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Energy bands in neon and argon
1
Si! Si! Si!
Si! Si! Si!
Volume 76A, number 5,6
PHYSICS LETTERS
14 April 1980
ENERGY BANDS IN NEON AND ARGON * M.A. KHAN1 and J. CALLAWAY Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA
Si
Received 24 October 1979 a 5!
The dependence of the energy bands in neon and argon on the choice of an exchange—correlation potential has been studied in a calculation using the LCGO method.
1. Introduction. There have been a number of bandstructure calculations for rare gas solids (RGS) during the last two decades. Since the band gap (Eg) of RGS can be obtained from the experimental data, it is the most interesting quantity to be studied from first principle calculations. These calculations have been carried out for Ne [1—4]and Ar [1—3,5—9]with different methods and different exchange potential approximations. Comparison of the widths of the energy gaps in the Ne and Ar shows that no matter which computational method is applied, the Hartree—Fock approximation invariably gives larger Eg values (as much as 4 eV in the case of Ne) than those observed experimentally. An electron—electron correlation correction is applied [3—7] to bring the bands into better agreement with experiment. In the following we give a short description of our computational procedure and some results for the energy bands of Ne and Ar. We also discuss the influence of the different types of exchange and exchange— correlation potentials on the forbiddengap, in particular, and on the energy bands in general. 2. Energy bands. In rare-gas solids, the highest occupied levels are completely filled and tightly bound. A tight-binding approach should produce all the phys°
Supported in part by the Division of Materials Research of the National Science Foundation. Permanent address: Laboratoire de Structure Eiectronique des Solides, 67000 Strasbourg, France.
ical characters of the valence states. The linear conibination of gaussian orbitals (LCGO) method [10] has been successfully applied to a series of transition metals [11] from vanadium to nickel and very good results have been obtained for the states in the 3d bands. In view of the fact that the LCGO method has given a very precise picture of the band structure for these elements, we decided to apply it to rare gas solids. The computational procedure has been described by Wang and Callaway [10]. The single particle wave equation [—V ÷U(r)] i~ (k, r) = E (k) i~i (k, r), n
n
(1)
is solved self-consistently. The ~~l1n(k,r) are the Bloch functions which are cxpressed as a linear combination of independent gaussian orbitals. The potential U(r~contains the nuclear and electronic Coulomb potential and also the exchange potential Vex(r) which in the Xa approximation is written as: V ‘r~= —6a Ui’3 14ir~ 1r~11/3 2 ex’~I [‘. I JP~~ where p(r) is the electron density. In the treatment of Kohn and Sham [12] and Gaspar [13] of the exchange potential a = 2/3 and according to Slater’s [141statistical average a = I. The Xa approximation treats a as an adjustable parameter [15,16]. The Coulomb interaction is also responsible for the polarization of the electrons, i.e. correlation between 441
Volume 76A, number 5,6
PHYSICS LETTERS
electrons by means of the Coulomb interaction is more general than considered in the Hartree—Fock scheme. There are some elegant ways to include the correlation effects in semi-conductors by polaron models [17,18]. These models determine the change in the energy levels due to creation of a hole in the valence band and an electron in the conduction band. This effect is treated as a perturbation upon the perfect band structure. The simplest way to include the correlation in band calculations is given by Hedin and Lundqvist [19]. The exchange—correlation potential U~can be expressed in terms of the exchange potential U according to the relation’ X
14 April 1980
Table I Energies of the seven lowest bands (in Ry) at F of Ne and Ar with the initial crystal potential. The energy gaps are given in eV also. The symmetries are indicated in the brackets. HL: Hedin—Lundqvist, KSG: Kohn—Sham--Gaspar, S: Siater. ~
______
Potential
Bands
Ne
HL
12 3 4
—60.2516 —2.5940 —0.9382 —0.1318
5 6 7 =
U (p)=~l(p)U(p) XC
—
Ar (1) (1) (15) (1)
1.1858 (25’) 1.5332 (15) 1.6590 (12) 0.8064 Ry 10.97 eV
—227.7454 —21.6045 —16.8676 —1.8176
(1) (1) (15) (1)
—0.7548 (15) --0.1231 (1) 0.4961 (25’) 0.63 17 Ry =8.691 eV
(3) KSG
X
U~(p)is the Kohn—Sharn exchange potential and 13(p) is written in a parametrized form as:
where
1 2 3 4
—60.0173 —2.4064 —0.7505 —0.0252
(1) (1) (15) (1)
5 76
1.2959 1.6418 1.7756
(25’) (15) (12)
—227.4700 —21.3747 —16.6322 —1.6472
(1) (1) (15) (1)
/3(r 5)
=
I
+ Bx ln(1 + 1/x) 13
where x = In the two
,A
r5/A, r5 =limitswhen (~,irp)1 r extreme
21 and B
=
0.7734.
5 —1.0 and r5 °°(i.e. when the electron density is very high and very low), we obtain the KSG and Slater approximations. The results obtained using the HL potential should lie somewhere between these two extreme cases. The starting charge density and the Fourier transforms of the potential were calculated with the GTO’s for free atoms as given by Huzinaga [201 and Huzinaga and Sukai [211 for Ne and Ar, respectively. For the final calculation of the bands we have added 3 “d” type orbitals in both cases. Thus for Ne we have 12 “s” type, 8 “p” type and 3 “d” type and for Ar 14 “s” type, 12 “p” type and 3 “d” type orbitals. The hamiltonian and overlap matrices have the dimensions of SiX 51 and 65 X 65 for Ne and Ar. First we studied the effect of different potential approximations on the energy levels at the center of the Brillouin Zone (BZ). In table I we give the values of the seven lowest energy levels of Ne and Ar at the center of the EZ (i.e. F). These energy levels have been calculated with three types of potentials (i.e. S, KSG and HL). These values are with the initial crystal potential (i.e. no self-consistency). We notice that the energy levels and the forbidden gaps are almost the same when the exchange potentials are those of KSG and HL types. In the former, no correlation is included while in the latter, the electron correlation is implicit—~
442
L’g
0.725 3 RY 9.864 eV —63.4619 —3.4059 —1.7302 0.2295
=
2 3 4 7 Eg
=
(1) (1) (15) (1) ~ 2 1.5630 (12) 1.5007 Ry 20.409 eV
—0.5874 (15) —0.0054 0.6222 (25’) (1) 0.5820 Ry =7.195 eV —231.7620 —23.5014 —19.0330 —2.3428
(1) (1) (15) (1) I1~ (1 0.3607 (25’) 1.0043 Ry =13.658 eV
ly contained. The calculated energy gaps 9.86 eV (10.97 eV) and 7.92 eV (8.69 eV) for Ne and Ar with KSG (HL) potential are roughly half as large as the experimentally obtained values of 21.14 eV and 14.16 eV [22]. We do obtain an acceptable value of the energy gap of 20.14 eV and 13.66 eV for Ne and Ar, respectively, when the Slater exchange potential is employed. But when the calculations are made self-consistent, these values become smaller. So we had to take a> 1 to obtain a correct self-consistent energy gap. We obtamed the energy gaps of 21 .06 eV and 13.59 eV when a 1 .25 for neon and n = 1.2 for argon. We show in the figures the self-consistent band schemes for Ne (fig. 1) and Ar (fig. 2) with a = 1. There are some relative displacements of the energy levels as one goes from a = Ito> 1; the highest valence a a a a a a a
Volume 76A, number 5,6
PHYSICS LETTERS
Ryd 5
2
1
2
I
5
I
2 2 15 1•62
~ 2’ 1
1
2
is 1 3’
Table 2 Self-consistent (Eg) energy gap (eV); me/rn for a conduction electron; and AF~ 5,the width of the highest valence band (eV), ofNeandArfor~=Iandrr>1. _____
2 2 1
3
I
3
~
1
4 12
12 25’
14 April 1980
____ —_________
~
________________ —~ -
Ne
Ar
—
—~-——-—
Si! Si! Si! Si! Si! Si! Si! 554 Si! Si! Si Si Si
as Si
SI
aa
Si
i!
a1
a1.25
a1
a1.2
0.97 16.23 0.41
1.01 21.06 0.20
1.06 12.12 1.02
1.10 13.59 0.68
Si SI 5! 5!
1.2 ‘S
m~/m Eg (eV) z~F’~5 (eV)
1 5’ 08
I
2
1 41
0
I
i
~‘
~
2 3 2’
I 1
In table 2 we give some characteristic features, including the width of the highest valence band, the effective mass of the conduction electron and the energy gap for Ne and Ar for two values of a. This table shows that the effective mass of the conduction electron at F1 is that of a free electron. Experimentally, the reduced masses of the excitons in RGS are found
2 2
1
1
-0.2
144___~.3
21151
x
a
V
W
Z
L
0
A
V
Fig. 1. Energy bands in Ne with ~
t
1.
band becomes narrower, whereas the electron effective mass in the conduction band at F remains almost unchanged. R5d__________________________________________
:
514~~l2
42
~
to be Ne lesstothan alsoobservation they decrease as we go from Xe one [22].and This is consistent with our result that the highest valence band in Ar is much wider (about three times) than that in Ne. For any detailed comparison of course it is necessary to include the effects of spin—orbit coupling and of polarization on the bands. 3. Conclusion. We have applied the LCGO method to the large gap semiconductors Ne and Ar. We have studied the effects of different local density exchange approximations on the band structure. The only way to obtain a satisfactory energy gap in Ne and Ar was to increase the parameter a of the Xci method to a value greater than one. The choices a = 1 .2 and 1.25 for Ar and Ne gave self-consistent band gaps which are in reasonable agreement with the experimental values. References [1] U. Rossler, Phys. Stat. Sol. 42(1970) 345. [21 L. Dagens and F. Perrot, Phys. Rev. B5 (1972) 641. [31 A.B. Kunz and D.J. Mickish, Phys. Rev. B8 (1973) 779 [4] R.N. Euwema, G.G. Wepfer, G.T. Serratt and D.L. Wilhite, Phys. Rev. B9 (1974) 5249.
~Q5
~
r
151
a
X
Z
W
Q
L
r
A
Fig. 2. Energy bands in Ar with a
=
1.
Z
1K
R.S. Knox and F. Bassani, Phys. Rev. 124 (1961) 652. [6] L.F. Matheiss, Phys. Rev. 133A (1964) 1399. [7] N.O. Lipari and W.B. Fowler, Phys. Rev. B2 (1970) 3354. [8] NO. Lipari, Phys. Rev. B6 (1972) 4071.
443
S
Volume 76A, number 5,6
PHYSICS LETTERS
191 Di. Mickish and A.B. Kunz, 1. Phys. C6 (1973) 1723. [101 C.S. Wang and J. Caliaway, Comput. Phys. Comrnun. 14 (1978) 327. 1111 D.G. Laurent, C.S. Wang and 1. Callaway, Phys. Rev. B17 (1978) 455; J. Callaway and CS. Wang, Phys. Rev. B16 (1977) 2095; CS. Wang and 1. Callaway, Phys. Rev. B15 (1977) 298. 1121 L.J. Sham and W. Kohn, Phys. Rev. 145 (1966) 561. [13] R. Gaspar, Acta Phys. Hung. 3 (1954) 263. [14] J.C. Slater, Phys. Rev. 81(1951) 385. [15] J.C. Slater, lB. Mann, F.M. Wilson and J.H. Wood, Phys. Rev. 184 (1969) 672.
14 April 1980
[16] iC. Slater, J.M. Wilson and ill. Wood, Phys. Rev. 179 (1969) 28. [17] Y. Toyazawa, Prog. Theor. Phys. 12 (1954) 421. [181 M. Inoue, C.K. Mahutte and S. Wang, Phys. Rev. B2 (1970) 539. [19] L. Hedin and B.!. Lundqvist, J. Phys. C4 (1971) 2064. [20] S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [211 S. Huzinaga and Y. Sakai, 1. Chem. Phys. 50 (1969) 1371. [22] U. Rossler, Rare gas solids, eds. M.L. Klein and J.A. Venable (Academic Press, New York, 1975) p. 505.
5 S S S S Si 51 SI
ss
SI SI
a a a
444
Si Si Si 55! 554 55! 55! Si!
at at as
at as
Si! Si! Si!
Si! Si! Si!
Volume 76A, number 5,6
PHYSICS LETTERS
14 April 1980
ENERGY BANDS IN NEON AND ARGON * M.A. KHAN1 and J. CALLAWAY Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA
Si
Received 24 October 1979 a 5!
The dependence of the energy bands in neon and argon on the choice of an exchange—correlation potential has been studied in a calculation using the LCGO method.
1. Introduction. There have been a number of bandstructure calculations for rare gas solids (RGS) during the last two decades. Since the band gap (Eg) of RGS can be obtained from the experimental data, it is the most interesting quantity to be studied from first principle calculations. These calculations have been carried out for Ne [1—4]and Ar [1—3,5—9]with different methods and different exchange potential approximations. Comparison of the widths of the energy gaps in the Ne and Ar shows that no matter which computational method is applied, the Hartree—Fock approximation invariably gives larger Eg values (as much as 4 eV in the case of Ne) than those observed experimentally. An electron—electron correlation correction is applied [3—7] to bring the bands into better agreement with experiment. In the following we give a short description of our computational procedure and some results for the energy bands of Ne and Ar. We also discuss the influence of the different types of exchange and exchange— correlation potentials on the forbiddengap, in particular, and on the energy bands in general. 2. Energy bands. In rare-gas solids, the highest occupied levels are completely filled and tightly bound. A tight-binding approach should produce all the phys°
Supported in part by the Division of Materials Research of the National Science Foundation. Permanent address: Laboratoire de Structure Eiectronique des Solides, 67000 Strasbourg, France.
ical characters of the valence states. The linear conibination of gaussian orbitals (LCGO) method [10] has been successfully applied to a series of transition metals [11] from vanadium to nickel and very good results have been obtained for the states in the 3d bands. In view of the fact that the LCGO method has given a very precise picture of the band structure for these elements, we decided to apply it to rare gas solids. The computational procedure has been described by Wang and Callaway [10]. The single particle wave equation [—V ÷U(r)] i~ (k, r) = E (k) i~i (k, r), n
n
(1)
is solved self-consistently. The ~~l1n(k,r) are the Bloch functions which are cxpressed as a linear combination of independent gaussian orbitals. The potential U(r~contains the nuclear and electronic Coulomb potential and also the exchange potential Vex(r) which in the Xa approximation is written as: V ‘r~= —6a Ui’3 14ir~ 1r~11/3 2 ex’~I [‘. I JP~~ where p(r) is the electron density. In the treatment of Kohn and Sham [12] and Gaspar [13] of the exchange potential a = 2/3 and according to Slater’s [141statistical average a = I. The Xa approximation treats a as an adjustable parameter [15,16]. The Coulomb interaction is also responsible for the polarization of the electrons, i.e. correlation between 441
Volume 76A, number 5,6
PHYSICS LETTERS
electrons by means of the Coulomb interaction is more general than considered in the Hartree—Fock scheme. There are some elegant ways to include the correlation effects in semi-conductors by polaron models [17,18]. These models determine the change in the energy levels due to creation of a hole in the valence band and an electron in the conduction band. This effect is treated as a perturbation upon the perfect band structure. The simplest way to include the correlation in band calculations is given by Hedin and Lundqvist [19]. The exchange—correlation potential U~can be expressed in terms of the exchange potential U according to the relation’ X
14 April 1980
Table I Energies of the seven lowest bands (in Ry) at F of Ne and Ar with the initial crystal potential. The energy gaps are given in eV also. The symmetries are indicated in the brackets. HL: Hedin—Lundqvist, KSG: Kohn—Sham--Gaspar, S: Siater. ~
______
Potential
Bands
Ne
HL
12 3 4
—60.2516 —2.5940 —0.9382 —0.1318
5 6 7 =
U (p)=~l(p)U(p) XC
—
Ar (1) (1) (15) (1)
1.1858 (25’) 1.5332 (15) 1.6590 (12) 0.8064 Ry 10.97 eV
—227.7454 —21.6045 —16.8676 —1.8176
(1) (1) (15) (1)
—0.7548 (15) --0.1231 (1) 0.4961 (25’) 0.63 17 Ry =8.691 eV
(3) KSG
X
U~(p)is the Kohn—Sharn exchange potential and 13(p) is written in a parametrized form as:
where
1 2 3 4
—60.0173 —2.4064 —0.7505 —0.0252
(1) (1) (15) (1)
5 76
1.2959 1.6418 1.7756
(25’) (15) (12)
—227.4700 —21.3747 —16.6322 —1.6472
(1) (1) (15) (1)
/3(r 5)
=
I
+ Bx ln(1 + 1/x) 13
where x = In the two
,A
r5/A, r5 =limitswhen (~,irp)1 r extreme
21 and B
=
0.7734.
5 —1.0 and r5 °°(i.e. when the electron density is very high and very low), we obtain the KSG and Slater approximations. The results obtained using the HL potential should lie somewhere between these two extreme cases. The starting charge density and the Fourier transforms of the potential were calculated with the GTO’s for free atoms as given by Huzinaga [201 and Huzinaga and Sukai [211 for Ne and Ar, respectively. For the final calculation of the bands we have added 3 “d” type orbitals in both cases. Thus for Ne we have 12 “s” type, 8 “p” type and 3 “d” type and for Ar 14 “s” type, 12 “p” type and 3 “d” type orbitals. The hamiltonian and overlap matrices have the dimensions of SiX 51 and 65 X 65 for Ne and Ar. First we studied the effect of different potential approximations on the energy levels at the center of the Brillouin Zone (BZ). In table I we give the values of the seven lowest energy levels of Ne and Ar at the center of the EZ (i.e. F). These energy levels have been calculated with three types of potentials (i.e. S, KSG and HL). These values are with the initial crystal potential (i.e. no self-consistency). We notice that the energy levels and the forbidden gaps are almost the same when the exchange potentials are those of KSG and HL types. In the former, no correlation is included while in the latter, the electron correlation is implicit—~
442
L’g
0.725 3 RY 9.864 eV —63.4619 —3.4059 —1.7302 0.2295
=
2 3 4 7 Eg
=
(1) (1) (15) (1) ~ 2 1.5630 (12) 1.5007 Ry 20.409 eV
—0.5874 (15) —0.0054 0.6222 (25’) (1) 0.5820 Ry =7.195 eV —231.7620 —23.5014 —19.0330 —2.3428
(1) (1) (15) (1) I1~ (1 0.3607 (25’) 1.0043 Ry =13.658 eV
ly contained. The calculated energy gaps 9.86 eV (10.97 eV) and 7.92 eV (8.69 eV) for Ne and Ar with KSG (HL) potential are roughly half as large as the experimentally obtained values of 21.14 eV and 14.16 eV [22]. We do obtain an acceptable value of the energy gap of 20.14 eV and 13.66 eV for Ne and Ar, respectively, when the Slater exchange potential is employed. But when the calculations are made self-consistent, these values become smaller. So we had to take a> 1 to obtain a correct self-consistent energy gap. We obtamed the energy gaps of 21 .06 eV and 13.59 eV when a 1 .25 for neon and n = 1.2 for argon. We show in the figures the self-consistent band schemes for Ne (fig. 1) and Ar (fig. 2) with a = 1. There are some relative displacements of the energy levels as one goes from a = Ito> 1; the highest valence a a a a a a a
Volume 76A, number 5,6
PHYSICS LETTERS
Ryd 5
2
1
2
I
5
I
2 2 15 1•62
~ 2’ 1
1
2
is 1 3’
Table 2 Self-consistent (Eg) energy gap (eV); me/rn for a conduction electron; and AF~ 5,the width of the highest valence band (eV), ofNeandArfor~=Iandrr>1. _____
2 2 1
3
I
3
~
1
4 12
12 25’
14 April 1980
____ —_________
~
________________ —~ -
Ne
Ar
—
—~-——-—
Si! Si! Si! Si! Si! Si! Si! 554 Si! Si! Si Si Si
as Si
SI
aa
Si
i!
a1
a1.25
a1
a1.2
0.97 16.23 0.41
1.01 21.06 0.20
1.06 12.12 1.02
1.10 13.59 0.68
Si SI 5! 5!
1.2 ‘S
m~/m Eg (eV) z~F’~5 (eV)
1 5’ 08
I
2
1 41
0
I
i
~‘
~
2 3 2’
I 1
In table 2 we give some characteristic features, including the width of the highest valence band, the effective mass of the conduction electron and the energy gap for Ne and Ar for two values of a. This table shows that the effective mass of the conduction electron at F1 is that of a free electron. Experimentally, the reduced masses of the excitons in RGS are found
2 2
1
1
-0.2
144___~.3
21151
x
a
V
W
Z
L
0
A
V
Fig. 1. Energy bands in Ne with ~
t
1.
band becomes narrower, whereas the electron effective mass in the conduction band at F remains almost unchanged. R5d__________________________________________
:
514~~l2
42
~
to be Ne lesstothan alsoobservation they decrease as we go from Xe one [22].and This is consistent with our result that the highest valence band in Ar is much wider (about three times) than that in Ne. For any detailed comparison of course it is necessary to include the effects of spin—orbit coupling and of polarization on the bands. 3. Conclusion. We have applied the LCGO method to the large gap semiconductors Ne and Ar. We have studied the effects of different local density exchange approximations on the band structure. The only way to obtain a satisfactory energy gap in Ne and Ar was to increase the parameter a of the Xci method to a value greater than one. The choices a = 1 .2 and 1.25 for Ar and Ne gave self-consistent band gaps which are in reasonable agreement with the experimental values. References [1] U. Rossler, Phys. Stat. Sol. 42(1970) 345. [21 L. Dagens and F. Perrot, Phys. Rev. B5 (1972) 641. [31 A.B. Kunz and D.J. Mickish, Phys. Rev. B8 (1973) 779 [4] R.N. Euwema, G.G. Wepfer, G.T. Serratt and D.L. Wilhite, Phys. Rev. B9 (1974) 5249.
~Q5
~
r
151
a
X
Z
W
Q
L
r
A
Fig. 2. Energy bands in Ar with a
=
1.
Z
1K
R.S. Knox and F. Bassani, Phys. Rev. 124 (1961) 652. [6] L.F. Matheiss, Phys. Rev. 133A (1964) 1399. [7] N.O. Lipari and W.B. Fowler, Phys. Rev. B2 (1970) 3354. [8] NO. Lipari, Phys. Rev. B6 (1972) 4071.
443
S
Volume 76A, number 5,6
PHYSICS LETTERS
191 Di. Mickish and A.B. Kunz, 1. Phys. C6 (1973) 1723. [101 C.S. Wang and J. Caliaway, Comput. Phys. Comrnun. 14 (1978) 327. 1111 D.G. Laurent, C.S. Wang and 1. Callaway, Phys. Rev. B17 (1978) 455; J. Callaway and CS. Wang, Phys. Rev. B16 (1977) 2095; CS. Wang and 1. Callaway, Phys. Rev. B15 (1977) 298. 1121 L.J. Sham and W. Kohn, Phys. Rev. 145 (1966) 561. [13] R. Gaspar, Acta Phys. Hung. 3 (1954) 263. [14] J.C. Slater, Phys. Rev. 81(1951) 385. [15] J.C. Slater, lB. Mann, F.M. Wilson and J.H. Wood, Phys. Rev. 184 (1969) 672.
14 April 1980
[16] iC. Slater, J.M. Wilson and ill. Wood, Phys. Rev. 179 (1969) 28. [17] Y. Toyazawa, Prog. Theor. Phys. 12 (1954) 421. [181 M. Inoue, C.K. Mahutte and S. Wang, Phys. Rev. B2 (1970) 539. [19] L. Hedin and B.!. Lundqvist, J. Phys. C4 (1971) 2064. [20] S. Huzinaga, J. Chem. Phys. 42 (1965) 1293. [211 S. Huzinaga and Y. Sakai, 1. Chem. Phys. 50 (1969) 1371. [22] U. Rossler, Rare gas solids, eds. M.L. Klein and J.A. Venable (Academic Press, New York, 1975) p. 505.
5 S S S S Si 51 SI
ss
SI SI
a a a
444
Si Si Si 55! 554 55! 55! Si!
at at as
at as