THEORETICAL ANALYSIS OF MULTIPHOTON INTERBAND ABSORPTION OF POLARIZED LIGHT IN CRYSTALS WITH A COMPLEX ZONE (PART 1) Текст научной статьи по специальности «Физика»

Section 6. Physics

https://doi.org/10.29013/ESR-21-3.4-48-51

Rasulov Voxob Rustamovich, associate professor of Fergana State University Rasulov Rustam Yavkachovich, professor of Fergana State University E-mail: r_rasulov51@mail.ru Kasimov Forrux, teacher of Andijan State University.

Muminov Islombek Arabboyevich, doctoral student of Fergana State University Qo'chqorov Mavzurjon Xurshidboyevich, teacher of the Kokand State Pedagogical Institute

THEORETICAL ANALYSIS OF MULTIPHOTON INTERBAND ABSORPTION OF POLARIZED LIGHT IN CRYSTALS WITH A COMPLEX ZONE (Part 1)

Abstract. A calculation was carried out to investegate the spectral dependence of the coefficient of interband multiphoton absorption of light in narrow-gap semiconductors of the In Sb type in the Kane model, where the contributions of intermediate states located in the subbands of light and heavy holes, in the subband of spin-orbit interaction of the valence band, and also in the conduction band in many a quantum process without taking into account the effect of coherent saturation. Since in semiconductors of the InSb type, the energy distance between the adjacent lower and upper conduction bands is significantly greater than the widths of the forbidden or spin-split bands. This makes it possible to carry out calculations in the two-band Kane approximation.

Keywords: semiconductor, coefficient of interband multiphoton absorption of light, two-band Kane's approximation.

Currently, the main research in the field of mul- little investigated both in theoretical and experi-tiphoton absorption of light is carried out in wide- mental aspects. The main reason for this is that the gap semiconductors, since a number of their physi- theoretical research of a number of photon-kinetic cochemical properties have been studied in depth phenomena in narrow-gap crystals requires the use and in more detail. In this respect, the multiphoton of not only the Luttinger-Kohn approximation, but effects occurring in narrow-gap crystals have been also the multiband Kane approach. In the case, theo-

retical calculations are performed using matrices of at least 6 x 6 or 8 x 8.

Next, we will determine the expression for the coefficient of multiphoton absorption of light ( K(N\ (0,T) ) using the perturbation theory [1]. In the calculations, we will take into account that H the effective carrier Hamiltonian consists of two components, one of which H0 is an unperturbed Hamiltonian, the eigenvalue E^ and the eigenfunc-tion I y/n) of which are known, in the second ( H' ), it takes into account the interaction of the carrier system with the electromagnetic field, i.e. ... electron-photon interaction, so

H = H0 + H '. (1)

In the general case, in perturbation theory for the diagonalization of the effective carrier Hamiltonian, a unitary operator of the form e— is chosen, under the action of which a diagonal (or quasi-diagonal) Hamiltonian is formed, i.e. H - e SHeS where S is some operator. In this case, the diagonal components of the matrix elements y/m IHI y/l, calculated using the wave functions of the current carriers in the Kane model [3; 4], are nonzero, and the off-diagonal ones are zero.

As a result, the multiphoton absorption coefficient conditioned by interband optical transitions in narrow-gap semiconductors of light can be calculated using the following relation

KNm (t)= x Kf4y

c ,m'c ;V,mV

N hE

Y WN) , ,

/ i c m ;ç ,mç ;V,mv

(2)

c ,mC ;ç ,mÇ ;V,mV

M{m\= -1- V

mm ~ / j

2 l ,m '

Here K(cN2c]Vmmv (<x>,T) is the partial light absorption coefficient corresponding to each interband optical transition, when calculating which it is necessary to sum over all intermediate states; K^m. (g>,T) is the resulting coefficient of multiphoton light absorption, which is determined by summing over all initial ( V,m'V^j(m'Vi =±3/2,± 1/2^ ),intermediate

(\ç,m'g) ) and final (|c,m')(m'± 1/2) ) states, 66

W

(N )

c,m'c ;ç,m' ;V,mC

is the probability of the transition of current carriers from the valence band to the conduction band through intermediate states, and it is defined as

w(n >

c,mc ;ç,m. ;V,mv

= — y |m(n >

h m '

c,mc ;ç,mç ;V,mv

(3)

(k) •[fe(k)-V (k)]• ô(Ec (k)-EVi (k)-Nhœ),

ç,m'ç) can be located and in the subbands of the subband

in

where intermediate states both in the conduction banc the valence band: |Vhh,m'hh) (m'hh =±3/2) of heavy holes and in the

subband of | Vlh,m'lh) (m'lh =±1/2) light holes, as well

as in the subband |SO,mSD) (m'sO =±1/2) of spinorbit splitting, the composite matrix elements of interband optical transitions are designated as MN2 ^ VK(k ) and for N = 1,2,3,4 are determined by the relations

M(1,= H ,

mm mm

Mmm=1 YHm h 2l

lm'

E - E E '- E,

H H H

ml lm" m"m

H H H

mm" m"l lm'

{Em'- E,) (Em.- E, ) (Em - E, ){ - Et )

(4a) (4b)

(4c)

"1 Y,HmlHll 'Hlm'

l ,l '

.(Em - E,) (Em - Ev) {Em,- E,) - Ev\

Mm=- x_1_

2 l,m",m" (Em" — El) (Em" — El )

H H H H H H H H

mm" m"mml mml lm' _|__ml lm" m"mm m"'m'

E„ - E,

E„, - E,

(4d)

-2 z

^ l,l',m'

HmlHll' Hlm' ' Hm' ' m

(Em'- El )( E{)

~ ^ HmlHlm'' Hm'' VHi f

4

1 1

V Em El Em' El' J

_ Hmm"Hm"lHll'Hl't

(Em -El')('-El)

1 1

V Em' El' Em El J

24

l ,l ',m' '

(Em - E, )(Em - Ev )(' - E, ) (Em' " E, )(Em' " E, )(' - El)

(Em - EV ) (Em.- El ) 1

(Em- E,) (Em.- Ev) 1

V Em El Em" El' J 1 1

(Em'- El.)(- Er) 3

1 1

V Em' El' Em" El J

+ -2

(Em"" E,) (Em- El' )

2 ,l',rHmlHll'Hll"Hl'm'

V Em El Em' El' У

1 1

V Em El Em' El' У

(Em - E, )(Em'- El' ) 3

V Em'' El Em" El' У

(Em " Et )(Em'" Er)

V Em'' El Em' ' El' J

(Em - E ) (Em - El ')(Em ~ El. ) (' - E, ) (Em' " E,, )( " E, )

/с (k)[fVl (k)] is distribution function of elec- netic wave, I is the light intensity, "ю is the expo-

trons (holes) in the band with energy Ec (k)\_E^ (k)] nent refraction of the medium at a frequency of ю .

in the conduction band (in branch l(hh,lh,SO) of the To simPlify further calculations, we assume that

valence band), v(cV ,SO) is the eigenvalue of the the ener§^ sPectrum of current carriers is sPheri-

projection of the angular momentum operator on the ca% symmetric, that is, proPortional to the square

z-axis of the momentum of the digital zone of the of the wave vector of the current carriers- We also

ordinal range [3; 4], Hmm are matrix elements of assume that for narrow-gap crystals the matrix ele-

A ■ p electron-photon interaction, calculated with re- ments of the type (e ■ p),, momentum operator is

sPect to the basis wave functions [3; 4] and it is de- quantitatively equal to m0PcV / h. As a result, we ob-termined usmg re2atiorn (4, a-d), in particular tain a relation that makes it possible to calculate the — (e ■ p) p is the momentum spectral and temperature dependences of the coef-v 'wf J 11 ficient of multiphoton absorption of light due to inoperator, A is the vector potential of the electromag- terband optical transitions

Hu -=■

W N , = _L

f „ Y

v mo?j

2nI

nc

((I?) hE - E g)-> ^(N?) x[ / (Ec (C?))-/(Ev, (€1))

, (5)

where k{Na) =

2 mr mv

c vl

mm + m„

1 m ^ 2

(Nhw-e) isthe wave J dcos(0) J dy\kdk2 £ |mn ;?>m, ;V>m, (k,0,p)| ,and

-10 0 i>m4

vector of the final state electron passing from the its comPonents are expressed by the following re-subband of the valence band to the conduction lations

band, where photons with energy Nho are absorbed, mc is the effective mass of current

carriers in the band c (V;), the value determined by the integral

^ i ,m; ,mv, \k ,JVl) (ee'p ) ?(2)

c ,m'c ;V, ,mVl

;V,mV \ kc ,mv,

c,mc ;V, ,m4l у cj

is

Y(e ■ p) , ,(г ■ p) ,V , [£cV (Л-ha]'1,

(6a), , ,(6b)

1

1

g,m

ft

(3)

* () ) = I (• p^ (• p)m^ (• pW,^ *[ V, (J )-^H^v (V )-, (6c)

(3)

m' (k'C'tn^ ) = Y ( ■ P) , • P) ,£ • P) ,, • p) ,V ,

ç,m!q ;Ç,m'ç

*[eÇ,V (! )-3^®]-1[e,,V (V )-2^®]-'[ec,V (V )-mT1,

(6d)

Below we will use the Kane model in narrow-gap ter of the Brillouin zone. This type of optical transi-

crystals. According to this model, (e ■ p ) . , ma' c * /c ,mc ;V; ,mV;

trix elements characterizing optical transitions oc-

tion occurs due to (( ■ -pj mixing zones [3-5]. As a result, the matrix element of the optical transition is

curring from the subbands of heavy and light holes proportional to ( /(m0Eg))( • p)m, ,v m , and its in the valence band to the conduction band do not cl'my'

depend on the wave vector of current carriers [2]. In particular, (e • p) , , , matrix elements will be less

V 1 / c ,mc ;c ,mc>

share in the multiphoton absorption coefficient of light absorption is relatively small.

If we take into account that, according to the than (e • p)cm, m, matrix elements for interband method for calculating the effective mass [3,4], we

have the relation pc,V = fm0PcV / h, then the coefficient of multiphoton absorption of light has the form

optical transitions in the region located near the cen-

K(N }(v,T ) =

V2E,

8^2 f PV

4k2N2 h2e2 v nE m2c

(Nhv/ Eg -1)1/2 (Nhv)/Eg)2N-1

f ( Ec (k^c,mV/j)) f ( EV (k^c,mV/j)) ^ ^(,rn) ;ç,m'i ;V,m'v (kimvj)

(7)

Using (7), we can analyze the spectral depen- cient for a specific case. This case will be covered in

dence of the multiphoton light absorption coeffi- our next work.

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2. Arifzhanov S. B., Ivchenko E. L. Multiphoton absorption of light in crystals with the structure of diamond and zinc blende // FTT. 1975. - Vol. 17. # 1. 81-89 b.

3. Bir G. L., Pikus G. E. Symmetry and deformation effects in semiconductors.- M.: Media, 2012.- 584 p.

4. Ivchenko E. L., Rasulov R. Ya. Symmetry and real band structure of semiconductors. - Tashkent. Fan, 1989.- 126 p.

5. Rasulov R. Ya., Akhmedov B. B., Muminov I. A., Umarov B. B. Crystals with tetrahedral and hexagonal lattices. Fergana. Classic. 2021.- 129 p.