CC BY
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https://doi.org/10.29013/ESR-21-5.6-47-50
Rasulov Voxob Rustamovich, associate professor of Fergana State University, 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 Qo'chqorov Mavzurjon Xurshidboyevich, teacher of the Kokand State Pedagogical Institute, Uzbekistan
Kodirov Nurillo Ubaydullo ogli, teacher of physics at the Lyceum at Fergana State University, Uzbekistan
INTERBAND THREE-PHOTON ABSORPTION IN CRYSTALS IN THE THREE-BAND KANE MODEL (PART 2)
Abstract. The spectral dependences of the coefficients of interband three-photon optical transitions for InSb and for some optical transitions are calculated and a numerical analysis of the coefficient of interband three-photon absorption of light is carried out, which taken into account the contribution to the three-photon absorption of optical transitions occurring during the simultaneous absorption of two photons in the Kane model.
Keywords: light absorption coefficient, photon, crystal, electron, holes.
In the first part of this work, interband three-pho- [1-5]. Then three-photon optical transitions from the ton optical transitions in crystals of the In Sb type are subband ofheavy holes in the valence band ( ty, m J ) classified and the spectral dependence of some optical to the conduction band ( | c, m|) ) generally have two transitions is analyzed. Next, we investigate the spec- types, which can be represented as a sum of different tral dependence of the coefficient of three-photon optical transitions depending on the initial state of interband absorption ofpolarized light in narrow-gap electrons participating in optical transitions: ifthe ini-crystals in the three-band Kane approximation. As in tial state of electrons is in subband ofheavy holes, then the first part of this work, the three-photon interband there are 16 different optical transitions, if the initial light absorption will be described by diagrams of the state of electrons is in the subband oflight holes, then type < | \j < ? \J where j, - de- there are 22. Therefore, below we calculate the spectral
, 1 1 ,. ^ i i -i ,t dependence of the coefficient of three-photon absorp-
scnbes one photon absorption, v [ - describes the r1 r .
, \ i tion oflight, let us consider for some of them,
successive absorption of two photons, and w - de- rr. r. , , , , ,. .
, r Ine coemcient or interband three-photon light
scribes the simultaneous absorption of two photons , , , ,
r r absorption is calculated as
K N ) (mTV K N ) (mT V W (N ) m
^summyW'1 ) ¿^ ^ c ,m'c ;V,mV \w->1 ) j ¿—I c,m'c ;ç,m'ç ;V,mV . (1)
c,m'c;V,mV c,m'c;ç,m' ;V,mV
Here K^.y^ (co,T) is partial interband three- \c,m'c)(m'c ± 1/2) ); K^(o),T) is total three-pho-
photon absorption coefficient oflight at which when ton absorption coefficient of light, W^-g^ Vm{r is
calculating it is necessary sum over the intermediate the probability of the optical transition between the
states (at a fixed ( ) and final states ( valence band and the conduction band, defined as
<Uv,m, - fiKU^ (k )f {fc (k )-fVt (k )]•*(( (k )-EVi (k )-Nhœ), (2)
h Ç,mç
where is the virtual state can be located both value ofthe total angular momentum operator in the
in the conduction band and in the subband ofheavy band with number v (c ,Vl ,SO ) .The matrix element
holes |Vhh,m'hh) (m'hh -±3/2) or in the subband of of the electron-photon interaction is defined as
light holes | V;h,m[h) (m'lh =±1/2) ,as well as in the h -spin-split subband |SO,m'sO) (m'sO = ±1/2) of the valence band, M^J-(k) is the composite ma-
im0rn
2nI
\1/2
V n*c J
(e • p) , where p is the opera-
tor is the momentum, A is the vector potential of the electromagnetic wave, I is the intensity of light, trix element of the optical transition under consid- and na is the refractive index of the medium at the eration, fc (kfVi (k)J are the distribution func- frequency a .
tions of current carriers, Ec (k)[eVi (k)] is the Since the probability of a multiphoton optical
c , . /t , \ . , . transition, which is used to determine the coefficient
energy spectrum or electrons (holes), is the eigen-
of multiphoton light absorption, is expressed as
W(N) , c ,mc ;V,mv
1
nh
f
V mo™ J
V nc J
(N hÉ - E g
«
(N )
c ,m'c ;V,mV vc ,mv
f ( Ec
where =
2m mv
c v
m + m.
V,
f ( EVi (c m
(nhm- Eg ), mc fa, )
type "A" and Mi% -,ç,m- V,mv y-c,mVl
ment of the optical transition of the type "B"
is
(3)
is the matrix ele-
«
(N )
c,m'c ;ç ,m' ;V,mV \ 'vc ,mv
effective mass in the zone c(V¡), the quantity determined by the integral of the type
1 2n œ
Jdcos(d) J dç\kdk2 X MS^^ {k,0,v)
M (N} , , (k c,mc \ç,mq ',V,mv y c,mvl
M (A ) (k Nffl)W m (B ]
c,mC \c,mC \V,m'v \ c,mv[ f c,mC ;c ,m'ç ;V,my \'vc
fc (k(mhh ) fhh (k(m
is the distribution function of
M{A ]
c,m'c ;ç ,m'(. ',V,mV \ ,xc
M{B ]
c,mC ;V,niy \ /vc
c \ c ,mhh
"'hh
c ,mhh
electrons (holes with energy E Let us consider the specific case. Let
MSC (k2j) is the composite matrix element consist of two terms, i.e.
M (N) (k M(A) (k (Nffl)U
c,m'c \q,m'q ',V,my \ c,mVl J c,m'c \q,m'q ',V,my \ c,mV{ '
M(A) , , ik(Nffl)Nll*M{B\ , ,(k{Nm)) + c,m'c;ç,m'ç ;V,mV \ c,mv, J c,m'c;ç,m'ç ;V,mV \ c,mv, J
+M (A \ , ,(k (B), , ,(k
c,m'c;ç,m'ç ;V,m'v \ c,mvl J c,m'c;ç,m'ç ;V,mV \ c,mv, J *
Whence we have that the total light absorption coefficient consists of the sum of the partial light absorption coefficients, determined by the quantities
M^) (k(Nffl)Nl where M(A} (k{Na
c,m'c;ç,m'ç ;V,mV I c,mVl J ) c,m'c;ç,m'ç ;V,mV I c,mv
is
and
M^ ;V,mHkcm,
, as
the matrix element of the optical transition of the well as the contributions to the total light absorption
e
c,m
-1
0 0
coefficient, determined by the quantity
M
(A )
cmC <>m'<- \'\ ,mv
M(B) I kv~ ) i e
c ,m'c ,m< 'Vm'y \ c ,ту{ J* * *
K(A'R'C) , (œ T ) = NhÉ W{A'B'C\ , ,
c,mc ;V,mv V 5 / t c,mc ;ç,mç ;V,mv *
(4)
w( AB. L- = —
f \2N
c ,m'c 'V,m\
nh
V mo® J
V nc J
where
3
c ,mVl
(N hÉ - E J1
M{ABV , (
c,m-'Vmv \ c,mv
f ( Ec
W(C), , = —
c ,m ;V,mv я
- f ( E
^V c ,mVl
r W 2nl^N
V m0®J
V nc
(N hÉ - Eg
f ( Ec
c ,mc ;ç ,mq ; V,mv \ c ,mvi
M (B ) i (N®>), M (A )
c m'c'çm'ç >v,mv \ c ,mvl J c m'c'çm'ç ; V,m'v\'vc ,mvl
f ( Ev (c,mv,
Mm ,v,i
cmc'qWç ;V,mv y c,mvl
calculating them separately, one can consider that the | v,-3/2) ^ | V, + 1/2)-— — | c,-1122),
numerical contribution of the partial absorption coef- I v,_з/2) ^ j v,-1/2) ^\c,-1/2), the coefficient of
ficients of light. Therefore, we will consider specific i , . , , j , ,. fl. i,f , ,
° r three-photon interband absorption or light tor crystals
optical transitions. For example, for an optical transi- ri- - i^.-
of cubic symmetry is determined by the relation
tion ofthe type j v,-3/2) ^ ^ |V,- 3/2) ^\c,-1/2), 7 7 7
E2
2 Eg
K(N=3) _ 3 K(3) (0) mcmhh
KcM' _ 4K (°'(m,. + m„„B/
'l | A -,
B_
x,.
V
a(N=3) + 2
Uc hh ,1
2| A-1 B
fc (mi )- fhh
b(N=3) c ,hh ,1
x-5r(3i) x
i c ,mhh
.(N=3) c ,hh ,1
-X» Xlh - Xhh - 2Хш (xlh - xhh - 2Xa )
(5)
and for transitions of type | v, -3/2) ^ \V, - 3/2)-|V,-3/2) ^ I c,—1/2), \v, -3/2) — |V, - 3/2)-^ Ic,-1/2), \v,-3/2) V,-1/2) V,-3/2)-
^ |c, —1/2), \v-3/2) ^ ^ |V,-1/2)-
^ |V,-1/2) ^ Ic,-1/2) we have
2m mV Nhm- E„ , l = lh,hh, SO, r{cNma) =-^^--g-, h - is wave
, 2 mc + mvt m0Eg
vector of current carriers, determined by the energy conservation law Ehh (k) - Ehh (k) - 3hrn = 0, for linearly polarized light
90
k(N:3 - 6K™ (o ) mmhh
(n=3) ь(N=3) =— c
uc ,hh ,1 10I- ' uc ,hh 1 lr,r ' Lc ,hh,1
135
180 (N=3) = 488 N=3) = 1 ,on c ,hht 0_ c ,hh,2 _ '
135 135 15
L c ,hh ,2
(mc + mhh )mo
x
1 70
а{ф=1 = 15, boi"circiikriypolarizedlight ^¡f =
fc (/Cimhh) ) fhh (mi) (xœ))r(m,!h) X
x((*î + + cNS^i )e'4
b(N=3) = 133 (n=3) = У96 a (N=3) = b (N=3) =
uc,hh,1 - 105' chh,1 ~ 1nc ' c,hh,2 ' uc,hh,2 ~ л nc '
396 105:
105
105
r(N=3) "c ,hh ,2
where Kg3) (0 ) = nm0
23
2ne
105,
Vn*chJ
k(3*) = 2mcm^l 3h®~ Eg c,mv ... , ...
mc + mv h
klE ¡9Pc B4, xffl= hrnl Eg, xi = Ei {k^ Eg,
s, =
2( A -1)2
2( A -1)
A B
1)
(hrnf (Em - Ehh - 2 ha)ha) (Elh - Ehh - 2 ha)(Elh - Ehh - ha)'
2hrn(Elh -Ehh -ha>) The spectral dependences of the coefficients of three-photon interband absorption of light for the above optical transitions are shown in Fig. 1. It can be seen from (Fig. 1) that the spectral dependence of KcNha / (0) depends on the type ofoptical transi-
tions. In particular, for the first and second types of optical transitions, with an increase in the frequency of light, it increases and passing through a maximum decreases for both linearly polarized light and circularly polarized light, but the maxima ofthe dependence differ from each other in value. The numerical values of the band parameters of In Sb were taken from [6]
1.
2.
4.
5.
Figure 1. Spectral dependence of the coefficient of three-photon interband light absorption in InSb crystals for the two cases considered in the text
References:
Rasulov R. Ya. Polarization optical and photovoltaic effects in semiconductors with linear and nonlinear absorption of light. Dissertation for thesis. of doctor's degree phys.-math. sciences. - St. Peters-burg.1993.- 206 p. (in Russian).
Ivchenko E. L. Two-photon absorption and optical orientation of free carriers in cubic crystals // Semi-conductors.1972. - Vol. 14. - Issue 12. - P. 3489-3485. (in Russian).
3. Arifzhanov S. B., Ivchenko E. L. Multiphoton absorption of light in crystals with the structure of diamond and zinc blende // Physics ofthe Solid State. 1975. - Vol. 17. - No. 1. - P. 81-89. (in Russian). Rasulov R. Ya. Linear circular dichroism in multiphoton interband absorption in semiconductors // Semiconductors. 1993.- T. 35.- No. 6.- P. 1674-1678. (in Russian).
Rasulov R. Ya. Linear-circular dichroism in multiphoton interband absorption in semiconductors // Physics of the Solid State.1993.- Vol. 35.- No. 6.- P. 1674-1677. (in Russian).
6. 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
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