# Terrace effects in grazing-incidence fast atom diffraction from a LiF(001)
surface

L. Frisco  Instituto de Astronomía y Física del Espacio (UBA-CONICET), Ciudad
Universitaria, (C1428EGA) Buenos Aires, Argentina.  M.S. Gravielle Author to
whom correspondence should be addressed.
Electronic address<EMAIL_ADDRESS>Instituto de Astronomía y Física del
Espacio (UBA-CONICET), Ciudad Universitaria, (C1428EGA) Buenos Aires,
Argentina.

###### Abstract

The effect produced by surface defects on grazing-incidence fast atom
diffraction (GIFAD) patterns is studied by considering the presence of
terraces in a LiF(001) sample. For helium atoms impinging along the
$\left\langle 110\right\rangle$ direction of the LiF surface, we analyze the
influence of a monolayer terrace with its edge oriented parallel or
perpendicular to the axial channel. We found that the presence of an outward
transverse step introduces a diffuse background above the Laue circle, which
displays additional peaked structures. For inward transverse steps, instead,
such a background is placed below the Laue circle, showing a much weaker
intensity. On the other hand, parallel steps give rise to asymmetric angular
distributions, which are completely confined to the Laue circle. Therefore,
these theoretical results suggest that GIFAD might be used to characterize
terrace defects.

##  I Introduction

Grazing-incidence fast atom diffraction (GIFAD) is an exceptionally sensitive
technique of surface analysis, which provides detailed information of the
electronic and morphological features of the surface Winter2011 ;
Debiossac2017 . During the 15 years since its first observation Schuller2007 ;
Rousseau2007 , the GIFAD method has been successfully applied to study the
topmost atomic layer of a wide variety of materials, ranging from insulators
Schuller2012 , semiconductors Debiossac2014  and metals Rios2013  to
adsorbate-covered metal surfaces Schuller2009b , ultrathin films Seifert2010 ,
organic-metal interfaces Seifert2013 ; Momeni2018 , and graphene layers
Zugarramurdi2015 . But in all cases, the use of well-ordered crystal targets
was considered an important prerequisite for the observation of interference
patterns Winter2011 .

In GIFAD the periodic ordering requirement of the crystal sample is
particularly crucial along the axial direction because projectiles probe long
distances of the surface along the incidence channel, about some hundred Å.
Hence, careful crystal manufacture and surface preparation represent central
issues in GIFAD experiments. Nevertheless, even under extremely good cleanness
conditions, real crystals have step defects that could affect the interference
patterns Farias2002 . In the case of alkali-halide crystals, such as NaCl,
KBr, and LiF, high resolution images provided by atomic force microscopy
commonly reveal the presence of terraces or steps on the topmost layer
Meyer1988 ; Prohaska1994 , which are unavoidable in the preparation process of
the sample Vidal2019 .

In this paper we address how the existence of terraces in a LiF(001) surface
might affect the diffraction patterns produced by grazing impact of fast He
atoms. The He/LiF(001) system has been extensively studied with GIFAD, being
currently considered as a prototype for this phenomenon. However, all the
theoretical descriptions assume an ideal perfect crystal surface diaz2022 ,
whereas defect contributions were only qualitatively discussed Winter2011 ;
Debiossac2021 .

The influence of surface terraces on GIFAD patterns is investigated here by
considering simple crystallographic geometries: a unique up- or down- step,
oriented perpendicular or parallel to the incidence channel. Different step
locations, relative to the focus position of the atomic beam, are analyzed.
Although such geometries represent an oversimplified description of actual LiF
surfaces, they will allow us to shed light on the expected contribution of
more complex step defects.

Our study is based on the use of the surface initial value representation
(SIVR) approximation Gravielle2014 ; Gravielle2015 , which is a semiquantum
method that offers a satisfactory description of GIFAD in terms of classical
projectile trajectories. The projectile-surface interaction, which is a key
ingredient in all GIFAD simulations, is described by means of the pairwise
additive (PA) potential of Refs. Miraglia2017 ; Gravielle2019 . This PA
potential is built as a sum of binary interatomic potentials that represent
the interaction of the atomic projectile with individual ionic centers in the
crystal, incorporating nonlocal contributions of the electron density, along
with the effect of the Madelung potential. Concerning our theoretical model,
it should be mentioned that the combination of the SIVR approach with this PA
potential has already shown to provide GIFAD patterns in very good agreement
with available experimental data for the He/LiF(001) system under different
incidence conditions Gravielle2014 ; Gravielle2015 ; Frisco2018 . Furthermore,
the use of PA models to represent the surface interaction makes it possible to
modify the crystallographic structure to include defects, such as terraces,
without an additional computational cost, which represents an important
advantage in comparison with ab initio calculations, like the ones based on
density functional theory.

The article is organized as follows. The theoretical model is summarized in
Sec. II, while results for step defects transverse and parallel to the axial
direction are presented and discussed in Secs. III. A and III.B respectively.
In Sec. IV we outline our conclusions. Atomic units (a.u.) are used unless
otherwise stated.

##  II Theoretical model

Within the SIVR approximation, the effective transition amplitude for elastic
atom-surface scattering reads Frisco2018 ; Gravielle2018

$\displaystyle\mathcal{A}^{\mathrm{(SIVR)}}(\mathbf{b})$ $\displaystyle=$
$\displaystyle\int d\mathbf{R}_{o}\ f_{s}(\mathbf{R}_{o}-\mathbf{b})\int
d\mathbf{K}_{o}\ f_{m}(\mathbf{K}_{o})$ (1) $\displaystyle\times
a^{\mathrm{(SIVR)}}(\mathbf{R}_{o},\mathbf{K}_{o}),$

where

$\displaystyle a^{\mathrm{(SIVR)}}(\mathbf{R}_{o},\mathbf{K}_{o})$
$\displaystyle=$ $\displaystyle\int\limits_{0}^{+\infty}dt\
\left|J_{M}(t)\right|^{1/2}e^{i\nu_{t}\pi/2}\ V_{\mathrm{PS}}(\mathbf{R}_{t})$
(2)
$\displaystyle\times\exp\left[i\left(\phi_{t}-\mathbf{Q}\cdot\mathbf{R}_{o}\right)\right]$

is the partial amplitude corresponding to the classical projectile trajectory
$\mathbf{R}_{t}\equiv\mathbf{R}_{t}(\mathbf{R}_{o},\mathbf{K}_{o})$, which
starts at the initial time $t=0$ in the position $\mathbf{R}_{o}$ with
momentum $\mathbf{K}_{o}$. The functions $f_{s}$ and $f_{m}$ describe the
spatial and momentum profiles, respectively, of the incident projectile wave-
packet, while the vector $\mathbf{b}$ denotes the initial position of the
wave-packet center. In Eq. (2),
$J_{M}(t)=\det\left[\partial\mathbf{R}_{t}/\partial\mathbf{K}_{o}\right]=\left|J_{M}(t)\right|\exp(i\nu_{t}\pi)$
is the Maslov factor (a determinant), $V_{\mathrm{PS}}(\mathbf{R}_{t})$
represents the projectile-surface interaction along the projectile path, and
$\mathbf{Q}=\mathbf{K}_{f}-\mathbf{K}_{i}$ is the projectile momentum
transfer, with $\mathbf{K}_{i}$ ( $\mathbf{K}_{f}$) being the initial (final)
projectile momentum and $K_{f}=K_{i}$. The SIVR phase at time $t$ reads
Gravielle2014

$\phi_{t}=\int\limits_{0}^{t}dt^{\prime}\
\left[\frac{\left(\mathbf{K}_{f}-\mathbf{K}_{t^{\prime}}\right)^{2}}{2m_{P}}-V_{\mathrm{PS}}(\mathbf{R}_{t^{\prime}})\right],$
(3)

where $m_{P}$ is the projectile mass and
$\mathbf{K}_{t}=m_{P}d\mathbf{R}_{t}/dt$ is the classical projectile momentum.

In this work, the projectile-surface potential is evaluated with the PA model
of Refs. Miraglia2017 ; Gravielle2019 . It is expressed as

$V_{\mathrm{PS}}(\mathbf{R}_{t})=\sum\limits_{\mathbf{r}_{\mathrm{B}}}v_{\mathbf{r}_{\mathrm{B}}}\left(\mathbf{R}_{t}-\mathbf{r}_{\mathrm{B}}\right)+U_{\mathrm{PS}}(\mathbf{R}_{{}_{t}}),$
(4)

where $v_{\mathbf{r}_{{}_{\mathrm{B}}}}\left(\mathbf{r}\right)$ describes the
short-range binary interaction between the projectile and the crystal ion
placed at the Bravais lattice site $\mathbf{r}_{{}_{\mathrm{B}}}$, as a
function of the relative vector $\mathbf{r}$, and
$U_{\mathrm{PS}}(\mathbf{R}_{{}_{t}})$ denotes the projectile polarization
term, which describes the long-range projectile-surface interaction due to the
rearrangement of the electron density of the atomic projectile.

The binary potentials $v_{\mathbf{r}_{{}_{\mathrm{B}}}}$ are expressed in
terms of the unperturbed electron densities of the projectile and the ionic
sites, incorporating not only non-local contributions of these electron
densities, but also the Madelung contribution, i.e., the effect of the ionic
crystal lattice on the electron density around each individual ionic site. In
turn, the potential $U_{\mathrm{PS}}$ depends on the surface electric field at
the position $\mathbf{R}_{{}_{t}}$ of the atomic projectile, reading

$U_{\mathrm{PS}}(\mathbf{R}_{{}_{t}})=-\frac{\alpha_{P}}{2}\left|\sum\limits_{\mathbf{r}_{\mathrm{B}}}\mathbf{E}_{\mathbf{r}_{\mathrm{B}}}(\mathbf{R}_{t}-\mathbf{r}_{\mathrm{B}})\right|^{2},$
(5)

where $\alpha_{P}$ is the dipole polarizability of the projectile
($\alpha_{P}=1.38$ a.u. for He atoms) and
$\mathbf{E}_{\mathbf{r}_{\mathrm{B}}}\left(\mathbf{r}\right)$ is the electric
field produced by the asymptotic charge of the crystal ion placed at
$\mathbf{r}_{{}_{\mathrm{B}}}$. In Eqs. (4) and (5) the summation on
$\mathbf{r}_{\mathrm{B}}$ covers all the occupied lattice sites of the crystal
sample, taking into consideration the presence of flat terraces, as well as
the rumpling of the topmost atomic layer of each terrace, while the sub-index
$\mathbf{r}_{\mathrm{B}}$ in $v_{\mathbf{r}_{\mathrm{B}}}$ and
$\mathbf{E}_{\mathbf{r}_{\mathrm{B}}}$ was included to distinguish the the two
different ions of the crystallographic basis.

From Eq. (1), the differential probability of scattering in the direction of
the solid angle $\Omega_{f}=(\theta_{f},\varphi_{f})$, with $\theta_{f}$
($\varphi_{f}$) being the final polar (azimuthal) angle measured with respect
to the surface (axial channel), can be obtained as:

$\frac{dP^{\mathrm{(SIVR)}}}{d\Omega_{f}}=K_{f}^{2}\int d\mathbf{b\
}\left|\mathcal{A}^{\mathrm{(SIVR)}}(\mathbf{b})\right|^{2},$ (6)

where the integral on $\mathbf{b}$ covers an area equal to a reduced unit cell
of the crystal surface. This integration, associated with the spot-beam effect
Frisco2018 , takes into account that it is experimentally impossible to
control the focus position of the atomic beam with nanoscale precision.
Details about the SIVR method can be found in Refs. Gravielle2014 ;
Gravielle2015 ; Frisco2018 ; Gravielle2018 .

##  III Results

With the aim of studying the effect produced by the presence of terraces in
alkali-halide crystal surfaces, we evaluated diffraction patterns for 4He
atoms scattered off LiF(001) along the $\left\langle 110\right\rangle$
direction considering a crystal sample with one step of height $H$ oriented
perpendicular or parallel to the incidence channel. The height of the step was
assumed to be equal to the distance between layers, that is, $H=\pm a/2$,
where $a$ is the lattice parameter ($a=4.02$ Å for LiF) and the sign $\pm$
indicates an outward ($+$) or inward ($-$) terrace parallel to the surface
plane (see Fig. 1). Notice that for alkali-halide surfaces, atomically flat
terraces with sizes ranging from $1000$ to $2000$ Å can be observed under the
usual cleanness conditions Prohaska1994 . But the heights of these terraces
are also variable and they can be higher than the monolayer height Meyer1988 .

The terrace was simulated by adding or removing a monolayer (according to the
sign of $H$) in the topmost half-plane of the crystal sample, as depicted in
Fig. 1. The topmost atomic layer of the terrace includes the rumpling, that
is, the different relaxation of the outermost F- and Li+ ions with respect to
the unreconstructed plane Miraglia2017 . However, our simplified step model
neglects the reconstruction effects at the edge of the terrace, describing the
crystal defect as a sharp step. Even though such an ideal cutting of the
crystal terrace represents a rough description of real surface defects, this
assumption can be consider as a zero-order approach to study the influence of
step defects in GIFAD. Moreover, recent topographic images of other alkali-
halide crystal - KBr - show sharp profiles of the monatomic steps Ito2016 .

Figure 1: (Color online) Depiction of (a) up- and (b) down- steps oriented
perpendicular to the axial channel ($\widehat{x}$).

In this article, we chosen a fixed impact energy, $E=$
$K_{i}^{2}/(2m_{P})=1.25$ keV, for which experimental GIFAD distributions are
available Schuller2009c . Two normal incidence energies
$E_{\perp}=E\sin^{2}\theta_{i}$ ($\theta_{i}$ denotes the glancing incidence
angle) are analyzed - $E_{\perp}=0.20$ and $0.46$ eV - which correspond
respectively to low and intermediate $E_{\perp}$ values for He/LiF GIFAD
Schuller2009 . Notice that for the latter normal energy, the experimental
projectile distribution reported in Ref. Schuller2009c  was adequately
described with our theoretical method without taking into account the presence
of crystal defects Frisco2018 . Then, such a good theory-experiment agreement
suggests that typical GIFAD experiments are carried out on large flat
terraces, being unaffected by step defects Debiossac2021 .

For different configurations and relative positions of the monolayer step,
two-dimensional projectile distributions, as a function of the final polar and
azimuthal angles, were calculated within the SIVR approach, as given by Eq.
(6). In the calculation of $\mathcal{A}^{\mathrm{(SIVR)}}$ [Eq. (1)], the
spatial and momentum profiles of the incident wave packet were determined as
given in Refs. Gravielle2015 ; Gravielle2018  by considering a collimating
scheme formed by a rectangular slit of area $0.40$ $\times 0.09$ mm2 (the
latter length in the transverse direction) placed at $30$ cm from the surface,
with an angular beam dispersion of $0.006$ deg. From these collimating
parameters, which are in agreement with current experimental setups Bocan2020
, we derived the transverse coherence lengths in the directions perpendicular
($\sigma_{y}$) and parallel ($\sigma_{x}$) to the incidence channel
Gravielle2015 ; Gravielle2018  \- $\sigma_{y}\simeq 13$ Å and
$\sigma_{x}\simeq 240$ ($160$) Å for $E_{\perp}=0.20$ ($0.46$) eV - which
verify the relation $\sigma_{x}\gg\sigma_{y}$. It reinforces the fact that
under grazing incidence, helium projectiles probe much longer distances along
the axial channel than in the perpendicular direction, making GIFAD patterns
more sensitive to transverse steps, oriented perpendicular to the channel,
than to those oriented in the parallel direction.

###  III.1 Effects due to a transverse terrace

In this subsection, we consider a LiF(001) surface with a monolayer step
oriented perpendicular to the axial direction. To investigate the influence of
the position of the border of the terrace on the diffraction patterns, we
determine the relative position of the transverse step by means of the
distance $d_{x}$ between the edge of the terrace and the mean focus point of
the incident beam, that is,

$d_{x}=x_{\mathrm{step}}-X_{\mathrm{F}},$ (7)

where $x_{\mathrm{step}}$ is the step position along the incidence channel
($\widehat{x}$) and $X_{\mathrm{F}}$ denotes the mean position along
$\widehat{x}$ of the focus point of the helium beam. The value of
$X_{\mathrm{F}}$ is defined as the average of the $x$\- positions
corresponding to the turning points $(X_{tp},Z_{tp})$ of projectile
trajectories specularly reflected from a perfect surface (see Fig. 1). For an
ideal LiF(001) crystal, these trajectories run on the flat regions of the
projectile-surface potential, i.e., along the F- or Li+ rows.

Using the relative step position defined by Eq. (7), positive distances
$d_{x}$ indicate that He projectiles should reach the turning point before
being affected by the step in the outgoing path, while negative values are
associated with steps affecting the incoming path of the incident atoms.
Clearly, this is only an overall description of the scattering process in the
presence of surface terraces because depending on the incidence conditions and
the $d_{x}$ value, the turning points of scattered projectiles could be
modified by the strong change in the surface potential introduced by the
presence of the outward or inward step.

We start analyzing projectile distributions for the higher normal energy -
$E_{\perp}=0.46$ eV - for which terrace effects are expected to be more
important. For this normal energy, corresponding to the incidence angle
$\theta_{i}=1.1\deg$, simulated diffraction patterns respectively derived from
a perfect LiF(001) surface and from a surface with an upward step placed at a
distance $d_{x}\simeq+120$ Å are compared in Fig. 2. Taking into account that
large terraces are usually present in the LiF samples used in GIFAD
experiments Debiossac2021 , this distance $d_{x}$ corresponds to a step
position relatively close to the beam focus point.

Figure 2: (Color online) Two-dimensional projectile distributions, as a
function of $\theta_{f}$ and $\varphi_{f}$, for $1.25$ keV 4He atoms scattered
off LiF(001) along the $\left\langle 110\right\rangle$ channel with
$E_{\perp}=0.46$ eV. SIVR simulations for (a) a perfect crystal surface and
(b) a surface with a transverse up-step placed at $d_{x}\simeq+120$ Å, with
$d_{x}$ as defined in Eq. (7), are displayed.

Both panels of Fig. 2 present similar interference structures, with equally
$\varphi_{f}$\- spaced Bragg maxima lying on a thin annulus associated with
the Laue circle Rousseau2007 ; Schuller2008 , which is defined by
$\theta_{f}^{(\text{{L}})}=\left(\theta_{i}^{2}-\varphi_{f}^{2}\right)^{1/2}$.
However, as a consequence of the presence of the upward step, the angular
distribution of Fig. 2 (b) shows outermost peaks noticeably extended towards
larger polar angles, along with different relative intensities of the inner
peaks, in comparison with those for a perfect surface (Fig. 2 (a)). Regarding
this latter effect, notice that the intensities of the Bragg maxima are
modulated by the intra-channel interference, associated with the profile of
the surface potential across the axial direction Schuller2008 ; Gravielle2014
, which suffers a local change at the edge of the terrace. Instead, the
$\varphi_{f}$ positions of the Bragg peaks are determined by the inter-channel
interference, which depends on the spacing between equivalent parallel
channels Schuller2008 ; Gravielle2014 , a parameter that is not altered by the
transverse step.

To thoroughly analyze the step effect on Bragg-peak intensities, in Fig. 3 (a)
we compare the azimuthally projected distributions corresponding to the two
panels of Fig. 2, obtained by integrating Eq. (6) over a reduced area on the
Laue circle Debiossac2016 . In Fig. 3 (a) we observe that the presence of the
transverse up-step introduces an almost constant background in the azimuthal
spectrum. Furthermore, the distortion of the surface potential caused by the
outward step strongly affects the peak intensities, markedly reducing the
relative intensity of the maxima of orders $5^{th}$ and $7^{th}$, as well as
that of the central peak ($0^{th}$ order) whose intensity becomes much lower
than those of the adjacent maxima, in contrast to what is observed for a
perfect surface. Also, the intensity of the outermost maxima (not shown in
Fig. 3 (a) ) is lowered as a result of the surface defect, whereas the
intensity of $6^{th}$\- order maxima is raised, making these latter peaks
visible.

Figure 3: (Color online) Azimuthal spectra, corresponding to the Laue annulus,
for the normal energy $E_{\perp}=0.46$ eV. Thin red line, SIVR simulations for
a transverse (a) up-step and (b) down-step, both placed at $d_{x}\simeq+120$
Å; thick gray line, SIVR results for a perfect crystal surface. Figure 4:
(Color online) Analogous to Fig. 2 (b), extending the $\theta_{f}$ range and
increasing the sensitivity of the intensity scale, as explained in the text.

At this point, it is important to recall that typical GIFAD patterns for
perfect crystal surfaces are essentially determined by the surface potential
averaged along the axial direction Zugarramurdi2012 ; Muzas2016 . But the
presence of a transverse step breaks the translation invariance associated
with this averaged surface potential, altering the effective slope of the
reflection plane around the edge of the terrace. This fact is evidenced in
Fig. 4, where we have extended the $\theta_{f}$ range of Fig. 2 (b), lowering
also the intensity scale by one order of magnitude to show the terrace
effects. The projectile distribution of Fig. 4 presents a diffuse background
for $\theta_{f}\succsim\theta_{f}^{(\text{{L}})}$, with additional peaks at
the central and outermost azimuthal angles. In this case, a large proportion
of the scattered projectiles ($\approx 54\%$ ) are deflected above the Laue
circle due to the steep increase of the surface potential at the step
position, while about $15\%$ of the incident projectiles penetrates in the
terrace bulk.

Figure 5: (Color online) Sample of projectile trajectories contributing to the
central region (i.e., in-plane trajectories) of Fig. 4, after running along
(a) F- and (b) Li+ rows of the initial half-plane (see Fig. 1). Both panels
display the normal position $Z$ as a function of the coordinate $X$ along the
$\langle 110\rangle$ channel. Gray and blue lines indicate trajectories ending
respectively on and off the Laue circle.

Since the most remarkable feature of the projectile distribution of Fig. 4 is
the central peak above the Laue circle, which is placed at $\theta_{f}\simeq
1.47\deg$, in Fig. 5 we show a sample of random projectile paths that
contribute to the central region of this angular distribution. Such
trajectories are confined to the scattering plane (i.e., in-plane
trajectories), initially running along the F- or Li+ rows of the topmost layer
in the initial half-plane (note that F- or Li+ rows switch at the terrace), as
respectively shown in Figs. 5 (a) and 5 (b). In both panels, the wide $x$\-
spread of the classical turning points is associated with the large
$\sigma_{x}$ value. Hence, projectile paths with turning points far away from
the step end on the Laue circle, without being affected by the presence of the
upward terrace. But those trajectories with turning points placed at distances
$\left|x_{\mathrm{step}}-X_{tp}\right|\lesssim 130$ ($170$) Å in Fig. 5 (a)
(Fig. 5 (b)), are deflected with a final polar angle
$\theta_{f}\succ\theta_{f}^{(\text{{L}})}$ due to the change in the surface
potential, which abruptly becomes repulsive at the step. We stress that in
Fig. 4, the on-Laue interference maxima, as well as the intense outermost
peaks above the Laue circle, are produced by quantum interference among
partial transition amplitudes corresponding to different projectile paths,
that is, they cannot be explained as points of accumulation of trajectories.
However, the central maximum above the Laue circle has a classical origin,
also being observed in the classical projectile distribution.

Figure 6: (Color online) Analogous to Fig. 2 (b) for a LiF surface with a
transverse up-step placed at (a) $d_{x}\simeq+67$ Å and (b) $d_{x}\simeq-67$
Å.

A similar central off-Laue maximum, associated with the presence of the
transverse terrace, can be seen in Fig. 6 (a) for $d_{x}\simeq+67$ Å, a step
position even closer to $X_{\mathrm{F}}$. But in this case, since the atomic
beam is almost focused on the step region, Bragg peaks start to wash out,
whereas the central peak above the Laue circle is clearly visible without
increasing the sensitivity of the intensity scale. The same up-step effects
are also observed in the projectile distribution of Fig. 6 (b) corresponding
to an upward step placed in front of the focus point, at $d_{x}\simeq-67$ Å.
Nevertheless, the intensity of the background and the off-Laue peaks markedly
decreases as the projectile trajectories meet the terrace edge in their
incoming paths.

Figure 7: (Color online) Analogous to Fig. 2 (b) for a LiF surface with a
transverse up-step placed at $d_{x}\simeq+200$ Å.

As expected, the step effects gradually disappear as the distance $d_{x}$
increases. The projectile distribution for an upward step placed at
$d_{x}\simeq+200$ Å, plotted in Fig. 7, looks similar to that for a perfect
crystal surface (Fig. 2 (a)), without any signature of the terrace effects,
except for the polar elongation of the outermost peaks with respect to those
derived from a perfect surface. Under these incidence conditions, most helium
projectiles ($\approx 70\%$) hit the detector plane with
$\theta_{f}\simeq\theta_{f}^{(\text{{L}})}$, while only a very small fraction
($\approx 2\%$) penetrates into the bulk at the step, indicating that the
projectile distribution tends to the one corresponding to a perfect LiF
surface.

The distance $d_{x}$ for which unperturbed GIFAD patterns can be obtained
depends on the normal energy. When $\theta_{i}$ decreases, so $E_{\perp}$
does, grazing projectiles probe longer distances along the axial direction,
being affected by transverse upward steps placed at longer distances $d_{x}$.
This fact can be observed in Fig. 8 (b), where the projectile distribution for
$E_{\perp}=0.20$ eV ( $\theta_{i}=0.7\deg$), produced by a crystallographic
configuration similar to that of Fig. 7, shows noticeable changes in the
relative peak intensities, with respect to those derived by considering a
perfect crystal surface, displayed in Fig. 8 (a).

Figure 8: (Color online) Analogous to Fig. 2 for the normal energy
$E_{\perp}=0.20$ eV and a transverse up-step placed at $d_{x}\simeq+200$ Å.
Figure 9: (Color online) Analogous to Fig. 2 (b) for a LiF surface with a
transverse down-step placed at $d_{x}\simeq+120$ Å.

Finally, we analyze the influence of a transverse down-step, as depicted in
Fig. 1 (b). In Fig. 9, we plot the angular distribution for $E_{\perp}=0.46$
eV derived by considering an inward step placed at $d_{x}\simeq+120$ Å. In
contrast to the up-step effects shown in Fig. 2 (b), the presence of the down
step does not affect the on-Laue distribution, which is similar to that for a
perfect surface (Fig. 2 (a)). This feature can be quantitative confirmed by
comparing the corresponding azimuthally projected spectra, which are quite
alike, as observed in Fig. 3 (b). In the case of Fig. 9, however, although
approximately $80\%$ of the trajectories are scattered from the LiF surface
with $\theta_{f}\simeq\theta_{f}^{(\text{{L}})}$, there are trajectories whose
outgoing paths are distorted by the decrease of the potential at the edge of
the terrace, being deflected below the Laue circle. It gives rise to a diffuse
background below the Laue circle, whose intensity is several orders of
magnitude lower than those of Bragg peaks.

###  III.2 Effects due to a parallel terrace

The terrace effects described in Sec. III. A change when the monolayer step is
oriented parallel to the axial direction. Analogously to the case of
transverse terraces, we determine the relative position of the parallel step
by means of the distance $d_{y}$ between the edge of the terrace and the mean
focus point, that is,

$d_{y}=y_{\mathrm{step}}-Y_{\mathrm{F}},$ (8)

where $y_{\mathrm{step}}$ is the step position across the incidence channel
($\widehat{y}$) and $Y_{\mathrm{F}}$ denotes the mean position of the focus
point of the incident beam along $\widehat{y}$. As a consequence of the
symmetry of the problem, only outward steps placed at positive distances
$d_{y}$ will be considered in this subsection.

Figure 10: (Color online) Analogous to Fig. 2 (b) for a LiF surface with a
parallel up-step placed along (a) a Li+ row (at $d_{y}=+7.1$ Å), and (b) a F-
row (at $d_{y}=+8.5$ Å) of the perfect crystal surface.

Taking into consideration that the transverse length of the surface area that
is probed by helium projectiles is much smaller than the axial one, in Fig. 10
we show two-dimensional angular distributions for surface steps along the
$\left\langle 110\right\rangle$ channel considering closer distances to the
focus point of the beam, that is, $d_{y}\simeq+7.1$ Å in Fig. 10 (a) and
$d_{y}\simeq+8.5$ Å in Fig. 10 (b). These steps are placed respectively on top
of Li+ and F- rows of the ideal perfect surface. We found that the angular
distributions of Fig. 10 are fully confined to the Laue circle, that is, all
scattered projectiles leave the surface with
$\theta_{f}\simeq\theta_{f}^{(\text{{L}})}$, while the fraction of
trajectories penetrating into the terrace bulk is lower than $1\%$.
Nonetheless, the presence of a parallel step in the area that is coherently
illuminated by the atomic beam introduces an azimuthal asymmetry in the GIFAD
patterns of Figs. 10 (a) and 10 (b), which display some interference maxima
with structures elongated along the Laue circle. This latter effect depends on
the exact position of the step, which determines the shape of the
equipotential curves in the region of the terrace edge. In Fig. 11 we plot the
equipotential curves, corresponding to surface potential averaged along the
axial direction, for the two cases of Fig. 10. Around the step region, the
equipotential contours vary if the terrace edge is along a Li+ or a F- row of
the perfect surface. Similar differences in the equipotential curves are also
observed for transverse terraces. But the influence of such border effects on
GIFAD patterns seems to be stronger for parallel than for transverse upward
steps.

Figure 11: (Color online) Equipotential curves of the axial potential
(averaged along the $\langle 110\rangle$ channel) for a LiF(001) surface with
a parallel up-step placed along (a) a Li+ row and (b) a F- row of the perfect
crystal surface. Energy scale in eV.

To investigate in more detail the asymmetry introduced by parallel steps, in
Fig. 12 we plot the azimuthal spectra corresponding to the distributions of
Fig. 10, contrasting them with that derived from a perfect crystal surface. In
both panels of Fig. 12, the intensities of the Bragg peaks of order $-3$ and
$+6$ increase as a consequence of the presence of the parallel step, while the
central region of the spectrum is not affected by the surface defect.
Furthermore, in Fig. 12 (a) the sharp increase of the $3^{rd}$-order maxima
smudges the $4^{th}$-order peak, both forming a broad intense peak on the left
side of the projectile distribution, whereas in Fig. 12 (b), the peak of order
$-4$ is not altered by the terrace border.

Figure 12: (Color online) Azimuthal spectra, corresponding to the Laue
annulus, for the cases of Fig. 10. Lines, analogous to Fig. 3.

##  IV Conclusions

In this work, the influence of surface defects on GIFAD patterns for the
He/LiF(001) system was investigated by considering the presence of a monolayer
step in the coherently illuminated region of the crystal target. Our
theoretical description was based on the use the SIVR method to represent the
grazing scattering process, combined with a PA atom-surface potential
Miraglia2017 . This potential model allowed us to easily incorporate outward
or inward monolayer terraces by adding or removing, respectively, an atomic
layer in a portion of the LiF sample.

In order to simplify the analysis, only two different orientations of the
terrace edge - transverse and parallel to the incidence channel - were
considered. For transverse up-steps placed close to the focus point of the
incident beam, at distances $d_{x}$ smaller than a few hundred Å, simulated
projectile distributions display the characteristic Bragg peaks on the Laue
circle, along with a diffuse background above the Laue circle, which presents
additional central and outermost peaks. The upward step also affects the
relative intensities of the on-Laue maxima, which are relevant for the use of
GIFAD as a surface analysis technique. For transverse down-steps, instead, the
projectile distribution approximates the one derived from a perfect crystal
surface, while the terrace border introduces only a very weak background below
the Laue circle, whose intensity is several orders of magnitude smaller than
those of the on-Laue maxima. As expected, these terrace effects depend on the
incidence conditions, gradually disappearing as $d_{x}$ increases.

On the other hand, as a consequence of the smaller transverse coherence length
of the beam in the direction perpendicular to the axial channel, the presence
of a parallel step affects the projectile distribution only if the step is
placed at a distance $d_{y}$ less than a few tens Å from the focus point of
the incident beam. In this case, the parallel up-step introduces an azimuthal
asymmetry in the angular spectrum, which is fully localized on the Laue
circle.

Summarizing, we found that terrace effects on GIFAD patterns strongly depend
on the orientation of the edge of the monolayer terrace, as well as on the
height (outward or inward) and the position of the step. Therefore, these
findings suggest that GIFAD may be a useful tool to study terrace defects on
alkali-halide surfaces. We hope that this study will be helpful to trigger
experimental research on this topic.

###### Acknowledgements.

M.S.G is grateful to H. Khemliche for the helpful discussions. We acknowledge
funding from ANPCYT (PICT-2017-1201; PICT-2017-2945; PICT-2020-1755;
PICT-2020-1434) of Argentina.

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