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Electronic charge densities at valence and conduction band edges of ZnSe and CdTe
Volume 114A, number 4
PHYSICS LETTERS
17 February 1986
E L E C T R O N I C C H A R G E D E N S I T I E S AT VALENCE AND C O N D U C T I O N BAND E D G E S OF ZnSe AND CdTe Renata M. W E N T Z C O V I T C H , Steven L. R I C H A R D S O N and Marvin L. C O H E N Department of Physics, University of California and Center for Advanced Materials, Lawrence Berkeley Laboratoo,. Berkeley, CA 94720, USA
Received 8 October 1985; revised manuscript received 27 November 1985; accepted for publication 2 December 1985
The empirical pseudopotential method (EPM) is used to calculate electronic charge densities at selected k points of the valence and conduction band edges of two II-VI semiconductors: ZnSe and CdTe.
In this paper we apply the empirical pseudopotential method (EPM) in the form used by Chelikowsky and Cohen [1 ] to compute the electronic charge densities at the F, L and X k points of the valence and conduction band edges of ZnSe and CdTe. These charge densities are decomposed into their angular momentum contributions and the observed differences in the bonding and antibonding behavior are discussed. The calculation was performed using the non-local EPM technique [1] with spin-orbit coupling included following the work of Weisz [2 ] with the modifications of Bloom and Bergstresser [3]. In this scheme, the wavefunctions of the one-electron hamiltonian are expanded in a set of 113 plane waves, while the local part of the pseudopotential is Fourier analyzed in terms of a small number of pseudopotential form factors, V(G), which have been empirically determined to reproduce experimental data such as band gaps and optical reflectivity. The non-local corrections to the pseudopotential were chosen to match those of ref. [1 ]. The secular equation for the EPM hamiltonian is solved to obtain wavefunctions and eigenvalues for the two compounds. The electronic charge density is calculated at selected k points in the Brillouin zone by taking the square of the wavefunctions at these points On,k(r) = I~n,k(r) 12 •
(1)
Finally, the w ave functions at a particular k point, ~bn,g(r), were decomposed into their angular momentum contributions"Xl,n,k, m rl" [4] 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
R Xl,n,k,rj rn " = 4rt f 0
dr r2(~n,klYl, m ) (YI,m I~n,k) ,
(2)
where ¢1 is the vector position of the ]th atom, R is equal to half the interatomic distance, and Yl, m is the spherical harmonic function. To find the percentage decomposition of s-, p-, and d-character, we normalized this quantity with the total contribution for the other angular momentum components of both ions at a given band n and k point. For the case of degenerate bands, the percentage decomposition was obtained from the appropriate averages. The electronic charge densities for the F, L, and X k points of the valence and conduction band edges of ZnSe and CdTe are shown in figs. 1, 2, and 3, respectively. The percentage decomposition of angular momentum for each of these states was calculated from eq. (2) and the results are listed in table I. Note that we performed the angular momentum decomposition on wavefunctions unperturbed by spin-orbit interaction, and we observe that these results are consistent with group theoretical predictions [5]. The charge distributions at F for the highest valence band (F~5) are illustrated in fig. 1. They show a strong p-like contribution directed along the covalent bond between the anion and the cation of the two semiconductors. The charge density at F in the conduction band (F~) is, however, antibonding as indicated by the minimum of charge density approxi203
Cd
(c)
(d)
/4.~ < ~ - ~ ~ { ....
J , , , ~ > ~ ~ O,
:
zn
/
Fig. 1. Charge density contours for electronic states at the F point inthe (110) plane. Note that spin-orbit interaction has been neglected. (a) ZnSe at highest valence band, (b) CdTe at highest valence band, (c) ZnSe at lowest conduction band, (d) CdTe at lowest conduction band. The contour intervals are in units of 0.5 in cases (a) and (b) and 1.0 electrons per primitive cell in cases (c) and (d).
i
Cd
0
:1 i I x
j/
el
,"
f
I
/
•
',, '
I'I) i
~
~.~./
Fig. 2. Charge density contours for electronic states at the L point in the (110) plane. Note that spin-orbit interaction has been neglected. (a) ZnSe at highest valence band, (b) CdTe at highest valence band, (c) ZnSe at lowest conduction band, (d) CdTe at lowest conduction band. The contour intervals are in units of 0.5 electrons per primitive cell.
Volume 114A, number 4
PHYSICS LETTERS
\
~ ¢
,/
/ . "I
/
c
, , ,
\ ,
•
Zn
[//
~
17 February 1986
\
\\
\
,
_ .
.
.
.
.
.
- - -
.
~
,
~ ' J
-
.
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~
Cd
"'~
,. O "
t ~ . ~
....
"
_
~-
(':m,\ a I#,~ _ / ./ .//./ /.
f
--
. n . c~~-
"--z
~
-
~ -
x;
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(c)
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,
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-
-
-
[
~
~
'.c~
e
, "~_//-~-
"\
~ . " ; ~
"'~
"\
-
I -1
X
Fig. 3. Charge density contours for electronic states at the X point, (27r/a)(001), in the (1]0) plane. Note that spin-orbit interaction has been neglected. (a) ZnSe at highest valence band, (b) CdTe at highest valence band, (c) ZnSe at lowest conduction band, (d) CdTe at lowest conduction band. The contour intervals are in units of 0.5 electrons per primitive cell.
Table 1 Percentage decomposition of angular momentum character for wavefunctions in the highest valence and first conduction bands of ZnSe and CdTe. v
v
I'15 Zn
v
L3 Se
Zn
X5 Se
Zn
Se
s Px py
t 6
t 84
t 15
t 79
Pz
J'
$
~
$
dxy dy z
t 9
T 1
T 5
T 1
* 25
*
c L1
F]
7*4 •
C X 1
Zn
Se
Zn
Se
37
63
49
34
1"
t
$
$
43
T 1
t 9
8
6
Zn
v F15
Se
Cd
v L3
Te
Cd
v X5
Te
Cd
C 1"1
Te
26
1
t 11 $
t 79 $
t 20 $
t 73 $
T 9
t 1
t 6
T 1
a*n + %v 6,9
+
c X 1
L~
Cd
Te
Cd
Te
Cd
41
59
46
25
t 14 ~
6
$
58
t 2
T 7
12
Te 22
1"
dx2_y2 "d3z2_ 1
23
8
205
Volume 114A, number 4
PHYSICS LETTERS
mately halfway along the bond. The appearance of a spherical s-like charge distribution about the anions is confirmed by the significant percentage decomposition of s-orbital character shown in table 1 * 1. We also find that there is more charge density centered around the anions and that this amount increases slightly as we go from CdTe to ZnSe, an observation which agrees with previous pseudopotential calculations in contrast to tight-binding predictions [6,7]. The situation at L in the valence band (L~), as shown in fig. 2, is similar to that at F~5, while the conduction band state has more of a p- and d-like character than the P~ state. We also find that while the percentage decomposition of p- and d-orbital character is quite similar in both semiconductors at L~, there is a significant difference in the amounts of s-, p- and d-orbital character at L~ in both compounds. A substantially different charge density distribution is observed at the X point in the conduction band. While the charge distribution at X in the valence band (X~) is similar to that at P[5, the charge density at X~ is much more uniformly spread throughout the unit cell with an appreciable enhancement in the interstitial region of the crystal along the (liO) plane. This behavior of the charge density at X~ has been seen and discussed in other calculations of charge distributions in group IV and I I I - V semiconductors in the context of interstitial impurities [8-11 ]. We note that because the angular momentum decomposition at X~ and X[ is quite similar in both compounds, one would expect the effect of an interstitial impurity on the electronic band structures of ZnSe and CdTe to be quite similar [8,9]. In conclusion, we have found that the charge density distribution at selected k points along the valence and conduction band edges is strongly dependent upon the symmetry of the wavefunction, ~Jn,~(r), and is quite similar in two prototypical I I - V I semiconductors. We expect that this sensitivity of the symmetry of a particular state will have a dominant influence on the relative weighting of such parameters as the pressure coefficients of direct and indirect band gaps in I I - V I semiconductors [12] and the intervalley electron-phonon matrix elements which depend on the :~i Strictly speaking, there is at least some f-character present in this state which we have not explicitly included in our angular momentum analysis. 206
17 February 1986
charge densities of the initial and final states of a particular transition [ 13]. We are grateful to Che-ting Chan, M.Y. Chou, Dr. Michel M. Dacorogna, Stephen Fahy, Professor Leo M. Falicov, and Mark S. Hybertsen for many helpful discussions. One of us (RMW) thanks the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico in Brazil for a Doctoral Fellowship, while (SRL) acknowledges the University of California at Berkeley for support through a Chancellor's Postdoctoral Fellowship for the 1983-85 academic years. Support for this work was provided by NSF Grant No. DMR8319024 and by the Director, Office of Energy Research, Office of Basic Energy Sciences, Material Science Division of the U.S. Department of Energy under Contract No. DE-ACO3-76SF00098.
References [1] J.R. Chelikowsky and M.L. Cohen, Phys. Rev. B 14 (1976) 556. [2] G. Weisz, Phys. Rev. 149 (1966) 504. [3] S. Bloom and T.K. Bargstresser,Solid State Commun. 6 (1968) 465. [4] C.T. Chan and S.G. Louie, Phys. Rev. B27 (1983) 3325; P.K. Lam and M.L. Cohen, Phys. Rev. B27 (1983) 5986. [5] F. Bassani, in: Semiconductors and semimetals, eds. R.K. Willardson and A.C. Beer (Academic Press, New York,
1966).
[6] T.P. Humphreys and G.P. Srivastava,Phys. Stat. Sol. 103 (1981) K85. [7] A. Baldereschi and K. Maschke, Inst. Phys. Conf. Ser. 43 (1979) 1167. [8] H.W.A. Rompa, M.F.H. Schurmans and F. Williams, Phys. Rev. Lett. 52 (1984) 675. [9] D.M. Wood, A. Zunger and R. de Groot, Phys. Rev. B31 (1985) 2570. [10] S.L. Richardson, M.L. Cohen, S.G. Louie and J.R. Chelikowsky, Phys. Rev. Lett. 54 (1985) 2549. [ 11 ] S.L. Richardson, M.L. Cohen, S.G. Louie and J.R. Chelikowski, to be published. [121 D.L. Camphausen, G.A.N. Connell and W. Paul, Phys. Rev. Lett. 26 (1971) 184; K.J. Chang, S. Froyen and M.L. Cohen, Solid State Commun. 50 (1984) 105; J.C. Phillips, Bonds and bands in semiconductors (Academic Press, New York, 1973). [13] S.L. Richardson, F.H. Pollak, M.L. Cohen, O. Glembocki and P. Parayanthal, to be published.
PHYSICS LETTERS
17 February 1986
E L E C T R O N I C C H A R G E D E N S I T I E S AT VALENCE AND C O N D U C T I O N BAND E D G E S OF ZnSe AND CdTe Renata M. W E N T Z C O V I T C H , Steven L. R I C H A R D S O N and Marvin L. C O H E N Department of Physics, University of California and Center for Advanced Materials, Lawrence Berkeley Laboratoo,. Berkeley, CA 94720, USA
Received 8 October 1985; revised manuscript received 27 November 1985; accepted for publication 2 December 1985
The empirical pseudopotential method (EPM) is used to calculate electronic charge densities at selected k points of the valence and conduction band edges of two II-VI semiconductors: ZnSe and CdTe.
In this paper we apply the empirical pseudopotential method (EPM) in the form used by Chelikowsky and Cohen [1 ] to compute the electronic charge densities at the F, L and X k points of the valence and conduction band edges of ZnSe and CdTe. These charge densities are decomposed into their angular momentum contributions and the observed differences in the bonding and antibonding behavior are discussed. The calculation was performed using the non-local EPM technique [1] with spin-orbit coupling included following the work of Weisz [2 ] with the modifications of Bloom and Bergstresser [3]. In this scheme, the wavefunctions of the one-electron hamiltonian are expanded in a set of 113 plane waves, while the local part of the pseudopotential is Fourier analyzed in terms of a small number of pseudopotential form factors, V(G), which have been empirically determined to reproduce experimental data such as band gaps and optical reflectivity. The non-local corrections to the pseudopotential were chosen to match those of ref. [1 ]. The secular equation for the EPM hamiltonian is solved to obtain wavefunctions and eigenvalues for the two compounds. The electronic charge density is calculated at selected k points in the Brillouin zone by taking the square of the wavefunctions at these points On,k(r) = I~n,k(r) 12 •
(1)
Finally, the w ave functions at a particular k point, ~bn,g(r), were decomposed into their angular momentum contributions"Xl,n,k, m rl" [4] 0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
R Xl,n,k,rj rn " = 4rt f 0
dr r2(~n,klYl, m ) (YI,m I~n,k) ,
(2)
where ¢1 is the vector position of the ]th atom, R is equal to half the interatomic distance, and Yl, m is the spherical harmonic function. To find the percentage decomposition of s-, p-, and d-character, we normalized this quantity with the total contribution for the other angular momentum components of both ions at a given band n and k point. For the case of degenerate bands, the percentage decomposition was obtained from the appropriate averages. The electronic charge densities for the F, L, and X k points of the valence and conduction band edges of ZnSe and CdTe are shown in figs. 1, 2, and 3, respectively. The percentage decomposition of angular momentum for each of these states was calculated from eq. (2) and the results are listed in table I. Note that we performed the angular momentum decomposition on wavefunctions unperturbed by spin-orbit interaction, and we observe that these results are consistent with group theoretical predictions [5]. The charge distributions at F for the highest valence band (F~5) are illustrated in fig. 1. They show a strong p-like contribution directed along the covalent bond between the anion and the cation of the two semiconductors. The charge density at F in the conduction band (F~) is, however, antibonding as indicated by the minimum of charge density approxi203
Cd
(c)
(d)
/4.~ < ~ - ~ ~ { ....
J , , , ~ > ~ ~ O,
:
zn
/
Fig. 1. Charge density contours for electronic states at the F point inthe (110) plane. Note that spin-orbit interaction has been neglected. (a) ZnSe at highest valence band, (b) CdTe at highest valence band, (c) ZnSe at lowest conduction band, (d) CdTe at lowest conduction band. The contour intervals are in units of 0.5 in cases (a) and (b) and 1.0 electrons per primitive cell in cases (c) and (d).
i
Cd
0
:1 i I x
j/
el
,"
f
I
/
•
',, '
I'I) i
~
~.~./
Fig. 2. Charge density contours for electronic states at the L point in the (110) plane. Note that spin-orbit interaction has been neglected. (a) ZnSe at highest valence band, (b) CdTe at highest valence band, (c) ZnSe at lowest conduction band, (d) CdTe at lowest conduction band. The contour intervals are in units of 0.5 electrons per primitive cell.
Volume 114A, number 4
PHYSICS LETTERS
\
~ ¢
,/
/ . "I
/
c
, , ,
\ ,
•
Zn
[//
~
17 February 1986
\
\\
\
,
_ .
.
.
.
.
.
- - -
.
~
,
~ ' J
-
.
/
•
~
Cd
"'~
,. O "
t ~ . ~
....
"
_
~-
(':m,\ a I#,~ _ / ./ .//./ /.
f
--
. n . c~~-
"--z
~
-
~ -
x;
/
(c)
\\ \ ,
. .
,\\\"\~\"Y"~"'/~ .'/I ," ,
,
\ 0 _ -.," S~ J / - /
/
/ ~ - ~ t ~
-
-
-
[
~
~
'.c~
e
, "~_//-~-
"\
~ . " ; ~
"'~
"\
-
I -1
X
Fig. 3. Charge density contours for electronic states at the X point, (27r/a)(001), in the (1]0) plane. Note that spin-orbit interaction has been neglected. (a) ZnSe at highest valence band, (b) CdTe at highest valence band, (c) ZnSe at lowest conduction band, (d) CdTe at lowest conduction band. The contour intervals are in units of 0.5 electrons per primitive cell.
Table 1 Percentage decomposition of angular momentum character for wavefunctions in the highest valence and first conduction bands of ZnSe and CdTe. v
v
I'15 Zn
v
L3 Se
Zn
X5 Se
Zn
Se
s Px py
t 6
t 84
t 15
t 79
Pz
J'
$
~
$
dxy dy z
t 9
T 1
T 5
T 1
* 25
*
c L1
F]
7*4 •
C X 1
Zn
Se
Zn
Se
37
63
49
34
1"
t
$
$
43
T 1
t 9
8
6
Zn
v F15
Se
Cd
v L3
Te
Cd
v X5
Te
Cd
C 1"1
Te
26
1
t 11 $
t 79 $
t 20 $
t 73 $
T 9
t 1
t 6
T 1
a*n + %v 6,9
+
c X 1
L~
Cd
Te
Cd
Te
Cd
41
59
46
25
t 14 ~
6
$
58
t 2
T 7
12
Te 22
1"
dx2_y2 "d3z2_ 1
23
8
205
Volume 114A, number 4
PHYSICS LETTERS
mately halfway along the bond. The appearance of a spherical s-like charge distribution about the anions is confirmed by the significant percentage decomposition of s-orbital character shown in table 1 * 1. We also find that there is more charge density centered around the anions and that this amount increases slightly as we go from CdTe to ZnSe, an observation which agrees with previous pseudopotential calculations in contrast to tight-binding predictions [6,7]. The situation at L in the valence band (L~), as shown in fig. 2, is similar to that at F~5, while the conduction band state has more of a p- and d-like character than the P~ state. We also find that while the percentage decomposition of p- and d-orbital character is quite similar in both semiconductors at L~, there is a significant difference in the amounts of s-, p- and d-orbital character at L~ in both compounds. A substantially different charge density distribution is observed at the X point in the conduction band. While the charge distribution at X in the valence band (X~) is similar to that at P[5, the charge density at X~ is much more uniformly spread throughout the unit cell with an appreciable enhancement in the interstitial region of the crystal along the (liO) plane. This behavior of the charge density at X~ has been seen and discussed in other calculations of charge distributions in group IV and I I I - V semiconductors in the context of interstitial impurities [8-11 ]. We note that because the angular momentum decomposition at X~ and X[ is quite similar in both compounds, one would expect the effect of an interstitial impurity on the electronic band structures of ZnSe and CdTe to be quite similar [8,9]. In conclusion, we have found that the charge density distribution at selected k points along the valence and conduction band edges is strongly dependent upon the symmetry of the wavefunction, ~Jn,~(r), and is quite similar in two prototypical I I - V I semiconductors. We expect that this sensitivity of the symmetry of a particular state will have a dominant influence on the relative weighting of such parameters as the pressure coefficients of direct and indirect band gaps in I I - V I semiconductors [12] and the intervalley electron-phonon matrix elements which depend on the :~i Strictly speaking, there is at least some f-character present in this state which we have not explicitly included in our angular momentum analysis. 206
17 February 1986
charge densities of the initial and final states of a particular transition [ 13]. We are grateful to Che-ting Chan, M.Y. Chou, Dr. Michel M. Dacorogna, Stephen Fahy, Professor Leo M. Falicov, and Mark S. Hybertsen for many helpful discussions. One of us (RMW) thanks the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico in Brazil for a Doctoral Fellowship, while (SRL) acknowledges the University of California at Berkeley for support through a Chancellor's Postdoctoral Fellowship for the 1983-85 academic years. Support for this work was provided by NSF Grant No. DMR8319024 and by the Director, Office of Energy Research, Office of Basic Energy Sciences, Material Science Division of the U.S. Department of Energy under Contract No. DE-ACO3-76SF00098.
References [1] J.R. Chelikowsky and M.L. Cohen, Phys. Rev. B 14 (1976) 556. [2] G. Weisz, Phys. Rev. 149 (1966) 504. [3] S. Bloom and T.K. Bargstresser,Solid State Commun. 6 (1968) 465. [4] C.T. Chan and S.G. Louie, Phys. Rev. B27 (1983) 3325; P.K. Lam and M.L. Cohen, Phys. Rev. B27 (1983) 5986. [5] F. Bassani, in: Semiconductors and semimetals, eds. R.K. Willardson and A.C. Beer (Academic Press, New York,
1966).
[6] T.P. Humphreys and G.P. Srivastava,Phys. Stat. Sol. 103 (1981) K85. [7] A. Baldereschi and K. Maschke, Inst. Phys. Conf. Ser. 43 (1979) 1167. [8] H.W.A. Rompa, M.F.H. Schurmans and F. Williams, Phys. Rev. Lett. 52 (1984) 675. [9] D.M. Wood, A. Zunger and R. de Groot, Phys. Rev. B31 (1985) 2570. [10] S.L. Richardson, M.L. Cohen, S.G. Louie and J.R. Chelikowsky, Phys. Rev. Lett. 54 (1985) 2549. [ 11 ] S.L. Richardson, M.L. Cohen, S.G. Louie and J.R. Chelikowski, to be published. [121 D.L. Camphausen, G.A.N. Connell and W. Paul, Phys. Rev. Lett. 26 (1971) 184; K.J. Chang, S. Froyen and M.L. Cohen, Solid State Commun. 50 (1984) 105; J.C. Phillips, Bonds and bands in semiconductors (Academic Press, New York, 1973). [13] S.L. Richardson, F.H. Pollak, M.L. Cohen, O. Glembocki and P. Parayanthal, to be published.