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Energy band structure of cadmium fluoride
Solid State Communications, Vol. 22, pp. 199—201, 1977.
Pergamon Press.
Printed in Great Britain
ENERGY BAND STRUCTURE OF CADMIUM FLUORIDE J.P. Albert, C. Jouanin and C. Gout Centre d’Etudes d’Electronique des Solides, associé au CNRS, Université des Sciences et Techniques du Languedoc, Place E. Bataillon, 34060 Montpellier, France (Received 2 November 1976 by F. Bassani) The electronic band structure of cadmium fluoride is calculated by a combined tight binding and pseudopotential method. The band structure is found to be rather similar in shape with that of CaF2. In particular it is shown that the occupied cationic d~bands do not perturb significantly the upper valence band predominantly composed of the F p-orbital. l’his explains the great similarity of the low energy part of the reflection spectra of CaF2 and CdF2. DURING the last few years, cadmium fluoride has received considerable to its unusual properties for an ionicattention high-gap owing [1—3]solid. A number of studies of its optical [3—6] and electrical [7—9] properties which speculate about its band structure in order to explain the known features of this material have been reported. We have thus performed a band structure calculation of CdF 2 with the hope of being then able to confirm or to rule out some of these speculations. CdF2 is found to crystallize in the fluorite structure with a lattice constant of 10.18 a.u. which is close to that of CaF2: 10.32 a.u. A simple electrostatic that the occu2~model ion liespredicts close below pied 4d bands p-bands of the of F the ion.Cd The comparison of the band
taken as the superposition of the individual free ion one. 3 approxiFor the calculated exchange part wemuffm-tin used the Slater p”centered at mation inside spheres each ion site with half nearest-neighbour radius. In order to calculate the matrix elements of the Hamiltonian between atomic orbitals u,~and ulL’ centered at R. and R, we made the following “spherical” approximation.
1. VALENCE BAND CALCULATION
R,) IHIu~’(r R,)) = ~1(EM+ Rs)~L~Iu~’(r R1)> —
—
—
—
orbital u1~. Contributions up to 3rd neighbour have been included in order to ensure a good convergence of the matrix elements. The results for the occupied bands of CdF2 are shown in Fig. 1 where the states have been computed at a number of points along the symmetry lines A and ~ in the first Brillouin zone. The higher valence band is chiefly composed of the bonding and antibonding linear combinations of the p F levels; its largest width is situated at I point and is given by the separation X~, X~(3.5 eV). We then find2~at about These 1.3 Ryd theare bands originating from d-levels. bands rather narrow and do the not Cd perturb significantly the upper valence bands. Indeed, the upward bending of the p-type ~ and ~ ~ bands is not due to the presence of the d~states because it still occurs when those are ignored (decoupled). In fact, these: features for the p-bands are also found is CaF 2 [13]thus which do not possess d~occupied states and seem characteristic of the fluorite structure. The greatest —
—
LCAO method is well known and weMore only give here The the essential features of our calculation. details can be found for example in reference [10]. Our basis set of atomic orbitals consisted of the 4d Cd2~and 2p, 2s F orbitals. The analytic form of these functions were those given by Richardson [11] 2~and Salez [121 for F. for Cd The coulombic part of the crystal potential was
—
+
-
structure of CaF2 with that of CdF2 can thus provide some useful informations about the influence of those d-levels on the valence band in the fluorite structure. Two methods have been used in order to calculate the band structure according to the type of levels considered. The linear combination of atomic orbitals (LCAO) method has been employed to calculate the (occupied) valence and core bands while the conduction band has been calculated by the pseudopotential (PP) method.
—
199
200
ENERGY BAND STRUCTURE OF CADMIUM FLUORIDE E (Ry) -1.0
2’ 5
.
-1.2
2 3
1 5’
1 331
25’
~
14
1
________
-2.8 2, ~ -3.0
1
1
2’ 1
<~> 3
.
L
r
A
x
Fig. 1. Valence band structure of CdF2. Some branches of the bands originating from the d~states have been drawn on the same line because of their closeness, interaction between the p- and d-states occurs between the two X1 levels which are repealed somewhat apart. We then find at 2.9 Ryd the levels originating from the F - s-states which do not interact significantly with the upper ones.
used by us in the band structures calculations of alkali fluorides [14] and magnesium fluoride [15]. These values are listed in Table 1. Table 1. Value of the Fparameters a8
ad
~
E (Fly) 1
=
Ad
-
12
~
5
~
0.5 1 2
Va +~(E—E~)Ic)
where V0 is the ionic potential and the sum extends over the occupied states Ic) with energy E~. For the negative ions the analytic form of Giulano and Ruggeri [17] is used. V8
=
~ VJF1
with V1(r)
A~
~~““~‘
7 1
V~ ps
A8
90 90Fig. 2 the 2.5 obtained 0.75conduction 0.5 bands 2 of We give in CdF2, the general shape of these bands is the same as in the alkali fluorides and CaF2 apart [13, 18] an upward repulsion of some levels when there exists an occupied state with the same symmetry. The d-type states (r,2, F~,X3) are thus repealed upwards and are decoupled from the s-states which constitute the bottom of the conduction band.
—
In order to calculate the conduction band, we choose the mixed pseudopotential method we have already used for the 2. alkali CONDUCTION fluorides [14] BAND and magnesium CALCULATION fluoride [15]. In this method, the crystal pseudopotential is taken to be equal to the superposition of the individual ionic pseudopotentials. For the positive ions the pseudopotential is constructed following the Philips and Kleinman approach [161:
Vol. 22, No. 3
=
—
e Nre
—
2 r + Ze (A,/~)
o. -
_____________ L
A
T
A
X
Fig. 2. Conduction bands of CdF2. 3. DISCUSSION AND CONCLUSION Our calculation shows then as improbable the possibilly for the conduction band to have its minimum at X point as was anticipated by Lee and Moser [2]. Indeed,
P 1 is the angular momentum projection operator, Z and N are the valence and nuclear charges respectively and the parameters a, and A, are angular momentum dependent. For those parameters we took the values already
the direct gap is found between F~and F, as in the alkali halides and CaF2. In fact, as for CaF2 [13, 18], the maximum of the valence band occurs at I point (Xi) in our calculation. However the occurence of an indirect
Vol. 22, No.3
ENERGY BAND STRUCTURE OF CADMIUM FLUORIDE
edge in CdF2 cannot be unambiguously assigned on these grounds only because of the high sensitivity of the relative positions of the key levels (F,5, X~)with respect to the matrix elements of the Hainiltonian. One must thus await more detailed experimental work to make a definite statement, Concerning the value of this gap, it is to be noted that in our model the theoretical value must be corrected by the polarization energy in order to be compared with the experimental one [19, 20]. We have not calculated thIs polarization correction but it can be noted that bringing the two gaps in agree.
201
ment implies for CdF2 a polarization energy of about 0.5 Ryd which is of the same order of magnitude as in the others fluorides [14, 15]. As it has been noted, the upp.er valence band of CdF2 is predominantly formed by the p F levels and is not much distorted by the cationlc d~states. This feature explains the great similarity in shape of the four strong low energy peaks in the reflection spectra of CaF2 and CdF2 near the band edge. A more detailed investigation of these two spectra in relation with their band structure will be presented elsewhere. -
REFERENCES 1.
KINGSLEY J.D. & PRENNER J.S.,Phys. Rev. Lett. 8,315 (1962).
2.
LEE T.H. & MOSER F., Phys. Rev. B3, 347 (1971).
3.
EISENBERGER P. & ALDERSTEIN M.G.,Phys. Rev. Bi, 1787, (1970) and references cited therein.
4. 5.
BERGER J.M., LEVEQUE G. & ROBIN J., C.R.Hebd. Seanc. Acad. Sci. Paris, 279, 512 (1974). BOURDILLON A.J. & BEAUMONT J.H.,J. Phys. C9, L473 (1976).
6.
FORMAN R.A., HOSLER W.R. & BLUNT R.F., Solid State Commun. 10, 19(1972).
7. 8.
KHOSLA R.P. & MATZ D., Solid State Commun. 6, 859 (1968). KHOSLA R.P.,Phys. Rev. 183, 695 (1969).
9.
WELLER P.F., Inorg. Chem. 5, 736 (1966).
10.
11.
CALLAWAY J. Energy Band Theory. Academic Press, New York and London (1964); BASSANI F. & PASTORI PARRAVICINI G. Electronic States and Optical Transitions in Solids. Pergamon, New York (1975). RICHARDSON J.W., BLACKMAN M.J. & RANOCHAK J.E.,J. Chem. Phys. 58, 3010 (1973).
12.
SALEZ C. (private communication).
13.
ALBERT J.P., JOUANIN C. & GOUT C., Phys. Rev. (in press).
14.
JOUANIN C., ALBERT J.P. & GOUT C., Nuovo Cim. 28B, 483 (1975).
15.
JOUANIN C., ALBERT J.P. & GOUT C.,J. Phys. 37, 595 (1976).
16. 17.
PHILIPPS J.C. & KLEIMMAN L., Phys. Rev. 116, 287 (1959). GIULANO E.S. & RUGGERI R.,Nuovo am. 6B, 53(1969).
18. 19.
ALBERT J.P., Unpublished Thesis, Montpellier (1975). FOWLERW.B.,Phys. Rev. 132, 1591 (1963).
20.
BASSANI F. & GIULANO E.S., Nuovo
am. 8B, 193 (1972).
Rdsumd Nous avons calculd la structure des bandes d’dnergie du fluorure de cadmium en combinant la mdthode des liaisons fortes et celle du pseudopotentiel. On trouve une structure de bande assez similaire
a
celle du CaF2. En particulier on montre que les bandes occupdes de type d du cation ne perturbent pas d’une manière significative la partie plus haute de la bande de valence qui est constitude principalement des orbitales p do fluor. Ceci explique la grande similitude des parties basses energies des spectres de reflection du CaF2 et du CdF2.
Pergamon Press.
Printed in Great Britain
ENERGY BAND STRUCTURE OF CADMIUM FLUORIDE J.P. Albert, C. Jouanin and C. Gout Centre d’Etudes d’Electronique des Solides, associé au CNRS, Université des Sciences et Techniques du Languedoc, Place E. Bataillon, 34060 Montpellier, France (Received 2 November 1976 by F. Bassani) The electronic band structure of cadmium fluoride is calculated by a combined tight binding and pseudopotential method. The band structure is found to be rather similar in shape with that of CaF2. In particular it is shown that the occupied cationic d~bands do not perturb significantly the upper valence band predominantly composed of the F p-orbital. l’his explains the great similarity of the low energy part of the reflection spectra of CaF2 and CdF2. DURING the last few years, cadmium fluoride has received considerable to its unusual properties for an ionicattention high-gap owing [1—3]solid. A number of studies of its optical [3—6] and electrical [7—9] properties which speculate about its band structure in order to explain the known features of this material have been reported. We have thus performed a band structure calculation of CdF 2 with the hope of being then able to confirm or to rule out some of these speculations. CdF2 is found to crystallize in the fluorite structure with a lattice constant of 10.18 a.u. which is close to that of CaF2: 10.32 a.u. A simple electrostatic that the occu2~model ion liespredicts close below pied 4d bands p-bands of the of F the ion.Cd The comparison of the band
taken as the superposition of the individual free ion one. 3 approxiFor the calculated exchange part wemuffm-tin used the Slater p”centered at mation inside spheres each ion site with half nearest-neighbour radius. In order to calculate the matrix elements of the Hamiltonian between atomic orbitals u,~and ulL’ centered at R. and R, we made the following “spherical” approximation.
1. VALENCE BAND CALCULATION
R,) IHIu~’(r R,)) = ~1(EM+ Rs)~L~Iu~’(r R1)> —
—
—
—
orbital u1~. Contributions up to 3rd neighbour have been included in order to ensure a good convergence of the matrix elements. The results for the occupied bands of CdF2 are shown in Fig. 1 where the states have been computed at a number of points along the symmetry lines A and ~ in the first Brillouin zone. The higher valence band is chiefly composed of the bonding and antibonding linear combinations of the p F levels; its largest width is situated at I point and is given by the separation X~, X~(3.5 eV). We then find2~at about These 1.3 Ryd theare bands originating from d-levels. bands rather narrow and do the not Cd perturb significantly the upper valence bands. Indeed, the upward bending of the p-type ~ and ~ ~ bands is not due to the presence of the d~states because it still occurs when those are ignored (decoupled). In fact, these: features for the p-bands are also found is CaF 2 [13]thus which do not possess d~occupied states and seem characteristic of the fluorite structure. The greatest —
—
LCAO method is well known and weMore only give here The the essential features of our calculation. details can be found for example in reference [10]. Our basis set of atomic orbitals consisted of the 4d Cd2~and 2p, 2s F orbitals. The analytic form of these functions were those given by Richardson [11] 2~and Salez [121 for F. for Cd The coulombic part of the crystal potential was
—
+
-
structure of CaF2 with that of CdF2 can thus provide some useful informations about the influence of those d-levels on the valence band in the fluorite structure. Two methods have been used in order to calculate the band structure according to the type of levels considered. The linear combination of atomic orbitals (LCAO) method has been employed to calculate the (occupied) valence and core bands while the conduction band has been calculated by the pseudopotential (PP) method.
—
199
200
ENERGY BAND STRUCTURE OF CADMIUM FLUORIDE E (Ry) -1.0
2’ 5
.
-1.2
2 3
1 5’
1 331
25’
~
14
1
________
-2.8 2, ~ -3.0
1
1
2’ 1
<~> 3
.
L
r
A
x
Fig. 1. Valence band structure of CdF2. Some branches of the bands originating from the d~states have been drawn on the same line because of their closeness, interaction between the p- and d-states occurs between the two X1 levels which are repealed somewhat apart. We then find at 2.9 Ryd the levels originating from the F - s-states which do not interact significantly with the upper ones.
used by us in the band structures calculations of alkali fluorides [14] and magnesium fluoride [15]. These values are listed in Table 1. Table 1. Value of the Fparameters a8
ad
~
E (Fly) 1
=
Ad
-
12
~
5
~
0.5 1 2
Va +~(E—E~)Ic)
where V0 is the ionic potential and the sum extends over the occupied states Ic) with energy E~. For the negative ions the analytic form of Giulano and Ruggeri [17] is used. V8
=
~ VJF1
with V1(r)
A~
~~““~‘
7 1
V~ ps
A8
90 90Fig. 2 the 2.5 obtained 0.75conduction 0.5 bands 2 of We give in CdF2, the general shape of these bands is the same as in the alkali fluorides and CaF2 apart [13, 18] an upward repulsion of some levels when there exists an occupied state with the same symmetry. The d-type states (r,2, F~,X3) are thus repealed upwards and are decoupled from the s-states which constitute the bottom of the conduction band.
—
In order to calculate the conduction band, we choose the mixed pseudopotential method we have already used for the 2. alkali CONDUCTION fluorides [14] BAND and magnesium CALCULATION fluoride [15]. In this method, the crystal pseudopotential is taken to be equal to the superposition of the individual ionic pseudopotentials. For the positive ions the pseudopotential is constructed following the Philips and Kleinman approach [161:
Vol. 22, No. 3
=
—
e Nre
—
2 r + Ze (A,/~)
o. -
_____________ L
A
T
A
X
Fig. 2. Conduction bands of CdF2. 3. DISCUSSION AND CONCLUSION Our calculation shows then as improbable the possibilly for the conduction band to have its minimum at X point as was anticipated by Lee and Moser [2]. Indeed,
P 1 is the angular momentum projection operator, Z and N are the valence and nuclear charges respectively and the parameters a, and A, are angular momentum dependent. For those parameters we took the values already
the direct gap is found between F~and F, as in the alkali halides and CaF2. In fact, as for CaF2 [13, 18], the maximum of the valence band occurs at I point (Xi) in our calculation. However the occurence of an indirect
Vol. 22, No.3
ENERGY BAND STRUCTURE OF CADMIUM FLUORIDE
edge in CdF2 cannot be unambiguously assigned on these grounds only because of the high sensitivity of the relative positions of the key levels (F,5, X~)with respect to the matrix elements of the Hainiltonian. One must thus await more detailed experimental work to make a definite statement, Concerning the value of this gap, it is to be noted that in our model the theoretical value must be corrected by the polarization energy in order to be compared with the experimental one [19, 20]. We have not calculated thIs polarization correction but it can be noted that bringing the two gaps in agree.
201
ment implies for CdF2 a polarization energy of about 0.5 Ryd which is of the same order of magnitude as in the others fluorides [14, 15]. As it has been noted, the upp.er valence band of CdF2 is predominantly formed by the p F levels and is not much distorted by the cationlc d~states. This feature explains the great similarity in shape of the four strong low energy peaks in the reflection spectra of CaF2 and CdF2 near the band edge. A more detailed investigation of these two spectra in relation with their band structure will be presented elsewhere. -
REFERENCES 1.
KINGSLEY J.D. & PRENNER J.S.,Phys. Rev. Lett. 8,315 (1962).
2.
LEE T.H. & MOSER F., Phys. Rev. B3, 347 (1971).
3.
EISENBERGER P. & ALDERSTEIN M.G.,Phys. Rev. Bi, 1787, (1970) and references cited therein.
4. 5.
BERGER J.M., LEVEQUE G. & ROBIN J., C.R.Hebd. Seanc. Acad. Sci. Paris, 279, 512 (1974). BOURDILLON A.J. & BEAUMONT J.H.,J. Phys. C9, L473 (1976).
6.
FORMAN R.A., HOSLER W.R. & BLUNT R.F., Solid State Commun. 10, 19(1972).
7. 8.
KHOSLA R.P. & MATZ D., Solid State Commun. 6, 859 (1968). KHOSLA R.P.,Phys. Rev. 183, 695 (1969).
9.
WELLER P.F., Inorg. Chem. 5, 736 (1966).
10.
11.
CALLAWAY J. Energy Band Theory. Academic Press, New York and London (1964); BASSANI F. & PASTORI PARRAVICINI G. Electronic States and Optical Transitions in Solids. Pergamon, New York (1975). RICHARDSON J.W., BLACKMAN M.J. & RANOCHAK J.E.,J. Chem. Phys. 58, 3010 (1973).
12.
SALEZ C. (private communication).
13.
ALBERT J.P., JOUANIN C. & GOUT C., Phys. Rev. (in press).
14.
JOUANIN C., ALBERT J.P. & GOUT C., Nuovo Cim. 28B, 483 (1975).
15.
JOUANIN C., ALBERT J.P. & GOUT C.,J. Phys. 37, 595 (1976).
16. 17.
PHILIPPS J.C. & KLEIMMAN L., Phys. Rev. 116, 287 (1959). GIULANO E.S. & RUGGERI R.,Nuovo am. 6B, 53(1969).
18. 19.
ALBERT J.P., Unpublished Thesis, Montpellier (1975). FOWLERW.B.,Phys. Rev. 132, 1591 (1963).
20.
BASSANI F. & GIULANO E.S., Nuovo
am. 8B, 193 (1972).
Rdsumd Nous avons calculd la structure des bandes d’dnergie du fluorure de cadmium en combinant la mdthode des liaisons fortes et celle du pseudopotentiel. On trouve une structure de bande assez similaire
a
celle du CaF2. En particulier on montre que les bandes occupdes de type d du cation ne perturbent pas d’une manière significative la partie plus haute de la bande de valence qui est constitude principalement des orbitales p do fluor. Ceci explique la grande similitude des parties basses energies des spectres de reflection du CaF2 et du CdF2.