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Angular momentum decomposition of k = 0 Bloch functions in group IV and zincblende crystals
Solid State Communications, Vol. 20, pp. 361-364, 1976.
Pergamon Press.
Printed in Great Britain
ANGULAR MOMENTUM DECOMPOSITION OF k = 0 BLOCH FUNCTIONS IN GROUP IV AND ZINCBLENDE CRYSTALS D.J. Chadi Xerox Palo Alto Research Center, Palo Alto, CA 94304, U.S.A. (Received 13 May 1976 by A.A. Maradudin) The angular momentum contents of k = 0 Bloch functions for a number of group IV and zincblende crystals are examined using a pseudopotential approach. The large amplitudes for d and )' character functions found even for valence electron wave functions are related to the overlap o f p and s symmetry states on different atoms. In zincblende crystals the strength of the d and f components increases around cations and decreases around anions for the valence bands. The f-like component is appreciable for the Vt conduction band particularly around a cation. 1. INTRODUCTION
xP(r) = ~ a(G) exp (iG.r)
A BLOCH FUNCTION ~k with wavevector k along a symmetry direction or at a special point of the Brfllouin zone is usually labeled according to its transformation properties under a group of crystalline symmetry operations3 A knowledge of the symmetry character of ~k can often be very useful in determining selection rules for various transitions. It can also be helpful in providing a connection between high symmetry (usually zone center) states in the crystals with atomic states from which they originate. For example, the lowest valence state at k = 0 in groups IV and zincblende crystals denoted by I'tv is associated with atomic s-states. Even though it is necessary to start with atomic s-states to get a I'~ state it is common to refer to this state as being s-like. This is because a Pt symmetry function remains unchanged under all the symmetry operations of the crystal even though it can be expressed as a superposition of different angular momentum states. In this paper we look at the character of k = 0 Bloch functions, as seen from an atomic site in diamond and zincblende structure crystals. This provides some interesting results which are useful in nonlocal, i.e. angular momentum dependent, pseudopotential calculations 2 and also for the choice of appropriate basis functions for tight-binding calculations, a The wavefunctions for the valence and the four lowest conduction bands are given in Section 2 and discussed in Section 3. 2. WAVEFUNCTIONS AT P To obtain the wavefunctions at k = 0 we use the empirical pseudopotential method ='t and expand the wavefunctions in a plane-wave basis:
(1)
For mnplicRy we use 27 plane-waves in the expansion. These include the (000), (111), (200) and (220) group of reciprocal lattice vectors. Symmetrized combinations of plane-waves can be used to reduce the size of the secular matrix to, at most, 5 x 5. 5 Spin-orbit interactions are not included in the calculations. Defining q = 2n/a
(2)
where a is the cubic lattice constant the I't and Fls valence and lowest conduction bands in zincblende crystals are given by ~t'(I"t) = ao + (at + a2) cos qx cos qy cos qz + ( a l - - a 2 ) s i n qx sine)' sinqz + a3(cos 2qx + cos 2qy + cos 2qz)
(3)
+ a4(cos 2qx cos 2qy + cos 2qx cos 2qz + cos 2qy cos 2qz) and
% ( r ' t s ) = ( ~ + ~2) cos qx cos qy sin qz + ( ~ - Jh) sin qx ~ qy cos qz + 1~3sin 2qz + ~4 sin 2qx sin 2qy (4) + i~s(COS2qx + cos 2qy) fm 2qz. The atoms in the basis are taken to be: anion at (0, O, O) and cation at 1/4(1, 1, 1),, Two other I'ts functions degenerate with ~'z(l"ts) are easily obtained through a permutation of coordinates in equation (4). Equations (3) and (4) also describe the valence and lowest conduction bands at k = 0 in diamond structure crystals provided the following parameters are set equal to zero: c~2 and a3 for the Ply state; ao, % and c~ for the Pa'e state;/h, and ~s for the Fz~'v state; and/31,/33 and/34 for the I'tsc state, where v and c denote valence and
361
362
BLOCH FUNCTIONS IN GROUP IV AND ZINCBLENDE CRYSTALS
Vol. 20, No. 4
Table 1. Expansion coefJ~cients for F l valence state (top number)and lowest F I (or Fa' ) conduc~ tion state (bottom number) for a number of group I V and zincblende crystals Expansion coefficients Compound
ao 0.956 0.0 0.938 0.0 0.897 0.355 0.911 0.327 0.839 0.529 0.886 - 0.301 0.822 -- 0.501
S i
Ge GaAs
- -
GaP
-
ZnSe
-
InSb CdTe
al
a2
a3
0.582 0.0 0.697 0.0 0.727 0.540 0.689 0.558 0.754 0.847 0.803 0.243 0.837 0A31
0.0 1.756 0.0 1.638 0.371 1.498 0_354 1.506 0.585 1.233 0.345 1.517 0.569 1.372
0.0 0.390 0.0 0.469 0.133 0.390 0.115 0.396 0.206 0.308 0.122 0A54 0.205 0.379
a4 -- 0.037 0.0 0.001 0.0 0.040 0.127 0.015 0.I 25 0.070 0.156 0.049 0.115 0.089 0.149
Table 2. Expansion coefficients for ['is (or F2s' ) valence state (top number)and lowest I'ls conduction state (bottom number) for a number of group I V and zincblende crystals Expansion coefficients Compound
/~I 1.715 0.0 1.704 0.0 1.578 0.592 1.575 0.644 1.455 0.877 1.595 0.494 1.481 - 0.749
Si Ge GaAs
- -
GaP
- -
ZnSe
- -
InSb
- -
CdTe
/~2 0.0 1.979 0.0 1.967 0.607 1.869 0.625 1.858 0.841 1.754 0.506 1.887 0.735 1.795
conduction bands. The values of the parameters appearing in equations (3) and (4) are shown in Tables 1 and 2 for a number of crystals. The normalization of the wavefunctions in equations (3) and (4) is given by:
! fZ
f qs'xPdr = 1
(5)
12
where [2 is the volume of the crystal or that of a unit cell. The pseudopotential form factors used in the calculations were obtained from reference 6. The use of form factors obtained in reference 4 lead to very similar results for the expansion coefficients.
/3a
/~4
0.699 0.0 0.717 0.0 0.718 -- 0.153 0.716 -- 0.107 0.716 -- 0.139 0.739 -- 0.136 0.745 - 0.159
0.287 0.0 0.259 0.0 0.261 -- 0.163 0.250 -- 0.199 0.218 -- 0.289 0.271 -- 0.135 0.254 -- 0.266
/3s 0.0 0.205 0.0 0.256 0.148 0.204 0.147 0.191 0.226 0.133 0.134 0.264 0.216 0.218
3. DISCUSSION 3.1 r l States: s and f character Let us consider the valence and conduction states at k = 0 starting with the 1~1 states. The various combinations of the cosine terms in equation (3) are s-like in character and for small r they have a c~ -- c2r 2 dependence on r. The sine term in equation (3) is, however, f-like in character and is proportional to xyz for small r. Table 1 shows that the amplitude of this term relative to the other terms is appreciable, particularly for the P~ conduction state.
Vol. 20, No. 4
The f-like character of the Pl band may, at f'LrSt sight, appear somewhat surprising. The valence and the lowest conduction bands in the diamond and zincblende structure crystals originate from bonding and antibonding combinations of the s and p levels of the constituent atoms. The f-like character of the 1"1 state can, however, be explained in a simple way. To do this we write the sine term in equation (3) as sm qx sin qy fm qz =
363
BLOCH FUNCTIONS IN GROUP IV AND ZINCBLENDE CRYSTALS
cos q ( x --xo) cos q ( y --Yo) cos q(z -- Zo) (6)
where x0 --Yo = Zo = a/4 and qxo = 7r/2 from equation (2). Therefore, the f-like function centered on the atom at (0, O, 0) is equivalent to an s-like function centered on the nearest-neighbor atom at (xe,Yo, zo). In a tight. binding picture the 1"1 state can, therefore, be regarded as being formed from a superposition of s-like functions centered on each atom in the crystal. It is when the s-function is seen from an origin displaced from its center of symmetry that higher angular momentum components appear. It is also interesting to study the change in the character of the wavefunctions as we go from a completely covalent crystal such as Ge to partially ionic crystals such as GaAs or ZnSe. In Ge aU atoms in the crystal are equivalent. The character of the wavefunction is therefore unchanged when the origin is moved from one atom to another. This can be easily verified using equation (3) (remembering that some of the parameters are zero by symmetry, as shown in Table I, for group IV crystals). In zincblende crystals the character of the wavefunction is affected when the origin is shifted from an anion to a cation. The parameters in Tables I and 2 describe the wavefunctions for the choice of the origin at an anion site. Compared to Ge, the 1"1vvalence and 1"1c conduction states in GaAs and ZnSe are more s and less f-like around the anion whereas the f-like character of the wavefunction is larger at the cation. The large f-like character of the 1"1 wavefunctions in diamond and zincblende crystals raises the question of the adequacy of a purely local, angular-momentum independent, pseudopotential. Since the atomic cores in these crystals (except those containing Hg) contains no f-electrons there is no cancellation of the attractive potential experienced by the FI electrons. The f-component of the wavefunctions should, therfore, experience a stronger attractive potential than the s-component for example. However, since an f-state has an • 3 dependence for small • a non-local f-potential in the core region, even though it may be appreciable in magnitude, is expected z to have only a small effect on the energies and wavefunctions. This is born out by nonlocal calculations which we have carried out for Ge, GaAs and ZnSe. A nonlocal potential of the type
(nonlocal) -- - ~ v3P3
for • • R
(7) = 0
for r ~ R
where P3 is a projection operator acting only on functions with f-symmetry and P3 is a constant were used in the calculations. For nonlocal potentials placed on anions, cations or on both, with v3 ~-- 1 Ry and touching sphere radii the changes in energies were a few hundredths of an eV. It should be noted that a nonlocal f-potential affects not only those states with r l (or 1"2') symmetry but those with 1"1s (or 1"2s') symmetry as well. 7 The effect of the f-potential on the triply degenerate states is also very small. The neglect of the ! ~ 3 component of the pseudopotential is therefore justified for valence and conduction bands in diamond and zincblende crystals. The mixing between s and f states can also be explained through a nonvanishing pseudopotential matrix element between these states. The local pseudopotential V(r) has the symmetry of a PI function and its expansion around an atomic site has the same form as that given by equation (3). This shows that V(r) itself has s and f-like characters about each atom. The t:like character arises from G - (I 11) and (31 I) (and higher) Fourier components of V(r). The fcomponent of V(r) has a strong influence on the character of the wavefunctions. It causes the mixing between s and fstates (also between p and d states) in group IV and zincblende crystals. The effect of this mixing on the wavefunctions and energy levels is particularly strong in diamonds a where the (111) Fourier component of V(r) is very negative. 3.2 Fls States: p and d character The approximate • dependence of valence and lowest conduction bands with 1,1s (or 1,as') symmetries is given by equation (4). The first term in equation (4) has Pz symmetry around the origin whereas the second term has dxy type symmetry. In analogy to the discussion of the mixing between s and f-states given above for the 1"1 states we may consider the dx:~ component as being equivalent to the overlap of a Pz function centered on a nearest-neighbor atom. In this way we can attribute a large part of the d character of the valence and conduction bands in tetrahedral semiconductors as arising from the overlap o f p functions. Another term with dxy symmetry in equation (4) is the one with coefficient 134.This term has dxy symmetry on all atoms in the crystal and represents in a sense the admixture of a "bona fide" dstate with the valence and conduction bands. The mixing between Pz and dxy symmetry functions can also be attributed to the fx~z component of the pseudopotential. Table 2 shows that as a crystal becomes more ionic
364
BLOCH FUNCTIONS IN GROUP IV AND ZINCBLENDE CRYSTALS
the Fls valence state becomes more p-like around the anion and more d-like around the cation. This state is, therefore, expected to be more sensitive to nonlocal 1 = 2 dependent potentials placed on cations than equivalent ones on anions. Nonlocal calculations confirm this s even though the electronic charge is concentrated more heavily on the anions. 9 The opposite trends for the character of the wavefunction hold for the F~s condunction state. 4. CONCLUSION We have examined the symmetry of the valence and lowest conduction bands at k = 0 in a number of group IV and zincblende crystals. Our analysis shows that the d and f-like character of the wavefunctions can be viewed, either in a tight-binding picture, as arising from the overlap of s and p orbitals centered on nearestneighbor atoms, or alternatively as arising from the flike component of the local pseudopotential. The fcharacter of the wavefunctions is most pro-
Vol. 20, No. 4
nounced for the F1 (or F2') conduction state. This aspect of the conduction wavefunction should be considered in the interpretation l ° ' n of experimental data from electron-energy-loss-spectroscopyn - ~ and photoelectron-yield-spectroscopy14 regarding the symmetry of empty surface states in Ge and GaAs. In these experiments the 3d core electrons of Ge and Ga were excited to empty surface states. The results have been analyzed l ° ' u by assuming the empty states to be predominantly s or p-like in character and neglecting the partial d or fcharacter of the wavefunctions. This assumption may be inadequate in view of the large mixing of d and f states with p and s states in the bulk. Tight-binding calculations for a number of diamond and zincblende structure crystals wluch make use of the results of the preceding sections will be reported elsewhere.a
Acknowledgement - I would like to thank R.M. Martin for useful discussions.
REFERENCES 1.
KOSTER G.F., Solid State Phys. (Edited by SEITZ F. & TURNBULL D.), Vol. 5, p. 173. Academic Press, New York 0957); BELL D.G., Rev. Mod. Phys. 26, 311 (1954).
2.
COHEN M.L. & HEINE V., Solid State Phys. (Edited by SEITZ F. & TURNBULL D.), Vol. 24, p. 37. Academic Press, New York.
3.
CHADI D.J. (to be published).
4.
COHEN M.L. & BERGSTRESSER T.K., Phys. Rev. 141,789 (1966).
5.
See, for example, the article by BASSANI F. in Semiconductors and Semimetals (Edited by WILLARDSON R.K. & BEER A.C.), Vol. l, pp. 32-39. Academic Press, New York (1966).
6.
CHELtKOWSKY J.R., CHADI DJ. & COHEN M.L., inPhys. Rev. B8 2786 0973).
7.
KANE E.O., in reference 5, pp. 84-85.
8.
CHADI D.J. (unpublished); CHELIKOWSKY J.R., Phys. Rev. BI4 (to be published).
9.
WALTER J.P. & COHEN M.L., Phys. Rev. B4, 1877 (1971).
10.
FREEOUF J.L.,Phys. Rev. Lett. 36, 1095 0976).
I 1.
LUDEKE R. & KOMA A., Phys. Rev. Lett. 34, 817 0975).
12.
LUDEKE R. & ESAKI L.,Phys. Rev. Lett. 33, 653 (1974).
13.
ROWE J.E. & IBACH H., Phys. Rev. Lett. 31,102 (1973).
14.
EASTMAN D.E. & FREEOUF J.L.,Phys. Rev. Lett. 33, 1601 0 9 7 4 ) ; ibid. 34, 1624 (1975).
Pergamon Press.
Printed in Great Britain
ANGULAR MOMENTUM DECOMPOSITION OF k = 0 BLOCH FUNCTIONS IN GROUP IV AND ZINCBLENDE CRYSTALS D.J. Chadi Xerox Palo Alto Research Center, Palo Alto, CA 94304, U.S.A. (Received 13 May 1976 by A.A. Maradudin) The angular momentum contents of k = 0 Bloch functions for a number of group IV and zincblende crystals are examined using a pseudopotential approach. The large amplitudes for d and )' character functions found even for valence electron wave functions are related to the overlap o f p and s symmetry states on different atoms. In zincblende crystals the strength of the d and f components increases around cations and decreases around anions for the valence bands. The f-like component is appreciable for the Vt conduction band particularly around a cation. 1. INTRODUCTION
xP(r) = ~ a(G) exp (iG.r)
A BLOCH FUNCTION ~k with wavevector k along a symmetry direction or at a special point of the Brfllouin zone is usually labeled according to its transformation properties under a group of crystalline symmetry operations3 A knowledge of the symmetry character of ~k can often be very useful in determining selection rules for various transitions. It can also be helpful in providing a connection between high symmetry (usually zone center) states in the crystals with atomic states from which they originate. For example, the lowest valence state at k = 0 in groups IV and zincblende crystals denoted by I'tv is associated with atomic s-states. Even though it is necessary to start with atomic s-states to get a I'~ state it is common to refer to this state as being s-like. This is because a Pt symmetry function remains unchanged under all the symmetry operations of the crystal even though it can be expressed as a superposition of different angular momentum states. In this paper we look at the character of k = 0 Bloch functions, as seen from an atomic site in diamond and zincblende structure crystals. This provides some interesting results which are useful in nonlocal, i.e. angular momentum dependent, pseudopotential calculations 2 and also for the choice of appropriate basis functions for tight-binding calculations, a The wavefunctions for the valence and the four lowest conduction bands are given in Section 2 and discussed in Section 3. 2. WAVEFUNCTIONS AT P To obtain the wavefunctions at k = 0 we use the empirical pseudopotential method ='t and expand the wavefunctions in a plane-wave basis:
(1)
For mnplicRy we use 27 plane-waves in the expansion. These include the (000), (111), (200) and (220) group of reciprocal lattice vectors. Symmetrized combinations of plane-waves can be used to reduce the size of the secular matrix to, at most, 5 x 5. 5 Spin-orbit interactions are not included in the calculations. Defining q = 2n/a
(2)
where a is the cubic lattice constant the I't and Fls valence and lowest conduction bands in zincblende crystals are given by ~t'(I"t) = ao + (at + a2) cos qx cos qy cos qz + ( a l - - a 2 ) s i n qx sine)' sinqz + a3(cos 2qx + cos 2qy + cos 2qz)
(3)
+ a4(cos 2qx cos 2qy + cos 2qx cos 2qz + cos 2qy cos 2qz) and
% ( r ' t s ) = ( ~ + ~2) cos qx cos qy sin qz + ( ~ - Jh) sin qx ~ qy cos qz + 1~3sin 2qz + ~4 sin 2qx sin 2qy (4) + i~s(COS2qx + cos 2qy) fm 2qz. The atoms in the basis are taken to be: anion at (0, O, O) and cation at 1/4(1, 1, 1),, Two other I'ts functions degenerate with ~'z(l"ts) are easily obtained through a permutation of coordinates in equation (4). Equations (3) and (4) also describe the valence and lowest conduction bands at k = 0 in diamond structure crystals provided the following parameters are set equal to zero: c~2 and a3 for the Ply state; ao, % and c~ for the Pa'e state;/h, and ~s for the Fz~'v state; and/31,/33 and/34 for the I'tsc state, where v and c denote valence and
361
362
BLOCH FUNCTIONS IN GROUP IV AND ZINCBLENDE CRYSTALS
Vol. 20, No. 4
Table 1. Expansion coefJ~cients for F l valence state (top number)and lowest F I (or Fa' ) conduc~ tion state (bottom number) for a number of group I V and zincblende crystals Expansion coefficients Compound
ao 0.956 0.0 0.938 0.0 0.897 0.355 0.911 0.327 0.839 0.529 0.886 - 0.301 0.822 -- 0.501
S i
Ge GaAs
- -
GaP
-
ZnSe
-
InSb CdTe
al
a2
a3
0.582 0.0 0.697 0.0 0.727 0.540 0.689 0.558 0.754 0.847 0.803 0.243 0.837 0A31
0.0 1.756 0.0 1.638 0.371 1.498 0_354 1.506 0.585 1.233 0.345 1.517 0.569 1.372
0.0 0.390 0.0 0.469 0.133 0.390 0.115 0.396 0.206 0.308 0.122 0A54 0.205 0.379
a4 -- 0.037 0.0 0.001 0.0 0.040 0.127 0.015 0.I 25 0.070 0.156 0.049 0.115 0.089 0.149
Table 2. Expansion coefficients for ['is (or F2s' ) valence state (top number)and lowest I'ls conduction state (bottom number) for a number of group I V and zincblende crystals Expansion coefficients Compound
/~I 1.715 0.0 1.704 0.0 1.578 0.592 1.575 0.644 1.455 0.877 1.595 0.494 1.481 - 0.749
Si Ge GaAs
- -
GaP
- -
ZnSe
- -
InSb
- -
CdTe
/~2 0.0 1.979 0.0 1.967 0.607 1.869 0.625 1.858 0.841 1.754 0.506 1.887 0.735 1.795
conduction bands. The values of the parameters appearing in equations (3) and (4) are shown in Tables 1 and 2 for a number of crystals. The normalization of the wavefunctions in equations (3) and (4) is given by:
! fZ
f qs'xPdr = 1
(5)
12
where [2 is the volume of the crystal or that of a unit cell. The pseudopotential form factors used in the calculations were obtained from reference 6. The use of form factors obtained in reference 4 lead to very similar results for the expansion coefficients.
/3a
/~4
0.699 0.0 0.717 0.0 0.718 -- 0.153 0.716 -- 0.107 0.716 -- 0.139 0.739 -- 0.136 0.745 - 0.159
0.287 0.0 0.259 0.0 0.261 -- 0.163 0.250 -- 0.199 0.218 -- 0.289 0.271 -- 0.135 0.254 -- 0.266
/3s 0.0 0.205 0.0 0.256 0.148 0.204 0.147 0.191 0.226 0.133 0.134 0.264 0.216 0.218
3. DISCUSSION 3.1 r l States: s and f character Let us consider the valence and conduction states at k = 0 starting with the 1~1 states. The various combinations of the cosine terms in equation (3) are s-like in character and for small r they have a c~ -- c2r 2 dependence on r. The sine term in equation (3) is, however, f-like in character and is proportional to xyz for small r. Table 1 shows that the amplitude of this term relative to the other terms is appreciable, particularly for the P~ conduction state.
Vol. 20, No. 4
The f-like character of the Pl band may, at f'LrSt sight, appear somewhat surprising. The valence and the lowest conduction bands in the diamond and zincblende structure crystals originate from bonding and antibonding combinations of the s and p levels of the constituent atoms. The f-like character of the 1"1 state can, however, be explained in a simple way. To do this we write the sine term in equation (3) as sm qx sin qy fm qz =
363
BLOCH FUNCTIONS IN GROUP IV AND ZINCBLENDE CRYSTALS
cos q ( x --xo) cos q ( y --Yo) cos q(z -- Zo) (6)
where x0 --Yo = Zo = a/4 and qxo = 7r/2 from equation (2). Therefore, the f-like function centered on the atom at (0, O, 0) is equivalent to an s-like function centered on the nearest-neighbor atom at (xe,Yo, zo). In a tight. binding picture the 1"1 state can, therefore, be regarded as being formed from a superposition of s-like functions centered on each atom in the crystal. It is when the s-function is seen from an origin displaced from its center of symmetry that higher angular momentum components appear. It is also interesting to study the change in the character of the wavefunctions as we go from a completely covalent crystal such as Ge to partially ionic crystals such as GaAs or ZnSe. In Ge aU atoms in the crystal are equivalent. The character of the wavefunction is therefore unchanged when the origin is moved from one atom to another. This can be easily verified using equation (3) (remembering that some of the parameters are zero by symmetry, as shown in Table I, for group IV crystals). In zincblende crystals the character of the wavefunction is affected when the origin is shifted from an anion to a cation. The parameters in Tables I and 2 describe the wavefunctions for the choice of the origin at an anion site. Compared to Ge, the 1"1vvalence and 1"1c conduction states in GaAs and ZnSe are more s and less f-like around the anion whereas the f-like character of the wavefunction is larger at the cation. The large f-like character of the 1"1 wavefunctions in diamond and zincblende crystals raises the question of the adequacy of a purely local, angular-momentum independent, pseudopotential. Since the atomic cores in these crystals (except those containing Hg) contains no f-electrons there is no cancellation of the attractive potential experienced by the FI electrons. The f-component of the wavefunctions should, therfore, experience a stronger attractive potential than the s-component for example. However, since an f-state has an • 3 dependence for small • a non-local f-potential in the core region, even though it may be appreciable in magnitude, is expected z to have only a small effect on the energies and wavefunctions. This is born out by nonlocal calculations which we have carried out for Ge, GaAs and ZnSe. A nonlocal potential of the type
(nonlocal) -- - ~ v3P3
for • • R
(7) = 0
for r ~ R
where P3 is a projection operator acting only on functions with f-symmetry and P3 is a constant were used in the calculations. For nonlocal potentials placed on anions, cations or on both, with v3 ~-- 1 Ry and touching sphere radii the changes in energies were a few hundredths of an eV. It should be noted that a nonlocal f-potential affects not only those states with r l (or 1"2') symmetry but those with 1"1s (or 1"2s') symmetry as well. 7 The effect of the f-potential on the triply degenerate states is also very small. The neglect of the ! ~ 3 component of the pseudopotential is therefore justified for valence and conduction bands in diamond and zincblende crystals. The mixing between s and f states can also be explained through a nonvanishing pseudopotential matrix element between these states. The local pseudopotential V(r) has the symmetry of a PI function and its expansion around an atomic site has the same form as that given by equation (3). This shows that V(r) itself has s and f-like characters about each atom. The t:like character arises from G - (I 11) and (31 I) (and higher) Fourier components of V(r). The fcomponent of V(r) has a strong influence on the character of the wavefunctions. It causes the mixing between s and fstates (also between p and d states) in group IV and zincblende crystals. The effect of this mixing on the wavefunctions and energy levels is particularly strong in diamonds a where the (111) Fourier component of V(r) is very negative. 3.2 Fls States: p and d character The approximate • dependence of valence and lowest conduction bands with 1,1s (or 1,as') symmetries is given by equation (4). The first term in equation (4) has Pz symmetry around the origin whereas the second term has dxy type symmetry. In analogy to the discussion of the mixing between s and f-states given above for the 1"1 states we may consider the dx:~ component as being equivalent to the overlap of a Pz function centered on a nearest-neighbor atom. In this way we can attribute a large part of the d character of the valence and conduction bands in tetrahedral semiconductors as arising from the overlap o f p functions. Another term with dxy symmetry in equation (4) is the one with coefficient 134.This term has dxy symmetry on all atoms in the crystal and represents in a sense the admixture of a "bona fide" dstate with the valence and conduction bands. The mixing between Pz and dxy symmetry functions can also be attributed to the fx~z component of the pseudopotential. Table 2 shows that as a crystal becomes more ionic
364
BLOCH FUNCTIONS IN GROUP IV AND ZINCBLENDE CRYSTALS
the Fls valence state becomes more p-like around the anion and more d-like around the cation. This state is, therefore, expected to be more sensitive to nonlocal 1 = 2 dependent potentials placed on cations than equivalent ones on anions. Nonlocal calculations confirm this s even though the electronic charge is concentrated more heavily on the anions. 9 The opposite trends for the character of the wavefunction hold for the F~s condunction state. 4. CONCLUSION We have examined the symmetry of the valence and lowest conduction bands at k = 0 in a number of group IV and zincblende crystals. Our analysis shows that the d and f-like character of the wavefunctions can be viewed, either in a tight-binding picture, as arising from the overlap of s and p orbitals centered on nearestneighbor atoms, or alternatively as arising from the flike component of the local pseudopotential. The fcharacter of the wavefunctions is most pro-
Vol. 20, No. 4
nounced for the F1 (or F2') conduction state. This aspect of the conduction wavefunction should be considered in the interpretation l ° ' n of experimental data from electron-energy-loss-spectroscopyn - ~ and photoelectron-yield-spectroscopy14 regarding the symmetry of empty surface states in Ge and GaAs. In these experiments the 3d core electrons of Ge and Ga were excited to empty surface states. The results have been analyzed l ° ' u by assuming the empty states to be predominantly s or p-like in character and neglecting the partial d or fcharacter of the wavefunctions. This assumption may be inadequate in view of the large mixing of d and f states with p and s states in the bulk. Tight-binding calculations for a number of diamond and zincblende structure crystals wluch make use of the results of the preceding sections will be reported elsewhere.a
Acknowledgement - I would like to thank R.M. Martin for useful discussions.
REFERENCES 1.
KOSTER G.F., Solid State Phys. (Edited by SEITZ F. & TURNBULL D.), Vol. 5, p. 173. Academic Press, New York 0957); BELL D.G., Rev. Mod. Phys. 26, 311 (1954).
2.
COHEN M.L. & HEINE V., Solid State Phys. (Edited by SEITZ F. & TURNBULL D.), Vol. 24, p. 37. Academic Press, New York.
3.
CHADI D.J. (to be published).
4.
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