Mathematical Problems in Engineering

Volume 2018, Article ID 3095257, 10 pages

https://doi.org/10.1155/2018/3095257

## Robust Optimal Design of Quantum Electronic Devices

^{1}Facultad de Ciencias, Universidad Autónoma de San Luís Potosí, San Luís Potosí, SLP, Mexico^{2}Departamento de Matemática Aplicada y Estadística, Universidad Politécnica de Cartagena, Cartagena, Spain

Correspondence should be addressed to Francisco Periago; se.tcpu@ogairep.f

Received 21 December 2017; Accepted 1 March 2018; Published 5 April 2018

Academic Editor: Ben T. Nohara

Copyright © 2018 Ociel Morales et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider the optimal design of a sequence of quantum barriers, in order to manufacture an electronic device at the nanoscale such that the dependence of its transmission coefficient on the bias voltage is linear. The technique presented here is easily adaptable to other response characteristics. There are two distinguishing features of our approach. First, the transmission coefficient is determined using a semiclassical approximation, so we can explicitly compute the gradient of the objective function. Second, in contrast with earlier treatments, manufacturing uncertainties are incorporated in the model through random variables; the optimal design problem is formulated in a probabilistic setting and then solved using a stochastic collocation method. As a measure of robustness, a weighted sum of the expectation and the variance of a least-squares performance metric is considered. Several simulations illustrate the proposed technique, which shows an improvement in accuracy over 69% with respect to brute-force, Monte-Carlo-based methods.

#### 1. Introduction

Nanoelectronic devices operate with extremely low intensity currents. Under these circumstances, it is desirable to have at our disposal mechanisms to produce and control electronic currents with high precision. Electronic beams are relatively easy to produce, but their filtering to obtain nanocurrents with specified properties is much more difficult. A widely used approach consists in directing the beam on a sequence of quantum barriers with an externally adjustable bias voltage applied throughout the device, obtaining a nanocurrent by quantum tunneling. One expects to be able to control the response of the device in such a way that intensity depends, say, linearly on the applied bias. From an engineering perspective, this setting naturally leads to an optimal design problem: what must be the width and height of the layers composing the barriers, supposedly fixed in number, in order to achieve this linear response? (Of course, the problem is quite general, admitting a more complex relation between the external voltage and the current, but here we deal with the linear case just for simplicity.)

There should be no need to stress the importance of the solution to this problem from a practical point of view, but it must be noticed right from the start that a closed-form, analytic solution is impossible to obtain in most cases. The use of numerical computations at some stage is unavoidable, and this leads to the question of which method to use in order to obtain a good approximation to the solution. In [1] the nonconstant potential energy profile is approximated by piecewise constant potentials. Then, the propagation matrix method [2, 3] is applied to compute the transmission coefficient and, finally, the gradient of a least-squares-type objective function (which is required by the numerical solution method) is computed using the adjoint method. It is important to point out that different discretization processes, which are used to approximate objective functions and/or its gradients, may lead to quite different results. Moreover, it has been observed in some optimization problems [4] that first approximating a cost functional and then computing the gradient of the approximated one in general differ from approximating the gradient of the exact cost functional. That is, the schemes “first discretize, then optimize” and “first optimize, then discretize” do not commute in general. Also, as it will be showed later on in this paper that optimizing for the same cost functional via its exact gradient gives different solutions compared to using an approximate one.

Another issue, which cannot be obviated in a realistic mathematical model, is the presence of uncertainties. There are several sources of uncertainty in the problem under consideration, one of the most important regarding the influence on the computed optimal design being the manufacturing uncertainties. Due to the smallness of the currents involved, and the narrow width of the quantum barriers needed, methods such as MBE or CVD are used to grow thin layers (in many cases, monolayers) of some material to build the barriers, two of the preferred ones being and (see [5, 6], for example). These methods allow the growth of even monolayers, but the difficulties inherent in the manufacturing process at a semicommercial scale lead almost inevitably to inaccuracies that ultimately lead to a potential configuration that may be different from the numerically computed, optimal one [7]. For these reasons, the problem of computing an optimal quantum profile which, in addition, is robust against those uncertainties is an important one. If there is some statistical information about the uncertainties, then the machinery of probability theory gives a framework in which we can include uncertainties (by using random variables and/or random fields) and model objective functions (by means of expectation and variance operators, among other choices). In [8], this approach has been used for the case in which the cost functional only includes the averaging of a least-squares performance metric and by using the standard Monte-Carlo method for its numerical resolution.

The present work addresses the problem of the optimal design of a quantum potential profile (modeling a nanoelectronic device) in order to obtain a transmission coefficient linearly depending on an externally applied bias voltage, in the presence of manufacturing uncertainties. The transmission coefficient is explicitly computed by using a semiclassical approximation based on the WKB method. As a consequence, an explicit formula for the gradient of the cost functional is obtained. The existence of many local minima for the problem under consideration has been already reported in [1, 8]. Although at a first glance the use of a search method for a global minimum (such as a genetic algorithm) could be reasonable, the dimension of the parameter space, depending on the number of layers, increases very quickly, and the optimization techniques for finding global minima become extremely complex due to the nonconvex character of the problem. Thus, a truly random search technique does not seem adequate, so in this paper we restrict ourselves to a gradient-based minimization algorithm.

A weighted sum of expectation and variance of a random least-squares performance metric is considered as a measure of robustness. The inclusion of the second-order statistical moment in the cost functional amounts to a reduction the dispersion of the random transmission coefficient and hence an increase in the robustness of the optimal design. Since the resulting integrand in the cost functional is smooth with respect to a random parameter, a sparse grid stochastic collocation method is proposed for the numerical approximation of the involved integrals in the random domain. This method preserves the parallelizable character of Monte-Carlo sampling. However, in contrast to Monte-Carlo (which is computationally very expensive, of order , with the number of random sampling points), the stochastic collocation method shows an exponential convergence with respect to the number of sampling points. Several simulations illustrate the proposed approach, which shows itself to be an improvement in accuracy over brute-force, Monte-Carlo-based ones of about .

#### 2. Setting of the Optimal Design Problems

Consider a nanoscale semiconductor electronic device composed of layers occupying positions . The local potential energy at the th layer is denoted by , . For , the potential energy is denoted by and for it is . It is assumed that a single electron propagating from is incident at and that a voltage bias is applied across the device. A linear approximation of the underlying Poisson’s equation [1, 9] leads to the following expression for the resulting potential energy profile:where is the vector of local layer potentials in the device, and is the characteristic function of the interval , .

The transmission coefficient of the device is defined as the ratio of current density transmitted from the device at and the incident one at . As explained in detail in [1], may be expressed aswhere and , for values of the energy . The cases and may be treated in a similar way. Here is the effective mass of the electron, denotes the electron charge, is Planck’s constant, is the electron energy, and finally solves the following boundary-value problem for the Schrödinger equation: Here denotes the unit imaginary number and the amplitude of the transmitted wave at .

##### 2.1. Deterministic Optimal Design

The (deterministic) optimal design problem considered in this paper is formulated as the following nonlinear data-fitting problem: given a desired transmission coefficient , which is defined for , and lower, , and upper, , bounds for the local layer potentials, with ,where is given by (3) with and , .

##### 2.2. Optimal Design under Manufacturing Uncertainties

As indicated in the Introduction, it is very convenient to analyze the robustness of optimal designs with respect to manufacturing uncertainties. These may be modeled by adding a vector of random variablesto the vector of local layer potentials. Here represents a random event and thus is a small unknown error in manufacturing the local potential . Hence, the cost functional considered in problem (5) becomes a random variable given byIn order to obtain a design of the potential energy profile less sensitive with respect to fabrication of unknown fluctuations, the new cost functional is considered:with being a weighting parameter. Here and var denote the expectation and variance operators, respectively. Then, the robust optimization problem is formulated aswhere is given by (8).

#### 3. Solving the Optimal Design Problems

The numerical resolution of the optimal design problems stated in the preceding section requires the computation of the transmission coefficient (3) and, therefore, the resolution of the boundary-value problem (4). This problem may be numerically approximated by standard numerical methods such as finite differences or finite elements. Another approach is proposed in [1] where, after approximating the potential , as given by (1), by piecewise constant potentials, problem (4) is transformed into a two-dimensional linear nonautonomous difference equation. Here we propose a different approach based on the WKB method [10]. From the point of view of optimization, WKB method is very appealing since, within its range of validity, it provides an explicit form for the solution to (4). From this, explicit expressions for the gradients of the cost functionals considered in problems (5) and (9) are derived. In addition, having an explicit expression for allows us to prove its smoothness with respect to and . From this, both existence of solutions to (5) and (9) and designing a computationally very efficient numerical resolution method will be derived in this section.

We begin by explicitly computing the transmission coefficient (3) and then describe the numerical resolution methods for problems (5) and (9).

##### 3.1. Explicit Computation of Transmission Coefficient

###### 3.1.1. Case of a Single Potential Barrier

For the sake of clarity, consider first the case of a single potential barrier as illustrated in Figure 1.