CC BY
36
https://doi.org/10.29013/ESR-20-5.6-77-80
Rasulov Voxob Rustamovich, associate professor of Fergana State University Rasulov Rustam Yavkachovich, professor of Fergana State University E-mail: [email protected] Eshboltaev Iqbol Mamirjonovich, senior teacher of Kokand State Pedagogical Institute.
Sultonov Ravshan Rustamovich, teacher of Kokand State Pedagogical Institute. Nasirov Mardonbek Xoldorbekovich, teacher of Ferghana Polytechnic Institute
ON THE THEORY OF FOUR PHOTONIC LINEAR CIRCULAR DICHROISM IN A HOLE-CONDUCTION SEMICONDUCTOR
Abstract. In this work the matrix elements of four photonic optical transitions accompanied between the subbands of the valence band of the cubic symmetry semiconductor were calculated, where the contribution of the simultaneous three photon interactions to the total matrix element of the four photonic optical transitions was taken into account and it was shown that this contribution depends on both the sign and the numerical value of the parameter in front of the cubic by the wave vector of holes, the term in the effective Hamiltonian.
Keywords: four photon optical transitions, matrix element, effective Hamiltonian, holes, photon.
Although single-photon (linear in intensity) ab- we consider below four photon absorption of polarsorption of polarized light in semiconductors, due to ized light in cubic symmetry semiconductors, due to optical transitions between the subbands of light and direct optical transitions between light and high hole high holes of the valence band, has been researchd subzones, where, differ [1-7], we take into account both theoretically and experimentally for a long time the simultaneous absorption of three photons. Next, [1; 2], but the question of linear-circular dichroism we will calculate the four-photon matrix element of (see, for example, [1-7]) of four photon absorption optical transitions between the subbands of high and of light, taking into account the simultaneous ab- light holes in the valence band of the semiconductor, sorption of three photon, remains open. Therefore, which is expressed by the ratio
( M(l)~M(l) .,M(1), .M(l)
rw _ _ M(4] -
mk ,m 'k
m "',m "-±1/2, ±3/2
M« _ M^,- V
mk ,m k m,m ¿^u
m,m m,m m ,m m ,m
v
(E - E ,r - fro)[E ,„r - E ,r - fro)[E ,r - E ,r - ho]
\ mk m k j \ m k m k / \ m k m k j
m« Ml m® , m« ,„M(2j M) ,
+ m , mm , mm , m + m , mm , mm , m +
(E „r - E ,r - 3fr®)(E ,r - E ,r - 2 ho] (E „r - E ,r - 3fr®)(E ,r - E ,r - ho]
\ m k m k /\ m k m k / \ m k m k /\ m k m k /
M^ .M{1\ MV , M(2) M(2} , M(3) MV ,
+ m ,m m ,m m ,m + m,m m ,m + m,m m ,m +
(E mr - E ,r - 2 ho)(E „r - E ,, - no) (E ,r - E ,t - 2 ha>) E - E ,k - ha
\ m k m k /\ m k m k / \ m k m k / m k m k
+ -
M{1) M(3} '
m ,m m ,m
E ,r - E ,r - 3hœ
m k m k
(1)
where the third term is described by the Feynman diagram mi the second by the diagram ., the third by the diagram -4—- -—A-, the
-, the fifth by the diagram
fourth diagram A/-
, the sixth by the diagram m
enth by the diagram m
Next, we will analyze optical transitions accompanied by sequential (non-simultaneous) absorption of four photons. These optical transitions are described by the following single diagram: 1+3/2) ^ |m), ^ |m')^- |m")^-|+l/2), where ^ schematically depicts a single-photon transition m", m', m = ±3 / 2, ± l / 2. Then, in the Luttinger-Kohn approximation, the following optical transitions are allowed, occurring between the subbands of light and high holes of a cubic symmetry semiconductor, corresponding to these transitions, are expressed as
M(l) M(l)
^ J-+1/2;+3/^J- +3/2; +3/2
6 (hm)
M
(i)
+3/2;+3/2
M(1) M[
+1/2\+1/2lvi +1/2;+3/2
r(i)
A
c h
Bk
2 (ha)
yf33e'ze -
18 (hrnf
M
(1)
+3/2;+3/2
A
6(A -1) + 2e 2
J
M(l) M(l)
^ J-+1/2;+3/^J- +3/2;+3/2
6 (hrn)
M
(1)
+3/2;+1/2
M(1) M(1)
^ J-+1/2;+3/^J- +3/2;+3/2
2 (hrnf
M
(1)
M(1) MK
-iKl+1/2;+3/2lv-1 +3/2;+3/2
f(1)
18 (ha)
+1/2;+1/2
M(1)
1 ^ +3/2;+1/2
M(l) M(l)
1V1+1/2\+1/2lvl +1/2 ;+3/2
6 (hrn)
M
(1)
+3/2;+1/2
(2)
M(1) M(1)
1V1+1/2\+1/2lvl +1/2 ;+3/2
18 (hrnf
M
(1)
+3/2;+1/2
M(1) M'
11 +1/2;+1/2 +1/2;+3/2
f(1)
6 (h®)
M (1)
the corresponding energy conservation law is determined by the Dirac function: $(Eih (()- Ehh (()-, Eih (( and Ehh (( are the energy spectrum of light and high holes, Mmm is the matrix element of a single-photon optical transition of the type | m) ^ | m'). Then the polarization dependence of the sum of these matrix elements is determined by the relation
4< A -
1) + - e 22 + 6e'
6 2 z j
-3
A
1
2(--1) + - e 22 + 2e Z
B 2 2
c
A
(3)
4(~ -1) ^ e2 - 2e'2 B2
where e'a (a = x, y, z) - is the polarization vector of the electromagnetic wave in the Cartesian coordinate system, where the Oz axis is chosen in the direction of the wave vector of the holes. Expressions for optical transitions | -3 / 2 ^ | m) ^
^ |m') ^ |m") ^ |+1/2) are defined in a similar way.
The matrix elements corresponding to the interband optical transitions of the type | ±3 / 2) ^ | m) ^ | m') ^ | m") ^ |+1 / 2 are equal to zero, i.e. such
K2e;2 (2e2 - 24e'2)
transitions are forbidden in the Lattiger-Kohn approximation.
Note that the sum of the squares of the matrix elements of the interband optical transitions due to the absorption of four separate photons, as well as for optical transitions of the type:
1+3/2) ^ |m|m")^|+1/2),
1+3/2) ^ |m|m")^|+1/2), 1+3/2) ^|m-1/2,
2
2
2
2
2
2
2
2
2
1+3/2) ^ |m) ^|+1/2) 1+3/2) ^\m1+1/2)
|+3/2) ^ |m) are given in [7-10],
where ^ schematically depicts the absorption of one photon, and ^ depicts the simultaneous absorption of two photons.
Then, squaring the modulus of the sum of the matrix elements of all the optical transitions under consideration and averaging over the solid angles of the wave vector of the holes, we can determine both the spectral and temperature dependences of the light absorption coefficient and the linearly circular absorption dichroism in a semiconductor of cubic symmetry. Below we present the numerical values of the linear-circular dichroism coefficient for individual transitions. In particular, for the above transitions, we have the following results:
M
(1-1-2 ;1-2-1;2-1-l)
±1/2,±3/2
4304
M
(1-1-2 ;1-2-1;2-1-1) +1/2,±3/2
189 4496
135
çAl
c h
A
c h
hœ
V
f ^ v ha j
- for cir-
cular polarization. The linear-circular dichroism coefficient (i.e., the ratio of the probability for linear polarization to the probability for circular polarization) for these transitions is equal to
(1-1-2 ;1-2-1 ;2-1-1) 272 , (1-1-2 ;1-2-1;2-1-1) 1345 )--and n -
n+1/2,±:
±3/2
281 '±1/2,±3/2 20452
where the superscript describes the order and number of absorbed photons.
The linear-circular dichroism coefficient for optical transitions of the type \J \Jis equal to
n(2+2) = —. For this transition, the orientation of the
25
moments of photoexcited holes occurs more intensively than the alignment of their momenta.
We consider four photonic optical transitions accompanied by the simultaneous absorption of three photons. In particular, in the Luttinger-Kohn approximation, a matrix element of an optical transition of the type :
64
M
(1-1-2 ;1-2-1 ;2-1-1) ±1/2,±3/2
327232
1 J+1/2,±3/2
945
4352
ch
B
2
y hrn j
and
135
A
c h
M — M'
+1/2;+1/2 +3/2;+3/2 ^4(1)
^ bl ^
y hœ j
r(3)
for linear,
M (1) — M (
+1/2 ;+1/2 +3/2;+3/2 ^ ^(3)
r(1)
'A, v
ch
3hrn
D'k
Myv +
1 ^ +1/2;+3/2 ^
hrn
M
+1/2;+3/2
V3h
m
Be'z {( + 7e;2 - el2 ) + ey ( + 6e^2 - e;2 )},
(4)
and for the optical transition type m
BkD
M(3L,(eA 2((-)e-
-1/2;+1/2^" +1/2;+3/2 (Elh - Ehh - h®)
ch
Sh
œ
(5)
where kw the wave vector of holes in the final state, determined from the conservation law corresponding to the optical transition under consideration.
The square of the modulus of the total matrix elements of optical transitions of the type lit
takes the form
I
m=±3/2;m'=±1/2
M
(4)
2 3 ( eA0 4
^ w b2^2
V
ch
V ha j
(2 + e ;2 )- 6
„1 |2 3, , 4\e„ ,\ — e '
+4 ^D ''
B 2k
V
Q A „12 „14 13 ,4 ,2 34e ±ez - — e !ez
+ 4
v B2k
„' I2 „'4 , |„' I2 „'2„l2 e l| e' + Ie l| e ;e x
Then, averaging the last relation over the wave vector of holes, we obtain:
(6)
2
2
8
X
2
2
e
z
M
(4)
\m=±3/2;m'=±1/2
linear
A
ch
hœ
21,5 + 4,9 ^D' + 0,5 r hœD'
for linear,
M
(4)
eA
^ B
ch
\m=±3/2;m =±1/2
for circular polarization.
In conclusion, we note that the coefficient oflin-early circular dichroism of four-photon absorption,
v ha j
B 2k
ha
2
V
B 2k
J
22,5 + 7,8—D' + 0,49 B k
haD ' B 2k
2
(7)
(8)
lowing values: X = 10.6 mkm is the wavelength of B = 5.66 x 10-38 J ■ m2 , then the ratio
light,
hœ
D' = 1,3 x 10 and contribution of taking into ac-
where the simultaneous absorption of three photons b 2k
is taken into account, depends on the band param- count the simultaneous absorption of three-photons
eter D'. For example, for p-GaAs we assume that in four-photon linear circular dichroism is not more
D' = 3,9 x 10-23 eV ■ sm3 it is positive, then for the fol- than 6%.
References:
1. Ivchenko E. L. Two-photon absorption of light and optical orientation of free carriers // Solid state physics. 1972.- Vol. 14.- No 12.- P. 3489-3494. (in Russian).
2. Rasulov R. Ya. Polarization optical photovoltaic effects in semiconductors with linear and nonlinear absorption of light. The dissertation for the degree of doctoral dissertation. Physicotechnical Institute named after Acad. Joffe - S.-P. 1993.- 168 p.
3. Rasulov R. Ya. Linear-circular dichroism in multiphoton interband absorption in semiconductors // Sol. St. Phys., 1993.- Vol. 35.- No. 6.- P. 12674-1677. (in Russian).
4. Rasulov R. Ya., Khoshimov G. Kh., Kholitdinov Kh. Linear-circular dichroism of nonlinear light absorption in n-GaP // Physics and Technology of Semiconductors, 1996.- Vol. 30.- No. 2.- P. 274-272. (in Russian).
5. 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.-Springer, 2017.- Vol. 59.- No. 3.- P. 463-468.
6. Rasulov V. R., Rasulov R. Ya., Eshboltaev I. Linear-Circular Dichroism of Four-Photon Absorption of Light in Semiconductors with a Complex Valence Band // Russian Physics Journal.- Springer, 2015.-Vol. 58.- No. 12.- P. 1681-1686.
7. Rasulov R. Ya. Linear circular dichroism in multiphoton interband absorption in semiconductors // Sol. St. Phys., 1993.- T. 35.- No. 6.- P. 1674-1678. (in Russian).
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