-High-Resolution Angle-Resolved Photoemission Study of the Al(100) Single Crystal

The free-electron-like surface-derived electronic state in Al(100) has been examined in detail by high-resolution angle-resolved photoemission spectroscopy (ARPES). We observed a kink structure in the energy band-dispersion at the energy of ∼ − 40 meV, which is derived from the electron-phonon interaction. Based on the quantitative analyses of the ARPES line shapes, we have obtained the imaginary and real parts of the self-energy. The electron-phonon coupling parameter has been evaluated to be λ el − ph = 0 . 67, which is much larger than the electron-electron coupling parameter λ el − el = 0 . 06. [DOI: 10.1380/ejssnt.2009.57]


I. INTRODUCTION
Recently, energy and momentum resolutions of angleresolved photoemission spectroscopy (ARPES) have been improved drastically. Now one can examine fine electronic structures of solids that are directly related to the physical properties of solids. Using photon energies in the vacuum ultraviolet and soft X-ray region, ARPES spectra are sensitive to the electronic states at surface. Based on the quantitative analyses of high-resolution ARPES spectral features, one can evaluate magnitudes of the electronphonon and electron-electron interactions at any points on the Fermi surfaces.
Aluminum is the textbook example of a trivalent nearly-free electron metal and a weak coupling superconductor. It has been well known that there exists a surfacederived state centered at theΓ point of the surface Brillouin zone (SBZ) in the (100) surface [1][2][3][4]. However, there has been no report on the Fermi surface mapping nor a detailed examination of the many-body interactions on the quasi-particles so far. We believe detailed quantitative examinations of the surface state in Al(100) will contribute to the fundamental understanding of the twodimensional Fermi liquid.
In this study, we have done high-resolution ARPES of Al(100) to clarify the surface-derived Fermi surface, and to evaluate magnitudes of the many-body interactions on the quasi-particles near the Fermi level (E F ). We have found a kink structure in the energy band dispersion, and evaluated the coupling parameters of the electron-phonon and electron-electron interactions. * This paper was presented at International Symposium on Surface Science and Nanotechnology (ISSS-5), Waseda University, Japan, 9-13 November, 2008. † Corresponding author: jiangjianll@hiroshima-u.ac.jp

II. EXPERIMENTAL
ARPES experiments were performed on the linear undulator beamline (BL-1) of a compact electron-storage ring (HiSOR) at Hiroshima University [6].
Highresolution ARPES measurements were carried out using an angular mode of the hemispherical electron-energy analyzer (ESCA200, GAMMADATA-SCIENTA). We set the total energy resolution at ∆E = 15 meV for highresolution measurements at hν = 43 eV, and 150 meV for the Fermi surface mapping at hν = 163 and 167 eV. The angular resolution was ∆θ = 0.3 • , yielding momentum resolution of ∆k = 0.017Å −1 for hν = 43 eV and ∆k = 0.034Å −1 for hν = 167 eV. The clean surface of Al(100) single crystal (purity 99.9999%) was obtained by repeated cycles of Ar + sputtering (5 keV) over 10 hours to remove oxide layers at surface, and annealing at 400 • C for 30 minutes to minimize surface roughness introduced by Ar + sputtering. The amount of impurities such as C, O, and S on the surface was evaluated below the detection limit of Auger electron spectroscopy. The sample was mounted on the liquid-He-flow-type 5-axis goniometer (i-GONIO LT, R-DEC Co.). By changing tilt and polar angles of the goniometer, we could perform the two dimensional Fermi surface mapping in k-space. The sample temperature was set at T = 30 K for high-resolution measurements, and T = 300 K for the Fermi surface mapping. The pressure of the main chamber was 1 × 10 −8 Pa. by dots in Fig. 1. The surface-derived Fermi surface in the first SBZ cannot be clearly observed at this photon energy probably due to weak matrix element [1]. Figure 2 exhibits the energy-band dispersion of the Al(100) surface state along theΓ-M direction taken at hν = 167 eV and 300 K. One can clearly see a freeelectron-like energy-band dispersion. By fitting with a parabolic function ε k = −ω 0 +(ω 0 /k 2 F )k 2 (blue dashed line in Fig. 2), we determined the Fermi energy (ω 0 ) and Fermi wave vector (k F ). We have determined the Fermi energy of the surface state as ω 0 = 2.63 eV, which coincides well with the calculated value of 2.62 eV [3]. The Fermi wave vector was evaluated to be k F = 0.94 ± 0.005Å −1 . By using the area of the Fermi surface S F = πk 2 F = 2.78Å −2 , the carrier density (n) of the surface state is calculated to be n = 2S F /(2π) 2 The Fermi wave vector k F = 0.94Å −1 was also obtained from the radius of the circular Fermi surface taken at hν = 163 eV in Fig. 1. We have changed photon energy from hν = 163 eV up to 185 eV with a step of ∆hν = 2 eV, and confirmed that the size of the Fermi wave vector (k F ) and the Fermi energy (ω 0 ) of the surface-derived state are independent of incident photon energy. It is a direct evidence that the surface state is localized at surface.
On the basis of the formula m * = 2 [d 2 ε(k)/dk 2 ] −1 , the effective electron mass is evaluated as m * = 1.27 m e , here m e stands for the free-electron mass. The value of m * obtained in the present study is larger than that obtained previously m * = 1.18m e [1]. As shown below, the effective mass is further enhanced due to the electron-phonon interaction near E F . Figure 3(a) shows a high-resolution ARPES image plot of the surface-derived state near the Fermi level taken at hν = 43 eV and at T = 30 K. It has a sharp spectral feature, which is suitable for the detailed line shape analyses to elucidate many-body interactions. In order to quantitatively analyze the spectral shape, we used momentum distribution curves (MDCs). We have fitted a MDC with a Lorentzian on a linear background, and obtained the peak position and linewidth (δk). The blue points in Fig. 3(a) indicate thus evaluated peak positions. Figure 3(c) shows the area surrounded by the blue square in Fig. 3(a). One can clearly recognize a kink structure at ∼ −40 meV below E F . Since the magnitude of the energy of the kink coincides well with the energy scale of the bulk Debye temperature of Al (Θ D = 426 K, k B Θ D = 37 meV), it is reasonable to assume that the structure is derived from the electron-phonon interaction [7,8].
In addition, there is another evidence for the origin of the kink structure if we look at the energy distribution curves (EDCs) near E F . Figure 3(b) shows three EDCs (1, 2, 3) obtained from cuts along broken lines in Fig. 3(a). LaShell and Jensen studied the Be(0001) surface state and determined the electron self-energy [5]. They indicated that the electron-phonon interaction produces a hump structure to the main peak in the ARPES spectra. Figure 3(b) shows that the spectral width becomes narrower as the peak approaches E F , and there appears the hump structure besides the main peak near E F . Observed spectral features are similar to those for Be(0001) surface state, indicating that the kink structure is derived from the electron-phonon interaction.
Next we examine the strength of many-body interactions in quantitative way. The ARPES spectral features are given by the single-particle spectral function A(k, ω), which is related to the imaginary part of the singleparticle Green's function, G(k, ω): where ε 0 k represents the energy of non-interacting band. In the present analyses, we have assumed that ε 0 k is linear near E F as shown by a solid line in Fig. 3(c). ReΣ(k, ω) and ImΣ(k, ω) are the real and imaginary parts of the self-energy (Σ) where direct information on the manybody interactions are included. ReΣ gives an energy shift from non-interaction band. ImΣ is related to the spectral width, Γ = |2ImΣ(k, ω)| [7,8]. In the present analyses, we used the MDC width (δ k ) to estimate the ImΣ(k, ω), namely, |2ImΣ(k, ω)| = δE = (dE/dk)δk , where dE/dk is the gradient of the energy band.
In the case that electron-scattering processes such as the electron-phonon, electron-electron, and electronimpurity interactions can be regarded as independent, the lifetime broadening (Γ = |2ImΣ(k, ω)|) of the quasiparticles is given by the sum of each contribution: Γ = Γ el−ph + Γ el−el + Γ 0 .
Figures 4(a) and 4(b) respectively show obtained Γ = |2ImΣ(k, ω)| and ReΣ(k, ω). Solid and dashed lines in Fig. 4(a) exhibit theoretical Γ el−ph and Γ el−el , respectively. Recently, Sklyadneva et al. have reported an ab initio study of the electron-phonon interaction in the surface electronic states of Al(100), on the basis of the density-functional theory using a linear response approach in the plane-wave pseudopotential representation [4]. In the calculation of Γ el−ph , we used theoretical Eliashberg function (α 2 F ) [4].
One can see experimental Γ = |2ImΣ(k, ω)| is explained well by the theoretical curves Γ el−ph and Γ el−el . Solid line in Fig. 4(b) indicates the theoretical ReΣ el−ph , which also agrees well with the experimental one. These results confirm that the kink structure originates from the electron-phonon interaction.
One can evaluate a dimensionless coupling parameter of the electron-phonon interaction as follows. Using the observed gradient of ReΣ el−ph at E F (dashed line in Fig. 4(b)), the parameter is evaluated to be λ el−ph = 0.67. We should note that the electron-phonon coupling parameter of the Al surface state is comparable with that for Be, λ = 0.7 [5]. The calculated electron-phonon coupling parameter is λ el−ph = 0.51 at theΓ point (at ω 0 = 2.62 eV), and increases with energy [4]. Although the coupling parameter at E F was not provided, calculated ReΣ el−ph (solid line in Fig. 4(b)) using theoretical α 2 F coincides well with the experimental one near E F . It clearly indicates experimental coupling parameter is in good agreement with the theoretical value.

IV. CONCLUSION
We have performed a high-resolution ARPES study of the surface-derived state in Al(100). We have observed a circular surface-derived Fermi surface centered at theΓ point. The area of the Fermi surface (S F = 2.78Å −2 ), the Fermi energy (ω 0 = 2.63 eV), the Fermi wave vector (k F = 0.94Å −1 ) and the carrier density (n = 1.4 × 10 15 cm −2 ) http://www.sssj.org/ejssnt (J-Stage: http://www.jstage.jst.go.jp/browse/ejssnt/) have been determined. We have observed a clear kink structure in the energy-band dispersion at ∼ −40 meV, which corresponds well to the energy scale of the bulk Debye temperature k B Θ D = 37 meV. We have evaluated the electron-phonon and electron-electron coupling parameters as λ el−ph = 0.67 and λ el−el = 0.06, respectively. We have found that λ el−ph is much stronger than λ el−el .