- Email: [email protected]
On the anisotropic optical response of Al(1 1 0)
Progress in Surface Science 74 (2003) 283–291 www.elsevier.com/locate/progsurf
On the anisotropic optical response of Al(1 1 0) A.I. Shkrebtii a, M.J.G. Lee a
b,*
School of Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, ON, Canada L1H 7L7 b Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON, Canada M5S 1A7
Abstract In a recent paper, good agreement was reported between the results of an ab initio fullpotential linear augmented plane wave (FP-LAPW) calculation of the linear optical response of Al(1 1 0) and experimental measurements of the reflectance anisotropy. In the present work, our FP-LAPW calculations are extended to develop a microscopic picture of the anisotropic optical response at the Al(1 1 0) surface. Evidence for an anisotropic intraband transition (Drude) contribution in the infrared is presented, and the anisotropy is explained by considering the redistribution of charge that occurs when an Al(1 1 0) surface is created. The interband transitions that make the dominant contribution to the reflectance anisotropy at higher energy are identified, and the symmetries and the surface or bulk character of the initial and final states are determined. Changes in the relative energies and occupations of electronic states due to surface charge redistribution are a possible mechanism for the interband contribution to the reflectance anisotropy. 2003 Elsevier Ltd. All rights reserved. Keywords: Metallic surfaces; Ab initio FP-LAPW method; Optical anisotropy; Al(1 1 0); Bulk termination effect; Charge transfer; Wave function symmetry
1. Introduction Reflectance anisotropy spectroscopy (RAS) has been applied to investigate a wide range of processes at clean and adsorbate-covered surfaces of cubic semiconductors [1]. The difference between the measured reflectivity at normal incidence, when light is polarized along the two principal axes of a surface, can be used as a non-invasive probe of the electronic structure of the surface, as was first proposed by Aspnes [2].
*
Corresponding author. Tel.: +1-416-978-2943; fax: +1-416-978-5848. E-mail address: [email protected] (M.J.G. Lee).
0079-6816/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.progsurf.2003.08.022
284
A.I. Shkrebtii, M.J.G. Lee / Progress in Surface Science 74 (2003) 283–291
Cubic semiconductors are optically isotropic in the bulk, and the observed reflectance anisotropy is due to modified optical transitions in the vicinity of the surface. Anisotropy may be induced by reduced symmetry, the presence of surface states, top-atom dynamics, surface reconstruction, or the adsorption of foreign atoms. When interpreted on the basis of microscopic theory, RAS is expected to be a useful probe of various surface properties and processes (see, e.g., [3–5] and references therein). The first reflectance anisotropy (RA) measurement at a metallic surface, Ag(1 1 0), was reported a decade ago [6], and detailed experimental studies of the RA at clean and adsorbate-covered (1 1 0) surfaces of copper, silver, and tungsten have recently been carried out [7–10]. Microscopic theory was applied in [11] to investigate the RA of Cu(1 1 0) and Ag (1 1 0). Recently, we and our co-workers reported a joint experimental and theoretical study of the RA of Al(1 1 0) [12]. The results of an ab initio self-consistent full-potential linear augmented plane wave (FP-LAPW) calculation of the linear dielectric function of clean Al(1 1 0) were shown to be in good agreement with the experimentally-measured RA of clean, oxidized, and Ni covered Al(1 1 0) surfaces. In the present paper, we address some important features of the experimental RA that were not discussed in [12]. These include the anisotropy of the intraband transition (Drude) term that dominates the optical response below about 1 eV, the charge shifts that accompany the formation of an Al(1 1 0) surface, and the polarization dependence of the interband transitions that are responsible for the RA. This paper is organized as follows. In Section 2, we describe briefly the measurement and calculation of the RA at an Al(1 1 0)–vacuum interface, and we outline the basis of our ab initio calculation. In Section 3.1, we present evidence for anisotropy in the Drude contribution, which we explain in terms of charge transfer to the [1 1 0] oriented chains of atoms at the Al(1 1 0) surface. In Section 3.2, we identify the principal interband transitions that are responsible for the RA in the interband regime. We describe the symmetries and the surface or bulk character of the initial and final states, and we relate the RA in the interband regime to surface-induced shifts of the energies of the electronic states of the metal.
2. Reflectance anisotropy of Al(1 1 0): experiment and theory The complex RA at a (1 1 0) surface is defined as the difference Dr between the reflectivities measured along the two principal axes, divided by the average reflectivity r of the surface [7–9]: Dr r r0 0 1 ¼ 2 110 ; r r1 1 0 þ r0 0 1
ð1Þ
where r1 1 0 and r0 0 1 are the complex reflectivities at normal incidence with light polarized parallel to the short [1 1 0] and the long [0 0 1] axis of the surface unit cell respectively, as shown in Fig. 1. Usually, both measurements and calculations determine the real part of the reflectance anisotropy, and the imaginary part is deduced
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Fig. 1. Perspective view of top four layers of clean Al(1 1 0) (1 · 1) surface. Primitive unit cell is marked by dashed lines. Arrows show principal crystallographic axes: longer edge of unit cell is along [0 0 1], and shorter edge is along [1 1 0]. Chains of atoms are oriented parallel to [1 1 0].
by Kramers–Kronig transformation. The experimental details have been reported more fully elsewhere [7–9,12]. The RA Dr=r can be expressed in terms of the surface and bulk dielectric functions [13]. The real part of the reflectance anisotropy is given by Dr 8pd e e0 0 1 Re Im 1 1 0 ¼ ; ð2Þ r k eb 1 where d is the thickness of the overlayer, k is the free-space wavelength of the light, e1 1 0 and e0 0 1 are the tensor components of the surface dielectric function (polarizability) calculated for light polarized parallel to the [1 1 0] and [0 0 1] directions, respectively (Fig. 1), and eb is the dielectric function of the bulk solid. Here, we summarize a few details of our computational method; for more complete details the reader is referred to [12,14]. The electronic structures and the wave functions of bulk aluminium and of the Al(1 1 0)–vacuum interface were calculated by solving the semi-relativistic Pauli equation by means of the FP-LAPW method using the WIEN2K code [15]. Exchange and correlation were treated in the generalized gradient approximation (GGA) [16]. The spin–orbit interaction, which is expected to be negligible for the valence electrons of aluminium, was not included in the calculation. No final state energy (scissors operator) shifts were applied. The calculated band structure and the energies of the interband peaks of the dielectric function of bulk Al are in very good agreement both with the results of pseudopotential plane wave calculations [17] and with current experimental data [18]. The crystal structure of Al is face-centred cubic. The Al–vacuum interface was represented by a supercell containing a slab consisting of 13 layers (six unit cells) of aluminium atoms aligned parallel to (1 1 0), surrounded by an empty region of equal volume to represent the vacuum. It was shown in [11] that an even smaller number of
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layers and a narrower vacuum region are sufficient to achieve a well-converged optical response. A damped oscillatory surface relaxation was assumed, represented by a contraction by 8.5% in the top Al(1 1 0) layer, an expansion by 4.8% in the second layer, and a contraction by 3.9% in the third layer [19].
3. Results and discussion In a recent paper, the RA of clean Al(1 1 0) was investigated both experimentally and theoretically as described above [12]. To emphasize the main result, and to point out some remaining discrepancies between experiment and theory, the measured real RA (solid line) and calculated interband (dashed line) real RA of Al(1 1 0) in the infrared and visible are compared in Fig. 2. 3.1. Anisotropy of Drude contribution There are two main contributions to the optical response of a metal: the intraband (Drude) term dominates below about 1 eV, and the term due to interband transitions dominates at higher energies. We begin by discussing the discrepancy between theory and experiment in the Drude regime. While the energy of the experimental RA peak is about 1.5 eV, irrespective of surface treatment, the experimental RA minimum occurs in the range 0.7–1.1 eV depending on the treatment of the surface [12], as shown in Fig. 2. Scaling the calculated RA curve to model the possible effect of surface defects is inconsistent with the observed behaviour of the RA minimum. It is commonly assumed that the Drude contribution to the optical response is isotropic. However, as seen from Fig. 2, the calculated interband contribution in the Drude regime is strong and positive, while the experimental RA is weak and nega-
Fig. 2. Experimental (solid line) and calculated interband (dashed line) real part of reflectance anisotropy of clean Al(1 1 0) surface.
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tive. The positive sign of the calculated interband contribution follows directly from the fact that, as discussed below, our slab calculation yields no electronic states of p1 1 0 symmetry that are less than 5 eV above the Fermi level. While a small shift, due to strain in the windows of the RAS equipment, may contribute to the negative RA observed in the Drude regime, the large remaining discrepancy between interband calculation and experiment is a strong indication of anisotropy in the Drude contribution. Although it is possible in principle to carry out a microscopic calculation of the anisotropy of the Drude contribution, it is not sufficient to know how the electron charge density is modified at the surface. One needs, in addition, a model for the frequency dependent microscopic conductivity of electrons localized at the surface. Instead, we consider a simple intuitive picture of the origin of the anisotropy. Fig. 3 shows the difference between the charge density in the surface layer of the relaxed Al(1 1 0) slab and that in the (1 1 0) plane of bulk Al, both calculated by the FP-LAPW method. It represents the shift of charge density that accompanies the creation of an Al(1 1 0) surface. The upper part of Fig. 3 is a grid plot of the charge shift, while a contour plot, projected onto the (1 1 0) unit cell, is shown at the lower part of Fig. 3. The atoms of the surface layer are located at the corners of the plot. An additional calculation showed that the charge density distribution in the central plane is essentially the same as that in bulk Al, which confirms that a 13 layer slab is sufficient to calculate the charge transfer at the surface. It follows that the creation of
Fig. 3. Difference of charge density between relaxed Al(1 1 0) layer and (1 1 0) plane of bulk Al. Grid plot of charge density difference is shown above, and contour plot projected onto (1 1 0) unit cell is shown below. Surface atoms are located at corners of plot.
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an Al(1 1 0) surface results in a transfer of charge to the [1 1 0] oriented chains of atoms at the surface. The charge transfer is expected to induce an anisotropic surface conductivity, enhancing the free electron (Drude) absorption for incident light polarized parallel to [1 1 0], and resulting in a negative Drude contribution to the real part of the RA of Al(1 1 0). The charge transfer can be understood as follows. While the creation of a (1 1 0) surface on the fcc lattice of bulk Al involves the removal of a half-space of atoms, the resulting charge transfer is dominated by the effect of removing the first overlayer, which is centered with respect to the atoms of the surface layer as shown Fig. 1. Because the centered atom in the first overlayer is closer to the longer [0 0 1] edge than to the shorter [1 1 0] edge, its removal brings about a greater increase in the potential along the longer edge of the surface cell than along the shorter edge, resulting in a transfer of charge from the longer edge to the shorter edge.
3.2. Anisotropy of interband contribution The dielectric function of bulk Al as calculated by the FP-LAPW method is reported elsewhere [12]. The result is in good agreement with the experimental data from Palik [18], demonstrating that density functional theory is capable of describing accurately the linear optical properties of bulk Al. Our calculation predicts a strong optical absorption peak at 1.5 eV, due to direct (vertical) transitions between occupied and unoccupied electronic states, whose k vectors lie close to the symmetry point K along the CK direction of the bulk Brillouin zone. Fig. 4 compares the band-resolved contribution of interband transitions to the imaginary (absorptive) part of the polarizability of an Al(1 1 0) slab for light polarized (a) along [0 0 1] (the longer side of the unit cell) and (b) along [1 1 0] (the shorter side of the unit cell). There are several strong peaks (Fig. 4a, A–C) in the energy range 1.4–1.8 eV for light polarized along [0 0 1], but none for light polarized along [1 1 0]. This indicates that in the interband regime the absorption of light is greater for light polarized along [0 0 1], which is consistent with the experimental observation that the real part of the RA is positive [12]. Our FP-LAPW calculation for an Al(1 1 0) slab shows that absorption peaks A and C, at 1.49 and 1.73 eV, respectively, are due to transitions from a p0 0 1 -like initial state band to an s-like final state band, while peak B at 1.62 eV is due to transitions from a (different) p0 0 1 -like initial state band to an s-like final-state band. That the initial (occupied) states are predominantly p0 0 1 -like and the final (unoccupied) states are predominantly s-like is consistent with the requirement that in electric dipole transitions the initial and final states be of opposite parity. Such transitions cannot be induced by light polarized in the [1 1 0] direction. Our FP-LAPW calculation for a 13 layer Al(1 1 0) slab shows that the lowest p1 1 0 -like valence state is more than 5 eV above the Fermi energy. It follows that there are no p1 1 0 -like states in the right energy range to act as initial or final states for electric dipole transitions with light polarized in the [1 1 0] direction. Therefore, after summing over all interband transitions and Lorenzian broadening is taken into account, our calculation yields a
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Fig. 4. Band-resolved plot of interband contribution to ImðeÞ derived from slab calculation of dielectric function of Al(1 1 0), for light polarized in [0 0 1] (upper) and [1 1 0] (lower) directions. Peaks A, B, and C, which make dominant contribution to calculated reflectance anisotropy, are discussed in text. Different lines denote contributions of different pairs of initial and final states.
strong negative peak in Im(e1 1 0 e0 0 1 ) that is consistent with the positive peak in the experimental RA. While the simple intuitive picture presented in Section 3.1, based on charge transfer to chains of surface atoms oriented along [1 1 0], provides a natural explanation for the RA observed in the Drude regime, it does not account directly for the RA in the interband regime. However, the enhanced charge density along [1 1 0] is expected to increase the energies of p1 1 0 -like states relative to p0 0 1 -like states. If the effect is to depopulate all of the p1 1 0 -like states by moving them from below to above the Fermi level, the result will be a strongly anisotropic polarization dependence of the matrix elements for interband transitions that is consistent with the experimental RA of Al(1 1 0). Our FP-LAPW calculations yield no true surface states (states for which the charge density is localized at the surface) at the Al(1 1 0) surface, although some surface resonances (states with enhanced charge density at the surface) are found. However, the initial and final states of the transitions that account for the interband contribution to the RA are all of intermediate or bulk character (i.e. they have a maximum charge density in the inner layers of the crystal and a reduced charge density at the surface). This is consistent with the experimental fact that the strongest feature in the RA spectrum of a clean Al(1 1 0) surface, the absorption peak at 1.5 eV
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for light polarized in the [0 0 1] direction, is essentially unchanged both in energy and in line shape, when the surface is oxidized, and also when it is covered by an overlayer of Ni. The experimental observations are evidence that surface states and surface resonances, as well as other surface features, such as top-atom relaxation and bonding to adsorbates, play little or no part in the RA of Al(1 1 0).
4. Conclusions The RA of Al(1 1 0) is caused by changes in the electronic states of the metal induced by the surface, which are known generically as bulk termination effects and are widely applied in surface optics [3–5,11,12]. A comparison between the calculated slab polarizability of Al(1 1 0) and the experimental data yields evidence for significant anisotropy in the Drude contribution to the RA. The increased absorption of light polarized in the [1 1 0] direction is interpreted as being due to anisotropic surface conductivity resulting from the transfer of charge to [1 1 0] oriented chains of atoms at the surface. In the interband regime, the anisotropy of the dielectric function deduced from a first-principles FP-LAPW calculation for a clean Al(1 1 0) slab is consistent with the experimentally observed RA. The strong polarization dependence is attributed to the absence of p1 1 0 -like electronic states with energies less than 5 eV above the Fermi level. The transfer of charge to [1 1 0] oriented chains of atoms at the surface shifts the relative energies, and hence the populations, of electronic states of p0 0 1 and p1 1 0 symmetry, resulting in polarization dependent matrix elements that may be responsible for bulk termination effects in interband transitions. The initial and final states of the optical transitions that make the dominant contribution to the RA in the interband regime are intermediate or bulk-like. This accounts for the experimental finding that the main peak in the RA spectrum is little affected in energy or line shape by oxidation or by the presence of an overlayer of Ni.
Acknowledgements The authors thank W. Richter, Th. Herrmann, N. Esser, R. Del Sole and O. Pulci for useful discussions. This work was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
References [1] [2] [3] [4]
W. Richter, Appl. Phys. A 75 (2002) 129. D.E. Aspnes, J. Vac. Sci. Technol. B 3 (1985) 1502. R. Del Sole, in: P. Halevi (Ed.), Photonic Probes of Surfaces, Elsevier, Lausanne, 1995, p. 133. A.I. Shkrebtii, N. Esser, W. Richter, W.G. Schmidt, F. Bechstedt, B.O. Fimland, A. Kley, R. Del Sole, Phys. Rev. Lett. 81 (1998) 721.
A.I. Shkrebtii, M.J.G. Lee / Progress in Surface Science 74 (2003) 283–291 [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
[16] [17] [18] [19]
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O. Pulci, G. Onida, R. Del Sole, L. Reining, Phys. Rev. Lett. 81 (1998) 5374. Y. Borensztein, W. Mochan, J. Tarriba, R. Barrera, A. Tadjeddine, Phys. Rev. Lett. 71 (1993) 2334. K. Stahrenberg, Th. Herrmann, N. Esser, W. Richter, Phys. Rev. B 61 (2000) 3043. Th. Herrmann, K. L€ udge, W. Richter, N. Esser, P. Poulopoulos, J. Lindner, K. Baberschke, Phys. Rev. B 64 (2001) 184424. K. Stahrenberg, T. Herrmann, N. Esser, J. Sahm, W. Richter, S.V. Hoffmann, Ph. Hofmann, Phys. Rev. B 58 (1998) 10207, 61 (2000) 13287(E). D.S. Martin, O. Zeybek, B. Sheridan, S.D. Barrett, P. Weightman, J.E. Inglesfield, S. Crampin, J. Phys.: Condens. Matter 13 (2001) 23916. P. Monachesi, M. Palummo, R. Del Sole, R. Ahuja, O. Eriksson, Phys. Rev. B 64 (2001) 115421. Th. Herrmann, M. Gensch, M.J.G. Lee, A.I. Shkrebtii, N. Esser, W. Richter, Phys. Rev. B, submitted for publication. R. Del Sole, Solid State Commun. 37 (1981) 537. K. Stahrenberg, Th. Herrmann, K. Wilmers, N. Esser, W. Richter, M.J.G. Lee, Phys. Rev. B 64 (2001) 115111. P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, WIEN2k, An augmented plane wave+local orbitals program for calculating crystal properties, Karlheinz Schwarz, Technical University of Wien, Austria, 2001. J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. K.-H. Lee, K.J. Chang, Phys. Rev. B 49 (1994) 236. E.D. Palik (Ed.), Handbook of Optical Constants of Solids II, Academic Press, San Diego, CA, 1991, pp. 435–448. B.W. Busch, T. Gustafsson, Surf. Sci. 415 (1998) L1074.
On the anisotropic optical response of Al(1 1 0) A.I. Shkrebtii a, M.J.G. Lee a
b,*
School of Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, ON, Canada L1H 7L7 b Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON, Canada M5S 1A7
Abstract In a recent paper, good agreement was reported between the results of an ab initio fullpotential linear augmented plane wave (FP-LAPW) calculation of the linear optical response of Al(1 1 0) and experimental measurements of the reflectance anisotropy. In the present work, our FP-LAPW calculations are extended to develop a microscopic picture of the anisotropic optical response at the Al(1 1 0) surface. Evidence for an anisotropic intraband transition (Drude) contribution in the infrared is presented, and the anisotropy is explained by considering the redistribution of charge that occurs when an Al(1 1 0) surface is created. The interband transitions that make the dominant contribution to the reflectance anisotropy at higher energy are identified, and the symmetries and the surface or bulk character of the initial and final states are determined. Changes in the relative energies and occupations of electronic states due to surface charge redistribution are a possible mechanism for the interband contribution to the reflectance anisotropy. 2003 Elsevier Ltd. All rights reserved. Keywords: Metallic surfaces; Ab initio FP-LAPW method; Optical anisotropy; Al(1 1 0); Bulk termination effect; Charge transfer; Wave function symmetry
1. Introduction Reflectance anisotropy spectroscopy (RAS) has been applied to investigate a wide range of processes at clean and adsorbate-covered surfaces of cubic semiconductors [1]. The difference between the measured reflectivity at normal incidence, when light is polarized along the two principal axes of a surface, can be used as a non-invasive probe of the electronic structure of the surface, as was first proposed by Aspnes [2].
*
Corresponding author. Tel.: +1-416-978-2943; fax: +1-416-978-5848. E-mail address: [email protected] (M.J.G. Lee).
0079-6816/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.progsurf.2003.08.022
284
A.I. Shkrebtii, M.J.G. Lee / Progress in Surface Science 74 (2003) 283–291
Cubic semiconductors are optically isotropic in the bulk, and the observed reflectance anisotropy is due to modified optical transitions in the vicinity of the surface. Anisotropy may be induced by reduced symmetry, the presence of surface states, top-atom dynamics, surface reconstruction, or the adsorption of foreign atoms. When interpreted on the basis of microscopic theory, RAS is expected to be a useful probe of various surface properties and processes (see, e.g., [3–5] and references therein). The first reflectance anisotropy (RA) measurement at a metallic surface, Ag(1 1 0), was reported a decade ago [6], and detailed experimental studies of the RA at clean and adsorbate-covered (1 1 0) surfaces of copper, silver, and tungsten have recently been carried out [7–10]. Microscopic theory was applied in [11] to investigate the RA of Cu(1 1 0) and Ag (1 1 0). Recently, we and our co-workers reported a joint experimental and theoretical study of the RA of Al(1 1 0) [12]. The results of an ab initio self-consistent full-potential linear augmented plane wave (FP-LAPW) calculation of the linear dielectric function of clean Al(1 1 0) were shown to be in good agreement with the experimentally-measured RA of clean, oxidized, and Ni covered Al(1 1 0) surfaces. In the present paper, we address some important features of the experimental RA that were not discussed in [12]. These include the anisotropy of the intraband transition (Drude) term that dominates the optical response below about 1 eV, the charge shifts that accompany the formation of an Al(1 1 0) surface, and the polarization dependence of the interband transitions that are responsible for the RA. This paper is organized as follows. In Section 2, we describe briefly the measurement and calculation of the RA at an Al(1 1 0)–vacuum interface, and we outline the basis of our ab initio calculation. In Section 3.1, we present evidence for anisotropy in the Drude contribution, which we explain in terms of charge transfer to the [1 1 0] oriented chains of atoms at the Al(1 1 0) surface. In Section 3.2, we identify the principal interband transitions that are responsible for the RA in the interband regime. We describe the symmetries and the surface or bulk character of the initial and final states, and we relate the RA in the interband regime to surface-induced shifts of the energies of the electronic states of the metal.
2. Reflectance anisotropy of Al(1 1 0): experiment and theory The complex RA at a (1 1 0) surface is defined as the difference Dr between the reflectivities measured along the two principal axes, divided by the average reflectivity r of the surface [7–9]: Dr r r0 0 1 ¼ 2 110 ; r r1 1 0 þ r0 0 1
ð1Þ
where r1 1 0 and r0 0 1 are the complex reflectivities at normal incidence with light polarized parallel to the short [1 1 0] and the long [0 0 1] axis of the surface unit cell respectively, as shown in Fig. 1. Usually, both measurements and calculations determine the real part of the reflectance anisotropy, and the imaginary part is deduced
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Fig. 1. Perspective view of top four layers of clean Al(1 1 0) (1 · 1) surface. Primitive unit cell is marked by dashed lines. Arrows show principal crystallographic axes: longer edge of unit cell is along [0 0 1], and shorter edge is along [1 1 0]. Chains of atoms are oriented parallel to [1 1 0].
by Kramers–Kronig transformation. The experimental details have been reported more fully elsewhere [7–9,12]. The RA Dr=r can be expressed in terms of the surface and bulk dielectric functions [13]. The real part of the reflectance anisotropy is given by Dr 8pd e e0 0 1 Re Im 1 1 0 ¼ ; ð2Þ r k eb 1 where d is the thickness of the overlayer, k is the free-space wavelength of the light, e1 1 0 and e0 0 1 are the tensor components of the surface dielectric function (polarizability) calculated for light polarized parallel to the [1 1 0] and [0 0 1] directions, respectively (Fig. 1), and eb is the dielectric function of the bulk solid. Here, we summarize a few details of our computational method; for more complete details the reader is referred to [12,14]. The electronic structures and the wave functions of bulk aluminium and of the Al(1 1 0)–vacuum interface were calculated by solving the semi-relativistic Pauli equation by means of the FP-LAPW method using the WIEN2K code [15]. Exchange and correlation were treated in the generalized gradient approximation (GGA) [16]. The spin–orbit interaction, which is expected to be negligible for the valence electrons of aluminium, was not included in the calculation. No final state energy (scissors operator) shifts were applied. The calculated band structure and the energies of the interband peaks of the dielectric function of bulk Al are in very good agreement both with the results of pseudopotential plane wave calculations [17] and with current experimental data [18]. The crystal structure of Al is face-centred cubic. The Al–vacuum interface was represented by a supercell containing a slab consisting of 13 layers (six unit cells) of aluminium atoms aligned parallel to (1 1 0), surrounded by an empty region of equal volume to represent the vacuum. It was shown in [11] that an even smaller number of
286
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layers and a narrower vacuum region are sufficient to achieve a well-converged optical response. A damped oscillatory surface relaxation was assumed, represented by a contraction by 8.5% in the top Al(1 1 0) layer, an expansion by 4.8% in the second layer, and a contraction by 3.9% in the third layer [19].
3. Results and discussion In a recent paper, the RA of clean Al(1 1 0) was investigated both experimentally and theoretically as described above [12]. To emphasize the main result, and to point out some remaining discrepancies between experiment and theory, the measured real RA (solid line) and calculated interband (dashed line) real RA of Al(1 1 0) in the infrared and visible are compared in Fig. 2. 3.1. Anisotropy of Drude contribution There are two main contributions to the optical response of a metal: the intraband (Drude) term dominates below about 1 eV, and the term due to interband transitions dominates at higher energies. We begin by discussing the discrepancy between theory and experiment in the Drude regime. While the energy of the experimental RA peak is about 1.5 eV, irrespective of surface treatment, the experimental RA minimum occurs in the range 0.7–1.1 eV depending on the treatment of the surface [12], as shown in Fig. 2. Scaling the calculated RA curve to model the possible effect of surface defects is inconsistent with the observed behaviour of the RA minimum. It is commonly assumed that the Drude contribution to the optical response is isotropic. However, as seen from Fig. 2, the calculated interband contribution in the Drude regime is strong and positive, while the experimental RA is weak and nega-
Fig. 2. Experimental (solid line) and calculated interband (dashed line) real part of reflectance anisotropy of clean Al(1 1 0) surface.
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tive. The positive sign of the calculated interband contribution follows directly from the fact that, as discussed below, our slab calculation yields no electronic states of p1 1 0 symmetry that are less than 5 eV above the Fermi level. While a small shift, due to strain in the windows of the RAS equipment, may contribute to the negative RA observed in the Drude regime, the large remaining discrepancy between interband calculation and experiment is a strong indication of anisotropy in the Drude contribution. Although it is possible in principle to carry out a microscopic calculation of the anisotropy of the Drude contribution, it is not sufficient to know how the electron charge density is modified at the surface. One needs, in addition, a model for the frequency dependent microscopic conductivity of electrons localized at the surface. Instead, we consider a simple intuitive picture of the origin of the anisotropy. Fig. 3 shows the difference between the charge density in the surface layer of the relaxed Al(1 1 0) slab and that in the (1 1 0) plane of bulk Al, both calculated by the FP-LAPW method. It represents the shift of charge density that accompanies the creation of an Al(1 1 0) surface. The upper part of Fig. 3 is a grid plot of the charge shift, while a contour plot, projected onto the (1 1 0) unit cell, is shown at the lower part of Fig. 3. The atoms of the surface layer are located at the corners of the plot. An additional calculation showed that the charge density distribution in the central plane is essentially the same as that in bulk Al, which confirms that a 13 layer slab is sufficient to calculate the charge transfer at the surface. It follows that the creation of
Fig. 3. Difference of charge density between relaxed Al(1 1 0) layer and (1 1 0) plane of bulk Al. Grid plot of charge density difference is shown above, and contour plot projected onto (1 1 0) unit cell is shown below. Surface atoms are located at corners of plot.
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an Al(1 1 0) surface results in a transfer of charge to the [1 1 0] oriented chains of atoms at the surface. The charge transfer is expected to induce an anisotropic surface conductivity, enhancing the free electron (Drude) absorption for incident light polarized parallel to [1 1 0], and resulting in a negative Drude contribution to the real part of the RA of Al(1 1 0). The charge transfer can be understood as follows. While the creation of a (1 1 0) surface on the fcc lattice of bulk Al involves the removal of a half-space of atoms, the resulting charge transfer is dominated by the effect of removing the first overlayer, which is centered with respect to the atoms of the surface layer as shown Fig. 1. Because the centered atom in the first overlayer is closer to the longer [0 0 1] edge than to the shorter [1 1 0] edge, its removal brings about a greater increase in the potential along the longer edge of the surface cell than along the shorter edge, resulting in a transfer of charge from the longer edge to the shorter edge.
3.2. Anisotropy of interband contribution The dielectric function of bulk Al as calculated by the FP-LAPW method is reported elsewhere [12]. The result is in good agreement with the experimental data from Palik [18], demonstrating that density functional theory is capable of describing accurately the linear optical properties of bulk Al. Our calculation predicts a strong optical absorption peak at 1.5 eV, due to direct (vertical) transitions between occupied and unoccupied electronic states, whose k vectors lie close to the symmetry point K along the CK direction of the bulk Brillouin zone. Fig. 4 compares the band-resolved contribution of interband transitions to the imaginary (absorptive) part of the polarizability of an Al(1 1 0) slab for light polarized (a) along [0 0 1] (the longer side of the unit cell) and (b) along [1 1 0] (the shorter side of the unit cell). There are several strong peaks (Fig. 4a, A–C) in the energy range 1.4–1.8 eV for light polarized along [0 0 1], but none for light polarized along [1 1 0]. This indicates that in the interband regime the absorption of light is greater for light polarized along [0 0 1], which is consistent with the experimental observation that the real part of the RA is positive [12]. Our FP-LAPW calculation for an Al(1 1 0) slab shows that absorption peaks A and C, at 1.49 and 1.73 eV, respectively, are due to transitions from a p0 0 1 -like initial state band to an s-like final state band, while peak B at 1.62 eV is due to transitions from a (different) p0 0 1 -like initial state band to an s-like final-state band. That the initial (occupied) states are predominantly p0 0 1 -like and the final (unoccupied) states are predominantly s-like is consistent with the requirement that in electric dipole transitions the initial and final states be of opposite parity. Such transitions cannot be induced by light polarized in the [1 1 0] direction. Our FP-LAPW calculation for a 13 layer Al(1 1 0) slab shows that the lowest p1 1 0 -like valence state is more than 5 eV above the Fermi energy. It follows that there are no p1 1 0 -like states in the right energy range to act as initial or final states for electric dipole transitions with light polarized in the [1 1 0] direction. Therefore, after summing over all interband transitions and Lorenzian broadening is taken into account, our calculation yields a
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Fig. 4. Band-resolved plot of interband contribution to ImðeÞ derived from slab calculation of dielectric function of Al(1 1 0), for light polarized in [0 0 1] (upper) and [1 1 0] (lower) directions. Peaks A, B, and C, which make dominant contribution to calculated reflectance anisotropy, are discussed in text. Different lines denote contributions of different pairs of initial and final states.
strong negative peak in Im(e1 1 0 e0 0 1 ) that is consistent with the positive peak in the experimental RA. While the simple intuitive picture presented in Section 3.1, based on charge transfer to chains of surface atoms oriented along [1 1 0], provides a natural explanation for the RA observed in the Drude regime, it does not account directly for the RA in the interband regime. However, the enhanced charge density along [1 1 0] is expected to increase the energies of p1 1 0 -like states relative to p0 0 1 -like states. If the effect is to depopulate all of the p1 1 0 -like states by moving them from below to above the Fermi level, the result will be a strongly anisotropic polarization dependence of the matrix elements for interband transitions that is consistent with the experimental RA of Al(1 1 0). Our FP-LAPW calculations yield no true surface states (states for which the charge density is localized at the surface) at the Al(1 1 0) surface, although some surface resonances (states with enhanced charge density at the surface) are found. However, the initial and final states of the transitions that account for the interband contribution to the RA are all of intermediate or bulk character (i.e. they have a maximum charge density in the inner layers of the crystal and a reduced charge density at the surface). This is consistent with the experimental fact that the strongest feature in the RA spectrum of a clean Al(1 1 0) surface, the absorption peak at 1.5 eV
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for light polarized in the [0 0 1] direction, is essentially unchanged both in energy and in line shape, when the surface is oxidized, and also when it is covered by an overlayer of Ni. The experimental observations are evidence that surface states and surface resonances, as well as other surface features, such as top-atom relaxation and bonding to adsorbates, play little or no part in the RA of Al(1 1 0).
4. Conclusions The RA of Al(1 1 0) is caused by changes in the electronic states of the metal induced by the surface, which are known generically as bulk termination effects and are widely applied in surface optics [3–5,11,12]. A comparison between the calculated slab polarizability of Al(1 1 0) and the experimental data yields evidence for significant anisotropy in the Drude contribution to the RA. The increased absorption of light polarized in the [1 1 0] direction is interpreted as being due to anisotropic surface conductivity resulting from the transfer of charge to [1 1 0] oriented chains of atoms at the surface. In the interband regime, the anisotropy of the dielectric function deduced from a first-principles FP-LAPW calculation for a clean Al(1 1 0) slab is consistent with the experimentally observed RA. The strong polarization dependence is attributed to the absence of p1 1 0 -like electronic states with energies less than 5 eV above the Fermi level. The transfer of charge to [1 1 0] oriented chains of atoms at the surface shifts the relative energies, and hence the populations, of electronic states of p0 0 1 and p1 1 0 symmetry, resulting in polarization dependent matrix elements that may be responsible for bulk termination effects in interband transitions. The initial and final states of the optical transitions that make the dominant contribution to the RA in the interband regime are intermediate or bulk-like. This accounts for the experimental finding that the main peak in the RA spectrum is little affected in energy or line shape by oxidation or by the presence of an overlayer of Ni.
Acknowledgements The authors thank W. Richter, Th. Herrmann, N. Esser, R. Del Sole and O. Pulci for useful discussions. This work was supported in part by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.
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