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Atomic approximation for alloys and their surfaces
Solid State Communications, Vol. 43, No. 6, pp. 491-494, 1982. Printed in Great Britain.
0038-1098/82/300491-04503.00/0 Pergamon Press Ltd.
ATOMIC APPROXIMATION FOR ALLOYS AND THEIR SURFACES D.W. Bullett School of Physics, Umve~ty of Bath, Bath BA2 7AY, England
(Received 26 January 1982 by C. W. McCombie) We report a simple, ab initio method for calculating the electronic structure of compositionally disordered alloys. Results are shown for Cu/Ni and Ag/Pd bulk systems, and the first calculations are reported for the surface electronic structure of random alloys, exemplified by {l 1 1 } surface states of Cu/Ni and Cu/Al alloys. IN RECENT YEARS, while great theoretical effort has been concentrated on refining the most sophisticated computational methods [I-10] for calculating the electronic structure of compositionally disordered alloys, simplified models have fallen into disrepute. Chief among the latter are the rigid-band model and the virtual-crystal approximation. Both are quite inappropriate to the wide range of random AB alloys in which the scattering potential (VB -- IrA) is strong. The alternative, equally simple, model introduced here overcomes the deficiencies of other simple methods and is applicable over the full concentration range, even in the presence of resonant scattering. The present atomic approximation for alloys (AAA) may be particularly useful for determining effects of short-range ordering in bulk alloys and for examining surface properties that are not easily amenable to the standard coherent-potential-approximation (CPA) [3-8] and average-t-matrix-approximation (ATA) [9, 10] approaches. In this brief letter I indicate, as examples, results for bulk Cu/Ni and Ag/Pd random alloys, for the {1 11 } surface states of a Cu/Ni alloy, and for the {11 1} zone-centre sp surface state of Cu/AI alloys. The validity of an ab initio atomic-orbital-based calculation in the two-centre approximation has been demonstrated already in applications to the surface properties of simple, noble and transition elements [ 1 l, 12]. From the accuracy of the results it is clear that the set of undistorted valence-level atomic orbitals on each atomic site do represent a quantitatively useful approcimation to the exact solution ~l of the localizedorbital pseudopotential equations [13]
Here S -l is the inverse overlap matrix and the Hamiltonian H is regarded as the sum of the isolatedatom Hamiltonian H t and the perturbation Vth introduced by each nearby atom 491
Y. atomic malls
The matrix elements (~[Vtk[ ~i) m the set ofimear equaUons (I) are calculated by direct integration over the atomic sphere appropriate to the atomic orbital 0k, with the perturbing potential ~k defined by overlapping the charge clouds of the two atoms concerned and using a local-density (a = 0.?) approximation for exchangecorrelation in the electron gas. These matrix elements define the (9 × 9) secular equation which determines the allowed Bloch energies E(k) and wave/unctions ~(k) in the perfect elemental crystal [13]. Surface properties of transition and noble elements may be mvestzgeted by constructing the (gn × 9n) determinant for a two. dtmensional crystal containing a thickness o f n atomic layers [ 1 l, 12]. The same formalism is appropriate for a compositionally disordered alloy. In the present analysls we consider a binary AB alloy in which an ordered array of lattice sites are randomly occupied by IrA or VB potenUals. For composition AxBl-=, at any lattice site there is a probability x of finding VA, (1 -- x) of finding VB. Short-range correlations could easily be incorporated at a later stage, and the extension to multi-component alloys is trivial. For an alloy of transition or noble metals there are now 2 sets of 9 possible orbltals for inclusion at each s~te. In the spirit of the virtual-crystal approximation, we can adopt a weighted average of the two components for the s- and p- orbitals. The same is not true for the narrow bands associated with the d-orbitals: here it would be quite wrong to replace the two narrow / = 2 resonances by a single resonance at some intermediate energy. Instead we must retain both sets ofd-orbitals at each lattice site, with their respective matrix elements weighted according to the appropriate probabilities. Since all sites in the random alloy are treated as
492
ATOMIC APPROXIMATION FOR ALLOYS AND THEIR SURFACES
exactly equivalent, k is still a good quantum number and the density of states for an AB alloy of any given composition can be calculated simply by diagonalising the (14 × 14) secular matrix at a sampled grid of k-points i n the irreducible Brilloum zone. This provides a local density of states on each of the A or B d-orbitals and the sp density of states; the total density of states is the weighted average. The difficulties introduced by negative regions in some of the ATA expressions [8-10] for the component densities of states PA(E) and PB(E) do not arise. We can easily estimate the energy broadening associated with the random fluctuations in the potential at each site. For the d-d alloys considered here, the indi. vidual d-d terms Vd are small compared to the energy splitting AEd of the two constituents, and perturbation theory gives an estunate of the broadening as a ~ ~/[12x(1 --x)] Vd with Vd "" 0.2 eV.
-6
-/+
-2
0
-~
-2
OeV
Fig. 1. Calculated density of states p(E) and its components Pcu(E), Psi(E) for the bulk Cu/Ni alloy system. Figures 1 and 2 display calculated densities of states in paramagneUc Cu/Ni and Ag/Pd random alloys over the full composltmn range. Each was calculated from 1 l0 representative k-points. Lattice constants for the alloys were assumed to vary linearly with composition. For pure Cu, we find a d-band extending from about 2 eV to 5 eV below EF, of a shape very reminiscent of earlier calculations [3]. As before [ 12], the valence p-level was adjusted to position EF correctly relative to the d-band. Dilute concentrations of Ni induce a narrow density-of-states peak at the NI d-resonance energy, 1.3 eV above the Cu d-band edge. At 10~ Ni the Ni resonance has a width at half-height of 0.7 eV. At about 307o Nl P(EF) starts to rise rapidly as Ni d-states begin to contribute (m a paramagnetic approximation), and by
Vol. 43, No. 6
/--,,, gO
as
::
9 95 1 i
-6
-~,
-2
0
-4
-2
OeV
Fig. 2. Calculated p(E) for the Ag/Pd alloy system. 50~ Ni the Ni peak ts starting to develop structure. Simultaneously with the development of the Ni-induced features, the Cu d-band becomes narrower as Ni atoms replace Cu atoms in the mean environment. At 95% Ni the Cu-induced feature shrinks to a width of about 0.7 eV, but, because the Cu atonuc d-resonance falls within the energy range of the pure Ni d-band, part of this low-energy peak is contributed by PNi(E). At 100~ Ni the shape and width (~ 4.6 eV) closely reproduce the band-structure results obtained by other methods [3]. Very similar effects can be seen in the results for Ag/Pd alloys (Fig. 2). We conclude that for both alloy systems the AAA method provides a correct account of the bulk electronic structure as a functlon of composition: Figures 1 and 2 reproduce the essential features of calculatlons by CPA [3-6, 8] and other methods [9, I0, 14] and of angle-integrated photoemission spectra [ 15, 16]. The {I 1 I} surface energy band dispersion for the binary substitutional alloy N1o.s4Cuo.ls has been observed by angle-resolved photoemission spectroscopy [ 17]. Cu segregates at the alloy surface, and the surface layer is thought to contain about 85% Cu, while deeper, layers have approximately the bulk composition [ 18, 19]. I modelled this system by a thin two-dimensional crystal containing a single Cuo.ssNlo.ls layer on top of five layers with composition Nio.mCuo.16. Figure 3 displays for the I'M and PK Brillouin zone directions those states which contain more than 50% of their weight in the Cu-rich surface layer. The calculated surface states agree closely in position, width and dispermon with those observed by photoemission [ 17];indeed, in some systems one may be able to deduce by this method the composition and structure of the surface layer.
ATOMIC APPROXIMATION FOR ALLOYS AND THEIR SURFACES
Vol. 43, No. 6 o
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Fig. 3. Surface states calculated for a CussNi=s {1 1 1} surface layer on a NimCu:6 alloy substrate, compared to the angle-resolved photoemission data of Heimann et al. [171. As a final example I show in Fig. 4 the effect on the zone-centre sp-like Cu {1 11} surface-state of alloying with 10at.% AI. Asonen and Pessa [20] have very recently reported photoemismon observatmns of this state, showing that Its energy in the alloy is conmder. ably lower than in pure Cu and depends on surface conditions. For the (1 x 1) -- (x/3 x X/3)R30° AI structure which originates from a one-third monolayer coverage orAl on the disordered (1 x I) bulk structure, the surface state appeared 1.2 eV below El¢ at the point; the state moved to -- 0.8 eV when the AI overlayer was removed [20]. In pure Cu(1 11) the eqmvalent state lies at - - 0.4 eV. The d-band edge remams relatively unmoved by alloying with 10% AI. The theoretical curves in Fig. 4 were calculated by the AAA for four different atomic arrangements m a 10 layer {11 1} crystal. In each case I have plotted the average energy of the even and odd surface bands, which are separated by about 0.3 eV for this thickness of crystal. Curve a for pure Cu shows the p-like surface state ~ 0.5 eV below E~, at the zone centre, rising away from ~ with an effective mass ~ 0.4 m. In curve b, for a Cus~Al:o alloy, the surface state band is displaced uniformly downwards by "" 0.25 eV. In curves c and d the surface layers of the CuseAl:e alloy have compomion CuevAl~ and CusoAlso respectively; relative to pure
e¥ Fig. 4. Upper figure: experimental E ( k , ) for {1 1 1} surface state near F for pure Cu and for CuscAllo with (x/3 x X/3)R30 ~ surface layer [20]. Lower figure. calculated surface state dispersion for Cu and various Cu~vAl:o alloys (see text). Cu the surface state moves to lower energy by 0.5 and 0.7 eV respectwely. The upper edge of the d-band remains essentially fLxed at 2.2eV below EF m all four cases. For the real Al-rich surface we might refer a (x/3 × ~/3)R30 ° surface layer with one AI atom for every two Cu atoms, in a regular array such that each Cu atom has 3 AI and 3 Cu nearest neighbours within its own layer. Thus the local environment of a surface Cu atom is equivalent to that o f a CusoAlso random surface layer on the Cu~Al:o substrate. The calculated surface-state energy shift for this atomic arrangement is conmstent with experimental shift
120]. We conclude that the atonuc approximation pro. rides an accurate representation of the electronic struc. ture of alloys, both within the bulk and at surfaces Apphcations to other alloy systems, other surfaces [21], and Fermi-surface properties will be reported elsewhere. REFERENCES
1. 2. 3.
H. Ehrenrmch & L. Schwartz, Solid State Phys. 31, 150 (1976). D.J. Sellmyer, Solid State Phya 33, 83 (1978). G.M. Stocks, R.W. Williams & J.S. Faulkner, Phys. Rev. B4, 4390 (1971).
494 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
ATOMIC APPROXIMATION FOR ALLOYS AND THEIR SURFACES
Vol. 43, No. 6
G.M. Stocks, R.W. Williams & J.S. Faulkner, J. 15. S.K. Hufner, G.K. Wertheun, & J.H. Wemlck, Phys. F3, 1688 (1973). Phys. Rev. !18, 4511 (1973). G.M. Stocks, W.M. Temmerman & B.L. Gyorffy, 16. A.D. McLachlan, J.G. Jenkin, R.C.G. Leckey & J Llesegang, J. Phys. F5, 2415 (1975). Phys. Rev. Lett. 41,339 (1978). W.M.Temmerman, B.L. Gyorffy & G.M. Stocks, 17. P. Heimann, J. Hermanson, H. Mlosga & H. J. Phys. FS, 2461 (1978). Neddermeyer,Solid State Coramun. 37, 519 B.E.A. Gordon, W.M. Temmerman & B.L. Gyorffy, (1981). 18. F.L. Williams & D. Nason, Surfi Scz. 45,377 J. Phys. FII, 821 (1981). (1974). A. Bansil, Phys. Rev. Lett. 41, 1670 (1978). A. Bansil, L. Schwartz & H. Ehrenrelch,Phys. Rev. 19. Y.S. Ng, T.T. Tsong & S.B. McLane, Phys. Rev. Lett. 42, 588(1979). BI2, 2893 (1975). 20. H. Asonen & M. Pessa, Phys. Rev. Lett 46, 1696 A. Bansil, Phys. Rev. !i20, 4025,4035 (1979). (1981). D.W.Bullett, Surf. Scz. 93, 213 (1980). 21. M. Pessa, H. Asonen, R.S. Rao, R. Prasad & A. D.W. Bullett, J. Phys. C14, 4521 (1981). Bansil, Phys. Rev. Lett. 47, 1223 (1981). D.W.Bullett, Solid State Phys. 35, 129(1980). C.B. Sommers, H.W. Myron & F.M. Mueiler, Solid State Commun. 37, 761 (1981).
0038-1098/82/300491-04503.00/0 Pergamon Press Ltd.
ATOMIC APPROXIMATION FOR ALLOYS AND THEIR SURFACES D.W. Bullett School of Physics, Umve~ty of Bath, Bath BA2 7AY, England
(Received 26 January 1982 by C. W. McCombie) We report a simple, ab initio method for calculating the electronic structure of compositionally disordered alloys. Results are shown for Cu/Ni and Ag/Pd bulk systems, and the first calculations are reported for the surface electronic structure of random alloys, exemplified by {l 1 1 } surface states of Cu/Ni and Cu/Al alloys. IN RECENT YEARS, while great theoretical effort has been concentrated on refining the most sophisticated computational methods [I-10] for calculating the electronic structure of compositionally disordered alloys, simplified models have fallen into disrepute. Chief among the latter are the rigid-band model and the virtual-crystal approximation. Both are quite inappropriate to the wide range of random AB alloys in which the scattering potential (VB -- IrA) is strong. The alternative, equally simple, model introduced here overcomes the deficiencies of other simple methods and is applicable over the full concentration range, even in the presence of resonant scattering. The present atomic approximation for alloys (AAA) may be particularly useful for determining effects of short-range ordering in bulk alloys and for examining surface properties that are not easily amenable to the standard coherent-potential-approximation (CPA) [3-8] and average-t-matrix-approximation (ATA) [9, 10] approaches. In this brief letter I indicate, as examples, results for bulk Cu/Ni and Ag/Pd random alloys, for the {1 11 } surface states of a Cu/Ni alloy, and for the {11 1} zone-centre sp surface state of Cu/AI alloys. The validity of an ab initio atomic-orbital-based calculation in the two-centre approximation has been demonstrated already in applications to the surface properties of simple, noble and transition elements [ 1 l, 12]. From the accuracy of the results it is clear that the set of undistorted valence-level atomic orbitals on each atomic site do represent a quantitatively useful approcimation to the exact solution ~l of the localizedorbital pseudopotential equations [13]
Here S -l is the inverse overlap matrix and the Hamiltonian H is regarded as the sum of the isolatedatom Hamiltonian H t and the perturbation Vth introduced by each nearby atom 491
Y. atomic malls
The matrix elements (~[Vtk[ ~i) m the set ofimear equaUons (I) are calculated by direct integration over the atomic sphere appropriate to the atomic orbital 0k, with the perturbing potential ~k defined by overlapping the charge clouds of the two atoms concerned and using a local-density (a = 0.?) approximation for exchangecorrelation in the electron gas. These matrix elements define the (9 × 9) secular equation which determines the allowed Bloch energies E(k) and wave/unctions ~(k) in the perfect elemental crystal [13]. Surface properties of transition and noble elements may be mvestzgeted by constructing the (gn × 9n) determinant for a two. dtmensional crystal containing a thickness o f n atomic layers [ 1 l, 12]. The same formalism is appropriate for a compositionally disordered alloy. In the present analysls we consider a binary AB alloy in which an ordered array of lattice sites are randomly occupied by IrA or VB potenUals. For composition AxBl-=, at any lattice site there is a probability x of finding VA, (1 -- x) of finding VB. Short-range correlations could easily be incorporated at a later stage, and the extension to multi-component alloys is trivial. For an alloy of transition or noble metals there are now 2 sets of 9 possible orbltals for inclusion at each s~te. In the spirit of the virtual-crystal approximation, we can adopt a weighted average of the two components for the s- and p- orbitals. The same is not true for the narrow bands associated with the d-orbitals: here it would be quite wrong to replace the two narrow / = 2 resonances by a single resonance at some intermediate energy. Instead we must retain both sets ofd-orbitals at each lattice site, with their respective matrix elements weighted according to the appropriate probabilities. Since all sites in the random alloy are treated as
492
ATOMIC APPROXIMATION FOR ALLOYS AND THEIR SURFACES
exactly equivalent, k is still a good quantum number and the density of states for an AB alloy of any given composition can be calculated simply by diagonalising the (14 × 14) secular matrix at a sampled grid of k-points i n the irreducible Brilloum zone. This provides a local density of states on each of the A or B d-orbitals and the sp density of states; the total density of states is the weighted average. The difficulties introduced by negative regions in some of the ATA expressions [8-10] for the component densities of states PA(E) and PB(E) do not arise. We can easily estimate the energy broadening associated with the random fluctuations in the potential at each site. For the d-d alloys considered here, the indi. vidual d-d terms Vd are small compared to the energy splitting AEd of the two constituents, and perturbation theory gives an estunate of the broadening as a ~ ~/[12x(1 --x)] Vd with Vd "" 0.2 eV.
-6
-/+
-2
0
-~
-2
OeV
Fig. 1. Calculated density of states p(E) and its components Pcu(E), Psi(E) for the bulk Cu/Ni alloy system. Figures 1 and 2 display calculated densities of states in paramagneUc Cu/Ni and Ag/Pd random alloys over the full composltmn range. Each was calculated from 1 l0 representative k-points. Lattice constants for the alloys were assumed to vary linearly with composition. For pure Cu, we find a d-band extending from about 2 eV to 5 eV below EF, of a shape very reminiscent of earlier calculations [3]. As before [ 12], the valence p-level was adjusted to position EF correctly relative to the d-band. Dilute concentrations of Ni induce a narrow density-of-states peak at the NI d-resonance energy, 1.3 eV above the Cu d-band edge. At 10~ Ni the Ni resonance has a width at half-height of 0.7 eV. At about 307o Nl P(EF) starts to rise rapidly as Ni d-states begin to contribute (m a paramagnetic approximation), and by
Vol. 43, No. 6
/--,,, gO
as
::
9 95 1 i
-6
-~,
-2
0
-4
-2
OeV
Fig. 2. Calculated p(E) for the Ag/Pd alloy system. 50~ Ni the Ni peak ts starting to develop structure. Simultaneously with the development of the Ni-induced features, the Cu d-band becomes narrower as Ni atoms replace Cu atoms in the mean environment. At 95% Ni the Cu-induced feature shrinks to a width of about 0.7 eV, but, because the Cu atonuc d-resonance falls within the energy range of the pure Ni d-band, part of this low-energy peak is contributed by PNi(E). At 100~ Ni the shape and width (~ 4.6 eV) closely reproduce the band-structure results obtained by other methods [3]. Very similar effects can be seen in the results for Ag/Pd alloys (Fig. 2). We conclude that for both alloy systems the AAA method provides a correct account of the bulk electronic structure as a functlon of composition: Figures 1 and 2 reproduce the essential features of calculatlons by CPA [3-6, 8] and other methods [9, I0, 14] and of angle-integrated photoemission spectra [ 15, 16]. The {I 1 I} surface energy band dispersion for the binary substitutional alloy N1o.s4Cuo.ls has been observed by angle-resolved photoemission spectroscopy [ 17]. Cu segregates at the alloy surface, and the surface layer is thought to contain about 85% Cu, while deeper, layers have approximately the bulk composition [ 18, 19]. I modelled this system by a thin two-dimensional crystal containing a single Cuo.ssNlo.ls layer on top of five layers with composition Nio.mCuo.16. Figure 3 displays for the I'M and PK Brillouin zone directions those states which contain more than 50% of their weight in the Cu-rich surface layer. The calculated surface states agree closely in position, width and dispermon with those observed by photoemission [ 17];indeed, in some systems one may be able to deduce by this method the composition and structure of the surface layer.
ATOMIC APPROXIMATION FOR ALLOYS AND THEIR SURFACES
Vol. 43, No. 6 o
-2"
•
0,4 •
oooooe
o
o o°
oo
eOolo,
~-.,._~_.. ~..
000. 0 oe
.:;;;:...e'':''::i::::" ":: ....... :i
o
,oeO°°o
493
=
0,2 .
0,2
•
k,,(.~"1) ,
O,&. !
i::
-~"
-O,t,
eV
- 0,8
oo o
.... ::::
riO0000
eV o hv =21,2eV o by= 16,8eV
4t8'
-2 j ~. OO0 oO <:0
oO
+
~ro
0,4 ,~, \0~2
,
0:2/a
' 0;/+
¢° ° ° ° 00 ~' o,, ~ . ~ o l,~
o 0-.:
<~
0
0<~oO K
f
R
Fig. 3. Surface states calculated for a CussNi=s {1 1 1} surface layer on a NimCu:6 alloy substrate, compared to the angle-resolved photoemission data of Heimann et al. [171. As a final example I show in Fig. 4 the effect on the zone-centre sp-like Cu {1 11} surface-state of alloying with 10at.% AI. Asonen and Pessa [20] have very recently reported photoemismon observatmns of this state, showing that Its energy in the alloy is conmder. ably lower than in pure Cu and depends on surface conditions. For the (1 x 1) -- (x/3 x X/3)R30° AI structure which originates from a one-third monolayer coverage orAl on the disordered (1 x I) bulk structure, the surface state appeared 1.2 eV below El¢ at the point; the state moved to -- 0.8 eV when the AI overlayer was removed [20]. In pure Cu(1 11) the eqmvalent state lies at - - 0.4 eV. The d-band edge remams relatively unmoved by alloying with 10% AI. The theoretical curves in Fig. 4 were calculated by the AAA for four different atomic arrangements m a 10 layer {11 1} crystal. In each case I have plotted the average energy of the even and odd surface bands, which are separated by about 0.3 eV for this thickness of crystal. Curve a for pure Cu shows the p-like surface state ~ 0.5 eV below E~, at the zone centre, rising away from ~ with an effective mass ~ 0.4 m. In curve b, for a Cus~Al:o alloy, the surface state band is displaced uniformly downwards by "" 0.25 eV. In curves c and d the surface layers of the CuseAl:e alloy have compomion CuevAl~ and CusoAlso respectively; relative to pure
e¥ Fig. 4. Upper figure: experimental E ( k , ) for {1 1 1} surface state near F for pure Cu and for CuscAllo with (x/3 x X/3)R30 ~ surface layer [20]. Lower figure. calculated surface state dispersion for Cu and various Cu~vAl:o alloys (see text). Cu the surface state moves to lower energy by 0.5 and 0.7 eV respectwely. The upper edge of the d-band remains essentially fLxed at 2.2eV below EF m all four cases. For the real Al-rich surface we might refer a (x/3 × ~/3)R30 ° surface layer with one AI atom for every two Cu atoms, in a regular array such that each Cu atom has 3 AI and 3 Cu nearest neighbours within its own layer. Thus the local environment of a surface Cu atom is equivalent to that o f a CusoAlso random surface layer on the Cu~Al:o substrate. The calculated surface-state energy shift for this atomic arrangement is conmstent with experimental shift
120]. We conclude that the atonuc approximation pro. rides an accurate representation of the electronic struc. ture of alloys, both within the bulk and at surfaces Apphcations to other alloy systems, other surfaces [21], and Fermi-surface properties will be reported elsewhere. REFERENCES
1. 2. 3.
H. Ehrenrmch & L. Schwartz, Solid State Phys. 31, 150 (1976). D.J. Sellmyer, Solid State Phya 33, 83 (1978). G.M. Stocks, R.W. Williams & J.S. Faulkner, Phys. Rev. B4, 4390 (1971).
494 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
ATOMIC APPROXIMATION FOR ALLOYS AND THEIR SURFACES
Vol. 43, No. 6
G.M. Stocks, R.W. Williams & J.S. Faulkner, J. 15. S.K. Hufner, G.K. Wertheun, & J.H. Wemlck, Phys. F3, 1688 (1973). Phys. Rev. !18, 4511 (1973). G.M. Stocks, W.M. Temmerman & B.L. Gyorffy, 16. A.D. McLachlan, J.G. Jenkin, R.C.G. Leckey & J Llesegang, J. Phys. F5, 2415 (1975). Phys. Rev. Lett. 41,339 (1978). W.M.Temmerman, B.L. Gyorffy & G.M. Stocks, 17. P. Heimann, J. Hermanson, H. Mlosga & H. J. Phys. FS, 2461 (1978). Neddermeyer,Solid State Coramun. 37, 519 B.E.A. Gordon, W.M. Temmerman & B.L. Gyorffy, (1981). 18. F.L. Williams & D. Nason, Surfi Scz. 45,377 J. Phys. FII, 821 (1981). (1974). A. Bansil, Phys. Rev. Lett. 41, 1670 (1978). A. Bansil, L. Schwartz & H. Ehrenrelch,Phys. Rev. 19. Y.S. Ng, T.T. Tsong & S.B. McLane, Phys. Rev. Lett. 42, 588(1979). BI2, 2893 (1975). 20. H. Asonen & M. Pessa, Phys. Rev. Lett 46, 1696 A. Bansil, Phys. Rev. !i20, 4025,4035 (1979). (1981). D.W.Bullett, Surf. Scz. 93, 213 (1980). 21. M. Pessa, H. Asonen, R.S. Rao, R. Prasad & A. D.W. Bullett, J. Phys. C14, 4521 (1981). Bansil, Phys. Rev. Lett. 47, 1223 (1981). D.W.Bullett, Solid State Phys. 35, 129(1980). C.B. Sommers, H.W. Myron & F.M. Mueiler, Solid State Commun. 37, 761 (1981).