Non-perturbative macroscopic theory of interfaces with discontinuous dielectric constant

TL;DR

This work develops a non-perturbative macroscopic theory for interfaces with discontinuous dielectric constants by introducing general boundary conditions (GBC) at the interface, governed by a single parameter $W$ that encodes short-range interfacial physics and ensures current conservation. By solving the Schrödinger equation exactly on both sides of the interface and enforcing the GBC, the authors derive a comprehensive framework for electron scattering, resonances, and surface states across energy regimes $E<0$, $0<E<V$, and $E>V$, with the central object being the energy-dependent function $ ext{Sigma}(E)$ that determines spectra and resonances via $ ext{Sigma}(E)=W$. The approach yields novel predictions including perfect transmission at certain energies, resonance widths tied to the mirror-force renormalization, and strong coupling between surface states and quantum-well levels under dielectric confinement, with concrete applications to photoemission and surface quantum wells, and potential extension to high-symmetry geometries. Overall, the theory provides a robust, exact, and non-perturbative description of dielectric confinement effects that are essential for understanding transport and optical properties of nanostructures at interfaces.

Abstract

Discontinuity of dielectric constants at the interface is a common feature of all nanostructures and semiconductor heterostructures. Near such interfaces, a charged particle creates a singular self-interaction potential which may be attributed to interaction with fictitious mirror charges. The singularity of this interaction at the interface presents an obstruction to a perturbative approach. In several limiting cases, this problem can be avoided by zeroing out the carrier wave function at the interface. In this paper, we have developed a non-perturbative theory which gives a self-consistent description of carrier propagation through an interface with a dielectric discontinuity. It is based on conservation of the current density propagating through the interface, and it is formulated in terms of general boundary conditions (GBC) for the wave function at the interface with a single phenomenological parameter W. For these GBC, we find exact solutions of the Schrödinger equation near the interface and the carrier energy spectrum including resonances. Using these results, we describe the photo effect at the semiconductor/vacuum interface and the energy spectrum of quantum wells (QWs) at the interface with the vacuum or a high-k dielectric. For a surface of liquid helium, we estimate the parameter W, and match the resulting electron spectrum with the existing experimental data and theoretical analysis.

Figures (14)

• Figure 1: The heterostructure formed between the semiconductor and the vacuum (a) or the dielectric matrix (b). The dashed line shows the step-like potential due to the band offset $V = 1$ eV created by a semiconductor/vacuum (a) or by a semiconductor/dielectric matrix (b) interface. Solid blue lines show the sum of the step-like potential and the self-interacting potential, $U_{\rm self}(z)$, created by mirror charge. The energy of the surface resonance levels (see Eq. \ref{['eq:En']}) created by the attractive potential $U_{\rm self}(z)$ in the vacuum or in the dielectric are shown by horizontal orange lines.
• Figure 2: Function $\sigma(x)$ for (a) real ($x=\alpha$) and (b) imaginary ($x=i\beta$) arguments. Vertical gray lines show poles of $\sigma(x)$.
• Figure 3: Dependence of $\Sigma(E)$ and $\Sigma_0(E)$ on the energy $E$, calculated for barrier height $V=1$ eV. Vertical gray lines show poles $E = E_n$. Panels (a) and (b) differ in vertical scale only. The range $E<0$ where the surface bound states might exist is marked by the rosa color; in the white region $0<E<V$ the under barrier reflection takes place; in the green region $E>V$ the scattering and reflection above the barrier occurs.
• Figure 4: (a, b). Calculation of the perfect transmission energy $E_{\rm tr}>V$ and surface parameter $W_{\rm tr}$ in structures with dielectric constants $\varepsilon_2=1$ and $\varepsilon_1=10$, and electron effective masses $m_2=m_0$, $m_1=0.2 m_0$. (c, d) Dependencies of $E_{\rm tr}$ and $W_{\rm tr}$ on $\varepsilon_1$ for $\varepsilon_2=1$, $m_2=m_0$ and different values of $m_1$. All calculations are conducted for the barrier height $V=1$ eV.
• Figure 5: (a, c) Dependence of the transmission coefficient $T_{\rm tr}$ on energy $E-V$ for the surface parameter $W=0$. (b, d) Dependence of the averaged transmission efficiency $T^{\rm eff}_{\rm tr}$ on the surface parameter $W$. In (a, c) the filled area shows the energy interval $V<E<V+k_{\rm B}T$ with $T=300$ K. The fill up the area in the (b) and (d) panels shows the room temperature distribution of the transmission coefficient $T_{\rm tr}$ in the same energy interval. The calculations are conducted for barrier height $V=1$ eV (a, b) and $V=0.2$ eV (c, d) and several dielectric constants $\varepsilon_1$ and $\varepsilon_2=1$.