CC BY
42
https://doi.org/10.29013/ESR-20-7.8-54-59
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
Nematov Xusnitdin, undergraduate, of Fergana State University
TWO-PHOTONE LINEAR-CIRCULAR DICHROISM IN NARROW-ZONE SEMICONDUCTORS
Abstract. The matrix elements of two-photon direct optical transitions between the valence and conduction bands in narrow-gap semiconductors in the three-band Kane model were calculated. All types of optical transitions were analyzed, which differ from the initial states of current carriers, leading to linear-circular dichroism in narrow-gap semiconductors with taking into account the effect of coherent saturation in two-photon optical transitions.
Keywords: narrow-gap semiconductor, two-photon absorption of polarized light, linear-circular dichroism, coherent saturation.
The absorption nonlinear in intensity of polarized radiation in semiconductors with a complex valence band was researched in [1-10], where direct intersubband optical transitions were taken into account and it was assumed that the nonlinearity in the dependence of the one-photon absorption coefficient on intensity arises due to the effect of coherent saturation of absorption, which is caused by a photoinduced change nonequilibrium hole distribution functions.
However, the question of two-photon absorption of light caused by direct between optical band transitions in narrow-gap semiconductors in the three-band approximation in the Kane model [9; 10] with taking into account the contribution of the effect of coherent absorption saturation depending on the de-
gree of radiation polarization, where the Kane model can be used, remained open, what this work is about.
As indicated in [2], in order to take this contribution into account, it is necessary to make a substitution under the sum sign in the expression for the probability of optical transitions from state lkj to l'kj state: instead of the nonequilibrium distribution function ofcurrent carriers fr (, (()) - f ( (( j, it is necessary to replace by
(( r- m
1+TT
M î
, where T? and
T k are the time of the exit of current carriers from
the saturation region into the zones with numbers l and l', f f, ((0) j ((f) is the equilibrium electron distribution function, M^ ^ is the composite
2
2
matrix element of the optical transition between the states | lkj and 11'k, with the help of which the linear-circular dichroism of multiphoton absorption of light is determined (see, for example, [1-9]).
Therefore, below we consider two-photon optical transitions described by Feynman diagrams
of the type and differing from each
other by the following initial states:
a)the initial states are in the subband of heavy holes, and the virtual states are in the subbands of the valence band. Then the matrix element of the optical transition | V, +3 / 2) ^ | m) ^ | c, +1/2) is de-
-i
C eA ^ c h
kp
(e -)2(A - B)ez,
e+e+B
in
(-ha) (Elh - Ehh - ha) the Kane model, for the optical transition of the type |V, +3/2} ^ |m) ^ |c,-1/2) is determined by the
M
(i)
M
(I)
M
(i)
M
(i)
sum
-ha
which is equal to
A
c h
V2pkBe'e -
Eh - Ehh - hm
Eh - Ehh - h ®
in the Kane
model, for the optical transition of the type |V,-3/2) ^ |m) ^ \c, +1/2) is determined by the
M
(i)
M
(i)
M(i) M(
1v±c ,-i/2;V,+i/2 V,+i/2;V,+3/2,
Elh - Ehh - hm Kane is equal to
r(i)
c ,-i/2;V,+3/2 ;V,+3/2;V,+3/2
termined by the sum
M(1) MK
c ,-1/2 ;V,-1/2 V,—1/2 3/2 1- 1- .1 T^ J 1 •
-, which in the Kane model is
f(i)
M(l) M(l)
1v±c ,+i/2 ;V,+3/2 ;V,+3/2 ;V,+3/2
-ha
sum-
-ha
which in the model
eA0 2
c h
iJlBkpe'e+
Eh - Ehh- hm
, where Mfm,.V m is the matrix
(h - Ehh - h™) equal to i ^A01 pk
for the
\V, -3/2) M «
e ' e ' B
optical
sum-
M(1)
c ,-i/2;V,-3/2lv±V ,-3/2;V ,-3/2
g + 2(A - B)ez, +_,
(-hœ) (Elh - Ehh - ha) transition of the type c, -1/2) is determined by the
M(i) MK
1v±c ,+i/2;V,+i/2 V,+i/2;V,+3/2, Elh - Ehh - hm
r(i)
(-ha) which is equal to
c ,m';V ,m
element of one-photon ( ^ ) transition of the type |c|V,m), where m' = ±1/2, m = ±3/2, p is the Kane parameter [11; 12], A and B are the band parameters of the semiconductor. The energy conservation law for these transitions is described by the function ô(Econd -Ehh -2ha).Matrix elements in the representation (c, +1/2),(c,-1/2) and (V, + 3/2), (V,- 3/2) are conveniently rewritten as the following matrix
M
(2)
eA0 c h
pk
2(A - B)ee ,
B
4lB
e'ze -
-ha
Em - Ehh - ha
Eh - Ehh - ha
ijle'e 'B
Eih - Ehh - ha
-i
b)the initial states are in the subband of light holes, and the virtual states are in the subbands of the valence band. Then the matrix element of the optical transition |V,+1/2) ^ |m) ^ |c, +1/2) is determined by
2(A - B)e '_ez -ha
type |V,-i/2) ^\m)-the
2b
Eh- Ehh- ha j
|c,-i/2) is determined by sum
M(l) M[i> M(i) My
1V1c,-i/2;V,-3/2 V,-3/2;V,-i/2 1 v±c,-i/2;V,-i/^JV,-i/2;-i/2 which
r(i)
r(i)
f(i)
the
m(1+1/2;v,+1/2M V ,+1/2;v,±i/2 , which in the Kane model is
sum
f(i)
M(i) M(i)
1v±c ,+i/2;V,+3/2 ;V,+3/2;V,+i/2 Ehh - Elh - hm
Ehh - Elh - hm
-ha
in the Kane model is equal to -i
eA0
pk x
;qual
-ha to
reA^2 c h
pk x
SBe
_Se'_B
i 2{A + B)e'+ez,
+
i e- 2(A + B)ez
V Ehh - Eih - ha
43 ha transitionofthe type | V,+i / 2)
e Ehh~ EB - h
J
|m)
v c h j for the optical
c, -1/2) is
S
-ha
, for the optical transition of the determined by the sum
M(l) M(l)
lvlc ,-i/2;V,+3/2 ;V, +3/2;V,+i/2 Ehh - Elh ~hm
2
2
2
2
M(1) M(
1v±c ,-1/2;V,+1/2 V,+1/2 ;V,+1/2
K1)
-hœ
which is equal to
which in the model is Kane is equal to
r eA Y pk(A + B)e'z2
c h
- 2-1
eA 2
ch
(A + B)pk^~ . The energy conserva-hœ
hœ
in the Kane model, for tion law for these transitions is described by the func-the ' " Voptical transition of the type tion S(EC - Elh -2hœ) .Matrix elements in the repre-\V,-1/2) ^ |m) ^ |c, +1/2) is determined by the sentati°n (c +1/2),(c-1/2) and
(V, +1/2),(V,-1/2) are conveniently rewritten as the following matrix
sum
M(l) M(l)
1v±c ,+1/2;V,-3/2lv±V ,-3/2;V ,-1/2
M(1) M(1)
1v±c ,+1/2;V,-1/2 V ,-1/2 ;—1/2
Ehh - Elh - h V
ha
M
(2)
'Al
ch
pk
43Be+2
2 (A + B)e'_ez'
y Ehh - Eh - hœ V3 hœ
_.2 ¡2 (A + B)e'2 V3 hœ
2 /2 (A + B)e'2 V3 (_hœ)
-i
SBt
2
2 (A + B)e+ez,
Ehh _ Em _ hœ V33
hœ
c) virtual states are in the conduction band, and the initial states are in the branch of light holes of the va-lence band. Then the matrix element of the optical transition |V.+1/2) ^ |m) ^ |c, +1/2) is deter-
M(l) M(l) mined by the sum M c(.+1/2;V.+1/2 , which in
is determined by the sum -1"Mc(1-1/2;c,-1/2Mi!^1/2;V,+1/2 ,
hœ
which in the Kane model is equal to
fi2k E*+^T-
2m
V "lc
\—pe'. For an optical transi-
Ec - Eh - hm
the 1
S
Kane
eA 2
ch
1 hœ of
r
model
h2k N
VEg + 2mc y
is equal pe +, for the optical tran-
'AÏ J_
V ch J hœ
tion, the type |V,-1/2)
mined by the sum Mc(1+1/2;c,+1/2MC,+1/2;V,-1/2
hœ
m
sition of the type
|V,-1/2) ^ |m) ^ |c,-1/2) by
1
is determined the sum
in
-i
the
' A^2
ch
1
hœ
Kane
f
h2k
g 2m
V '"c J
model 2
|c, +1/2) is deter, which is equal to -pe'z . Here
hœ
Mí1-)1/2;c,-1/2MC1-)1/2;V,-1/2, which is equal in the Kane
model i
V3
A
ch j
1
h®
h2k
cal transition, the type V,+1/2)
Eg + 2m
cj
pe - . For an opti-
>|m) ^ |c, -1/2)
M
(2)
_!_ [ A
S v ch
2
1
hœ
|m) = |c,± 1/2) , Eg - is the band gap. The energy conservation law for these transitions is described 8(Ec - Elh -2hœ) by the function. Matrix elements in the representation (c, +1/2),(c,-1/2) and (V, +1/2),(V,-1/2) are conveniently rewritten as a matrix
V2e'
h2k ^ '
Eg + 2m
pk
c J
-i42e
ie
d) virtual states are in the conduction band, and the initial states are in the branch of light holes in the valence band. Then the matrix element of the optical
transition | V.+1/2) ^ | mj ^ | c. +1 / 2)is determined
M(l) M(l)
by the sum 1 c,+1/2;c,+1/2Mc.+1/2;V.+1/2 , which is equal to
for the optical transition of the type \V.-1/2) ^ |m) ^ |c.-1/2) is determined by the
sum -1"M(1-)1/2;c.-1/2Mc(1-)1/2;V.-1/2 , which is equal to
nm
Ec - Eh - hœ
s
A
ch
1
hœ
Eg+
2m
V '"c
pe - in the Kane model.
-i
eA ch
1
hœ
ti2k2^
Eg+
g 2m
V "lc
l^pe'z in the Kane model,
For an optical transition |V.+1/2) ^ |m) ^ |c.-1/2) the type is determined by the sum
2
2
2
—M(l) M(l)
. ly±c ,-i/2;c,-i/2 c,-i/2;V,+i/2,
hrn
which in the Kane model model is equal to
-i
is equal to
eA 2
V ch
i
hrn
f
h2k
VEg + 2mc y
eA0 V ch J
i
hrn
pi2k 2 a Eg+h1
g 2m
V "lc J
;pez•
I~pe'z • For an op-
Here, Eg is theenergy gap. The energy conservation law for these transitions is described by the function
tical transition |V,-1/2) ^\m) ^ lc, +1/2), the type r \ A/r . . , . . .1
1 /i/i /> /t- o(Ec -Ehh -2h(Q).Matrix elements in the representa-
|V,-1/2) ^\m) ^\c, +1/2) is determined by the tion (c, +1/2),(c,-1/2)and (V, +1/2),(V,-1/2)are
sum -^Mc(i)i/2;c,+i/2Mc,+i/2;V,-i/2
hrn
f(i)
M
(2)
, which in the Kane conveniently rewritten as the following matrix
V2e '
i
S
eA 2
V ch
i
hrn
h2k ^ '
Eg + 2m
pk
c J
sum
'eV2
V ch y
-iy/2e'z ie-
i
e) virtual states are in the spin-split band, and the initial states are in the heavy hole branch of the valence band. Then the matrix element of the optical + m(1) M1
r ^ ly±c,+1/2;A,-1/2 A,-1/2;V,-3/2
E A- Ehh -h&
M(i) M(i) + lvlc,+i/2;A,+i/2 A,+i/2;V,-3/2 T
r(i)
transition |V, +3/2) ^ |m) ^ |c, +1/2) is determined by the sum \
{ ch
'eA ^
v ch
, which in the Kane model
p (-ie -H *'- 2e'zI *').
Î) which
M(i) M(i) + M(i) M(
1v±c ,+i/2 ;A,+i/2 A ,+i/2;V,+3/2 1 v±c ,+i/2;A,-i/2lv'1 A,-i/2;V,+3/2
f(i)
J EA-Ehh -hm r(i)
E A- Ehh
is equal to —¡= V3
For an optical transition, the type | V, -3/2) ^ | m ^ | c, +1/2) is determined by the su
p
is equal
eA0 V ch J
eAn
E A- Ehh -h®
(-ie'zH*' + 2e+I*' [in the Kane model,
v ch ) ea-Ehh-ha
+ M (1) M (1)
T lvlc,ai/2;A,-i/2 A,-i/2;V,-3/2
y I M (i) Ma
1 1v±c,ai/2;A,ai/2 A,ai/2;V,-3/2 ^
fti)
for the optical transition of the type |V,-3/2) ^ |m) ^ |c,-1/2) is determined by the
is equal to
eA0 c h
, which in the Kane model
(2ie'zI ' + e+H ').
EA- Ehh-h®
sum
ch
V
E A- Ehh -h®
M(i) M(l) +
1c,-i/2;A,+i/2 A,+i/2;V,-3/2 ^
+ M(i) M(i)
^ -'"c,-i/2;A,-i/2 A,-i/2;V,-3/2
V3
2
eA0 V ch y
p
E A- Ehh -h®
Here |m) = |SO, ±1/ 2), EA isthe energy spectrum ofcurrentcarriersinthe spin-split band, G ', F ', H ' -determined by relations (24.20) in [12], * is the sign of complex conjugation. The energy conservation (-ie' H*' -2e'Iinthe law for these transitions is described by the function z S(EC -Ehh -2h(o).Matrix elements in the represen-
, which is equal to
Kane m°dd. for an optical transition, the type tation (c, +1/2 ), (c, -1/2) and (V, + 3/2 ),(V, - 3/2) |V,-3/2) ^ |m) ^ |c, +1/2) is determined by the
||m(2,) il ~
m ,m ¡7
eA0 2
v ch J
p
EA- Ehh -h®
are conveniently rewritten as the following matrix
-e ' H *' - i 2e ' I *' -e ' H + 2ie ' I * "
d) virt u al^ state s are ^n the spin-sp1it band, ^nd the initial state s are in the E ran ch o f l i ght tiole s of thv valence band. Then the matrix element of the optical transition |V,+1/^ ^ ^ |c, +1/2) is determined by the sum
2e'zI '- ie + H ' 2e-1 ' + ie'zH
(i) A/f(i)
p
+ Mc,+i/2;A,-i/2MA,-i/2;V,+i/2 eA 2
i
V6
V ch
EA- Elh -hm
, which is equal to
ie +(h ")- ez (G '- F ')
eA0 v ch J
EA - Elh -h ®
M^1^ M[i +
lvlc,+i/2;A,A,+i/2;V,+i/2 ^
r(i)
in the Kane model, for the optical transition of the type |V,-1/^ ^\m ^ |c,-1/^ is determined by
thesum
eA
v ch J ea -Eh-ha
M(i) M(1) +
1vic,-i/2;A,-i/2lviA,-i/2;V,-i/2 ^
2
i
X
X
2
i
i
i
+ M(1) M(1)
^ ly±c,-1/2;Д,+1/2^" A,+l/2;V,-l/2
(1)
1/2;Д,-
c h
X X-
p
У
EA- Elh - Ä©
which is equal to
V3e - H'- ie'z (G '-F ')
sum
e_\ ch
\2
ЕА-ЕШ -h(0
M(1) Mw +
1v±c,+1/2;Л,+1/2 A,+1/2;V,-1/2 ^
f(l)
л/6
in the Kane model. for an optical transition, the type
f eAn v
is determined by the sum
+ M(1) M(1)
^ 1v±c ,+1/2;Л,-1/2 Л,-1/2^,-1/2
is
c h
1
E A" Elh " h®
M(1) M(1) + M(1) M(1)
1 c ,-1/2;A ,+1/2^" A,+1/2;V,+1/2 ^ "c ,-1/2; A ,-1/2^" A,-1/2;V,+1/2
which in the Kane model is equal to
2
il
ch
p
EA- Elh - h®
, which in the Kane model qual to
[(G '- F ')e -+ ieZH "] .
1
eA
p
-\e'zH ie -(G '-F ')]
л/б I ch J (EA- Eh -hay
For an optical transition, the type |V,-1/2) ^ |m) ^ |с, +1/2) is determined by the
Here |m) = |SO, ± 1/2). The energy conservation law for these transitions is described by the function 8(Ec - Eh - 2hœ) . Matrix elements in the representation (c, +1 /2),(c,-1/2) and (V, +1 /2),(V,-1 /2) are conveniently rewritten as the following matrix
M
(2)
1 I eAn
p
E A- Ehh - hV
1 46 v c h
From the energy conservation law, the wave vector of photoexcited electrons participating in optical transitions from the subband of light holes to the
conduction band is determined by the expression
12 (c, L )
k(2œ) = J—(lhœ-Eg^, where mc are the masses
of electrons in the conduction band, mL is the effective mass of holes in the subband L, L = Ih (L = hh )
for light (heavy) holes, ¡л+, L ) = .
mc + mL
From the latter relations, it is easy to obtain expressions for the energies of the intermediate state for light, heavy holes:
а) if the transition comes from heavy holes, then
EL=hh (k(,L=hh ) = '
Se+H* - ie'z (G' - F') -ie- (G - F')■+ e'zH*' ie'zH' - e + (G ' - F') eZ (G' - F') + iV3e-
pending on the frequency of light and the band parameters, will be written as:
1) at an optical transition from light holes to the conduction band:
,(c, Ih)
Eh (Cih ) - Ehh (k(2i=)h ) = ¡ж (2hö - Eg ),
,,(c ,lh)
E (k^) - Ehh(kf^ш) = Eg + (ihm - Eg ).
c ,L=lh' hh^ c ,L=Ih^ g
Ec (k(2r=ih ) - Eh (k(2th ) = 2ht;
M
m„
E (k(2ffl) ) —
nlh\Kc,L—hh> ~
mc + m, m ■ m
hh
hh
mh (m+mhh )
comes from
(2hm- Eg ),
(2h©- Eg ). b) if the
2) at optical transition from heavy holes to the conduction band:
. (c.hh)
Eh (k£h) - Ehh (k^lh) = (2n® - Eg ).
Ec (kC2 tlhh) - Ehh (k (2 th) = 2ha>.
M
(c ,hh)
transition
EL=hh(k(2r=lh)= ... mi'm I. )(2ha-Eg),
light holes, then
mhh (m + mh )
Eh (kfZh ) = ■
-(lha- Eg ). Then the energy denominators in the composite matrix elements, de-
m + m
Ec (kc2?=hh) - Elh(k(2?=hh) = Eg + ^^( - Eg ).
Thus, the interband matrix elements of two-photon optical transitions in narrow-gap semiconductors have been calculated, which can be used to calculate the two-photon linear - circular dichroism caused by interband optical transitions in the Kane model.
This work was partially funded by the grant OT-02-66.
1
e
2
2
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