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Section 5. Physics
https://doi.org/10.29013/ESR-21-11.12-35-39
Rasulov Voxob Rustamovich, associate professor of Fergana State University Rasulov Rustam Yavkachovich, professor of Fergana State University, Uzbekistan E-mail: r_rasulov51@mail.ru Eshboltaev Ikbolzhon Mamirzhonovich, associate professor of the Kokand State Pedagogical Institute,
Uzbekistan
Qo'chqorov Mavzurjon Xurshidboyevich, teacher of the Kokand State Pedagogical Institute.
Mamatova Mahliyoxon Adxamovna, doctoral student of Fergana State University, Uzbekistan
MATRIX ELEMENTS OF THREE PHOTONIC OPTICAL TRANSITIONS IN CRYSTALS OF CUBIC SYMMETRY. OPTICAL TRANSITIONS FROM THE SPIN-ORBITAL SPLITTING BAND TO THE CONDUCTION BAND
Abstract. The types of three-photon optical transitions in crystals of cubic symmetry, occurring between the bands of spin-orbit splitting and the conduction band, are determined. Expressions are obtained for the matrix elements of the above optical transitions depending on the degree of polarization of the exciting light.
Keywords: interband three-photon optical transitions, matrix elements, spin-orbital splitting band, conduction band.
The first works devoted to the study of two-photon interband transitions in crystals were carried out in the early 1960 s, shortly after the appearance of lasers [1-3]. When calculating the matrix elements of two-photon transitions in crystals, perturbation theories were used in the field of an unpolarized electromagnetic wave [2; 3], where the two-band Kane model was used.
In [4-10], both theoretically and experimentally, linear-circular dichroism (LCD) of two- and three-
photon absorption of light in crystals of cubic symmetry in the region of the center of the Brillouin zone was investigated. However, a theoretical study of the multiphoton absorption of polarized light in crystals of cubic symmetry, occurring between the bands of spin-orbital splitting and the conduction band in the three-band Kane approximation, remained open, to which this work is devoted.
Next, we will consider in detail three-photon optical transitions originating from the state | SO,-1/2)
of the spin-orbit splitting subband into \c, -1/2) and differ from each other in the arrangement of vir-conduction bands, which consist of the following tual states and their order, where | A, m) ^ | B, m) de-different optical transitions
XI SO, ±1/2) =
m
ZI SO, ±1/2)
m ,m'
ZI SO, ±1/2)
m ,m'
ZI SO, ±1/2)
m ,m'
ZI SO, ±1/2)
m ,m'
ZI SO, ±1/2)
m ,m'
ZI SO, ±1/2)
m ,m'
ZI SO, ±1/2)
m ,m'
ZI SO,-1/2)
m ,m'
ZI SO,-1/2)
m ,m'
M
scribes the optical transition from state |A ,m) to | B, m), which occurs with the simultaneous absorption (1) of two photons, m,m' = ±3/2 for the subband of heavy holes (| hh,m) ), m,m' = ±1 / 2is for subbands of light holes |h, m)) and spin-orbit splitting || SO, m)) and conduction bands ||c ,m sum
over m,m' in (1), we obtain a sum consisting of 10
SO, m H SO, m') H |c, ± 1/2),
nonzero terms, each of which describes a separate type of optical transitions. Further calculations are made according to the golden rule of quantum mechanics [14].
We note here that three-photon optical transitions occurring from the state
| SO, ±1/2) spinorbit splitting subband to the conduction band | c, +1/2) are determined in a similar way.
Then each term (1) describes a composite matrix element. For example, for an optical transition of the type ZI SO, -1/2) ^ | hh, m) ^ \hh,m') ^ \c, -1 / 2), the composite matrix element has the form
y,m) ^ |c,± 1/2), y ,m) ^\V ,m')^ | c, ± 1/2), c,m) ^ |c,m') ^ |c, ± 1/2)
V ,m) ^ |c,m | c, ± 1/2), c ,m) ^ |V ,m | c, ± 1/2),
V ,m) ^|SO,m')^ | c, ± 1/2),
SO,m ^ \V,m')
c,m) — | SO,m') -
SO, m — |c,m') -
►|c, ± 1/2), |c, -1/2),
|c, -1/2),
(1)
M
(1)
M
(1)
c ,-1/2;V,-3/2 V,-3/2;V ,-3/2^ V,-3/2;SO,-1/2
M(1) M(1) M(1) 1V1c ,-1/2;V,-3/2 V,-3/2;V ,-1/21>a V,-1/2;SO,-1/2
(Ehh - Eso - 2h®)(Ehh - Eso - (Ehh - Eso - 2h®)(Em - Eso -
+
1
(m(1) M(1) M(1) + M(1) M(1) M
^ lvlc,-1t2;V,+1t2lvl V,+1/2;V,+3/2 V,+3/2;SO,-1/2 ^ 1 1 c,-H2-V,-H2^1 V,-1/2;V,-3/^^ V,-
fl1)
f(1)
rl1)
(1)
(1)
0 U (2)
M(1) M(1) M(1) + M(1) M(1) M(1) 1
c,-1/2;V,+1/2 V,+1/2;V,+1/2 V,+1/2;SO,-1/2 "c,-1/2;V,-1/^JV,-1/2;V,-1/2lv-'V,-1/2;SO,-1/2 I'
( "£so ~2h))(Ehh "Eso " M +_1_
(Elh - Eso - 2h(°)(Elh - Eso -
where |A,m) ^ \B,m') Elh (Eso ) is the energy spec- interband optical transition |A,m) ^ |B,m') from
trum of current carriers in zones |lh,m)(|SO,m)), state |A,m)to |B,m'). In Kane's model [11; 12], this
Mm, is the matrix element of a single-photon matrix element takes the form
-73pB'k *<
3\e -I
{Ehh -ESO -2)
A B
-1
{Ehh -ESO {Elh -ESO
+
{Elh -ESO -2h®){Ehh -ESO
+
(A +1)(3|e: I2 + 4e:2 )
(Elh - ESO - 2h®)(Elh - ESO - h®)
(3)
1
1
1
where Pc is the Kane parameter, A, B are the band parameters of the crystal, which are used to determine the effective masses of light (mh ) and heavy holes (mhh) or the Luttinger-Kohn parameters yv y2
by equalities
2m,,
■ = A + B = ■
2mr
<Yi + 2r
2"
h2 h2
= A -B =-(y1 -2y2), e'x,e', e'z are the pro-
2m
2m„
jections of the polarization vector e along the x',y' , z' axis, associated with the direction of the wave vector of electrons k (k ||z' ), e± = e'x± ie'. Then: for linearly polarized light
|e^ |2 = cos2 (e ■ k j, |e+12 = 1 -|2 = 1 - cos2 (q • k j, for circularly polarized light where
(q ■k),|e; |2 -1 -1< |2 + pctrc cos(( ■k) =
1
I i |2 1-2
el =- sin
I z\ 2
1
1 + cos2
(( ■k ) + Pcrccos (( k )
In a similar way, we can define expressions for the remaining types of transitions corresponding to each term in formula (l).
Below we classify the above optical transitions depending on the states of virtual transitions and present expressions for the matrix elements of three-photon interband optical transitions:
1.1. When both virtual states lie in the valence subbands is determined by the expression
i n^pBk2 e' fasoUle; |2 +ttso12e;2), (4) 2^3 mc (ho) z\ 1 +1 z
where
ftso11 =
-2ha)
A
ha) (
-ha)
1.2. When both virtual states lie in the conduction band has the form
f h Y Pk e, (5)
-I
s
Pk e
v m y
(hm)(2hrn)
1.3. When both virtual states lie in the spin-orbit splitting subband has the form
. (6)
S (ha)2
1.4. When the first virtual state lies in the valence state, and the second conduction band has the form
i h2 PBk2
2yß mc (ha)
i
where R so41 = 3
Rso 41 e + Wso 42e
(7)
1 1
- ESO - hœ Eh - E SO
4
Elh ~ ESO ~ -hrn
V hh
^5042 -
1.5. When the first virtual state lies in the conduction band and the second in the valence band, then in the approximation quadratic in the wave vector in the effective carrier Hamiltonian, some of the optical transitions are forbidden, and some are nonzero and equal to _ i PcB2k2e'z 2^3 (2ftffl)
where ^5051 = 3
4
^5052 -
e' I2 Rso51 + le'2 |^so52), (8)
V Elh ESO h® Ehh ESO h(t) J
eh - eso - h®
(Eh
2ha)(Ehh
3( - +1) B
ha)
(Elh - ESO - 2h®)(Elh - ESO - hffl)
1.6. When the first virtual state lies in the valence band and the second in the spin-orbit splitting subband is determined by the relation
s e'((so61 le' |2 + Rso62|e' f), (9)
31
where Rso61 =-+-,
Ehh - ESO - 2h® Eh - ESO - 2h
Rso21 -
4(A +1) B
(Elh - ESO - 2h®)(Elh - ESO - hm)
Mso62 - -
4
Eh - ESO - 2hm
1.7. When the first virtual state lies in the conduction band, and the second in the spin-orbit splitting subband is defined as
1
3
1
B
3
j__p< (10)
3^3 (2ha)(2ha>)
1.8. When the first virtual state lies in the spinorbit splitting subband, and the second in the conduction band has the form
2i h2 PcAkX3 . (11)
V3 mc (ha)2
1.9. When the first virtual state lies in the spin-orbit splitting subband, and the second in the valence band has the form
6^3 (2hrn)
^5091 le ' I +^5092 le 'I
(12)
where
^5091 - 3
v ehh - eso - 2hm
eh - eso - 2ha> J
^5092 -
Elh - ESO - 2h®
As can be seen from the last relations, the polarization dependences of the matrix elements of transitions of the type (2, 3, 8) are described by one expression, and the remaining optical transitions differ from each other, the graphs of which are shown in (Fig. A1). Note that using the square of the modulus of matrix elements, i.e. expressions (4-11), it is easy to analyze the polarization dependence of both the three-photon absorption coefficient of light and its linear-circular dichroism, to which the next part of this work will be devoted.
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