CC BY
11
UDC 535.8; 537.9; 539.33
DOI 10.21685/2072-3040-2018-3-9
V. D. Krevchik, A. V. Razumov, I. M. Moyko, T.-R. Li, Y.-H. Wang
DOUBLE PHOTO-IONIZATION OF TWO-ELECTRON IMPURITY CENTERS IN QUASI-ZERO-DIMENSIONAL STRUCTURES IN AN EXTERNAL MAGNETIC FIELD
Abstract.
Background. Impurity atoms with two electrons are the simplest systems in which double ionization by a single photon is possible. In this case, influence of an external magnetic field on the double photoionization spectra can lead to new effects associated with the optical absorption dichroism. The aim of this work is to take into account influence of an external magnetic field on the first ionization potential of a two-electron impurity center in a quantum dot, as well as theoretical study of the double photoionization features for two-electron impurity centers in a quasi-zero-dimensional structure in an external magnetic field.
Materials and methods. Influence of an external magnetic field has been taken into account in framework of the perturbation theory. Calculation of the binding energy and the first ionization potential of a two-electron atom has been carried out by a variation-al method, where the second ionization potential has been taken as an empirical parameter. Expressions for the light impurity absorption coefficients are obtained in the dipole approximation taking into account the quantum dots radius dispersion.
Results. An analytical expression for the first ionization potential of a two-electron impurity center under an external magnetic field is obtained by a variational method within the semi-empirical model framework. Coefficients of the light impurity absorption have been calculated in the dipole approximation, in cases of the light longitudinal and transverse polarization with respect to the magnetic field direction, during photoionization of a two-electron impurity by a single photon.
Conclusions. It is shown that magnetic field has a stabilizing effect on two-electron impurity centers in a semiconductor quantum dot. Dichroism of the optical absorption has been appeared in the absorption band edge shift and in appearance of the additional peaks in the absorption spectral curve in case of the light polarization transverse to the external magnetic field direction (e ^ B) as also in disappearance of the double-hump profile in case of the longitudinal light polarization (e || B).
Key words: two-electron impurity centers, ionization potential, double pho-toionization, dichroism, quantum dot.
В. Д. Кревчик, А. В. Разумов, И. М. Мойко, Тян-Ронг Ли, Ю-Хуа Ванг
ДВОЙНАЯ ФОТОИОНИЗАЦИЯ ДВУХЭЛЕКТРОННЫХ ПРИМЕСНЫХ ЦЕНТРОВ В КВАЗИНУЛЬМЕРНЫХ СТРУКТУРАХ ВО ВНЕШНЕМ МАГНИТНОМ ПОЛЕ
© 2018 Krevchik V. D., Razumov A. V., Moyko I. M., Li T.-R., Wang Y.-H. Данная статья доступна по условиям всемирной лицензии Creative Commons Attribution 4.0 International License (http://creativecommons. org/licenses/by/4.0/), которая дает разрешение на неограниченное использование, копирование на любые носители при условии указания авторства, источника и ссылки на лицензию Creative Commons, а также изменений, если таковые имеют место.
Аннотация.
Актуальность и цели. Примесные атомы с двумя электронами представляют собой простейшие системы, в которых возможна двойная ионизация одним фотоном. При этом влияние магнитного поля на спектры двойной фотоионизации может приводить к новым эффектам, связанным с дихроизмом оптического поглощения. Целью данной работы является учет влияния внешнего магнитного поля на первый потенциал ионизации двухэлектронного примесного центра в квантовой точке, а также теоретическое исследование особенностей двойной фотоионизации двухэлектронных примесных центров в квазинульмерной структуре во внешнем магнитном поле.
Материалы и методы. Учет влияния внешнего магнитного поля проводился в рамках теории возмущений. Расчет энергии связи и первого потенциала ионизации двухэлектронного атома осуществлялся вариационным методом, где в качестве эмпирического параметра брался второй потенциал ионизации. Выражения для коэффициентов примесного поглощения света получены в ди-польном приближении с учетом дисперсии радиуса квантовых точек.
Результаты. В рамках полуэмпирической модели вариационным методом получено аналитическое выражение для первого потенциала ионизации двухэлектронного примесного центра в условиях внешнего магнитного поля. В ди-польном приближении рассчитаны коэффициенты примесного поглощения света в случае продольной и поперечной по отношению к направлению магнитного поля поляризации света, при фотоионизации двухэлектронной примеси одним фотоном.
Выводы. Показано, что магнитное поле оказывает стабилизирующее действие на двухэлектронные примесные центры в полупроводниковой квантовой точке. Дихроизм оптического поглощения проявляется в смещении края полосы поглощения и к появлению на спектральной кривой поглощения дополнительных пиков в случае поперечной по отношению к направлению внешнего магнитного поля (e L B) поляризации света и к исчезновению «двугорбого»
профиля в случае продольной (e || B) поляризации света.
Ключевые слова: двухэлектронные примесные центры, потенциал ионизации, двойная фотоионизация, дихроизм, квантовая точка.
Introduction
Semiconductor nanostructures are unique objects in which the charge carriers energy spectrum can be controlled, thereby creating device-oriented structures with predetermined characteristics. An important feature of such systems is the strong dependence of their physical properties on the presence of impurities, which inevitably arise in the nanostructures manufacturing process. Two-electron impurity states in quantum dots (QDs) are of special interest. This is due to the fact that, because of the size restriction in all three directions, conditions for the two-electron impurity states existence in a QD are more favorable in comparison with the bulk semiconductor case. In this case, two energy levels, which are formed in the band gap, are divided by the correlation energy AE = E2 - E : where E and E2 are the first and second ionization potentials of the impurity center.
If for the single-electron (hydrogen-like) impurities study in low-dimensional semiconductor systems there is a fairly large number of papers [1, 2], then for the two-electron or helium-like impurities study there are a much less papers. Interest in this type of impurities is associated with possibility for observing
one of the fundamental reactions - process of the double photoionization of an impurity atom by a single photon [3]. In addition, it attracts the possibility of controlled modulation of the binding energy for two-electron impurity centers (TIC) and, accordingly, of control for the optical transitions energies by means of an external magnetic field. This is important for practical applications, in particular, in the development of qubits, as well as for photosensors with controlled sensitivity.
Purpose of this work is a variational calculation of the first ionization potential of the quantum dot TIC in an external magnetic field, as well as a theoretical study of the double photoionization features for TIC in a quasi-zero-dimensional structure in an external magnetic field.
1. Variational calculation of the first ionization potential for a two-electron impurity in a semiconductor quantum dot in an external magnetic field
We will calculate the first ionization potential Ei for TIC with an effective nuclear charge equal to zero (Z = 0) in a semiconductive QD in an external magnetic field. Let us use the semiempirical model TIC, developed in [4]. The short-range potential in this model is approximated by a potential well with depth Vo , radius d of which is much smaller than the radius of the localized state. As an empirical parameter, we take the energy of a doubly ionized impurity, i.e. second ionization potential E2. We will assume that the impurity atom is located at the center of QD. Influence of the magnetic field on the ground state for TIC will be taken into account in framework of the perturbation theory. The two-electron wave function satisfies the Schrodinger equation
H (pi, P2 )Y(pi, P2 ) = E^(pi, P2), (1)
где
H (Pb p2 ) = H (pi) + H (p2) + h7mad |pi-p 2 + VB (pi,p2 ) H(pf ) = -h2Aj2m*ad - h2V(pf)/2ma2d ,
pd = d/ad , V(pi) = ]V(0, pi ~ pd , Vo* = Vo/Ed , pi = ri/ad , Г - electron coordi-
l0, pi > pd
2 / * 2 21 * 2
nates at i = 1,2, ad = £h /me , Ed = h /2m ad - effective Bohr radius and
Bohr energy respectively, Vb(pi,p2) = m*®B(p2 +p2)4-ihЮв(д/Эф)2 -
perturbation operator; m - effective electron mass, Юв = \e\B / m - cyclotron frequency.
The value Ei will be sought by a variational method with wave functions taken as a product of one-electron wave functions ¥(pi):
¥(pi,p2) = ^(pi )^(p2), (2)
Using results of [5], where, within framework of the spherically symmetric potential well model (the model of "rigid walls"), an expression for the electron
wave function localized at a short-range potential in a QD has been obtained; for a single-electron wave function ¥ (p;-), we obtain
Pi-
sh (( 1 -pdn 1) sin (хоРг)
sh ((n"1
/ * —1 —1 sh (( —Pi П
sin (XoPd )
Pi ^ Pd
(3)
sh ((n 1
Pi ^Pd •
With account of (3), the trial two-electron wave function is written in the
form
*(PbP2 ) =
B 2
P1P2
sh2 ((1 —Pdn"1 )sin (X0P1 )sin (X0P2)
sh2 ((ROn"1
sin2 (XoPd )
> Pi ^ Pd
/ * "1 " 1 \ / * "1 "1 sh (Ron " P1n )sh (Ron "P2n
(4)
sh2 (R*n"1
Pi ^Pd >
here Xo = V V* -n-2 , Vo* = VolEd , n = V Ed/\E2\, RQ = R^ , RQ - the QD
radius, B = ^ 2n_y (th (Ron-1)- R*n_1cosec ((n-1
In framework of the perturbation theory, the energy of TIC in a magnetic field (in the Bohr units) is determined by
ев (Ro,n) £(o)k\n) e(1)(Ro,n
+-
(5)
Ed Ed Ed here e(0) (Rq,n))Ed - unperturbed energy of the two-electron impurity center [3]:
e(0 )(R*, n)
Ed
■ = "23 n"2 n2 xsh "4 (Ron"1 )x
x(th (Ron"1)" Ron"1cosec((n"1)) 2 j 2"1 (2n"1R* " sh (2n"1R))
+n
+
+2ln
4n"1Ro " 8ch3 (n"1Ro) sh (n"1R*) Chi (2n"R) +
(2exp(1 + C )n"1Ro*) sh (2n" R) + (Chi (4n" R)" ln (2)) sh (4n"R) -
"Shi (2n"1Ro) (1" 2ch (2n"R) - ch (4n"R)) + ch (4n"R) Shi (4n" 1Ro'
(6)
где Chi(x) и Shi(x) - the integral hyperbolic cosine and sine respectively, C = 0,577 - the Euler's constant.
According to the perturbation theory, the first approximation correction
e(i)(( , П)/Ed will be determined by the average value of perturbation potential in the state Ш(0) (pi, p2):
s(i)
((, n)
= /ш(0)
Ed
*(0)(pb p2 )Vb (pi, p2 )V(0)(pi, p2 )).
(7)
With the expression account for the unperturbed two-electron wave function (4), for (7) we have
,(i)
((*, n)
2 02 * 2 2 -2 п 2 m wBadn
Ed
-x
п 2 п п 2п
xj d0i sin 0i J dф1 J d02 sin 02 J dФ2
sh4 ((n-1) (th ((n-1) - R*n-1 cos ec (R*n-1 ^
0(pd-pi )x
0
0 0
pd pd
0
4 / * -1 -1
sh (( -pdn
4
sin
(X0pd)
x0(pd -p2) J d pi J d p2 (p2 +p2 )sin2 (X0pi )sin2 (X0p2 ) + 0(pi -pd)x
0 0
* *
x0(p2 - pd) J dpi J dp2 (p2 + p2 )sh2 (^0-Л-1 - pin-1 )sh2 (^0-Л-1 - p2Tl pd pd
The integrals in (8) are as follows: pd pd
J dpi J dp2 (p2 + p2 )sin2 (X0pi )sin2 (X0p2) = 2-53-1 X-4 x
-Г
. (8)
x
0 0 .2„2 ,,¿„.4.4
3 + 6%2pd + 16%4pd -24x2Pd cos(2x0pd) + (3 -6x2pd )cos(4x0pd) +
2„2
+12X0pd sin (2X0pd)- 32x0pd sin (2X0pd) + 6X0pd sin (4X0pd)
(9)
J dpi J dp2 (pi2 + p2 )sh2 (( - pin-1 )sh2 (( - p2n-1) = 2-53-1 n4 x
pd pd x
-3 + 24R*2n-2 +16R*4n-4 - 24R0n-2pd -16Vpd - -
-16RVpd + 16пЛ4 - 24п-2 (( -Pd )ch (2n-1 (( -Pd )) +
+(3 + 6n-2pd )ch (4n-1 (r*-Pd))-24n-1R*sh (2n-1 ((*-Pd ))-
-8n-3R*3sh(2n-1 (r* - Pd)) + 12n-1Pdsh(2n-1 (( - Pd)) --24n-3Ro*P> (2n-1 (( - Pd)) + 32n-3P3dsh (2n-1 (( - Pd)) +
+6n 1Pdsh(4П 1 ((R* -Pd
Then expression for the correction to energy (7) is written as
(10)
e(1)(Ro, n
* 2 2 2 -2 m ®Badп П
Ed
6sh4 (R*n-1)(th(R*n-1) - R*n-1csch((n-1
-x
X
+
sh4 (( -Pd n-1
X4 sin4 (XoPd )
x
-3 + 6x2P2 + 16x4Pd -24x2P2 cos(2xoPd) +
2 „2
(3 - 6X2Pd )cos(4XoPd ) + 12XoPd sin (2XoPd )- 32XoPd sin(2XoPd ) + +6X0Pd sin (4xoPd)] + n4 [-3 + 24Ro*2n-2 + 16R* V - 24R*n-2Pd --16R**3n-4Pd - 6n-2Pd - 16Ro*n-4Pd + 16n-4Pd - 24n-2 (r* - Pd )Pd x xch (2n-1 ((*-Pd )) + (3 + 6n-2Pd )ch (4n-1 ((-Pd)) + -24n-1R**sh (2n-1 (r* - Pd)) - 8n-3R*3sh (2n-1 (( - Pd)) + +12n-1Pdsh (2n-1 (r* - Pd)) - 24n-3R*Pdsh (2n-1 (Ro* - Pd ]
+32n 3Pdsh(2n 1 (R*-Pd)) + 6n 1Pdsh(4n 1 ((-Pd
By making in (11) the limiting transition (d ^0), for £(1)(Rq,n))
(11)
Ed we
obtain
,(1)
((*, n)
* 2 2 -2
m щ п n
Ed
6sh4Ed (R**n-1)(th (R**n-1) - RoVcsch((*n-1
x
x
-3 + 24R*V2 +16 RIV4 + 3ch (-1R*
-8Л-1^0 (3 + П-2 R*2 )sh (R
(12)
Taking into account (6) for the TIC energy (5) we have
eB((,f) 3 2 2 -4i * i\
V ■ = -23 f-2n2 xsh-4 (Ron-1 )x
E,
d
x( th(R*n-1)-R*n-1cosec((n-1)) 2 J2-1 (2n-1R* -sh(-1R*))2 +
+n
4n-1R* - 8ch3 (n-1R*) sh (n-1R*) Chi (2n-1R*) +
+2ln
(2exp(1 + C )n-1R0 )sh (R) + (Chi (R)- ln (2)) sh (R )-
-Shi (2n-1R*) (1 - 2ch (2n-1R*) - ch (4n-1R*)) + ch (4n-1R*) Shi (■4n-1R*
* 2 2 -2 m юв п n
6sh4Ed (( 1)(th(( 1)-R*n 1csch(( 1
x
-3 + 24 R*2n2 +
+16 R*4^-4 + 3ch (4n" 1R*) 8n-1R* (3 + n-2 R*2) sh (2^" 1R*
(13)
Minimization of the expression (13) requires to find an extreme value fj. which requires to solve the transcendental equation
d£B (( П
3f
The explicit expression for (14) is:
= 0.
(14)
27 п2П -2
-x
sh4 ((-1)(th ((-1) - R*in-1cosec ((-1
x
R*n 2csc( R)-R*n 3ctg( 1R0*)csc( R)-Щ 2sech( 1R*)
+
+-
27п2П-3 - 28п2П-4cth ((-1)
sh'
((-1)(th ((-1)-R*fj-1cosec ((-1
^ (2 R*fj-1 -sh (-1<
+
x
16R*4 + 24R*2n2 - 3n4 (1 - ch (4R*n-1)) - 8R*n (R*2 + 3n2) sh (2R*n) +2R*n
+
^¡л A / ^¡A О \ / 1 \ / ^¡Л Л \
16R*3 - 12R*n2 + 4R* (R*2 + 3n2 )csch (2Д*п-1) + 6n ((2 + n2 ) x
x sh (2R*n-1) - 3ii3 sh (4 R*n-1
R*csc (Ro*ii 1) - n tgh ((n 1
= 0. (15)
n)
If we take into account, that the minimum value of the functional Ed , achieved with an extreme value of the parameter n = n, is the sum
eв ((n)
---El + E2, (16)
Ed Ed Ed
we can find the first potential as a function of the second, taken from the experiment
A = £b ((n) E2. (17)
Ed Ed Ed
Figure 1 shows the relationship between the first and second ionization potentials for TIC with the zero nuclear charge (Z = 0) in semiconductive QD, taking into account the influence of an external magnetic field, in the Bohr energy Ed units, obtained from (17) by numerical calculations. Curves 1, 2, 3 correspond to different values of the magnetic field induction B . Comparison of curves 1, 2 and 3 shows that in a magnetic field the threshold value of the second ionization potential for a two-electron impurity decreases, and thus the magnetic field has a stabilizing effect on the TIC in a semiconductive QD, which is caused by the Landau levels dynamics in an external magnetic field and by suppression of the Coulomb electron interactions.
2. The absorption dichroism under double photoionization of two-electron impurity centers for a quasi-zero-dimensional structure in an external magnetic field
Let us consider the photoionization process for TIC in an external magnetic field in a semiconductive QD. Wave function of the initial state is determined by the expression
22 n-2
*(P1, P2 )=:—*--*—— x
th ((n 1 )-Ron 1cosec ((n 1
sh ((1 - p1n-1) sh ((1 - py n-1)
X—--t^-v--. (18)
2 * —1 P1Pysh (i^on
Wave function of the final state is taken as a product of the wave functions for electrons in a spherical QD
ф(PbР2 ) = ^n,l,m(PbФьn,l,m(Р2,Ф2,^ •
(19)
у
3
у
2
Fig. 1. Relationship between the first E and second E ionization potentials of a two-electron impurity center with the zero nuclear charge (Z = 0) in a semiconductive QD for different values of an external magnetic field induction B:
1 - B = 0; 2 - B = 2 T; 3 - B = 3 T
Influence of the magnetic field on the electron final states in a QD will be taken into account in framework of the perturbation theory. Then, in the second order of the perturbation theory, the electron wave function will have the form [6]:
-)*2T
w,e,ф) = *п0/>,e,ф)+ z r e,ф), (20)
n - Xn,r
n'l'm
where Y(0L (r, 0, ©) - the zero-order wave function:
n,l,m V ' '
^00l,m(r, e, ф) = Y m ( ф)
J 3 l+-2
( X ^
Xn,l *
R*
a3/2 ah
^¿0*47J - ((n,l)
(21)
l+-2
here i = 1, 2; Yim - the normalized ball functions; l, m - the orbital and magnetic quantum numbers; Jv(x) - the first order Bessel function with form v; kni = ^ni /R); t,„i - n -th root for Bessel function of the l -th order.
In the second order of perturbation theory, the electron energy spectrum in an external magnetic field can be written as [6]:
R*2 V , , J2
^ „(0) , Tr ^ R0 \Vn,l,m\n'X,m
E = E ; + Vn
+
I
n,l,m;n,l,m ■ 2 ~ 2
n'l'm' п - Xn',l'
(22)
here Xn' /' - root of the Bessel function with half-integer order l +1 / 2,
E(0) = X21Eh jRo2 - zero approximation to electron energy in a dimensionally quantized band, Vn 1 mn> 1 m> - matrix element of the perturbation operator, which in spherical coordinates has the next view
Vn,l,m;nim 1, n'=0
h®Bm
4п((h - X2',l ) ) +3 (n,, ) ) +3 ((n ',l )
x
x
R0Xn',lJl+1 ((0Xn',l ) Jl+3 ((0Xn,l ) -. 2 2
R0XXn,lJl+1 ((0Xn,l ) J, + 3 ((0Xn',l ) 2
l+-2
/ \k * 2 * (-1) mh ®BR0
Г 3 Л l+3
X ,2
2k+1
+
I
k=0 k!ГI l + k + - 12l+4+2k
x
x
F
(l - m + 4)((- m + 3)(( - m + 2)((- m +1)
-k,-l - k - 3,l + 5 ;Хп2',1-2 ^
Г Г Jf2,
i
(2l +1) (2l -1)2 (2l - 3)
(2l + 2k + 2)Г| l + ^
2 F
3 5 X2', ^ -k, -l - k--,l + - 2,1
2 ^ X^
(2l + 3) (2l -1) (2l + 2k + 4)
- m)((- m -1)(( + m + 2)(( + m +1)
+
F
Г 3 5 f2 Л
-k,-l - k - 3,l + - XJn',l+2
+-
2' 2' X2,
2l + 2k + 6
-x
x
1
(l + m + 4 )(l + m + 3)(l + m + 2)((+ m +1)
(2l + 5)(2l + 3)2 (2l +1)
(23)
The effective Hamiltonian for interaction with the light wave field in case of the longitudinal polarization e^ with respect to the magnetic field direction
( e^ ± B ) is determined by the next expression:
H
(s ) _
int B ~ '
-iX0h
f 2 * Л 2лй a
10
1/2
£Mm
*2
exp (iqJr )(eb Vr ),
(24)
where À0 - coefficient of the local field, taking into account the amplitudes difference for the local and average macroscopic fields; Io - the light intensity; œ - the
absorbed light frequency; £ - the static dielectric permeability of the QD material;
*
a - the fine structure constant with account of dielectric permeability.
The matrix element, which determines magnitude of the oscillator strength for the dipole optical transitions of electrons from the ground state of TIC (18) to the discrete spectrum state ¥n m i (p,9,0) of QD (20), in case of the light polarization along the magnetic field direction, will be written as
M(s >_ i Л oJ». 10 V ш
(En,l,m -E1 )x
+
,l,m (01,Ф1 ) (P2 )|(,r1 )|^(PbP2 )) + (En,l,m -E2 )),l,m (P1,0ЪФ1 )* (P2 ))еЛ,r2 )|*F(PbP2 ))
Calculation of the matrix element (25) leads to the following integrals
2n
J exp(-imty)d9 = <
I 0, if m Ф 0, [2n, if m _ 0.
(25)
(26)
As it can be seen from (26), that the selection rule for the magnetic quantum number m is such that in case of e^ || B optical transitions from the impurity level are possible only in the QD states with value m = 0 .
Taking into account (18)-(26), expression for the square modulus of the matrix element (25) can be written as
M
(s )
a
2 цш^
to adR02sh6 ((V
x
25 n-3
x
th (( 1
-R**r\ 1cosec ((n 1
x
2^lR*-2 + Vnl,0;n,l,m + S V ty^2 +1 E1\/Ed +|E2\/Ed
n'l'm
J2 y 2 n - X n'l'
x
2
x
-2 I-
k-3 (1 + n-2 )kn1 + in-1 (kn1 - i 2n-1
Jl+3/2 (^nl )
x
xch (R*n-1) ^ R* (kn1 - in-1) + i^kn1 - in-1 (kn1 + i 2n-1 )ch ((n-1 )x
-\Jkn1 + in-1 (kn1 - i 2n-1 )sh (((п-1 )x -yjkn1 - in-1 (kn1 + i 2n-1 )sh ((On-1 )x
2 "* (kn1 + 'n 1
2 * (kn1 - "П 1
xCi
/П Ro* (kn1 + i'n 1
1 nk,
n1
-2
2
n-2 + k21) x
x n
-1
(n 2 + 4)R0*cos(kn1R0)-(n 2 + 2kn1 + ))sin(kn1Ro)
+
+(n
-2
(knl -1) + k2nl (kni + 1))ssh (Ro*n-1)}
+
+
1
->*2
R Vn,l,0;n',l',m' \
K\(1 +n 2) kn' 1 +'n 1 (kn'1 2n 1
п2 - X 2, ,
n'l'm п л n ',l'
Jl'+3/2 (^n'l )
x
xch (Ro*n-1 )Si ^ 2R* (kni - in-1) + фп' 1 - in-1 (kn'1 + i 2n-1 )ch (( )x
v
Г I—h
xSi
2R0 (kn'1 + in 1) +V kn'1 + in 1 (kn'1 - i2n 1 )sh (Ro*n 1 )x
xCi
Г 1 * 2 Ro
R0 (kn'1 - in 1) +V kn'1 - in 1 (kn'1 + i 2n 1 )sh (Ro*n 1 )x
xCi
*
kn' 1 + 'n
-1
Ji ( + * )-2 x
x n
-1
11 2 + kn21 )R0 cos (kn'1R0) -(n 2 + 2kn'1 + k2'1 )R0 sin (( *
+
+ (n-2 (г -1) + til' (kn'l' + 1))ssh(( )
(27)
here Ci (x) h Si (x) - the integral cosine and sine respectively.
We assume that the sizes dispersion u for QDs arises in process of the phase decomposition of a supersaturated solid solution and is satisfactorily described by the Lifshitz-Slezov formula [7]:
34 eu2 exp [-1/(1 - 2 u/3)]
p (u )_
3
, u <T, 11 2
2 3 (u + 3)з (3/2-u) 3
(28)
3
u > —, 2
where u = Ro / Ro , Ro and Ro - the QD radius and its mean value respectively; e - base of natural logarithm.
Coefficient of the light impurity absorption a(s )(œ) with longitudinal polarization to the magnetic field direction, taking into account the QDs size dispersion, is determined by the expression
3/2
a(s ) (ш) _ Si duP (u ) M(s) 2 ô(En ,0,1 +|E1 + |e2 - h ш), (29)
П10
n0
where No - the QDs concentration in dielectric matrix; ô(x) - the Dirac delta function.
Taking into account (27) and performing integration in (29), the light impurity absorption coefficient a(s ) (œ) can be written as
a
(s )
N K
(- )_ SS
P(un ) N0n2Ed
X 2Wô*
—* \ 2
x
_1l_0 a2П((un ) sh6((n
,-1
x
( / -* \2. ,2
„2 1-* \-2 ^ (R0 un ) yn,l,0;n',l',m'\ ...
((un ) + Vn,l ,0;n,l ,0 + S --2 - - 2 ,-L+l EM Ed +| E2|/Ed
n'l'm П Xn',l'
2
x
x
26 n-3
—* -1 \ —* -1 / —* -1 R0unn I-R0unn cosec(R0unn
x
x
kn13(1 +n 2) Vkn1 +i'n 1 (kn1 -i2n
Jl+32 (^nl )
x
5
Г -* л _
xch ((n-1) J2^^2 (kn1 - in-1) + i\lkn1 - in-1 (kn1 + i 2n-1 )ch (пП-1
x
( r~=i
xSi
xCi
2RnUn(kn1 + in-1) Wkn1 + in-1 (kn1 -i2n-1 )sh(lnn-1 )x
(kn1 - in 1) +Vkn1 - in 1 (kn1 + i2n 1 )sh((Vl 1 )x
xCi
П Ro*un (kn1 + 'П 1
' nk,
2 / -2 , 2 \-2 n2 + k21) x
n1
x n
,-1
(n 2 + kП1 )R*un cos (kn1R0*un )-(n 2 + 2kn1 + 4 ))п sin (kn1R0*u
+
+(n
-2
(kni -1) + k2i (kni + 1))sh(nf-1)}
+
+
1
—* \2
un ) Vn,l,m;n',l',m'
kn'1 ( +n 2 ) kn'1 +'"n 1 (kn'1 -i2n Г
П2 - X 2 ,
n'l'm' n X n',l'
Jl'+3/2 ((>n'l )
x
xch (R^ u,n 1 )
i2^ (, -in-'
+i^kn'1 - in 1 (kn'1 + i 2n 1 )ch ((Чп 1 )x
xSi
( —
2 R0un
kn '1 + '"n 1
+ + in 1 (kn'1 - i 2n 1 )sh ((n n 1 )x
xCi
2R(*Un(kn'1 - in-1) +Vkn'1 - in-1 (kn'1 + i2n-1 )sh ((4n-1 )x
xCi
2Ъип(и. , ,.„-1
kn'1 + in"
' nk.
— (n 2 + k,'1) 2 x
n'1
x{n
-1
(n 2 + k2'1 ))п cos(kn'1R0*un )) 2 + 2kn'1 + k2"1 )R0un sin(kn'1R0un
+
+(n
-2
(kn'l' -1) + k2'l' (kn'l' + 1))i^h (Inf 1)}
(30)
where
х
2(X-|E^/Ed -|E2\/Ed -УпЛМ-1/Ес1)
V S |Vn,1,-1;n',l>'| IEd (Х/ ) + V n'l'm'
l
"2 х
+
R02 S |Vn,1,-1;n',l>'| /Ed (X V n'l'm
2
+
+4
(X -1Ei I - |E2 - Vn ,1,-1;n ,1,-1)/EdR02
1
1 ^ 2
; X = Ш1 Ed .
Next we consider the double photoionization process for TIC in an external magnetic field in a semiconductive QD in case of the light transverse polarization to the external magnetic field direction.
The effective Hamiltonian H(n b for interaction with the light wave field, which are characterized by a wave vector q and by the unit vector of polarization e^t transverse to the magnetic field direction can be written as follows
H(t) . |2лй2а*
Hint B _-i h ^
( i\e\B ^
0\l—^— 70exP(qtr) xxtVr)—[e^t,r]
*2 m ю
v
2h
(31)
/
r(t)
The matrix elements ^ , determining the electron optical transitions
from the ground state of TIC to the QD discrete spectrum states for case of the transverse polarization e^t can be represented in the dipole approximation as the
sum of two terms:
m (t )= m()+ m(),
(32)
where
r(t)
M\4 _ i I
V ю
(En,l,m -E1 )х
+
х(^П,l,m (Pb°1'Ф1 ) (P2 )|(xt,r1 )|*F(PbP2 )) + (En,1,m - E2 )),l,m (Pb©1,Ф1 )* (P2 )| (eXt,r2 )|*F(PbP2 ))
(33)
and
(t) _ ^ /2па 1 h юв
M2 _-^0
ю
,l,m (Pl,0bФ1 (P2 )|[,r1 ]z |^(PbP2 ))
+
+
(^2,l,m l0ЪФ1 )* (P2 )[,r2]z |^(PbP2 ))
(34)
Calculations in (33) and (34) lead to the following expression for the square modulus of the matrix element (32)
M
(t)
2 a* n^2 ^o Ed
25 n-3
ю
_w w u_x_-_
adR*2slh6 ((n-1) (th (R*n-1) - R**n-1cosec((n-1))
x
x
2^2lR0 2 + Vn,l,m;n,l,m + 1
0*2 V r r j2 ■fl0 rn,l,m;n,l,m
п2 - X 2 ,
nl'm п ^ n ',l'
+
+ Ы/ Ed +| E^ Ed + Й Юв/Ed
x
x
kn13 (1 +П 2) 2\/kn1 +'П 1 (kn1 -i2П ^
Jl+3/2 (^nl)
x
xch(R*n-1) ^R* l -'П-1) + i^kn1 - if-1 (k,1 + i2n-1 )ch ((n-1 )x
"\]kn1 + in-1 (k,1 - i 2n-1 )sh (R*n-1 )x -■\jkn1 - if-1 (k,1 + i 2n-1 )sh (R**n-1 )x
lPR'
V nk,1
/ПRo* (kn1 + i'n 1
(kn1 - '"n 1
+
+
Щ R0* (kn1 + i'n 1
-2
(n-2+4) x
x{n
-1
n 2 + k,1) R* cos (
11 2 + 2kn
n1 + k,1 )R*sin (kn1Rh) (l2 - 1) + k2nl (kni + 1))sh((1 )} +
+
+
1
*2
R0 Vn,l,m;n',l',m' \
kn'l (1 +n 2) ^kn'1 +''n 1 ( 1 - '2n 1)
n2 - X 2 ,
n'l'm' n X n',l'
Jl
Г+3/2 (^n'l Ы
x
2
{ *
xch ((n-1 ) (kn'1 - in-1 ) + Фп! - in-1 ( + i 2n-1 )ch ((-1 )x
xSi
'2R* ( + in-1 ) + V kn'1 + in-1 (kn'1 - i 2n-1 )sh (R0n-1 )x
xCi
/ Л
^ (kn'1 - in 1 ) +V kn'1 - in 1 (kn'1 + i 2n 1 )sh (( 1 )x
xCi
( I—*
kn'1 + 'П
-1
ÎE (+k>2")-2 x
x{n
-1
(n 2 + kl'1 )R0cos (kn'1R0 )-(n 2 + 2kn'1 + k2'1 )R0sin (kn'1R0 +(n-2 (kn'l' - 1) + k^'l' (kn'l' + 1))sh((n-1 )}
+
In this case, calculation in (33) and (36) of the following integral
2i\ t . x fn, ifm = ±1,
I d9cos 9exp (-im^) = <
0
0, ifm ф ±1
(35)
(36)
gives a selection rule for the magnetic quantum number m.
Assuming that the QDs-size distribution function is still the Lifshitz-Slezov
function (28), expression for the light absorption coefficient aB) (œ) can be written as
aB) (ш) _ 2n1N0 S S S|m|,1JduP (u )
0 n1, n m_-1 0
M
(t )
ô(En,l,1 + + E2 -Пш) ,(37)
where 5| m| 1 - the Kronecker symbol, taking into account the selection rule for a magnetic quantum number m (36):
P (Un )
/ ч 7 2-,N K +1
a (X 2 a N°?'Ed S S S «m,±1
ad h
jî—î ¿—t m,±1 *2 61 ~* ' -1\ n_1l_0 m_-1 XR0 sh (( unn )
x
x
/ — * ' \2 I |2
2 / —* , \-2 ((un ) |^и,1 ,m;n',l',m'|
2^nl ((un ) + Vn,l,m;n,l,m + S -»---+
n'l'm'
-л-2 y-2
3
+ | E1I/Ed +| E2I/Ed + h Юв/Ed
x
x-
n
-3
(th (l n 1)- R*u2n 1cosec (In n 1))
x
x
' -2 I-
k-3 (1 + n-2 )kn1 + in-1 (kn1 - i 2n-1)
Ji+3/2 kk>ni) Л
x
l —* /
xch (k*u2n-1) V~n~(kn1 -in-1) + i<Jkn1 - in-1 (kn1 +i2n-1 )ch ('n-^
v
Г Г"=*
x
xSi
IRt (km + z'n-1) +V k,1 + in-1 (kn1 - i 2n-1)
sh
Г — * Л
R0un
n
V /
x
xCi
ГI2Rou2
n'kn1 - 'n
-1
+
Vkn1 - in 1 (kn1 + i2n 1 )sh ((R*u2n 1 )x
(
xCi
n
- Ro*u2(kn1 + 'n
-1
~Y
x n
-1
'nkn1
+ k,1 ))°0иП cos(kn1R0*u2)-(n-2 + 2kn1 + 4))'sin(kn1R0*u2 +(n-2 (kni - 1) + kh (kni + 1))sh(In-1)} +
-2
2
n-2 + kh) x
+
+
1,
n'l'm'
'l V
un J Vn,l,m;n',l',m' ^
-2
kn'3 l + n 2 )kn'1+'n 1 l 1 2n 1 '
-л-2 if 2
Jl'+3/2 (^n'l Ы
x
I -* /
xch ((и, n-1 ) J2^ (kn'1 - i'n-1) + iyjkn'1 - in-1 ((
kn'1 +' 2n 1 )x
( !~=i
xch (R**un n 1 )Si
I( + i'n-1) + V kn'1 + z'n-1 (kn'1 - i 2n-1 )sh
Г _ * Л RoU
n
v /
x
xCi
Г r=h ,
2R0 un
n^ (kni - in-1) + V kn'1 - in-1 (kn'1 + i 2n-1 )sh (Ro*u2n-1 )x
2
xCi
( / -* ,
\2R0un
Л"
kn1 + г'п
-1
2-(п 2 + k/1)2 x
x n
-i
nkn'1
П-2 + kJl )) cos (ДзЧ )-(n-2 + 2kn'1 + k/1 )x ,-2/, , .2 . . , inLU/D*
xRounsin ((ЯомП)+(n (Vr -1)+Vr (Vr+1))ssh( (Пп
(38)
where
x
2(-|E1\/Ed -|E21/Ed -Vn1lM-1/Ed )
V S |Vn,1,-1;n',l>'| IEd (('/ -n2) + V n'l'm
1
"2 x
+
f
2
R S |Vn,1,-1;n',l',m^ Ed (nr V n'l'm'
22
+
1 >
1/2
+
4 ((-| E1 -| E2I - Vns1s-1.ns1s-1 - П щ )/ EdR*2
Figure 2 shows the spectral dependences of the light absorption coefficients associated with double photoionization of TIC in an external magnetic field.
It can be seen that in case of e^ ± B (curve 2) the light impurity absorption threshold shifts to the short-wave region of the spectrum due to increase in energy of the TIC ground state. In addition to the main "double-humped" profile on the absorption curve, additional peaks appear (Zeeman doublet).
The same figure shows dynamics of the impurity absorption band with an increase in the magnetic field induction B in case of e^ || B (cf. curves 1 and 3): the absorption threshold shifts to the long-wavelength region of the spectrum, while the absorption coefficient decreases and the right peak on the spectral curve disappears, which associated with effect of the electronic correlations suppressing in a magnetic field.
Thus, the optical absorption dichroism is observed in the double photoioni-zation spectra of TIC in a magnetic field, which is associated with change in the selection rules for a magnetic quantum number m : m = 0 in case e^ || B and m = ±1 in case e^ ± B .
Conclusions
Theoretical study of the magnetic field influence effect on the binding energy for TIC has been carried out by variational method in framework of the pertur-
bation theory. It is shown that with an increase in an external magnetic field, an increase in the TIC binding energy occurs and magnetic field has a stabilizing effect on TIC in semiconductive QD due to effect of the electronic correlations suppressing. In the dipole approximation, in framework of the effective mass method, the light impurity absorption coefficients in case of double ionization for TIC by a single photon in an external magnetic field, have been calculated.
X = йю/ E
Fig. 2. Spectral dependence of the light absorption coefficients in tne aouoxe photoionization of two electron impurity centers in a quasi-zero-dimensional structure at E = 0,03 eV; R = 1, for different directions of the light polarization relative to an external magnetic field: B: 1 - B = 0, 2 - B = 2 T ( e 1B ), 3 - B = 2 T ( e H B )
It is shown that the light optical absorption dichroism is associated with change in the selection rules for a magnetic quantum number. In case of e^ || B, the magnetic field influence is appeared in an increase of the photon energy threshold value and in disappearance of the right peak on the spectral curve due to suppression of electron correlations in a magnetic field.
A characteristic feature of the double photoionization spectrum in case of e^ 1B is the Zeeman double doublet: four additional peaks appear on the main "humps" of the absorption curve, while the light impurity absorption band edge is shifted to the spectrum long-wave region.
References
1. Krevchik, V. D. The light impurity absorption in structures with quantum dots / V. D. Krevchik, R. V. Zaitsev // Solid state physics. - 2001. - Vol. 43, № 3. - P. 504507.
2. Galiev, V. I. Spectra of energy and optical absorption for small impurities in a semi-conductive quantum dot / V. I. Galiev, A. F. Polupanov // Semiconductive Physics and Technology. - 1993. - Vol. 27, № 7. - P. 1202-1210.
3. Krevchik, V. D. Double photoionization of two-electron impurity centers in quasi-zero-dimensional structures / V. D. Krevchik, A. V. Razumov, P. S. Budyansky // Proceedings of the higher educational institutions. Volga region. Physics and Mathematics. -2014. - № 3. - P. 289-240.
4. Grinberg, A. A. On the energy spectrum of multiply charged impurity centers in semiconductors / A. A. Grinberg, E. D. Belorusets // Solid state physics. - 1978. -Vol. 20. - P. 1970-1978.
5. Levashov, A. V. Energy spectrum and optical properties of the "quantum dot - impurity center" complex / A. V. Levashov, V. D. Krevchik // Semiconductive Physics and Technology. - 2002. - Vol. 36, № 2. - P. 216-220.
6. Krevchik, V. D. Effect of a magnetic field influence on the recombination radiation
associated with A+ -centers in quantum dots / V. D. Krevchik, A. V. Razumov, P. S. Budyansky // Proceedings of the higher educational institutions. Volga region. Physics and Mathematics. - 2015. - № 3 (35). - P. 125-143.
7. Lifshits, I. M. On kinetics of the diffusion decay of supersaturated solid solutions / I. M. Lifshits, V. V. Slezov // Journal of Experimental and Theoretical Physics. - 1958. -Vol. 35, № 2 (8). - P. 479-492.
References
1. Krevchik V. D., Zaitsev R. V. Solid state physics. 2001, vol. 43, no. 3, pp. 504-507.
2. Galiev V. I., Polupanov A. F. Semiconductive Physics and Technology. 1993, vol. 27, no. 7, pp. 1202-1210.
3. Krevchik V. D., Razumov A. V., Budyansky P. S. Proceedings of the higher educational institutions. Volga region. Physics and Mathematics. 2014, no. 3, pp. 289-240.
4. Grinberg A. A., Belorusets E. D. Solid state physics. 1978, vol. 20, pp. 1970-1978.
5. Levashov A. V., Krevchik V. D. Semiconductive Physics and Technology. 2002, vol. 36, no. 2, pp. 216-220.
6. Krevchik V. D., Razumov A. V., Budyansky P. S. Proceedings of the higher educational institutions. Volga region. Physics and Mathematics. 2015, no. 3 (35), pp. 125143.
7. Lifshits I. M., Slezov V. V. Journal of Experimental and Theoretical Physics. 1958, vol. 35, no. 2 (8), pp. 479-492.
Кревчик Владимир Дмитриевич
доктор физико-математических наук, профессор, декан факультета приборостроения, информационных технологий и электроники, Пензенский государственный университет (Россия, г. Пенза, ул. Красная, 40)
E-mail: physics@pnzgu.ru
Разумов Алексей Викторович
кандидат физико-математических наук, доцент, кафедра общей физики и методики обучения физике, Пензенский государственный университет (Россия, г. Пенза, ул. Красная, 40)
E-mail: Razumov_alex@mail.ru.
Krevchik Vladimir Dmitrievich Doctor of physical and mathematical sciences, professor, dean of the faculty of instrument engineering, information technology and electronics, Penza State University (40 Krasnaya street, Penza, Russia)
Razumov Aleksey Viktorovich Candidate of physical and mathematical sciences, associate professor, sub-department of general physics and methods of physics teaching, Penza State University (40 Krasnaya street, Penza, Russia)
Мойко Ирина Михайловна аспирант, Пензенский государственный университет (Россия, г. Пенза, ул. Красная, 40)
E-mail: irmojko@yandex.ru Ли Тян-Ронг
доктор философии, доцент, Институт функциональных и экологических материалов, Университет Ланчжоу (КНР, Ланчжоу, ул. Саут Тяншуй 222)
E-mail: litr@lzu.edu.cn
Ванг Ю-Хуа доктор философии, Университет Ланчжоу, директор Международного Управления, директор Управления по связям с Гонконгом, Макао и Тайванем, Профессор кафедры исследования люминесцентных материалов, Школа Физических Наук и Технологии, директор Центральной Лаборатории Специальных Функциональных Материалов и Структурного Дизайна, Министерство Образования (КНР, Ланчжоу, ул. Саут Тяншуй 222)
E-mail: wyh@lzu.edu.cn
Moyko Irina Mikhaylovna Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia)
Li Tian-Rong
PhD, associate professor,
Institute of Functional and Environmental
Materials, Lanzhou University
(222 South Tianshui South Road, Lanzhou,
P. R. China)
Wang Yu-Hua
PhD, Lanzhou University Director of International Office Director of the Office of Hong Kong, Macao and Taiwan Affairs Professor of Luminescent Materials Studies in School of Physical Science and Technology Director of the Key Laboratory of Special Function Materials and Structure Design, the Ministry of Education (222 South Tianshui South Road, Lanzhou, P. R. China)
УДК 535.8; 537.9; 539.33
Double photo-ionization of two-electron impurity centers in quasi-zero-dimensional structures in an external magnetic field / V. D. Krevchik, A. V. Razumov, I. M. Moyko, T.-R. Li, Y.-H. Wang // Известия высших учебных заведений. Поволжский регион. Физико-математические науки. - 2018. -№ 1 (45). - С. 111-133. - DOI 10.21685/2072-3040-2018-3-9.
