- Email: [email protected]
Electronic structure in β′-AgMg
~ )
Solid State Communications, Vol. 81, No. 8, pp. 667-670, 1992. Printed in Great Britain.
003 8- 1098/92 $5.00 +. 00 Pergamon Press plc
ELECTRONIC STRUCTURE IN /3'-AgMg R.G. Jordan'f, A.M. Begleyt§, Yan Liu~, S.L. Qiut and R.I.R. Blyth$¶ Alloy Research Center, Department of Physics, Florida Atlantic University, Boca Raton, Florida 33431-099t, USA. :~ Surface Science Research Centre, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, UK. Received October 21 1991 by J. Tanc We have calculated the density of states in/3'-AgMg (CsCl-structure) using the SCFLMTO-ASA method. We have determined also the x-ray photocurrent from the valence bands and the Fermi surface and we obtain very good agreement with previous experimental measurements. We conclude that the SCF-LMTO-ASA method provides a good description of the valence s-, p- and d-band structure in/3'-AgMg.
t3'-AgMg forms the beta-brass (CsC1) structure with an electron/atom (e/a) ratio of 3/2. It is a member of a special class of alloys, commonly referred to as Hume-Rothery phases (or electron compounds), whose structures are strongly correlated with the e/a ratio [1]. It is not surprising therefore that over the years there has been considerable interest in investigating the electronic structure in a number of such phases, which has resulted in an extensive coverage in the scientific literature. Early examples where, for instance, joint experimental and theoretical approaches have been used to good effect, include the studies of the Fermi surfaces in CuZn [2], AgZn [3] and Pdln[4]; more recent reports include detailed studies of the k-resolved photoemission from 13'-NiAl [5], a study of the/3'- to ~-phase transition in AgZn [6] and an ab initio investigation of the phase stability of Cu-Zn alloys [7]. However, there appears to have been few detailed investigations of the electronic structure in ~'-AgMg. Dunsworth et al [8] reported a study of the de Haas-van Alphen effect and they carried out a non self-consistent, relativistic LMTO band structure calculation that gave a Fermi surface in good agreement with their experimental results. More recently, Blyth et al[9] calculated the density of states in 3'-AgMg as part of a more general study of the electronic structure in a number of Mg-rich transition metal alloys. However, they did not discuss their results for ~3'-AgMg in any quantitative detail. Apart from these reports we are not aware of any more band structure
calculations and comparisons with experimental measurements for this material. As far as other relevant experiments are concerned, Weightman et M [10] determined the valence d-bandwidth in 3'-AgMg using x-ray photoelectron spectroscopy (XPS) but their principal interest was to examine the influence of the bandwidth on Auger spectra. They also measured the shift of the Ag ads/= core level on alloying. In this paper we describe self-consistent calculations of the electronic structure in ~'-AgMg. In order to assess quantitatively the validity of our results we have determined also the Fermi surface and the x-ray photocurrent from the valence bands. We compare the former, which is derived from s- and p-electrons, with the resuits of Dunsworth et al [8]; we compare the latter with the x-ray photoemission measurements of Weightman et al [10]. Although XPS lacks the wavevector/~-resolution of angle-resolving uv techniques, it is nevertheless a useful probe of the electronic structure since, in this particular case, it provides important information about the changes in the valence d-band structure - - such as its shape, position and width - - and the shift of core level binding energies on alloying. We calculated the electronic structure in ~'-AgMg using the self-consistent field, linearized muffin-tin orbital method within the atomic sphere approximation (SCF-LMTO-ASA) [11]. The LMTO-ASA method is a fast technique - - at the expense of only a small loss in accuracy - - and it has been used with great success to determine the electronic structure in a variety of ordered systems [12], including other Mg-based/~'-phases [9,13]. In our calculations we only included components up to
Present address: Collegeof Engineeringand AppliedScience, State Universityof New York,StonyBrook, New York 11794, USA. ¶ Presentaddress: Los AlarnosNationalLaboratory,MS K765, Los Alamos,New Me.'dco87545, USA. 667
ELECTRONIC STRUCTURE I N f l ' - A g M g
668
g = 2 in the basis functions since we expect the higher components to play a negligible role. The core states were treated fully relativistically and were relaxed during the iteration process. The valence states were deter mined using the scaiar relativistic approximation with the spin-orbit coupling included variationally; further technicai details can be found in ref. [14]. For convenience, we assumed equal atomic sphere radii for Ag and Mg [15] and we used the yon Barth-Hedin form for the local exchange and correlation [16]. When deter mining the equilibrium lattice constant we carried out the self-consistency cycles on a grid of 165 k-points in the irreducible wedge of the simple cubic Brillouin zone. When calculating the density of states we increased the number of/~-points in the wedge to 680. We included the combined correction term [111 in the final iteration and when determining the density of states and the Fermi surface, 1
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F i g u r e 1. (a) The site decomposed densities of states and the total densities of states in ~'-AgMg. The dashed curve is the Ag contribution, the chain curve is the Mg contribution and the solid line in the total density of states. (b) The total density of states in Ag.
Vol. 81, No. 8
The equilibrium lattice constant we calculate is 3.243.~, which is some 1.6 - 2% smaller than the values of 3.297 - 3.311.~. obtained experimentally [8,17], see also ref. [9]. However, such a difference is typical of that obtained for transition metals and alloys using calculations involving the local density approximation. From the volume dependence of the pressure we obtain a value of 0.89 Mbar for the bulk modulus. We also carried out a set of similar S C F - L M T O - A S A calculations for pure Ag (fcc) and Mg (hcp). (The shapes of the resulting densities of states, the lattice constants, etc were all in very good agreement with those previously reported [18].) Using the total energies so determined we find that the heat of formation of ~'-AgMg is 0.37 e V / a t o m . As far as we are aware, there are no experimental data on the bulk modulus nor the heat of formation of fl'-AgMg. The site decomposed total densities of states in fi'AgMg near the equilibrium volume are shown in figure l(a); the total density of states is very similar to that shown by Blyth et al [9]. (Note that the Fermi energy, at zero binding energy, is towards the left of the figure.) Not surprisingly the most significant contribution is from the Ag 4d-related states, which determine both its overall shape and width. The width (at ~ 1 state/eV/cell) is ~ 2.2 eV and is considerably larger than that obtained by Dunsworth et al[8] ( ~ 1.7 eV), see their figure 4. The origin of this discrepancy in bandwidth is not entirely clear. It is not a consequence of our use of a smaller lattice constant since we obtain a reduction of only 0.1 - 0.2 eV in the width when we use the same lattice constant as Dunsworth et all8]. On the other hand our value of 0.525 s t a t e s / e V / c e l l for the total density of states at E~ is in reasonable agreement with theirs (0.543). Our calculations indicate that about 0.1e is transferred from the Mg atomic sphere to the Ag atomic sphere. Since the d-states are much more localized than the s- and p-states, it is possible that they are more sensitive to the effects of self-consistency. In figure l(b) we show our calculated density of states in pure Ag. A comparison between figures l(a) and l(b) shows that the Ag 4d-bandwidth in fl'-AgMg is about 40% or so smaller than that in pure Ag. This is a consequence of the reduced overlap of the Ag 4d wavefunctions in ~ - A g M g compared with pure Ag; for, not only is the Ag-Ag distance about 12% greater in the former, there are also fewer Ag-Ag nearest neighbor bonds. It appears also that the energy positions of the b o t t o m edges of the d-bands (i. e., at the greatest binding energies) are approximately the same in both cases and so it is the position of the top of the d-band that is most significantly affected on alloying. Weightman et al [10] carried out some high resolution x-ray photoemission measurements of the valence bands in Ag and ¢£-AgMg using A1 K~ radiation and we show their results in figure 2. The widths of the spectra are 3.6 ± 0.1 eV and 2.0 ± 0.1 eV, for Ag and
Vol. 81, No. 8
ELECTRONIC STRUCTURE INfl'-AgMg I
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x-ray photocurrent using the method described by Winter et al [19]. In this (essentially) ab initio scheme the 1-electron photoeurrent is expressed as a sum over the local densities of states modulated by the appropriate photon-electron matrix elements linking the initial and final states. Therefore, using the densities of states and potential functions from the LMTO calculation as input, we can achieve a more quantitative comparison between the photoemission measurements and the electronic structure calculations. The only unknown (i.e., adjustable) quantity is the hole lifetime.
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F i g u r e 2. Photoemission spectra from (a) Ag and (b) j3'-AgMg obtained using XPS with A1 Ka radiation. (These data were obtained from Dr. P.T. Andrews and appear in ref. [10]. The solid lines have been added for clarity.)
#'-AgMg respectively [10], i.e., there is a reduction of some 44% on alloying. Our calculated reduction in the d-bandwidth (and the widths themselves, namely, 3.75 eV and ~ 2.2 eV for Ag and fl'-AgMg, respectively) compare very well with these values. In addition, our calculations are entirely consistent with their observation that the narrowing results predominantly from a shift in energy of the upper d-band edge whilst the positions of the lower band edges remain roughly the same. However, such direct comparisons with experimental spectra ignore the possible effects of the photonelectron matrix elements. Therefore, we calculated the )-...
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In figure 3 we show the computed x-ray photocurrent for fl'-AgMg using A1 K~ radiation. (Note that since this is the elastic photoeurrent there is no 'background' contribution.) We included a Lorentzian broadening function with a FWHM of 0.6 eV in the calculation, representing a combination of the quoted experimental resolution [10] and a (constant) inverse lifetime for the hole. The agreement with the experimental spectrum is very good, implying that lifetime effects do not play a major role in determining the shape and width of the spectrum. However, the pea.k positions in the calculated photocurrent are ,-~ 0 . 6 - 0.7 eV closer to the Fermi energy than those measured experimentally; similar shifts occur in the case of other ordered noble metal alloys [20-22]. Presumably, the discrepancy results from our neglect of self-energy effects. (In this particular ease also, it appears that the photon-electron matrix elements play only a relatively minor role in determining the shape of the spectrum since simply folding a Lorentzian function into the density of states produces a similar curve.) We calculated also the changes in the Ag core level eigenvalues, with respect to the Fermi energies, on alloying. We find that compared with pure Ag the Ag core levels in fl'-AgMg are shifted by about +0.9 eV to increased binding energy. In their experiment Weightman e* al [10] measured a shift of +0.5 eV in the binding
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F i g u r e 3. The calculated x-ray photocurrent from /3'AgMg with A1 Ka radiation.
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F i g u r e 4. The calculated Fermi surface shown in the unfolded irreducible wedge of the Brillouin zone
670
ELECTRONIC STRUCTURE INfl'-AgMg
energy of the Ag 3ds/~ core level on alloying. Our calculated value of +0.93 eV is in line with this result, bin. we should point out that such a simple comparison ignores any differences in the relaxation effects, etc. [23] associated with the core holes in the pure metal and the alloy. In figure 4 we show the calculated Fermi surface in the unfolded irreducible Brillouin zone wedge. Its topology is very similar to that calculated by Dunsworth et al I8] and is in very good agreement with their experimental results. In summary, therefore, we have cMculated the density of states, the 1-electron x-ray photocurrent and the Fermi surface in /3'-AgMg. Although the valence d-bandwidth we obtain is somewhat larger than that determined in a previous (non self-consistent) calculation we find that the x-ray photocurrent, which is dominated by the contribution from the Ag-related dstates, is in very good agreement with published photoemission measurements. In addition, the Fermi surface we obtain is in good agreement with that reported in an earlier theoretical and experimental study. This suggests that the valence s-, p- and d-band structure in /3'-AgMg is well described by the SCF-LMTO-ASA method. Clearly, more discriminating experiments such as angle-resolved uv photoemission measurements - - would be required in order to probe the bands and test these results in /~-point by k-point detail. A c k n o w l e d g e m e n t s : We thank Dr. P.T. Andrews (University of Liverpool) for providing us with a copy of his experimental data and Drs. W. Temmerman and G.Y. Guo (SERC Daresbury Laboratory) for their help and advice in using the LMTO codes. We acknowledge the support provided by Florida State University through the allocation of CRAY supercomputer resources. YL is grateful to the Physics Department, Florida Atlantic University for financial support and RIRB acknowledges SERC for the provision of a Research Studentship.
man J. Phys.: Condens. Matter 1, 3315 (1989). Also N.M. Harrison Ph.D Thesis University of Birmingham, England (1990). 7 P.E.A. Turchi, M. Sluiter, F.J. Pinski, D.D. Johnson, D.M. Nicholson, G.M. Stocks and J.B. Staunton Phys. Rev. Left. 67, 1779 (1991). 8 A.E. Dunsworth, J.-P. Jan and H.L. Skriver J. Phys. F: Metal Phys. 8, 1427 (1978). 9 R.I.R. Blyth, P.T. Andrews, N. Heritage and P A R . Birtwistle - accepted for publication in J. Phys.:
Condens. Matter. 10 P. Weightman, P.T. Andrews and A.C. Parry-Jones J. Phys. C: Solid State Phys. 12, 3635 (1979). P. Weightman and P.T. Andrews J. Phys. C: Solid State Phys. 13, 3529 (1980). 11 O.K. Andersen Phys. Rev. B12, 3060 (1975). Also H.L. Skriver The LMTO Method (Springer-Verlag Berlin, 1984). 12 O.K. Andersen, O. Jepson and D. G16tzel in Highlights of Condensed Matter Theory, eds. F. Bassani, F. Fumi and M.P. Tosi (North H o l l a n d - Amstcrdam, 1985). 13 See, for example, A.E. Dunsworth, J.-P. Jan and H.L. Skriver J. Phys. F: Metal Phys. 9, 261 (1979). 14 W.M. Temmerman, P.A. Sterne, G.Y. Guo and Z. Szotek Mol. Sire. 4, 153 (1989) and W.M. Temmerman and P.A. Sterne J. Phys.: Condens. Matter 2, 5529 (1990). 15 We found that the use of different atomic sphere radii for Ag and Mg had only a minor effect on our results. 16 U. von Barth and L. Hedin J. Phys. C: Solid State Phys. 5, 1629 (1972). 17 See, for example, W.B. Pearson A Handbook of Lat-
tice Spacings and Structure of Metals and Alloys 18
References 1 See, for example, K. Girgis in Physical Metallurgy, Part 1, eds. R.W. Cahn and P. Haasen (North Holland - Amsterdam, 1983) p. 220. Also T.B. Massalski, same volume, p. 153. 2 H.L. Skriver and N.E. Christensen Phys. Rev. B8, 3778 (1973) and references therein. 3 H.L. Skriver Phys. Status Solidi 58, 721 (1973). 4 A.E. Dunsworth Phys. Rev. B12, 2030 (1975) and references therein. 5 S.-C. Liu, J.W. Davenport, E.W. Plummer, D.M. Zehner and G.W. Fernaado Phy.s. Rev. B43, 1582 (1990). 6 N.M. H~trrison, P.J. Durham and W.M. Tcmmcr-
Vol. 81, No. 8
19
20 21
22 23
(Pergamon Press - London, 1958). See, for example, V.L. Moruzzi, J.F. Janak and A.R. Williams Calculated Electronic Properties of Metals (Pergamon Press - New York, 1978) and D.A. Papaconstantopoulos Handbook of the Band Structure of Elemental Metals (Plenum Press- New York, 1986). H. Winter, P.J. Durham and G.M. Stocks J. Phys. F: Metal Phys. 14, 1047 (1984). H. Winter, P.J. Durham, W.M. Temmerman and G.M. Stocks Phys. Rev. B33, 2370 (1986). P. Weinberger, A.M. Boring, R.C. Albers and W.M. Temmerman Phys. Rev. B38, 5357 (1988). R.G. Jordan, D.M. Zehner, N.M. Harrison, P.J. Durham and W.M. Temmerman Z. Phys. B. 75, 291 (1989). R.G. Jordan, A.M. Begley, Y. Jiang and M.A. Hoyland J. Phys.: Condens. Matter 3, 1685 (1991). A.R. Williams and N. Lang Phys. Rev. Left. 40, 954 (1978).
Solid State Communications, Vol. 81, No. 8, pp. 667-670, 1992. Printed in Great Britain.
003 8- 1098/92 $5.00 +. 00 Pergamon Press plc
ELECTRONIC STRUCTURE IN /3'-AgMg R.G. Jordan'f, A.M. Begleyt§, Yan Liu~, S.L. Qiut and R.I.R. Blyth$¶ Alloy Research Center, Department of Physics, Florida Atlantic University, Boca Raton, Florida 33431-099t, USA. :~ Surface Science Research Centre, University of Liverpool, P.O. Box 147, Liverpool L69 3BX, UK. Received October 21 1991 by J. Tanc We have calculated the density of states in/3'-AgMg (CsCl-structure) using the SCFLMTO-ASA method. We have determined also the x-ray photocurrent from the valence bands and the Fermi surface and we obtain very good agreement with previous experimental measurements. We conclude that the SCF-LMTO-ASA method provides a good description of the valence s-, p- and d-band structure in/3'-AgMg.
t3'-AgMg forms the beta-brass (CsC1) structure with an electron/atom (e/a) ratio of 3/2. It is a member of a special class of alloys, commonly referred to as Hume-Rothery phases (or electron compounds), whose structures are strongly correlated with the e/a ratio [1]. It is not surprising therefore that over the years there has been considerable interest in investigating the electronic structure in a number of such phases, which has resulted in an extensive coverage in the scientific literature. Early examples where, for instance, joint experimental and theoretical approaches have been used to good effect, include the studies of the Fermi surfaces in CuZn [2], AgZn [3] and Pdln[4]; more recent reports include detailed studies of the k-resolved photoemission from 13'-NiAl [5], a study of the/3'- to ~-phase transition in AgZn [6] and an ab initio investigation of the phase stability of Cu-Zn alloys [7]. However, there appears to have been few detailed investigations of the electronic structure in ~'-AgMg. Dunsworth et al [8] reported a study of the de Haas-van Alphen effect and they carried out a non self-consistent, relativistic LMTO band structure calculation that gave a Fermi surface in good agreement with their experimental results. More recently, Blyth et al[9] calculated the density of states in 3'-AgMg as part of a more general study of the electronic structure in a number of Mg-rich transition metal alloys. However, they did not discuss their results for ~3'-AgMg in any quantitative detail. Apart from these reports we are not aware of any more band structure
calculations and comparisons with experimental measurements for this material. As far as other relevant experiments are concerned, Weightman et M [10] determined the valence d-bandwidth in 3'-AgMg using x-ray photoelectron spectroscopy (XPS) but their principal interest was to examine the influence of the bandwidth on Auger spectra. They also measured the shift of the Ag ads/= core level on alloying. In this paper we describe self-consistent calculations of the electronic structure in ~'-AgMg. In order to assess quantitatively the validity of our results we have determined also the Fermi surface and the x-ray photocurrent from the valence bands. We compare the former, which is derived from s- and p-electrons, with the resuits of Dunsworth et al [8]; we compare the latter with the x-ray photoemission measurements of Weightman et al [10]. Although XPS lacks the wavevector/~-resolution of angle-resolving uv techniques, it is nevertheless a useful probe of the electronic structure since, in this particular case, it provides important information about the changes in the valence d-band structure - - such as its shape, position and width - - and the shift of core level binding energies on alloying. We calculated the electronic structure in ~'-AgMg using the self-consistent field, linearized muffin-tin orbital method within the atomic sphere approximation (SCF-LMTO-ASA) [11]. The LMTO-ASA method is a fast technique - - at the expense of only a small loss in accuracy - - and it has been used with great success to determine the electronic structure in a variety of ordered systems [12], including other Mg-based/~'-phases [9,13]. In our calculations we only included components up to
Present address: Collegeof Engineeringand AppliedScience, State Universityof New York,StonyBrook, New York 11794, USA. ¶ Presentaddress: Los AlarnosNationalLaboratory,MS K765, Los Alamos,New Me.'dco87545, USA. 667
ELECTRONIC STRUCTURE I N f l ' - A g M g
668
g = 2 in the basis functions since we expect the higher components to play a negligible role. The core states were treated fully relativistically and were relaxed during the iteration process. The valence states were deter mined using the scaiar relativistic approximation with the spin-orbit coupling included variationally; further technicai details can be found in ref. [14]. For convenience, we assumed equal atomic sphere radii for Ag and Mg [15] and we used the yon Barth-Hedin form for the local exchange and correlation [16]. When deter mining the equilibrium lattice constant we carried out the self-consistency cycles on a grid of 165 k-points in the irreducible wedge of the simple cubic Brillouin zone. When calculating the density of states we increased the number of/~-points in the wedge to 680. We included the combined correction term [111 in the final iteration and when determining the density of states and the Fermi surface, 1
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F i g u r e 1. (a) The site decomposed densities of states and the total densities of states in ~'-AgMg. The dashed curve is the Ag contribution, the chain curve is the Mg contribution and the solid line in the total density of states. (b) The total density of states in Ag.
Vol. 81, No. 8
The equilibrium lattice constant we calculate is 3.243.~, which is some 1.6 - 2% smaller than the values of 3.297 - 3.311.~. obtained experimentally [8,17], see also ref. [9]. However, such a difference is typical of that obtained for transition metals and alloys using calculations involving the local density approximation. From the volume dependence of the pressure we obtain a value of 0.89 Mbar for the bulk modulus. We also carried out a set of similar S C F - L M T O - A S A calculations for pure Ag (fcc) and Mg (hcp). (The shapes of the resulting densities of states, the lattice constants, etc were all in very good agreement with those previously reported [18].) Using the total energies so determined we find that the heat of formation of ~'-AgMg is 0.37 e V / a t o m . As far as we are aware, there are no experimental data on the bulk modulus nor the heat of formation of fl'-AgMg. The site decomposed total densities of states in fi'AgMg near the equilibrium volume are shown in figure l(a); the total density of states is very similar to that shown by Blyth et al [9]. (Note that the Fermi energy, at zero binding energy, is towards the left of the figure.) Not surprisingly the most significant contribution is from the Ag 4d-related states, which determine both its overall shape and width. The width (at ~ 1 state/eV/cell) is ~ 2.2 eV and is considerably larger than that obtained by Dunsworth et al[8] ( ~ 1.7 eV), see their figure 4. The origin of this discrepancy in bandwidth is not entirely clear. It is not a consequence of our use of a smaller lattice constant since we obtain a reduction of only 0.1 - 0.2 eV in the width when we use the same lattice constant as Dunsworth et all8]. On the other hand our value of 0.525 s t a t e s / e V / c e l l for the total density of states at E~ is in reasonable agreement with theirs (0.543). Our calculations indicate that about 0.1e is transferred from the Mg atomic sphere to the Ag atomic sphere. Since the d-states are much more localized than the s- and p-states, it is possible that they are more sensitive to the effects of self-consistency. In figure l(b) we show our calculated density of states in pure Ag. A comparison between figures l(a) and l(b) shows that the Ag 4d-bandwidth in fl'-AgMg is about 40% or so smaller than that in pure Ag. This is a consequence of the reduced overlap of the Ag 4d wavefunctions in ~ - A g M g compared with pure Ag; for, not only is the Ag-Ag distance about 12% greater in the former, there are also fewer Ag-Ag nearest neighbor bonds. It appears also that the energy positions of the b o t t o m edges of the d-bands (i. e., at the greatest binding energies) are approximately the same in both cases and so it is the position of the top of the d-band that is most significantly affected on alloying. Weightman et al [10] carried out some high resolution x-ray photoemission measurements of the valence bands in Ag and ¢£-AgMg using A1 K~ radiation and we show their results in figure 2. The widths of the spectra are 3.6 ± 0.1 eV and 2.0 ± 0.1 eV, for Ag and
Vol. 81, No. 8
ELECTRONIC STRUCTURE INfl'-AgMg I
0
I
2
I
4
I
6
x-ray photocurrent using the method described by Winter et al [19]. In this (essentially) ab initio scheme the 1-electron photoeurrent is expressed as a sum over the local densities of states modulated by the appropriate photon-electron matrix elements linking the initial and final states. Therefore, using the densities of states and potential functions from the LMTO calculation as input, we can achieve a more quantitative comparison between the photoemission measurements and the electronic structure calculations. The only unknown (i.e., adjustable) quantity is the hole lifetime.
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BINDING ENERGY (eV)
F i g u r e 2. Photoemission spectra from (a) Ag and (b) j3'-AgMg obtained using XPS with A1 Ka radiation. (These data were obtained from Dr. P.T. Andrews and appear in ref. [10]. The solid lines have been added for clarity.)
#'-AgMg respectively [10], i.e., there is a reduction of some 44% on alloying. Our calculated reduction in the d-bandwidth (and the widths themselves, namely, 3.75 eV and ~ 2.2 eV for Ag and fl'-AgMg, respectively) compare very well with these values. In addition, our calculations are entirely consistent with their observation that the narrowing results predominantly from a shift in energy of the upper d-band edge whilst the positions of the lower band edges remain roughly the same. However, such direct comparisons with experimental spectra ignore the possible effects of the photonelectron matrix elements. Therefore, we calculated the )-...
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In figure 3 we show the computed x-ray photocurrent for fl'-AgMg using A1 K~ radiation. (Note that since this is the elastic photoeurrent there is no 'background' contribution.) We included a Lorentzian broadening function with a FWHM of 0.6 eV in the calculation, representing a combination of the quoted experimental resolution [10] and a (constant) inverse lifetime for the hole. The agreement with the experimental spectrum is very good, implying that lifetime effects do not play a major role in determining the shape and width of the spectrum. However, the pea.k positions in the calculated photocurrent are ,-~ 0 . 6 - 0.7 eV closer to the Fermi energy than those measured experimentally; similar shifts occur in the case of other ordered noble metal alloys [20-22]. Presumably, the discrepancy results from our neglect of self-energy effects. (In this particular ease also, it appears that the photon-electron matrix elements play only a relatively minor role in determining the shape of the spectrum since simply folding a Lorentzian function into the density of states produces a similar curve.) We calculated also the changes in the Ag core level eigenvalues, with respect to the Fermi energies, on alloying. We find that compared with pure Ag the Ag core levels in fl'-AgMg are shifted by about +0.9 eV to increased binding energy. In their experiment Weightman e* al [10] measured a shift of +0.5 eV in the binding
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F i g u r e 3. The calculated x-ray photocurrent from /3'AgMg with A1 Ka radiation.
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F i g u r e 4. The calculated Fermi surface shown in the unfolded irreducible wedge of the Brillouin zone
670
ELECTRONIC STRUCTURE INfl'-AgMg
energy of the Ag 3ds/~ core level on alloying. Our calculated value of +0.93 eV is in line with this result, bin. we should point out that such a simple comparison ignores any differences in the relaxation effects, etc. [23] associated with the core holes in the pure metal and the alloy. In figure 4 we show the calculated Fermi surface in the unfolded irreducible Brillouin zone wedge. Its topology is very similar to that calculated by Dunsworth et al I8] and is in very good agreement with their experimental results. In summary, therefore, we have cMculated the density of states, the 1-electron x-ray photocurrent and the Fermi surface in /3'-AgMg. Although the valence d-bandwidth we obtain is somewhat larger than that determined in a previous (non self-consistent) calculation we find that the x-ray photocurrent, which is dominated by the contribution from the Ag-related dstates, is in very good agreement with published photoemission measurements. In addition, the Fermi surface we obtain is in good agreement with that reported in an earlier theoretical and experimental study. This suggests that the valence s-, p- and d-band structure in /3'-AgMg is well described by the SCF-LMTO-ASA method. Clearly, more discriminating experiments such as angle-resolved uv photoemission measurements - - would be required in order to probe the bands and test these results in /~-point by k-point detail. A c k n o w l e d g e m e n t s : We thank Dr. P.T. Andrews (University of Liverpool) for providing us with a copy of his experimental data and Drs. W. Temmerman and G.Y. Guo (SERC Daresbury Laboratory) for their help and advice in using the LMTO codes. We acknowledge the support provided by Florida State University through the allocation of CRAY supercomputer resources. YL is grateful to the Physics Department, Florida Atlantic University for financial support and RIRB acknowledges SERC for the provision of a Research Studentship.
man J. Phys.: Condens. Matter 1, 3315 (1989). Also N.M. Harrison Ph.D Thesis University of Birmingham, England (1990). 7 P.E.A. Turchi, M. Sluiter, F.J. Pinski, D.D. Johnson, D.M. Nicholson, G.M. Stocks and J.B. Staunton Phys. Rev. Left. 67, 1779 (1991). 8 A.E. Dunsworth, J.-P. Jan and H.L. Skriver J. Phys. F: Metal Phys. 8, 1427 (1978). 9 R.I.R. Blyth, P.T. Andrews, N. Heritage and P A R . Birtwistle - accepted for publication in J. Phys.:
Condens. Matter. 10 P. Weightman, P.T. Andrews and A.C. Parry-Jones J. Phys. C: Solid State Phys. 12, 3635 (1979). P. Weightman and P.T. Andrews J. Phys. C: Solid State Phys. 13, 3529 (1980). 11 O.K. Andersen Phys. Rev. B12, 3060 (1975). Also H.L. Skriver The LMTO Method (Springer-Verlag Berlin, 1984). 12 O.K. Andersen, O. Jepson and D. G16tzel in Highlights of Condensed Matter Theory, eds. F. Bassani, F. Fumi and M.P. Tosi (North H o l l a n d - Amstcrdam, 1985). 13 See, for example, A.E. Dunsworth, J.-P. Jan and H.L. Skriver J. Phys. F: Metal Phys. 9, 261 (1979). 14 W.M. Temmerman, P.A. Sterne, G.Y. Guo and Z. Szotek Mol. Sire. 4, 153 (1989) and W.M. Temmerman and P.A. Sterne J. Phys.: Condens. Matter 2, 5529 (1990). 15 We found that the use of different atomic sphere radii for Ag and Mg had only a minor effect on our results. 16 U. von Barth and L. Hedin J. Phys. C: Solid State Phys. 5, 1629 (1972). 17 See, for example, W.B. Pearson A Handbook of Lat-
tice Spacings and Structure of Metals and Alloys 18
References 1 See, for example, K. Girgis in Physical Metallurgy, Part 1, eds. R.W. Cahn and P. Haasen (North Holland - Amsterdam, 1983) p. 220. Also T.B. Massalski, same volume, p. 153. 2 H.L. Skriver and N.E. Christensen Phys. Rev. B8, 3778 (1973) and references therein. 3 H.L. Skriver Phys. Status Solidi 58, 721 (1973). 4 A.E. Dunsworth Phys. Rev. B12, 2030 (1975) and references therein. 5 S.-C. Liu, J.W. Davenport, E.W. Plummer, D.M. Zehner and G.W. Fernaado Phy.s. Rev. B43, 1582 (1990). 6 N.M. H~trrison, P.J. Durham and W.M. Tcmmcr-
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(Pergamon Press - London, 1958). See, for example, V.L. Moruzzi, J.F. Janak and A.R. Williams Calculated Electronic Properties of Metals (Pergamon Press - New York, 1978) and D.A. Papaconstantopoulos Handbook of the Band Structure of Elemental Metals (Plenum Press- New York, 1986). H. Winter, P.J. Durham and G.M. Stocks J. Phys. F: Metal Phys. 14, 1047 (1984). H. Winter, P.J. Durham, W.M. Temmerman and G.M. Stocks Phys. Rev. B33, 2370 (1986). P. Weinberger, A.M. Boring, R.C. Albers and W.M. Temmerman Phys. Rev. B38, 5357 (1988). R.G. Jordan, D.M. Zehner, N.M. Harrison, P.J. Durham and W.M. Temmerman Z. Phys. B. 75, 291 (1989). R.G. Jordan, A.M. Begley, Y. Jiang and M.A. Hoyland J. Phys.: Condens. Matter 3, 1685 (1991). A.R. Williams and N. Lang Phys. Rev. Left. 40, 954 (1978).