The Fermi surface dimensions of disordered Cu3Au as determined by angle resolved photoemission spectroscopy

Solid State Communications. Vol. 107, No. 8. pp. 385-390. 1998

Pergamon

@ 1998 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0038-1098198 $19.00 + 00

PII: 800384098(98)002794J

THE FERMI SURFACE DIMENSIONS OF DISORDERED CusAu AS DETERMINED RESOLVED PHOTOEMISSION SPECTROSCOPY

BY ANGLE

J.A. Con Foe,“,* APJ. Stampfl, a B. Mattern, b A. Ziegler, b M. Hollering, b L. Ley, b J.D. Riley a and R.C.G. Leckey ” ;*School of Physics, La Trobe University, Bundoora Campus, Victoria 3083, Australia b Institut fur Technische Physik II, Universitat Erlangen-Niirnberg, 91058 Erlangen, Germany (Received 18 May 1998; accepted 27 May 1998 by R. G Clark)

The bulk Fermi surface of the disordered phase of CusAu has been experimentally mapped using angle resolved constant initial state (CIS) photoemission spectroscopy. The shape of the Fermi surface in the (112) plane has been determined using only those transitions which could be identified from the Fermi surface to primary cone free electron bands. A Fourier series formula, using a linear combination of coefficients for the Fermi surfaces of Cu and Au has been found to be in good agreement with the experimentally determined Fermi surface of disordered CusAu. Furthermore, the neck radius, determined absolutely in k-space, was found to be smaller than the neck radius of Cu by - 10%; a difference which may be explained by the presence of Au and the sensitivity of the neck region to changes in the crystal potential. @ 1998 Published by Elsevier Science Ltd Keywords: A. metals, D. electronic band structure, E. photoelectron spectroscopies, E. synchrotron radiation.

1. INTRODUCTION The Fermi surfaces of metals and alloys, and in particular their linear dimensions, may be thought of as characteristic quantities which describe the electronic structure and the form of the crystal potential [l]. Such dimensions should be sensitive to both chemical composition and atomic structure as they describe the kspace extent of the top-most conduction electrons. In this paper we have studied the effect of alloying on the Fermi surface of Cu by mapping the Fermi surface of substitutionally disordered CusAu. Both metals are face centred cubic (FCC), the Fermi surface of the alloy being derived from a common sp-like band which is not expected to change greatly due to disorder or composition [2]. The difference in the normalised belly radius between Cu and Au is small along the I1X * Current address: Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, U.S.A. and Australian Nuclear Science and Technology Organisation (ANSTO), Lucas Heights, NSW 2234, Australia.

or [ 1101 direction, however, along the IAX or [ 1001 direction Au has a belly radius which is - 6% larger than that of Cu. The normalised neck radius (i.e. the dimension of the gap centred about the L point in the (111) plane) of Cu and Au are also different: Cu is larger by - 5%. Furthermore the neck radius of the alloy is predicted to be 5-8% smaller than the same value for Cu [2-4]. We show that a decrease of - 10% is actually observed which may be explained by the presence of Au in the crystal which strongly affects the neck region of the disordered alloy. Both Cu and CusAu have been extensively studied: the latter for its alloying properties [5-71, its structural properties [2,8-l 11,and for the bulk and surface order-disorder transition it displays [ 121. Determination of the Fermi surface of Cu has been highly successful using a variety of techniques (see, for example [13]), the most successful of these techniques uses the de Haas-van Alphen (dHvA) effect. The accuracy of this technique is -0.2% [14] which is more accurate than current electronic structure calculations. The 385

386

FERMI SURFACE DIMENSIONS

OF DISORDERED

disadvantage of this technique is the restrictive experimental conditions which must be met: long range order, low temperatures and near perfect crystals. These conditions generally exclude the use of this technique on alloy materials such as CusAu. Nevertheless, dHvA measurements have been performed on ordered Cu3Au after overcoming the materials difficulties [3,4]. The work presented here is, to our knowledge, the only study of the disordered alloy which has yielded linear dimensions [ 15,181. It has been previously shown that it is possible to map the Fermi surface using photoemission [20-221. The relative accuracy of this technique is poorer than dHvA (0.2% compared with 410% for photoemission), however advantages such as experimental flexibility and the ability to determine the Fermi surface directly from the acquired data permit a wider variety of materials to be measured than traditional Fermiology techniques. In the present work the linear dimensions of the disordered Fermi surface of CujAu are compared with those previously determined for the Cu Fermi surface using CIS spectroscopy [22]. The changes evident about the neck region are discussed in terms of the electronic band structure, the crystal potential and photoemission processes.

2. EXPERIMENTAL Angle resolved photoemission spectra were acquired using an electron energy dispersing analyser of toroidal geometry [23] on the TGM4 beam line at the synchrotron storage ring in Berlin (BESSY) [24]. The combined energy resolution of the monochromator and analyser (which was determined by the width of the Fermi edge) varied from AE = 0.2to AE = 0.4 eV for photon energies between 10 to 100 eV The polar angle resolution of the analyser was determined to be r2” ,see [22,23] for details, and the azimuthal resolution was f lo as determined from the geometric dimensions of the analyser. Both materials were cleaned by repeated cycles of Ar+-ion sputtering (1000 eV beam energy) for 1S-30 minutes followed by annealing at - 500°C for the same duration. The disordered structure was established on the CusAu( 111) sample by annealing at temperatures above the transformation temperature of 390°C. The structure and cleanliness of the Cu and CusAu surfaces were established by the observation of the expected LEED patterns [25] and characteristic surface states [11,26] in the vicinity of the Fermi edge. The Cu Fermi surface was determined by acquiring CIS spectra at the Fermi level for photon energies be-

Vol. 107, No. 8

CusAu

s x

x x

ho [eV]

110 105 100

k,, [A-‘1

Fig. 1. Stack plot of constant initial state spectra in the (112) plane of CusAu. Open circles indicate transitions that fall onto free electron primary cone final states; circles and crosses indicate transitions from surface related features. tween 10 and 100 eV in 1 eV steps and polar angles between +90”. Data acquired from the (110) and (00 1) planes was sufficient to determine the Fermi cross section in the irreducible wedge of the Brillouin zone thus yielding the linear dimensions of the neck and belly. Further details of these results can be found in [22]. In order to determine the CUJAU Fermi linear dimensions and cross-section, CIS spectra at the Fermi level were acquired between 35 I hw I I IO eV in steps of 1 eV for polar angles between +90” in steps of - 1’ for the (112) plane. Raw data acquired from both Cu and CusAu were processed in the same way. The angle spectra were normalised so that the intensity of the highest and lowest observed levels for each photon energy were the same. All spectra were acquired with p-polarised light at normal incidence to the surface; as a result spectra measured at equivalent negative and positive polar angles had nearly the same intensity distributions.

3. RESULTS The CIS spectra collected from disordered CU~AU, see Fig. 1, were obtained using a previously reported

Vol. 107, No. 8

0

FERMI SURFACE DIMENSIONS

1

a1 k,,[A- 1

OF DISORDERED

Cu3Au

387

2

Fig. 2. The Fermi cross-section (thick solid lines) for the (112) plane plotted in the extended Brillouin zone. Experimental positions are shown by circles. photoemission technique [2 1,221 which restricts measurements to a high symmetry plane of the three dimensional bulk Brillouin zone so that only photoemission transitions from the Fermi surface in that plane are generally measurable. Most transitions can be identified as being associated with lifetime broadened primary cone free electron final state bands thus allowing the cross section of the Fermi surface in that plane to be readily determined. The FCC Brillouin zone slice and Fermi crosssection which were probed by the measurements are shown in Fig. 2 for the (112) plane of disordered CusAu. Many of the major peaks in Fig. 1 can be identified as free electron primary cone transitions from that part of the Fermi cross-section shown in Fig. 2. Spectra were acquired over a large range of polar angles: the x-axis in Fig. 1 has been converted to the parallel component of the k-vector [27] given

Fig. 3. The experimental Fermi surface cross-section of CusAu in the (112) plane as determined by photoemission. Experimental points (open circles) have been mapped back onto the irreducible Brillouin zone. Thick lines indicate the calculated position of the Fermi surface. been mapped back into k-space in Fig. 2 where a best fit to the calculated Fermi cross-section was obtained using an inner potential, VOof 4.0 eV with respect to the Fermi level [29]. Circles with crosses mark the position of points which were identified as surface related features [30]: such states are not generally expected to coincide in k-space with bulk states and so do not fall onto the calculated Fermi contour in Fig. 2 when they are mapped into k-space.

4. DISCUSSION

The results of mapping the primary cone transitions into the reduced Brillouin zone for the full set of data acquired are given in Fig. 3. These points were translated from the extended zone using bulk G vectors and fall onto the Fermi surface cross-section within the intrinsic photoemission k-space error of Ak - +O. IA-’ k sine, where EK is the electron ki- [22,32,33]. The Fermi cross-section was determined by kll = JF netic energy with respect to the vacuum level of the using the Fourier series formula given by Halse [34] alloy. Transitions are expected to generally obey the to fit the dHvA measurements of Cu and Au. Specifenergy conservation law, E(k) - Er(k) = ttw, where ically, the seven Fourier coefficients reported by CoE is the final state energy with respect to the Fermi leridge and Templeton for Cu and Au 1141were used level 1281.Thus, by acquiring CIS spectra I(kll), such to generate the CuaAu Fermi surface cross-section asas those shown in Fig. 1 with the initial state cho- suming that the bands were shifted in k-space by an amount determined by the percentage of Cu and Au sen at the Fermi energy, E,c, as a function of photon energy, transitions located at different k-space points in the alloy (i.e. each new coefficient was determined on the uppermost band crossing the Fermi level were using CcU’*” = 0.75Cc” + 0.25C;“” for i=1,7). Deimel observed as peaks at the appropriate values of kll. and co-workers [3,4] used this rigid band model to inPlotting peak positions in the kll-plane traces directly terpret their dHvA data taken from ordered (simple cubic) CuxAu. the Fermi surface cross-section. Table 1 summarises the linear dimensions obtained Open circles mark the position of peaks in Fig. 1 which have been identified as transitions to free elec- from Cu and CusAu along the TKX direction. The extron final state primary cone bands. These points have perimentally determined values are compared to the

388

FERMI SURFACE DIMENSIONS

Table 1. Fermi surface linear dimensions of the belly radius along the KX direction in units of ?. The lattice constants for Au, Cu and CusAu are 4.08A-‘, 3.6147A-’ and 3.7428A-’ respectively Technique ARCIS* Rigid band LMTO dHvA*

Au

cu 0.74

CusAu 0.73 0.7416 0.7377 0.742 0.761+ 0.7361 0.7429 0.7366 0.7431 * Angle-resolved constant initial state spectra (ARCIS). Value derived using free electron final states. t Derived from ordered LMTO calculation. # Top dHvA value from [l4], bottom value from [34].

Table 2. Fermi surface linear dimensions of the neck radius in units of $, The lattice constants for Au, Cu and CusAu are 4.08A-‘, 3.6147A-’ and 3.7428A-’ respectively Technique ARCIS*

Au -

cu 0.20 0.18

CusAu 0.13

0.1349 0.16 0.141 0.1406 0.1396 0.1474 0.1424+ 0.1396 0.1474 * Value derived using free electron final states. t Value from ordered CusAu [4]. Top Cu dHvA value from [l4], bottom value from [34]. Rigid band LMTO dHvA

linear muffin-tin orbital (LMTO) values and those obtained from dHvA measurements. The LMTO calculated distances for the disordered alloy were determined by firstly calculating the linear dimensions of the ordered phase and then remapping these dimensions back into the disordered unit cell. Our calculation [35] of the ordered phase is in good agreement with the previously reported LMTO results of Skriver and Lengkeek [8]. We find that the unfolded LMTO bands yield dimensions which are in agreement with the Fourier series formula along the high symmetry directions examined. For Cu and CUJAU along the TKX direction (belly radius) the experimentally determined dimensions are the same to within experimental error as those values obtained from the band structure calculation and dHvA results. Along this direction the dimensions of the Au Fermi surface are similar to Cu and the disordered alloy. Agreement between the CujAu experimental data and calculation is poorer, however, than observed for Cu. This can be attributed to, the simple approach of folding out the LMTO calculation for the ordered SC zone to obtain the disordered FCC Fermi surface dimensions, and to the rigid band values which are based only on the stoichiometric combination of the Fourier series describing the Fermi surfaces of Cu and Au. Table 2 gives the neck radii of Cu and CusAu: the neck radii are by symmetry directly comparable for

OF DISORDERED

Cu3Au

Vol. 107, No. 8

the ordered and disordered unit cells of the alloy . We therefore also include the neck radius of ordered CUJAU as determined by dHvA measurements [3,4]. The neck radius dimensions are expected to be similar across the order-disorder transition to within a few percent [36]. Included for comparison are the neck radius dimensions of Au as determined by dHvA measurements and our LMTO calculation. The neck radius of the disordered alloy is similar to that of Au. Our results agree with the values calculated for the alloy using the Fourier series expansion to within experimental error. The discrepancy observed between the experimental and theoretical values for the neck radius of both materials is probably due to a number of effects. It is known for Cu that the widths of the surface and bulk bands near the L point (neck region) are strongly temperature dependent [32,37]. Furthermore the mean free path is only expected to be - 4 - 5A for those photoelectrons that were acquired around the Lpoint region (i. e. 70 5 Rw 5 75 eV) thereby smearing out the electron k vector to - +0.2A-‘. The Cu neck radius (as determined by dHvA) is 10% larger than the dimensions of the CUJAU neck obtained from this experiment. In interpreting this result it is worth noting that along the IKX and TX directions the Fermi surface is distant from the Brillouin zone boundary. The neck region arises from the contact of the Fermi surface with the Brillouin zone boundary being very sensitive to changes in the the crystal potential there [38]. For the disordered alloy the normalised belly radius along the IKX direction (Table l), is about the same as Cu [39] while the neck radius is more Au-like in character (Table 2). This behaviour can be explained by considering the shape of the bands passing through the Fermi level. The sp-like band passing through the Fermi level around the belly is quite steep whereas around the neck region it is quite flat [38]: small shifts in energy, due to changes in the crystal potential, should be pronounced around the neck. Therefore we conclude that the alloy Fermi surface may be generally viewed as a stoichiometric combination of the Fourier series describing the Fermi surfaces of Au and Cu: the presence of Au, results in a reduced neck dimension for the alloy, whereas its presence has minimal influence along the IKX direction since both Cu and Au have similar Fermi surface dimensions there. These results confirm earlier reports which suggested that the Fermi surface of Cu and CujAu should be similar [2]. We find that although the belly radius is unchanged, alloying affects the neck radius of the Fermi surface by a small but detectable amount.

Vol. 107, No. 8

FERMI SURFACE DIMENSIONS 5. SUMMARY

The Fermi surface of the disordered phase of CusAu has been mapped. A rigid band model was used to successfully describe the measured Fermi surface cross-section and was found to be in substantial agreement with the one obtained from our unfolded LMTO calculation. A comparison between the linear dimensions of the Fermi surfaces of disordered Cu3Au and Cu indicates they are similar with the belly radius unchanged upon alloying. We find, however, that the neck radius of the alloy is smaller than that observed in Cu and more Au-like. The neck region is most sensitive to the chemical character of the alloy, a region which is expected to be easily changed by the detailed form of the crystal potential. Acknowledgements-We

would like to thank Professor N.E. Christensen for providing his LMTO program. This work is supported by the Australian Research Council under contract number A69701004and by the German Ministry of Education and Research under contract no. 05 SWEADAB3. J. Con Foo would like to thank La Trobe University for providing a postgraduate scholarship to carry out this work. REFERENCES

OF DISORDERED

Cu3Au

389

13. Cracknell, AI’., Electron States and Fermi Surfaces of Elements, (Edited by K. Hellwege and 0. Madelung), Landoldt-Bornstein, New Series, Springer-Verlag, Berlin, Group III, Vol. 13c, 1984. 14. Coleridge, ET. and Templeton, I.M., Phys. Rev. B, 25, 1982,7818.

15. Some early positron annihilation measurements [16,17] have been reported for both phases, however poor statistics did not permit the resolution of anything other than gross features. 16. Morinaga, H., Phys. Lett., 32A, 1970. 75. 17. Morinaga, H., Phys. Lett., 34A, 1971, 384. 18. We have previously published results from this material [19], however an error in determining the belly radius from the ordered LMTO calculation lead to an erroneous interpretation of this data. 19. Con Foo, J.A., Tkatchenko. S., Stampfl, A.I?J., Ziegler, A., Mattern, B., Hollering, M., Denecke, R., Ley, L., Riley, J.D. and Leckey, R.C.G., Surface and Interface Analvsis,

24, 1996, 535.

20. Aebi, I’., Osterwalder, J., Fasel, R., Naumovic, D. and Schlapbach, L., Surf Sri., 307/309, 1994,9 17. 21. Stampfl, A.P.J., Con Foo. J.A., Leckey, R.C.G., Riley, J.D., Denecke, R. and Ley. L., Surf Sci.,

3311333, 1995, 1272. 22. Con Foo, J.A., Stampfl, ARJ., Riley, J.D., Leckey, R.C.G., Ziegler, A., Mattern, B.. Hollering, M., Denecke, R. and Ley, L., Phys. Rev. B, 53, 1996,

N.W. and Mermin, N.D., Solid State Physics, Holt, Rinehart and Winston, New York,

9649. 23. Leckey, R.C.G. and Riley, J.D., Appl. Surf Sri.,

1976. 2. Arola, E., Rao, R.S., Salokatve, A. and Bansil, A., Phys. Rev. B, 41, 1990, 7361. 3. Deimel, RF’., Higgins, R.J. and Goodall, R.K., Phys. Rev. B, 24, 1981, 6197. 4. Deimel, PI? and Higgins, R.J., Phys. Rev. B, 25, 1982,7117. 5. Eberhardt, W., Wu, S.C., Garrett, R., Sondericker, D. and Jona, F., Phys. Rev. B, 31, 1985, 8285. 6. Wertheim, G.K., Mattheiss, L.F. and Buchanan, D.N.E., Phys. Rev. B, 38, 1988, 5988. 7. Stuck, A., Osterwalder, J., Greber, T., Htifner, S. and Schlapbach, L., Phys. Rev. Lett., 65, 1990,

22123, 1985, 196. 24. Theis, W., Braun, W. and Horn, K.. BESSY Yearbook, BESSY, Berlin, 199 1, p. 462 25. Potter, H.C. and Blakely, J.M., J I/at. Sci. Technol.,

I.

Ashcroft,

3029. 8. Skriver, H.L. and Lengkeek, HP., Phys. Rev. B, 19, 1979, 900. 9. Buck, T.M., Wheatley, G.H. and Marchut, L., Phys. Rev, Lett., 51, 1983, 43. 10. Jordan, R.G. and Ginatempo, B., Mat. Res. Sot Symp. Proc., 186, 1991, 193. 11. Lau, M., Lbbus, S., Courths, R., Halilov, S., Gollisch, H. and Feder, R., Ann. Physik, 2, 1993,450. 12. Alvarado, S.F., Campagna. M., Fattah, A. and Uelhoff. W.. Z. Phvs. B, 66, 1987, 103.

12, 1975, 635. 26. Jordan, R.G. and Sohal, G.S., J. Phys. State Phys., 15, 1982, L663.

C: Solid

27. The perpendicular component of the k-vector is not conserved across the solid/vacuum interface. 28. Himpsel, F.J., Nuclear Instruments and Methods, 208, 1983,753. 29. Free electron final state bands may be determined

using E(k)=% - Vo, where Vo is with respect to the Fermi level of the sample. 30, Surface states/resonances for photon energies 35 I Rw I 40 and 100 5 Rw I 110 eV were identified by their characteristic E(kll) dispersion [ll]. We believe the features within the energy range 45 I fiw I 50 and 90 I Rw 5 95 eV are due to emission from previously unreported surface resonance states: their E(kll ) dispersion is very similar to the surface dispersion predicted for Cu [31]. 31. Dempsey, D.G. and Kleinman, L., Phys. Rev. B, 16, 1977,5356.

390

FERMI SURFACE DIMENSIONS

32. Knapp, J.A., Phys. Rev, B 19, 1979, 2844. 33. This value is the lower limit of the error which occurs for tzw - 10 eV. Larger errors up to 0.2A-’ are expected for the photon energy range measured. 34. Halse, M.R., Phil. Trans. R. Sot. Lond., 265, 1969, 507. 35. We used an ab-initio self-consistent

LMTO computer code provided by N. E. Christensen.

OF DISORDERED

Cu3Au

Vol. 107, No. 8

36. Jordan, R.G., Sohal, G.S., Gyorffy, B.L., Durham, RJ., Temmerman. W.M. and Weinberger, P., J Phys. F Met. Phys., 15, 1985, L135. 37. McDougall, B.A., Balasubramanian, T. and Jensen, E., Phys. Rev. B, 51, 1995, 13891. 38. Jepsen, O., Gliitzel, D. and Mackintosh, A.R., Phys. Rev. B, 23, 198 1, 2684. 39. The rigid band model predicts only a - 2% dif-

ference between the belly radius of Cu and CusAu along the TAX direction.