The polariton mechanism of luminescence plays an important role in the formation of low-temperature emission spectra of semiconductor crystals in the vicinity of exciton resonances. To date, many fundamental questions concerning the polariton luminescence (PL), which are related to the specific features of polariton dispersion, conditions of radiation transmission through a boundary in the vicinity of resonance, and the character of spatial and energetic distribution of polaritons, have been understood  -  . A detailed study of the exciton energy structure of crystals shows that, along with the phenomena of exciton-polariton interaction (polariton effect) and spatial dispersion (dependence of the permittivity tensor on the wave vector ), the effects related to exciton decay may play an important role in the formation of low-temperature photoluminescence spectra. In particular, exciton decay may cause build-up of the emission of radiative surface modes (purely spatially decaying waves, propagating into crystal bulk at ) and the interference luminescence of coherently emitting states of different dispersion branches with close ω and values. A PL theory was developed in   for the case where dissipative exciton decay may violate the applicability criteria for the kinetic Boltzmann equation
This theory proved to be valid at an arbitrary relation between and (where and are, respectively, the wave vector and the absorption coefficient for polaritons of upper dispersion branch 2). The contributions of the waves of lower polariton branch 1 and strongly damped wave 2 to the PL in the vicinity of longitudinal exciton frequency (branch 3) were successively calculated and the experimental spectra of ZnP2, CdS, and СdTe crystals were adequately analyzed based on this theory. However, the specific calculation and experiment were performed for the simplest case of normal polariton emission from a crystal into vacuum, where the contribution of longitudinal excitons to PL is zero. At the same time, it has been known for a long time (see, e.g.,    ) that energy transfer can be implemented in crystals by purely longitudinal excitons in the presence of spatial dispersion (SD). These excitons manifest themselves in low-temperature photoluminescence spectra of CdS crystals only in the case of oblique emission   . Nevertheless, their contributions to the EPL have not been quantitatively analyzed.
In this study, the microscopic PL theory for CdS-type crystals with SD is generalized with allowance for the contributions of the emission of purely longitudinal excitons in the vicinity of frequency . The EPL is considered for the case where inequality (1) is violated simultaneously for the additional waves and but remains valid for the polaritons of the lower dispersion branch . The developed EPL theory, in contrast to the conventional PL theory based on the use of polariton distribution function, allows one to take into account the interference of coherently emitting states of transverse polaritons and longitudinal excitons in the presence of decay (this is undoubtedly a new effect, which was not considered in      ), i.e., go beyond the applicability limits of the kinetic equation for waves 2 and 3, which is also determined, along with (1), by a more stringent condition:
If inequality (2) is valid, one can neglect the interference contribution of coherently emitting polariton states of different dispersion branches. Below, using this theory, we analyzed the angular dependence of experimental EPL spectra of CdS crystals with hexagonal point symmetry for s- and p-polarized radiation.
2.1. Statement of the Problem
Let us consider a dipole-active triplet exciton, which is characterized by the resonance frequency , effective mass M, and longitudinal-transverse splitting , in a crystal with isotropic permittivity tensor   
where is the background permittivity; is the mechanical
exciton energy; and is the exciton decay, which is determined by scattering or capture processes. It is known that normal waves of two types can be excited in these crystals: transverse ( , where c is the speed of light in vacuum) and longitudinal ( ). For most semiconductors (for example, meV and meV for CdS) and in the vicinity of the resonance , which is determined by the condition
the dispersion equations of transverse polaritons ( ) and longitudinal excitons ( ) can be presented in a simplified form:
Note that polaritons 1 and 2 are described by formula (5) in the ranges and , respectively ( ).
Figure 1 shows calculated exciton-polariton dispersion curves in the case of isolated exciton resonance with isotropic effective mass for the following values: 1) 0.075, 2) 0.5, 3) 1.0, 4) 1.457, 5) 2.0, and 6) 5.0 meV. As can be seen in the figure, the dispersion curves of exciton-polariton modes are significantly transformed with an increase in decay . This transformation is especially pronounced near the longitudinal frequency for modes 2, 2' and 3, 3', which can propagate only in the frequency range of and are purely decaying (i.e., radiative surface) when . The characteristic frequency is determined by the condition of total internal reflection for waves
Figure 1. Calculated energy spectrum of normal waves depending on the exciton decay ħГ in a cubic crystal with SD for the following parameters of the lowest exciton resonance A (n = 1) in CdS: ħω0 = 2552.4 meV, ħωLT = 2.0 meV, М = 0.9 m0 (m0 is the free-electron mass), and εb = 9.4  . The dashed lines correspond to noninteracting mechanical excitons and photons: 1, 1', 2, 2' are transverse polaritons and 3, 3' are longitudinal excitons.
Here, θ is the exit angle of radiation from the crystal into vacuum and is the refractive index of the crystal for wave β ( at ). We should note the following. First, the occurrence of an imaginary part of wave vector at (Figure 1, curves 2', 3') at indicates a certain spatial decay of propagating modes, which leads (in some frequency range ) to violation of inequalities (1) and (2). Second, in the frequency range , where condition (1) is always violated for waves 2 and 3, a real additive to wave vector arises at finite Г (curves 2 and 3); this additive indicates that radiative surface modes 2 and 3 cease to be purely decaying and are involved in the transfer of exciton excitation energy in the crystal. The latter circumstances are undoubtedly of great importance for the formation of the EPL spectrum of crystals in the emission geometry of purely longitudinal excitons, whose analysis is the object of our study.
Let us analyze the spectral intensity of radiation with polarization λ, propagating in vacuum near the crystal surface along the unit vector (which lies in the yz plane, while the internal normal to the crystal boundary is directed along the z axis; , where , ):
Here, at and at . The partial intensity describes the interference contribution of waves 2 and 3 to the EPL. We assume that low-temperature luminescence is excited via stationary illumination of crystal by natural light in the intrinsic absorption spectral region ( ), where free electrons and holes are generated. Being thermalized, these charge carriers are coupled (as a result of the Coulomb interaction) into excitons, which, in turn, are multiply scattered from phonons, impurities, and other lattice defects to relax (in both energy and momentum) into the vicinity of resonance ωo, where, in terms of the conventional PL theory, some nonequilibrium stationary polariton distribution function is settled. Near the frequency ωL, the densities of states
where is the crystal volume and is the polariton group velocity, are significantly different for the polaritons belonging to branch 1 and branches . For example, at , for CdS ( ). Then, naturally, the wave scattering within the branches can be disregarded in the vicinity of frequency . For the same reason, one can neglect the generation of waves 2 and 3 with small values (Figure 1(a)), which occurs due to the scattering by LO phonons from higher lying exciton states and due to their possible induction by multiple specular reflection of waves 1 from the internal crystal boundary  . Thus, the distribution function f of polaritons is formed mainly due to their relaxation over the states of the transverse mode .
Since the successive determination of the function is beyond the scope of our study, this function is considered as specified. Naturally, isotropization of the distribution function over directions and polarization states occurs in a cubic crystal as a result of multiple scattering. Furthermore, when carrying out a calculation in a narrow frequency range near , it is sufficient to restrict oneself to the simplest factorized form
which contains some smooth frequency dependence . Here, L is the effective distribution depth for the polaritons of branch 1, which is determined by the processes of multiple scattering of free carriers and excitons-polaritons. The coordinate dependence on only z characterizes the problem as homogeneous along the surface plane.
The partial contribution is determined from the formula
is the energy transmittance of wave β with polarization λ, emerging from the crystal into vacuum, and is the corresponding amplitude transmittance. In expression (8), is the normal component of the energy fluence to the surface and is the coupling coefficient between the energy fluence and the squared modulus of the electric field amplitude for the normal wave β:
2.2. Theoretical Calculation
The intensity of s-polarized EPL,
does not contain the interference term, because condition (2) is assumed to be satisfied. The partial contribution is determined in the kinetic approximation from formula (8), where λ = s. To calculate at finite exciton decays, i.e., with allowance for the arbitrary relation between Rek2 and α2, we will use the Keldysh diagram technique and find the Green's function for polaritons of branch 2 with polarization λ at a specified distribution function for polaritons of branch 1, neglecting the scattering in branch 2. Figure 2 shows the corresponding diagram for the function (here, β = 2). The solid lines present the exciton Green’s functions, renormalized with allowance for the exciton-photon and exciton-lattice interactions
where is the polariton decay, determined by the scattering or capture processes. The function is expanded in a sum of two pole terms:
Figure 2. Diagram for the Green’s function : at and at .
When calculating , the functions and are assigned to the upper and lower exciton lines in Figure 2. It is not necessary to take into account the diagram with external lines and (i.e., to consider the 1®1 scattering), because the function is assumed to be specified. Using the diagram technique rules, we obtain
Here, is the amplitude of a light wave with frequency ω and transverse component (note that is a good quantum number), equal to the transverse component of the wave vector of this wave in vacuum; is the amplitude transmittance; is the refractive index of wave β; is the polarization vector; is its projection onto the X axis; Zj is the coordinate of the jth scattering center; and C is a constant.
Substituting (9b) into (9a), replacing summation over j with integration over according to the rule , and integrating over kz and , we finally arrive at
where ; the values of and n2 are calculated with allowance for the decay .
In the case of elastic scattering from static defects,
where Ni is the defect concentration and is the Fourier transform of the perturbing potential. For an isotropic potential, . Note that, at , the k2(ω) value is small in comparison with k1(ω), and the term in (9c) can be replaced with .
In the case of p-polarized radiation, to calculate the partial contributions , and the interference component to the total intensity , using formula (8),
We will consider the Green’s functions (for the transverse polaritons of branch 2) and (for longitudinal excitons). The diagram for these functions is shown in Figure 2, where and . The solid line in the case of longitudinal exciton corresponds to the Green's function
where is determined from formula (6), and is a solution to the dispersion equation for longitudinal excitons:
Using the corresponding considerations and calculation technique for the s polarization, we arrive at
and is obtained from by replacing with . It was taken into account that the electric field strength and the exciton part of crystal polarization for longitudinal excitons are related as follows: . Expression (14) describes the interference contribution of the waves of branches 2 and 3 to the intensity of external p-polarized radiation; this contribution is caused by scattering from the same polariton center of branch 1 into coherently emitting states 2 and 3 under conditions where inequality (2) is also violated, along with Equation (1).
Now we have to calculate the amplitude transmittances of wave β into vacuum. To this end, we used, along with the boundary Maxwell conditions, Pekar additional boundary conditions with a “dead layer.” For the case where p-polarized wave 2 falls on the crystal surface from the bulk, we find
The transmittance is obtained from by the symmetric replacement . When a longitudinal wave falls on the crystal surface from the bulk, we determine the amplitude coefficient of conversion of longitudinal-exciton normal wave into external light as follows:
3. Discussion of the Results of Numerical Calculation and Comparison with Experimental Data
The partial ( , ) and integrated ( , ) luminescence spectral intensities were numerically calculated using formulas (8 - 10) and (13 - 15) as basic ones. Auxiliary formulas of the (16), (17) type were applied to calculate the amplitude transmittances and refractive indices . The only variable parameter of the theory is the effective distribution depth L for lower branch polaritons. The value was assumed to be known from experiment. Numerical calculations were performed with the following values of the main parameters of exciton resonance for CdS crystal: , , ( is the free-electron mass), background permittivity , and “dead layer” thickness . These values of parameters are in good agreement with the experimental results on exciton reflection of light and exciton luminescence at T = 2 K   .
Figure 3 shows the theoretical spectral dependences , and , calculated with different values of the G and L parameters. The intensity and half-width of the curves and, correspondingly, turned out to depend strongly on . Similar parameters of the curves and at L > 1 μm are barely sensitive to variations in L, which is due to the relatively large absorption coefficient for wave 3 .
Moreover, the half-width D3 of is practically completely determined by the G value. According to the numerical calculation, . Note that the contribution of to sharply decreases with an increase in ; at the
Figure 3. Spectral dependences of the total (I) and partial (I2p, I3p) intensities and the interference contribution Ip23, calculated for the parameters of exciton resonance А(n = 1) in CdS at the exit angle θ = 800 and different values of the exciton decay, hГ= 0.075 (1 - 3) and 0.15 (1' - 3') meV, and the distribution depth of branch-1 polaritons, L = 0.2 (1, 1'), 1.0 (2, 2'), and 2.0 (3, 3') μm.
same time, increases and significantly broadens. As a result, at the same L value, a rise in G leads to a slight decrease in the maximum of the dependence at the frequency and its broadening. This is explained as a result of SD suppression for longitudinal waves by mechanical-exciton decay.
The quantum theory presented in Section 2 was used to describe the experi mental EPL spectra of CdS crystals in “isotropic” emission geometries (s polarization, when lies in the хz plane and the optical axis , and p polarization, when ; here, the z axis is perpendicular to the emitting crystal surface). Spectra were recorded on a setup developed by Professor A.V. Sel’kin (we kindly acknowledge his support). The setup was based on a DFS-24 spectrometer and operated in the photon-counting mode at a minimum spectral width of the slit (0.04 meV), under conditions providing maximum spectral resolution. EPL excitation was performed at a wavelength λ = 476.5 nm by an Ar+ laser beam focused on the crystal surface into a spot ~0.4 × 4 mm2 in size at a luminous flux power of 7 mW.
Figure 4(a) and Figure 4(c) demonstrate the general view of the luminescence profiles for the CdS crystal in the s-emission geometry at (a) and (c) . The dashed line in Figure 4(a) presents the experimentally measured (from the 2 LO replica spectrum) occupancy function for the polariton states of branch 1; it was used to determine the distribution function , which is also shown in Figure 4(a) by a dotted line. The theoretical EPL spectra in Figure 4(a') and Figure 4(c') (curves 0), calculated at , , and , are in good agreement with experiment (triangles). The partial contribution of (dotted lines 2) plays an
Figure 4. (a) (b) (c) Experimental and (а') (b'), (c') theoretical EPL spectra of CdS crystals (T =2 K) for the detection geometries corresponding to the (a) (а') (с) (c') s and (b) (b') p polarizations. The dashed line in panel shows a 2LO-replica spectrum of the occupancy function for polariton states of branch 1; this spectrum was used to calculate the distribution function f1 (w), which is shown by a dotted line. (0) Total intensity and (triangles) experimental data, (1 - 3) the contributions of the transverse polaritons (branches 1, 2) and longitudinal excitons (branch 3), and (23) the interference contribution Iр23.
important role in the formation of total intensity in the frequency range .
We should especially note the following. As can be seen in Figure 4(a') and Figure 4(c'), using renormalization of dashed curve 1 (describing the contribution of lower branch polaritons), one can obtain agreement between this curve and experiment with a certain error; however, within this error, it is not correct to put a question about the exact value of contribution . Therefore, the good description of experimental results based on the kinetic theory in the studies devoted to the PL in the vicinity of longitudinal exciton frequency seems doubtful.
Figure 4(c) shows an EPL profile for a CdS crystal in the p-polarized emission geometry at ; here, along with the fundamental maximum , one can see a small additional maximum , located at the short-wavelength side of (at the frequency ). Previously the occurrence of this maximum was explained by only the “deexcitation” of purely longitudinal excitons. However, according to the results of numerical calculations presented in Figure 4(c'), the line is formed in a complicated way. The spectral dependences of the total intensity (solid line 0) and the partial contributions (curve 1), (2), (3), and the interference component (23) were calculated for the known CdS parameters  at the same G, L, and l values as the curves in Figure 4(a') and Figure 4(c'). The frequency range is noteworthy; here, within the kinetic approximation, longitudinal excitons (as well as transverse polaritons 2) cannot be transformed into external photons, i.e., yield luminescence into vacuum. According to the wave theory, which is consistent with the experiment (compare curve 0 and triangles in Figure 4(c'), the exciton decay induces emission of radiative surface modes and their interference. This, undoubtedly, is a new mechanism of EPL formation in the crystals having spatial dispersion. Strictly speaking, the frequencies are somewhat different for waves 2 and 3. At in the vicinity of , the dispersion relations for waves 2 and 3 can be expressed accurate to the small parameter using the formulas
Hence, (18) and (19) yield the following relations for the characteristic frequencies and :
In the case of ultimately large exit angles ( ), these relations yield the following expressions for the CdS parameters at T = 2 K:
As was shown by the experiment and numerical calculation (Figure 4(c')), the emission line is peaking specifically near the characteristic frequency of transverse polariton 2: . Due to the influence of exciton decay G, the real maximum of the frequency dependence shifts to shorter wavelengths with respect to the frequency and approaches the frequency . Figure 4(c') shows also that the profile of the line cannot be obtained using only the contribution . It should be especially noted that, even for the practically minimal value for CdS crystals, the kinetic approximation cannot be used to describe the partial contributions and in the vicinity of the emission line , because and inequality (1) is not satisfied at for the longitudinal wave, whereas for the transverse mode is a critical frequency, below which the value is purely imaginary at for the given direction in vacuum.
Figure 5 shows a comparison of the theoretical (curves 0) and experimental
Figure 5. Comparison of the theoretical (curves 0) and experimental (triangles) EPL spectra for the (а) (с) (е) s- and (b) (d) (f) p-polarized emission from CdS crystals at different exit angles: θ = 300, 450, and 600. The designations are the same as in Figure 4.
(triangles) spectra of CdS crystals and the calculated frequency dependences of the partial contributions (curves 1, 2, 3, 23) in the emission geometries corresponding to the s and р polarizations at different values of the exit angle , , and . One can see that the relative contributions and to the luminescence depend weakly on θ; however, the contribution (curves 3) from normal wave 3 significantly increases with a rise in θ and plays a key role in the formation of the line.
The rise in the contribution with an increase in θ is directly related to the manifestation of SD in luminescence spectra. In the absence of SD, waves 3, as well as waves 1, cannot exist in the frequency range . At , longitudinal waves 3 undergo total internal reflection and, therefore, the contribution of to is zero. This situation occurs also at , a fact explaining the sharp decrease in with an increase in frequency in the range . It is noteworthy that the interference component gives a small negative contribution to the total intensity , and its absolute value decreases with an increase in θ, when modes 2-2' and 3-3' diverge in Figure 1. However, without the contribution of , it is difficult to obtain quantitatively a small minimum at ωL in the EPL spectrum that would be practically independent of θ. Let us analyze in more detail the blue shift of the line with respect to that occurs with an increase in angle θ at and . In this case, taking into account that , the dispersion relation for the states of branch 2 (near ), emitting into vacuum, can be written as
Naturally, for the θ values at which , wave 2 undergoes spatial decay even at . As follows from (22), this occurs in the frequency range at and, at , in the frequency range below , the formula for which can be derived from (22):
This expression differs from (20) by only the replacement of with . Thus, with an increase in θ, the bottom of the energy band of “light” transverse polaritons emitting into vacuum shifts upwards; i.e., the longitudinal transverse splitting somewhat increases (see (23)). This effect manifests itself mainly in the shift of the emission line with an increase in θ. For two exit angles, and , formula (24) yields the relation
Using this relation, we obtain a value of 3.2 for CdS ( ) at , , whereas Figure 5(b), (f )demonstrates a somewhat different value: 2.8. The cause of this small inconsistency is that the condition is not exactly satisfied in the case under consideration.
1) A theory of exciton-polariton luminescence of cubic crystals with spatial dispersion, when the applicability criteria of the kinetic Boltzmann equation for the geometry of longitudinal-exciton emission are violated, was developed. A new mechanism of EPL formation was proposed, which involves coherent scattering of polaritons from well-defined initial quantum states into emitting (final) states of normal modes with a short lifetime, for which the wave vector is not a well-defined quantum number.
2) The formation of EPL spectrum has a number of new specific features. Due to the transformation of the normal-wave spectrum with allowance for the real exciton decay, radiative surface modes are involved in the energy transfer through the crystal boundary, and the presence of SD leads to their interference interaction. The contributions of and to sharply decrease with an increase in , which can be explained by the SD suppression due to mechanical-exciton decay for longitudinal waves. At the same time, the contribution of increases and becomes significantly broadened, as a result which an increase in G at the same L value leads to a weak decrease in the amplitude of the peak at and its broadening.
3) The contribution of purely longitudinal excitons to the EPL and their interference with polaritons of the upper dispersion branch are low (~10−20%); nevertheless, they must be taken into account to obtain quantitative agreement with experiment.
4) The above-described method for analyzing EPL spectra makes it possible to determine the optical parameters of crystal (for example, the value proceeding from the frequency of the line). It can be used with small variations to study the inelastic scattering spectra of polaritons in the vicinity of longitudinal exciton frequency (taking into account the finite decay and SD).
We are grateful to A.V. Sel’kin for supplying experimental results.
 Permogorov, S.A., Travnikov, V.V. and Sel’kin, A.V. (1972) Effects of a Spatial Dispersion in Reflexion Spectra at Inclined Falling of Light on Crystal Border. Physics of the Solid State, 14, 3642-3651.
 Abdukadirov, A.G., Ivchenko, Е.L., Sajin, М.I., Sel’kin, A.V. and Yuldashev, N.Kh. (1989) Поляритонная люминесценция в окрестности частоты продольного экситона с учетом затухания. Optics and Spectroscopy, 67, 845-850.
 Akhmadaliev, B.J., Polvonov, B.Z. and Yuldashev, N.Kh. (2014) Influence of Exciton Decay on the Polariton Luminescence Spectra of CdTe Crystal. Optics and Spectroscopy, 116, 244-248.
 Bagaev, V.S., Klevlov, Yu.V., Kolosov, S.A., Krivobok, V.S. and Shepeli, A.A. (2010) Optical and Electrophysical Properties of Defects in High-Purity CdTe. Physics of the Solid State, 52, 37-42.