ACTA PHYSICA POLONICA A
Vol. 122 (2012)
Electron and Exciton Quasi-Stationary
No. 1
s-States
in Open Spherical Quantum Dots
∗
M.V. Tkach , Ju.O. Seti and O.M. Voitsekhivska
Chernivtsi National University, Kotsiybynsky Str. 2, Chernivtsi, 58012, Ukraine
(Received December 10, 2011; in nal form February 10, 2012)
The theoretical calculation of spectral parameters of electron and exciton quasi-stationary s-states in open
spherical quantum dot is performed within the eective mass approximation and rectangular potentials model.
The conceptions of probability distribution functions (over quasi-momentum or energy) of electron location inside
of quantum dot and their spectral characteristics: generalized resonance energies and widths are introduced. It
is shown that the generalized resonance energies and widths, obtained within the distribution functions, satisfy
the Heisenberg uncertainty principle for the barrier widths varying from zero to innity. At the same time, the
ordinary resonance energies and widths dened as complex poles of scattering S -matrix, do not satisfy it for the
small barrier widths and, therefore, are correct only for the open quantum dots with rather wide potential barriers.
PACS: 71.15.Dx, 73.21.La, 73.22.Dj, 73.90.+f
1. Introduction
resonance energies (GREs) and generalized resonance
The modern experimental abilities of infrared range
cascade lasers, resonance tunnel diodes and separable
quantum dots production and wide perspectives of their
utilization in microbiology and medicine [14] constantly
stimulate the interest of theoretical investigations of open
nanostructures. The explaining of all physical phenomena in them is connected with the description of electron
and exciton quasi-stationary spectra and interaction of
these quasi-particles with classic and quantized elds.
widths (GRWs) valid at arbitrary potential barrier width
are introduced and studied.
It is also proven that
the universal characteristic of electron or exciton quasi-stationary states (QSSs) in open spherical QD is the
probability distribution function (over quasi-momentum
or energy) of quasi- particle located inside of QD. The dependences of electron and exciton GREs and GRWs on
the barrier width is studied for InAs/GaAs/InAs nanostructure.
The electron spectrum in open quantum lms, wires
and dots is studied using dierent theoretical methods
2.
[511]. In the framework of the eective mass approxima-
S -matrix
and probability distribution
functions of quasi-particles located
tion and rectangular potential barriers model, the quasi-
inside of open spherical QD
-stationary electron spectrum in open spherical quantum dot (QD) is usually studied within the scattering
S -matrix
method [7, 8, 11] because it allows the exact
solution of the Schrödinger equation. The complex poles
of
S -matrix
dene the resonance energies (REs) and res-
onance widths (RWs) of electron in open spherical QD
with wide barriers rather well [11].
Nevertheless, it is already established that the quasi-stationary electron (exciton) spectrum in open spherical
QD with thin and super thin barriers, the most perspec-
The open spherical QD (Fig. 1) consisting of semiconductor core (0) inside of the shell (1) embedded into the
outer medium (2) is under study. The radius of core-well
r0 and thickness of barrier-shell is ∆. In spherical coordinate system with the beginning in QD center, the
is
electron or hole eective masses and potential energies
are xed by the expressions
tive for the practical utilization, cannot be dened by the
complex poles of
S -matrix
[11].
{
m(r) =
{
In this paper, the new characteristics of electron and
exciton states in open spherical QDs:
the generalized
U (r) =
m0 ,
,
m1 ,
0, 0 ≤ r ≤ r0 , r0 + ∆ ≤ r < ∞,
U, r0 ≤ r ≤ r1 = r0 + ∆.
Using the Hamiltonian of the system
∗ corresponding author; e-mail:
[email protected]
(207)
(1)
M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska
208
ξ(k) =
m1 kr0 cot(kr0 ) − m0 χr0 + m0 − m1
.
m1 kr0 cot(kr0 ) + m0 χr0 + m0 − m1
S(k)-matrix
(8)
(6), of course, coincides to the one obtained
in Refs. [7, 8, 11] in other analytical form, but its expression through the real
Z(k)
function has the advantages
which would be clear further. Especially, the expression
(6) is valid for
in
Z(k)
k > k0 (E > U )
at the condition
χ → iχ
function.
Now, we introduce the probability distribution functions
W (k)
or
W (E)
(over quasi- momentum or energy)
of quasi-particle located inside of open spherical QD (in
r1 = r0 + ∆ radius)
∫
1 r1
2
W (k) =
|X(kr)| dr,
r1 0
∫
1 r1
2
|XE (r)| dr,
W (E) =
r1 0
the sphere of
Fig. 1. Geometrical and potential energy schemes of
open spherical QD.
H=−
where
~2
1
∇
∇ + U (r),
2 m(r)
(2)
we solve the stationary Schrödinger equation exactly and
Ψℓmk (r, θ, φ) = Rℓk (r)Yℓm (θ, φ).
Yℓm (θ, φ) the spherical
0, 1, 2, . . . ; m = 0, ±1, ±2, . . .) and the
(3)
functions
radial ones
(ℓ
=
Rℓk (r)
are taken as linear combinations of the Hankel functions
Rℓk
 (0)
]
(0) [
+
Rℓ (kr) = aℓ h−

ℓ (kr) + hℓ (kr) ,




0 ≤ r ≤ r0 ,



[
]


 R(1) (kr) = a(1) h− ( iχr) + S (1) h+ (i χr) ,
ℓ
ℓ
ℓ
ℓ
ℓ
=
 r0 ≤ r ≤ r1 = r0 + ∆,




]

(2)
(2) [

+

Rℓ (kr) = aℓ h−

ℓ (kr) + Sℓ (k)hℓ (kr) ,


r0 + ∆ ≤ r < ∞,
(4)
k = ~−1
(5)
their densities of currents at all nanostructure interfaces
together with the normalizing one, dene all unknown
Sℓ (k)-matrix
two close energy values (E and
E1 ) we obtain
1 ~2
1 [
′
(r)
XE1 (r)XE
W (E) =
lim
r1 2m0 E1 →E E1 − E
]
− XE′ 1 (r)XE (r) r=r1 .
where the
[7, 8, 11].
Z(k) =
1−
m0
m1
[
],
1+ξ(k) exp(−2χ∆)
1 − χr1 1−ξ(k)
exp(−2χ∆)
to the
S -matrix
through
(13)
(
)
Z(k)
.
k
(14)
Taking into account the analytical form of
for the probability distribution function at
Z(k) (7), we
{√
k ≤ k0 :
]
m0 k [ 2
ξ + exp(−4χ∆)
m1 χ
)}
(
√
m0 ′
m0
ξ −
2ξ∆
+ 2 exp(−2χ∆) χ
m1
m1
/(
[
]{
m0
1 + Z(k)2 χr1
[ξ + exp(−2χ∆)]
m1
})
m1 − m0
+
[ξ − exp(−2χ∆)] ,
m1
W (k) =
(6)
(7)
]−1 d
k [
1 + Z 2 (k)
πr1
dk
get the exact and convenient for calculations expression
where
kr1
(12)
S(k) = e2 i δ(k) ,
s-states (ℓ = 0). The exact
S0 (k) ≡ S(k) matrix in under
spectrum (E ≤ U, k ≤ k0 ) is
1 + i Z(k)
,
1 − i Z(k)
2
sin(kr + δ),
π
phase (δ ) is related
=
the expression
conveniently written in the form
S(k) = e−2 i kr1
where
√
(2)
XE (r)
Further, avoiding the sophisticated expressions, we are
barrier region of energy
(11)
r ≥ r1
According to the general theory [12, 13], for
(2)
U (r) = 0, XE (r) function is written as
going to observe only the
analytical expression for
(10)
The calculation of these functions is analytically per-
W (k) =
The continuity conditions of radial wave functions and
coecients and scattering
XE = rRE (r).
after some transformations it is obtained
where
√
√
2m0 E, χ = (k02 − k 2 )m1 /m0 ,
√
k0 = ~−1 2m0 U .
X(kr) = rR(kr),
formed exactly. Really, using Schrödinger Eq. (2) for the
obtain the complete set of wave functions
Here,
(9)
where
kr1
π
{
[
]
m0
ξ =2
r0 kχr02 1 + cot2 (kr0 )
m1
′
(15)
Electron and Exciton Quasi-Stationary s-States . . .
( √
)
}
√
m1 k 2
m1 − m0 m1 k
− 1+
χr0 cot(kr0 ) +
m0 χ2
m1
m0 χ
/{[
]2 }
m0
m1 − m0
kr0 cot(kr0 ) −
χr0 −
.
m1
m1
Let us, rst of all, analyze the main properties of
and
W (k)
W (E)) stays valid also for k ≥ k0
χ → iχ. Further, it is proven that just
function allows introducing the GRE
and GRW conceptions which are true independently of
the width of the open spherical potential barrier.
and GRWs of QSSs in such a way that the latter would
(∆ is given in the units of InAs lattice constant
χ∆ > 1
e2
.
(17)
ε|re − rh |
Here, Eg band gap energy, He (re ), Hh (rh ) the electron and hole Hamiltonians, expression (2) and ε diHex = Eg + He (re ) + Hh (rh ) −
obtained from the complex poles of
The Schrödinger equation with Hamiltonian (17) cannot be solved exactly. Thus, the approximated method
is used. When the sum of uncoupling electron and hole
resonance energies in the respective exciton QSS is much
are shown in Fig. 2.
W (K = kr0 ) and
It is clear that their prop-
erties are dierent for dierent barrier widths:
small
≤ ∆ ≪ r0 ), relative (∆ ≤ r0 ); and big (∆ > r0 ). Let
us observe W (K) and W (E) peculiarities for the small
∆ magnitudes (Fig. 2). When the potential barrier is
absent (∆ = 0), from the expression for probability distribution function
W (K)|∆=0 =
[
]
1
1
sin(2K)
(1 − j0 (K)) =
1−
,
π
π
2K
tion over quasimomentum for the exciton located in open
spherical QD is given by the expression
W (ke , kh ) = W (ke )W (kh )
]
[
[
]
ke kh d ke−1 Ze (ke ) /dke d kh−1 Zh (kh ) /dkh
= 2 2
.
π r1
[1 + Ze2 (ke )]
[1 + Zh2 (kh )]
(18)
Using (5) and (18), we obtain the probability distribution function over the energy for the exciton located in
open spherical QD
(19)
and Fig. 2 one can see that
average value
W̄ ∆=0
1
= lim
A→∞ A
GRWs
in
spherically-
= 0).
and
W (E)
functions
∫
∞
W (K)dK =
0
1
π
(20)
<
= π2 sin2 kn< r0 )
minimum (Wn
2
>
and maximum (Wn
= π2 sin kn> r0 ) values, (n =
<
0, 1, 2, . . . , ∞) at kn = r0−1 K2n , kn> = r0−1 K2n+1 , where
consistently
K2n
and
taking
K2n+1
are the even and odd roots of the equa-
tion
W ′ (K)|∆=0 = 0,
or Kcot(K) = 1.
(21)
The oscillations of W (K) and W (E) functions in K - or
E -scales, respectively, create the continuous sequence of
peaks (ne = 1, 2, 3, . . . ∞), each characterized by its maximum and width.
W (Ee , Eh ) = W (Ee )W (Eh ),
W (K)
perform the quasi-periodical oscillations respectively the
it can be assumed that the probability distribution func-
and
according
The electron distribution functions
W (E)
bigger than the energy of their interaction in these states,
GREs
S -matrix
K = kr0 ,
-particles are mainly located.
exciton
and
to Refs. [12, 13].
electric constant of 0 and 2 media where the quasi-
the
a)
would coincide to the usual REs and RWs
(0
The Hamiltonian of exciton has the form
-symmetric states (ℓ
W (k)
functions for the electron. They would give
the opportunity to introduce the conceptions of GREs
at
Expression (15) for the distribution function
xing
W (E)
be valid for the whole innite range of barrier widths
(16)
(and equivalent to it
(E ≥ U ) when
W (E) distribution
209
As far as it is true for any width, it
is reasonable to introduce two main spectral characterisw
2 > 2
tics of n-th electron peak: GRE: En = ~ (kn ) /(2m0 ),
e
e
>
corresponding to the maximum of Wne (Kn ), and GRW:
e
(+)
(−)
(±)
Γnw = Ene − Ene , where Ene energies are the roots of
e
3. Electron and exciton quasi-stationary
s-states
in open spherical QD
the equation, dened by the natural condition (Fig. 2):
2W (E) = W (Kn>e ) + W (Kn<e ).
(22)
From Fig. 2 one can see that for increasing
The calculation of the probability distribution func-
∆,
the
heights of all under barrier peaks are growing in the vicin-
tions of electron and exciton located in open spherical
ity of resonances due to the decrease of
QD is performed according to the developed theory for
between the resonances. Since, at the increasing
InAs/GaAs/InAs nanostructure with physical parame-
peaks have at rst the quasi-Lorentz shape and then con-
ters presented in Table.
sistently transform into
TABLE
Physical parameters of InAs/GaAs/InAs nanostructure.
InAs
GaAs
me
0.022
0.067
mh
0.41
0.5
a [Å]
6.058
5.653
ε∞
15.1
10.9
ε0
16.3
12.9
Eg [eV]
0.35
1.52
K ∼ Kne .
δ -like
W
in intervals
∆
these
functions with maxima at
At limit case ∆ → ∞, GRWs of under barrier
w
w
QSSs tend to zero (Γn → 0) and their GREs (En ) coe
e
c
incide to the stationary electron energy spectrum (En )
e
in closed spherical QD, as it must be according to the
physical considerations.
Analyzing now the spectral parameters:
GREs and
GRWs of the electron QSSs in open spherical QD, we
must note that they satisfy Heisenberg uncertainty prin-
M.V. Tkach, Ju.O. Seti, O.M. Voitsekhivska
210
Γnwe → Γnse → 0
and
Enwe → Ens e → Enc e ,
where
Enc e
energies of bound stationary states of electron in closed
spherical QD.
The GREs, REs and GRWs, RWs of exciton QSSs are
calculated within the approximated method, using the
following considerations. When the electronhole interaction is neglected, the energies and widths of exciton
QSS
s-states
are dened, depending on the method, by
the formulae
En(0)w,s
= Eg + Enw,s
+ Enw,s
;
e nh
e
h
Γn(0)w,s
= Γnw,s
+ Γnw,s
,
e nh
e
h
(23)
w
w
where En n , Γn n the GREs and GRWs of electron
e h
e h
(e) and hole (h) QSSs, obtained before within the probas
s
bility distribution function and En n , Γn n the REs
e h
e h
and RWs of the same states, obtained as the complex
Fig. 2. Evolution of electron probability distribution
functions (W (K = kr0 ) and W (E)) at the small barrier
width of open spherical QD and r0 = 50a.
poles of
S -matrix.
The Coulomb potential energy of electronhole interaction does not create the additional potential barrier for
the transition of both quasi-particles from QD. Thus, we
ciple at the whole range of barrier widths (0
≤ ∆ < ∞).
assume that it is only renormalizing the energy of exciton
Also, for the big barrier widths they must be equal to the
QSSs without changing their widths in the rst approx-
REs and RWs dened from the complex poles of scatter-
imation.
ing
S -matrix.
The renormalized energies of QSSs |ne nh ⟩ in
open nanostructure cannot be calculated using the wave
functions normalized at
δ -function.
Therefore, the calcu-
lation of electronhole binding energy (∆Ene nh ) is performed as in Ref. [14]. Instead of the single open spherical
QD we observe the equivalent two-well closed spherical
QD with so big width of the outer shell-well that the energies and widths of former resonance states with good
exactness coincide to the respective REs and RWs in open
QD. Herein, we use the perturbation theory and in the
rst approximation the corrections to the exciton REs
are written as
∫
∫ ∞
e2 ∞
2
2
dre
drh |Rne (re )| |Rnh (rh )|
∆Ene nh = −
ε 0
0
{
−1
re , re ≥ rh ,
(24)
× re2 rh2
rh−1 , rh ≥ re ,
Fig. 3. Dependences of electron spectral parameters
Γnw,s
(a) and Enw,s
(b) on the barrier width ∆.
e
e
In Fig. 3, where
on
∆
Enwe , Γnwe
and
Ens e , Γnse
dependences
are shown, one can see that GREs and GRWs sat-
isfy all abovementioned demands.
From Fig. 3a,b it is
clear that for the under barrier energies (E
< U)
for the
barrier of one monoshell (a) order or bigger, the QSSs
w
s
w
s
spectral parameters (En , En and Γn , Γn ), dened
e
within
W
function and
e
S -matrix
e
e
are better coinciding
between each other, the bigger is the width
∆.
When
the barrier width is smaller than a and tends to zero,
s
s
the REs (En ) also tend to zero and RWs (Γn ) increase
e
e
s
s
s
tending to innity. It causes that αn (∆) = En /Γn
e
(dash curves at inset in Fig. 3a) at small
e
∆
e
become
smaller than 1/2, contradicting the Heisenberg princis
ple (Ene tne = ~Ene /Γne = ~αn ≥ ~/2). As far as
e
w
w
GREs (En ) and GRWs (Γn ) are concerned, at ∆ tend-
Rne (re ), Rnh (rh ) electron (e) and hole (h) wave
functions in the states ne and nh of two-well closed spher-
where
ical QD, approximating open spherical QD [14].
w,s
Thus, the spectral characteristics: energies (En n )
e h
w,s
and widths (Γn n ) are xed by the expressions
e h
Enw,s
= Eg + ∆Ene nh + Enw,s
+ Enw,s
;
e nh
e
h
Γnw,s
= Γnw,s
+ Γnw,s
.
e nh
e
h
(25)
s
The numeric calculations of exciton REs (En n ), RWs
e h
s
w
w
(Γn n ) and GREs (En n ), GRWs (Γn n ) are performed
e h
e h
e h
s
for open spherical QD InAs/GaAs/InAs. Herein, En ,
e
s
s
s
En and Γne , Γn are xed by the complex poles of
h
h
and Sh -matrixes.
w
energies, xed by the formulae: En
the respective
Se -
The generalized
= ~2 (kn>e )2 /2me ,
(+)
(−)
w
Γne = Ene − Ene ,
e
tudes always satisfying Heisenberg principle (solid curves
Enwh = ~2 (kn>h )2 /2mh
(+)
(−)
Γnwh = Enh − Enh
at the inset).
spectral parameters of probability distribution function
e
e
ing to zero, the both parameters tend to the nite magniFinally, let us note that at
∆ → ∞
and widths:
are dened as the corresponding
Electron and Exciton Quasi-Stationary s-States . . .
4. Conclusions
for exciton in QD (Fig. 4, without accounting of the
band gap energy
Eg ).
In Fig. 4a the spatial shape of
W (ke , kh ) in k -space is
peak of W (Ee , Eh ) dis-
probability distribution function
presented.
211
In Fig. 4b the rst
tribution function together with the respective terms of
w
w
w
w
generalized energies (Ee1 , Eh1 ) and widths (Γe1 , Γh1 ) are
shown.
The spectral parameters (REs and RWs) of spherically symmetric electron, hole and exciton QSSs in open
spherical QD calculated within the method of complex
S -matrix poles are true in
widths (∆) except the small
the wide range of barrier
≪ r0 ), where these
magnitudes contradict the Heisenberg uncertainty prinones (∆
ciple.
The probability distribution functions (W (k),
W (E))
of electron located inside of open spherical QD and their
spectral parameters (GREs and GRWs) correctly describe the spherically symmetric QSSs in the whole innite range of barrier widths satisfying the Heisenberg
principle.
The proposed method of electron, hole and exciton
GREs and GRWs calculation, after several modications,
can be also used for the multishell open quantum dots,
wires and lms. Thus, it would be actual for the evalua-
Fig. 4. Probability
distribution
functions
W (ke , kh ) (a), W (Ee , Eh ) (b) and spectral parameters of electron and hole quasi-stationary states.
tion of spectral characteristics of QSSs in resonance tunnel devices produced at the base of open nanostructures
with thin barriers.
References
Fig. 5. Dependences of exciton spectral parameters
(b) on the barrier width ∆.
(a) and Enw,s
Γnw,s
e nh
e nh
The typical dependences of generalized resonance (w)
and resonance (s) spectral parameters of several exciton
QSSs on the barrier width (∆) are given in Fig. 5 (with-
Eg ). The gure
proves that the exciton spectral parameters (the same
out accounting of the band gap energy
for the electron and hole), determined by both methods,
coincide well at
∆ ≥ 2a ≥ 1
the small barrier widths.
nm and strongly dier for
The GREs of exciton QSSs
weakly depend on barrier thickness and GRWs increase,
approaching some nite magnitudes for the smaller barrier widths. For the rather big widths, the REs and RWs,
obtained from the complex poles of
true magnitudes while at small
∆
S -matrix
give the
these magnitudes are
not correct because they contradict the Heisenberg uncertainty principle.
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