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Theory of surface states
Surface Science 0 North-Holland
55 (1976) 93-108 Publishing Company
THEORY OF SURFACE STATES II. The copper and tungsten (001) surfaces
S.J. GURMAN * Cavendish Laboratory, University of Cambridge, Cambridge CB3 OHE, England
Received
2 August
1975
The results of a calculation of the surface state energies in the s-d hybridization gap for copper and tungsten (001) surfaces are reported. Surface states are found to exist over a large part of the two-dimensional surface Brillouin zone in both cases, resulting in a band of localized states about 1 eV wide situated near the bottom of the d-band. The energies of these states are largely independent of the details of the surface potential.
1. Introduction It has been known for a number of years that electrons can exist in states localized near crystal surfaces [ 1 ] . If we have a semi-infinite crystal, then the requirements for a localized state to exist at its surface are: (1) the electron has insufficient energy to escape into the vacuum; (2) it has momentum parallel to the surface and energy such that it encounters a band gap of the crystal and so cannot escape into the bulk; (3) wavefunctions can be matched at the surface to give a valid electron state. Surface state calculations in the past have concentrated mainly on semiconductors [2] , because such states strongly influence many properties of devices made from these materials. The calculation of surface state energies for semiconductors has shown that a narrow band of localized states exists in the valence band-conduction band gap, and these results have been verified by experiment [3]. Calculations have also been performed for simple metals, and a recent self-consistent calculation for aluminium [4] has shown a significant density of surface states on all three simple crystal faces. Of more recent interest are the d-band surface states [5,32] which occur in gaps produced by hybridizing d-bands, and this paper reports the results of a calculation of surface state energies in such a gap in two typical transitions metals, copper and tungsten. In the first paper in this series [6] , referred to below as I, we discussed the * Present address: Science WA4 4AD, England.
Research
Council,
Daresbury
Laboratory,
Daresbury,
Warrington
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existence of surface states in band gaps in very general terms. The results, derived assuming only that there was two-dimensional symmetry and a mirror-plane parallel to the surface, showed that surface states should be a common occurrence on all types of crystalline solids, and we derived general criteria for their existence. The major gap in the band structure of transition metals is that caused by the hybridization of the s- and d-bands and, by the criteria derived in I, we should expect surface states to exist in it. We scanned the entire two-dimensional surface Brillouin zone (SBZ) of copper and tungsten (which are typical examples of the two main crystal structures found in transition metals: fee and bee respectively), in search of such states. In each case a substantial density of surface states was fount in the s-d hybridization gap. These states were found to be insensitive, within reasonable limits, to the details of the surface barrier potential, because of their tightly-bound d-band nature. Experimental investigation of the difference in electron behaviour for clean and contaminated crystal surfaces gives some evidence supporting our calculated results, particularly in the case of tungsten.
2. Details of calculation In these numerical calculations, we use the method of matching of wavefunctions at the crystal surface to determine the allowed energies of localized states, rather than the reflection matrix method of I, since these calculations antedate the development of that method. The use of the reflection matrix method to calculate both the extended and localized states of a thin film of copper will be described elsewhere [7]. We assume that the crystal is semi-infinite, and still possesses the full infinite periodicity in the directions parallel to the surface. Thus in these directions we can still use Bloch’s theorem, and the component of crystal momentum parallel to the surface, k,, , is a good quantum number and is conserved (modulo g, a reciprocal lattice vector of the two-dimensional surface net). To find the energy of a surface state, we take a fixed value of k,, and scan across the energy gap until the matching equation is satisfied. The crystal wavefunctions which we must use in the matching are those appropriate to the semi-infinite crystal, i.e. those wavefunctions which obey the Schrodinger equation within the crystal, and are bounded at infinity in the directions parallel to the surface: thus k,, is restricted to real values. The wavefunctions must also be bounded as z -+ 0~)deep within the crystal. The allowed solutions must also obey a boundary condition at the surface, but this requirement we leave until later as the matching condition. Thus we need to include all the wavefunctions of the infinite crystal, and in addition all those waves which behave asymptotically as e-@. These are not allowed states of the infinite crystal, since they are not bounded as z -+ -00. We have removed this requirement by truncating the crystal near z = 0. These decaying waves appear as Bloch waves with a complex K,, the imaginary part of
S.J. GurmanfTheory of surface states. Ii
9s
which, 4, must be positive. 4 gives the rate of decay of the electron density due to these waves as we go deeper into the crystal. In the band gaps of the crystal, these decaying waves are the only ones present, and it is they which make up the surface state. The need to include the decaying waves rules out all the usual methods of bandstructure calculation. The presence of the full symmetry in the directions parallel to the surface, and, for most simple crystal faces, the simple arrangement of atoms in layers parallel to the crystal surface, lead us to use the layer scattering method of Kambe [9]. This method has been extensively developed and programmed for LEED calculations [lo] , It is particularly convenient for surface state calculations since it calculates eigenfunctions and eigenvalues (K,) for a fixed input E and kl,: precisely the form of result which we require in the surface state problem. The layer scattering method is further simplified by the fact that for the (001) faces of fee and bee lattices the layer is a mirror plane of the crystal. Kambe’s method uses the muffin tin approximation for the crystal potential, the scattering properties of a single atom being represented by a few phase shifts. For convenience in performing the energy scan, we parametrize these phaseshifts as functions of energy using the scheme of Cooper et al. [ Ill. We consider the crystal as consisting of a stack of planes of muffin tin potentials, the layers lying parallel to the crystal surface and being of infinite extent. In the cubic metals the repeat unit of the crystal structure in the z-direction is a single layer and for the (001) surface each layer is a mirror plane of the crystal. Each layer is a two-dimensional Bravais net, i.e. there is only one atom per unit cell, Because we have used the muffin tin approximation the potential between the layers is constant, and in this region we expand the Bloch functions in a series over the reciprocal lattice vectors of the two-dimensional surface net,g. The coefficients in this expansion will be z-dependent. We calculate the scattering of these plane wave components by a single layer in the form of reflection and transmission matrices and then combine the layers to form the semi-infinite crystal. The resulting scattering eigenproblem is solved to obtain the Bloch eigenfunctions and eigenvalues of the crystal. Because each layer is periodic, k,, is conserved (modulo g) in the scattering and we therefore obtain the eigenvalue K, at a given E and k,,. Details of the setting-up and solving of the eigenproblem are given by Pendry t101. The solution of the eigenproblem gives us an expression for the Bloch waves in the region of constant potential between layers. If we truncate the crystal in such a region by a surface, we must match these Bloch waves to those of the vacuum to obtain an allowed state. To simplify the matching problem we assume a plane surface barrier, i.e. one that is independent of coordinates parallel to the surface, rr. Such a barrier does not mix components with different values of g. This assumption of a plane barrier is reasonable for fairly close-packed regular layers such as that forming the fcc(OO1) surface. It is less good for surfaces like the bcc(01 l), where there are “channels” between the surface atoms in the 01 direction. For semiconductors, with their loosely packed surfaces, it is likely to be an even poorer assumption. Any
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of surface states. I1
non-planar effects at the surface will have the periodicity of the lattice (unless there is any reconstruction) and hence will only mix different g values and not introduce any new values of g. The simplest form which we can use for the surface barrier is the simple step up to the vacuum level [the Abrupt Potential Model (APM)] .In this model we take the total potential to be W
= vu W,
- z) + I&t&
O(z - z(r) >
(1)
the step of height Vu being located at zo. The zero of energy is at the muffin-tin zero. A reasonable position for the step is half-way between layers [ 151 . The step barrier is obviously not an exact repre~ntation of the surface potential, and as a better approximation we use a barrier with a smooth variation parallel to the surface, but still maintain the independence from rll. The form we use is V(z) = Vo( 1 + ePz)- ’ .
17_)
The width parameter fl may be expected to be of the order of a reciprocal screening length: in copper this is 1 au. This form approximates the self-consistent surface potential determined by Lang and Kohn [ I.51 . In using this form, V replaces the V. of eq. (1): we thus truncate the smooth barrier and match at zo, which we choose to be a point where V is small but where we are still outside the layer of muffin tins, after integrating the vacuum wavefunctions through the barrier numerically as far as zo. The total wavefunction in the crystal is a sum over all the Bloch waves with a given energy and k, and we must match this to the vacuum wavefunction. If we have n plane wave components in each Bloch wave, then we need n Bloch waves to achieve a unique matching at all points of the surface. The layer scattering method [lo] gives us 2n eigenfunctions, and in the band gap exactly n of these have the required property of decaying as z + *, and it is these which we use in the matching. The details of the matching procedure are given by Forstmann and Pendry [S] . If we have a mirror plane parallel to the surface, the wavefunctions in the crystal can be made real, and the problem of finding the surface state energy reduces to that of finding the zero of a real determinant: this we do by scanning in energy across the gap and plotting the value of the matching determinant.
3. Surface states on the copper (001) surface Copper forms a fee crystal with a cube edge (I = 6.81735 au. The (001) surface Bravais net is therefore the simple square of side a/d/2 aligned at 45” to the cube edges. The reciprocal net is also a simple square. The 2-D surface Brillouin zone is shown, superimposed on a (001) projection of the bulk zone in fig. I. In our calculation we take each layer as having the bulk crystal structure and set all layer spacings equal: the LEED calculations of Laramore [14] indicate that any dilation or contraction of the layer spacing at the surface is less than 5% of the bulk spacing.
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-K,
Fig. 1. Face-centred cubic (001) surface Brillouin zone (dashed line) superimposed on a (001) projection of the threedimensional crystal zone. All inequivalent points of the SBZ lie in the shaded triangle. The K axes are those of the infinite three-dimensional crystal.
To describe the scattering properties of an atom we have used the phase shifts of Burdick [ 121 as parametrized by Cooper et al. [I l] . We limit ourselves to three phase shifts. The height of the surface barrier was set at 0.47 hart (with the zero of energy at the muffin tin zero) equal to the value found by Burdick. To check the accuracy of our calculation we calculated the bandstructure for k,, = 0, which is the bulk rX direction, from the muffin tin zero to the Fermi energy. Using nine g values in the expansion gave results which agreed with those of Cooper et al. to better than 0.003 hart. Increasing the expansion set to thirteen vectors changed none of the energies at r or X by more than 0.002 hart. All the surface state energies were calculated using nine g vectors with occasional check runs using thirteen: at no point in the surface Brillouin zone did the two values of the surface state energy relative to the bottom of the gap differ by more than 0.001 hart. We have already given a brief discussion of the form of the surface barrier. From the point of view of the actual calculations, the simplest form of barrier to use is the plane step placed midway between layers, the APM. We have used this model in the calculation of the surface state energies over the entire surface Brillouin zone, to determine the density of surface states. In addition we have calculated the surface state energies at a few representative points in the zone for various positions and shapes of the barrier to determine the sensitivity of the surface states to the barrier. The surface states in the s-d gap were found to be highly insensitive to the barrier details, and so we may expect the results calculated using the APM to be an accurate reflection of the surface states on the (001) surfaces, whose actual barrier potential will be smoothed over a finite width.
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4. Surface state energies in the APM
Using a step potential
barrier of height V, = 0.47 hart placed half a layer spacing
states/ hart SBZ
a
loo -
Total
0.08
0.12
E hart
N (El
b
30 -
Surface !wer
0.08
t
OJO
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0.08
O.lO
0.12
c ‘hyer
Third Wer
Fourth
t-iel_ LJL 0.08
0.x)
0.12
0.08
0.x)
0.12
Fig. 2. (a) Total surface state density per surface cell for the copper (001) surface. (b) Surface state densities per atom for atoms in the first four atomic layers for the copper (001) surface. EF = 0.28 hart.
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T
Fig. 3. Surface states an the copper (001) surface. (a) Energy contours, in hartrees. (b) Localization contours, in units of the layer spacing. There are no surface states around @. The dotted line is at a localization length of 10 atomic layer spacings taken as the limit of the surface state region.
from the last layer of atoms, and nine g values in the wavefunction expansion the entire irreducible section of the surface Brillouin zone (SBZ) was scanned for surface states in the energy region of the s-d hybridization gap. The values of k,, were taken on a square grid of spacing 0.05 au-l and further points graphically interpolated between these to give about 300 surface state energies to form the density of states histogram. The results are shown in figs. 2 and 3. In fig. 2b we give the local density of states, i.e. the density of states per surface cell averaged in the z-direction over the given layer. In addition we give the integrated local density of states in fig. 2a, which is the sum over all layers in the crystal of the individual layer local densities. On the copper (001) surface, a surface state was found in every pure s-d gap (i.e. one which is not crossed at any energy by a propagating band), as was predicted by the reflection matrix method of paper I, and in nearly every other absolute gap. Surface states exist over about 75% of the SBZ in the s--d gap: i.e. there are about 1.5 electrons/surface cell in localized states in this energy range. There are no states around H since in this region there is no absolute energy gap. In general the states were found to lie very close (usually less than 0.01 hart) to the bottom of the gap. This is the position predicted by the two band model. Our result for the Fpoint (k,, = 0) agrees with that of Forstmann and Pendry [S] , and with that of Marcus and Jepsen ]32] . The surface states form a band about 0.040 hart wide, most of the states being in the single peak (fig. 2) centred 0.17 hart below the Fermi level. The decay constant, or localization length, which is the distance in which the electron density in the state falls by a factor, e, was found on average to be 10 au, or about three layer
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spacings, although for some states it falls as low as 3 au. The variation of the localization length across the SBZ is shown in fig. 3b. We see that the localization of the states is approximately constant across the zone, rising very sharply near the points where the surface states disappear. In the histograms of the density of surface states we see the decay of the surface state density with depth in the crystal. About 80% of the surface state density lies within the first four atomic layers, within about 8 A of the surface. The peak density of surface states in the first atomic layer, within which about 30%, of the density is localized, is 35 electrons/hart atom, compared to a bulk (propagating states) value of around 40 in this energy range. An atom in the first layer has 0.6 electron in localized states: these electrons are drawn from the propagating s- and d-bands involved in the hybridization. If we calculate the electron density in the bands in a simple tight binding model we find that the surface state density is compensated by the decrease in electron density in the propagating band. There is therefore little or no charging at the surface, and self-consistently requirements are not likely to alter the surface state energies significantly.
5. Sensitivity
of surface states to the barrier
The precise form of the potential near the surface is not known with any accuracy for the transition metals. The height of the barrier can be established from work function measurements and band calculations, but the shape and position are in doubt. Some theoretical results, applicable to the simple metals have been published [ 1.51, and these can be extended to electron densities typical of transition metals [33]. The resulting barrier shapes are approximately of the form (2) with /3 approximately equal to an inverse screening length. Because of the uncertainty surrounding the details of the barrier, it is important to investigate how sensitive the surface states are to the barrier: only those results which are relatively intensitive can be taken as representative of the actual conditions. We shall consider the effects of changing the barrier height, position and width on the surface states in the s-d hybridization gap at a general value of k,, . We choose a point where the gap between the two hybridizing bands is not crossed by another propagating band: a typical point is at k,, = (0.47,O.l) where the gap extends from 0.1123 to 0.1169 hart. Using the APM, we investigated the sensitivity of the surface state energy to the barrier height and position. With the barrier placed half a layer spacing from the last atom, changing the height from 0.35 to 0.60 hart did not alter the surface state energy to within the accuracy of measurement: the state remained at E, = 113.10 + 0.02 mhart. Changing the position of the barrier with V. set at 0.47 hart gave a similar result: changes in E, were very small until we moved the barrier to within a quarter of a layer spacing from the centre of the atomic layer, where the surface state energy started to fall. With this lack of sensitivity to gross changes in the surface barrier, we would not
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expect any of the effects due to the replacement of the step barrier by a smooth barrier of the form of eq. (2) to be large. This is borne out by the numerical results. With the centre of the barrier placed halfway between layers, the change in surface state energy consequent upon going from a step barrier (13-l = 0) to a wide smooth barrier (‘J-l = 0.7 au) was very small, only about 0.1 mhart, or 2% of the gap width. The screening length in copper is about 1 au, so we would expect a self-consistent barrier to have a 0-l of about this value. We see that using such a barrier will give surface state energies almost identical to those obtained using the APM. This is largely because the s-d surface states lie well below the vacuum level, and so electrons in them do not see a large increase in the classically-allowed region when we change from a step to a smooth barrier: we might expect states nearer to the vacuum zero to be more sensitive. The states in the gap we have considered are rather less localized than the average. in copper, with a localization length of S--6 layers, which will tend to diminish their sensitivity to the barrier. However, calculations at several points in the SBZ with widely varying degrees of localization all give the same result: the surface state energy in the s-d gap is virtually independent of the barrier details over the range of physical interest. If we examine the form of the surface state wavefunction we can see the reason for the insensitivity of its energy to the barrier: in the wavefunction expansion, the high g components dominate. These have large imaginary values of I& and hence decay rapidly as we move away from the centre of the pseudolayer. The intensity of the surface state wavefunction halfway between layers is about a quarter of its peak value. We therefore have a low density of localized electrons between the layers, where the barrier is located and so changes in the barrier will have little effect on the surface state until we move it close to the layer. If we calculate analytic~ly the g expansion of the m = 0 3d wavefunction (given by Co&on [16] : J/3$(‘) = r2 eFcyT%@,
4) ,
with (Y4 2.7 au-‘) we find that it is very similar to our numerical face state is almost entirely d-like.
(3) form: i.e. the sur-
6. Surface states on the tungsten (001) surface Tungsten exists in the form of a bee crystal, and we take it as a typical transition metal of this structure. As well as determining the surface state distribution on the (001) face of tungsten itself, this calculation points up the differences in the surface state distribution across the SBZ from that of copper: these differences are largely due to the different band symmetries imposed by the different crystal structures. From the general theory of paper I and the two examples we treat here, we obtain an idea of the surface state distribution to be expected on any transition metal with fee or bee structure. Since the copper calculation has shown the surface states in the
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Fig. 4. Body-centred cubic (001) surface Brillouin zone (dashed line) superimposed on a (001) projection of the three-dimensional zone. All inequivalent points of the SBZ lie in the shaded triangle. The K axes are those of the infinite crystal.
s-d hybridization gap to be insensitive to the barrier details, we consider only the Abrupt Potential Model here, using a step barrier placed half a layer spacing from the last atomic layer. We omit relativistic effects, in particular the spin-orbit interaction: this omission has little effect on the s-d gap [ 171. We have not considered the surface states in the gap close to the Fermi energy, since this will be strongly influenced by the spin-orbit interaction which lifts the degeneracy between the A,. A21 and A, bands at r. A tight binding calculation by Sturm and Feder [ 1S] shows the existence of a surface state in this gap at p: this state is in fact a surface resonance, since it will interact with the propagating A, band, but calculations show that it has a strongly localised character [ 181. The bee tungsten crystal has a cube edge a = 5.8911 au. The (001) surface net is therefore a simple square, as is the reciprocal net. The Z-D Brillouin zone is shown superimposed --on a (001) projection of the bulk zone in fig. 4. The irreducible section is the triangle I‘XM. We use the phase shifts of Jennings and McRae [ 191, which are those appropriate to Mattheiss’ [ 171 V, , parametrized along the lines suggested by Cooper et al. [ 111. The barrier height was fixed at 0.590 hart, as found by Mattheiss. Again as an accuracy check, we calculated the band structure fork,, = 0, the bulk I’H direction. We found good agreement with ~attheiss with twenty-one g vectors in the wavefunction expansion and fair agreement using thirteen. Most of the surface state energies were calculated using thirteen vectors to keep computing time down: check runs using twenty-one vectors showed that the changes in surface state energy consequent upon increasing the set were always less than 0.005 hart and were largely independent of k,,. All the layer spacings were set equal to the bulk value: the LEED results of Jennings and McRae [19] show that there is little or no dilation of the surface layer.
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state energies in the APM
Using thirteen g vectors in the wavefunction expansion, we scanned the entire irreducible section of the SBZ for surface state solutions of the matching equations in the energy range of the s-d hybridization gap, using a step barrier placed half a layer
N(E) states/ hart SBZ
I 60.
a
Total
40 -
1 0.25
0.27
0.29
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E hart
N E)
D
20
surface layer
law
10
t
0.25
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l-aThird hyer
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027
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a31
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Fourth law
029
031
C
0.27
Fig. 5. (a) Total surface state density per surface cell for the tungsten (001) surface. 0~) Surface state densities per atom for atoms in the first four atomic layers for the tungsten (001) surface. EF = 0.42 hart.
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spacing from the last atomic layer. We found one surface state in virtually all the absolute s-d gaps. The main feature determining the surface state distribution in the SBZ is the crystal structure, which requires a degeneracy of the A, and A, bands at HI,. This leads to a large overlap of these two bands over a large part of the SBZ around r. There is consequently no gap and no surface states in this region (fig. 6). Similarly there are no states around R due to the degeneracy of A, and A, bands at P,. The interaction between any localized and propagating states at the same energy will be zero from symmetry considerations at r and m and along Tm, and so true surface states can exist at these points even though the s-d gap is crossed by another propagating band, and presumably these points are surrounded by small regions of surface resonances, but the vast majority of true surface states are in the region around R. The region of true surface states covers about 60% of the SBZ. As in the case of copper, this region was scanned on a grid of spacing 0.05 au-l and further surface state energies found by graphical interpolation between these points. The results were used to plot the density of states histograms of fig. 5 and the contour plots of fig. 6. The surface states are concentrated in a single peak of width 0.02 hart centred 0.14 hart below the Fermi energy. The degree of localization is fairly constant across the zone, varying between two and four atomic layers (3-6 a), except where it rises rapidly at the edges of the region of surface states. The states in the main peak of surface state density are the most localized (see fig. 6) and so this peak is strongly localized in the surface layer. The average localization length was found to be about three atomic layers (“5 A) leading to about 80% of the surface state density lying in the first four atomic
i Fig. 6. Surface states on the tungsten (001) surface. (a) Energy contours, in hartrees. (b) Localization contours in units of the layer spacing. There are no surface states near f and 9: The dashed line is at a localization length of 10 atomic layer spacings, taken as the limit of the surface state region.
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layers, within 6 a of the surface. About 30% of the density lies in the first atomic layer, giving a peak density of 20 electrons/hart atom there. An atom in the surface layer has 0.4 electron, out of a total of six, in localized states. The surface states lie in the energy range of the two lowest d-band peaks in the bulk density of states [ 171 and the high density of bulk states will tend to make the surface states difficult to observe. However, if the peak is largely due to states near the zone edge, it will be strongly attenuated near the surface by the matching requirements on the propagating bands which remove electron density from the band edges near the surface [20] and this may improve the visibility of the surface states. In summary, the numerical results for the tungsten (001) surface show a significant density of surface states, which are grouped around the L%point due to the crystal symmetries.
8. Experimental
evidence for surface states
In order to find surface states experimentally, we need a technique which probes only the first few atomic layers. For this reason, the most useful are the methods which use energetic electrons. These are surface sensitive because of the low mean free path of the excited electrons. The electrons used in UPS, Auger spectroscopy and in energy loss measurements, with kinetic energies between 10 and 100 eV have mean free paths below 10 A [21] . The most surface sensitive techniques of all are Ion Neutralization Spectroscopy [22] and Field Emission [28]. However, results from INS are few and give little clear data on surface states, whilst field emission cannot probe deep enough below the Fermi energy to find the s-d hybridization surface states, although it has provided evidence for the existence of surface states in the gap just below the Fermi level due to spin-orbit interaction [28] . In all these methods, to detect surface dependent structure we compare results from clean and contaminated surfaces: this obviously causes problems since the adsorbate introduces many extraneous effects as well as destroying the intrinsic surface states by removing the surface periodicity and so mixing states with different k,, . 8. I. Copper (001) The early paper of Forstmann and Pendry [5] quoted in support of their results UPS data of Spicer [23] showing an anomalous peak at -6 eV. This structure was not observed by Eastman and Cashion [24], who believed it to be due to contamination in the earlier experiments [24]. The Auger results of Powell and Mandl [25] with an energy resolution of -0.5 eV show a small peak at the bottom of the d-band which disappears on contamination. This peak lies 2.6 eV below the main d-band peak which we assign as due to the double peak in the theoretical density of states at 0.37 and 0.32 ryd. Thus the surface dependent structure lies about 5 eV below the Fermi energy, close to our calculated surface state energy. Powell and Mandl
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also observe two characteristic energy losses, which disappear upon contamination, at 4.5 and 7.0 eV. The higher loss can be ascribed to the surface plasmon [26], and we consider the lower as due to transitions from the surface state band to states near the Fermi energy, probably those near Q and Z. Such transitions conserve k,,. 8.2. Tungsten (001) For tungsten, as for copper, there is little direct experimental evidence for intrinsic surface states; except for the state close to the Fermi level. This has been observed in UPS [27], field emission [28] and optical reflectivity experiments [29]. For the state in the s-d gap we depend upon UPS and energy loss measurements. Several directional UPS results show a strong removal of states by surface contamination at about 4.5 eV below the Fermi level. This is particularly pronounced in the results of Egelhoff and Perry [30] taken for various values of k,, along i%. Surface states are symmetry allowed along all of i%, the band extending from -3.0 eV (G) to -5.0 eV at the minimum near (0.2,0.2). The removal of these states is complete by an exposure of 0.25 monolayers suggesting that they are due to intrinsic surface states removed by the destruction of surface periodicity, rather than that the dip is due to the removal of electrons from this energy range by the formation of bonding orbitals with the contaminant. We note that the dip moves to higher energies with increasing k,, in accordance with the bandstructure between k,, = (0.2,0.2) and m. Electron energy loss spectra show surface sensitive losses at 1 S, 4.3 and 10 eV which disappear upon contamination [3 1] We identify the surface plasmon at 10 eV and assign the other two to transitions to states just above the Fermi level from the spin-orbit and s-d gap surface states respectively. For the s-d gap states the final state will probably be near C or A since such transitions conserve k,,. Edwards and Probst found that the intensity of this loss peak was greater with the primary beam aligned along the (01) direction of the surface cell than those for the -- beam aligned along (11); our results show a higher density of surface states near I’X than near i;R.
9. Conclusions Our motivation for this work was the result of a two-band model calculation [6] which predicted that a surface state should always exist in an absolute band gap within the zone, within very general limitations. The numerical results, found using an accurate crystal potential and a reasonable approximation for the surface barrier potential have borne out this prediction: a surface state exists everywhere in the two-dimensional zone where the s-d gap is not completely crossed by another propagating band. Because of this, the region of existence of surface states (and hence their total number) is largely controlled by the crystal symmetry which fixes the degeneracies at symmetry points and hence controls the overlap of the various bands.
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This point is demonstrated by the difference in surface state distribution between the fee copper and bee tungsten results. The calculated surface state energies were found to be largely independent of the details of the surface barrier, in marked contrast to the NFE case [4] and we may therefore regard these results, found using a step barrier, as accurately representing the true state of affairs on the (001) surface. The results show a substantial density of surface state electrons, localized within the first three or four atomic layers, in a band about 1 eV wide located near the bottom of the d-band. The experimental evidence for these states is not as conclusive as we might wish, but is strongly indicative of the presence of such a band which disappears upon contamination of the surface. The predictions of the two band model are general and the degree with which they agree with the numerical results suggests that similar surface state bands will exist in the s-d hybridization gap of most, if not all, other transition metals, and that consequently the surface and bulk densities of states may be expected to differ markedly.
References [ 1) W. Shockley, Phys. Rev. 56 (1939) 317; E. Goodwin, Proc. Cambridge Phil. Sot. 35 (1939) 205, 221, 232. [2] Reviews are given by: R.O. Jones, in: Surface Physics of Semiconductors and Phosphors, Eds. C.A. Scott and C.E. Reed (Academic Press, New York, 1974); S.G. Davison and J.D. Levine, Solid State Phys. 25 (1970) 1. [3] J.E. Rowe and H. Ibach, Phys. Rev. Letters 31 (1973) 102; 32 (1974) 421. [4] E. Caruthers, L. Kleinman and G.P. Alldredge, Phys. Rev. B8 (1973) 4570; B9 (1974) 3325. [S] F. Forstmann and J.B. Pendry, Z. Physik 235 (1970) 74; M. Tomasek and P. Mikusik, Phys. Rev. B8 (1973) 410. [6] 3.B. Pendry and S.J. Gurman, Surface Sci. 49 (1975) 87. [7] S.J. Gurman, J. Phys. F5 (1975) L194. [8] F. Forstmann, Z. Physik 235 (1970) 69. (91 K. Kambe, Z. Naturforsch. 22a (1967) 322. [lo] J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974). [II] B. Cooper, E. Kreiger and B. Segall, Phys. Rev. B4 (1971) 1734. [12] G.A. Burdick, Phys. Rev. 129 (1963) 138. [ 131 V. Heine, Solid State Phys. 24 (1970) 1. [14] G.E. Laramore, Phys. Rev. B9 (1974) 1204. [15] N.D. Lang and W. Kohn, Phys. Rev. Bl (1970) 4555. [ 161 C. Coulson, Valence (Oxford Univ. Press, 1961). [17] L. Mattheiss, Phys. Rev. 139 (1965) A1893. [18] K. Sturm and B. Feder, Solid State Commun. 14 (1974) 1317. [19] P. Jennings and E.G. McRae, Surface Sci. 23 (1970) 363. [20] M.J. Kelly, Thesis, Univ. of Cambridge (1974). [21] C.J. Powell, Surface Sci. 44 (1974) 29. [22] H.D. Hagstrum and G.E. Becker, Phys. Rev. 159 (1967) 572. [23] W.E. Spicer, in: Proc. Symp. on Optical Properties and Electronic Structure of Metals and Alloys, Ed. F. Abel&s (North-Holland, Amsterdam, 1966).
108
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of surjixe states.
II
DE. Eastman and J.K. Cashion, Phys. Rev. Letters 24 (1970) 310; DE. ~ast~n,Fhys. Rev. B3 (1930) 1769. [ZSJ C.J. PoweB and A. Mandl, Phys. Rev. 36 (1972) 4418. [26] L.H. Jenkins and M.F. Chung, Surface Sci. 26 (1971) 151. [27] B. Feuerbacher and B. Fitton, Phys. Rev. Letters 29 (1972) 786. [28] J.W. Gadzuk and E.W. Plummer, Rev. Mod. Phys. 43 (1973) 487. 1291 G.W. Rubloff, J. Anderson, M.A. Passler and P.J. Stiles, Phys. Rev. Letters 32 (1974) 667. [30] W.F. Egelhoff and D.L. Perry, Phys. Rev. Letters 34 (1975) 93. [31] D. Edwardsand F.M. Probst, J. Chem. Phys. 55 (1971) 5175. f 321 P. Marcus and D.W. Jepsen, in: Computational Methods for Large Molecules and Locahzed States in Solids, Eds. F. Herman, A.D. McLean and R.K. Nesbet (Plenum, New York, 1973). The authors have since revised the results of this paper and now conclude that CutOOl) does exhibit a surface state. [33] N.D. Lang, private communication to J.B. Pendry (ref. [ 101, p. 239). [24]
55 (1976) 93-108 Publishing Company
THEORY OF SURFACE STATES II. The copper and tungsten (001) surfaces
S.J. GURMAN * Cavendish Laboratory, University of Cambridge, Cambridge CB3 OHE, England
Received
2 August
1975
The results of a calculation of the surface state energies in the s-d hybridization gap for copper and tungsten (001) surfaces are reported. Surface states are found to exist over a large part of the two-dimensional surface Brillouin zone in both cases, resulting in a band of localized states about 1 eV wide situated near the bottom of the d-band. The energies of these states are largely independent of the details of the surface potential.
1. Introduction It has been known for a number of years that electrons can exist in states localized near crystal surfaces [ 1 ] . If we have a semi-infinite crystal, then the requirements for a localized state to exist at its surface are: (1) the electron has insufficient energy to escape into the vacuum; (2) it has momentum parallel to the surface and energy such that it encounters a band gap of the crystal and so cannot escape into the bulk; (3) wavefunctions can be matched at the surface to give a valid electron state. Surface state calculations in the past have concentrated mainly on semiconductors [2] , because such states strongly influence many properties of devices made from these materials. The calculation of surface state energies for semiconductors has shown that a narrow band of localized states exists in the valence band-conduction band gap, and these results have been verified by experiment [3]. Calculations have also been performed for simple metals, and a recent self-consistent calculation for aluminium [4] has shown a significant density of surface states on all three simple crystal faces. Of more recent interest are the d-band surface states [5,32] which occur in gaps produced by hybridizing d-bands, and this paper reports the results of a calculation of surface state energies in such a gap in two typical transitions metals, copper and tungsten. In the first paper in this series [6] , referred to below as I, we discussed the * Present address: Science WA4 4AD, England.
Research
Council,
Daresbury
Laboratory,
Daresbury,
Warrington
94
S.J. Gurman/Theor~~ of surface states. II
existence of surface states in band gaps in very general terms. The results, derived assuming only that there was two-dimensional symmetry and a mirror-plane parallel to the surface, showed that surface states should be a common occurrence on all types of crystalline solids, and we derived general criteria for their existence. The major gap in the band structure of transition metals is that caused by the hybridization of the s- and d-bands and, by the criteria derived in I, we should expect surface states to exist in it. We scanned the entire two-dimensional surface Brillouin zone (SBZ) of copper and tungsten (which are typical examples of the two main crystal structures found in transition metals: fee and bee respectively), in search of such states. In each case a substantial density of surface states was fount in the s-d hybridization gap. These states were found to be insensitive, within reasonable limits, to the details of the surface barrier potential, because of their tightly-bound d-band nature. Experimental investigation of the difference in electron behaviour for clean and contaminated crystal surfaces gives some evidence supporting our calculated results, particularly in the case of tungsten.
2. Details of calculation In these numerical calculations, we use the method of matching of wavefunctions at the crystal surface to determine the allowed energies of localized states, rather than the reflection matrix method of I, since these calculations antedate the development of that method. The use of the reflection matrix method to calculate both the extended and localized states of a thin film of copper will be described elsewhere [7]. We assume that the crystal is semi-infinite, and still possesses the full infinite periodicity in the directions parallel to the surface. Thus in these directions we can still use Bloch’s theorem, and the component of crystal momentum parallel to the surface, k,, , is a good quantum number and is conserved (modulo g, a reciprocal lattice vector of the two-dimensional surface net). To find the energy of a surface state, we take a fixed value of k,, and scan across the energy gap until the matching equation is satisfied. The crystal wavefunctions which we must use in the matching are those appropriate to the semi-infinite crystal, i.e. those wavefunctions which obey the Schrodinger equation within the crystal, and are bounded at infinity in the directions parallel to the surface: thus k,, is restricted to real values. The wavefunctions must also be bounded as z -+ 0~)deep within the crystal. The allowed solutions must also obey a boundary condition at the surface, but this requirement we leave until later as the matching condition. Thus we need to include all the wavefunctions of the infinite crystal, and in addition all those waves which behave asymptotically as e-@. These are not allowed states of the infinite crystal, since they are not bounded as z -+ -00. We have removed this requirement by truncating the crystal near z = 0. These decaying waves appear as Bloch waves with a complex K,, the imaginary part of
S.J. GurmanfTheory of surface states. Ii
9s
which, 4, must be positive. 4 gives the rate of decay of the electron density due to these waves as we go deeper into the crystal. In the band gaps of the crystal, these decaying waves are the only ones present, and it is they which make up the surface state. The need to include the decaying waves rules out all the usual methods of bandstructure calculation. The presence of the full symmetry in the directions parallel to the surface, and, for most simple crystal faces, the simple arrangement of atoms in layers parallel to the crystal surface, lead us to use the layer scattering method of Kambe [9]. This method has been extensively developed and programmed for LEED calculations [lo] , It is particularly convenient for surface state calculations since it calculates eigenfunctions and eigenvalues (K,) for a fixed input E and kl,: precisely the form of result which we require in the surface state problem. The layer scattering method is further simplified by the fact that for the (001) faces of fee and bee lattices the layer is a mirror plane of the crystal. Kambe’s method uses the muffin tin approximation for the crystal potential, the scattering properties of a single atom being represented by a few phase shifts. For convenience in performing the energy scan, we parametrize these phaseshifts as functions of energy using the scheme of Cooper et al. [ Ill. We consider the crystal as consisting of a stack of planes of muffin tin potentials, the layers lying parallel to the crystal surface and being of infinite extent. In the cubic metals the repeat unit of the crystal structure in the z-direction is a single layer and for the (001) surface each layer is a mirror plane of the crystal. Each layer is a two-dimensional Bravais net, i.e. there is only one atom per unit cell, Because we have used the muffin tin approximation the potential between the layers is constant, and in this region we expand the Bloch functions in a series over the reciprocal lattice vectors of the two-dimensional surface net,g. The coefficients in this expansion will be z-dependent. We calculate the scattering of these plane wave components by a single layer in the form of reflection and transmission matrices and then combine the layers to form the semi-infinite crystal. The resulting scattering eigenproblem is solved to obtain the Bloch eigenfunctions and eigenvalues of the crystal. Because each layer is periodic, k,, is conserved (modulo g) in the scattering and we therefore obtain the eigenvalue K, at a given E and k,,. Details of the setting-up and solving of the eigenproblem are given by Pendry t101. The solution of the eigenproblem gives us an expression for the Bloch waves in the region of constant potential between layers. If we truncate the crystal in such a region by a surface, we must match these Bloch waves to those of the vacuum to obtain an allowed state. To simplify the matching problem we assume a plane surface barrier, i.e. one that is independent of coordinates parallel to the surface, rr. Such a barrier does not mix components with different values of g. This assumption of a plane barrier is reasonable for fairly close-packed regular layers such as that forming the fcc(OO1) surface. It is less good for surfaces like the bcc(01 l), where there are “channels” between the surface atoms in the 01 direction. For semiconductors, with their loosely packed surfaces, it is likely to be an even poorer assumption. Any
S.J. Gurman/Theory
96
of surface states. I1
non-planar effects at the surface will have the periodicity of the lattice (unless there is any reconstruction) and hence will only mix different g values and not introduce any new values of g. The simplest form which we can use for the surface barrier is the simple step up to the vacuum level [the Abrupt Potential Model (APM)] .In this model we take the total potential to be W
= vu W,
- z) + I&t&
O(z - z(r) >
(1)
the step of height Vu being located at zo. The zero of energy is at the muffin-tin zero. A reasonable position for the step is half-way between layers [ 151 . The step barrier is obviously not an exact repre~ntation of the surface potential, and as a better approximation we use a barrier with a smooth variation parallel to the surface, but still maintain the independence from rll. The form we use is V(z) = Vo( 1 + ePz)- ’ .
17_)
The width parameter fl may be expected to be of the order of a reciprocal screening length: in copper this is 1 au. This form approximates the self-consistent surface potential determined by Lang and Kohn [ I.51 . In using this form, V replaces the V. of eq. (1): we thus truncate the smooth barrier and match at zo, which we choose to be a point where V is small but where we are still outside the layer of muffin tins, after integrating the vacuum wavefunctions through the barrier numerically as far as zo. The total wavefunction in the crystal is a sum over all the Bloch waves with a given energy and k, and we must match this to the vacuum wavefunction. If we have n plane wave components in each Bloch wave, then we need n Bloch waves to achieve a unique matching at all points of the surface. The layer scattering method [lo] gives us 2n eigenfunctions, and in the band gap exactly n of these have the required property of decaying as z + *, and it is these which we use in the matching. The details of the matching procedure are given by Forstmann and Pendry [S] . If we have a mirror plane parallel to the surface, the wavefunctions in the crystal can be made real, and the problem of finding the surface state energy reduces to that of finding the zero of a real determinant: this we do by scanning in energy across the gap and plotting the value of the matching determinant.
3. Surface states on the copper (001) surface Copper forms a fee crystal with a cube edge (I = 6.81735 au. The (001) surface Bravais net is therefore the simple square of side a/d/2 aligned at 45” to the cube edges. The reciprocal net is also a simple square. The 2-D surface Brillouin zone is shown, superimposed on a (001) projection of the bulk zone in fig. I. In our calculation we take each layer as having the bulk crystal structure and set all layer spacings equal: the LEED calculations of Laramore [14] indicate that any dilation or contraction of the layer spacing at the surface is less than 5% of the bulk spacing.
S.J. GurmanlTheory
of surface states. II
iii
91
-K,
Fig. 1. Face-centred cubic (001) surface Brillouin zone (dashed line) superimposed on a (001) projection of the threedimensional crystal zone. All inequivalent points of the SBZ lie in the shaded triangle. The K axes are those of the infinite three-dimensional crystal.
To describe the scattering properties of an atom we have used the phase shifts of Burdick [ 121 as parametrized by Cooper et al. [I l] . We limit ourselves to three phase shifts. The height of the surface barrier was set at 0.47 hart (with the zero of energy at the muffin tin zero) equal to the value found by Burdick. To check the accuracy of our calculation we calculated the bandstructure for k,, = 0, which is the bulk rX direction, from the muffin tin zero to the Fermi energy. Using nine g values in the expansion gave results which agreed with those of Cooper et al. to better than 0.003 hart. Increasing the expansion set to thirteen vectors changed none of the energies at r or X by more than 0.002 hart. All the surface state energies were calculated using nine g vectors with occasional check runs using thirteen: at no point in the surface Brillouin zone did the two values of the surface state energy relative to the bottom of the gap differ by more than 0.001 hart. We have already given a brief discussion of the form of the surface barrier. From the point of view of the actual calculations, the simplest form of barrier to use is the plane step placed midway between layers, the APM. We have used this model in the calculation of the surface state energies over the entire surface Brillouin zone, to determine the density of surface states. In addition we have calculated the surface state energies at a few representative points in the zone for various positions and shapes of the barrier to determine the sensitivity of the surface states to the barrier. The surface states in the s-d gap were found to be highly insensitive to the barrier details, and so we may expect the results calculated using the APM to be an accurate reflection of the surface states on the (001) surfaces, whose actual barrier potential will be smoothed over a finite width.
S.J. Gurman/Theory
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of surface states. II
4. Surface state energies in the APM
Using a step potential
barrier of height V, = 0.47 hart placed half a layer spacing
states/ hart SBZ
a
loo -
Total
0.08
0.12
E hart
N (El
b
30 -
Surface !wer
0.08
t
OJO
0.12
0.08
O.lO
0.12
c ‘hyer
Third Wer
Fourth
t-iel_ LJL 0.08
0.x)
0.12
0.08
0.x)
0.12
Fig. 2. (a) Total surface state density per surface cell for the copper (001) surface. (b) Surface state densities per atom for atoms in the first four atomic layers for the copper (001) surface. EF = 0.28 hart.
S.J. Gurman/Theory of surface states. II
99
T
Fig. 3. Surface states an the copper (001) surface. (a) Energy contours, in hartrees. (b) Localization contours, in units of the layer spacing. There are no surface states around @. The dotted line is at a localization length of 10 atomic layer spacings taken as the limit of the surface state region.
from the last layer of atoms, and nine g values in the wavefunction expansion the entire irreducible section of the surface Brillouin zone (SBZ) was scanned for surface states in the energy region of the s-d hybridization gap. The values of k,, were taken on a square grid of spacing 0.05 au-l and further points graphically interpolated between these to give about 300 surface state energies to form the density of states histogram. The results are shown in figs. 2 and 3. In fig. 2b we give the local density of states, i.e. the density of states per surface cell averaged in the z-direction over the given layer. In addition we give the integrated local density of states in fig. 2a, which is the sum over all layers in the crystal of the individual layer local densities. On the copper (001) surface, a surface state was found in every pure s-d gap (i.e. one which is not crossed at any energy by a propagating band), as was predicted by the reflection matrix method of paper I, and in nearly every other absolute gap. Surface states exist over about 75% of the SBZ in the s--d gap: i.e. there are about 1.5 electrons/surface cell in localized states in this energy range. There are no states around H since in this region there is no absolute energy gap. In general the states were found to lie very close (usually less than 0.01 hart) to the bottom of the gap. This is the position predicted by the two band model. Our result for the Fpoint (k,, = 0) agrees with that of Forstmann and Pendry [S] , and with that of Marcus and Jepsen ]32] . The surface states form a band about 0.040 hart wide, most of the states being in the single peak (fig. 2) centred 0.17 hart below the Fermi level. The decay constant, or localization length, which is the distance in which the electron density in the state falls by a factor, e, was found on average to be 10 au, or about three layer
100
S.J. Gurman/Theor_v of surface states. II,
spacings, although for some states it falls as low as 3 au. The variation of the localization length across the SBZ is shown in fig. 3b. We see that the localization of the states is approximately constant across the zone, rising very sharply near the points where the surface states disappear. In the histograms of the density of surface states we see the decay of the surface state density with depth in the crystal. About 80% of the surface state density lies within the first four atomic layers, within about 8 A of the surface. The peak density of surface states in the first atomic layer, within which about 30%, of the density is localized, is 35 electrons/hart atom, compared to a bulk (propagating states) value of around 40 in this energy range. An atom in the first layer has 0.6 electron in localized states: these electrons are drawn from the propagating s- and d-bands involved in the hybridization. If we calculate the electron density in the bands in a simple tight binding model we find that the surface state density is compensated by the decrease in electron density in the propagating band. There is therefore little or no charging at the surface, and self-consistently requirements are not likely to alter the surface state energies significantly.
5. Sensitivity
of surface states to the barrier
The precise form of the potential near the surface is not known with any accuracy for the transition metals. The height of the barrier can be established from work function measurements and band calculations, but the shape and position are in doubt. Some theoretical results, applicable to the simple metals have been published [ 1.51, and these can be extended to electron densities typical of transition metals [33]. The resulting barrier shapes are approximately of the form (2) with /3 approximately equal to an inverse screening length. Because of the uncertainty surrounding the details of the barrier, it is important to investigate how sensitive the surface states are to the barrier: only those results which are relatively intensitive can be taken as representative of the actual conditions. We shall consider the effects of changing the barrier height, position and width on the surface states in the s-d hybridization gap at a general value of k,, . We choose a point where the gap between the two hybridizing bands is not crossed by another propagating band: a typical point is at k,, = (0.47,O.l) where the gap extends from 0.1123 to 0.1169 hart. Using the APM, we investigated the sensitivity of the surface state energy to the barrier height and position. With the barrier placed half a layer spacing from the last atom, changing the height from 0.35 to 0.60 hart did not alter the surface state energy to within the accuracy of measurement: the state remained at E, = 113.10 + 0.02 mhart. Changing the position of the barrier with V. set at 0.47 hart gave a similar result: changes in E, were very small until we moved the barrier to within a quarter of a layer spacing from the centre of the atomic layer, where the surface state energy started to fall. With this lack of sensitivity to gross changes in the surface barrier, we would not
S.J. GurmanfTheory
of surface
states. II
101
expect any of the effects due to the replacement of the step barrier by a smooth barrier of the form of eq. (2) to be large. This is borne out by the numerical results. With the centre of the barrier placed halfway between layers, the change in surface state energy consequent upon going from a step barrier (13-l = 0) to a wide smooth barrier (‘J-l = 0.7 au) was very small, only about 0.1 mhart, or 2% of the gap width. The screening length in copper is about 1 au, so we would expect a self-consistent barrier to have a 0-l of about this value. We see that using such a barrier will give surface state energies almost identical to those obtained using the APM. This is largely because the s-d surface states lie well below the vacuum level, and so electrons in them do not see a large increase in the classically-allowed region when we change from a step to a smooth barrier: we might expect states nearer to the vacuum zero to be more sensitive. The states in the gap we have considered are rather less localized than the average. in copper, with a localization length of S--6 layers, which will tend to diminish their sensitivity to the barrier. However, calculations at several points in the SBZ with widely varying degrees of localization all give the same result: the surface state energy in the s-d gap is virtually independent of the barrier details over the range of physical interest. If we examine the form of the surface state wavefunction we can see the reason for the insensitivity of its energy to the barrier: in the wavefunction expansion, the high g components dominate. These have large imaginary values of I& and hence decay rapidly as we move away from the centre of the pseudolayer. The intensity of the surface state wavefunction halfway between layers is about a quarter of its peak value. We therefore have a low density of localized electrons between the layers, where the barrier is located and so changes in the barrier will have little effect on the surface state until we move it close to the layer. If we calculate analytic~ly the g expansion of the m = 0 3d wavefunction (given by Co&on [16] : J/3$(‘) = r2 eFcyT%@,
4) ,
with (Y4 2.7 au-‘) we find that it is very similar to our numerical face state is almost entirely d-like.
(3) form: i.e. the sur-
6. Surface states on the tungsten (001) surface Tungsten exists in the form of a bee crystal, and we take it as a typical transition metal of this structure. As well as determining the surface state distribution on the (001) face of tungsten itself, this calculation points up the differences in the surface state distribution across the SBZ from that of copper: these differences are largely due to the different band symmetries imposed by the different crystal structures. From the general theory of paper I and the two examples we treat here, we obtain an idea of the surface state distribution to be expected on any transition metal with fee or bee structure. Since the copper calculation has shown the surface states in the
102
S.J. Gurman/Theory
of surface states. Ii
Fig. 4. Body-centred cubic (001) surface Brillouin zone (dashed line) superimposed on a (001) projection of the three-dimensional zone. All inequivalent points of the SBZ lie in the shaded triangle. The K axes are those of the infinite crystal.
s-d hybridization gap to be insensitive to the barrier details, we consider only the Abrupt Potential Model here, using a step barrier placed half a layer spacing from the last atomic layer. We omit relativistic effects, in particular the spin-orbit interaction: this omission has little effect on the s-d gap [ 171. We have not considered the surface states in the gap close to the Fermi energy, since this will be strongly influenced by the spin-orbit interaction which lifts the degeneracy between the A,. A21 and A, bands at r. A tight binding calculation by Sturm and Feder [ 1S] shows the existence of a surface state in this gap at p: this state is in fact a surface resonance, since it will interact with the propagating A, band, but calculations show that it has a strongly localised character [ 181. The bee tungsten crystal has a cube edge a = 5.8911 au. The (001) surface net is therefore a simple square, as is the reciprocal net. The Z-D Brillouin zone is shown superimposed --on a (001) projection of the bulk zone in fig. 4. The irreducible section is the triangle I‘XM. We use the phase shifts of Jennings and McRae [ 191, which are those appropriate to Mattheiss’ [ 171 V, , parametrized along the lines suggested by Cooper et al. [ 111. The barrier height was fixed at 0.590 hart, as found by Mattheiss. Again as an accuracy check, we calculated the band structure fork,, = 0, the bulk I’H direction. We found good agreement with ~attheiss with twenty-one g vectors in the wavefunction expansion and fair agreement using thirteen. Most of the surface state energies were calculated using thirteen vectors to keep computing time down: check runs using twenty-one vectors showed that the changes in surface state energy consequent upon increasing the set were always less than 0.005 hart and were largely independent of k,,. All the layer spacings were set equal to the bulk value: the LEED results of Jennings and McRae [19] show that there is little or no dilation of the surface layer.
S.J. Gurman/7Beory
7. Surface
of surface
states. II
103
state energies in the APM
Using thirteen g vectors in the wavefunction expansion, we scanned the entire irreducible section of the SBZ for surface state solutions of the matching equations in the energy range of the s-d hybridization gap, using a step barrier placed half a layer
N(E) states/ hart SBZ
I 60.
a
Total
40 -
1 0.25
0.27
0.29
0.31
E hart
N E)
D
20
surface layer
law
10
t
0.25
0.27
0.29
0.31 E hart
0.25
0.27
l-aThird hyer
0.25
027
0.29
a31
0.29
0.31
Fourth law
029
031
C
0.27
Fig. 5. (a) Total surface state density per surface cell for the tungsten (001) surface. 0~) Surface state densities per atom for atoms in the first four atomic layers for the tungsten (001) surface. EF = 0.42 hart.
104
S.J. GurmanlTheory
of surface states. II
spacing from the last atomic layer. We found one surface state in virtually all the absolute s-d gaps. The main feature determining the surface state distribution in the SBZ is the crystal structure, which requires a degeneracy of the A, and A, bands at HI,. This leads to a large overlap of these two bands over a large part of the SBZ around r. There is consequently no gap and no surface states in this region (fig. 6). Similarly there are no states around R due to the degeneracy of A, and A, bands at P,. The interaction between any localized and propagating states at the same energy will be zero from symmetry considerations at r and m and along Tm, and so true surface states can exist at these points even though the s-d gap is crossed by another propagating band, and presumably these points are surrounded by small regions of surface resonances, but the vast majority of true surface states are in the region around R. The region of true surface states covers about 60% of the SBZ. As in the case of copper, this region was scanned on a grid of spacing 0.05 au-l and further surface state energies found by graphical interpolation between these points. The results were used to plot the density of states histograms of fig. 5 and the contour plots of fig. 6. The surface states are concentrated in a single peak of width 0.02 hart centred 0.14 hart below the Fermi energy. The degree of localization is fairly constant across the zone, varying between two and four atomic layers (3-6 a), except where it rises rapidly at the edges of the region of surface states. The states in the main peak of surface state density are the most localized (see fig. 6) and so this peak is strongly localized in the surface layer. The average localization length was found to be about three atomic layers (“5 A) leading to about 80% of the surface state density lying in the first four atomic
i Fig. 6. Surface states on the tungsten (001) surface. (a) Energy contours, in hartrees. (b) Localization contours in units of the layer spacing. There are no surface states near f and 9: The dashed line is at a localization length of 10 atomic layer spacings, taken as the limit of the surface state region.
S.J. Gunnon/Theory
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105
layers, within 6 a of the surface. About 30% of the density lies in the first atomic layer, giving a peak density of 20 electrons/hart atom there. An atom in the surface layer has 0.4 electron, out of a total of six, in localized states. The surface states lie in the energy range of the two lowest d-band peaks in the bulk density of states [ 171 and the high density of bulk states will tend to make the surface states difficult to observe. However, if the peak is largely due to states near the zone edge, it will be strongly attenuated near the surface by the matching requirements on the propagating bands which remove electron density from the band edges near the surface [20] and this may improve the visibility of the surface states. In summary, the numerical results for the tungsten (001) surface show a significant density of surface states, which are grouped around the L%point due to the crystal symmetries.
8. Experimental
evidence for surface states
In order to find surface states experimentally, we need a technique which probes only the first few atomic layers. For this reason, the most useful are the methods which use energetic electrons. These are surface sensitive because of the low mean free path of the excited electrons. The electrons used in UPS, Auger spectroscopy and in energy loss measurements, with kinetic energies between 10 and 100 eV have mean free paths below 10 A [21] . The most surface sensitive techniques of all are Ion Neutralization Spectroscopy [22] and Field Emission [28]. However, results from INS are few and give little clear data on surface states, whilst field emission cannot probe deep enough below the Fermi energy to find the s-d hybridization surface states, although it has provided evidence for the existence of surface states in the gap just below the Fermi level due to spin-orbit interaction [28] . In all these methods, to detect surface dependent structure we compare results from clean and contaminated surfaces: this obviously causes problems since the adsorbate introduces many extraneous effects as well as destroying the intrinsic surface states by removing the surface periodicity and so mixing states with different k,, . 8. I. Copper (001) The early paper of Forstmann and Pendry [5] quoted in support of their results UPS data of Spicer [23] showing an anomalous peak at -6 eV. This structure was not observed by Eastman and Cashion [24], who believed it to be due to contamination in the earlier experiments [24]. The Auger results of Powell and Mandl [25] with an energy resolution of -0.5 eV show a small peak at the bottom of the d-band which disappears on contamination. This peak lies 2.6 eV below the main d-band peak which we assign as due to the double peak in the theoretical density of states at 0.37 and 0.32 ryd. Thus the surface dependent structure lies about 5 eV below the Fermi energy, close to our calculated surface state energy. Powell and Mandl
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also observe two characteristic energy losses, which disappear upon contamination, at 4.5 and 7.0 eV. The higher loss can be ascribed to the surface plasmon [26], and we consider the lower as due to transitions from the surface state band to states near the Fermi energy, probably those near Q and Z. Such transitions conserve k,,. 8.2. Tungsten (001) For tungsten, as for copper, there is little direct experimental evidence for intrinsic surface states; except for the state close to the Fermi level. This has been observed in UPS [27], field emission [28] and optical reflectivity experiments [29]. For the state in the s-d gap we depend upon UPS and energy loss measurements. Several directional UPS results show a strong removal of states by surface contamination at about 4.5 eV below the Fermi level. This is particularly pronounced in the results of Egelhoff and Perry [30] taken for various values of k,, along i%. Surface states are symmetry allowed along all of i%, the band extending from -3.0 eV (G) to -5.0 eV at the minimum near (0.2,0.2). The removal of these states is complete by an exposure of 0.25 monolayers suggesting that they are due to intrinsic surface states removed by the destruction of surface periodicity, rather than that the dip is due to the removal of electrons from this energy range by the formation of bonding orbitals with the contaminant. We note that the dip moves to higher energies with increasing k,, in accordance with the bandstructure between k,, = (0.2,0.2) and m. Electron energy loss spectra show surface sensitive losses at 1 S, 4.3 and 10 eV which disappear upon contamination [3 1] We identify the surface plasmon at 10 eV and assign the other two to transitions to states just above the Fermi level from the spin-orbit and s-d gap surface states respectively. For the s-d gap states the final state will probably be near C or A since such transitions conserve k,,. Edwards and Probst found that the intensity of this loss peak was greater with the primary beam aligned along the (01) direction of the surface cell than those for the -- beam aligned along (11); our results show a higher density of surface states near I’X than near i;R.
9. Conclusions Our motivation for this work was the result of a two-band model calculation [6] which predicted that a surface state should always exist in an absolute band gap within the zone, within very general limitations. The numerical results, found using an accurate crystal potential and a reasonable approximation for the surface barrier potential have borne out this prediction: a surface state exists everywhere in the two-dimensional zone where the s-d gap is not completely crossed by another propagating band. Because of this, the region of existence of surface states (and hence their total number) is largely controlled by the crystal symmetry which fixes the degeneracies at symmetry points and hence controls the overlap of the various bands.
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This point is demonstrated by the difference in surface state distribution between the fee copper and bee tungsten results. The calculated surface state energies were found to be largely independent of the details of the surface barrier, in marked contrast to the NFE case [4] and we may therefore regard these results, found using a step barrier, as accurately representing the true state of affairs on the (001) surface. The results show a substantial density of surface state electrons, localized within the first three or four atomic layers, in a band about 1 eV wide located near the bottom of the d-band. The experimental evidence for these states is not as conclusive as we might wish, but is strongly indicative of the presence of such a band which disappears upon contamination of the surface. The predictions of the two band model are general and the degree with which they agree with the numerical results suggests that similar surface state bands will exist in the s-d hybridization gap of most, if not all, other transition metals, and that consequently the surface and bulk densities of states may be expected to differ markedly.
References [ 1) W. Shockley, Phys. Rev. 56 (1939) 317; E. Goodwin, Proc. Cambridge Phil. Sot. 35 (1939) 205, 221, 232. [2] Reviews are given by: R.O. Jones, in: Surface Physics of Semiconductors and Phosphors, Eds. C.A. Scott and C.E. Reed (Academic Press, New York, 1974); S.G. Davison and J.D. Levine, Solid State Phys. 25 (1970) 1. [3] J.E. Rowe and H. Ibach, Phys. Rev. Letters 31 (1973) 102; 32 (1974) 421. [4] E. Caruthers, L. Kleinman and G.P. Alldredge, Phys. Rev. B8 (1973) 4570; B9 (1974) 3325. [S] F. Forstmann and J.B. Pendry, Z. Physik 235 (1970) 74; M. Tomasek and P. Mikusik, Phys. Rev. B8 (1973) 410. [6] 3.B. Pendry and S.J. Gurman, Surface Sci. 49 (1975) 87. [7] S.J. Gurman, J. Phys. F5 (1975) L194. [8] F. Forstmann, Z. Physik 235 (1970) 69. (91 K. Kambe, Z. Naturforsch. 22a (1967) 322. [lo] J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974). [II] B. Cooper, E. Kreiger and B. Segall, Phys. Rev. B4 (1971) 1734. [12] G.A. Burdick, Phys. Rev. 129 (1963) 138. [ 131 V. Heine, Solid State Phys. 24 (1970) 1. [14] G.E. Laramore, Phys. Rev. B9 (1974) 1204. [15] N.D. Lang and W. Kohn, Phys. Rev. Bl (1970) 4555. [ 161 C. Coulson, Valence (Oxford Univ. Press, 1961). [17] L. Mattheiss, Phys. Rev. 139 (1965) A1893. [18] K. Sturm and B. Feder, Solid State Commun. 14 (1974) 1317. [19] P. Jennings and E.G. McRae, Surface Sci. 23 (1970) 363. [20] M.J. Kelly, Thesis, Univ. of Cambridge (1974). [21] C.J. Powell, Surface Sci. 44 (1974) 29. [22] H.D. Hagstrum and G.E. Becker, Phys. Rev. 159 (1967) 572. [23] W.E. Spicer, in: Proc. Symp. on Optical Properties and Electronic Structure of Metals and Alloys, Ed. F. Abel&s (North-Holland, Amsterdam, 1966).
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DE. Eastman and J.K. Cashion, Phys. Rev. Letters 24 (1970) 310; DE. ~ast~n,Fhys. Rev. B3 (1930) 1769. [ZSJ C.J. PoweB and A. Mandl, Phys. Rev. 36 (1972) 4418. [26] L.H. Jenkins and M.F. Chung, Surface Sci. 26 (1971) 151. [27] B. Feuerbacher and B. Fitton, Phys. Rev. Letters 29 (1972) 786. [28] J.W. Gadzuk and E.W. Plummer, Rev. Mod. Phys. 43 (1973) 487. 1291 G.W. Rubloff, J. Anderson, M.A. Passler and P.J. Stiles, Phys. Rev. Letters 32 (1974) 667. [30] W.F. Egelhoff and D.L. Perry, Phys. Rev. Letters 34 (1975) 93. [31] D. Edwardsand F.M. Probst, J. Chem. Phys. 55 (1971) 5175. f 321 P. Marcus and D.W. Jepsen, in: Computational Methods for Large Molecules and Locahzed States in Solids, Eds. F. Herman, A.D. McLean and R.K. Nesbet (Plenum, New York, 1973). The authors have since revised the results of this paper and now conclude that CutOOl) does exhibit a surface state. [33] N.D. Lang, private communication to J.B. Pendry (ref. [ 101, p. 239). [24]