The linear circular dichroizm of one photon absorbtion of polarization light in p-Te Текст научной статьи по специальности «Физика»

Section 11. Physics

Rasulov Rustam Yavkachovich, Professor of Fergana State University E-mail: r_rasulov51@mail.ru Karimov Ibrohim Nabiyevich, Professor of Andijan State University Mamatova Mahliy, Researcher of Fergana State University Omonova Hilola, Undergraduate of Fergana State University Sultanov Ravshan Rustamovich, Researcher of Fergana State University

THE LINEAR CIRCULAR DICHROIZM OF ONE PHOTON ABSORBTION OF POLARIZATION LIGHT IN p-Te

Abstract. The spectral and temperature dependences of the one photon absorption coefficient of polarized radiation are calculated and the linear-circular dichroism in hole conductivity tellurium is investigated. In this case, the coherent saturation of the final state of holes is taken into account. Keywords:

Nonlinear absorption of light in a semiconductor with a degenerate valence band, due to direct optical transitions between subbands of heavy and light holes and depending on the state of polarization of the radiation, was researched in [1-8]. In these works, it is signed that nonlinearity in the dependence of the one photon absorption coefficient on intensity occurs due to resonance absorption saturation. This saturation is due to the photoinduced change in the distribution functions of light and heavy hole s in the region of the momentum space near the surface Ehh (k) - Ehl (k) - ña = 0 corresponding to the resonance condition. Here Ehh (k)(Ehl(k)j is the energy spectrum of heavy (light) holes, a is the frequency of light.

In [8], a multiphoton linear-circular dichroism (LCD) in p-Ge was investegated in the developed nonlinearity mode, when n-photon processes with n = (l + 5) a comparable contribution to absorption occur. In [9], four-photon processes in semiconductors due to optical transitions between the subbands of the valence band were investigated. However, one photon linear-circular dichroism in a semiconductor with a "camel's back" band structure remained open, to which this article is devoted.

Here we consider the one photon linear-circular dichroism of the absorption of polarized light, taking into account the effect of coherent saturation [3; 4] in the hole conduction tellurium, due to direct optical transitions between the valence band subbands. In this case, we assume that the photon energy satisfies the inequalities hrn<< Eg, ASO, where Eg is the width of the forbidden zone, ASO is the spin-orbit splitting

of the valence band.

Due to the smallness of the wave vector of the photon compared to the wave vector of the electron (hole) formed as a result of absorption, when calculating the absorption coefficient of light, we can assume \q\ << |k| and put \q\ = 0 where q(k) is the wave vector of the photon (hole).

Then, following [3], the one photon absorption coefficient can be written as

K(œ,T) = -L yÇ (/WrfK^f 5(E2kk-Ek-Hv), (1)

where M„r ,r (k ) is the matrix element of a one photon optical

2k ;1k

transition from state

to

|2fc), I =

nm® A2

2nc

is the light

intensity, Ek is the energy spectrum of holes in the subzone

l (for p-Te l = Mj, M2), fk is their nonequilibrium function depth of the distribution of holes with energy Ek, and na is the

refractive index of light at frequency a . The remaining quantities are well known.

Since the matrix element of the momentum operator is determined by the Hamiltonian of the current carriers, which has the form [10]

H = H+X , (2)

a

where relativistic small band parameters were not taken into account, ua (a = x, y, z) are Pauli matrices. H 0 = Ak\ + Bk^, Ax = A2, Ay = 0, Az = fikz, 2A2 is spin-orbital splitting of

the valence band at the point M (P) of the Brillouin zone).

The wave functions of holes in the subbands M1 (and M2) of the valence band are a superposition of states with the projection of the angular momentum on the axis Z (mz = ±3 / 2)

^m',= x cm; K), (3)

mz =±3/2

c3/2 = C-3/2 = C =V(1+n)/2,

. Here we must bear in mind that the

where

n=PK (A2 +P2kl)-choice of coefficients Cl (l = 1,2) corresponds A> 0, i.e. Cj xC2=2-1 x A2(A2 + f32klyj/2 and contains A2 not, where

the index "2" should be attributed to the upper hole subband, and "1" to the lower one. Therefore, in order to make the transition from the point M to P the Briluyuen zone, we introduce the parameter r = -A2/l A21 in C2, i.e. C2 = r^J(1 -n)/2 . Since

for the lower valence band the spectrum is E1 ~ -^A2 + p2kl, therefore the signs C1 and C2 should be different (Note that the extremes of the conduction band and the valence band Te are located at the point M and P the Brillouin zone).

According to (2), the energy spectrum of holes in subbands M1 and M2 is described by the formula:

em©2 ±(2+PXT. (4)

Here Xv = Ak[ + Bk2z, A = h2 / (2m±),

B = h2/ (2 m,,), m± and m,, are the transverse and longitudinal effective masses of holes in the subbands M1 and M2, equal with the opposite sign, to the effective masses of electrons,

B = 0.326x 10-14 eV• cm2,

A = 0.363x10-14 eV-cm2

Ar = A = 63.15 x10-

eV, p2 = 0.6 x 10-15 eV2 • cm2

Then one of the subbands of the valence band has a "hump", the

2e 2A2

K (ca,T) = ■

-2A2

j(A2- - A.)

of which

fr2 A2 A

= A- ^ + v 4 A fr vector corresponds:

2

V y

determined by the ratio = —2.37 meV to which the wave

1

Pv V4A

-A2 =±2.16-106 cm

k . =±

z,min

In fig. 1 it is shown the one-dimensional energy spectrum of holes in the valence zone of tellurium. We note here that if we increase the values of the transverse effective mass of holes to 16%, then the depth of the hump increases to 9%.

Then the matrix element of the momentum operator p has the form

(e

\C i )

Mv

/ dH (jfc )

( Mr ê ■ \ '

\ d k

M =

ôA1(k )

+H Mr

Z

P=x, y ,z

d ka

d d ka

M

(5)

Z Aa^ a

M

a=x, y, z

with the help of which, it is possible to classify optical transitions, where e is the light polarization vector.

Next, we consider one photon absorption of light and its linearly circular dichroism in tellurium. In a spherically symmetric approximation in the energy spectrum of current carriers, one photon absorption intensity nonlinear in intensity in semiconductors with a degenerate band structure was studied in [2], where it was shown that the absorption coefficient of light decreases with increasing intensity. However, this question for the p-Te remains open.

For one photon absorption of polarized radiation caused

by intersubband optical transitions in the case when light propagates across to the main symmetry axis of tellurium, i.e. for case e = (0,0,ez )

m

eP 21 =-T eJ n

Pv a2

VA2 + Pk

(6)

Then in a spherical approximation, i.e. at Erç,M () = Em2m (k 1, K) = A2A2 + Byki ± A2, (7)

[10]. the coefficient of one photon absorption of light is

exp

V kBT

exp

where we took into account that, from the energy conservation law S\E r -E r -2ha), the wave vector of holes:

\ 3k 1k }

( - A2 )A2 1 exp ( -WV 1, (8)

kBT (A2 - A1 )) t. kBT (A2 - A1 )) In the hump-like approximation (see formula (4)), we

have

k =

ihrn-2A2

'(A - A).

K (c°,t) =

kBT

1

hrn-A (

, kBT

„kBT

cnfi B kH

(9)

a=x, y, z

E

2

F

E( ki). meV

-4

150

100

\ 50 kz,108

-1 0 1 4

-50

cm

-i

where k(a) =

2K

of holes, determined by the ratio of e Fp = the concentration of holes, 1

Figure 1. Energy spectrum of holes in the valence zone of tellurium

for tellurium. Then the probability of a one photon intersub-band optical transition for linear polarization is as follows

</W(1) = 4 (I / Is), (12)

ha

| -1, Ef is the chemical potential

efp _ 2n2pfih2

3m

S, p is

S = { dkz

exp

kBT

WM - f

2m

+

+exp

kBT

VA2+fY k2

h2 2m

where I = 1,2;,

/ W0 = 4rc (I /2Is ), a I

W(1) =■

f (EJ(1 -e

(13) b

(10)

The coefficient of one photon absorption of light in approximation (4) when light propagates across to the main axis of symmetry of tellurium, i.e. when ez = 0, ex ^ 0, ey ^ 0 determined in a similar way.

Next, we will investigate the linear-circular dichroism ofone photon absorption of light in p-Te taking into account the Rabi effect [10], which for an arbitrary light intensity I is determined by the probability of an intersubband optical transition

naB (ha)2 %

E» = Ek(k^ = 0,kZ = kZC) , 00 = 0(kz = kZC),

2 A = E2 (k = 0) - El(k = 0) is the energy gap between the subbands (the valence zone of tellurium and the conduction band of gallium phosphide) in the center of the Brillouin zone,

yjl + x +4x

4n (x) = —

4x

1

/1 + x ~—j= ln

x

yfl + x ~4x

((y)=4l

4 y

1 y -1 •

1 + arcsin

fy

11 + y

(x = I / Is), (14) (y = x/2), (15)

W(1) =

4n aI n iab2k.

"I f (Ek )(1 -

-ptirn

k 5(Ef-E,f-t©)

where Is = a2TxT2I0, I () =

Й3 4

a na 8nab2

e(1 + 2/(S!s))1

;(12)

, e = (A2 + b xr, f (e )

is the distribution function of current carriers with energy E, is the thin structure parameter (e2 / c), nc is the refractive index of medium light at a frequency c , Ti is the time of exit from the resonance region of current carriers in l [2] before the linear in the wave vector of the term in the effective Hamilto-nian (see, for example, [11]), fl = 1/ kBT, T is the temperature, kBis the Boltzman constant, GaP [12]; b = PY and $ = jiYkz

presented in (fig. 2). From fig it is seen that in the region of small light intensity, when the Rabi effect can be neglected, the probabilities of one photon optical transitions do not depend on the degree of polarization, but with increasing light intensity, linear-circular dichroism occurs even with one photon light absorption. According to the last relations, the factor of linear - circular dichroism n = WjD / W^ within the limits of high intensity, that is I >> I0, does not depend on intensity and is equal to /4 «1.1 and in frequency A / h one photon linear - circular dichroism in semiconductors with a "camel's back" zone structure ofthep-Te and n-GaP type taking into account the Rabi effect for the corresponding optical transitions.

K||,extr = K KT), (16)

Figure 2. Dependence W / Wn on the intensity of the exciting light depending

In conclusion, we note that the absorption coefficient of light K (a>,T) at a fixed temperature with increasing frequency of the exciting light, satisfies the condition, passes through an extreme value defined by the expression %a> A, where are the roots of the next cubic equation

on its degree of polarization in the p-Te

dependence K| ((0,T) has a maximum at a point

2P

2yx3 - x2 +

I - 2y ,t>

\

x +1 = 0,

A,

1

and does not depend on the parame-

m I I y

ters s and m , and for the case of smaller values s

Consider in more detail the "behavior" of K (co,T) in the

experimentally interesting region of the frequency of illumina-

m

tion: ha — 2A2 < 2A2. Then for large values e " '

m.

©(2) = — 11--— . As can be seen from the last relation, for

2 V 2y(3

the appearance of the maximum value in the spectral depen-

m

dence K (g),Tj in the case e({——, the following condition

m,,

7^2

the for the temperature should be satisfied T .

kD

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