The insulator-metal transition volumes for BaS, BaSe, and BaTe are calculated for the first time, using the self-consistent augmented spherical wave technique. The metallized transition volumes are smaller than those corresponding to the NaCl yields CsCl structural transitions, but, 10 to 15% larger than those obtained by the Herzfeld dielectric theory. The calculated equilibrium energy gaps in the NaCl structure underestimate the measured ones by 50 to 60%

Carlsson, A. E.

Wilkins, J. W.

English

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411
- r 1149,^-- 2-,157
Band-Overlap Metallization of BaS, BaSe, and BaTe
rft
A. E. Carlsson" and J. W. Wilkins
Laboratory of Atomic and Solid State Physics
and
The Materials Science Center
Cornell University, Ithaca, New York 14853
Abstract
The insulator-metal transition volumes for BaS, BaSe, and BaTe are
calculated for the first time, using the self-consistent augmented-
spherical-wave technique. The metallization transition volumes are 1)
smaller than those corresponding to the NaC AyCsC R structural transitions,
but, 2) 10-15% larger than those obtained by the Herzfeld dielectric
theory. The calculated equilibrium energy gaps in the NaC t structure
underestimate the measured ones by 50-60X.
(NASA-CR-1733 96) BARD-OVERLAP RETALLIZATION	 N84-22458
OF Bas, BaSe AND BaTe (Cornell Univ.) 19 p
SC 102/1!F A01	 CSCL 20L
unc la s
G3/76 18945
- 
_	
`sca r s ._ car. sir AW s.__j=.W ^a
-1-
I. Introduction	 -
The availability of diamond anvil pressure cells and synchrotron X-ray
radiation permit precise optical and structural studies of 'the pressure-
induced metallization of closed-shell compounds. The metallization is
usually described i within the one-electron approximation as the broadening
and subsequent overlap of a filled valence band and an empty conduction
band. The closed-shell Ba chalcogenides are particularly promising
compounds for metallization: 1) the fractional compression for metalli-
zation should be less than for isoelectronic compounds and elements, such
as CsI and Xe, which have larger band gaps, and 2) the metallization
pressure should be less than for other divalent chalcogenides (e.g., MgTe)
which have larger bulk moduli.
In this paper we describe self-consistent band structure calculations
for BaS, BaSe, and BaTe under compression. The calculations predict band
overlap transition volumes, illustrate the chemical trends in these
volumes, and will provide a test of the one-electron approximation used to
calculate the band gaps when experimental data on the metallization volumes
t
becomes available. All of the compounds are studied in both the equil-
ibrium NaCk(B1) crystal structure and the CsC t(B2) crystal structures,
	 g	 i
because they are known 2 . 3 ," to transform under pressure into the CsCt
structure. This is the first systematic study of the insulator-metal tra p -
.	 1
sitions in this series of compounds, taking into account the effects of the
t
structural transit ions. The volume-dependent band gap of BaO has pre-
viously been calculated 5 by the APW method, only in the NaC kstructure.
i
BaO is not included here because our code does not work correctly for the
highly compressed Ba atom close to the band overlap transition.
.,_
-2-
Furthermore, accurate treatment of strongly localized orbitals such as-0 2p
is difficult within density functional theory.
II. Method of Calculation
The calculations are performed using the self-consistent aug-
mented-spherical-wave method. 6 In this method the one-electron Schrodinger
equation is solved approximately using an energy-independent basis set
obtained by numerical integration of the Schrodinger equation inside atom-
centered spheres, with boundary conditions which take into account the full
symmetry of the crystal. The self-consistent one-electron potential is
generated with a local-density approximation ?
 for exchange and correlation
of the Hedin-Lundqvist form. 8 Relativistic effects are neglected. We now
comment briefly on the accuracy of the two primary approximations made in
the calculations:
A) The local-density approximation (LDA). It is well known tht
for many closed-shell systems the local-density approximation yields theo-
retical zero-pressure band gaps which are smaller than the experimental
gaps by up to a factor of two. In systems with bands that are very narrow'
compared to the band gap it is possible to remove a large part of the dis-
crepancy by the use of the self-interaction correction (SIC) scheme.9
Unfortunately it is not known how to extend the SIC scheme accurately and
unambiguously to problems in which the bands are broad compared to the band
gap, such as the metallization process of interest here. 10 On the other
hand, since the local-density approximation is expected to improve as the
system becomes more metallic or free-electron like, it will be more
-3-
accurate close to the - insulator-metal transition than at the equilibrium
volume. The discontinuity 11 0 12 between.the exact potentials for valence
and conduction band electrons should decrease with the band gap, and in
fact vanishes 12 in a simple model calculation. Furthermore, recent
exchange-only calculations 13 for atoms have shown that the differences
between eigenvalue spacings obtained using the local density potential and
the exact ground state potential are small. Thus a fairly accurate
estimate of the insulator-metal transition volume may result even if the
zero-pressure band gap is inaccurate.
B) The use of the ASW basis set. We use the following basis
orbitals: Ba 6s5p5d4f, S 3s3p3d4f, Se 4s4p4d4f, and Te 5s5p5d4f. In our
ASW calculations l" for fcc Xe, and CsI in the CsC x structure, we obtain
gaps which differ from APW results 15 , 16 by less than 0.2 eV. The ASW
method is also similar in accuracy to the linear muffin-tin orbital (LMTO)
method with the "combined correction" term. 17 The band structure of Si in
the diamond 18 and fcc 19 crystal structures has been calculated with the
LMTO method. In the diamond structure, empty spheres were included,
resulting in eightfold coordination of the spheres. Comparison of the LMTO
results with full-potential linear-augmented-plane-wave method 20 and
self-CLnsistent pseudopotential 19 results again reveals discrepancies of
less than 0.2 eV. Thus the use of the ASW method for solving the
one-electron.part of the problem does not appear to cause major errors in
closely packed structures 'such 'as CsC1. However, the NaC.E structure
results are expected to be more sensitive to the app roximations in the ASW
l
method, because of the lower . coordination number. The sensitivity tests to
be presented in Section III confirm these expectations.
-4-
1II. Results
The relevant particulars of the band structures are as follows:
In the NaC t structure the gap is always indirect, between the top of the
chalcogen valence p band at t and the bottom of the Ba 5d band at X. The
shape of this d-band is similar to that in a fcc noble or transition metal,
1
as would be expected from the fact that the Ba atoms constitute a fcc sub-
lattice. The Ba 6s level, doubly occupied in the atom, is empty in the
solid because the two 6s electrons are transferred to the chalcogen valence
p band. The results for the CsC t structure are similar, except that at
zero pressure the top of the valence band is at M and the bottom of the
conduction band at r.
Little quantitative significance can be attached to the calculated
band gaps in the NaCt structure, which are too sensitive to the fraction
fBa of the unit cell filled by the Ba sphere. The value of fBa for
each volume and crystal structure is chosen by approximately minimizing 21
the calculated total energy with respect to.fBa. This is a more appro-
priate variational principle than, for example, minimizing the band gap,
since density functional theory must obtain the correct total energy but
not necessarily the correct band gap. As seen in Table I, a change in
fBa of O.u5 in either direction from the starred value (the value closest
to minimizing the total energy), which corresponds to a change of only 6%
in the Ba sphere radius, can change the gap for BaTe in the NaC t structure
by as much. as 0.2 eV. The difference between results obtained using an spd
basis set and the spdf basis set described above, in this and subsequent
tables, illustrates the effects of the Ba 4f levels. These should be
-5-
significant because of the large atomic number of Ba. As seen in the
results, the 4f effects are small but not negligible. Unless otherwise
noted, all stated results are obtained using the spdf basis'.
In contrast to the NaC R case, the corresponding changes in the CsC A
structure gaps are practically negligible. The higher coordination number
in the CsCt structure provides better justification for sphericalizing the
Wigner-Seitz cell. We may therefore consider the results obtained for the
CsC.t structure to be a fairly accurate test of local density theory rather
than of the particular approximation used to solve the effective
one-electron Schrodinger equation. To obtain a similar test for the NaC R
structure, calculations must be performed using a larger number of ASW
spheres or more sophisticated (and computationally costly) band structure
techniques, such as the linear-augmented- plane-wave method17.
Despite the sensitivity of the NaC A structure band gaps to fBa, a
reasonable upper bound to the band gap produced by solving the one-electron
equations exactly might be obtained by maxi.mizing the gap with respect to
fBa, (although we emphasize that there is no justification for believing
that this procedure gives the correct band gap). The result i ng gaps
Eghmax are compared with the experimental values" 
Eexpt 
n Table 2. The
theoretical gaps are low by 50-60%, a.discrepancy larger than the 30-50%
error usually associated with local density band gaps. However, the exper-
imental trend of increasing band gap with decreasing atomic number is re-
produced by the calculations. Furthermore, the absolute magnitude of the
error, roughly 2 eV for all three compounds, is not particularly large when
compared to results for similar compounds. Recent band calculations 14#16
-6-
J
for CsI, a compound isoelectronic3
by 2 112 eV, with a fractional er
The last column of the table
4	 which minimize the total energy.
experimental gaps.
to BaTe, have y1Llded band gaps too small
ror of 40%.
th
shows the gaps E go*
 at the values of fBa
All are much smaller than the
We now turn to the band overlap transitions and the NaC I+CsC R
structural transitions. The volume dependence of the band gaps in the CsC x
structure is shown in Fig. 1. All of the gaps drop monotonically and
transform from direct to indirect under compression, except for BaTe which
remains indirect. The chemical trend in the gaps is unaffected by the
compression. The calculated values Eth (Vstruc) of the band gaps in the
g	 CsC t
CsC$ structure at the volumes 2'3' 4 Vstruc immediately after the structural
CsC A
transition are given in Table 3. All of the gaps arepositive, indicating
s
that the band overlap transition occurs after the structural transition.
This is consistent with recent band gap measurements `' on BaTe and BaSe and
previous visual inspections 2 of BaS in the CsC t structure. The volumes
VBOCR at which the band overlap transitions occur are given in the third
column of the table. As expected, the amount of compression required to
induce the transition increases with the . zero pressure band gap (cf. Table
2). Although-experimental measurements of VBOC a have not yet been made, it
is useful to compare our results with the volumes 
Vdiel 
obtained from a
simple dielectric theory l originally due to Herzfeld. As seen in the
Table, the band theoretic results . are consistently higher than the
-1-
ORIGMAL p.a,G'!—C 19
OF POOR 
QUAL, 4
dielectric theory estimates, by 10-15x, although both estimates show the
same chemical trend. Band calculations 14 9 is $ 16 for CsI, Xe and Ar have
predicted band overlap transition volumes higher than the dielectric theory
estimate by roughly the same amount.
At this stage our starting belief that the metallization volume (and
hence pressure) would be well predicted is somewhat battered. As shown in
Table 4, the calculated equilibrium volumes Vth 
t 
in the NaCt structure are
16-11% lower than the experimental values23 Vexpt. The results for the
NaC A
CsCL structure are expected to be more accurate than the NaCl results,
because as mentioned earlier the larger coordination number provides more
justification for sphericalizing the Wigner-Seitz cell. However, as seen
in Table 4, the calculated equilibrium volume 
V
CsC I in the CSC I structure
is still roughly 10% lower than an experimental value obtained `' by
backwards extrapolation of the equation of state measured beyond the
structural transition. Since the difference between VBO 
R and Vth s 
in
Bare is less than the error in VNaC R' the calculated pressure must be off
by more than a factor of two in the harmonic approximation, and proba3ly
much more when anharmonic effects are included. Nevertheless, we hive
displayed the calculated pressures in the caption to Fig. 1. To obtain
accurate estimates of the transition pressures in the Ba chalcogenides, it
will be necessary to devise one-electron potential which give more accurate
equilibrium volumes than the local density approximation.
-v-
Acknowledgements
We are greatly indebted to A. R. Williams and V. L. Moruzzi for
allowing us to use their band structure code and for helpin g
 us to
implement it.
	
We thank A. L. Ruoff, and T. A. Grzybowski, and
A. K. McMahan for several helpful conversations. This work was partly
supported by the Materials Science Center under a grant from the National
Science Foundation, and by the National Aeronautics and Space
Administration under contract NAG-2-159.
3
z- .3
-9-
Footnotes and References
*Present address: Department of Physics, Washington University. St. Louis,
Mo. 63130.
1. For a general discussion, see M. Ross and A. K. McMahan, in Physics of
Solids Under High Pressure, edited by J. S. Schilling and
R. N. Shelton (North-Holland, New York, 1981), p. 161.
2. S. Yamaoka, 0. Shimomura, H. Nakazawa, and 0. Fukunaga, Solid State
Comm. 33, 87 (1980) - BaS.
3. T. A. Grzybowski and A. L. Ruoff, Phys. Rev. 627, 6502 (1983) - BaSe.
4. T. A. Grzybowski and A. L. Ruoff, in Proceedings of the IX AIRAPT
International High Pressure Conference, to be published - BaSe and
BaTe.
5. M. S. Bukowinski and J. Hauser, Geophys. Res, Lett, 7, 689 (1980)6
6. A. R. Williams, J. Kubler, and C. D. Gelatt, Phys. Rev. B 19, 6094
(1979).
7. W. Kohn and L. J. Sham, Phys. Rev. 140, A 1133 (1965).
8. L. Hedin and B. I. Lundqvist, J. Phys. C 4, 2064 (1971).
9. J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5648 (1981).
1U. See, however, M. R. Norman and J. P. Perdew, Phys. Rev. B28, 2135
(1983).
11. J. P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 (1983).
12. L. J. Sham and M. Schluter, Phys. Rev. Lett 51, 1888 (1983).
13. D. C. Langreth and M. J. Mehe, Phys. Rev. 828, 1809 (1983).
. 
__'ft.ix " •
-10-
14. For Csl at a lattice constant of 8.6299 bohrs, the ASW gap is 3.78
eV. The band overlap transition volume is 45% of the equilibrium
volume and the transition pressure 600-700 Kbar. For Xe at a lattice
constant of 10.24 bohrs the ASW gap is 5.67 eV.
15. M. !toss and A. K. McMahan, Phys. Rev. 621, 1658 (1980).
16. J. Aidun and M. S. T. Bukowinski, Solid State Comm. 47, 855 (1983); to
be published.
17. 0. K. Andersen, Phys. Rev. B12, 3060 (1975).
18. D. Glbtzel, B. Segall, and 0. K. Andersen, Solid State Comm. 36,403
(1980).
19. A. K. McMahan, private communication. These calculations are
described in A. K. McMahan, M. T. Yin, and M. L. Cohen, Phys. Rev.
824, 7210 (1981).
20. 0. R. Hamann, Phys. Rev. Lett 42, 662 (1980).
21. For each compound, in the CsC t structure, the total energy is
minimized at the observed equilibrium volume VNap R in the NaCt
structure, and at a volume close to the band overlap transition. For
other volumes, f Ba is obtained by linear extrapolation or
interpolation. Because of the large number of calculations required
in this procedure, we employed the spd rather than the spdf basis to
obtain the optimal sphere radii. The resulting expressions for 
f 6
are: .BaSp fBa	 0.48 =.0.23 (,_V/Vet);t); BaSe, fBa = 0.46 - 0.16
(1-V/Veap^); BaTe, f Ba 0.43 -. 0.04 ( 1 - V/Ve,pt). In the NaCt	 (
1
-11-
structure, fBa 
is obtained by interpolation or extrapolation from
values at 
VNaCt 
and 0.95 V xPt The expressions for fBa are:
Bas, fBa = 0.32 - 0.13 (1-V/VNap t); 	 BaSe, fBa - 0.31 - 0.12
(1-V/V xpt); BaTe, fBa	 0.30
charge in the Ba sphere varies
structure and -1.7 to -2.2 lei
22. W. E. Strehlow and E. L. Cook,
23. J. H. Donnay, G. Donnay, E. G.
(Washington, Amer. Cryst. Soc.
0.10 (1-V /VNap t) . The resulting total
from -1.3 to -1.9 lei in the CSC A
in the NaC A structure.
J. Phys. Chem. Ref. Data 2, 163 (1973).
Cox, and M. V. King, Crystal Data
1963).
-12-
Table Captions
Table 1. Sensitivity of calculated band gap in BaTe to the fraction fBa
of unit cell volume occupied by Ba sphere, for fixed volume
(518 a 30 /atom pair) in both MC A and CsC t crystal structures.
Starred-vol,imes minimize the total energy.
Table 2. Compar;son of calculated and experimental band gaps in NaC t
structure. Eth obtained using the sphere radii which minimize
g o*
the total energy.
Table 3. Quantities relevant to structural and band overlap transitions.
Vexp 
t is the measured equilibrium volume in the NaC t structure..NaC
The Herzfeld criterion ) predicts the metall,zation volume Vdiel
at which the dielectric constant is infinite in terms of the
expt
zero-pressure dielectric constant c and experimental volume VNaCt
by (c-1)/(e+2) : Vdiel/Vexpt. Other quantities are defined in
NaC t
the-text.
Table 4. Calculated and measured equilibrium volumes in the NaC t and CSCt
structures. V xpt given in a 3/.atom pair.NaCL
...	 -^
--	 -13-
Table 1
fBa
NaC I
E9	 (0) CsC AE9	 (eV)
spdf spd spdf spd
0.25 0.75 0.64
0.30 0.92 0.82*
0.35 1.12 1.01 0.73 0.64
0.41 1.30 1.18 0.72 0.64*
0.48 1.40 1.28 0.11 0.65
y^
-14-
ORKANAL PAGE M
OF POOR QUALITY
Table 2
E9h ,max (eV)	 E9xpt(eV)
	
E9h*(eV)
o
BaS	 2.1	 3.9	 1.6
BaSe	 :.. 8 	 3.6	 1.3
BaTe	 1.4	 3.4	 0.9
-15-
ORIGINAL PACE 19
OF POOR QUALfiY
Table 3
Vstruc BO	 Vdiel
th st ruc CsC t_ VCSC t
Eg ( VCsC t Vexpt Vexpt
	 V	 PP	 Vex ptNaC t Na.0 t	 NaC t
spdf spd spdf	 spd
BaS	 1.0 0.8 0.77a 0.60
	 0.62
	 0.55
BaSe	 U.8 0.6 0.78b 0.63	 0.66
	 0.58
BaTe
	 0.2 O.0 0.78b 0.74	 0.77
	 0.63
a Ref. 2	 b Ref. 4
Table 4
Vth Vexpt Vth
NaC t CSC t CSC R
Vexpt Vexpt Vexpt
NaC t NaC t NaC t
spdf	 spd spdf spd
0.83	 '	 0.84 -- 0.76 0.78
0.84	 0.85 -- 0.79 0.80
0.84	 0.84 0.91 0.80 0.81
Vexpt
NaC X
BaS
	
440
Ba Se
	
483
BaTe
	
578
I	 rT
-16-
	 L,
ORIGINAL PAGE IJ
OF POOR QUALITY
I
T"
-17-
Figure Caption
Fig. 1. Volume dependence of band gaps of Ba chalcocenides in the
structure. VexptNaCR is the measured equilibrium volume in t
structure (Ref. 23). Filled circles denote calculated po
Pressures at band-overlap transitions are 320 Kbar for Ba
Kbar for BaSe, and 40 Kbar for BaTe.
ORIGINAL E `,'^;^ r-#
OF POOR QUAL1 Y
1.0
E9(eV)
0
BoX
	 •^I
X = S/5e/
	
Tee!
•
•
0.5
	 expt	 1.0
v^ UNaC I
Fig. 1


Band-overlap metallization of BaS, BaSe and BaTe

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