K. HUANG AND G. WYLLIE 27 ON CONDITIONS FOR THE EXISTENCE OF SURFACE STATES BY C. A. COULSON AND G. R. BALDOCK Received 13th February, 1950 The existence or non-existence of surface states on the boundary of a mono- valent metal is investigated fully for simple cubic models in I, 2 and 3 dimen- sions, using the molecular modification of Bloch wave functions. Limiting values are found for the necessary changes required a t the surface atoms, both for their Coulombic (electronegativity) terms cc and their resonance integrals /I. It is shown that surface states may be induced by the perturbation caused by the approach of a polar, or ionic, group ; and that such surface states are more easily produced close together than separately. In certain non-cubic systems, e.g. graphite, surface states exist even without any necessary changes in cc or /.I on the boundary.Finally, a series of recent theorems on the charge distribu- tion in large molecules is applied to a study of the alternating charge distribution sometimes associated with surface states. From the earliest recognition that there were sometimes surface states associated with a metal, their importance for all types of surface reactions has been appreciated. But very few attempts have been made to see under what conditions such states appear. It is our intention in this note to present some preliminary results which we have just obtained in this field. Since a more complete account of this work will be presented later, we shall be content here merely to summarize our28 EXISTENCE O F SURFACE STATES conclusions, without giving many details of the procedure by which they have been obtained.After we have described the model and our notation, it will be con- venient to report our results first for the simple cubic lattice in one, two or three dimensions, and then for other lattices. We conclude with a short discussion of the validity and relevance of the idea of surface states, using for this purpose some known results in the theory of molecular structure. Notation.-As is usual, we treat a metal by the Bloch method or, more properly, by the “ molecular ” modification of it which recognizes the existence of one or more boundaries of the metal, and therefore replaces the progressive waves of Bloch’s treatment by stationary waves such as (I) below, in which all the coefficients are real.This is simply the familiar molecular-orbital method extended to much larger molecules than those normally found in chemistry, and occupying one, two or three dimensions. This link with molecules will enable us to make use of several quite general theorems recently established in this field (the so- called LCAO approximation) .1 As in the work of Goodwin and others, the metal is supposed to con- sist of a large, but finite, number of positively charged atoms which pro- vide an effective field for the conduction electrons. Each conduction electron is governed by the same Hamiltonian H . Its motion is described by stationary de Broglie waves, which are solutions of the wave equation €I+ = E+. The tight-binding approximation is used to represent the appropriate single-electron wave functions, so that, if the atoms are suitably ordered, any such orbital is where the c, are constants to be determined by a variational calculation and 4, is the atomic orbital which would be used by the conduction electron if it were in the presence of atom I only. We are implicitly neglecting any of the hybridization recently proposed for metals by Pauling, and we concentrate for the moment on systems where all, or nearly all, of the atoms contribute one metallic electron.This model is essentially the same as that used by Goodwin, though we differ from him in the way we deal with the boundary of the array and also in our inclusion of certain overlap integrals. Let us put 16 = Z C r 4 r , - (1) x, = J+:~+,dr, P I S = J4:H+*dr, ‘ (2) sTs = J ~ ? + J T .Then a, may be called, in molecular language, the Coulomb term of atom Y , and p,, and S,, are the resonance and overlap integrals between atoms Y and s. Without loss of generality we may take each 4, to be real : this makes both j3,, and S,, real also, and symmetric in r and s. The coefficients c, and the energy E of the orbital ( I ) are now related by means of the secular equations : . ( 3 ) The summation in (3) is taken over all the atoms of the metal except Y , and there is one equation for each atom r . If we decide to retain prs Coulson and Longuet-Higgins, Proc. Roy. SOC. A , 1947, 191, 39; 1947, Goodwin, Proc. Camb. Phil. SOC., 1939, 35, 205. Mott and Jones, Theory of the Properties of Metals and Alloys (Oxford c,(a, - E ) + z ’ c , ( ~ ~ ~ - ES,,) = 0, r = I, 2, .. . . s 192, I5 ; 1948, 193,4478 456; 1948, 195, 188. Univ. Press, 1936). Pauling, Proc. Roy. SOC. A , 1949, 196, 343.C. A. COULSON AND G. R. BALDOCK 29 between adjacent atoms only, relatively few terms survive in (3). As a rule we shall do this by supposing that interactions between all other pairs of atoms are negligibly small. In at least one of our calculations, however, we consider interactions between next-to-nearest neighbours also. Some explanation is needed regarding the inclusion of the overlap integrals STs. It is true, as Chirgwin and Coulson 6 have recently shown, that there are many cases in which charge distributions and interatomic distances are entirely unaffected by its inclusion.But the energies are always slightly changed and, particularly in the upper half of the con- duction band, as Coulson and Kushbrooke 6 showed for graphitic layers, the shape of the band is strongly dependent on the value of this overlap integral. Now if all the atoms are entirely equivalent, we may put for all Y , a, = a s,, "' = s '}for all neighbouring pairs YS. (4) But it soon follows that for simple cubic crystals, this precludes the possibility of surface states. We shall therefore accept (4) for internal atoms, and be prepared to characterize the surface atoms by a different value of a,, and different values of Brs and S,,. The details of these new values vary from model to model, and are described for each case as it occurs. The questions that we ask are : For what values of the parameters a, 8, S are there solutions of the secular equations that may be interpreted as surface states ? And do the energies of these states lie inside or out- side the band of continuous distribution of enzrgy ? But there are other related questions.Although the existence of surface states is undoubtedly an aid to catalytic action, it is by no means inconceivable that, even if such states do not exist at first, they are never- theless brought into existence by the approach of some other group, particularly polar groups, toward the metal surface. Thxe is a close parallel here to some work by Coulson and Longuet-Higgins,l who studied the perturbations induced in large aromatic molecules by the approach of some charged ion.In our present case, we could approximately de- scribe such activity by a change in the ar of one surface atom. And we ask the question: if one surface atom is changed in this way, will there be a surface state induced in the metal, which was not there before ? A supplementary question, to which we can give only an incomplete answer, is : in such a case, how will two such surface states, associated with two distinct surface atoms, behave towards each other ? Questions of this sort are clearly very important in discussing the variation of cata- lytic activity with fraction of surface occupied. Results.-We must now define a surface state more precisely. If, in one of the single-electron wave functions (I), c, + o as the distance of atom Y from atom m increases, and if this occurs at a rate independent of the total number N of atoms in the crystal (N being assumed large), then $ is said to be a point state associated with atom m.This imFlies that an electron in the orbital t,b is effectively Iestricted to a region close to atom m. The extent of this restriction depends on the rapidity of the convergence of the sequence (c7) to zero. This convergence is not necessarily monotonic, and there may be local increases as well a s de- creases near to atom m. We shall also encounter orbitals for which G, is relatively high in the regions of atoms m, n, 9, . . ., tending to zero away from these atoms. 5 Chirgwin and Coulson, Proc. Roy. soc. A , 1950 (in press). 6 Coulson and Rushbrooke, Proc. Roy. SOC. Edin., A , 1948, 62, 350.30 EXISTENCE O F SURFACE STATES Such states will be termed multiple point states associated with atoms m, n, p , .. . . The definitions of line states and surface states can obviously be formulated in precisely analogous ways. the one- dimensional linear chain, the two-dimensional square lattice, and the simple cubic lattice. For these models we shall outline some of the con- ditions under which these special states may occur. Throughout the following summary, the assumptions (4) are made for all the atoms except for those specifically mentioned. (I) LINEAR CHAIN.-Let there be N atoms in a chain numbered consecutively from I to N , and let We shall first consider three simplified types of crystal: a, = aN = u’ Bla = /3N-l, for the end atoms, = q/3 for the end bonds, where /3 is the resonance integral for all other adjacent pairs and 7 is a multiplier which may be either greater than or less than unity.In most cases q will be greater than I. We have thought it proper to allow for the end resonance integrals having a different value from the internal ones, partly because detailed calculations for the linear case of polyene hydrocarbon chains7 shows that the end links are appreciably shorter than the others, and also because in some unpublished work we have shown that variations may also be anticipated near the surface, and also in the surface, cf three-dimensional aggregates. In this model we include possible next-nearest-neighbour interaction by writing where w will usually be quite small. type we write p 2 4 = 1395 = - * - = BN-3, N-1 = For the end interactions of this p 1 3 = f i N - 2 , N = Wq8* Following the suggestion of Mulliken for molecules, and developed and by de Heer,e we suppose that all overlap by Chirgwin and Coulson integrals S,, are proportional to the corresponding Bra.I t is convenient to define y = p - a s , h = ( a - a’)/y, * ( 5 ) cr = X / ( I + AS), I - p = q 2 / ( I + AS). - (6) and to put These substitutions enable us to express the condition for point states quite simply in terms of cr, p, and W. If u = 0, the end atoms have the same Coulomb term as the internal atoms ; if in addition p = 0, then 9 = I, so that fi12 = /323, etc. (i) u = p = 0. There are no point states, whatever the value of W . (ii) o I. If w2 is neglected we find that the condition for point states depends only on u and p, and not at all on w (second- neighbour interaction). There is a point state whose energy is below the continuous band If (I = o the conditions become T] > 4 2 .If T] = I the conditions become u < - I / ( I + S ) , if CJ + p < - I and one with energy above the band if a - p > I. or a > I / ( I - S ) , Both these states are dcubly degenerate. They are double point and the states correspond to those found by Goodwin. states associated with the end atoms. 7 Coulson, Proc. Roy. SOC. A , 1939, 169, 413. Bde Heer, Phil. Mag., 1950 (in press).C. A. COULSON AND G. R. BALDOCK 31 (iii) u, = cc’. We now consider the possibility of point states occurring when nothing but the Coulomb term of the mth atom is changed.This atom may perfectly well be an internal or boundary atom. In this case the overlap integral S is ignored. The condition for a point state associated with atom m is I A ] > ~ / m . Thus in the interior of the chain, any deviation of a, from cc gives rise to a point state. (11) SQUARE LATTIcE.-(i) In a square lattice the next-to-nearest neighbours are separated by a distance 1 / z times the lattice constant. If we take into account the corresponding resonance integral wfi (without changing any of the other parameters), we find that no point or line state emerges for any value of W . (ii) We now consider the effect of altering the Coulomb term of only one atom (m), on the boundary of the rectangular array, from a to a’. In this case we shall neglect the overlap integral S altogether and write A = ( E - a‘)/fl instead of ( 5 ) .We assume that atom m is not near the corner of the rectangle, and we find that if I A 1 > 2.8 (approx.) there is a state whose energy lies outside the continuous band. This state is a point state associated with atom m. (iii) Suppose now that two atoms, m and n, separated by p atomic distances dong one of the edges, have an altered Coulomb term a’. If p is large, the condition for the existence of double point-states associ- ated with these atoms is I A I > 2.8. As p is reduced, the limiting value of I A I steadily decreases. For example, when p = 2 , so that the altered atoms are separated by only one intervening atom, the condition is I A 1 > 2.3 ; and when p = I , so that they are neighbours, it is I A I > 2 .Since I h I measures the difference in electronegativity of the affected atoms, we infer that induction of surface states is made easier by the presence of a perturbation already existing on the surface ; and that such surface (point) states will tend to be associated together in groups rather than distributed at random over the whole surface. It is possible that we have here a little insight into the nature of the active centres so frequently postulated for catalytic activity. (111) SIMPLE CUBIC LATTICE.-(i) Let us denote any atom in the crystal by Ir, s, t ) , where I < Y < L , I < s < M , ~ < t g N . Evidently there are LMN atoms in the crystal. We shall consider the effect of a change in the parameters for all theatoms on the two opposite faces r = I , Y = L of the crystal.alsl = aJ& = a’ (for all s, t). Also we suppose that the resonance integrals between all neighbouring atoms in these two planes have the value x/3, and that the resonance integrals such as 2g between one atom in the surface and the adjacent one inside the crystal, have the value qfi. All other non-vanishing reson- ance integrals are equal to /3, and we suppose that in every case the overlap integral is proportional to the corresponding resonance integral. We define A, CT, p as in the case of the linear chain (eqn. ( 5 ) , (6)), and we also introduce Y and p defined by For these surface atoms I - Y = K r ) / ( I + AS) p = u + 2v (cos I$ + cos +), where r+ = h / ( M + I) K = I , 2 , . . . M 2 = I, 2 , .. . N. 3 = Zn/(N + I ) The conditions for surface states will now depend on p and p.32 EXISTENCE OF SURFACE STATES There is a band of surface states of low energy if p + p < - I, and one of high energy if p - p > I. These bands consist of levels depending on p and p, and since p may take MN different values for fixed u and v the conditions may be fulfilled for part of one band but not for all of it. Each level is doubly degenerate and all the states are double surface states associated with the faces Y = I, Y = L. There is one such orbital for each of the modified surface atoms. These bands of surface states may ovzrlap the continuous band of energy levels, (ii) If only one atom (m) on the surface has a different Coulomb term u’ and if we neglect overlap, we find the condition for an energy level out- side the continuous band is 1 X ] > 4.5 (approx.).This state is a point state associated with atom m. IV. OTHER CRYSTAL STRucTunEs.--Our previous examples have all been associated with the simple cubic structure, in which the unit cell only contains one atom. But as soon as we go to more complicated structures in which two or more atoms comprise the unit cell, the situation is quite different. For now surface states may appear even without any perturbation of the edge atoms. This is because not all surface atoms are equivalent geometrically (as they were in the earlier cubic crystal), and the differences in environment produce the same effect as a change in a in the simpler model. Fig. I shows a part of a single An example will illustrate this effect.n FIG. I. layer of graphitic carbon, in which the atoms are arranged in regular hexagonal form. The electronic wave functions for such a layer have been discussed fully by Bradburn, Coulson and Rushbrooke,Q who ob- tained the energies in terms of the numbers n and m which determine the size of the layer. These energies are obtained from the secular determinant, which may be factorized, corresponding to nodes of the wave function parallel to the m and n directions. A combination of eqn. (9) and (12) in their paper, together with some rather tedious analy:,is which we shall not reproduce, shows that in a large layer such as Fig. I there are 2nJ3 surface states. Application of the argument developed by Coulson 10 shows that in a layer of this kind all the energies are in the range E , & 38, which is known to be the range covered by the continuum of two-dimensional orbitals.Thus none of the surface states lie outside the continuum. In Fig. I the surface states are associated with the atoms on the extreme left- and right-hand edges, and not those on the top or Bradburn, Coulson and Rushbrooke, Proc. Roy. Soc. Edin. A , 1948, 62, 336. l o Coulson, Proc. Camb. Phil. Soc., 1950, 46, 202.C. A. COULSON AND G. R. BALDOCK 33 bottom. This is linked with the fact that in a graphitic layer the unit cell contains two atoms. Those atoms on the left are all crystallo- graphically equivalent, and so are those on the right, but those a t the top and bottom are not. It may be presumed that similar surface states will appear in other such lattices. We are making further calculations to investigate this in more detail.One conclusion, however, is quite clear from the graphite example: surface states are quite liable to occur even without any changes in Coulomb terms or resonance integrals, but simply on account of the geometrical structure of the boundary. General Molecular Considerations.-It is perhaps worth stressing that certain quite general results may be obtained by treating a crystal as a large molecule and using the general theory developed by Coulson and Longuet-Higgins.1 For example : ( I ) The existence of surface states does not necessarily imply an uneven distribution of charge. Thus, in the graphite example in (117) we are dealing with an " alternant molecule ", and $here will therefore be a Drecisely equal number of electrons associated with each atom.The excess of charge near the edges due to the surface states is compensated by a corresponding excess in the body of the crystal due to the other electrons. (ii) In the same way, as Chirgwin and Coulson have ~ h o w n , ~ if there are no changes in Coulomb terms, but only in resonance integrals, the net charge is uniform. (iii) But if the Coulomb term of one atom is changed in the direction of increasing its electronegativity, it will carry a final net negative charge, and as we proceed away from this atom, other atoms will have an alter- nating positive and negative charge. In the case of a surface layer with increased or-values, the alternation of charge occurs as between one layer and its neighbouis. There are still one or two further molecular considerations that we may. " carry over " to crystal lattices. (iv) There is considerable uncertainty regarding the validity of using the same Hamiltonian operator H for each electron. These operators are not strictly self-consistent, and as Chirgwin and Coulson have shown in some unpublished work, we ought really to use a different operator for each electron, The resulting errors are not likely to be so serious in crystals as in molecules. (v) A more serious difficulty arises from the fact that the so-called " lowest configuration I' is not, by itself, a sufficiently good wave function. As Miss Jacobs l1 has shown for naphthalene, there is often considerable configurational interaction, whi'ch results in a slight unevenness of charge, more particularly a t the edges of the molecule, i.e. at the surface of the crystal. We are investigating this matter more carefully a t the moment, in this Department using a simplified model. Conclusion.-To some extent all the models that we have discussed above are largely formal. And consequently our conclusions are not immediately applicable t o any real problem. But we believe that they do illustrate tendencies and situations likely to be found in actual surfaces, and may serve to " point the way " to further specific calculation. Wheatstone Physics Laboratory, Thus : King's College, London. 11 Jacobs, Proc. Physic. SOC. A , 1g4g,62, 710. B