Quantum Physics of Thin Metal Films

Transcription

1 Quantum Physics of Thin Metal Films Tai C. Chiang ( 江台章 ) University of Illinois at Urbana-Chamaign My lectures are organized as follows (Fig. 1): I will begin with a discussion of angleresolved photoemission spectroscopy from solids. This is a powerful (and the only) technique available for general band structure mapping. Next I will talk about a particle in a box, quantization, and phase shift. These topics are not necessarily familiar to most people, especially the phase shift, and so I will spend some time explaining it in detail. Next I will show how to apply these concepts to gain an understanding of the electronic structure of thin films, with an emphasis on atomically uniform thin films. I will then talk about one-dimensional shell effects, quantum oscillations of properties, substrate effects, and diffraction at interfaces. Please feel free to ask questions any time. 1. Angles-resolved photoemission from solids Fig. 1 The technique of angle-resolved photo-emission has been around for a very long time. I will begin with a few historical remarks (Fig. 2). An important question about photo-emission is: What does it measure? A simple answer is that it measures electronic structure; more ge-nerally, it measures the band structure or quasi-particle spectral functions of solids. But there are many details you need to know before we can talk about thin films. Specifically there has been a long standing question about photoemission: does it measure the bulk or surface properties? This question has been posed repeatedly in the literature and during conferences. I hope to give you some answers. The light source for photoemission can be a synchrotron or a lamp a synchrotron is just a fancy Fig

2 "lamp." The light source sends a beam onto the sample, and electrons are ejected into vacuum. One measures the energy and angular distributions of the electrons, from which one can deduce the states of the electrons before photoexcitation (or the properties of the holes or quasiparticles left behind). Let me begin with a few historical remarks (Fig. 3). The photoelectric effect was discovered in 1887 by Hertz. His experiment involved two spark gaps. When a sufficiently high voltage is applied across a Fig. 3 spark gap, a spark can form spontaneously. Hertz noticed a correlation between two nearby spark gaps: when one spark gap went off, it could induce a spark in the second gap. He theorized that light emitted by the first spark gap struck the second spark gap, causing electron emission; the emitted electrons caused the second spark gap to go off. The concept of the energy quantum associated with light was first introduced in 1900 by Max Planck. In order to explain the black body radiation spectrum, he had to make the assumption that electromagnetic radiation is quantized and the energy of the quantum (photon) is hν (Planck constant times frequency). It is interesting to note that while he proposed the quantum concept, he himself had difficulties accepting it completely (see his original paper). Five years later (1905), Albert Einstein formulated a theory for the photoelectric effect. His photoelectric equation states that the energy of photon minus the work function gives the maximum kinetic energy of the photoemitted electrons. Eleven years later Robert Millikan did an experiment to prove that Einstein was correct. His experiment involved some pretty sophisticated equipment: a machine shop in vacuo. It is perhaps the first UHV experiment in history (see next slide). Shortly afterwards, it was recognized that the photoelectric effect, or photoemission, could be a powerful probe of the electronic and atomic structures of solids. Already at that early stage of the development of the photoemission techniques, there was a debate regarding whether surface or bulk effects were being measured because the photoexcited electrons were known to have a very short escape depth, typically 5 to 10 Angstroms or so. This short escape depth means that only 2 or 3 atomic layers are being probed. Photoemission is thus very sensitive to the surface condition, and surface cleanliness is important. Many early photoemission measurements were troubled by surface contamination. UHV (ultrahigh vacuum) techniques and equipment including copper gaskets, ion pumps, ion gauges, etc. became well developed in the 1950s. Synchrotron radiation sources became available in the 1960s. These technical advances were key to the development of the photoemission technique. Today, we have third-generation light sources, multi-channel and area detectors, etc. The overall energy resolution can be better than 1 mev, as demonstrated in recent experiments by Professor Shin's group. The angular resolution is also much improved. Looking toward the future, free electron lasers, energy-recovery linacs, and other novel light sources, as well as new types of detectors, are likely to provide enhanced or additional capabilities, including nanofocus or nano-imaging, femtosecond time resolution, 3D data taking, detailed investigations of coherence and nonlinear effects, etc. 215

3 A picture of Millikan s machine shop in vacuo is shown in Fig. 4. It is a glass chamber equipped with a lathe driven by electromagnets outside the vacuum chamber. There are three metal targets (alkali metals). Each target is rotated to face the lathe cutter. After a freshly cut surface is prepared, it is rotated to face the window on the right. A beam of light through the window ejects electrons from the target, and the photoelectrons are collected by an electrode. The electrode is biased to form a retarding field analyzer, and so the details of the photoelectric effect can be verified. The alkali metal shavings from the cutting fall down to the bottom of the chamber. They are very reactive and remove the residual gases in the chamber. The setup is essentially a getter pump. It is interesting to note that Millikan proved that Einstein's equation was correct, but he had trouble believing in the quantum concept (see his original paper). Today, photoemission is often performed with a synchrotron as the light source. An example is shown in Fig. 5. The picture on the upper right corner shows the direct beam (zero order light) from a beamline at the Synchrotron Radiation Center in Stoughton, Wisconsin. Figure 6 is a schematic diagram for the geometry involved in angle-resolve photoemission. The analyzer determines the energies and the emission angles of the electrons. What one measures is the wave vector (or momentum) k of the photoelectron in vacuum. But what one wants is the wave vector k inside the crystal before photoexcitation. For a well ordered surface, the system has 2- dimensional translational symmetry, and so k // is conserved when the photoexcited electron Fig. 4 Fig. 5 Fig

4 crosses the surface. On the other hand, k is not necessarily conserved. This is known as the " k problem." The solution to this problem will be discussed below. In angle-resolved photoemission, one typically employs low energy photons (long wave length limit). Their momenta are practically zero (compared to the Brillouin zone size) and can be ignored. Optical transitions involve the initial state, the final state, p A, and A p. For direct band-to-band Fig. 7 optical excitations, k is conserved (vertical transitions). Thus, the measured k // of the photoelectron equals the k // of the initial state. On the other hand, the measured k of the photoelectron generally differs from the k of the initial state. Let us take a look at a couple of examples (Fig. 7): Ag(100) and Ag(111). For simplicity, consider normal emission only (emission perpendicular to the surface). k // = 0 for the photoelectron and the initial state. We only need to worry about k z or k. Shown in Fig. 7 is the band structure of Ag. The many bands between -4 and -8 ev are the d-state derived bands, and we will not worry about these for now. Ag has 1 n = 5 electron, which gives rise to a free-electron-like sp band. This band crosses the Fermi level along [110] and [100], and it hybridizes with the d bands where they overlap. Along [111], the Fermi level lies within a gap of the sp band, and the system looks like a one-dimensional semiconductor. Let us consider normal emission from Ag(100) as shown in Fig. 8. Since k // = 0, the relevant direction of the band structure is the [100] direction (along k ). The band diagram on the right is an amplified version to show details near the Fermi level. A lower band and an upper band are separated by a band gap at the zone boundary. The Fermi level crosses the lower band. Direct vertical transitions at photon Fig

5 energies of 11, 14, and 17 ev are shown in the right panel of Fig. 8. On the left are normal emission data obtained at various photon energies. The horizontal axis is the binding energy of the initial state (with the Fermi level at 0). The data show photoemission peaks at binding energies in agreement with the band diagram. Changing the photon energy allows one to probe different points in k space. While the data are consistent with the band diagram, the data alone do not yield the value of k ; this is the " k problem." At higher photon energies when the direct transition peak is away from the Fermi level, one sees a Fermi edge in the emission. The "background" from the Fermi level to the direct transition peak is mostly caused by surface photoemission (or indirect transitions, to be discussed later). There is also phonon-assisted photoemission that can give rise to a background (especially at high temperatures). The situation is similar for normal emission from Ag(111) as shown in Fig. 9. The system is like a one-dimensional semiconductor along this direction, with the Fermi level being very close to the lower edge of the gap. Again, the data show an initiate state peak that moves with photon energy in agreement with the band diagram, but the data along do not yield information about k. Different from the (100) case is that there is a surface state as indicated. Surface states can exist within bulk gaps (where Bloch states cannot propagate, or must be damped). For Ag(111), this surface state (a Shockley surface state) happens to be very close to the lower edge of the gap and is just below the Fermi level. A bulk gap does not necessarily support a surface state; it depends on the crystal potential and the surface boundary condition. Other features of interest in the spectra include the vacuum level cutoff. As in the Ag(100) case, there is background-like indirect emission. Also, note that the direct transition peak is asymmetric. It seems to have a long tail on the lower binding energy side. These features are caused by surface photoemission (or indirect transitions, to be explained later). Surface photoemission is very important for thin films where quantum well peaks are often excited by this mechanism. Fig

6 Fig. 10 The k problem was a key issue of interest in the s. It hampered straightforward band mapping, a quest of importance to testing the foundation of the quantum theory of solids. We (the IBM group at that time) decided to take a look at this problem using cleaved GaAs(110) as the test system (Fig. 10). The cleaved surface can be nearly defectfree. The band structure of GaAs is relatively simple, and so this system is quite ideal for the experiment. Recall the earlier slides about photoemission from Ag(100) and Ag(111). At low photon energies, the relevant band diagrams are simple just one initial band and one final band. At higher photon energies, there can be a great many final bands caused by Brillouin zone folding of the nearly free electron bands. At ~30 ev and above for most solids, the final bands form a "spaghetti." With lifetime broadening, these bands merge into essentially a continuum. Thus, the details of the band structure at high energies become unimportant; the crystal potential, of the same order as lifetime broadening, can be ignored. However, not all final states in this continuum are important. Only those that can effectively couple to the detector need to be considered. A clever way to visualize what states are important is to reverse the sense of time. Conceptually, one shoots a beam of electrons backwards from the detector toward the sample, as in a LEED (low energy electron diffraction) experiment. The electron beam gets diffracted by the crystal potential. At sufficiently high energies (compared to the crystal potential), the dominant beam inside the crystal is the (0, 0) beam (undiffracted beam), which can be well described by a plane wave. For normal emission, this plane wave is just exp( ik f z). The wave vector kf (inside the crystal) is related to the wave vector k in vacuum by refraction. Refraction comes about because of a shift in average electrostatic potential between the inside of the crystal and vacuum. The average crystal potential relative to vacuum, called the inner potential, is ~10-15 ev typically. For off normal emission, refraction can change the direction of the beam. 219

7 Reversing the sense of time for this LEED state yields the time-reversed LEED state, which is just the photoemission final state. Thus, the dominant photoemission final state is just a plane wave state inside the crystal that is coupled to the state in vacuum via refraction by the inner potential. From the measured wave vector in vacuum, one can easily determine the wave vector kf in the solid. This kf is conserved during the photoexcitation process by direct transition, and so it is just the k of the initial state. As the photon energy increases, k in vacuum increases, and so is kf in the extended zone. The directtransition peak positions should trace out the band dispersion relations. This is the case for the data from GaAs(100) (see the figures). There are four dispersing peaks corresponding to the four valence bands in GaAs. At ~25-30 ev photon energy, kf is near the zone center. As the photon energy increases, the peaks move in accordance with the band dispersion relations. At ~60 ev, the third band goes through its minimum, Σ min 1. At ~80 ev, the bands reach the zone boundary. At even higher photon energies, the band dispersion relations repeat. Additional peaks A and Fig. 11 Fig. 12 A' in Fig. 10 are Auger transitions associated with Ga and As. Other stationary peaks are caused by indirect transitions and can be related to features in the one-dimensional density of states as labeled in the figure. The symbols in the lower panel in Fig. 10 are the experimentally determined dispersion relations. The solid curves are from a calculation (see Fig. 11). The agreement is excellent. Thus, the k problem is solved. Figure 11 shows the experimental band dispersion relations of GaAs along the three major symmetry directions based on normal and off-normal emission data. The dashed curves are pseudopotential calculations from Pandey and Phillips performed before the experiment. The agreement is excellent. This is one of the relatively rare instances where the theory is done before the experiment. It is a testimonial to the success of the theory. Figure 12 is the band structure of Cu determined by Yves Petrov and his colleagues based on the same idea. The method works very well. 220

8 Fig. 13 What about the peak width in angle-resolved photoemission spectra? Let us take a look at the Ag(100) case again. The peaks are about 1 ev wide and the line shapes are asymmetric. These are actually intrinsic features, not caused by experimental resolution or sample problems. Let us focus on the line width first. The figure on the right shows a schematic band diagram including lifetime broadening of the initial state (photoexcited hole or quasiparticle) and the final state (photoexcited electron in the solid). The quasiparticle can decay, and its lifetime is often the quantity of interest. The photoexcited electron can be scattered by electrons, phonons, or impurities. Electron-electron scattering dominates here, and it accounts for the short escape depth (or a short lifetime and a large lifetime width). The diagram shows that the total spectral width corresponds to the overlapping region Fig

9 between the broadened final band and the vertically translated, broadened initial band. The result is a width ~1 ev, and it is dominated by the final state lifetime broadening in this case. Generally, quasiparticle lifetime widths are small near the Fermi level. Thus, direct-transition peak widths are often dominated by final state lifetime broadening. To compute the photoemission linewidth, we use Fermi's golden rule (Fig. 14). It involves the matrix element squared, a delta function for energy conservation, an angular delta function (assuming perfect angular resolution), a spectral weight function (a Lorentzian) for the initial state, and another one for the final state. The integration is over all possible initial and final states. The integral can be worked out if we take a linear approximation for the band structure and assume that the matrix element is a constant within the region of interest. The result is a Lorentzian with a width γ given by the formula shown in Fig. 15, where Γ i and Γ f are the initial and final state widths, v denotes the group velocity, and θ is the emission angle. For normal emission, θ = 0. The formula is simpler. Still, the peak width is not a simple combination of Γ i and Γ f, and extracting these quantities from the experiment is not necessarily straightforward. Fig. 15 For initial states near the Fermi level, Γ f >> Γ i, typically. At normal emission, the peak width is often dominated by Γ f. This is the case for the 13 ev spectrum for Ag(100) (see figure). For off-normal geometries, the situation can become quite interesting. As shown in the figure, the spectrum taken at ev and at an emission angle of 80 degrees has a peak width much narrower. Note that the binding energy of the peak is similar in the two cases, and the initial state lifetime width due to electron-electron scattering is expected to be similar. 222

10 Fig. 16 The very narrow peak width follows from the formula given in Fig. 16. With the photon energy and emission angle chosen, it can be shown that the perpendicular component of the initial state group velocity is zero ( vi = 0 ). The peak width does not depend on Γ f, and is proportional to Γ i. The factor of proportionality C depends on v i //, and can be less than unity if v i // < 0. Thus, the measured peak width can be "compressed" to less than the quasiparticle lifetime width! This compression does not violate quantum mechanics, and is simply the result of the kinematical constraints associated with the angle-resolved photoemission geometry. Fig

11 To accentuate the compression effect, sinθ should be made as large as possible. With θ chosen to be 80 degrees in the experiment, the compression factor is about 2 for the case under consideration. In conclusion, the measured peak width is related to the quasiparticle lifetime width, but the relationship is generally nontrivial. The above analysis leaves behind a question the direct transition peak is asymmetric, not a Lorentzian. Let us review the Ag(111) case discussed earlier. In each spectrum, the direct transition peak has a long tail on the low binding energy side. As a result, the "background" intensity appears higher on the low binding energy side a curious phenomenon. Note that the secondary electron background arising from inelastic scattering generally rises toward higher binding energies; this is opposite to what we see here. Fig. 18 Here is a detailed view of the 8 ev spectrum. Prominent features include a surface state peak, the direct transition peak, the indirect transition region, a cutoff of the indirect transition at the band edge, and an inelastic background (Bkg). The indirect transition is mostly on the low binding energy side of the direct transition peak. The solid curve is the results of numerical modeling to be described next. Fig

12 Let us take a detailed look at the optical transition matrix element (Fig. 19). It involves Ap and p A. A straightforward derivation yields a "bulk" matrix element Mb and a "surface" matrix element Ms. Mb involves A and is the usual momentum matrix element. In the long wave length limit (at low photon energies), it gives rise to direct, vertical (momentum conserving) transitions in the bulk. Ms involves A (divergence of the vector potential). This term was recognized fairly early on to be important, but many textbooks ignored it in any case because of difficulties in actual calculations. It was sometimes argued that a gauge could be chosen to make A zero. This is actually not possible near a surface where there is a dielectric discontinuity. See the slide the boundary condition for the perpendicular component of the vector potential across the surface involves the dielectric function of the solid; this leads to a discontinuity. Upon differentiation to obtain A, a delta function emerges. The matrix element Ms is thus proportional to the dielectric discontinuity and the initial and final state wave functions at the surface. It gives rise to surface photoemission any initial state with a nonzero amplitude at the surface can contribute to the photoemission signal. Unlike the bulk term, Ms does not require momentum conservation. Ms is responsible for the indirect transition continuum in the spectrum. The total matrix element is the sum of Mb and Ms, and there can be interference between the two contributions. Mb by itself gives rise to a direct transition peak, while Ms by itself gives rise to a continuum spanning the entire valence band width. Interference gives rise to an asymmetric peak as seen in the experiment. The line shape is somewhat like that of a Fano resonance. On the low binding energy side of the direct transition peak, Mb and Ms interfere constructively, leading to a high continuum intensity. On the high binding energy side, they interfere destructively. A note of caution: the dielectric discontinuity at the surface is actually not infinitely sharp. The delta function mentioned above is only an approximation. A better approximation would be a peak with a width about the size of an atom. Figure 20 is a summary of our numerical modeling. The coefficient C is proportional to the dielectric discontinuity. A detailed analysis shows that there should be a factor of ~ 2 π. Since this estimate is very rough, we treat C as a fitting parameter instead. The results from the fitting show that this estimate is pretty close. The other key points of the modeling are listed in Figure 20. Fig

13 Fig. 21 The fit for the 8 ev spectrum is shown as the solid curve (left figure in Fig. 21). In addition to the direct and indirect transitions, we have added a Shirley background function to simulate the effect of inelastic scattering. It is constructed to be proportional to the integrated area to the right under the curve. In other words, the inelastic intensity at each energy is assumed to be proportional to the integrated spectral weight at higher energies. The proportionality factor is chosen as a fitting parameter. Also included in the fitting is a surface state peak. The fit describes the data well. The value of C from the fitting is shown in Fig. 21. Shown on the right in Fig. 21 is an analysis of the two contributions Mb and Ms. If we include Mb only in the calculation, a symmetric peak is observed. If we include Ms only in the calculation, we should get something like a one-dimensional density of states. The calculated density of states, shown in the figure, exhibits a sharp spike at the band edge. But this is not seen in the calculated spectrum. Why? Fig

14 Presented in Fig. 22 are the calculated probability density functions for the initial states at 1, 0.1, and 0.01 ev below the band edge. In each case, the probability density function shows rapid oscillations modulated by an envelope function. The rapid oscillations are related to the band edge wave vector. The envelope functions are related to how far away the energy is from the band edge. Recall that the term Ms is proportional to the amplitude of the initial state at the surface. As the electron energy approaches the band Fig. 23 edge, the wavelength of the envelope function becomes very long, and there is very little probability left near the surface. This depletion of the probability density at the surface effectively suppresses the spike in the onedimensional density of states, resulting in a rounded cutoff of the indirect-transition spectrum at the band edge. This depletion of the probably at the surface near a band edge is not a universal feature for all materials. For Ag(111), the Shockley surface state is very close to the band edge. The total charge density including both the surface state and the bulk states should be more or less Fig. 24 uniform all the way up to the surface. Since the surface state is concentrated near the surface, the bulk states with energies close by must be correspondingly depleted near the surface. As mentioned earlier, the factor C in the surface photoemission term must be roughly 2 π times the dielectric function minus 1. The dielectric function has a complex value depending on frequency. Figure 23 shows the values of C extracted from fitting the data taken from Ag(111) at different photon energies. The corresponding values of 2π ( ε 1) are also shown for comparison, where the values of ε are taken from data compiled from earlier optical measurements. As seen in the figure, the real parts are in close agreement. The imaginary parts differ by an offset. Overall, the agreement is not bad considering the approximations employed in the model. Let us take a look at the Ag(100) case in Fig. 24. The spectrum taken at 14 ev shows an asymmetric line shape just as in the Ag(111) case. This asymmetry was evident in data contained in some old papers, but it did not seem to arouse curiosity at that time. 227

15 Fig. 25 Figure 25 is an experimental verification of the surface nature of the asymmetric lineshape. The bottom spectrum taken from Cu(100) shows an asymmetric lineshape as the Ag(100) case. The solid curve is a fit. The top spectrum is taken from Cu(100) covered by 2 monolayers (ML) of Ag, where the lineshape is symmetric. The dielectric discontinuity at the surface is now replaced by one at the interface: εcu ε Ag. This is much smaller and so the indirect transition term Ms is very much suppressed. The interference effect becomes negligible, and the peak becomes symmetric. Naturally, one wonders about the dielectric discontinuity at the surface of the 2 ML Ag film, which should give rise to indirect transitions for the initial states that have a finite amplitude at the surface. It turns out that this emission mechanism is important for quantum well states, which will be discussed later. In the present case, the Ag film is so thin that no quantum well states exist within the region of interest. Fig. 26 Recall that the model employed time-reversed LEED states in the calculation. For the corresponding LEED state, an electron beam is incident on the crystal. The transmitted 228

16 wave into the crystal decays exponentially with a characteristic mean free path. The timereversed LEED state employed in the calculation includes this mean free path as a fitting parameter. It controls the lifetime width of the direct transition peak. The results from fitting to the data for Ag(111) and Ag(100) are shown in Fig. 26; the two data sets cover different ranges in photon energy, and yet the results join smoothly. The mean free path at low photon energies are quite long, almost 50 Å or so. The reason for the longer mean free path at low photon energies is that there is less phase space for electron scattering. Fig. 27 Figure 27 summarizes the key points discussed so far. Photoemission probes both bulk and surface states. The mean free path (or escape depth) can be as short as a few Angstroms (a couple of atomic layers), and yet one can still detect substantial emission from the bulk states, as seen in examples presented so far. The optical transition matrix element has two contributions: a bulk term involving the momentum matrix element, and a surface term arising from a nonzero divergence-a at the surface. The surface term by itself leads to continuum emission that often resembles a k_parallel-resolved, local density of states at the surface (assuming that the final state wave function does not change drastically over the range of interest). This surface term does not involve the escape depth directly. The bulk and surface terms can interfere, giving rise to an asymmetric direct-transition peak. A nonzero divergence-a can arise wherever there is a dielectric discontinuity. So, one has to be careful in dealing with non-uniform systems, such as thin films. Thin films will be the next topic of discussion. Typically we try to avoid direct transitions by choosing the photon energies appropriately. Surface photoemission then dominates, and the results should be approximately a k // -resolved, local density of states at the surface, which could include any surface states in the system. Again, photoemission escape depth is not directly relevant for surface photoemission. As will be discussed later, quantum well states can be observed in rather thick films. The limit on quantum interference is related to the quasiparticle coherence length, not the escape depth after photoexcitation. 229