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Surface reconstruction of a two-band crystal
Applications of Surface Science 1 (1977) 44—5 8 © North-Holland Publishing Company
SURFACE RECONSTRUCTION OF A TWO-BAND CRYSTAL II. Model resufts I’ Si. CUNNINGHAM, W. HO
*
and W.H. WEINBERG
**
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA and
L. DOBRZYNSKI Laboratoire de Physique des So/ides, Institut Supéricur d’Electronique do Nord, 3, rue Francois Baës, 59046 Lille Cedex, France Received 20 April 1977
The change in the total electronic energy which occurs upon reconstruction of the (001) face of a model two-band crystal with the CsCI structure is presented. The energy changes due to a (2X 1) and a c(2 X 2) reconstruction are considered. The results are obtained by a Green’s function perturbation technique developed in the previous paper. The Green’s functions appropriate for this reconstructed surface are analytic. The question of whether or not reconstruction occurs on a surface is shown to be dependent upon the relative magnitudes of the electronic energy and the strain energy.
1. Introduction In the first paper in this series, we developed the Green’s function formalism needed to study the effect of surface reconstruction on the electronic energy [1]. In this paper, we demonstrate the formalism by explicitly calculating the effect of reconstruction on a model two-band crystal. In a previous paper, we studied the ideal unreconstructed (001) surface of a crystal having the CsC1 structure [2] .We used the LCAO formalism, assumed the tight binding approximation, and derived analytic expressions for both the bulk and the surface Green’s functions. in this paper, we use the previously derived surface Work supported by AROD Grant No. DAFICO4-75-0170. *
**
Present address: Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19174, USA. Camille and Henry Dryfus Foundation Teacher—Scholar and Alfred P. Sloan Foundation Fellow.
S.L. Cunningham eta!.
/ Surface reconstruction of two -band crystal. II
45
Green’s functions to describe the unreconstructed surface.The change in the density of states due to the reconstruction is determined conveniently by the phase shift technique. The change in the electronic energy is then obtained by a simple integration. In the few cases where they occur, we find the surface states by the resolvent technique. Two different reconstructed structures are considered here. The surface atoms are rearranged arbitrarily in two different ways forming the (2 X 1) and the c(2 X 2) structures. In the (2 X 1) case, we study separately motions parallel and perpendicular to the surface. In addition, we study the change in the electronic energy as a function of the band gap. In all cases considered, we find a decrease in the total electronic energy for a filled band. Depending upon the relative strength of this energy compared to the unfavorable strain energy created by moving the surface atoms out of their bulk equilibrium positions, reconstruction can occur. In section 2 we present the Green’s function for the (1 X 1) surface of the model system. Then we determine the change in the density of states for (I) a (2 X 1) reconstruction where the atoms are displaced in the z-direction, (2) a (2X I) reconstruction where the atoms are displaced in the y-direction, and (3) a c(2 X 2) reconstruction. The atomic positions of each of these reconstructions are shown in I. Then we evaluate the change in the electronic ground state energy for each of the cases as a function of the magnitude of the change in the hopping integral. Finally, this energy is compared with the elastic strain energy, and parameter values are found which determine whether reconstruction will occur or not. In section 3, we summarize our conclusions.
2. Results and discussion The band structure for the bulk crystal referenced to the unreconstructedsurface Brillouin zone (SBZ), within the tight binding approximation and assuming only first neighbor interactions, ‘y, is given by Ek!
2cos2ø~/2cos2 ±~(E~ + 256 ‘y
0y12
cos2 ø~/2)u/2 ,
(1)
where 4i~’a 0k,
(2)
a0 is the crystal lattice constant, and Eg is the band gap which for this structure is equal to the magnitude of the difference in orbital energies, E1 and E2, between the two types of atoms [2]. There are two branches, and / takes on the values I and 2. The zero of the energy is chosen arbitrarily to be the average of Li and E2. For purposes of illustration, the parameters ‘y and Eg are fitted to the photoemission data for Cs! obtained by DiStefano and Spicer [3]. They found the width of the valence band to be 2.5 ±0.3 eV and the band gap at 300 K to be 6.2 eV.
46
S.L. Cunningham et a!.
/ Surface reconstruction
of two-band cr,vstal. II
Using these values, we find
y= 0.58eV,
(3)
EglY= I0.7.
(4)
The Green’s functions for crystals with the (001) unreconstructed surface obtained in our previous paper are given as the difference of two bulk Green’s functions [2]. They take on a rather simple form for the surface layer, l~= 1~= I, namely, 2(t + 1), (5) G(1 I, ll,4~)=dt/f G(12, 12,~
(6)
2, 5)~’at/f
G(l1, ~
(7)
,
G(12, ~
(8)
,
where
(9) d=E—E
(10) (11)
2, 4ycos~x/2cos~y/2, f
~(ad—
2f2)/2f2
1)1/2, ~+ i(l
—
(12)
,
~2)1/2
~> 1
I~I <1
(13)
2.1. (2 Xl) Reconstruction, z-diplacement The assumed displacement geometry for this situation and the corresponding SBZ are shown in I. To proceed, we must first derive the unperturbed surface Green’s functions appropriate for the (2 X 1) reconstruction. There are four basis atoms in the (2 X 1) reconstructed surface. Thus, the bulk band structure referenced to the new SBZ consists of four branches and is given by [2] ‘~kj= ±~(E + 25672 cos2 ~/2 cos2 ~~J/2 cos2 Ø~/
2)h/2 (14) where we have chosen the plus (minus) sign for / = 1 (2), and 2 ø~/2sin2 ~y/2 cos2 ø~/2)1/2 (15) + 25672 cos for / = 3(+) and 4(—). These equations result naturally from simply folding the bulk bands appropriate to the unreconstructed surface in eq. (1). ,
S.L. Cunningham eta!. / Surface reconstruction of two -band crystal. II
47
To derive the needed Green’s functions, we follow the procedure in I and express the Green’s function as a sum over the (2 X I) SBZ as follows G(l~,l’~’,E)-~~ ~ a
0
+ G(l~j3,l~’13,4~— sgn(Øy)irey, E)
exp 1
[~P5-_sgn(t~y)irey]
a0
•
x(ll’)
,
(16)
where sgn (cby) denotes the sign of the y-component of~,and is the unit vector in the y-direction. The sum over the (2 X 1) SBZ in eq. (16) still includes the same terms as the sum over the (I X 1) SBZ. Since we have chosen to distort the surface in the y-direction, the unit distance in the x-direction remains the same as the (1 Xl) surface, while that in the y-direction becomes twice as long. When referring to the same pair of atoms, we have simply
X~(ll’)=X~(LL’)
(17)
for the x-component of X(ll’) and X(LL’). However, the corresponding equality for the y-component does not always hold. We equate the coefficients of the sums over the (1 XI) SBZ with the terms in eq. (16) and obtain immediately GA(L2A,L~AP,~5,E) ~exp + G(l5~,~
—
~
[x~(ll’) _xy(LL’)]) [G(l2~,l~’,~5,E)
sgn(Øy) ~ey~E) exp
(_~sgn(~y) ~
xY(ll’))]
,
(18)
where we have used the fact that (19)
Nn=~Ns.
We can further simplify eq. (18) by evaluating the exponentials. Eq. (18) may then be dissected into the following O(L~13,L~f3’,45,E)= ~[G(l~1J, l’,4~,E)+ G(l~f3,l~’j3’,~5 sgn(~y)irey,E)] (20) —
=
G(L~I3+ 2,L~p’+ 2,4~,E),
G(L5i3, Lj3’ ÷~ —
if)
(21)
~ exp(—it~~) [G(l~l’3, 1~l3’,i~, if)
G(lj3, l~f3’,4~ sgn(~y)7rey~E)] —
,
(22)
G(L~/3+2,L~f3’,.fi5,E)=~ exp(i~,)[G(l~13,lj’3’,45,E) —G(l~I3,!i3’,45—sgn(~y)irey,E)]
,
(23)
where ~3,13’1,2, and Xy(LL’)O,
(24)
S.L. Cunningham et a!. / Surface reconstruction of two-band crystal. II
48
since all the atoms are in the same enlarged unit cell. Thus, we see that the layer-bylayer reconstructed surface Green’s functions are obtained as a sum or difference of two known (I X 1) surface Green’s functions. The perturbation matrix is obtained as in I. The periodic change in the nearest neighbor hopping integrals is shown in I. The perturbation is confined to the first and second planes both of which are in the first layer and is given by
-
V(1A, L4
,
,4~)= 2e cos 2x —
0
v~
0
v” exp(—i2~1,)
0
—v’
—v c~
—v’~
—v
0
V
exp(i2~) 0
.
(25)
where the rows (and colums) are labeled by the index L~ 1 and A = 1, 2, 3, and 4, and
v
=
exp(i~~/2).
(26)
We recall that e, in units ofy, is the change in y due to reconstruction. The Green’s function G that is needed is therefore reduced to a (4 X 4) matrix with 1~= l~= L~= = 1 in eqs. (20)—(23). Once V and G are known, the effects of reconstruction on the electronic structure of the surface can be studied within the formalism developed in 1. The sum over the SBZ is performed using the special points sampling procedure [4]. There are no surface states for this reconstructed surface structure. hi fig. 1, we 0.04
~n(E)
(2xl) z-displacement 0.7
0.02
~ -0.04 ‘—-0.1 ——0.1 Fig. 1. Change in the density of states per (2 X 1) reconstructed surface unit cell for displacements of atoms normal to the surface. Singularities at the band edges are due to the flatness of the bands.
S.L. Cunningham eta!. I Surface reconstruction of two-band crystal. II
49
present the change in the density of states. States are shifted toward the bottom (top) of the valence (conduction) band, leading to a decrease in the electronic energy. States are also pushed toward the band gap region in a very small energy range. These singularities in the density of states are always present in our calculation since the band edges are flat [5]. To insure conservation of states, the integral of ~n up to the Fermi level must be zero. This is one way of stating the Friedel sum rule, and it is satisfied at the zeros of the total phase shift [2]. In all cases this sum rule is satisfied with the Fermi level in the middle of the band gap. 2.2. (2 X 1) Reconstruction, y-displacement
The unperturbed energy bands and Green’s function for the y-displacement reconstructed surface are the same as those for the z-displacement. The displacements of the atoms and their effects on y are shown in I. Since the structures are different for y- and z-displacements, the perturbation matrices are different. For the y-dis placement, the perturbation matrix becomes
4x
V(lA, 1A’,4 -
5)
=
2 cos
—
2
0 —v
—u”
~
—v
0 2~y) 0
*
0 v’~exp(—i2Ø3,) —vO 0
v
v 0
*
.
(27)
ueXP(i
The matrix elements are between the same basic vectors as eq. (25). Again, no surface states due to this type of reconstruction are found. In fig. 2,
An(E)
a
004
(2x1) ~r
0.2 0.7
0.02
~
~‘
-004 —‘-10 --‘-ID Fig. 2. Change in density of states per (2 X 1) reconstructed surface unit cell for in-plane displacement of the atoms. It is qualitatively the same as fig. 1.
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. ii
50
we show the change in the density of states. Qualitatively, it is very similar to th~ one for z-displacement shown in fig. 1. However, the exact structure is different and will result in a different gain in electronic energy for the same set of model parameters. This type of surface reconstruction has been considered previously [6]. 2.3. c(2 X 2) Reconstruction The geometry for this case and the corresponding SBZ are shown in I. The bulk band structure corresponding to this SBZ is given by 2 ~~/2 cos2 ~/2 cos2 ~~/2)1’2 ±~(E~ + 256 72 COS for / = l(+) and 2(—), but in contrast with eq. (15), we have ,
EkI
=
±~(E + 256 ~2 sin2 ~~/2 sin2 ~/2 cos2 ~~/2)~2
(28)
(29)
,
for! = 3(+) and 4(—).
The surface Green’s functions are derived in the same way as for the (2 X 1) case. We first convert the sum in eq. (1) to a sum over the c(2 X 2) SBZ, G(l~,1’~’,E) =
~ —
X exp
~
~ exp (i ~
~
x(l1’))
sgn(~~5~)rre~ — sgn(Ø~)rre~,E)
(~[45-_sgn(~)rre~_sgn(ø~)ire~]
x(ll’))]
.
(30)
Equating the coefficients of the sum over the two different SBZ’s, we obtain
~
G(L~A,L~A’,~5,if)~exp x(ll’)) ~ + G(1~I3,l~l3’,45 — sgn(~~)ire~ — sgn(t~~)ire~,E)
X exp
(_~sgn(Ø~) ~
x~(ll’) i sgn(Ø~)~x~(ll’))] —
,
(31)
where, again, Nn=~Ns,
X(LL’)
=
0
(32) ,
(33)
since all the atoms are in the same enlarged c(2 X 2) unit cell. Eq. (31) can be dissected similarly as was done in the (2 Xl) z-displacement case. However, the result is more lengthy and will not be given. The perturbation matrix is given by
S.L. Cunningham et al. / Surface reconstruction of two-band crystal. II
51
V(IA, 1A’,4
5)=
0
—l ~
0
—l ~~i(~x+~y)
0
—l ÷~1(øx~y) 0
~
—1 ÷~x~y) ~
C
0
1
1 _~i(øx+0y) 0 (34)
where, again, the rows (and columns) are labeled by L5 = I and A = 1, 2, 3, and 4, For this case, surface states are found to lie both below the valence band and above the conduction band for arbitrarily small e. The fact that reconstruction can create new surface states is not new. For the one-band model, simulating the transition metals, Dobrzynski and Mills found that reconstruction created surface states lying symmetrically below and above the energy band but only when the perturbation exceeded a certain critical value [7]. Thus, reconstruction can add new features to surface electronic spectra. In fig. 3, we show the positions of the surface states in relation to the surface energy bands. In each case the energy bands are cross sections along a particular segment in the SBZ. In fig. 3, the single cross-hatched areas denote a continuum of single bulk states, whereas the double cross-hatched area shows where two continua of bulk states overlap due to the folding back of the SBZ. In fig. 3a, the surface states are located near the zone boundary and merge into the energy bands for smaller wave vectors. At a higher value of icy, shown in fig. 3b. surface states exist (a) ——Surface Sfates
~
0.8 -~ C (2x2) 5
0.3 0.7
_______________________
-~
~
0.1
0.2
Fig. 3a
k~a0
0.16
0.18
0.20
52
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. II (b)
57t >-.
—-Surface States = 0
k~
C (20.3 x 2)
5.5 ~
Eq
~ U~
0 7
53j-
0075
-568
0.15
-
-5.70
-5.72
-
Fig. 3b Fig. 3. Relation of surface states to energy bands of the c(2 x 2) reconstruction along two segments in the SBZ. Surface states appear symmetrically below and above the valence and the conduction bands, respectively. Double cross-hatched areas denote overlap due to folding of the bands. (a) Surface states merge into the energy bands. (b) Surface states exist for all values of k~.No surface states are found inside the band gap.
for all values of k~.Surface states, when they exist, lie close to the band edges, but no surface states are found inside the band gap. For surface states lying below (above) an energy band, the partial phase shift jumps from 0 to ir (—ii) [2]. This introduces additional sharp features in the total phase shift. In fig. 4, we show the change in the density of states. The extra features in the phase shift lead to the oscillations near the band edges which are shown with an expanded energy scale. The positive peak below the band edge shows an increase in the density of states due to the presence of the surface states. As before, the net effect of reconstruction is to push states toward the bottom (top) of the valence (conduction) band. In comparison with fig. 2, we see that the changes in the density of states for the (2)< 1) and the c(2X 2) reconstruction are quite different.
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. II
53
~n(E)
-
-002
0
05~
-0.04 -0.5
—-0.3
-
—-10
--
— -
—-(0
IC
Fig. 4. Change in the density of states per c(2 X 2) surface unit cell. Rapid oscillations are due to the surface states which lie close to the band edges. In the expanded scale the positive finite peak in ~n is due to the surface states.
2.4. Ground state energy
The change in the ground state electronic energy is given by eq. (12) in I. The integration is carried out numerically for all the cases considered with the Fermi level set at the middle of the band gap (EF = 0). The results are plotted in fig. 5. There are three features to notice. First, for the values of e considered, our numerical resuits show a quadratic dependence of ~U on c. This was also found to be true for small values of e with the one-band model [7]. Second, for the (2X I) reconstruction, the results show that for all values of e and for a band gap of l0.7y, the lowering in the electronic energy is greater for an in-plane displacement than for a displacement perpendicular to the surface. For the same gap, the lowering is greater still for the in-plane motion of the c(2 X 2) reconstruction. Third, with all other parameters equal, the results show an inverse relation between the decrease in the electronic energy and the band gap. In fig. 6, we show the explicit dependence of /~Uon the band gap for two values of e. The decrease in the electronic energy is largest in the case of zero band gap (bcc metal). This can be understood in terms of the interaction between states with the same k but different in energy, i.e., the smaller the energy difference, the stronger the interaction. Therefore, the interaction between states in different bands increases as the band gap decreases and is the strongest in the zero band gap case. As a result, states are pushed more toward the bottom (top) of the valence (conduction)
54
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. 11 C
0.1
0.2
-0.01
—0.02
I
\~ \\...
C d
e
cb e
7 (2xI), Eg’IO. (2~’
0
b
A1J
0.3
I
—-—
d
a
(2x11, E~~0 (2X1)z, Eg~0.7 c(2X2>, E ‘(0.7
9
-0.03 Fig. 5. Change in the electronic ground state energy, ~U, as a function of the change e in the nearest neighbor hopping integral, -y. Subscript z in case (d) denotes displacement normal to the surface. In cases without subscripts, displacements are taken to be in the plane of the surface. The decrease in U is quadratic in .
band with decreasing energy gap, and the decrease in electronic energy is largest at zero band gap. In addition, as the gap increases the bands become more narrow, while the absolute maximum of the phase shift remains approximately constant. In the limit of infinite gap, the bands approach zero width, and ~U also goes to zero. From fig. 5, we see that reconstruction using our simple model always lowers the C
,
5
,
ID X
~0.2 -0.02 Fig. 6. Change in the electronic ground state energy, ~U, as a function of the energy gap, Eg, for two values of . Surface is reconstructed into a (2 x 1) structure with displacement in the plane of the sqrface.
S.L. Cunningham et a!. / Surface reconstruction of two-band crystal. II
electronic energy of the system, and the decrease becomes larger for increasing values of e (and hence, by assumption, for increasing values of atomic displacements). This is because the present calculation does not include the elastic strain energy. In I, we obtained the relation between the strain energy Q and the parameter . Inserting the values of the constants and using the mass of the iodine atom for M (as in Cs!), we obtain (35) (Q/y) 0.93 (a/c2)(e/y)2
.
We see that this simple model results in Q (as well as ~U) being quadratic in e. Consequently, whether the total energy ~ is positive or negative is determined entirely by the choice of constants a and c. In fig. 7, we present the two cases where Q < ~.Uand Q> ~U. For this figure, can be given approximately by [8] =
—0.30 (e/y)2
.
(36)
Thus, we see in comparison with eq. (35), that if a/c2 <0.32, then the total energy ~ will be negative as in fig. 7a. This situation leads to reconstruction. On the other hand, if a/c2 >0.32, the total energy ~1~’t will be positive as in fig. 7b,andno reconstruction will occur.
(a) —~U
0.03
12X1)
Eg -(0.7
-
0.02 C’vf~_ -~
0.01 —
-0.03
Fig. 7a
—
—
56
S.L. Cunningham eta!. /Surface reconstruction of two-band crystal. II (2x1)
(b)
0.03
//
5q107
/Q 0.02
-1;
C-—-
//
0.01
// ,,
/
,
____~‘ --
-—--
‘z:~ -0.03
-
~
Fig. 7b
Fig. 7. Change in the total electronic energy, ~Et, as a function of for the (2)) 1) reconstruction. Strain energy Q depends quadratically on e.The surface force constant is taken to be 1/2 the bulk value. Displacement is in the plane of the surface. (a) c = ~/3 in eq. (35); (b) c = -.J3/2.
3. Conclusions We have found that the presence of art empty conduction band is of fundamental importance in understanding reconstruction on semiconductor or insulator surfaces. Contrary to the results of Dobrzynski and Mills which showed that surface reconstruction has no effect on a one-band crystal when the band is filled [7], we find that if a second band is present, the magnitude of the decrease in the electronic energy upon reconstruction is comparable to that obtained on the bcc metal with a half-filled band. In other words, the fact that the electronic band structure may contain gaps is not of fundamental importance as far as the change in electronic energy due to reconstruction is concerned. What is important, however, is that the calculation of this energy change includes the effect of these empty conduction bands. This conclusion is not surprising in light of the simple chemical ideas of repulsion of levels upon coupling. The process of reconstruction discussed in this paper is not like those discussed elsewhere and which carry the labels Peierls transition, giant Kohn anomaly, or Jahri—Teller distortion [9]. For example, for the (2 X I) reconstruction on Si(1 11), the half-filled surface band in the gap of the unreconstructed crystal is split into two bands, one filled and one empty, due to the reconstruction [10]. The process discussed in this paper, on the other hand (and also by Dobrzynski and Mills [7]), does not require the system to open any new gaps in the density of states, or under-
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. II
57
go a metal—insulator transition. Rather, in our mechanism, if the system is initially a metal (or an insulator), it remains a metal (or an insulator) after reconstruction. We have found that the gain in the electronic energy is inversely proportional to the band gap, i.e., the smaller the band gap, the larger the effect of reconstruction. Thus, this mechanism will have its greatest effect on metals and narrow gap semiconductors and its least effect on large gap insulators, if all other parameters are equal. We have found that different reconstructed structures result in different changes in the electronic energy in spite of the fact that all of the model parameter values are identical and the fact that the model is very simple. Thus, it is possible, in principle, to use this method to predict the reconstructed structures on systems that have not yet been investigated experimentally. This is due to the fact simple models are often successful in predicting trends even though they fail at giving reliable absolute energies. This feature of being able to make predictions is not shared by all the other theoretical approaches to reconstruction. Most important, we have found that our simple model always gives a decrease in the electronic energy. This statement is true no matter how many bands (2 or more) are included and no matter what the reconstructed structures may be. This is because the process of reconstruction always allows new states to interact which did not interact before the reconstruction. This new interaction always pushes the states apart in energy. Thus, the lower energy states which are filled are always pushed down, and the higher energy states which are empty are pushed up. This always leads to a gain in the electronic energy. The reason this conclusion is important is that it now adds a burden to the theorist in future calculations. It will not be sufficient, as in the past, to show that any particular chosen reconstructed surface structure lowers the electronic energy [11]. It will also be necessary both to prove that the model used is sufficiently accurate to distinguish between various structures and that other structures do not have a lower electronic energy than the chosen structure. Meeting all these necessary criteria will not be easy.
Acknowledgment We acknowledge support of this research by the Army Research Office (Durham) under Grant No. DAHCO4-75-0 170.
References [11 S.L. Cunningham, W. Ho, W.H. Weinberg and L. Dobrzynski, previous paper, hereafter referred to as I. 121 W. Ho, S.L. Cunningham, W.H. Weinberg and L. Dobrzynski, Phys. Rev. B12(l 975) 3027. 131 T.H. DiStefano and W.E. Spicer, Phys. Rev. B7 (1973) 1554.
58
S.L. Cunningham eta!. / Surface reconstruction of two.band crystal. II
[4) S.L. Cunningham, Phys. Rev. BlO (1974) 4
98g. [5] L. Van Hove, Phys. Rev. 89(1953)1189. [61 W. Ho, S.L. Cunningham, W.H. Weinberg and L. Dobrzynski, Solid State Commun. 18 (1976) 429. 171 L. Dobrzynski and DL. Mills, Phys. Rev. B7 (1973) 2367. [81 A second degree polynomial fit to six data points was used to derive eq. (36). The constant term is zero and the linear term is negligible compared to the quadratic term for these values of e. [91 E. Tosatti and P.W. Anderson, Solid State Commun. 14 (1974) 773; Japan. J. AppI. Phys., Suppl. 2, Pt. 2 (1974) 381. 110] K.C. Pandey and J.C. Phillips, Phys. Rev. Letters 34(1975)1450; J.A. Appelbaum and DR. Hamann, Phys. Rev. B12 (1975) 1410. [11) D. Haneman, Phys. Rev. 121 (1961) 1093; N.R. Hanson and D. Haneman, Surface Sd. 2 (1964) 566; D.L. Heron and D. Haneman, Surface Sci. 21(1970)12.
SURFACE RECONSTRUCTION OF A TWO-BAND CRYSTAL II. Model resufts I’ Si. CUNNINGHAM, W. HO
*
and W.H. WEINBERG
**
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA and
L. DOBRZYNSKI Laboratoire de Physique des So/ides, Institut Supéricur d’Electronique do Nord, 3, rue Francois Baës, 59046 Lille Cedex, France Received 20 April 1977
The change in the total electronic energy which occurs upon reconstruction of the (001) face of a model two-band crystal with the CsCI structure is presented. The energy changes due to a (2X 1) and a c(2 X 2) reconstruction are considered. The results are obtained by a Green’s function perturbation technique developed in the previous paper. The Green’s functions appropriate for this reconstructed surface are analytic. The question of whether or not reconstruction occurs on a surface is shown to be dependent upon the relative magnitudes of the electronic energy and the strain energy.
1. Introduction In the first paper in this series, we developed the Green’s function formalism needed to study the effect of surface reconstruction on the electronic energy [1]. In this paper, we demonstrate the formalism by explicitly calculating the effect of reconstruction on a model two-band crystal. In a previous paper, we studied the ideal unreconstructed (001) surface of a crystal having the CsC1 structure [2] .We used the LCAO formalism, assumed the tight binding approximation, and derived analytic expressions for both the bulk and the surface Green’s functions. in this paper, we use the previously derived surface Work supported by AROD Grant No. DAFICO4-75-0170. *
**
Present address: Department of Physics, University of Pennsylvania, Philadelphia, Pennsylvania 19174, USA. Camille and Henry Dryfus Foundation Teacher—Scholar and Alfred P. Sloan Foundation Fellow.
S.L. Cunningham eta!.
/ Surface reconstruction of two -band crystal. II
45
Green’s functions to describe the unreconstructed surface.The change in the density of states due to the reconstruction is determined conveniently by the phase shift technique. The change in the electronic energy is then obtained by a simple integration. In the few cases where they occur, we find the surface states by the resolvent technique. Two different reconstructed structures are considered here. The surface atoms are rearranged arbitrarily in two different ways forming the (2 X 1) and the c(2 X 2) structures. In the (2 X 1) case, we study separately motions parallel and perpendicular to the surface. In addition, we study the change in the electronic energy as a function of the band gap. In all cases considered, we find a decrease in the total electronic energy for a filled band. Depending upon the relative strength of this energy compared to the unfavorable strain energy created by moving the surface atoms out of their bulk equilibrium positions, reconstruction can occur. In section 2 we present the Green’s function for the (1 X 1) surface of the model system. Then we determine the change in the density of states for (I) a (2 X 1) reconstruction where the atoms are displaced in the z-direction, (2) a (2X I) reconstruction where the atoms are displaced in the y-direction, and (3) a c(2 X 2) reconstruction. The atomic positions of each of these reconstructions are shown in I. Then we evaluate the change in the electronic ground state energy for each of the cases as a function of the magnitude of the change in the hopping integral. Finally, this energy is compared with the elastic strain energy, and parameter values are found which determine whether reconstruction will occur or not. In section 3, we summarize our conclusions.
2. Results and discussion The band structure for the bulk crystal referenced to the unreconstructedsurface Brillouin zone (SBZ), within the tight binding approximation and assuming only first neighbor interactions, ‘y, is given by Ek!
2cos2ø~/2cos2 ±~(E~ + 256 ‘y
0y12
cos2 ø~/2)u/2 ,
(1)
where 4i~’a 0k,
(2)
a0 is the crystal lattice constant, and Eg is the band gap which for this structure is equal to the magnitude of the difference in orbital energies, E1 and E2, between the two types of atoms [2]. There are two branches, and / takes on the values I and 2. The zero of the energy is chosen arbitrarily to be the average of Li and E2. For purposes of illustration, the parameters ‘y and Eg are fitted to the photoemission data for Cs! obtained by DiStefano and Spicer [3]. They found the width of the valence band to be 2.5 ±0.3 eV and the band gap at 300 K to be 6.2 eV.
46
S.L. Cunningham et a!.
/ Surface reconstruction
of two-band cr,vstal. II
Using these values, we find
y= 0.58eV,
(3)
EglY= I0.7.
(4)
The Green’s functions for crystals with the (001) unreconstructed surface obtained in our previous paper are given as the difference of two bulk Green’s functions [2]. They take on a rather simple form for the surface layer, l~= 1~= I, namely, 2(t + 1), (5) G(1 I, ll,4~)=dt/f G(12, 12,~
(6)
2, 5)~’at/f
G(l1, ~
(7)
,
G(12, ~
(8)
,
where
(9) d=E—E
(10) (11)
2, 4ycos~x/2cos~y/2, f
~(ad—
2f2)/2f2
1)1/2, ~+ i(l
—
(12)
,
~2)1/2
~> 1
I~I <1
(13)
2.1. (2 Xl) Reconstruction, z-diplacement The assumed displacement geometry for this situation and the corresponding SBZ are shown in I. To proceed, we must first derive the unperturbed surface Green’s functions appropriate for the (2 X 1) reconstruction. There are four basis atoms in the (2 X 1) reconstructed surface. Thus, the bulk band structure referenced to the new SBZ consists of four branches and is given by [2] ‘~kj= ±~(E + 25672 cos2 ~/2 cos2 ~~J/2 cos2 Ø~/
2)h/2 (14) where we have chosen the plus (minus) sign for / = 1 (2), and 2 ø~/2sin2 ~y/2 cos2 ø~/2)1/2 (15) + 25672 cos for / = 3(+) and 4(—). These equations result naturally from simply folding the bulk bands appropriate to the unreconstructed surface in eq. (1). ,
S.L. Cunningham eta!. / Surface reconstruction of two -band crystal. II
47
To derive the needed Green’s functions, we follow the procedure in I and express the Green’s function as a sum over the (2 X I) SBZ as follows G(l~,l’~’,E)-~~ ~ a
0
+ G(l~j3,l~’13,4~— sgn(Øy)irey, E)
exp 1
[~P5-_sgn(t~y)irey]
a0
•
x(ll’)
,
(16)
where sgn (cby) denotes the sign of the y-component of~,and is the unit vector in the y-direction. The sum over the (2 X 1) SBZ in eq. (16) still includes the same terms as the sum over the (I X 1) SBZ. Since we have chosen to distort the surface in the y-direction, the unit distance in the x-direction remains the same as the (1 Xl) surface, while that in the y-direction becomes twice as long. When referring to the same pair of atoms, we have simply
X~(ll’)=X~(LL’)
(17)
for the x-component of X(ll’) and X(LL’). However, the corresponding equality for the y-component does not always hold. We equate the coefficients of the sums over the (1 XI) SBZ with the terms in eq. (16) and obtain immediately GA(L2A,L~AP,~5,E) ~exp + G(l5~,~
—
~
[x~(ll’) _xy(LL’)]) [G(l2~,l~’,~5,E)
sgn(Øy) ~ey~E) exp
(_~sgn(~y) ~
xY(ll’))]
,
(18)
where we have used the fact that (19)
Nn=~Ns.
We can further simplify eq. (18) by evaluating the exponentials. Eq. (18) may then be dissected into the following O(L~13,L~f3’,45,E)= ~[G(l~1J, l’,4~,E)+ G(l~f3,l~’j3’,~5 sgn(~y)irey,E)] (20) —
=
G(L~I3+ 2,L~p’+ 2,4~,E),
G(L5i3, Lj3’ ÷~ —
if)
(21)
~ exp(—it~~) [G(l~l’3, 1~l3’,i~, if)
G(lj3, l~f3’,4~ sgn(~y)7rey~E)] —
,
(22)
G(L~/3+2,L~f3’,.fi5,E)=~ exp(i~,)[G(l~13,lj’3’,45,E) —G(l~I3,!i3’,45—sgn(~y)irey,E)]
,
(23)
where ~3,13’1,2, and Xy(LL’)O,
(24)
S.L. Cunningham et a!. / Surface reconstruction of two-band crystal. II
48
since all the atoms are in the same enlarged unit cell. Thus, we see that the layer-bylayer reconstructed surface Green’s functions are obtained as a sum or difference of two known (I X 1) surface Green’s functions. The perturbation matrix is obtained as in I. The periodic change in the nearest neighbor hopping integrals is shown in I. The perturbation is confined to the first and second planes both of which are in the first layer and is given by
-
V(1A, L4
,
,4~)= 2e cos 2x —
0
v~
0
v” exp(—i2~1,)
0
—v’
—v c~
—v’~
—v
0
V
exp(i2~) 0
.
(25)
where the rows (and colums) are labeled by the index L~ 1 and A = 1, 2, 3, and 4, and
v
=
exp(i~~/2).
(26)
We recall that e, in units ofy, is the change in y due to reconstruction. The Green’s function G that is needed is therefore reduced to a (4 X 4) matrix with 1~= l~= L~= = 1 in eqs. (20)—(23). Once V and G are known, the effects of reconstruction on the electronic structure of the surface can be studied within the formalism developed in 1. The sum over the SBZ is performed using the special points sampling procedure [4]. There are no surface states for this reconstructed surface structure. hi fig. 1, we 0.04
~n(E)
(2xl) z-displacement 0.7
0.02
~ -0.04 ‘—-0.1 ——0.1 Fig. 1. Change in the density of states per (2 X 1) reconstructed surface unit cell for displacements of atoms normal to the surface. Singularities at the band edges are due to the flatness of the bands.
S.L. Cunningham eta!. I Surface reconstruction of two-band crystal. II
49
present the change in the density of states. States are shifted toward the bottom (top) of the valence (conduction) band, leading to a decrease in the electronic energy. States are also pushed toward the band gap region in a very small energy range. These singularities in the density of states are always present in our calculation since the band edges are flat [5]. To insure conservation of states, the integral of ~n up to the Fermi level must be zero. This is one way of stating the Friedel sum rule, and it is satisfied at the zeros of the total phase shift [2]. In all cases this sum rule is satisfied with the Fermi level in the middle of the band gap. 2.2. (2 X 1) Reconstruction, y-displacement
The unperturbed energy bands and Green’s function for the y-displacement reconstructed surface are the same as those for the z-displacement. The displacements of the atoms and their effects on y are shown in I. Since the structures are different for y- and z-displacements, the perturbation matrices are different. For the y-dis placement, the perturbation matrix becomes
4x
V(lA, 1A’,4 -
5)
=
2 cos
—
2
0 —v
—u”
~
—v
0 2~y) 0
*
0 v’~exp(—i2Ø3,) —vO 0
v
v 0
*
.
(27)
ueXP(i
The matrix elements are between the same basic vectors as eq. (25). Again, no surface states due to this type of reconstruction are found. In fig. 2,
An(E)
a
004
(2x1) ~r
0.2 0.7
0.02
~
~‘
-004 —‘-10 --‘-ID Fig. 2. Change in density of states per (2 X 1) reconstructed surface unit cell for in-plane displacement of the atoms. It is qualitatively the same as fig. 1.
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. ii
50
we show the change in the density of states. Qualitatively, it is very similar to th~ one for z-displacement shown in fig. 1. However, the exact structure is different and will result in a different gain in electronic energy for the same set of model parameters. This type of surface reconstruction has been considered previously [6]. 2.3. c(2 X 2) Reconstruction The geometry for this case and the corresponding SBZ are shown in I. The bulk band structure corresponding to this SBZ is given by 2 ~~/2 cos2 ~/2 cos2 ~~/2)1’2 ±~(E~ + 256 72 COS for / = l(+) and 2(—), but in contrast with eq. (15), we have ,
EkI
=
±~(E + 256 ~2 sin2 ~~/2 sin2 ~/2 cos2 ~~/2)~2
(28)
(29)
,
for! = 3(+) and 4(—).
The surface Green’s functions are derived in the same way as for the (2 X 1) case. We first convert the sum in eq. (1) to a sum over the c(2 X 2) SBZ, G(l~,1’~’,E) =
~ —
X exp
~
~ exp (i ~
~
x(l1’))
sgn(~~5~)rre~ — sgn(Ø~)rre~,E)
(~[45-_sgn(~)rre~_sgn(ø~)ire~]
x(ll’))]
.
(30)
Equating the coefficients of the sum over the two different SBZ’s, we obtain
~
G(L~A,L~A’,~5,if)~exp x(ll’)) ~ + G(1~I3,l~l3’,45 — sgn(~~)ire~ — sgn(t~~)ire~,E)
X exp
(_~sgn(Ø~) ~
x~(ll’) i sgn(Ø~)~x~(ll’))] —
,
(31)
where, again, Nn=~Ns,
X(LL’)
=
0
(32) ,
(33)
since all the atoms are in the same enlarged c(2 X 2) unit cell. Eq. (31) can be dissected similarly as was done in the (2 Xl) z-displacement case. However, the result is more lengthy and will not be given. The perturbation matrix is given by
S.L. Cunningham et al. / Surface reconstruction of two-band crystal. II
51
V(IA, 1A’,4
5)=
0
—l ~
0
—l ~~i(~x+~y)
0
—l ÷~1(øx~y) 0
~
—1 ÷~x~y) ~
C
0
1
1 _~i(øx+0y) 0 (34)
where, again, the rows (and columns) are labeled by L5 = I and A = 1, 2, 3, and 4, For this case, surface states are found to lie both below the valence band and above the conduction band for arbitrarily small e. The fact that reconstruction can create new surface states is not new. For the one-band model, simulating the transition metals, Dobrzynski and Mills found that reconstruction created surface states lying symmetrically below and above the energy band but only when the perturbation exceeded a certain critical value [7]. Thus, reconstruction can add new features to surface electronic spectra. In fig. 3, we show the positions of the surface states in relation to the surface energy bands. In each case the energy bands are cross sections along a particular segment in the SBZ. In fig. 3, the single cross-hatched areas denote a continuum of single bulk states, whereas the double cross-hatched area shows where two continua of bulk states overlap due to the folding back of the SBZ. In fig. 3a, the surface states are located near the zone boundary and merge into the energy bands for smaller wave vectors. At a higher value of icy, shown in fig. 3b. surface states exist (a) ——Surface Sfates
~
0.8 -~ C (2x2) 5
0.3 0.7
_______________________
-~
~
0.1
0.2
Fig. 3a
k~a0
0.16
0.18
0.20
52
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. II (b)
57t >-.
—-Surface States = 0
k~
C (20.3 x 2)
5.5 ~
Eq
~ U~
0 7
53j-
0075
-568
0.15
-
-5.70
-5.72
-
Fig. 3b Fig. 3. Relation of surface states to energy bands of the c(2 x 2) reconstruction along two segments in the SBZ. Surface states appear symmetrically below and above the valence and the conduction bands, respectively. Double cross-hatched areas denote overlap due to folding of the bands. (a) Surface states merge into the energy bands. (b) Surface states exist for all values of k~.No surface states are found inside the band gap.
for all values of k~.Surface states, when they exist, lie close to the band edges, but no surface states are found inside the band gap. For surface states lying below (above) an energy band, the partial phase shift jumps from 0 to ir (—ii) [2]. This introduces additional sharp features in the total phase shift. In fig. 4, we show the change in the density of states. The extra features in the phase shift lead to the oscillations near the band edges which are shown with an expanded energy scale. The positive peak below the band edge shows an increase in the density of states due to the presence of the surface states. As before, the net effect of reconstruction is to push states toward the bottom (top) of the valence (conduction) band. In comparison with fig. 2, we see that the changes in the density of states for the (2)< 1) and the c(2X 2) reconstruction are quite different.
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. II
53
~n(E)
-
-002
0
05~
-0.04 -0.5
—-0.3
-
—-10
--
— -
—-(0
IC
Fig. 4. Change in the density of states per c(2 X 2) surface unit cell. Rapid oscillations are due to the surface states which lie close to the band edges. In the expanded scale the positive finite peak in ~n is due to the surface states.
2.4. Ground state energy
The change in the ground state electronic energy is given by eq. (12) in I. The integration is carried out numerically for all the cases considered with the Fermi level set at the middle of the band gap (EF = 0). The results are plotted in fig. 5. There are three features to notice. First, for the values of e considered, our numerical resuits show a quadratic dependence of ~U on c. This was also found to be true for small values of e with the one-band model [7]. Second, for the (2X I) reconstruction, the results show that for all values of e and for a band gap of l0.7y, the lowering in the electronic energy is greater for an in-plane displacement than for a displacement perpendicular to the surface. For the same gap, the lowering is greater still for the in-plane motion of the c(2 X 2) reconstruction. Third, with all other parameters equal, the results show an inverse relation between the decrease in the electronic energy and the band gap. In fig. 6, we show the explicit dependence of /~Uon the band gap for two values of e. The decrease in the electronic energy is largest in the case of zero band gap (bcc metal). This can be understood in terms of the interaction between states with the same k but different in energy, i.e., the smaller the energy difference, the stronger the interaction. Therefore, the interaction between states in different bands increases as the band gap decreases and is the strongest in the zero band gap case. As a result, states are pushed more toward the bottom (top) of the valence (conduction)
54
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. 11 C
0.1
0.2
-0.01
—0.02
I
\~ \\...
C d
e
cb e
7 (2xI), Eg’IO. (2~’
0
b
A1J
0.3
I
—-—
d
a
(2x11, E~~0 (2X1)z, Eg~0.7 c(2X2>, E ‘(0.7
9
-0.03 Fig. 5. Change in the electronic ground state energy, ~U, as a function of the change e in the nearest neighbor hopping integral, -y. Subscript z in case (d) denotes displacement normal to the surface. In cases without subscripts, displacements are taken to be in the plane of the surface. The decrease in U is quadratic in .
band with decreasing energy gap, and the decrease in electronic energy is largest at zero band gap. In addition, as the gap increases the bands become more narrow, while the absolute maximum of the phase shift remains approximately constant. In the limit of infinite gap, the bands approach zero width, and ~U also goes to zero. From fig. 5, we see that reconstruction using our simple model always lowers the C
,
5
,
ID X
~0.2 -0.02 Fig. 6. Change in the electronic ground state energy, ~U, as a function of the energy gap, Eg, for two values of . Surface is reconstructed into a (2 x 1) structure with displacement in the plane of the sqrface.
S.L. Cunningham et a!. / Surface reconstruction of two-band crystal. II
electronic energy of the system, and the decrease becomes larger for increasing values of e (and hence, by assumption, for increasing values of atomic displacements). This is because the present calculation does not include the elastic strain energy. In I, we obtained the relation between the strain energy Q and the parameter . Inserting the values of the constants and using the mass of the iodine atom for M (as in Cs!), we obtain (35) (Q/y) 0.93 (a/c2)(e/y)2
.
We see that this simple model results in Q (as well as ~U) being quadratic in e. Consequently, whether the total energy ~ is positive or negative is determined entirely by the choice of constants a and c. In fig. 7, we present the two cases where Q < ~.Uand Q> ~U. For this figure, can be given approximately by [8] =
—0.30 (e/y)2
.
(36)
Thus, we see in comparison with eq. (35), that if a/c2 <0.32, then the total energy ~ will be negative as in fig. 7a. This situation leads to reconstruction. On the other hand, if a/c2 >0.32, the total energy ~1~’t will be positive as in fig. 7b,andno reconstruction will occur.
(a) —~U
0.03
12X1)
Eg -(0.7
-
0.02 C’vf~_ -~
0.01 —
-0.03
Fig. 7a
—
—
56
S.L. Cunningham eta!. /Surface reconstruction of two-band crystal. II (2x1)
(b)
0.03
//
5q107
/Q 0.02
-1;
C-—-
//
0.01
// ,,
/
,
____~‘ --
-—--
‘z:~ -0.03
-
~
Fig. 7b
Fig. 7. Change in the total electronic energy, ~Et, as a function of for the (2)) 1) reconstruction. Strain energy Q depends quadratically on e.The surface force constant is taken to be 1/2 the bulk value. Displacement is in the plane of the surface. (a) c = ~/3 in eq. (35); (b) c = -.J3/2.
3. Conclusions We have found that the presence of art empty conduction band is of fundamental importance in understanding reconstruction on semiconductor or insulator surfaces. Contrary to the results of Dobrzynski and Mills which showed that surface reconstruction has no effect on a one-band crystal when the band is filled [7], we find that if a second band is present, the magnitude of the decrease in the electronic energy upon reconstruction is comparable to that obtained on the bcc metal with a half-filled band. In other words, the fact that the electronic band structure may contain gaps is not of fundamental importance as far as the change in electronic energy due to reconstruction is concerned. What is important, however, is that the calculation of this energy change includes the effect of these empty conduction bands. This conclusion is not surprising in light of the simple chemical ideas of repulsion of levels upon coupling. The process of reconstruction discussed in this paper is not like those discussed elsewhere and which carry the labels Peierls transition, giant Kohn anomaly, or Jahri—Teller distortion [9]. For example, for the (2 X I) reconstruction on Si(1 11), the half-filled surface band in the gap of the unreconstructed crystal is split into two bands, one filled and one empty, due to the reconstruction [10]. The process discussed in this paper, on the other hand (and also by Dobrzynski and Mills [7]), does not require the system to open any new gaps in the density of states, or under-
S.L. Cunningham eta!. / Surface reconstruction of two-band crystal. II
57
go a metal—insulator transition. Rather, in our mechanism, if the system is initially a metal (or an insulator), it remains a metal (or an insulator) after reconstruction. We have found that the gain in the electronic energy is inversely proportional to the band gap, i.e., the smaller the band gap, the larger the effect of reconstruction. Thus, this mechanism will have its greatest effect on metals and narrow gap semiconductors and its least effect on large gap insulators, if all other parameters are equal. We have found that different reconstructed structures result in different changes in the electronic energy in spite of the fact that all of the model parameter values are identical and the fact that the model is very simple. Thus, it is possible, in principle, to use this method to predict the reconstructed structures on systems that have not yet been investigated experimentally. This is due to the fact simple models are often successful in predicting trends even though they fail at giving reliable absolute energies. This feature of being able to make predictions is not shared by all the other theoretical approaches to reconstruction. Most important, we have found that our simple model always gives a decrease in the electronic energy. This statement is true no matter how many bands (2 or more) are included and no matter what the reconstructed structures may be. This is because the process of reconstruction always allows new states to interact which did not interact before the reconstruction. This new interaction always pushes the states apart in energy. Thus, the lower energy states which are filled are always pushed down, and the higher energy states which are empty are pushed up. This always leads to a gain in the electronic energy. The reason this conclusion is important is that it now adds a burden to the theorist in future calculations. It will not be sufficient, as in the past, to show that any particular chosen reconstructed surface structure lowers the electronic energy [11]. It will also be necessary both to prove that the model used is sufficiently accurate to distinguish between various structures and that other structures do not have a lower electronic energy than the chosen structure. Meeting all these necessary criteria will not be easy.
Acknowledgment We acknowledge support of this research by the Army Research Office (Durham) under Grant No. DAHCO4-75-0 170.
References [11 S.L. Cunningham, W. Ho, W.H. Weinberg and L. Dobrzynski, previous paper, hereafter referred to as I. 121 W. Ho, S.L. Cunningham, W.H. Weinberg and L. Dobrzynski, Phys. Rev. B12(l 975) 3027. 131 T.H. DiStefano and W.E. Spicer, Phys. Rev. B7 (1973) 1554.
58
S.L. Cunningham eta!. / Surface reconstruction of two.band crystal. II
[4) S.L. Cunningham, Phys. Rev. BlO (1974) 4
98g. [5] L. Van Hove, Phys. Rev. 89(1953)1189. [61 W. Ho, S.L. Cunningham, W.H. Weinberg and L. Dobrzynski, Solid State Commun. 18 (1976) 429. 171 L. Dobrzynski and DL. Mills, Phys. Rev. B7 (1973) 2367. [81 A second degree polynomial fit to six data points was used to derive eq. (36). The constant term is zero and the linear term is negligible compared to the quadratic term for these values of e. [91 E. Tosatti and P.W. Anderson, Solid State Commun. 14 (1974) 773; Japan. J. AppI. Phys., Suppl. 2, Pt. 2 (1974) 381. 110] K.C. Pandey and J.C. Phillips, Phys. Rev. Letters 34(1975)1450; J.A. Appelbaum and DR. Hamann, Phys. Rev. B12 (1975) 1410. [11) D. Haneman, Phys. Rev. 121 (1961) 1093; N.R. Hanson and D. Haneman, Surface Sd. 2 (1964) 566; D.L. Heron and D. Haneman, Surface Sci. 21(1970)12.