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Section 7. Physics
https://doi.org/10.29013/ESR-20-7.8-49-53
Rasulov Voxob Rustamovich, associate professor, of Fergana State University Rasulov Rustam Yavkachovich, professor, of Fergana State University E-mail: [email protected] Axmedov Bahodir Bahroovich, doctoral student of Fergana State University.
Muminov Islombek Arabboyevich, doctoral student, of Fergana State University Polvonov Baxtiyor Zaylobidinovich, associate professor, of Ferghana Polytechnic Institute
LINEAR-CIRCULAR DICHROISM OF ONE-PHOTON ABSORPTION OF LIGHT IN NARROW-ZONE SEMICONDUCTORS. CONTRIBUTION OF THE EFFECT OF COHERENT SATURATION
Abstract. The matrix elements of optical transitions, the spectral and temperature dependences of the one-photon absorption coefficient of polarized radiation were calculated, and the linear-circular dichroism in narrow-gap semiconductors was investigated taking into account the effect of coherent saturation.
Kewords: narrow-gap semiconductor, absorption of polarized light, linear-circular dichroism, coherent saturation.
The nonlinear in intensity of both one- and multiphoton absorption of polarized radiation in semiconductors with a complex valence band was researched in [1-9], where direct intersubband optical transitions were taken into account and it was believed that the nonlinearity in the dependence of the one-photon absorption coefficient on the intensity arises due to the effect of coherent saturation of absorption, which is due to a photo-induced change in the nonequilibrium distribution functions of holes.
However, the contribution of the effect of coherent saturation of absorption ofpolarized radiation [3; 4] to single-photon linear-circular dichroism caused by direct between optical band transitions in narrowgap semiconductors, where the Kane model [10] can be used, remains open, to which this work is devoted. Therefore, we will consider this contribution in the region of light intensity, where perturbation theory is applicable. As indicated in [9], in order to take this contribution into account, it is necessary to make a substitution under the sum sign in the expression for
Ik) to
the probability of optical transitions from state l 'kj : instead of the nonequilibrium distribution function of current carriers f, AE,, (k ) ) - f, IE, (k ) ), it is
lHWW
l'k
necessary
\Jl 'k
to
epl
replace
by (( - f
- f(
J Ik
(o)
Ik
1 +
, where T r and
k
2n Hœ
K«\œ,T) = f^f£( -f,)
where I = ^^
2nc
I, l', k
is the light intensity, E^ is the energy spectrum of current carriers in the subband l, nœ is the refractive index of light at the frequency ffl. The rest of the values are generally known.
Next, we will consider single-photon absorption of light and its linear-circular dichroism in narrow-gap semiconductors. In a spherically symmetric approximation in the energy spectrum ofcurrent carriers, one-photon nonlinear radiation absorption in semiconductors with a degenerate band structure was researched in [2], where it was shown that the absorption coefficient of light decreases with increasing intensity.
Further calculations will be performed in the two-band approximation (Kanev's model) [10; 11], where the energy spectrum of current carriers is spherically symmetric and the effective masses of heavy and light holes are determined by the relations (see formula (14.5) in [11])
(mj m ) = -1 (m J m, ) = 4PCV/(3m0E J. (2)
As we can see, the two-band approximation incorrectly conveys the curvature of the heavy-hole
T„r are the times of exit of current carriers from the
l k
saturation region in the bands with numbers l and l' , flf0\ (ff0)) is the equilibrium electron distribution function, Mr~ k is a composite matrix element of the optical transition between the states ltj and l'k'^j. Following [3], the one-photon absorption coefficient can be written in the form
it f 5(En - Ek - H (1)
branch (the correct curvature will be obtained only if the distant zones are taken into account). According to [11], in this approximation (mjm*) = 1 -( + 2Aso )lm0B /((Aso + EJ)(3)
Note here that the Luttinger-Kohn parameters (see, for example, [11]) are related to the Kane parameter by the relations
D = -h2 P2V/(3m2Eg),
A -B = h2 /(2m0), (4)
, B h2
A + — =-
2 2m„
2 r>2
h2 p
CV
2m0>Eg
where the Kane parameters are determined by the relation -iPCVôap = J dxdydzS (r )P X p (r ), Xp(r ) is one of the basis functions defined according to (13.18) [11], where spin-orbit mixing with other bands is not taken into account.
The spectral dependence of the light absorption coefficient in narrow-gap semiconductors for direct between-band optical transitions, neglecting the non-parabolicity in the energy spectrum of current carriers, according to the formula:
K (1) = 0
■ 2 2 4n e
I
4n2e2
crnnamo
P2 — cv 12
Ca>m0n* s J ,m ;k
—2 —— 2
61 le', + \e\
eP
cs Vm
PÎSM + [|e +1 + e
(()
8(Ekk - EV,k - Ä®» =
2
+ 24
' (0) Pel
M
(5)
where the reduced density of states (excluding spin index (0) indicates the neglect of non-parabolicity
degeneracy) (and k■ p mixing). It was assumed here that the final
p(% (®) = Xs (() - E* () -H =1 ^states of the current carriers are empty, and the initial
2 it '2
Hi is reducedeffective mass: ^ = (m- + m-1) InSb ¡1 = m*,ß2 = m* / 2;k(0) = 2^ (ha - Eg ))h
-1 states are completely occupied. Since e J = 2/3 and
in the :—¡2 1 1
1/2 \e.\ = 1/3 then
2
2
z
K (1) = 1K0
e2 4n24n p2 2
XM(0)h
-2 e2 Eg
cn„h hrn
ki0)
'l + ^
(7)
cna (2n) comQ 31=1>2
(since in InSb kQ°Vk{0) = Note that taking into account non-parabolicity in
the energy spectrum will lead to the following relation
K(i)_ 4^e2
0
cmnam0 13
K \ 2 «2 (kl )Pc ,»+
24
- + —
9 9
a
Pc ^ (®)
Here k1 and k2 they satisfy the law of conservation of energy:
1 [Eg +n(ki )] = h®> n(k2 ) = , (9)
where (k1 ) = (©- Eg )2 > vf (k2 ) = (ho) >
t>2k2 ^2 T
EQ + QEg = (2hv-Eg > EQ + 2 Eg = (»©),
g g 2m v g' g g m
(ki ) =
n(k2 )
n(ki ) + Eg hrn
(8)
k2 =
m
2n(k,) )
(( - Eg )))f = k?'^.
For the reduced density of states, taking into account non-parabolicity, we have 4n
k =
m
'ha- 2 -(ha- Eg )/(h2Eg )~T = kf0)
ha
Pc* (®) = ^Tk
(2n)3
V
there-
J
fore,
( \ (o)/ \ ki n(ki ) ( \ (ow \ k2 hc pc* (œ)=Pc* ' (œ)=(co)-
k(0) Ea
So,
K
e2 Eg
hcn,, hm
ki0)
f V/2 hm
v ^ y
f
2yl2
1 2
- + -
33
v
E2
v hm y
h®
hm + E„
\ 1/2
2E
v g y
(10)
(11)
It can be seen from (11) that K^increases Note that the contribution of the coherent sat-
with increasing of frequency of light, where uration effect to the light absorption coefficient,
% = Eg / ha . In particular, as the frequency of light which is taken into account, is determined by the
doubles, K(1) (£ ) it triples. expression
K (>,T ) = I (fhh - L )ô(Ehh - Ec + hœ){
" 1 ¡t;5=±1/2,m=±1/2,±3/2
M £);v,m(2) 2
a M 2;v,m(2) 2
tfrn2
), (12)
where fhh (fc) is the distribution function of holes and other quantities are well known. For example, (electrons), Ehh (Ec) is the energy spectrum ofholes for an optical transition |V> ±3/2) ^ |C> ±1/2), the
(electrons), the sign (...) means averaging over the quantity K(1)(ft)>T) is determined by the relation solid angles of the wave vector of current carriers,
16e2 3cm h2n.
-AÖ • k% • P2 • F(ßXm) .3(m)-[fhh(E^J)- f (Eck%)
(13)
where
2
F (ß,l,®) = [l - exp (\ßhw]\ exp [ß(-Ehh (k(%hh ))
Zffl = 4
a„
-j-2 2 n œ
f
— + m„
'A-12 p 2,
c n
& = ^ Ej,
sorption coefficient. And for 0 we have: for linearly polarized light
^ =
lin
C+1
= j_1 d/u
2
1 - /
(14)
J m
, 0"1 = kBT, 3(o) = |
>
1 + C, 1«±1
It is seen from (13) that the linear-circular di-chroism of the light absorption coefficient depends on the value 3(co) . In particular, disregarding the contribution of the effect of coherent saturation (i.e., at Zm= 0 ) K(1)(©,T) to both for linearly and circularly polarized light 3(£ffl = 0) = 4, i.e. in this case, there is no linear-circular dichroism of the light ab-
for elliptically polarized light
c+1 f 2(1+^'2)p-
L dv ■ 2 r, n> (15)
^ ■ =
arc
1+ Z.
(1 + v'2 ) PcrcV
where P is the degree of circular polarization of
arc
light, the sign "± " corresponds to the <j± polarizations, ^ = cos0', ^ = cos0,where 0(0') the angles between vectors « (q) and k, q is the wave vector of the photon. In case Pcirc = 1 we have that: for linearly polarized light
^lin = Z
-5/2
Z3/2 + z2
TI CO CO
arcsin
z.
\ 1/2
1 + z
arcsin
« y
z«
\ 1/2
1 + z
« y
(16)
for circularly polarized light
2 (zrVC+1 -ZfflarcsinVc)
X ■ =■
arc
z
5/2
(17)
From (16) and (17) it can be seen that with increasing light intensity, it grows >> 1, and within the limits of high intensity, that is, it does not depend on the intensity and equal to ©1.1 This means that in the region of high intensities, the linear-circular dichroism has saturation.
Thus, one-photon linear-circular dichroism in narrow-gap semiconductors arises only when the effect of coherent saturation is taken into account. Following [9], it is possible to calculate the multiphoton linear - circular dichroism caused by interband optical transitions, to which a separate work will be devoted.
This work was partially funded by the grant OT-F2-66.
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2
e
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