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Section 12. Physics
Rasulov Voxob Rustamovich, Researcher of Fergana State University E-mail: r_rasulov51@mail.ru Rasulov Rustam Yavkachovich, professor of Fergana State University Eshboltaev Iqbol Mamirjonovich, resarcher of Kokand State Pedagogical Institute. Mamadaliyeva Nargiza Zokirjon qizi, Tolaboyev Dinmuhammad, undergraduate of Fergana State University
PHOTON DRAG EFFECT IN p-Te
Absract: The spectral and temperature dependences of the current of the photon drag effect in the tellurium of hole conduction are calculated. In this case, the momentum of the photon is taken into account both in the law of conservation of energy and in the law of conservation of momentum. The calculation of the photocurrent was carried out in the approximation of the hole momentum relaxation time.
Keywords: photocurrent, photon drag, photon, relaxation time of hole momentum.
The effect of dragging by photon (PhDE) in telhri^ (c31|z),a PhDE current is generated both along the principal
caused by the transfer of the photon momentum to the electronic subsystem, was experimentally observed in [3; 4]. A theoretical interpretation of the experimental results [3; 4] is given in [5] in the spherically symmetric band approximation and in [6], taking into account both the quadratic and linear in the wave vector (k ) contributions to the effective Hamil-tonian of hole. As indicated in [7], taking into account the dependence of the square of the matrix element of the optical transition on leads to an additional contribution to the current of the PhDE.
Expanding the current density of the PhDE (j ), proportional to the intensity of light, along the polarization vector and the photon wave vector (q ), we have the following relation [8]
ja= I^aprSepeYqS > (1)
where I is the intensity, e is the polarization vector of light, and &aPrS is the PhDE tensor (a,p,y,8 = x,y,z).
In what follows, we consider the theory of a linear PhDE produced in homogeneous crystals when they are illuminated by linearly polarized light. Then (JaprS it is real and has nonzero components in crystals of arbitrary symmetry. Therefore, the EFM arises in media, both with a center of symmetry, and without an inversion center. For example, in tellurium, in the propagation of linearly polarized light along the main axis
axis of the crystal and in the direction transverse to C3 jz = > jx = te^ sin20', jy = fo^ cos20' (6)
Here 9' is the angle between the plane of polarization of light and the axis of rotation of the second order, directed along the x axis.
In microscopic theory, the expression for the PhDE current in the relaxation time approximation has the form
j = -e X Vni fn
(as )
(2)
where v is the velocity operator, e is the elementary charge, is the asymmetric (nonequilibrium) part of the hole distribution function in the zone n. In the future, we calculate the calculation in approximation in the relaxation time t^ and take into account the following Keldysh diagrams where the wavy line is a photon, the solid line is a hole.
In addition to taking into account the dependence of the optical transition probability on the photon momentum (both in the energy conservation law and in the momentum conservation law), we also take into account the following contribution to the PhDE current, which is related to the dependence of the magnetic field strength vector H
H = iA(q x e). (3)
tlK
Here A = eAev is the vector potential of the light wave. Then we have the following additional term in the effective hole Hamiltonian H [10]
ig (â(q x e )) = ipog (Hâ) (4) e h
HH = (-1) qer' (V1 ->r cos0 +nsin0) (6)
ch 2m„ V '
where
u, eA h2
H =---ig[
ch 2m„
where g is the g-factor of the holes, HQ =
2m0c
is the Bohr
magneton, ûa are the Pauli matrices. Further, we have the following relations useful for further calculations
ey,(ncosO-yJ 1 -rf sint
(l\H 'I l ') = H'n,= ieeAgq
-(-1)) ( cosO
0)
M.
2,k +q ;1k f
coso + ez, sin
; ef J
e' = {cosOcosp, - sm^,sm0sin^},
q' = q {-sin0,O,cos0|,
( 2 2 2 \-1/2
A2 + ¡3Vkz) , ¡3V is zone parameter of tellurium, 2A2 is the energy gap at the point M of the Brillouin zone.
Then the square of the matrix element of the interband optical transition wich depends (linear) on q is written as
(5)
2m
cos
ez Q '^vVW2 - 2Akz,)
dex,ez, + eLz,sind + i(e xe')x, (n'cosö-^/ 1 -n'2 • sind
(7)
where Qz' =
n = v(K ^ K) (1 -v'2)-1/2 en
A=
2
A + A,
dk,
A1,2 - "
2m
-, m12 is effec-
2
(( * l' = 1,2),
tive mass of holes. It is seen from (7) that after angular integration, the quantity proportional becomes zero, i.e. in tellurium
this additional contribution does not arise to the linear PhDE due to non-vertical optical transitions, and only a circular PhDE [11] can arise and this contributionin Te disappears in the case fiVkz\|A2. Then in the spherical approximation in the
energy spectrum of the holes: E^ =(-1) A2 + Atk2 the longitudinal current of the PhDE in tellurium (without taking into account the g-factor of the holes) is defined as
Û = K-1)1' 2eK"hq A
where
l =1,2
e X ß2v flK
5 ha m* A2 - Al
W
1+2
3
d ln rl (E°l ) d ln EÎ
A ha- 2 A.,
A2 - A1
XT
{ r
1 - exp
V v
ha
'XT
K = -
e2 kBT ß2ko
J J
(9)
K „ = K (e\\C3 )=n0 , / x
V " 3) 3cnmh1a(A2 - Ai )
is the light absorption coefficient in tellurium during the optical transition of holes between the subbands m1 and m2, e; = El (k = k J,k2 = (ha - A22) X ( A2 - Ai )-1,
4PX
2nnch ha A
exp
ha . , -2 . -AXo +A2
KT
where A,, , =
h2
\j(h®f -
(8)
(11)
-, ko =
4A2
-, m „ and m, lon-
k2 =
^ fx„ = e
^ A2 -AXh
XBTe kBT
(A2 - A, ) K + 4k2 (A2 - A, )A2
4ß
For completeness of the problem, we give below an expression for the intersubband absorption coefficient of light when the "camels back" of the subband of the valence band in tellurium is taken into account, i.e.
ax2+A±(ki+k2 )+(-1)'V A2+ßVki
and in type e||C3 in type
Ex =
(10)
2m ' 0 2BV
iU ' V
gitudinal and transverse effective masses of holes, «is the Fermi energy, other quantities commonly known.
Figure 1 shows the spectral dependence of the absorption coefficient of linearly polarized radiation (K / K0 ) in in tellurium for two temperatures, where the following values of the band parameters are chosen: A2 = 63,15MeV, A±= 0,326-10-14 eV • sm ~2, A = 0,363-10-14 eV • sm~\
132 = 0,6 • 10-15 eV2 • sm-, K0 = ¡5ve(2A± • 274nna )-1.
It can be seen from (fig. 1) and (fig. 2) that with increasing temperature the extreme value of the light absorption coefficient in p-Te decreases by more than two orders of magnitude and shifts towards smaller frequencies.
2
2
h
K(x) , arb.umte
0.035
0.030
0.025
0020
0.0 L5
0.010
f
2 3 4 5
X
Figure1. The spectral dependence of the light absorption coefficient in p-Te for kBT /2A2 = 0,2 , where x = ha/kBT. The values of the zone parameters are given in the text
Figure 2. The spectral dependence of the light absorption coefficient in p-Te for kBT /2A2 = 2 , where x = ha/kBT. The values of the zone parameters are given in the text
In the approximation (10), the contribution to the PhDE due to the account of (6) is described by the tensor (for e || C3)
a
(() = (_!)' 16nel Kv SlcBTA2hagTiKh
ha
2
Of V
moß-
(ha)2 -4A2
where 5 is band parameter of in tellurium.
It can be seen from the latter that the contribution to the PhDE in in tellurium, which arises from taking into account the -g factor, increases with increasing temperature, the quantitative value of which depends on the value of 5 and g factor.
Calculations show that the extreme value of the theoretical spectral dependence of the current of the PhDE is 1.2 times smaller than the experimental value. This, apparently, is connected with neglecting the anisotropy in the energy spectrum
of the electrons. Naturally, in this case, the spectral and temperature dependences ofthe current of the PhDE must be calculated numerically. This case requires separate consideration.
This work is partially funded by the OT-F2-66 project.
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