DIMENSIONALLY QUANTIZATION OF THE ENERGY SPECTRUM OF HOLES IN A P-TE QUANTUM WELL Текст научной статьи по специальности «Физика»

Section 3. Physics

https://doi.org/10.29013/AJT-22-7.8-26-30

Rasulov Voxob Rustamovich, PhD, associate professor of Fergana State University Rasulov Rustam Yavkachovich, Professor of Fergana State University Farmonov Islam Elmar-ugli, Master of Fergana State University Holmatova G. M., Master of Fergana State University

DIMENSIONALLY QUANTIZATION OF THE ENERGY SPECTRUM OF HOLES IN A P-Te QUANTUM WELL

Absract. The dimensionally quantization in a potential well grown on the basis of a gyrotropic crystal (for example, p-Se or p-Te) is theoretically investigated.

Expressions are obtained for the wave functions of holes depending on the dimensionally quantization number.

It was shown that the dimensionally-quantized spectrum of holes in gyrotropic crystals depends on the ratio of the hole energy to the height of the potential barrier. In particular, the energy spectrum of holes in the well consists of a set of dimensionally-quantized levels that do not intersect with each other due to the presence of an energy gap between subbands of the valence band.

Keywords: dimensionally quantization, wave function, holes, energy spectrum.

I. Introduction problem is hampered by the complexity of the band

Recently, considerable attention has been structure of the crystal. drawn to dimensionally quantization (DQ), which In particular, in [5-7], such a problem in the

has applications in optoelectronics [1]. For semi- case of a rectangular dimensionally quantized well

conductors with a simple band structure, the study (DQW) with a fixed thickness was solved numerical-

of interlevel optical transitions in structures for an ly. However, even a small variation in the thickness

arbitrary potential was carried out in [2; 3]. At the or depth of the RQW can greatly change the final

same time, interlevel optical transitions in semi- result, which makes it difficult to analyze intermedi-

conductor structures with hole conductivity are ate calculations. In [8], on the basis of perturbation

of interest because of the nonzero absorption for theory, analytical expressions [9] were obtained; the

light of arbitrary polarization, which have practical energy spectrum, the wave function of holes, and

applications [4]. A theoretical study of this kind of the intersubband absorption of polarized radiation

in an infinitely deep quantum well of a semiconductor were studied. The calculations were carried out in the Luttinger-Kohn approximation [10; 11] for semiconductors with a zinc blende lattice.

However, the theoretical study of dimensional quantization in a potential well (DQW) grown on the basis of a gyrotropic crystal (for example, p-Se or p-Te) remains open, which is the subject of this communication.

Note that the study of a number of phenomena, in particular, optical or photovoltaic effects in dimensionally-quantized structures (QW) requires knowledge of the energy spectrum and wave functions of electron current carriers.

II. Basic ratios

For a quantum well with a potential U (z), the effective Hamiltonian for a quantum well with a potential U(z) is the effective Hamiltonian of electrons in p-Te in the form

H - Ho +X , (1)

_____ a=x ,z

where Ho = Akl + Bk2z, Ax = A, Az = (3kz and it is assumed that the phases of t M12 he function are chosen so that the coefficient at kz is real, 2A - spinorbit splitting of the valence band at the M(P) point of the Brillouin zone), k2 = kl + k^, A,B,3V are band parameters ofp-Te, kL — k±(sin^, cos^o) is two-dimensional wave vector directed alon g th e interfa ce.

The wave functions in the upper valence bands ( M'1 and M'z) are a superposition of states with the projection of the angular momentum on the axis Z (mz =±3/2)

*M>, - £ C£| mz(, (2)

m7 =±3/2

where C312 -C®2 = C1 = ^(1 + rç)/2, C%2 = —C322 =

\-1/2

\1/2

= C2 =yj(1 —rç)/2, V=pkz(A2 + PXT

The spectrum of holes in the valence band in a bulk gyrotropic crystal has the form

El (,ky,kz ) = Akl + Bk2 - (-1)-1 (A2 +PX ) (k = 1,2). (2') Here A = h2 /(2m±),B = h2 /(2m||), mL and m|| are the transverse and longitudinal effective masses

of holes in the subbands M'1 and M'z, which are

equal with the opposite sign to the effective masses

of electrons.

Then choosing the dimensionally quantization

1 d

axis Oz and assuming that kz —--from (1) we

z i dz

have where

H -A

R2 = A

H = H0 + R2k2,

1 0 0 1

- B

k2,

1 0 0 1

07 -iA

1 0 0 -1

(3)

■f+U (7 ),

07

(4)

III. Hole states in a tellurium quantum well

The unperturbed energy levels EJ0) and the

,(0)

¥2

(0)

(0)

in the sub-

wave function of electrons

bands M^ (£ = 2,1) the conduction band of p - Te are determined from the following matrix differential equation with band parameters A,B:

H oVf = ^f, E 0"

(5)

where E^ =

0 E

. Then we have

A

¥2

(0)

(0)

sl

dz2

A3¥20) "

A¥i0) _ ¥

+ U (z )

¥2

(0)

¥1

(0)

(6)

To simplify the problem, we assume that

U(z) = U0 = const. Then the last equation will look like o

^ _! U (z)-e Vf+iSv dll =o,

b l j2 b dz b 1 (7)

dz

A„(0)

B dz B

dV^_ir[/(z) _elv;» - AMi =c

dz2 b l 1 1 b dz b 2

or

Ô ¥

(0)

Ôz2

■ k"e¥+0) + iK

d¥(0) Ôz

-îk2a¥(0) - 0, (8)

where k"e = — y±0) = ± iy^.

1 (U0 - E) ,

A B

KPV =

B

We will seek solution (8) in three approximations, which we analyze in more detail below.

1. Approximation. From (8) it is easy to obtain the following system of equations

' ayf

^2

d¥(0) dz

-km0) - 0,

(9)

K

Ka

A W(0) - 0,

whose solution we write in the form

y/(°} = D± exp(k±z) + C± exp(-k±z), (10) where k+—ke , k_- ka, unknown quantities D±, C are determined from the boundary conditions of the problem, which will be discussed further. Note that k_ the real value, k+ parameter can be both real (at U0 )E ) and imaginary (at U0 (E ) values. Then the wave function the first case will be exponen-

tial, and in the second case it will be trigonometric. If k± the value is real, then we can assume that D±- 0. This means that current carriers (holes in p-Te) with energy E(U0 will behave like a de Broglie (plane wave), but in other cases they will not.

Further, we assume that the holes are in the potential well. Then

yf) = D+ exp (îkz ) + C+ exp {-Ikz ),

(0) ^ y- - C exp

V

kPv J

where it is taken into account that U0 = 0, k+ - iy]ë/b — îk . Then, from the conditions of or-thonormality and finiteness of the wave functions of holes at the interfaces of the well, we have that

cos(kz)

(11)

^0)

(z ) = 2k1/2-

"ll/2

(12)

[2kû + sin (2Ka ) From the boundary condition of the problem, we obtain expressions for the size-quantized energy spectrum of holes

M -L 1 V

(13)

(2n+l) 2 , ,

E- -A-n (n - 0,1,2,...)

4a2

2. Approximation. Now we look for the solution of equation (8) in the form

¥

(0)

= Dexp(kz) y/(0> = D* exp, (14)

where k* and D* are the complex conjugate wave vector and a parameter whose analytical form can be determined from the above boundary conditions. Then it is easy to obtain the following relations, useful for further calculations

(■-k2 + k2e ) + ^

D =i .L\ r '\-kk?v + *2)J D (15)

D'm l 2 2\2 i * 2\2 Dre, \-K2 + ke ) +\-kkpv +k2a )

where Dre and Dm are the real and imaginary values of the quantity. It can be seen from (15) that the form of the wave function (14) depends on the physical nature of the wave vector. Therefore, consider the following cases:

a) let the wave vector be a real value, then

D =-2 ^ D , (16)

'm 2,1 re

^re

where gre = (-k2 +k2e)/. Then the wave functions of holes take the form

rf (z ) =

2 +1 + 2K

Dre exp (kz ). (17)

?re + 1

Then the energy spectrum of current carriers is determined from the following transcendental equation

V B J±

(ee1 + 1

K

Pv

V ' ee

+ 4

B2

(re1 + 1 2(

-KB ±

( 2 + 1Ka ee

1/2

(18)

It can be seen from (18) that the energy spectrum takes real values when the following inequalities are fulfilled

2

E > B

2çn

Çee2 + 1

2Çee

Çe<

Çee2 + 1

K

Çe,

+ 1

«l >

V^ ee

Çee

Pv

J

V

1/2

Ç 2 + 1Kpv ee

V' ee

(19)

b) let the wave vector be an imaginary quantity, then the relationship between the quantities Dm and D is described as

= -2-

Ç n

Çim ^ 1

De ,

(20)

and the wave functions of holes are determined by

2

^ (z ) = 1% (1+ b )exP (™ ) i + bn

¥-0) (z ) = Dn

1 -

(±2 )

b22

2 Л

ex

p(kz), (21)

where çim = (к2 +к2Е - ккру) !к\ . fc, -

к& к KpV

2 2 кЕ-к

Now let's try to solve the system of equations (7) in general form. To do this, after transforming in (7), it is easy to obtain the Schrodenger equation in the form

a4w(0) , f

dW ôz4

U -E

B

^ ôv20)

V

ôz2

1

--ГM2 -B2

( - E)2 } - 0. (22)

Consider the following cases:

(2 >K-+K+

«0 ^ Po > , f

where p04 = — ¡Л2-(Uo -E2)

Pi =

2

U -E

B

v

a) for holes located in the well (U0 = 0) ,equation (22) takes the form

-2 (0)

-K^f = 0, (23)

g У2 ау2'

0 ^ ' 4

dz4 1 az2

whose solution we seek in the form

= В2 • e" + В2 • e" z + В3 • e", (24) where K2 = B 1 (E + B~lp2v), = B 2(л2 -E2) ,

«1 , «2 = , N±

(«2 )+v(K2 ) + 4«

and,inwhatfollows we will take into account that

(^2 )<^(K2 )2 + 4^4 . Then, from the conditions of finiteness of the wave functions at the interface boundaries, we have

- e- 2i sin(K_- a)

(0) i \ _ B2 Wl z _ 2 ch(tt+- a) - cos(K_- a)

where B2 is determined from the normalization condition of^20) (z ).

The energy spectrum of holes in a potential well is determined by the relation

sin• a)- cos• a) +—exp(-K+ • a) = 0. (26)

b) for holes outside the well (U0 ^ 0) ,the wave function of holes is determined by expression (25) and the energy spectrum by relation (26), but the following substitutions must be made: K1 ^ p1 ,

-K+-z

e

-iK--z

) + 2 2'

iK--z

e

-K+-z

(25)

c) in case of resonance, i.e. when the energy of holes is numerically equal to the height of the potential barrier, then the wave function of holes is determined by expression (25) and the energy spectrum by relation (26), but the following substitutions must be made: Kj2 ^ K01

K0 ^ K00, where

It can be seen from relation (22) that the wave function ofholes in the potential well has two terms, one of which is exponentially decaying, and the rest are oscillating.

Conclusion

Thus, it was shown that the dimensionally-quan-tized spectrum of holes in gyrotropic crystals has a complex form and depends on the ratio of the hole energy to the potential barrier height. In particular, the energy spectrum of holes in the well consists of a set of dimensionally-quantized levels that do not intersect with each other due to the presence of an energy gap between the sus M' and M'2.

Expressions are obtained for the wave functions and energy spectra of electrons for various cases, differing from each other in the 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 valence band of a gyrotropic crystal.

Pv

К

*01 B '

K40 - B 2 (A2 - E2 )

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