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Quantum-size effects in thin solid xenon films
Journal of Electron Spectroscopy and Related Phenomena 98–99 (1999) 121–131
Bagus Special Issue
Quantum-size effects in thin solid xenon films M. Gru¨ne a, T. Pelzer a, K. Wandelt a,*, I.T. Steinberger b a
Institut fu¨r Physikalische und Theoretische Chemie der Universita¨t Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany b Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Received 17 October 1997; accepted 7 April 1998
Abstract Ultraviolet photoelectron spectra of solid xenon films on top of Ru(0001) and Cu(100) substrates revealed peak patterns due to confinement of electrons in quantum wells formed by the metal substrate–Xe insulator–vacuum system. The data (along with previous ones for Pt(100) substrates) were analyzed, assuming simple tight-binding dispersion in the direction GL, perpendicular to the film, expressly taking into account the polarization of the first monolayer adjacent to the metal. The good fit obtained implied that the sets of experiments sampled the 5p3=2 mj 1=2 and 5p1=2 valence bands at several points of the actual dispersion curves, furnishing an experimental determination of these curves along the GL direction. The absolute value of the effective mass mⴱ for the electrons at the top of the bands (in terms of the free electron mass) was found to be near unity for both bands, depending somewhat on the substrate. These substrate-dependent variations are attributed to the effect of the substrate on the potential and the wave function in the Xe films, combined with imperfections in the texture and structure of the films. 䉷 1999 Elsevier Science B.V. All rights reserved. Keywords: Quantum size effects; Ultraviolet; Ultraviolet photoelectron spectroscopy; Xenon; Insulating films
1. Introduction Quantum wells in semiconductors have been intensively studied for the past twenty years or so. The study was made possible by virtue of the development of sophisticated modern technology (chiefly, molecular beam epitaxy) for the preparation of materials and structures by layer-by-layer growth. Beyond the basic interest from both the physics and materials science point of view, these studies are also of foremost technological importance, enabling the development of new electronic and opto-electronic devices. In the semiconductor case, the quantum well usually comprises a slice of semiconductor of type * Corresponding author. Tel.: ⫹ 49-228-732253/2654; Fax: ⫹ 49-228-732551; e-mail: [email protected]
A sandwiched between two semiconductors of type B, with the band gap of A being smaller than that of B. The width of the A-type material is, in most cases, of the order of 10 monolayers. In principle, one might also envisage other, asymmetrical quantum well arrangements like A–B–vacuum or metal–semiconductor–vacuum. The latter system would normally also involve the formation of an accumulation or exhaustion layer of one of the carriers and, therefore, an electric potential which varies continuously along the direction perpendicular to the metal/semiconductor interface. If the semiconductor is replaced by a highly insulating substance, like a rare gas solid, then the low concentration of free carriers prevents band bending effects. Moreover, it is relatively easy to grow rare-gas adsorbates epitaxially, layer by layer, on a number of single crystal surfaces, using
0368-2048/99/$ - see front matter 䉷 1999 Elsevier Science B.V. All rights reserved. PII: S0368-204 8(98)00281-3
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well-established surface science control techniques. Thus the metal–rare gas solid–vacuum quantum well is, in principle, similar to the metal–semiconductor–vacuum quantum well, but it is much simpler to prepare and can be regarded as the simplest prototype of the latter. Low energy (0–10 eV) electron transmission spectra in thin films (up to 10 monolayers) of argon and methane showed a series of peaks [1]. The number of the peaks increased with the number of layers. Analysis of the data showed that the electrons traverse the samples ballistically and the peak positions correspond to constructive interference between the electrons reaching the metal substrate directly and those that had undergone a reflection at both interfaces (metal–adsorbed film and adsorbed-film–vacuum). This quantumsize resonance effect enabled the determination of the dispersion of the respective conduction bands. McNeill et al. recently reported a two-photon angular resolved photoemission study of quantum well states in Xe and Kr films on top of Ag(111) [2]. One of the important results presented by McNeill et al. was the very precise measurement of the Xe bulk conduction band [2]. Experiments referring to valence bands were performed using xenon multilayers adsorbed on top of atomically clean Pt(100)hex [3,4], Ag(111) or Cu(100) [4,5] surfaces. The experiments comprised ultraviolet photoelectron spectroscopy (UPS) of the 5p1=2 and 5p3=2 levels of the adsorbate. In all cases, there was an appreciable shift of the peaks between the first and second layer. Moreover, the intensity of the peak of the first layer decreased in step with the growth of the second layer: when the second layer was completed, the peak corresponding to the first one disappeared completely. The evolution of the third layer quenched the peak of the second layer and gave rise to two other peaks, symmetrically positioned with respect to the quenched one. In a sample consisting of N monolayers N ⫺ 1 peaks were considered; these were arranged with mirror symmetry with respect to a fixed photon energy. This energy was that of the central peak in samples comprising an even number of layers. Evolution of the pattern corresponding to N monolayers always involved quenching of all patterns that appeared for thinner samples. The total width of the pattern increased with thickness: for
N 9 (observed with Pt(100)hex as substrate) the total width was about 1.2 eV [3]. The results clearly indicate that instead of assigning single peaks to individual layers, one has to consider peak patterns and assign each pattern to a multilayer sample as a whole. The patterns have been successfully interpreted [3] on the basis of a quantum-well model, assuming a well of infinite wall height and simple tight-binding dispersion of the valence bands. However, the model explicitly postulates that a sample Nd thick (d being the thickness of one monolayer) should accommodate N ⫺ 1 standing waves of wave numbers k with k np /Nd, n 1,2,…,N ⫺ 1, implying that there are no waves of wavelengths equal to the width of just two monolayers. A qualitative argument supporting this postulate referred to the first, polarized layer adjoining the metal surface as not being part of the Xe film under consideration. In this paper, we present further experimental results on xenon multilayers, namely on top of Ru(0001) and on top of Cu(100). The valence-band quantum-size effects in xenon on top of various substrates are compared, taking into account work function differences. A calculation shows that a simple tight-binding dispersion formula can be fitted very well to the experimental data. The calculation takes into account the polarization of the first layer as a step in a compound quantum well and involves walls of finite heights.
2. Experimental The experiments were carried out in an ultra-high vacuum chamber, with a base pressure lower than 1 × 10 ⫺8 Pa. The substrate could be cooled down by a closed-loop He refrigerator and heated up to 1000 K by thermal radiation from a filament (for Cu), or up to 1600 K by electron bombardment (for Ru); reaching the lowest temperature was found to be essential for the layer-wise growth of the Xe multilayers, especially in the case of the Cu(100) substrate. The chamber was equipped with standard surface analytical tools, such as LEED, AES, TDS, UPS, a Kelvin probe and an ion gun for Ar ⫹ sputtering. In all experiments, the angle of incidence (with respect to the surface normal) of the UV light was kept fixed at about 60⬚; the electrons were collected in a direction
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Fig. 1. Ultraviolet photoelectron spectra of xenon films on a Ru(0001) substrate.
perpendicular to the plane of the sample, with an angular resolution of 2⬚. Assuming that the Xe layers grow regularly with the f.c.c. structure, (this point will be further elucidated in Section 4) with (111) layers parallel to the substrate, we may state that the direction GL in the k-space was sampled in each case. Most experiments were performed by means of a He I photon source; the photoelectron energies were determined by means of a hemispherical energy analyzer. With the 5 eV pass energy employed in most experiments, the energy resolution was about 70 meV. In some experiments (especially for high Xe coverages), the pass energy was set to 2 eV in order to improve the resolution at the expense of the signal-to-noise ratio. The preparation of Ru started with annealing in oxygen at 1150 K until no carbon could be seen by AES. The respective contributions of C and Ru to the peak at 273 eV were estimated by the maximum-tobaseline/minimum-to-baseline ratio. To remove residual C from the surface, a series of O2-adsorption and
123
CO-desorption cycles followed, each with a saturation dose of O2, until a CO recombination TDS-peak could no longer be detected. Eventual residual oxygen was then removed by sputtering with Ar ⫹ ions of 1 kV at room temperature for 10 min, with subsequent annealing at 1550 K for several minutes. After this procedure, AES showed neither oxygen nor carbon, and the LEED picture was that of a clear and wellordered Ru(0001) surface. Cu was sputtered at 900 K for 30 min. At this temperature, impurities segregate to the surface. Sputtering was continued while the sample cooled down to room temperature, followed by a flash to 800 K. After this procedure, AES showed no oxygen and the LEED pattern was that of a clean Cu(100) surface. The quality of each substrate was also controlled by PAX [6]. The Xe 5p1=2 peak showed no shift (apart from the known commensurate–incommensurate transition on Ru) and a FWHM of only 200 to 280 meV. Commercially available Xe of 99.99% purity (Messer-Griesheim) was attached via a leak valve to the chamber. The Xe doses were measured with an ion gauge without further correction and are given in Langmuir (L). The dose was calibrated in terms of the number of monolayers deposited by establishing the dose necessary to complete one monolayer (ML), using the PAX method. In the course of the measurements it turned out that integral multiples of this dose corresponded to the maximum height of a series of UPS peaks characteristic to the same integral number of monolayers, indicating that the sticking coefficient (presumably very near to unity) did not vary with layer thickness. In the case of the Ru(0001) substrate, we checked the calibration also by Xe TDS, which gives wellresolved peaks for the first two Xe layers. After cleaning and annealing the substrate as described above, it was cooled down to the minimum temperature that could be reached (typically 44 K), flashed up to a few hundred K in order to remove impurities eventually adsorbed during the long first cooling down of the sample, and cooled down quickly again for the start of xenon deposition. We deposited Xe at first in very small steps (2–3 L) in order to determine the dose for one monolayer and, subsequently, in steps of 0.2 or 0.25 monolayers for the thicker samples. In the case of the Cu(100) substrate,
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one series of measurements was performed as described above and a second series was performed starting with the deposition of three monolayers—this made it possible to extend the scope of measurements to appreciably thicker films (up to 7 ML) than previously reported [4,5]. The work functions of the pure metals were obtained from the width of He–I excited UPS spectra, measured from turning point to turning point. The work function changes induced by one monolayer of xenon were determined both by this UPS method and by a Kelvin probe vibrating parallel to the surface [7,8].
3. Results In Fig. 1, a set of spectra selected from the series with the Ru(0001) substrate is presented, starting from a complete monolayer, with dose increments of about 0.5 ML. The peaks observed in each spectrum can be divided into two groups, on the two sides of the broken vertical line at 6.34 eV. The group on the left of the line is associated with the valence bands that evolved from the 5p3=2 atomic states of Xe and the one on the right is associated with the 5p1=2 valence band. Arrows point to the approximate positions of the principal peaks. In the higher-energy series one sees that for the 1 monolayer film (19 L) there is one 5p1=2 peak at about 6.48 eV. With increasing coverage, this peak decreases in amplitude and is totally absent in films made up of more than two monolayers (40 L and above, in Fig. 1). The 40 L film exhibits two main peaks marked by arrows and the 60 L peak three, the highest-energy peak appearing only as a shoulder. It follows that for these samples the number of peaks is equal to the number of monolayers. However, only three peaks appear in the 80 L film, and four in the 100 L one, seemingly violating this rule. Nevertheless, in these spectra the most prominent peak (at 7.51 eV and 7.56 eV, respectively) could totally obscure a weak and broad band under its high-energy flank. As most spectra were taken on samples that did not comprise an integral number of monolayers, some smaller peaks, associated with the beginnings of the next monolayer are also seen, but not marked, e.g., the 6.8 eV peak in the 40 L spectrum or the 7.1 eV peak in the 60 L and the 100 L spectra. The pattern is, on the whole, similar to that presented in the paper by
Schmitz-Hu¨bsch et al. [3] for Xe films on the Pt(100)hex substrate: distinct sets of peaks are characteristic to samples consisting of integral numbers of monolayers, the sets disappear upon completion of the next monolayer. However, in the paper quoted, only N ⫺ 1 peaks were assigned to N-monolayer samples: peaks analogous to the ones marked with down arrows (the ‘extra peaks’) in our Fig. 1 did appear there as well, but were not considered to belong to the main sets of peaks. The broken vertical line at 6.34 eV in Fig. 1 indicates not only the borderline between the two groups, but also the approximate position of a peak that appears in all spectra of films at least two monolayers thick, analogous to the peak ‘1’ in Fig. 5 by Paniago et al. [5]. The amplitude of the peak increases slowly and monotonically with the film thickness. Its fixed position and its presence in most spectra distinguish this peak from all other peaks. In Ref. [5] it has been rather tentatively assigned to a Xe surface state, but, as the authors point out, further study is needed for verification. The behaviour of the lower-energy groups of peaks in the spectra is similar to that of the higher-energy groups, but, in this case, the above-mentioned peak, at about 6.34 eV (marked with a broken line), seems to interfere with the observation of the ‘extra’ peak in the thicker samples. A similar set of spectra for the Cu(100) substrate is shown in Fig. 2. The spectra are taken from two different series of experiments: for the lower coverages, the dose was increased gradually, as in the Ru(0001) case, starting from 1 L. The thickest film of this series was made by admitting 91 L. It turned out that the quality of the spectra in the thickest films of the series was rather low: the peaks were illdefined and washed out. The deterioration was probably due to irregular growth conditions (see also Section 4). Therefore, as stated above, a new series of films was prepared, this time starting with a dose of 36 L. Inspection of the Xe/Cu(100) spectra reveals clear similarities with the Xe/Ru(0001) spectra shown in Fig. 1. The difference in exposure to reach saturation of one monolayer is due to slightly different substrate temperatures. The evolution of a given set of peaks with coverage, in parallel with the gradual disappearance of the set belonging to the sample with the last integral number of layers, is best seen by subtracting from each
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Fig. 3. UPS incremental difference spectra of xenon films on a Ru(0001) substrate.
Fig. 2. Ultraviolet photoelectron spectra of xenon films on a Cu(100) substrate.
spectrum another one corresponding to somewhat lower coverage. Examples of such difference spectra are shown in Fig. 3 for the Ru(0001) substrate, with increments of 4 L, and in Fig. 4 for Cu(100) with increments of either 3 or 3.25 L. One can observe the two groups of peaks (at higher and lower energies) in both graphs. Peak patterns have fixed positions until they disappear after completion of the next monolayer. The main features of the spectra stand out, indeed, they are clearer in the difference spectra representations than in the raw spectra of Figs. 1 and 2. The determination of the peak positions started by inspecting both the raw spectra and the difference spectra: only such peaks were considered that could be seen in the raw spectra as well. If neighbouring peaks were of approximately equal heights, their
position was ascertained from that difference spectrum where the pattern in question had the largest amplitudes. If, however, neighbouring peaks differed in height appreciably, a fit of 2–3 adjacent peaks in the original spectrum by Lorentzians was necessary. The accuracy of the determination of the peak positions was better for thinner samples and for larger peaks; it was typically about 10 meV for the strongest peaks in the thinnest films and deteriorated to about 100 meV for the weakest peaks in the thickest films. The number N of completed monolayers versus the peak positions is presented in Fig. 5. It includes the results for the Cu(100) and the Ru(0001) substrate shown above, as well as those for Pt(100)hex from Ref. [3], the latter in order to enable direct comparison with the other substrates and with the model to be presented below. For the Cu(100) substrate, the results reported for Xe films up to 4 monolayers [4,5], are in good accord with the present results. The pattern of peak positions is very similar for all three substrates except for horizontal displacements
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the positions of these peaks involves a large experimental error, since these peaks are broad and weak, close to the ‘constant energy’ peaks of the first set around 7.02 eV, marked by triangles. 4. Discussion
Fig. 4. UPS incremental difference spectra of xenon films on a Cu(100) substrate.
and differences in the total number of monolayers with resolvable data. Each pattern can be divided into two groups, as stated already with reference to Figs. 1–4. For N ⱖ 2, most of the pattern within a given group has an approximate mirror symmetry with respect to a central energy value that recurs in all samples of even-numbered layers [3,4]. There are two sets of peaks for each substrate which do not fit into the symmetric patterns. The members of one set are marked by open triangles—they lie at positions very near to the borderline between the two symmetric groups. As stated above, these peaks are irrelevant to the present study. The peaks of the other set (the ‘extra peaks’), marked by open circles, appear on the highenergy side of each group. There are three such ‘extra’ peaks both for Xe/Cu(100) and for Xe/Ru(0001) (at N 2 in the 5p3=2 group and at N 2 and 3 in the 5p1=2 group) and five for Xe/Pt(100)hex (at N 2, 3, 4, 6 and 7 in the 5p1=2 group). In the Cu(100) pattern the peaks at N 5 and 6 at 7.13 and 7.15 eV, respectively, seem to belong to the 5p1=2 series, but the determination of
The extensive results presented above on Xe films on top of Ru(0001) and Cu(100) substrates underline the main qualitative conclusions which had been reported recently about thin Xe films [3–5]: (a) the peak patterns observed in UPS are characteristic of each film as a whole and not of individual monolayers; (b) the peak patterns are formed as a result of the quantization of the energy of valence-band electrons, associated with the crystal momentum perpendicular to the film (quantum-size effect). It will now be shown that a simple model calculation, explicitly taking into account the polarization of the monolayer adjacent to the substrate, is in quantitative agreement with the results. The E(k) dispersion relations in the GL direction of the k-space are deduced from the fit of the model to the experimental data. A comparison of the results for the various substrates will conclude Section 4. 4.1. Calculation Ultraviolet photoelectron spectroscopy (UPS) of the 5p1=2 and 5p3=2 states of xenon adsorbates on a metal surface shows that the binding energy EBF of these states with respect to the Fermi level of the metal is dependent on the amount of adsorbed xenon. In particular, EBF is substantially smaller for submonolayer coverage than for xenon atoms residing on top of a completed xenon monolayer [6]. The change is typically of a few tenths of an eV, depending on the metal and its surface plane. Beyond the first monolayer, the changes are much smaller. This variation of the binding energy with the distance from the metal plane has been interpreted in two different ways: (a) as an initial-state effect, due to the change in the work function caused by the adsorbed xenon [9,10], and (b) as a final-state effect, due to the image force acting on the hole that had been created by the photoionization process [11]. In Fig. 6, highly schematic energy diagrams are presented for both models. In accord with the
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For both cases (a) and (b) VD ⫺ Emax E0V ⫺ a
1
where E0V is the appropriate ionization potential of a single atom and a is a relaxation energy due both to the presence of the adsorbate and the metal surface. Thus a may be influenced by the distance from the metal interface; we shall assume that a is constant for N ⬎ 1. The binding energy EBF , referred to the Fermi energy of the metal, will be, in model (a): EBF VD ⫹ VB ⫺ fM ⫺ E
2a
and in model (b): EBF VD ⫺ fM ⫺ E
Fig. 5. Number of layers contained in the film versus UPS peak positions (binding energies) for films grown on different substrates. Dots and triangles, experimental; vertical lines, calculated. See text.
2b
In the following calculation the crystal film on top of the metal surface is treated as a compound quantum well (Fig. 6). The wave functions and the dispersion are consistent with the tight-binding approximation [12] in the xenon crystal and the boundary conditions. Interactions between the valence bands are neglected and, instead of making detailed assumptions about the wave functions and the potential, the constants of the tight-binding model are regarded as parameters adjusted so as to fit the experimental data. In regions A and D we assume exponentially decaying plane waves (cf. Fig. 6). In region A (x ⬍ 0)
cA x A exp kA x
3
In region D (x ⬎ Nd) simplifying approximation employed, all the change of the potential is shown as occurring step-like between the first and second layer. It will be shown that for the main result (the dispersion) it is irrelevant whether the initial-state or the final-state model is adopted. In Fig. 6, E is the electron energy (constant in the whole structure), f M the work function of the pure substrate and VB the potential jump between the first and the second layer. In the initial-state model VB ⫺ Df, where Df is the work function change caused by one completed Xe monolayer. VD is a convenient reference level within the sample: VD ⫺ Emax (Emax being the upper limit of E as a function of kC, see below) is the energy difference between the vacuum level and the valence band probed.
cD x D exp ⫺kD x
4
In region C (d ⬍ x ⬍ Nd), according to the Bloch theorem
cC x C sin kC x ⫹ d2 f x
5
where f(x) is a periodic function with the periodicity of the lattice, representing (unspecified) Wannier functions. It follows from the periodicity of f(x) that f nd f d
n 0; …N
5a
With the further assumption that f(x) has mirror symmetry with respect to the atom centres df x 0 n 0; …N 5b dx xnd Eq. (5a) and Eq. (5b) are only approximately correct
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Fig. 6. Energy diagram for the model employed in the calculation (a) initial state scheme; (b) final state scheme (see text).
at the xenon–metal and xenon–vacuum interfaces (n 0 and n N). In region B, (0 ⬍ x ⬍ d) treated as a part of the xenon crystal, we have
electron mass. For model (a), V1 VB, for model (b), V1 0. In region C, according to the tight-binding model assumed,
cB x B sin kB x ⫹ d1 f x
E ⫺gC 1 ⫹ cos kC d
6
The wave functions are real; they and their first derivatives must be continuous at x 0, d and Nd. These conditions lead to the elimination of A,B,C,D and f(d) and we obtain: kB k ⫺ B tan N ⫺ 1kC d tan kB d ⫹ arctan kA kC ⫹ arctan
kC kD
7
In order to find the allowed energy levels and kvalues, the appropriate dispersion relations will be invoked. The height of the barrier that the valence electrons in the insulator ‘see’ at the interface with the metal should be strongly dependent on the band structure of both substances at the interface. Fortunately, it turns out that the results are almost insensitive to the value of this parameter as long as it is not too small. For convenience, we assume that the top of the barrier is at the Fermi energy of the metal, and that the electron effective mass in the metal equals the free electron mass. We have then in region A p 2m0 ⫺E ⫹ VD ⫹ V1 ⫺ f kA 8 ប É is Planck’s constant divided by 2p and m0 the free
9
2g C is the width of the appropriate valence band. In region B, the dispersion relation should be the same as in region C: E ⫺gB 1 ⫹ cos kB d ⫹ V 0
10
0
Here V accounts for the change in the reference level (cf. Fig. 6). Its value is fixed by the condition that the difference between the binding energy in the second (and higher) layers and that in the first one must be VB: V 0 VB ⫺ 2 gB ⫺ gC
11
We allowed in Eq. (11) gB 苷 gC , since the vicinity of the metal may strongly affect the value of g [12]. g B is a convenient fitting parameter. Finally in the vacuum (region D) p 2m0 ⫺E ⫹ VD kD 12 ប Eliminating k A, k B and k D from Eq. (7) by means of Eqs. (8)–(12), an equation in kC is obtained, with g B and g C as adjustable parameters. Since the solid-state quantum well consisted of N atomic layers, we sought N solutions (with the lowest kC values) of this equation. The binding energies EBF were then calculated from the set of kC values by means of Eq. (9) and Eq. (2a) (or Eq. (9) and Eq. (2b), see below) and compared with the experimental binding energies. The parameters g B, g C and VD were then varied
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Table 1 The work function f of the pure metal surface, its change Df due to one monolayer of adsorbed xenon, the fitting parameters g B, g C and VD (see text), the effective mass mⴱC at the top of the bands, the distance VD ⫺ Emax from the top of the band to the vacuum level and the parameter a . Units: eV, except for mⴱC , which is given in terms of the free electron mass.
f
Df
Band
Cu(100)
4.74
⫺ 0.33
Ru(0001)
5.52 a
⫺ 0.62 b
Pt(100)hex
5.72 c
⫺ 0.54 c
5p1/2 5p3/2 5p1/2 5p3/2 5p1/2 5p3/2
gB ⫺ 1.65 ⫺ 1.65 ⫺ 2.17 ⫺ 2.17 ⫺ 2.17 ⫺ 2.17
gC
mⴱC
VD
VD ⫺ Emax
a
⫺ 0.48 ⫺ 0.48 ⫺ 0.58 ⫺ 0.53 ⫺ 0.62 ⫺ 0.53
⫺ 1.24 ⫺ 1.24 ⫺ 1.04 ⫺ 1.13 ⫺ 0.96 ⫺ 1.13
12.62 11.17 12.60 11.16 12.48 10.99
11.66 10.21 11.45 10.10 11.23 9.93
1.77 1.92 1.98 2.03 2.20 2.20
Ref. [6]. According to Ref. [27] f 5.37 eV Ref. [8] c Ref. [7]. According to Ref. [28] f 5.84 eV a
b
until satisfactory agreement between theory and experiment (vertical lines and dots, respectively, in Fig. 5) was obtained. It turned out that the value of g B ensuring the agreement was sensitively dependent on VB (not used as a fitting parameter, known, however, only with an accuracy of about ^ 0.1 eV), but only slightly affected by the values of g C and VD. During the fitting procedure, the observed extra peaks (marked by down arrows in Figs. 1 and 2) were not taken into account. Even so, Fig. 5 shows that the solutions do apply, within the experimental error, for these peaks as well. Table 1 summarizes the values of the parameters VD, g B and g C used in the fitting procedure, along with f and Df , obtained from the literature. It also includes the effective mass m * at the top of the valence bands, calculated from the relationship
mⴱC ⬅
ប2 m0
1 d2 E=dk2
k0
ប2 1 m 0 gC d 2
13
The data in Table 1 are based on model (a) of Fig. 6. The changes necessary to implement model (b) are given above in Eq. (2a), Eq. (2b) and Eq. (8). The effect of these changes is totally insignificant, since the solutions of Eq. (7) are not noticeably influenced by them: the arctan(kC/k D) term in Eq. (7) varies much more slowly with kC than the tan(N ⫺ 1)kCd term. As for the table, on the basis of model (b) the values of VD have to be substituted by VD ⫹ VB and those of a by a ⫺ VB. No other changes have to be made. It was stated above that for a sample of thickness Nd the N lowest solutions in kC of Eq. (7) were
considered. The first N ⫺ 1 of these was found to be very near (with a deviation of ^ 5% at most) to the values of the wave vector, as given for a quantum well of width Nd with infinite walls and tight-binding dispersion [3]. This agreement is due to the fact that the actual barriers are, indeed, rather high. The Nth kCvalue would be, in that model, kC p /d, but it is disregarded there. In the present model, however, kC ⬎ p /d for the Nth solution and the energies calculated from this value of kC are in reasonable agreement with experiment (the ‘extra peaks’ in Figs. 1 and 2). The peculiarity of having a solution beyond the edge of the first Brillouin zone is, of course, due to the fact that the first layer (adjacent to the metal) breaks the translational symmetry. 4.2. Dispersion of the valence bands Fig. 7 shows the dependence of both the experimental and calculated peak energies on the values of (k × d)/p (k kC). The energies E 0 are referred to the vacuum level, i.e. E 0 E ⫺ VD, in order to facilitate comparison with conventional dispersion curves in crystalline solids. For a given substrate and band, the data pertaining to all samples are presented in the same graph. In other words, each graph is determined by only three adjustable parameters; VD, g B and g C — the values of these parameters appear in Table 1. The good agreement between the experimental points and the calculated curve fully justifies the use of the dispersion relation as given by Eq. (9). A peculiarity of the plots of Fig. 7 is the existence of experimental points beyond kC p /d, as stated above.
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Fig. 7. The dispersion of the xenon films in the direction G L , for the three substrates. Solid lines, calculation; open circles, experiment. In each graph the upper curve refers to the 5p1/2 band, the lower one to the 5p3/2 mj 1/2 band.
We emphasize again that these results hold equally for both models of Fig. 6. To the authors’ best knowledge, the results presented in this paper (including those from Ref. [3]) lead to the first realization for an insulator from the suggestion by Loly and Pendry [13], to determine valence band structure in a given direction of the kspace by means of quantum-size effects in epitaxially grown thin films. The overall shape of the graphs of Fig. 7 is very similar to that obtained by diverse band structure calculations. Disregarding, for the moment, the small differences between the various cases, we see that the effective electron mass mⴱC in both bands is about ⫺1 (see Table 1) and the width of the bands DE is about 1 eV. The obtained value of 兩mⴱ 兩 is much smaller, and of DE much larger, than in the early
band structure calculations [14,15] but they are close to results of newer calculations [16,17] supported by experimental results [16,18,19]. Returning to Table 1, it has to be pointed out that, as a rule, the values of mⴱC for the 5p1/2 band are more reliable than for the 5p3/2 mj 1/2 band, since the former were obtained from fitting more experimental peaks. No attempt was made to analyse the peaks pertaining to the 5p3/2 mj 3/2 valence band, as only very few of these could be discerned. Table 1 also shows a dependence of the values of a and mⴱC on the substrate. According to the simple tight-binding model [12], both parameters depend on integrals that involve the difference between the atomic and solidstate potentials DU(r) as well as the wave function F (r). Obviously, the values of both DU(r) and F (r) should be strongly dependent on the distance from the interface (cf. Ref. [20]). It follows that, for very thin films, the values of a and mⴱC may appreciably depend on the nature of the substrate. We determined each a and mⴱC value from the UPS spectra of a series of samples, with up to 7 monolayers for Xe/Cu(100), 6 monolayers for Xe/Ru(0001) and 9 monolayers for Xe/Pt(100)hex. In the analysis, the thicker samples carry more weight, simply because the number of peaks increases with the number of monolayers. Within the experimental accuracy, we could not perceive any systematic differences in the parameters depending on the number of monolayers with N ⱖ 2. All this casts doubt on attributing the variations in a and mⴱC on the substrate solely to a direct effect of the surface on the potential differences DU(r) and on the wave function F (r). Indirect effects that should be considered are those of crystal structure and texture. There exists a plethora of experimental results indicating that one monolayer of Xe grows in all cases considered as a close-packed hexagonal layer, with an interatomic spacing not necessarily equal to that in bulk Xe (see, e.g., Refs. [21–23]). There is, however, little information about further growth on top of monolayers except for the well-established fact of layer-by-layer growth. On the other hand, the cohesive energy difference between the f.c.c. and h.c.p. bulk structures is very small, of the order of 0.1% [24]. As a result, even in bulk crystals of Xe and Kr, h.c.p. and f.c.c. regions often coexist and stacking faults and dislocations abound [25,26]. It is likely that similar effects can arise also
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in the case of epitaxial growth on top of single crystal metal surfaces, strongly influenced by the surface, e.g., by the mismatch. The parameters of the function EV(k) along G L, as determined in this work, depend, in turn, on these structure and texture variations, through changes of symmetry and of interlayer distances. 5. Conclusions It has been demonstrated that the E(k) dispersion of solid Xe films along a given direction can be determined in detail experimentally by means of UPS peak patterns observed in thin films due to quantum confinement. The method should be applicable to other insulators or semiconductors as well, provided layer-by-layer growth can be ascertained and both the film surface and the film–metal interface planes are nearly perfect.
Acknowledgements The authors appreciate most valuable discussions with L.D. Shvartzman (Jerusalem) and J. Henk (Uppsala).
References [1] G. Perluzzo, G. Bader, L.G. Caron, L. Sanche, Phys. Rev. Letters 55 (1985) 545. [2] J.D. McNeill, R.L. Lingle Jr., R.E. Jordan, D.F. Padowitz, C.B. Harris, J. Chem. Phys. 105 (1996) 3883. [3] T. Schmitz-Hu¨bsch, K. Oster, J. Radnik, K. Wandelt, Phys. Rev. Letters 74 (1995) 2595. [4] K. Oster, Ph.D. thesis, University of Bonn (1996). [5] R. Paniago, R. Matzdorf, G. Meister, A. Goldmann, Surface Science 325 (1995) 336.
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[6] K. Wandelt, J. Hulse, J. Ku¨ppers, Surface Science 104 (1981) 212; K.J. Wandelt, Vac. Sci. Technol. A2 (1984) 802. [7] B. Pennemann, Ph.D. thesis, University of Bonn (1991). [8] F. Zbikowski, private communication (1996). [9] R. Miranda, S. Daiser, K. Wandelt, G. Ertl, Surface Science 131 (1983) 61. [10] I.T. Steinberger, K. Wandelt, Phys. Rev. Letters 58 (1987) 2494. [11] C. Kaindl, T.-C. Chaing, D.E. Eastman, F.J. Himpsel, Phys. Rev. Letters 45 (1980) 1808; T. Mandel, G. Kaindl, M. Domke, W. Fischer, W.D. Schneider, Phys. Rev. Letters 55 (1985) 1638. [12] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt, Rinehart and Wilson, Saunders College, Philadelphia, 1976. [13] P.D. Loly, J.B. Pendry, J. Phys. C: Solid State Phys. 16 (1983) 423. [14] H.R. Reilly, J. Phys. Chem. Solids 28 (1967) 2067. [15] U. Ro¨ssler, Phys. Status Solidi 42 (1970) 345. [16] K. Horn, A.M. Bradshaw, J. Noffke, K. Hermann, in: Proceedings of the Sixth Vacuum Ultraviolet Radiation Physics Conference, Charlottesville, Extended Abstracts, Abstract 132, U.S. Naval Research Laboratory, Washington, D.C., 1980. [17] J. Henk, private communication (1996). [18] B. Kessler, A. Eyers, K. Horn, N. Mu¨ller, B. Schmiedeskamp, G. Scho¨nhense, U. Heinzmann, Phys. Rev. Letters 59 (1987) 331. [19] B. Kessler, N. Mu¨ller, B. Schmiedeskamp, U. Heinzmann, Solid State Commun. 90 (1994) 523. [20] J. Henk, R. Feder, J. Phys. C: Condens. Matter 6 (1996) 191. [21] R.H. Roberts, J. Pritchard, Surf. Sci. 54 (1976) 687. [22] K. Christmann, J.E. Demuth, Surf. Sci. 120 (1982) 291. [23] J.A. Venables, J.L. Seguin, J. Suzanne, M. Bienfait, Surf. Sci. 145 (1984) 345. [24] J.A. Venables, C.A. English, K.F. Niebel, G.J. Tatlock, J. Physique 35 (C7) (1974) 113. [25] Y. Sonnenblick, Z.H. Kalman, I.T. Steinberger, J. of Crystal Growth 58 (1982) 143. [26] J.A. Venables, B.L. Smith, in: M.L. Klein, J.A. Venables (Eds.), Rare Gas Solids Vol. II, Academic Press, London, 1977, Chapter 10. [27] H. Shi, K. Jacobi, Surface Science 278 (1992) 281. [28] H.H. Rotermund, S. Jakubith, A. Kubala, A. von Oertzen, G. Ertl, J. Electron. Spectrosc. Relat. Phenom. 52 (1990) 811.
Bagus Special Issue
Quantum-size effects in thin solid xenon films M. Gru¨ne a, T. Pelzer a, K. Wandelt a,*, I.T. Steinberger b a
Institut fu¨r Physikalische und Theoretische Chemie der Universita¨t Bonn, Wegelerstrasse 12, D-53115 Bonn, Germany b Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel Received 17 October 1997; accepted 7 April 1998
Abstract Ultraviolet photoelectron spectra of solid xenon films on top of Ru(0001) and Cu(100) substrates revealed peak patterns due to confinement of electrons in quantum wells formed by the metal substrate–Xe insulator–vacuum system. The data (along with previous ones for Pt(100) substrates) were analyzed, assuming simple tight-binding dispersion in the direction GL, perpendicular to the film, expressly taking into account the polarization of the first monolayer adjacent to the metal. The good fit obtained implied that the sets of experiments sampled the 5p3=2 mj 1=2 and 5p1=2 valence bands at several points of the actual dispersion curves, furnishing an experimental determination of these curves along the GL direction. The absolute value of the effective mass mⴱ for the electrons at the top of the bands (in terms of the free electron mass) was found to be near unity for both bands, depending somewhat on the substrate. These substrate-dependent variations are attributed to the effect of the substrate on the potential and the wave function in the Xe films, combined with imperfections in the texture and structure of the films. 䉷 1999 Elsevier Science B.V. All rights reserved. Keywords: Quantum size effects; Ultraviolet; Ultraviolet photoelectron spectroscopy; Xenon; Insulating films
1. Introduction Quantum wells in semiconductors have been intensively studied for the past twenty years or so. The study was made possible by virtue of the development of sophisticated modern technology (chiefly, molecular beam epitaxy) for the preparation of materials and structures by layer-by-layer growth. Beyond the basic interest from both the physics and materials science point of view, these studies are also of foremost technological importance, enabling the development of new electronic and opto-electronic devices. In the semiconductor case, the quantum well usually comprises a slice of semiconductor of type * Corresponding author. Tel.: ⫹ 49-228-732253/2654; Fax: ⫹ 49-228-732551; e-mail: [email protected]
A sandwiched between two semiconductors of type B, with the band gap of A being smaller than that of B. The width of the A-type material is, in most cases, of the order of 10 monolayers. In principle, one might also envisage other, asymmetrical quantum well arrangements like A–B–vacuum or metal–semiconductor–vacuum. The latter system would normally also involve the formation of an accumulation or exhaustion layer of one of the carriers and, therefore, an electric potential which varies continuously along the direction perpendicular to the metal/semiconductor interface. If the semiconductor is replaced by a highly insulating substance, like a rare gas solid, then the low concentration of free carriers prevents band bending effects. Moreover, it is relatively easy to grow rare-gas adsorbates epitaxially, layer by layer, on a number of single crystal surfaces, using
0368-2048/99/$ - see front matter 䉷 1999 Elsevier Science B.V. All rights reserved. PII: S0368-204 8(98)00281-3
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well-established surface science control techniques. Thus the metal–rare gas solid–vacuum quantum well is, in principle, similar to the metal–semiconductor–vacuum quantum well, but it is much simpler to prepare and can be regarded as the simplest prototype of the latter. Low energy (0–10 eV) electron transmission spectra in thin films (up to 10 monolayers) of argon and methane showed a series of peaks [1]. The number of the peaks increased with the number of layers. Analysis of the data showed that the electrons traverse the samples ballistically and the peak positions correspond to constructive interference between the electrons reaching the metal substrate directly and those that had undergone a reflection at both interfaces (metal–adsorbed film and adsorbed-film–vacuum). This quantumsize resonance effect enabled the determination of the dispersion of the respective conduction bands. McNeill et al. recently reported a two-photon angular resolved photoemission study of quantum well states in Xe and Kr films on top of Ag(111) [2]. One of the important results presented by McNeill et al. was the very precise measurement of the Xe bulk conduction band [2]. Experiments referring to valence bands were performed using xenon multilayers adsorbed on top of atomically clean Pt(100)hex [3,4], Ag(111) or Cu(100) [4,5] surfaces. The experiments comprised ultraviolet photoelectron spectroscopy (UPS) of the 5p1=2 and 5p3=2 levels of the adsorbate. In all cases, there was an appreciable shift of the peaks between the first and second layer. Moreover, the intensity of the peak of the first layer decreased in step with the growth of the second layer: when the second layer was completed, the peak corresponding to the first one disappeared completely. The evolution of the third layer quenched the peak of the second layer and gave rise to two other peaks, symmetrically positioned with respect to the quenched one. In a sample consisting of N monolayers N ⫺ 1 peaks were considered; these were arranged with mirror symmetry with respect to a fixed photon energy. This energy was that of the central peak in samples comprising an even number of layers. Evolution of the pattern corresponding to N monolayers always involved quenching of all patterns that appeared for thinner samples. The total width of the pattern increased with thickness: for
N 9 (observed with Pt(100)hex as substrate) the total width was about 1.2 eV [3]. The results clearly indicate that instead of assigning single peaks to individual layers, one has to consider peak patterns and assign each pattern to a multilayer sample as a whole. The patterns have been successfully interpreted [3] on the basis of a quantum-well model, assuming a well of infinite wall height and simple tight-binding dispersion of the valence bands. However, the model explicitly postulates that a sample Nd thick (d being the thickness of one monolayer) should accommodate N ⫺ 1 standing waves of wave numbers k with k np /Nd, n 1,2,…,N ⫺ 1, implying that there are no waves of wavelengths equal to the width of just two monolayers. A qualitative argument supporting this postulate referred to the first, polarized layer adjoining the metal surface as not being part of the Xe film under consideration. In this paper, we present further experimental results on xenon multilayers, namely on top of Ru(0001) and on top of Cu(100). The valence-band quantum-size effects in xenon on top of various substrates are compared, taking into account work function differences. A calculation shows that a simple tight-binding dispersion formula can be fitted very well to the experimental data. The calculation takes into account the polarization of the first layer as a step in a compound quantum well and involves walls of finite heights.
2. Experimental The experiments were carried out in an ultra-high vacuum chamber, with a base pressure lower than 1 × 10 ⫺8 Pa. The substrate could be cooled down by a closed-loop He refrigerator and heated up to 1000 K by thermal radiation from a filament (for Cu), or up to 1600 K by electron bombardment (for Ru); reaching the lowest temperature was found to be essential for the layer-wise growth of the Xe multilayers, especially in the case of the Cu(100) substrate. The chamber was equipped with standard surface analytical tools, such as LEED, AES, TDS, UPS, a Kelvin probe and an ion gun for Ar ⫹ sputtering. In all experiments, the angle of incidence (with respect to the surface normal) of the UV light was kept fixed at about 60⬚; the electrons were collected in a direction
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Fig. 1. Ultraviolet photoelectron spectra of xenon films on a Ru(0001) substrate.
perpendicular to the plane of the sample, with an angular resolution of 2⬚. Assuming that the Xe layers grow regularly with the f.c.c. structure, (this point will be further elucidated in Section 4) with (111) layers parallel to the substrate, we may state that the direction GL in the k-space was sampled in each case. Most experiments were performed by means of a He I photon source; the photoelectron energies were determined by means of a hemispherical energy analyzer. With the 5 eV pass energy employed in most experiments, the energy resolution was about 70 meV. In some experiments (especially for high Xe coverages), the pass energy was set to 2 eV in order to improve the resolution at the expense of the signal-to-noise ratio. The preparation of Ru started with annealing in oxygen at 1150 K until no carbon could be seen by AES. The respective contributions of C and Ru to the peak at 273 eV were estimated by the maximum-tobaseline/minimum-to-baseline ratio. To remove residual C from the surface, a series of O2-adsorption and
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CO-desorption cycles followed, each with a saturation dose of O2, until a CO recombination TDS-peak could no longer be detected. Eventual residual oxygen was then removed by sputtering with Ar ⫹ ions of 1 kV at room temperature for 10 min, with subsequent annealing at 1550 K for several minutes. After this procedure, AES showed neither oxygen nor carbon, and the LEED picture was that of a clear and wellordered Ru(0001) surface. Cu was sputtered at 900 K for 30 min. At this temperature, impurities segregate to the surface. Sputtering was continued while the sample cooled down to room temperature, followed by a flash to 800 K. After this procedure, AES showed no oxygen and the LEED pattern was that of a clean Cu(100) surface. The quality of each substrate was also controlled by PAX [6]. The Xe 5p1=2 peak showed no shift (apart from the known commensurate–incommensurate transition on Ru) and a FWHM of only 200 to 280 meV. Commercially available Xe of 99.99% purity (Messer-Griesheim) was attached via a leak valve to the chamber. The Xe doses were measured with an ion gauge without further correction and are given in Langmuir (L). The dose was calibrated in terms of the number of monolayers deposited by establishing the dose necessary to complete one monolayer (ML), using the PAX method. In the course of the measurements it turned out that integral multiples of this dose corresponded to the maximum height of a series of UPS peaks characteristic to the same integral number of monolayers, indicating that the sticking coefficient (presumably very near to unity) did not vary with layer thickness. In the case of the Ru(0001) substrate, we checked the calibration also by Xe TDS, which gives wellresolved peaks for the first two Xe layers. After cleaning and annealing the substrate as described above, it was cooled down to the minimum temperature that could be reached (typically 44 K), flashed up to a few hundred K in order to remove impurities eventually adsorbed during the long first cooling down of the sample, and cooled down quickly again for the start of xenon deposition. We deposited Xe at first in very small steps (2–3 L) in order to determine the dose for one monolayer and, subsequently, in steps of 0.2 or 0.25 monolayers for the thicker samples. In the case of the Cu(100) substrate,
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one series of measurements was performed as described above and a second series was performed starting with the deposition of three monolayers—this made it possible to extend the scope of measurements to appreciably thicker films (up to 7 ML) than previously reported [4,5]. The work functions of the pure metals were obtained from the width of He–I excited UPS spectra, measured from turning point to turning point. The work function changes induced by one monolayer of xenon were determined both by this UPS method and by a Kelvin probe vibrating parallel to the surface [7,8].
3. Results In Fig. 1, a set of spectra selected from the series with the Ru(0001) substrate is presented, starting from a complete monolayer, with dose increments of about 0.5 ML. The peaks observed in each spectrum can be divided into two groups, on the two sides of the broken vertical line at 6.34 eV. The group on the left of the line is associated with the valence bands that evolved from the 5p3=2 atomic states of Xe and the one on the right is associated with the 5p1=2 valence band. Arrows point to the approximate positions of the principal peaks. In the higher-energy series one sees that for the 1 monolayer film (19 L) there is one 5p1=2 peak at about 6.48 eV. With increasing coverage, this peak decreases in amplitude and is totally absent in films made up of more than two monolayers (40 L and above, in Fig. 1). The 40 L film exhibits two main peaks marked by arrows and the 60 L peak three, the highest-energy peak appearing only as a shoulder. It follows that for these samples the number of peaks is equal to the number of monolayers. However, only three peaks appear in the 80 L film, and four in the 100 L one, seemingly violating this rule. Nevertheless, in these spectra the most prominent peak (at 7.51 eV and 7.56 eV, respectively) could totally obscure a weak and broad band under its high-energy flank. As most spectra were taken on samples that did not comprise an integral number of monolayers, some smaller peaks, associated with the beginnings of the next monolayer are also seen, but not marked, e.g., the 6.8 eV peak in the 40 L spectrum or the 7.1 eV peak in the 60 L and the 100 L spectra. The pattern is, on the whole, similar to that presented in the paper by
Schmitz-Hu¨bsch et al. [3] for Xe films on the Pt(100)hex substrate: distinct sets of peaks are characteristic to samples consisting of integral numbers of monolayers, the sets disappear upon completion of the next monolayer. However, in the paper quoted, only N ⫺ 1 peaks were assigned to N-monolayer samples: peaks analogous to the ones marked with down arrows (the ‘extra peaks’) in our Fig. 1 did appear there as well, but were not considered to belong to the main sets of peaks. The broken vertical line at 6.34 eV in Fig. 1 indicates not only the borderline between the two groups, but also the approximate position of a peak that appears in all spectra of films at least two monolayers thick, analogous to the peak ‘1’ in Fig. 5 by Paniago et al. [5]. The amplitude of the peak increases slowly and monotonically with the film thickness. Its fixed position and its presence in most spectra distinguish this peak from all other peaks. In Ref. [5] it has been rather tentatively assigned to a Xe surface state, but, as the authors point out, further study is needed for verification. The behaviour of the lower-energy groups of peaks in the spectra is similar to that of the higher-energy groups, but, in this case, the above-mentioned peak, at about 6.34 eV (marked with a broken line), seems to interfere with the observation of the ‘extra’ peak in the thicker samples. A similar set of spectra for the Cu(100) substrate is shown in Fig. 2. The spectra are taken from two different series of experiments: for the lower coverages, the dose was increased gradually, as in the Ru(0001) case, starting from 1 L. The thickest film of this series was made by admitting 91 L. It turned out that the quality of the spectra in the thickest films of the series was rather low: the peaks were illdefined and washed out. The deterioration was probably due to irregular growth conditions (see also Section 4). Therefore, as stated above, a new series of films was prepared, this time starting with a dose of 36 L. Inspection of the Xe/Cu(100) spectra reveals clear similarities with the Xe/Ru(0001) spectra shown in Fig. 1. The difference in exposure to reach saturation of one monolayer is due to slightly different substrate temperatures. The evolution of a given set of peaks with coverage, in parallel with the gradual disappearance of the set belonging to the sample with the last integral number of layers, is best seen by subtracting from each
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Fig. 3. UPS incremental difference spectra of xenon films on a Ru(0001) substrate.
Fig. 2. Ultraviolet photoelectron spectra of xenon films on a Cu(100) substrate.
spectrum another one corresponding to somewhat lower coverage. Examples of such difference spectra are shown in Fig. 3 for the Ru(0001) substrate, with increments of 4 L, and in Fig. 4 for Cu(100) with increments of either 3 or 3.25 L. One can observe the two groups of peaks (at higher and lower energies) in both graphs. Peak patterns have fixed positions until they disappear after completion of the next monolayer. The main features of the spectra stand out, indeed, they are clearer in the difference spectra representations than in the raw spectra of Figs. 1 and 2. The determination of the peak positions started by inspecting both the raw spectra and the difference spectra: only such peaks were considered that could be seen in the raw spectra as well. If neighbouring peaks were of approximately equal heights, their
position was ascertained from that difference spectrum where the pattern in question had the largest amplitudes. If, however, neighbouring peaks differed in height appreciably, a fit of 2–3 adjacent peaks in the original spectrum by Lorentzians was necessary. The accuracy of the determination of the peak positions was better for thinner samples and for larger peaks; it was typically about 10 meV for the strongest peaks in the thinnest films and deteriorated to about 100 meV for the weakest peaks in the thickest films. The number N of completed monolayers versus the peak positions is presented in Fig. 5. It includes the results for the Cu(100) and the Ru(0001) substrate shown above, as well as those for Pt(100)hex from Ref. [3], the latter in order to enable direct comparison with the other substrates and with the model to be presented below. For the Cu(100) substrate, the results reported for Xe films up to 4 monolayers [4,5], are in good accord with the present results. The pattern of peak positions is very similar for all three substrates except for horizontal displacements
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the positions of these peaks involves a large experimental error, since these peaks are broad and weak, close to the ‘constant energy’ peaks of the first set around 7.02 eV, marked by triangles. 4. Discussion
Fig. 4. UPS incremental difference spectra of xenon films on a Cu(100) substrate.
and differences in the total number of monolayers with resolvable data. Each pattern can be divided into two groups, as stated already with reference to Figs. 1–4. For N ⱖ 2, most of the pattern within a given group has an approximate mirror symmetry with respect to a central energy value that recurs in all samples of even-numbered layers [3,4]. There are two sets of peaks for each substrate which do not fit into the symmetric patterns. The members of one set are marked by open triangles—they lie at positions very near to the borderline between the two symmetric groups. As stated above, these peaks are irrelevant to the present study. The peaks of the other set (the ‘extra peaks’), marked by open circles, appear on the highenergy side of each group. There are three such ‘extra’ peaks both for Xe/Cu(100) and for Xe/Ru(0001) (at N 2 in the 5p3=2 group and at N 2 and 3 in the 5p1=2 group) and five for Xe/Pt(100)hex (at N 2, 3, 4, 6 and 7 in the 5p1=2 group). In the Cu(100) pattern the peaks at N 5 and 6 at 7.13 and 7.15 eV, respectively, seem to belong to the 5p1=2 series, but the determination of
The extensive results presented above on Xe films on top of Ru(0001) and Cu(100) substrates underline the main qualitative conclusions which had been reported recently about thin Xe films [3–5]: (a) the peak patterns observed in UPS are characteristic of each film as a whole and not of individual monolayers; (b) the peak patterns are formed as a result of the quantization of the energy of valence-band electrons, associated with the crystal momentum perpendicular to the film (quantum-size effect). It will now be shown that a simple model calculation, explicitly taking into account the polarization of the monolayer adjacent to the substrate, is in quantitative agreement with the results. The E(k) dispersion relations in the GL direction of the k-space are deduced from the fit of the model to the experimental data. A comparison of the results for the various substrates will conclude Section 4. 4.1. Calculation Ultraviolet photoelectron spectroscopy (UPS) of the 5p1=2 and 5p3=2 states of xenon adsorbates on a metal surface shows that the binding energy EBF of these states with respect to the Fermi level of the metal is dependent on the amount of adsorbed xenon. In particular, EBF is substantially smaller for submonolayer coverage than for xenon atoms residing on top of a completed xenon monolayer [6]. The change is typically of a few tenths of an eV, depending on the metal and its surface plane. Beyond the first monolayer, the changes are much smaller. This variation of the binding energy with the distance from the metal plane has been interpreted in two different ways: (a) as an initial-state effect, due to the change in the work function caused by the adsorbed xenon [9,10], and (b) as a final-state effect, due to the image force acting on the hole that had been created by the photoionization process [11]. In Fig. 6, highly schematic energy diagrams are presented for both models. In accord with the
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For both cases (a) and (b) VD ⫺ Emax E0V ⫺ a
1
where E0V is the appropriate ionization potential of a single atom and a is a relaxation energy due both to the presence of the adsorbate and the metal surface. Thus a may be influenced by the distance from the metal interface; we shall assume that a is constant for N ⬎ 1. The binding energy EBF , referred to the Fermi energy of the metal, will be, in model (a): EBF VD ⫹ VB ⫺ fM ⫺ E
2a
and in model (b): EBF VD ⫺ fM ⫺ E
Fig. 5. Number of layers contained in the film versus UPS peak positions (binding energies) for films grown on different substrates. Dots and triangles, experimental; vertical lines, calculated. See text.
2b
In the following calculation the crystal film on top of the metal surface is treated as a compound quantum well (Fig. 6). The wave functions and the dispersion are consistent with the tight-binding approximation [12] in the xenon crystal and the boundary conditions. Interactions between the valence bands are neglected and, instead of making detailed assumptions about the wave functions and the potential, the constants of the tight-binding model are regarded as parameters adjusted so as to fit the experimental data. In regions A and D we assume exponentially decaying plane waves (cf. Fig. 6). In region A (x ⬍ 0)
cA x A exp kA x
3
In region D (x ⬎ Nd) simplifying approximation employed, all the change of the potential is shown as occurring step-like between the first and second layer. It will be shown that for the main result (the dispersion) it is irrelevant whether the initial-state or the final-state model is adopted. In Fig. 6, E is the electron energy (constant in the whole structure), f M the work function of the pure substrate and VB the potential jump between the first and the second layer. In the initial-state model VB ⫺ Df, where Df is the work function change caused by one completed Xe monolayer. VD is a convenient reference level within the sample: VD ⫺ Emax (Emax being the upper limit of E as a function of kC, see below) is the energy difference between the vacuum level and the valence band probed.
cD x D exp ⫺kD x
4
In region C (d ⬍ x ⬍ Nd), according to the Bloch theorem
cC x C sin kC x ⫹ d2 f x
5
where f(x) is a periodic function with the periodicity of the lattice, representing (unspecified) Wannier functions. It follows from the periodicity of f(x) that f nd f d
n 0; …N
5a
With the further assumption that f(x) has mirror symmetry with respect to the atom centres df x 0 n 0; …N 5b dx xnd Eq. (5a) and Eq. (5b) are only approximately correct
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Fig. 6. Energy diagram for the model employed in the calculation (a) initial state scheme; (b) final state scheme (see text).
at the xenon–metal and xenon–vacuum interfaces (n 0 and n N). In region B, (0 ⬍ x ⬍ d) treated as a part of the xenon crystal, we have
electron mass. For model (a), V1 VB, for model (b), V1 0. In region C, according to the tight-binding model assumed,
cB x B sin kB x ⫹ d1 f x
E ⫺gC 1 ⫹ cos kC d
6
The wave functions are real; they and their first derivatives must be continuous at x 0, d and Nd. These conditions lead to the elimination of A,B,C,D and f(d) and we obtain: kB k ⫺ B tan N ⫺ 1kC d tan kB d ⫹ arctan kA kC ⫹ arctan
kC kD
7
In order to find the allowed energy levels and kvalues, the appropriate dispersion relations will be invoked. The height of the barrier that the valence electrons in the insulator ‘see’ at the interface with the metal should be strongly dependent on the band structure of both substances at the interface. Fortunately, it turns out that the results are almost insensitive to the value of this parameter as long as it is not too small. For convenience, we assume that the top of the barrier is at the Fermi energy of the metal, and that the electron effective mass in the metal equals the free electron mass. We have then in region A p 2m0 ⫺E ⫹ VD ⫹ V1 ⫺ f kA 8 ប É is Planck’s constant divided by 2p and m0 the free
9
2g C is the width of the appropriate valence band. In region B, the dispersion relation should be the same as in region C: E ⫺gB 1 ⫹ cos kB d ⫹ V 0
10
0
Here V accounts for the change in the reference level (cf. Fig. 6). Its value is fixed by the condition that the difference between the binding energy in the second (and higher) layers and that in the first one must be VB: V 0 VB ⫺ 2 gB ⫺ gC
11
We allowed in Eq. (11) gB 苷 gC , since the vicinity of the metal may strongly affect the value of g [12]. g B is a convenient fitting parameter. Finally in the vacuum (region D) p 2m0 ⫺E ⫹ VD kD 12 ប Eliminating k A, k B and k D from Eq. (7) by means of Eqs. (8)–(12), an equation in kC is obtained, with g B and g C as adjustable parameters. Since the solid-state quantum well consisted of N atomic layers, we sought N solutions (with the lowest kC values) of this equation. The binding energies EBF were then calculated from the set of kC values by means of Eq. (9) and Eq. (2a) (or Eq. (9) and Eq. (2b), see below) and compared with the experimental binding energies. The parameters g B, g C and VD were then varied
M. Gru¨ne et al. / Journal of Electron Spectroscopy and Related Phenomena 98–99 (1999) 121–131
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Table 1 The work function f of the pure metal surface, its change Df due to one monolayer of adsorbed xenon, the fitting parameters g B, g C and VD (see text), the effective mass mⴱC at the top of the bands, the distance VD ⫺ Emax from the top of the band to the vacuum level and the parameter a . Units: eV, except for mⴱC , which is given in terms of the free electron mass.
f
Df
Band
Cu(100)
4.74
⫺ 0.33
Ru(0001)
5.52 a
⫺ 0.62 b
Pt(100)hex
5.72 c
⫺ 0.54 c
5p1/2 5p3/2 5p1/2 5p3/2 5p1/2 5p3/2
gB ⫺ 1.65 ⫺ 1.65 ⫺ 2.17 ⫺ 2.17 ⫺ 2.17 ⫺ 2.17
gC
mⴱC
VD
VD ⫺ Emax
a
⫺ 0.48 ⫺ 0.48 ⫺ 0.58 ⫺ 0.53 ⫺ 0.62 ⫺ 0.53
⫺ 1.24 ⫺ 1.24 ⫺ 1.04 ⫺ 1.13 ⫺ 0.96 ⫺ 1.13
12.62 11.17 12.60 11.16 12.48 10.99
11.66 10.21 11.45 10.10 11.23 9.93
1.77 1.92 1.98 2.03 2.20 2.20
Ref. [6]. According to Ref. [27] f 5.37 eV Ref. [8] c Ref. [7]. According to Ref. [28] f 5.84 eV a
b
until satisfactory agreement between theory and experiment (vertical lines and dots, respectively, in Fig. 5) was obtained. It turned out that the value of g B ensuring the agreement was sensitively dependent on VB (not used as a fitting parameter, known, however, only with an accuracy of about ^ 0.1 eV), but only slightly affected by the values of g C and VD. During the fitting procedure, the observed extra peaks (marked by down arrows in Figs. 1 and 2) were not taken into account. Even so, Fig. 5 shows that the solutions do apply, within the experimental error, for these peaks as well. Table 1 summarizes the values of the parameters VD, g B and g C used in the fitting procedure, along with f and Df , obtained from the literature. It also includes the effective mass m * at the top of the valence bands, calculated from the relationship
mⴱC ⬅
ប2 m0
1 d2 E=dk2
k0
ប2 1 m 0 gC d 2
13
The data in Table 1 are based on model (a) of Fig. 6. The changes necessary to implement model (b) are given above in Eq. (2a), Eq. (2b) and Eq. (8). The effect of these changes is totally insignificant, since the solutions of Eq. (7) are not noticeably influenced by them: the arctan(kC/k D) term in Eq. (7) varies much more slowly with kC than the tan(N ⫺ 1)kCd term. As for the table, on the basis of model (b) the values of VD have to be substituted by VD ⫹ VB and those of a by a ⫺ VB. No other changes have to be made. It was stated above that for a sample of thickness Nd the N lowest solutions in kC of Eq. (7) were
considered. The first N ⫺ 1 of these was found to be very near (with a deviation of ^ 5% at most) to the values of the wave vector, as given for a quantum well of width Nd with infinite walls and tight-binding dispersion [3]. This agreement is due to the fact that the actual barriers are, indeed, rather high. The Nth kCvalue would be, in that model, kC p /d, but it is disregarded there. In the present model, however, kC ⬎ p /d for the Nth solution and the energies calculated from this value of kC are in reasonable agreement with experiment (the ‘extra peaks’ in Figs. 1 and 2). The peculiarity of having a solution beyond the edge of the first Brillouin zone is, of course, due to the fact that the first layer (adjacent to the metal) breaks the translational symmetry. 4.2. Dispersion of the valence bands Fig. 7 shows the dependence of both the experimental and calculated peak energies on the values of (k × d)/p (k kC). The energies E 0 are referred to the vacuum level, i.e. E 0 E ⫺ VD, in order to facilitate comparison with conventional dispersion curves in crystalline solids. For a given substrate and band, the data pertaining to all samples are presented in the same graph. In other words, each graph is determined by only three adjustable parameters; VD, g B and g C — the values of these parameters appear in Table 1. The good agreement between the experimental points and the calculated curve fully justifies the use of the dispersion relation as given by Eq. (9). A peculiarity of the plots of Fig. 7 is the existence of experimental points beyond kC p /d, as stated above.
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M. Gru¨ne et al. / Journal of Electron Spectroscopy and Related Phenomena 98–99 (1999) 121–131
Fig. 7. The dispersion of the xenon films in the direction G L , for the three substrates. Solid lines, calculation; open circles, experiment. In each graph the upper curve refers to the 5p1/2 band, the lower one to the 5p3/2 mj 1/2 band.
We emphasize again that these results hold equally for both models of Fig. 6. To the authors’ best knowledge, the results presented in this paper (including those from Ref. [3]) lead to the first realization for an insulator from the suggestion by Loly and Pendry [13], to determine valence band structure in a given direction of the kspace by means of quantum-size effects in epitaxially grown thin films. The overall shape of the graphs of Fig. 7 is very similar to that obtained by diverse band structure calculations. Disregarding, for the moment, the small differences between the various cases, we see that the effective electron mass mⴱC in both bands is about ⫺1 (see Table 1) and the width of the bands DE is about 1 eV. The obtained value of 兩mⴱ 兩 is much smaller, and of DE much larger, than in the early
band structure calculations [14,15] but they are close to results of newer calculations [16,17] supported by experimental results [16,18,19]. Returning to Table 1, it has to be pointed out that, as a rule, the values of mⴱC for the 5p1/2 band are more reliable than for the 5p3/2 mj 1/2 band, since the former were obtained from fitting more experimental peaks. No attempt was made to analyse the peaks pertaining to the 5p3/2 mj 3/2 valence band, as only very few of these could be discerned. Table 1 also shows a dependence of the values of a and mⴱC on the substrate. According to the simple tight-binding model [12], both parameters depend on integrals that involve the difference between the atomic and solidstate potentials DU(r) as well as the wave function F (r). Obviously, the values of both DU(r) and F (r) should be strongly dependent on the distance from the interface (cf. Ref. [20]). It follows that, for very thin films, the values of a and mⴱC may appreciably depend on the nature of the substrate. We determined each a and mⴱC value from the UPS spectra of a series of samples, with up to 7 monolayers for Xe/Cu(100), 6 monolayers for Xe/Ru(0001) and 9 monolayers for Xe/Pt(100)hex. In the analysis, the thicker samples carry more weight, simply because the number of peaks increases with the number of monolayers. Within the experimental accuracy, we could not perceive any systematic differences in the parameters depending on the number of monolayers with N ⱖ 2. All this casts doubt on attributing the variations in a and mⴱC on the substrate solely to a direct effect of the surface on the potential differences DU(r) and on the wave function F (r). Indirect effects that should be considered are those of crystal structure and texture. There exists a plethora of experimental results indicating that one monolayer of Xe grows in all cases considered as a close-packed hexagonal layer, with an interatomic spacing not necessarily equal to that in bulk Xe (see, e.g., Refs. [21–23]). There is, however, little information about further growth on top of monolayers except for the well-established fact of layer-by-layer growth. On the other hand, the cohesive energy difference between the f.c.c. and h.c.p. bulk structures is very small, of the order of 0.1% [24]. As a result, even in bulk crystals of Xe and Kr, h.c.p. and f.c.c. regions often coexist and stacking faults and dislocations abound [25,26]. It is likely that similar effects can arise also
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in the case of epitaxial growth on top of single crystal metal surfaces, strongly influenced by the surface, e.g., by the mismatch. The parameters of the function EV(k) along G L, as determined in this work, depend, in turn, on these structure and texture variations, through changes of symmetry and of interlayer distances. 5. Conclusions It has been demonstrated that the E(k) dispersion of solid Xe films along a given direction can be determined in detail experimentally by means of UPS peak patterns observed in thin films due to quantum confinement. The method should be applicable to other insulators or semiconductors as well, provided layer-by-layer growth can be ascertained and both the film surface and the film–metal interface planes are nearly perfect.
Acknowledgements The authors appreciate most valuable discussions with L.D. Shvartzman (Jerusalem) and J. Henk (Uppsala).
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