## Abstract

The energy levels of bound states of an electron in a quantum well with BenDaniel-Duke boundary condition are studied. Analytic, explicit, simple, and accurate formulae have been obtained for the ground state and the first excited state. In our approach, the exact, transcendental eigenvalues equations were replaced with approximate, tractable, algebraic equations, using algebraic approximations for some trigonometric functions. Our method can be applied to both type I and type II semiconductors and easily extended to quantum dots. The same approach was used for the semi-quantitative analyze of two toy models of Janus nanorods.

### Keywords

- type I and type II semiconductors
- BenDaniel-Duke boundary conditions
- Janus nanorods
- toy models

## 1. Introduction

In the last three decades, nanophysics became a domain of increasing interest and intense research, due to the huge number of new effects produced at nanoscale level, in quantum wells (QWs), quantum dots (QDs), Janus nanoparticles, etc. These new effects are fascinating from the perspective of both applied and theoretical physics. The semiconductors provide the largest area of challenging subjects, due to their applications in nanoelectronic devices, multifunctional catalysis, (bio-)chemical sensors, data storage, solar energy conversion, etc.

An attractive aspect of nanophysics is the fact that a quite large number of interesting problems can be approached using quite simple theoretical tools, sometimes at the level of one-particle quantum mechanics. In some cases, the properties of nanostructures like quantum wells, quantum dots, or quantum rods can be explained by just solving the Schrodinger equation with simple potentials. For instance, the basic physical properties of a heterostructure consisting of a thin layer of a semiconductor

But, simple as the theoretical tools needed for its investigation are, this problem of quantum mechanics involves two important issues: the position-dependent mass (PDM) quantum physics and semiconductor heterostructures. Let us shortly comment on these points.

The roots of the position-dependent effective mass concept are to be found in the pioneering works of Wannier (1937) and Slater (1949) (see Ref. 1 in [1]). Recent papers give explicit methods to obtain explicit solutions of the Schrödinger equation with PDM, for various forms of this dependence and for several classes of potentials [2, 3, 4].

However, in practical situations usually encountered in the physics of semiconductor junctions of two materials,

The transition from the complex problem of a real semiconductor (for instance, Kane theory) to the simple problem of a particle moving in a square well with BenDaniel-Duke boundary conditions is indicated, for instance, in Chapter III of Bastard’s book [5]. This simple problem provides, however, a realistic description of states near the high-symmetry points in the Brillouin zone of a large class of semiconductors. “It [i.e., ‘the simple problem’] often leads to analytical results and leaves the user with the feeling that he can trace back, in a relatively transparent way, the physical origin of the numerical results.” ([5], p. 63).

The boundary conditions for the wave functions or envelope functions at interfaces generate the eigenvalue equations for energy; of course, different boundary conditions generate different eigenvalue equations. They are transcendental equations, involving algebraic, trigonometric, hyperbolic, or even more complicated functions. With few exceptions (for instance, the Lambert equation [7]), their solutions, which cannot be expressed as a finite combinations of elementary functions, are not systematically studied.

However, in some situations, quite accurate analytical approximate solutions can be obtained. When a transcendental equation mixes algebraic and trigonometric functions, it might be possible to approximate the trigonometric functions with algebraic expression, and to transform, in this way, the exact transcendental equation into an approximate algebraic one. In its simplest form, for instance, in approximations like

In this chapter, we shall obtain approximate analytical results for the energy of electronic bound states in quantum wells and in simple models of Janus semiconductor nanorods. As the concept of Janus nanoparticle is less popular than the concept of QWs or QDs, we shall give here some short explanations.

Their name derives from the Roman god Janus: his head had two opposite faces. A Janus nanodot can be a sphere composed of two semispheres of different materials. A Janus nanorod can be a nanorod having the left half and the right half made of different materials. Due to their intrinsic duality, the opposite parts of Janus particles can be functionalized differently [10]. Janus particles with an electron-donor and -acceptor side may be used in photovoltaics. As the Janus nanoparticles have lower symmetry than their homogenous counterparts, their theoretical description is more difficult. In this chapter, we shall propose toy models for semiconductor Janus nanorods.

The structure of this chapter is the following. We shall firstly formulate the basic theory for the quantum mechanical problem of a quantum well, composed of a thin semiconductor sandwiched between two massive ones. This heterojunction can de-modeled by a quantum well (QW), essentially a finite square well, with BenDaniel-Duke boundary conditions. Such a problem was recently discussed by several authors, like Singh et al. [11, 12], who replaced the trigonometric functions entering in the transcendental equations for the bound states energy by the first few terms of their series expansion; in this way, the equations become simple, tractable algebraic ones. Our approach is different, being based on a more sophisticate “algebraization” of trigonometric functions, as proposed by de Alcantara Bonfim and Griffiths [8]. We shall obtain explicit formulas (series expansions) for the ground state energy and for the first excited state, very accurate if the well is not too shallow. Our results can be applied to both type I and type II semiconductors.

In the last part of our chapter, we shall study two toy models for semiconductor Janus nanorods; for the simplest one, we shall obtain analytical expressions for some energy eigenvalues of electronic bound states.

## 2. Basic theory

We shall solve the Schrodinger equation for an electron moving in a square well, described by the potential:

considering that its mass is position dependent. More exactly, the mass inside the well,

So, the Schrodinger equation for bound states is:

Its physically acceptable solutions, that is, the wave functions, have to satisfy two conditions: (1) the continuity of the wave function and (2) the continuity of the probability currents density at the interface. The first one is encountered in all quantum mechanical problems, but the second one is specific to the case of the position-dependent mass [6], defined by the Eq. (2), and takes the form:

Eq. (4) is known as the BenDaniel-Duke boundary condition. The notations

The

Due to the parity of the potential,

The symmetric wave functions, describing the even states, are:

The ground state wave function is, of course,

The continuity of these functions in

So, the wave function outside the well is:

The wave functions are normalized if:

These results generalize the formula (24) in [11] and the Eqs. (25.3e, o) in [13].

It is convenient to use the potential strength

and to define also

It is easy to see that:

Let us mention that, if the mass is position-independent, that is, if

If the mass is position dependent, according to (2), the eigenvalue equations obtained from the Schrodinger equations, using BenDaniel-Duke boundary conditions have the form:

We shall consider that both

or, equivalently:

For

If

In this chapter, we shall obtain precise analytical approximations for the energy of the first two states, that is, for the ground state and for the first excited state, considering the cases

## 3. Approximate analytical solutions for eigenvalue equations

### 3.1. The first even state (the ground state)

According to Eq. (29), the dimensionless momentum of the first even state, which is also the ground state, is the smallest positive root of the equation:

We shall discuss separately the cases

#### 3.1.1. The case β > 1

It is useful to introduce the new parameters

because the eigenvalue equation can be written in a simpler form:

In the most physically interesting cases,

We shall replace the exact, transcendental Eq. (31) with an approximate, algebraic equation, using one of the formulae proposed in [8] for

The precision of this approximation on various subintervals of

The algebraic approximation of the eigenvalue equation, we get with (34) is:

with

(35) can be written as:

Following the approach outlined in [19] and applied to this problem in [18], introducing the notation:

and considering that the well is not too shallow:

we obtain for the physically interesting root the expression:

If the depth of the well increases indefinitely,

It is useful to write (39) in terms of more physical parameters,

so Eq. (39) takes the form:

and:

It is a simple exercise to check that the first three terms of the previous formula coincides with the first three terms of the power series given by Eq. (17) of [8].

If the parameter

#### 3.1.2. The case β < 1

If

with the following definitions for the parameters:

Using the de Alcantara Bonfim-Griffiths algebraization for

a cubic equation:

Following the same steps as in the previous case, we find that the parameters

in Eq. (39). The final result,

### 3.2. The first odd state

#### 3.2.1. The case β > 1

The exact eigenvalue equation for the first odd state, which is also the first excited state, can be written as:

As the shape of the function

A detailed discussion of the precision of this approximation is given in [18] (see Fig. 3 and Eq. (88)). Following, exactly the same steps as in the case of the ground state, we find that

For

For both cases—

The relative errors of the formulas (39) and (50), with respect to the exact roots of the corresponding algebraic equations, are very small—of about

As already mentioned, one of the physical motivations of the calculation of the energy of bound states in heterostructures is to explain their photoluminescence properties. In several cases (see for instance [20]), the authors use Barker’s formula for the energy levels in a square well [15]. Much more precise analytical expressions for these energy are available in the literature [8, 9], for the case of constant mass; in this paper, we propose similar formulas, considering the case of position-dependent mass.

### 3.3. Higher-order states

In the previous subsections, we analyzed the ground state

but the eigenvalue equation, obtained in this way, is a sextic equation (which cannot be reduced to a cubic equation in

### 3.4. Graphical illustration of our main results

In order to illustrate graphically some of our results, let us notice that, using Eqs. (17)–(22), we can write the following relations for the energy:

where

We shall plot our main results, that is, the series expansions of the dimensionless wave vectors,

### 3.5. Applications to other nanostructures

Our calculations can be easily applied to type II semiconductors heterostructures, when one of the effective mass of the charge carrier is negative:

and can be solved following exactly the same approach.

As already mentioned, the wave function in the Schrodinger Eq. (3) can be interpreted as an envelope function. This approximation works well when the materials constituting the heterostructures are perfectly lattice-matched and they crystallize in the same crystallographic structure (in the most cases, the zinc blend structure). Its application is restricted to the vicinity of the high-symmetry points in the host’s Brillouin zone

As there are some similarities between QWs and QDs, our results are also relevant for these devices. The simplest remark is that the eigenvalues equations for the first odd state in a QW are identical to that corresponding to the * inter alia,*for the interpretation of photoluminescence spectra and photon harvesting of QDs.

## 4. The infinite square well with two semiconductor slabs

### 4.1. The symmetric case

Let us consider an infinite 1D square well, delimited by two rigid walls situated in

with:

We want to investigate how the energies of the electronic bound states will be affected, compared to the situation when in the infinite well there is only one slab, with effective electron mass

we have, with (57):

The electronic wave function is obtained solving the Schrodinger equation, as in the case of a finite well, studied in Section 2:

The boundary conditions for the wave function give:

and the continuity in the origin:

The BenDaniel-Duke boundary condition means:

or:

Together with the orthonormality condition for the wave function, Eqs. (61)–(63) and (65) form a system of five equations for five quantities,

Replacing in (66), the values of

we get:

With

it can be written as:

If

so the solutions corresponding to the infinite well with an homogenous medium inside the walls.

Eq. (70) is a transcendental one, and its solutions cannot be expressed as a finite combination of elementary functions. A quite popular analytical approximation for the tangent function has been proposed by de Alcantara-Bonfim [8] and generalized by the present author [9]:

In order to see how this approximation works, let us consider the first two roots of Eq. (68), if

So,

We find that

However, due to the rapid variation of the tangent functions near its singularities, this approximation method must be used with utmost care, as it can easily give unacceptable results (this is the case of the first root, for

### 4.2. The asymmetric case

Let us consider now the case of a rectangular infinite asymmetric well, with the potential:

with

Defining the wave vector

noticing that:

and following exactly the same steps as in the symmetric case, we obtain the following eigenvalue equation:

with the notations:

If

## 5. Conclusions

In this chapter, we obtained approximate analytical solutions for the eigenvalue equation of the first two bound states in a semiconductor quantum well, in a particular case of position-dependent mass of the charge carrier—in fact, the simplest one, corresponding to BenDaniel-Duke boundary conditions. This position dependence can be characterized by

We also proposed two models for a semiconductor Janus nanorod—a system, which was not yet treated analytically.

Our results can be easily extended to more realistic (e.g., linear) position dependence of the mass carrier and to other nanosystems. For instance, the eigenvalue equations for the wave vectors of bound energy levels of a finite barrier rectangular-shaped quantum dot, Eq. (36) in [21], are quite similar to ours—(22), (23), but somewhat more complicated. The ground state energy of electrons and holes in a core/shell quantum dot is given by Eq. (21) of [22], an equation similar to ours, just mentioned previously. Such results are important, inter alia, for the interpretation of photoluminescence spectra of heterojunctions.

## Acknowledgments

The author acknowledges the financial support of the IFIN-HH-ANCSI project PN 16 42 01 01/2016 and of the IFIN-HH-JINR grant 04-4-1121-2015/2017.