Journal of Nanotechnology

Volume 2016, Article ID 3794109, 8 pages

http://dx.doi.org/10.1155/2016/3794109

## Nanosensing Backed by the Uncertainty Principle

Department of Physics, North Carolina Central University, 1801 Fayetteville Street, Durham, NC 27707, USA

Received 4 December 2015; Accepted 5 January 2016

Academic Editor: Wen Zeng

Copyright © 2016 I. Filikhin 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

Possibility for a novel type of sensors for detecting nanosized substances (e.g., macromolecules or molecule clusters) through their effects on electron tunneling in a double nanoscale semiconductor heterostructure is discussed. We studied spectral distributions of localized/delocalized states of a single electron in a double quantum well (DQW) with relation to slight asymmetry perturbations. The asymmetry was modeled by modification of the dot shape and the confinement potential. Electron energy uncertainty is restricted by the differences between energy levels within the spectra of separated QWs. Hence, we established a direct relationship between the uncertainty of electron localization and the energy uncertainty. We have shown in various instances that a small violation of symmetry drastically affects the electron localization. These phenomena can be utilized to devise new sensing functionalities. The charge transport in such sensors is highly sensitive to minuscule symmetry violation caused by the detected substance. The detection of the electron localization constitutes the sensor signal.

#### 1. Introduction

A new generation of nanosensors is underway and is expected to revolutionize modern engineering fields, including bioengineering and drug delivery systems in nanomedicine, to name a few. Semiconductor heterostructures, such as quantum dots and rings, are of high interest for the development of these nanosensors, as well as many other predicted nanodevices. Of particular importance is the electron tunneling that occurs between the nanosized elements of such devices. Double quantum systems facilitate the study of tunneling related to barrier penetration effects in double well potentials [1]. When the elements of a quantum system get coupled, the energy barrier splitting of degenerate levels occurs, due to their common wave function. As a result, two nearly degenerate eigenstates are formed, which are a linear combination of the wave functions of the electron in isolated dots. Likewise, electron spectrum of double quantum dots formed by a set of quasidoublets is well described within the one-dimensional formalism given in [1] for double quantum wells. An example of theoretical analysis for double quantum dots has been presented in [2, 3]. These are supported by experimental studies of different aspects (spin effects, coupling distance, electron-phonon coupling, hybrid nanostructures, etc.) which are tightly connected to, and controlled by, charge tunneling in the DQDs. These aspects are extensively developing; see, for instance, [4–6]. Recently, we have reported in [7] that single electron localized/delocalized states and the tunneling in DQWs and DQDs are highly sensitive to the violation of reflection symmetry of the dots. It should be noted that this aspect is relevant to quantum dot reality, as fabrication technologies produce dot arrays with imperfect shape and distributions. These imperfections inherently induce a chaotic behavior in the QDs. It has been demonstrated that chaos strongly influences charge transport and other properties of QDs [8, 9]. The relevance of electron tunneling and chaos in real QD arrays is obvious for next generation nanodevices, for instance, future single molecule nanosensors, quantum computing devices, and advanced solar cells, to name a few.

In the present work we study the spectral distribution of electron localized/delocalized states and tunneling in DQWs. The average coordinate is used to characterize the localization of a single electron. We discuss the case of identical QWs constituting a DQW, as an example, when the uncertainty principle is manifested. Uncertainty of electron localization occurs when the difference of electron energies in the left and right QW is very small; that is the case of almost identical QWs. The symmetry violation caused by differences in the geometry and/or the confinement potentials in left and right QWs is thoroughly discussed.

#### 2. Effective Model for InAs/GaAs Quantum Dots

We consider quantum dots composed of InGaAs on a GaAs substrate. The fabrication of such kind of quantum dots is reported in [10, 11], for example. In practice, QDs have average lateral size and height of 41 and 1.6 nm, respectively, with variations within 23% and 28%, respectively. Hence, in our model the heterostructured QD dimensions were varied within these limits. The QDs were laterally distributed (two-dimensional array) for minimizing the computational recourses. The underlying quantum problem is modeled utilizing the -perturbation single subband approach, which is mathematically formulated by the Schrödinger equation as follows [12]:

Here is the single band -Hamiltonian operator , is the electron effective mass, which depends on the position of the electron, and is the band gap potential, which is null inside the QW, , and constant, equal to , outside the QW. The value of is defined by the conduction band offset for the bulk. The band gap potential for the conduction band is chosen as eV [13]. The magnitude of is calculated as , where and are the band gaps of QD and the substrate, respectively. The coefficient can be different for the conduction and valence bands. Here, the dimensionless constant values are taken from [14]. For the conduction band , and for the valence band . Using experimental values eV and eV, eV was obtained for the conduction band. The bulk effective masses of InAs and GaAs are and , respectively [15], where is the free electron mass. is the effective potential simulating the strain effect; it is attractive and acts inside the QW. The magnitude of the potential can be chosen [12] to reproduce experimental data. In presented work, the magnitude of for the conduction band is 0.21 eV. With this value, the results of the 8th band -calculations of [16] are well reproduced [17].

#### 3. Wave Function of Two Level System

Single electron spectrum of a two-level system is defined as a set of quasidoublets [1]. The one-dimensional wave functions of each quasidoublet can be expressed as follows [1]:whereRelation (2) shows that the wave function decomposed onto the basis set (). The parameter is a coupling coefficient of the quantum system elements. It depends on the wave function overlap for the “unperturbed states” and of the left and right quantum dots considered to be separated. is energy difference of separated QWs. It is defined for each energy level in the spectrum. An illustration of the band gap model with the effective potential is presented in Figure 1.