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Resonant Light Absorption by Semiconductor Quantum Dots
The cross-section of light absorption by size-quantized semiconductor quantum dots (QD) is calculated in the case of a resonance with an exciton in cubical crystals of 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 .
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 equals to the exciton frequency . The resonant peak width is determined by the exciton damping , which consists of nonradiative and radiative damping. An important role of the radiative damping was proved for the first time in [6–8]. Light reflection by some structures, consisting of quantum wells, wires, and dots was considered in .
The light elastic scattering on a QD of arbitrary form and sizes in a resonance with excitons is investigated in , 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 .
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 , 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., ).
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 , 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 and charge 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 in these expressions leads again to the limitation by the lowest order on the light-electron interaction. But substitution of the genuine fields inside of the object leads to the precise results.
The induced electric and magnetic fields are represented with the help of the vector and scalar potentials The retarded potentials are [19, page 447] where 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 and substituting (2) into (1), we obtain an integral equation for the induced field 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 . The main contribution into the average current density at temperature is as follows: where determinations of  are used: is the average value on the ground state, is the commutator of operators and , is the operator of current density in the interaction representation, , , , is the classic electric field.
We assume . It was assumed also in (3) that currents and charges are absent at infinite distances, and that the electromagnetic field equals 0 at , 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  and suppose that the stimulating light frequency or carrying frequency of a pulse irradiation are close to the semiconductor energy gap . Sizes of a semiconductor object are much more than a lattice constant . The distances, on which the smooth multiplier from the wave function varies, are much greater 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 where the determinations are used: are the electron charge and mass, respectively, is the interband matrix element of quasimomentum operator, corresponding to the exciton with indexes , is the envelope exciton wave function at , where is the electron (hole) radius-vector, is the exciton energy counted from the ground state energy, is the nonradiative exciton damping. The set 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 ). Finally, is the Fourier transform of the electric field The main and conjugated contributions in (5) are chosen so that where is the wave vector of stimulating light, the function describes the form of a stimulating pulse . The average density of charge induced inside of a semiconductor object may be determined with the help of (4) and continuity equation
4. The Case of the Exciton in a QD
Let us consider an exciton consisting of an electron from twofold degenerated conductive band and a hole from twofold degenerated valence band chipped of by the spin-orbital interaction in class crystals. The exciton (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 and the hole wave functions are Having combined (8) and (9) in pairs, we obtain fourfould degenerated excitonic state for which the interband matrix elements of momentum operator result in where the scalar is introduced; are the unite vectors along the crystallographic axes.
We assume that the QD radius 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 exciton, when other excitons and excitonic energy levels may be neglected. Our energy level is nondegenerated, then the index takes on four values from 1 to 4, and only the values , according to (10), depend on . Without light-electron interaction we have though the function may be chosen as a real one. After summation on in (4) with using (10), we obtain where
Using (2), we obtain electric and magnetic fields on large distances from a QD where 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 , determined by the genuine electric field inside of the QD and consisting of stimulating and induced fields where is the circular polarization vector. In the case of a monochromatic irradiation but at a pulse irradiation the frequency is widespread in the interval near , and is vice inverse to the pulse duration .
Using the formulas for retarded potentials, we obtain the induced field where Equation (20) is an integral equation for . Then
Equation (23) may be considered as a system of three equations for the components . 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 in 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 depends only on the module , that is, The condition (25) is fulfilled, if the function is spherically symmetrical or the QD sizes are much less than the wave length of the stimulating light when . 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 corresponds to the lowest excitonic energy level , and Then where The value (30) of an exciton energy correction coincides completely with the results of  and preceding results of [16–18], where it was considered as the result of the long-range exchange electron-hole interaction (see ). 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 by and by .
For the case (19) of the monochromatic irradiation we obtain where .
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 where With the help of (31) we obtain The module of the total flux of scattered light per time unite is Obtaining (35) from (34) we have used the relations where is the scattering angle. An integration is performed. Having divided on the density of the stimulating light flux and using , we obtain the total cross-section of light scattering The differential cross-sections have been determined in .
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 . 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 of excitons, and it equals 0 at . The same result was obtained at a monochromatic irradiation of a QW [7, 24]. The cause is that at 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 [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., ). 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 in the case of a monochromatic irradiation. Using (31) and we obtain where correspond to the right (left) circular polarization of the stimulating light.
Since (40) and (41) are applicable at , it is obviously that due to the factor only angles can contribute into the constant energy flux. However, there is the factor in the RHS of (41), which equals 0 at . Therefore, the constant energy flux on the large distances from a QD corresponds only to the vector .
Let us calculate the energy flux going through a plane perpendicular to the stimulating light direction in a time unite on large distances behind of a QD. A surface element equals , and , , where is the angle between and . Having integrated on the angle which gives , using (36) and going from the variable to the variable , we obtain
Further we substitute variable by the variable and obtain At and we obtain In (46), the terms going to 0 at , and rapidly oscillating with are neglected. Analogically we calculate the contribution into the energy flux and find that at the value .
10. The Cross-Section of Light Absorption
Since , the flux may be represented as two parts where is determined by (35), Obviously that the energy flux compensates the total flux of the scattered energy, and corresponds to the energy, absorbed by the QD per time unite at the monochromatic irradiation. Having divided on the density of the stimulating energy flux, we obtain the total cross-section of light absorption and reduces to 0 at , 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 ), and the scattering cross-section is of the second order on that interaction (it contains the factor ). The ratio of the scattering and absorption cross-sections equals . The absorption cross-section has the maximum value at comparable values and At in the resonance and as functions of are represented in Figure 1.
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 in cubic crystals of 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 by the damping and the exciton energy by the energy , where is the radiative damping, 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 and excitonic spherical envelope wave functions.
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