## Abstract

The interface optical phonons arise near the hetero-interface of a quantum nanostructure. Moreover, its spectrum and dispersion laws may differ from ones for excitations arising in the bulk materials. The study of such excitations can give fundamentally new information about the optical and transport properties of nanostructures. The interaction of charged particles with polar optical phonons can lead to the large radius polaron creation in the materials with high ionicity. This chapter deals with the results of our theoretical investigations of the polaron states in quantum wells, quantum wires, and quantum dots. The charged particle and exciton interaction with both bulk and interface optical phonons are taken into account. The original method has been developed taking into consideration an interface phonon influence. The enhancement conditions are found for both strong and weak interactions. It is established that the barrier material dielectric properties give a decisive contribution to the polaron binding energy value for strong electron-phonon interaction. The manifestation of strong polaron effects is a pronounced demonstration of the interface optical phonon influence on optical and transport properties of nanostructures.

### Keywords

- interface optical phonons
- quantum well
- quantum wire
- quantum dot
- electron-phonon interaction
- polaron

## 1. Introduction

The electron-phonon interaction proves to be rather weak for most of the phonon branches. Such interaction can be taken into account in the framework of perturbation theory. The interaction of charged particles with polar optical phonons turns out to be fundamentally different [1]. The effective constant of the electron-phonon interaction may exceed unity in materials with high ionicity. Moreover, the formation of a new type of elementary excitations, which is a bound state of charged particles and polar optical phonons, is possible even in bulk materials. This is the so-called large-radius polaron. The conditions for the appearance of such polaron are most favorable in quantum-dimensional structures. First, the additional branches of polar optical phonons, which are the interface phonons, appear in such structures. Second, the effective interaction of charged particles with polar optical phonons increases with the decreasing structure dimensionality. This significantly expands the range of materials for the nanostructure design where the large-radius polaron formation is possible. The large-radius polaron appearance significantly changes the optical and transport properties of nanostructures. Even the manifestation of polaron superconductivity may take place [2]. Available theoretical studies of large radius polaron in quantum nanostructures consider the charged particle interaction with only one polar phonon mode [2, 3, 4, 5, 6, 7, 8, 9, 10]. This approach seems to be inconsistent for us. The phonon spectrum modification turns out to be very significant in quantum nanostructures. Therefore, it is necessary to take into account the interaction with all phonon branches in the large-radius polaron investigations.

In this chapter, conditions of strong electron-phonon interaction observation are investigated theoretically in the quantum well, quantum wire, and quantum dot. Particular attention is paid to the theory of charged particle interaction with interface optical phonons playing a decisive role in quantum wells and quantum wires. The contribution of interface phonons to the interaction energy value turns out to be comparable with that of bulk phonons in the quantum dot case. The conditions necessary for the strong electron-phonon interaction are obtained for all types of nanostructures. Analytic expressions for the polaron binding energy are found for nanostructures considered. In some cases, the results for the weak electron-phonon interaction are discussed. This helps to understand better the interaction in a region of intermediate values of the coupling constant where obtaining the analytical result is impossible.

The total Hamiltonian of the system is given by:

where the electron Hamiltonian

Here

In this case, the contribution of the electron-phonon interaction can be taken into account by perturbation theory.

The adiabatic approximation turned out to be an effective method for solving the problem in the case of a strong electron-phonon interaction. Within the framework of this approach, the motion of the charged particles (electrons and holes) is considered to be fast, and vibrations of the atoms of the crystal lattice are supposed to be slow. The Hamiltonian from Eq. (1) can be averaged over the wave function of fast motion

Eq. (4) contains the energy of charged particles
_{,} which is the functional of the wave function in a general case. Here

where the index * r* denotes different phonon branches;

*is the quantum number that takes various values for the different nanostructures;*n

Here the interaction parameters

As a result, we get

As follows from Eq. (8), the spectrum of all phonon branches remains unchanged in the adiabatic approximation. The value

## 2. Symmetric quantum well

In general, for the case of a quantum well, the interaction of charged particles with the phonons of the well, barriers, and interface phonons must be taken into account. Let us consider the case of complete localization of charged particles within a quantum well. In this case, the interaction of such particles with barrier material phonons can be neglected. Nevertheless, the effect of barriers is very important. This is determined by the structure and properties of interface phonon spectrum. To describe the properties of interface phonons, we will use the continuum model proposed in [11]. The spectrum of the symmetric mode of interface phonons is determined from the solution of the following equation:

where * L* is the quantum well characteristic size;

*is two-dimensional wave vector;*q

Here
* L* that is less than the polaron radius

The exact definition of the polaron radius

When the inequality from Eq. (11) is satisfied, the electron wave function

where

The electron-phonon interaction parameters

Here

where

It is this quantity from Eq. (11) on which the adiabatic approximation used in our work is based. In the next order in the parameter from Eq. (11), some corrections to polaron binding energy Eq. (14) appear. These corrections can be expressed in terms of the dimensionless wave function of two-dimensional polaron

where

A similar consideration can be repeated for a hole polaron. The main contribution to the binding energy of a hole polaron is determined by an expression analogous to Eq. (14). It looks like this:

Usually, for semiconductor materials, the hole mass

The condition from Eq. (18) plays an important role in the study of a polaron exciton. The interaction of an exciton with optical phonons has a number of additional features. Polarization of the medium, created by an electron and a hole, partially compensates each other. The degree of this compensation essentially depends on the ratio of the radii of the electron and hole polarons,

Here

Eqs. (19) and (20) are valid for narrow quantum wells, the width of which satisfies the inequality

The possibility of strong coupling of an exciton with polar optical phonons depends on the relationship between

When the opposite relationship is satisfied, that is

The second contribution in Eq. (23) is small, compared to the first one in the parameter

For most II–VI compounds, the exciton radius
* Å*. The electron polaron radius

*and hole polaron one*Å,

*. Therefore, the strong exciton-phonon interaction condition from Eq. (22) can be satisfied. This means that the quasi-two-dimensional polaron formation is possible in sufficiently narrow quantum wells of width*Å

*.*Å

The heterovalent quantum wells based on II–VI/III–V materials are more promising target for the experimental study of polaron effects in the case of strong electron-phonon interaction. For such structures, growth technologies have been developing successfully in recent times [17]. In the III–V compounds, effective masses of quantum well carriers are small. The optical dielectric function of the barriers based on II–VI materials is also rather small. Thus, given above values of exciton and polaron radii increase by 2 − 3 times. Hence, a quasi-two-dimensional polaron in heterovalent quantum wells can be observed for the well widths
* Å*. Quantum wells of more complex configuration (e.g., I–VII/III–V) can also become a promising object for the polaron study when strong electron-phonon interaction takes place.

## 3. Cylindrical quantum wire

In the quantum wires under consideration, the spectrum of interface phonons depends on the one-dimensional wave vector

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

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

*, surrounded by*CdSe

*barriers for*ZnSe

The adiabatic parameter of this problem is the ratio between the quantum wire radius

Below, an exact analytic expression is obtained for determining the polaron radius. The inequality Eq. (25) means that the main contribution to the binding energy of a polaron is determined by the wave vector values which are small as:

According to Eq. (25), the electron wave function for n-th size quantization level can be written as:

Here

where

Generally, the value of polaron binding energy

Here

The interaction of an electron with interface phonon mode of the frequency close to barrier frequency

Eq. (30) contains the optical dielectric function of the barriers

The electron polaron binding energy as the functional of unknown yet wave function

The solution of nonlinear Eq. (32) has the form:

By substituting the wave function from Eq. (33) into Eq. (32), we obtain the polaron binding energy as:

Thus, the polaron radius

The substitution of the material parameters [18] for * ZnSe/CdSe/ZnSe* quantum wire into Eq. (35) leads one to expect that the strong polaron effects should be observed at a wire radius

*.*Å

The condition for polaron exciton appearance in a quantum wire is analogous to that considered above for a quantum well, Eq. (18). The basic requirement is a significant difference between the hole and the electron masses. If the radius of a quantum wire corresponds to the conditions from Eqs. (21) and (22), a complete compensation of the contributions from the electron and hole does not occur, and a strong electron-phonon interaction is possible.

## 4. Spherical quantum dot

In this chapter, we study the structures in which the quantum dot and matrix materials have different phonon modes and its dielectric functions are described by Eq. (10). We have used the approximation presented in [19], where the interface phonon spectrum is described by the following equation:

where

For our problem, the coefficients

where

In this case, the energy of electron size quantization level polaron shift has the form:

As follows from Eq. (43), the bulk and interface phonon contributions to the polaron binding energy are summed. It will be seen from the further consideration that the interface phonon contribution can exceed the surface phonon one under certain conditions. The results obtained make it possible to calculate the polaron shifts for any size quantization level. Consider a polaron shift for a particle with a spherical wave function. For example, it could be an electron in the ground state. The polaron shift can be obtained analytically [20] and is equal to:

It follows from Eq. (44) that taking into account matrix polarization leads to an increase in the polaron effect. It should be noted that there is a noticeable polaron shift even for quantum dots based on a nonpolar material. This is due to the presence of interface phonons that create the polarization in surrounding matrix. Note also that for the quantum dot case, the contributions of charged particle interaction with bulk and interface phonons are of the same order of magnitude in the adiabatic parameter.

This is the main difference between this problem and the quantum well and quantum wire considered earlier. For these structures, the largest contribution in the adiabatic parameters from Eq. (11) and Eq. (25) is caused by the interface phonons. The inequality (45) is satisfied, for example, for * CdSe* quantum dots in a

*matrix when the dot radius*ZnSe

*.*Å

Another significant feature of the polaron in quantum dots is a significant suppression of the polaron exciton state. The exciton polaron shift turns out to be zero for the localization of the electron and hole with wave functions of the same symmetry inside the quantum dot. The nonzero interaction of an exciton with polar optical phonons arises for different symmetries of the electron and hole wave functions only. This is possible if the quantum dot is made of a material where interband transitions are forbidden (e.g., _{2}) or if the valence band complex spectrum is taken into account. The latter is typical for most III–V and II–VI semiconductor compounds. It is shown in [20] that taking into account the valence band degeneracy in the Luttinger Hamiltonian model leads to a noticeable difference between the polaron shift for the electron and hole. In this case, the exciton-phonon interaction can turn out to be strong at the quantum dot.

## 5. Weak electron: phonon interaction

The interaction of charge particles with polar optical phonons can be weak in nanostructures based on materials with low ionicity. When the condition Eq. (3) is satisfied, the electron-phonon interaction described by the Hamiltonian

The frequency from Eq. (46) is

Eqs. (46) and (47) are similar to the known results from two-dimensional polaron theory [14]. However, the effective coupling constant is equal to:

It is seen from Eq. (48) that, just as in the case of a strong-coupling polaron, the effective electron-phonon interaction constant is determined by the effective electron mass inside the quantum well and the barrier material dielectric properties. This value is analogous to the Frohlich constant, but it is not a characteristic of any particular material and is determined by the quantum well properties. In a specific approximation, when the condition
* ZnO-ZnMgO* quantum well, obtained in [23]. In estimating the effective mass, the authors of [23] have used the Frohlich constant for

*instead of the effective constant from Eq. (48). Using the effective constants of Eq. (48) greatly improves the agreement between theory and experimental data.*ZnO

## 6. Conclusions

It is shown that the interface phonons play an important role in the polaron state formation in quantum nanostructures. In quantum wells and quantum wires, the polaron binding energy is determined mainly by the interaction of charged particles with interface optical phonons. In the quantum dots, the contribution due to the interaction with interface phonons is additive with the energy of interaction with bulk phonons. Moreover, for nanostructures based on the same materials, the polaron binding energy increases with the structure dimensionality reduction.

Thus, the results obtained show that the barrier material ionicity degree plays a fundamental role for forming the large radius polarons in quantum wells and quantum wires. Meanwhile, the quantum well itself can be based on low ionicity material. The interaction of charged particles with interface optical phonons is the reason that the polaron effects are enhanced significantly.

The condition for observing a polaron exciton is the essential difference between the electron and hole effective masses. In this case, only partial compensation of the phonon interaction with charged particles occurs, and the achievement of a strong electron-phonon interaction is possible.

Thus, the appearance of strong polaron effects is a clear demonstration of the interface phonon influence on optical and transport properties of nanostructures.

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