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Simple model of surface and chemisorption states on the (111) plane of ZnS type crystals
SURFACE
SCIENCE
SIMPLE
MODEL
3 (1965) 333-347 o North-Holland
OF SURFACE
ON THE (111) PLANE J. KOUTECKP Institute
of Physical
Publishing
Co., Amsterdam
AND CHEMISORPTION
STATES
OF ZnS TYPE CRYSTALS and M. TOMASEK
Chemistry of the Czechoslovak Prague, Czechoslovakia
Academv
of Sciences,
Received 18 May 1965 The surface and chemisorption states of a simple model of a semi-infinite ZnS type (sphalerite) crystal, limited by the (111) plane were studied. The existence conditions of various states localised near the surface were studied and the classification of these states was carried out. The differences between characteristics of surfaces which are formed by more electronegative atoms and surfaces formed by more electropositive atoms were pointed out. In the case of more electronegative surfaces the energy band of Shockley surface states is placed closely above the top of the highest valence band, the formation of Tamm surface states with the energies below the valence band is facilitated and the formation of Tamm surface states with the energies above the conductivity bands becomes more difficult. As far as the more electropositive surface is concerned, the energy band of Shockley states is placed near the bottom of the conductivity band, the formation of Tamm states with energies below the lowest valence band is made more difficult and the formation of Tamm states with the energies above the highest conductivity band facilitated. In the gaps between individual valence or conductivity energy bands of volume states, further energy bands of surface states of a new kind can originate. The earlier studied model of the diamond lattice is a special case of the model investigated in this work. From the studied model follows, that a semi-infinite crystal can become a degenerated p- or n-semi-conductor if, in the forbidden energy gap, Shockley surface states do not exist at least for one of the two possible (111) delimiting planes.
1. Introduction Recently several very simple models of finite crystals were calculated from the viewpoint of the origin of electron surface states l). Special attention was paid to surface and chemisorption states of diamond-like latticess-5). The considered models envisaged that the covalent character of bonds between neighbouring atoms was especially expressed. A generalization of similar calculations for the ZnS lattice should be carried out, in which the covalent character of bonds are also predominant. Intermetallic compounds, such as InSb, GaSb, InAs, GaAs which are of considerable importance in semiconductor physics, as well as interesting compounds of BN, Sic, etc. types belong to this group. In this work a very simplified semi-infinite model of the ZnS type (sphalerite) 333
334
.I. KOUTECKi’
AND
M. TOMASEK
lattice limited by the (111) plane is considered. Surface and chemisorption one-electron states localized in the proximity of this limiting plane are studied. The tight binding approximation is used and similar assumptions are made as in preceding papers. The only difference as compared with the preceding work lies in the fact that in every elementary cell of the “diamond” lattice two atoms of different types are considered. At the same time the nearest neighbours of an A type atom are B type atoms and vice versa. In the study of the considered model of a binary crystal, emphasis is again placed on general conclusions. The model is too simplified and therefore considerably distant from reality. A further question is whether the ideal (111) plane is in general sufficiently stable in structures of this type. 2. Model For each of the A type atoms and B type atoms we consider four equivalent localized orbitals - sp3 hybrides (cf. ref.2)). The axes of maximum density of atomic orbitals of the given atom are orientated in the direction of the line connecting this atom with its neighbours. The resonance integrals between hybrids belonging to different atoms and lying along these connecting lines will be denoted as y. Resonance integrals between hybrids belonging to the same atom are further considered. Because there are two types of atoms, we distinguish two resonance integrals of this type, yA and yB, according to the hybrid orbitals being localized on atom A or B. Similarly, we have to distinguish Coulomb integrals txA and cla of hybrid atomic orbitals of different kinds. The limiting plane (111) is selected so that it cuts only one single bond of the surface atom (cf. fig. 1). We assume that the surface atom is of the B type. Each surface atom of the crystal has one hybrid orbital vertical to
Fig. 1. Bonding conditions of a surface B atom in chemisorption on the (111) plane of a ZnS type crystal. The lined plane denotes crystal surface. The empty circles stand for A atoms, the full circle stands for B atom and the shaded circle represents a chemisorbed atom.
SIMPLE
MODEL
335
OF SURFACE
the surface which in the case of the pure surface drifts towards
the vacuum
and in case of chemisorption (when the surface is completely covered) forms a bond with the chemisorbed atom. A single atomic orbital with the Coulomb integral u is considered for the chemisorbed atom. The sp3 orbital of the surface atom projecting from the crystal is generally somewhat different from the analogical orbital of the atom inside the crystal. In short we shall call this orbital the surface orbital. The mentioned difference is expressed by attribution of the Coulomb integral LX’to this surface orbital. The resonance integral between the surface orbital directed out of the crystal and the atomic orbital of the chemisorbed atom is designated as /I.
3. Method of calculation The method of calculation is completely analogous to the calculations in refs.273). The one-electron function in the considered approximation can be written in the form
$j;<,,
=
z
i
dj;pm~(51,52)Up;r,,5~(Y
-
“3’3)
+
dj(rl,t2)”
(l)
m,=lp=l
where up;t;,,c2(r - m3a3) = N-’
c
ei(m1r1+m2c2’qp(r- mIaI - m2a2 - m3a3)
m,,mz (2) u = N-’
c ei(“‘5L+m2523~(r - mlal m,.m2
- m2a2 - m3a3 - x)
m,, m2, m3 are integers. ‘pp are atomic hybrid orbitals. Atomic hybrid orbitals with 1 to 4 belong to A type atoms and atomic orbitals with indices 5 to 8 to B type atoms. x is the atomic orbital of the chemisorbed atom. a,, a2, a3 are primitive translations of the ZnS lattice. a, and a2 are placed in the (111) plane. The vector x is perpendicular to the plane mlal +mza, and its length is equal to the distance of the chemisorbed atom from this plane. 5i are components of the wave vector; 5i ~(0,27c). Nis the number of elementary cells in the plane parallel to the (111) plane of the cyclic crystal, which according to Born-Karman conditions serves as a model of a two-dimensional infinite crystal. The limiting plane of the crystal is defined by the relation m,=l. For the calculation of the coefficients in the wave functions of surface states, we obtain entirely the same equations as in refs.2) and 3).
di;p,m3 =-
YLp,~,; 10
dj;al + PLpmz;sldj dj = /JLdj;Bl
+
u’Lpm3;81 dj,gl (3)
336
J. KOUTECK?
AND
M. TOM&K
where
a)-‘,
L = (W W is the energy
of the localized
(4)
state. The determinant M.4 _ 2 I
A=
A is defined
as
z (5) MB 1
where (Mc)jk = (dc - W - YC)6j, +
YC
Zj, = ye-ir4-j8k,4-j+l C=A,B; We shall introduce
=
j,k=l-4;
co=O.
new variables
xl=(~-aA+~A)j~~ Yl
(6)
xZ=(w-ccB+~B)/~? YAb
2
YZ =
h/Y
>
6 =
x=+(xl+x2)
4 (%
-
cIA)/Y
+
3 (h
-
Y2).
(7)
Subtracting the fourth row of the determinant A from its first three rows and the fifth row of the determinant A from its last three rows we obtain -x,0 0
y-‘A
=
0 0
-x1
0
0
1’1
Yl 0
0 0
0 ei63
-x1
(-
Yl
eic2 0
_
e-i<0
0
0
Xl
_
e-Xo
0
e-ih
0
Xl
_
e-%0
0
0
0
0
0
y2
y2
y2
Xl
0
+ eiEo
e-i61
,-X0
Yl) (-
x2
_ ei5o
eitl
0
f
YJ
-x20
x2
0
_
eib
x’2
0
0
_
e%o
x2
0
= (x1x2 -
e-?
Xl
o2
[wP2
where we have introduced
the designation
A,,, is the subdeterminant p-th column.
originatjng
-
YlY2
1‘q”l
from A by omitting
0 -x2
0
0
-x
(8)
the p’-th row and
SIMPLE
MODEL
337
OF SURFACE
4. The energy spectrum of volume states The roots of the equation A=0
(10)
i.e. the roots of equation XIX2 = 1
(lOa>
and ‘Pl’pZ =
YlY2
IAl2
(lob)
give the energies of volume states. The band limits of volume states are obtained by the equation (lob) if we put IAl =0 or IAl =4 in this equation. In the first case the energy is determined by the equation
with the roots x = ox1,2 = 2y, + [(2y, + 8)’ + 1-j+
(12)
or by the equation (P2
=
(13)
0
with the roots x = Ox3e4 = 27, f [(2y, - 6)2 + 11% In the second
case the energy is determined
(141
by the equation
xix2 = 1
(19
with the roots x =
4x1.2
=
[l + S’]”
f
(16)
or by the equation x1x2
-
1 -
4hx2
+
~2x1)
+
16~1~2 = 0
-
rd
(17)
with the roots x =
4x3,4 =
2(Yl
+
72)
f
uxr2
+
81’
+
v.
(18)
From this we easily obtain a clear picture of the shape of energy spectrum of the volume state energies for the special case y1 = y2 = 7’: The energy band of bonding volume states I is placed for 6 > 0 in the interval for 6 ~0 in the interval The energy band of bonding
volume
states II is placed
for 6 > 0 in the interval for 6 < 0 in the interval The energy band
of volume
(Ox’, 4x3) and (‘x3, 4x3).
antibonding
for 6 > 0 in the interval for 6 <0 in the interval
(4x1, ‘x3) and (4x1, Oxi). states III is placed (‘x4, 4x4> and (Ox’, 4x4>.
338
J. KOUTECKq
The energy band
of volume
AND
M. TOM.@EK
antibonding
energies
for 6 >O in the interval for 6~0 in the interval
IV is placed
(4~2, Ox’) and (4x2, ‘x4).
Besides the above mentioned energy bands of volume states there exist two further energy bands of volume states. The energies of these bands do not at all depend on the wave vector. Both bands are twofold degenerated, one of them belongs to the group of bonding energy bands, the second one to the group of antibonding bands. Their position in the energy spectrum of volume states is given by eqs. (1Oa) or (16). The bands I and II and the bands III and IV are touching for 6 = 0 because (P1=(P2.
The bands
II and 111 are touching
if the condition
4j2 2 1 + b2 (19) is fulfilled. The intervals of forbidden energy bands are designated as follows: B is placed between the bands II and III, C, is placed between the bands I and II, C, is placed between the bands III and IV. The open interval x> 4x3 is called D 1, the open interval defined by the condition x < 4x2 is called D,. 5. The calculation
of states localized
near the surface
The energy of surface or chemisorption electronic states is obtained from the condition for the existence of a solution of the system of equations, consisting in this case of eq. (3a) for m3=0, p= 1, and eq. (3b):
In eq. (20) we have introduced
y as a unit of energy and in agreement
with
this we have defined (T = fiiy,p
d/y, w = (iv - x)/y =
=
x -
rJ-
;(aA+qJ
Yl
-
+
72 (21)
2’
The quantity y will be used as a unit of energy also in the following text. of Subdeterminants A 11 and A, 1 which are essential for the calculation integrals in the Eq. (20) can be transformed to the following form A 11 = (1 A,,
=
(~1x2
x1x2) -
[(XZWl 11
[‘~1~2hx2
+
72)
((P2 -
l>a
+
YlX2) +
-
YlY2X2
e-iTs(xlx2
-
IQ/‘] YlY2
Iaj2)l
(22)
SIMPLE
MODEL
OF SURFACE
339
STATES
where 2 a
=
C
eitj
j=O Xl
=
‘pl
+
x2
Y2X1r
=
(P2
+
YlX2
’
By substituting (22) and (8) into eq. (4) and by carrying integration we obtain
~,o;,,=
A,,(z,
bo;
[YlY2
81
(A 11x2
= -l
+
(23) out the appropriate
zlj-1[YlY2/~J(l
-w2)2]-1
la/
(1
-
72 (XIX2
-
xlx2)2l-
1 (22
q2cx2xi1
-
Yl
21>Ial
(24)
1
Zl))
where zi,2 = &[l
f (1 - 4jK2)“]
(25)
and lc = [WPI If we substitute
relation
- #I’
+
hY21
[Iul
w21-
(24) in the basic equation
(261
l*
(20) we finally obtain
F = p + o’/w
(27)
where (“2%
f’=
+
Y2IX2
-
YlY2X2
A
(aI2
x1x2 - w2 lai2 + z1 laj w2hx2
- 1) = B-C’
(28)
The second expression for the function F given in eq. (28) points at the connection with the theory of surface states from refs.2) and 3). In this expression we have A = (1 - z~z;,x, B = 1 - x1x2z;z;’ C = z1 (x1x2 - l)/jul 4=
X2CPl + lulyl
Y2 9
0 =2=-%-
O’ x2
lq
72’
-_xl;,yl.
(29)
z2
In the case of Shockley surface states, where a change of the potential inside the crystal is not considered (p = 0), a formally identical relation is obtained, according to eq. (27), as was derived in ref.2). zyz; = 1.
(30)
For the existence of surface states of the Shockley type, the fulfilment of eq. (30) is not a sufficient condition because at the same time the relation B=I=C must be fulfilled.
(31)
340
J.
KOUTECKq
AND
M. TOh&EK
In a more general case of surface states when p 4 0, (T= 0, the energies localized states are given by the equation F=p.
The formula
(28) is simplified
of
(32)
for [al = 0 to
F = (~1
(33)
+ Y&XI.
It can be easily shown that function F for la1 =0 is an increasing function which has no discontinuity in the interval (4x4, 4x1). Our further interest is the behaviour of function F for very large absolute values of x1 or x2. From definitions (28), (9), (23) and (25) there follows this asymptotic behaviour of function F: F - x2 - 4y, + 3y,.
(34)
/
3 X Fig. 2 corresponds to the more electronegative surface. Function y = F(x, la/) is shown for yr = ya = y’ = 0.25 and 6 = 0.5. The limit curves F(x, Ia\) for Ial = 0 (thick solid lines -) and Ial = 3 (dashed lines ----) deliniate the shaded region, where surface and chemisorption electronic states can appear. These regions are also limited by thin solid lines (-), formed by the endpoints of function F(x, Ial), lying on the limits of onedimensional energy bands of volume states for various values of (a(. Limits of 3-dimensional energy bands of volume states are drawn as thin vertical solid lines ( ~-). On the top of the figure, the designation of various allowed and forbidden 3-dimensional energy bands of volume states may be found. -2
-7
2
SIMPLE
MODEL
OF SURFACE
341
STATES
For the discussion of surface states the behaviour of function F in those points x, which form the limits of energy bands of volume states, is of great importance : a) If equation (11) is fulfilled and if [al 21 is valid, it becomes obvious (compare (25) and (26)) that z1 = - lal-I. From this follows that in the point Oxi F=$ = (6 + 2y, + [(2y, + 6)2 + l]+t>-’ = y, (35)
4 5 F 4
3
2
1
0
-1
/
-2 L
-1
Fig. 3
corresponds
0
1
2
to the more electronegative surface. Function y = F(x, Ial) is shown for 71 = yz = y’ = 0.25 and 6 = 1.
is valid and that in the point
‘x2
F=~;‘={6+2y,-[(2y,+6)~+l]*j-‘=y, (36) is valid. b) If relation (13) is fulfilled and if lal
(37)
and in the point x = Ox4 F=x2=-~+22yl-[(2y,-6)2+1]~=y4.
(38)
342 c)
J.
KOUTECKq
AND
M. TOM.&EK
If eq. (15) is fulfilled, then in the point x=~x~ F = x2 = - 6 + (1 + s’>+ = y,
(39)
is valid for arbitrary Jai and in the point x=~x* F = x2 = - 8 - [l + a*]+ = y, is valid also for an arbitrary
(40)
[al.
9 & F
/ 2
7 3 I
0
-2
-3 -1 k -2
Fig. 4
corresponds
-1
0
1
to the more electropositive surface. Function y = F(x, 1al) is shown for yl = yz = y’ = 0.25 and 6 = - 0.5.
d) If the eq. (17) is fulfilled, and y1 = y2 = y’, then in the point 4x3 relations F = Y,
(41)
is valid for Ial = 3, and in the point 4x4 the relation (42) F = Y6 is obtained for Ial = 3. According to eq. (32) we obtain the energies of surface states as the intersections of the function y=F(x) with the straight line y=p. According to eq. (27) we obtain the energies of chemisorption states as the intersections
SIMPLE
of the function
MODEL
OF SURFACE
STATES
343
y = F(x) with the hyperbola (43)
In figs. 2 to 5 the function F(x) for selected sets of parameters is shown. In these figures the allowed and forbidden energy bands of volume states are indicated at the same time.
-4
p’ AL
C
-2
Fig. 5
corresponds
-7
0
7
to the more electropositive surface. Function y = F(x, [al) is shown for yl = yz = y’ = 0.25 and 6 = - 1.
The function x=F(x, Ial) forms a family of curves for various values [al. This family of curves covers the shaded regions of the described figures. The limits of these regions are given on one hand by limit curves for la] =O, or Ial = 3. Further limits of these regions are formed by those points in which the function E’(x, Ial) ends on the limits of forbidden energy bands of volume states for a given (~1. 6. Discussions
of results
Three types of surface states can be distinguished. Surface states, the energy of which is placed in gap B, are obviously Shockley surface states.
344
I.
KOUTECK+
AND
M. TOMP;SEK
These can again be interpreted as an expression of dangling bonds caused by the interruption of one bond of each surface atom. Surface states, placed below or above the system of energy bands of volume states, e.g. in the interval of energies D, or D,, are Tamm states. These states are connected with the change of the Coulomb integral of the surface orbital. There exist further surface states the energies of which are placed in the gaps C, or CZ. Parts of these can be situated inside the energy bands of volume states. From the figures, it becomes obvious, that there exist other energy bands of surface states which are completely situated inside the energy bands of volume states. This last type of states is obviously connected with changes in the character of bonds between atoms which are placed just below the surface of the crystal. It is expected for Shockley surface states, that they occur only with small changes of the Coulomb integral of the surface hybrid atomic orbital, whereas Tamm surface states need on the contrary a large change of this integral. From the figures it is obvious that the conditions for the existence of Shockley surface states eliminate the simultaneous existence of Tamm surface states and vice versa. If y1 =yZ = y’ is valid then from eqs. (39) and (41) it follows that the maximum value of the function F in the interval B is exactly equal to the minimum value of this function in the interval Dr. If the above named condition is fulfilled then it follows from the eqs. (40) and (42) that the minimum value of the function F in interval B is exactly equal to the minimum value of this function in interval D,. From this it follows that for y1 = yZ = y’ the simultaneous existence of Shockley and Tamm surface states is excluded. It is interesting to compare surface states, the energy of which is placed in the interval B with surface states in the case of a linear chain, in which atoms of two kinds alternatively occur and in which an alternation of bond by the strength is envisaged (cf. refs. 116). If the stronger bond is interrupted formation of a semi-infinite chain, then there exists a maximum and minimum value of the change of the Coulomb integral of the border atom for which Shockley states still occur. In the model, which is under discussion in the presented work, the stronger of the considered interactions is also interrupted by the formation of a surface by the (111) plane. This interaction is characterized by the resonance integral y. The origin of the band of forbidden energies B is connected with this interaction because this energy gap separates bonding and antibonding states from the point of view of this interaction. Really, the condition for the existence of Shockley states the energy of which is situated in B is for yl=yZ=y’: Y6
<
P
<
Y5.
(44)
SIMPLE
MODEL
OF SURFACE
STATES
345
In interval C, for 6 >O function F has a discontinuity and therefore in this interval surface states can exist, if for the change of the Coulomb integral of the surface orbital either p>Y3 or p 0, there exists a maximum and a minimum value of the function F and therefore must be in these limits (Y4,Yz) if the energy of surface states should lie in interval C,. For 6~0, there exist surface states in interval C1 if Y1Y4 or p
x2
(45)
(compare eqs. (37) to (40)) it is evident that the position of intersections of the function F with the straight line Y=O can be estimated from eq. (45). The energies of Shockley surface states, which are situated in interval B, lie somewhat to the right (for larger values of x) from the point x=6. From this follows that when going from 161to -161, a shift in the position of the energy band of Shockley states from larger values of x to smaller values of x occurs, i.e. there occurs a shift of Shockley surface state energies from lower to higher energies. This transition corresponds physically to the transition from the plane (11 I), formed by more electronegative atoms to the parallel surface plane of the same crystal, which is formed by more electropositive atoms. From conditions (44) it follows that Shockley surface states must always be obtained in our model if p =0 holds. However, with very large positive values of 6 the energy band of Shockley state is very close to the top of the valence band II. In the case of the limiting plane formed by more electropositive atoms (6 ~0) a very small difference between the electronegativity of atoms of both kinds A and B is sufficient to cause a shift of the
346
J. KOUTECK+
AND M. TOM%EK
energy band of Shockley surface states close to the bottom of the conduction band III or even causes that this band touches the bottom of the conductivity band III. From eq. (44) it follows further that the larger is the value 6, i.e. the more electronegative are the surface atoms, the lower are the positive values p, at which Shockley surface states may still occur in region B. On the other hand, the larger is 6, the smaller positive values of p are sufficient for the occurence of Tamm surface states in interval D,. The larger is the value of 6, the smaller negative p can lead to the creation of Shockley states with energies in interval B and the smaller negative values of p lead to the appearing of Tamm states in interval D,. Also these existence conditions for the occurrence of Shockley surface states with energies in region B and Tamm states with energies in regions D, and D, are analogical to the existence conditions for Shockley and Tamm surface states in the case of a linear chain, composed of atoms of two kinds. The greater the absolute values 161, the narrower are the energy bands of volume states as well as energy bands of surface states, and the forbidden energy bands of volume states are expanded. This fact is natural because with the increasing value of 161 the energy barriers between atoms of the same kind increase. With large values (61 electrons are localized predominantly on the more electronegative atoms. The preceding considerations on the conditions for the existence of Shockley and Tamm surface states can also be summarized as follows: If relation 6p > 0
(46)
is valid, the existence of Shockley surface states is made more difficult as compared with the diamond lattice. On the contrary, Tamm states are becoming more probable. From this it is obvious that an increase in the difference between the potential in the region of the surface orbital of atom B and the potential in the region of the analogical orbital of atom A in comparison with this difference inside the crystal, causes that the existence of Tamm states is made easier. This circumstance is easily understandable because Tamm states are connected with the localization of electron near the surface in consequence of the existence of a deep potential well near the surface. Shockley states, as known, show an opposite behaviour. If on the contrary 6p -c0 holds, the appearance of Shockley surface states becomes easier and the appearance of Tamm states becomes more difficult. Let us assume that the number of electrons in the crystal is equal to the number of atomic orbitals considered and at the same time equal to the number of electrons in the crystal which can occupy molecular orbitals originating from these atomic orbitals. The band of Shockley surface states
SIMPLE
MODEL
OF SURFACE
STATES
347
with energies in interval B is then only half filled. Let us follow the changes which occur if the value of quantity p increases continuously. If p exceeds the value yz, the band of surface states immerses into the band II. Then, at the same time, a number of holes equal to the number of surface atomic orbitals enter the band. The Fermi surface will therefore coincide with the top of the valence band II. Thus we obtain a model of a degenerated semiconductor of the p type. If, on the contrary, the energy band of the surface states disappears from the interval B to the band III, a number of electrons equal to the number of surface orbitals enters the conductivity band III. We obtain thus a model of a degenerated semi-conductor of type n with Fermi surface, situated on the bottom of the lowest conductivity band. Continuous transition of the energy band of surface states with energies lying between the highest valence and the lowest conductivity bands into one of these bands under the influence of a change of parameters is probably a more general phenomenon, which is not connected only with the considered model.
1) 2) 3) 4) 5) 6)
J. Kouteckf, Adv. Chem. Phys. 9, p. 85. J. Koutecky and M. TomBSek, Phys. Rev. 120 (1960) 1212. J. Kouteckf, Surface Science 1 (1964) 280. J. Koutecky and M. Tom%ek, Czech. J. Phys. B12 (1962) 48. D. Pugh, Phys. Rev. Letters 12 (1964) 390. A. T. Amos and S. G. Davison, Physica 30 (1964) 905.
SCIENCE
SIMPLE
MODEL
3 (1965) 333-347 o North-Holland
OF SURFACE
ON THE (111) PLANE J. KOUTECKP Institute
of Physical
Publishing
Co., Amsterdam
AND CHEMISORPTION
STATES
OF ZnS TYPE CRYSTALS and M. TOMASEK
Chemistry of the Czechoslovak Prague, Czechoslovakia
Academv
of Sciences,
Received 18 May 1965 The surface and chemisorption states of a simple model of a semi-infinite ZnS type (sphalerite) crystal, limited by the (111) plane were studied. The existence conditions of various states localised near the surface were studied and the classification of these states was carried out. The differences between characteristics of surfaces which are formed by more electronegative atoms and surfaces formed by more electropositive atoms were pointed out. In the case of more electronegative surfaces the energy band of Shockley surface states is placed closely above the top of the highest valence band, the formation of Tamm surface states with the energies below the valence band is facilitated and the formation of Tamm surface states with the energies above the conductivity bands becomes more difficult. As far as the more electropositive surface is concerned, the energy band of Shockley states is placed near the bottom of the conductivity band, the formation of Tamm states with energies below the lowest valence band is made more difficult and the formation of Tamm states with the energies above the highest conductivity band facilitated. In the gaps between individual valence or conductivity energy bands of volume states, further energy bands of surface states of a new kind can originate. The earlier studied model of the diamond lattice is a special case of the model investigated in this work. From the studied model follows, that a semi-infinite crystal can become a degenerated p- or n-semi-conductor if, in the forbidden energy gap, Shockley surface states do not exist at least for one of the two possible (111) delimiting planes.
1. Introduction Recently several very simple models of finite crystals were calculated from the viewpoint of the origin of electron surface states l). Special attention was paid to surface and chemisorption states of diamond-like latticess-5). The considered models envisaged that the covalent character of bonds between neighbouring atoms was especially expressed. A generalization of similar calculations for the ZnS lattice should be carried out, in which the covalent character of bonds are also predominant. Intermetallic compounds, such as InSb, GaSb, InAs, GaAs which are of considerable importance in semiconductor physics, as well as interesting compounds of BN, Sic, etc. types belong to this group. In this work a very simplified semi-infinite model of the ZnS type (sphalerite) 333
334
.I. KOUTECKi’
AND
M. TOMASEK
lattice limited by the (111) plane is considered. Surface and chemisorption one-electron states localized in the proximity of this limiting plane are studied. The tight binding approximation is used and similar assumptions are made as in preceding papers. The only difference as compared with the preceding work lies in the fact that in every elementary cell of the “diamond” lattice two atoms of different types are considered. At the same time the nearest neighbours of an A type atom are B type atoms and vice versa. In the study of the considered model of a binary crystal, emphasis is again placed on general conclusions. The model is too simplified and therefore considerably distant from reality. A further question is whether the ideal (111) plane is in general sufficiently stable in structures of this type. 2. Model For each of the A type atoms and B type atoms we consider four equivalent localized orbitals - sp3 hybrides (cf. ref.2)). The axes of maximum density of atomic orbitals of the given atom are orientated in the direction of the line connecting this atom with its neighbours. The resonance integrals between hybrids belonging to different atoms and lying along these connecting lines will be denoted as y. Resonance integrals between hybrids belonging to the same atom are further considered. Because there are two types of atoms, we distinguish two resonance integrals of this type, yA and yB, according to the hybrid orbitals being localized on atom A or B. Similarly, we have to distinguish Coulomb integrals txA and cla of hybrid atomic orbitals of different kinds. The limiting plane (111) is selected so that it cuts only one single bond of the surface atom (cf. fig. 1). We assume that the surface atom is of the B type. Each surface atom of the crystal has one hybrid orbital vertical to
Fig. 1. Bonding conditions of a surface B atom in chemisorption on the (111) plane of a ZnS type crystal. The lined plane denotes crystal surface. The empty circles stand for A atoms, the full circle stands for B atom and the shaded circle represents a chemisorbed atom.
SIMPLE
MODEL
335
OF SURFACE
the surface which in the case of the pure surface drifts towards
the vacuum
and in case of chemisorption (when the surface is completely covered) forms a bond with the chemisorbed atom. A single atomic orbital with the Coulomb integral u is considered for the chemisorbed atom. The sp3 orbital of the surface atom projecting from the crystal is generally somewhat different from the analogical orbital of the atom inside the crystal. In short we shall call this orbital the surface orbital. The mentioned difference is expressed by attribution of the Coulomb integral LX’to this surface orbital. The resonance integral between the surface orbital directed out of the crystal and the atomic orbital of the chemisorbed atom is designated as /I.
3. Method of calculation The method of calculation is completely analogous to the calculations in refs.273). The one-electron function in the considered approximation can be written in the form
$j;<,,
=
z
i
dj;pm~(51,52)Up;r,,5~(Y
-
“3’3)
+
dj(rl,t2)”
(l)
m,=lp=l
where up;t;,,c2(r - m3a3) = N-’
c
ei(m1r1+m2c2’qp(r- mIaI - m2a2 - m3a3)
m,,mz (2) u = N-’
c ei(“‘5L+m2523~(r - mlal m,.m2
- m2a2 - m3a3 - x)
m,, m2, m3 are integers. ‘pp are atomic hybrid orbitals. Atomic hybrid orbitals with 1 to 4 belong to A type atoms and atomic orbitals with indices 5 to 8 to B type atoms. x is the atomic orbital of the chemisorbed atom. a,, a2, a3 are primitive translations of the ZnS lattice. a, and a2 are placed in the (111) plane. The vector x is perpendicular to the plane mlal +mza, and its length is equal to the distance of the chemisorbed atom from this plane. 5i are components of the wave vector; 5i ~(0,27c). Nis the number of elementary cells in the plane parallel to the (111) plane of the cyclic crystal, which according to Born-Karman conditions serves as a model of a two-dimensional infinite crystal. The limiting plane of the crystal is defined by the relation m,=l. For the calculation of the coefficients in the wave functions of surface states, we obtain entirely the same equations as in refs.2) and 3).
di;p,m3 =-
YLp,~,; 10
dj;al + PLpmz;sldj dj = /JLdj;Bl
+
u’Lpm3;81 dj,gl (3)
336
J. KOUTECK?
AND
M. TOM&K
where
a)-‘,
L = (W W is the energy
of the localized
(4)
state. The determinant M.4 _ 2 I
A=
A is defined
as
z (5) MB 1
where (Mc)jk = (dc - W - YC)6j, +
YC
Zj, = ye-ir4-j8k,4-j+l C=A,B; We shall introduce
=
j,k=l-4;
co=O.
new variables
xl=(~-aA+~A)j~~ Yl
(6)
xZ=(w-ccB+~B)/~? YAb
2
YZ =
h/Y
>
6 =
x=+(xl+x2)
4 (%
-
cIA)/Y
+
3 (h
-
Y2).
(7)
Subtracting the fourth row of the determinant A from its first three rows and the fifth row of the determinant A from its last three rows we obtain -x,0 0
y-‘A
=
0 0
-x1
0
0
1’1
Yl 0
0 0
0 ei63
-x1
(-
Yl
eic2 0
_
e-i<0
0
0
Xl
_
e-Xo
0
e-ih
0
Xl
_
e-%0
0
0
0
0
0
y2
y2
y2
Xl
0
+ eiEo
e-i61
,-X0
Yl) (-
x2
_ ei5o
eitl
0
f
YJ
-x20
x2
0
_
eib
x’2
0
0
_
e%o
x2
0
= (x1x2 -
e-?
Xl
o2
[wP2
where we have introduced
the designation
A,,, is the subdeterminant p-th column.
originatjng
-
YlY2
1‘q”l
from A by omitting
0 -x2
0
0
-x
(8)
the p’-th row and
SIMPLE
MODEL
337
OF SURFACE
4. The energy spectrum of volume states The roots of the equation A=0
(10)
i.e. the roots of equation XIX2 = 1
(lOa>
and ‘Pl’pZ =
YlY2
IAl2
(lob)
give the energies of volume states. The band limits of volume states are obtained by the equation (lob) if we put IAl =0 or IAl =4 in this equation. In the first case the energy is determined by the equation
with the roots x = ox1,2 = 2y, + [(2y, + 8)’ + 1-j+
(12)
or by the equation (P2
=
(13)
0
with the roots x = Ox3e4 = 27, f [(2y, - 6)2 + 11% In the second
case the energy is determined
(141
by the equation
xix2 = 1
(19
with the roots x =
4x1.2
=
[l + S’]”
f
(16)
or by the equation x1x2
-
1 -
4hx2
+
~2x1)
+
16~1~2 = 0
-
rd
(17)
with the roots x =
4x3,4 =
2(Yl
+
72)
f
uxr2
+
81’
+
v.
(18)
From this we easily obtain a clear picture of the shape of energy spectrum of the volume state energies for the special case y1 = y2 = 7’: The energy band of bonding volume states I is placed for 6 > 0 in the interval for 6 ~0 in the interval The energy band of bonding
volume
states II is placed
for 6 > 0 in the interval for 6 < 0 in the interval The energy band
of volume
(Ox’, 4x3) and (‘x3, 4x3).
antibonding
for 6 > 0 in the interval for 6 <0 in the interval
(4x1, ‘x3) and (4x1, Oxi). states III is placed (‘x4, 4x4> and (Ox’, 4x4>.
338
J. KOUTECKq
The energy band
of volume
AND
M. TOM.@EK
antibonding
energies
for 6 >O in the interval for 6~0 in the interval
IV is placed
(4~2, Ox’) and (4x2, ‘x4).
Besides the above mentioned energy bands of volume states there exist two further energy bands of volume states. The energies of these bands do not at all depend on the wave vector. Both bands are twofold degenerated, one of them belongs to the group of bonding energy bands, the second one to the group of antibonding bands. Their position in the energy spectrum of volume states is given by eqs. (1Oa) or (16). The bands I and II and the bands III and IV are touching for 6 = 0 because (P1=(P2.
The bands
II and 111 are touching
if the condition
4j2 2 1 + b2 (19) is fulfilled. The intervals of forbidden energy bands are designated as follows: B is placed between the bands II and III, C, is placed between the bands I and II, C, is placed between the bands III and IV. The open interval x> 4x3 is called D 1, the open interval defined by the condition x < 4x2 is called D,. 5. The calculation
of states localized
near the surface
The energy of surface or chemisorption electronic states is obtained from the condition for the existence of a solution of the system of equations, consisting in this case of eq. (3a) for m3=0, p= 1, and eq. (3b):
In eq. (20) we have introduced
y as a unit of energy and in agreement
with
this we have defined (T = fiiy,p
d/y, w = (iv - x)/y =
=
x -
rJ-
;(aA+qJ
Yl
-
+
72 (21)
2’
The quantity y will be used as a unit of energy also in the following text. of Subdeterminants A 11 and A, 1 which are essential for the calculation integrals in the Eq. (20) can be transformed to the following form A 11 = (1 A,,
=
(~1x2
x1x2) -
[(XZWl 11
[‘~1~2hx2
+
72)
((P2 -
l>a
+
YlX2) +
-
YlY2X2
e-iTs(xlx2
-
IQ/‘] YlY2
Iaj2)l
(22)
SIMPLE
MODEL
OF SURFACE
339
STATES
where 2 a
=
C
eitj
j=O Xl
=
‘pl
+
x2
Y2X1r
=
(P2
+
YlX2
’
By substituting (22) and (8) into eq. (4) and by carrying integration we obtain
~,o;,,=
A,,(z,
bo;
[YlY2
81
(A 11x2
= -l
+
(23) out the appropriate
zlj-1[YlY2/~J(l
-w2)2]-1
la/
(1
-
72 (XIX2
-
xlx2)2l-
1 (22
q2cx2xi1
-
Yl
21>Ial
(24)
1
Zl))
where zi,2 = &[l
f (1 - 4jK2)“]
(25)
and lc = [WPI If we substitute
relation
- #I’
+
hY21
[Iul
w21-
(24) in the basic equation
(261
l*
(20) we finally obtain
F = p + o’/w
(27)
where (“2%
f’=
+
Y2IX2
-
YlY2X2
A
(aI2
x1x2 - w2 lai2 + z1 laj w2hx2
- 1) = B-C’
(28)
The second expression for the function F given in eq. (28) points at the connection with the theory of surface states from refs.2) and 3). In this expression we have A = (1 - z~z;,x, B = 1 - x1x2z;z;’ C = z1 (x1x2 - l)/jul 4=
X2CPl + lulyl
Y2 9
0 =2=-%-
O’ x2
lq
72’
-_xl;,yl.
(29)
z2
In the case of Shockley surface states, where a change of the potential inside the crystal is not considered (p = 0), a formally identical relation is obtained, according to eq. (27), as was derived in ref.2). zyz; = 1.
(30)
For the existence of surface states of the Shockley type, the fulfilment of eq. (30) is not a sufficient condition because at the same time the relation B=I=C must be fulfilled.
(31)
340
J.
KOUTECKq
AND
M. TOh&EK
In a more general case of surface states when p 4 0, (T= 0, the energies localized states are given by the equation F=p.
The formula
(28) is simplified
of
(32)
for [al = 0 to
F = (~1
(33)
+ Y&XI.
It can be easily shown that function F for la1 =0 is an increasing function which has no discontinuity in the interval (4x4, 4x1). Our further interest is the behaviour of function F for very large absolute values of x1 or x2. From definitions (28), (9), (23) and (25) there follows this asymptotic behaviour of function F: F - x2 - 4y, + 3y,.
(34)
/
3 X Fig. 2 corresponds to the more electronegative surface. Function y = F(x, la/) is shown for yr = ya = y’ = 0.25 and 6 = 0.5. The limit curves F(x, Ia\) for Ial = 0 (thick solid lines -) and Ial = 3 (dashed lines ----) deliniate the shaded region, where surface and chemisorption electronic states can appear. These regions are also limited by thin solid lines (-), formed by the endpoints of function F(x, Ial), lying on the limits of onedimensional energy bands of volume states for various values of (a(. Limits of 3-dimensional energy bands of volume states are drawn as thin vertical solid lines ( ~-). On the top of the figure, the designation of various allowed and forbidden 3-dimensional energy bands of volume states may be found. -2
-7
2
SIMPLE
MODEL
OF SURFACE
341
STATES
For the discussion of surface states the behaviour of function F in those points x, which form the limits of energy bands of volume states, is of great importance : a) If equation (11) is fulfilled and if [al 21 is valid, it becomes obvious (compare (25) and (26)) that z1 = - lal-I. From this follows that in the point Oxi F=$ = (6 + 2y, + [(2y, + 6)2 + l]+t>-’ = y, (35)
4 5 F 4
3
2
1
0
-1
/
-2 L
-1
Fig. 3
corresponds
0
1
2
to the more electronegative surface. Function y = F(x, Ial) is shown for 71 = yz = y’ = 0.25 and 6 = 1.
is valid and that in the point
‘x2
F=~;‘={6+2y,-[(2y,+6)~+l]*j-‘=y, (36) is valid. b) If relation (13) is fulfilled and if lal
(37)
and in the point x = Ox4 F=x2=-~+22yl-[(2y,-6)2+1]~=y4.
(38)
342 c)
J.
KOUTECKq
AND
M. TOM.&EK
If eq. (15) is fulfilled, then in the point x=~x~ F = x2 = - 6 + (1 + s’>+ = y,
(39)
is valid for arbitrary Jai and in the point x=~x* F = x2 = - 8 - [l + a*]+ = y, is valid also for an arbitrary
(40)
[al.
9 & F
/ 2
7 3 I
0
-2
-3 -1 k -2
Fig. 4
corresponds
-1
0
1
to the more electropositive surface. Function y = F(x, 1al) is shown for yl = yz = y’ = 0.25 and 6 = - 0.5.
d) If the eq. (17) is fulfilled, and y1 = y2 = y’, then in the point 4x3 relations F = Y,
(41)
is valid for Ial = 3, and in the point 4x4 the relation (42) F = Y6 is obtained for Ial = 3. According to eq. (32) we obtain the energies of surface states as the intersections of the function y=F(x) with the straight line y=p. According to eq. (27) we obtain the energies of chemisorption states as the intersections
SIMPLE
of the function
MODEL
OF SURFACE
STATES
343
y = F(x) with the hyperbola (43)
In figs. 2 to 5 the function F(x) for selected sets of parameters is shown. In these figures the allowed and forbidden energy bands of volume states are indicated at the same time.
-4
p’ AL
C
-2
Fig. 5
corresponds
-7
0
7
to the more electropositive surface. Function y = F(x, [al) is shown for yl = yz = y’ = 0.25 and 6 = - 1.
The function x=F(x, Ial) forms a family of curves for various values [al. This family of curves covers the shaded regions of the described figures. The limits of these regions are given on one hand by limit curves for la] =O, or Ial = 3. Further limits of these regions are formed by those points in which the function E’(x, Ial) ends on the limits of forbidden energy bands of volume states for a given (~1. 6. Discussions
of results
Three types of surface states can be distinguished. Surface states, the energy of which is placed in gap B, are obviously Shockley surface states.
344
I.
KOUTECK+
AND
M. TOMP;SEK
These can again be interpreted as an expression of dangling bonds caused by the interruption of one bond of each surface atom. Surface states, placed below or above the system of energy bands of volume states, e.g. in the interval of energies D, or D,, are Tamm states. These states are connected with the change of the Coulomb integral of the surface orbital. There exist further surface states the energies of which are placed in the gaps C, or CZ. Parts of these can be situated inside the energy bands of volume states. From the figures, it becomes obvious, that there exist other energy bands of surface states which are completely situated inside the energy bands of volume states. This last type of states is obviously connected with changes in the character of bonds between atoms which are placed just below the surface of the crystal. It is expected for Shockley surface states, that they occur only with small changes of the Coulomb integral of the surface hybrid atomic orbital, whereas Tamm surface states need on the contrary a large change of this integral. From the figures it is obvious that the conditions for the existence of Shockley surface states eliminate the simultaneous existence of Tamm surface states and vice versa. If y1 =yZ = y’ is valid then from eqs. (39) and (41) it follows that the maximum value of the function F in the interval B is exactly equal to the minimum value of this function in the interval Dr. If the above named condition is fulfilled then it follows from the eqs. (40) and (42) that the minimum value of the function F in interval B is exactly equal to the minimum value of this function in interval D,. From this it follows that for y1 = yZ = y’ the simultaneous existence of Shockley and Tamm surface states is excluded. It is interesting to compare surface states, the energy of which is placed in the interval B with surface states in the case of a linear chain, in which atoms of two kinds alternatively occur and in which an alternation of bond by the strength is envisaged (cf. refs. 116). If the stronger bond is interrupted formation of a semi-infinite chain, then there exists a maximum and minimum value of the change of the Coulomb integral of the border atom for which Shockley states still occur. In the model, which is under discussion in the presented work, the stronger of the considered interactions is also interrupted by the formation of a surface by the (111) plane. This interaction is characterized by the resonance integral y. The origin of the band of forbidden energies B is connected with this interaction because this energy gap separates bonding and antibonding states from the point of view of this interaction. Really, the condition for the existence of Shockley states the energy of which is situated in B is for yl=yZ=y’: Y6
<
P
<
Y5.
(44)
SIMPLE
MODEL
OF SURFACE
STATES
345
In interval C, for 6 >O function F has a discontinuity and therefore in this interval surface states can exist, if for the change of the Coulomb integral of the surface orbital either p>Y3 or p
x2
(45)
(compare eqs. (37) to (40)) it is evident that the position of intersections of the function F with the straight line Y=O can be estimated from eq. (45). The energies of Shockley surface states, which are situated in interval B, lie somewhat to the right (for larger values of x) from the point x=6. From this follows that when going from 161to -161, a shift in the position of the energy band of Shockley states from larger values of x to smaller values of x occurs, i.e. there occurs a shift of Shockley surface state energies from lower to higher energies. This transition corresponds physically to the transition from the plane (11 I), formed by more electronegative atoms to the parallel surface plane of the same crystal, which is formed by more electropositive atoms. From conditions (44) it follows that Shockley surface states must always be obtained in our model if p =0 holds. However, with very large positive values of 6 the energy band of Shockley state is very close to the top of the valence band II. In the case of the limiting plane formed by more electropositive atoms (6 ~0) a very small difference between the electronegativity of atoms of both kinds A and B is sufficient to cause a shift of the
346
J. KOUTECK+
AND M. TOM%EK
energy band of Shockley surface states close to the bottom of the conduction band III or even causes that this band touches the bottom of the conductivity band III. From eq. (44) it follows further that the larger is the value 6, i.e. the more electronegative are the surface atoms, the lower are the positive values p, at which Shockley surface states may still occur in region B. On the other hand, the larger is 6, the smaller positive values of p are sufficient for the occurence of Tamm surface states in interval D,. The larger is the value of 6, the smaller negative p can lead to the creation of Shockley states with energies in interval B and the smaller negative values of p lead to the appearing of Tamm states in interval D,. Also these existence conditions for the occurrence of Shockley surface states with energies in region B and Tamm states with energies in regions D, and D, are analogical to the existence conditions for Shockley and Tamm surface states in the case of a linear chain, composed of atoms of two kinds. The greater the absolute values 161, the narrower are the energy bands of volume states as well as energy bands of surface states, and the forbidden energy bands of volume states are expanded. This fact is natural because with the increasing value of 161 the energy barriers between atoms of the same kind increase. With large values (61 electrons are localized predominantly on the more electronegative atoms. The preceding considerations on the conditions for the existence of Shockley and Tamm surface states can also be summarized as follows: If relation 6p > 0
(46)
is valid, the existence of Shockley surface states is made more difficult as compared with the diamond lattice. On the contrary, Tamm states are becoming more probable. From this it is obvious that an increase in the difference between the potential in the region of the surface orbital of atom B and the potential in the region of the analogical orbital of atom A in comparison with this difference inside the crystal, causes that the existence of Tamm states is made easier. This circumstance is easily understandable because Tamm states are connected with the localization of electron near the surface in consequence of the existence of a deep potential well near the surface. Shockley states, as known, show an opposite behaviour. If on the contrary 6p -c0 holds, the appearance of Shockley surface states becomes easier and the appearance of Tamm states becomes more difficult. Let us assume that the number of electrons in the crystal is equal to the number of atomic orbitals considered and at the same time equal to the number of electrons in the crystal which can occupy molecular orbitals originating from these atomic orbitals. The band of Shockley surface states
SIMPLE
MODEL
OF SURFACE
STATES
347
with energies in interval B is then only half filled. Let us follow the changes which occur if the value of quantity p increases continuously. If p exceeds the value yz, the band of surface states immerses into the band II. Then, at the same time, a number of holes equal to the number of surface atomic orbitals enter the band. The Fermi surface will therefore coincide with the top of the valence band II. Thus we obtain a model of a degenerated semiconductor of the p type. If, on the contrary, the energy band of the surface states disappears from the interval B to the band III, a number of electrons equal to the number of surface orbitals enters the conductivity band III. We obtain thus a model of a degenerated semi-conductor of type n with Fermi surface, situated on the bottom of the lowest conductivity band. Continuous transition of the energy band of surface states with energies lying between the highest valence and the lowest conductivity bands into one of these bands under the influence of a change of parameters is probably a more general phenomenon, which is not connected only with the considered model.
1) 2) 3) 4) 5) 6)
J. Kouteckf, Adv. Chem. Phys. 9, p. 85. J. Koutecky and M. TomBSek, Phys. Rev. 120 (1960) 1212. J. Kouteckf, Surface Science 1 (1964) 280. J. Koutecky and M. Tom%ek, Czech. J. Phys. B12 (1962) 48. D. Pugh, Phys. Rev. Letters 12 (1964) 390. A. T. Amos and S. G. Davison, Physica 30 (1964) 905.