Effect of the Spatial Dispersion on the Shape of a Light Pulse in a
  Quantum Well
Effect of the Spatial Dispersion on the Shape of a Light Pulse in a Quantum Well
L. I. Korovin, I. G. Lang
A. F. Ioffe Physical-Technical Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia
S. T. Pavlov†‡
†Facultad de Fisica de la UAZ, Apartado Postal C-580, 98060 Zacatecas, Zac., Mexico; and
‡P.N. Lebedev Physical Institute, Russian Academy of Sciences, 119991 Moscow, Russia; pavlov@sci.lebedev.ru
Reflectance, transmittance and absorbance of a symmetric light pulse, the carrying frequency of
which is close to the frequency of interband transitions in a quantum well, are calculated. Energy
levels of the quantum well are assumed discrete, and two closely located excited levels are taken into
account. A wide quantum well (the width of which is comparable to the length of the light wave,
corresponding to the pulse carrying frequency) is considered, and the dependance of the interband
matrix element of the momentum operator on the light wave vector is taken into account. Refractive
indices of barriers and quantum well are assumed equal each other. The problem is solved for an
arbitrary ratio of radiative and nonradiative lifetimes of electronic excitations. It is shown that
the spatial dispersion essentially affects the shapes of reflected and transmitted pulses. The largest
changes occur when the radiative broadening is close to the difference of frequencies of interband
transitions taken into account.
PACS numbers: 78.47. + p, 78.66.-w
Irradiation of the low-dimensional semiconductor sys-
tems by light pulses and analysis of reflected and trans-
mitted pulses allow to obtain the information regarding
the structure of energy levels as well as relaxation pro-
cesses.
The radiative mechanism of relaxation of excited en-
ergy levels in quantum wells arises due to a violation of
the translation symmetry perpendicular to the he quan-
tum well plane1,2. At low temperatures, low impurity
doping and perfect boundaries of quantum wells, the
contributions of the radiative and nonradiative relax-
ation can be comparable. In such situation, one can-
not be limited by the linear approximation on the elec-
tron - light interaction. All the orders of the interac-
tion have to be taken into account3,4,5,6,7,8,9. Alterations
of asymmetrical10,11,12,13 and symmetrical13,14,15 light
pulses are valid for narrow quantum wells under condi-
tions k d ≪ 1 (d is the quantum well width, k is the mag-
nitude of the light wave vector corresponding to the car-
rying frequency of the light pulse) and an independence
of optical characteristics of a quantum well on d. How-
ever, a situation is possible when the size quantization
is preserved and for wide quantum wells if k d ≥ 1 (see
corresponding estimates in16). In such a case, we have to
take into account the spatial dispersion of a monochro-
matic wave 9,19 and waves composing the light pulse16.
Our investigation is devoted to the influence of the spa-
tial dispersion on the optical characteristics (reflectance,
transmittance and absorbance) of a quantum well irradi-
ated by the symmetric light pulse. A system, consisting
of a deep quantum well of type I, situated inside of the
space interval 0 ≤ z ≤ d, and two semi-infinite barriers,
is considered. A constant quantizing magnetic field is
directed perpendicular to the quantum well plane what
provides the discrete energy levels of the electron sys-
tem. A stimulating light pulse propagates along the z
axis from the side of negative values z. The barriers are
transparent for the light pulse which is absorbed in the
quantum well to initiate the direct interband transitions.
The intrinsic semiconductor and zero temperature are
assumed.
The final results for two closely spaced energy levels
of the electronic system in a quantum well are obtained.
Effect of other levels on the optical characteristics may be
neglected, if the carrying frequency ω ℓ of the light pulse
is close to the frequencies ω 1 and ω 2 of the doublet levels,
and other energy levels are fairly distant. It is assumed
that the doublet is situated near the minimum of the
conduction band, the energy levels may be considered in
the effective mass approximation, and the barriers are
infinitely high.
In the case h̄ K⊥ = 0 (h̄ K⊥ is the vector of the quasi-
momentum of electron-hole pair in the quantum well
plane) in a quantum well, the discrete energy levels are
the excitonic energy levels in a zero magnetic field or
energy levels in a quantizing magnetic field directed per-
pendicularly to the quantum well plane. As an example,
the energy level of the electron-hole pair in a quantizing
magnetic field directed along the z axis (without taking
into account the Coulomb interaction between the elec-
tron and hole which is a weak perturbation for the strong
magnetic fields and not too wide quantum wells) is con-
sidered.
I. THE ELECTRIC FIELD
Let us consider a situation when a symmetric excit-
ing light pulse propagates through a single quantum well
along the z axis from the side of negative values of z.
Analogously to13,14,15, the electric field is chosen as
E 0 (z, t) = e ℓE 0 e
−iω ℓ p
http://arxiv.org/abs/0704.1697v1
Θ(p)e−γ ℓ p/2 + [1−Θ(p)]eγ ℓ p/2
+ c.c., (1)
where E 0 is the real amplitude, p = t− ν z/ c,
e ℓ =
(ex ± i e y)
are the unite vectors of the circular polarization, ex e y
are the real unite vectors, Θ(p) is the Heaviside function,
1/γ ℓ determines the pulse width, c is the light velocity in
vacuum, ν is the refraction index, which is assumed the
same for the quantum well and barriers (the approxima-
tion of a homogeneous media). The Fourier-transform of
(1) is as follows
E 0(z, ω) = e
ikz [e ℓ E 0(ω) + e
ℓ E 0(−ω)] ,
E 0(ω) =
E 0 γ ℓ
(ω − ω ℓ)2 + (γ ℓ/2)2
, k =
. (2)
The electric field in the region z ≤ 0 consists of the
sum of the exciting and reflected pulses. The Fourier-
transform may be written as
ℓ (z, ω) = E 0 (z, ω) + ∆E
ℓ (z, ω),
where ∆E ℓ(z, ω) is the electric field of the reflected pulse
∆E ℓ(z, ω) = e ℓ ∆E
ℓ(z, ω) + e ∗ℓ ∆E
ℓ(z,−ω). (3)
In the region z ≥ d, there is only the transmitted pulse,
and its electric field is
r(z, ω) = e ℓ E
r(z, ω) + e ∗ℓ E
r(z,−ω). (4)
It is assumed below that the pulse, having absorbed in
the quantum well, stimulates the interband transitions
and, consequently, the appearance of a current. In barri-
ers, the absorption is absent. Therefore, for the complex
amplitudes ∆E ℓ(z, ω) and E r(z, ω) in barriers for z ≤ 0
and z ≥ d, we obtain the expression
d z 2
+ k 2 E = 0. (5)
The expression for the electric field inside of the quantum
well (0 ≤ z ≤ d) has a form
d z 2
+ k 2 E = −4πi ω
J(z, ω), (6)
where J(z, ω) is the Fourier-transform of the current den-
sity, averaged on the ground state of the system. The
current is induced by the monochromatic wave of the
frequency ω. In the case of two excited energy levels,
J(z, ω) is expressed as follows
J(z, ω) =
γ rj Φ j(z)
dz′ Φ j(z
′)E(z ′, ω),
ω̃ j = ω − ω j + iγ j/2. (8)
where γ j is the nonradiative damping of the doublet,
γ r j is the radiative damping of the levels of the doublet
in the case of narrow quantum wells, when the spatial
dispersion of electromagnetic waves may be neglected.
In particular, the doublet system may be represented
by a magnetopolaron state18. In such a case,
γ r,j = γ r Q j , γ r =
h̄ cν
p 2c v
h̄ ω g m 0
m 0 c
where m 0 is the free electron mass, H is the magnetic
field, e is the electron charge, p c v is the matrix element
of the momentum, corresponding to the circular polar-
ization, p2c v = |p c v x |2 + |p c v y |2. The factor
Q j =
± h̄ (Ω c − ωLO)
h̄2(Ω c − ωLO)2 + (∆E pol)2
determines the change of the radiative timelife at a deflec-
tion of the magnetic field from the resonant value when
the resonant condition Ω c = ωLO is carried out. ∆E pol
is the polaron splitting, Ω c and ωLO are the cyclotron
frequency and optical phonon frequency, respectively. In
the resonance, Q j = 1/2 and γ r1 = γ r2.
When calculating J(z, ω), it was assumed that the
Lorentz force, determined by the external magnetic field,
is large in comparison with the Coulomb and exchange
forces in the electron-hole pair. In that case, the vari-
ables z (along magnetic field) and r⊥ (in the quantum
well plane) in the wave function of the electron-hole pair
may be separated. This condition is carried out for the
quantum well on basis of GaAs for the magnetic field,
corresponding to the magnetopolaron formation9. Be-
sides, if the energy of the size quantization exceeds the
Coulomb and exchange energies, the electron-hole pair
may be considered as a free particle. Then, in the ap-
proximation of the effective mass and infinitely high bar-
riers, the wave function, describing the dependence on z
, accepts a simple form
Φ j(z) =
πm cz
πm vz
, 0 ≤ z ≤ d, (9)
and Φ j(z) = 0 in barriers, where m c (m v) are the
quantum numbers of the size-quantization of an electron
(hole).
In the real systems, the approximation (9) is not al-
ways carried out. However, the taking into account
the Coulomb and exchange interactions will result only
into some changes of the function Φ j(z), what does not
change qualitatively the optical characteristics, as it was
shown for the monochromatic irradiation20.
Indices j = 1 and j = 2 in Φ j(z) correspond to the
pairs of quantum numbers of the size-quantization in a
direct interband transition. m
c ( v)
corresponds to the
index j = 1, and m
c ( v)
corresponds to the index j = 2.
In interband transitions, the Landau quantum numbers
are conserved. The total electric field E is included into
the RHS of (7), what is connected with the refuse from
the perturbation theory on the coupling constant e2/h̄c.
In further calculations, an equality of quantum num-
bers m
c v = m
c v is assumed. Then,
Φ 1(z) = Φ 2(z) = Φ(z),
and the current density in the RHS of (7) takes the form
J(z, ω) =
i ν c
ω − ω 1 + iγ 1
ω − ω 2 + iγ 2
×Φ(z)
dz′Φ(z′)E(z′). (10)
With the help of the indicated simplifications, as it
was shown in9,18,20, the field amplitudes in the Fourier-
representation ∆E ℓ(z, ω) and E r(z, ω) result in
∆E ℓ(z, ω) = −iE0(ω)(−1)m c+m ve−ik(z−d)N ,
E r(z, ω) = E0(ω)e
ikz(1− iN ), (11)
where E0(ω) is given in (3). Here, the frequency depen-
dence is determined by the function
N = ε (γ r1 ω̃2 + γ r2 ω̃1)/2
ω̃1 ω̃2 + iε(γ r1 ω̃2 + γ r2 ω̃1)/2
. (12)
The function N includes the value
ε = ε′ + iε′′, (13)
which determines influence of the spatial dispersion on
the radiative broadening (ε′ γ r) and shift (ε
′′ γ r) of the
doublet levels. ε′ and ε′′ are equal9,20:
ε′ = Re ε = 2B2
1− (−1)m c+m v cos kd
, (14)
ε′′ = Imε = 2B
(1 + δmc,mv )(mc +mv)
2 + (mc −mv)2
8mcmv
−(−1)mc+mvB sinkd− (2 + δmc,mv)(kd)
8π2mcmv
, (15)
B = 4π
2 m c m v kd
[π 2 (m c +m v)2 − (kd)2 ] [(kd)2 − π 2 (m c −m v)2 ]
(if kd → 0, ε → 1 (m c = m v, an allowed transition )
and ε → 0 (m c 6= m v, a forbidden transition)).
II. THE TIME-DEPENDENCE OF THE
ELECTRIC FIELD OF REFLECTED AND
TRANSMITTED LIGHT PULSES
With the help of the standard formulas, let us go to
the time-representation
∆E ℓ(z, t) ≡ ∆E ℓ(s) =
dω e−iω s ∆E ℓ(z, ω), s = t+ νz/c, (16)
E r(z, t) ≡ E r(p) = 1
dω e−iω p E r(z, ω),
p = t− νz/c. (17)
The vectors ∆E ℓ(s) and E r(p) have the form
∆E ℓ(s) = e ℓ ∆E
ℓ(s) + c.c.,
r(p) = e ℓ E
r(p) + c.c.. (18)
It is seen from expressions (11) and (12) that the denom-
inator in integrands of (16) and (17) is the same. It may
be transformed conveniently to the form
ω̃1 ω̃2 + i(ε/2) (γ r1 ω̃2 + γ r2 ω̃1)
= (ω − Ω 1) (ω − Ω 2), (19)
where Ω 1 and Ω 2 determine the poles of the integrand
in the complex plane ω. They are equal
Ω1,2 =
ω1 + ω2 −
(γ1 + γ2)−
(γr1 + γr2)
[ω1 − ω2 −
(γ1 − γ2)−
(γr1 − γr2)]2
−ε2γr1γr2
. (20)
Thus, in the integrands of 16) and (17), there are 4 poles:
ω = ω ℓ± iγ ℓ/2 and ω = Ω 1,2. The pole ω = ω ℓ+ iγ ℓ/2
is situated in the upper half plane, others are situated in
the lower half plane.
Integrating in the complex plane ω, we obtain that the
function ∆E ℓ(z, t), determining, according to (17), the
electric field vector of the reflected pulse ∆E ℓ(z, t), has
the form
∆E ℓ(z, t) = −iE 0(−1)m c+m veikd
×{R1[1−Θ(s)] + (R2 +R3 +R4)Θ(s)}, (21)
where
R1 = exp(−iω ℓs+ γ ℓs/2)
γ̄ r1/2
ω ℓ − Ω1 + iγ ℓ/2
γ̄ r2/2
ω ℓ − Ω2 + iγ ℓ/2
R2 = exp(−iω ℓ − γ ℓs/2)
γ̄ r1/2
ω ℓ − Ω1 − iγ ℓ/2
γ̄ r2/2
ω ℓ − Ω2 − iγ ℓ/2
R3 = − exp(−iΩ1s)(γ̄ r1/2)
ω ℓ − Ω1 − iγ ℓ/2
ω ℓ − Ω1 + iγ ℓ/2
R4 = − exp(−iΩ2s)(γ̄ r2/2)
ω ℓ − Ω2 − iγ ℓ/2
ω ℓ − Ω2 + iγ ℓ/2
, (22)
where
γ̄ r1 = ε
′ γ r1 +∆γ, γ̄ r2 = ε
′ γ r2 −∆γ,
ε′ γ r1(Ω2 − ω2 + iγ2/2)
Ω1 − Ω2
ε′ γ r2(Ω1 − ω1 + iγ1/2)
Ω1 − Ω2
. (23)
The function E r(z, t), corresponding to a transmitted
light pulse, is represented in the form
E r(z, t) = E 0
T1[1−Θ(p)] + (T2 + T3 + T4)Θ(p)
Ω1 − Ω2
where
T1 = exp(−iω ℓp+ γ ℓp/2)M(ω ℓ + iγ ℓ/2)
ω ℓ − Ω1 + iγ ℓ/2
ω ℓ − Ω2 + iγ ℓ/2
T2 = exp(−iω ℓp− γ ℓp/2)M(ω ℓ − iγ ℓ/2)
ω ℓ − Ω1 − iγ ℓ/2
ω ℓ − Ω2 − iγ ℓ/2
T3 = − exp(−iΩ1 p)M(Ω1)
ω ℓ − Ω1 − iγ ℓ/2
ω ℓ − Ω1 + iγ ℓ/2
T4 = exp(−iΩ2 p)M(Ω2)
ω ℓ − Ω2 − iγ ℓ/2
ω ℓ − Ω1 + γ ℓ/2
. (25)
The function M has the structure
M(ω) = (ω − ω1 + iγ1/2)(ω − ω2 + iγ2/2)−
(ε′′/2)[γr1(ω − ω2 + iγ2/2) + γr2(ω − ω1 + iγ1/2)].
When the electric field of stimulating light pulse E 0(z, t)
(determined in (2)) is extracted from E r(z, t) , i.e., it is
assumed
r(z, t) = E 0(z, t) + ∆E r(z, t), (26)
then, ∆E r(z, t) will differ from ∆E ℓ(z, t) only by substi-
tution of the variable s = t + νz/c by p = t − νz/c and
by absence of the factor (−1)m c+m v exp(ikd).
Thus, being taken into account, the spatial dispersion
provides a renormalization of radiative damping γ r i. In
nominators of formulas (21), the renormalization leads to
multiplication of γ r i on the real factor ε
′, i.e., decreases
the value γ r i (diagrams of functions ε
′ and ε′′ are rep-
resented in9). In denominators, γ r i is multiplied on the
complex function ε, that means the appearance, together
with the change of the radiative broadening, of a shift of
resonant frequencies. In the limit kd = 0, expressions
(21) - (23) coincide with obtained in (14).
=1 =0.0005 eVeV
=0.00005 eV
=0.0005 eV
t, s, p {ps)
6420-2  
FIG. 1: The reflectance R, transmittance T , absorbance A,
and stimulating pulse P as time dependent functions for three
magnitudes of the parameter kd in the case of a long stimulat-
ing pulse (γ ℓ ≪ ∆ω) γ r ≪ γ, γ ℓ .∆ω = 6.65.10
eV, ω ℓ =
ReΩ1 = Ω res.
t (s, p) ps
=0.005 eV
 =0.00005 eV
  =0.0005 eV
2.10-4
1.10-4
-1.10-4
-2.10-4
-1  0 21
FIG. 2: Same as in Fig.1 for an exciting pulse of a middle
duration (γ ℓ ≃ ∆ω) γ r ≪ γ ≪ γ ℓ .
III. THE REFLECTANCE, TRANSMITTANCE
AND ABSORBANCE OF STIMULATING LIGHT
PULSE
The energy flux S(p), corresponding to the electric field
of stimulating light pulse, is equal
S(p) =
(E 0(z, t))2 = ezS0P(p), (27)
where S0 = cE
0/2πν, ez is the unite vector along the
light pulse. The dimensionless function
P(p) = (E
0(z, t))2
= Θ(p)e−γℓp + [1−Θ(p)]eγℓp (28)
determines the spatial and time dependence of the energy
flux of stimulating pulse. The flux, transmitted through
the quantum well, has a form
(E r(z, t))2 = ezS0T (p), (29)
the reflected energy flux has a form
ℓ = − ezc
(E ℓ(z, t))2 = −ezS0R(s). (30)
The dimensionless functions T (p) and R(s) correspond
to parts of transmitted and reflected energy fluxes of the
stimulating pulse. The dimensionless absorbance is de-
fined as
A(p) = P(p)−R(p) − T (p) (31)
(since for reflection z ≤ 0, the variable in R is s = t −
|z|/c).
The dependencies of the reflectance R, transmittance
T , absorbance A, and stimulated momentum P on the
variable p (or s for R) for the case m c = m v = 1 are
represented in figures. It was assumed also that
γ r 1 = γ r 2 = γ r, γ 1 = γ 2 = γ. (32)
It follows from (21) and (24) that the resonant frequen-
cies are ω ℓ = ReΩ1 and ω ℓ = ReΩ2. The calculations
were performed for
ω ℓ = ReΩ1 = Ωres. (33)
Let us go from the frequency ω ℓ to
Ω = ω ℓ − ω1, (34)
then the resonant frequency is
Ωres =
−∆ω + ε′γ r +Re
(∆ω)2 − ε2γ2r
. (35)
It depends on three parameters: ∆ω = ω1 − ω2, γ r and
kd, since the complex function ε depends on kd (see (15)).
Functions R, T ,A and P are homogeneous functions
of the inverse lifetimes and frequencies ω1, ω2, ω ℓ. There-
fore, a choice of the measurement units is arbitrary. For
the sake of certainty, all these values are expressed in eV .
The time dependence of the optical characteristics of a
quantum well is represented in figures for the different
magnitudes of kd. The curves, corresponding to kd = 0,
were obtained in14. It was assumed in calculations that
∆ω = 0.065eV , what corresponds to the magnetopolaron
state in a quantum well on basis of GaAs and to the width
d = 300Å of the quantum well18,19,21.
t (p, s) ps
 = 0.0005 eV
 = 0.00005 eV
  = 0
151050
FIG. 3: Same as in Fig.1 for an exciting pulse of a middle
duration (γ ℓ ≃ ∆ω) γ r ≪ γ ≪ γ ℓ .
t (s, p) ps
= 0.005 eV
= 0.00005 eV
  = 0
4.10-4
-6.10-4
-4.10-4
2.10-4
-2.10-4
FIG. 4: Same as in Fig.1 for an exciting pulse of a middle
duration (γ ℓ ≃ ∆ω) γ = 0.
IV. THE DISCUSSION OF RESULTS
Fig.1 corresponds to a long (wide in comparison to
∆ω) stimulating pulse and a small radiative broadening
(γ r ≪ γ, γ ℓ). In this case, the transmittance T domi-
nates. The shape of the curve weakly differs from P and
weakly depends on kd. The dependence on the spatial
dispersion is seen at the curves R and A. For example,
the reflectance R at kd = 3 is two times less than at
kd = 0. However, the magnitude R is shares of percent.
Fig.2 corresponds to a stimulating light pulse of a mid-
dle length, when γ ℓ ≃ ∆ω and γ r ≪ γ ≪ γ ℓ. There
appear peculiarities: a light generation (negative absorp-
 = 0.00666
t (p, s) ps
21-1 0
FIG. 5: Same as in Fig.1 for four magnitudes of the parameter
kd in the case, when ∆ω is close to γ r, γ ℓ ≫ γ r ≫ γ.
tion) after the light pulse transmission and oscillations
of R,A and T . The generation is a consequence of the
fact that the electronic system has no time to irradiate
the energy during the propagation of such pulse. Oscil-
lations is a consequence of beatings with the frequency
(under condition ω ℓ = ReΩ1)
Re (Ω1 − Ω2) = Re
(ω1 − ω2)2 − (ε′ + iε′′)2γ2r. (36)
A noticeable effect of the spatial dispersion takes place
in the reflectance R during transmission of the pulse, as
well as after its transmission. The spatial dispersion af-
fects the transmittance T and absorbanceA after passing
through a quantum well, when these values are small.
In Fig. 3 and 4, the optical characteristics are repre-
sented at γ = 0 and a long stimulating pulse (γ ℓ ≪ ∆ω,
Fig.3) and a pulse of a middle duration, when γ ℓ ≃ ∆ω
(Fig.4). Since, in that case, the real absorption is ab-
sent, one have to accept the function A, defined in (31),
as an energy part, stored up by a quantum well for the
time being due to the interband transitions (if A > 0), or
an energy part, which is generated by the quantum well
during and after propagation of the pulse (A < 0). The
same concerns to Fig.2, however, the part of the stored
energy there, which disappears if γ → 0, corresponds to
the real absorption. The oscillation period in Fig.2 and
4 does not depend on the parameter kd, since, at chosen
magnitudes of the parameters ∆ω and γ r the beating
frequency (36) is almost equal to ω1 − ω2, and compara-
tively small changes of the functions ε′ and ε′′ does not
affect practically on the beating frequency.
In Fig.5, where ∆ω is near γ r (6.65.10
−3eV and
6.66.10−3eV , respectively), the stimulating light pulse is
5 times shorter, than in Fig.3 and 4, and γ ℓ ≫ γ r ≫ γ.
In that case, the spatial dispersion affects strongly on
the optical characteristics. In the interval 0 ≤ kd ≤ 3,
the reflectance increases 8 times approximately, and the
transmittance decreases 6 times. Such a sharp change is
due to the dependence of γ̄ r1 and γ̄ r2 on kd. For ex-
ample, at kd = 0 γ̄ r1 = −17303.9, γ̄ r2 = 193066, 6, and
at kd = 3 Re γ̄ r1 = 1960.21, Re γ̄ r2 = 442, 718. And at
the same time, R and T ≤ 1, since they are the result
of substraction of large magnitudes and therefore these
differences are sensitive to changes of kd.
In14, it was shown that, at kd = 0, there are the sin-
gular points on the time axis, where T = A = 0 and
R = P , or R = A = 0 and T = P (total reflection or
total transmission). It is seen from the figures that the
singular points are preserved and in the case kd 6= 0,
there is only a small shift of them. In Fig.5 the point of
the total transmission appears at kd = 0. At kd = 0.5,
this point disappears, and at kd = 1.5 and kd = 3.0 the
point of the total reflection appears. If kd = 1.5, then
R = P , A+ T = 0 (A < 0). If kd = 3.0, then as before
R = P , but A = T = 0. Thus, growing of the parameter
kd changes the type of a singular point.
Thus, the spatial dispersion of the electromagnetic
waves, forming the light pulse, noticeably affect the op-
tical characteristics of a quantum well. This influence is
especially strong, when γ r ≃ ∆ω.
Let us note in conclusion that the results obtained
above are valid at equal refraction indices of barriers and
quantum well. Otherwise, one is to take into account re-
flection of boundaries of a quantum well. However, this
problem is outside the scopes of present article.
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http://arxiv.org/abs/cond-mat/0104262
http://arxiv.org/abs/cond-mat/0004178
http://arxiv.org/abs/cond-mat/0002229
http://arxiv.org/abs/cond-mat/0006364
http://arxiv.org/abs/cond-mat/0203390
http://arxiv.org/abs/cond-mat/0001248
http://arxiv.org/abs/cond-mat/0403302
http://arxiv.org/abs/cond-mat/0001248
