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Band Structure and Optical Properties of Diamond
Article  in  Physical Review Letters · September 1968
DOI: 10.1103/PhysRevLett.21.715
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Thomas Bergstresser
University of California, Berkeley
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VOI.UME 16, NUMBER 9 PHYSICAL REVIEW LETTERS 28 I'"EBRUARY 1966
=d. In this respect, it might be interesting
to see if Al-A10+-Ag/In/Ag diodes (with relatively
thin silver films) produce significantly
sharper structure in analogy to multiple-beam
interferometers, i.e., two silver mirrors rather
than one. The present model proposes e-e
(or h-h) composite states into which electrons
(or holes) tunnel preferentially because such
states satisfy a macroscopic quantum condition
of the "beat" or envelope" momentum &P = (2w/
d)qh. This point of view seems to differ significantly
from the physical picture underlying
the McMillan-Anderson calculation, although
both yield the same energy spectrum. There
appears to be some reason to believe that simultaneous
solutions of the three-dimensional Gor'kov
equations (lamina of thickness d) may lead to
a regime qualitatively similar to the one dis-
cussed.
The authors are indebted to G. W. Lehman
for valuable discussions, and to R. R. Hargrove
for preparing the diodes studied.
*Work supported by the Division of Research, Metallurgy
and Materials Programs, U. S. Atomic Energy
Commission, under Contract No. AT-(11-1)-GEN-8.
~W. J. Tomasch, Phys. Rev. Letters 16, 16 (1966).
2W. J. Tomasch, Phys. Rev. Letters 15, 672 (1965).
W. L. McMillan and P. W. Anderson, Phys. Rev.
Letters 16, 85 (1966).
4The fact that the single atypical structural feature
exhibited by the thinner In films of Ref. 1 could be incorporated
into the main series by plotting e&(q) was
first pointed out in Ref. 3.
5J. R. Schrieffer, Rev. Mod. Phys. 36, 200 (1964).
J. Bardeen, L. N. Cooper, and J. R. Schrieffer,
Phys. Rev. 108, 1175 (1957).
7P. N. Dheer, Proc. Roy. Soc. (London) A260, 333 (1961).
Although the choice of nodes at the surface has a
certain appeal, this condition is not mandatory. Alternative
boundary conditions can produce the same spec-
trum.
~T. Wolfram and G. W. Lehman, to be published.
BAND STRUCTURE AND OPTICAL PROPERTIES OF DIAMOND*
W. Saslow, t T. K. Bergstresser, and Marvin I . Cohen)
Department of Physics, University of California, Berkeley, California
(Received 27 January 1966)
Recently several authors' ' have discussed
new measurements of the optical properties
of diamond, and they have raised many questions
about the measurements and about the
theoretical interpretation of the data. To answer
a few of these questions, we present in
this Letter a calculation of the electronic band
structure of diamond. The band structure was
calculated by means of the empirical pseudopotential
method (EPM)'~' which has been used' '
successfully to interpret the optical properties
of a large number of semiconductors and insulators.
The analysis of the resulting diamond
band structure yields some new interpretation
of the structure in the optical reflectivity.
Within the scope of this interpretation, the
calculated band gaps agree with experiment
to within -0.01 Ry near the fundamental band
gap and to within -0.05 Ry over a range of
1.0 Ry.
The EPM involves choosing pseudopotential
form factors which give band structures consistent
with the experimental measurements.
These form factors are first constrained to
give a few of the principal band gaps in agreement
with experiment and then used to determine
the electronic band structure at many
points in the Brillouin zone. The pseudopotential
form factors V~ used for diamond are
(in Rydbergs) V,~, = —0.811, V»o -—0.337, V»,
=0.132, and V„,=0.041. The V„,form factor,
which is identically zero for a linear
superposition of spherical atomic potentials,
is included here to account for the distribution
of valence charge'~" arising from tetrahedral
bonding. This tetrahedral distribution
of charge accounts for the presence of the
otherwise forbidden (222) reflection in x-ray
data "~"
The band structure of diamond appears in
Fig. 1. The calculation of the energy bands
is convergent to -0.003 Ry. The lattice constant
was taken to be 3.57 A." In Table I
we list the principal energy gaps of this band
structure and the corresponding experimental
values. The location of the conductionband
minimum is also included in Table I.
The error in the experimental energies at which
the peaks occur in the optical constants is large
both because of the inherent broadness of the
354
V&&UME 16, +UMBER 9 PH YSICAL RE VIE%' LETTERS 28 FEBRUARY 1966
20
10
Table I. Theoretical and experimental values for
the location of the conduction band minimum and the
principal band gaps of diamond. Except for the I"25~-
4& gap which was determined using absorption measurements,
all experimental splittings were obtained
only from plots of the imaginary part of the frequency
dependent dielectric function.
Theory
(eV)a
Experiment
(eV)a
0
—10
+minimum
I'25'-&~
I'25'-I'i5
X4-Xg
Ig-I g
I'25'-I'~2
(o.vv, o, o)
5 4
7.3
12.9
10.9
16.5
(o.v6+.o2, o, o)b
5.48c
7 3d, e
9.5-10.5
16-1Vf e
—20
—30
~ (~/2 t/2 ~/2) I' = —(0,0,0) X = —(1,0,0)
aExcept for &minimum
bP. J.Dean, E. C. Lightowlers, and D. R. Wright,
Phys. Rev. 140, A352 (1965).
C. D. Clark, P. J.Dean, and P. V. Harris, Proc.
Roy. Soc. (London) A277, 312 (1964); C. D. Clark,
J. Phys. Chem. Solids 8, 481 (1959).
Ref. 2.
Ref. 4.
fRef. 1.
FIG. 1. The electronic band structure of diamond.
structure and because of the broadness and
shifts introduced by the Kramers-Kronig transformations
of the data. The agreement between
theory and experiment is very good, especially
when one notes that the gaps are large and
the errors in these gaps are a small percentage
of the total splitting.
The most prominent peak in the reflectivity
spectrum is associated, as in other semiconductors
and insulators having the diamond structure,
with the X4-X~ transition. The theoretical
X4-X, splitting appears to be 0.3-0.6 eV
too large. Since this transition may itself constitute
only a nominal part of the optical structure
around this peak, as in Si,"a density of
states or e, calculation may be necessary to
fix this energy difference more precisely.
The structure in the 16- to 17-eV range was
previously assigned'&' to the L point in the zone.
As is shown in Fig. 1, the L splittings are lower
in energy and the only ostensible candidate
for this transition is r„-r»i, we tentatively
make this assignment. The L,I-L, transition
is assumed to give rise to the hump-like structure
in the experimental spectrum'~'~4 at 9.5-
10.5 eV. The L,I-L, transition, which lies higher
in energy, is most likely masked by the X4X~
peak.
We assign the first direct threshold to r»II
j, and put this edge at 7~ 3 eV . Some me asurements
' have revealed that the structure near
this threshold appears to be very sensitive to
temperature, and this sensitivity has stimulated
interpretation of the optical structure in terms
of hybrid excitons. ' We presume that further
study of this region is warranted. At present
we cannot predict the form of the I"25/ ry, peak
or its temperature dependence since there are
several critical points in this region which may
give rise to optical fine structure and these
critical points are very sensitive to the choice
of form factors. In addition, our calculation
is not sufficiently accurate to confirm the existence
of hybrid excitons, but our assignment
of the I"»~ ry5 edge -7.3 eV restricts the size
and shape of the proposed exciton. The I'»iI'»
edge cannot be raised without moving the
&, minimum closer to the X point unless the
X4-X, splitting is made larger than 13 eV. We
have forced the conduction-band minimum to
agree with the value obtained by Dean et al."
If this constraint is relaxed and ~~ is moved
out beyond (0.80, 0, 0), only then can the I'»~ry5
gap be increased beyond 8 eV for a fixed
X4-Xq splitting. The dependence of the principal
gaps on the values of the form factors is
355
VOLUME 16,NUMBER 9 PHYSICAL REVIEW LETTERS 28 I'EBRUARV 1966
V1ii V22o Vsri V222
I'25' -I'i5
I'2s' -I'i2'
X4-Xg
I'25I -Xg
L3'-L g
—0.116
-0.132
-0.094
-0.154
-0.018
-0.052
-0.162
0.018
-0.012
0.192
0.222
-0.190
0.176
0.274
0.258
-0.265
-0.154
-0.102
-0.206
—0.152
given in Table II.
The simplicity of the (1s)' cores of carbon
have inspired orthogonalized plane-wave" band
calculations, but it was not obvious a priori
that the EPM method would be successful for
diamond. The success of this method presumably
relies on the applicability of the Phillips
cancellation theorem. "However, the absence
of P core states mea. ns that the kinetic energy
of the valence P electrons is not cancelled. "
The apparent success of the EPM method for
diamond probably arises from the fact that the
valence P states have small probability of being
in the core because of the form of the p
wave function, and therefore complete cancellation
is not imperative for these states.
One of us (MLC) benefitted from conversa. tions
with Dr. Frank Herman and Professor
J. C. Phillips.
*Work supported by the National Science Foundation.
)National Science Foundation Predoctoral Fellow.
f.Alfred P. Sloan Fellow.
Table II. The change of the principal energy gaps in
eV for a change in form factor of +0.01 Ry.
~W. C. Walker and J. Osantowski, Phys. Rev. 134,
A153 (1964).
H. R. Philipp and E.A. Taft, Phys. Rev. 136, A1445
(1964); 127, 159 (1962).
J.C. Phillips, Phys. Rev. 139, A1291 (1965).
4D. M. Roessler, thesis, University of London, 1966
(unpublished).
5M. L. Cohen and T. K. Bergstresser, Phys. Rev.
141, 789 (1966).
6J. C. Phillips, Phys. Rev. 112, 685 (1958); D. Brust,
J. C. Phillips, and F. Bassani, Phys. Rev. Letters 9,
94 (1962); D. Brust, M. L. Cohen, and J.C. Phillips,
Phys. Rev. Letters 9, 389 (1962); D. Brust, Phys. Rev.
134, A1337 (1964).
T. K. Bergstresser, M. L. Cohen, and E. W. Williams,
Phys. Rev. Letters 15, 662 (1965).
M, L. Cohen and J.C. Phillips, Phys. Rev. 139,
A912 (1965).
~L. Kleinman and J. C. Phillips, Phys. Rev. 125,
819 (1962).
~OK. H. Bennemann, Phys. Rev. 133, A1045 (1964).
M. Renninger, Z. Krist. 97, 107 (1937).
~2M. Renninger, Acta Cryst. 8, 606 (1955).
~3R. W. G. Wyckoff, Crystal Structures (Interscience
Publishers, New York, 1963).
E. O. Kane, to be published.
~
C. D. Clark, P. J.Dean, and P. V. Harris, Proc.
Roy. Soc. (London) A277, 312 (1964); C. D. Clark,
J. Phys. Chem. Solids 8, 481 (1959).
~GP. J.Dean, E. C. Lightowlers, and D. R. Wright,
Phys. Rev. 140, A352 (1965).
~YF. Herman, Phys. Rev. 93, 1214 (1954); Proceedings
of the International Conference on the Physics of
Semiconductors, Paris (Dunod, Paris, 1964), p. 3.
~
P. W. Anderson, Concepts in Solids (W. A. Benjamin,
Inc. , New York, 1963), p. 66.
~SL. Kleinman and J. C. Phillips, Phys. Rev. 116,
880 (1959).
RICHARDSON-SCHOTTKY EFFECT IN INSULATORS*
P. R. Emtage and J.J.O'Dwyerf
Westinghouse Research Laboratories, Pittsburgh, Pennsylvania
(Received 31 January 1966)
The Richardson-Schottky formula for thermionic
emission from a metallic cathode into
the conduction band of an insulator is frequently'
stated as
4vem(aT)' -(y, ay)/aT—J, e
field strength immediately in front of the cathode.
It has recently been pointed out by Simmons'
that this expression is invalid when the
mobility of the electrons in the dielectric is
low, for if one determines the density of current
carriers in the insulator, n, from the
relationship
In this expression yo is the work function, and
the Schottky term is given by
ay =(e'I' /e)'",
C
where e is the dielectric constant, and F~ the
J=ne pI', (3)
one may then find that n becomes so large that
back-diffusion from the dielectric to the metal
will occur. Unfortunately Simmons's discus-
356
VOLUME 21, NUMBER 10 PHYSICAL REVIEW LETTERS 2 SEP/EMBER 1968
ERRATA
BAND STRUCTURE AND OPTICAL PROPERTIES
OF DIAMOND. W. Saslow, T. K. Bergstresser,
and Marvin L. Cohen [Phys. Rev. Letters
16, 354 (1966)].
Three points in Fig. 1 were drafted 1 eV too
high. They are the I » point in the conduction
band, and the two adjacent points along the A
line. A correct picture of the band structure is
given in Fig. 1 of Luis R. Saravia and D. Brust,
Phys. Rev. 170, 683 (1968).
RECENT p-PRODUCTION EXPERIMENTS AND
THE PREDICTIONS OF CHILL SYMMETRY.
D. A. Geffen and T. Walsh [Phys. Rev. Letters
20 1536 (1968)].
The following additions in proof to Refs. 1 and 3
were inadvertently omitted. Reference 1: See
also J. Schwinger, Phys. Rev. 167, 1546 (1968),
for a similar analysis of the Novosibirsk data.
Reference 3: R. Arnowitt, M. H. Friedman, and
P. Nath, Phys. Rev. Letters 19 1085 (1967).
It is worth mentioning that the latter authors
were the first to note in print the possibility of a
virtual p-mass dependence of the pnm vertex following
from an effective Lagrangian.
TEST OF THE &S=&Q RULE IN LEPTONIC DECAYS
OF NEUTRAL E MESONS. Bryan R. Webber,
Frank T. Solmitz, Frank S. Crawford, Jr.,
and Margaret Alston-Garnjost [Phys. Rev. Letters
21 498 (1968)].
In the abstract, "Im(~) = -0.88+0.08" should be
replaced by "Im(x) = -0.08 +0.08."
In the fourth line of the second column of p.
498, "four-constant" should be "four-constraint. "
The second sentence after Eq. (7) should read,
"They are also insensitive to our choice for t5 t."
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