COEFFICIENT OF INTERBAND TWO-PHOTON ABSORPTION OF LIGHT AND ITS LINEAR-CIRCULAR DICHROISM Текст научной статьи по специальности «Физика»

https://doi.org/10.29013/ESR-21-1.2-39-43

Rasulov Voxob Rustamovich, associate professor of Fergana State University Rasulov Rustam Yavkachovich, professor of Fergana State University E-mail: [email protected] Axmedov Baxodir Baxromovich, doctoral student of Fergana State University Muminov Islombek Arabboyevich, doctoral student of Fergana State University Qo'chqorov Mavzurjon Xurshidboyevich, teacher of the Kokand State Pedagogical Institute

COEFFICIENT OF INTERBAND TWO-PHOTON ABSORPTION OF LIGHT AND ITS LINEAR-CIRCULAR DICHROISM

Abstract. Expressions are obtained for the spectral dependence of the coefficient of the interband two-photon absorption of light in narrow-gap semiconductors in the Kane model.

The dependences of the partial coefficient of interband two photon absorption of light are analyzed, which differ from each other by the types of optical transitions, depending on the degree of polarization of the light, and carried out a quantitative analysis of the linear-circular dichroism factor of two photon absorption of light and it is shown that the main contribution to the linear-circular dichroism is made by optical transitions from the subband of light holes to the conduction band.

Keywords: initial state, virtual state, interband two photon absorption of light, Kane's model, optical transitions, semiconductor.

Nonlinear multiphoton absorption of polarized light caused by optical transitions between subband light and heavy holes in the valence band of a semiconductor, and its linear-circular dichroism are considered in [1-8]. However, the question of the linear-circular dichroism of the interband two-photon absorption of light, as well as the spectral and temperature dependences of the absorption coefficient of light in narrow-gap semiconductors, remains open, to which this work is devoted.

Below, we obtain an expression for the spectral dependence of the coefficient of interband two-photon absorption of light in narrow-gap semiconductors in the Kane model. In further calculations, we use the calculation method proposed in [5-8]).

Note that the coefficient of multiphoton absorption of light consists of partial components, which by their nature depend on the zone in which the current carriers are located both in the initial and in the virtual state.

In further (intermediate) calculations, instead of

Z((l - fcond )(Econd - EL - 2^W)F(k) , we use the

* 1 expression-F(k L )k2 L, where kc L is the wave

(2p)3 c'L c'L '

vector determined from the energy conservation

law: Ec - El - 2hw = 0. In particular, in the spherical

approximation in the energy spectrum of current

h2k2

carriers, i.e. in the case of EL = Ef) +--, the wave

2mL

vector of current carriers participating in interband

optical transitions is defined as

k2L = ^(hw -Ex),where ^M = is

ci L h2 1 g} mc + mL

the reduced effective mass, mL is the effective mass of current carriers in the zone (or subband) with the number L. In particular, for L = c for the conduction band, then Ef) = Eg, L = ih (hh) for the subband of light (heavy) holes in the valence band EL0) = 0.

Note that the frequency dependence of the denominators in the matrix elements is determined by the energy conservation law, the type of optical transitions and virtual states under consideration. For example, if the virtual states are in the valence band, and the initial one is in the subband of heavy holes, then the denominator in the matrix element of this transition is determined by the

h2

are taken into ac-

ratios A - B = ——, A + B =-

2mhh 2mih

count.

In what follows, we will calculate the partial two-photon absorption coefficients, which differ from each other from the types of optical transitions, i.e. from the initial, intermediate and virtual states:

a) the initial state is in the heavy-hole subband of the valence band. In this case, the coefficient between the two-photon zone light absorption is determined by the expression

(c .hh^X

k

(2)

C,±1/2;V,±3/2 '

. 1 (^

—ho — , h I (2n )3 h3

is determined by the expression

Ehh - Elh - hw = -^^lZBl fhw-Eg )+ hw , if

this transition occurs from the subband of light holes, then the denominator in the matrix element

of this transition is determined as

mc mhh - mlh

Eh- Ehh - hw = ———l

m.

mr + m

"(2h® "Eg)

'hh

A i ^

V^hoo - Efhh x (1)

JVC ,±1/2; V ,±3/2'

mih mhh + mc

Â(2)

C,±1/2 ;V,±3/2

1

(2hw - Eg^j ,where the /

here Eg. = %(( - fcond)S(Econd -EL -2hw), pcV is

k

the Kane parameter [9], e± = e'x ± ie'y and here (and further) it was assumed that Oz k, symbol (...) means averaging over the solid angles of the wave vector of current carriers,

4p \ B2

2(A - B)e 'ez

(-hw) (h- Ehh -hw)

\ A-B ^

v B ,

a(2) +

C,±1/2 ;V,±3/2 ^

15 fhw)

which in the spherical approximation in the energy spectrum of current carriers takes the form

) d 4iB

- hw)

œ hw S2

E - v lh Ehh - hw 0

e 'e '

(h - Ehh -hw)

b (2± C,±1/2 ;V,±3/2

(2)

Â(2,sfer )

C,±1/2;V,±3/2

B2

15(hw)

16m,,

(mhh - mih )

a(2) + 2 C,±1/2;V,±3/2 ^

hw(mhh + mc )

b(2) C.+1/2 ;V,+3/2

(3)

aC,±1/2;V,±3/2 2

aC,±1/2;V,±3/2 3

where for linearly (circularly) - polarized light

a(2) = 9 ) -(2)

C,±1/2;V,±3/2 y

= 13) b(2) = 3 (b(2)

C,±1/2;V,±3/2 ^ C,±1/2;V,±3/2 ^C,±1/2;V,±3/2

In this case, the coefficient of linear-circular dichro-ism for these optical transitions depends on the frequency of light and band parameters;

*(2)

*C,±1 ( b (2)

m (mhh- mh )(2hw- Eg )- hw

b) if the initial state is in the subband of light holes, then we get

= 13 ).

K(2)

C,±1/2;V,±1/2

here

32p2 1„ -hw-x

( A Y P?vk

c ,lh

c h

-Â(2) (4)

C,±1/2;V,±1/2(4)

Â(2)

C,±1/2 ;V,±1/2

1

4p

3Be+

+ 2

(A + B)e '_ez

(Ehh - Eh -hw) (-hw)

2>/2(A + B)

2

(-hw)

2

2

2

2

B2

15(ha)

A+B B

3 ha

è Ehh - Eih - ha.

î(2) C,±1/2 ;V,±1/2 '

(5)

which in the spherical approximation in the energy spectrum of current carriers takes the form

f \2 ^

â(2,sfer )

C,±1/2;V,±1/2

h4 (mhh - mih )2

15 (( amhhm m )

2m

Y

hh

è mhh - m,h 0

+

3 ha

m mhh- mih è mhh m + mh

(lh a- Eg )ha

i(2) C,±1/2;V,±1/2 '

(6)

where for linearly (circularly) - polarized light a) if the initial state is in the subband of heavy

a,

(2)

C,±1/2;V,±1/2

= 8 (

(2)

C,±1/2;V,±1/2

= 7 ), the linear-circular holes of the valence band, then, without taking into

dichroism coefficient for these optical transitions does account the contribution of the effect of coherent

not depend on the light frequency and is equal to 8/7. saturation in kC2±1/2;V,±3/2,

Now let the virtual states of the current carriers be in the conduction band. Then:

we

have

K (2) = 2h ai X

C,±1/2;V,±3/2 » T^c ,hh

h I

' A I4 Ç Pcvk h2 J

v ch 0 è ha mc 0

1

15

î(2) C,±1/2;V,±3/2 '

(7)

where for linearly (circularly) - polarized light b) if the initial state is in the subband of light

,(2)

C,±1/2 ;V,±3/2

= 2 ( ai

2)

±1/2;V,±3/2

= 3 ), the coefficient of holes of the valence band, then

linear-circular dichroism for these optical transitions is constant and equal to 2/3;

K

(2)

C,±1/2;V,±1/2

= — 2ha—X

c ,1 h

i eA° I4 Ç Pvk h2 J

è c h 0 è ha mc 0

3

(2)

C,±1/2;V,±1/2 '

(8)

3

(2)

C,±1/2;V,±1/2

/ M +1

a 1 I 4 a 1 Ç eA0 Î — — W è ch 0 ha mc cV l „'„' l2 Ieze +1

\Î lit., , ha S

+

+

V2e:2 2 \

1 1 4 aa 1 Ç eA0 J 2f" Pvk 1 ha mc 2 .fie:12

J + 4 h2a2 S è c h 0

(9)

from which, without taking into account the contribution of the effect of coherent saturation in Kq±1/2;V,±1/2 , we obtain that for light with linear (circular) polarization 3C2 ±1/2.V ±1/2 = 8/15 (3c±1/2-V±1/2 = 7/15^ ,andthe coefficient of linear-circular dichroism is 7/8.

Now let the virtual states of charge carriers be in the extended spin-orbital zone:

a) if the initial state is in the subband of heavy holes of the valence band, then we get that

K(2) = 2h a X

±1/2 ;V, ±3/2 , 1c,hh

h I

BkP

A J _i__

ch J V2 (Ea- Ehh - ha)

F(2)

C,±1/2;V,±3/2

(10)

here

2

2

1

2

e 'e '

C,±1/2;V,±3/2

11 + 4 aw

h2a2

eAn

BkP

cV

ch ) V2 (ea - Ehh - ha)

+

1 + 4 a<a

-j-2 2 h a

eA^ c h

BkP

cV

+

(11)

V2 (ea Ehh -

from which, without taking into account the contri- b) if the initial state is in the subband of light bution of the effect of coherent saturation in holes of the valence band, then the coefficient of kC",±1/2;V±3/2 , we obtain that for light with linear (cir- two photon absorption of polarized light is deter-cular) polarization, the coefficient of linear-circular mined as dichroism is 2/3;

K(2) - 2h ai X x

C,±1/2;V,±1/2 ZjfiUJ ^c,lh *

h I

eA

BkP^

c h ) V6 (Ea- Ehh - ha)

F

(2)

C,±1/2;V,±1/2 '

(12)

3e±2 + 4e 2,

F

(2)

C,±1/2;V,±1/2

1 + 4 a

h2w2

eA

BkP

ch ) 46 (Ea- Ehh - ha)

+

3e ±2 + 4e 2,

(13)

+

/ e,e |2 |

\i a 1 i 4 a " 1 Ç eAo ) P"k 1 ha mc I „'„' l2 Ieze +1

lit, , ha S1 ch )

from which, without taking into account the contribu-

(2)

tion of the effect ofcoherent saturation in KC ;1/2.V ±3/2 , we obtain that for light with linear (circular) polarization, the coefficient oflinear-circular dichroism is 3/2.

Note that the total coefficient of two-photon light absorption is determined by the sum of the above-mentioned partial coefficients of two-photon light absorption.

Thus, the main contribution to the linear-circular dichroism of two-photon absorption of light comes from optical transitions proceeding from the subband of light holes to the conduction band.

Next, we calculate the spectral dependence of the total coefficient of two-photon absorption of light in the Kane model and use the following expressions

for the energy spectra of current carriers in the parabolic approximation

E, ( )-E. + ^ +

k2 P2v

2

+ 3 a

2m„

Eg ( +A)

hh

(( )=

2mn

Eh (()

h2k2 2k P.

2 T)2

cV

- -A +

h2k2

2m0 k 2P

3E

cV

(14)

2m0 3(A + Eg ) Eg (A) is the width of the forbidden (spin-orbital) band, PcV is the Kane parameter [9]. Then the spectral dependence of the coefficient of two-photon absorption oflinearly polarized light in the region of small values of the wave vector of current carriers will be written as

2

2

1

2

t 2

e

2

1

2

2

2

1

^ (w) = K?i 3(2

(2,1 ) ,V

^ 2hW

è Eg 0

(15)

here K(0) = Pv

e ,V

he 2n2E 3

Eg «

for the case l = 1,

E » E for the case l = 2,

3

(2,1) e ,V

(X)

4X

3/2

3

(2,2) e ,V

(X)

15^6 (X+1)3 32X3/2

480

36

(x+1)1/2 ,(X + 2)3

(3X+1)2 (x+1):

(x+1)1/2 , (X+ 2)3/

15 (x+1) î (3X+1) where X=( -ha- Eg )/ Eg . Calculations show that under illumination of InSb with linearly polarized light, both in the case of Eg ESO and

E g » ESO, the spectral dependence of K(2V (w) increases with increasing frequency, reaches a maximum, and then decreases. This is due to the complexity of the band structure of the semiconductor in the Kane model, which is reflected in the matrix

(x+1)5

9 (X+1)4 + 40 (X+1)2 + 96

(X+1)4 + 2 (x+1)2 + 6)

(16)

(17)

elements and in the energy spectra. This gives rise to complex dependences of the density of states and energies of both the final and initial states of photoexcited current carriers on the frequency of light. If we restrict ourselves to the spherical approximation in the energy spectrum, then K® (®) will increase with increasing frequency under the condition Eg « EsO.

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