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Phase changes and configuration mixing in Sm chalcogenides
Volume 58A, number 1
PHYSICS LETTERS
26 July 1976
PHASE CHANGES AND CONFIGURATION MIXING IN Sm CHALCOGENIDES * S.D. MAHANTI, T.A. KAPLAN and M. BARMA Department of Physics, Michigan State University, East Lansing, Michigan 48824, USA Received 1 June 1976 We investigate a model, based on the localized-electron mechanism discussed by Kaplan and Mahanti, which incorporates the lattice energy and the lowering of the conduction band with increasing pressure. The results are compared with experiment.
Samarium chalcogenides have attracted considerable experimental [1—71and theoretical [8—10]attention in the last few years. Some of these compounds undergo a transition to a mixed-configuration phase on applying pressure. At 300 K SmS [1—3]undergoes a a first-order phase transition at pressure P -‘6 kbar, with a decrease in volume of “-‘10%. In the mixed phase, each Sm ion is in a mixture off6 and f5d configurations. The amount of admixture has been determined from the lattice constant [1],isomer shift [2], magnetic susceptibility [3], and XPS data [5] ; a ratio off6 to f5d of -‘3:7 has been found. While the low-P phase is a semiconductor, the mixed phase is a peculiar metal, its resistivity p at room temperature being —‘100 times larger than that of copper, increasing with decteasing temperature T, and the lowT specific heat being o~T and anomalously high [6]. SmSe and SmTe change smoothly from the semiconducting to the “metallic” phase [1]. In all but one (9] ~ of the models of the Sm-chalcogenides, all the d electrons (-‘0.7/Sm atom) are in a broad one-electron type of band in -the mixed phase, mixing arising from the scattering from d to f states and vice versa. Here we show that it may not be necessary to assume that all, or even most, of the d electrons are band-like to obtain a theoretical understanding of these systems. Recently, two of us [9] proposed a mechanism for configuration mixing in Sm-chalcogenides involving simultaneous excitation off6 to f5 (localized d) states at two Sm sites. The existence of such localized excitations (Frenkel exci* Supported by NSF Grant No.GH-34565. ~ A localized picture for most of the d-electrons in SmB6
was also discussed in ref. [11]. These authors did not consider a coherent mixing of the two configurations, in contrast to ref. [9].
tons) was suggested by optical absorption data [7, 12] in the semiconducting phase which showed intense sharp peaks, interpreted [12]as intra-atomic f6 f5 d transitions. Also, the mechanism for this mixing, namely the Coulomb matrix element responsible for the Van der Waals interaction was estimated [9]to be large enough to give significant f6 f5d mixing. The transition from the unmixed (pure f6) to a mixedconfiguration state (symbolically, Af6 + Bf5 d) was found [9]to be second order on the basis of a simple model which neglected the lattice energy. Here we present a model, based on the localized picture [9] which incorporates a) the lattice energy and b) the descent [7,13] of the conduction band with increasing P. We consider only the ground state ~ and work within a two-level model [9]involving an f2 spin-singlet and an fd spin-singlet at each Sm-site. In a mean-field approximation (not Hartree—Fock) we write the crystal wave function B = [Aa~ 4ad~t af~ladlt)] 0) .(l) 1~a~ +_(af creates an electron in Wannier function p at site I with spin a, and IA 12 + 1B12 = I. We calculate (~,HØ) where H is the usual non-relativistic Hamiltonian minus the ionic kinetic energy. Assuming that (A) the ionic positions always form the NaCl lattice, (B) we can retain in the Sm-Sm interactions only terms of leading order in (Sm-Sm separation)—1, and (C) we may expand all matrix elements to second order in V—V 2 where Vis the volume per Sm and V2 is the value of VatP = 0, we find the energy per Sm atom, —
—
II
~2
—
Our comparison of T = 0 theory with experiments at 300 K is reasonable in view of the insensitivity of P~to temperature change for SmS, found in ref. [1].
43
Volume 58A, number 1
PHYSICS LETTERS
E(n, V)(1 —n)~k2(V—V2)2 +n[~k 2+c] —W(V)n(l —n). (2) 3(V— V3) Here n = 1B12, the number of d electrons in (1), and W is the quantity 4z1 ~Iestimated to be ~0.5 eV in ref. [9]. V 3 is the location of the energy minimum when n = 1. k2 and k3 are constants and c E(l, V3) E(0, V2). Eq. (2) above is the extension of the mean-field approximation to eq. (2) of ref. [9]with2 the of the of “lattice” energy, the retention quantity ~e(V) ref. [9]is E(l, }k2(V V) E(0,V2) V). Let n* be the value of the variational parameter n which minimizes eq. (2) and let F’ = _dE(n*, V)/dV. Depending on the parameter values, dF’/dV might be positive in which case Maxwell’s equal-area construction is used to obtain the physical pressure P from F’.
26 July 1976
IC
0.9 SmS
SmSe
SmTe
0.8
—
—
—
The parameters k2, V2 and V3 are taken from experi3~ ment radius (V3 l~ 0.83 ionic andV2 k was determined from the Sm 2 = B/V2 where B is the P = 0 bulk modulus). k3 and c are also determined from experimental values of ~ I d&/dFI~=0and ~e(V2), the excitonic energy ohserved optically at P = 0, via k3
—
k 2~R ‘~“V 2
C =
~
~k
‘V 2’
‘~—
—
V3)~ ‘
‘k3’‘2”V “~ 2 ‘.
V3/\2
20
30
40 P (Kbar) ~0 60
70
80
90
00
Fig. 1. Pressure-volume curves obtained with parameter values
given in the text.
the one in SmS. In fact, with the parameters above, SmTe undergoes a second order transition while the change in SmSe is weakly first order (~ V/V 2 -‘0.01). These results are quite sensitive to the parameter values. In particular, the transition in SmSe becomes second order on increasing W or decreasing ~by 10%. The overall agreement between results and experiment is quite good in view of our approximations and simplifications. In particular, assumption (C) above will be ‘~
poor at sufficiently high F where the hard core repulsion becomes important.
(4\
enable A prediction experiment of to thedistinguish present model it from [9] the which “band” might
‘
‘
In deriving (3) we assumed that n” = 0 iii the neighborhood ofF = 0, consistent with experiment [2,5] The parameter values for (SmS, SmSe, SmTe) are: B(500, 400, 400) kbar; ~(12, 12, 12) meV/kbar; &(V2) (0.6, 0.75, 1.0) eV; ~P(0.3, 0.4, 0.5) eV. For simplicity we took W(V) = ii’, independent of V and chose it to get rough agreement with experimental F— V curves. These values of ~ have the order of magnitude estimated previously [91, and the trend through the series 5, Se, Te might result from the enhancement of ~ through the polarizability of the anions. The results are shown in fig. 1. We see a first order transition for SmS at ~c 14 kbar with ~ V/V2 -‘0.12 and find a change in ~ from 0 to ‘-0.7. For SmSe and SmTe the fairlyrapid decrease in V at ‘—30 kbar and at ‘—‘50 kbar respectively is much less rapid than -
t~One can argue plausible that the filled Ss-5p shell essentially determine 4ion the[(4f)5]. size of both the Sm~’~ and the Sm~
i~
~3) ,.
—
°•~
[(4fl5 5d]
models, is that in the mixed state each Sm ion possesses an electric dipole moment d1. Ordering of the d1’s is determined essentially by dipolar interactions. The existence of such self-consistently induced moments has been discussed by Javnes and Wigner [14]‘in connection with the ferroelectricity in BaTiO3. We note however that the nature of the ordering of a set of dipoles is unknown, and is likely to be much more complex than that in a simple ferro- or antiferroelectric. The metallic behavior in collapsed SmS is interpreted in terms of the partial lowering of a broad conduction band at the first order transition [15]. In our model the number of electrons b occupying Bloch states near the bottom of this band is small (b 0.1/Sm atom). l’his picture is based on the following: (a) The effective mass m* at the band bottom (which occurs at the X-point [16]) in the semiconducting phase is estimated from refs. [16, 17] to lie between band lowering 0.4due andto0.8 thefree lattice electron collapse masses. across (b)the The
Volume 58A, number 1
PHYSICS LETTERS
26 July 1976
1st order transition is ‘--‘0.6 eV (ref. [4] we used ~V/V2 12%). (c) If ~ were zero, the chemical potential p would be 0.6 eV above the band bottom (from (b), assuming the gap 0 as P -~ ~c —0); with W *0 it can be seen (assuming orthogonal band and localized states) that p is reduced by ~ = (~ ~~*2 the energy lowering from E(0, V) in the collapsed phase. (b) and (c) give p 0.5 eV; with m* = 0.8, we get b 0.15.2/V Usingb = 0.15 and a p~2cm sec. gives p -‘130 mobility [17] of’--lO cm at 300°Kin rough agreement with experiment [6]. Occupation of the band by —‘0.1 electrons/Sm leaves holes in the localized, mixed state, eq. (1). Polaron effects on these holes might lead to a large enough density~of states at p to account for the low-T specific heat [6]. Further, these holes may scatter the band electrons giving rise to the increase [6] in p at low T. Experimentally, the mixed (ground) state is nonmagnetic. An important problem [6] is the reconciliation of the lack of magnetism with the known Kramers degeneracy of the f5 ground state in an octahedral crystal field. In our localized model ~ one must consider the modification of the Sm3~ ground state caused by the localized d-electron (giving an even number of electrons at each site) plus the effect of W. Within the mean-field approximation it seems likely that each Sm-ion will have a nonmagnetic ground state. One possibility leading to such a result is the following: Wcauses the lowest f5 d multiplet to mix strongly with the higher lying f6 J 0 state; it can be seen that such mixing leads to a’ nondegenerate single-ion ground state and this would be nonmagnetic because of time-reversal invariance of the mean-field Hamiltonian. Another question is concerned with the stability of the localized excitons against screening by the
small number of conduction electrons. Even if all of
1~One should not take literally the nonmagnetic nature of our 2-state model.
[17] J.W. McClure, J. Phys. Chem. Soi. 24 (1963) 871; A.V. Golubkov et al., Soy. Phys. Sol. St. 7 (1966) 1963. [18] J. Schweitzer, to be published, Phys. Rev. B.
—
-~
—
w)~*
+
the latter were to cluster inside the d shell of each Sm, only a small fraction of the f-shell charge would be screened, insuring stability. Further, support exists [18] for the possible coexistence of these excitons and band electrons, both derived from the same one-electron rtates, although the generalization of this calculation [18] to a more realistic Hamiltonian is needed. We thank M. Campagna, M.N. Cohen, C.L. Foiles, T. Penney and S. von Molnar for very helpful discussions. -
References [1] A. Jayaraman et al., Phys. Rev. Letters 25 (1970) 1430; A. Jayaraman et al., Phys. Rev. Bil (1975) 2783. [2] J.M.D. Coey et al., AlP Conf. Proc. no. 24 (1974) p. 38. [3] M.B. Maple (1971) 511.and D. Wohlleben, Phys. Rev. Letters 27 [4] T. Penney and F. Holtzberg, Phys. Rev. Letters 34 (1975) 322.
[5] M. Campagna et ai., Phys. Rev. Letters 33 (1974) 165. [6] S.D. Bader et al., Phys. Rev. B7 (1973) 4786. [7] E. Kaldis and
P. Wachter, Sol. St. Comm. 11(1972) 907. [8] L.L. Hirst, J. Phys. Chem. Solids 35 (1974) 1285. [9] TA. Kaplan and S.D. Mahanti, Fhys. Lett. 51A (1975) 265. [10] C.M. Varma and V. Heine, Phys. Rev. Bli (1975) 4763.
[11] J.C. Nickerson et al., Phys. Rev. B3 (1971) 2030. [12] F. Holtzberg and J.B. Torrance, ALP Conf. Proc. no.5 (1971) p. 860. [13] A. Jayaraman et al., Phys. Rev. B9 (1974) 2513. [14] E.T. Jaynes and E.P. Wigner, Phys. Rev. 79 (1950) 213. [15] T.A. Kaplan, S.D. Mahanti and Phys. Soc. 20 (1975) 383.
M. Barma, Bull. Am.
[16] H.L. Davis, Proc. 9th Rare Earth Res. Conf., Va. Poly Inst. and State Un., Blacksburg, Va. (1971).
45
PHYSICS LETTERS
26 July 1976
PHASE CHANGES AND CONFIGURATION MIXING IN Sm CHALCOGENIDES * S.D. MAHANTI, T.A. KAPLAN and M. BARMA Department of Physics, Michigan State University, East Lansing, Michigan 48824, USA Received 1 June 1976 We investigate a model, based on the localized-electron mechanism discussed by Kaplan and Mahanti, which incorporates the lattice energy and the lowering of the conduction band with increasing pressure. The results are compared with experiment.
Samarium chalcogenides have attracted considerable experimental [1—71and theoretical [8—10]attention in the last few years. Some of these compounds undergo a transition to a mixed-configuration phase on applying pressure. At 300 K SmS [1—3]undergoes a a first-order phase transition at pressure P -‘6 kbar, with a decrease in volume of “-‘10%. In the mixed phase, each Sm ion is in a mixture off6 and f5d configurations. The amount of admixture has been determined from the lattice constant [1],isomer shift [2], magnetic susceptibility [3], and XPS data [5] ; a ratio off6 to f5d of -‘3:7 has been found. While the low-P phase is a semiconductor, the mixed phase is a peculiar metal, its resistivity p at room temperature being —‘100 times larger than that of copper, increasing with decteasing temperature T, and the lowT specific heat being o~T and anomalously high [6]. SmSe and SmTe change smoothly from the semiconducting to the “metallic” phase [1]. In all but one (9] ~ of the models of the Sm-chalcogenides, all the d electrons (-‘0.7/Sm atom) are in a broad one-electron type of band in -the mixed phase, mixing arising from the scattering from d to f states and vice versa. Here we show that it may not be necessary to assume that all, or even most, of the d electrons are band-like to obtain a theoretical understanding of these systems. Recently, two of us [9] proposed a mechanism for configuration mixing in Sm-chalcogenides involving simultaneous excitation off6 to f5 (localized d) states at two Sm sites. The existence of such localized excitations (Frenkel exci* Supported by NSF Grant No.GH-34565. ~ A localized picture for most of the d-electrons in SmB6
was also discussed in ref. [11]. These authors did not consider a coherent mixing of the two configurations, in contrast to ref. [9].
tons) was suggested by optical absorption data [7, 12] in the semiconducting phase which showed intense sharp peaks, interpreted [12]as intra-atomic f6 f5 d transitions. Also, the mechanism for this mixing, namely the Coulomb matrix element responsible for the Van der Waals interaction was estimated [9]to be large enough to give significant f6 f5d mixing. The transition from the unmixed (pure f6) to a mixedconfiguration state (symbolically, Af6 + Bf5 d) was found [9]to be second order on the basis of a simple model which neglected the lattice energy. Here we present a model, based on the localized picture [9] which incorporates a) the lattice energy and b) the descent [7,13] of the conduction band with increasing P. We consider only the ground state ~ and work within a two-level model [9]involving an f2 spin-singlet and an fd spin-singlet at each Sm-site. In a mean-field approximation (not Hartree—Fock) we write the crystal wave function B = [Aa~ 4ad~t af~ladlt)] 0) .(l) 1~a~ +_(af creates an electron in Wannier function p at site I with spin a, and IA 12 + 1B12 = I. We calculate (~,HØ) where H is the usual non-relativistic Hamiltonian minus the ionic kinetic energy. Assuming that (A) the ionic positions always form the NaCl lattice, (B) we can retain in the Sm-Sm interactions only terms of leading order in (Sm-Sm separation)—1, and (C) we may expand all matrix elements to second order in V—V 2 where Vis the volume per Sm and V2 is the value of VatP = 0, we find the energy per Sm atom, —
—
II
~2
—
Our comparison of T = 0 theory with experiments at 300 K is reasonable in view of the insensitivity of P~to temperature change for SmS, found in ref. [1].
43
Volume 58A, number 1
PHYSICS LETTERS
E(n, V)(1 —n)~k2(V—V2)2 +n[~k 2+c] —W(V)n(l —n). (2) 3(V— V3) Here n = 1B12, the number of d electrons in (1), and W is the quantity 4z1 ~Iestimated to be ~0.5 eV in ref. [9]. V 3 is the location of the energy minimum when n = 1. k2 and k3 are constants and c E(l, V3) E(0, V2). Eq. (2) above is the extension of the mean-field approximation to eq. (2) of ref. [9]with2 the of the of “lattice” energy, the retention quantity ~e(V) ref. [9]is E(l, }k2(V V) E(0,V2) V). Let n* be the value of the variational parameter n which minimizes eq. (2) and let F’ = _dE(n*, V)/dV. Depending on the parameter values, dF’/dV might be positive in which case Maxwell’s equal-area construction is used to obtain the physical pressure P from F’.
26 July 1976
IC
0.9 SmS
SmSe
SmTe
0.8
—
—
—
The parameters k2, V2 and V3 are taken from experi3~ ment radius (V3 l~ 0.83 ionic andV2 k was determined from the Sm 2 = B/V2 where B is the P = 0 bulk modulus). k3 and c are also determined from experimental values of ~ I d&/dFI~=0and ~e(V2), the excitonic energy ohserved optically at P = 0, via k3
—
k 2~R ‘~“V 2
C =
~
~k
‘V 2’
‘~—
—
V3)~ ‘
‘k3’‘2”V “~ 2 ‘.
V3/\2
20
30
40 P (Kbar) ~0 60
70
80
90
00
Fig. 1. Pressure-volume curves obtained with parameter values
given in the text.
the one in SmS. In fact, with the parameters above, SmTe undergoes a second order transition while the change in SmSe is weakly first order (~ V/V 2 -‘0.01). These results are quite sensitive to the parameter values. In particular, the transition in SmSe becomes second order on increasing W or decreasing ~by 10%. The overall agreement between results and experiment is quite good in view of our approximations and simplifications. In particular, assumption (C) above will be ‘~
poor at sufficiently high F where the hard core repulsion becomes important.
(4\
enable A prediction experiment of to thedistinguish present model it from [9] the which “band” might
‘
‘
In deriving (3) we assumed that n” = 0 iii the neighborhood ofF = 0, consistent with experiment [2,5] The parameter values for (SmS, SmSe, SmTe) are: B(500, 400, 400) kbar; ~(12, 12, 12) meV/kbar; &(V2) (0.6, 0.75, 1.0) eV; ~P(0.3, 0.4, 0.5) eV. For simplicity we took W(V) = ii’, independent of V and chose it to get rough agreement with experimental F— V curves. These values of ~ have the order of magnitude estimated previously [91, and the trend through the series 5, Se, Te might result from the enhancement of ~ through the polarizability of the anions. The results are shown in fig. 1. We see a first order transition for SmS at ~c 14 kbar with ~ V/V2 -‘0.12 and find a change in ~ from 0 to ‘-0.7. For SmSe and SmTe the fairlyrapid decrease in V at ‘—30 kbar and at ‘—‘50 kbar respectively is much less rapid than -
t~One can argue plausible that the filled Ss-5p shell essentially determine 4ion the[(4f)5]. size of both the Sm~’~ and the Sm~
i~
~3) ,.
—
°•~
[(4fl5 5d]
models, is that in the mixed state each Sm ion possesses an electric dipole moment d1. Ordering of the d1’s is determined essentially by dipolar interactions. The existence of such self-consistently induced moments has been discussed by Javnes and Wigner [14]‘in connection with the ferroelectricity in BaTiO3. We note however that the nature of the ordering of a set of dipoles is unknown, and is likely to be much more complex than that in a simple ferro- or antiferroelectric. The metallic behavior in collapsed SmS is interpreted in terms of the partial lowering of a broad conduction band at the first order transition [15]. In our model the number of electrons b occupying Bloch states near the bottom of this band is small (b 0.1/Sm atom). l’his picture is based on the following: (a) The effective mass m* at the band bottom (which occurs at the X-point [16]) in the semiconducting phase is estimated from refs. [16, 17] to lie between band lowering 0.4due andto0.8 thefree lattice electron collapse masses. across (b)the The
Volume 58A, number 1
PHYSICS LETTERS
26 July 1976
1st order transition is ‘--‘0.6 eV (ref. [4] we used ~V/V2 12%). (c) If ~ were zero, the chemical potential p would be 0.6 eV above the band bottom (from (b), assuming the gap 0 as P -~ ~c —0); with W *0 it can be seen (assuming orthogonal band and localized states) that p is reduced by ~ = (~ ~~*2 the energy lowering from E(0, V) in the collapsed phase. (b) and (c) give p 0.5 eV; with m* = 0.8, we get b 0.15.2/V Usingb = 0.15 and a p~2cm sec. gives p -‘130 mobility [17] of’--lO cm at 300°Kin rough agreement with experiment [6]. Occupation of the band by —‘0.1 electrons/Sm leaves holes in the localized, mixed state, eq. (1). Polaron effects on these holes might lead to a large enough density~of states at p to account for the low-T specific heat [6]. Further, these holes may scatter the band electrons giving rise to the increase [6] in p at low T. Experimentally, the mixed (ground) state is nonmagnetic. An important problem [6] is the reconciliation of the lack of magnetism with the known Kramers degeneracy of the f5 ground state in an octahedral crystal field. In our localized model ~ one must consider the modification of the Sm3~ ground state caused by the localized d-electron (giving an even number of electrons at each site) plus the effect of W. Within the mean-field approximation it seems likely that each Sm-ion will have a nonmagnetic ground state. One possibility leading to such a result is the following: Wcauses the lowest f5 d multiplet to mix strongly with the higher lying f6 J 0 state; it can be seen that such mixing leads to a’ nondegenerate single-ion ground state and this would be nonmagnetic because of time-reversal invariance of the mean-field Hamiltonian. Another question is concerned with the stability of the localized excitons against screening by the
small number of conduction electrons. Even if all of
1~One should not take literally the nonmagnetic nature of our 2-state model.
[17] J.W. McClure, J. Phys. Chem. Soi. 24 (1963) 871; A.V. Golubkov et al., Soy. Phys. Sol. St. 7 (1966) 1963. [18] J. Schweitzer, to be published, Phys. Rev. B.
—
-~
—
w)~*
+
the latter were to cluster inside the d shell of each Sm, only a small fraction of the f-shell charge would be screened, insuring stability. Further, support exists [18] for the possible coexistence of these excitons and band electrons, both derived from the same one-electron rtates, although the generalization of this calculation [18] to a more realistic Hamiltonian is needed. We thank M. Campagna, M.N. Cohen, C.L. Foiles, T. Penney and S. von Molnar for very helpful discussions. -
References [1] A. Jayaraman et al., Phys. Rev. Letters 25 (1970) 1430; A. Jayaraman et al., Phys. Rev. Bil (1975) 2783. [2] J.M.D. Coey et al., AlP Conf. Proc. no. 24 (1974) p. 38. [3] M.B. Maple (1971) 511.and D. Wohlleben, Phys. Rev. Letters 27 [4] T. Penney and F. Holtzberg, Phys. Rev. Letters 34 (1975) 322.
[5] M. Campagna et ai., Phys. Rev. Letters 33 (1974) 165. [6] S.D. Bader et al., Phys. Rev. B7 (1973) 4786. [7] E. Kaldis and
P. Wachter, Sol. St. Comm. 11(1972) 907. [8] L.L. Hirst, J. Phys. Chem. Solids 35 (1974) 1285. [9] TA. Kaplan and S.D. Mahanti, Fhys. Lett. 51A (1975) 265. [10] C.M. Varma and V. Heine, Phys. Rev. Bli (1975) 4763.
[11] J.C. Nickerson et al., Phys. Rev. B3 (1971) 2030. [12] F. Holtzberg and J.B. Torrance, ALP Conf. Proc. no.5 (1971) p. 860. [13] A. Jayaraman et al., Phys. Rev. B9 (1974) 2513. [14] E.T. Jaynes and E.P. Wigner, Phys. Rev. 79 (1950) 213. [15] T.A. Kaplan, S.D. Mahanti and Phys. Soc. 20 (1975) 383.
M. Barma, Bull. Am.
[16] H.L. Davis, Proc. 9th Rare Earth Res. Conf., Va. Poly Inst. and State Un., Blacksburg, Va. (1971).
45