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Quantum Beating Patterns in the Surface Energy of Pb Film Nanostructures
Peter Czoschke, Hawoong Hong, Leonardo Basile and Tai-Chang Chiang
Frederick Seitz Materials Research Laboratory, University of Illinois at Urbana-Champaign
ES(N)
Free-Electron Calculation
Introduction
Ultrathin metal films grown on semiconductor substrates exhibit a rich variety of growth behavior and morphology
at different temperatures due to quantum size effects. We have used the surface x-ray diffraction station at Sector
33ID (UNICAT) to study the nanoscale structural evolution of Pb films grown on Si(111) at 110 K as they are
annealed to 280 K. The film morphology passed through various different metastable phases before reaching a state
of local equilibrium, at which point the coverage of different height Pb structures was analyzed and related to the
thickness-dependent surface energy. Rich patterns are seen in the resulting energy landscape, consistent with a model
calculation based on a free-electron gas confined to a one-dimensional quantum well.
pN  e
 E S ( N ) / k BT
cos[ 2k F ( N  N )t0 ]
ES ( N )  A

( N  N )
Structure Stability
Extended X-Ray Reflectivity
Surface Morphology
Experiment
Theory (Fit)
5
pN''
5
280 K
0
Film Evolution (Annealing)
Intensity (arb. units)
253 K
200 K
174 K
110 K
pN (% Surface Area Covered)
10
266 K
266 K
5
0
40
20
253 K
5
6
7
8
9
l (Si(111)hex r.l.u.)
10
11
12
Extended x-ray reflectivity of a sample with an initial
thickness of 11 AL of Pb after annealing to the temperatures
indicated. Curves are fits to a kinematic model which reveal
the relative coverage of different film thicknesses present on
the sample surface. The small fringes near the half-order
point between the Pb Bragg peaks in the data at higher
temperatures are due to quasibilayer variations in the
stability of different thicknesses.
15
20
25
Pb Layers N (AL)
30
-20
5
The relative stability of different thicknesses (the
discrete second derivative of pN) shows bilayer
variations with the preference for even/odd
thicknesses alternating approximately every 9 AL.
Assuming that the pN values follow a Boltzmann
distribution, this “beating” effect is accurately
reproduced by a two-parameter (A and N)† fit to the
free-electron form of the surface energy.
20
200 K
0
20
174 K
0
10
0
Results
0
40
†N
10
15
20
25
Pb Thickness N (AL)
30
From the fit to the experimental data, the
details of the thickness-dependent surface
energy are obtained. Their functional form and
overall energy scale are consistent with
independent first-principles calculations done
for related systems.
is a phase factor to account for the presence of the Si substrate.
20
Summary
110 K
0
4
20
-5
ES(N) (meV)
280 K
3
Surface Energy
0
2
6
10
14
18
22
26
30
34
Pb Layers N (AL)
The fractional surface area covered by exactly N Pb layers,
pN, obtained from fits to the x-ray data. After deposition,
the film is relatively smooth due to the layer-by-layer
growth mode. As the sample is annealed, the film
bifurcates into the preferred thicknesses of 10 and 12 AL.
Near room temperature, the film has become very rough
with variations that reveal details of the surface energy.
 Quantum confinement of electrons in a metal film leads to oscillations in the surface
energy. We have observed this effect experimentally with surface x-ray diffraction,
effectively measuring the surface energy through the film structure.
 We observed the surface morphology evolve from a smooth film, through the “magic
thickness” phase, to a highly roughened state near room temperature and have
correlated such behavior with details of the global energy landscape.
 In Pb(111) films, such quantum effects produce bilayer oscillations with a “beating”
pattern that results in a strong preference for even/odd thicknesses.
P. Czoschke, et al., Physical Review Letters (submitted)