quantum physics of thin metal films - university of...

2 AAPPS Bulletin April 2008, Vol. 18, No. 2

Highlight of the Issue

Quantum Physics of Thin Metal Films

Tai Chang Chiang

Department of Physics, University of Illinois at Urbana-Champaign,1110 West Green Street, Urbana, IL 61801, USA


Thin metal films have been studied for decades and are widely employed in a myriad of applications. Recent advances in preparing atomically uniform films have transformed this field of research; measurements can now be performed with precisely known film thicknesses in terms of the number of atomic layers. It is shown that the physical properties of ultra thin films can deviate substantially from the bulk counterparts, and such differences are related to the electronic structure that can be determined directly from angle-resolved photoemission. Specifically, the film property variations as a function of film thickness and boundary conditions can be largely understood in terms of a “one-dimensional shell effect” akin to the periodic property variations of the elements in the periodic table. More com-plicated electronic effects can arise from coherent coupling of the electrons in the film to the substrate electronic states and from diffraction of the electrons by the substrate atomic structure. This review is a discussion of some key concepts that underlie the quantum phenomena in thin metal films.

1. INTRODUCTIONSurfaces, thin films, and self-organized surface structures can exhibit interesting and useful properties markedly different from their bulk counterparts due to sym-metry reduction, geometric confinement of electrons, and boundary effects. The underlying quantum physics is a corner-stone for nanoscale science [1] which is a broadly based interdisciplinary enterprise highly relevant to the advancement of materials, devices, and technologies. Thin films, in particular, are of fundamental

interest; the simple geometry facilitates a detailed exploration of the connection between atomistic details and macroscopic physical and chemical properties. Thin films also provide a research path bridging surfaces and bulk materials. Through sys-tematic studies of films with varying thick-nesses, effects pertaining to the surface, the interface, and the bulk can be identified and characterized in detail. A thorough understanding of the basic scientific issues in such systems enables advanced materi-als concepts through artificial layering, interfacial engineering, layer alloying, and doping. Furthermore, preparation and processing of thin films by deposition and annealing can lead to novel self-assembled and self-organized structures that function as quantum dots, wires, and stripes. A de-tailed investigation of thin film effects and phenomena is essential for establishing a basis for understanding the evolution, kinetics, energetics, and properties of these nanoscale systems. The vast opportunities for scientific and technological advances have fostered a strong interest in the basic physics of thin metal films.

A key issue in thin film physics is quan-tum confinement of the electrons and the resulting quantization of the electronic wave vector along the direction perpendic-ular to the film surface [2]. The continuum states characteristic of the bulk are thus replaced by a discrete set of quantum well states or subbands with their energies de-pendent on the boundary conditions. The resulting modifications to the electronic structure affect the physical properties, leading to atomic-layer-by-atomic-layer variations that can be quite dramatic [3]. Experimental results on property

variations reported in the literature include studies of surface energy, thermal stability, work function, surface adsorption, elec-tron-phonon coupling, superconducting transition temperature, etc.

The subband electronic structure in thin films can be mapped directly by angle-resolved photoemission spectroscopy. Recent advances in thin film preparation have made it possible to create atomically uniform films, thus facilitating highly pre-cise measurements of the electronic ener-gies as a function of system configuration including the film thickness and interfacial structure. Examples will be presented below to show how this information can be directly linked to physical property variations. To a first approximation, the electronic structure of a thin film can be described in terms of the standard model – a particle confined in a quantum box – as commonly described in quantum mechan-ics textbooks. As the film thickness in-creases, the number of occupied subbands (below the Fermi level) increases. Each time a subband crosses the Fermi level, the system properties must change in response to the new electronic configuration. Such Fermi level crossing happens periodically as a function of the film thickness. The re-sult is a damped oscillatory modulation of properties as a function of film thickness, akin to the periodic property variations of the elements in the periodic table. This “one-dimensional shell effect” has been observed in a number of cases.

While the particle-in-a-box model works well in most cases, there are situa-tions where this model does not adequately describe the experimental results for a

AAPPS Bulletin April 2008, Vol. 18, No. 2 3


number of reasons. For example, the de-tailed atomic structure of the substrate can be important, and the interfacial atomic potential can diffract or scatter the elec-trons in the film. Coherent coupling of the electronic states in the film to the substrate states can also be important, leading to partial confinement or hybridization. This area of research is progressing rapidly, and there is a large body of existing literature. Space limitations do not allow a detailed discussion here. This paper contains a brief review of the basics, with selected examples to illustrate the essential ideas.

2. SPECTROSCOPY OF THE ELECTRONIC STRUCTURE OF FILMS

Quantum well states in films appear as discrete peaks in angle-resolved photo-emission spectra for electron emission along the surface normal, the direction of confinement [2, 4, 5, 6, 7, 8, 9, 10]. Shown in Fig. 1 is a schematic diagram for the normal-emission geometry. Experimen-tal results taken from Ag films grown on Fe(100) are shown in Fig. 2 [11, 12]. The discrete peaks in the spectra correspond to quantum well states, or standing waves of electrons formed by coherent multiple reflections between the two boundaries of the film. The bottom spectrum is for a film with a thickness of 38 monolayers (ML). Adding 0.5 ML to this film yields the middle spectrum, which exhibits two sets of quantum well peaks. One set is at the same positions as the 38 ML case, and the other corresponds to a thickness of 39 ML. Adding another 0.5 ML to this film for a total coverage of 39 ML (top spec-

trum), the peaks corresponding to 38 ML are completely suppressed, and only the 39 ML peaks remain. This discrete layer behavior, observed over a wide range of thicknesses to over 100 ML, establishes that the film is uniform on an atomic scale within the region probed by photoemission (~1 mm).

Such atomically uniform films had been thought to be impossible. Film growth by deposition is inherently a random process, which tends to yield roughness that usually scales with the film thickness. Creating atomically uniform films requires tailoring the growth process to follow a pathway that favors the formation of a uniform thickness. Determining the absolute thick-ness of a thin film can be problematic, but with uniform films, quantum-well spectroscopy provides a solution. Fig. 3 presents normal-emission spectra for Ag coverages of N = 1-15 ML [13]. The peaks move discontinuously, and the spectrum for thickness N shows no emission from peaks associated with thicknesses of N±1. This discrete atomic layer resolution al-lows absolute calibration of film thickness by atomic layer counting.

The energies of the quantum well states are determined by the Bohr-Sommerfeld quantization rule:

2kNt + Φ = 2nπ, (1)

where k is the magnitude of the wave vector of the Bloch electron along the surface normal direction, t is the mono-layer thickness, Φ is the sum of the phase shifts at the two film boundaries, and n is a quantum number. The quantum numbers n = 1-4 are labeled for the quantum well peaks in Fig. 3. Both the wave vector and the phase shift depend on energy. With quantum well peak positions determined for a number of diffeent film thicknesses, one could solve the above equation to yeild the bulk band structure E(k). This is a very accurate method for band structure determination [2, 13].

Fig. 4 presents another case in which atomically uniform films have been suc-cessfully prepared [14, 15]. The system is Pb films grown on Si(111). Unlike the case of Ag on Fe(100) in which there is a very good lattice match, Pb and Si have very different lattice constants. Neverthe-less, the films are atomically uniform as verified by the discrete peak evolution in the normal-emission phtoelectron spectra. The in-plane crystallographic directions of the Pb film are parallel to those of the Si substrate, but the lattices are otherwise incommensurate. Pb is a free-electron-

Fig. 1: Schematic diagram for photoemission from a film along the surface normal direc-tion.

Fig. 2: Normal-emission spectra taken from 38, 38.5, and 39 ML of Ag on Fe(100). The 38.5 ML spectrum shows two sets of peaks, one set at the 38 ML positions, and the other at the 39 ML positions.

Fig. 3: Normal-emission spectra of Ag films on Fe(100) with thicknesses N = 1-15. The quantum numbers (1-4) are indicated.



like metal. Its electronic structure can be viewed, to first order, as a jellium (or a Fermi sea of electrons). It costs very little energy to move the electrons around. The dangling bonds on the Si(111) surface can be easily terminated by the mobile electrons in the Pb jellium, and so epitaxial constraint is unimportant for this type of interfacial electronic bonding. As a result, the Pb film simply adopts its own natural lattice constant to minimize the strain energy in the film [16].

3. SURFACE PROPERTIES OF THIN FILMS – WORK FUNC-TION AND CHEMISORPTION

Per density functional theory, the ground state of a system is a unique functional of the electron density. Each time a sub-band edge crosses the Fermi level as the film thickness varies, the electron density function changes, and the physical prop-erties should change correspondingly. Such changes generally follow a damped oscillatory behavior as a function of N with a functional form resembling Friedel oscillations. The system should become bulklike in the limit of a very large film thickness. The oscillation period can be found from Eq. (1). Taking the difference between two consecutive crossings (∆n = 1) at the Fermi level yields

∆Nt = πkF =

2λF . (2)

Thus, the oscillation period is just one half of the Fermi wavelength. This is the same

oscillation period for the giant magneto-resistance (GMR) effect in certain multi-layer systems [17, 18, 19]. For Ag(100), the period is 5.8 ML. This is a dominant contribution to the variations in physical properties, but there can be others.

A measurement of the work function of Ag/Fe(100) for N = 0-15 reveals such quantum oscillations, as shown in Fig. 5 [20]. The upper panel of the figure shows the energy positions of the quantum well states deduced from a fit to the experi-mental data. As photoemission measures only occupied states, the points above the Fermi level are deduced from this model fit. The fit is excellent for the states below the Fermi level. Crossings of the Fermi level for the different subbands (n = 1-4) are marked by arrows. Based on fairly gen-eral arguments, the work function should exhibit a dip (or cusp) at each crossing [21, 22]. This prediction corresponds closely to the experimentally observed work func-tion variations. Also shown are results from first-principles calculations of the work function. A complication is that the Ag and Fe lattices are slightly mismatched. The figure shows two calculations. The

one labeled “unstrained” was for a system in which the Fe substrate was slightly strained to conform to an unstrained Ag film. The results are in fairly good agree-ment with the experiment; specifically, the dips at the first two crossings are well reproduced. One possible source for the discrepancy at N = 2 and below is strain effects. Films this thin are likely strained to conform to the substrate lattice, while thicker films are strain relieved. The theo-retical results labeled “strained” were for a system in which the Ag film was slightly strained to conform to an unstrained Fe substrate. Ignoring an overall shift, the point at N = 2 becomes much lower than the point at N = 3, with a difference very close to the experiment. Thus the good agreement between theory and experiment is extended down to N = 2.

The work function is just one of the many surface properties that are closely coupled to the electronic structure. By implication, chemisorption and catalytic properties of thin film surfaces can also exhibit quantum oscillations. These effects have been reported in the literature [23]. Nanoscale engineering of catalytic mate-rials has been a topic of intense interest. Uniform thin films are not necessarily fit for catalytic applications, but they do pro-vide a basis for understanding the general phenomenon of property modulation by varying the system dimensions.

4. THERMAL STABILITY AND MORPHOLOGICAL EVOLU-TION

Thermal stability is an important practi-cal issue for thin film applications. Since atomic motion associated with thermal instability is generally thermally activated with an exponential dependence on the energy difference between different con-figurations, a slight difference in electronic energy can have a large effect on the stabil-ity. For Ag films on Fe(100), the measured maximum stability temperatures for dif-ferent thicknesses are presented in Fig. 6(a) [24]. In the experiment, each Ag/Fe film was ramped up in temperature until its morphology changed. Films with thick

Fig. 4: Photoelectron intensity at normal emission as a function of film thickness and binding energy for Pb films prepared on a Pb-terminated Si(111) surface.

Fig. 5: (a) Energies of quantum well states at normal emission as a function of N for Ag on Fe, deduced from a fit to the experimental quantum well peak energies below the Fermi level. The arrows indicate Fermi level cross-ings of subbands. (b) Measured and computed work functions as a function of N.



nesses of N = 1, 2 and 5 ML were stable to temperatures over 800 K, while other films for N up to 15 began to bifurcate at T ~ 400 K into adjacent-integer-monolayer thicknesses N ± 1. Of special interest is the case of N = 5. This thickness was so stable that the film survived the highest annealing temperature available during the experiment, at which the sample was observed to glow in the chamber. Chang-ing the thickness by just one monolayer to N = 4 or 6 made the film unstable at about room temperature. The effect is very dramatic.

Each quantum well state of Ag/Fe as seen in Fig. 3 corresponds to a subband which disperses as a function of the in-plane momentum k||. From photoemission results, we can compute the total electronic energy A(N) of the system by summing over the occupied states. Cutoff at the Fermi level of subband occupancy gives rise to monolayer-by-monolayer variations in the total electronic energy, thus affecting the thermal stability. The quantity relevant to stability against N → N ± 1 bifurcation is the energy difference

∆(N) ≡ 21 [A(N+1)+A(N-1)]-A(N). (3)

This is proportional to the discrete second derivative of A(N). A large positive ∆(N) corresponds to a stable film thickness. Fig. 6(b) shows the results of a calculation based on photoemission measurements (not applicable for N = 1). Indeed, N = 2 and 5 should be particularly stable, in agreement with the experiment. A first-principles total-energy calculation has confirmed the finding [25].

The free energy as a function of film thickness can be readily surveyed by measuring the roughness of a film that has been annealed to high temperatures, as demonstrated in a study of Pb films deposited on Si(111) [26, 27]. From Eq. (2) and taking into account the discrete atomic ayer structure of films, Pb films should have a quantum oscillation period of 2.2 ML, which implies a nearly bilayer modulation of properties. Thus, films with

even N are expected to be markedly dif-ferent from films with odd N. However, over a sufficiently wide range of N, the phase of the even-odd oscillations can reverse because the period is not exactly 2 ML. The result is a beating pattern with a period of 9 ML superimposed on the bilayer oscillations.

Fig. 7 shows the annealing behavior of a Pb film on Si(111) with an initial cover-age of 11 ML. Plotted is the film thickness distribution pN expressed in terms of a per-centage of the surface coverage deduced from synchrotron x-ray reflectivity data. At the base deposition temperature of 110 K, the initial film thickness distribution is narrow, but the film is not atomically uni-form. It turns out that 11 ML is an unstable thickness (compared to the neighboring thicknesses 10 and 12 ML); so it is difficult to prepare an atomically uniform film at this particular thickness. As the sample is progressively annealed to higher tempera-tures as indicated in the figure, the film thickness distribution goes through several stages. The first stage is “bifurcation,” and the result is a film largely made of the more stable thicknesses 10 and 12 ML. The next stage is the “uniform-height-island” stage with the surface dominated by islands 12-ML high separated by a wetting layer. This comes about because of phase separation

of the system into a state corresponding to a local minimum in the surface energy at thickness N = 12 and a state corresponding to the global minimum at N = 1 (wetting layer). The thickness 10 ML is actually more stable than 12 ML. However, the system is prevented from forming uniform 10 ML islands because the deep minimum in the surface energy at N = 1 favors the formation of taller islands to increase the area covered by the wetting layer. Similar uniform-height islands have been observed by STM and x-ray diffraction for growth at intermediate temperatures at which the surface mobility is sufficiently high for self organization [28, 29].

Annealing to higher temperatures re-sults in a broadening of the thickness dis-tribution. At 280 K, the residual preference for 12 ML from the uniform height phase has disappeared, and the film has reached local equilibrium. The resulting broad distribution is traditionally described in terms of a roughness. However, an inspec-tion of the data reveals a structure within this thickness distribution. Superimposed on the dotted “background roughness” related to entropy effects are bilayer oscil-lations with even-odd crossovers occur

Fig. 6: (a) Temperature T at which a Ag film on Fe with an initial thickness of N becomes unstable. (b) Calculated energy difference ∆(N) against bifurcation.

Fig. 7: Island height distribution for a Pb film with an initial coverage of 11 ML after anneal-ing to various temperatures. The final distribu-tion shows bilayer oscillations with even-odd crossovers at 9 layer intervals.



ring every nine atomic layers as indicated by the triangles. The relative population differences can be related to a Boltzmann factor. From the measurements, the surface energy for different film thicknesses can be extracted. The results are well described by a functional form based on a free elec-tron model:

Es(N) = Bsin (2kFNt + φ)

Nα + C, (4)

where kF is the Fermi wave vector, t is the monolayer thickness, B is an amplitude parameter, φ is a phase shift that depends on the interface properties, α is a decay exponent, and C is a constant offset. A plot of this function with the parameters chosen for a best fit to the Pb/Si data is shown in Fig. 8 (the constant offset is ignored).

5. STRUCTURAL RELAXATION OF THIN FILMS

An issue of great interest for smooth films is the internal layer structure. The neighboring atomic layer spacings could deviate from the bulk value in response to the modified electronic structure near the boundaries [30, 31]. Specifically, confinement leads to Friedel-like charge oscillations within the film with a period of 2.2 atomic layers. This period, given by one half of the Fermi wavelength, is the same as that governing the film property variations as a function of film thickness. A model calculation for a freestanding 7-ML Pb film is shown in Fig. 9(a) as an illustration. The electronic density wave exerts a force on the atomic planes, lead-ing to lattice distortions. The force can be calculated, to first order, from either the electrostatic field or the local charge

gradient, as shown in Fig. 9(b). The two methods of calculation yield very similar answers within the film. This approxi-mately bilayer relaxation effect is quite large for Pb films on Si(111), as verified by synchrotron x-ray diffraction experiments. Results from first-principles calculations are in good accord [32].

6. ELECTRON-PHONON COU-PLING AND SUPERCONDUC-TIVITY

Electron-phonon coupling plays a cen-tral role in many important and useful physical effects and phenomena, includ-ing superconductivity, charge density waves, and structural phase transitions [33]. Angle-resolved photoemission has emerged as a powerful tool for measuring this quantity [34, 35]. At sufficiently high temperatures and at energies not too close to the Fermi level, the lifetime width ∆E of a given electronic state as measured by photoemission shows a linear dependence on temperature T caused by phonon scat-tering. The slope of this linear dependence is related to the electron-phonon coupling strength. The so-called electron-phonon mass enhancement parameter λ is deter-mined by the following relationship:

λ = 2πkB

1dT

d∆E , (5)

For thin films, this quantity shows oscil-

latory variations as a function of film thickness for the same reason as discussed above – a one-dimensional shell effect.

Measurements of λ have been carried out for Ag(100) films grown on Fe(100) [36, 37]. The results are summarized in Fig. 10. The top panel shows the bind-ing energies of the quantum well states of interest. The middle panel shows the corresponding λ determined from pho-toemission. The bottom panel shows the results from a simple model calculation, which agree well with the experiment. As expected, λ exhibits oscillatory variations as a function of N. Furthermore, there is a ~1/N decay pattern that overlays the oscillations, leading to an enhancement of λ at small N. This enhancement can be attributed to interface effects. A large λ is often associated with a high superconduct-ing transition temperature within the BCS model. The results suggest an interesting possibility of enhanced or novel supercon-ducting behavior in thin films. Ag in the bulk form is not superconducting. For Ag on Fe, the chances of finding supercon-ductivity are probably slim because Fe is

Fig. 8: Surface energy for Pb films on Si(111) de-duced from an x-ray analysis of film roughness.

Fig. 9: (a) Calculated charge density of a 7-ML Pb film. (b) Force on the atomic layers calculated from the electrostatic field or charge gradient. The two calculations agree closely within the film, but differ outside.

Fig. 10: Top, energies of quantum well states for Ag on Fe as a function of N. Middle and bottom, measured and calculated electron-phonon coupling parameters.



ferromagnetic. Other substrates may be better candidates.

While enhanced or novel superconduc-tivity is yet to be discovered, oscillatory variations in the superconducting transi-tion temperature TC has been found in Pb films deposited on Si(111) [38, 39, 40]. The results from an experiment are shown in Fig. 11 [38], where bilayer oscil-lations are evident. The samples used in this experiment were Pb films deposited on Si(111), then coated with a Au over-layer for protection in order to transfer the sample from the growth chamber through air to the low temperature transport mea-surement equipment. The Au overlayer could introduce a phase shift, affecting the oscillation pattern.

7. BEYOND THE SIMPLE PAR-TICLE-IN-A-BOX MODEL: SUBSTRATE EFFECTS

The simplest picture for a thin film quantum well is that of a pair of paral-lel electron mirrors that reflect electrons back and forth to form standing waves in a manner similar to the optical modes in a Fabry-Pérot interferometer [12]. The ac-tual electronic structure of a thin film can be much richer. The main differences are: (1) electrons in the substrate can couple to those in the film, resulting in a coupled structure that can be difficult or impracti-cal to describe in terms of a single-particle phase shift analysis, as given by Eq. (1) [41, 42]; (2) the film can support surface

states, which interact with the rest of the system to create a complicated spectral weight function [43]; and (3) the cor-rugation potential at the film-substrate interface can lead to “multi-beam” mixing for incommensurate interfaces, resulting in complex electronic effects [44].

As an example, Fig. 12 displays photo-emission results for Ag film thicknesses of 8, 8.6, and 9 ML on Ge(111) [41]. Each panel shows a Shockley surface state SS just below the Fermi level and several quantum well subbands with roughly para-bolic dispersion curves. Looking closer, a band splitting is evident within the circle in Fig. 12(b); the two bands represent a linear combination of the bands in Figs. 12(a) and 12(c). Here, a thickness change by 1 ML causes a noticeable shift of the band, and the band splitting is caused by the presence of two thicknesses, 8 and 9 ML, in the film. This demonstrates atomic layer resolution in this system and, more importantly, atomic layer uniformity at integer monolayer thicknesses.

The three concave curves in Fig. 12 indicate the bulk band edges of Ge. The roughly parabolic dispersions of the quan-tum well subbands show a break (or kink) as they cross the band edges. The square box in Fig. 12(a) indicates such a kink. This portion of the data is shown in detail in Fig. 13(a), together with a model cal-culation in Fig. 13(b). Also shown are the corresponding energy distribution curves in Fig. 13(c). The quantum well peak is seen to split into two peaks near the cross-over point. Thus, the usual quasiparticle picture based on a phase analysis does not work here. The cause of the splitting and band distortion is a hybridization interac-tion across the interface between the elec-tronic states in the Ag film and those in Si below the band edge. The fit in Fig. 13 uses a Hamiltonian similar to that employed in the Anderson-Newns model.

The interface between a film and its substrate is not a flat mirror-like boundary, although it is often a very good first ap-proximation. The atomic structure can lead to interesting and important consequences. Specifically, a new kind of quantum well state (second-kind) has been observed in Ag films on Ge(111) [44]. In this case, the reflections at the interface are umklapp retroreflections caused by the substrate surface corrugation potential. The retro-reflections reverse the directions of the incident electrons at an oblique angle as opposed to specular reflections that bounce the electrons off to directions symmetric with respect to the surface normal. On the right-hand side of Fig. 14 are ray dia-grams illustrating the interference paths for the formation of quantum well states of the first kind and the second kind. The quantization conditions are also indicated. As observed by angle-resolved photoemis-sion, the retroreflections cause these new states to have a characteristic photoelec-tron emission pattern that is centered about directions away from the surface normal, providing a clear experimental distinction from the usual states. The results for a 13 ML Ag film on Ge(111), in an emission plane along the ГM direction, are shown on the left-hand side of Fig. 14. The sec

Fig. 11: Measured superconducting transition temperatures of uniform Pb films of various thicknesses on Si(111).

Fig. 12: Subband structure of Ag films on Ge(111) as observed by photoemission for three different thicknesses. The horizontal axis is the emission angle. The subbands are roughly parabolic in shape. SS indicates a Shockley surface state. The three concave curves cor-respond to Ge band edges. The band splitting highlighted by the circle in (b) at a noninteger layer thickness demonstrates atomic layer resolution. The kink highlighted by the box in (a) is caused by a hybridization interaction with the substrate states.



ond-kind emission pattern consists of a set of roughly parabolic bands centered about the M point of the Ge substrate. Also shown are the results of a model calculation, which agree well with the experiment. Note that the data in Fig. 12, acquired along a different direction (ГK ), do not show the same pattern.

A more subtle effect is that the substrate doping level can also influence the quan-tum well electronic structure of films. The two top panels in Fig. 15 compare angle-resolved photoemission data with in-plane dispersion along the ГK direction taken from Ag films of a thickness of N = 8 ML deposited on n-type Si(111) substrates with a doping level of (a) n = 2 ×1015/cm3 (lightly doped) and (b) n = 5 × 1018/cm3 (highly doped) [42]. The data from the lightly doped sample show a surface state (SS) of the Ag film and a set of quantum well subbands labeled by the quantum number ν = 1-3. These subbands exhibit “kinks” near the top Si valence band edge as explained above. An example of such a kink is indicated by an arrow. The data for the highly doped sample show similar features and, additionally, fringes near the Si valence band edge. Fig. 15 (c) presents an enlarged view of the region contained

within the rectangular box in Fig. 15(b) to show details of the fringes. Experiments carried out on p-type substrates show no such fringes.

We have performed a calculation using simple, but fairly realistic, model wave functions for the system. Fig. 16 presents a plot of the electronic potential for a Ag film of 8 ML on the highly doped Si at in-plane wave vector kx = 0.22 Å−1 (cor-responding to a polar emission angle of ~6°). The Si substrate has a gap in which the Fermi level EF lies. At the Ag-Si interface, the Fermi level is pinned near midgap. Band bending in the depletion region of Si gives rise to an approximately linear dependence of the valence band edge, as indicated in Fig. 16. Propagating electronic states in Si exist only below this edge. There is no gap in Ag at this kx , and all Ag states below the Fermi level are propagating in nature. The wave functions for the first five states, counting from the Fermi level, are shown. The first one (ν = 1) lies completely within the Si band gap. The other four states, at lower energies,

Fig. 13: (a) Data from the rectangular box in Fig. 12(a). (b) Results from a fit. The solid and dashed curves show the dispersion relations of the Ge band edge and the uncoupled quantum well state, respectively. (c) Corresponding energy distribution curves at different emission angles showing peak splitting.

Fig. 14: Right: schematic ray diagrams for the interference paths correspond-ing to quantum well states of the first kind (top panel) and the second kind (bottom panel). The first kind involves two specular (S) reflections, one each at the surface and the interface, while the second kind involves two S reflections at the surface and a pair of conjugate umklapp (U) reflections at the interface. The quantization condition for each case is indicated, where D = Nt denotes the film thickness. Left: angle-resolved photoemission data taken from a 13 ML Ag film on Ge(111) along ГM (top panel) and the same overlaid with labels and results from a model calculation (bottom panel). The set of approximately parabolic bands centered about the M point of Ge are quantum well states of the second kind. The quantum numbers n are indicated. Q1, Q2, and Q3 are quantum well states of the first kind. SS is a Shockley surface state.

Fig. 15: Angle-resolved photoemission data for 8 ML of Ag grown on (a) lightly doped n-type Si and (b) highly doped n-type Si. (c) is an enlarged view of the region contained within the rectangular box in (b). The photon energy used was 22 eV.



penetrate into the Si depletion region to various depths. The relatively shallow slope of the potential within the Si causes the electronic states with different ν’s to pile up near the Si band edge, giving rise to the closely spaced fringes. As kx increases, the Si band edge moves down, and more states become confined within the Ag film, as seen in Fig. 15. Fig. 17(b) presents the calculated dispersion relations of the confined quantum well states, which agree well with the data shown in Fig. 17(a). The curve in Fig. 17(a) represents the top band edge of Si.

For lightly n-doped Si substrates, the slope of the potential in Si is essentially zero. No fringes are expected, and none are observed. Likewise, no fringes are ex-pected or observed for p-doped substrates. Note that the ν = 2 and 3 states in Fig. 15(a) for the lightly doped sample, instead of bending over to form fringes, simply continue into the continuum region of Si, with a kink for each at the Si band edge as discussed above. The states within the Si band continuum are actually quantum well resonances, as they are not fully confined. The Ag-Si boundary causes partial reflec-tion; the resulting interference effect gives rise to broadened, quasi-discrete states. Such quantum well resonances are also present in Fig. 15(b) for the highly doped sample at energies beyond the range of band bending (or confinement).

The Ag films and the Si substrates are lattice mismatched and incommensurate. Nevertheless, the wave functions in Ag and Si can be matched across the inter-face plane. The resulting state is coher-ent throughout the entire system. The combination of a quantum well (Ag film) and a quantum slope (Si substrate) yields a rich electronic structure. The results demonstrate that coherent wave function engineering, as is traditionally carried out in lattice-matched epitaxial systems, is entirely possible for incommensurate systems. This can substantially broaden the selection of materials useful for coher-ent device architecture.

8. CONCLUDING REMARKSNanoscale phenomena are of fundamen-tal scientific interest and technological importance. Ultrathin films, with one na-noscale dimension that can be controlled with atomic layer precision, provide an excellent model platform for experiment-ing with size and boundary effects. This review presents examples to illustrate the basic quantum physics of thin films. A central issue is quantum confinement, which leads to discretization of the elec-tronic states. This in turn leads to a one-dimensional shell effect that modulates

the physical properties of thin films as a function of thickness. Such modulations might include damped periodic oscilla-tions superimposed on N-α-type trends as well as possibly more complicated behavior at very small film thicknesses. Property variations in thermal stability, work function, electron-phonon coupling, superconducting transition temperature, etc. have been demonstrated.

Also presented in this review are examples illustrating the effects of the substrates and the film boundaries on the film electronic structure. Studies have been performed in which modification of the film-substrate interface at the atomic scale leads to substantial changes of the film properties [45, 46]. These findings are expected based on the sensitivity of quantum well wave functions to boundary conditions. As a result of this sensitivity, the phase of the quantum oscillations as-sociated with the one-dimensional shell effect can be tuned.

Property tuning in thin films is useful for various applications. The discussion in this paper is limited to simple model systems. More exploratory work is needed, including systems made of materials with inherently strong electron correlation ef-fects. Multiple layer stacking and doping, prepared with atomic-scale precision, are areas ripe for detailed exploration. In view of the continued shrinking of device dimensions, quantum and coherent effects are expected to become an important is-sue in design considerations. These ef-fects also provide opportunities of novel device concepts. There is indeed a great potential for scientific and technological advances.

ACKNOWLEDGMENTSThis review is largely based upon work

supported by the U.S. Department of Energy (grant DE-FG02-07ER46383). We acknowledge the Petroleum Research Fund, administered by the American Chemical Society, and the U.S. National Science Foundation (grant DMR-05-03323) for partial support of personnel and

Fig. 16: Plot of the highly doped Si valence band and the first five quantum well wave functions at kx = 0.22 Å

−1 for a Ag film thick-ness of 8 ML. Fig. 17: Photoemission data for 8 ML Ag on

highly doped Si(111). The horizontal axis is the in-plane momentum kx. (a) The curve indicates the position of the Si valence band edge. (b) The curves show the calculated energy dispersion relations of the confined quantum well states.



the beamline facilities at the Synchrotron Radiation Center, where much of the photoemission work was performed. The Synchrotron Radiation Center is supported by the U.S. National Science Foundation (grant DMR-05-37588). Much of our x-ray diffraction work was carried out at the Advanced Photon Source, Argonne National Laboratory, which is supported by the U.S. Department of Energy (con-tract W-31-109-ENG-38).

REFERENCES[1] U.S. Department of Energy Report

on “Nanoscale Science, Engi-neering and Technology Research Directions,” available at http://www.sc.doe.gov/bes/reports/files/NSET_rpt.pdf.

[2] T.-C. Chiang, Surf. Sci. Rep. 39, 181 (2000).

[3] T.-C. Chiang, Science 306, 1900 (2004).

[4] F. J. Himpsel, J. E. Ortega, G. J. Mankey, and R. F. Willis, Adv. Phys. 47, 511 (1998).

[5] S-Å. Lindgren and L. Walldén, Handbook of Surface Science, Vol. 2, Electronic Structure, ed. S. Hol-loway, N. V. Richardson, K. Horn, and M. Scheffler (Elsevier, Amster-dam, 2000).

[6] M. Milun, P. Pervan, D. P. Woodruff, Rep. Prog. Phys. 65, 99 (2002).

[7] L. Aballe, C. Rogero, P. Kratzer, S. Gokhale, and K. Horn, Phys. Rev. Lett. 87, 156801 (2001).

[8] I. Matsuda, T. Ohta, and H. W. Yeom, Phys. Rev. B 65, 085327 (2002).

[9] A. Mans, J. H. Dil, A. R. H. F. Ettema, and H. H. Weitering, Phys. Rev. B 66, 195410 (2002).

[10] Y. Z. Wu, C. Y. Won, E. Rotenberg, H. W. Zhao, F. Toyoma, N. V. Smith, and Z. Q. Qiu, Phys. Rev. B 66, 245418 (2002).

[11] J. J. Paggel, T. Miller, and T.-C. Chiang, Phys. Rev. Lett. 81, 5632 (1998).

[12] J. J. Paggel, T. Miller, and T.-C. Chi-ang, Science 283, 1709 (1999).

[13] J. J. Paggel, T. Miller, and T.-C.

Chiang, Phys. Rev. B 61, 1804 (2000).

[14] M. H. Upton, T. Miller, and T.-C. Chiang, Appl. Phys. Lett. 85, 1235 (2004).

[15] M. Upton, C. M. Wei, M. Y. Chou, T. Miller, and T.-C. Chiang, Phys. Rev. Lett. 93, 026802 (2004).

[16] H. Hong, R. D. Aburano, D.-S. Lin, H. Chen, T.-C. Chiang, P. Zschack, and E. D. Specht, Phys. Rev. Lett. 68, 507 (1992).

[17] J. E. Ortega and F. J. Himpsel, G. J. Mankey, and R. F. Willis, Phys. Rev. B 47, 1540 (1993).

[18] A. Danese and R. A. Bartynski, Phys. Rev. B 65, 174419 (2002).

[19] Z. Q. Qiu and N. V. Smith, J. Phys. Condensed Matter 14, R169 (2002).

[20] J. J. Paggel, C. M. Wei, M. Y. Chou, D.-A. Luh, T. Miller, and T.-C. Chiang, Phys. Rev. B 66, 233403 (2002).

[21] C. M. Wei and M. Y. Chou, Phys. Rev. B 66, 233408 (2002).

[22] F. K. Schulte, Surf. Sci. 55, 427 (1976).

[23] X. Ma et al., Proc. Nat. Acad. Sci. 104, 9204 (2007).

[24] D.-A. Luh, T. Miller, J. J. Paggel, M. Y. Chou, and T.-C. Chiang, Science 292, 1131 (2001).

[25] C. M. Wei and M. Y. Chou, Phys. Rev. B 68, 125406 (2003).

[26] P. Czoschke, L. Basile, H. Hong, and T.-C. Chiang, Phys. Rev. Lett. 93, 036103 (2004).

[27] P. Czoschke, H. Hong, L. Basile, and T.-C. Chiang, Phys. Rev. B 72, 075402 (2005).

[28] M. Hupalo, S. Kremmer, V. Yeh, L. Berbil-Bautista, E. Abram, and M. C. Tringides, Surf. Sci. 493, 526 (2001).

[29] H. Hong, C.-M. Wei, M. Y. Chou, Z. Wu, L. Basile, H. Chen, M. Holt, and T.-C. Chiang, Phys. Rev. Lett. 90, 076104 (2003).

[30] P. Czoschke, H. Hong, L. Basile, and T.-C. Chiang, Phys. Rev. Lett. 91, 226801 (2003).

[31] P. Czoschke, H. Hong, L. Basile,

and T.-C. Chiang, Phys. Rev. B 72, 035305 (2005).

[32] C. M. Wei and M. Y. Chou (unpub-lished results).

[33] G. Grimvall, The Electron-Phonon Interaction in Metals (North Hol-land, New York, 1981).

[34] T. Balasubramanian, E. Jensen, X. L. Wu, and S. L. Hulbert, Phys. Rev. B 57, R6866 (1998).

[35] M. Hengsberger, D. Purdie, P. Sego-via, M. Garnier, and Y. Baer, Phys. Rev. Lett. 83, 592 (1999).

[36] D.-A. Luh, T. Miller, J. J. Paggel, and T.-C. Chiang, Phys. Rev. Lett. 88, 256802 (2002).

[37] J. J. Paggel, D.-A. Luh, T. Miller, and T.-C. Chiang, Phys. Rev. Lett. 92, 186803 (2004).

[38] Y. Guo et al., Science 306, 1915 (2004).

[39] M. M. Özer, Y. Jia, Z. Zhang, J. R. Thompson, and H. H. Weitering, Science 316, 1594 (2007).

[40] D. Eom, S. Qin, M.-Y. Chou, and C. K. Shih, Phys. Rev. Lett. 96, 027005 (2006).

[41] S.-J. Tang, L. Basile, T. Miller, and T.-C. Chiang, Phys. Rev. Lett. 93, 216804 (2004).

[42] N. J. Speer, S.-J. Tang, T. Miller, and T.-C. Chiang, Science 314, 804 (2006).

[43] S.-J. Tang, T. Miller, and T.-C. Chiang, Phys. Rev. Lett. 96, 036802 (2006).

[44] S.-J. Tang, Y.-R. Lee, S.-L. Chang, T. Miller, and T.-C. Chiang, Phys. Rev. Lett. 96, 216803 (2006).

[45] D. A. Ricci, T. Miller, and T.-C. Chiang, Phys. Rev. Lett. 95, 266101 (2005).

[46] D. A. Ricci, T. Miller, and T.-C. Chiang, Phys. Rev. Lett. 93, 136801 (2004)

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