# Direct Interband Light Absorption in Strongly Prolated Ellipsoidal Quantum Dots’ Ensemble

- KG Dvoyan
^{1}Email author, - DB Hayrapetyan
^{1, 2}and - EM Kazaryan
^{1}

**Received: **4 August 2008

**Accepted: **11 November 2008

**Published: **3 December 2008

## Abstract

Within the framework of adiabatic approximation, the energy levels and direct interband light absorption in a strongly prolated ellipsoidal quantum dot are studied. Analytical expressions for the particle energy spectrum and absorption threshold frequencies in three regimes of quantization are obtained. Selection rules for quantum transitions are revealed. Absorption edge and absorption coefficient for three regimes of size quantization (SQ) are also considered. To facilitate the comparison of obtained results with the probable experimental data, size dispersion distribution of growing quantum dots by the small semiaxe in the regimes of strong and weak SQ by two experimentally realizing distribution functions have been taken into account. Distribution functions of Lifshits–Slezov and Gaussian have been considered.

### Keywords

Ellipsoidal quantum dot Interband light absorption Selection rules Quantum dot ensemble## Introduction

Development of the novel growth techniques, such as the Stranski–Krastanov epitaxial method etc., makes possible to grow semiconductor quantum dots (QDs) of various shapes and sizes [1–3]. As is known, the energy spectrum of charge carriers in QDs is completely quantized and resembles the energy spectrum of atoms (artificial atoms) [4]. In recent years, many theoretical and experimental works have evolved, where ellipsoidal, pyramidal, cylindrical, and lens-shaped QDs were considered [5–13]. As a result of diffusion, the confining potential, formed during the growth process, in most cases can be approximated with a high accuracy by a parabolic potential. However, an effective parabolic potential may arise in a QD in view of features of its external shape [14]. In particular, the case in point is a QD having the shape of a strongly prolated ellipsoid of revolution [15].

Investigations of the optical absorption spectrum of various semiconductor structures are a powerful tool for determination of many characteristics of these systems: forbidden band gaps, effective masses of electrons and holes, their mobilities, dielectric permittivities, etc. There are many works devoted to the theoretical and experimental study of the optical absorption both in massive semiconductors and size-quantized systems. The presence of size quantization (SQ) essentially influences the absorption mechanism. In fact, the formation of new energy levels of the SQ makes possible new interlevel transitions.

In this paper, the electron states and direct interband absorption of light in a strongly prolated ellipsoidal QD (SPEQD) at three regimes of SQ is considered. Absorption edge and absorption coefficient for three regimes of SQ are also considered. To facilitate the comparison of obtained results with the probable experimental data, size dispersion distribution of growing QDs by the small semiaxe in the regimes of strong and weak SQ by two experimentally realizing distribution functions have been taken into account. Distribution function of Lifshits–Slezov has been considered in the first model and distribution function of Gauss has been considered in the second case.

## Theory

### Regime of Strong Size Quantization

*a*

_{1}and

*c*

_{1}are the small and large semiaxes of the SPEQD, respectively.

*Z*-direction. This allows one to use the adiabatic approximation [16]. The Hamiltonian of the system in the cylindrical coordinates has the form

*μ*

_{p}is the effective mass of the particle, is the effective Rydberg energy, is the effective Bohr radius of the particle,

*e*is the particle charge, and

*κ*is the dielectric constant. The wave function can be sought in the form

*z*-coordinate, the particle motion is localized in a two-dimensional potential well with the effective variable width

*α*

_{n+1,m}are the zeros of the first-kind

*J*

_{ m }(

*r*) Bessel function. For the lower levels of the spectrum, the particle is mainly localized in the region |

*z*| <<

*a*. Based on this, we expand

*ɛ*

_{1}(

*z*) into a series

### Regime of Intermediate Size Quantization

*Z*-direction. Therefore, we restrict ourselves to the case of a one-dimensional exciton. It is clear that in this SQ regime, the energy of electron motion predominates over the energy of heavy-hole motion (from the condition

*μ*

_{e}<<

*μ*

_{h}). Based on the above, the electronic potential acted on the hole can be averaged over the electron motion and written as

_{n,m,N}(

*Z*′) is the electron wave function. The condition

*a*

_{h}<<

*a*

_{1}allows us to expand potential (11) into a series near the point

*Z*= 0, where is the effective Bohr radius of the hole. Finally we obtain for expression (11) in dimensionless quantities

### Regime of Weak Size Quantization

## Direct Interband Light Absorption

*μ*

_{e}<<

*μ*

_{h}. The absorption coefficient is defined by the expression [18]

*ν*and

*ν*′ are the sets of quantum numbers corresponding to the electron and heavy hole,

*E*

_{g}the forbidden band gap of a massive semiconductor, Ω the incident light frequency,

*A*a quantity proportional to the square of modulus of the matrix element of the dipole moment taken over the Bloch functions [17]. Finally, in the regime of strong SQ, for the quantity

*K*and the absorption threshold, we obtain

here
and
Formula (21) characterizes the dependence of the effective forbidden band gap on the semiaxes *a*_{1} and *c*_{1}. With increasing semiaxes, the absorption threshold decreases, but the dependence on the small semiaxis becomes stronger. Consider now the selection rules for transitions between the levels with different quantum numbers. For the magnetic quantum number, the transitions between the levels with *m* = −*m*′ are allowed, while for the quantum number of the fast subsystem the transitions with *n* = *n*′. For the oscillatory quantum number, the transitions for the levels with *N* = *N*′ are allowed. Note that the analytical form of expression (20) is given with allowance for the above-mentioned selection rules.

*ν*transforms into a set of closely spaced lines corresponding to different values of

*ν*′. In this regime of SQ, the absorption coefficient has the form

Here,
is an integral, which is calculated numerically,
and
. In this case, the transitions between the levels with*m* = −*m*′ and*n* = *n*′ are allowed. It should also be noticed that taking into account the effective one-dimensional Coulomb interaction leads to the destruction of the previous symmetry of the task and to the full removal of selection rules for the oscillatory quantum number*N*.

*E*is the energy (18) in dimensional quantities. It should be noted that

*φ*(0) ≠ 0 only for the ground state when

*l*=

*m*= 0 (

*l*is the orbital quantum number). Finally, in the regime of weak SQ, we get for the absorption coefficient and absorption threshold the expressions

Here, denotes an integral, which is calculated numerically, and The most important feature of this case is the fact that with changing semiaxes of the SPEQD the excitonic level shift is determined by the total mass of the exciton.

## Direct Interband Light Absorption with Account of Dispersion of QDs Geometrical Sizes

*P*(

*u*), we obtain for the absorption coefficient corresponding formula:

## Discussion

As is seen from formula (10), the energy spectrum of CCs in SPEQD is equidistant. This result is related only to the lower levels of the spectrum. Numerical calculations for the case of strong SQ were performed for a*GaAs* QD with the following parameters:*μ*_{e} = 0.067*m*_{e},*μ*_{e} = 0.12*μ*_{h},*κ* = 13.8,*E*_{R} = 5.275 meV,
and
are the effective Bohr radii of the electron and hole,*E*_{g} = 1.43 eV is the forbidden band gap of a massive semiconductor. In the strong SQ regime, the frequency of transition between the equidistant levels (for the value*n* = 0), at fixed values*a*_{1} = 0.5*a*_{e}and*c*_{1} = 2.5*a*_{e}, is equal to*ω*_{00} = 3.32 × 10^{13} s^{− 1}, which corresponds to the infrared region of the spectrum. For the same values of quantum numbers, but with the values*a*_{1} = 0.4*a*_{e}and*c*_{1} = 2*a*_{e}, we obtain*ω*_{10} = 5.19 × 10^{13} s^{− 1}, which is half as much again the preceding case. As is seen from formula (10), with increasing semiaxes the particle energy is lowered. Note that this energy is more “sensitive” to changes of the small semiaxis, which is a consequence of the higher contribution of SQ into the particle energy in the direction of the axis of ellipsoid revolution. With increasing semiaxes, the energy levels come closer together, but remain equidistant.

In the regime of intermediate SQ, the influence of the electron–hole Coulomb interaction is exhibited by means of the coefficients*α* and*β* in formulas (12) to (14). Note that with the limiting transition*α* → 0,*β* → 0, we arrive at the results of the regime of strong SQ.

*GaAs*(for equal values of the large semiaxis and cylinder height), respectively, on the small semiaxis, quantum wire radius, and radius of the cylindrical QD. As seen, the curve of the electron ground state energy in SPEQD is disposed higher, which is caused by the larger contribution of SQ into the particle energy as compared to other two cases.

*K*on the frequency of incident light, for the ensemble of SPEQDs in strong SQ regime. As it mentioned above, instead of distinct absorption lines, account of size dispersion will give a series of fuzzy maximums. Note that both in the model of Gaussian and in the model of Lifshits–Slezov, QDs distributions a single distinctly expressed maximum of absorption is observed. When the light frequency is increased, the second weakly expressed maximum is seen. Further increase of the incident light frequency results in a fall of absorption coefficient.

*n*=

*n*′ = 1 transition family, and weak expressed picks are the result of the transitions between equidistant levels. The second weaker pick corresponds to the

*n*=

*n*′ = 2 transition family. It is also obvious from Fig. 7that the intensity corresponding to the above-mentioned family is weaker than the first maximum. This is the result of the small volume of the overlapping integral. The above-mentioned means, that the probability of transition is decreased.

*K*on the frequency of incident light, for the ensemble of SPEQDs in weak SQ regime. As can be seen from the picture taking into account the Coulomb interaction leads to the appearing of secondary well-expressed maximums of absorption. By the other word, the difference with the previous case has the quantitative character.

## Conclusion

In this work, we obtained that the electron energy is equidistant inside SPEQD in all three SQ regime cases. The impact of the dispersion of geometrical sizes for the QDs ensemble on direct light absorption is also investigated.

The SPEQDs, as more realistic nanostructures than quantum wires, have various commercial applications, in particular, in large two-dimensional focal plane arrays in the mid- and far infrared (M&FIR) region they have important applications in the fields of pollution detection, thermal imaging object location, and remote sensing as well as infrared imaging of astronomical objects.

These optimized quantum structures can be formed by direct epitaxial deposition using a self-assembling QDs technique, e.g., described in US Patent # 6541788 entitled “Mid infrared and near infrared light upconverter using self-assembled quantum dots” as well as by usage of MBE, MOCVD, or MOMBE deposition systems.

This theoretical investigation of SPEQDs can be effectively used for direct applications in photonics as background for simulation model. For further investigations, it is also important to develop a scheme for optimization of growth of SPEQDs needed for second harmonic generation.

## Declarations

### Acknowledgments

This work was carried out within the framework of the Armenian State Program “Semiconductor Nanoelectronics” and ANSEF Grant # PS NANO-1301, 2008.

## Authors’ Affiliations

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