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Optical constants of germanium and gray tin the k . p method
Solid State Communications, Vol. 5, pp. 513-516, 1967. Pergamon Press Ltd. Printed in Great Britain
OPTICAL CON~ANTS OF GERMANIUM AND GRAY TIN THE jç,~METHOD* C.W. Higginbotham, Fred H. Pollak and Manuel Cardonat Brown University, Physics Department, Providence, Rhode Island (Received 26 April 1967 by E. Burstein)
The imaginary parts of the dielectric constants of germanium and gray tin have been calculated from the previously obtained ~. p band structures of these materials. Spin-orbit coupling effects have been included in the gray tin calculation. The results of these calculations are compared with experimental data.
A ~. p HAMILTONIAN referred to 15 orbital state~at ~, = 0 has been shown to provide a good representation of the band structure of diamond- andzincblende-type materials in the region of fundamental optical transitions. 1 From this band structure it is possible to obtain the band energies and their eigenvectors thoughout the entire Brillouin zone. The matrix elements and density of states can then be calculated to obtain the optical constants of the material. For the heavier materials in this class the spin-orbit coupling becomes important. In the k. p representation this effect is easily included. ~ Calculations of the imaginary part of the dielectric constant £ 2 have been carried out for germanium without spin-orbit coupling effects and for gray tin including these effects. To the best of our knowledge these are the first calculations optical constants of solids from their bandofstruc-
~ is given by (in atomic units): M ~k’ 2 ~ = 1 .1.1’ ‘—~ I 2~ / ~ ~2 ~ V E
J
—
3i
Ic
~-.J
where M ~‘ (~cJis the matrix element of p between occupied and empty states (j and j’ iespectively) at the point k of reciprocal space. These matrix elements have been calculated throughout the zone and found not to be very ~ dependent for a given set of bands. 1 Thus it is a good approximation to consider them constant ~, making equation (1) = £2
ture which include spin-orbit coupling. Calcula-
(w)
=
4 2 —fl-— 3 w2
M
2
N
(w)
,
(2)
where N 4 (w) is the combined density of states for direct optical transitions and M an average matrix element which can be considered to dependon w.
tions without spin-orbit effects, based on a pseudopotential Hamiltonian, have been published 3 byBrust. Direct interband transitions are mainly
The combined density of states N 4 (w) was obtained by a method originated by Gilat, et al. ~ The first Brillouin zone is divided into a uniform cubic mesh in ~ space. Within each of these cubes the constant energy surfaces are approximated by planes. The density of states is then given by the volume of k-space between consecutive constant energy surfaces. If the cube is at the edge of the zone, the result is multiplied by a weighting factor which is simply the
reponsible for the structure observed in the optical constants of insulators, semiconductors, and semimetals in the visible and ultraviolet regions of the spectrum. Their contribution to
—
_________________
*Supported by the National Science Foundation under grants No. GP 5909 and GP 4815. t A. P. Sloan Research Fellow.
513
514
GERMANIUM AND GRAY TIN
fraction of the volume of the cube inside the zone.
30 25
For the germanium calculation a 15 x 15 orbital k. Hamiltonian was diagonalized at the centres of the cubes of our mesh to obtain the energy eigenvalues and the components of energy gradients. About 750 diagonalizations were performed throughout the irreducible section of the 1 zone previously calculated p parameters. Sinceusing the zone can be reduced by k. syrnmetry to 48 equivalent irreducible sections, this is the same as sampling 36, 000 points in the complete Brillouin zone. The combined density of states N 4 (w) was then calculated as described above. From a consideration of N4(w) for separate pairs of bands and the constant energy surfaces of these transitions, five main critical points in N4 (w) were identified and the corresponding point in k-space located. It was then assumed that the average matrix elements l~i ~ ~ at the energies of these five points were equal to the matrix elements of the corresponding singularities. The matrix elements at other energies were found by linear interpolation between these points. The results of the calculation are shown in Fig. 1. The solid line is the calculated e~(n) while the dashed line is the experimental 5 obtained from curve the Kramers-Kronig analysis of the normal-incidence of Philipp and Ehrenreich reflection data. The overall shape of the calcu-
Vol. 5, No. 7
20
i~
5
~
GERMANIUM
-
/
I’
o
I
2
3
.0 E (.V)’5 4
E (.V)
—~
—~
—~
~
6
FIG. 1 Calculated and experimental values of the imaginary part of the dielectric constant of germanium 2. No spin-orbit coupling effects have been included in the calculation. at E(E2—~ E3). We find for germanium that the main contribution to this peak comes from critical points at 2ir 17 7 ~). 3\ A slight lowering of the X 4 X1 gap used for the determination of the ~. p parameters would most probably give a good fit of the E2 peak to the experimental data. The E~peak, due to L3’ L3 and t~ ~ 2, transItions, is stronger than the corresponding experimental peak. A number of possible reasons for this discrepancy can ~e given, 1. e. the average matrix element I MI used for this peak may not be correct and also the k. ~Hamiltonian may not give an accurate representation of the band structure at high energies. 1 A lowering of the E2 peak, by means of readjustment of the ~ p parameters as described above, would lower romewhat the intensity of the calculated E~peak. The correctness of the strength of the experimental ~2 peak can also be questioned. The germanium calculation was made chiefly to test the computer program before the spin-orbit effects were ineluded. Because of the agreement with experiment it was decided that repeating the calculation with better ~ç.~ parameters would not be instructive. Spin-orbit coupling effects were included in the gray-tin calculation by doubling the number —
~ .~.,
.~,
—~
lated curve is in good agreement with experiment The lowest direct edge (E0) shown in the insert, is seen to be nearly parabolic. No attempt has been made to compare this edge with experimental data since it is strongly affected by spin-orbit interaction neglected in this calculation. £2 starts rising steeply at the L~3’ L1 edge (1.9 eV), which is of the M0 variety. The E1 3 critical (A peak corresponds to an M1 point in the [lii 3 direction Qf ~-space 3 A ~). It has a high-energy shoulder which will probably be destroyed when spin-orbit coupling is included, The E~shoulder between 3. 1 and 3. 8 eV, which reproduces rather accurately the experimental line shape, starts with an M3 critical point (r~ I’1~)at 3. 1 eV and includes an M1 critical point in tue £100 3 direction close to = 0. The calculated E2 peak, while reproducing well the experimental line shape, is shifted towards higher energies. This is due to our having fitted the X4 X1 gap to the position of the E2 peak in the determination 6 theof main the ~..p conparameters. by Kane tribution to theAsE2shown peak for silicon does not occur at X but at clusters of critical points near X and
5
—~
—~
Vol. 5, No. 7
GERMANIUM AND GRAY TIN
tion of the split peaks with a relatively small number of diagonalizations. Unfortunately there r
30
~A
(x,-~x,)
6-”A6 ________
) \ J GRAY
25
2
(r7+-~r,)
TIN
L4-.L,-~L6,.
~L6~L6+
L4~L,~
r9~r~
15]
[
I0.?~
515
I
.
E~ E1+~1 E~E~+~
t
+2 + +
3
$
~I~~L5
+ 4+ $
$5
6
E ( eV)
FIG. 2 Calculated and experimental values of the imaginary part of the dielectric constant of gray tin. Spin-orbit interac-
are no experimental determinations of £2 with which to compare the results of the peaks of our calculated spectrum. The general shape of the curve in Fig. 2 agrees with the experimental e2(w) of InSb, 6 with due allowance for the difference between the energy gaps of this semiconductor and those of gray tin. The split A peaks and the X peak dominate the curve, with some added structure to the high-energy side of electroreflectance the latter. The positions spectrum of the of gray peaks tininare the shown by arrows at the bottom of Fig. 2 and the peaks identified. It can be seen that these peaks agree quite well with the structure found in the calculated ~2 (w) . The splitting of the E1, E1 + A 1 peaks is due specifically to spin-orbit coupling and also the splitting of E~into a quadruplet (onlyseen three components of this quadruplet have been in the electroreflectance spectrum~). The weak structure associated with the r,r6- transitions appears in Fig. 2 becaus~of the use of an average matrix element ._.~
included in the calcuof basis states at ~ = 0 and introducing the matrix elements A ~‘and A 15, which were defined in a previous paper. 2 These matrix elements are not independent and hence the inclusion of spin-orbit effects requires only one additional adjustable parameter. The values of these and the other k. p parameters were taken from Ref. 2. Because’bf the large increase in computer time (approximately quadruple when the rank of the matrix is doubled) only about 200 diagonalizations of the 30 X 30 Hamiltonian were performed, corresponding to about 10, 000 points throughout the Brillouin zone. Gray tin was chosen for the calculation with spin-orbit coupling because of its large Spinorbit interaction, which facilitates the resolu-
I M I and is spurious. While these transitions are actually forbidden by symmetry, they may appear in the electroreflectance spectrum since the electric field may lift the parity selection rule. The calculations were performed with an IBM 360 Model 50 computer located at Brown University. The diagonalization subroutine was based on the Jacobi method. The calculations in Fig. 1 took a total of 3 hr of computer time while those in Fig. 2 took 6 hr. The sharpness of some of the A and L peaks in Figs. I and 2 may have been enhanced by the piecewise replacement of the constant-energy surfaces by planes. We are in the process of using a quadratic fit to the constant-energy surfaces in order to assess the magnitude of the computational error.
References 1. •2.
CARDONA M. and POLLAK F.H.,
Phys. Rev. 142,
530 (1966).
POLLAK F.H., HIGGINBOTHAM C.W. and CARDONA M., Proc. Int. Conf. Semiconductors Physics, Kyoto, p.20 (1966).
3.
BRUST D., Phys. Rev. 134, A1337 (1964).
4.
GILAT G. and DOLLING G., Phys. Left. 8, 304 (1964); GILAT G. and RAUBENIHEIMER L. J., Ph~. Rev. 144, 390 (1966).
5.
PHILIPP H.R. and EHRENREICH H.,
Phys. Rev. 129, 1550 (1963).
516
GERMANIUM AND GRAY TIN
6.
KANE E.O.,
7.
CARDONA M., McELROY P., POLLAX F. H. 4, 319 (1966).
Vol. 5, No. 7
Phys. Rev. 146, 558 (1966). and SHAKLEE K. L., Solid State Comm.
La partie imagtnaire de la constante diélectrique du germanium et du silicium a dtd calculee a partir des bandes d’energte de ces matériaux obtenues par la methode ~ p. Les effects du couplage spin-orbite ont été consideFes pour l’êtain gris. Les resultats sont compares avec des données experimentales.
OPTICAL CON~ANTS OF GERMANIUM AND GRAY TIN THE jç,~METHOD* C.W. Higginbotham, Fred H. Pollak and Manuel Cardonat Brown University, Physics Department, Providence, Rhode Island (Received 26 April 1967 by E. Burstein)
The imaginary parts of the dielectric constants of germanium and gray tin have been calculated from the previously obtained ~. p band structures of these materials. Spin-orbit coupling effects have been included in the gray tin calculation. The results of these calculations are compared with experimental data.
A ~. p HAMILTONIAN referred to 15 orbital state~at ~, = 0 has been shown to provide a good representation of the band structure of diamond- andzincblende-type materials in the region of fundamental optical transitions. 1 From this band structure it is possible to obtain the band energies and their eigenvectors thoughout the entire Brillouin zone. The matrix elements and density of states can then be calculated to obtain the optical constants of the material. For the heavier materials in this class the spin-orbit coupling becomes important. In the k. p representation this effect is easily included. ~ Calculations of the imaginary part of the dielectric constant £ 2 have been carried out for germanium without spin-orbit coupling effects and for gray tin including these effects. To the best of our knowledge these are the first calculations optical constants of solids from their bandofstruc-
~ is given by (in atomic units): M ~k’ 2 ~ = 1 .1.1’ ‘—~ I 2~ / ~ ~2 ~ V E
J
—
3i
Ic
~-.J
where M ~‘ (~cJis the matrix element of p between occupied and empty states (j and j’ iespectively) at the point k of reciprocal space. These matrix elements have been calculated throughout the zone and found not to be very ~ dependent for a given set of bands. 1 Thus it is a good approximation to consider them constant ~, making equation (1) = £2
ture which include spin-orbit coupling. Calcula-
(w)
=
4 2 —fl-— 3 w2
M
2
N
(w)
,
(2)
where N 4 (w) is the combined density of states for direct optical transitions and M an average matrix element which can be considered to dependon w.
tions without spin-orbit effects, based on a pseudopotential Hamiltonian, have been published 3 byBrust. Direct interband transitions are mainly
The combined density of states N 4 (w) was obtained by a method originated by Gilat, et al. ~ The first Brillouin zone is divided into a uniform cubic mesh in ~ space. Within each of these cubes the constant energy surfaces are approximated by planes. The density of states is then given by the volume of k-space between consecutive constant energy surfaces. If the cube is at the edge of the zone, the result is multiplied by a weighting factor which is simply the
reponsible for the structure observed in the optical constants of insulators, semiconductors, and semimetals in the visible and ultraviolet regions of the spectrum. Their contribution to
—
_________________
*Supported by the National Science Foundation under grants No. GP 5909 and GP 4815. t A. P. Sloan Research Fellow.
513
514
GERMANIUM AND GRAY TIN
fraction of the volume of the cube inside the zone.
30 25
For the germanium calculation a 15 x 15 orbital k. Hamiltonian was diagonalized at the centres of the cubes of our mesh to obtain the energy eigenvalues and the components of energy gradients. About 750 diagonalizations were performed throughout the irreducible section of the 1 zone previously calculated p parameters. Sinceusing the zone can be reduced by k. syrnmetry to 48 equivalent irreducible sections, this is the same as sampling 36, 000 points in the complete Brillouin zone. The combined density of states N 4 (w) was then calculated as described above. From a consideration of N4(w) for separate pairs of bands and the constant energy surfaces of these transitions, five main critical points in N4 (w) were identified and the corresponding point in k-space located. It was then assumed that the average matrix elements l~i ~ ~ at the energies of these five points were equal to the matrix elements of the corresponding singularities. The matrix elements at other energies were found by linear interpolation between these points. The results of the calculation are shown in Fig. 1. The solid line is the calculated e~(n) while the dashed line is the experimental 5 obtained from curve the Kramers-Kronig analysis of the normal-incidence of Philipp and Ehrenreich reflection data. The overall shape of the calcu-
Vol. 5, No. 7
20
i~
5
~
GERMANIUM
-
/
I’
o
I
2
3
.0 E (.V)’5 4
E (.V)
—~
—~
—~
~
6
FIG. 1 Calculated and experimental values of the imaginary part of the dielectric constant of germanium 2. No spin-orbit coupling effects have been included in the calculation. at E(E2—~ E3). We find for germanium that the main contribution to this peak comes from critical points at 2ir 17 7 ~). 3\ A slight lowering of the X 4 X1 gap used for the determination of the ~. p parameters would most probably give a good fit of the E2 peak to the experimental data. The E~peak, due to L3’ L3 and t~ ~ 2, transItions, is stronger than the corresponding experimental peak. A number of possible reasons for this discrepancy can ~e given, 1. e. the average matrix element I MI used for this peak may not be correct and also the k. ~Hamiltonian may not give an accurate representation of the band structure at high energies. 1 A lowering of the E2 peak, by means of readjustment of the ~ p parameters as described above, would lower romewhat the intensity of the calculated E~peak. The correctness of the strength of the experimental ~2 peak can also be questioned. The germanium calculation was made chiefly to test the computer program before the spin-orbit effects were ineluded. Because of the agreement with experiment it was decided that repeating the calculation with better ~ç.~ parameters would not be instructive. Spin-orbit coupling effects were included in the gray-tin calculation by doubling the number —
~ .~.,
.~,
—~
lated curve is in good agreement with experiment The lowest direct edge (E0) shown in the insert, is seen to be nearly parabolic. No attempt has been made to compare this edge with experimental data since it is strongly affected by spin-orbit interaction neglected in this calculation. £2 starts rising steeply at the L~3’ L1 edge (1.9 eV), which is of the M0 variety. The E1 3 critical (A peak corresponds to an M1 point in the [lii 3 direction Qf ~-space 3 A ~). It has a high-energy shoulder which will probably be destroyed when spin-orbit coupling is included, The E~shoulder between 3. 1 and 3. 8 eV, which reproduces rather accurately the experimental line shape, starts with an M3 critical point (r~ I’1~)at 3. 1 eV and includes an M1 critical point in tue £100 3 direction close to = 0. The calculated E2 peak, while reproducing well the experimental line shape, is shifted towards higher energies. This is due to our having fitted the X4 X1 gap to the position of the E2 peak in the determination 6 theof main the ~..p conparameters. by Kane tribution to theAsE2shown peak for silicon does not occur at X but at clusters of critical points near X and
5
—~
—~
Vol. 5, No. 7
GERMANIUM AND GRAY TIN
tion of the split peaks with a relatively small number of diagonalizations. Unfortunately there r
30
~A
(x,-~x,)
6-”A6 ________
) \ J GRAY
25
2
(r7+-~r,)
TIN
L4-.L,-~L6,.
~L6~L6+
L4~L,~
r9~r~
15]
[
I0.?~
515
I
.
E~ E1+~1 E~E~+~
t
+2 + +
3
$
~I~~L5
+ 4+ $
$5
6
E ( eV)
FIG. 2 Calculated and experimental values of the imaginary part of the dielectric constant of gray tin. Spin-orbit interac-
are no experimental determinations of £2 with which to compare the results of the peaks of our calculated spectrum. The general shape of the curve in Fig. 2 agrees with the experimental e2(w) of InSb, 6 with due allowance for the difference between the energy gaps of this semiconductor and those of gray tin. The split A peaks and the X peak dominate the curve, with some added structure to the high-energy side of electroreflectance the latter. The positions spectrum of the of gray peaks tininare the shown by arrows at the bottom of Fig. 2 and the peaks identified. It can be seen that these peaks agree quite well with the structure found in the calculated ~2 (w) . The splitting of the E1, E1 + A 1 peaks is due specifically to spin-orbit coupling and also the splitting of E~into a quadruplet (onlyseen three components of this quadruplet have been in the electroreflectance spectrum~). The weak structure associated with the r,r6- transitions appears in Fig. 2 becaus~of the use of an average matrix element ._.~
included in the calcuof basis states at ~ = 0 and introducing the matrix elements A ~‘and A 15, which were defined in a previous paper. 2 These matrix elements are not independent and hence the inclusion of spin-orbit effects requires only one additional adjustable parameter. The values of these and the other k. p parameters were taken from Ref. 2. Because’bf the large increase in computer time (approximately quadruple when the rank of the matrix is doubled) only about 200 diagonalizations of the 30 X 30 Hamiltonian were performed, corresponding to about 10, 000 points throughout the Brillouin zone. Gray tin was chosen for the calculation with spin-orbit coupling because of its large Spinorbit interaction, which facilitates the resolu-
I M I and is spurious. While these transitions are actually forbidden by symmetry, they may appear in the electroreflectance spectrum since the electric field may lift the parity selection rule. The calculations were performed with an IBM 360 Model 50 computer located at Brown University. The diagonalization subroutine was based on the Jacobi method. The calculations in Fig. 1 took a total of 3 hr of computer time while those in Fig. 2 took 6 hr. The sharpness of some of the A and L peaks in Figs. I and 2 may have been enhanced by the piecewise replacement of the constant-energy surfaces by planes. We are in the process of using a quadratic fit to the constant-energy surfaces in order to assess the magnitude of the computational error.
References 1. •2.
CARDONA M. and POLLAK F.H.,
Phys. Rev. 142,
530 (1966).
POLLAK F.H., HIGGINBOTHAM C.W. and CARDONA M., Proc. Int. Conf. Semiconductors Physics, Kyoto, p.20 (1966).
3.
BRUST D., Phys. Rev. 134, A1337 (1964).
4.
GILAT G. and DOLLING G., Phys. Left. 8, 304 (1964); GILAT G. and RAUBENIHEIMER L. J., Ph~. Rev. 144, 390 (1966).
5.
PHILIPP H.R. and EHRENREICH H.,
Phys. Rev. 129, 1550 (1963).
516
GERMANIUM AND GRAY TIN
6.
KANE E.O.,
7.
CARDONA M., McELROY P., POLLAX F. H. 4, 319 (1966).
Vol. 5, No. 7
Phys. Rev. 146, 558 (1966). and SHAKLEE K. L., Solid State Comm.
La partie imagtnaire de la constante diélectrique du germanium et du silicium a dtd calculee a partir des bandes d’energte de ces matériaux obtenues par la methode ~ p. Les effects du couplage spin-orbite ont été consideFes pour l’êtain gris. Les resultats sont compares avec des données experimentales.