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Electronic structure of doped f semiconducting compounds
Physica 102B (1980) 88-90 © North-Holland Publishing Company
ELECTRONIC STRUCTURE OF DOPED f SEMICONDUCTING COMPOUNDS C. D E M A N G E A T , M.A. K H A N and J.C. P A R L E B A S Laboratoire de Magn~tisme et de Structure Electronique des Solides, (LA CNRS N ° 306) Universit~ Louis Pasteur, lnstitut Le Bel 4, rue Blaise Pascal 67070, Strasbourg Cedex, France
We present a tight-binding formalism to calculate the local electronic structure on the nearest neighbours to a substitutional anion impurity embedded in a SmS-type compound. Our treatment, which carefully treats the df admixture, is a good framework for further explicit contact with the experimentally observed properties of valence transition in several impurity problems.
H °= ~
There are many examples of f semiconducting compounds which exhibit intermediate valence characteristics under alloying. These properties are generally associated with a transfer of f electrons to the ds conduction bands. In this paper we focus our attention on impurities, like As, P, O, etc. in SmS-type compounds, which induce valence transitions even at low concentration [1-3]. We inquire into the general question of the local electronic structure on the nearest neighbours to such a substitutional anion impurity. A first model [4] has been proposed on the qualitative effect of fd hybridization in the prt,~ence of one impurity embedded in a semiconducting material where the Fermi energy EF lies in the very small gap between a narrow f band and a broad ds conduction band (CB). Here we settle a more quantitative formalism which takes into account a generalized SlaterKoster fit to first principles calculations for the host bands structure [5]. The present treatment should be a convenient framework for explicit contact with available data, for example in the case of doped SmS compounds. Before solving the impurity system let us calculate (within a simple but realistic model) the electronic structure of a pure semiconducting SmS-type compound. The corresponding Hamiltonian is written in the Bloch representation from a generalization of the Anderson Hamiltonian to a lattice of f impurities:
Imktr)er~(mktrl+ d ~[]:k)(fkl
We notice that an analogous model has been used by Jullien and Coqblin [6] in order to study the df hybridization in actinide metals and alloys. The first term represents the p valence and sd conduction bands formed from the spd tightbinding orbitals Im;t~) with spin o- and centred on the samarium site A when m = 1, xy, yz, zx, x 2 _ y2, 3z 2 _ r symmetries and on the sulfur site A when m = x, y, z symmetries. The second term of eq. (1) describes a zero width "spinless band" of energy e f. It is well known that because of their compactly localized and highly correlated nature, f energy levels cannot be obtained directly from conventional one-electron bands structure calculation methods. Since at most one electron can be easily taken out of the f shell to CB, the localized quasi-particle level will be assumed to contain one (spinless) state per cation. The energy gap between e ~ and CB is then the effective energy required to excite an f electron. The last term of eq. (1) is actually different from a one-electron mixing between f and spd states, but we will consider it as a phenomenological parameter responsible for the final non-zero f bandwidth. We remark that the admixture between f and d states is as non88
C. Demangeat et al./Electronic structure of doped f semiconductors
negligible as the semiconducting gap is smaller; also, it is possible to show that it can be enhanced by fd correlations [7]. In the present paper we only consider the fd-type of admixture. Taking into account spin degeneracy for spd states, the eigensolutions {Inkcr), E°(k)} of H ° for a given k vector of the first Brillouin zone are obtained by diagonalization of a {10 x 10} matrix expressed in the {Ilk), ]akar)} set with a = s, p, d symmetries:
H°lnktr) = E°(k )lnk~).
(2)
The following parameters (in Slater-Koster notation) appear in eq. (2): sstr, pptr, pplr, ddtr, ddlr, ddS, sptr, sdtr, pdtr, pdTr, e s, e p, eel, et2~ for a states: they are deduced from a Slater-Koster fit to a K K R bands structure calculation of SmS compounds [5]. It is important to note that part of our dd overlap parameters implicitly comes from mixing of the d orbitals with the anion p orbitals. Besides, e f is adjusted in order to obtain the order of magnitude of the experimental gap between e f and the bottom of C.B. Also, we introduce a phenomenological parameter fdtr between f and d states which is consistent with a very small bandwidth. The different densities of states (DOS) are then computed. Now let us consider a single impurity substituted at an anion site 0 in a SmS-type compound. We study the electronic impurity levels localized on the six nearest cation neighbouring sites of site 0. In agreement with experimentally observed collapses of such allows [1-3], the impurity is essentially presumed to increase the overlap integrals ddtr, dd~r and dd8 between ImRtr) orbitals (m = xy, yz, zx, x 2 - y2, 3z 2 - r 2) around 0. Thus, the perturbed Hamiltonian is given by: n = n ° + <~,>,...]mRtr)8/3'~;(rnR'trl,
(3)
m,m',tr
where the notation (R, R') indicates that the summation of eq. (3) is only over nearest neighbouring sites among the six sites surrounding 0: u, 8/3rr' being the dd hopping change between two such sites. For simplicity, we will only retain diagonal 8/3r~, matrix elements which are easily expressed in term of three "vectors" 8/3~, 8/3~',
89
and 8/37 relative to (0, 1, 1), (1, 0, 1), and (1, 1, 0) directions of RR'. Also, we neglect all the interatomic dd Green's functions in order to keep our results as simple as possible. Actually, in eq. (3) we should also take into account a similar df hopping change noted 5~3mr; instead of that we consider only {8/37'} changes, but we consider them as being effectively enhanced by 8/3 mf effects combined with host fm mixing. The above model is then solved in closed form after some manipulation. In particular the d local DOS on a site R nearest neighbour to 0 is expressed by
n~(E) Im
2 0 0 - (E)[1 - 2 N " (E){G °" (E)} 2] [1 - 4 D m ( E ) { G ° " ( E ) } 2] ' (4)
with / V ' ( E ) = [(8/3~)~ + (8/37) ~ + 2(8/37) ~] 3
+ 4O°m(E) l~ 3137,. 3
3
D ' ( E ) = ~ (8/37')2 + 4G°m(E)I~ 8/37'. In eq. (4) we note G°m(E) the intraatomic d host Green functions, i.e. (mRtrl(E +- H°)-llmRtr ). Retaining only fd hybridized Green's functions between nearest neighbours R and R ' we easily obtain the f local DOS:
n~R(E) = --~r-~ Im{G01(E)-
488fl~[ G°fd(E)] 2 1 + 28/3~G°~'(E)
16[G°fd(E)]2[8/3~ + 4(8/3 IS)EG°~'(E)] ~, + 1 - 2G°~(E)[8/3~ + 4(8/3 ~)2G°~g(E)] J (5) where the superscripts 4 and 5 mean (x 2 - y2) and ( 3 z 2 - r 2) symmetries; G°t(E) is the intrasite f Green's function, i.e. (fRI(E÷-H°)-IIfR) and G°fd(E) is a particular hybridized Green's function, i.e. (fR I(E ÷ - H°)-'15; R'; tr). We are presently engaged in a detailed numerical investigation of eqs. (4) and (5) in the case of doped SmS compounds, but let us just present here preliminary (but illustrative) results when the contributions of each of the m orbitals
90
C. Demangeat et al.lElectronic structure of doped f semiconductors
are supposed to be equal, i.e. when we are only dealing with s-type orbitals [8]. In that case the following intriguing phenomenon may occur. A bonding state can be easily extracted from CB and fall below CB even below e f (similarly an antibonding state falls above CB). As a consequence, and through the essential fd admixture, the previously considered bonding state at energy below e f is able to drive a local f "quasistate" above the Fermi level. This process causes a drastic change in the valence of the samarium ions around 0 in agreement with the experimental tendency [1-3]. Clearly, the df mixing should play an appreciable role in all impurity problems as the present. Finally, we would like to point out that it is possible to relate the change in the hopping integrals to the corresponding macroscopic volume variation through a dipolar tensor
calculation (see, for example, [9]). Further studies in this direction are in progress. References [I] F. Holtzberg, P. Pena, T. Penney and R. Tournier, in: Valence Instabilities and Related Narrow-Band Phenomena, R,D. Parks, ed. (Plenum, New York, 1977) p. 507. [2] O. Pena and R. Tournier, ICM' 1979, to appear in J. Magn. Magn. Mat. [3] G. Krill and J.M. Leger, to be published. [4] R. Camley, J.C. Parlebas, K.R. Subbaswamy and D.L. Mills, J. de Physique C5 (1979) 372. [5] H.L. Davis, 9th R.E. Research Conf. (Field Editor), vol. 1 (1971) p. 3. [6] R. Jullien and B. Coqblin, Phys. Rev. B 8 (1973) 5263. 17] D.I. Khomskii and A.N. Kocharjan, Solid State Commun. 18 (1976) 985. [8] J.C. Parlebas and C. Demangeat, 3~mes Journ6es RCP 520, Grenoble (1980). [9] G. Moraitis and F. Gautier, J. Phys. F 9 (1979) 2025.
ELECTRONIC STRUCTURE OF DOPED f SEMICONDUCTING COMPOUNDS C. D E M A N G E A T , M.A. K H A N and J.C. P A R L E B A S Laboratoire de Magn~tisme et de Structure Electronique des Solides, (LA CNRS N ° 306) Universit~ Louis Pasteur, lnstitut Le Bel 4, rue Blaise Pascal 67070, Strasbourg Cedex, France
We present a tight-binding formalism to calculate the local electronic structure on the nearest neighbours to a substitutional anion impurity embedded in a SmS-type compound. Our treatment, which carefully treats the df admixture, is a good framework for further explicit contact with the experimentally observed properties of valence transition in several impurity problems.
H °= ~
There are many examples of f semiconducting compounds which exhibit intermediate valence characteristics under alloying. These properties are generally associated with a transfer of f electrons to the ds conduction bands. In this paper we focus our attention on impurities, like As, P, O, etc. in SmS-type compounds, which induce valence transitions even at low concentration [1-3]. We inquire into the general question of the local electronic structure on the nearest neighbours to such a substitutional anion impurity. A first model [4] has been proposed on the qualitative effect of fd hybridization in the prt,~ence of one impurity embedded in a semiconducting material where the Fermi energy EF lies in the very small gap between a narrow f band and a broad ds conduction band (CB). Here we settle a more quantitative formalism which takes into account a generalized SlaterKoster fit to first principles calculations for the host bands structure [5]. The present treatment should be a convenient framework for explicit contact with available data, for example in the case of doped SmS compounds. Before solving the impurity system let us calculate (within a simple but realistic model) the electronic structure of a pure semiconducting SmS-type compound. The corresponding Hamiltonian is written in the Bloch representation from a generalization of the Anderson Hamiltonian to a lattice of f impurities:
Imktr)er~(mktrl+ d ~[]:k)(fkl
We notice that an analogous model has been used by Jullien and Coqblin [6] in order to study the df hybridization in actinide metals and alloys. The first term represents the p valence and sd conduction bands formed from the spd tightbinding orbitals Im;t~) with spin o- and centred on the samarium site A when m = 1, xy, yz, zx, x 2 _ y2, 3z 2 _ r symmetries and on the sulfur site A when m = x, y, z symmetries. The second term of eq. (1) describes a zero width "spinless band" of energy e f. It is well known that because of their compactly localized and highly correlated nature, f energy levels cannot be obtained directly from conventional one-electron bands structure calculation methods. Since at most one electron can be easily taken out of the f shell to CB, the localized quasi-particle level will be assumed to contain one (spinless) state per cation. The energy gap between e ~ and CB is then the effective energy required to excite an f electron. The last term of eq. (1) is actually different from a one-electron mixing between f and spd states, but we will consider it as a phenomenological parameter responsible for the final non-zero f bandwidth. We remark that the admixture between f and d states is as non88
C. Demangeat et al./Electronic structure of doped f semiconductors
negligible as the semiconducting gap is smaller; also, it is possible to show that it can be enhanced by fd correlations [7]. In the present paper we only consider the fd-type of admixture. Taking into account spin degeneracy for spd states, the eigensolutions {Inkcr), E°(k)} of H ° for a given k vector of the first Brillouin zone are obtained by diagonalization of a {10 x 10} matrix expressed in the {Ilk), ]akar)} set with a = s, p, d symmetries:
H°lnktr) = E°(k )lnk~).
(2)
The following parameters (in Slater-Koster notation) appear in eq. (2): sstr, pptr, pplr, ddtr, ddlr, ddS, sptr, sdtr, pdtr, pdTr, e s, e p, eel, et2~ for a states: they are deduced from a Slater-Koster fit to a K K R bands structure calculation of SmS compounds [5]. It is important to note that part of our dd overlap parameters implicitly comes from mixing of the d orbitals with the anion p orbitals. Besides, e f is adjusted in order to obtain the order of magnitude of the experimental gap between e f and the bottom of C.B. Also, we introduce a phenomenological parameter fdtr between f and d states which is consistent with a very small bandwidth. The different densities of states (DOS) are then computed. Now let us consider a single impurity substituted at an anion site 0 in a SmS-type compound. We study the electronic impurity levels localized on the six nearest cation neighbouring sites of site 0. In agreement with experimentally observed collapses of such allows [1-3], the impurity is essentially presumed to increase the overlap integrals ddtr, dd~r and dd8 between ImRtr) orbitals (m = xy, yz, zx, x 2 - y2, 3z 2 - r 2) around 0. Thus, the perturbed Hamiltonian is given by: n = n ° + <~,>,...]mRtr)8/3'~;(rnR'trl,
(3)
m,m',tr
where the notation (R, R') indicates that the summation of eq. (3) is only over nearest neighbouring sites among the six sites surrounding 0: u, 8/3rr' being the dd hopping change between two such sites. For simplicity, we will only retain diagonal 8/3r~, matrix elements which are easily expressed in term of three "vectors" 8/3~, 8/3~',
89
and 8/37 relative to (0, 1, 1), (1, 0, 1), and (1, 1, 0) directions of RR'. Also, we neglect all the interatomic dd Green's functions in order to keep our results as simple as possible. Actually, in eq. (3) we should also take into account a similar df hopping change noted 5~3mr; instead of that we consider only {8/37'} changes, but we consider them as being effectively enhanced by 8/3 mf effects combined with host fm mixing. The above model is then solved in closed form after some manipulation. In particular the d local DOS on a site R nearest neighbour to 0 is expressed by
n~(E) Im
2 0 0 - (E)[1 - 2 N " (E){G °" (E)} 2] [1 - 4 D m ( E ) { G ° " ( E ) } 2] ' (4)
with / V ' ( E ) = [(8/3~)~ + (8/37) ~ + 2(8/37) ~] 3
+ 4O°m(E) l~ 3137,. 3
3
D ' ( E ) = ~ (8/37')2 + 4G°m(E)I~ 8/37'. In eq. (4) we note G°m(E) the intraatomic d host Green functions, i.e. (mRtrl(E +- H°)-llmRtr ). Retaining only fd hybridized Green's functions between nearest neighbours R and R ' we easily obtain the f local DOS:
n~R(E) = --~r-~ Im{G01(E)-
488fl~[ G°fd(E)] 2 1 + 28/3~G°~'(E)
16[G°fd(E)]2[8/3~ + 4(8/3 IS)EG°~'(E)] ~, + 1 - 2G°~(E)[8/3~ + 4(8/3 ~)2G°~g(E)] J (5) where the superscripts 4 and 5 mean (x 2 - y2) and ( 3 z 2 - r 2) symmetries; G°t(E) is the intrasite f Green's function, i.e. (fRI(E÷-H°)-IIfR) and G°fd(E) is a particular hybridized Green's function, i.e. (fR I(E ÷ - H°)-'15; R'; tr). We are presently engaged in a detailed numerical investigation of eqs. (4) and (5) in the case of doped SmS compounds, but let us just present here preliminary (but illustrative) results when the contributions of each of the m orbitals
90
C. Demangeat et al.lElectronic structure of doped f semiconductors
are supposed to be equal, i.e. when we are only dealing with s-type orbitals [8]. In that case the following intriguing phenomenon may occur. A bonding state can be easily extracted from CB and fall below CB even below e f (similarly an antibonding state falls above CB). As a consequence, and through the essential fd admixture, the previously considered bonding state at energy below e f is able to drive a local f "quasistate" above the Fermi level. This process causes a drastic change in the valence of the samarium ions around 0 in agreement with the experimental tendency [1-3]. Clearly, the df mixing should play an appreciable role in all impurity problems as the present. Finally, we would like to point out that it is possible to relate the change in the hopping integrals to the corresponding macroscopic volume variation through a dipolar tensor
calculation (see, for example, [9]). Further studies in this direction are in progress. References [I] F. Holtzberg, P. Pena, T. Penney and R. Tournier, in: Valence Instabilities and Related Narrow-Band Phenomena, R,D. Parks, ed. (Plenum, New York, 1977) p. 507. [2] O. Pena and R. Tournier, ICM' 1979, to appear in J. Magn. Magn. Mat. [3] G. Krill and J.M. Leger, to be published. [4] R. Camley, J.C. Parlebas, K.R. Subbaswamy and D.L. Mills, J. de Physique C5 (1979) 372. [5] H.L. Davis, 9th R.E. Research Conf. (Field Editor), vol. 1 (1971) p. 3. [6] R. Jullien and B. Coqblin, Phys. Rev. B 8 (1973) 5263. 17] D.I. Khomskii and A.N. Kocharjan, Solid State Commun. 18 (1976) 985. [8] J.C. Parlebas and C. Demangeat, 3~mes Journ6es RCP 520, Grenoble (1980). [9] G. Moraitis and F. Gautier, J. Phys. F 9 (1979) 2025.