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2
NANOSYSTEMS: Krevchik V.D., Razumov A.V., et al. Nanosystems:
PHYSICS, CHEMISTRY, MATHEMATICS Phys. Chem. Math., 2022,13 (3), 250-264.
http://nanojournal.ifmo.ru
Original article DOI 10.17586/2220-8054-2022-13-3-250-264
Influence of dissipative tunneling on the photodielectric effect associated with the excitation of impurity complexes A+ + e in a quasi-zero-dimensional structure
Vladimir D. Krevchik1, Alexey V. Razumov1, Mikhail B. Semenov1, Alexander V. Levashov1, Alexey V. Shorokhov1'2
1 Penza State University, Physics Department, Penza, Russia 2University of Jyvaskyla, Seminaarinkatu, Finland
aalex.shorokhov@gmail.com, bmisha29.02.1@gmail.com
Corresponding author: Alexey Vladimirovich Shorokhov, alex.shorokhov@gmail.com
PACS 73.40.Gk, 03.65.Xp
Abstract Effect of tunneling decay for the quasi-stationary A+-state, in an impurity complex A+ + e (a hole, localized on a neutral acceptor, interacting with an electron, localized in the ground state of a quantum dot) on the photodielectric effect, associated with the excitation of impurity complexes A+ + e in a quasi-zero-dimensional structure, has been studied in the zero-radius potential model in the one-instanton approximation. Calculation of the binding energy of a hole in an impurity complex A+ + e was performed in the zero radius potential model in the adiabatic approximation. It is shown that as the probability of dissipative tunneling increases, the binding energy of a hole in a complex A+ + e decreases, which is accompanied by an increase in the effective localization radius of the impurity complex and, accordingly, an increase in the magnitude of the photodielectric effect. The spectral dependence of the photodielectric effect has been calculated in the dipole approximation taking into account the dispersion of the quantum dot radius. A high sensitivity of the magnitude of the photodielectric effect to such parameters of dissipative tunneling as the frequency of the phonon mode, temperature, and coupling constant with the contact medium, has been revealed.
Keywords photodielectric effect, quantum dot, adiabatic approximation, impurity complex, relative permittivity, dissipative tunneling, adiabatic potential, quasi-zero-dimensional structure
Acknowledgements The work was supported by a grant from the Ministry of Science and Higher Education of the Russian Federation 0748-2020-0012.
For citation Krevchik V.D., Razumov A.V., Semenov M.B., Levashov A.V., Shorokhov A.V. Influence of dissipative tunneling on the photodielectric effect associated with the excitation of impurity complexes A+ + e in a quasi-zero-dimensional structure. Nanosystems: Phys. Chem. Math., 2022,13 (3), 250-264.
1. Introduction
Dielectric permittivity, as a measure of polarizability, is a fundamental parameter that determines the properties of various materials. Reversible control of the permittivity of materials in a wide range is an urgent problem. In this regard, of interest is the photodielectric effect (PDE), associated with a change in the permittivity of a material under the action of photoirradiation. In recent years, interest in PDE in various materials has increased due to a wide range of practical applications [1-8]. Thus, in article [1], a photo sensor developed on the basis of PDE in CdS is presented, which changes its electrical capacitance upon photoirradiation. The significant change in photocapacitance discovered by the authors of [1] opens up certain prospects for photosensor applications. In [2], PDE was experimentally studied in the compound LaAl0.99Zn0.01O3_5. It was found that photoirradiation significantly increases the dielectric permittivity in the compound under study in a wide frequency range. It is expected that the results of [2] will contribute to the development of photocapacitors that allow remote control of the dielectric response using photoirradiation. In article [3], the systematic tuning of the PDE in Ba(Al1_xZnx)2O4-,s was experimentally studied by varying the Zn concentration (x) and the intensity of the incident light. Results of the study in [3] can form the basis for the development of innovative phototunable functional devices. From the point of view of the possibility of controlling the PDE in a wide range, semiconductor nanostructures with impurity centers are promising. The presence of the quantum size effect makes it possible to control the binding energy of impurity centers, as well as the energy spectrum of the nanostructure, by varying its characteristic size.
Interest in the photodielectric effect (PDE) associated with the excitation of impurity complexes A+ + e in quantum dots (QDs) (a hole, localized at the A+-center, forms a quasi-stationary A+-state; an electron is localized in the ground
© Krevchik V.D., Razumov A.V., Semenov M.B., Levashov A.V., Shorokhov A.V., 2022
state of the QD) is due to the possibility of its usage as a method for spectroscopic studies of impurities in quasi-zero-dimensional semiconductor structures [9]. Spectral studies of impurities using PDE, in contrast to photothermal ionization, do not require thermal ionization of excited states. Therefore, measurements can be carried out at arbitrarily low temperatures, which eliminates the effect of temperature on the width of the optical absorption line. From a fundamental point of view, PDE is of interest as a nonlinear optical effect with a lower threshold than conventional nonlinear optical effects, and in the case of QD, this threshold can be controlled by changing of its (QD's) characteristic size. The resonant frequencies v0, which characterize the PDE dispersion, can be in the submillimeter range, for example, as estimates show, for InSb QDs with radius R0 « 70 nm, v0 ~ 1011 c-1. Consequently, when a semiconductor quasi-zero-dimensional structure is irradiated with energy quanta hv0, the refractive index of submillimeter waves should change. Therefore, PDE is an effective mechanism for the influence of IR-radiation on the propagation of submillimeter waves in semiconductor quasi-zero-dimensional structures with impurity complexes A+ + e. As will be shown below, the presence of tunneling decay of quasi-stationary A+ - states in a QD provides additional degrees of freedom for controlling of the PDE. The purpose of this work is to study theoretically influence of the tunnel decay of a quasi-stationary A+-state on the PDE, associated with the excitation of impurity complexes A+ + e in a quasi-zero-dimensional structure with semiconductor QDs.
2. The binding energy of a quasi-stationary A+-state in an impurity complex A+ + e in a semiconductor
quantum dot
Figure 1 shows the energy structure of the "quantum dot with an impurity complex A+ + e - external matrix" model under consideration. Quasi-stationary A+-state can be formed by attaching an additional hole to a neutral acceptor. The finite lifetime of a hole is associated with the decay of the A+-state.
Fig. 1. Energy structure of the "quantum dot with an impurity complex A+ + e - external matrix" model. Electron is localized in the ground state of the quantum dot, the hole is in the quasi-stationary state of A+ -center; AE = h/r - magnitude of the broadening of the electronic energy level of the quantum dot, this broadening is associated with the radiative lifetime t; AE = hr0 is the broadening of the impurity level, which is associated with the tunneling decay of the A+-state; DT - dissipative tunneling; the electron optical transition (EOT) is the optical transition of an electron from the ground state of the QD to the excited state; the dotted line shows the change in the potential profile of the adiabatic potential of the electron due to the change in its quantum state; R0 is the QD radius.
We will assume that the process of decay of the quasi-stationary A+-state is mainly due to the dissipative tunneling of the hole. In the one-instanton approximation, the tunnel decay probability r0 can be represented as r0 = B exp (-S), where the action S and the pre-exponential factor B are defined as (in the Bohr units) [10]:
B
2у/ЩДЦ1 + 6*
y/nh
A*
в*cosh() - 1
+ D*
в* cosh( в2) - 1
+
A*
'в! _f* 2 f01
в*
sinh(~2)
в1 cosh(^L_f01)
1
+ D*
в* cosh^^ _f*2)
2
sinh^^)
1
A*
1
в*
»[4* _f*ij
• u в* sinh
+ D*
cosh ( в2—T
• к /'в* sinh (
1
A*
в*
cosh
■Iе*]
1
+ D*
cosh
■ !,( в* апЫ
and
S = a* 2(6* + 1)(3 - 6*)т0* - (6* + 1)2(T0)2 (6* + 1)2 / (1 - X2)
2в*
27'
coth (в*
sinh (в*
I cosh ((в* - THy7^) - cosh v7^) } + cosh ((в* - т(
1 — X'
coth (в VX2) - -
sinh (в* -у/Щ)
cosh (в* - т*) у X2 - cosh (в* у X2) г + cosh ((в* - т(
where
A* D*
2e
*2„ *2
2e
*2„ *2
■ъ.
(+)
(_)
Yi(.p. J Y6.p. '
в* = 7тГв^2ио*7(.^У a*eT, в* = у/ъв=фЩ1&) j a*£T, 7*i = 2Vt!TO= V/Y(_)To*y V2,
(+)T *' 5.p. T0
V2,
702 = 2V72T0 =
6* = 6/a,
в* = УЦ*/ a*eT,
т0*' = arcsinh ((1 - 6*) sinh в */(1 + 6*)) + в * , 7' = ^(eL2a*2/4U0* + 1 + e4a*2/4eL2U0*)2 - eL2a*2/U(
eT = kT/Eh, e* = hVC/Eh, eL = hwL/Eh,
X i — — 1 2
2
1
eL2a*2/4U0* + 1 + e4a*2/4eL2U0* - ¡iV2/^
eL2a*2/4U* + 1 + e4a*2/4eL4* + ^К2а*2/4Ц* + 1 + e4a*2/4eL2U0f - ¡L2a*2/4U(
(1)
(2)
Figure 2 shows dependence of the probability of dissipative tunneling on the parameters ¡L, ¡T and ¡C, which characterize, respectively, frequency of the phonon mode, temperature, and constant of interaction with the contact medium. As it can be seen from Fig. 2, that, as the parameters e*L and e*T increase, the probability of dissipative tunneling of a hole increases too (due to approaching to the top of the potential barrier), and an increase in the parameter e*C leads to blocking of the tunneling decay (due to an increase in the degree of viscosity of the contact medium (or of the heat-bath).
x
i 2
*
в
2
2
2
1 2
в*
в*
2 2
01
2
в*
2
2
7 ) —
1
1
L
L
p
1
1
2
a
Fig. 2. Dependence of the probability of dissipative tunneling at R0 = 70 nm, n = 3.5 on the next parameters (in relative units): a - and sC, under sT = 1; b -sL and sT, under sC = 3
Let the electron be initially in the s-state (l = m = 0), then the hole is acted upon by a spherically symmetric potential:
2 Ro 2 vn,(,m (r) = - h^m^ d?e, (3)
4-K££o J | r - Te |
0
where e - the electron charge; s - permittivity of the QD material; s0 is the electric constant; (re) is the electron
wave function in QD; n is the radial quantum number of an electron; m = 0, ±1, ±2 ... - magnetic quantum number; l = 0,1, 2 ... is the orbital quantum number. In the first order of perturbation theory, for the ground state of an electron
(m = 0, l = 0), potential (3) can be represented as:
Vn,o,o (r) = -
e2^
4neeoRo
+
* 2 2 r2
2 :
(4)
where ^n = Yo - Ci(2nn) + ln (2nn), yo = 1.781 is the Euler constant; Ci(x) is the integral cosine; the frequency is determined by the following relation:
ftwn = [(2ftVn2e2) / (3mhRo3e)]:
(5)
The energy levels of such an oscillator are given in the form
En, o, o
, n2, n3 4neeoRo n
and the corresponding one-particle wave functions are written as
e ^ + ni + n2 + n3 + 3 ) ,
(6)
IX
ni ,n2,n3
(x,y,z) = Cnexp -
X2 + y2 + z2
"2al
Hni ( - ) HnJ — ) HnJ -
«J V aw V an
(7)
-1/2
2ni+n2+n3n1!n2!n3!n3/2an ; an = w ft/(m^wn); Hn(x) are the Hermite polynomials; n1; n2, n3 -
where Cn
quantum numbers, corresponding to the energy levels of a harmonic oscillator (6).
Using, further, the zero-radius potential procedure (see, for example, [11]), we obtain an equation, that determines dependence of the binding energy of a hole in a complex A+ + e on the QD and dissipative tunneling parameters.
The short-range impurity perturbation potential can be written correctly in the form of a pseudopotential [12] as The short-range potential can be represented as a boundary condition at the point (xa, ya, za) given by a pseudo-potential of the form
V (x,y, z; Xa, ya, Za) = 76 (x - Xa) 6 (y - ya) 6 (z - Za) X
d d d 1+(x - xa) dx + (y - ya) dy + (Z - Za) ^
(8)
where 7 = 2nft2/ (amh) is the zero-range potential intensity; a is determined by the energy Ej of the bound state of the same A+-center in the bulk semiconductor; pa, za are the coordinates of the A+-center in QD.
To determine the binding energy of a hole in the complex A+ + e, it is necessary to construct a one-particle Green's function G(r, Ra; EAn) to the Schrodinger equation with a Hamiltonian, containing potential (8):
G (r, Ra; EAn) = -
^n1,n2,n3 (xa, ya, za) ^1,n2,n3 (x, y, z)
1^n3 -EAn - elf + + ^ (n1 + n2 + n3 + 3
(9)
where EAn = -ft2An/(2mh).
For the Green's function, taking into account (7), we obtain
G (r,Ra,£An) = -
3/2a2
E
ni, n2,n3
Hni an Hni On Hn2 an Hn2 an
2ni !n1!
2n2 !n2!
an HnJ an
2n3 !n3!
ft^n ( -eAn -
e2^
£l 0 Rw,
+ ft + 3 + n1 + n2 + n3
(10)
where eAn = EA„/ftwn. Further, considering that
1
-■eAn -
elo^Wn
+ fWn + 3 + n1 + n2 + n3
dt exp
-M -SAn -
2« s
n
e2A
iro 3 N
„ , + T--1---+ n1 + n2 + n3
eRoftwn ftwn 2 ,
(11)
n
2+Ra
1
2
X
1
X
the expression for the Green's function will take the following form
G (r,R0,eA„
1
n3/2a" ftw,
_
dt exp
—M —£An —
2 C} s n
e2ß
+ + 3 2
E
Hm (Oy^ (fn
ni!
£
"2
_t"2
frj M fn
n2!
E
2
tna \ Hnj -M Hnj f-
n3!
. (12)
The summation in (12) over quantum numbers can be performed using the Moeller formula for the generating function of Hermite polynomials [13]
2
k=0
z \k Hk (x) Hk (y)
1
k!
vT—Z2
exp
2xyz — (x2 + y2) z2 1 - z2
(13)
Finally, for the Green's function we will have
DO
X
£R0
_tn
e
X
X
2
2
X
DO
G r,R0,eAJ = —
1
n3/2a" hu>r
exp
r2 + R;
2a"
dt exp
—t —eAn —
e2ß" + + 3
-2i\ _3/2
x (1 — e 2t) exp
2e_t (rfij — e_2t (r2 + Rf)
a" (1 — e_2t)
. (14)
After separating the divergent part in (14) (see, for example, [11]), we obtain
X
G (r,R0,eAn
1
exp
2 —eAn —
e2ßn
£rq h^r
+ iEo + ^ |r_R°1 + hwn + 2
n3/2 a"
2n
R -
(2n)3/2
dt exp
—M —eAn —
e2 ß"
eR0fiw,
+ + 3 2
2e-
exp
(rrRa) — e_2t (r2 + RO)j _^ (1 — e_2t)-
■3/2
(1 — e_2t )
2-3/2
(r-Ra)2
tVt
. (15)
Acting on both sides of relation (15) by the operator T , we obtain an equation, that determines dependence of the binding energy EAn of a hole in a complex A+ + e on the parameters of the QD, as well as on the parameters of dissipative tunneling and the quantum number n.
2ß
-n" — -Rn + 4 + R
+
2
'7n n
dt exp
t ( s 2 , A ■ sr^ * 2y" ß" . o — 2 1 —Ynnn + 4 «7nro--R^ + 3
0
exp —
2t\/2t
D * D* rq ra
1_e
1+e-
(1 — e
_2t)3/2
(16)
where 7n = V/6RSRS/(2nn); n = V|Ei|/Efc; nn ^EA„/Eft; ^ = hro/(4Eh).
Further, it is necessary to find the explicit form of the potential V„ ;jm(rh), defined by formula (3) in the case, when the electron is in the excited p-state (l = 1, m = 0) of QD. Expression (3) can be represented as
(rh) =
4nee0
sin 0d0 / d<^>
rh 2 2 RQ 2 2
(re, 0, y)| re dre + i | (re, 0, <£>)| re dre
%/rh2 + re2 — 2rhre cos 0 J Vrh2 + re 2 — 2rhre cos 0
(17)
a
n
X
2 a 2 t
X
2
a
X
n
Y
X
2
n
n
2
e
The wave function of an electron, in the p-state with m = 0, is given by the next expression
J 3
^n,1,o (re, 0, <£>)
V Ro
V^reRoJ5 (Xn,1)
cos 0,
(18)
then, after integration in (17) we obtain
Vn,1,o(rh)
e2 X
n,1
120nRorh3££o (3Xn,1 cos (Xn,1) + (Xn,12 - 3) sin (X„,1))2
15R3 (R + Xn,12rfc^ X
,'2Xn,1rhA onv D4 /2Xn,1rh"
X sin | —^- I - 30Xn1Rorh cos
Ro
Ro
) + Xn,1rh |^3r2 (1 + 2Xn,12) - 5R2 (8Xn,12 - 3) -
- 3cos (2Xn,1) (5Ro + r^) + 6Xn,1 5RoXn,1 ( Ci (2Xn,1) - C^2XRp) - ln (R^) ) -
- sin(2Xn,1) (5R2 + rh2)
. (19)
Note, that in the case of p-state with m = 0, the minimum of potential (19) is shifted relative to the QD center (see Fig. 1). Position of the minimum is determined from the solution of the following transcendental equation, which is obtained from the equality to zero of the derivative of expression (19) with respect to rh:
2Xn,1rh ,
15R4 cos
( 2Xn,1rh
Ro
+ 5R2Xn,1rh
- (cos (2Xn,1) + 2Xn,1 sin (2Xn,1) - 1 - 2X^1)^ min4
- 15R3 sin ( 2Xn,Rrh mm ) (ro - xn,1 rh min2) = 0. (20)
Expanding expression (19) into a Taylor series near rhmin and restricting ourselves to the quadratic term, we obtain
Vn,1,o(rh) = -SR-^o -
m-h^n ° (rh - rh min) 2
(21)
where the quantities y^0 and w^0 are defined as follows
=
X
n,1
30R0rh min (3Xn,1 cos (Xn,1) + (xn,1 - 3) sin (Xn,1))
6r5 minXn, 1 3rh minXn,1
15r3 minXn,1R0 + 40rh minXn,1R0 + 3rh minXn,1 cos (2Xn,1) + 15rh minXn,1R0 cos (2Xn,1) +
+ 30rh minXn,1R4 co^2rh "R^1 ) - 30rh minXn,1R0Ci (2Xn,1) + 30rh minXn,1R0Ci
3 X3 R2Ci (2X ) , 30r3 X3 R2Ci (2rh minXn,1
V Ro
- 30r3 minXn,1R0 l^^Rf) +6r5 minXn,1 sin(2Xn,1) +30r3 minXn,1 sin(2Xn,1) -
- 15r2 minXn ,1R3 sin (2rh minXn,1 1 - 15R0 sin
min n
Ro
5 • (2rh minXn, 1
0 sin( R0
. (22)
The binding energy of a hole localized at the A+ center, in the case when the electron is in the p-state with m = 0, is determined from the solution of the transcendental equation
n =
+
p2 -■qn
2ft
p,0
o
+ 4 iro + 37n +
7n,0n 0
dt exp
-(„p)2 + 4ir* — -M + A
2 I (nn) +4iio R* + 7n
exp
li2 if /^1-e-t
1+e-
'2t\J2t
(1 - e-2t)3/2
(23)
2
2
2
/n
2
where
=
In
5rh5mmRg3 (3Xn,i cos (Xw,i) + {Xh - 3) sin (Xw,i)) Xn,1
r * 5 X -, + 2r * 5 X 3 _
' h minXn,1 + 2' h minXn,1
5
5
— ^r *3 y3 i? * 2 _ „
5'h minXn,1R 0 'h minXn
Xn,1 cos (2Xn,1) + 15rh minXn,1R* 2 (rh2minXn,1 - 4R* 2)
• / 2rh minXn,1
2rh minXn, 1
- 2rh5minXn,1 sin (2Xn,1 ) - 15R* 3 sin ( h "£* ^ ) + 2R* 2)
0
0
\ -1
and correspondingly rh min/rh min/«h, n«0 = V7 |Ep,0,An |/Eh.
The wave function of a hole localized at the A+-center in the p-state is defined as:
where
£ P -
£An —
*An ('h)
F I 3 + fAn I r ( 1 + £An 1 I 4 + 2 M 4 + 2
2n2a3„ 4 + ^ - ^ 3 + ^
rh2
p,n
-3/4
W
5'h5minR5 (3Xn,1 cos (Xn,1) + (Xn,1 - 3) sin (X„,1))-Xn.1
-P
"An . 1 2 ' 4
p,n
1/2
(24)
(25)
X rh5minXn,1 + 2rh5minXi3,1 - 5rh3minXn, 1R0 2 - '^minXn,1 cos (2Xn,1) +
+ 15rh minXn,1R0 2 (rh2minXn,1 - 4R0 2)
cos
2'
X
n,1
0
- 2rh5minXn,1 sin(2Xn,1) -
Xn
- 45rh2minXn,1 R03 sin [ h "T* n,M + 30R*5 sin
0
2rh minXn,1
0
1/2
R* n2 - 2^0
ahYn.
(26)
where ap,n
Thus, the wave function of the excited state of the electron-hole pair will have the following form:
$ out (re,rh)= <1 (re) *£;n (rh) . (27)
Figures 3(a) and 3(b) show dependence of the binding energy of the quasi-stationary A+-state in the impurity complex A+ + e on the dissipative tunneling parameters sc and sT for InSb-based QDs at Ro = 70 nm, calculated using formula (23).
It can be seen, that with an increase in the parameters ■ L and ■ T , which characterize the frequency of the phonon mode and temperature, respectively, the binding energy of the quasi-stationary A+-state decreases (see Fig. 3(b)) due to an increase in the probability of dissipative tunneling (see Fig. 2(a,b)), as a result, the wave function of the quasi-stationary A+-state "spreads", which leads to an increase in the effective radius of the localized A+-state. An increase in the parameter ■ C , which characterizes the constant of interaction with the contact medium (with the heat-bath), is accompanied by the blocking of tunnel decay and a corresponding increase in the binding energy of the quasi-stationary A+-state (Fig. 3(a)).
The obtained relations (16), (23), and (25) will be used in the next section to calculate the process of photoexcitation of impurity complexes A+ + e in a quasi-zero-dimensional structure.
3. Photodielectric effect in a quasi-zero-dimensional structure with impurity complexes A+ + e in the presence
of dissipative tunneling
Let us consider the process of photoexcitation for complexes A+ + e in QDs under conditions of dissipative tunneling. We will assume that the A+-center is localized at the point Ra = (0,0,0), (see Fig. 1). The energy spectrum of an electron in a size-quantized band will be determined by an expression of the next form
X« ,Eh
,J h (28)
R* 2 R0
Here Xn,; — Xn,; / v^Eh; Xn l is the root of the l-th order Bessel function.
The effective Hamiltonian Hint of interaction with the field of a light wave can be written as
Hi
— i H Ai
2nH2c
-/0 exp (i ' r) (eAs Vf),
(29)
2
2
'
h
x
1
m * 2w
Fig. 3. Dependence of the binding energy of the quasi-stationary A+ - state in the impurity complex A+ + e in InSb-based QDs on the parameters of dissipative tunneling at R0 = 70 nm and n = 3.5: a -on the parameters eL and , at = 1; b -- from parameters and , at = 3.
where A0 is the local field coefficient; a* is the fine structure constant, taking into account the static relative permittivity e; 10 is the intensity of light with frequency w, wave vector <fs and the unit polarization vector e,\s; Vr is the Hamilton operator.
The matrix element which determines the magnitude of the oscillator strength of the dipole optical transitions of an electron and a hole from the ground state $in(re, rh) to the excited states $OMi(re, rh) of the QD, is written as
MAa = m I ^^ IE
»•M ±°i}
- E1,1,° - ( EAn^ - EA1
X ( (rh)| (rh)) (V>„,1 (re) | (eAs, re) | Vi,° (re)> . (30)
Taking into account the explicit form of the wave functions (18) and (25), the matrix element (30) can be written in the form
Mf,A = i
---° (En,1, ° - E1,1,° - (Ep„ - EA1)) X
8n
R°2 J3/2 (X°,1) J5/2 (X„,1)
r( 4 + ¥ r 4 + Y
2n2<„ U (4 + - ^4 + ^m
F Z'3 I fAr^ r ( 1 +
M 4 + 2 M 4 + 2
2n2ap,„ (V (4 + ^P) - ^ 4 + ^
x W ss
W- ^ XS ^ e-2j1/^ R J 2 (S X-1) X
n 2n
x^ d0 sin 0 J (0,y>) Y1jm (0,^)cos (31)
After calculating the integrals in (31) and taking into account the expression for the energy spectrum of charge carriers (23) for the matrix element (31), we obtain (in the Bohr units)
Mf, A (X ) = i ah Eh (25n)1/2 X„,1
a*/°Eh
(iiiR-Il - (,,„2 - „»))
X (3X„,1 cos X„,1 +sin X„,1 (X„,12 - 3)) '
r 3 + ^ r 4 + ¥
r I 3 I f An \ p / 1 I fAn
1 I 4 + 2 M 4 + 2
H3 +11) - n4 + ^J J (n3 + ^J - n3 +
(X»,12 cos X»,1 (X»,12 - n2) +sinX»,1 (3X»,12 - n2)) (y„y„) R° J3/2 (X°,1) J5/2 (X„,1) (n2 - X„,12)2
-3/4
œ œ
X
i=° j=°
(-1)j (Y»-1 + Y»-1)j (Y»)-7r (-(4 +
2j j !i!
r; 2 ( 2+i+j)r(-2)
r(1 + %) (3U d+i+j r (1 + "f) Y» R°
p 2 (1+i+j)^
13 . .3 eAP» 5 . .3 R;2\ 2 4 2 2 2 y„ / 3 + i + j 3 + f^j
1
e
p
x 2F2 ( 1 + i + + ^, 2 + i +
1
R
;2
R
4 2 ' 2
; 2 (1+i+j)^ ( 1
p Y»
+
VtA( 4 + ^AT
F( 4"
(4 + "AT) (2)
(1 + i + j)r 4 + "AT
2fJ 1 + i + j, 4 + f ;2 + i + j, 2; +
2Y
+
Y»R°
2 ( 1+i+j)r( 2)
13
R
;2
( 1 ) f3 fp \2fM2+i+2+i+j;^^
(2 + i+j 3 + "^J V2 4222 y„
(32)
where R* = Ro/«h; k = mh/me; nS = ^EA1/Eh; nS = \/EAn/Eh; pEg(ai,... ap; bi... bg; z) is the generalized hypergeometric function.
x
x
X
p
2
r
h
2 ' 4
s;n
X
X
X
X
X
X
X
p
Let us further assume that the dispersion u of QD sizes arises during the phase decomposition of a supersaturated solid solution and is satisfactorily described by the Lifshitz-Slezov formula [14]
34eu2 exp [—1/ (1 - 2u/3)] 3
P (u)=4 23 (u + 3)3 (3/2 - u) V 3 (33)
u> 2,
where u = R0/R0, R0 and R0 - the QD radius and its average value, respectively; e is the base of the natural logarithm.
With relatively small changes in the permittivity e(Ae < 3e), change in the value e during photoexcitation in the dipole approximation, taking into account the dispersion of the QD radius, is determined from the following formula (see [15-17]):
3/2
(w) = ''' 0 0 y" a„ (w) Tna„P (u) du, (34)
u < ô ;
u + 3)3 (3/2 — u) 3 2
11
Ae (w) = E J (w) Tna«P (u) du,
o
where /0 is the radiation intensity; w is the incident light frequency; an is the polarizability of excited states of an electron in QD; rn is the lifetime of excited electronic states; an(w) is the partial photon absorption cross section for QD with the complex A+ + e.
Since the value 1/r„ determines the total probability, per unit time, for spontaneous emission of photons during quantum transitions of an electron from the p-state (with different quantum numbers n) to the ground state, then, assuming that the system is characterized only by the radiative lifetime, the expression for r„ will be written as
1
— = (35)
Tn
where the total transition probability is given by an expression of the next form
=yJ E out (re,rh) |V| $in (re,rh))|2 dp (Eout), (36)
the quantity dp(Eout) characterizes density of the final states number for the system and is defined as
, ^ n V (W)2 d^
dP (E°"t)= 1' L 3 , (37)
(2n) ftc3
where W = (Ein - Eout)/ft = (^(Xp^ - n2)/2m*ahR2 - - EA1)) /ft; d^ is an infinitesimal element of the solid angle, and the interaction operator of a particle with an electromagnetic field (in the one-photon approximation) has the next form
e
V =--A (r) p. (38)
mec
Here A(r) is the vector potential operator defined by the formula (in the Coulomb gauge)
A (r) = E (^TW^y72 eV (flk,.+ a^e-^) . (39)
Respectively, and a\ a are the operators of annihilation and creation of a photon with a wave vector k and polarization a, respectively. In the case of the dipole approximation, the expression for the total probability, in the Bohr units, taking into account the QD size dispersion, will be determined as follows
"e2E3 n,i - n2) _ / 2_ nS2 ^ |p ,2
3ft4c3 y (Ru)2 "
(nPn - nil) J |Pn,i,o|2 , (40)
where the value Pn,1j0 is determined by the next expression
PP = < (rl)l *Ah (rl)) ($n,i,0 (re) | (el, re) | *1,0 (re)) . (41)
Taking into account the previously obtained expression for the matrix element (32), the expression for (41) can be written as
PP (R» =
4Xn,i (X„,12 cosXn,i (X„,12 - n2) +sinXn,i (3X„,12 - n2))
iîgW J3/2 (Xo,i) J5/2 (Xn,l) (n2 — Xn, l2)2 (as
p,
3/2
r( 4 + r (f + ^f-^ ( 3 +% ) - ^ f 4 + %
Fl 3 + fAn \ r / 1 + fAn M 4 + 2 M 4 + 2
^ 4 + > - ^ 4 + >
(-1)3 (q-p + q-p)3
X S 3=0 23a2i"j!i! (3Xn,i cos Xn,i +sinXn,i (Xp,i2 - 3))
r (-D 4 + %
r( 4+¥ (i;
(R
12 ( 2
r (-D
3
2F2 - + i -d + i+j r(4 + ^ 12
3 + eAn-5 + i +
4+ 2 '2+ 4 +2' a2
u
+
qp,n \/M0
(«0
^(i+i+j)^ (1
r( D
1 e
2
(2 + i + j)r(} + ff) ^4 + +4
.. (Rgu)
+
+
+
i + £in ^s;n ^ 4 + 2
rf 4 + ff )( D
+
(«S
>2(i+i+j)
r (-D
3e
(1 + i + j)r( 4 + ^
2F2 I 1 + i + j,-7 + ^'2 + i
An . <
. 3 (Ru)2
xp,n \/L0
(«S
)2 ( 2+<
-r-T^ 2 FJ 2+i-
(2 + i+j) r( 4 + ¥ V2
42
>7 + ^ + i
13
4
2 2 2
21 q2 2 qp,n
, (Ru)2
p,n
+
(42)
Accordingly, for the lifetime of an electron at an excited level, taking into account the dispersion of the QD sizes, we obtain the following expression
Tp («SU
3ft4 c3
4e2E3 ( fc(XR;U)n2) - (nPn2 - nAi2)) |PP (RSu) I
(43)
The polarizability «^(—0^) electronic states in a QD with a complex A+ + e in the Bohr units is determined by an expression of the next form
aP (Ru)
4ah3
Eh fc^-n2) (RoM)2
npn - nAi
(44)
The partial photons absorption cross sections for QDs with the complex A+ + e are determined by the following formula
2_ r < (w,Ro) = tt- |Mf,A (w,Ro)|2---0-—, (45)
0 *2T0 + fR2(xn;1 -n2) _(ep _ Es , hw
4 + I 2me*R02 (EAn EA1) ftw
Where Mf,A(w) is determined by formula (32). Rewriting (45) in the Bohr units, we obtain
ap (X,fiou) =
2n
|Mf,A (X )|2 ro
ft2IoEh ro2 f fc(xn,1-n2) p 2 s 2 N ^ + ^ (fi0«)2 - (nAn2 - nAi2) - X)
2
(46)
Thus, expression (34) takes the form
Ae (X)
N0/012n2ft4c3qh5a* e2Eh5X 2
N 3/2
/ P (u) du-
|pp (Ro u |2
^^ - (nPn2 - nAi2)
(RÎ0«)2
1 0
If + ( k(xn,l-n2)
4 + V (R>)2
- (nPn2 - nAi2) - xj
(47)
x
x
X
p
p
œ œ
X
X
p
2
p
u
p
1
2
2
p
X
2
X
or, taking into account (32), for (47) we finally obtain
N
Ae (X) = -,2 ^ 5 V2- E
n=1
3/2
e2Eh5X 2
du -
P (u)
*(*2Li-n2)
(RJ0*«)2
- (nPn2 - nil2)
4Xn,i (X„,i2 cos Xn,i (X„,i2 - n2) +sinXn,i (3X„,i2 - n2))
R*0u J3/2 (Xo,i) J5/2 (Xn,i) (n2 - X„,i2)2 (aSi„aPi„)3/2
A
r( 3 + r (j + ^y-^ (3 + %) - ^ (4 + %
r (3 + fin 1 V (i +
1 ^ 4 + 2 ) 1 ^ 4 + 2
Tft+íWTi+ln
00 00
xEE
i=0 j=0
(-1)j (a-n + a—n)j
_(-i) (4 + %) j
2ja2i„j !i!(3Xn,i cos Xn,i + sin Xn,i (X„,i2 - 3)) ^ r ^i + fi^j (
(Ro
2 ( f +i+j)
r (-2)
2F2 77 + i + j, 7 + -£t;x + i + j, ~;
in .
3. (RR
u
(§ + i + j) r( i + - V2 ' ■ "4 ' 2 '2 " a2
+
ap,n (R?ou ,2
i2(i+i+j) r (2)
(2 + i + jr(f + ff) 2F211 + i +j4 + • 2 +' + j
(RRo
'p,n 2
u
p,n
+
+
+
a I i + fin '-''sjn I 4 \ 2
r( 4 + ff) (2) i
(RR0
) 2 (i+i+j)
r (-2)
3. (R
(1 + i+j )r( 4 + fi-
2fM 1+i+j,4 + 2n;2+i+j,2 a2
+
p,n
+
■^n \/L0
(RRO
)2 (1+j+j)r(2)
13
(1 ) (3 s" \ 2F212 + i+j'7 + ^;2>2 + i+j;
(2 + i+j r(4 + ^J V2 4222
. (RRo
p,n
o 1 0
r02 + ( fc(xn,!-n2) (np 2 _ nS 2)_X
4 + I rso^2 (nin nii ) X
r. (48)
(-R0")2
Figure 4(a,b,c) shows the spectral dependence of the relative change in permittivity Ae/e for the quasi-zero-dimensional structure with InSb QDs for various dissipative tunneling parameters eL, e*C and eT, calculated by formula (48). As can be seen from Fig. 4(a,b), an increase in the phonon mode value and temperature (respectively, parameters eL and eT) also leads to a significant increase in Ae/e, which is associated with an increase in the effective radius of the excited states of the impurity complex A+ + e under conditions of tunneling decay of the quasistationary A+ state.
An increase in the degree of "viscosity" of the contact medium or of the heat-bath (parameter eC ) leads to the suppression of the PDE due to an increase in the localization of the wave function of the quasi-stationary A+-state (see Fig. 4(c)). From Fig. 4(a,b,c) it is also seen that the PDE spectral dependence curves contain a peak that appears when the photon energy becomes comparable with the average energy of the optical transition. In this case, position of the peak depends on the parameters of dissipative tunneling, which, as shown above, have a significant influence on the effective localization radius of the quasi-stationary A+ state.
4. Conclusion
Influence of dissipative tunneling on the PDE, associated with the excitation of impurity complexes A+ + e during intraband optical transitions of an electron in a QD, has been theoretically studied, taking into account change in the profile of the adiabatic potential of an electron.
In contrast to [15], where the case of a localized A+-state is considered, in this paper we consider the case of a quasi-stationary A+-state (the impurity level is located between the bottom of the adiabatic potential and the energy level of the ground state of the hole, see Fig. 1). In this case, the width of the potential barrier is finite, and the influence of the tunnel decay of the quasi-stationary-state should be taken into account. As a result, new possibilities appear for controlling the PDE in a quasi-zero-dimensional structure by varying such parameters of dissipative tunneling as temperature, the frequency of the phonon mode, and the constant of interaction with the contact medium, which differs significantly from the case of a localized A+-state [15], when the PDE is controlled only by changing the average quantum dot radius.
In the dipole approximation, an analytical formula is obtained for the spectral dependence of the change in the permittivity of a quasi-zero-dimensional structure upon photoexcitation of impurity complexes A+ + e under conditions
x
2
X
X
f 2
X
X
p
u
X
p
2
p
u
u
2
2
p
u
X
2
X
Fig. 4. Spectral dependence of the relative change in the permittivity of a quasi-zero-dimensional structure with InSb QDs at Ej = 3.5 meV; R° = 70 nm, for different values of the dissipative tunneling parameters:
a) 1 - eL = 1, eT = 1, eC = 3; 2 - eL = 2, eT = 1, eC = 3;
b) 1 - eL = 1, eT = 1, eC = 3; 2 - eL = 2, eT = 3, eC = 3;
c) 1 - eL = 1, eT = 1, eC = 3; 2 - eL = 1, eT = 1, eC = 4.
of tunnel decay of a quasi-stationary A+-state, taking into account the dispersion of the QD radius. It is found that varying the dissipative tunneling parameters leads to a change in the effective localization radius of the quasi-stationary A+-state in the QD and, as a consequence, to a noticeable change in the permittivity of the quasi-zero-dimensional structure. It is shown that the spectral dependence of the PDE is characterized by a peak that appears when the photon energy becomes comparable with the average energy of the optical transition. The position of the peak depends on the parameters of dissipative tunneling. It is revealed that an increase in the degree of "viscosity" of the contact medium is accompanied by the suppression of the PDE due to the strengthening of the localization of the wave function of the quasi-stationary A+-state. Thus, the controllability of the PDE in a quasi-zero-dimensional structure with impurity complexes A+ + e in the presence of dissipative tunneling has been demonstrated.
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Submitted 26 April 2022; revised 16 May 2022; accepted 17 May 2022
Information about the authors:
Vladimir D. Krevchik - Penza State University, Physics Department, Krasnaya str., 40, Penza, 440026, Russia; ORCID 0000-0002-3522-8326; physics@pnzgu.ru
Alexey V Razumov - Penza State University, Physics Department, Krasnaya str., 40, Penza, 440026, Russia; razumov_alex@mail.ru
Mikhail B. Semenov - Penza State University, Physics Department, Krasnaya str., 40, Penza, 440026, Russia; ORCID 0000-0003-4348-0000; misha29.02.1@gmail.com
Alexander V Levashov - Penza State University, Physics Department, Krasnaya str., 40, Penza, 440026, Russia; physics@pnzgu.ru
A.V. Shorokhov - Penza State University, Physics Department, Krasnaya str., 40, Penza, 440026, Russia; University of Jyvaskyla, Seminaarinkatu, 15, PO BOX 35, Fl-40014, Finland; alex.shorokhov@gmail.com
Conflict of interest: the authors declare no conflict of interest.
