CC BY
11
https://doi.org/10.29013/ESR-20-1.2-93-96
Rasulov Voxob Rustamovich, PhD, assistant professor Rasulov Rustam Yavkachovich, DSc, Professor of Fergana State University E-mail: [email protected].
Sultonov Ravshan Rustamovich, Researcher of Fergana State University Eshboltaev Ikbol Mamirjonovich, PhD, lecturer at the Kokand Pedagogical Institute Muminov Islombek Arabboyevich, Researcher of Fergana State University
MATRIX ELEMENTS OF TWO AND THREE-PHOTON ABSORPTION OF POLARIZED RADIATION IN A CUBIC SYMMETRY SEMICONDUCTOR
Abstract. A quantum-mechanical calculation of the matrix elements of two and three photon absorption of polarized radiation due to optical transitions between the subbands of the valence band of a semiconductor of cubic symmetry is carried out. In this case, both the absorption of single photons and the simultaneous absorption of two photons are taken into account.
The mechanism of two, three, and four photon linear circular dichroism of light absorption in a p-GaAs semiconductor is revealed.
Keywords: quantum-mechanical analysis, matrix elements, two and three photonic optical transitions, light absorption, semiconductor.
The advent of lasers and masers made it possible to research nonlinear optical phenomena and the multiphoton linear circular dichroism of light absorption in a semiconductor [1-2].
At present, multiphoton linear-circular and circular-circular dichroism has been researched in semiconductors by absorbing light of different frequencies and polarizations [3-6], which is caused between optical band transitions, i.e. two and three photon absorption of unpolarized light, due to optical transitions between the valence band and the conduction band of the semiconductor, are researched.
In the above researches, the processes of light absorption due to multiphoton optical transitions between the subbands of one, for example, the va-
lence or conduction bands of a semiconductor have remained open, and the simultaneous absorption of two photons has not been taken into account [4-6]. To fill this research gap, we first discuss the phenomenology of two and three photon linear circular dichroism in cubic symmetry semiconductors.
In further quantum-mechanical calculations of the coefficient of two and three photon absorption of light (linearly circular dichroism), we follow [4-6].
Note that the research of some optical parameters of a semiconductor, for example, when calculating a single and multiphoton absorption coefficient of polarized radiation or a polarization-dependent photocurrent, it is necessary to calculate the matrix elements of the optical transitions under consider-
ation. Therefore, we will further analyze the matrix elements of optical transitions for specific cases.
M (N=3) 1 x ±3/2,±1/2
(k )
+
M(N=3) (k) ±3/2,+1/2 vW
eAn
B3
where h[2) (e') = h[2) (k ^ e'), H^ (k )
eft y (ft®)
Following [4-6], the matrix element of two photon optical transitions can be represented as
(1)
-40e '2 +84e '4
-6ez e, +- e,
z i 2 i
is the effec-
tive Hamiltonian of holes in the Luttinger-Kohn representation [7; 8], ex,,e ,,ez. are the components of the vector e', where ex,,e , are the projections of the light polarization vector e on the axis x',y' perpendicular to the wave vector of holes ( k ). Note that the first term (1) describes a two quantum interband subband optical transition occurring by the absorption of two single photons, and the second term (1) characterizes the contribution of two photonic optical transitions flowing through virtual states located far from the valence band to the composite matrix element. M(2) ,r therefore, the second term (1) de-
mk ,mk v 7
scribes the simultaneous absorption of two photons.
Since we are interested in optical transitions of the type |±3/2) ^ | ±1 / 2, we therefore present the expressions for the matrix elements of photonic optical transitions ( M(N\
subbands of the semiconc
occurring between the uctor valence band. In the
calculations of
M
(N )
, we pay attention to the multiphoton optical transitions depicted by the following N = 2 for N = 3
^ ^ ^ \J ^ I \J , where 1 the diagram
describes a single-photon, and the diagram describes the simultaneous absorption of two photons.
Next, we calculate the matrix elements for various types of two photonic optical transitions, depending on the degree of polarization of light.
First, we group the optical transitions by their physical nature, i.e. let us consider both the sequential absorption of two or three photons, and thesimultaneous absorption of two photons. Calculations show that the matrix element oftwo photon optical transitions of the type I ±3/2)->| a-H±1/2) described by the
diagrams 11 is equal to
e h
B^H- e12, and for
2 +
the matrix element of two photon optical transitions of the type | ±3 / 2) ^ | ±1 / 2) described by the diagram
M is equal to -I —0 I Byf3e'ze'T and for |+3/2) ^
|±1/2) is -
A.
e h
B-— e- • As a result, the square
2 -
of the modulus of the matrix element of two photon optical transition described by the sum of the diagrams
ILM,
is described by the expression
4
M (N=2) ±3/2,±1/2
(k )
=75
Mo e h
B2 e ' e„
(2)
y
and the square of the modulus of the matrix element of two photon optical transition type
|±3 / 2->| m->| +1 /2), 1+3/2) ^ |±1 / 2) is defined as
M (N=2) ±3/2,+1/2
(k )
eAo e h
4
B2 e '
(3)
and after angular averaging we have
m (N :2L(k )
1 20
eAo
e h
4
B2
(4)
±3/2,+1/2 v
[ 8 for linear polarization, [7 for circular polarization, Then the coefficient of linear circular dichroism calculated for the above optical transitions is 8/7.
If we take into account the coherent saturation effect [7-9], then the contribution of this effect to the matrix element of the above optical transitions is defined as
I
s
m'=±1/2, m=±3/2
Mm (k )
m m" '
M'NKk )
' I -
m'=±1/2, m=±3/2
1 + 4
a
h rn
MNm (k )
Mw (k )
m m v,v/
(5)
We note here that in order to determine the probabilities of optical transitions or the light absorption coefficient, angular averaging of expressions (2, 4, 5) over the solid angles ofthe hole wave vector is required.
2
2
1 >> 4
а
tía2
MN (k )
m m " '
, it is therefore convenient to
These angular averages for N = 2,3,... taking into account the effect of coherent saturation (taking into account (3a)) cannot be analytically solved. Consid- integrate the hole vectors over the solid angles of the ering that the condition is satisfied for the experimen- wave by expanding the radical (5) in a series In par-tally interesting region of light intensity ticular, for one photon optical transitions, we have
I «
m'=±1/2, m=±3/2
M(N-1](k )
mm v '
9 а
4 hV
eAo
ch
Л8
B4
1296 le '±ez, Г +(36ez 2 le +12 + |e '2|
and after angular integration we obtain the following relations
■ 1 5
\ m =±1/2, m=±3/2
M (N-1](k )
mm v /
9 а
4 h2®2
eA
ch
B4
315
29792 for linear polarization, 30395 forcircularpolarization.
(б)
(7)
It can be seen from the last expression that the contribution of the coherent saturation effect to the coefficient of two photonic linear circular dichroism in p-GaAs is 0.98.
Next, we will calculate the matrix elements of the three photon optical transition described by a diagram ill ., which can be represented as I ±3/2) ^ |m) ^ |m') ^ I ±1/2). This matrix element
(Bk )
transition described by the diagram J^_which
can be represented as | ±3 / 2) ^ | m) ^ | ±1 / 2), is determined by the expression
V3
2
A
ch ) hrn
2(A -1) + 2e'2 +1 e B z 2
- 4(A - 1)e'2
is equal to 2^/3ÍeAo
v ch
(ho)
2 e+
Í О \
. I |2 3 , |2
4\e, J — e'
v
s
У
where |m), \ m') are the numbers of intermediate states located in the subbands of the valence band of the semiconductor. | m) ^ | m') describes a single photon optical transition between |m) and |m') states. In the calculations, a summation will be made according to the numbers of these states. If we denote the optical transition occurring during the simultaneous absorption of two photons as | n) ^ | m), then the matrix element of the three photon optical
e + (136e'2 - 13ef ).Ma-
Similarly, the expression for the matrix element of
the diagram IM is definedThen the resulting matrix element of the optical transitions described by the diagrams |±3/2) ^\m) ^|m')^|±1/2) , |±3/2) ^ |m) |±3/2) ^ |m) ^|±1/2)is
, V3 ( eA0 Y expressed as--—0 I -
8 ^ ch ) ha trix elements for optical transitions of a type
I±3/2) ^ |m) ^ |+1/2),|±3/2) ^ \m) ^\m') ^ |+1/2),
I ±3 / 2) ^ | m) ^ J+1 / 2) are defined in a similar way.
Thus, for the square of the modulus of the matrix element of optical transitions of the type I ±3 / 2) ^ I m) ^ I m') ^ I ±1 / 2), we have the following expression
s
m'=±1/2, m=±3/2
M
(N=3)
(k )
A Bk ch
24
(ha)
. I |2 3 i , |2 4\e, J — e'
+
. I |2 3 ,|2 4 e.\ — e '
After angular averaging over the solid angles of the wave vector of holes in the last expression, we have: for linearly polarized light
({
M(1-1-1)
1"+1/2;+3/2
+
M(1-1-1)
1-1/2;-3/2
405
16
eA0 ch
л
B3 . ha
M
(1-1-1) +1/2;+3/2
+
M
(1-1-1) -1/2;-3/2
297
eA 6
linear pol
and for circularly polarized light
ch
circ.pol. \ J
whence we obtain that the coefficient of linear circu-B 3 _!_ lar dichroism of three-photon light absorption in p-ha GaAs semiconductors is greater than one, i.e.
n
(1-1-1) _
±1/2;±3/2
= 44/15.
2
2
1
2
2
3
s
6
4
For optical transitions of type
MXM
we have the following relations
where from
M (N=3) ±3/2,±1/2
M (N=3) 1 x ±3/2,±1/2
(k )
(k )
+
+
M±3/2^1/2 (k )
m(n=3) (k) ±3/2,+1/2 vW
2 9 f eA0 ^ |e '21 c h
(hay
eAn
B3
-40e '2 + 84e '4
-40e '2 + 84e '4
■6e Z 2e ;2 + 2 e ;4
CJ2J2 . j f4
-6ez e, +—e,
z 1 2 1
• v J (hay ±
Thus, we obtained an expression for the matrix ing the last optical transitions into account gives a elements for two and three photonic optical tran- nonzero contribution to two and three-photon linear sitions, where the simultaneous absorption of two circular dichroism in semiconductors with a com-photons was taken into account. It is shown that tak- plex valence band.
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