The Role of the Carrier Mass in Semiconductor Quantum Dots

TL;DR

This paper re-examines effective mass theory for semiconductor quantum dots, highlighting the impact of carrier mass discontinuity on electronic structure and surface charge density, and introduces a new scale to better predict energy shifts.

Contribution

It introduces a novel quantum scale, $\sigma$, accounting for mass discontinuity effects, improving the accuracy of effective mass theory in quantum dot modeling.

Findings

Surface charge density scales as 1/σ near the dot surface.

Ground state energy shift is weaker than quadratic in σ.

Model predictions agree with experimental valence band data.

Abstract

In the present work we undertake a re-examination of effective mass theory (EMT) for a semiconductor quantum dot. We take into account the fact that the effective mass ($m_i$) of the carrier inside the dot of radius R is distinct from the mass ($m_o$) in the dielectric coating surrounding the dot. The electronic structure of the quantum dot is determined in crucial ways by the mass discontinuity factor $\beta \equiv m_i/m_o$. In this connection we propose a novel quantum scale, $\sigma$, which is a dimensionless parameter proportional to $\beta^2 V_0 R^2$, where $V_0$ represents the barrier due to dielectric coating. The scale $\sigma$ represents a mass modified ``strength'' of the potential. We show both by numerical calculations and asymptotic analysis that the charge density near the nanocrystallite surface, $\rho (r = R)$, can be large and scales as $1/\sigma$. This fact suggests a significant role for the surface in an EMT based model. We also show that the upshift in the ground state energy is weaker than quadratic, unlike traditional EMT based calculations, and chart its dependence on the proposed scale $\sigma$. Finally, we demonstrate that calculations based on our model compare favorably with valence band photoemission data and with more elaborate theoretical calculations.