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Band structure and optical properties of aluminum
Solid
State
Vol. 8, pp. 4 13-416,
Communications,
BAND STRUCTURE
AND OPTICAL David
Lawrence
Radiation
Laboratory,
University
(Received
24 November
Pergamon
1970.
PROPERTIES
Printed
Press.
in Great
Britain
OF ALUMINUM*
Brust of California,
Livermore,
California
94550
1969 by R.H. Silsbee)
Using Ashcroft’s potential, a calculation of the interband contribution to e,(w) for aluminium is made. The large peak near 1.6eV can be understood by examination of an appropriate reflection plane in the Brillouin zone. In addition, another peak should exist near 0.5 eV. The optical effective mass is computed to be 1.45 m, .
THE was
et al.’
WORK of Ehrenreich one of the first
structure
calculations
of simple
polyvalent
was
result
their
peak
in e,(w)
band
transitions
efforts with
which near
1.5 eV was
associated
and c in the Brillouin considerable
zone,
in an improved experiment
in this
there
agreement
peak.
Using
theory
range
from about
necessary
about
and
a quadratic
interpolation
sample
generated
test
form factors data.
to get the band
pseudopotential This choice was were
Our subsequent
the Fermi to ep (w).
originally
structure
of aluminium,
was
was
fit to Fermi
discussion
will
purposes
that
V(K) = 0. Actually,
V(4) = O.O562Ry, the resulting
band
and all
good
structure
breaking *Work Atomic
performed under the auspices Energy Commission.
of the U.S.
a fixed
out that energy
made
at
for all
This
using large procedure.
a convergence In
4 x 4, 9 x 9, and
these dimensions considerations.
The
for the present
errors The
were
points
eigenvalues.
with
inadequate
(convergence
were
considerably
9 x 9, however, -- 0.05 eV).
gave
The nine
are the lowest kinetic energy states part of the zone studied. Of course, secular
equation
of the band
structure,
of a 9 x 9 secular
4 13
were
of 0.2eV).
results
using
determined
scheme.a
on the
it
in the zone.
by a Monte-Carlo
(convergence
plane waves in the 1/48th
other
first
eigenvalues
110,000
equations; by energy
judged
in excess
surface shape is quite closely related The potential coefficients are then
V(3) = O.O179Ry,
trials
4 x 4 was
surface
show
were
be pointed
performed
13 x 13 secular being suggested
parameters were made as Ashcroft’s
points
elements
It should
an mass.
results,
energy
at an additional
particular, In order
contribution,
points. To achieve good histogram eigenvalues and matrix elements
calculated
use of a result discussed elsewhere* allows accurate calculation of the optical effective
convergent
to compute
matrix
of these statistics,
1.35-4.OeV. In addition, the present calculations indicate the presence of an infrared peak at 0.5 eV in the interband contribution to E,(W). The
Ashcroft’s chosen.3
of other
for the interband
2000 inequivalent
Dipole
the
results
between
in the spectral
was
was
the observed
communication
expression
To get satisfactorily
W
near
although of this
to those
e”,(w), to the complex part of the frequencydependent dielectric constant is given elsewhere.7
to inter-
points
between
intensities
outlined
The
interest
the prominent
related
with
discrepancy
and calculated methods
that
similar
1,4-6
properties
Of particular
showed
quite
calculations.
band
the optical
metals.
is generally
on aluminum
to correlate
equation
results
in symmetry
but with
it is negligible.
the choice
BAND STRUCTURE
414
l
I 70 -
AND OPTICAL
1
PROPERTIES
I
Vol. 8, No. 6
OF ALUMINUM
c
l . l
l
i.c
0
60 l
0
50 0
y40
3
l\
-0
I
1,
l
3
I .
w
2
’ :.
30
’ \ ’ \ ? f
0
20 0
1
0
rr
10
,\_ 2 RW
0.2
4
3 -
I
-1
I 1
0
-
T
l
- 2 /< 0.3
0.4
I
I
0.5
0.6
0.7
k
ev
FIG. 1. Imaginary part of the dielectric constant due to interband transitions. The experimental curve (dashed line) was derived by Ehrenreich et al.’ using reflectivity data. A Drude background term was subtracted; i.e. E:(W) = E,(W) - e;(w), where E{(W) is deduced using a free electron expression. The theoretical curve (dots) is derived by the method described in the text, and does not include lifetime broadening effects.
FIG. 2. Energy bands in the rXWK reflection plane of the Brillouin zone of aluminum. The solid line refers to the locus (0.93, k, 0.0). A plus sign indicates even-parity vis-&vis . reflection in the plane, whereas a minus sign indicates odd parity. Some important transitions contributing to the 1.6eV in e,“(w) are shown by arrows.
should be examined.
The energy bands lying
near the Fermi level are drawn along parallel Figure 1 shows
the computed
spectrum
for
E:(W). The experimental results were derived by Ehrenreich et al.’ using the best reflectivity measurements available to them. To extract E:(W) from e2(w), they made a Drude fit to the free electron part of the absorption using m*, (optical effective mass) and @lifetime) as adjustable parameters. Then E:(O) = Ed - e;(w). As Fig. 1 shows, and experiment
the agreement between is reasonable
theory
with regard to the
peak near 1.6eV; particularly when it is noted that no lifetime broadening has been incorporated. To understand
the origin of this peak fully,
Fig. 2
lines
in the I’XWK reflection
important contributing by arrows.
plane.g
transitions
Some
are indicated
Note how nearly parallel
the two
responsible understand.
bands are. This is quite simple to It is necessary to observe that the --- plane waves (1, 1, 1) and (1, 1, 1) are degenerate over the entire reflection plane (that is the portion under study). primarily
These
by the V(4) component
two are split of the potential.
Thus the band gap separating the resulting parallel bands is ry 2 1 V(4) 1 - 1.53 eV, and a large portion of the reflection to the optical
plane contributes
peak with a rather large dipole
BAND STRUCTURE
Vol. 8, No. 6
AND OPTICAL
matrix element. It is interesting to note in this connection that the transitions at the W point are not particularly close toho(peak) in the present calculation. In particular, W, -+ W, = 1.95 eV; and W, I + W, = 0.89eV with W,, > E,. Therefore, these transitions are not important for an understanding of this peak. Another interesting result is the prominent --peak at 0.5eV. Study of the plane waves (1, 1, 1) and (2, 0, 0) discloses that they have a plane of degeneracy. Since they are coupled by the coefficient V(3), we expect - and indeed find a peak in e:(w) near 2 ) V(3) 1 = 0.49eV. Both in this case and that of the 1.6eV peak the relevant transitions have strong matrix elements. In addition to e:(w), we may compute the optical effective mass which appears in the Drude free electron term.‘~‘” In order to do this, we take advantage of a simple relationship between the optical effective mass and E:(W) which is band theoretically proved elsewhere* (that it should be true can easily be seen from an inspection of the sum rules for Ed’ ); namely,
PROPERTIES
OF ALUMINUM
415
value of 1.5 required to fit the infrared reflectivity spectrum.’ The use of a Drude model with frequency-independent parameters has recently been criticized by Powell.” Since many particle corrections are apparently small,12 it would appear from our calculations that the Drude model is at least reasonable for the infrared. it is of interest to reiterate the close equality between the positions of the first two peaks in E:(W) and the coefficients V(3) and V(4). The positions of the peaks can serve to determine the potential and, therefore, help in delineating the Fermi surface. In addition, measurements of the optical spectra under pressure would be extremely interesting. A computational study of differential optical line shapes would, however, require a high-resolution study as has been done for semiconductors. l3 As a final comment, it should be pointed out that lifetime broadening would decrease the amplitude of the two peaks. The good agreement with the tail portion (ho > 2eV) of E:(W) would probably not be spoiled. l4
(1)
We also note that Hughes, et ~1.‘~ have used Ashcroft’s potential and found the large peak near 1.6 eV.
where N is the concentration of valence electrons per unit volume (three per aluminum atom). Direct integration over the computed spectrum gives m*,/m = 1.45. This is in good agreement with the
Acknowledgements - The author is grateful to Drs. Fred Wooten, Tony Huen, and Marvin ROSS, for useful discussions. He would also like to thank Dr. Phillip Best for drawing his attention to reference 11.
m0 m*,
_ 1 _ mo s e:(W) wdw 2rl2N,2 ’
REFERENCES 1.
EHRENREICH
2.
BRUST
3.
ASHCROFT
N.W., Phil.
4.
HARRISON
W.A., Phys.
5.
HEINE V., Proc.
6.
ROSS M. and JOHNSON K.W., to be published.
7.
BRUST D., Phys.
8.
BRUST D., P_hys. Rev.
9.
For a map of the Brillouin
10.
H., PHILIPP
H.R. and SEGALL
B., Phys.
Rev.
132, 1918 (1963).
D., to be published. Mug. 8, 2855 (1963). Rev.
R. Sot.
COHEN M.H., Phil.
Rev.
118, 1182 (1960).
(London)
A240, 361 (1957).
134, Al337
(1964).
139, A489 (1965). zone,
the reader
Mug. 3, 762 (1958).
may wish to see Fig. 7 of reference
7.
BAND STRUCTURE
416 11.
POWELL
C.J.,
12.
BEEFERMAN
13.
SARAVIA
14.
BRUST D. and KANE E.O.,
15.
HUGHES A.J.,
AND OPTICAL
PROPERTIES
OF ALUMINUM
Vol. 8, No. 6
to be published. L.W. and EHRENREICH
L.R.
H., Bull. Am. phys. Sot.,
and BRUST D., Solid State Commun. Phys.
Rev.
Series II, 14, 397 (1969).
7, 669 (1969).
176, 894 (1968).
JONES D. and LETTINGTON
A.H., J. Phys.
Chem. 2, 102 (1969).
A l’aide du potentiel d’Ashcroft, on calcule la contribution entrebande a Ed pour l’aluminum. Le grand pit pres de 1.6eV s’explique en examinant un plan de riflexion convenable dans la zone Brillouin. De plus, un autre pit doit se trouver pris de 0.5 eV. La masse effective optique se calcule a 1.45 m,.
State
Vol. 8, pp. 4 13-416,
Communications,
BAND STRUCTURE
AND OPTICAL David
Lawrence
Radiation
Laboratory,
University
(Received
24 November
Pergamon
1970.
PROPERTIES
Printed
Press.
in Great
Britain
OF ALUMINUM*
Brust of California,
Livermore,
California
94550
1969 by R.H. Silsbee)
Using Ashcroft’s potential, a calculation of the interband contribution to e,(w) for aluminium is made. The large peak near 1.6eV can be understood by examination of an appropriate reflection plane in the Brillouin zone. In addition, another peak should exist near 0.5 eV. The optical effective mass is computed to be 1.45 m, .
THE was
et al.’
WORK of Ehrenreich one of the first
structure
calculations
of simple
polyvalent
was
result
their
peak
in e,(w)
band
transitions
efforts with
which near
1.5 eV was
associated
and c in the Brillouin considerable
zone,
in an improved experiment
in this
there
agreement
peak.
Using
theory
range
from about
necessary
about
and
a quadratic
interpolation
sample
generated
test
form factors data.
to get the band
pseudopotential This choice was were
Our subsequent
the Fermi to ep (w).
originally
structure
of aluminium,
was
was
fit to Fermi
discussion
will
purposes
that
V(K) = 0. Actually,
V(4) = O.O562Ry, the resulting
band
and all
good
structure
breaking *Work Atomic
performed under the auspices Energy Commission.
of the U.S.
a fixed
out that energy
made
at
for all
This
using large procedure.
a convergence In
4 x 4, 9 x 9, and
these dimensions considerations.
The
for the present
errors The
were
points
eigenvalues.
with
inadequate
(convergence
were
considerably
9 x 9, however, -- 0.05 eV).
gave
The nine
are the lowest kinetic energy states part of the zone studied. Of course, secular
equation
of the band
structure,
of a 9 x 9 secular
4 13
were
of 0.2eV).
results
using
determined
scheme.a
on the
it
in the zone.
by a Monte-Carlo
(convergence
plane waves in the 1/48th
other
first
eigenvalues
110,000
equations; by energy
judged
in excess
surface shape is quite closely related The potential coefficients are then
V(3) = O.O179Ry,
trials
4 x 4 was
surface
show
were
be pointed
performed
13 x 13 secular being suggested
parameters were made as Ashcroft’s
points
elements
It should
an mass.
results,
energy
at an additional
particular, In order
contribution,
points. To achieve good histogram eigenvalues and matrix elements
calculated
use of a result discussed elsewhere* allows accurate calculation of the optical effective
convergent
to compute
matrix
of these statistics,
1.35-4.OeV. In addition, the present calculations indicate the presence of an infrared peak at 0.5 eV in the interband contribution to E,(W). The
Ashcroft’s chosen.3
of other
for the interband
2000 inequivalent
Dipole
the
results
between
in the spectral
was
was
the observed
communication
expression
To get satisfactorily
W
near
although of this
to those
e”,(w), to the complex part of the frequencydependent dielectric constant is given elsewhere.7
to inter-
points
between
intensities
outlined
The
interest
the prominent
related
with
discrepancy
and calculated methods
that
similar
1,4-6
properties
Of particular
showed
quite
calculations.
band
the optical
metals.
is generally
on aluminum
to correlate
equation
results
in symmetry
but with
it is negligible.
the choice
BAND STRUCTURE
414
l
I 70 -
AND OPTICAL
1
PROPERTIES
I
Vol. 8, No. 6
OF ALUMINUM
c
l . l
l
i.c
0
60 l
0
50 0
y40
3
l\
-0
I
1,
l
3
I .
w
2
’ :.
30
’ \ ’ \ ? f
0
20 0
1
0
rr
10
,\_ 2 RW
0.2
4
3 -
I
-1
I 1
0
-
T
l
- 2 /< 0.3
0.4
I
I
0.5
0.6
0.7
k
ev
FIG. 1. Imaginary part of the dielectric constant due to interband transitions. The experimental curve (dashed line) was derived by Ehrenreich et al.’ using reflectivity data. A Drude background term was subtracted; i.e. E:(W) = E,(W) - e;(w), where E{(W) is deduced using a free electron expression. The theoretical curve (dots) is derived by the method described in the text, and does not include lifetime broadening effects.
FIG. 2. Energy bands in the rXWK reflection plane of the Brillouin zone of aluminum. The solid line refers to the locus (0.93, k, 0.0). A plus sign indicates even-parity vis-&vis . reflection in the plane, whereas a minus sign indicates odd parity. Some important transitions contributing to the 1.6eV in e,“(w) are shown by arrows.
should be examined.
The energy bands lying
near the Fermi level are drawn along parallel Figure 1 shows
the computed
spectrum
for
E:(W). The experimental results were derived by Ehrenreich et al.’ using the best reflectivity measurements available to them. To extract E:(W) from e2(w), they made a Drude fit to the free electron part of the absorption using m*, (optical effective mass) and @lifetime) as adjustable parameters. Then E:(O) = Ed - e;(w). As Fig. 1 shows, and experiment
the agreement between is reasonable
theory
with regard to the
peak near 1.6eV; particularly when it is noted that no lifetime broadening has been incorporated. To understand
the origin of this peak fully,
Fig. 2
lines
in the I’XWK reflection
important contributing by arrows.
plane.g
transitions
Some
are indicated
Note how nearly parallel
the two
responsible understand.
bands are. This is quite simple to It is necessary to observe that the --- plane waves (1, 1, 1) and (1, 1, 1) are degenerate over the entire reflection plane (that is the portion under study). primarily
These
by the V(4) component
two are split of the potential.
Thus the band gap separating the resulting parallel bands is ry 2 1 V(4) 1 - 1.53 eV, and a large portion of the reflection to the optical
plane contributes
peak with a rather large dipole
BAND STRUCTURE
Vol. 8, No. 6
AND OPTICAL
matrix element. It is interesting to note in this connection that the transitions at the W point are not particularly close toho(peak) in the present calculation. In particular, W, -+ W, = 1.95 eV; and W, I + W, = 0.89eV with W,, > E,. Therefore, these transitions are not important for an understanding of this peak. Another interesting result is the prominent --peak at 0.5eV. Study of the plane waves (1, 1, 1) and (2, 0, 0) discloses that they have a plane of degeneracy. Since they are coupled by the coefficient V(3), we expect - and indeed find a peak in e:(w) near 2 ) V(3) 1 = 0.49eV. Both in this case and that of the 1.6eV peak the relevant transitions have strong matrix elements. In addition to e:(w), we may compute the optical effective mass which appears in the Drude free electron term.‘~‘” In order to do this, we take advantage of a simple relationship between the optical effective mass and E:(W) which is band theoretically proved elsewhere* (that it should be true can easily be seen from an inspection of the sum rules for Ed’ ); namely,
PROPERTIES
OF ALUMINUM
415
value of 1.5 required to fit the infrared reflectivity spectrum.’ The use of a Drude model with frequency-independent parameters has recently been criticized by Powell.” Since many particle corrections are apparently small,12 it would appear from our calculations that the Drude model is at least reasonable for the infrared. it is of interest to reiterate the close equality between the positions of the first two peaks in E:(W) and the coefficients V(3) and V(4). The positions of the peaks can serve to determine the potential and, therefore, help in delineating the Fermi surface. In addition, measurements of the optical spectra under pressure would be extremely interesting. A computational study of differential optical line shapes would, however, require a high-resolution study as has been done for semiconductors. l3 As a final comment, it should be pointed out that lifetime broadening would decrease the amplitude of the two peaks. The good agreement with the tail portion (ho > 2eV) of E:(W) would probably not be spoiled. l4
(1)
We also note that Hughes, et ~1.‘~ have used Ashcroft’s potential and found the large peak near 1.6 eV.
where N is the concentration of valence electrons per unit volume (three per aluminum atom). Direct integration over the computed spectrum gives m*,/m = 1.45. This is in good agreement with the
Acknowledgements - The author is grateful to Drs. Fred Wooten, Tony Huen, and Marvin ROSS, for useful discussions. He would also like to thank Dr. Phillip Best for drawing his attention to reference 11.
m0 m*,
_ 1 _ mo s e:(W) wdw 2rl2N,2 ’
REFERENCES 1.
EHRENREICH
2.
BRUST
3.
ASHCROFT
N.W., Phil.
4.
HARRISON
W.A., Phys.
5.
HEINE V., Proc.
6.
ROSS M. and JOHNSON K.W., to be published.
7.
BRUST D., Phys.
8.
BRUST D., P_hys. Rev.
9.
For a map of the Brillouin
10.
H., PHILIPP
H.R. and SEGALL
B., Phys.
Rev.
132, 1918 (1963).
D., to be published. Mug. 8, 2855 (1963). Rev.
R. Sot.
COHEN M.H., Phil.
Rev.
118, 1182 (1960).
(London)
A240, 361 (1957).
134, Al337
(1964).
139, A489 (1965). zone,
the reader
Mug. 3, 762 (1958).
may wish to see Fig. 7 of reference
7.
BAND STRUCTURE
416 11.
POWELL
C.J.,
12.
BEEFERMAN
13.
SARAVIA
14.
BRUST D. and KANE E.O.,
15.
HUGHES A.J.,
AND OPTICAL
PROPERTIES
OF ALUMINUM
Vol. 8, No. 6
to be published. L.W. and EHRENREICH
L.R.
H., Bull. Am. phys. Sot.,
and BRUST D., Solid State Commun. Phys.
Rev.
Series II, 14, 397 (1969).
7, 669 (1969).
176, 894 (1968).
JONES D. and LETTINGTON
A.H., J. Phys.
Chem. 2, 102 (1969).
A l’aide du potentiel d’Ashcroft, on calcule la contribution entrebande a Ed pour l’aluminum. Le grand pit pres de 1.6eV s’explique en examinant un plan de riflexion convenable dans la zone Brillouin. De plus, un autre pit doit se trouver pris de 0.5 eV. La masse effective optique se calcule a 1.45 m,.