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Localized-electron mechanism for configuration mixing in Sm compounds
Volume 5 1A,number 5
PHYSICS LETTERS
24 March 1975
LOCALIZED-ELECTRON MECHANISM FOR CONFIGURATION MIXING IN Sm COMPOUNDS*
Department
T.A. KAPLAN and S.D. MAHANTI of Physics, Michigan State University, East Lunsing, Michigan 48824, USA Received 4 March 1975
A new mechanism for (4fJ6 - (4f)5 Sd configuration mixing in SmBe and SmS is described. It involves the simultaneous intra-atomic excitation of two Sm sites.-
The compounds, SmB6 at normal pressure, and SmS under pressure e 6 kbar) have drawn considerable attention recently [l-4] . A remarkable feature common to these systems is the existence of a “mixedvalence” or, preferably [3] , mixed-configuration ground state, the configurations being Sm2+ (4f6) and Sm2+(4f55d or 4f56s). Although refs. [1,2,4] to Hirst’s theory [5] of this mixing has been made, quantitative calculations within this theory have been limited to a single impurity (Sm2’) in a conductionelectron sea. Hirst’s mechanism for interconfigurational mixing (ICM) is the single-electron-transfer matrix element between a 4f Wannier state to a Sd or 6s Bloch state. In this letter we point out a new mechanism which can cause ICM, we estimate its magnitude and discuss some of its consequences. Our new mechanism is based on localized-electron excitations* ’ . An apparent difficulty in finding such a mechanism is the fact that f and d are of opposite parity and a Sm site is inversion-invariant. Our mechanism overcomes this difficulty by virtue of its being a two-electron excitation. The essential matrix element involved is that, &j E (2i 2j IUIli lj),
(1)
of the Coulomb electron-electron interaction u involving 4f Wannier functions Wli, wlj at sites i and j, and 5d (or 6s) Wannier functions W2i, wzj at sites i and j.
* Research supported by NSF Grant No. GH-34565. *l Previous workers [6] interpreted the optical absorption peaks observed in SmS in terms of localized f + d excitations. The narrowness (the order of several tenths of a volt) of these peaks suggests that a localized treatment of the dstates is a reasonable first approximation.
Whereas an analogous one-particle matrix element (2ilhl li) will vanish by inversion symmetry, (1) does not vanish. The matrix element (1) involves the transition li+2i, lj+2j. We consider for simplicity a model in which there is one f-state and one d-state at each site, wti are real, and there are 2 electrons per site. We regard f2 and fd as being analogous to f6 and f5 d, respectively. In our model, we consider only crystal states Q which involve atoms that are either in f2 or the fd singlet. Thus if N= number of atoms, there are 2N such a’s, so that the matrix of the electronic Hamiltonian H, in the a,basis, can be represented by a spin-Hamiltonian which is a function of N spin-l/2 spin operators Tl , ... TN. Neglecting all inter-atomic matrix elements other than (1) or its equal (li 2j I uI2i lj), the spin Hamiltonian turns out to be 91=-Ae
CTiz i
+4 Ctij Q
Tix 5x.
(2)
The Ising operator came from 4 Tix I& = Ti+ q+ +Ti+Ti_ +h*C*; AE=h22_hll-~1111 +u1212+u1221 where h, = (vi Ih I vi), uvCcM=(vipiIulxiGi), andh is the usual bare one-electron hamiltonian. The correspondence Clli+ (f2)i, /Ii + (fd singlet)i between the eigenstates of Tis and the electronic states defines the relation between (2) and electronic properties. For example, the electron density operator in this model is
+
[Wli(‘)2”“2i(J’)21 ($- Ti~)‘~W,,ir)W2i(f)Ti,}
For our initial rough purposes we consider the ground state of (2) in the mean field approximation. Again 265
PHYSICS LETTERS
Volume 51A, number 5
for simplicity we assume Eii = - I( I for nearest neighbor i, j, zero otherwise. Then the mean field ground state is \kM, = II&iq + B&); its energy, EM, = (‘kMF, %qMF) E N(Ae/2)eMF is given in terms of n s 42 l[ l/Ae by -1 ‘MF =
-1
- (2n)-‘(n
n
n>l
significant effect on the transition. It is also important to go beyond the mean-field approximation (nonzero (Tix) would imply long-range-ordering of atomic electric dipoles* 3 see eq. (3)). Such steps have been taken in a preliminary way and our results will be reported elsewhere.
(4)
(Z = coordination number); also B2 = 0 for n < 1, B2 = (q - 1)/(27,9 for n > 1. We will see below that 77 tends to increase with pressure p. Hence our model can show an essential property of SmS, namely a change from a single configurational state, B2 = 0, to a mixed configuration state, 0 < B2 < 1, with increasing p. The order of magnitude of .$ can be estimated by noting that to leading order in inverse interatomic distance Ri’, fii is the dipolar interaction between the charge densities e~~~(~)w~~(~) and ew&)w&); so 14‘1= e2x2/Ri, ex = dipole moment of either charge. Taking Rij = 4 A (-n.n. distance in SmS) and x = 0.3 A (- distance to the maximum in the 4f radial charge density), one obtains 141~ 0.01 eV and hence the pertinent parameter 42 14‘1 x 0.1 eV. An alternate estimate, based on the recognition that 5~ is the essential type of matrix element responsible for the Van der Waals (Re6) interaction, gives larger but similar numbers. Optical data suggest [6] that for SmS at atmospheric pressure Ae is between 0.05 and 0.5 eV., depending on the interpretation, and decreases appreciably with p. Thus, our mechanism must be considered in any realistic theory of the configuration mixing. The removal of our simplifying assumptions is an important next step to test this mechanism quantitatively. For example, ~ii is closer to a dipolar interaction than our constant value for nearest neighbors*’ ; there is more than two orbitals and more than two electrons per site; the dependence of the lattice energy on the electron configuration is expected to have a
266
24 March 1975
We thank Dr. M. Barma and Dr. M. Campagna for very helpful discussions.
*’ Our simplitied ferroelectric ground state would give, by symmetry, zero lowering of energy if the dipolar nature of + was considered for a cubic crystal, and hence there would be no phase transition. However, it is easy to construct an antiferroelectric state (with dipoles mutually parallel within any [ 1001 sheet) that gives, using nearestneighbor dipolar Q, energy lowering which is the same order as that found in the simplified model. *3 For our model (2), with nearest-neighbor Er)for a one-dimensional lattice, Pfeuty [ 81 has shown that ( rix) # 0 for 1) > 2, in qualitative agreement with the mean field prediction.
References Ill M.B. Maple and D. Wohlleben, Phys. Rev. Lett. 27 (1971) 511.
PI M. Campagna et al., Phys. Rev. Lett. 32 (1974) 885. 131 J.C. Nickerson et al., Phys. Rev. B3 (1971) 2030. 141 J.M.D. Coey, S.K. Ghatak and F. Holtzberg, AIP Conf. Proc., Magnetism and magnetic mat’ls, San Francisco (1974). I51 L.L. Hirst, Phys. Kondens. Mater. 11 (1970) 255. 161 E. Kaldis and P. Wachter, Solid St. Commun. 11 (1972) 907; F. Holtzberg and J.B. Torrance, AIP Conf. Proc. No. 5 (1971) 860. 171 B. Batlogg, J. Schoenes and P. Wachter, Phys. Lett. 49A (1974) 13. 181 Pfeuty, Ann. Phys. 57 (1970) 79.
PHYSICS LETTERS
24 March 1975
LOCALIZED-ELECTRON MECHANISM FOR CONFIGURATION MIXING IN Sm COMPOUNDS*
Department
T.A. KAPLAN and S.D. MAHANTI of Physics, Michigan State University, East Lunsing, Michigan 48824, USA Received 4 March 1975
A new mechanism for (4fJ6 - (4f)5 Sd configuration mixing in SmBe and SmS is described. It involves the simultaneous intra-atomic excitation of two Sm sites.-
The compounds, SmB6 at normal pressure, and SmS under pressure e 6 kbar) have drawn considerable attention recently [l-4] . A remarkable feature common to these systems is the existence of a “mixedvalence” or, preferably [3] , mixed-configuration ground state, the configurations being Sm2+ (4f6) and Sm2+(4f55d or 4f56s). Although refs. [1,2,4] to Hirst’s theory [5] of this mixing has been made, quantitative calculations within this theory have been limited to a single impurity (Sm2’) in a conductionelectron sea. Hirst’s mechanism for interconfigurational mixing (ICM) is the single-electron-transfer matrix element between a 4f Wannier state to a Sd or 6s Bloch state. In this letter we point out a new mechanism which can cause ICM, we estimate its magnitude and discuss some of its consequences. Our new mechanism is based on localized-electron excitations* ’ . An apparent difficulty in finding such a mechanism is the fact that f and d are of opposite parity and a Sm site is inversion-invariant. Our mechanism overcomes this difficulty by virtue of its being a two-electron excitation. The essential matrix element involved is that, &j E (2i 2j IUIli lj),
(1)
of the Coulomb electron-electron interaction u involving 4f Wannier functions Wli, wlj at sites i and j, and 5d (or 6s) Wannier functions W2i, wzj at sites i and j.
* Research supported by NSF Grant No. GH-34565. *l Previous workers [6] interpreted the optical absorption peaks observed in SmS in terms of localized f + d excitations. The narrowness (the order of several tenths of a volt) of these peaks suggests that a localized treatment of the dstates is a reasonable first approximation.
Whereas an analogous one-particle matrix element (2ilhl li) will vanish by inversion symmetry, (1) does not vanish. The matrix element (1) involves the transition li+2i, lj+2j. We consider for simplicity a model in which there is one f-state and one d-state at each site, wti are real, and there are 2 electrons per site. We regard f2 and fd as being analogous to f6 and f5 d, respectively. In our model, we consider only crystal states Q which involve atoms that are either in f2 or the fd singlet. Thus if N= number of atoms, there are 2N such a’s, so that the matrix of the electronic Hamiltonian H, in the a,basis, can be represented by a spin-Hamiltonian which is a function of N spin-l/2 spin operators Tl , ... TN. Neglecting all inter-atomic matrix elements other than (1) or its equal (li 2j I uI2i lj), the spin Hamiltonian turns out to be 91=-Ae
CTiz i
+4 Ctij Q
Tix 5x.
(2)
The Ising operator came from 4 Tix I& = Ti+ q+ +Ti+Ti_ +h*C*; AE=h22_hll-~1111 +u1212+u1221 where h, = (vi Ih I vi), uvCcM=(vipiIulxiGi), andh is the usual bare one-electron hamiltonian. The correspondence Clli+ (f2)i, /Ii + (fd singlet)i between the eigenstates of Tis and the electronic states defines the relation between (2) and electronic properties. For example, the electron density operator in this model is
+
[Wli(‘)2”“2i(J’)21 ($- Ti~)‘~W,,ir)W2i(f)Ti,}
For our initial rough purposes we consider the ground state of (2) in the mean field approximation. Again 265
PHYSICS LETTERS
Volume 51A, number 5
for simplicity we assume Eii = - I( I for nearest neighbor i, j, zero otherwise. Then the mean field ground state is \kM, = II&iq + B&); its energy, EM, = (‘kMF, %qMF) E N(Ae/2)eMF is given in terms of n s 42 l[ l/Ae by -1 ‘MF =
-1
- (2n)-‘(n
n
n>l
significant effect on the transition. It is also important to go beyond the mean-field approximation (nonzero (Tix) would imply long-range-ordering of atomic electric dipoles* 3 see eq. (3)). Such steps have been taken in a preliminary way and our results will be reported elsewhere.
(4)
(Z = coordination number); also B2 = 0 for n < 1, B2 = (q - 1)/(27,9 for n > 1. We will see below that 77 tends to increase with pressure p. Hence our model can show an essential property of SmS, namely a change from a single configurational state, B2 = 0, to a mixed configuration state, 0 < B2 < 1, with increasing p. The order of magnitude of .$ can be estimated by noting that to leading order in inverse interatomic distance Ri’, fii is the dipolar interaction between the charge densities e~~~(~)w~~(~) and ew&)w&); so 14‘1= e2x2/Ri, ex = dipole moment of either charge. Taking Rij = 4 A (-n.n. distance in SmS) and x = 0.3 A (- distance to the maximum in the 4f radial charge density), one obtains 141~ 0.01 eV and hence the pertinent parameter 42 14‘1 x 0.1 eV. An alternate estimate, based on the recognition that 5~ is the essential type of matrix element responsible for the Van der Waals (Re6) interaction, gives larger but similar numbers. Optical data suggest [6] that for SmS at atmospheric pressure Ae is between 0.05 and 0.5 eV., depending on the interpretation, and decreases appreciably with p. Thus, our mechanism must be considered in any realistic theory of the configuration mixing. The removal of our simplifying assumptions is an important next step to test this mechanism quantitatively. For example, ~ii is closer to a dipolar interaction than our constant value for nearest neighbors*’ ; there is more than two orbitals and more than two electrons per site; the dependence of the lattice energy on the electron configuration is expected to have a
266
24 March 1975
We thank Dr. M. Barma and Dr. M. Campagna for very helpful discussions.
*’ Our simplitied ferroelectric ground state would give, by symmetry, zero lowering of energy if the dipolar nature of + was considered for a cubic crystal, and hence there would be no phase transition. However, it is easy to construct an antiferroelectric state (with dipoles mutually parallel within any [ 1001 sheet) that gives, using nearestneighbor dipolar Q, energy lowering which is the same order as that found in the simplified model. *3 For our model (2), with nearest-neighbor Er)for a one-dimensional lattice, Pfeuty [ 81 has shown that ( rix) # 0 for 1) > 2, in qualitative agreement with the mean field prediction.
References Ill M.B. Maple and D. Wohlleben, Phys. Rev. Lett. 27 (1971) 511.
PI M. Campagna et al., Phys. Rev. Lett. 32 (1974) 885. 131 J.C. Nickerson et al., Phys. Rev. B3 (1971) 2030. 141 J.M.D. Coey, S.K. Ghatak and F. Holtzberg, AIP Conf. Proc., Magnetism and magnetic mat’ls, San Francisco (1974). I51 L.L. Hirst, Phys. Kondens. Mater. 11 (1970) 255. 161 E. Kaldis and P. Wachter, Solid St. Commun. 11 (1972) 907; F. Holtzberg and J.B. Torrance, AIP Conf. Proc. No. 5 (1971) 860. 171 B. Batlogg, J. Schoenes and P. Wachter, Phys. Lett. 49A (1974) 13. 181 Pfeuty, Ann. Phys. 57 (1970) 79.