CC BY
11
Section 5. Physics
https://doi.org/10.29013/ESR-21-1.2-25-29
Akhmadaliyev Bozorboy Joboraliyevich, Fergana Polytechnic Institute, Uzbekistan E-mail: [email protected]
SPECIFIC FEATURES OF THE DISPERSION OF MIXED EXCITON-POLARITON MODES IN UNIAXIAL CRYSTALS OF THE CdS TYPE
Abstract. The dispersion of mixed exciton-polariton modes of single-axis crystals of the CdS type has been theoretically considered in the case where the boundary conditions of Boltzmann's kinetic equation do not apply. It is shown that different mechanisms of irradiated states occur.
Keywords: semiconductor, mixed exciton-polariton, transverse-longitudinal splitting, weak po-lariton effect, anomalous dispersion, inhomogeneous mode.
We consider the energy spectrum of mixed exci- known [1-3], is its strong anisotropy: in an arbitrary
ton-polaritons in a uniaxial CdS crystal near an iso- direction of wave propagation, different from k 1 C
lated dipole-active excitons resonance An = 1, the and k // C, at a fixed frequency, can be excited two
optical transition to which is allowed only in the transverse polariton modes (T1, T2) and two exciton-
E±C polarization of light (where C - is the optical polariton modes mixed type (M1, M2), the variances
axis of the crystal). A feature of such a spectrum, as is of which are schematically shown in (Fig. 1).
Figure 1. A schematic representation of the dispersion curves of emitting mixed (M1, M2) and transverse (T1, T2) polariton modes with a fixed direction of radiation into vacuum at an exit angle 0 without taking into account the attenuation. Dashed line - photons in the crystal, dotted - transverse and longitudinal excitons. The inset on the left shows the geometry of recording mixed-mode radiation (CXP geometry, i.e., C // X and p-polarized radiation is recorded in the XZ plane)
X
The dielectric tensor of interest to us s.. (w,k), of a crystal with spatial dispersion (SD) in the coordinate system shown in the inset to (Fig. 1) is characterized by only two components si (w,k) and s //(w,k), corresponding to the polarizations of light E1C and E//C. Near the resonance frequency «0, we can assume that s / /(w,k) » sb = const and the entire dependence of e.. on « and k is included in s 1 (w, k):
s 1 (w,k) = sb 1
1 + -
w
lt
wT (k ) -w- ir /2
(1)
where wT(k) = w0 + ñkl/2M// + ñ(k2-k¡)/2M±, (2)
eb L - is the component of the background permittivity tensor for polarization E LC, mlt = - -is the longitudinal-transverse splitting, hw0 and hwL - are the bottom energies of the bands of transverse and longitudinal excitons, kx - is the vector projection k onto the axis C, ML and M-1 - are the components of the inverse effective mass tensor exciton for the directions of propagation k L C and k L C.
When writing (1) and further, the inequalities
w
lt'
- u0\ << which are valid for many semiconductor crystals in the actual spectral region for resonant exciton luminescence, are considered to be satisfied. Taking into account these inequalities and (1) the dispersion equations for the considered modes
c 2k2/ w2 =s1 (w, k), (mods T1, T2)
2k 2
' (w, ^(mods M1,M2)
w
-c2(k2 -k2 )/e
w,
= e,
can be represented as dependencies
(t )
w = w\ -wTy
(ck / w0 )
-e
(m )
w = w\ =wM
lt
-e
.r
-i
2
r
--1—, 2
(3)
(4)
| + wLT, wlt =wltK / ebK
(5)
(ck /w0 j
where
wm (k) = wr (k)
and it is accepted that sb = sb 1» sb//.
Dispersion curves of the polaritons of the upper branches T2 and M2 in (Fig. 1) are described by ex-
pressions (3), (4 the wave vector
at r ^ 0 in the range of values of , and the curves T1 and M of the polariton branches - by the same expressions at k > <Jsbk0. Expressions (3), (4) with a real frequency « can be considered in the region of complex vectors (Fig. 2, curves M'1 and M'2), even at r = 0, corresponding to solutions for surface radiation modes with Re k2 < 0 and Im k2 = 0 (in this case (Im k • Re k) = 0), which are excited at the crystal -vacuum interfaces (surface- radiation modes). At r = 0, such waves are not associated with energy transfer in the medium and therefore do not directly contribute to external radiation. However, when r * 0, surface radiation modes are partially included in the energy transfer in the medium (see also Fig. 2), therefore, these modes should be taken into account in the crystal luminescence [4-6].
Areas of dispersion of inhomogeneous surface-radiation waves are actually shown in (Fig. 1) in the area to the left of the point of intersection of curves T2 and M2. To the right of this point at r = 0, the dispersion curves depict well-defined quantum (polariton) states of the crystal. The point of intersection itself corresponds to zero values of the normal components of the wave vectors of the modes at the interface.
When analyzing the process of light emission taking into account the dissipative damping hT, it is necessary to take into account that, near the frequen-
for the polaritons of the up-
cy WL,
the values
Im kc
ReL
and
per branches becomes comparable with the modes under consideration are, generally speaking, inhomogeneous (branches M' and M'2 in (Fig. 2) characterize the spatial attenuation ofwaves). In addition, in the same region of the spectrum, the real parts of the wave vectors of the mixed modes M and M2 take close values, especially when it comes to propagation directions close to k 1 C, which is clearly demonstrated by the numerically calculated dispersion curves in (Fig. 2). This leads to the fact that the criteria
Figure 2. The calculated dispersion curves of mixed exciton-inhomogeneous (a) and exciton-polariton (b-f) modes near in a uniaxial CdS crystal, corresponding to the angles of exit from the crystal to vacuum Q = 5° (a), 15° (b), 30° (c), 45° (d), 60° (e) and 80° (f) for the following values of optical parameters: h r = 0.1 meV, h o0 = 2552.4 meV, h oLT = 2 meV, eb1 = 9.4, ML= 0.9m0, ML= 2.85 m 0. At 0^0, the curves M1, M\ and M2, M'2 correspond to longitudinal excitons and transverse photons
Re hp >>a=2|Im k p|,
kb' kb" ^^ b', ^^ b" (where P' ^ P", and a - is the absorption coefficient for the polariton of the dispersion branch rj = p, P', P") the applicability of the kinetic equation near the frequency o>L is clearly violated for the polaritons of the T2, M2, and M1 branches.
The left inset to (Fig. 1) shows the geometry of recording the emission of mixed modes M and M2. In such a geometry of the experiment, it becomes possible to change the "longitudinal-transverse" splitting coLT (see (4), (5) and Fig. 2, a-f) by choosing the angle 6 of radiation exit into vacuum (i.e., the angle sin6>
0M b inside the crystal, sinö = kx / k(
m ß
yM ß
= kMßX / kMß, where ß = 1,2)
0 '
In this regard, writing the dispersion equation for mixed modes in the form (4) turns out to be the most convenient. In this case, the formal equivalence of the dispersion equations for transverse (3) and mixed (4) waves is achieved. The difference between (4) and (3) lies only in the fact that (4) contains another "resonant" frequency (o>L instead of o>0) and another "longitudinal-transverse" splitting ( cc stead of u>LT).
It should be added here that under experimental conditions the radiation spectrum is recorded at a fixed value of 6 (or coLT ), i.e. with a fixed projection kx of the wave vector (the same for all emitting modes k0x ° kMpx). This leads to the fact that for the emitting modes the angles 66Mp between kMp and C inside the crystal turn out to be variable (depending on the frequency «) and the dispersion equation (4) determines the states emitting light in the external direc-
tion specified by the angle 6 (the corresponding solutions of this equation are presented in (Figures 1) and 2 by curves M1 and M2). At nonzero values of hr, the angles 6M^ become complex. In addition, when the condition Re6Mp > Re6*Mjb is met, mixed polaritons cannot escape into vacuum due to total internal reflection. When hr is large enough, the concept of wave intensity inside a radiating medium loses its meaning.
As can be seen from (Fig. 2), even at a moderate value hr = 0.1 meV of a mechanical exciton, the dispersion curves M1, M1 'and M2, M2' for emitting states of mixed modes, depending on the exit angle 6, have a number of features. Firstly, at small angles of radia-tionexit 6 <10°, the polariton effect is practically absent due to its suppression by damping ( WLT << r ), and we have near the frequency œL only inhomoge-neous surface-radiation modes. Secondly, starting from the value 6 ~ 15°, a noticeable effective longitudinal-transverse splitting (weak polariton effect) is observed, which increases with increasing angle 6, reaching a value at 6 = 80°. Thirdly, when 6 varies in the range 30°-90°, we have an intermediate light-
in- exciton interaction with hcö
'lt
hr = 0.1
Fourthly, at 6 > 150, mixed polaritons practically do not show anomalous dispersion.
In conclusion, it can be concluded that the mechanism of emission into vacuum of states of mixed exciton-polariton modes in the vicinity of the frequency of the longitudinal exciton strongly depends on the ratio of the values of mechanical damping of the exciton hr and the effective longitudinal-transverse splitting h coLT, which is uniquely related to the angle of radiation exit in vacuum.
References:
1. Ahmadaliev B., Yuldashev N. Radiation of exciton-polariton mixed modes in crystals of type CdS in a vicinity of critical value of decay mechanical excitons. Norwegian Journal of development of the International Science // 2017.- Vol. 1.- No. 12.- P. 72-85.
2. Axmediyev N. N. Pol prontranstvennoy dispersii v pogloshchenii sveta eksitonami // JETF 1980.-T. 79.- No. 4 (10).- P. 1534-1543.
3. Akhmadaliev B. Zh., Polvonov B. Z., Yuldashev N. Kh. Influence of Exciton Decay on the Polariton Luminescence Spectra of CdTe Crystal // Optics and Spectroscopy,- Moscow, 2014.- Vol. 116.- No. 2.-P. 244-248.
4. Akhmadaliev B. Zh., Yuldashev N. Kh., Yulchiev I. I. Surface-Radiative Modes and Longitudinal Excitons in the Spectra of Exciton-Polariton Luminescence // Optika i Spektroskopiya, 2018.- Vol. 125.-No. 3.- P. 330-338.
5. Abukadyrov A. G., Sazhin M. I., Sel'kin A. V. and Yuldashev N. Kh. Polariton luminescence of mixed modes in crystals with spatial dispersion // Zh. Eksp. Teor. Fiz. 1990.- 97.- No. 2.- P. 644-662.
6. Yuldashev N. Kh. Eksiton - polyaritonnaya lyuminestsentsiya i perenos rezonansnogo izlucheniya v kristallah.- Fergana: "Farg'ona", 2002.- 214 p.
