# Interface Phonons and Polaron Effect in Quantum Wires

- A. Yu Maslov
^{1}Email author and - O. V. Proshina
^{1}

**5**:1744

**DOI: **10.1007/s11671-010-9704-0

© The Author(s) 2010

**Received: **22 June 2010

**Accepted: **13 July 2010

**Published: **11 August 2010

## Abstract

The theory of large radius polaron in the quantum wire is developed. The interaction of charge particles with interface optical phonons as well as with optical phonons localized in the quantum wire is taken into account. The interface phonon contribution is shown to be dominant for narrow quantum wires. The wave functions and polaron binding energy are found. It is determined that polaron binding energy depends on the electron mass inside the wire and on the polarization properties of the barrier material.

### Keywords

Quantum wire Electron–phonon interaction Interface phonons Polaron## Introduction

The electron–phonon interaction in semiconductor heterostructures is of greater interest in comparison to bulk materials. This is due to the fact that the quasi-particle space localization leads to the modifications of the energy spectrum. The all-important factor is the rise of new vibration branches of optical spectrum, namely, the interface optical phonon [1]. In addition, the intensity of electron–phonon interaction is changed. The interaction of charge particles with polar optical phonons should exhibit the most intensity. This interaction is of considerable importance in the understanding of the properties of heterostructures based on material with high ionicity. It can lead to self-consistent bond state of a charge particle and phonons, that is, the large radius polaron [2].

Currently, an investigation on the part played by interface phonons has attracted considerable interest in polaron state formation study. The heterostructures of different symmetry are under investigation. The contributions to polaron binding energy both of interface and of bulk optical phonons are the same value order in the quantum dots [3–5]. Taking into account, interface phonons are essential for quantitative analysis of the polaron states. It does not lead to new qualitative effects. Alternatively, the interface phonon role dominates in polaron binding energy for quantum well case [6, 7]. In response to this fact, the strong electron–phonon interaction can be realized in the quantum wells based on non-polar material with high iconicity barrier material. In addition, from the results, it follows that profound polaron effects should be expected, e.g., in the Si/SiO_{
2
} compounds. Although there are no polar optical phonons in the material of such quantum well, these may be produced at the heteroboundary. As a result, the strong interaction of charged particles with interface phonons becomes possible. Conversely, the essential depression of electron–phonon interaction is possible when the quantum well is made of polar material and for the barriers is taken non-polar material.

In recent years, varied technologies of semiconductor quantum wire growth with assorted barriers are progressing rapidly. The most success has been achieved for the quantum wires based on III–V compounds [8–12]. Some advances have been made in the formation of II–VI semiconductor wire structures [13, 14]. It is in these structures that the polaron states can arise. At the same time, no extended theoretical study of the polaron states in such structures is available. Proper allowance must be made for the interaction of charge particles with interface optical phonons for an understanding of this problem. In this paper, we develop a theory of polarons in the quantum wires, taking into account the interaction of charged particles with all branches of the optical phonon spectrum.

### Interface Phonons in the Quantum Wire

*i*(

*i*= 1, 2). The dielectric function is given by:

*I*

_{ m }is the

*m*-th order modified Bessel function of the first kind,

*K*

_{ m }is the

*m*-th order modified Bessel function of the second kind,

*k*is the wave vector,

*ρ*

_{0}is the quantum wire radius. The spectrum of interface phonons is determined by solution of Eq. 3. In Fig. 1 is shown the wave-vector dependence of the interface phonon frequencies. This dependence is calculated for the quantum wire based on

*CdSe*surrounded by

*ZnSe*barriers with

*m*= 0 in Eq. 3. The material parameters are taken from [17].

_{ n }(

*m*) is

*n*-th order root of the equation

*J*

_{ m }(μ) = 0,

*J*

_{ m }is the

*m*-th order Bessel function of the first kind. The interaction parameters

*α*

_{ m }(

*k*) have the form:

## The Polaron in the Quantum Wire

*ρ*

_{0}. Let the quantum wire be surrounded with compositionally identical barriers. In order to separate the effect of exactly dielectric irregularities, we assume that the potential well for electrons is rather deep, so that the penetration of the wave functions under the barrier can be disregarded. In this case, the interaction of charged particles with barrier phonons is weak. We write the Hamiltonian of the system as

*M*is the electron effective mass. If the interaction of an electron with polar optical phonons is strong, the polaron binding energy can be determined with the use of adiabatic approximation. In so doing, the electron subsystem is fast and phonon subsystem is slow. The adiabatic parameter here is the ratio of the quantum wire radius

*ρ*

_{0}to the polaron radius

*a*

_{0}:

*a*

_{0}is obtained below. The condition (Eq. 13) implies that the main contribution to the polaron binding energy is given by small values of the wave vector

*k*such that

*n*

^{(e)},

*m*

^{(e)}define not disturbed electron state in the quantum wire. In the case of total electron localization in the cylindrical quantum wire, the wave function has the form:

Here
is *n*^{(e)}-th root of *m*^{(e)}-th order Bessel function. The wave function
is to be obtained by solving self-consistent problem. In so doing, the total wave function from Eq. 15 is perceived to be normalized.

*M*is the electron mass inside the quantum wire. The form of phonon Hamiltonian from Eq. 11 remains unchanged. Averaged Hamiltonian of electron–phonon interaction can be written as:

*α*

_{ m }(

*k*) from Eq. 5 averaged with the electron wave function from formula (Eq. 15). We obtain average Hamiltonian

*ΔE*

_{ e }depends on electron size-quantization level number and on optical-phonon spectrum properties. These phonons are localized in the quantum wire and at the heteroboundary. After the procedure of angle averaging which is expressible in explicit form, we obtain this energy

*ΔE*

_{ e }as:

The energy (Eq. 22) is defined by the electron interaction with phonon modes correspond to *m* = 0 only. This equation (Eq. 22) contains the contribution to polaron energy for all size-quantization levels. This contribution is caused by the interaction of localized electron with confined and interface phonons. It can be used for numerical analysis of electron–phonon interaction characteristic properties. However, the electron energy and wave function can be obtained analytically on condition the unequality (Eq. 14) is satisfied.

## Results and Discussion

*k*which describe the value of electron–phonon interaction is of the order reciprocal to polaron radius

*a*

_{0}The logarithmic function changes weakly in this region. Therefore, we can consider with the same accuracy in parameter (14) that the energy is equal to:

*χ*(

*z*). It can be written as:

*χ*(

*z*):

*E*

_{ pol }:

*a*

_{0}is obtained by solving the transcendental equation. It has the form:

It is this quantity from Eq. 29 which contains the adiabatic parameter (Eq. 13). Substituting material parameters [17] into Eq. 29 for the quantum wire *ZnSe/CdSe/ZnSe* leads one to expect that the strong polaron effects for these structure should be observed when the quantum wire radius *ρ*_{0} < 40 *Å.*

It might be well to point out that both the polaron binding energy (Eq. 28) and polaron radius (Eq. 29) depend on effective electron mass inside the quantum wire and barrier dielectric properties. This clearly demonstrates the prevailing role of the interaction of an electron with interface optical phonons. The availability of the surface phonons leads to widening the range of materials in which the strong polaron effect should be expected. The strong electron–phonon interaction may exist near the interface between polar and non-polar materials. Among other things the significant electron–phonon interaction can result from the interface phonon influence in Si/SiO_{
2
} heterostructures.

The results obtained show that the intensity of electron–phonon interaction is determined significantly by interface optical phonons in narrow quantum wires corresponding to the condition (Eq. 13). These interface phonons are localized basically in the heteroboundary vicinity. And its field penetrates also into the barriers region. By this is meant that the interface phonons can produce the effective canal of excitation transfer in the structures with several quantum wires. Related ways should be allowed for the transport theory development in quantum nanostructures.

This work was supported by Russian Foundation for Basic Research, grant 09-02-00902-a and the program of Presidium of RAS “The Fundamental Study of Nanotechnologies and Nanomaterials” no. 27.

## Declarations

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

## Authors’ Affiliations

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