Resonant Light Absorption by Semiconductor Quantum Dots

Abstract

The cross-section of light absorption by size-quantized semiconductor quantum dots (QD) is calculated in the case of a resonance with an exciton Γ₆×Γ₇ in cubical crystals of 𝑇_𝑑 class. The interference of stimulating and induced electric and magnetic fields is taken into account. The cross-section of light absorption is proportional to the exciton nonradiative damping 𝛾.

1. Introduction

Study of the resonant reflection and absorption of light by low-dimensional semiconductor objects is a simple and reliable way of the determination of exciton parameters [1–5]. When the size-quantized semiconductor objects (quantum wells (QW), quantum wires, and quantum dots (QD) are irradiated by light, elastic light scattering and absorption intensify resonantly if the light frequency 𝜔_𝑙 equals to the exciton frequency 𝜔₀. The resonant peak width is determined by the exciton damping Γ=𝛾_𝑟+𝛾, which consists of nonradiative 𝛾 and radiative 𝛾_𝑟 damping. An important role of the radiative damping 𝛾_𝑟 was proved for the first time in [6–8]. Light reflection by some structures, consisting of quantum wells, wires, and dots was considered in [9].

The light elastic scattering on a QD of arbitrary form and sizes in a resonance with excitons is investigated in [10], where the quantum perturbation theory is used. However, the quantum method does not allow to get out of the lowest order on the light-electron interaction and to calculate corrections to the exciton energy due to that interaction. Such approximation is acceptable only under condition 𝛾_𝑟≪𝛾.

In the present work, a semiclassical method [9, 11–14] is applied for calculation of electric and magnetic fields, while a description of electrons remains quantum one. The semiclassical method allows to calculate precisely electric and magnetic fields on large distances from a QD, that is, to take into account all the orders on the light-electron interaction. It allows to introduce into the theory the nonradiative damping 𝛾, to calculate the light absorption and the exciton energy corrections due to the long-range exchange electron-hole interaction. At last, the method admits consideration of the monochromatic and pulse irradiation (see, e.g., [15]).

There are two variants of the semiclassical method. The first of them assumes using of boundary conditions for electric and magnetic fields on the semiconductor object boundaries (see, e.g., [11, 16]). However, using of boundary conditions in the case of spherical QDs, for instance, realizes into cumbersome calculations [11], and calculations become much more complicate for other forms. Therefore, we use here the second variant of the semiclassical method—the method of retarded potentials—allowing to avoid using of boundary conditions at all.

2. The Method of Retarded Potentials

First of all, we calculate the current 𝐣(𝐫,𝑡) and charge 𝜌(𝐫,𝑡) densities induced by the electric field inside of the object [12, 17, 18] and averaged on the ground state of the crystal. Using of stimulating electric field 𝐄₀(𝐫,𝑡) in these expressions leads again to the limitation by the lowest order on the light-electron interaction. But substitution of the genuine fields 𝐄(𝐫,𝑡) inside of the object leads to the precise results.

The induced electric and magnetic fields are represented with the help of the vector 𝐀(𝐫,𝑡) and scalar 𝜑(𝐫,𝑡) potentials 1Δ𝐄(𝐫,𝑡)=−𝑐𝜕𝐀(𝐫,𝑡)−𝜕𝑡𝜕𝜑(𝐫,𝑡),𝜕𝐫Δ𝐇(𝐫,𝑡)=rot𝐀(𝐫,𝑡). (1) The retarded potentials are [19, page 447] 1𝐀(𝐫,𝑡)=𝑐𝑑³𝑟^′𝐣(𝐫^′,𝑡−𝜈|𝐫−𝐫^′|/𝑐)|𝐫−𝐫^′|,1𝜑(𝐫,𝑡)=𝜈²𝑑³𝑟^′𝜌(𝐫^′,𝑡−𝜈|𝐫−𝐫^′|/𝑐)|𝐫−𝐫^′|, (2) where 𝜈 is the light refraction coefficient (we assume it is identical inside and outside of a QD). Having used the current and charge densities with the precise field 𝐄(𝐫,𝑡)=𝐄₀(𝐫,𝑡)+Δ𝐄(𝐫,𝑡) and substituting (2) into (1), we obtain an integral equation for the induced field Δ𝐄(𝐫,𝑡) inside of the object. In some cases this equation may be solved. Having calculated the precise field inside of the object, we obtain the fields outside with the help of (2) and (1).

3. The Induced Current Density

The quantum theory of conductivity of low-dimensional objects is elaborated in [17, 18, 20]. The average current and charge densities induced by a weak spatially inhomogeneous electromagnetic field are calculated for the spatially inhomogeneous systems. It was shown that the averaged current and charge densities contain two contributions, first of which is expressed through the electric field, and the second is expressed through the first spatial derivatives of the electric field. Situations when the second contribution may be neglected are discussed in [20]. The main contribution into the average current density at temperature 𝑇=0 is as follows: 𝑗_𝛼𝑖(𝐫,𝑡)=ℏ𝑑³𝑟^′𝑡_−∞𝑑𝑡^′×⟨0|[𝑗_𝛼̂(𝐫,𝑡),𝑑_𝛽(𝐫^′,𝐭^′)]|0⟩𝐸_𝛽(𝐫^′,𝐭^′)+𝑐.𝑐., (3) where determinations of [11] are used: ⟨0|⋯|0⟩ is the average value on the ground state, [𝑎,𝑏̂] is the commutator of operators 𝑎 and 𝑏̂, 𝐣(̂𝐫,𝑡) is the operator of current density in the interaction representation, 𝐝(𝐫)=∑_𝑖𝐫_𝑖𝜌_𝑖(𝐫), 𝐫_𝑖=𝐫_𝑖−⟨0|𝐫_𝑖|0⟩, 𝜌_𝑖(𝐫)=𝑒𝛿(𝐫−𝐫_𝑖), 𝐄(𝐫,𝑡) is the classic electric field.

We assume 𝑇=0. It was assumed also in (3) that currents and charges are absent at infinite distances, and that the electromagnetic field equals 0 at 𝑡→−∞, what corresponds to adiabatic switching on of fields. We suppose an interaction between charged particles, a possible presence of an external potential and quantizing magnetic field.

We apply (3) to consideration of low-dimensional size-quantized semiconductor objects [7] and suppose that the stimulating light frequency or carrying frequency of a pulse irradiation are close to the semiconductor energy gap ℏ𝜔_𝑔. Sizes 𝑅 of a semiconductor object are much more than a lattice constant 𝑅≫𝑎. The distances, on which the smooth multiplier from the wave function varies, are much greater 𝑎 and comparable to the sizes of the object. Then the effective mass approximation is applicable and the size-quantization condition is satisfied. The last statement means that the exciton energy spectrum is discrete one. Then [7, 8], the average density of induced current in QWs, QDs, and quantum wires is determined by the expression 𝐣(𝐫,𝑡)=𝑖𝑒²2𝜋ℏ𝜔_𝑔𝑚²_0𝜂×{𝐩^∗_𝑐𝑣𝜂𝐹_𝜂(𝐫)^+∞_−∞𝑒𝑑𝜔^{−𝑖𝜔𝑡}𝜔−𝜔_𝜂+𝑖𝛾_𝜂×/2𝑑𝐫^′𝐹^∗_𝜂(𝐫^′)(𝐩_𝑐𝑣𝜂ℰ(𝐫^′,𝜔))+𝐩_𝑐𝑣𝜂𝐹^∗_𝜂(𝐫)^+∞_−∞𝑒𝑑𝜔^{−𝑖𝜔𝑡}𝜔+𝜔_𝜂+𝑖𝛾_𝜂×/2𝑑𝐫^′𝐹_𝜂(𝐫^′)(𝐩^∗_𝑐𝑣𝜂ℰ(𝐫^′,𝜔))}+𝑐.𝑐., (4) where the determinations are used: 𝑒=−|𝑒|,𝑚₀ are the electron charge and mass, respectively, 𝑝_𝑐𝑣𝜂 is the interband matrix element of quasimomentum operator, corresponding to the exciton with indexes 𝜂, 𝐹_𝜂(𝐫) is the envelope exciton wave function at 𝐫_𝑒=𝐫_ℎ=𝐫, where 𝐫_𝑒(𝐫_ℎ) is the electron (hole) radius-vector, ℏ𝜔_𝜂 is the 𝜂 exciton energy counted from the ground state energy, 𝛾_𝜂 is the nonradiative exciton damping. The set 𝜂 includes indexes of semiconductor valence and conduction bands, and 6 indexes, characterizing an exciton in the effective mass approximation (these 6 indexes describe the function 𝐹_𝜂(𝐫)). Finally, ℰ(𝐫,𝜔)) is the Fourier transform of the electric field 1𝐄(𝑟,𝑡)=2𝜋^∞_−∞𝑑𝜔𝑒^{−𝑖𝜔𝑡}ℰ(𝐫,𝜔)+𝑐.𝑐. (5) The main and conjugated contributions in (5) are chosen so that ℰ₀(𝐫,𝜔)=2𝜋𝐸₀𝐞_𝑙𝑒^𝑖𝐤𝐫𝐷₀(𝜔), (6) where 𝐤 is the wave vector of stimulating light, the function 𝐷₀(𝜔) describes the form of a stimulating pulse [12]. The average density of charge induced inside of a semiconductor object may be determined with the help of (4) and continuity equation div𝐣(𝐫,𝑡)+𝜕𝜌(𝐫,𝑡)𝜕𝑡=0. (7)

4. The Case of the Exciton Γ_𝟔×Γ_𝟕 in a QD

Let us consider an exciton consisting of an electron from twofold degenerated conductive band Γ₆ and a hole from twofold degenerated valence band Γ₇ chipped of by the spin-orbital interaction in 𝑇_𝑑 class crystals. The exciton Γ₆×Γ₇ (see [10, 11]) is the most simple object in contrast to excitons containing light and heavy holes. All the measurable values do not depend on the direction of vectors relatively of crystallographic axes, that is, a crystal plays a role of an isotropic medium.

According to determinations of [21, page 73], the electron wave functions have the structure Ψ_𝑐1=𝑖𝑆↑,Ψ_𝑐2=𝑖𝑆↓, (8) and the hole wave functions are Ψ_ℎ1=1√31(𝑋−𝑖𝑌)↑−√3Ψ𝑍↓,_ℎ2=1√31(𝑋+𝑖𝑌)↓+√3𝑍↑. (9) Having combined (8) and (9) in pairs, we obtain fourfould degenerated excitonic state for which the interband matrix elements of momentum operator result in 𝐩_𝑐𝑣1=𝑝_𝑐𝑣√3(𝐞_𝑥−𝑖𝐞_𝑦),𝐩_𝑐𝑣2=𝑝_𝑐𝑣√3(𝐞_𝑥+𝑖𝐞_𝑦𝐩),_𝑐𝑣3=𝑝_𝑐𝑣√3𝐞_𝑧,𝐩_𝑐𝑣4𝑝=−^𝑐𝑣√3𝐞_𝑧, (10) where the scalar 𝑝_𝑐𝑣=𝑖⟨𝑆|𝑝_𝑥|𝑋⟩ (11) is introduced; 𝐞_𝑥,𝐞_𝑦,𝐞_𝑧 are the unite vectors along the crystallographic axes.

We assume that the QD radius 𝑅 is much less than the bulk exciton Bohr radius (strong-confinement regime). Under these conditions the binding energy of the exciton, determined by the Coulomb interaction, may be neglected. From the four exciton states, to which the interband matrix elements (10) of the quasimomentum correspond, it is easy (by drawing up the linear combinations) to obtain three bright and one dark excitonic states, about which the speech goes, for example, in [11, 22, 23]. Below we show that the corrections to the bright exciton energy, caused by long-range exchange interaction of electrons and holes, occur in our theory automatically and coincide with the results of [11, 22, 23].

5. Electric and Magnetic Fields on Large Distances from a QD

Let us consider light scattering by a QD near the resonance with some energy level of Γ₆×Γ₇ exciton, when other excitons and excitonic energy levels may be neglected. Our energy level is nondegenerated, then the index 𝜂 takes on four values from 1 to 4, and only the values 𝐩_𝑐𝑣𝜂, according to (10), depend on 𝜂. Without light-electron interaction we have 𝜔_𝜂=𝜔₀,𝛾_𝜂=𝛾,𝐹_𝜂(𝐫)=𝐹(𝐫), (12) though the function 𝐹(𝐫) may be chosen as a real one. After summation on 𝜂 in (4) with using (10), we obtain 𝐣(𝐫,𝑡)=𝑖𝑒²𝑝²_𝑐𝑣3𝜋ℏ𝜔_𝑔𝑚²₀𝐹(𝐫)^∞_−∞𝑑𝜔𝑒^{−𝑖𝜔𝑡}𝐓(𝜔)𝐿(𝜔)+𝑐.𝑐., (13) where 𝐓(𝜔)=𝑑³1𝑟ℰ(𝐫,𝜔)𝐹(𝐫),𝐿(𝜔)=𝜔−𝜔₀+1+𝑖𝛾/2𝜔+𝜔₀.+𝑖𝛾/2 (14)

With the help of (7) we obtain the induced charge density 𝑒𝜌(𝐫,𝑡)=²𝑝²_𝑐𝑣3𝜋ℏ𝜔_𝑔𝑚²₀^∞_−∞𝑑𝜔𝜔𝑒^{−𝑖𝜔𝑡}(𝑑𝐹(𝐫)𝑑𝐫)𝐓(𝜔)𝐿(𝜔)+𝑐.𝑐. (15) The nonresonant term from (14) is neglected below.

Using (2), we obtain electric and magnetic fields on large distances from a QD Δ𝐄_𝑐(𝐫,𝑡)|_𝑟→∞=𝐸⁺(𝐫,𝑡)𝐞⁺_𝑠+𝐸⁻(𝐫,𝑡)𝐞⁻_𝑠+𝑐.𝑐.,Δ𝐇_𝑐(𝐫,𝑡)|_𝑟→∞=𝐻⁺(𝐫,𝑡)𝐞⁺_𝑠+𝐻⁻(𝐫,𝑡)𝐞⁻_𝑠+𝑐.𝑐., (16) where 𝐸^±𝑒(𝐫,𝑡)=−²2𝜋ℏ𝜔_𝑔𝑚²₀𝑐²𝑟^∞_−∞𝑑𝜔𝜔𝑒^{𝑖(𝑘𝑟−𝜔𝑡)}𝑄(𝐤_𝑠,𝜔,𝐞^±_𝑠𝐻),_±(𝐫,𝑡)=∓𝑖𝜈𝐸^±(𝐫,𝑡),𝑄(𝐤_𝑠,𝜔,𝐞^±_𝑠2)=3𝑝²_𝑐𝑣𝑃(𝐤_𝑠)(𝐓(𝜔)𝐞^∓_𝑠𝑑)𝐿(𝜔),𝑃(𝐤)=³𝑟𝑒^{−𝑖𝐤𝐫}𝐹(𝐫), (17)𝐤_𝑠=(𝜔𝜈/𝑐)(𝐫/𝑟) is the wave vector of scattered light.

To obtain precise induced electric and magnetic fields on large distances from a QD, we have to calculate the vector 𝐓(𝜔), determined by the genuine electric field inside of the QD and consisting of stimulating and induced fields 𝐄(𝐫,𝑡)=𝐸₀𝐞_𝑙^∞_−∞𝑑𝜔𝑒^{𝑖(𝐤𝐫−𝜔𝑡)}𝐷₀(𝜔)+Δ𝐄(𝐫,𝑡), (18) where 𝑘=𝜔𝜈/𝑐,𝐞_𝑙 is the circular polarization vector. In the case of a monochromatic irradiation 𝐷₀(𝜔)=𝛿(𝜔−𝜔_𝑙), (19) but at a pulse irradiation the frequency 𝜔 is widespread in the interval Δ𝜔 near 𝜔_𝑙, and Δ𝜔 is vice inverse to the pulse duration Δ𝑡.

Using the formulas for retarded potentials, we obtain the induced field ℰ(𝐫,𝜔)=2𝜋𝐞_𝑙𝐸₀𝑒^𝑖𝐤𝐫𝐷₀(𝜔)−2𝑒²𝑝3ℏ𝑐^2𝑐𝑣𝜔𝜔_𝑔𝑚²₀𝑐1×𝐿(𝜔)[𝐓(𝜔)+𝑘²𝑑(𝐓(𝜔))𝑑𝑑𝐫𝑑𝐫]Φ(𝐫), (20) where Φ(𝐫)=𝑑³𝑟^′𝐹_𝜂(𝐫^′)𝑒^{𝑖𝑘|𝐫−𝐫′|}|𝐫−𝐫^′|. (21) Equation (20) is an integral equation for ℰ(𝐫,𝜔). Then 1Φ(𝐫)=(2𝜋)³𝑑³𝑞𝑒^𝑖𝐪𝐫𝑃(𝐪){4𝜋𝑃𝑞²−𝑘²+2𝑖𝜋²𝑘𝑃𝛿(𝑞−𝑘)},=1𝑎−𝑏2(1+1𝑎−𝑏+𝑖𝛿𝑎−𝑏−𝑖𝛿),𝛿→+0. (22)

Let us substitute (22) in (20), multiply both sides on 𝐹(𝐫), and integrate on 𝐫. We obtain the equation for the vector 𝐓(𝜔): 𝐓(𝜔)(1+𝐶(𝜔)𝑑³𝑞𝐽(𝐪))=2𝜋𝐸₀𝐞_𝑙𝒟₀(𝜔)𝑃^∗+(𝐤)𝐶(𝜔)𝑘²𝑑³𝑞𝐪(𝐪𝐓(𝜔))𝐽(𝐪), (23) where determinations 2𝐶(𝜔)=3(𝑒²)𝑝ℏ𝑐^2𝑐𝑣𝜔𝜔_𝑔𝑚²₀𝑐𝐿(𝜔),𝐽(𝐪)=|𝑃(𝐪)|²[12𝜋²𝑃𝑞²−𝑘²+𝑖4𝜋𝑘𝛿(𝑞−𝑘)], (24) are introduced.

Equation (23) may be considered as a system of three equations for the components 𝑇_𝑥(𝜔),𝑇_𝑦(𝜔),𝑇_𝑧(𝜔). Having solved this system, we can substitute the results into expressions for induced fields on large distances from the QD. Thus, the problem of light scattering in the resonance with an exciton Γ₆×Γ₇in a QD of arbitrary form is solved in principal at any relation of the light wave length and QD's sizes.

6. The Special Case of a Spherical Symmetrical Envelope Wave Function

Let us consider a special case when the system of three equations is reduced to one equation. It happens if the function 𝑃(𝐤) depends only on the module 𝑘, that is, 𝑃(𝐤)=𝑃(𝑘). (25) The condition (25) is fulfilled, if the function 𝐹(𝐫) is spherically symmetrical or the QD sizes are much less than the wave length of the stimulating light when 𝑃(𝐤)=𝑃(0). For instance, (25) is satisfied in the case of a spherical QD limited by the infinitely high rectangular potential barrier. Then the envelope wave function 1𝐹(𝑟)=2𝜋𝑅sin²(𝜋𝑟/𝑅)𝑟²𝜃(𝑅−𝑟) (26) corresponds to the lowest excitonic energy level 𝜔₀, and 2𝑃(𝑘)=𝑘𝑅^𝜋₀𝑑𝑥sin𝑘𝑅𝑥𝜋sin²𝑥𝑥,𝑃(0)=1. (27) Then 𝐓(𝜔)=2𝜋𝐸₀𝐞_𝑙𝐷₀(𝜔)𝑃^∗(𝑘)(𝜔−𝜔₀+𝑖𝛾/2)𝜔−𝜔₀+𝑖(𝛾+𝛾_𝑟)/2, (28) where 𝛾_𝑟=8𝜈9𝑒²𝑝𝑐ℏ^2𝑐𝑣𝜔²𝜔_𝑔𝑚²₀𝑐²|𝑃(𝑘)|²,𝜔₀=𝜔₀+Δ𝜔,Δ𝜔=−4𝑒²𝑝9𝜋ℏ𝜔_𝑔𝜔𝑚^2𝑐𝑣2₀𝜈^2∞₀𝑑𝑞𝑞²(𝑃(𝑞))²×(𝑞²−3𝑘²)𝑃𝑞²−𝑘². (29)(30) The value (30) of an exciton energy correction coincides completely with the results of [6] and preceding results of [16–18], where it was considered as the result of the long-range exchange electron-hole interaction (see [8]). Having substituted (28) in (16) and (17), we obtain electric and magnetic fields on the large distances from the QD applicable in the cases of the monochromatic or pulse irradiation.

The precise results for the fields distinguish from the results of the lowest approximation on the light-electron interaction only by the substitution 𝜔₀ by 𝜔₀ and 𝛾 by Γ=𝛾+𝛾_𝑟.

For the case (19) of the monochromatic irradiation we obtain Δ𝐄((𝐫),𝑡)|_𝑟→∞3=−4𝐸₀𝛾_𝑟𝑘_𝑙𝑟[(𝐞_𝑙𝐞⁺_𝑠)𝐞⁺_𝑠+(𝐞_𝑙𝐞⁺_𝑠)𝐞⁻_𝑠]×𝑒^{𝑖(𝑘𝑙𝑟−𝜔𝑙𝑡)}𝜔_𝑙−𝜔₀+𝑖Γ/2+𝑐.𝑐.,Δ𝐇((𝐫),𝑡)|_𝑟→∞=3𝑖𝜈4𝐸₀𝛾_𝑟𝑘_𝑙𝑟[(𝐞_𝑙𝐞⁻_𝑠)𝐞⁺_𝑠−(𝐞_𝑙𝐞⁺_𝑠)𝐞⁻_𝑠]×𝑒^{𝑖(𝑘𝑙𝑟−𝜔𝑙𝑡)}𝜔_𝑙−𝜔₀+𝑖Γ/2+𝑐.𝑐., (31) where 𝑘_𝑙=𝜔_𝑙𝜈/𝑐.

7. The Pointing Vector on Large Distances from the QD and the Total Cross-Section of Light Scattering at the Monochromatic Irradiation

The Pointing vector on large distances from the QD equals 𝐒|_𝑟→∞=𝐒₀+𝐒_inter𝑓+𝐒_scat, (32) where 𝐒₀=𝑐𝐄4𝜋⁰×𝐇₀=𝑐𝜈𝐸2𝜋²⁰𝐞_𝑧,𝐒_inter𝑓=𝑐4𝜋[𝐄₀×Δ𝐇+Δ𝐄×𝐇₀𝐒],_scat=𝑐4𝜋Δ𝐄×Δ𝐇. (33) With the help of (31) we obtain 𝐒_scat=9𝜋4𝑆₀𝛾²_𝑟(𝑘_𝑙𝑟)²𝐫𝑟|𝐞_𝑙𝐞⁻_𝑠|²+|𝐞_𝑙𝐞⁺_𝑠|²(𝜔_𝑙−𝜔₀)²+Γ²/4. (34) The module of the total flux of scattered light per time unite is Π_scat=3𝜋2𝑆₀𝛾²_𝑟(𝑘_𝑙)²1(𝜔_𝑙−𝜔₀)²+Γ²/4. (35) Obtaining (35) from (34) we have used the relations |𝐞⁺_𝑙𝐞⁻_𝑠|²=|𝐞⁻_𝑙𝐞⁺_𝑠|²=14(1+cos𝜃)²,|𝐞⁺_𝑙𝐞⁺_𝑠|²=|𝐞⁻_𝑙𝐞⁻_𝑠|²=14(1−cos𝜃)², (36) where 𝜃 is the scattering angle. An integration 𝜃 is performed. Having divided Π_scat on the density 𝑆₀ of the stimulating light flux and using 𝑘_𝑙=2𝜋/𝜆_𝑙, we obtain the total cross-section of light scattering 𝜎_scat=3𝜆2𝜋^2𝑙𝛾²_𝑟/4(𝜔_𝑙−𝜔₀)²+Γ²/4. (37) The differential cross-sections have been determined in [8].

8. The Light Absorption: The Role of the Nonradiative Damping

The results for the cross-sections of light scattering on QD, obtained with the help of the quasiclassic method, coincide with the lowest approximation on the light-electron interaction with the results of the quantum perturbation theory [10]. However, the quasiclassic method allows to obtain also the cross-section of light absorption by a QD. In the case of a monochromatic irradiation the absorption is stipulated by the nonradiative damping 𝛾 of excitons, and it equals 0 at 𝛾=0. The same result was obtained at a monochromatic irradiation of a QW [7, 24]. The cause is that at 𝛾=0 a dissipation of energy, spent on an exciton creation, is absent. And this energy returns at an exciton annihilation. In the case of a pulse irradiation the integral absorption equals 0 at 𝛾=0 [15, 25–27].

It was shown at calculations of light absorption by a QW that the interference of stimulating and induced electromagnetic fields must be taken into account (see, e.g., [25]). The same is true for the case of QDs.

9. The Interference Contribution into Energy Fluxes

Let us calculate an interference contribution into the Pointing vector at 𝑟→∞ in the case of a monochromatic irradiation. Using (31) and 𝐄^±₀(𝐫,𝑡)=𝐸₀𝐞^±_𝑙𝑒^{𝑖(𝐤𝑙𝐫−𝜔𝑙𝑡)}𝐇+𝑐.𝑐.,_±0(𝐫,𝑡)=𝐸₀𝜈[𝐞_𝑧×𝐞^±_𝑙]𝑒^{𝑖(𝐤𝑙𝐫−𝜔𝑙𝑡)}+𝑐.𝑐., (38) we obtain 𝐒_inter𝑓=𝐒_𝑧+𝐒_⟂,𝐒_𝑧3=−4𝛾_𝑟𝑘_𝑙𝑟𝑆₀𝐞_𝑧|𝐞⁺_𝑙𝐞⁻_𝑠|²(𝑒^{𝑖(𝐤𝑙𝐫−𝑘𝑙𝑟)}𝜔_𝑙−𝜔₀𝐒−𝑖Γ/2+𝑐.𝑐.),_⟂=34𝛾_𝑟𝑘_𝑙𝑟𝑆₀(𝐞^±_𝑙(𝐞^∓_𝑙𝐞^±_𝑠)(𝐞^∓_𝑠𝐞_𝑧)𝑒^{𝑖(𝐤𝑙𝐫−𝑘𝑙𝑟)}𝜔_𝑙−𝜔₀−𝑖Γ/2+𝑐.𝑐.), (39)(40)(41) where +(−) correspond to the right (left) circular polarization of the stimulating light.

Since (40) and (41) are applicable at 𝑟→∞, it is obviously that due to the factor 𝑒^{𝑖(𝐤𝑙𝐫−𝑘𝑙𝑟)} only angles 𝜃→0 can contribute into the constant energy flux. However, there is the factor (𝐞^∓_𝑙𝐞_𝑧) in the RHS of (41), which equals 0 at 𝜃=0. Therefore, the constant energy flux on the large distances from a QD corresponds only to the vector 𝐒_𝑧.

Let us calculate the energy flux 𝚷_𝑧=𝑑𝑠𝐒_𝑧, (42) going through a plane perpendicular to the stimulating light direction 𝑧 in a time unite on large distances behind of a QD. A surface element equals 𝑑𝑠=𝜌𝑑𝜌𝑑𝜑, and 𝜌=𝑧tan𝜃, 𝑟=𝑧/cos𝜃, where 𝜃 is the angle between 𝐤_𝑙 and 𝐫. Having integrated on the angle 𝜃 which gives 2𝜋, using (36) and going from the variable 𝜌 to the variable 𝜃, we obtain 𝚷_𝑧=−3𝜋8𝛾_𝑟𝑧𝑘_𝑙𝐞_𝑧𝐒₀𝜔_𝑙−𝜔₀−𝑖Γ/2₀^𝜋/2𝑑𝜃sin𝜃×(1+cos𝜃)cos𝜃²𝑒^{𝑖𝑘𝑙𝑧(1−1/cos𝜃)}+𝑐.𝑐. (43)

Further we substitute variable 𝜃 by the variable 𝑡=(cos𝜃)⁻¹−1 and obtain 𝚷_𝑧=−3𝜋8𝛾_𝑟𝑧𝑘_𝑙𝐞_𝑧𝑆₀𝜔_𝑙−𝜔₀×−𝑖Γ/2_∞0𝑑𝑡(2+𝑡)1+𝑡²𝑒^{−𝑖𝑘𝑙𝑧𝑡}+𝑐.𝑐. (44) At 𝑧→∞𝑧^∞₀𝑑𝑡(2+𝑡)1+𝑡²𝑒^{−𝑖𝑘𝑙𝑧𝑡}→4𝑧^∞₀𝑑𝑡𝑒^{−𝑖𝑘𝑙𝑧𝑡}=−4𝑖𝑘_𝑙(1−𝑒^{−𝑖𝑘𝑙𝑧}), (45) and we obtain 𝚷_𝑧=3𝜋2𝛾_𝑟𝑘²_𝑙𝐞_𝑧𝑆₀(𝑖𝜔_𝑙−𝜔₀−𝑖Γ/2+𝑐.𝑐.)=−3𝜋2𝐞_𝑧𝑘²_𝑙𝑆₀𝛾_𝑟Γ(𝜔_𝑙−𝜔₀)²+Γ²./4 (46) In (46), the terms going to 0 at 𝑧→∞, and rapidly oscillating with 𝑧 are neglected. Analogically we calculate the contribution 𝚷_⟂=∫𝑑𝑠𝐒^±_⟂ into the energy flux and find that at 𝑧→∞ the value 𝚷_⟂→0.

10. The Cross-Section of Light Absorption

Since Γ=𝛾+𝛾_𝑟, the flux 𝚷_𝑧 may be represented as two parts 𝚷_𝑧=−𝐞_𝑧Π_scat−𝐞_𝑧Π_abs, (47) where Π_scat is determined by (35), Π_abs=3𝜋2𝑆₀𝛾_𝑟𝛾/𝑘²_𝑙(𝜔_𝑙−𝜔₀)²+Γ²/4. (48) Obviously that the energy flux −𝐞_𝑧Π_scat compensates the total flux of the scattered energy, and Π_abs corresponds to the energy, absorbed by the QD per time unite at the monochromatic irradiation. Having divided Π_abs on the density 𝑆₀ of the stimulating energy flux, we obtain the total cross-section of light absorption 𝜎_abs=3𝜆2𝜋^2𝑙𝛾_𝑟𝛾/4(𝜔_𝑙−𝜔₀)²+Γ²/4 (49) and reduces to 0 at 𝛾=0, as it was supposed in advance.

Comparing (49) and (37) we find that in the lowest approximation on the light-electron interaction the absorption cross-section is of the first order on that interaction (it contains the factor 𝑒²/𝑐ℏ), and the scattering cross-section is of the second order on that interaction (it contains the factor (𝑒²/𝑐ℏ)²). The ratio of the scattering and absorption cross-sections equals 𝛾_𝑟/𝛾. The absorption cross-section has the maximum value at comparable values 𝛾_𝑟 and 𝛾. At 𝛾_𝑟=𝛾 in the resonance 𝜔_𝑙=𝜔₀𝜎^res_scat=𝜎^res_abs=3𝜆8𝜋^2𝑙. (50)𝜎^res_scat and 𝜎^res_abs as functions of 𝛾_𝑟/𝛾 are represented in Figure 1.

Figure 1 

𝜎resscat and 𝜎resabs as functions of 𝛾𝑟/𝛾 in the resonance 𝜔0=𝜔𝑙. 𝜎0=(3/2𝜋)𝜆2𝑙.

11. Conclusion

Thus, the electric and magnetic fields induced at the light irradiation of QDs are calculated in the resonance of the stimulating light and excitons with the help of the semiclassical method of the retarded potentials. The fields on the large distances are calculated exactly.

The concrete calculations are performed for the excitons Γ₆×Γ₇ in cubic crystals of 𝑇_𝑑 class. It is shown that in the case of small or spherically symmetric QDs the light-electron interaction realizes into the substitution of the nonradiative excitonic damping 𝛾 by the damping Γ=𝛾+𝛾_𝑟 and the exciton energy 𝜔₀ by the energy 𝜔₀=𝜔₀+Δ𝜔, where 𝛾_𝑟 is the radiative damping, Δ𝜔 is the correction to the exciton energy due to the long-range exchange interaction of electrons and holes.

The light scattering and absorption cross-sections by QDs are obtained for arbitrary QDs under condition 𝑅≪𝜆_𝑙 and excitonic spherical envelope wave functions.