[Paper Review] Dramatic Impact of Dimensionality on the Electrostatics of PN Junctions

Hesameddin Ilatikhameneh, Tarek A. Ameen+4 more|arXiv (Cornell University)|Apr 18, 2017

Thermal properties of materials20 references3 citations

TL;DR

This paper proposes a novel analytic model for PN junction electrostatics in 1D, 2D, and 3D systems, accounting for neutral region effects and image charges. It reveals that dimensionality drastically increases depletion width sensitivity: while 3D junctions show square-root dependence on ε/N, 2D and 1D exhibit linear and exponential dependencies, respectively, enabling highly sensitive 1D sensors with strong bias-dependent response.

ABSTRACT

Low dimensional material systems provide a unique set of properties useful for solid-state devices. The building block of these devices is the PN junction. In this work, we present a dramatic difference in the electrostatics of PN junctions in lower dimensional systems, as against the well understood three dimensional systems. Reducing the dimensionality increases the depletion width significantly. We propose a novel method to derive analytic equations in 2D and 1D that considers the impact of neutral regions. The analytical results show an excellent match with both the experimental measurements and numerical simulations. The square root dependence of the depletion width on the ratio of dielectric constant and doping in 3D changes to a linear and exponential dependence for 2D and 1D respectively. This higher sensitivity of 1D PN junctions to its control parameters can be used towards new sensors.

Motivation & Objective

• To address the lack of validated analytic models for low-dimensional PN junctions that include neutral region effects and finite thickness.
• To quantify the dramatic change in electrostatic behavior—especially depletion width scaling—when moving from 3D to 2D and 1D systems.
• To develop and validate analytic equations that match both numerical simulations and experimental measurements across dimensions.
• To demonstrate the potential of 1D PN junctions as highly sensitive sensors due to their exponential dependence on control parameters.

Proposed method

• A new analytic approach is developed that incorporates image charges to model the electric field in neutral regions, improving upon the standard depletion approximation.
• Exact analytic expressions for the potential profile are derived in 2D and 1D using modified Green's functions and Bessel function integrals, valid for all thicknesses and radii.
• The model accounts for fringing fields and long-range charge screening via a thickness- and radius-dependent screening factor β, which transitions from 2 (thin) to 1 (thick, 3D-like).
• Approximate equations for depletion width are derived under the condition T ≪ W_D and R ≪ W_D, with error <10% when T < W_D/7 or R < W_D.
• The model is validated against finite-element numerical simulations in NEMO5 and experimental data from KPFM and optical measurements.
• The potential profile equations include exact solutions involving inverse hyperbolic and logarithmic terms, with β adjusting for finite thickness and screening.

Experimental results

Research questions

• RQ1 How does the depletion width of a PN junction scale with dielectric constant and doping in 1D and 2D systems compared to 3D?
• RQ2 What is the role of neutral region screening and image charges in low-dimensional PN junctions, and how can they be analytically modeled?
• RQ3 How does finite thickness (in 2D) or radius (in 1D) affect the potential profile and depletion width in low-dimensional junctions?
• RQ4 To what extent do analytic predictions match numerical simulations and experimental measurements in low-dimensional PN junctions?
• RQ5 Can the exponential dependence of 1D depletion width on bias and material parameters enable new sensing applications?

Key findings

• The depletion width in 1D PN junctions scales exponentially with ΔVε/N, leading to extreme sensitivity to bias and material parameters.
• In 2D, the depletion width shows a linear dependence on ΔVε/N, significantly increasing sensitivity compared to the 3D square-root scaling.
• The analytic model matches numerical simulations and experimental data with high accuracy, especially when finite thickness and screening effects are included.
• For very thin 2D flakes or narrow 1D nanowires, the depletion width exceeds the 3D value by orders of magnitude due to reduced screening.
• The screening factor β decreases from 2 (for zero thickness) to 1 (for thick, 3D-like junctions), quantifying the transition from 2D to 3D behavior.
• The model predicts that 1D junctions with small radii exhibit a strong exponential response to voltage bias, making them ideal for ultra-sensitive photodetectors and sensors.