npF ~ - U/a ~, F ( 2 k F I R i - R i l )
(6) with kF being the Fermi momentum and
F ( x ) = (x cos x -- sin x)[x 4. As in the RKKY-theory the interaction between scatterers is mediated by the conduction electrons. The oscillating function F(2 kFR) has its first and deepest minimum at Ro ~ 4.5[kF. This favors the f-occupied sites to arrange themselves such that the distance between them is nearly Ro, if the lattice fits this condition. Hence, the f-sites are likely to form clusters by adjusting the bonding distances nearly Ro, in order to minimize the energy. A difference with respect to the usual RKKYinteraction should be pointed out. While the usual RKKYqnteraction between magnetic moments is originated by spin density oscillations in the electron gas, the interaction given by equation (6) is caused by
(7)
Vol. 27, No. 10
charge density oscillations, which are screened by the Coulomb interaction between d-electrons. In order to retain the interaction, equation (6), we must assume that kFRo < (UaaPF) -I/2. It should be mentioned that hybridization between f- and d-electrons also tends to suppress the interaction; on the other hand, the interaction itself, equation (6), reduces the hopping of the f-electrons and hence the width of the f-level. A criterion for the stability of a state is difficult to give, unless eF >> Urn ~ Via, Vta being the hybridization. The size of the clusters sensitively depends on N r and the crystal symmetry. At concentrations where N r is of the order of No we should also consider the formation of order in a more extended region. We discuss below the extreme case of formation of a superlattice for N t = ~N. For small N e we have a small kF and a large Ro. In this case almost all the ions have a f-electron and the problem can be regarded as interactions between f-holes, which arrange themselves in order to minimize the energy. Formation of excitons between f-holes and d-electrons is also expected in this case. The above discussion is valid for the groundstate. The temperature range for which the spatial correlations are important is determined by the excited configurations. Except for special cases, like the superlattice formation presented below, the excitations essentially form a continuum above the groundstate. A small hybridization may allow transitions between different configurations and the new groundstate becomes a combination of energetically favorable inhomogeneous states. At high temperatures a large number of configurations play a role and effective field theories are applicable. Let us discuss now the case of formation of a superlattice. We consider the exaggerated case of a triple periodicity in a one-dimensional lattice with N t = ~N. The new Brillouin zone is reduced to one-third of the original one. The original band is divided into three bands within the reduced zone scheme. Energy gaps of 2Urn/3 are opened at k = + rr/3a and + 21r13a (a = lattice constant), i.e. at the center of the zone and its boundary. The available conduction electrons, i.e. N e = I N can just be accommodated in the first band, if we consider the spin degeneracy. This is clearly the groundstate configuration of the system for given N r = 1[3N. Low energy excited configurations for fixed N t can be obtained by creating "dislocations" in the ordered state. It remains to analyze the stability condition of the ordered configuration with respect to changes in N r. If we localize an additional f-electron or remove one from the superlattice, the problem can be regarded as the ordered superlattice with I N -T-1 conduction electrons scattered by the additional f-electron or hole. An additional f-electron or hole introduces three simple
Vol. 27, No. 10
SPATIAL CORRELATIONS OF RARE-EARTH IONS
"dislocations" in the superlattice, which can be distri. buted in several possible spatial arrangements. Since the detailed calculation of the energy change Ae is tedious, we limit ourselves to a perturbational estimation:
Ae = [e(nc = ~ + x ) - - e ( n c = ~)]/IxlN = - - ( E - - e F + ~Ura) sign x + ~Ura -
v
aor + . . . .
(8)
where ev is the Fermi-energy for the periodic n c = case, x N is the excess of conduction electrons over ~N and a is a positive number of the order of one. The system is then stable in the superlattice configuration if Ae > 0, i.e. if (9)
I f -- e F q- ~Ufdl < ~Ufd --olU~d~ F.
In the one-electron picture, neglecting the second order term, equation (9) is equivalent to the condition that the shifted f-level lies within the gap of the shifted d-band. This provides the range for E in which the system will be locked at Nf = N/3 within the superlattice configuration. A similar situation appears in a two-dimensional triangular lattice, where the superlattice forms triangles x/3 times larger in size than the original ones. In this case, however, the Fermi surface does not match
993
exactly the boundaries of the first Brillouin zone of the reduced zone scheme and a larger Ufd is required to produce the above effect. We do not find such idealized cases in simple three dimensional lattices. However, the formation of a superlattice in a certain plane (charge density waves) may reduce the energy of the system even if the Fermi surface does not strictly coincide with the Brillouin zone boundary, according to the H u m e Rothery rule for alloys. The density of states at the Fermi level and the scattering of the electrons determine the low temperature resistivity. In the presence of order in an extended region the density of states will be reduced at certain energies. If the Fermi level happens to lie in one of these possible minima, the available number of electrons for conduction is reduced. In this case, on the other hand, the electrons are scattered coherently by the potential favoring the conductivity. For a quantitative discussion of the magnetic and transport properties the lifetime of the f-levels, not considered here, must be taken into account. Although no experimental evidence for spatial correlations has yet been reported, it is interesting to notice that the average number off-occupied ions is Kept around 1/3 in some of the valence-fluctuating systems, like ~-Ce, Sml_xY~,S (x >~ 0.15) and SmB6 [7].
REFERENCES 1.
VARMA C.M., Rev. Mod. Phys. 48, 219 (1976).
2.
FALICOV L.M. & KIMBALL J.C., Phys. Rev. Lett. 22, 997 (1969).
3.
RAMIREZ R. & FALICOV L.M.,Phys. Rev. B3, 2425 (1971).
4.
GON~ALESDASILVA C.E.T.&FALICOVL.M.,J. Phys. CS,906(1972);BRINGERA.,Z. Phys. B21,21 (1975).
5.
RAJAN V.T., AVIGNON M. & FALICOV L.M., Solid State Commun. 14, 149 (1974).
6.
The spatial correlation in mixed valence systems is contrasted to the correlation in alloys, since the scattering centers are not fixed but can choose cooperatively the most favorable configuration.
7.
FRANCESCHI E. & OLCESE G.L.,Phys. Rev. Lett. 22, 1299 (1969); PENNY T. & HOLTZBERG F.,Phys. Rev. Lett. 34, 322 (1975); NICKERSON J.C., WHITE R.M., LEE K.N., BACHMANN R., GEBALLEL T.H. & HULL G.W., JR.,Phys. Rev. 133, 2030 (1971).