Electronic structure of ideal and relaxed surfaces of ZnO: A prototype

PHYSICAL REVIEW 8

VOLUME 24, NUMBER 12

I 5

DECEM HER

1981

Electronic structure of ideal and relaxed surfaces of ZnO: A prototype ionic wurtzite semiconductor and its surface properties

I. Ivanov*

and

J. Pollmann

Institutfur I'hysik, Universitat Dortmund, 4600 Dortmund 50, West Germany (Received 18 May 1981) We report the results of a detailed theoretical investigation of electronic properties of unrelaxed and relaxed ZnO surfaces. The surface electronic structure is evaluated using the scattering theoretical method on the basis of an accurately fit empirical tight-binding Hamiltonian for bulk ZnO. Surface band structures and wave-vector-resolved as weH as wave-vector-integrated layer densities of states for the polar Zn- and O-terminated (0001/0001) surfaces and for the nonpolar (1010) and (1120) surfaces are fully discussed. In agreement with experiment, we find no bound surface states in the gap energy region. Instead of anion- and cation-derived dangling-bond surface states familiar at the ZnO surface ionic resonances occur which lie well within from more covalent zinc-blende semiconductors the projections of the bulk valence and conduction bands. In addition, coualent back-bond surface states are found which are derived from Zn 4s-0 2p mixed bulk states. Finally we find surface states near the top of the projected conduction bands which are the antibonding counterparts of the covalent back-bond states. The nature and origins of all these surface features are discussed in detail. We have studied, as well, the relaxed (1010) surface using the relaxation model proposed by Duke et al. in order to identify characteristic relaxation-induced effects. We find that surface relaxation affects the ionic resonances only marginally while the more covalent back-bond and anti-backand electron-energy-loss bond states are stronger influenced. Comparing our results with ultraviolet-photoelectron spectroscopy data we find very good agreement. In the course of the discussion of our results we develop a very general picture of typical surface electronic properties of tetrahedraHy coordinated, ionic semiconductors which is found to be in good accord with the experimentally observed trends.

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I. INTRODUCTION The theory of electronic properties of semiconductor surfaces has been a very active field in the past decade (see, e.g. , Refs. 1 and 2). Almost all investigations, however, have concentrated on the low-index faces of diamond- and zinc-blende-type materials for the obvious reason that these semiconductors are mogt important technologically. The surfaces of wurtzite-type semiconductors are of considerable interest for technological applications, as well, e.g. , as for space-charge layers, Schottky barriers, varistors, and optical storage devices' or as cataTherefore, they have been intensively lysts. studied experimentally in the past several In order to be able to make optimal use years. ' of these materials' properties in device fabrication, a detailed knowledge of their surface electronic properties is highly desirable. Nevertheless, no detailed theoretical study of the surface electronic structure of these materials" has been published, up to date, It was only very recently that both I ee and Joannopoulos" as mell as the present authorss very briefly reported about first surface electronic structure calculations for the ideal, nonpolar (1010) surface of a wurtzite-type semiconductor, namely ZnQ. In our previous communication'4 we presented, as mell, a few results on the polar ZnQ surfaces. In Ref. 35 we gave a short discussion of the main influences of relaxation effects on the nonpolar (1010) surface.

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In this paper, we present a full account of our investigations of the surface electronic structure of ideal and relaxed ZnO surfaces. We have chosen ZnO as a prototype wurtzite-type material for a number of reasons. (a) There exists a wealth of experimental information on the low-index faces of ZnO (Befs. 3-31). (b) ZnO is a compound semiconductor, whose ionicity (J; = 0. 616 on the Phillips ionicity scale" ) resides exactly at the borderl. ine between covalent and ionic semiconductors. It lies at the "ionic extreme" of the tetrahedraH. y coordinated compound semiconductors whose zincblende or wurtzite structure leads to their classification as covalently bonded bulk materials. Studying the surface electronic properties of ZnO, therefore, yields interesting information about ionicity-induced features in the surface electronic structure of tetrahedraHy coordinated, ionic compound semiconductors. (c) The nonpolar ZnO(1010) surface seems to be a natural model system for well-defined reversible exthermodynamically perimental studies. 4 The atomic geometry of this natural cleavage face is mell understood from the results of low-energy electron diffraction (LEED) measurements and a dynamical LEED analysis of the data. We can, therefore, use this well-established relaxation model in our studies of relaxation effects on the electronic structure. (d) A detailed study of the electronic properties of ldeRl Rnd relRxed, cleRn ZnQ surfaces is an important and necessary precursor" for investigations of defects (e.g. , 0 vacancies) and

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1981 The American Physical Society

I.

7276

IVANOV

AND

adsorption processes Ie.g. , 0, or CO on (1010)] at the ZnO surfaces. A detailed knowledge of such systems might turn out to be a key for an understanding of the catalytic action of, e.g. , ZnO(1010) in the CO, catalysis. Our calculations have been carried out using the scattering theoretical method' which we will refer to in the following as STM. The STM, by now, has been shown to be a very versatile and efficient tool in surface, interface, and surfaceor interface-defect theory. ~ In this rq. ethod, the surface is treated as a two-dimensionally periodic perturbation of an otherwise perfect bulk solid, which is localized in the direction perpendicular to the surface. Once the bulk band structure, the Bloch functions, and thus the one-particle bulk Green's function are given, the STM solves the surface problem (lack of periodicity perpendicular to the surface) exactly. Boundstate energies, i.e. , the surface band structure, total cha. nges in the density of states (DOS) and local densities of states (LDOS) resolved with respect to the k„wave vector or with respect to can be diparticular orbitals or symmetries rectly computed. The first step in the theory is the determination of the bulk electronic structure of the material under study. %'e use the empirical tight-binding method (ETBM) for a reliable and very realistic description of the most important bulk valence and conduction bands of ZnQ. The bulk band structure and density of states resulting from our ETBM treatment are discussed in detail in Sec. II. It turned out that inclusion of all second-nearestneighbor O-O and Zn-Zn interactions is necessary to obtain reliable bulk bands. The ideal surfaces are then "created" by "removing" sufficiently many layers in order to decouple the two resulting semi-infinite solids (for details see Refs. 2, 37, or 38). The surface electronic structure obtained in this way for the polar and nonpolar ZnO low-index faces is discussed in

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Sec. III. Comparing our results for the ideal surfaces with experimental electron-energy-loss (EELS) and ultraviolet-photoelectron spectroscopy (UPS)

data yields very satisfactory agreement. This is amazing at a first glance since it is well known that the polar faces are nonideal" and the nonpolar (1010) face undergoes an ionic type of relaxation. In order to resolve this issue, we have studied in detail the effects of surface relaxation for the nonpolar (1010) surface, for which the relaxation geometry is known best from exThe results of that study are disperiment. cussed in Sec. IV. The outcome of the discussion leads to the conclusion that in a strongly ionic

"" '

'

J.

POLI, MANN

material surface relaxation effects are much less than in a purely covalent semiconductor. This result then explains why our theoretical LDOS's for the ideal surfaces, can already yield a very satisfactory qualitative interpretation of experimentally determined features at the real surface s. In Sec. V we give a detailed comparison of our theoretical results with angle-integrated UPS and EELS data. Possible transitions between occupied and empty states at or near the nonpolar (1010) and the polar (0001 and 0001) surfaces are identified and found to agree remarkably well with the EELS peaks. From our LDOS's we can discriminate between surface-induced and mainly bulkderived transitions. Our identification of strongly surface-induced transitions is in excellent agreement with the results of experimental determinations of the surface sensitivity of some EELS peaks by contaminationxx x' and primary energy variation" tests. The summary in Sec. VI concludes the paper. pronounced

II. EMPIRICAL TIGHT-BINDING DESCRIPTION OF BULE ZnO

The starting point for our STM treatment of surfaces is the description of the bulk material. A basic understanding of the bulk electronic properties, in particular, establishes a useful background for a meaningful interpretation of the outcome of involved surface electronic structure calculations. Let us discuss, therefore, in detail our empirical tight-binding Hamiltonian and the resulting ZnO bulk band structure first. ZnO is a tetrahedrally coordinated semiconductor, which crystallizes in the hexagonal wurtzite structure. The bulk unit cell contains two Zn cations and two O anions. The crystal can be viewed as a sequence of 0-Zn double layers which are stacked along the c axis, i.e. , along (0001). The structure parameters are the second-nearestneighbor distance e = 3.25 A and the ratio c/e =1.602 (see Ref. 44). We use throughout this paper the symmetry notation of Ref. 45. The theory of the ZnO bulk band structure has an "Olympic history. Since 1969, every four years there has been one publication on this subject. Bossier" reported in 1969 the first bulk band structure of ZnO which was evaluated by a calculation. In 1973 Korringa-Kohn-Rostoker Bloom and Ortenburger~' reported an ernPirical pseudopotential calculation and in 1977 Chelikowsky" published the first self consistently termined bulk band structure using a nonlocal pseudopotential approach. It is now again four years later that we report about bulk ZnO properties, which we have determined from an acZnQ

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EI, ECTRONIC STRUCTURE OF IDEAL AND RELAXED. . . curately fit second-nearest-neighbor ETBM Hamiltonian. One of the major controversies in the discussion of the bulk bands in the early 1970's has been the energetic position of the Zn3d' bands. That controversy finally has been settled by both ' and experimental" theoretical results which clearly show that the Zn3d bands lie some 3 eV below the s-p valence bands with their DOS peak near -7.8 eV belork the top of the valence bands. The notion that the Zn 3d states have no significant influence on the bonding in ZnO was, as well, confirmed by a systematic study of the Zn3d energy level positions in zinc-chalcogenide crystals. It was found that in the four compounds ZnO, ZnSe, ZnS, and ZnTe the Zn3d DOS peak has the same half-width and the same energy separation from the vacuum level. These observations lend strong support to the fact that the closed Zn3d shell can be visualized as a full "corelevel" shell although it is energetically separated from the other valence bands in ZnO by only 3 eV. In order to arrive at a meaningful and realistic ETBM Hamiltonian, we start out from the atomic term values that are shown in Fig. 1(a). For a reliable ETBM bulk band-structure fit it is important to have sufficiently many parameters to describe the essential physics correctly. At the same time, however, it is necessary to retain as few parameters as possible so that their physical significance remains clear and the fit is not enIt turned out in our caltirely overdetermined. culations, that the Zn0 bulk valence and conduction bands (from —6 to +10 eV) can very accurately be fitted with a five-parameter Hamiltonian. In this Hamiltonian we take into account only the Zn 4s and the 0 2' orbitals [see Fig. 1(b)]. With two Zn and two 0 atoms per unit cell we thus have to diagonalize an 8x 8 Hamiltonian matrix to obtain the bulk band structure. The 02' states lie deep below the top of the valence bands as was shown experimentally" and theoretically" and they are found to form a narrow band near -20 eV below p, . These orbitals do not hybridize to any significant extent with the

'

"

0

Zn

1

V, p

1)—

4s (8.

2p (14. 2)

1)—

3d (17.

states. 4'

The Zn3& bands are nicely separated from the g-p valence bands, as well, as was found in UPS experiments" ~ and in the self-consistent pseudopotential calculation. We have carried out test calculations including the Zn 3d orbitals and have found, in agreement with the previous results, only a very small admixture of the Zn3d states into the lowest part of the s-p valence bands (along the I'-M line). Since the d bands are not involved anyway in the chemical bonding" they will be influenced only marginally by the surface creation. We, therefore, decided to omit the closed

"

""

"

-E„=O

--20

2

Vpp~

0 - Ep=-

""

0 2p

0

Z fl

2

Vppo 1

V, p

Zs (29.1)

l.

FIG. Atomic energy levels of the Zn and 0 atoms as given in the Herman-Skillman tables; the energy reference is the vacuum level in (a) . (b) shows the g-p valence- and conduction-band groups resulting from the Zn4s and 02p electrons shcematically. The origin and character of the different band groups is indicated by the different hatchings. In addition, the tight-binding matrix elements which affect a particular band group mostly are given next to that band group (for more details, see text).

Zn 3d "core-level" shell altogether in order to arrive at a much more efficient and, in particular, physically transparent description of the most ZnO bulk valence and conduction bands in terms of a five-parameter ETBM Hamiltonian.

important

The physical significance of these parameters (see Fig. 1). Taking

is easily demonstrated into account all firstinteractions, U', „(Zn-Zn),

and second-nearest-neighbor

F»(O), E„(Zn), U,', (Zn-O), U», (O-O), and U~, „(O-O). We use we have

the usual nomenclature of the tight-binding matrix elements (see, e. g. , Ref. 59). The first parameter entering the calculations is the difference between the on-site energy levels F„(Zn) —E»(O). That difference can be determined from fitting the gap energy correctly. The upper part of the valence bands is mainly 02& derived. The 0-0 second-neare st-neighbor interactions and

p',

&~2

„, therefore, are entirely determined

by

a

careful. fit of the 02p valence bands. The lower part of the conduction bands is mainly Zn4+ derived, so that again a second-nearest-neighbor interaction, the Zn-Zn V,', matrix element, determines these bands. Finally we are left with Zn-0 p,' interaction. the first-nearest-neighbor As can be seen in the schematic Fig. 1(b) that interaction eventually determines both the total conduction and valence bandwidths so that it can

I.

IVANOV

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AND

be chosen, as well, on the basis of good and trans'This very schematic parent physical reasons. discussion has made clear that there is certainly no physical/y meaningful way of fitting the ZnO bulk band structure with a first-nearest-neighbor ETBM Hamiltonian, only. %'e thus started out with an "educated guess" for the five parameters on the basis of the above discussion. The "fine tuning" was done by minimizing the deviations of our bulk band structure from the self-consistent pseudopotential results of Chelikowsky ' and by checking our DQS with experimental~ UPS and mainly x-ray photoelectron The five parameters, spectroscopy (XPS) data. representing our ZnQ ETBM Hamiltonian are given in Table I. The bulk band structure and the bulk density of states are given in Fig, 2. Qur band structure is found to be in good agreement with the previous calculations, ~' ' in particular, with the self-consistent band structure of Chelikowsky. 8 Small deviations exist only for the lowest valence band along the I"-M-L, and K-I' lines. In Ref. 48 this band is slightly higher in energy than ours and it has more dispersion along I - jg and I -K. In order to ease the comparison of the calculated DQS with the XPS spectrum we have included in Fig. 2 a 0.3-eV lifetime-broadened DGS as well. The broadened DGS is in excellent agreement with

""

J.

POI I, MANN

I.

ZnG in

eV. The en-

ergy levels of the Zn4s and 02p states are and E&&=-2, 1 eV, respectively.

E~"=5.7 eV

TABLE

ETBM parameters

for

.

84z,

—8»

y, &(zn-O)

y2, (O-O)

y~

(G-0)

y2 (Zn- yn)

""

the XPS spectrum, as can be seen in Fig. 2. The comparison of the DQS with the XPS data, of course, can only prove that our ETBM Hamiltonian accurately descri, bes the occupied states, i.e. , the valence bands. From the comparison of our conduction bands with Chelikowsky's results" we can conclude, in addition, that our Hamiltonian also describes the conduction bands very accurately. As a matter of fact, it turns out that it is much easier to determine a realistic ETBM Hamiltonian for ZnQ, which describes both the valence and conduction bands correctly, than for typical III-V or group-IV semiconductors, like GaAs or Si. There are simple reasons for this fact. Firstly, the gap in ZnO is fairly large (3.3 eV) so that only a small amount of g-p hybridization occurs. Secondly, therefore, the conduction and valence bands are mainly determined by different sets of on-site

—22

A

f6

LN

I

A

HK

DOS I ui b.

units)

FIG. 2. Bulk hand structure of ZnO for various high-symmetry lines in the irreducible part (see inset) of the hulk BriQouin zone and density of states. The full line showers the total density of states and the dotted line gives the cation contribution to the DOS. The difference between the two curves is thus the anion contribution to the DOS. For further comparison of the DOS vrith XPS data (dashed line) a lifetime-broadened (y= 0.3 eV) DOS has been included, as dwell. Note that the shadings, characterizing the differen. t band groups in Fig. 1+), have been included in this figure bebveen the band structure and the DOS plots.

IDEAL AND RELAXED. . . and interaction matrix elements so that they can essentially be fitted "separately. The third and most important reason for the good conductionband fit is the large gap between the Zn4g-Q 2p antibonding conduction bands and the higher conduction bands (which we did not include in our calculations) which begin at +12.5 eV (see Ref. 48). In order to elucidate the origins of the various bands we have included in Fig. 2 the sehematie shadings from Fig. 1(b) between the band structure and the DQS. As can be seen, the upper valence bands (from -3 up to 0 eV), which usually show a considerable g-p mixing in more covalent

"

cubic zinc-blende semiconductors, now become mainly ionic p-like states. Their dispersion, therefore, mainly stems from the second-nearestneighbor Q-Q inter actions. The unbroadened DQS in Pig. 2 clearly reveals this fact. Similarly, the conduction bands from 3.3 up to 7. 5 eV are mainly Zn4g derived and their dispersion is thus entirely determined by The lower part of the valence bands and the uppermost part of the conduction bands have more pronounced covalent character. For example the I', bonding and antibonding states are Q 2P, -Zn 48 mixtures. The I"", bonding state is 83% 02p, -1ike and 17% Zn4s-like while the I"', antibonding state has the opposite wavefunction admixture ratio. To conclude this section, we have determined a reliable ETBM Hamiltonian for bulk ZnQ which is completely specified in terms of five parameters, whose physical significance has been discussed. This bulk Hamiltonian now can serve as a reallstle basis and thus as a good starting point for our sur fac e electronic struc ture theory. %e have seen that the bulk states separate essentially into three different groups of bands: (a) the ionic 02p valence bands from —3 to 0 eV, (b) the ionic Zn4s conduction bands from 3.3 to 7. 5 eV~ and (c) the somewhat covalent bonding and antibonding Q 2pZn4@ bands in the energy regions from -5.0 to -3.0 eV and 7.5 to 8.5 eV, respectively.

P'„.

material. The new boundary conditions give rise to most of the characteristic features in the electronic stx'ueture of a particular surface. Relaxation and x eeonstruction effects, though being very important for quantitative analyses, very often turn out to yield no really new qualitative effects. Surface relaxation often shifts the occupied ideal surface states down and the empty ideal surface states uP in energy. Surface reconstruction, in addition, leads to backfolding and consecutive splitting and shifting of-the ideal surface states. Therefore, the very basis of a detailed understanding of the electronic properties of a, particular real surface is a complete knowledge of the nature and origins of the surface states and resonances at the corresponding ideal surface. A. Formalism

The formal treatment of ideal surfaces in the STM has been described in detail in Refs. 2 and 37. A full account of the theory, as it applies to wurtzite-type materials is described in Ref. 38. Here we briefly summarize the basic equations which we have used to study the ZnQ surfaces. In the STM, a surface is introduced as a twodimensionally periodic short-range pertux bation 0 of the perfect crystal. The surface creating perturbation U is localized in the direeti. on perpendicular to the surface and the bulk material is described by a realistic ETBM Hamiltonian (see Sec. II). The electronic properties of the perturbed solid are determined by treating the localized perturbation U' in the framework of potential scattering theory. Since U is a highly localized perturbation, the eigensolutions of the full Hamiltonian & =00+ U can be calculated exactly. The solutions may be separated into two categories: (a) the bound surface states and (b) the resonances. The bound states energetically lie within the forbidden gaps of the bulk spectrum, i.e. , of the spectrum of Ho. Their energies are determined by the zeros of the following determinant

III. THE IDEAL SURFACES OF ZnG

Ideal surfaces constitute a well-defined and very useful starting point for the investigation of charaetexistic surface electronic properties of a particular material. An ideal surface is defined as an abrupt termination of the bulk crystal at a particular plane, keeping all the intra- and interatomic interactions unchanged. In this approximation, no new parameters are necessary for the calculation of the electronic structure of the free surfaces. The electronic properties of an ideal surface simply result from the new boundary conditions at the soLid-vacuum interface as opposed to the periodic boundary conditions in the bulk

detP I

-6'(Z)v ii=0,

which in turn represent the discrete poles of the scattering matrix. Here G'(p) is the bulk Green's function defined as G

(E) =lim (F +ie —FI, ) q~+

(2)

The resonances within the band continua, i.e. , within the quasicontinuous spectrum of &„are most easily identified by calculating either the surface -induced change s in the density of state s or the various layer densities of states. The total density of states of the semi-infinite solid is given by

I.

7280

IVANOV

where G(E), the surface Green's function, obtained from the Dyson equation: G

AND

can be

=G'+G'UG.

(4)

Since It is a localized perturbation, Eq. (4) can easily be solved (see, e.g. , Refs. 2 and 37) so that we can obtain wave-vector-, atom-, or orbital-resolved layer densities of states from Eq. (3) by evaluating appropriate partial traces in a layer orbital basis. Layer orbitals constitute a symmetry-adapted set of basis functions. They are defined as lin'ear combinations of atomic orbitals which are centered on a particular layer. Since semi-infinite solids retain a two-dimensional periodicity parallel to their surface, these layer orbitals can be defined as two-dimensional Bloch sums in the following way: 1

e'~'f~i+ ui

ln, ii, m;q&=~

In,

p,

, m,

2

j),

'(5)

with

'The orthonormal atomic orbitals y have been used to represent the bulk Bloch functions as well. The two-dimensional Bloch vector is labeled Q in order to emphasize that the number of g values in the surface Brillouin zone is mostly different from the number of possible kit values, with kit being the surface-parallel component of a threedimensional bulk Bloch vector k. The atomic orbitals y„ in the gath surfa. ce-parallel plane are localized at the basis atom ~ in the two-dimenThe two-disional Bravais lattice unit cell mensional Bravais lattice, spanned by the vectors (pi} has N, unit cells in the normalization area. The perturbation matrix U is for all ideal surfaces in PxinciPle the same. We "remove" from the periodic bulk solid as many layers (parallel to the pa, rticular surface to be studied) as necessary to decouple completely the two resulting semi-infinite solids. The removal is accomplished by shifting the on-site elements of the electrons to be removed to infinity (for details see Refs. 2, 37, and 38). In consequence, the perturbation matrix is diagonal in the layer orbital representation and is given by

j.

(n, p. , m,' q

I

U

I

n ',

p.

', m", q&

6, g

=lim u6„, 6„„, ~ oo Q

r

6

„. (7)

The sum over z only runs over the layers to be

J.

POLLMANN

removed (typically two layers for our applications of the formalism to ZnO). The fI matrix, therefore, essentially is a very small unit matrix acting only on the layers to be removed. Representing Eq. (1) in the layer orbital basis yields avery simple equation for the bound surface states. They are given by the solution of detll&n,

y, m. 'qlG'(E) ln' Ji' nl'

q&Ii=0

(8)

where ~„and nz„' only run over the removed layers. The energy values E, (Q) for each Q vector in the surface Brillouin zone for which Eq. (7) is satisfied, represent the surface band structure (SBS). Layer densities of states can easily be obtained from Eqs. (3) and (4). In order to simplify the notation, we will occasionally use a "supe~index" ) as an abbreviation of ~, p, , and gpss so that

(I)~(n, ti, m). The layer density of states is then given as N, (4,

E) =--1m&, , , (4, ~),

where the matrix elements of the surface Qreen's function are obtained from Eq. (4). The equation can be rewritten in the layer orbital basis using the p matrix representation given in Eq. (7) as Gll =Gll

2

lyly

Gll, (G )i,t', Gl'„ l.

(11)

Here G' is the bulk Green's function submatrix in the subspace affected by U. This particularly simple form of the solutions of the Dyson equation is a direct consequence of our very efficient layer removal method for surfaces. We see that the layer density of states separates into the bulkderived contribution -ImG'„and the second term directly represents the surface-induced change in the LDOS. Note that Eq. (10) yields the LDOS resolved with respect to the wave vector g, the orbital ~, the atom p, , and the layer nz, since ) stands for ~, g, yg. More integral information is simply obtained by carrying out appropriate sums. The density of states for a particular layer, is given by N (E) = ——Im

p g g „„„(q,E) . G

(12)

q

The formulas (8), (10), and (11) represent a very transparent description of ideal surfaces. In particular, they allow for a very efficient evaluation of the surface electronic structure. All that is needed in the theory are diagonal and offdiagonal matrix elements of the bulk Qreen's function in the layer orbital representation. This outcome of the theory is not too astonishing since up

ELECTRONIC STRUCTURE OF IDEAL AND RELAXED. . . to now, according to our definition of ideal surfaces, we have not yet changed any of the quantities that describe the bulk electronic features. All that has been done in this section was to take into account exactly the new boundary conditions imposed by the solid-vacuum interface. The key quantity in the theory of electronic properties of ideal surfaces, therefore, is the one-particle Qreen's function which is given for the various

surfaces in the Appendix.

sec-

In the following

tions we will now discuss our xesults for the four most important low-index faces of ZnQ. B. The nonpolar (1010) and (1120) surfaces

Let us begin our discussion with the nonpol. ar (1010) and (1120) natural cleavage planes. Since there exists a very close relationship between surface geometry and surface electronic properties (see, e.g. , Refs. 2, 39, and 61) it is very helpful to have the surface geometry in mind when we discuss the surface electronic structure. 1. The(1010) surface The surface unit cell and the irreducible part of the surface Brillouin zone are shown in Fig. 3. Each surface unit cell contains one anion (0) and one cation (Zn). These two ions are first-nearest neighbors with respect to the bulk lattice. The in-surface bond between the anion and the cation is directed along the crystal's c axis. Each surface layer anion (cation) is bonded, in addition, to two subsurface layer cations (anions). The physical surface creation of a (1010) face thus necessitates breaking of two bonds pex layer unit cell. Comparing the binding environment at the (1010) surface with the bulk binding environment, we see that each surface layer anion (cation) is missing one of the four nearest-neighbor cations (anions) and four of the twelve next-nearest-neighbor anions (cations) it usually has in the bulk. Note, as well,

that the cations and anions in the subsurface plane lose two of their twelve next-near est neighbors upon surface creation. In our formalism a (1010) surface is simply created by removing a doub]. e layex' from the bulk crystal. Removal of only one double layer parallel to the (1010) plane is sufficient and necessary in order to create two completely decoupled surfaces since we describe the bulk material by a second-nearest-neighbor ETBM Hamiltonian. 'The U matrix is Bxa in size only, since there are eight orbitals pex surface unit cell in a double layer (one Zn4s and three 0 2p orbitals in each of the two planes in the double layer). The bulk Green's-function submatrix, whose determinant is needed for evaluating the bound-state energies [see Eq. (8)] and whose inverse is needed for calculating the I DOS [see Eq. (11)], is merely 8 x 8 in size. It, nevertheless, allows us to determine exactly the electronic properties of the semiinfinite ZnO crystal terminated by an ideal (1010) surface. The Green's-function matrix elements are given in the Appendix. The surface band structure is shown in Fig. 4 together with the projected bulk bands. Bound surface states are shown as full lines and pronounced resonances are indicated by dashed lines which are superimposed on the bulk band projections. The lower part of the projected valence bands results from the g-p mixed covalent bulk states. The upper part results from the group of four 0 2p-dexived bulk valence bands which are strongly anionic in nature. The lowex part of the projected conduction bands (from 3.3 to roughly 7. 5 eV) is mainly derived from the bulk Zn4s conduction bands and is therefore cationic in nature.

/Il/I/////////////

8

. //'/

6

S

—0

Q.

(1010) ////////

(1010)

-2

/

llllly/1//~/////r~&&& I

////////

rrxsxzri~w~igiyg~~~g/p/ggygg~~

rr////

I I I I I

r «/w

h

—P) kX

&

FIG. 3. Surface unit cell and irreducible part of the for the $010) ZnO surface.

2D Brillouin zone

FIG. 4. Projected bulk band structure and surface for the unrelaxed, nonpolar (1010) Zno surface. The layer unit cell and the irreducible part of the surface Brillouin zone are shorn as insets. Bound states (full linea) and resonances (dashed bnes) are labeled according to their character (see text).

band structure

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AND

Finally, the uppermost part of the projected conduction bands can be visualized as the antibonding counterpart of the s-p mixed projected valence bands between -5 and —3.0 eV. In consequence of the very different nature of the projected bulk band groups in Fig. 4, the resulting surface states are very different in character as well, as will be discussed below. The first and most striking result to be recognized in Fig. 4 is the complete lack of gap surface states. This result is in full agreement with the experimental data. In UPS (Refs. 3 and 13) and in EELS (Refs. 11 and 17) experiments at clean ZnO surfaces neither occupied nor empty gap surface states (with densities of more than 10" cm-') have been reported. The lack of gap surface states, which we find for the other ZnO surfaces, as well, is a consequence of the large This finding is a very ionicity of the material basic key to the understanding of ZnO surface properties as opposed to surface features at diamond- and zinc-blende-type materials. Surface creation in ZnO does not necessitate breaking gp' bulk bonds, as in the case for zinc-blende-type semiconductors (see the schematic Fig. 14). The task of creating a surface rather consists in separating ions and in consequence there is no need for a strong charge rearrangement (see again Fig. 14). As a result, atomic- or ioniclike features stand out. This conclusion is confirmed by the surface band structure, shown in Fig. 4. (We will come Jack to this point in some more detail in Sec. IIID. ) In order to ease the discussion of the various surface induced features which lie mostly very near the projected bulk band edges, we give in

J.

POLLMANN

.. .O

O.

ZnO x

'(

~

(1010}nonpolur

'

I

I

I

I

I

I

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-4-2

0

2

4

6

8

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or

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Fig. 5 wave-vector-resolved layer densities of states on the first three layers in direct comparison with the corresponding LDOS at a bulk layer. The LDOS's are given for the I', X', 34, and X points in the surface Brillouin zone. It should be noted that the LDQS's in Fig. 5 have been calculated with 0.3-eV lifetime-broadening in the Green's functions. Let us first concentrate on the features labeled & and $' in Figs. 4 and 5. The Q2p-derived states P form a group of four surface states which are most pronounced near the X point. These features exist throughout the surface Brillouin zone, partly as bona fide bound states and mostly as resonances. They are entirely anion derived and may, therefore, be classified as anionic resonances rather than a, s anion-derived dangling-bond states. The center of gravity of the & bands nearly coincides with the 0 2p on-site level p» which emphasizes the ionic nature of these features even more. The dispersion of the & bands is fairly

-6-4-2

I

0

2

4

6

8

ENERGY

-6

{eV)

FIG. 5. Wave-vector-resolved layer densities of states LDOS) on the. first three layers in comparison with the bulk LDOS for some high-symmetry points in the surface Brillouin zone (0.3 eV Lorentzian broadened). The surface-induced changes in the LDOS's as compared to the bulk curve are highlighted by the hatchings. The labeling of the prominent peaks is according to Fig. 4. I',

small in consequence of the fact that this dispersion originates from second-nearest-neighbor in-

teractions.

"

The second ionic surface feature in Figs. 4 and the Zn4g resonance labeled $. This weak resonance only occurs near the zone center (see Fig. 5, as well) and is similar to a surface resonance at a simple metal surface. It energetically coincides with the Zn4g atomic on-site energy level p~ in the crystal. The resonances S and p represent the new surface-induced features in the electronic structure near the projected band edges. In a certain sense, these ionic resonances are the "remnants" of the anion- and cation-derived dangling bonds, well known from (110) surfaces of zinc-blende semiconductors. Since both the Zn and the O surface layer ions are missing four of their usual twelve bulk second-nearest neighbors, they somewhat "relax" to their atomic energy levels and give rise to an increase of the surface layer density of states near the on-site 4s and 2p energies, respectively (see Fig. 5). The lower part of the projected valence bands 5

is

ELECTRONIC STRUCTURE OF IDEAL AND RELAXED. . . and the upper part of the projected conduction bands are more covalent in nature, as we have mentioned already. In consequence, the band groups B and A in Fig. 4 are typical covalent back-bond and anti-back-bond states. They originate from the projected covalent bulk bands and result from the changes in bonding strength of the Zn-Q nearest-neighbor bonds in the surface layer and between the surface and the first subsurface layer. Since each surface layer atom loses one nearest and four second-nearest neighbors, the Zn-Q bonding strength of the in-surface-layer Zn-O bonds bebonds and the backward-directed tween the surface and the subsurface layer changes considerably. As a consequence, the covalent g states partly shift uP in energy relative to the bulk bands from which they are derived. The opposite behavior is found for the antibonding counterparts Note, that not only the strongly dispersive g band between g and X' has an antibonding counterpart g. The same is true for the other covalent states near -4.7 eV which occur'between X' and I' only as resonances. The corresponding antibonding states are resonances altogether. The covalent back-bond states J3 show an appreciable Zn4S admixture in their wave functions. At the surface layer, e.g. , this admixture amounts to points, 27%, 35%, and 16% for the X, M, and respectively. When we further analyze the orbital character of particular surface states, we find, e.g. , that the p state at the M point consists of 0 2p, , 35% Zn4s character and 45%, 10%, and

"

I'

2p„and 2p„character, respectively.

1'

This analysis

clearly reveals that the & state at the 3f point is directly connected with the in-surface layer Zn-0 bonds which are parallel to the crystal & axis. (We have chosen the z axis of our coordinate system parallel to the c axis. ) The high s and P, contributions (35% and 45%, respectively) in the g state wave function thus convincingl. y demonstrate that this state originates from the in-surface layer Zn-Q bonds. These bonds are very efficiently coupled to the corresponding subsurface layer bonds via the first-nearest-neighbor coupling y', , (Zn-0) (see Fig. 3). Therefore, the fl band

gad'.

sh sast I gdspe s.o b t liminary experimental angle-resolved ultraviolet photoemission spectroscopy (AHUPS) investigations have confirmed the strong dispersive behavior of this bRnd. Analyzing the orbital character for the p bands,

we find similar characteristics as for the jp states, however with an inverse reciprocal ratio of 8- to p-state contributions. The antibonding features are essentially localized on the first two layers (see Fig. 5). Note that these states may very well

serve as surface-induced

final states for electron

728$

energy loss induced transitions. A full characterization of the surface electronic properties, i.e. , an identification of the atom and orbital contributions to the various features and their localization behavior can be accomplished LDOS's in by analyzing wave-vector-resolved detail. For the high-symmetry points such LDOS's are given in Fig. 5 and we have used the detailed information already in our interpretation of the results for the (1010) surface. The figure shows the LDOS's per layer unit cell, so that they contain both the anion and the cation contributions. These LDQS's were obtained by summing orbitaland atom-resolved densities according to Eq. (12) (without the g sum). The most prominent structure in Fig. 5 is the Q 2p resonance which is to be seen at all high-symmetry points. Due to the lifetime broadening in the Green's functions, the various jp states at X' are not resolved separately emany longer. The broadening, nevertheless, phasizes the main surface-induced features particularly clearly. Note that the Zn4s resonance only occurs at T' while the covalent 9 and g features are found throughout the Brillouin zone. The localization properties of the surface-induced features can easily be identified in Fig. 5. The shaded areas indicate the local changes in the density of states which are induced by the surface. The very strong localization of these features is characteristic for an ionic materiRl since there ale no pronounced charge real rangements connected with surface creation (see again Fig. 14). Up to now, we have analyzed wave-vector-resolved information about the (1010) surface. The "classical" way of obtaining a complete picture of,the surface electronic properties is to calculate the wave-vector-integrated layer density of states [see Eq. (12)j. This quantity is given in Fig. 6 for the first three layers in comparison with the bulk density of states. We have calculated this information in order to be able to compare our results with angular integrated UPS data of Gopel gt g). The experimental spectrum is given in Fig. 6, as well. We will come back to a detailed comparison of theory and experiment in See. V. We note in Fig. 6 that much of the detailed structure seen in Fig. 5 has been averaged out by the Brillouin-zone sum. Nevertheless, the main surface-induced trends are retained in that figure (except for the S resonance which occurs only near the I' point and is, therefore, not resolved). We clearly see the ~ resonance and the back-bond and anti-back-bond states g and g which are shifted up and down in energy, respectively, as compared to the bulk states from which they are derived. The surface-induced ~y~~g~s in the integrated LDOS are found to be restricted to the

"

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POLLMANN LDOS (1010)ZnQ

l

Zno

M

{1010) PS Herr

C

cf

Vl

O' C3j

l I

Bulk

-6

-4

-2

0

2

I

I

I+

6

8

ENERGY (eV)

FIG. 6. %ave-vector-integrated LDOS' s (summed over the surface Brillouin zone) for the first three layers and a bulk layer of the unrelaxed ZnOQ010) surface. The He II UPS spectrum of Ref. 13 is given for comparison (dashed line). The surface-induced changes are hatched.

I I I

I I I I I

I I

first

two

layers.

The above-presented classical complete picture of surface electronic properties consists of very integral information. The experimental trend nowadays is directed towards high angular resoAngle-resolved UPS and lution measurements. EELS experiments have become very popular. It would, therefore, be highly desirable if surface electronic structure theory could obtain a comPlete and yet angle-resolved picture of the surface electronic properties. To this end, we present in Figs. 7 and 8 a very compact, complete and, nevertheless, detailed representation of the ZnO(1010) surface electronic properties. Figure 7 shows, as a reference, the full information about the electronic properties of a ZnO(1010) bulk layer. It shows the proj ected bulk band structure and wavevector-resolved LDOS's for a (1010) bulk layer along the high-symmetry lines of the surface Brillouin zone (SBZ). The projected band structure (PBS) in the lower part of the picture only reveals in which energy regions bulk states do exist. The LDOS plot (upper part), in addition, shows the corresponding density of bulk states at the various g points in the SBZ. Now Fig. 8 very precisely reveals the main surface-induced effects. Not only the energetic positions and the dispersion of bound states and pronounced resonances (lower part; see also Fig. 4) LDOS's at the but also the wave-vector-resolved surface layer are given. (We do not show the corresponding pictures for some lower-lying layers as well, since the ZnO surface features are highly localized as we have seen. ) From the figure we

I

FIG. 7. Comprehensive view of the bulk electronic properties of a (1010)-parallel layer of ZnO. Both the projected bulk band structure gower part) as well as wave-vector-resolved layer densities of states (upper part) are given for the high-symmetry lines of the surface Brjllouin zone. obtain direct information about the density of available states near the surface which may be studied spectroscopically by either ARUPS or angle-resolved energy loss (ABELS) measurements. Figures 7 and 8, e. g. , clearly reveal that a normal emission ABUPS experiment at the ZnO(10T0) face cannot yield useful information about surface -induced features. However, scanning through the area near the X point by an appropriate choice of geometry in the experimental setup ShouM yieM detailed information about the P resonances (note the high state density) and the antibonding A states when ARUPS or ARELS experiments are carried out. As a

ELECTRONIC STRUCTURE OF IDEAL AND RELAXED. . . LDQS (1010}ZnO 1stlayer

I

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p

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perimental interest. The surface unit cell of the (1120) face and the irreducible part of the surface Brillouin zone are shown as insets in Fig. 9. have shown that, within LEED investigations" experimental errors, this surface neither relaxes nor reconstructs. The surface unit cell (like the bulk unit cell) conta. ins two anions and two cations. The 0-Zn-0 units form zigzag chains along the c ~is. Every anion (cation) is bonded to two cations (anions) in the chain, i. e. , in the surface plane, and to one cation (anion) in the subsurface layer. 'The surface layer ions are missing, therefore, one first-nearest neighbor and five secondnearest neighbors. The neighboring chains in the surface layer are separated by a third-nearestneighbor distance. The main interactions near the surface, therefore, occur along the zigzag chains and along the back bonds from the surface to the subsurface layer. The projected bulk bands and the surface band structure for the (1120) fa. ce are shown in Fig. 9. The PBS is similar to the corresponding result in Fig. 4 for the (10TO) face. The stomach gap around the M point is somewhat smaller in Fig. 9 than in Fig. 4. At the (1120) surface we find bound states only in very small areas of the SBZ. The states near -2 eV are a,gain anion-derived Pstates. The covalent surface states p and p only occur and g. 'They again mainly result from the near decrease of the Zn-0 bonding strength in the surface layer chains. The electronic structure of the (1120) surface thus resembles very much the cor-

I

I I

I I I

I

I

FIG. 8. Comprehensive view of the surface electronic structure of the unrelaxed Zno(1010) surface. Both the surface band structure as well as the wave-vector-resolved layer densities of states (for the top layer) are lines of the surface Brilshown along the high-symmetry louin zone. Full lines show true bound states and resonances S and I' and the covalent back-bond and antiback-bond surface states B and A are labeled accordingly.

matter of fact, angular-resolved UPS measurements at Zno(10TO) are currently being carried out 64 and we hope that our results will turn out to be very helpful in the analysis of the data. A discussion comparing the theoretical results presented in this section with experiment is given in Sec. V. 2. The (1120)surface

There exists another nonpolar natural cleavage face of ZnO which has not attracted very much ex-

)

Zno

0-

(1120j

LLI

-2

-6r

x

FIG. g. Projected bulk band structure and surface band structure for the (1120) Zno surface. The layer unit cell and the irreducible part of the surface Brillouin zone are shown as insets.

I.

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AND

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POLI

MANN

pl'opel'iles of 'tile (1010) fRce. In Rddlto our knowledge there are no detailed extion, perimental investigations of the microscopic electronic properties of this surface available. Therefore, we do not want to discuss the (1120) surface in more detail at this point. 1'espolldlllg

(0001j / I0001)

0

0

C. The polar (0001) and (0001) surfaces

/

~ /

The ZnQ crystal consists, as xnentioned already in Sec. II, of double layers which are stacked perpendicular to the hexagonal crystal g axi, s. Each double 1.ayer contains one plane of cations and one plane of anions. While the doub1. e layers are separated by a nearest-neighbor distance the two planes in each double layer are only separated by one third of a nearest-neighbor distance. Each anion (cation) is bonded to three cations (anions) on the other plane of the same double layer and to one cation (anion) in the next double layer. The nearest-neighbor bonds connecting the double 1ayexs are para1. lel to the crystal. c axis and thus perpendicular to the double 1ayers. Breaking these bonds leads to two different ZnQ surfaces. Qne of the two semi-infinite solids is terminated by a Zn layer while the other is terminated by an 0 layer. They are referred to as the (0001) and (0001) surface, respectively. The geometry of these two surfaces 18 tlM same. Thelx' physical properties, obviously, will be different since one face contains cations only and the other contains anions only in the outermost layer. The surface net is hexagonal [as for the (111) and (111) surfaces of diamond and zinc-blende semiconductors]. The surface unit cell and the irreducible part of the surface Brillouin zone are shown in Fig. 10. There is one atom per surface unit cell. The neighboring sulfRce lRy'el' Rnlolls (CR'tiolls) RX'e second-nearest neighbors with respect to the bulk lattice. Each surface layer anion (cation) is misscations ing one of the four first-nearest-neighbor (anions) and three of the twelve second-nearestneighbor anions (cations) it usuaHy has in the bulk. Note, a.s weH, that the cations (anions) in the subsurface plane (i.e. , in the second plane of the double layer) lose three of their usual. twelve second-nearest neighbors upon surface creation. In oux' formalism these sux'faces ar e slxnply created by removing one double layer from the bulk crystal. In this way we actually create a neutral macrocrystal with a, Zn- and an Q-texxninated face mhich are infini. tely far apax't from one another. 5 The U matrix is 4x4 only, since thexe are four orbitals per surface unit cell in a double layer (one Zn48 and three 0 2@ on the corresponding »ye»). The bulk Green's-function submatrix, whose deterxninant me need for calculating the bound states and whose inverse is needed for the

/ /

FIG. 10. Surface unit cell and irreducible part of the for the po]. ar (0001) and (0001) Zno surfaces. 2D BrQlouin zone

LDQS's is merely 4x4 in size. It, nevertheless, yields exactly the surface electronic properties on the Zn- and the Q-terminated surface simultaneously. Separating the features derived from the Zn-terminated surface from those derived from the 0-terminated face is trivial (see Appendix). The Green'8-function matrix elements for the polar surfaces are given, as mell, in the Appendix. The surface band stxucture for both the Zn- and the Q-terminated surfaces is shown in Fig. 11. The projected bulk band structure is, of course, for both polar faces the same. The tmo surfaces, homever, give rise to distinctly different surface features. We have, therefore, plotted the resulting bound states and xesonances for both faces in the same gxaph. In Fig. 12 me shorn, in addition,

faces pP/g

-2

FIG. 11. Surface band structure of the polar ZnO surfaces. The bulk projections for the Zn-terminated II0001) and the 0-terminated (0001) faces are identical. %e have, therefore, plotted the surface states of thoro different faces in the same graph. The Zn face mainly yields the features A, S, and 8, while the oxygen resonances P occur predominantly at the 0 face.

0-

ZnO polar

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0 foce

face

Zn I

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]+

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ENERGY (eV}

FIG. 12. %ave-vector-resolved LDOS* s on the first three layers at the Zn-terminated face (left column) The and at the 0-terminated face (right column). bulk LDOS is given by dashed lines for direct comparison. The surface-induced positive changes in the LDOS are hatched.

wave-vector-resolved layer densities of states on the first three layers of both surfaces. The LDOS's are given for the I', M, and g points in the surface Brillouin zone. Note that these LDQS's on the various layers are "naturally" atom resolved, since the surface-parallel planes contain either Zn or 0 atoms, respectively. %'e have again used 0. 3-e+ lifetime-br oadened Green 8 functions. In agreement with experiment we find no surface states in the gap energy region for both polar surfaces which result from the large ioniclty in this case as we]1 (see Fig. 14 and the discussion). The predominant features at the Zn-terminated surface are the back-bond (B) and anti-back-bond Q) surface states. They exist in a large region of the surface Brillouin zone as true bound surface states and continue as sharp resonances into the band projections. The energetic position, the origin and nature of these surface states allows us to classify them as covalent. They are derived from the g-p mixed l, ower bulk valence- and upper The bulk conduction-band states, respectively. dispersion of these bands originates from the in-

teraction of the nearest-neighbor Zn-0 back bonds. Since every surface-Layer Zn atom is missing one nearest-neighbor Q atom, the bonding strength of the mixed q-p states decreases near the surface so that the B states move up and the antibonding g states move down in energy relative to the projected s-p mixed bulk states. This resuLt is simiLar to the behavior of the p and g states at the (10TO) surface. The energetic shifts of the g and& states relative to the bulk band projections can clearly be seen, as well, in Fig. 12, especially for the 4f and K points. %e see that the changes in the layer densities of states are essentially localized at the first two layers (that is typical for zone-edge states" "). The orbital character is directly revealed in the figure since we have only s orbitals on the Zn layers and only p orbj. tais on the 0 Layers. Note that the p and& states, therefore, are q-Like on the first and p-like on the second layer. It is interesting to notice that these states which are mainly localized at the back bond in the first double layer, in a certain sense have reciprocal wave-function conti lbutlons on the two layex s. The occupied back bond has a predominant p admixture from the 0 layer while the empty anti-back-bond has a predominant s admixture from the Zn layer. In addition to the covalent surface states the Znterminated surface yields as well, more or less pronounced changes in the density of states within the Zn 4g and 0 2p bands, as can be seen in Fig. 12. In particular, near the zone center, we find again a Zn 4p-derived resonance, which is now more pronounced than at the (1010) surface. The 1,DOS of the S resonance near 1" (left upper panel of Fig. 12) shows oscillations around the bulk layer DOS which are very typical for s-band metal resonances. In Fig. 12 we see that there even occur p resonances at the Zn-terminated (0001) face. They originate in this case from the subsurfacelayer 0 atoms which lose three of their twelve next-nearest neighbors upon surface creation. At the 0-terminated polar (000T) face the P resonance is the most predominant feature. The surface layer 0 atoms missing three of their twelve second-nearest neighbors now give rise to even more pronounced resonances, which extend throughout the SBZ, partly as bona fide surface states and mostly as resonances. Figure 13 summarizes the surface electronic feature's of both polar Zn0 faces in the "classical" LDQS's are way. The wave-vector-integrated again "naturally" resolved with respect to the Zn and 0 contributions. The Zn face is dominated by the covalent g and g states and shows a very strong S resonance (if we have in mind that it occurs only near T') which is far more extended than

—

—

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face

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ENERGY (eV) FIG. 13. %ave-vector-integrated LDOS' s on the first three layers of the Zn-terminated (upper panel) and 0-terminated gower panel) ZnO surfaces in direct comparison wi.th the corresponding bulk LDOS' s (dashed lines). The DOS changes are again hatched. Note the 'natural" separation in anion (0) and cation (Zn) contributions.

the other features. The 0 face is fully dominated by the 0 2p-derived P resonances. In the third layer, the surface induced changes are already very small. D. Summary of the ideal surface properties

Let us briefly summarize the main features in the sq. rface electronic structure of ZnO and compare thein with typical surface electronic properties of more covalently bonded semiconductors. The key to an understanding of the ZnO surfaces, as opposed to Ge and QaAs, for example, is the large ionicity of the material. The essential difference between the surfaces of a diamond-type, a zinc-blende-type, and a wurtzite-type serniconductor (being increasingly more ionic) can most easily be shown schematically for the polar faces, e.g. , for the (111/Il'K) and (0001/0001) faces, respectively. In the schematic Fig. 14 we bond with its show the surface-perpendicular nearest neighborhood for the nonpolar Ge, QaAs, In order to creand ZnQ surfaces, respectively. ate physica]ly one of these surfaces, that particular bond needs to be broken (for the other lowindex faces more than one such bond per unit cell becomes disrupted). We have included, as well, schematic plots of the total valence charge density along this bond. %e can clearly see in the figure that cleaving Ge means cutting through the bond charge while cleaving ZnO means separating

———DBa

VQ "L%%~%~

R

FIG. 14. Pictorial representation of the essential difference between a diamond-type, a zinc-blende-type, and a wurtzite-type semiconductor surface (with increasing ionicity along the line). The upper three panels show the total valence charge along the surfaceperpendicular bond for (111) Ge, (ill/111) GaAs, and (0001/0001) ZnO. The charge density is drawn in a (110) and (1010) plane for the various structures, respectively. The heavy lines show bonds lying in the drawing plane. The angle between the other bonds is bisected by the drawing plane. These bonds are indicated by dashed lines. Note the staggered (Ge, GaAs) of the six nearestand eclipsed (ZnO) arrangements neighbor bonds with respect to the surface-perpendicular bond. It is obvious from the figure that the task of the "surface creating wedge" is very different in the three cases. In consequence the resulting surface features are very different, as shown schematically in the lower three panels. &„and &, characterize the valence- and conduction-band edges, respectively, and DB stands for dangling bond, which can be anion-derived (DBg) or cation derived (DBc). The ionic resonances are labeled R& and Rg since they are either cationic or anionic in nature, respectively (for more details, see text).

ions. Creating a Ge(111) surface, therefore, leaves behind a dangling charge lobe, i.e. , a free gp' hybrid whose energy lies in the middle between the bonding arid the nonbonding bands. Thus a band of dangling-bond states within the projected gap results, as is indicated in the schematic graph in the lower part of Fig. 14 (for the detailed Ge and QaAs surface band structures, see, e.g. , Ref. 39). Cleaving the ZnQ crystal is accomplished by separating anions and cations against their electrostatic ion-ion attraction. No dramatic change in the charge density around the ions needs to occur. In consequence, the properties of the electrons near the surface become more ionic- or even atomiclike and the surface thus gives rise to pronounced ionic resonances which energetically occur near the on-site term values. No danglingbond states in the gap result since there is no dangling charge lobe. The anionic and cationic

ELECTRONIC STRUCTURE OF IDEAL AND RELAXED. . . resonances lie well within the projected valence bands, respectively (see Fig. 14). As far as the ionicity is concerned, GaAs lies right in the middle between Ge and ZnQ, with an ionicity of f,. =0.31 (see Ref. 36). Creating a (ill/51K) GaAs surface still necessitates cutting through a bond charge so that dangling-bond gap state s result. But the ionic charac ter of GaAs splits the anion- and cation-derived dangling-bond bands and shifts them to the corresponding band edges. In a sense, the anion- and cation-derived dangling bonds in GaAs are "halfway" between a pronounced covalent dangling-bond gap state (as in Ge) and a strong ionic resonance (as in ZnO). This behavior is schematically shown in the lower part of Fig. 14. We have seen that in an ionic material like ZnQ the surface-induced changes in the electronic properties are strongly localized near the surface. This shows that they are particularly closely related to the changes in the local binding environment that occurs when a surface is created. In Table II we have summarized the coordinations of the surface and the subsurface layer atoms for the various surfaces which reveal these changes directly. Note that in bulk ZnO each atom has four nearest neighbors and tggelve second-nearest neighbors. From the remarkable changes in the coordinations of the surface- and the subsurfacelayer atoms we could have expected from the very beginning that each of the four discussed ZnQ surfaces gives rise to the ionic resonances $' and p and to the covalent back-bond and anti-back-bond surface states @ and p, respectively. In Secs. III 8 and III C we have discussed in detail how pronounced these various features occur at the various surfaces. For the convenience of the reader these results are schematically summarized in Table and conduction

III.

I.et us conclude this section with a few preliminary remarks concerning the comparison of our theoretical results for the ideal surfaces with experimental data for the real surfaces. We have

TABLE II. Coordinations of the ZnO ions in the surface layer and in the first subsurface layer. The bulk coordination is four for the first-nearest neighbors (NN) and 12 for the second-nearest neighbors.

Surface-layer Coordination of ZnO ions

(1010) (1120) (0001/OOOO)

ions

1st NN

2nd NN

Subsurface-layer ion s

1st

NN

2nd NN

10

11 9

7289

TABLE III. Summary of the various surface features P, and B which occur at the four studied surfaces in the energy ranges given in the table. The size of the letters is meant to indicate the spectral weight which the various features have at the different surfaces.

A, $,

0010)

Ml'graf Icing&

7- 8 5.7

01~0) (000~) (000 Zn

BV

0

A

BV

- -2. eV --3 tO-5BV 1

carried out such a comparison previously and found amazingly good general agreement. That finding lent support to the notion that relaxation or reconstruction effects on surface electronic properties are less pronounced in ionic materials than in more covalent zinc-blende semiconductors. While the dangling bonds at a surface of a covalent semiconductor are chemically reactive and very sensitive to any kind of structural changes in the surface layer, the ionic resonances at the ZnQ surface should not react strongly to geometry

'

changes. We, therefore, can expect weaker relaxation-induced effects in ZnQ than, for example, in GaAs. In order to clarify this point we have investigated the relaxed ZnO(1010) surface whose relaxation geometry seems to be well established. IV. RELAXATION EFFECTS ON THE ZnO(1010) SURFACE

I.et us now discuss the effects of surface structural changes on the electronic properties of the nonpolar !1010) surface. As mentioned already in the introduction to Sec. III, surface relaxation or surface reconstruction usually give rise to downward shifts of occupied states and upward shifts of empty surface states. The question then is, for a zeal surface, how large these shifts are and how sensitive they are to atomic rearrangements. It is well known, by now, that surfaces of diamond-type crystals become ionic by reconstruction and surfaces of heteropolar covalent zincblende-type crystals become more ionic by relaxation in order to be able to lower the occupied surface states and to raise the empty surface states (see, e.g. , Ref. 2). Now ZnO is fairly ionic from the very beginning. It is, therefore, interesting to investigate @Ozg this material lowers its energy by a surface geometry change. Since there is one anion and one cation per surface unit cell, one can expect that the surface merely relaxes

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and does not reconstruct. This atomic relaxation of the (1010) surface has been investigated in detail by Duke et gl. who proposed the following relaxation model. All top-layer ions are supposed to move vertically do&onward; the 0 ions by 0.05 + O. i A. Note that in consequence of this relaxation three different Zn-0 bond Lengths in the surface double layer result. %'e can expect that by relaxation charge will be transferred from the cation (Zn) to the anion (0), i. e. , from the "down atom" to the "up atom. " This relaxation model, therefore, seems to fit nicely into the general charge trans-fer reLaxation scheme proposed in Ref. 2 for semiconductor surface relaxations and reconstructions.

"

A. Formal treatment of surface relaxation

In the framework of the scattering theoretical method, surface relaxation effects can easily be taken into account, as has been discussed in Ref. 2. In addition to the removal of sufficiently many layers for creating decoupled ideal surfaces [see Eq. (7) and the Appendix] we have to take into account the changes in the interaction matrix elements due to relaxation. Both exponential as well as d ' scaling of the matrix elements have been used (where d is a nearest-neighbor distance) We have apvery successfully in the past. plied the d ' scaling to the matrix elements V,', , and p'„„in order to represent the rep,'~, laxation model of Duke et gl. Since our bulk Hamiltonian retains all first- and second-nearestneighbor interactions, the geometry changes in the surface layer yield a nonvanishing perturbation matrix U on the first three layers, i.e. , on the first double layer and the first plane of the second double layer. The U matrix now takes the form

""

p'„,

.

U, „creates the ideal surfaces and U, describes the surface relaxation. With four orbitals per layer unit cell (one Zn 4s and three 0 2p orbitals) It,„ is an 8 x 8 and IL„~ is a 12x 12 matrix so that we still are working with very small perturbation and Green's-function matrices. The bound states follow from Eq. (1) and the resonances and layer densities of states are calculated from Eq. (4) which can simply be rewritten as

where

G = G'+O'U(1

~

—O'U)

-'G'.

(14)

For the relaxed surface, therefore, Eqs. (8) and (11) have to be replaced by the corresponding Eqs. (1) and (14) represented in the layer orbital basis.

J.

POLLMANN

This can equally well be done. Since the matrices involved are now 12x12 or 20x 20, respectively, it takes more computer time, of course, to calculate the electronic structure of the relaxed surface. We do not want to go further into the calculational details at this point. B. Relaxation-induced

effects

In order to identify the characteristic influences of the relaxation on the surface electronic structure in a systematic way, we have studied the (1010) surface changing the Zn and 0 surface atom positions continuously between the ideal lattice positions and their positions in the suggested relaxation model. ' In general, we find that the ionic resonances and the covalent surface states react very differently to surface relaxation. Most remarkably, the ionic resonances are almost unaffected by relaxation. We have expected this result, since the ionic charge density should not be influenced too much by relatively small movements of the ions. (Remember that the ionic features obtain their characteristics from secondnearest neighbor interactions. ) The P and $ resonances are "pinned" near the on-site Zn 4s and 0 2p energy levels, respectively. Qn the other hand, we find an appreciable change in energy position of the covalent back-bond and anti-back-bond states. Since these features are closely related to the Zn —0 nearest-neighbor bonds in the first double layer, they react sensitively to bond-length changes. In Fig. 15 we show how the energy positions of the 9 and P states at the M point vary when we shift the surface layer Zn atoms vertically down (see Fig. 4 for comparison). In this "computer experiment" we have kept the 0 atoms fixed at their ideal surface locations. The studied geometry is shown as an inset in Fig. 15. The shaded "bands" are the projected valence and conduction bands at the M point (see Fig. 4). Note the change in energy scale, as compared to all previous figures. This scale fictitiously exaggerates the relaxation effects to a certain extent. We see in Fig. 15 that the back-bond state 8 moves do&en in energy by about 0. 5 eV and the anti-backbond state & moves zp by about the same amount for the fully relaxed (-0.45 A) Zn atoms. This down- and upward shift of occupied and empty states, respectively, is very familiar from the

zinc-blende semiconductor sur faces. However, in those materials the covalent surface states are dangling-bond states and occur in the projected gap. Relaxation then shifts them out of the gap into the valence- and conduction-band projections, as is well known for GaAs (Ref. 69) or InSb, for example. At Zn0 surfaces, the covalent states occur much farther down in the projected valence

"

ELECTRONIC STRUCTURE DF IDEAL AND RELAXED. . . (1010jzno

relaxed surface

point

M

////

8.0

8

' cB

7.8

r////

A I

6

PI Oj

7. 6

4

ZOO

4

4—

I

7.

s

/

dz„

.4—

~1010)

s

-4,

E-. ~, '-

~//////yap

-4. 6 LLI

-4.8 proj

VB

-5.0

-52

X'

-54 I

0.0

t

0.

1

i

I

0.2

I

l

0.3

(

I

0.4

I

t

0.5

I

0.6

dz„{A)

I

FIG. 15. Study of relaxation-induced shifts of the covalent surface states/A and B at the point as a function of the vertical downward shift dz~ of the top-layer Zn atoms. The 0 atoms have been held fixed in their unrelaxed positions in this case. The shaded "bands" represent the projected bulk valence and conduction bands at the, M point. Note the energy scale change g, s compared to the other figures in this paper. The inset shows a side view of the studied geometry.

bands and higher up in the projected conduction bands so that their shifts do not strongly change the characteristics of the surface properties. Nevertheless, the crystal again gains energy by the same familiar mechanism, namely shifting down occupied bonding and shifting up empty antibonding states in energy. We would like to mention, finally, that the A and II states (see Fig. 15) are found to behave very much like a two-level system reacting to a change in the mutual interaction between the two levels. The interaction predominantly involved in this process is the Zn4S-0 2p matrix element V,'~(Zn-Q). The full effect of the surface relaxation on Zn0 (10IO) is shown in Fig. 16. The geometry of the relaxation model" is shown as an inset. Comparing Figs. 16 and 4, we realize that only marginal changes have occured in the p or S resonances. The covalent surface states, as well, have not dramatically changed (we are now using the old energy scale again). Nevertheless, we can identify a distinct downward shift of the main g band and an upward shift of the main g band. It is interesting to notice that the main p band has now become fully resonant with bulk states. We observe, as well, that there are three 8 and& features in

FIG. 16. Surface band structure and projected bulk for the relaxed ZnO(1010) surface (see Fig. 4 for comparison). Bound surface states and pronounced resonances are again shown by full lines and

band structure

dashed-lines, respectively. The insets show a top and a side view of the relaxation geometry proposed in Ref.

31.

certain areas of the Brillouin zone at the relaxed surface (between P and M). They can simply be related to the three different types of Zn-0 bonds at the relaxed surface. The strongly dispersive resonance, obvious1. y, originates from the Zn-0 in-surface-layer bonds (see the corresponding discussion in Sec. IIIB and Fig. 4). The same is true for the corresponding anti-back-bond band. 'The flat band near -4.8 eV extending from I' to X and the corresponding antibinding resonance above +8 eV are related to the two 0-Zn back bonds (per unit cell) which undergo only small changes, when the 0 atoms are moved downward by 0.05 p . These state s are, there fore, almo st the same as at the ideal surface. Finally, the other two Zn-0 back bonds (per unit cell) become considerably stronger, when the Zn atoms are moved downward by 0.45 A. In consequence, these bonds are stronger than any bulk bond and, therefore, they give rise to a band of surface states which lies bolos' the bulk valence-band projection. This band occurs for short wavelengths only (from X to M) as is typical for featur'es that are induced by local changes of bonding configurations. The total effect of the surface relaxation can be summarized by noting that the strong cationic relaxation accompanied by a very small anionic relaxation has led to an appreciable downward shift of occupied surface states, so that the electronic energy has been lowered noticeably by a pronounced cationic relaxation.

I.

7292

IVANOV

AND

In conclusion of our discussion of the relaxed surface we would like to stress once more a very general point that was already mentioned in the introduction to Sec. III. We think that the essential and qualitative physics of a particular real surface can be learned from a careful analysis of the corresponding ideal surface. The detailed knowledge of the ideal surface properties can be used as an extremely helpful guideline in the interpretation of the more subtle quantitative details of the real surface. We hope that we have convincingly demonstrated this fact for the case of the ZnO(1010) surface. ZnO(IOTO)

V. COMPARISON WITH EXPERIMENT

Numerous experimental investigations of polar and nonpolar Zn0 surfaces have been carried out in the past. ' ' This compilation of the literature is not meant to be complete. Detailed discussions of the experiments and extended reference lists may be found in recent review articles by Heiland and Luth' and by Qopel. ~ In the following we compare our results with surface-sensitive UPS and EELS data. The measurements were performed on clean surfaces under well-defined and reproducible conditions. Let us focus our attention first on the UPS data which yield information about the occupied states. All UPS data reported so far 3, ~3, &5, &6, 58 clearly show that there are no occupied surface states above the top of the bulk valence bands. The surface is found to modify the valence-band spectrum but no gap states occur. This result was obtained, as well, in our calculations for all four studied surfaces and we have given a physical interpretation of these findings in the previous sections. The surface-induced changes in the angular-integrated experimental spectra are not very dramatic. Nevertheless, distinct trends have been identified, e.g. , by Gospel, Bauer, and Hansson who studied the clean (1010) surface. ~' We have included their UPS spectrum from Ref. 13 in our Fig. 6. The VPS (HeII) spectrum has maximal surface sensitivity. Therefore, the authors of Ref. 13 could identify two essential surface-induced effects by comparing their VPS (He II) data with their XPS (AIKo. ) spectra. Firstly, it was found that the energy positions of the two prominent valence-band peaks change. The covalent q-p peak shifts up in energy by O. I eV and the ionic 0 2p peak shifts down in energy by 0.2 eV. The net effect is a decrease of the energy separation between these two peaks by 0.3 eV, as compared to the bulk spectrum. This general trend has been followed up by primary energy variations and was found by other investigators, as well. Secondly, Gopel, Bauer, and Hansson report an

"

J.

POLLMANN

0

2p peak near -2 eV. These two changes in the experimental spectrum are mainly due to the surface states, as was proved by primary energy variation and contamination tests. The two experimentally determined effects

increase of the main

can, as well, be seen in our wave-vector-integrated LDOS's in Fig. 6. %'hen we approach the surface our LDOS's show exactly the trends found in experiment. The quantitative comparison between the experimental spectrum and our theory for the ideal surface (Fig. 6) is very instructive. The downward shift of the 0 2p peak by roughly 0.2 eV is identical in both theory and experiment. It indicates the charge rearrangement in consequence of a weakening in the binding strength of 0 surface layer atoms which have lost four of their usual twelve second-nearest-neighbor 0 atoms. The calculated upward shift of the covalent peak at the first two layers by roughly 0.3 eV is considerably larger than the measured shift. This quantitative discrepancy is a consequence of the surface relaxation which was not taken into account in the calculation of the LDOS's in Fig. 6. In Fig. 17 we show the wave-vector-integrated LDOS's for the relaxed ZnO(IOTO) surface. Comparing Figs. 17 and 6 we see that the general relaxation induced effects are well confirmed by the LDOS's. The ionic resonances are almost not affected while the covalent gjl and g states move down and up in energy, respectively, as compared

relaxed surface I

l

)

I

t

I

I

Vl

I

I

ZnO

(1010) I

p

PS Herr

C

C5

Bu/k

-6 -4. -2

0

2

4

6

8

ENERGY (ev) FIG. 17. Wave-vector-integrated LDOS's on the first three layers of the relaxed g.010) Zno surface. The corresponding bulk 1ayer LDOS and the experimental UPS He&& spectrum are given for comparison. The surface-induced upward and downward shifts of the B and P peaks, respectively, are emphasized by the hatchings.

ELECTRONIC STRUCTURE OF IDEAL AND RELAXED. . .

7298

to the ideal surface. In Fig. 17 we have again included the UPS (Hell) spectrum of Ref. 13. The agreement between the LDOS's on the first two layers (i.e. , the double layer) and experiment is excellent. We even see the enhancement of the density of states in the Q 2p peak near -2 eV as compared to the bulk peak. It may be amazing, on a first glance, that "simple" layer densities of states are in such good agreement with experiment. It should be noted, however, that the use of He11 radiation in the experiment (a) guarantees a very high surface sensitivity (very short escape depth near minimum) and (b) leads to final states with relatively high energy (about +30 eV). Therefore, we can expect that mostly surface doublelayer features have been sampled and the final states are free-electron-like to a reasonable ap-

TABLE IV. Transition energies for polar and nonpolar ZnO surfaces as determined theoretically in comparison with the energetic positions of losses in the EELS data. The values labeled a and b are from Refs. 11 and 17, respectively. The assignment of characteristic transitions is made according to the nomenclature introduced in Sec. III. The underlined values originate from transitions which are particularly surface sensitive.

proximation. Finally, matrix element effects are averaged out in the angular-integrated spectrum to a large extent. Therefore, not only the peak positions but also the relative height of the peaks in the theoretical LDQS's on the first double layer agree well with the experimental spectrum. More detailed experimental information on the valence-band spectra will emerge from angularresolved UPS measurements which are currently being carried out. Results are, however, not yet available. A second group of experiments we want to use for comparison are the electron energy loss studies by Dorn, LQth, and Buchel, by Gopel, and %'e have determined from our by Margoninski. results for the polar and nonpolar surfaces the most probable transition energy values for transitions from occupied states in the valence-bandenergy region to empty states in the conductionband-energy region. These values have been determined by simply estimating the peak positions in the joint density of states taking into account conservation of momentum parallel to the surface. The parallel-momentum transfer in typical EELS experiments is of the order of 1/100 of a primitive, reciprocal-lattice vector" so that "vertical" transitions seem to be a reasonable assumption. Our values for the transition energies are given in Table IV in comparison with the experimentally determined peak positions in the EELS spectra. Our values are not meant to be the results of a, quantitative electron energy loss theory. We rather want to use these values in order to identify the origins of the various experimentally determined peaks and to discuss their surface sensitivity. In the table, the transition energies are given for the polar faces and for the nonpolar (1010) surface. Qur results for the ideal surfaces deviate to a certain extent from the numbers given

(0001)

"

"

"

ZnO

Theory

Experiment Ideal

(1010)

7.4 9.1

7.5 9.1 10.5

(0001) Zn face

7.3

P-S

94

P-A

10.2

B-S

7.3 9.8 10.3

11.8

B-A

12.7

7.5 9.0

P-S

112

7.1 9.6 10.2

12.9

12.6

B-A

7.5 9.1 10.7

7.4 9.0 9.8 11.4

10.9 12.6

12.6

0 face

Relayed

11.8

P-A

B-S

P-S P-A

B-S B-A

~W. Gopel, J. Vac. Sci. Technol. 16, 1229 (1979). bR. Born, H. Liith, and M. Biichel, Phys. Rev. B 16, 4675 (1977).

in our previous communication. ' These differences are due to the changes in the bulk Hamiltonian which we had to make in order to obtain a more reliable bulk description (see Ref. 60). The theoretically determined values for the nonpolar face are in reasonable general agreement with

experiment. The transitions are classified according to the nomenclature of the most prominent surface features as introduced in Sec. III. In this way, we arrive at an identification of possible initial and final states involved in the loss process (see Table III). On the basis of our ana, — lysis, e.g. , the 10.5/10. 9-eV loss tentatively can be identified as resulting from transitions between the back-bond states 8 and the Zn 4g resonance. Such an identification was not possible at the time when the experimental spectra were reported" since our calculations were the first to show that there exist 8 and g covalent back-bond and antiback-bond states at the ZnQ surfaces. From our results for the wave-vector-resolved and wavevector-integrated LDQS's it is obvious that bulklike states considerably contribute to the measured peaks in the EELS spectra. Certain transitions, however, would not occur in the bulk and are only found at the surface. We have carefully analyzed our results and arrived at the conclusion

I.

IVANOV

AND

that all those transition energies which are underlined in the table are strongly surface sensitive. This finding is in very good agreement with the experimental identifaction of the surface sensitivity of the various peaks. Exactly the underlined transition energies have been found to be most sensitive to primary energy variations and to contamination tests. Note, that the Zn4g resonance 9 is involved in all transitions which have been identified as most surface sensitive. This fact becomes clear from our previous results. While the 8, P, and g features change and shift the bulk density of states, the $ feature gives rise to a new state density in an energy region where the bulk IX3S is almost zero. Therefore, these transitions are much more unlikely in bulk-derived spectra. They give rise to peaks or shoulders in the EELS spectra, taken with 35 eV primary energy and are found to vanish in the corresponding 80 eV spectra (for more details see

Refs. 11 and 17). We have given in Table IV the transition energies for both the ideal and the relaxed (1010) sur-

%'e note again, that taking into account the atomic relaxation at the surface according to the model of Duke et al. improves the general agreement between theory and experiment. The comparison of our values with the experimental data for the polar faces has more qualitative character. Nevertheless, a reasonable general agreement is found and a tentative identification of origin, nature, and surface sensitivity of the loss peaks can be made. Remember, however, that the theoretical numbers result from our calculations for the clean, ideal surfaces while the experimental spectra may be influenced by defects These complicaand steps to a certain extent. tions at the zeal polar surfaces are certainly the main reason for the quantitative deviations between theory and experiment in Table IV.

faces.

"

VI. SUMMARY

In conclusion, we have presented a detailed analysis of the surface electronic structure of the prototype wurtzite semiconductor ZnQ. The bulk material has been described by a realistic ETBM Hamiltonian. We have fully discussed this Harniltonian in order to make transparent how basic physical properties of a particular bulk material can be incorporated in an ETBM description when the parameters entering the Hamiltonian are determined carefully in a physically meaningful way. %'e have classified the resulting bulk bands according to their predominantly ionic or covalent character, respectively. The various surfaces have been treated using the scattering theoretical method, which again allowed for a very efficient

J.

POI I MANN

treatment of semi-infinite solids. Surface band structures and wave-vector-resolved as well as wave-vector-integrated layer densities of states have been discussed at length in order to identify characteristic electronic features at surfaces of a wurtzite-type heteropolar-ionic semiconductor. We found that these materials do not give rise to bound surface states in the gap. %'e rather identified pronounced anion- and cation-related ionic resonances which lead to a noticeable increase in the state density near the atomic (ionic) on-site levels in the crystal. In addition, more covalent back-bond and anti-back-bond surface states occured, which are similar to corresponding features at zinc-blende semiconductor surfaces, The pronounced ionic character of ZnQ leads to a strong localization of surface electronic features in the outermost double layer. The structureand ionicity-induced trends have been discussed in comparison with corresponding characteristic properties of more covalent materials. %'e have, as well, compared our results with various experimental data. Good general agreement between theory and experiment was found. This very good overall agreement which was even quantitative in a few cases, in our opinion, lends further support to the notion that a carefully determined empirical tight-binding bulk Hamiltonian can indeed be used as a reliable starting point for the determination of surface-induced electronic properties using the scattering theoretical method. W'e believe that we have been able to draw a fairly general picture of typical surface electronic properties of tetrahedrally coordinated ionic semiconductors which crystallize in the wurtzite structure. ACKNOWLEDGMENT

Financial support by the Deutsche Forschungsgemeinschaft under Grant No. Po 215/1 is gratefully acknowledged. APPENDIX: EVALUATION

OF THE GREEN'S-FUNCTION MATRIX ELEMENTS

In this appendix we briefly summarize the formulas for the Qreen's-function matrix elements needed for the calculation of bound-state energy levels or layer densities of states [see Eqs. (8) and (ll), respectively]. In the layer orbital representation we need only very small submatrices of Q since the surface-creating perturbation U can be represented by a very small matrix. In order to create any one of the four different surfaces discussed in this paper, we need to remove two planes of atoms from the perfect bulk crystal since the latter is described by a second-nearest-neigh-

EI, ECTRONIC STRUCTURE OF IDEAL AND RELAXED. . . The surface unit cells bor ETBM Hamiltonian. of the polar faces contain one atom each and the surface unit cells of the nonpolar faces contain two atoms or four atoms for the (10TO) and (1120) surfaces, respectively. The size of the Green'sfunction submatrices which need to be set up is, therefore, 4 x 4, 8 x 8, and 16 x 16 for the three different cases. The bulk Green's-function matrix elements are given in the layer orbital representa-

tion by

(I;q

~G

(E) (I'q) I'

q&, g «'q~"'&&""~I' +is — „(R) h

follows. (a) (1070) surface:

—m for m even 4 7t'

—gpss+

4 (b) (2120)

(Al)

V,

— for 12

~ odd.

surface:

3 p=—

—

with (p ~nk) = '. g

k

Q(

(r) =

n

(k)

CV

Q

27 g = 3 m for all rn.

etk (Ri+~p)

x 0, (r —R,. ~,) .

(c) (0002)/(OOOT) surfaces:

(A2)

The g„k are the bulk Bloch functions. The coefficients q are obtained by diagonalizing the bulk The geometry-related differences Hamiltonian. between the various surfaces simply enter through the scalar products (I;g ~nk). The matrix elements defined in (Al) can be reduced to a one-dimensional integral given as G~~i(q

E) =2

-8

e

m

m'

F

(x q Z)tfx

(AS)

with

Z+ ie

- Z„(x, q)

(A4)

The g vector, of course, ranges over different Brillouin zones for the different surfaces. Remember that ) is a composite index standing for ~, p, and m [see relation (8)]. The constant P and the function g are given for the various surfaces as

*Present address: IBM, T. J. Watson Research Center, Yorktown Heights, New York, 10598.

I

(Advances in in estkorperprobleme, Solid State Physics), edited by J. Treusch (Vieweg, Braunschweig, 1978), Vol. XVIII, p. 155. (Advances in Pollmann, in FesthorperproMeme, Solid State Physics), edited by J. Treusch (Vieweg, Braunschweig, 1980), Vol. XX, p. 117 G. Heiland and H. Luth, in Chemisorption Systems, Vol. III of Chemical Physics of Solid Surfaces and Heterogeneous Catalysis, edited by D. A. King (Elsevier, Amsterdam, in press). %. Gopel, in Festkorperprobleme, (Advances in Solid Treusoh (Vieweg, BraunState Physics), edited by schweig, 1980), Vol. XX, p. 177. H. Moorman, D. Kohl, and G. Heiland, Surf. Sci. 80,

~M. Schluter,

J.

J.

—ppg for 7r

4

pg

—en+ — for 8

even nz

odd.

states at the two polar faces follow simultaneously from Eq. (8) using the 4x4 Green'sfunction matrix. The discrimination between states stemming from the Zn- or 0-terminated face, respectively, can easily be accomplished by a few test calculations with three removed layers or by studying the layer densities of states. Note that for the evaluation of the surface band structure [Eq. (8)] the solutions of (1) for E &E„(q,x) are sea. rched. In this case, we can set e identical to zero in Eq. (A4) so that we have to deal with a Hermitian matrix in Eq. (8). This fact speeds up the calculations considerably. For fur'The bound

ther formal details see Ref. 38.

261 (1979). H. Moorman, D. Kohl, and G. Heiland, Surf. Sci. 100, 302 (1980). W. Mokwa, D. Kohl, and G. Heiland, Surf. Sci. 99, 202

(1980). and G. Neuenfeldt, Surf. Sci. 55, 362 (1976). 9%. Gopel, Z. Phys. Chem. (Frankfurt am Main) 106, 211 (1977). ~ W. Gopel, Ber. Bunsenges. Phys. Chem. 82, 744 (1978). ~~W. Gopel, J. Vac. Sci. Technol. 16, 1229 (1979). ~2&. Hotan, %. Gopel, and R. Haul, Surf. Sci. 83, 162

%. Gopel

(1979). ~3%'. Gopel,

R. S. Bauer,

and G. Hansson,

Surf. Sci. 99,

138 (1980).

~4'.

Gopel and V. Lampe, Phys. Rev. ~~R. Dorn and H. Luth, Appl. Phys.

J.

B 22, 6447 (1980). 47, 5097 (1976).

I.

7296

IVANOV

AND

Luth, G. W. Rubloff, and W. D. Grobman, Solid State Commun. 18, 1427 (1976). ~~R. Dorn, H. Luth, and M. Buchel, Phys. Rev. B 16, 4675 {1977}. i8H. Liith, in Festborperprobleme, (Advances tn Solid Treusch (Vieweg, BraunState physics), edited by schweig, 1981), Vol. XXI. p. 117. ~ H. Froitzheim and H. Ibach, Z. Phys. 269, 17 {1974). 2 Y. Margoninski, Surf. Sci. 94, L167 (1980). 2~J. Onsgaard, S. M. Barlow, and T. E. Gallon, Phys. C 12, 925 {1979). 22E. N. Lassettre, A. Skerbele, M. A. Dillon, and K. J. Chem. Phys, 48, 5066 (1968). Boss, M. Nitzan, Y. Grinshpan, and Y. Goldstein, Phys. Rev. B 19, 4107 (1979). MS. Joite, H. Hoinkes, H. Kaarmann, and H. Wilsch, Surf. Sci. 84, 462 (1979). +P. E. Chandler„P. A. Taylor, and B. Hopkins, Surf. Sci. 82, 500 (1979). Glaser, in Proceedings of the Eighth InterH. national Vacuum Congress, Cannes, 2980, Suppl. a la Revue "Le Vide, les Couches Minces, No. 201, edited by F. Abeles and M. Croset (Soc. Francaise du Vide, ~6H.

J.

J.

J.

J.

J.

"

1980), p. 723.

C. B. Duke

and A.

R. Lubinsky, Surf. Sci. 50, 605

(1975).

.

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press}. J. C. Phillips, Bonds

and Bands in Semiconductors, (Academic, New York, 1973). Pollmann and S. T. Pantelides, Phys. Rev. B 18, 5524 {1978). I. Ivanov, Ph. D. thesis, Dortmund, 1980 (unpublished). 9I. Ivanov, A. Mazur, and Pollmann, Surf. Sci. 92, 365 {1980). Pollmann and S. T. Pantelides, Phys. Rev. B 20, 4~J. Pollmann and S. T. Pantelides, Solid State Commun.

J.

J.

J.

J.

POLLMANN

1740 {1979). 30, 621 (1979); Phys. Rev. B 21, 709 (1980). 42J. Pollmann,

A. Mazur, and M. Schmeits, Surf. Sci.

99, 165 (1980). 43J. Pollmann, Solid State Commun. 34, 587 (1980). +H. Schultz and K. H. Thiemann, Solid State Commun. 32, 783 (1979). 45J. L. Birman, Phys. Rev. 115, 1493 (1959). 4~U. Bossier, Phys. Rev. 184, 733 (1969). 47S. Bloom and I. Ortenburger, Phys. Status Solidi B 58, 561 (1973). 4~J. R. Chelikowsky, Solid State Commun. 22, 351 (19773. 49R. L. Hengehold, R. Almassy, and F. L. Pedrotti, Phys. Rev. B 1, 4784 (1970). 5 R. Klucker, H. Nelkowski, Y. S. Park, M. Skibowski, and T. S. Wagner, Phys. Status Solidi B 45, 265 (1971). 5~D. W. Langer and C. Vessely, Phys. Rev. B 2, 4885

J.

J.

(1970).

C. J. Vesely

and D. W. Langer,

Phys. Rev. B 4, 451

(1971).

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