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Voxob Rustamovich Rasulov Researcher of Fergana State University. E-mail: [email protected] Rustam Yavkachovich Rasulov Professor of Fergana State University Eshboltaev I.M.
Resarcher of Kokand State Pedagogical Institute.
Ahmedov B.
Researcher of Fergana State University.
Mamadaliyeva N.Z. Graduate student of Kokand State Pedagogical Institute.
INVESTIGATION OF DIMENSIONAL QUANTIZATION IN A SEMICONDUCTOR WITH A COMPLEX ZONE BY THE PERTURBATION THEORY METHOD
Abstract: The problem of the energy spectrum and the wave function of electrons in subbands of the conduction band in n-GaP with allowance for the dimensional quantizationwais theoretically considered. Keywords: energy spectrum, quantum well, wave function, dimensional quantization.
quantized well, requires knowledge of the energy spectrum and wave functions of electrons.
For a quantum well with potential U(z), we represent the effective Hamiltonian of electrons in n-GaP as
H = H 0
where
Recently, optical transitions between levels in a dimensional quantized well (DQW), which are used in infrared photoconverters [1], have attracted considerable attention. For semiconductors with a simple zone, the calculation of interlevel transitions for an DQW of an arbitrary potential was carried out earlier in [2, 3]. At the same time, the interlevel optical transitions in the DQW of hole conduction are of interest because of the nonzero absorption for light of arbitrary polarization, which have practical application [4]. A theoretical research of this type of problem is made difficult by the complexity of the band structure of a semiconductor. In particular, in [5-7] such a problem was solved numerically in the case of a rectangular DQW with a fixed thickness. However, even a small variation of the thickness or depth of the DQW can greatly change the final result, which makes it difficult to analyze intermediate calculations. In [8], on the basis of the perturbation theory, analytical expressions were obtained [9]. The energy spectrum of the holes was studied, and the inter-subband absorption of polarized 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 lattice.
However, a theoretical research 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, which was researched in this work.
Note that the research of a number of phenomena, in particular optical or photovoltaic effects in a dimensionally
-Rfc
(1)
A "1 0 " " A3 0 "
2 0 -1 0 Ai _
R2 =
(2)
I2! + U (z ), yz
B3 D sin^cos^ D sin^cos^ B1 A31, B31, D, P are n-GaP band parameters, k2L = kl + k2y, k± = (kx, ky ) (or kx = k± cos^, ky = k± sin^) is two-dimensional wave vector directed along the interface, r± = (x, y). Below, we assume that the wave function of electrons in the D QW plane is Tœeip(!'k1fL ).
The unperturbed energy levels E^ (0) and the wave func-
(0) .
W3
(0)
,(0)
tion of electrons y/(
tion band 3,1)
following matrix differential equation H0,°' _ E,
. Then we have
in the subbands of the conduc-
at n - GaP are determined from the
" (0) _ B , (0) where
E3 0 0 e
>3(0) "
I 2 _-W<0) _
sL
' dz2
+ p—
dz
"%30) "
%r _
-Wi
(0)"
W3
(0)
+ U (z )
W3
(0)"
Wi
(0)
(3)
where the third term describes the transformation of an electron with mass m1(3) to mass m3(1).
Section 12. Physics
From the last equation it is clear that there are two cases. 1-case. In this case, we will assume that the effective masses of the electrons in both subbands are the same, i.e. A3 = A = A. Then the last equation will be
W3
(0)
-W(
(0)
sL
'dz2
aw30) 1 P a
AW((0) J dz
-EwT'
-W(
(0)
W3
(0)
+ U (z )
W3
(0)
W(
(0)
(4)
dZ--P dZ--1 [u (z) - e]z+ 1 =o, (6)
dz A dz A[ Jb" A 2b + W
If we assume that U(z) = U0 = const and make the following notation = -1 (U0 - E), K = "7~, 2x = P, then we will A A 2 4
have
32C dZ-
A
-- 2iX^-KZ-+KZ_' = 0.
(7)
ÀWi(0).
Then we have a system of equations
-, - P , + A [u (z ) - , + 7 V = 0,
dz2 dz
Solution (7) Z- = C ■ exp(az) is simplified if we assume that Z- (z) function is a real quantity, the characteristic equation for which has roots
dz2 A dz
3E,„(tl> D d,„(0)
A2
(5)
a±= ix±J-Z2 + 41 K-KC
+P Mi+1 u (z) - e vr - i V=0.
dz A dz AL 1 ]Y 1 A r 1
Next, we make the notation of type ^3<0) + !Vi<0) = Z(0) and assume that E3 = E1 = E = E(k2) - Bk2. Then we get the equation for Z(0)
(8)
To simplify the solution of the problem, we assume that C = C*, C - is a real quantity. Then
«± = ¡X + ^j-X2 + 4( -K2a) (9)
and we have that
If K(k2a . Than
Z-(z) = exp(iXz){c+ • exp(zj+ 4( -k2)) + C_ • exp(-x2 + 4( -k2)
Z- = exp (i^z) C( • cos (z(/+ 4 (k2a-k2e ) ) + sin (z(/ + 4 (-K ))
=
Considering the boundary conditions of type
£±(z = -a/2) = 0 , £±(z = +a/2) = 0 , if condition cos (a /2X)± i sin (a /2%) 0 is satisfied, then the relationship between C( and C2 is defined as
C( = ±iC2tg(a / 2.^x1 + 4( -k2e)). Inthiscase, from the normalization condition
CeI =
and the expression for C(z) is
sin ^ EJ x2+4 (-KE ) cos I EJ X + 4 (-KE )
(10)
(11)
(12)
Z-(z ) = exp (ixz )
sin ff z + E W + 4 (-kE ) )-sin ff z - E W + 4 (-kE )
1
2 ^ 2v " ) ^ 2,
sin [J z - 2 ^ + 4 (K-K ) ^ + sin [[ z + + 4 (K-4 )
whence the electron wave functions are determined by the ratios
= 12 j[cos(^z) - sin(^z)] • sin ^^ z + 2 jjx2 + 4 (K-K ) ) - [cos(xz) + sin(xz)] • sin [J z - 2 + 4 ((-k2b )
( .
i
2
At = ±a /2) = 0,^10)(z = ±a /2) = 0, we obtain expressions for the energies of the dimensionally-quantized states of electrons at the point X of the Brillouin zone, i.e. with y
[œs(xz ) + sin(^z )]■sin I [z + E Wx2 + 4 ((-Kl ) l + [cos(Xz ) - sin( Xz )]-sin \\z - E Ux2 + 4 (-Kl )
) = exp (-z %)x C ( cos I zj 4 ( -K )-X2 ) + ^ 3 sin ( 4 (K -K )-;
(13)
(14a) (14b)
(16)
A P2 E (2m + () E
E = U 0-------a2
0 2 (6A 4
A P2
E = U 0-----An2n2 aE,
0 2 (6A
(15)
where the first ratio corresponds to even to the inversion of the coordinates of states, and the second to odd states, an integer.
Note that in the case when 4 ( -k2 )-x2 ) 0 , then the wave function can be represented as
'■E '^kj A I 1 C3si^ ^ 4\'^E '^kj 'X
Then, from the normalization condition of the wave function, it is easy to obtain expressions for C(,C3 in the form
C -e = sh(ax) ::
( 2x[x2+4 ((-K }-x2 ] x{4(kE-K) + x2 • cos((4(-k2a)-x2 )+ (17a) +X ^ 4 (kE2 -k\ )-x2 • sin (a J 4 (k;: -k^ )-xE
C-2 =
sh (ax)
2x[x2 + 4 ((-K )~X2 2-case. Let ^x2 - 4 ( +
X{-4( -K) + x2 ■ cos((4(( -K)-Z2 ) +X ■J4(-K)-x2 ■ sin((4( -k2a)-x2 )}. (17b)
be a real value. Then the Schrodinger equation has the form
32C(°)
dz2
■2%
dZ
(0)
+ 2X
dz
dZ2(0)
(-K )Zi(0) = 0,
( +K2)0) =
(18)
0,
Z1(0) = exP (-z x)
2) )J
(19)
I dz2 dz
whose solution we represent in the form
F exp ((x2 - 4 (+K) )-
+F2 exp X2 - 4 ( From the latter one can see if x2 - 4*1)0 ,i.e. KEr )ka , then (19) has the form
exp (-z x)
where k2 = K; -(-1)
(1+«)'2
F1 exp (z^X—K2 ) +
+F; exp (-z<JX2 - 4k2 ) k2a . In this case, the probability den
(20)
sity for finding electrons falls exponentially. From the boundary condition d^Z/t^
= 0 we have
Z(0> = F1eip (-z z)x
eip (zV x2 - 4K )- eip (-zV x2 - 4k2 )
(21)
d
and the condition —Z
dz
(0)
0 gives the ratio by which one
lets determine the energy dispersion taking into account the dimensional quantization
2 k
2 -
eip (la^X^-ÂK2 ) = -
X
+ 2 1 -
2k X
where kI
= Y"(U0-E«), = 7T,
2k
(22)
X
E = E - Bk2. Due to the
A^ 0 A A 2
lack of an experiment on dimensionally quantization in n-
GaP, we do not carry out a numerical calculation (22).
3-case. In this case, we will not pay attention to the case where the conversion of an electron from one zone into another occurs, i.e. assume that P = 0. Then when
U(z> = U0 = const we have the following system of equations
fdV(0>
dV1- - *vi°>=0,
(23)
21 Here ^ = —
dz2
dV30)
I dz2
K- (-1)
- k023v30) = 0
(1+Ç)/2
— U
Solution (23) is (24)
sought in the form
(0) n ik0£z -ik0£z
) = B(e s + D(e H where D^, B^ are constants, determined by the boundary conditions of the problem under consideration. From the condition that the derivative of the wave functions to zero at the interfaces of the well interface, it is easy to obtain E^ (k±,n^,a)
Ec = Bk2 +
_L I +( -1>(1+L)/2- + U0 ( = 1,2,3,...). (25) a ) 2
Thus, itwas shown that the dimensionally-quantized spectrum of electrons in a semiconductor, the conduction band of which consists of two subzones, between which there is an energy gap, consists of a 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 parameters 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 k±, 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 work will be devoted.
The work is partially financed by the grant OT-F2-66.
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