CC BY
12
https://doi.org/10.29013/ESR-21-5.6-42-46
Rasulov Voxob Rustamovich, associate professor of Fergana State Universityy
Uzbekistan Rasulov Rustam Yavkachovich, professor of Fergana State University, Uzbekistan E-mail: r_rasulov51@mail.ru Muminov Islombek Arabboyevich, doctoral student of Fergana State University,
Uzbekistan
Eshboltaev Ikbolzhon Mamirzhonovich, Associate Professor of the Kokand State Pedagogical Institute,
Uzbekistan
Qochqorov Mavzurjon Xurshidboyevich, teacher of the Kokand State Pedagogical Institute,
Uzbekistan
LINEAR-CIRCULAR DICHROISM OF INTERBAND THREE-PHOTON ABSORPTION IN CRYSTALS (PART 1)
Abstract. Interband three-photon optical transitions in crystals of the InSb and type are classified, and the spectral dependence of some optical transitions is analyzed. The coefficient of linear-circular dichroism of interband three-photon absorption in a crystal in the Kane model is calculated.
Keywords: interband three-photon optical transitions, spectral dependence, crystal.
The first works devoted to the research of two- absorption ofpolarized light in narrow-gap crystals in
photon interband transitions in crystals were carried the three-band Kane approximation remained open.
out in the early 1960 s, shortly after the appearance Three-photon interband light absorption can be
of lasers [1-3]. In calculating the matrix elements described by diagrams of type ; i j ,
of two-photon transitions in crystals, perturbation j $ \J , where X describes one photon
theories were used in the field of an unpolarized elec- : i
tromagnetic wave [2; 3], where the two-band Kane absorption, XX describes the successive absorption
model was used. of two photons, and 'y describes the simultaneous
In [4-7], both theoretically and experimentally, absorption oftwo photons. Then three-photon optical
linear-circular dichroism (LCD) of two- and three- transitions from the valence band ( \V!, m ¡) ) to the con-
photon absorption of light in crystals of cubic sym- duction band (\c,m[) ) generally have four types,
metry in the region of the center of the Brillouin zone which can be represented as the sum of the following
was investigated. However, the question of spectral optical transitions depending on the initial state ofelec-
and temperature researches of multiphoton interband trons participating in optical transitions: a) the initial
state of electrons is in the subband ofheavy holes
X {I Vhh, ±3/2) ^ V, m) c, ± 1/2) + \Vhh, ±3/2) ^ \V,, m) ^ \V,, ml) c, ± 1/2) +
m,, m =±1/2, ±3/2
+ 1 Vhh, ±3/2) c, m, c, m')^\c, ± 1/2) + |Vhh, ±3/2) ^ |SO, m, |SO, m\ c, ± 1/2) + + | Vhh, ±3/2) ^ V, m, c, m)^\c, ± 1/2) + \Vhh, ±3/2) ^ |c, mt | V,, m\ c, ± 1/2) + + | Vhh, ±3/2) ^ V, mr | SO, m;>^| c, ± 1/2) + | Vhh, ±3/2) SO, m, | V,, m) ^ |c, ± 1/2) + + |Vhh,±3/2) c,m,SO,m)^\c,± 1/2) + \Vhh,±3/2) ^|SO,m,c,m)^ |c,± 1/2)}, X {| Vhh, +3/2) ^ \V, m) c, ± 1/2) + \Vhh, +3/2) ^ \V, m) ^ |V,, ml) c, ± 1/2) +
m,, m; =±1/2,±3/2
+ 1 Vhh, +3/2) c,m) c,m')^ |c,± 1/2) + |Vhh, +3/2) ^ |SO,m) ^\SO,m') c,± 1/2) + + | Vhh, +3/2) ^ V,,m) c,m;)^|c,± 1/2) + \Vhh, +3/2) ^ |c,m) ^ | V,,m) ^\c,± 1/2) + + | Vhh, +3/2) ^ |V,, m) ^ | SO, m;)^| c, ± 1/2) + | Vhh, +3/2) SO, m) ^ | V,, ml) ^ |c, ± 1/2) + + |Vhh, +3/2) c,m) SO,m)^\c,± 1/2) + |VM, +3/2) ^|SO,m) c,m)^ |c,± 1/2)};
b) the initial state of electrons is in the subband of heavy holes
x {| Vh, ±1/2)" ^ "I V, m)" ^ "I c, ± 1/2)+| Vlh, ±1/2) ^ V, m V, m)^\ c, ± 1/2)+
m,, m;=±1/2,±3/2
+ 1 Vhh,±1/2) ^ |c,ml|c,m)^\c,± 1/2) + |V,h,±1/2) ^ |SO,m,|SO,m)^ |c,± 1/2) +
+ | Vm, ±1/2) ^ V, m: c, m)^\c, ± 1/2) + |Vft, ±1/2) c, mt | V,, c, ± 1/2) +
+ | Vlh, ±13/2) ^\V, m, |SO, m;)^| c, ± 1/^ + |Vh, ±1/2) SO, ml \V,, m\)^\c, ± 1/2) + + | Vlh,±1/2) c,m) SO,m;)^|c,± 1/^ + | Vh,±1/2) ^ |SO,m) c,c,± 1/^},
X {| Vh,+1/2) ^ V, m,) c, ± 1/2) + \Vlh,+1/2) ^ V, m,) ^ V, m\) c, ± 1/2) +
m,, m| =±1/2,±3/2
+ 1 Vhh,+1/2) ^ |c,ml|c,m)^\c,± 1/^ + |V,h,+1/2) ^ |SO,m,|SO,m,|c,± 1/^ + + | Vm,+1/2 ^\V, m, c, m;)^| c, ± 1/2 +1 Vh,+1/2 c, m, |V,, m\ c, ± 1/^ + + |Vhh,+13/2 ^ \V,m,|SO,m')^ |c,± 1/2 + |Vh,+1/2 HSO,m,|V,m';)^\c,± 1/2 + + |Vhh,+1/2 H |c,m,)h|SO,m';)H\c,± 1/2 + |Vh,+1/2 H |SO,m,)h|c,m[) c,± 1/2};
c) the initial states of electrons are in the subband of heavy holes
^ fl SO, ±1/2 ^ \V, m,) c, ± 1/2 + | SO, ±1/2) ^ , m) ^ \Vl, m) ^\c, ± 1/2 +
m,, m/=±1/2,±3/2
±1 SO, ±1/2 H c, m,) c, m[)^\c, ± 1/2) + | SO, ±1/2) ^ |SO, m) ^ |SO, m') c, ± 1/2 + ±1 SO, ±1/2) H |V, m) c, \c, ± 1 /2) + | SO, ±1/2) c, m) H \Vt, m\) c, ± 1/2 + + | SO, ±13/2) H \V, m) H |SO, c, ± 1/2 + 1 SO, ±1/2) SO, m) H \V, m) H\c, ± 1/2) + + 1 SO, ±1/2 H c, m) SO, m)H\ c, ± 1/2 + | SO, ±1/2) H |SO, m,) c, m;)^| c, ± 1/2},
X {I SO,+1/2) ^ , m,) c, ± 1/2 + | SO,+1/2) ^ , m) ^ \Vt, m\)^\c, ± 1/2 +
Ml, =±1/2,±3/2
I c, ± 1/2) + | SO,+1/2) — I SO, mt) — |SO, m\) —|c, ± 1/2) + ► |c,± 1/2) + |SO,+1/2) — |c,m,) — \Vl,m,') —|c,± 1/2) +
+ | SO,+1/2) — |c, mj — |c, m\)-+ | SO,+1/2) — |V,, m) — I c, m\) + 1 SO,+1/2) — |V,, m,} — I SO, m') — \ c, ± 1/2) + | SO,+1/2) — | SO, m,} — | V,, m\) — \ c, ± 1/2) + + 1 SO,+1/2) — I c, m) — I SO, m¡) — |c, ± 1/2) + | SO,+1/2) — |SO, m) — | c, m) — \ c, ± 1/2)},
where each component differs from each other in the order of virtual states, which can appear both in the subbands of the valence band (| Vl, m) and in the conduction band (| c, m') ) or in the spin-split band ( | SO, m') ), mj, m'1=±3/2 for subbands of heavy holes, ml, ml = ± 1 / 2 for subbands oflight holes, conduction bands and spin-split-off bands, ml or m' the eigenvalue of the total momentum operator. | A, ma ) ^ | B, mb) characterizes the optical transition from state |A, ma) to \ B, m^j, occurring by the simul-
taneous absorption of two photons (see, for example, [5-7]), | A, ma) B, mb) characterizes a one-photon optical transition from the \A,ma) to \B,mb)state.
Note that some of the above matrix elements may turn out to be equal to zero in the zero, linear, and quadratic approximation in the wave vector in the energy spectrum (in the effective Hamiltonian).
In what follows, we choose the following Luttinger Cohn basis functions [8]. Then the effective Hamiltonian in the above sequence of basic functions takes the form
fl2k2 Ec + n k 2mc 0 42 k+ "Si 3 z V6 0 S K
0 h2k2 Ec + n k 2mc 0 46 k+ ŒP*. 3 z -r*- 42 S K S k
à- 0 F H I 0 H S 42i
L - Pk \3 z -iPk 46 - H* G 0 I G - F V2 V 2
Pk V6 k+ L - Pk \3 z I* 0 G -H ^ r^H ' V 2 F - G V2
0 V2 k+ 0 I* -H* F Si * H *
iPk V3 kz 43 H * V2 G - F 42 pH 2 -42i F + g . --A 2 0
s k+ - iPk 43 k Si ' Hh * 2 F - G S H V2 0 F + G . --A 2
where each component differs from each other by the order of the band parameters [8]. In particular, the dimensionless Luttinger constants yl,y2,y3 are related to the band parameters a, b, d as
2m,
Ti ^ A
r B
—T =_ ■ 2m 2
2m,
T =
D
2^
, the nu-
merical values of which are given in [9]. Th en the energy spectrum of light and heavy holes takes the
form Elh (k) - Ev - (A + B)k2 , Ehh (k) = Ev —
— ( A — B)k2, where the effective masses of light and heavy holes do not depend on the direction of the wave vector and are determined by the relations:
■ = A + B = ■
<Yi + 2/2),
2mh 2mo " ' 2mhh
■ = A - B -
2m,
- (y1 - 2y2 ). In this case, the matrix elements of
2
the pulse operator are determined by e -VH(k) , where e is the vector of the polarization of the light.
Note that there are 16 types of optical transitions that differ from each other in virtual states. Therefore, below we will consider individual optical transitions. For example, the matrix element of an optical transition described by the following
~^PcB2k 2< V2 c +
1
(-2 hco)
4(A -1)2ef B
+
3\e'_\
diagrams is determined by the relation, and for transitions of the type \V,-3/2) ^ | V,-3/2) ^ |V,-3/2) c,-1/2), \V,-3/2) ^ |V,-3/2) ^
V, -1/2) c, - 1/2),|V, -3/2) ->| V, -1/2) ^
^ |V,-3/2) c,-1/2)\V,-3/2) ^|V,-1/2) ^
V,-1/2) ^ |c,-1/2) we have
(_M (Eh _Ehh _M
+
2e
2
(( _ Ehh _ 2h®)
2(f -1) + 2(A+1)
(_M (( _ Ehh _M
2
Then the spectral dependences of the coefficients of the linear-circular dichroism of these optical transitions, determined using the probabilities of these transitions, are shown in Fig. 1. Figure 1 shows that the spectral dependence of the linear-circular dichroism coefficient ( ^ (œ) ) depends on the type of opti-
a)
cal transitions. In particular, for the first type of optical transition, it increases with an increase in the frequency oflight, and for the second type of optical transition, with an increase in the frequency of light, it first decreases and, passing through the minimum,
increases.
b)
Figure 1. Spectral dependence of the linear-circular dichroism coefficient for three-photon interband light absorption in crystals of cubic symmetry for two cases
References:
1. Miller A., Johnston A., Dempsey J., Smith J., Pidgeon C. R. and Holah G. D. Two-photon absorption in InSb and Te // - J. Phys.1978. No12.-Pp. 4839-4849.
2. Comparee C. R. Pidgeeon, Wherrett B. S., Johnston A. M., Dempsey J. and Miller A. Two-photon absorption in zinc-blende semiconductors // - Phys. Rev. Lett. 1979. - Vol. 42. 1979. - P. 1785-1788 and references therein.
3. Braunstein R. and Ockman N. Optical double-photon absorption in CdS // - Phys. Rev. A.1964. -Vol. 34. - P. 499-507.
4. Ivchenko E. L. Two-photon absorption and optical orientation of free carriers in cubic crystals // Semi-conductors.1972. - T. 14. - Issue 12. - No. - P. 3489-3485. (in Russian).
5. Rasulov R. Ya. Linear circular dichroism in multiphoton interband absorption in semiconductors // Physics of the Solid State.1993.- T. 35.- No. 6.- P. 1674-1678. (in Russian).
6. Rasulov R. Ya. Linear-circular dichroism in multiphoton interband absorption in semiconductors // Solid State Physics.1993.- Vol. 35.- No. 6.- P. 1674-1677.
7. Rasulov V. R. Rasulov R. Ya., Eshboltaev I. Linearly and circular dichroism in a semiconductor with a complex valence band with allowance for four-photon absorption of light // Physics of the Solid State. 2017.- Vol. 59.- No. 3.- P. 463-468.
8. Ivchenko E. L., Rasulov R. Ya. Symmetry and real band structure of semiconductors. - Tashkent. Fan. 1989.-126 p. (in Russian).
9. Vurgaftman I.and Meyer J. R., Ram-Mohan L. R. Band parameters for III-V compound semiconductors and their alloys // Journal of Applied Physics. - Vol. 89. - No. 11. 2001. - P. 5815-5875. URL: https://doi.org/10.1063/L1368156.
