Download Calculation of surface optical properties: from qualitative

Thin Solid Films 455 – 456 (2004) 764–771
Calculation of surface optical properties: from qualitative understanding
to quantitative predictions
W.G. Schmidta,*, K. Seinoa, P.H. Hahna, F. Bechstedta, W. Lub, S. Wangb, J. Bernholcb
a
¨ Festkorpertheorie
¨
¨ Jena, Max-Wien-Platz 1, 07743 Jena, Germany
Institut fur
und Theoretische Optik, Friedrich-Schiller-Universitat
b
Department of Physics, North Carolina State University, Raleigh, NC 27695-8202, USA
Abstract
In the last couple of years there has been much methodological and computational progress in the modeling of optical properties
from first principles. While the calculation of non-linear optical coefficients is still hampered by numerical limitations—
demonstrated here for the case of bulk GaAs—linear optical spectra can now be calculated accurately and with true predictive
power, even for large and complex surface structures. This allows on one hand for a much better understanding of the origin of
specific features such as surface optical anisotropies. We find that in particular microscopic electric fields at the surface induce
slight deformations of bulk-like wavefunctions and thus give rise to optical anisotropies even from sub-surface layers. On the
other hand, from the comparison of measured and calculated spectra, one can now confidently reach conclusions on the surface
geometry. This short review focuses on the simulation of reflectance anisotropy spectroscopy. The clean, hydrogenated and uracilcovered Si(001) surface is used to illustrate the microscopic origin of surface optical anisotropies and the present state-of-the-art
in computational modeling of optical spectra.
䊚 2003 Elsevier B.V. All rights reserved.
PACS: 78.68.qm; 78.66.Fd; 78.66.Db; 78.66.Qn
Keywords: Reflectance anisotropy spectroscopy; Density-functional calculations; Semiconductor surfaces; Second harmonic generation
1. Reflectance anisotropy spectroscopy
Optical spectroscopies are extremely valuable for the
in situ, non-destructive and real-time surface monitoring
under challenging conditions as may be encountered,
e.g. during epitaxial growth. Reflectance anisotropy
spectroscopy (RAS) is one of the plethora of surface
sensitive spectroscopies probing linear optical coefficients, which has been particularly successful in exploring surface structures.
Already in 1966 Cardona, Pollak and Shaklee
observed that rotating a Si(110) sample around the
surface normal changed the reflectivity as a function of
the azimuthal angle w1x. However, it took about twenty
years before the phenomenon of surface-induced optical
anisotropies started to be investigated systematically w2–
5x. The high potential of RAS for growth-monitoring
*Corresponding author. Tel.: q49-3641-635910; fax: q49-3641635182.
E-mail address: [email protected] (W.G. Schmidt).
was demonstrated by Kamiya et al. w6x by measurements
on GaAs(001) in ultra-high vacuum as well as in
atmospheric pressure of H2, He and N2. It was noted
that each of the well-known GaAs(001) surface reconstructions such as (4=4), (2=4) and (4=2) gives rise
to a characteristic, fingerprint-like RAS spectrum. The
correlation between reflection high-energy electron diffraction (RHEED) patterns and the RAS spectra allowed
them to establish a database for GaAs(001) surface
reconstructions w7x. The surface reconstruction can thus
be determined in situ, even at ambient conditions. For
many semiconductors a change of the surface reconstruction during layer growth occurs. RAS can, therefore
be easily used to monitor growth and layer completion.
This application has proven highly successful w8,9x and
is now exploited commercially.
However, surface optical spectroscopies give only
indirect information about the surface structure and
chemistry. The intuitive interpretation of the spectra in
terms of bond polarizabilities is difficult and often leads
to wrong results even for relatively simple surfaces such
0040-6090/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved.
doi:10.1016/j.tsf.2003.11.263
W.G. Schmidt et al. / Thin Solid Films 455 – 456 (2004) 764–771
as Si(001) (see discussion in Ref. w10x). Reliable calculations of surface optical spectra are needed to fully
exploit their diagnostic potential.
One of the first major attempts to calculate surface
optical properties was made in 1971 by McIntyre and
Aspnes w11x. They approximated the inhomogeneity of
the surface dielectric tensor by a two-step function, i.e.
modeled the surface by a three-layer system consisting
of vacuum, thin film and substrate. This model was
later extended to include surface anisotropy w12x and
remains very popular because of its simplicity.
A more general approach was taken in 1979 by
Bagchi, Barrera and Rajagopal w13x. They started by
considering a semi-infinite electron gas. The truncation
of the bulk leads to a modification of the optical
properties in the surface region. Based on the assumption
that this region is much thinner than the light wavelength, the light propagation equations were solved. A
few years later Del Sole w14x generalized this method
to the case of larger anisotropies in real crystals and
obtained an expression for the surface contribution to
the reflectance, DRyR, where R is the reflectance according to the Fresnel equation. For s-light polarized along
i and normal incidence one obtains
S
W
DRi
4v T D´iiŽv. T
X
Žv.s ImU
T
T
R
c
V ´bŽv.y1 Y
(1)
where ´b is the bulk dielectric function, and
|
y|dzdz9dz0dz-´ Žv;z,z9.´
D´ijs dzdz9wy´ijŽv;z,z9.ydijdŽzyz9.´0Žv;z.z~
x
iz
|
(2)
y1
zz
Žv;z9,zz0.´zjŽv;z0,z-..
Here ´ijŽv;z,z9. is the non-local macroscopic dielectric
tensor of the solid-vacuum interface accounting for all
many-body and local-field (LF) effects w15x. The second
term in Eq. (2) is difficult to calculate because of the
inversion required to obtain ´y1
and because of the
zz
four-fold integration. It contains off-diagonal terms ´iz
of the susceptibility of the semi-infinite crystal. Manghi
et al. w16x investigated the influence of these terms for
GaAs(110) and GaP(110) surfaces. No relevant contributions of the off-diagonal terms to the reflectance were
noticed and they are therefore, usually neglected. The
first term of Eq. (2) can be evaluated by replacing the
semi-infinite crystal by an artificial super-cell, large
enough to represent the vacuum as well as the surface
and bulk regions of the crystal under investigation.
Provided that: (i) the slab is large enough to properly
describe the surface region of the crystal, i.e. the surface
as well as surface-modified bulk wavefunctions and (ii)
the off-diagonal terms of the dielectric tensor are small
compared to the diagonal ones, a simple expression for
765
the surface contribution to the reflectivity can be derived
w16x
S
W
4pahs
ii Žv. T
DRi
4v T
U
X
Žv.s ImT
T.
R
c
V ´bŽv.y1 Y
(3)
Here ahs
ii Žv. with isx,y is the diagonal tensor component of the averaged half-slab polarizability. It is
important to note that under the conditions mentioned
above, Eq. (3) contains in principle all surface contributions to the optical reflectance. This includes contributions due to surface electronic states, atomic
relaxations, the influence of the surface potential on the
bulk wavefunctions as well as the surface LF effects,
i.e. the influence of the surface-modified microscopic
fluctuations of the electric field on the macroscopic
dielectric response, and many-particle effects such as
the electronic self-energy and the electron–hole attraction. Of course, it is not an easy task to include all these
contributions in practical calculations of the slab polarizability a. Up to very recently, many approximations
were required to calculate a. One possibility consists in
combining the atomic structure optimization within density-functional theory (DFT) with electronic structure
calculations done in the tight-binding (TB) approximation. This method has for example been applied to clean
w17x and Sb-covered GaAs(001) surfaces w18x as well
as the AsyInP(110) interface w19x. Other encouraging
results were obtained by combining DFT calculations
with the scissors-operator approximation w20x to the
electronic self-energy. This method has been applied to,
among others, bare and As-terminated Si(001) surfaces
w21,22x. Pulci et al. w23x used a linear parametrization
of the quasiparticle shifts calculated within the GW
approximation with respect to surface localization to
calculate the RAS for the GaAs cleavage face.
While the examples discussed above allow for a
meaningful interpretation of the experimental spectra,
the quantitative agreement is often unsatisfactory. This
has caused a considerable amount of confusion and
speculation about the origin of surface optical anisotropies in the literature. Even the validity of the slab
approach for the calculation of surface optical spectra
was questioned. In particular the fact that surface optical
spectra often show resonances near the energies of bulk
critical point (CP) energies has been considered puzzling. It has led to the development of a series of models
which—by choosing suitably fitted parameters—were
often successful in reproducing specific features of the
measured spectra:
1. The surface local-field effect was one of the first
explanations for optical anisotropies. Although exact
expressions for the LF contributions to the surface
optical response have been derived decades ago
w15,24x, the complexity of the problem required until
766
W.G. Schmidt et al. / Thin Solid Films 455 – 456 (2004) 764–771
recently an approximate treatment based on modeling
the crystal surface by a lattice of polarizable entities,
which obey a Clausius-Mossotti-like relation w25–
27x.
2. A photon-induced localization of the electron wave
packets constituting the initial and final states of the
optical excitation was made responsible for derivative-like features in the optical spectra of passivated
Si(001) and Si(113) surfaces w28–32x.
3. Optical anisotropies of the GaAs(001) and (111)
surfaces near the E1 and E0 CP energies of bulk GaAs
were attributed to a so-called surface termination
effect, i.e. the termination of the electron wavefunctions at the crystal surface w33x.
4. In addition to those models there were studies that
emphasized the influence of the surface electric field
and strain induced dichroism at the bulk CPs of the
dielectric function on the surface optical response
w34–37x.
The surface termination effect has already been shown
by Del Sole and Onida w38x to be irrelevant for RAS.
However, a comprehensive, quantitative picture on how
strongly the surface optical anisotropies are related to
surface electric fields, surface induced strain, surface
defects, local fields and many-body effects such as selfenergy and electron–hole attraction effects could only
be obtained very recently, thanks to the advent of
supercomputing and advanced numerical algorithms.
2. Origin of surface optical anisotropies
The optical anisotropy of the Si(001) surface has
been intensively studied by both experiment
w22,28,30,40–44x and theory w10,21,22,45–47x, due to
its technological importance and its model character for
semiconductor surface science. In order to determine the
origin of the spectral features of the Si(001)c(2=4)
surface, Schmidt et al. w39x performed ab initio calculations based on a massively parallel real-space finitedifference implementation w48x of the DFT. Self-energy
effects were taken into account using the scissors operator. A linear cut-off function was used to separate the
contributions to the RAS from electronic transitions
within the uppermost four atomic layers and from the
bulk-like layers underneath. Fig. 1 shows that surfacestate related transitions are mainly responsible for the
optical anisotropies below the E1 CP energy. This holds
in particular for the features T1 and T2. The appearance
of optical anisotropies due to transitions between surface
states is not surprising: In a number of instances such
as InP, GaP and SiC surfaces, numerical modeling
succeeded in relating characteristic anisotropy peaks to
specific surface states w49–53x.
More difficult to understand is the appearance of
optical anisotropies such as T3, which are not related to
¯
Fig. 1. RAS wReµŽrØ110Ø
yrw110x.yNrM∂x for Si(001)c(2=4), calculated for
the fully relaxed configuration (solid lines), a geometry where the
relaxation was restricted to the uppermost two atomic layers (dashed
lines) and an anisotropically strained slab (dotted lines). Shown are
contributions to the RAS from electronic transitions within the uppermost four atomic layers, from the layers underneath, and the total
signal. From Ref. w39x
transitions between surface states. It has led to the
suggestion that subsurface stress due to the dimerization
of the surface atoms may be largely responsible for RAS
contributions from deeper layers w36x. In order to test
this hypothesis additional calculations were performed
for a geometry, where only the uppermost two atomic
layers were allowed to relax, and the remaining atoms
occupy their ideal bulk positions. Fig. 1 shows the RAS
spectrum of the configuration with surface-restricted
relaxation (dashed lines) in comparison with the results
for the fully relaxed geometry (solid lines). The surfacerelated RAS minima T1 and T2 become more pronounced. Only small modifications occur for the RAS
arising from electronic transitions below the uppermost
four atomic layers.
A second possibility to probe the influence of surface
stress on the optical spectra consists in slightly changing
the lateral dimensions of the surface unit cell. The
dimerized surface is stretched along the dimer bonds
and compressed across the dimer rows w54x. Accordingly, the Si(001)c(2=4) surface was relaxed in a unit cell
whose lateral dimensions are shrunkyenlarged by 0.5%
along w110x y w110x. The RAS spectra calculated for this
configuration are shown by dotted lines in Fig. 1. The
optical anisotropies arising from subsurface transitions
are slightly affected. In particular the T3 feature is
enhanced. It may thus be partially attributed to reconstruction-induced stress in the layers beneath the surface.
The calculations thus show that both the structural
relaxation in the bulk and anisotropic strain fields
W.G. Schmidt et al. / Thin Solid Films 455 – 456 (2004) 764–771
Fig. 2. Local part of the effective potential calculated for
Si(001)c(2=4) shown in two planes perpendicular to the surface
normal.
modify the bulk-related optical anisotropy. Its salient
features, however, are not affected. This indicates that
the surface induces optical anisotropies in the bulk
predominantly via an electronic coupling.
In Fig. 2 the local part of the effective potential
entering the DFT Kohn–Sham equations
GGA
VŽr.sVlocal
ps Žr.qVHŽr.qVXC Žr.
(4)
is shown in two planes perpendicular to the surface
normal which cut the uppermost and seventh atomic
layer of Si(001). Here Vlocal
is the local part of the ionic
ps
GGA
pseudopotentials and VH and VXC
denote the Hartree
and exchange and correlation potential in generalized
gradient approximation (GGA), respectively. As can be
seen from the figure, both the absolute magnitude and
the ‘shape’ of the potential differ strongly between the
first and seventh atomic layer. In order to quantify this
effect we average the potential in the planes perpendicular to the surface normal. The resulting quantity
NVMŽz.s
1
dxdyVŽx,y,z.,
A A
|
767
where L is the bilayer thickness and xyy are parallel to
the 110y110 direction. The difference Eq. (6) approaches zero only below the seventh atomic layer beneath the
surface (cf. Fig. 3). Obviously, the surface-induced
anisotropy of the microscopic electric fields reaches far
into the bulk. It therefore affects wavefunctions, which
cannot be categorized as electronic surface states, but
should rather be addressed as electric-field modified
bulk wavefunctions. The surface-induced deformations
of these bulk-like wavefunctions are weighted differently
by the x and y components of the optical transition
operator, leading to an anisotropic optical response from
layers beneath the surface. Although these anisotropies
might thus to be thought to be related to surface local
fields, this is not true in a strict sense, because they
already show up in the (0,0)-element of the microscopic
dielectric tensor in reciprocal space.
Superimposed on the surface-induced anisotropies of
the electron wavefunctions may be deformations due to
electric fields induced by, e.g. the pinning of the Fermi
level at the sample surface. These deformations may be
anisotropic too, as has been shown by first-principles
calculations for GaAs(001) surfaces (cf. Fig. 6 in Ref.
w55x). The linear electro-optic effect modifies mainly
the optical anisotropy from the bulk atomic layers,
resulting in changes of the RAS signal, which are
strongly reconstruction dependent, however. The electric
field affects the atomic lattice as well as the electron
wavefunctions and their eigenvalues. It has been found,
however, that the changes of the combined density of
states due to the electric field as well as the fieldinduced lattice relaxations are negligible compared to
the deformations of the wavefunctions, resulting in
modified transition matrix elements w55x.
The surface potential may not only be altered by
electric fields, but also by surface defects such as steps.
(5)
where A is the area of the surface unit cell, is shown in
Fig. 3. The averaged potential NVM in the uppermost two
atomic layers differs strongly from the value in the
deeper layers. In order to quantify the anisotropy of the
potential we consider the layer-averaged microscopic
electric fields perpendicular to the surface normal. In
detail, we calculate
N)=xV)y)=yV)MŽz.
1
L
zqLy2
|
zyLy2
dz9
s
S
≠VŽx,y,z9.
T ≠VŽ,y,z9.
1
dxdyU
y
T
A A
≠x
≠y
V
|
)
) )
W
T
),
X
T
Y
(6)
Fig. 3. Averaged effective potential and its anisotropy for
Si(001)c(2=4) shown along the surface normal. Dashed lines indicate the positions of the atomic layers.
768
W.G. Schmidt et al. / Thin Solid Films 455 – 456 (2004) 764–771
3. Many-body effects in surface optical spectra
observed experimentally w43x. However, the calculated
A9 and B9 peaks are redshifted in comparison with the
experiment by 0.5 eV and 0.8 eV, respectively. Selfenergy corrections calculated in GW approximation shift
the excitation energies to larger values. Furthermore, the
B9 optical anisotropy is much increased. This is not only
related to the non-uniform self-energy corrections of the
DFT-LDA eigenvalues, but also a consequence of the v
scaling of the surface contribution to the reflectance (cf.
Eq. (1)). Surface LF effects can be expected from both
the microscopic fluctuations of the electric field within
the bulk, and from the truncation of the bulk itself. The
numerical calculation, however, shows that LF effects
lead to surprisingly small changes of the spectrum in
the particular case studied here. A distinct change of the
RAS spectrum, however, results from the inclusion of
the attractive electron–hole interaction. A strong reduction of the B9 optical anisotropy and an increase of the
integrated A9 peak area occurs. At the same time, the
A9 and B9 optical anisotropies are shifted to lower
energies by approximately 0.2 eV.
The stepwise inclusion of many-particle effects in the
calculation leads to a considerable and systematic
improvement of the agreement with the experiment. The
remaining discrepancies are mainly due to the numerical
limitations of the study w61x, in particular with respect
to the slab thickness and Brillouin zone sampling.
The full inclusion of many-body effects in the calculation of surface optical spectra leads to the limits of
today’s computer capacity. However, numerically wellconverged spectra calculated within the independentparticle approximation and using the scissors-operator
approach to account for the band-gap underestimation
The inclusion of many-body effects—beyond those
already contained in the approximations for exchange
and correlation, which enter the Kohn–Sham equations—in the calculation of surface optical spectra is a
fairly recent achievement. In the late 1990s it has
become possible to correct the single-particle excitation
energies for quasiparticle effects, using approximations
to the self-energy within the GW approach w23,49,57x.
The solution of the Bethe–Salpeter equation (BSE)
allows electron–hole attraction, i.e. excitonic, and LF
contributions to the surface optical response to be taken
into account. While the first applications of the BSE to
surfaces were restricted to the description of few strongly
localized surface states w58,59x, advanced algorithms
together with a massive parallelization of the computations made it recently possible to model the excitonic
and LF effects in a wide spectral range w60,61x.
In Fig. 5 the influence of many-body effects on the
calculated optical anisotropy of the monohydride
Si(001)(2=1) surface w61x is shown. The spectrum
calculated in the independent-particle approximation, i.e.
the DFT-LDA spectrum shows in principle the features
¯
Fig. 5. RAS spectra wReµŽrØ110Ø
yrw110x.yNrM∂x calculated (within DFTLDA, in GW approximation, in GWA with the effects of local fields
included, and in GWA with the effects of local fields and the electron–
hole attraction included) for the monohydride Si(001)(2=1) surface
from Ref. w61x are compared with the measured data w43x.
¯
Fig. 4. Step-induced optical anisotropy wReµŽrØ110Ø
yrw110x.yNrM∂x calculated for SA, SB and DB steps of the Si(001) surface. From Ref.
w39x
The modified potential may on one hand lead to defectlocalized electronic states, which are directly involved
in electronic transitions, thus modifying the surface
optical response. On the other hand, additional optical
signals can be expected from perturbed bulk-like wavefunctions localized far beneath the uppermost substrate
layers. Defect-induced optical anisotropies may be of
similar magnitude as the signals from the perfectly
ordered surface. This is demonstrated in Fig. 4, where
we have plotted the calculated optical anisotropies due
to the three most commonly observed step configurations on Si(001) surfaces w56x: SA steps and rebonded
steps of type SB and DB w39x. The calculated step-related
anisotropies (which are in excellent agreement with
experiment w41x) explain the distinct differences in the
RAS spectra measured for different Si(001) surface
samples w22,40–42x.
W.G. Schmidt et al. / Thin Solid Films 455 – 456 (2004) 764–771
769
Fig. 6. Left: UracilySi(001) adsorption configurations. White, gray, dark gray, dark, and small symbols indicate C, N, Si, O, and H atoms,
¯
yrw110x.yNrM∂x for the respective UracilySi(001) adsorption configurations, calculated within DFT-GGA. A
respectively. Right: RAS wReµŽrØ110Ø
scissors operator has been used to account for self-energy effects.
are in many instances sufficient for the reliable prediction of surface optical anisotropies. This is because RAS
spectra are difference spectra, which are furthermore
normalized to the bulk dielectric function (cf. Eq. (1)).
Many systems, for example surface defects or very large
reconstructions such as the ones formed by In nanowires
on Si(111) surfaces w62x can presently only be described
within the single-particle approximation.
4. Outlook
Organic functionalization of semiconductors has
become important for the development of new semiconductor-based devices. The physical and chemical
properties of hybrid organicyinorganic materials depend
crucially on the structural and chemical details of the
interface. In principle it should be possible to obtain
such information from optical spectroscopies such as
RAS. However, at present there are only relatively few
experimental and theoretical studies that explore optical
anisotropies of organically modified semiconductors
(see, e.g. Refs. w63–65x).
The adsorption of uracil on Si(001), investigated
experimentally by scanning tunneling microscopy and
high-resolution electron energy-loss spectroscopy w66x,
leads to what can be considered a prototypical interface
between a polyfunctional molecule and a semiconductor
surface. The (001) surface of silicon is the starting point
for the fabrication of most microelectronic devices.
Uracil (C4H4N2O2) is a small molecule featuring one
C_ C double bond, two amino and two carbonyl groups
and may thus bond to the surface in various ways. In
addition, its tautomerism and electrostatic effects have
been found to be important for the interface formation
w66,67x. In order to probe the sensitivity of RAS with
respect to different uracil bonding configurations we
calculate spectra for the most relevant dative (D1, D2
in Fig. 6) and covalently bonded (C1, C2) interface
configurations (details will be published elsewhere).
These spectra (Fig. 6) indicate that the adsorption of
organic molecules strongly modifies the optical response
of the Si(001) surface (cf. Fig. 1) and leads to a signal
that is very sensitive to the details of the interface. In
particular for photon energies of approximately 3.3 eV,
i.e. slightly below the E1 CP, a strong dependence of
the surface optical anisotropy on the adsorption configuration can be stated: oxygen insertion into the uppermost Si dimers (D2, C2) leads to a strong negative
anisotropy. However, also the structures where uracil
remains intact, and is dative (D1) or covalently bonded
to the substrate (C1) can be clearly distinguished by
means of RAS.
RAS spectra simulated for cyclopentene adsorption
configurations on Si(001) w65x indicate that measurements of reflectance difference data can even discriminate between ‘cis’ and ‘trans-type’ adsorption
configurations. The high sensitivity of RAS with respect
to the details of the surface chemical bonding in conjunction with the accuracy achieved in the numerical
prediction of such spectra thus suggests RAS as a
powerful tool also for monitoring the formation of
organic thin films.
Thus far we have focused on the modeling of linear
optical spectra. However, non-linear optical processes
such as second harmonic generation (SHG) are also
increasingly being used for surface and interface analysis
w71x. Unfortunately, the microscopic interpretation of the
measured spectra is severely hampered by numerical
difficulties. This is demonstrated in Fig. 7, where we
compare recently calculated values for the secondharmonic susceptibility x2xyzŽy2v;v,v. of bulk GaAs
with experiment w68x. Obviously, the theoretical results
obtained within a full-potential linearized augmented
plane-wave method w70x and from plane-wave pseudopotential calculations w69x neither agree well with each
other nor with the present calculations utilizing a realspace finite-difference approach w48x. The agreement
between theory and experiment is also unsatisfactory.
770
W.G. Schmidt et al. / Thin Solid Films 455 – 456 (2004) 764–771
Acknowledgments
Grants of computer time from the DoD Challenge
¨
Program, the Leibniz-Rechenzentrum Munchen,
and the
¨
Hochstleistungsrechenzentrum Stuttgart are gratefully
acknowledged.
References
Fig. 7. Spectral dependence of )x2xyz) of GaAs as measured w68x and
calculated here and in Refs. w69x (a) and w70x (b), respectively.
The reasons for the discrepancies are on the one hand
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nearly all technical parameters, in particular the sampling of the Brillouin zone (in the present calculations
a set of 32 000 randomly chosen points was needed to
obtain converged results). On the other hand, non-linear
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present calculations a scissors-operator approach was
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5. Conclusions
The difference between the optical response of the
surface compared to that of the bulk may be related to
the possible presence of surface electronic states, but
can also arise from surface induced anisotropies in the
crystal potential extending far below the uppermost
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calculated non-linear optical properties, however, still
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