# Fourier Transform-Based Post-Processing Drift Compensation and Calibration
Method for Scanning Probe Microscopy

M. Le Ster<EMAIL_ADDRESS>S. Pawłowski  I. Lutsyk  P. J.
Kowalczyk<EMAIL_ADDRESS>University of Lodz, Faculty of Physics
and Applied Informatics, Department of Solid-State Physics, Pomorska 149/153,
Lodz, 90-236, Poland

###### Abstract

Scanning probe microscopy (SPM) is ubiquitous in nanoscale science allowing
the observation of features in real space down to the angstrom resolution. The
scanning nature of SPM, wherein a sharp tip rasters the surface during which a
physical setpoint is maintained via a control feedback loop, often implies
that the image is subject to drift effects, leading to distortion of the
resulting image. While there are _in-operando_ methods to compensate for the
drift, correcting the residual linear drift in obtained images is often
neglected. In this paper, we present a reciprocal space-based technique to
compensate the linear drift in atomically-resolved scanning probe microscopy
images without distinction of the fast and slow scanning directions;
furthermore this method does not require the set of SPM images obtained for
the different scanning directions. Instead, the compensation is made possible
by the a priori knowledge of the lattice parameters. The method can also be
used to characterize and calibrate the SPM instrument.

Scanning probe microscopy, calibration, drift compensation, reciprocal space

##  1 Introduction

Scanning probe microscopy (SPM) has revolutionized nanotechnology and our
understanding of nature [1], allowing for the real space observation and
characterization of nano-sized objects (and under certain conditions, local
manipulation), such as individual atoms [2], molecules [3], nanoparticles [4],
surfaces of crystalline bulk materials [5], 1D crystals such as nanotubes [6],
as well as larger macromolecules such as DNA [7] and viruses [8], and even
larger objects of biological nature [9]. Among the many SPM techniques, the
most popular today are scanning tunneling microscopy (STM) [5, 10], wherein a
biased atomically-sharp metallic probe is brought near the surface of the
sample maintaining a quantum tunneling current setpoint; and atomic force
microscopy (AFM) [11] where the deflection of the cantilever (attached to the
sharp tip-probe), in contact or near-contact with the surface, is measured via
optical methods. Beyond topography-based SPM (yielding height maps), there are
many other SPM techniques tangential to STM and AFM dedicated to additional
electronic characterizations that are worth mentioning such as scanning
tunneling spectroscopy [12] (mapping the electronic density of states),
Kelvin-probe force microscopy [13] (mapping the electronic work function) and
magnetic force microscopy [14] (sensitive to the surface magnetization).

The inherent scanning nature of SPM mean that the image acquisition is
simultaneously obtained with the rastering of a piezoelectric tube carrying
the tip-sensor. While this has the advantage of achieving extremely high
resolution images (with lateral resolutions down to the angstrom scale, or
even below [15]) and the ability to characterize local physical properties,
SPM often suffers from drift effects of various origins, such as piezoelectric
hysteresis and scanner creep [16] (due to mechanical relaxation of the
piezoelectric components) and thermal drift [17] (due to differing thermal
expansion coefficients across different constituent parts of the SPM
instrument, whether in the sample stage or in the scanning stage elements),
inducing image distortion.

While there exist _in-operando_ methods to compensate the drift effects
(_e.g._ , relying on the monitoring a feature in two SPM images obtained
sequentially [18]), it is possible that distortion still occurs in the final
image due to subsequent changes in the drift velocity vector, its typical
values being in the nm/s range at room temperature [16]. A number of post-
processing methods (_i.e._ , after image acquisition) to compensate for
thermal drift have previously been developed using a number of different
approaches; Fourier-transform cross-correlation [19], phase correlation [20],
or relying on images obtained in different scanning directions [21], and
lastly addressing non-linear distortion effects [22]. Another possible
scenario where post-processing correction is required occurs when sudden
changes in the tip-sample interactions lead to undesirable effects such as an
unforeseen loss of sharpness or a substantial noise increase, before _in-
operando_ drift compensation methods are applied. For this reason it is often
desirable to have post-processing drift compensation methods in order to
ensure proper mapping, _i.e._ , spatial integrity of the final image.

In this paper, we present a technique based on the Fourier transform of the
acquired SPM image that allows for post-processing compensation of the linear
drift which may exist in images whether or not _in-operando_ efforts have been
undertaken to curb drift effects. The reciprocal space nature of the technique
requires that the real space images contains periodic features in order to
easily correct the skewed raw image. A similar method has been developed
previously [23], however its implementation requires several additional steps
(such as the determination of the drift velocity vector) and the knowledge of
the fast and slow scanning directions, irrelevant in our method. Another
strength of the method is that it can be applied on a single image (without
requiring the different images obtained for different scanning directions)
obtained using an SPM instrument that has not been calibrated along the
$(x,y)$ scanning directions. After introducing the model and its
implementation, we benchmark and illustrate the method on two images that
contain periodic modulations: _(i)_ an atomically-resolved graphite sample
imaged by STM and _(ii)_ a calibration grating sample imaged by AFM.

##  2 Model and Implementation

Figure 1a shows a simulated ideal drift-free SPM image consisting of a
triangular lattice (see inset for real space base vectors), which can be of
various origins: primitive unit cell at the surface of a crystalline material
(bulk or 2D material, typical scale: $\sim 3~{}\mathrm{\AA}$), superlattice
(such as moiré patterns [24], surface reconstruction [10], or charge-density-
wave, _e.g._ , in 1T-TaS2 [25], typical scale: $\sim 10-100~{}\mathrm{\AA}$),
or due to artificial nano-patterning (_e.g._ , via lithography, typical scale:
$10-10000$ nm). Figure 1b shows an equivalent SPM image, this time obtained
under significant linear drift conditions, resulting in a strongly distorted
image (compare insets in Figs. 1a and 1b to visualize distortion of the
primitive real space vectors). The unit cell vectors
$(\mathbf{r}_{1},\mathbf{r}_{2})$ are known _a priori_ by means of theory or
other experimental methods; alternatively the sample may be a calibration
sample.

Figure 1: (a) Ideal SPM image of a triangular lattice (in arbitrary units of
spatial dimensions). (b) Drift-containing SPM image of a triangular lattice.
(c) FFT of the ideal SPM image in (a). (d) FFT of the drift-containing SPM
image in (b). (e) Linear transformations between the different vector bases.
Insets in (a, b) are higher resolution cropped images showing the ideal
($\mathbf{r}_{1},\mathbf{r}_{2}$) and observed
($\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2}$) primitive real space
vectors, respectively. Arrows in (c, d) correspond to the primitive reciprocal
lattice vectors ($\mathbf{k}_{1},\mathbf{k}_{2}$) and
($\mathbf{k}^{\prime}_{1},\mathbf{k}^{\prime}_{2}$) for the ideal and observed
images, respectively.

Figures 1c and 1d are the fast Fourier transforms (FFTs) of the ideal and
drifted real space images in Figs. 1a and 1b respectively. The sharp features
resolved in FFTs correspond to the primitive reciprocal lattice vectors
$\mathbf{k}_{1}$ and $\mathbf{k}_{2}$ ($\mathbf{k}^{\prime}_{1}$ and
$\mathbf{k}^{\prime}_{2}$).

###  2.1 Matrix definitions

The essence of the model is to obtain the linear matrix $M_{r}$ that linearly
transforms the observed lattice
($\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2}$) into the ideal lattice
($\mathbf{r}_{1},\mathbf{r}_{2}$) which would be resolved in a drift-free
imaging conditions on a calibrated instrument. The distorted image (under the
$M_{r}$ linear transformation) will then display a corrected lattice
$(\mathbf{r}_{1},\mathbf{r}_{2})$. The matrix scheme associating the different
vector bases is detailed in Fig. 1e. The matrix $M_{r}$ transforms
($\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2}$) into
($\mathbf{r}_{1},\mathbf{r}_{2}$), and the reciprocal space equivalent $M_{k}$
transforms ($\mathbf{k}^{\prime}_{1},\mathbf{k}^{\prime}_{2}$) into
($\mathbf{k}_{1},\mathbf{k}_{2}$). The inverse matrices $M_{r}^{-1}$ and
$M_{k}^{-1}$ are the inverse operations, transforming
$(\mathbf{r}_{1},\mathbf{r}_{1})$ into
$(\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{1})$ and
$(\mathbf{k}_{1},\mathbf{k}_{1})$ into
$(\mathbf{k}^{\prime}_{1},\mathbf{k}^{\prime}_{1})$, respectively.

Instead of directly computing the real space matrix $M_{r}$, the inputs in the
method are the observed and ideal _reciprocal_ lattice parameters, in order to
obtain a reciprocal space matrix $M_{k}$. The real space matrix $M_{r}$ can
then be obtained by matrix inversion and transposition. The reason this is
that the observed reciprocal lattice parameters
$(\mathbf{k}^{\prime}_{1},\mathbf{k}^{\prime}_{2})$ are easy to extract from
the raw data (_i.e._ by simple observation of the FFT) as opposed to the real
space vectors $(\mathbf{r}^{\prime}_{1},\mathbf{r}^{\prime}_{2})$ whose
characterization is subject to a higher uncertainty and possibly local
variation across the same image. The ideal reciprocal lattice vectors
$\mathbf{k}_{1}$, $\mathbf{k}_{2}$ can be calculated using standard formulae
[26].

Solving for a general matrix $M_{k}$ such that
$\mathbf{k}_{1}=M_{k}\times\mathbf{k}^{\prime}_{1}$ and
$\mathbf{k}_{2}=M_{k}\times\mathbf{k}^{\prime}_{2}$, we can obtain the
reciprocal space matrix $M_{k}$:

$\begin{split}M_{k}&=\frac{1}{\Delta}\begin{pmatrix}k_{1x}k^{\prime}_{2y}-k_{2x}k^{\prime}_{1y}&k_{2x}k^{\prime}_{1x}-k_{1x}k^{\prime}_{2x}\\\
k_{1y}k^{\prime}_{2y}-k_{2y}k^{\prime}_{1y}&k_{2y}k^{\prime}_{1x}-k_{1y}k^{\prime}_{2x}\end{pmatrix},\\\
\Delta&=k^{\prime}_{1x}k^{\prime}_{2y}-k^{\prime}_{2x}k^{\prime}_{1y}.\end{split}$
(1)

While a correction of the reciprocal space data (_i.e._ , FFT) can be achieved
using $M_{k}$, the motivation concerns the drift correction of _real space_
images, and we are now concerned with the matrix $M_{r}$. It can be shown that
the matrix $M_{r}$ is derived from $M_{k}$ as follows:

$M_{r}=(M_{k}^{T})^{-1}$ (2)

where $M^{T}$ denotes the transpose of $M$ (see supplementary information for
a formal demonstration).

###  2.2 Implementation

We briefly describe the implementation of the model using the warpAffine
function from the openCV library [27] to perform the linear transformation of
the raw SPM image. The warpAffine function returns the straightened image
(with no dimension metadata) and takes the following inputs: $(a)$ the image
array ($n^{\prime}\times m^{\prime}$ pixels, no specified pixel or image
dimensions), $(b)$ an affine $(2\times 3)$ matrix (of which the third column
is a translation vector in pixel coordinates), $(c)$ the final image size (in
pixels) and $(d)$ an optional border mode. The four inputs as well as the
final image dimensions are described in the following paragraphs.

#### Image array.

We operate primarily on the $(n^{\prime}\times m^{\prime})$ real space image
array obtained by the SPM instrument (we also operate on the FFT image later).

####  $(2\times 3)$ matrix.

The input matrix $M$ is a $(2\times 3)$ matrix representing an affine
transform, _i.e._ , $M\times X=M^{\prime}\times X+T$ with $M^{\prime}$ the
$(2\times 2)$ linear matrix, $X$ the vector to modify and $T$ a translation
vector. We find that it is convenient to use a linear $(2\times 2)$ matrix
that is _normalized_ , which we define in the following. Any linear matrix $M$
can be thought of as a combination of a biaxial scaling matrix $M_{s}$, a
rotation matrix $M_{\theta}$ and a shear matrix $M_{\tau}$ as follows:

$\begin{split}M&=M_{s}\times M_{\theta}\times M_{\tau}\\\
M_{s}&=\begin{pmatrix}s_{x}&0\\\ 0&s_{y}\end{pmatrix}\\\
M_{\theta}&=\begin{pmatrix}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{pmatrix}\\\ M_{\tau}&=\begin{pmatrix}1&\tau\\\
0&1\end{pmatrix}\end{split}$ (3)

where $s_{x}$ and $s_{y}$ are scaling factors along the $x$ and $y$
directions, $\theta$ is the rotation angle and $\tau$ the shear parameter. The
order of the matrix multiplication is not crucial (one could alternatively
scale, then shear, and finally rotate; or in another order), however it is
important to keep the order of operation consistent since matrix
multiplication is not commutative. The _normalized_ matrix $\tilde{M}$ is
then:

$\tilde{M}=M_{\theta}\times M_{\tau}.$ (4)

It is worth noting that $\det(\tilde{M})=1$, _i.e._ , a linear operation under
a normalized matrix preserves areas. The $(2\times 3)$ is then a composite
matrix made of the $(2\times 2)$ _normalized_ matrix $\tilde{M_{r}}$, and of a
$(1\times 2)$ translation vector $T$ which we define in the following. The
translation vector $T$ ensures that that every pixel coordinates $(p,q)$ of
the final image are such that $p,q\geq 0$, and that $(0,0)$ is a corner of the
output image. With $C=\\{{0\choose 0},{n^{\prime}-1\choose 0},{0\choose
m^{\prime}-1},{n^{\prime}-1\choose m^{\prime}-1}\\}$ the set of corner pixel
coordinates of the raw image, the translation vector $T$ is then:

$T=-\begin{pmatrix}\min{\left\\{\left(\tilde{M}_{r}\times
C_{i}\right)\cdot{1\choose 0}\right\\}}\\\
\min{\left\\{\left(\tilde{M}_{r}\times C_{i}\right)\cdot{0\choose
1}\right\\}}\end{pmatrix}$ (5)

Given a non-normalized matrix $M=\begin{pmatrix}a&b\\\ c&d\end{pmatrix}$, its
rotation, shear and scaling parameters defined in eq. (3) can be retrieved:

$\begin{split}\theta&=\mathrm{atan2}(c,d)\\\
\tau&=\frac{a\sin\theta+b\cos\theta}{a\cos\theta-b\sin\theta}\\\
s_{x}&=\frac{a}{\cos\theta+\tau\sin\theta}=\frac{b}{\tau\cos\theta-\sin\theta}\\\
s_{y}&=\frac{c}{\sin\theta}=\frac{d}{\cos\theta}\end{split}$ (6)

with $\mathrm{atan2}$ the $2$-argument arctangent function. If $s_{x}$ or
$s_{y}$ diverge using the left-hand equalities ($\cos\theta+\tau\sin\theta=0$
or $\sin\theta=0$), the right-hand expression may be used. The normalization
of a matrix $M$ is achieved by first extracting its $\theta$ and $\tau$
components with eqs. (6) and using eq. (3) with $s_{x}=s_{y}=1$.

#### Output dimensions.

The dimensions of the output image $(n,m)$ can be obtained by considering the
difference between the maximum and minimum pixel coordinates (for both column
and row indices) upon transformation of the corner pixel coordinates with
$\tilde{M}_{r}$. The updated dimensions (in pixel units) are then:

$\begin{split}\begin{pmatrix}n\\\
m\end{pmatrix}=&\begin{pmatrix}\max{\left\\{\left(\tilde{M}_{r}\times
C_{i}\right)\cdot{1\choose 0}\right\\}}\\\
\max{\left\\{\left(\tilde{M}_{r}\times C_{i}\right)\cdot{0\choose
1}\right\\}}\end{pmatrix}+T\\\ \end{split}$ (7)

The dimensions (in pixel units) are then rounded to the closest integer
number.

#### Border mode.

An image that is warped such that $\tau\neq 0$ (_i.e._ , that undergoes shear)
will not be rectangular; rather, its corners will describe a parallelogram.
The border mode argument set to BORDER_CONSTANT will fill the values of the
undefined pixels in the border. To minimize artifacts in the FFT of the image
(due to the presence of discontinuity at the border), it is desirable to set
the value to the average value of the SPM image.

#### Spatial dimensions.

Finally, aside from the warpAffine operation (which returns a dimensionless
image), it is important to consider the _spatial_ dimensions of the image to
properly assign the dimensions of the output image, crucial for accurate
compensation. Using an approach identical to the dimensions in pixel space,
where $D=\\{{0\choose 0},{L^{\prime}_{x}\choose 0},{0\choose
L^{\prime}_{y}},{L^{\prime}_{x}\choose L^{\prime}_{y}}\\}$ the set of corner
spatial coordinates, the dimensions $L_{x}$, $L_{y}$ of the final image after
the warping procedure are:

$\begin{split}\begin{pmatrix}L_{x}\\\
L_{y}\end{pmatrix}=&\begin{pmatrix}\max{\left\\{\left(M_{r}\times
D_{i}\right)\cdot{1\choose 0}\right\\}}\\\ \max{\left\\{\left(M_{r}\times
D_{i}\right)\cdot{0\choose 1}\right\\}}\end{pmatrix}\\\
-&\begin{pmatrix}\min{\left\\{\left(M_{r}\times D_{i}\right)\cdot{1\choose
0}\right\\}}\\\ \min{\left\\{\left(M_{r}\times D_{i}\right)\cdot{0\choose
1}\right\\}}\end{pmatrix}\end{split}$ (8)

##  3 Results

###  3.1 Scanning Tunneling Microscopy

Figure 2: (a) Raw STM image of HOPG, at the location of a step edge (diagonal
line). (b) Drift-corrected STM image in (a) (color scale in Å). (c) Line
profiles extracted from lines in (a) and (b) of the same colors (vertically
offset for clarity). (d) FFT of the raw data in (a). (e) FFT of the corrected
data in (b). (f) Warped FFT of the raw data in (d). The hexagons in FFTs (d-f)
connecting the reciprocal lattice points of graphite allow to visualize the
presence of drift in (d) and its compensation in (e, f).

We illustrate the model using an experimental STM image shown in Fig. 2a. The
two main areas (dark brown and orange) in the image correspond to different
atomically flat planes of the highly-ordered pyrolithic graphite (HOPG)
crystal (space group: $P6_{3}/mmc$). The two areas of the image are offset
vertically at the step edge by $\sim 6$ Å, _i.e._ , 2 atomic planes parallel
to the $c$ axis. The corrugation on the top-left area of the image corresponds
to the unit cell of graphite, of well-known lattice constants
($r_{1}=r_{2}=2.46~{}\mathrm{\AA}$, forming a triangular lattice along the
(0001) plane [28]). This SPM image may be relevant for the investigation of
the physics related to the presence of defects along the edge of the atomic
terrace (see _e.g._ , the brighter features located at kinks and the standing
waves near the edge of the top terrace). Compensating for the drift in such
image may be of significant importance for correct real space
characterization.

The line profile, measured from the orange line in Fig. 2a, shown in Fig. 2c
(with the same color) indicates a lattice parameter of $\simeq 2.60$ Å along
this particular direction, suggesting either a calibration error (related to
the piezoelectric tube which drives the tip motion across the surface) or the
presence of thermal drift. The FFT of the raw data is shown in Fig. 2d. The
primitive reciprocal lattice vectors $\mathbf{k}^{\prime}_{1}$ and
$\mathbf{k}^{\prime}_{2}$ are clearly resolved (as well as the composite
reciprocal lattice vector $\mathbf{k}_{1}-\mathbf{k}_{2}$), and it is clear
that they do not form a regular hexagon (see dashed lines), which is to be
expected in an SPM image suffering from linear drift.

Knowing the expected lattice parameters $(\mathbf{r}_{1},\mathbf{r}_{2})$, the
ideal primitive reciprocal vectors $(\mathbf{k}_{1},\mathbf{k}_{2})$ are
calculated; the observed primitive reciprocal vectors
$(\mathbf{k}^{\prime}_{1},\mathbf{k}^{\prime}_{2})$ are obtained from FFT in
Fig. 2b. The reciprocal matrix $M_{k}$ is then easily obtained using eq. (1),
the real space matrix $M_{r}$ is computed using eq. (2). The normalized matrix
$\tilde{M_{r}}$, the translation component $T$ from eq. (5), and finally the
expected dimensions in pixel and spatial coordinates eqs. (7, 8) are also
computed. The method is then applied on the raw STM image in Fig. 2a,
obtaining the image in Fig. 2b. The line profile extracted from the same
location as previously (now in blue) is shown in Fig. 2c; the apparent lattice
parameter is now in agreement with the theoretical value of graphite,
confirming the success of the method. Note that the corrected image is skewed,
resulting from a non-zero shear ($\tau=0.066$). The other real space linear
transformation parameters are $s_{x}=0.914$, $s_{y}=1.118$ and
$\theta=3.84^{\circ}$. Figure 2e shows the FFT of the drift-corrected image of
Fig. 2b (where the border of Fig. 2b is filled with $0$ values). The sharp
features in the FFT corresponding to the primitive reciprocal lattice now
describe a regular hexagon, further confirming a satisfactory drift
compensation.

Figure 2f is obtained by applying the warpAffine operation on the FFT image in
Fig. 2d using a normalized $M_{k}$ matrix instead. There are no structural
differences to be observed with Fig. 2e; however the artifacts present in the
FFT differ slightly. This is particularly evident in case of large shear
values, where significant edge effects (due to the border populated with $0$
values) can be observed in the FFT of the skewed image. In this example
however, this effect is relatively weak, and either FFT images can be used for
further analyses if required.

###  3.2 Atomic Force Microscopy

We now focus on an AFM image of a commercially available SiO2/Si calibration
grating sample (TGQ1 from NT-MDT), shown in Fig. 3a. The sample is
characterized by the presence of SiO2 pillars with bare Si material in between
them (dark blue). In contrast to the previous case, the topography modulation
in the SPM image here results from the artificial periodic pillar structure.
The nominal distance between the pillars, arranged in a square lattice, is
$3.00\pm 0.05$ µm. Quick inspection of the raw AFM image indicates a scaling
error; the pillars appear to be separated by a smaller apparent distance,
_i.e._ , $\simeq 2$ µm in both directions. The FFT of the raw image (see inset
in Fig. 3a) further confirms this in reciprocal space, where the two primitive
reciprocal vectors $\mathbf{k}^{\prime}_{1}$ and $\mathbf{k}^{\prime}_{2}$
associated with the periodic pillar structure are arranged $\sim 0.5$ µm-1
from the origin. This is consistent with a significant calibration error,
which we correct using the method, as shown in the following.

Figure 3: (a) Raw AFM image of a nano-patterned Si sample. (b) Corrected AFM
image (color scale in µm). Insets are the FFT of the images shown in main
panels.

The as-measured primitive reciprocal coordinates (from the FFT in the inset of
Fig. 3a) and the nominal real lattice parameters ($3.0\times 3.0$ µm2 in a
square lattice) are input into the model. Figure 3b shows the AFM image after
correction. The image dimensions is now consistent with the nominal periodic
spacing. The associated transformation parameters are $s_{x}=1.509$,
$s_{y}=1.613$, $\tau=0.017$ and $\theta=0.17^{\circ}$. The non-zero shear
parameter $\tau$ indicates the presence of trace drift in the AFM image
(albeit about 5 times weaker than in the first example in Fig. 2). The scaling
factors $s_{x}$ and $s_{y}$ can then be used to calibrate the scanning stage
of the AFM instrument.

##  4 Discussion and conclusion

As shown in the two examples above, the FFT-based distortion method reliably
corrects for shear (typically resulting from drift) and scaling errors
(resulting from improper or absent calibration of the SPM scanning stage). The
method allows to retrieve the different scaling, rotation and shear parameters
contributing to a general linear transformation. The correction factors
$s_{x}$ and $s_{y}$ indicate how to scale the $x$ and $y$ directions of the
piezo-electric scanning stage appropriately in order to obtain a 1:1 scale
correspondence between the image and the object.

We believe that our method and the simplicity of its implementation should
encourage scientists using SPM imaging for crystalline or patterned samples to
correctly straighten their obtained microscopy images, and/or to calibrate
their instrument using such approach. The method could also be applied to
reciprocal space (FFT) data exclusively, _e.g._ , in quasi-particle
interference experiments where FFTs of STM data measured at different bias
voltages allows to characterize the dispersion behaviour of scattering
electrons in the first Brillouin zone [29].

More generally, we believe that this method is not restricted to SPM imaging
techniques and could be used in other correction scenarios; as long as the
image contains periodic information resulting from a object with known
periodic features, it can be straightened using the present method. One can
imagine an example scenario where a photograph of an object containing a
periodic structure (_e.g._ , textile sample) taken at an angle with respect to
its normal axis (thus leading to an apparent linear distortion, supposing a
long focal length) could be straightened using the method described in this
paper.

##  5 Statements

###  5.1 Methods and Code Availability

The graphite sample was purchased commercially (SP-1 grade HOPG, from SP)
cleaved in air and immediately inserted in the ultra-high vacuum (UHV)
chamber, outgassed for $\sim 2$ h to about $450^{\circ}$C to get rid of
adsorbed contaminants, and imaged with STM at room temperature using an
Omicron VT-AFM with Pt/Ir tips. The SiO2/Si grating calibration sample was
purchased from NT-MDT. AFM imaging was performed using a Terra AFM [30]
instrument. An example python script with detailed and commented steps for
drift correction is freely available on GitHub at
https://github.com/maximelester/driftcorrection.

###  5.2 Author Contributions and Interest Statement

M.L.S. designed the project, model, and code; wrote the manuscript. S.P.
conducted the AFM experiment. P.J. and I.L. prepared the graphite sample and
obtained the STM image, assisted with the development of the model, concept
and writing of the manuscript. The authors declare no competing financial
interests.

###  5.3 Acknowledgments

This work was supported by the National Science Center, Poland (M.L.S.:
2022/47/D/ST3/03216; I.L.: 2019/35/B/ST5/03956; P.J.K.: 2018/31/B/ST3/02450)
and IDUB 6/JRR/2021 (M.L.S).

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# Symbol Rate and Carries Estimation in OFDM Framework: A high Accuracy
Technique under Low SNR

Zetian Qin, Yubai Li, Benye Niu, Qingyao Li and Renhao Xue

###### Abstract

Under low Signal-to-Noise Ratio (SNR), the Orthogonal Frequency-Division
Multiplexing (OFDM) signal symbol rate is limited. Existing carrier number
estimation algorithms lack adequate methods to deal with low SNR. This paper
proposes an algorithm with a low error rate under low SNR by correlating the
signal and applying an Fast Fourier Transform (FFT) operation. By improving
existing algorithms, we improve the performance of the OFDM carrier count
algorithm. The performance of the OFDM’s useful symbol time estimation
algorithm is improved by estimating the number of carriers and symbol rate.

###### Index Terms:

Correlation method, OFDM signal, symbol rate, number of carriers, symbol time.

##  I Introduction

In the next generation network (6G), signal processing technologies will be
further developed [1, 2, 3, 4]. Among them, orthogonal frequency division
multiplexing (OFDM) is a highlighted signal transmission technology [5]. OFDM
divides channels into non-interference sub-channels, which can provide
transmission according to their unique characteristics. Therefore, OFDM has an
irreplaceable advantage in providing reliable communication [6], in the case
of extreme low Signal-to-Interference plus Noise Ratio (SINR) [7]. Therefore,
OFDM technology is widely used in dense cellular networks [8] and ultra-long
distance transmission [9, 10].

For non cooperative communication [11], there are three steps to process when
receiving unknown signals. (i) Determine the modulation mode of unknown
received signal. If the modulation mode of the received signal is OFDM
modulation, we should carry out the parameter blind estimation. (ii) Extract
the characteristic quantities of the received signal and use these
characteristic quantities to estimate the carrier frequency, symbol rate,
symbol time and other parameters. (iii) Decode the unknown received signal by
using the parameters obtained by the estimation method and restore the
original symbols. At present, the technology of modulation recognition is
relatively mature [12, 13, 14, 15, 16]. In engineering, the technology of
restoring signal symbols is very complete [17, 18]. However, one of the
challenges of OFDM is how to extract the required parameters from a blind
signal. Because of its complex modulation mode, a lot of key parameters need
to be estimated [19, 20, 21], such as the symbol rate, useful symbol time, and
the number of carries.

For the above key parameters, existing literature has proposed estimation
methods. For useful symbol time, [22, 23, 24] provided different estimation
methods, respectively: (i) [22] is based on the auto-correlation feature of
the OFDM signal’s Cyclic Prefix (CP), (ii) the authors in [23] used high-order
cumulants combined with wavelet transform, (iii) the authors in [24] obtained
the CP length and useful symbol time through maximum likelihood. Spectrum
symmetry algorithm in [24] and cyclic spectrum algorithm in [23] are used to
obtain the carrier frequency. The second-order cyclic cumulant is used to
derive the the number of carriers [25]. However, the performance of high-order
cumulants is poor at low Signal-to-Noise Ratio (SNR). Although the above
methods have provided a mature scheme for parameter estimation of OFDM, their
performance will not be reliable when dealing with low SNR [22, 23, 24, 25].
To overcome this challenge, this paper analyzes the relationship between
number of carries of OFDM signal spectrum and data length. Under low SNR,
multi segment data superposition is used to reduce the influence of noise in
the power spectrum and design a traversal algorithm to estimate the total
symbol time of OFDM signal.For the oversampled signal, this use traditional
method to estimate the oversampling rate and design a alternative algorithm to
estimate the number of carries of OFDM signal. Compared with the traditional
method, the substitution method and traversal method designed in this paper
have better performance under low SNR.

TABLE I: Summary of Notations Notation | Description
---|---
$\sigma_{s}^{2}$; $N_{s}$ | Average power of signal; Total number of points of a single OFDM symbol.
$N_{u}$; $N_{g}$ | Points of a single useful OFDM symbol; Points of CP in a single OFDM symbol.
$N_{O}$; $N_{P}$ |  Amount of OFDM symbols; Parameter used to estimate $N_{s}$.
$N_{os}$ | Amount of points of a symbol after oversampling.
$N_{ch}$; $N_{cn}$ |  Number of data segments with $N_{o}\cdot N_{P}$ data points; Number of carriers of received signal.
$i$; $k$ | Time domain abscissa; Starting point of time domain accumulation operation.
$j$; $i_{m}$ |  Imaginary unit; Number of symbols of each piece of data in (4).
$L_{M}$; $L^{\prime}$; $L$ |  Points of using data; Length of the sliding window; Value set in advance between $N_{s}$ and $L_{M}$.
$k_{\rm{all}}$; $k_{\max}$ |  Abscissa of all peaks in $Y(n,N_{P})$; Maximum value of $k_{\rm{all}}$.
$N_{\rm{all}}$; $L_{p}$ |  Amounts of $k_{\rm{all}}$; Peak spacing.
$N_{\rm{use}}$; $L_{\rm{cur}}$ | Recorded peak; Temporary peak spacing.
$k_{\rm{cur}}$; $N_{\rm{cur}}$ | Abscissa of current statistics; Amount of temporary counted peaks.
$k_{i}$; $k_{j}$ |  Abscissa of $k_{\rm{all}}$.
$N_{\max}$; $N_{\min}$ |  Possible maximum and minimum value of $N_{P}$.
$P_{u}$; $N_{PL}$ |  The array used to store useful $N_{P}$; Length of array $P_{u}$.
$q$; $L_{p}$ |  Oversampling rate; Distance between the peaks of $F_{r}\left(k,\tau\right)$.

##  II Calculation of symbol rate by traversal method

Based on the traditional model, this paper makes some improvements. First, we
introduce the traditional parameter estimation method. According to [26], the
useful symbol length of OFDM is estimated by auto-correlation of the signal.
The model is given as follows,

$\mathbb{E}\left\\{r\left(i\right)r^{\ast}\left(i+k\right)\right\\}=\left\\{\begin{aligned}
&\sigma_{s}^{2},\ &k=0,\\\ \sigma_{s}^{2}&\cdot\frac{N_{g}}{N_{s}},\
&k=N_{u},\\\ &0,\ &\mathrm{others,}\end{aligned}\right.$ (1)

where $\sigma_{s}^{2}=\mathbb{E}\left\\{\left|s(n)\right|^{2}\right\\}$,
$N_{g}$ are points of cyclic prefix (CP) in a single OFDM symbol, $*$
represents the conjugate of the signal, $k$ is the delay of $r\left(i\right)$,
$N_{s}$ is the number of points of a single OFDM symbol, and $N_{u}$ is the
number of useful symbols. $N_{u}$ can be found by (2). Note that the OFDM
useful symbol time can be given as $N_{u}$ divided by the sampling rate. (1)
shows that
$\mathbb{E}\left\\{r\left(i\right)r^{\ast}\left(i+k\right)\right\\}$ reaches
its second largest value when $k$ is equal to $N_{u}$. Therefore, we traverse
$k$ and find the second peak of
$\mathbb{E}\left\\{r\left(i\right)r^{\ast}\left(i+k\right)\right\\}$ by the
following formula [26]:

$\displaystyle
N_{u}=\mathop{\arg\max}\limits_{k}\left(\frac{\left|\sum_{i=1}^{L_{M}-L^{\prime}}r\left(i\right)r^{\ast}\left(i+k\right)\right|}{\frac{1}{2}\sum_{i=1}^{L_{M}-L^{\prime}}\left(\left|r\left(i\right)\right|^{2}+\left|r^{\ast}\left(i+k\right)\right|^{2}\right)}\right),$
(2) $\displaystyle k=1,2,...,L,$

where $L_{M}$ are points of using data, $L$ is a value set in advance between
$N_{s}$ and $L_{M}$, the denominator represents the energy of the received
signal, $L^{\prime}$ is the length of the sliding window, and the molecule
represents the sum of the auto-correlation of the received signal.

For the estimation method of the symbol rate, we generally estimate the total
OFDM symbol time, because its reciprocal is the symbol rate. The formula in
[26] is as follows,

$\begin{split}L\left(k\right)=\frac{\sum_{i=1}^{L^{\prime}}r\left(i+k\right)r^{\ast}\left(i+k+N_{u}\right)}{\sum_{i=1}^{L^{\prime}}\left(\left|r\left(i+k\right)\right|^{2}+\left|r\left(i+k+N_{u}\right)\right|^{2}\right)},\\\
k=1,2,...,L_{M}-N_{u}-L^{\prime},\end{split}$ (3)

where $k$ indicates the location of the data point, $i$ represents the data
position of the sliding window, $r\left(i+k\right)$ represents the $i^{th}$
data of the sliding window. $N_{s}$ is the distance between the midpoint of
adjacent peaks of $L\left(k\right)$.

For OFDM signals, when the number of data points used is $N_{s}\cdot N_{o}$,
where $N_{o}$ are the number of symbols, and the initial point of data used is
the initial point of a single symbol, each sub-carrier has a corresponding
peak on the frequency spectrum. The spacing between corresponding peaks
$N_{o}$, which is shown in the Fig. 1.

Figure 1: FFT of the received signal.

After performing auto-correlation operation on the data points of the received
signal and then FFT operation, the result is as shown in Fig. 2 .

Figure 2: Convolution of frequency domain signal.

In order to further reduce the influence of noise, we divide the data used
into multiple segments. These data segments start with the initial point of a
single OFDM symbol. The following formula can be obtained by summing each
segment of data after auto-correlation operation and then FFT operation.

$\displaystyle
y\left(i,N_{P}\right)=\frac{1}{N_{ch}}\sum_{i_{m}=1}^{N_{ch}}{r\left(N_{P}\cdot\left(i_{m}-1\right)+i\right)}$
(4)
$\displaystyle\times{r^{\ast}\left(N_{P}\cdot\left(i_{m}-1\right)+i\right)},$
$\displaystyle i=1,2,...,N_{o}\cdot N_{P},$
$Y\left(n,N_{P}\right)=\left|\sum_{i=1}^{L_{M}}y(k,N_{P})e^{-jni}\right|,$ (5)

where $N_{ch}$ is the number of chips, $N_{o}$ is number of chips observed.
When $N_{P}$ is the number of points of a single symbol, the peak whose
abscissa is an arithmetic progression will appear on $Y\left(n,N_{P}\right)$.
At low SNR, the peaks still exist, while the amplitudes of peaks become lower.
When $N_{P}$ is not equal to $N_{s}$, peaks that belong to arithmetic
progression account for a reduced proportion of all peaks.

Figure 3: Comparison diagram under different SNR and $N_{P}$.

Then we design a method to find $N_{s}$. According to (4) and (5), the
traversal algorithm is proposed: (i) traverse $N_{P}$, (ii) find the abscissa
of the peak of $Y\left(n,N_{P}\right)$, (iii) Use algorithm 1 to find out the
number of peaks with fixed spacing and the total number of peaks, (iv) use
algorithm 2 to find $N_{P}$ which satisfy the judgment conditions and (v)
calculate the mean value of all $N_{P}$ that meet the judgment conditions, and
judge the calculation result as $N_{s}$. The specific algorithm processes are
shown in algorithm I and algorithm 2.

(A) Initialize: Record the abscissa of all peaks in $Y(n,N_{P})$ (denoted as
$k_{\rm{all}}$), the maximum value of $k_{\rm{all}}$ (denoted as $k_{\max}$),
amounts of $k_{\rm{all}}$ array $N_{\rm{all}}$, peak spacing $L_{p}=0$,
recorded peak $N_{\rm{use}}=0$, temporary peak spacing $L_{\rm{cur}}=0$, the
abscissa of current statistics $k_{\rm{cur}}=0$, amount of temporary counted
peaks $N_{\rm{cur}}=0$.

(B) Find the number and spacing of the peaks whose abscissa satisfies the
arithmetic sequence:

for $k_{i}=1;k_{i}<\frac{N_{\rm{all}}}{2};k_{i}=k_{i}+1$ do

for $k_{j}=k_{i}+1;k_{j}<\frac{N_{\rm{all}}}{2}+2;k_{j}=k_{j}+1$ do

if $k_{\rm{all}}(k_{j})-k_{\rm{all}}(k_{i})>3$ then

$N_{\rm{cur}}=2$.

$L_{\rm{cur}}=k_{\rm{all}}(k_{j})-k_{\rm{all}}(k_{i})$.

$k_{\rm{cur}}=k_{\rm{all}}(k_{j})$.

while $k_{\rm{cur}}<k_{\max}$ do

$k_{\rm{cur}}=k_{\rm{cur}}+L_{\rm{cur}}$.

if $L_{\rm{cur}}$ is a member of $k_{\rm{all}}$ then

$N_{\rm{use}}=N_{\rm{use}}+1$.

if $N_{\rm{cur}}>N_{\rm{use}}$ then

$N_{\rm{use}}=N_{\rm{cur}}$.

$L_{p}=L_{\rm{cur}}$.

end if

end if

end while

end if

end for

end for

return $\left[N_{\rm{use}},N_{\rm{all}},L_{p}\right]$.

Algorithm 1 $\left[N_{\rm{use}},N_{\rm{all}},L_{p}\right]$ = functionA
$\left(Y\left(n,N_{P}\right)\right)$

(A) Initialize: Initialize value $N_{P}$ and its possible minimum and maximum
value $N_{\min}$ and $N_{\max}$. Record the array used to store useful $N_{P}$
(Denoted as $P_{u}$). Let $N_{P}=N_{\min}$, $N_{o}=6$.

(B) Calculate the distance and number between the peaks of $Y(n,N_{P})$ under
the current $N_{P}$:

Generate $Y\left(n,N_{P}\right)$.

$\left[N_{\rm{use}},N_{\rm{all}},L_{p}\right]$ = functionA
$\left(Y\left(n,N_{P}\right)\right)$.

(C) Determine whether the data is useful:

if $\frac{N_{\rm{use}}}{N_{\rm{all}}}>0.5$ and $N_{all}>N_{min}$ and
$L_{p}=N_{o}$ then

Store $N_{P}$ in $P_{u}$, $N_{P}=N_{P}+1$.

if $N_{P}<N_{\max}$ then

Regenerate $Y\left(n,N_{P}\right)$ .

$\left[N_{\rm{use}},N_{\rm{all}},L_{p}\right]$ = functionA
$\left(Y\left(n,N_{P}\right)\right)$.

Go to step (C).

end if

else

if There are continuous values in $P_{u}$ tail then

Remove discontinuous values in $P_{u}$.

Go to step (D).

else

$N_{P}=N_{P}+1$.

if $N_{P}<N_{\max}$ then

Regenerate $Y(n,N_{P})$ and $k_{\rm{all}}$.

$\left[N_{\rm{use}},N_{\rm{all}},L_{p}\right]$ = functionA
$\left(Y\left(n,N_{P}\right)\right)$.

Go to step (C).

end if

end if

end if

(D) Determine the final value:

$N_{PL}$ is the length of array $P_{u}$.

return $N_{s}=\sum_{k=1}^{N_{PL}}\frac{1}{N_{PL}}P_{u}\left(k\right)$.

Algorithm 2 Traversal method

At last, the improved method achieves 75% accuracy and its Amplitude Error
(AE) is between 0.1 and 0.2 under low SNR.

##  III Calculation of number of carries and useful symbol time by
substitution method

For the over-sampled signal, its data points need to be separated before using
the above method to estimate $N_{s}$. According to [26], the number of
carriers and the oversampling rate are estimated by oversampling the signal
and performing a correlation operation, which can be expressed as follows,

$F_{r}\left(\omega,\tau\right)=\lim_{L_{M}\to\infty}\sum_{i=0}^{L_{M}}\mathbb{E}\left\\{r\left(i\right)r^{\ast}\left(i+\tau\right)\right\\}\exp\left(-j\tfrac{2\pi}{L_{M}}\omega
i\right),$ (6)

where $L_{M}$ is the amount of using data, $j$ is imaginary part,
$r\left(i\right)$ is an over-sampled signal, $F_{r}$ represents power spectral
density. Generally, when $\tau=1$, the power spectral density is used in (7).
After the above analysis, the estimation formula of oversampling rate is as
follows,

$q=\frac{L_{M}}{L_{p}},$ (7)

where $L_{p}$ is the distance between the peaks of $F_{r}\left(k,\tau\right)$.
After calculating $q$, the signal’s data points are separated to restore the
original signal. The symbol length $N_{os}$ of the original signal can be
obtained by using the algorithm 1 to process the OFDM received signal without
over sampling. The estimation result formula of OFDM carrier number is as
follows,

$N_{cn}=2^{\left(\left\lfloor\log_{2}{\left(N_{os}\right)}\right\rfloor\right)},$
(8)

where $N_{cn}$ is the OFDM carrier number, $\left\lfloor\cdot\right\rfloor$
represents rounding down to an integer, $q$ is the oversampling rate. After
calculating the number of carriers, the length of the OFDM useful symbol can
be obtained as follows,

$N_{u}=q\cdot N_{cn}.$ (9)

Figure 4: FFT of correlation signal after oversampling when the SNR(dB) is 0
and -40.

The image calculated by (6) is less affected under low SNR. Use $N_{s}$ by
algorithm 1 under low SNR can improve the accuracy of the algorithm and reduce
the error.

(A) Initialize:

Input received signal $r\left(i\right)$.

(B) Generate $F_{r}$ and calculate $q$:

$F_{r}\left(k,\tau\right)=\lim_{M\to\infty}\sum_{n=0}^{M}\mathbb{E}\left\\{r\left(i\right)\cdot
r^{\ast}\left(i+\tau\right)\right\\}\cdot$
$\exp\left(-j\tfrac{2\pi}{M}kn\right)$ , $q=\frac{L_{M}}{L_{P}}$.

(C) Calculate $N_{os}$.

Under-sample $r\left(i\right)$ and use the result by 1 to calculate $N_{os}$.

(D) Determine the number of carriers:
$N_{cn}=2^{\left(\left\lfloor\log_{2}{\left(N_{os}\right)}\right\rfloor\right)}$.

(E) Determine $N_{u}$ according to (8):

return $N_{u}=q\cdot N_{cn}$.

Algorithm 3 Substitution method

##  IV Experiment and results

Figure 5: Comparison of symbol length algorithm accuracy.

In this experiment, the modulation mode is 16 Quadrature amplitude modulation
(QAM). The number of carriers is 128, the proportion of CP is 0.25. The
carrier frequency is 10M. The sampling rate is 40M. The number of symbols is
20. The channel is the Gaussian channel. Set the SNR from -40dB to 10dB. The
number of tests under each SNR is $10^{5}$. When SNR is lower than -5dB, we
choose 1 to calculate the $N_{s}$. When SNR is higher than -5dB, we choose (3)
to calculate the $N_{s}$.

Figure 6: Comparison of useful symbol length and carrier number algorithm
accuracy.

By observing the results of the experiments, it can be found that under high
SNR, the accuracy of $N_{s}$ and $N_{u}$ are not as good as (2) and (3), and
their AEs are about 0.1. The AEs of two algorithms of carrier number have the
same performance. At low SNR, the accuracy of the new estimation method for
$N_{s}$ and $N_{u}$ is between 0.9 and 1, and the AE is between 0 and 0.15.
Compared with the original algorithm, the performance is significantly
improved. The accuracy of the carrier number can reach 1 and the AE is 0.

Figure 7: Comparison diagram of symbol length algorithm AE.

The accuracy and AE of $N_{s}$ fluctuate under low SNR. In the experiment,
$N_{s}$ is set to 160. Members in $P_{u}$ include $N_{s}$, but their mean
value may not be equal to ns. In this case, it is challenging to determine
which value is the accurate number of OFDM symbol points. The proposed
algorithm improves the AE performance under low SNR.

Figure 8: Comparison diagram of symbol length and carrier number algorithm AE.

##  V Conclusion

This paper proposed an improved algorithm for the total OFDM symbol time
$N_{s}$ (reciprocal of symbol rate). Under low SNR, the estimation accuracy
has been improved. However, the OFDM useful symbol time that used in the the
oversampling rate algorithm is greatly affected by noise. In order to avoid
the situation above and improve the performance, the useful symbol time is
replaced with the total symbol time. OFDM signal has the characteristic that
the number of carriers is exponential of 2. The proposed algorithm can achieve
high accuracy. After estimating the number of carriers, the OFDM useful symbol
time can be deduced.

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# Aluminum functionalized silicene: a potential anode material for alkali
metal ion batteries

Bruno Ipaves  Instituto de Física, Universidade de São Paulo, São Paulo, SP,
Brazil  João F. Justo  Escola Politécnica, Universidade de São Paulo, São
Paulo, SP, Brazil  Lucy V. C. Assali  Instituto de Física, Universidade de São
Paulo, São Paulo, SP, Brazil<EMAIL_ADDRESS>
###### Abstract

We have investigated the possibility of using aluminum functionalized silicene
trilayers (ABC-Si4Al2) as an anode material for alkali metal ion batteries
(AMIBs). First, we studied the thermodynamic stability of ABC-Si4Al2 using ab-
initio molecular dynamics simulations, showing that this material remains
stable up to 600 K. Then, we explored the properties of alkali metal atoms
(Li, Na, K) adsorption in ABC-Si4Al2, finding several available sites with
high adsorption energies. Moreover, we computed the diffusion properties of
those atoms along high-symmetry paths using the nudged elastic band method.
The results indicated diffusion barriers as low as those in graphite,
especially for Na (0.32 eV) and K (0.22 eV), which allows those ions to
migrate easily on the material’s surface. Our studies also revealed that the
full loaded Li4Si4Al2, Na2Si4Al2, and K2Si4Al2 systems provide low open-
circuit voltage, ranging from 0.14 to 0.49 V, and large theoretical capacity
of 645 mAh/g for Li- and 322 mAh/g for Na- and K-ion batteries, values that
are close to the ones in other anode materials, such as graphite, TiO2, and
silicene-based systems. Those results indicate that aluminum functionalized
few-layer silicene is a promising material for AMIBs anodes, particularly for
Na- and K-ion batteries.

###### keywords:

American Chemical Society, LaTeX

IR,NMR,UV

##  1 Introduction

The current challenges of energy transition toward renewable sources, such as
solar and wind power systems, have been intrinsically associated with the
search for efficient technologies for energy storage. Considering energy
storage in the electric form, the lithium-ion battery (LIB) has been a leading
option, which has been integrated into several electronic devices since
entering the market around thirty years ago 1.

Although the LIB technology has been widely used, it still carries some
limitations when applied to large-scale systems, such as slow charging and
degradation after only a few hundred cycles 2. Moreover, lithium is not an
abundant chemical element on Earth 3. Therefore, it is crucial to investigate
alternative materials for batteries, in particular, new materials for
electrodes. This represents an essential step toward improving the performance
of rechargeable batteries to meet the world’s increasing demand for renewable
energies.

Sodium-ion (SIBs) and potassium-ion (KIBs) batteries have received attention
in recent years since they possess working principles, and physical and
chemical properties similar to LIBs. Moreover, the sodium (Na) and potassium
(K) chemical elements are more abundant and less expensive than lithium (Li)
4, 5. On the other hand, the larger Na and K ions, as compared to Li,
represents a major challenge for the SIBs and KIBs, once they present high
diffusion barriers, high lattice strain, and irreversible structural
degradation 1. For example, in LIBs the anode is usually made of graphite,
having a theoretical capacity of 372 mAh/g by forming LiC6 6. However, this
value is reduced to 279 mAh/g when the K ion is used (KC8) 7. Additionally,
due to a thermodynamic instability in interpolating Na ions, graphite is
rarely used as an anode in SIBs 3, 5.

Within this context, two-dimensional (2D) materials have emerged as potential
components of alkali metal ion batteries (AMIBs). These materials carry
suitable properties for those applications, such as large surface-to-volume
ratio and elasticity, which allows efficient transport and superior ion
adsorption, as well as small volume expansion of the electrode materials 8, 9.
Among several two-dimensional materials investigated, since the discovery of
graphene, silicene has emerged as a promising candidate for applications in
batteries. Silicene is made of silicon, one of the most abundant chemical
elements on Earth, that is also environmentally friendly 10. Furthermore, it
has been determined theoretically that the silicene monolayer has a 954 mAh/g
capacity for LIBs and SIBs 11, 12, larger than that of graphite.
Correspondingly, experimental investigations have shown that crystalline
silicene (c-silicene) exhibits great performance as an anode in LIBs and KIBs
with storage capacities respectively of 721 mAh/g 13 and 180 mAh/g 14.

Several strategies have been explored to get better materials for AMIB
electrodes, such as the chemical doping and composition, in order to improve
the ion conductivity and diffusibility within the solid 15. For example, it
has been shown that Si3C, a silicene-based compound in which some Si atoms
have been replaced by C ones, could be a potential anode material for AMIBs
16. Recently, other investigations have shown that TiS2 doped with B, C, N, O,
F, or P atoms could be used as anodes, improving the overall performance of
AMIBs 17. Additionally, theoretical investigations of the BSi3 and AlSi3
systems, in which the silicene monolayer was doped with B and Al atoms,
respectively, have shown that they have better properties for Na- and K-ion
batteries, when compared to pristine silicene 18. Therefore, doping 2D systems
with different chemical elements represents a promising strategy to increase
the performance of the AMIBs.

The physical properties of functionalized few-layer silicene compounds,
specifically bilayers (Si2X2) and trilayers (Si4X2) (with X = B, N, Al, P),
have recently been explored by us using first-principles investigations 19,
showing that many functionalizations are energetically favorable, displaying
large negative enthalpy of formation. The resulting compounds have shown
metallic or semiconducting behavior, indicating several potential
applications. In particular, the systems with metallic behavior, such as
silicene functionalized with Al, has properties that make them good candidates
to work as battery electrodes. The aluminum doping enlarges the lattice
parameter of the original silicene material by about 7% 19, which may increase
the adsorption of Li-, Na-, and K-ions, which is one of the requirements for
superior anodes 20.

Here, we performed a theoretical investigation on the properties of aluminum-
doped silicene trilayer ABC-Si4Al2, and its potential application as an AMIB
anode. First, we explored the thermodynamic stability of this system as a
function of temperature, showing that ABC-Si4Al2 remains structurally stable
up to 600 K. Since the rate performance of AMIBs is associated with ionic
diffusion properties 15, we then calculated the adsorption energies of Li, Na,
and K atoms in several sites of ABC-Si4Al2, as well as their diffusion
barriers along several high-symmetry pathways. Finally, we investigated the
effects of alkali metal concentration and estimated the open-circuit voltage
and the specific theoretical capacity of ABC-Si4Al2 as an anode material for
AMIBs. The results indicated that ABC-Si4Al2 provides migration energy
barriers and open-circuit voltage as low as in commercial anode materials,
such as graphite. Moreover, the ABC-Si4Al2 maintains a good capacity for
lithium, which is similar to that of pristine crystalline silicene and larger
than that of graphite. Interestingly, the capacity for potassium is enhanced
when compared to that of pristine crystalline silicene, and for sodium, we
found the same capacity, showing that aluminum functionalized silicene
trilayers (ABC-Si4Al2) is a promising anode material for AMIBs.

##  2 Computational Details

The first-principles calculations were performed within the framework of the
density functional theory 21, 22, with the Quantum ESPRESSO computational
package 23, 24. We used the exchange-correlation potential based on the
generalized gradient approximation to describe the electronic interations, as
proposed by Perdew, Burke, and Ernzerhof (PBE) 25. In order to properly
describe the effects of the dispersive van der Waals interaction 26, 27, 28,
the Dion et al. scheme 29, optimized by Klimeš et al. (optB88-vdW) 30, was
used.

The projector augmented-wave (PAW) method 31 was applied to describe the
interactions between ions and electrons, with a cutoff energy of 1100 eV. To
compute the electronic states, the Brillouin zones were sampled by a $16\times
16\times 1$ Monkhorst-Pack $k$-point grid 32. Self-consistent iterations were
performed until the total energy difference between two consecutive steps was
lower than 0.1 meV/atom, while the structural optimization was performed until
forces were lower than 1 meV/Å in any ion, where relaxations and distortions
were considered in all ions, without symmetry constraints.

The 2D structures were built with hexagonal simulation supercells, considering
periodic boundary conditions. We used a lattice parameter of 25 Å in the
perpendicular direction to the nanosheet ($z$-axis), which is large enough to
avoid interactions among cell images in that direction. The methodology
presented so far has already been used in several investigations, providing an
appropriate description of similar 2D systems 33, 19.

The adsorption energies ($E_{ads}$) and the diffusion barriers of the M = Li,
Na, or K alkali adatoms were obtained by constructing a $3\times 3\times 1$
supercell of the ABC-Si4Al2 system and, after the structural optimization, the
adsorption energies were calculated by the equation

$E_{ads}({\rm{M}})=\Big{[}E_{t}\big{(}{\ce{Si4Al2M}_{n}}\big{)}-E_{t}\big{(}{\ce{Si4Al2}}\big{)}-nE_{t}\big{(}{\rm
M}\big{)}\Big{]}\Big{/}n\,,$ (1)

where $E_{t}\big{(}{\ce{Si4Al2M}_{n}}\big{)}$ is the total energy of the host
nanosheet with $n$M alkali adatoms, $E_{t}\big{(}{\ce{Si4Al2}}\big{)}$ is the
total energy of an isolated ABC-Si4Al2 trilayer, and $E_{t}\big{(}{\rm
M}\big{)}$ is the total energy, per atom, of the M alkali metal in a body-
centered cubic crystalline structure, computed within the same methodology
described previously.

For the diffusion barriers, we used the nudged elastic band (NEB) method 34 as
implemented in the Quantum ESPRESSO package 24. Two adjacent sites with the
largest negative $E_{ads}$ were chosen as initial and final configuration
reference states, where seven images were considered along the minimum energy
pathway (MEP).

The open-circuit voltage (OCV) is related to the $E_{ads}$ by

${\rm OCV}=-\frac{\Delta G_{\\!f}}{ne}\approx-\frac{E_{ads}}{ne},$ (2)

where $\displaystyle\Delta G_{\\!f}=E_{ads}+P\Delta V_{\\!f}-T\Delta S_{\\!f}$
is the Gibbs free energy and, at room temperature, $\displaystyle P\Delta
V_{\\!f}$ and $\displaystyle T\Delta S_{\\!f}$ can be neglected 35. Therefore,
OCV was obtained using equation (2) by computing $E_{ads}$ as defined in
equation (1), when inserting adatoms into ABC-Si4Al2 trilayer.

The theoretical specific capacity ($C$) was obtained by using

${C}=\frac{n_{max}\times z\times{\rm F}\times 10^{3}}{A_{\ce{Si4Al2}}},$ (3)

where $\rm{F}=26.810$ Ah/mol is the Faraday constant, $z=1$ is the charge of
the intercalating ion (Li, Na, or K), $n_{max}$ is the maximum number of M
alkali adatoms that the host can store, and $\displaystyle A_{\ce{Si4Al2}}$ is
the atomic mass of the host.

Additionally, we used ab-initio molecular dynamics (AIMD) simulations, as
implemented in the Vienna ab-initio simulation package (VASP) 36, to explore
the thermodynamic stability of the ABC-Si4Al2 host system. To allow possible
structural reconstructions, a $5\times 5\times 1$ supercell, with 150 atoms,
was used. The time step of the simulations was set to 1 fs and a Nose–Hoover
thermostat (NVT) ensemble was employed to carry out the calculations at
several temperatures (from 0 to 1000 K) for 5 ps.

##  3 Results

Silicene bilayers and trilayers, with several surface functionalizations
(atomic species B, N, Al, P) and stacking configurations, were recently
investigated by us using first-principles calculations 19. According to the
phonon spectra results and Born stability criteria, only some of the
functionalized systems were found to be mechanically stable. In addition, most
of these stable structures exhibit insulating or semiconducting electronic
behavior, which is not entirely suitable for applications associated with
battery electrodes.

On the other hand, among the stable systems studied, the silicene trilayer
functionalized with aluminum atoms in the ABC stacking configuration, labeled
as ABC-Si4Al2, is metallic and its lattice parameter is larger than the
pristine few-layer silicene 19, which could allow a large set of adsorption
sites. Those properties already suggest that ABC-Si4Al2 system is a promising
candidate for applications as battery electrodes. Figure 1 presents a
schematic representation of the ABC-Si4Al2 structure. The optimized lattice
parameters of the primitive cell are $a$ = $b$ = 4.143 Å, with an $h$ = 6 Å
thickness.

Figure 1: (a) Top view, (b) primitive cell, and (c) side view of the ABC-
Si4Al2 and the binding sites for Li-, Na-, and K-alkali metal atoms. The
hollow, top, and bridge sites are represented respectively by H, T, and B
symbols. The dashed lines indicate the unit cell limits. The blue and grey
spheres represent respectively Si and Al atoms, while the green spheres
represent the binding sites for adsorption of the alkali metal ions.

###  3.1 Thermodynamic Stability

The thermodynamic stability of the ABC-Si4Al2 structure was investigated with
AIMD simulations, using a 150-atoms supercell and lattice parameters of $a$ =
$b$ = 20.715 Å, at several temperatures from 0 to 1000 K. Figure 2(a) shows
the total energy of the system at 300 K during 5 ps. After about 1 ps, the
system reaches thermal equilibrium, and then the energy oscillates around an
equilibrium value. The figure also shows a snapshot of the final structure,
indicating that the system keeps its structural integrity, i.e., the atoms
only oscillate around their crystalline equilibrium positions, with no broken
bonds or defect formation. Similar behavior is observed at temperatures up to
600 K. At higher temperatures, there is a clear change in the behavior of the
system, as it starts to present broken bonds. Figure 2(b) shows the total
energy of the system at 1000 K during 5 ps, as well as a snapshot of the final
structure, pointing out structural degradation, with a large concentration of
broken bonds.

Figure 2: Total energy over 5 ps during AIMD simulations at (a) 300 K and (b)
1000 K, along with the respective snapshots of the final structures (top and
side views) at the end of the simulation; (c) variation in the average total
energy, relative to the reference value at T = 0 K, of the last 2 ps of the
simulations as a function of temperature. The coloring of spheres in (a) and
(b) is consistent with those in figure 1.

By inspecting the final structure in the simulation, it is already possible to
estimate that 600 K is the temperature threshold at which the system can
maintain its structural integrity. Below this threshold, the atoms essentially
oscillate around their crystalline equilibrium positions, without breaking
bonds, while over this threshold, some broken bonds could be observed. In
order to determine this threshold more precisely, we investigated the behavior
of the average total energy of the system in equilibrium as a function of the
temperature. Figure 2(c) shows the average total energy of the last 2 ps of
the simulations as a function of temperature. It can be observed that the
average total energy increases linearly with temperature below 600 K and,
taking into account the virial theorem, it is clear that the average kinetic
and potential energies are very close, indicating structural integrity. Over
600 K, figure 2(c) shows that the average total energy still increases
linearly with temperature, but faster, pointing out that there are broken
bonds and defects, as well as some atomic diffusion.

Those results demonstrate that the ABC-Si4Al2 structure is considerably stable
for temperatures up to 600 K, and could be used as electrodes for applications
below that threshold. Nevertheless, the nanosheet suffers considerable
structural degradation at higher temperatures, and close to 1000 K, they are
unsuitable for those applications.

###  3.2 Ion Adsorption Properties

In the previous section, we showed that the ABC-Si4Al2 compound keeps its
structural stability up to 600 K. We now explore the possibility of using this
material as an anode in AMIBs. For that, two important properties were
investigated: how the alkali metal ions (Li, Na, or K) are adsorbed on the
host surface, as well as incorporated inside the ABC-Si4Al2 system, and how
they diffuse within the surface. We used a $3\times 3\times 1$ supercell, with
optimized lattice parameters of $a$ = $b$ = 12.429 Å, which corresponds to a
chemical stoichiometry of ABC-M1Si36Al18 (M=Li, Na, K). This supercell is
large enough for the simulation of one alkali atom adsorbed/inserted in the
system, while avoiding interaction among images.

We first computed the adsorption energies of alkali metals in several high-
symmetry binding sites on the trilayer surface, as shown in figure 1 (b), and
subsequently in two sites inside the trilayer structure, as presented in
figure 1 (c). We considered the hollow site (H) at the center of a hexagon,
the top site (T) directly above each atom, and the bridge site (B) at the
midpoint of a Si-Al bond. The bulk interstitial adsorption was also considered
in two sites: the layer hollow site (H1), at the center of a hexagonal ring,
and the interstitial hollow site (H2), inside the structure at the center of a
hexagon, as presented in figure 1 (c).

After performing structural optimization, the alkali atoms moved from the B to
the TAl site and from the H1 to the H2 one, indicating that the ions are
unlikely to be adsorbed at the B and H1 sites. Therefore, using equation (1),
we calculated the adsorption energy (Eads) at the remaining sites, which are
presented in Table 1 for Li-, Na-, and K-adatoms in the H, T, and H2 sites.
The negative values of Eads indicate that the sites are energetically
favorable for adsorption. The values in the table are in the same range as
those in silicene-based materials, previously reported in the literature as
potential candidates for anodes in AMIBs 18, 13, 16.

Table 1: Adsorption properties of M atoms (M=Li, Na, K) in ABC-M1Si36Al18. The table shows the adsorption energy $E_{ads}$ (in eV), the charge transfer $\rm{\Delta Q}$ (in $|e|$), and the equilibrium distances $d_{\rm M}$ (in Å) between the adatom and the closest surface atom in the system. Atom | Site | $E_{ads}$ | $\rm{\Delta Q}$ | $d_{\rm M}$
---|---|---|---|---
Na | H | $-0.291$ | 0.811 | 1.905
Li | H | $-0.483$ | 0.859 | 1.177
Li |  H2 | $-0.470$ | 0.836 | 0.878
K | H | $-0.681$ | 0.803 | 2.423
K |  TAl | $-0.481$ | 0.818 | 3.105
K |  TSi | $-0.279$ | 0.826 | 3.072

Considering all adsorption sites investigated, the H is the most energetically
favorable site to adsorb Li, Na, or K atoms with Eads(K) ($-0.681$ eV) $<$
Eads(Li) ($-0.483$ eV) $<$ Eads(Na) ($-0.291$ eV). Similar behavior has been
observed in pristine silicene monolayer, in which the H is the most favorable
site for Li and Na adsorption 11, 37. The T sites are also suitable for
adsorption of K atoms, as we found negative adsorption energies of $-0.481$ eV
and $-0.279$ eV for TAl and TSi, respectively. On the other hand, the
adsorption energies for both Li and Na are positive at the T sites, indicating
that they are not favorable for adsorption. Finally, the interstitial H2 site
is only energetically favorable for Li adsorption, with $E_{ads}=-0.470$ eV,
while for Na and K ions we found positive values. The result regarding the H2
site could be explained by considering the alkali metal atomic (ionic) radius
and, hence, there is not enough space in this site to host Na or K ions. In
particular, among all the ABC-M1Si36Al18 systems studied here, the ones
related to K atoms seem to be the most promising to work in ion batteries,
i.e., as an anode in KIBs, since the K ions can be adsorbed in several sites
of the ABC-Si4Al2 trilayer host and, at the same time, they are the systems
that present greater stability when compared to those that adsorb Li or Na,
presenting the lowest values of $E_{ads}$.

In addition to the Eads, table 1 lists the equilibrium distance ($d_{M}$) and
the charge transfer ($\rm\Delta Q$). The $d_{M}$ is defined as the vertical
distance (in the $z$-direction) between the adatom and the closest neighboring
surface atom. We observed that $d_{M}$ increases with the increasing atomic
(ionic) radius and mass of the adatoms, where $d_{\rm K}>d_{\rm Na}>d_{\rm
Li}$ for the adsorption sites on the surface. The $d_{M}$ depends on the
adsorption position, e.g., regarding TAl and TSi sites, the closest surface
atoms are the Al and the Si atoms, respectively. Nevertheless, the closest
surface atom is the Al for the H site, while it is the Si atom for the H2 one
inside the structure. We also computed the $\rm\Delta Q$ from the ions toward
the ABC-Si4Al2 host, using the Bader analysis 38, 39, 40, 41. The values
presented in table 1 are around +0.8 $|e|$, showing that the valence charge of
the ions is almost fully transferred to the Si4Al2 host and, hence, indicating
the ionization of the adatoms.

###  3.3 Ion Diffusion Properties

We then investigated the diffusion of the alkali metal atoms on ABC-Si4Al2
since the charge and discharge rates of the AMIBs are directly associated with
the ion mobility 15. Herein, we used the NEB method and a $3\times 3\times 1$
supercell, in which two H adjacent sites were chosen as the initial and final
configurations since it is the most energetically favorable site, i.e., it
presents the largest adsorption energy value.

Three different high-symmetry pathways on the ABC-Si4Al2 surface were
considered, as illustrated in figure 3 (a). In Path 1, a single alkali atom
moves from an H site to another H one passing through a B site. In Path 2, the
atom passes through a TSi site, while in Path 3, it passes above a TAl one.
Particularly, for the Li atom, we considered an additional path, moving it
from the H to the H2 site once the $E_{ads}$ is negative in the H2 site for
the Li atom, which shows that it could be incorporated inside the structure.
Figure 3 (b) illustrates this additional path (Path 4). At the end of NEB
calculations, Path 1 converged toward Path 3, indicating that the ions diffuse
to the top of the aluminum/silicon sites, and then to the other H site. An
equivalent diffusion path was also observed in pristine silicene monolayer 11,
which is consistent with what we previously observed regarding the alkali
atoms moving from the B site to the TAl one after accomplishing the structural
optimization.

Figure 3: (a) Top and (b) side view of the ABC-Si4Al2 structure, indicating
the binding sites (diffusion coordination 1, 4 and 7) and paths for Li/Na/K
adatoms. The black, red, and orange arrows in (a), as well as the purple one
in (b), represent the high-symmetry paths explored in this investigation
(paths 1, 2, 3, and 4, respectively, as discussed in the text). The figure
also shows the energy barriers associated with the diffusion of (c) Li; (d)
Na; and (e) K; in ABC-Si4Al2. The grey, blue, and brown spheres represent the
Al, Si, and alkaline metal ions (Li, Na, K), respectively. Moreover, in (b)
the green spheres represent the Li ion.

The energy profiles along different paths are presented in figures 3 (c), (d),
and (e) for Li, Na, and K ions, respectively. Comparing the same path, the
calculated diffusion barriers ($E_{b}$) decrease as the atomic (ionic) radius
and mass of the diffusing atoms increase: $E_{b}({\rm Li})>E_{b}({\rm
Na})>E_{b}({\rm K})$. At this point, it is worth noting that the diffusion
coordination 4 is the TSi site on Path 2, the TAl site on Path 3, and the H1
site on Path 4. For Path 3, it can be observed in figures 3 (c), (d), and (e)
that the peak is not at the diffusion coordination 4; instead, there is a
valley in that coordination that is due to the interaction between the alkali
ions and the aluminum atom, which was also observed in pristine silicene 11,
12. The $E_{b}$ values regarding the ABC-Si4Al2 trilayer surface, Paths 2 and
3, are in the 0.53-0.86 eV, 0.32-0.50 eV, and 0.22-0.40 eV ranges for the
systems with Li, Na, and K adatoms, respectively. For the Li atom, the lowest
$E_{b}$ value of 0.28 eV is found in Path 4, suggesting that lithium atoms
alloy with the ABC-Si4Al2 structure during the lithiation.

Table 2 presents the Li, Na, and K ions diffusion barriers values in the ABC-
Si4Al2 system, as compared with values reported in the literature for others
candidate materials to serve as AMIBs anodes. The surface diffusion barriers
$E_{b}$ corresponding to Path 3 for Li, Na, and K atoms are 0.53, 0.32, and
0.22 eV, respectively. These migration energy barriers values are close to the
ones found in other silicon-based systems proposed as anode materials, such as
AlSi3 18, Si3C 16, and Si2BN 42, as shown in table 2.

Table 2:  Diffusion barrier energy ($E_{b}$), in eV, theoretical capacity
($C$), in mAh/g, and Open-Circuit Voltage (OCV), in V, for Li, Na, K ions in
ABC-Si4Al2 trilayer system, as compared to results reported in the literature
for other potential candidates to work as AMIB anodes. Herein, the $C$ and the
OCV values are computed for the Li4Si4Al2, Na2Si4Al2, and K2Si4Al2 systems.

Anode |  | Diffusion barrier energy ($E_{b}$) |  | Theoretical Capacity ($C$) |  | Open-Circuit Voltage (OCV)
---|---|---|---|---|---|---
Material |  | Li | Na | K |  | Li | Na | K |  | Li | Na | K
ABC-Si4Al2 |  | 0.53-0.86 | 0.32-0.50 | 0.22-0.40 |  | 645 | 322 | 322 |  | 0.49 | 0.16 | 0.14
Graphite |  | 0.22-0.40 43 |  |  |  | 372 6 |  | 279 7 |  | 0.11 44 |  |
TiO2 |  | 0.35-0.65 45, 46, 47 | 0.52 45 |  |  | 200 46 |  |  |  | 1.5-1.8 44 |  |
Silicene |  | 0.23 11 | 0.16 12 | 0.14 48 |  | 954 11 | 954 12 |  |  |  | 0.30-0.50 12 |
c-silicene |  |  |  |  |  | 721 13 |  | 180 14 |  |  |  |
VS2 |  | 0.22 44 | 0.62 49 |  |  | 466 44 | 233 49 |  |  | 0.93 44 | 1.32 49 |
AlSi3 18 |  |  | 0.29 | 0.17 |  |  | 1928 | 964 |  |  |  |
Si3C 16 |  | 0.47 | 0.34 | 0.18 |  | 1394 | 1115 | 836 |  | 0.58 | 0.50 | 0.71
Si2BN 42 |  | 0.48 | 0.32 |  |  | 1158 | 933 | 836 |  | 0.46 | 0.29 |

Our results indicate that the diffusion barrier of K ion on ABC-Si4Al2 is
lower than those of Li and Na atoms, reinforcing which was previously observed
that the K atom possesses more sites to be adsorbed, as well as presents the
highest negative adsorption energies and equilibrium distances, when compared
to the respective values of Li and Na atoms. Moreover, the diffusion barriers
of the systems studied here are similar to those in anode materials based on
graphite (0.22-0.40 eV) 43 and TiO2 (0.35-0.65 eV) 45, 46, 47, implying that
the ABC-Si4Al2 could be used as LIB, SIB, or KIB anodes.

To further compare the diffusion properties among those paths, we have
obtained the diffusion constant (D) of an alkali metal atom, estimating it by
the Arrhenius equation 50:

$D\sim exp\left(-\frac{E_{b}}{k_{B}T}\right),$ (4)

where $E_{b}$ and $k_{B}$ are the diffusion barrier and Boltzmann’s constant,
respectively, and $T$ is the environment temperature. Therefore, at room
temperature, we estimate that the diffusion constant of Path 3, for a single
Li, Na, and K, could be more than 103 times faster than Path 2. Moreover,
regarding Path 3, the K mobility on ABC-Si4Al2 is more than 105 times faster
than that of the Li ion and 49 times faster than that of Na one, being as fast
as in commercial graphite anodes. Accordingly, we expect good high-rate
performance for the ABC-Si4Al2 as the anode material, primarily in KIBs.

###  3.4 Open-Circuit Voltage and Theoretical Capacity

The open-circuit voltage (OCV) is the maximum voltage available for a battery,
while the theoretical capacity ($C$) is the maximum amount of ions that can be
stored in the host 35. This section explores the adsorption energy ($E_{ads}$)
as a function of the number of Li, Na, or K atoms adsorbed on ABC-Si4Al2, in
order to estimate the OCV and the $C$ by using equations 2 and 3,
respectively.

Herein, we studied the ABC-Si4Al2 single- and double-side surface adsorptions,
as well as the incorporation of Li atoms in the H2 sites (figure 1).
Accordingly, we considered several configurations of ABC-Li_xSi4Al2, ABC-
Na_ySi4Al2, and ABC-K_ySi4Al2 systems ($\rm{x=0.11,1,2,3,4}$, and
$\rm{y=0.11,1,2}$) with the adatoms initially placed in the most favorable
adsorption positions. Adsorption energies are presented in figure 4, while the
corresponding optimized structures are shown in figure 5 (a)-(h). It is worth
mentioning that the first points in figure 4 represent the ${\rm x=0.11}$
concentration and the $E_{ads}$ energy of a single-side adsorption of M1
alkali metal adatom in the H site of a 3$\times$3$\times$1 supercell
(ABC-M1Si36Al18), which was reported in the previous section.

Figure 4: Adsorption energies ($E_{ads}$) as function of M concentration (x)
in ABC-M_xSi4Al2 (M = Li, Na, K).

Figure 4 shows that the adsorption energy continuously decreases as the M
concentration increases, except for Li for $\rm{x}>3$. During ion adsorption,
there are two important effects: the repulsion force between neighboring atoms
and the structural changes. The repulsion force is mainly responsible for the
adsorption energies of all ions decrease in the $0<{\rm x}<3$ range, while the
structural changes are dominant when ${\rm x}>3$, increasing the adsorption
energy 51. Interestingly, the $E_{ads}$ for Li, Na, and K remained negative
throughout the range explored here, indicating the ABC-Si4Al2 is a suitable
anode material for Li-, Na-, or K-ion batteries.

Figures 5 (a), (e), and (g) report the resulting configurations of single-side
surface adsorption of M1 adatom on the ABC-Si4Al2, in which the H site was
fully covered by Li, Na, or K atoms (ABC-M1Si4Al2). Figures 5 (b), (f), and
(h) show the double-side surface adsorption with the H site fully covered
(ABC-M2Si4Al2). For the Li-ion, we also have ABC-Li3Si4Al2 and ABC-Li4Si4Al2
systems that are shown in figures 5 (c) and (d), respectively. The most stable
configuration of ABC-Li3Si4Al2 has all H2 positions fully loaded and one
surface side with all H sites fully covered by Li atoms.

Figure 5: ABC-M_xSi4Al2 (M = Li, Na, K) optimized structures for several
alkaline metal concentration values (x).

Although it is possible to find several other phases at low concentrations by
using different supercell sizes, we focused on the full loaded structure to
estimate the OCV and the $C$. Accordingly, the OCV and $C$ were estimated for
Li4Si4Al2, Na2Si4Al2, and K2Si4Al2 systems, with the respective values
presented in table 2. The OCV for Li-, Na-, and K-related systems are,
respectively, 0.49 V, 0.16 V and 0.14 V, which are comparable with values in
commercial anode materials, such as graphite (0.11 V) and TiO2 (1.5-1.8 V) 44,
as well as other candidates for anodes, such as silicene (0.30-0.50 V) 11, 12,
VS2 (0.93-1.32 V) 44, 49, Si3C (0.50-0.71 V) 16, and Si2BN (0.29-0.46 V) 42.
The low voltage values, which we found for ABC-Si4Al2, are desirable
characteristics for anode electrodes 1.

The theoretical capacity of the ABC-Si4Al2 is 645 mAh/g for LIBs, and 322
mAh/g for both SIBs and KIBs, which are higher than the capacity in commercial
anode materials, such as graphite (372 mAh/g for LIBs 6 and 279 mAh/g for KIBs
7), and TiO2 (200 mAh/g for LIBs 46). Moreover, the Li capacity on ABC-Si4Al2
of 645 mAh/g is slightly smaller than an experimental value of 721 mAh/g found
in pristine crystalline silicene as anode material for LIBs 13. Remarkably,
the K capacity on ABC-Si4Al2 of 322 mAh/g is higher than 180 mAh/g when
pristine crystalline silicene was experimentally investigated as anode
material for KIBs 14. The crystalline silicene presents an experimental
thickness of $\approx$ 6.5 Å and lattice parameter of $\approx$ 3.4 Å 13, 14.
This system appears to be structurally similar to the few-layer pristine
silicene nanosheets made of silicene trilayer vertically combined. Indeed, we
recently investigated two stacking configurations (AA′A and ABC) of pristine
silicene trilayers bonded covalently, and the values obtained for the
thickness and the lattice parameter were $\approx$ 7.0 Å and 3.858 Å,
respectively 19. Due to the aluminum doping, the ABC-Si4Al2 trilayer exhibits
a smaller thickness of 6.0 Å and a larger lattice parameter of 4.143 Å 19.
This lattice expansion could be the reason for the increase of the K
adsorption, i.e., the higher capacity of K on ABC-Si4Al2, as compared to
crystalline silicene. Therefore, our results indicate that ABC-Si4Al2 has a
superior theoretical capacity for alkali metal ion storage, primarily for
KIBs.

##  4 Concluding Remarks

In summary, the ABC-Si4Al2 was systematically investigated as a potential
anode material for Li, Na, and K ion batteries. Our previous investigation has
shown that this structure is dynamically and mechanically stable, and herein
the simulations have shown that ABC-Si4Al2 is also thermodynamically stable up
to 600 K. We explored the adsorption and diffusion properties of alkali metal
ions (Li, Na, K) on ABC-Si4Al2, finding large adsorption energies and low
diffusion barriers values. Moreover, the ABC-Si4Al2 provides good theoretical
capacity and low open-circuit voltage. The lowest OCV value was obtained for
the K adatom and the highest theoretical capacity was found for Li one. Our
estimated surface diffusion barriers, OCV, and specific capacity are
comparable to the respective values in commercial anode materials, such as
graphite and TiO2, and other candidate materials, such as pristine silicene.
We also found a slightly smaller capacity for LIBs and a higher one for KIBs
as compared with the experimental values of pristine crystalline silicene.
Those results suggest that doped few-layer silicene is a potential material
for anode batteries, with a good high-rate performance and theoretical
capacity for alkali metal storage (Li, Na, K), primarily for SIBs and KIBs.
Furthermore, the functionalization of few-layer silicon-based systems could be
a route to developing novel electrodes for AMIBs.

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