[1]Carlo da Cunha [1]School of Informatics, Computing, and Cyber-Systems,
Northern Arizona University, Flagstaff, 86011, AZ, USA 2]Department of
Electronics and Mechanical Engineering, Chiba University, Chiba, 263-8522,
Chiba, Japan 3]School of Electrical, Computer and Energy Engineering and
Center for Solid State Electronics Research, Arizona State University, Tempe,
85287, AZ, USA 4]School of Computing and Augmented Intelligence, Arizona State
University, Tempe, 85281, AZ, USA

# Machine Learning Methods for Background Potential Estimation in 2DEGs

<EMAIL_ADDRESS>Nobuyuki Aoki  David Ferry  Kevin Vora  Yu Zhang  *  [  [
[

###### Abstract

In the realm of quantum-effect devices and materials, two-dimensional electron
gases (2DEGs) stand as fundamental structures that promise transformative
technologies. However, the presence of impurities and defects in 2DEGs poses
substantial challenges, impacting carrier mobility, conductivity, and quantum
coherence time. To address this, we harness the power of scanning gate
microscopy (SGM) and employ three distinct machine learning techniques to
estimate the background potential of 2DEGs from SGM data: image-to-image
translation using generative adversarial neural networks, cellular neural
network, and evolutionary search. Our findings, despite data constraints,
highlight the effectiveness of an evolutionary search algorithm in this
context, offering a novel approach for defect analysis. This work not only
advances our understanding of 2DEGs but also underscores the potential of
machine learning in probing quantum materials, with implications for quantum
computing and nanoelectronics.

###### keywords:

machine learning, scanning gate microscopy, quantum point contact,

##  1 Introduction

Quantum-effect devices and materials, characterized by exotic quantum
properties, may hold the foundation for next-generation technologies. Within
this landscape, two-dimensional electron gases (2DEGs) emerge as fundamental
structures, often crafted through the use of two-dimensional materials [1, 2]
or heterostructures [3]. They play an important role in novel technologies
such as quantum computing, providing a precisely controlled environment to
exploit quantum phenomena. Moreover, 2DEGs show intriguing phenomena such as
the quantum Hall effect [4], with direct applications in precision metrology
[5].

The presence of impurities and defects can significantly impact the electronic
properties of 2DEGs [6, 7]. These anomalies introduce energy levels within the
bandgap and scatter electrons, directly influencing carrier mobility and
conductivity. High-electron-mobility transistors (HEMTs) [8] and other devices
rely on precise control of carrier concentration and mobility, both of which
are susceptible to defects and impurities [9]. Uncontrolled defects can also
introduce decoherence, limiting the quantum coherence time of qubits and
raising reliability concerns in electronic devices.

However, detecting the background potential caused by these defects and
impurities poses a considerable challenge. 2DEGs are remarkably thin, and
defects often occur at the nanometer scale or smaller. Moreover, these
imperfections may not manifest as visible features but can exert only indirect
effects on electronic properties.

Scanning gate microscopy (SGM) [10] emerges as a valuable technique to address
this challenge [11]. SGM employs a sharp charged tip to perturb a 2DEG while
concurrently monitoring its conductivity. This yields a conductance map across
the device, reflecting carrier flow in response to the perturbation. This
technique has been used to image compressibility features of the quantum Hall
regime [12], to estimate ballistic carrier trajectories in graphene [13], and
to visualize wavefunction scars in quantum dots [14]. While the information
provided by SGM may require nuanced interpretation [15, 16, 17], it can be
leveraged to indirectly infer details about the background potential [18, 19,
20].

In this work, we undertake a comparative study of three machine learning
techniques applied to estimate the background potential of a 2DEG using SGM
data. To focus on a well-defined and localized region of the 2DEG, we utilize
a quantum point contact (QPC) [21] as a prototypical device [22, 23, 24]. QPCs
find wide-ranging applications in nanoelectronics and quantum computing,
underscoring the significance of understanding the background potential for
optimizing device performance.

Our estimation of the background potential from the SGM conductance map hinges
on three different techniques: a picture-to-picture translator based on
generative adversarial neural networks (Pix2Pix GAN), cellular neural networks
(CNN) [25, 26, 27], and an evolutionary search algorithm (ES) [28]. Other
attempts to solve this inverse problem include the use of deep convolutional
neural networks [29] and deep reinforcement learning [30]. In all techniques,
a conductance map is obtained from SGM, and the inverse problem of estimating
the background potential is solved using different machine learning
techniques.

We gauge the performance of these techniques through a series of calculated
parameters. First, we calculate the entropy of the estimated potential images.
This is a technique commonly used to assess the quality of microscopy images.
For example, images with high entropy values contain more information, while
images with low entropy values are often associated with uniform and
repetitive patterns [31, 32]. The spatial variability of the estimated
potential is studied with surface roughness analysis, while the fractal
dimension offers information about the potential at different scales. Fractal
dimension studies appear as an important concept in quantum-effect devices.
For instance, it is used to study signatures of chaotic behavior in quantum
dots [33, 34, 1]. Complementing these techniques, we also studied the
distribution of sizes of potential features using morphological granulometry.
The importance of performing a granulometric analysis relies on the fact that
variations in the potential can impact the behavior of the device
significantly. Finally, we also estimate the structure of a quenched structure
of the potential to identify static features such as those associated with
lattice defects.

Our results, drawn from a limited pool of experimental data, suggest that,
given the constraints of data availability and acquisition costs, the
evolutionary search technique yields the most promising outcomes.

This study showcases the effectiveness of machine learning techniques in
estimating the background potential of 2DEGs, offering valuable insights that
can guide future research in addressing similar inverse problems.

The structure of this manuscript is outlined as follows: First we describe the
experimental methods to fabricate the 2DEG, the QPC and how the SGM
measurement were obtained. We then describe how we used Green’s functions [35]
and perturbation theory [25] to obtain image pairs of different potential
configurations and the corresponding SGM signal. Next we describe the three
machine learning techniques: Pix2Pix GAN, CNN, and ES. We then show the
results for image entropy, roughness, fractal dimension, and morphological
granulometry. We complete the discussion of the results with an estimate of
the quenched disorder potential.

##  2 Methods

###  2.1 Experimental

A two-dimensional electron gas (2DEG) was created by a modulation-doped
heterostructured consisting of In0.53Al0.47As/In0.53Ga0.47As/In0.53Al0.47As
layers on an (001) InP substrate [3, 9, 1]. This 2DEG is located $45$ nm below
the surface, which is coated with $30$ nm of polymethylmethacrylate (PMMA) to
avoid short-circuits between the tip and the device.

Shubnikov-de Haas measurements revealed the occupation of two subbands with
electron densities of $7.2\times 10^{11}$ cm-2 and $2.1\times 10^{11}$ cm-2.
This corresponds to an electron mobility of $7.4\times 10^{4}$ cm2/V$\cdot$s
and a mean-free path of $1.2$ $\mu$m [3, 23, 9, 22]. A quantum point contact
(QPC) was etched on top of the heterostructure using electron beam
lithography. The QPC has a physical opening of $600$ nm and the etched
trenches have a radius of approximately $800$ nm. A voltage between $0$ V and
$-7.2$ V was applied to the in-plane gates defined by the etched trenches to
control the electrical width of the QPC. The material structure as well as a
Hall bar lithographically defined on its surface, and the QPC are shown in
Fig. 1.

Figure 1: A sketch of the InGaAs/InAlAs heterostructure, a Hall bar defined
lithographically on its surface, and the quantum point contact etched on the
Hall bar.

Scanning gate measurements were conducted by rastering a tip coated with $15$
nm of PtIr $40$ nm above the PMMA layer while changes in conductance were
monitored. A current of $10$ nA was applied across the QPC, while the voltage
was measure with a lock-in amplifier operating in quasi-DC. All measurements
were performed at a system temperature of $280$ mK, and the tip was biased so
that the maximum change in conductance was $0.2\times 2e^{2}/h$.

###  2.2 Computational

Three distinct methods for estimating the background potential from SGM data
were used: i) Pix2Pix GAN, ii) Cellular Neural Networks, and iii) Evolutionary
Search.

####  2.2.1 Green’s Functions

In the weak-perturbation regime, the tip potential shifts the eigenstates of
the system according to first-order perturbation theory [25]:

$\Delta\varepsilon_{n}=\braket{n}{V}{n},$ (1)

where $n$ is an eigenstate, and $V$ is the tip-induced potential. If the tip-
induced potential is sufficiently sharp, it can be approximated by:

$V(\mathbf{r})=V_{0}\delta(\mathbf{r}-\mathbf{r}_{0}),$ (2)

where $\mathbf{r}_{0}$ is the position of the tip on the plane of the 2DEG.
This leads to a shift in eigenenergy given by:

$\Delta\varepsilon_{n}=V_{0}|\varphi_{n}(\mathbf{r}_{0})|^{2},$ (3)

where $\varphi_{n}(\mathbf{r}_{0})$ is the wavefunction evaluated at
$\mathbf{r}_{0}$.

Consequently, changes in conductance are given by:

$\Delta G(\mathbf{r}_{0};\varepsilon)\approx\frac{\partial
G}{\partial\varepsilon_{F}}\sum_{n}\Delta\varepsilon_{n}\delta(\varepsilon-\varepsilon_{n})\propto
D(\mathbf{r}_{0}).$ (4)

Here, we specifically focus on states indexed by $n$ that are in proximity to
the Fermi energy, and introduce the concept of the local density of states
(LDOS), denoted by $D$.

The tip-induced potential on the 2DEG, however, is given by:

$V_{tip}(\mathbf{r},\mathbf{r}_{tip})=\frac{d_{0}^{2}}{\|\mathbf{r}-\mathbf{r}_{tip}\|^{2}+d_{0}^{2}}.$
(5)

In this expression, the parameter $d_{0}$ is held contant at a value
corresponding to 80 lattice constants. Additionaly, $\mathbf{r}_{tip}$
signifies the location of the tip within the plane of the 2DEG. We apply a
truncation to the tip potential, limiting its influence to a radius that
aligns closely with $14$ lattice constants, a value approximating the physical
tip radius.

Consequently, changes in conductance, in the weak-perturbation regime, can be
approximated by:

$\Delta
G(\mathbf{r}_{0};\varepsilon)\approx\sum_{\mathbf{r}}V_{tip}(\mathbf{r},\mathbf{r}_{tip})D(\mathbf{r}).$
(6)

To find $D(\mathbf{r})$, a potential is generated, the corresponding
Hamiltonian $\mathbf{H}$ is calculated, and then the corresponding Green’s
function [35] is found. Instead of computing the Green’s function for the
entire potential, the images were segmented into slices, and the Green’s
function was computed to each slice $n$ using:

$\mathbf{G}_{nn}(\varepsilon)=\left[\left(\varepsilon+i\eta\right)\mathbf{I}-\mathbf{H}_{n}\right]^{-1},$
(7)

where $\eta\rightarrow 0$ ensures that the retarded Green’s function is
obtained. The slices are then merged using Dyson’s equation:

$\mathbf{G}_{ab}=\mathbf{G}^{0}_{ab}+\sum_{mn}\mathbf{G}^{0}_{am}\mathbf{V}_{mn}\mathbf{G}_{nb},$
(8)

where the upscript $0$ indicates the unperturbed Green’s function for the
slice, and $\mathbf{V}$ is the coupling matrix between slices. The local
density of states (LDOS) is given by:

$D_{mn}=-\frac{1}{\pi}Im\left[G_{nn;mm}\right].$ (9)

Given the computed carrier density, the Fermi wavelength is
$\sqrt{2\pi/n_{2D}}\approx 30$ nm, which is comparable to the physical tip
radius. All images were computed on a $600\times 600$ grid, which corresponds
to an area of $675\times 675$ nm2. This implies a discretization spacing of
$1.125$ nm, which gives a wavenumber of $0.21$ nm-1, which is one order of
magnitude away from the first Brillouin zone located at $5.58$ nm-1. A saddle
potential was included to mimic the behavior of the side-gate potential.

####  2.2.2 Pix2Pix GAN

Spatial variations in the background potential should be close to the Bohr
radius ($4\pi\varepsilon\hbar^{2}/m^{*}e^{2}$) [36], which, for InGaAs, is
approximately $18$ nm, corresponding to eight lattice constants. Random
potentials were created, where each site is occupied with a probability
$p\in[0,0.04]\%$. If the site is occupied, then a truncated Coulomb-like
potential is created around a radius of five lattive constants. The maximum
potential at the site is $10$ meV. This impurity representation fully captures
the expected variations in the background potential. A total of $1600$
potentials were created, and, using image augmentation corresponding to
flipping and salt-and-pepper noise, this number was increased to $12800$.

The Pix2Pix GAN [37] is a deep learning architecture that learns to translate
images from one domain to another, such as converting sketches into realistic
images. Given our dataset of input images, we would like Pix2Pix GAN to
predict the target images available in the output dataset. Next, we describe
the Pix2Pix GAN and its underlying concepts, including the generator and
discriminator networks. The Pix2Pix model is a type of conditional GAN, or
cGAN, where the generation of the output image is conditional on input. The
discriminator is provided both with an input image and the target image and
must determine whether the target is a plausible transformation of the input
image while the generator is responsible to generate the target images.

The generator is an encoder-decoder model which uses a U-Net architecture
[38]. The model takes an input image and generates a target image. It does
this by first downsampling or encoding the input image down to a bottleneck
layer, then upsampling or decoding the bottleneck representation to the size
of the output image. The U-Net architecture means that skip-connections are
added between the encoding layers and the corresponding decoding layers,
forming a U-shape. The generator is trained via adversarial loss, which
encourages the generator to generate plausible images in the target domain.
The generator is also updated via the mean absolute error (MAE) loss measured
between the generated image and the expected output image. This additional
loss encourages the generator model to create plausible translations of the
input image.

The discriminator is a deep convolutional neural network that performs image
classification. Specifically, conditional-image classification. It takes both
the input image and the target image as input and predicts the likelihood of
whether the target image is real or a fake translation of the source image.
The discriminator design is based on the effective receptive field of the
model, which defines the relationship between one output of the model to the
number of pixels in the input image. This is called a PatchGAN model and is
carefully designed so that each output prediction of the model maps to a
$70\times 70$ square or patch of the input image. The benefit of this approach
is that the same model can be applied to input images of different sizes, e.g.
larger or smaller than $256\times 256$ pixels. The output of the model depends
on the size of the input image but may be one value or a square activation map
of values. Each value is a probability for the likelihood that a patch in the
input image is real. These values can be averaged to give an overall
likelihood or classification score if needed. The discriminator is updated in
a standalone manner, so the weights are reused in this composite model but are
marked as not trainable. The composite model is updated with two targets, one
indicating that the generated images were real (cross entropy loss), forcing
large weight updates in the generator toward generating more realistic images,
and the executed real translation of the image, which is compared against the
output of the generator model (L1 loss).

Implementation Details: We first pre-process the data and implement a Keras
(with TensorFlow backend) version of the original implementation111Available
at https://github.com/phillipi/pix2pix. Each image is loaded, rescaled, and
split into the input and target elements. Each image pairs are resized to the
width and height of $256\times 256$ pixels. Once loaded, we can save the
prepared arrays to a new file in compressed format for later use. Typically,
GAN models do not converge; instead, an equilibrium is found between the
generator and discriminator models. As such, we cannot easily judge when
training should stop. Therefore, we can save the model and use it to generate
sample image-to-image translations periodically during training, such as every
$10$ training epochs. We can then review the generated images at the end of
training and use the image quality to choose a final model. We run for
approximately $100$ epochs for the presented results.

####  2.2.3 Cellular Neural Networks

Images are segmented into $N\times N$ discrete cells whose states
$\\{X_{\mathbf{i}}\mid i_{x,y}=1,2,\ldots,N\\}$ evolve according to [26]:

$\frac{dX_{\mathbf{i}}(t)}{dt}=-X_{\mathbf{i}}(t)+\sum_{\mathbf{j}\in\mathcal{N}(\mathbf{i})}A_{\mathbf{ij}}Y_{\mathbf{j}}(t)+\sum_{\mathbf{j}\in\mathcal{N}(\mathbf{i})}B_{\mathbf{ij}}U_{\mathbf{j}}(t)+Z,$
(10)

where $\mathbf{A}$ and $\mathbf{B}$, known as cloning templates, are tensors
used to propagate specific information behaviors within a local neighborhood
of cells across the image. $\mathbf{U}$ is an input to the cell, $Z$ is an
offset, typically set to $1$, and $\mathbf{Y}$ is the output of the cell,
given by $Y_{\mathbf{i}}(t)=\text{tanh}\left(X_{\mathbf{i}}(t)\right)$. Also,
$\mathcal{N}(\mathbf{i})$ denotes the neighborhood of cell $\mathbf{i}$. The
Moore neighborhood, defined as
$\mathcal{N}_{\mathbf{i}}=\\{\mathbf{j}:|i_{x}-j_{x}|\leq 1,\
|i_{y}-j_{y}|\leq 1\\}$, was used in all computations.

To obtain the steady-state output of the CNN, we discretize Eq. 10 and use
Euler’s method to obtain update rule for the cell states:

$\mathbf{X}^{n+1}=(1-\Delta)\mathbf{X}^{n}+\Delta\boldsymbol{\xi}^{n},$ (11)

where $\Delta=10^{-3}$ is a unit time period, and:

$\boldsymbol{\xi}^{n}=\mathbf{A}\cdot\mathbf{Y}^{n}+\mathbf{B}\cdot\mathbf{U}+\mathbf{J}.$
(12)

In this last equation, the dot represents the contraction between a tensor and
a multi-dimensional array. $\mathbf{J}$ is a full-rank array of ones. The
dynamical system was evolved until the maximum difference between
$\mathbf{X}^{n+1}$ and $\mathbf{X}^{n}$ was less than $10^{-3}$.

For training, the input $\mathbf{U}$ is set to the theoretically computed SGM
image, while the expected steady-state output is the potential used to compute
the corresponding SGM image. The dynamical system is evolved until steady-
state, and then the cloning templates are updated using gradient ascent
according to:

$\mathbf{A}^{n+1},\mathbf{B}^{n+1}=\mathbf{A}^{n},\mathbf{B}^{n}-\eta\nabla_{\mathbf{A},\mathbf{B}}L(\mathbf{A}^{n},\mathbf{B}^{n}),$
(13)

where $\eta=10^{-3}$ is the learning rate, and $L$ is the loss, defined as the
correlation between the obtained state $\mathbf{X}$ and the expected
experimental SGM image.

The number of images and their preparations were the same used in Pix2Pix GAN.

####  2.2.4 Evolutionary Search

The evolutionary algorithm [39, 28] starts with a population of
$\\{\mathbf{U}_{i}\mid i=1,2,\ldots,20\\}$ individual corresponding to
different potentials. At each interation, the SGM images corresponding to each
test solution are computed together with the correlation $r_{i}$ between them
and the experimental SGM image. The individual with the highest correlation is
declared the winner, and the rest of the swarm move towards it according to:

$\mathbf{U}_{i}^{n+1}=\alpha\mathbf{U}_{i}^{n}+(1-\alpha)\mathbf{U}_{winner}^{n}+\theta(r_{i})\boldsymbol{\eta},$
(14)

where $\alpha=0.95$ is a hyperparameter,
$\boldsymbol{\eta}\sim\mathcal{N}(0,1)$, and $\theta(r)$ is given by:

$\theta(r)=\left\\{\begin{array}[]{cl}\theta_{0}e^{-\gamma r}&\text{if
}r>0,\\\ \theta_{0}&\text{otherwise},\end{array}\right.$ (15)

where $\theta_{0}=0.1$ meV and $\gamma=5$. This function garantees that the
adjustment is small if the correlation is high, mimicking the dynamics of
simmulated anneling.

This relaxation procedure was performed until the highest correlation was at
least $72\%$. Although this may be considered a threshold, it takes a
considerably long computation time using current technology. Nonetheless, this
correlation between SGM signals corresponds to a correlation of $78\%\pm 10\%$
between the estimated and expected potentials for Gaussian fields.

##  3 Results

Figure 2 shows a typical SGM image and the corresponding background potential
using the techniques studied in this work. Surprisingly, the background
estimates exhibit notable discrepancies depending on the employed method.
Since no ground truth is available, finding the method that gives the most
appropriate result becomes a formidable task.

Figure 2: a) Experimental SGM image and the estimated background potential
using b) Pix2Pix GAN, c) CNN, and d) Evolutionary Search. All images
correspond to an area of $675\times 675$ nm2.

###  3.1 Image Entropy

To quantify the estimation difference, we computed the image entropy for the
whole set of images for each estimation technique. The image entropy is
defined as:

$S=-\sum_{\langle image\rangle}p_{i}\log_{2}(p_{i}),$ (16)

where $p_{i}$ is the probability of finding a pixel of the image with
intensity $i$. This probability was computed from the histogram of the image
using $256$ bins.

The importance of using the image entropy to qualify differences in estimation
relies on the fact that the image entropy quantifies the level of complexity
in the distribution of the intensities within the image. For example, if it is
equally probable to find any intensity in a binary image, $p_{i}=1/2$ and the
entropy is one, representing complete uncertainty. On the other hand, if only
one intensity $i^{\prime}$ is likely to be found, then
$p_{i}=\delta_{i,i^{\prime}}$, and the entropy is zero, representing complete
knowledge.

For our computations, we found the entropies shown in Table 1. Background
potentials estimated using Pix2Pix showed the smallest average entropy among
images, whereas the other two techniques displayed similar entropies. This is
probably due to the nature of the estimation processes. Pix2Pix works by,
given an SGM image, creating a potential image that minimizes an adversarial
loss on a set of theoretically generated images. The model is then used to
estimate real images, but there is no guarantee that experimental images have
the same probability distribution as that of the theoretical set. This is
furthered enforced by a very limited theoretical dataset. CNN is a more direct
approach that does not depend explicitly on the probability distributions,
rather, it produces a non-linear mapping between SGM and potential images. On
the other hand, it is not garanteed that the experimental images have the same
mapping as that obtained from the thoeretical set. Finally, evolutionary
search works by progressively obtaining a potential that, given a physical
model, produces an SGM image that is the close to the experimental one.
Consequently, this last technique is not limited by the finite dataset or a
specific mapping generated by a finite set. As a result, evolutionary search
produces potential backgrounds that are more complex and richer in details.

Table 1: Image entropies computed for Pix2Pix GAN, Cellular Neural Networks (CNN), and Evolutionary Search (ES) | GAN | CNN | ES
---|---|---|---
Minimum | 3.11 | 5.60 | 5.94
Average | 3.75 | 6.00 | 6.06
Maximum | 4.37 | 6.56 | 6.20
|  |  |

###  3.2 Potential Roughness

The root mean square average of the estimated potential, or roughness, is
defined as:

$R=\sqrt{\frac{1}{L_{x}L_{y}}\sum_{\begin{subarray}{c}x=1\\\
y=1\end{subarray}}^{L_{x}L_{y}}\left(P(x,y)-\langle P\rangle\right)^{2}},$
(17)

where $L_{x}$ and $L_{y}$ are the width and height of the image, $P(x,y)$ is
the pixel-value at coordinates $(x,y)$, and $\langle P\rangle$ denotes the
pixel average of the image.

The surface roughness quantifies the short-wavelength spatial variability of
the estimated potential, and, consequently, gives an idea of potential
texture. Potential roughness plays an important role for quantum coherent
devices, since it may cause variations and broadening of the energy levels of
confined electrons. Moreover, potential roughness may increase electron
scattering and dephasing, which leads to reduced coherence times. These
effects can directly affect the scanning gate microscopy patterns. The
transmission-dependent roughness measures for the different techniques is
shown in Fig. 3.

Figure 3: RMS roughness as a function of transmission for potentials estimated
via Pix2Pix GAN (blue disks), cellular neural networks (orange filled
squares), and evolutionary search (green stars).

Pix2Pix GAN estimates potentials with little roughness and variability of
roughness, which can not correspond to a realistic scenario because as the
transmission is reduced, the background potential moves closer to the Fermi
level and its effects should be more pronnounced. Potentials estimated by CNN
and ES are rougher, and follow a similar trend for transmission values smaller
than approximately $2$. The roughnesses of potentials estimated by ES,
however, show a smaller variability, which is an expected result. Changing the
transmission less than unity should not cause a significant movement of the
background potential closer to the Fermi level.

####  3.2.1 Fractal Dimension

The fractal dimension is a related concept to the surface roughness. While
surface roughness measures irregularities and deviations from a smooth
potential, the fractal dimension measures how a pattern fills space at
different scales in a self-similar fashion. The Hausdorff-Besicovitch
dimension can be understood through a similar transformation. For example, if
amplifying an object by a factor $n$, we obtain $n^{D}$ copies of the same
object, we say that $D$ is its dimension. Mathematically, the Hausdorff-
Besicovitch dimension is given by:

$D_{HB}=\frac{\log(N)}{\log(n)},$ (18)

where $N$ is the number of obtained copies.

For calculating the fractal dimension of images, however, we use the
Minkowski-Bouligand dimension. It consists of covering the image with a grid,
and, for each box of the grid applying a renormalization measure $\mu$ that
indicates whether that box is necessary to cover the pattern under analysis.
The most used measure is simply the presence of a filled pixel. Our images,
however, are gray-scale. Therefore, we normalized the images as:
$M\rightarrow\left(M-\min(M)\right)/\left(\max(M)-\min(M)\right)$ and used a
measure $\mu_{B}=\min\left(1,\sum_{b\in B}b\right)$ that counts the total
intensity inside a box. The number of boxes necessary to cover the image at
some scale $\epsilon$ gives an idea of the intricacy, or level of detail of
the image at that scale. Similarly to the Hausdorff-Besicovitch, the
Minkowski-Bouligand dimension is obtained from:

$D_{MB}=-\lim_{\epsilon\rightarrow
0}\frac{\log\left(N_{\epsilon}\right)}{\log(\epsilon)},$ (19)

where $N_{\epsilon}$ is the number of boxes. This dimension is obtained from a
linear regression of $\log(N_{\epsilon})\times\log(\epsilon)$.

Figure 4 shows the Minkowski-Bouligand dimension for the potentials estimated
using Pix2Pix GAN, cellular neural networks, and evolutionary seach. While CNN
and ES estimate potentials with a fractal dimension close to $2$, Pix2Pix GAN
estimates potentials with an average dimension close to $1.5$ with a much
greater variability. It is also interesting to note that potentials estimated
by ES have a complexity that is nearly constant and does not depend much on
the electronic transmission. On the other hand, CNN offers potentials with a
similar complexity, but is particularly sensitive to low electronic
transmission values.

Figure 4: Box dimension as a function of transmission for potentials estimated
via Pix2Pix GAN (blue disks), cellular neural networks (orange filled
squares), and evolutionary search (green stars).

####  3.2.2 Morphological Granulometry

The estimated potentials display a set of grains that can be studied by a
morphological analysis. To perform this computation, first the dilation of an
image $M$ is calculated according to a kernel $K$ as:

$(M\oplus K)(x,y)=\max_{(i,j)\in K}\left\\{M(x+i,y+j)\right\\}.$ (20)

Similarly, the erosion of the same image is given by:

$(M\ominus K)(x,y)=\min_{(i,j)\in K}\left\\{M(x+i,y+j)\right\\}.$ (21)

The opening of the image is then given by the dilation of the erosion of the
image:

$M\bullet K=(M\ominus K)\oplus K.$ (22)

This morphological operation removes structures smaller than the kernel from
the image.

The granulometry $G_{n}(M)$ is then computed by finding the cardinality of the
opening of the image at different kernel sizes $n$. The grain size
distribution is finally given by the difference of successive granulometries:

$S_{k}(M)=G_{k}(M)-G_{k+1}(M).$ (23)

In all our calculations, we used a square kernel, and used the mean of the
elements as the cardinality function.

The results obtained with each technique are shown in Fig. 5. Our results
indicate that the grain distribution exhibits a higher degree of dispersion
with a maximum value around $31.64$ nm2, corresponding to a dot with a $3.17$
nm radius. The spectrum for the potential obtained using CNN is also dispersed
with a maximum around $1.26$ nm2, corresponding to a $0.635$ nm radius dot.
Finally, the spectrum for the potential obtained using evolutionary search
peaks around $11.39$ nm2, which corresponds to a $1.90$ nm radius dot. The
Bohr radius for InGaAs is around $20$ nm [36], which has an area of $1,256.64$
nm2. Although all techniques estimate more granular potentials, the spectrum
obtained using evolutionary search demonstrates reduced variation in the
vincinity of this value, whereas the spectra obtained with the other two
techniques decay to zero.

Figure 5: Average grain distribution spectra for the potentials estimated by
Pix2Pix GAN (blue discs), cellular neural networks (orange filled squares),
and evolutionary search (green stars). The shaded regions cover the minimum
and maximum values for each grain size.

####  3.2.3 Quenched Disorder Potential

Disorder in the background potential can be influenced by external factors
such as the electrostatic field of the side gates. For example, impurities can
be charged by the side gate-induced electric field in the channel, or
piezoelectric effects can induce strain variations. Also, surface trap states
in the dielectric or in the channel can create localized structures that are
not quenched. On the other hand, there are some potentials, such as those
caused by lattice defects and surface roughness, that are quenched and do not
vary due to any external factor. Some charged impurities and the alloy
potential caused by the random distribution of different atomic species are
also quenched.

By varying the voltages on the side-gates, we can modify the effective
aperture of the quantum constriction, leading to changes in the transmission
through it. Our challenge arises from the fact that we estimate the potential
based on this transmission map, which is influenced by external parameters.
This complicates the estimation of the quenched potential features, which
ideally should remain unaffected by external parameters. To estimate the
quenched potential, we look for features that remain consistent and are
minimally affected across different transmissions.

To achieve this, we compute the pixel-wise precision, which is given by the
inverse of the variance, computed for all images along the transmission
dimension. This measure is high when there is greater certainty in the data
set, and low other otherwise. Figure 6 shows the estimted alloy potential
obtained for the different techniques.

Figure 6: Estimated alloy potential for a) Pix2Pix GAN b) cellular neural
networks, and c) evolutionary search.

The three techniques estimate static alloy potentials that are correlated by
only $7$ %. Although they are statistically uncorrelated, both Pix2Pix GAN and
CNN try to infer a potential given training examples. This is highlighted by
more intense regions on the top and bottom of the images corresponding to the
gate regions. This kind of inference assumes that the the SGM image
corresponds to a potential, at least, similar to those in the training set.
Moreover, the estimated potential may not necessarily reproduce the obtained
SGM signal, since there is no physical law conditioning the inference. On the
other hand, ES tries to find, through a physical law, the potential that
creates an SGM signal closest to the experimentally obtained. However, it
requires a physical law, and the inference assumes that the physical
phenomenon is strictly modeled by that law. In our case, we are assuming that
the SGM signal was produced by a one-electron wavefunction.

##  4 Conclusion

In conclusion, our study examined the estimation of background potentials in
two-dimensional electron gases (2DEGs) using three distinct machine learning
techniques: Pix2Pix GAN, cellular neural networks (CNN), and evolutionary
search (ES). The evaluation encompassed several crucial aspects of the
estimated potentials, shedding light on their characteristics and
implications.

Entropy analysis revealed that Pix2Pix GAN consistently yielded potentials
with the lowest average entropy among the three techniques, while CNN and ES
exhibited similar entropy values. This discrepancy can be attributed to the
nature of the estimation processes, with Pix2Pix GAN relying on a theoretical
dataset that may not perfectly match the experimental data distribution.

Regarding potential roughness and its variation, Pix2Pix GAN produced
potentials with limited roughness and variability. This outcome, however,
departs from realistic scenarios where the background potential’s proximity to
the Fermi level should result in more pronounced effects as the transmission
decreases. In contrast, CNN and ES generated rougher potentials that align
more closely with expected trends, with ES demonstrating a notably reduced
variation in roughness.

The examination of fractal dimensions exposed interesting differences. Pix2Pix
GAN estimated potentials with an average dimension close to 1.5 and
significant variability. In contrast, CNN and ES provided potentials with a
fractal dimension near 2. Notably, ES-generated potentials displayed
consistent complexity, independent of electronic transmission levels, while
CNN exhibited sensitivity to lower transmission values.

Granulometry analysis showcased dispersed grain distributions for all
techniques, with peak values indicating granule sizes. The spectra derived
from CNN and ES showed dispersion, with peak values corresponding to specific
radii. Interestingly, ES demonstrated less variation near the Bohr radius,
indicating a narrower distribution.

Finally, an analysis of the quenched potential revealed that Pix2Pix GAN and
CNN primarily inferred potentials based on training examples. This inference
approach assumes a degree of similarity between the SGM image and those in the
training set. However, it does not guarantee the accurate reproduction of the
SGM signal, as no physical law governs the inference process. In contrast, ES
sought to find a potential that best matched the experimentally obtained SGM
signal based on physical laws, relying on a strict modeling assumption.

In summary, our investigation into background potential estimation techniques
for 2DEGs using SGM data elucidated the strengths and limitations of each
approach. While Pix2Pix GAN and CNN aim at modeling the general relationship
between the SGM signal and the potential without access to a conversion
function, ES require a physical assumption to guide the search. Therefore ES
can operate on a single stance at a time and fine-tune estimate the potential
reversely given the SGM signal. Pix2Pix GAN, while producing low-entropy
potentials, may not align with realistic scenarios. CNN and ES, by contrast,
demonstrated rougher potentials more consistent with expectations. Our
findings provide valuable insights into the selection of suitable techniques
for addressing inverse problems in this domain, offering guidance for future
research and applications in quantum electronics and nanotechnology.

## References

  *   * da Cunha et al. [2016] Cunha, C., Mineharu, M., Matsunaga, M., Matsumoto, N., Chuang, C., Ochiai, Y., Kim, G.-H., Watanabe, K., Taniguchi, T., Ferry, D.K., Aoki, N.: Conductance fluctuations in high mobility monolayer graphene: Nonergodicity, lack of determinism and chaotic behavior. Sci. Rep. 6, 33118 (2016)
  * Matsumoto et al. [2017] Matsumoto, N., Mineharu, M., Matsunaga, M., Chuang, C., Ochiai, Y., Oto, K., Kim, G.-H., Watanabe, K., Taniguchi, T., Ferry, D.K., Cunha, C.R., Aoki, N.: Shubnikov–de haas measurements on a high mobility monolayer graphene flake sandwiched between boron nitride sheets. Journal of Physics: Condensed Matter 29(22), 225301 (2017) https://doi.org/10.1088/1361-648X/aa6d36
  * Aoki et al. [2005] Aoki, N., Cunha, C.R., Akis, R., Ferry, D.K.: Scanning gate microscopy investigations on an InGaAs quantum point contact. Appl. Phys. Lett. 87, 223501 (2005)
  * v. Klitzing et al. [1980] Klitzing, K., Dorda, G., Pepper, M.: New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett. 45, 494–497 (1980) https://doi.org/10.1103/PhysRevLett.45.494
  * v Klitzing and Ebert [1985] Klitzing, K., Ebert, G.: Application of the quantum hall effect in metrology. Metrologia 21(1), 11 (1985) https://doi.org/10.1088/0026-1394/21/1/004
  * Takagaki and Ferry [1992] Takagaki, Y., Ferry, D.K.: Conductance of quantum point contacts in the presence of disorder. Phys. Rev. B 46, 15218 (1992)
  * Rashkeev et al. [1996] Rashkeev, S.N., Zagoskin, A.M., Shekhter, R.I., Wendim, G.: Soft disorder effects in the conductance quantization in quantum point contacts: indirect backscattering statistics. Solid State Comm. 97(4), 279–283 (1996)
  * Han et al. [1990] Han, J., Ferry, D.K., Newman, P.: Ultra-submicrometer-gate algaas/gaas hemts. IEEE Electron Device Letters 11(5), 209–211 (1990) https://doi.org/10.1109/55.55252
  * Aoki et al. [2006] Aoki, N., Cunha, C.R., Akis, R., Ferry, D.K., Ochiai, Y.: Sgm measurements on a disordered InGaAs QPC. In: Kono, J., Leotin, J. (eds.) Proceedings of the 12th International Conference on Narrow Gap Semiconductors, vol. 187, pp. 473–478. Taylor & Francis, Tolouse, France (2006)
  * Topinka et al. [2000] Topinka, M.A., LeRoy, B.J., Shaw, S.E.J., Heller, E.J., Westervelt, R.M., Maranowski, K.D., Gossard, A.C.: Imaging coherent electron flow from a quantum point contact. Science 289(5488), 2323–2326 (2000) https://doi.org/10.1126/science.289.5488.2323 https://www.science.org/doi/pdf/10.1126/science.289.5488.2323
  * Aoki et al. [2014] Aoki, N., Cunha, C.R., Akis, R., Ferry, D.K., Ochiai, Y.: Scanning gate imaging of a disordered quantum point contact. Journal of Physics: Condensed Matter 26(19), 193202 (2014) https://doi.org/10.1088/0953-8984/26/19/193202
  * Aoki et al. [2005] Aoki, N., Cunha, C.R., Akis, R., Ferry, D.K., Ochiai, Y.: Imaging of integer quantum hall edge state in a quantum point contact via scanning gate microscopy. Phys. Rev. B 72, 155327 (2005) https://doi.org/10.1103/PhysRevB.72.155327
  * Morikawa et al. [2015] Morikawa, S., Dou, Z., Wang, S.-W., Smith, C.G., Watanabe, K., Taniguchi, T., Masubuchi, S., Machida, T., Connolly, M.R.: Imaging ballistic carrier trajectories in graphene using scanning gate microscopy. Applied Physics Letters 107(24), 243102 (2015) https://doi.org/10.1063/1.4937473 https://pubs.aip.org/aip/apl/article-pdf/doi/10.1063/1.4937473/13177484/243102_1_online.pdf
  * Ferry et al. [2011] Ferry, D.K., Burke, A.M., Akis, R., Brunner, R., Day, T.E., Meisels, R., Kuchar, F., Bird, J.P., Bennett, B.R.: Open quantum dots—probing the quantum to classical transition. Semicond. Sci. Technol. 26, 043001 (2011)
  * Jalabert et al. [2010] Jalabert, R.A., Szewc, W., Tomsovic, S., Weinmann, D.: What is measured in the scanning gate microscopy of a quantum point contact? Physical Review Letters 105, 166802 (2010)
  * Ly et al. [2017] Ly, O., Jalabert, R., Tomsovic, S., Weinmann, D.: Partial local density of states from scanning gate microscopy. Phys. Rev. B 96, 125439 (2017)
  * Pala et al. [2008] Pala, M.G., Hackens, B., Martins, F., Sellier, H., Bayot, V., Huant, S., Ouisse, T.: Local density of states in mesoscopic samples from scanning gate microscopy. Phys. Rev. B 77, 125310 (2008)
  * Liu et al. [2015] Liu, P., Martins, F., Hackens, B., Desplanque, L., Wallart, X., Pala, M.G., Huant, S., Bayot, V., Sellier, H.: Formation of quantum dots in the potential fluctuations of ingaas heterostructures probed by scanning gate microscopy. Phys. Rev. B 91, 075313 (2015) https://doi.org/10.1103/PhysRevB.91.075313
  * Jalilian et al. [2011] Jalilian, R., Jauregui, L.A., Lopez, G., Tian, J., Roecker, C., Yazdanpanah, M.M., Cohn, R.W., Jovanovic, I., Chen, Y.P.: Scanning gate microscopy on graphene: charge inhomogeneity and extrinsic doping. Nanotechnology 22(29), 295705 (2011) https://doi.org/10.1088/0957-4484/22/29/295705
  * Pioda et al. [2007] Pioda, A., Kičin, S., Brunner, D., Ihn, T., Sigrist, M., Ensslin, K., Reinwald, M., Wegscheider, W.: Discrete charging of traps visualized by scanning gate experiments on a quantum point contact. Phys. Rev. B 75, 045433 (2007) https://doi.org/10.1103/PhysRevB.75.045433
  * van Wees et al. [1988] Wees, B.J., Houten, H., Beenakker, C.W.J., Williamson, J.G., Kouwenhoven, L.P., Marel, D., Foxon, C.T.: Quantized conductance of point contacts in a two-dimensional electron gas. Phys. Rev. Lett. 60, 848 (1988)
  * Aoki et al. [2007] Aoki, N., Cunha, C.R., Morimoto, T., Akis, R., Ferry, D.K., Ochiai, Y.: Scanning gate imaging of transport within an InGaAs QPC. AIP Conf. Proc. 893, 715 (2007)
  * da Cunha et al. [2006] da Cunha, C.R., Aoki, N., Akis, R., Ferry, D.K., Ochiai, Y.: Imaging of quantum interference patterns within a quantum point contact. Physica E: Low-dimensional Systems and Nanostructures 34(1), 682–685 (2006) https://doi.org/10.1016/j.physe.2006.03.060
  * da Cunha et al. [2006] Cunha, C.R., Aoki, N., Morimoto, T., Ochiai, Y., Akis, R., Ferry, D.K.: Imaging of quantum interference patterns within a quantum point contact. Appl. Phys. Lett. 89(24), 242109 (2006)
  * da Cunha et al. [2022] Cunha, C.R., Aoki, N., Ferry, D.K., Lai, Y.-C.: A method for finding the background potential of quantum devices from scanning gate microscopy data using machine learning. Machine Learning: Science and Technology 3(2), 025013 (2022) https://doi.org/10.1088/2632-2153/ac6ec7
  * Chua and Yang [1988] Chua, L.O., Yang, L.: Cellular neural networks: theory. IEEE Transactions on Circuits and Systems 35(10), 1257–1272 (1988) https://doi.org/10.1109/31.7600
  * Orzo [2000] Orzo, L.: Optimal cnn templates for deconvolution. In: Proceedings of the 2000 6th IEEE International Workshop on Cellular Neural Networks and Their Applications (CNNA 2000) (Cat. No.00TH8509), pp. 111–116 (2000). https://doi.org/10.1109/CNNA.2000.876830
  * da Cunha et al. [2023] da Cunha, C.R., Aoki, N., Ferry, D.K., Velasquez, A., Zhang, Y.: An investigation of the background potential in quantum constrictions using scanning gate microscopy and a swarming algorithm. Physica A: Statistical Mechanics and its Applications 614, 128550 (2023) https://doi.org/10.1016/j.physa.2023.128550
  * Percebois and Weinmann [2021] Percebois, G.J., Weinmann, D.: Deep neural networks for inverse problems in mesoscopic physics: Characterization of the disorder configuration from quantum transport properties. Phys. Rev. B 104, 075422 (2021) https://doi.org/10.1103/PhysRevB.104.075422
  * Nguyen et al. [2021] Nguyen, V., Orbell, S.B., Lennon, D.T., Moon, H., Vigneau, F., Camenzind, L.C., Yu, L., Zumbühl, D.M., Briggs, G.A.D., Osborne, M.A., Sejdinovic, D., Ares, N.: Deep reinforcement learning for efficient measurement of quantum devices. npj Quantum Information 7(1), 100 (2021) https://doi.org/10.1038/s41534-021-00434-x
  * Amâncio et al. [2021] Amâncio, M.A., Pinto, E.P., Matos, R.S., Nobre, F.X., Brito, W.R., Fonseca Filho, H.D.: Nanoscale morphology and fractal analysis of tio2 coatings on ito substrate by electrodeposition. Journal of Microscopy 282(2), 162–174 (2021) https://doi.org/10.1111/jmi.12990 https://onlinelibrary.wiley.com/doi/pdf/10.1111/jmi.12990
  * da Fonseca Filho et al. [2022] Fonseca Filho, H.D., Pires, M.P., Souza, P.L., Matos, R.S., Prioli, R.: Investigation of the morphological and fractal behavior at nanoscale of patterning lines by scratching in an atomic force microscope. Microscopy Research and Technique 85(3), 1046–1055 (2022) https://doi.org/10.1002/jemt.23974 https://analyticalsciencejournals.onlinelibrary.wiley.com/doi/pdf/10.1002/jemt.23974
  * Takagaki et al. [2000] Takagaki, Y., ElHassan, M., Shailos, A., Prasad, C., Bird, J.P., Ferry, D.K., Ploog, K.H., Lin, L.-H., Aoki, N., Ochiai, Y.: Magnetic-field-controlled electron dynamics in quantum cavities. Phys. Rev. B 62, 10255–10259 (2000) https://doi.org/10.1103/PhysRevB.62.10255
  * Ochiai et al. [2007] Ochiai, Y., Ujiie, Y., Yumoto, N., Harada, S., Morimoto, T., Aoki, N., Bird, J.P., Ferry, D.K.: Chaotic behavior in the magneto-resistance of quantum dot and quantum point contact. Progress of Theoretical Physics Supplement 166, 127–135 (2007) https://doi.org/10.1143/PTPS.166.127 https://academic.oup.com/ptps/article-pdf/doi/10.1143/PTPS.166.127/5467176/166-127.pdf
  * Zhang and Liu [2019] Zhang, X.W., Liu, Y.L.: Electronic transport and spatial current patterns of 2d electronic systems: A recursive green’s function method study. AIP Adv. 9, 115209 (2019)
  * Kulbachinskii et al. [2012] Kulbachinskii, V.A., Yuzeeva, N.A., Galiev, G.B., Klimov, E.A., Vasil’evskii, I.S., Khabibullin, R.A., Ponomarev, D.S.: Electron effective masses in an ingaas quantum well with inas and gaas inserts. Semicond. Sci. Technol. 27, 035021 (2012)
  * Isola et al. [2017] Isola, P., Zhu, J.-Y., Zhou, T., Efros, A.A.: Image-to-image translation with conditional adversarial networks. In: 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5967–5976 (2017). https://doi.org/10.1109/CVPR.2017.632
  * Shelhamer et al. [2017] Shelhamer, E., Long, J., Darrell, T.: Fully convolutional networks for semantic segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 39(4), 640–651 (2017) https://doi.org/10.1109/TPAMI.2016.2572683
  * da Cunha [2023] Cunha, C.R.: Machine Learning for the Physical Sciences: Fundamentals and Prototyping with Julia, 1st edn. CRC Press, Boca Raton, FL (2023)