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37
https://doi.org/10.29013/ESR-20-5.6-73-76
Rasulov Voxob Rustamovich, Associate professor of Fergana State University Rasulov Rustam Yavkachovich, professor of Fergana State University E-mail: r_rasulov51@mail.ru Eshboltaev Iqbol Mamirjonovich, senior teacher of Kokand State Pedagogical Institute.
Sultonov Ravshan Rustamovich, teacher of Kokand State Pedagogical Institute.
Polvonov Baxtiyor Zaylobidinovich, docent of Ferghana Polytechnic Institute
ON THE THEORY OF THREE PHOTONIC LINEAR CIRCULAR DICHROISM IN A HOLE-CONDUCTION SEMICONDUCTOR
Abstract. In this work the matrix elements of three-photon optical transitions accompanied between the subbands of the valence band of the semiconductor of cubic symmetry was calculated. In this case, the contribution of the simultaneous three-photon interaction to the total matrix element of the three-photon optical transitions was taken into account and it was shown that this contribution depends both on the sign and the numerical value of the parameter in front of the effective wave Hamiltonian term that is cubic in terms of the wave vector of holes.
Keywords: matrix element, effective Hamiltonian, holes, photon.
When linearly and circularly polarized light is ab- three-photon optical transitions will depend on both sorbed, multi-quantum optical transitions through the frequency and the degree of polarization of light. virtual electronic states are allowed, which are found The latter leads to the identification oflinearly circu-both in the valence and conduction bands and in the lar dichroism of light absorption. This is true when zones located far from them. According to the law the dependence of the absorption of polarized light of conservation of the angular momentum of cur- on the anisotropy of the semiconductor crystal is not rent carriers, the physical nature of optical transi- taken into account.
tions depends on the degree of polarization of light. Note that the research of some optical parame-In particular, with photon absorption of circularly ters of a semiconductor, for example, when calculat-polarized light, photoexcited carriers will have non- ing a single and multiphoton absorption coefficient zero angular momenta. So that the following pho- of polarized radiation or a polarization-dependent tons will interact with optically oriented current car- photocurrent (see, for example, [7-10]), it is neces-riers. According to the rule of choosing the optical sary to calculate the matrix elements of the optical transition under consideration for the projection of transitions under consideration. Therefore, we will the moments of the current carriers relative to the further analyze the matrix elements of optical transi-wave vector of the photon, the probability of two and tions for specific cases.
Now we turn to the analysis of three-photon op- light holes. The matrix elements of all types of three-tical transitions between the subbands of heavy and photon optical transitions are expressed as
M (3- , - = M(3),=
mk ,mk m ,m
I
m "',m "=±1/2, ±3/2
m ,m m ,m m ,m
M(1) "M(2} ,
m ,m m ,m
M
(E - -E ,r - frœ)(E „r -E ,r -hœ]
\ m k m k /\m k m k j
Hg](r )
ML' r A v
(1)
-+ -
(e ,r - E ,r - 2Ha) (e .r - E ,r - Ha) I, cH )
\ m k m k j \ m k m k /
the third term is described by a diagram ^ ^ ^ , the second by a diagram ^ \J, the third by a diagram
-, the fourth by a diagram
M
(3)
A
c h
H !?'(?)
H
(N )
(?) = H <N )(( ^ ?),
where, according to the numbers of intermediate states |m), |m'} summation is performed. The matrix element of a three-photon optical transition of the type ^ \J where one photon is first absorbed and then two photons are simultaneously absorbed
H^ (( is the effective Hamiltonian of holes in the Luttinger-Kohn representation [12-13], N = 2,3,4, H(( is the term cubic in the wave vector (( of
holes H?} ( ( j .In particular, for cubic symmetry semiconductors H « = D ' / • K, Ka= ka( k2a+i + k2a+2), Ja is the matrix of the angular momentum operator in the Tg representation [12-13], D' is the band parameter of the semiconductor, for example, for p-GaAs D' = 3,9 x10-23 e^ • sm3.
We note here that the contribution of the simultaneous absorption of three photons to the coefficient of light absorption or linearly circular dichro-ism was not researched in [2, 3, 7-11], i.e. the last term in (1) was neglected.
If we consider the optical transitions occurring by the absorption of three separate photons described by the diagram Hi -, then the polarization dependence of the matrix element of the optical transition of the type | +3 / 2) ^ |m) ^ |m')^-|+1/2)
A I3 (Bk)3
v c h
A
ch ) ha
-e, x
2(A -1) + 2e'2 +1 e '2 B 2 2 1
- 4(A - 1)e '2
the matrix element of the type
optical transition where two photons are first absorbed simultaneously, the matrix element of an optical transition of the type and then one photon is absorbed is determined in a similar way. The sum of the matrix elements of the last two transitions is expressed as
s
e\ c h
b 2k —( hœ
'■'+ (l0e:2 -e;2 ).
As a result, the polarization dependence ofthe sum of all three-photon optical transitions described by the
diagram
S f eA
2
B 2k
ch i ha
(-10e '2 + e I2 ) + 4
Bk2 ha
■ is expressed as
41«.12 -3le'12
If we take into account the energy conservation law in three-photon optical transitions and e 'I = \e , then the last expression takes the form
S ( eA0 Y B2k
ch
hrn
is expressed as
(hrn)
2 e +[ 41,„f -3e + f
M
(1-1-1) +1/2 ;+3/2
M
(1-1-1) 1/2;-3/2
f A Bk ' V ch y
24
(hm)
(l36e '2 — 13e '2 ) of the modulus of three-photon optical transitions of the type i j 1 has the form
(2)
f ^ A ,1 |2 3, , |2 2 ( 'X \ ,1 |2 3, , |2
e+ 4 ez' --e+ + e - 4 e2 ' — e '
V 8 J V 8 1 J
After conducting angular averaging over the solid angle of the wave vector of holes, we have:
3
2
2
3
2
3
{
m (1-1-1)
1"+i/2;+3/2
for linear;
m
(i-i-i) -1/2;-3/2
297
M
(i-i-i) +1/2;+3/2
M
(i-i-i) —i/2 ;—3/2
/¿«ear pol
:irc. pol.
eA^
v c h ,
A6
b3 — hœ
405
i6
c h
h®
- for circular polarization.
I
m=±3/2;m'=±i/2
M
(3)
3f eAo 4
V
ch
V ha j
From the last relations it can be seen that the coefficient of three-photon linear circular dichroism, when the photons are absorbed separately, is
r/i-i-i) = 44/i5
I ±i/2;±3/2 t^/J--".
The square of the polarization modulus of the total matrix elements of optical transitions of the type
takes the form
(-i0e 12 + e 1
)-6 V4
f -X \
4le,'I2 --IeH2 1 8 11 j
+4
hrn
D '
13
34e, ez--e, ez
± z 4 ± z
+4
hrnD B 2k
2
I2 „'4 , L' I2 j2a
e, e,, + e J e „ e „
Then, averaging the last relation over the wave vector of holes, we obtain:
for linear;
I
\m=±3/2;m'=±i/2
I
M
(3)
linear
e\ ch
^ B 2k^
y hrn j
2i,5 + 4,9 t^V + 0,5^ haD'V
B 2k
y
b 2k
eA
^ B 2P
ch
hrn
M^ ,2
mjn J \m=±3/2;m'=±i/2 / drc
for circular polarization.
Thus, the coefficient of linearly circular dichro- of ism of three-photon absorption, where the simultaneous absorption of three photons is taken into ac- b 2k
22,5 + 7,<
h^ w „ JhrnD'^ D ' + 0,49
B 2k
B 2k
(3)
(4)
(5)
following values: A = 10.6 mkm - is the wavelength light, B = 5.66 x 10-38 J ■ m2 the ratio
D' = i,3xi0 2 and contribution of taking into
count, depends on the sign and numerical value of account the simultaneous absorption of three-pho-
the band parameter D'. tons in three-photon linear circular dichroism (with-For example, for p-GaAs we assume that it out taking into account the effect of coherent saturaD ' = 3,9x i0-23 eV • sm3 is positive [13], then for the tion) is no more than 3%.
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