Quantum Mechanical Analysis of Electron Transport in Nano-scale Semiconductors

Authors: Dr. Jane Doe, Prof. John Smith
Date: May 2026
Subject: Quantum Physics & Nanotechnology

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Abstract

This report presents a theoretical analysis of electron transport through quantum well structures. We derive the stationary Schrödinger equation and apply it to a one-dimensional potential barrier, demonstrating the quantum tunnelling effect.

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1. Theoretical Framework

The behavior of electrons in the semiconductor is governed by the time-independent Schrödinger equation:

$$-\frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}) + V(\mathbf{r})\psi(\mathbf{r}) = E\psi(\mathbf{r})$$

Where:

• $\hbar$ is the reduced Planck's constant ($1.0545718 \times 10^{-34} \text{ J}\cdot\text{s}$)
• $m$ is the effective mass of the electron
• $\psi(\mathbf{r})$ is the wave function
• $V(\mathbf{r})$ is the potential energy

1.1 Boundary Conditions

At the interface ($x=0$), the wave function and its first derivative must be continuous:

1. $\psi_{L}(0) = \psi_{R}(0)$
2. $\frac{d}{dx}\psi_{L}(0) = \frac{d}{dx}\psi_{R}(0)$

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2. Electrodynamic Modeling

To analyze the electromagnetic field interaction, we utilize Maxwell's Equations in a vacuum:

$$\begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0} \\ \nabla \cdot \mathbf{B} &= 0 \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} &= \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{aligned}$$

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3. Experimental Workflow

The simulation and fabrication process is illustrated in the diagram below:

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4. Results & Discussion

The transmission probability $T$ for a rectangular barrier of width $a$ and height $V_0$ ($E < V_0$) is given by:

$$T = \left[ 1 + \frac{V_0^2 \sinh^2(\kappa a)}{4E(V_0 - E)} \right]^{-1}$$

Where $\kappa = \sqrt{2m(V_0 - E)}/\hbar$.

Energy (eV)	Transmission $T$	Remarks
0.1	$1.2 \times 10^{-4}$	Low Tunnelling
0.5	$3.5 \times 10^{-2}$	Moderate
0.9	$0.85$	High Transmission

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5. Conclusion

The findings confirm that nano-scale transport is dominated by quantum effects, necessitating the use of the Schrödinger equation for accurate device modeling.