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Section 4. Physics
https://doi.org/10.29013/ESR-21-9.10-22-26
Rasulov Voxob Rustamovich, PhD, associate professor of Fergana State Universityy
Fergana, Uzbekistan E-mail: r_rasulov51@mail.ru Rasulov Rustam Yavkachovich, Professor of Fergana State University, Fergana, Uzbekistan Eshboltaev Iqboljon Mamijonovich, PhD, associate professor of Kokand State Pedagogical Institute,
Fergana, Uzbekistan Quchqarov Mavzurjon Xursanboyevich, Resyarcher of Kokand State Pedagogical Institute,
Fergana, Uzbekistan Sultonov Ravshan Rustamovich, PhD, Senior Lecturer of Kokand State Pedagogical Institute,
Fergana, Uzbekistan
INVESTIGATION OF SIZE QUANTIZATION IN A GYROTROPIC SEMICONDUCTOR
Abstract. The problem of the energy spectrum and wave function of electrons in the subbands and the valence band of tellurium is considered theoretically with allowance for size quantization.
Keywords: energy spectrum, quantum well, wave function, dimensional quantization.
Recently, optical transitions between levels in a di- study of this type of problem is made difficult by the
mensional quantized well (D QW), which are used in complexity of the band structure of a semiconductor.
infrared photoconverters [1], have attracted consider- In particular, in [5-7] such a problem was solved
able attention. For semiconductors with a simple zone, numerically in the case of a rectangular D QW with a
the calculation of interlevel transitions for an D QW of fixed thickness. However, even a small variation of the
an arbitrary potential was carried out earlier in [2; 3]. thickness or depth of the DQW can greatly change the
At the same time, the interlevel optical transitions in final result, which makes it difficult to analyze interme-
the DQW ofhole conduction are ofinterest because of diate calculations. In [8], on the basis ofthe perturba-
the nonzero absorption for light of arbitrary polariza- tion theory, analytical expressions were obtained [9].
tion, which have practical application [4]. A theoretical The energy spectrum of the holes was studied, and
the intersubband absorption ofpolarized radiation in an infinitely deep semiconductor quantum well was studied. The calculations were carried out in the Luttinger - Cohn approximation [10; 11] for semiconductors with a zinc blende grating.
However, a theoretical study of dimensional quantization in a potential well grown on a semiconductor base with a complex zone, one subzone of which has a "hump-like structure" (for example, n-GaP or p-Te) remains open, to which the present work
Note that the study of a number of phenomena, in particular optical or photovoltaic effects in a di-
The unperturbed energy levels EJ0) and the
v
wave function of electrons y/f ^ - 2
bands M^ (£ = 2,1) of the conductionband in p - Te are determined from the following matrix differential equation
Htff = Esyf?\ (4)
(0)
in the sub-
where E^ =
0
V
(0)
-Vi
(0)
0
sL
dz2
. Then we have
(0)
A3V3
AiVi0)
+ U (z )
V
(0)
Vi
(0)
V0)
3Г3 (0)
EiVi
(5)
For nonzero values of the two-dimensional wave
mensionally quantized well, requires knowledge of . 7 2 r.i . j .1 c
' n b vector k± or the energy spectrum and the wave function
the energy spectrum and wave functions of electrons. For a quantum well with a potential U (z), we rep-
resent the effective Hamiltonian of electrons in p-Te in the form
h = ho +X (1)
a=x ,z
where H0 - Ak2L + Bk], Ax = A, Az = (3kz and it is assumed that the phases of the function are ch osen so that the coefficient at is real, 2A is the spin-orbit splitting of the valence band at the point M(P) of the Brillouin zone).
Then choosing the dimensional quantization axis and assuming that from (1) we have
of current carriers, one can expand in a series in k±, i.e. E(k, z) = Ei(fc1 =0) + ^1 8E
V! dkV,
and
к>ХмТк;(6)
к, -0
|h,N,±)=Vç (к,z) (0,z) +
z
1
v! dk:
K-ZxfX
(7)
k, -0
H = H0 + R2k 2,
where
H0 -A
0 1 1 0
+ B
1 0 0 1
i"тт + PV
dz
1 0 0 -1
+ U (z ), R2 - A
1 0 0 1
(2)
-1 d i — + dz
(3)
Тогда имеем As indicated in [3, 8], the relationship between the coefficients from equation (6) and functions from (7) at the same £ are related by
(qh\ qe
-j —-, where means averaging
over the state. £ Then we have
/ dH\ 8E
dk,
dk,
2( kA)t^Y^W (7)
Substitution of (2) and (3) in (7) gives
2k ,R
\ — S^k- (8)
. V—0
V —1
kL = kx + ky, A,B, PV are band parameters p-Te, kL — kL (sin^, cos^) is the two-dimensional wave vector directed along the interface
(3)
Whence equating terms of the same order in the two-dimensional wave vector of electrons in both sides of Eq. (8), we have
(0) R2| 40)),
(1) R|X?) + 2(X? RIX?) + 2(X?RIX?) + ^X |W01x22 "
Л? - 0, =
(4)
= —2
(x? R2I xf)+{xf] R2I 2f)+{xf] R212 +{x? R2122)+(2 R212i)
(9)
2
3
v-0
v-0
2
V —1
2
4
Now let us determine xf1 from the equation: [H0 - Ef (0)? ] ;o) - 0 , where the unperturbed E^ (0) energy levels and y wave functions are determined from two independent stationary Schrodinger equations
2,„(0) a,„(0)
Af (0) - B f -Pvif + [U(z) - t2 f - 0,
a^20) - b 2
From the latter, the following equation can be obtained
y . (0),,„ V0)
ff+[U (z)-Êf - 0.
(10)
-kM0) + Kpy
i^y0 - 0, (11)
az2 "" ^ dz where it was assumed that U(z) = U0 = const and the following designationswere drawn: -0) =^2° — ZV1(0),
kf = —
E В
1 (Uo " Ê),
=-, kp = P
a ri Pv b
B
Note that the solution of equation (11) can be carried out in three approximations, which are analyzed in more detail below.
1-approximation. Let both the real and imag2-nary components of equation (11) be equal to zero separately. Then we have
dz2
d¥(0)
-Kyf = 0,
(12)
K
dz
K
A W(0) = 0, Pv
whence we immediately obtain that
i i \
(0) ^ y _ - C exp
v y
y/^ = D1 exp (kez ) + D2 exp (-kez ),
where
Kl = B(U0 " ,
KPv =
(13) Pv
K2 = &
B\ " / A b pv b Q, D1,D2 are unknown coefficients determined
from the boundary condition of the problem under consideration.
2-approximation. We seek the solution of equation (11) in form
vf - Dexp(kz), y/0) = D* exp(k*z) . ( 14)
Then, assuming that D — Re(D) + iIm(D) - De + iDm and substituting (14), we obtain
expressions connecting the real and imaginary parts of the coefficient D :
D^ = i •
(■-k2 +kE ) + ^
K KPv + KA ,
(-K2 + K ))(
K KPv + K
)
-D„.
Whence it can be seen that if the value is real, then Dm = - 2—^—De, if the value is imaginary,
^re ^ 1
then D = -2 D , where c = (-k2 + kI )
im 2 ^ j re * re y E j
/
/ (
KKpv +
) Ç m (
= ( +K2e-KKpv
)/
K
3-approximation. From the system of equations (10) we have the following relation, which connects the functions ^20) and y/t(0):
^ = B ^u(z)_E2V« + iSoMl. (15)
Yt A dz2 aL w 2 ™2 A Sz
A Ôz
After some transformation, we have equations for
W2
(0)
a4 (0) 1
d y(2 [ +1 B
dz4
U - E-ÊV.
U t B
2\ a2 (0) d ^2)
dz2
1
(16)
V "J
( ¥ (A/"(u 0" E2 i}--(0) - 0.
It can be seen from (20) that in this case, three cases can also be analyzed:
1-case, i.e. when a microparticle is in a pit, i.e. U0 = 0. Then (16) takes the form
eV20) , „2 3V20'
■+K2 -Kv20) = 0.
(17)
dz4 1 ôz2 where K2 -B 1 (t + B-'ft), K0 -B 2(A2 -1^.
The solution to equation (17) can be represented as ^20) - X Bv-exp(av ■ z), (18)
where
v-1,2,3,4
a1 - -a3
1 3 a2 ~ ~a4 ~ iK,
K±^2 ±^K2 ) + >/(K2 )2 + 4^0 . Since «1 >0, «3<0,
therefore, in (20) we can assume that B1 = 0, i.e.
^20) = m2 • exp(itt_ ■ z) + m3 • exp(-K+ • z) + B4 •
• exp(-itt _• z). (19)
It can be seen from the last relations that the first term is exponentially decaying, and the last two terms are oscillating.
To determine the values В3, В2, B4, we use the fol-
lowing boundary conditions: = v20) (a) = 0. Then we have
vf (z = 0) =
В4 =
[exp(№- ■ a) - exp(-K+ ■ a)]
В3 = В2 ^-1 +
[exp(-№- ■ a) - exp(-K+ ■ a)]
[exp(2z'K_ • a + • a) -exp(-K+ • a) -2isin(z'K- • a)]exp(-K+ • a) 2[ch(K+ • a) -cos(K_ • a)]exp(-K+ • a)
2nd case. Let be Un ^ 0. Then we transform
equation (18) as
a„2 a2^20)
Pi ■
az4
az2
-^20) - 0,
(20)
where
Pi =
2
U0 -E
0B
- J2 {A2 "(U 0 - E 2 ) . Then the solution to the last
Bm (m = 2,3,4) where the following replacements should be made: fp\ ^ ^2 P4 ^ .
Resonant case, i.e. when (U0 - E2 ) = 0 U0 = E .
Then equation (16) takes the form
a4,,7(0) p2 ~(0)
aw dW2 _K40,20) = 0, (23),
equation will b(0 defined as
\j/[0) - В3 • exp(-a1 • z) + B2- exp(a2
+ B4 • exp(-â2 • z), where
■ z) +
az4 "+K()1 az2
whose solution takes the form wT - G3 • exp((p( • z) + G2 • exp(ip+ • z) + Ç • exp((ip+
(21)
where k± —
" V2
±K2i +
(4i + 4^q0 f
X
(24)
a3
. (22)
Here, expressions am (m = 1,...,4) and Bm (m = 2,3,4) are defined as am (m = 1,...,4) and
_ , K0i B ' N°4 -B2(A2-E2).
It can be seen from the last relations that in this case, too, the first term in relation (24) exponentially decays, and the last two terms are oscillating.
Using boundary conditions of the type ^20) (z = 0 ) = ^20) (a ) - 0, it is easy to obtain the relations by which the unknown C2, C3, C4 coefficients are determined
C 4 _ C 2
C -C + •
[exp(2i^+ • a + k_ • a) -exp(-K_ • a) -2isin(^+ • a)]exp(-K_ • a) 2[ch(K_ • a) -cos(^+ • a)]exp(-K_ • a)
• a + k_ • a) -exp(-K_ • a) -2isin(K+ • a)]exp(-K_ • a)
[exp(2i
k,
2[ch(K_ • a)-cos(k+ • a)]exp(-K- • a)
Thus, it was shown that the dimensionally-quan-tized spectrum of electrons in a semiconductor, the conduction band of which consists of two subzones, between which there is an energy gap, consists ofa set of dimensionally quantized levels that do not intersect each other due to the presence of an energy gap. Expressions are obtained for the wave functions and energy spectra of electrons for different cases, differing from each other by relations for the characteristic wave vectors, which, in turn, depend on the band pa-
rameters of the semiconductor and on the energy gap between the subbands of the conduction band.
In conclusion, we note that this problem can be solved by the perturbation theory method, where we can consider as a perturbation the terms in the effective Hamiltonian containing kL, where it is necessary to expand the energy spectrum and the wave function of electrons in a two-dimensional wave vector. This case requires separate consideration, to which the following message will be devoted.
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