Direct Interband Light Absorption in Strongly Prolated Ellipsoidal Quantum Dots’ Ensemble
© to the authors 2008
Received: 4 August 2008
Accepted: 11 November 2008
Published: 3 December 2008
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.
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) . 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 . In particular, the case in point is a QD having the shape of a strongly prolated ellipsoid of revolution .
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.
Regime of Strong Size Quantization
Regime of Intermediate Size Quantization
Regime of Weak Size Quantization
Direct Interband Light Absorption
here and Formula (21) characterizes the dependence of the effective forbidden band gap on the semiaxes a1 and c1. 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.
Here, is an integral, which is calculated numerically, and . In this case, the transitions between the levels withm = −m′ andn = 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 numberN.
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
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 aGaAs QD with the following parameters:μe = 0.067me,μe = 0.12μh,κ = 13.8,ER = 5.275 meV, and are the effective Bohr radii of the electron and hole,Eg = 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 valuen = 0), at fixed valuesa1 = 0.5aeandc1 = 2.5ae, is equal toω00 = 3.32 × 1013 s− 1, which corresponds to the infrared region of the spectrum. For the same values of quantum numbers, but with the valuesa1 = 0.4aeandc1 = 2ae, we obtainω10 = 5.19 × 1013 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 coefficientsα andβ in formulas (12) to (14). Note that with the limiting transitionα → 0,β → 0, we arrive at the results of the regime of strong SQ.
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.
This work was carried out within the framework of the Armenian State Program “Semiconductor Nanoelectronics” and ANSEF Grant # PS NANO-1301, 2008.
- Harrison P: Quantum Wells, Wires and Dots, Theoretical and Computational Physics. Wiley, NY; 2005.View ArticleGoogle Scholar
- Bastard D: Wave Mechanics Applied to Semiconductor Heterostructures. Les editions de physique, Paris; 1989.Google Scholar
- Kazaryan EM, Petrosyan SG: Physical Principles of Semiconductor Nanoelectronics. Izd. RAU, Yerevan; 2005.Google Scholar
- Bayer M, Stern O, Hawrylak P, Fafard S, Forchel A: Nature. 2000, 405: 923. COI number [1:CAS:528:DC%2BD3cXks1Wmu7o%3D]; Bibcode number [2000Natur.405..923B] 10.1038/35016020View ArticleGoogle Scholar
- Boze C, Sarkar CK: Physica B. 1998, 253: 238. Bibcode number [1998PhyB..253..238B] 10.1016/S0921-4526(98)00407-4View ArticleGoogle Scholar
- Califano M, Harrison P: J. Appl. Phys.. 1999, 86: 5054. COI number [1:CAS:528:DyaK1MXms1eqsr0%3D]; Bibcode number [1999JAP....86.5054C] 10.1063/1.371478View ArticleGoogle Scholar
- Chen H, Apalkov V, Chakraborty T: Phys. Rev. B. 2007, 75: 193303. Bibcode number [2007PhRvB..75s3303C] 10.1103/PhysRevB.75.193303View ArticleGoogle Scholar
- Wang ZM, Holmes K, Mazur IY, Ramsey KA, Salamo GJ: Nanoscale Res. Lett.. 2006, 1: 57. Bibcode number [2006NRL.....1...57W] 10.1007/s11671-006-9002-zView ArticleGoogle Scholar
- Hayrapetyan DB: J. Contemp. Phys.. 2007, 42: 293.Google Scholar
- Juharyan LA, Kazaryan EM, Petrosyan LS: Solid State Commun.. 2006, 139: 537. COI number [1:CAS:528:DC%2BD28XotlWlsr0%3D]; Bibcode number [2006SSCom.139..537J] 10.1016/j.ssc.2006.07.012View ArticleGoogle Scholar
- Tshantshapanyan AA, Dvoyan KG, Kazaryan EM: J. Mater. Sci: Mater. Electron.. 2008.Google Scholar
- Dvoyan KG, Kazaryan EM: Phys. Status Solidi. b. 2001, 228: 695. COI number [1:CAS:528:DC%2BD38XhtFKjuw%3D%3D]; Bibcode number [2001PSSBR.228..695D] 10.1002/1521-3951(200112)228:3<695::AID-PSSB695>3.0.CO;2-PView ArticleGoogle Scholar
- Dvoyan KG, Kazaryan EM, Petrosyan LS: Physica E. 2005, 28: 333. Bibcode number [2005PhyE...28..333D]View ArticleGoogle Scholar
- Maksym P, Chakraborty T: Phys. Rev. Lett.. 1990, 65: 108. COI number [1:CAS:528:DyaK3cXltVemu7o%3D]; Bibcode number [1990PhRvL..65..108M] 10.1103/PhysRevLett.65.108View ArticleGoogle Scholar
- Hayrapetyan DB, Dvoyan KG, Kazaryan EM, Tshantshapanyan AA: Nanoscale Res. Lett.. 2007, 2: 601. Bibcode number [2007NRL.....2..601D] 10.1007/s11671-007-9079-zView ArticleGoogle Scholar
- Galitsky VM, Karnakov BM: Kogan VI Practical Quantum Mechanics. Nauka, Moscow; 1981.Google Scholar
- Anselm AI: Introduction to Semiconductors Theory. Nauka, Moscow; 1978.Google Scholar
- Efros AlL, Efros AL: Sov. Phys. Semicond.. 1982, 16: 772.Google Scholar
- Lifshits IM, Slezov VV: Sov. Phys. JETP. 1958, 35: 479. COI number [1:CAS:528:DyaG1MXhsFI%3D]Google Scholar
- Leonard D, Krishnamurthy M, Reaves CM, Denbaars SP, Petroff PM: Appl. Phys. Lett.. 1993, 63: 23. 10.1063/1.110199View ArticleGoogle Scholar