-
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
Highlight of the Issue
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
Quantum Physics of Thin Metal Films
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.
-
4 AAPPS Bulletin April 2008, Vol. 18, No. 2
Highlight of the Issue
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.
-
AAPPS Bulletin April 2008, Vol. 18, No. 2 5
Quantum Physics of Thin Metal Films
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.
-
6 AAPPS Bulletin April 2008, Vol. 18, No. 2
Highlight of the Issue
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.
-
AAPPS Bulletin April 2008, Vol. 18, No. 2 7
Quantum Physics of Thin Metal Films
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.
-
8 AAPPS Bulletin April 2008, Vol. 18, No. 2
Highlight of the Issue
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.
-
AAPPS Bulletin April 2008, Vol. 18, No. 2 9
Quantum Physics of Thin Metal Films
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.
-
10 AAPPS Bulletin April 2008, Vol. 18, No. 2
Highlight of the Issue
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)