Low-dimensional semiconductor systems used in modern nanodevices for generating and detecting electromagnetic radiation require an accurate and efficient approach to determining their geometric and functional parameters. This paper proposes a methodology that allows the analysis of large datasets obtained from experimental and theoretical studies of nanostructures in order to identify precise parameters of their spatial confinement. Automation of data processing and nanosystem parameter identification is implemented using machine learning methods and a convolutional neural network, enabling efficient and reliable characterization of nanosystem parameters of arbitrary symmetry. The developed software tool will significantly streamline the work of specialists in nanotechnology and nanostructured material synthesis.
deviations of measured quantities from their model values. The main approaches to determining
the geometric parameters of nanostructures from experimentally measured current or conductivity
values are inverse problems of quantum scattering theory. However, although many different
numerical methods of this kind exist even for single quantum wells, they usually do not yield
unambiguous results [
The relevance of this problem lies in the fact that an approach that allows one to establish the geometric confinement of nanostructures based on experimental measurements would allow one to significantly optimize the operation of existing nanodevices. In this paper, we aim to solve this problem by applying machine learning methods to the analysis of large data arrays obtained from experimental studies and measurements of typical values of electron current and conductivity in nanosystems. Using these data and generalizing them with a developed mathematical model describing the electronic states in nanostructures, a convolutional neural network will be trained. When working with new experimental data, the developed neural network will unambiguously and reliably determine the geometric parameters, thus solving the general problem.
Currently, there is relatively little specialized software designed for modeling the properties of
nanostructures. However, the following software systems, which can be classified by their level of
abstraction, should be highlighted. In particular, software systems that use quantum-mechanical
methods to describe interactions at the atomic level are worth mentioning: DFT, VASP, and
Quantum ESPRESSO [
Using the aforementioned software, and as further illustrated in Fig. 1, it is possible to model the
fundamental properties of nanostructures with a small number of layers. This allows a qualitative
assessment of the spectral parameters of electrons in nanostructures and the calculation of the
potential diagram of such a nanosystem. However, it should be concluded that none of the
indicated software systems even approximately possesses the functionality required to solve
inverse problems of identifying the geometric confinement of nanostructures, despite their fairly
rich functionality for manipulating the input parameters of nanostructures and the materials from
which they are formed. In light of the existing challenges associated with this topic, two relevant
studies [
We are beginning to address the stated problem of establishing a general approach, according to which we are working with input data networks that characterize the electronic spectrum , = 1, 2, .. in a nanostructure. We cannot directly apply datasets obtained from experimental spectral measurements, as such datasets do not directly correspond to the geometric structure , = / ( – is the characteristic size of the nanostructure) of nanostructures. Therefore, we link spectral datasets to basic mathematical models that most accurately describe the spectral characteristics of electrons in such systems. where () is the geometric confinement to be identified, is the electronic spectrum, is the effective mass of the electron in the given structure.
Next, we present the Schrödinger equations and boundary conditions in discretized form on a one-dimensional uniform grid: ℏ2 −α ( Ψi+1−2 Ψi+ Ψi−1)+U i Ψi− E Ψi=0 , α = 2 mh2 , i=1 , ... , N −1 ; Ψ0=Ψa , Ψ N =Ψb , Ψ1−Ψ0 =g0 , Ψ N −Ψ N−1 =gN ,
h h Ψ0=Ψ N , Ψ−1=Ψ N−1 .
As a result, the spectral problem is reduced to a matrix equation describing the relationship between the energy spectrum and the geometric confinement of the nanostructure on a discrete grid: {∂ Ψ( p)( x , t )
| ∂ x x=xp i ℏ ∂ Ψ ( x , t ) ∂ t =[− ℏ2 ∂2 +U ( z )] Ψ ( x , t );
2 m ∂ x2 Ψ( p)( x , t )|x=xp= Ψ( p+1)( x , t )|x=xp , = ∂ Ψ( p+1)( x , t ) ∂ x
.
, i=1 , ... , N −1 , n≥0 ; Ψ0=Ψna , ΨnN =Ψb ,
n n {Ψ1−h Ψ0 =g0n , ΨnN −hΨ N−1 =gnN ,
n n n Ψ0=Ψ N , Ψ−n1=Ψ N−1 .
n n n H Ψ = E Ψ ,
Ψ =( Ψ1 , ... , Ψ N−1)T , 0 ...
0 −α 2α +U N−1 ).
We will use the resulting table dependencies = () as input datasets for training the neural network. Another mathematical model we will use takes into account the kinetics of processes in nanostructures. It is based on the full Schrödinger equation and its limiting conditions. This mathematical model is represented as follows:
Approximating this mathematical model on a two-dimensional discrete variable grid (, ), we will have the following Crank-Nicholson type schemes for a multilayer system: i ℏ Ψin+Δ1 −tΨin =−α2 [( Ψin++11−2 Ψin+1+ Ψin−1)+( Ψin+1−2 Ψin+ Ψin−+11)]+U i Ψin+1+ Ψin , 2 (2) (3) (4) (5) Representing the result in a convenient matrix form, we will obtain:
A Ψn+1=B Ψn , Ψn=( Ψ1n , ... , ΨnN−1)T , 1−2λ (2α +U 1) 2λ α 0 ... 0
−λ α 0 1+λ (2α +U 2) −λ α −λ α ⋱ ⋱ ⋱ ⋮ 0 2λ α 0 1−2λ (2α +U 2) 2λ α 2λ α ⋱ ⋱ ⋱ ⋮ 0 ⋮ ⋱ ⋱ 1+λ (2α +U N−2) −λ α ⋮ ⋱ ⋱ 1−2λ (2α +U N−2) 2λ α 0 ...
0 −λ α 0 ...
0 −2λ α 1+λ (2α +U N−1) 1−2λ (2α +U N−1) ), ). (6) (7)
By now substituting into this difference scheme the data obtained from experimental measurements of the energy spectrum (), we again obtain datasets (in form = (, )) convenient for use in training a neural network.
The data obtained by substituting the experimental results into models (5) and (6) underwent minimal preprocessing. The primary goal of the data processing was to organize the values into columns: the first contained the coordinates of the difference scheme corresponding to the partition of the structure’s localization region, while the remaining columns contained the spectral values . An example of such tabular data and its visualization is presented in Figures 2a and 2b, respectively.
H (U )= E Ψ ,
We begin working with the input data directly based on model (5), (6). We will accept the input data as vectors, which leads to the following matrix equation:
where () is a tridiagonal Hamiltonian that depends on the vector of potentials = (1, 2, ...,−1). In our case, the experimental spectral data is a set of energy levels En. As a result, we need to construct a mapping as follows: or is it the same as a dependency relationship: f NN : {En , ti }→ {xi }, {U i }iN=−11=U i ({xi }) . (8) (9)
To preprocess and normalize the data, we used the Wolfram Mathematica 13.2 environment, which implements this as follows: data = Import["spectra_table.csv"]; xGrid = data[[2 ;; , 1]]; Edata = data[[2 ;; , 2 ;;]]; scaledE = Standardize /@ Transpose[Edata]; scaledE = Transpose[scaledE]; scaledX = (xGrid - Mean[xGrid])/StandardDeviation[xGrid]; dataset = MapThread[Association["Input" -> #1, "Target" -> #2] &, {scaledE, scaledX}]; {train, val} = TakeDrop[dataset, Round[0.8 Length[dataset]]]; where in our case scaledXList is a list of corresponding labels for training the neural network: coordinates and -vectors for each dadaset. The neural network architecture was designed to use the vector itself to reconstruct the vectors and .
For this, we use a multi-layer perceptron (MLP) (NetChain / NetGraph). In Wolfram
Mathematica, we implemented this as follows:
inputDim = Length[scaledE[[
LinearLayer[256], ElementwiseLayer[Ramp], (* ReLU *) DropoutLayer[0.2], LinearLayer[128], ElementwiseLayer[Tanh],
LinearLayer[outputDim] }, "Input" -> inputDim ]; When training the neural network, the process was controlled by a loss function in the form: In Wolfram Mathematica, this parameter was taken into account automatically using the NetTrain directive:
LMSE= 1 ∑M|x(jtrue)− x( pred)|2 .
j M j=1 (10) trainedNet = NetTrain[net, train, ValidationSet -> val, TrainingProgressReporting -> "Panel", BatchSize -> 32, MaxTrainingRounds -> 200, TargetDevice -> "GPU", LossFunction -> MeanSquaredLossLayer[] ];
After the initial good approximation was made, the predicted values of and were updated by minimizing the discrepancy between the experimental energy data and the eigenvalues () obtained in the model (5), (6), where the objective function is as follows: where is the weighting coefficient and the regularizer has the form:
K J (U )=∑ [λ j H (U )− Eejxp ]2+ γ R (U ) , j=1
K R (U )=∑ (U j+1+U j)2 .
j=1 (11) (12)
Now the implementation of the construction of a tridiagonal matrix, the formation of the objective function and minimization using a combined scheme looks as follows: makeH[U_List, alpha_] := Module[{n = Length[U]}, SparseArray[{
Band[{1, 1}] -> 2 alpha + U, Band[{1, 2}] -> -alpha,
Band[{2, 1}] -> -alpha
}, {n, n}]
];
lossFunc[Uvec_?VectorQ] := Module[{H, eig},
H = makeH[Uvec, alphaVal];
eig = Sort[Eigenvalues[H]];
Total[(eig[[1 ;; K]] - Eexp)^2] + gamma Total[(Differences[Uvec])^2]
];
Ustart = NetPredict[trainedNet, singleEInput];
res = FindMinimum[
lossFunc[U],
{U, Ustart, StepMonitor :> Null},
MaxIterations -> 200
];
Uopt = U /. res[[
Figure 3 schematically presents the principle of training a neural network using the vector representation of the spectrum as a function of the discrete grid coordinates .
Let us first focus on the training process of the neural network used to identify the parameters of
the nanosystems in all the samples studied.
network was directly applied to the task of identifying precise nanostructure parameters from
experimental data based on the stationary Schrödinger equation in difference form. This model was
trained using experimentally derived data from published experimental papers [
A key factor in selecting the datasets used to train the neural network was their consistency with experimental results obtained for various nanostructures possessing diverse physic-chemical parameters and geometric confinement. The only unifying feature among them is that all of these structures are two-dimensional in terms of geometry.
To test the performance of the neural network trained on experimental datasets, experimental
measurement data were used that did not overlap with the input data employed for training. Key
factors for evaluating the neural network’s performance included its speed and its ability to
accurately identify parameters compared to other available models. The reference model, currently
the standard, is based on self-consistent solutions to the Schrödinger-Poisson system of equations
(Schrödinger-Poisson solver). The theoretical foundation of this method was, for example,
developed in our previous paper [
To demonstrate the performance of the trained neural network in identifying nanostructure
parameters, five samples composed of different materials were randomly selected. The
experimental studies from which the verification data were obtained correspond to papers [
Further, Figures 4a and 4b present the results of calculating the full spatial confinement for two different nanostructure samples using both the neural network and the direct computational method. Moreover, both samples were fabricated from entirely different types of semiconductors: arsenide (a) and nitride (b). As shown, both methods produce virtually identical results; however, the neural network-based approach, due to its speed, is more efficient and streamlines the computational process. This represents a significant advantage for direct applications in nanoelectronics, as it considerably facilitates the automation of computational work with samples.
An approach to the automated identification of geometric and spatial confinement in low-dimensional structures using machine learning methods is proposed.
The developed neural network, trained on input experimental data corresponding to measured electronic spectra and aligned with a difference scheme based on a mathematical model of the low-dimensional system, enables efficient identification of nanostructure shapes with arbitrary input physical parameters. Its performance is nearly one order of magnitude higher than that of the Schrödinger-Poisson model.
The obtained results demonstrate that the proposed methodology provides efficient processing of experimental samples and reliable identification of their parameters. These results show promise for the development of software with broad applications in specialized areas of nanotechnology and electronics.
The author’s have not employed any Generative AI tools.