CC BY
6
https://doi.org/10.29013/ESR-21-3.4-52-56
Rasulov Voxob Rustamovich, associate professor of Fergana State University Rasulov Rustam Yavkachovich, professor of Fergana State University E-mail: r_rasulov51@mail.ru Muminov Islombek Arabboyevich, doctoral student of Fergana State University. Qo'chqorov Mavzurjon Xurshidboyevich, teacher of the Kokand State Pedagogical Institute
Kodirov Nurillo,
teacher of physics at the Lyceum at Fergana State University
THEORETICAL ANALYSIS OF MULTIPHOTON INTERBAND ABSORPTION OF POLARIZED LIGHT IN CRYSTALS WITH A COMPLEX ZONE (Part 2)
Abstract. For specific cases, the spectral and temperature dependences of the coefficient of interband multiphoton absorption of light in narrow-gap direct-gap semiconductors are calculated. Optical transitions occurring in the presence of even and odd photons were calculated in the spherically symmetric approximation in the energy spectrum of current carriers.
Keywords: semiconductor, optical transition, photon, two-band Kane's approximation.
The first works on two-photon interband transi- dence of both the probability of multiphoton optical
tions in crystals were carried out in the early 1960 y., shortly after the appearance of lasers [1-3]. When calculating the matrix elements of two-photon transitions in crystals, perturbation theories were used in the field of an unpolarized electromagnetic wave [2; 3], where the two-band Kane model was used.
In [4-8], both theoretically and experimentally, the linear-circular dichroism (LCD) of two- and three-photon absorption of light in crystals of cubic symmetry was investegated, but the question of spectral and temperature researches of multiphoton interband absorption of polarized light in narrowgap crystals in the three-band Kane approximation remained open.
In the first part of this article, a general expression is obtained for the spectral and temperature depen-
transitions and the coefficient of multiphoton absorption of polarized light in narrow-gap crystals. Based on the results obtained in the first part of this work, below we will carry out a theoretical analysis of cases when this odd and even number of photons is involved. It should be noted that in further calculations (to simplify the solution of the problem) we will not focus on multiphoton optical transitions with simultaneous absorption of two photons, i.e. we will assume that photons are absorbed separately. Ifwe pay attention to the fact that in quantitative calculations the conservation law for multiphoton absorption of light energy is described by a function
( h 2k2 h2k2 ^
8--h E h---Nhrn with which it is possi-
^ 2mc 2mv J ble to determine the dependence of the wave vector
of electrons in the final state on the band parameters and on the frequency of light, then for expression (6) mentioned in part 1, we get:
c >mVj J
)
cml ;V,mV
/ c,mC ;V, mV
(-1)'
(N-1)/2~ N-1
r N -1 ^ v 2 y
.(1)
(fia)
P4
2N-1tj8N-7
n E„
f ( ^ (fcC2,m.-,™>))-f ( mc-
.(2)
where
C 2 N-1 ~ ^
i „2 \
V hc
2N-1 N 8N-6^2(N-1)21-2N
(2N - 1)hÉ
«
r
f3 [(N -1)!]4 (2N - 1)hÉ -
v
(3)
Figure 2 shows the plots of function
(2N - 1)hÉ E„
describing the spectral depen-
In particular, the spectral and temperature dependence of the coefficient of multiphoton absorption of light in the presence of an odd number of photons is determined by the expression K(N-1) (œ,T) = C2n-F2N-1 ((2N - 1)hffl) / Eg ,)h2(N-1) x
dences of the light absorption coefficient for three-and five-photon optical transitions corresponding to the optical transitions described in Figure 1. It can be seen from (Fig. 2) that in narrow-gap crystals the spectral dependence of the three- and five-photon light absorption coefficients passes through a maximum, and with an increase in the photon energy, the maximum values of the functions FN (x ) are shifted towards low frequencies. This is due to the fact that, in narrow-gap crystals, not only the matrix elements
of optical transitions, but also the densities of states
(2N - 1)hÉ
or current carriers depend on the ratio-.
p E«
A quantity characterizing the absorption of an even number of photons.
(2N - 1)hÉ E„
^(2N? Am0?)
c,mc ;V,mV \ c,mVi )
2N
2(N -1)
• p )
c ,m! ;V ,mV
( p) ,.
v £ /c ,m„;
2(N - 1)m0(mc +mV )hk
(2Nffl) c ,mv.
c ,mc';V ,mV
mcmv (ha>)2N 1
f (ec (?))-f ( Mc?))]-(4)
Figure 1. Schematic representation of optical transitions between three types of interband (a), between subzones of one zone (b)
Figure 2. Graphs of functions FN (x ), describing the spectral dependence of the light absorption coefficient in the case when the number of single photons is odd:
x = N hœ / E„
In this expression, the spectral dependence strongly depends on the frequency of light (with respect to case (2N-l) ) because the quantity ^ivV^m depends on the wave vector kkf^ in the presence of an even number ofphotons, while for an odd number of photons such dependence (4). As a result, the coefficient of multiphoton absorption of light with the participation of an even number (2N ) of photons is determined as follows
(2nW ^ r 2Nha), 2N-i P4N-3
K > ((0,T) = C2NF2N
where
C
r2N ha^
T" y
h2
2N 778N-5 , (5)
n E„
2N
e
V nc J
xr4N N-10 N
N n 2
4V2[(N - l)!]2 f2N-2 f -\(6)
Fn - (Nha / Eg -1)3/2 / (Nha / Eg )
\4 N-1
(7)
In (fig. 3) it is shown that plots of F2N (x) functions from x =-, which describe the light ab-
Eg §
sorption coefficient for two- and four-photon (see
( 2N ht?
Fig. 4) optical transitions
x = -
E
. It can be
g J
seen from Fig. 3 that in narrow-gap crystals the spectral dependence of the two- and four-photon absorption coefficients of light passes through a maximum and it shifts to the region of low frequencies as the photon energy increases.
a)
b)
Figure 3. Graphs of the function FN (x) for different values N : x = N ha! Eg .
For a more detailed analysis of the results in (Fig. 5) shows the graphs of function FN (x) for different values: x = N ha ! Eg, where the same dependences are obtained, shown in (Fig. 2, 3).
Figure 4. Schematic representation of four-photon optical transitions: three interband and one intraband (a), one interband and three intraband transitions (b).
Now, based on the above results, we present the spectral dependence of the coefficients of two- and three-photon absorption of light.
K(N=2)(x ) = K 20(2x )-5(2x -1)3/2, K(N-3) (x) - K30 (3x -1)1/2 (3x), (8)
Here
h2P 3
K = 5 5 —V K30 5,5 n3E7g
Figure 5. Function dependence FN (x ) on for different x = Nhœ/ E
x - 3hœ/ E
g
K20 - 0.4^
. In particular, K 30 = 73.8
m
GWt2
as-
sumes the value InSb (see Table 1). In quantitative calculations, the fine structure constant is taken into account equal to e2/[hc) = 1/137 . Figures 6 and 7 show, as an example, the spectral dependences of the coefficients of two and three-photon absorption of light in a narrow-gap semiconductor In As.
It can be seen from these results and (Figures 5-7) that with an increase in the frequency of light, the coefficient of multiphoton absorption of light first increases and reaches a maximum, and then decreases. This is due to the specificity of the Kane model used to research the band structure in narrow-gap crystals. In particular, in the Kane model, some off-diagonal matrix elements of the momentum operator do not depend on the wave vector of current carriers, which does not occur in the Luttinger-Kohn model.
Figure 6. Spectral dependence of the coefficient of interband two-photon absorption KN _2^2h©/E«)/K20 in a narrow-gap semiconductor
Figure 7. Spectral dependence of the coefficient of interband two-photon absorption
K(N-3)(3hœ/E « )/ K 30
g 30
semiconductor
Table 1. - Numerical values of the band parameters of some semiconductors
Crystal InSb GaAs InAs
Eg (V ) 0.235 1.519 0.417
Ep (eV ) 23.3 28.8 21.3
A50 (eV ) 0.81 0.34 0.39
Yi 34.8 6.98 20.0
Y 2 15.5 2.06 8.5
Y3 16.5 2.93 9.2
mso/mo 0.11 0.172 0.14
n 3.95 3,42 3.42
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