TWO-PHOTONE LINEAR-CIRCULAR DICHROISM IN NARROW-ZONE SEMICONDUCTORS Текст научной статьи по специальности «Медицинские технологии»

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: r_rasulov51@mail.ru 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.

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e

2

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