Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well
Electron spectrum in cylindrical quantum dot HgS embedded into ZnS medium is calculated using the variational method with variational parameter in Hamiltonian. The dependence of energy spectrum on the quantum well sizes is established. The electron spectrum calculated in the framework of infin...
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| Цитувати: | Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well / V.A. Holovatsky, O.M. Voitsekhivska, M.J. Mikhalyova, M.M. Tkach // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 863-871. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1209882017-06-14T03:06:08Z Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well Holovatsky, V.A. Voitsekhivska, O.M. Mikhalyova, M.J. Tkach, M.M. Electron spectrum in cylindrical quantum dot HgS embedded into ZnS medium is calculated using the variational method with variational parameter in Hamiltonian. The dependence of energy spectrum on the quantum well sizes is established. The electron spectrum calculated in the framework of infinitely deep potential well is compared to the one obtained within the variational method. It is shown that the first method gives satisfactory results for the ground level only and at rather big sizes of quantum well. На основі варіаційного методу з варіаційним переметром у гамільтоніані розраховано електронний спектр у циліндричній квантовій точці HgS, розміщеній у середовищі ZnS. Встановлена залежність енергетичного спектру, розрахованого в рамках моделі безмежно глибокої потенціальної ями, задовільно працює лише для основного рівня і при досить великих розмірах квантової ями. 2000 Article Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well / V.A. Holovatsky, O.M. Voitsekhivska, M.J. Mikhalyova, M.M. Tkach // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 863-871. — Бібліогр.: 4 назв. — англ. 1607-324X DOI:10.5488/CMP.3.4.863 PACS: 79.60.jv http://dspace.nbuv.gov.ua/handle/123456789/120988 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
Electron spectrum in cylindrical quantum dot HgS embedded into ZnS
medium is calculated using the variational method with variational parameter in Hamiltonian. The dependence of energy spectrum on the quantum
well sizes is established. The electron spectrum calculated in the framework of infinitely deep potential well is compared to the one obtained within
the variational method. It is shown that the first method gives satisfactory
results for the ground level only and at rather big sizes of quantum well. |
| format |
Article |
| author |
Holovatsky, V.A. Voitsekhivska, O.M. Mikhalyova, M.J. Tkach, M.M. |
| spellingShingle |
Holovatsky, V.A. Voitsekhivska, O.M. Mikhalyova, M.J. Tkach, M.M. Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well Condensed Matter Physics |
| author_facet |
Holovatsky, V.A. Voitsekhivska, O.M. Mikhalyova, M.J. Tkach, M.M. |
| author_sort |
Holovatsky, V.A. |
| title |
Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well |
| title_short |
Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well |
| title_full |
Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well |
| title_fullStr |
Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well |
| title_full_unstemmed |
Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well |
| title_sort |
electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2000 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120988 |
| citation_txt |
Electron spectrum in confined cylindrical nanoheterosystem with finite depth of potential well / V.A. Holovatsky, O.M. Voitsekhivska, M.J. Mikhalyova, M.M. Tkach // Condensed Matter Physics. — 2000. — Т. 3, № 4(24). — С. 863-871. — Бібліогр.: 4 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT holovatskyva electronspectruminconfinedcylindricalnanoheterosystemwithfinitedepthofpotentialwell AT voitsekhivskaom electronspectruminconfinedcylindricalnanoheterosystemwithfinitedepthofpotentialwell AT mikhalyovamj electronspectruminconfinedcylindricalnanoheterosystemwithfinitedepthofpotentialwell AT tkachmm electronspectruminconfinedcylindricalnanoheterosystemwithfinitedepthofpotentialwell |
| first_indexed |
2025-07-08T18:58:39Z |
| last_indexed |
2025-07-08T18:58:39Z |
| _version_ |
1837106328013635584 |
| fulltext |
Condensed Matter Physics, 2000, Vol. 3, No. 4(24), pp. 863–871
Electron spectrum in confined
cylindrical nanoheterosystem with
finite depth of potential well
V.A.Holovatsky, O.M.Voitsekhivska, M.J.Mikhalyova,
M.M.Tkach
Chernivtsi National University,
2 Kotsiubinsky Str., 274012 Chernivtsi, Ukraine
Received November 9, 2000
Electron spectrum in cylindrical quantum dot HgS embedded into ZnS
medium is calculated using the variational method with variational param-
eter in Hamiltonian. The dependence of energy spectrum on the quantum
well sizes is established. The electron spectrum calculated in the frame-
work of infinitely deep potential well is compared to the one obtained within
the variational method. It is shown that the first method gives satisfactory
results for the ground level only and at rather big sizes of quantum well.
Key words: electron, quantum dot, nanocrystal
PACS: 79.60.jv
Introduction
The progressive development in physics of 2D heterosystems with quantum wells
and their practical utilization in semiconductor lasers attracts many scientists to
investigate the systems with even smaller dimension, namely: quantum wires and
quantum dots. In quantum dots (QD) – “artificial atoms”, the charge carriers are
confined in all three directions and have the totally discrete energy spectra. Since the
lasers produced on the basis of QD would have the superhigh temperature stability
of threshold density of current and generational frequency, small time of ground
level occupation, and, consequently, the high working frequencies [1–3].
Several methods of QD creation have been known so far. The molecular-beam
epitaxy method is used to produce complicated QD of spherical shape [4]. The
transversal etching of quantum wires produces the cylindrical quantum dots. The
pyramidal QD with a rectangular base are obtained in [3].
It is clear that the QD of spherical and cubic symmetry are the best theo-
retically investigated systems because when calculating the electron spectrum the
Schrödinger equation has an exact solution for the potential well with finite depth
c© V.A.Holovatsky, O.M.Voitsekhivska, M.J.Mikhalyova, M.M.Tkach 863
V.A.Holovatsky et al.
2h
ρ
0
m0
m1
r0
θ r
φ
0
z
yx
Figure 1. Geometry of cylinder approximated by sphere.
and infinite depth as well. The Schrödinger equation for the cylindrical quantum dot
has an exact solution only in case of infinitely deep potential well in the direction
of cylinder axis. For the QDs of a more complicated shape (pyramid, ellipse eth.)
there are only the approximated methods of energy spectrum calculation. Therefore,
the theory of quasiparticles (electron, hole, exciton, phonon) spectra is developed,
generally, for the spatially confined spherical, cylindrical and rectangular heterosys-
tems. Most of the scientists use the dielectric continuum model for the description
of contacting media and effective mass approximation for the quasiparticles. The
results of exciton spectra investigation in complicated spherical heterosystems [4]
prove the possibility of applying these methods for QD of 30− 40 Å sizes.
In this paper the spectrum and wave functions of quasiparticle in cylindrical
potential well are obtained within the Bethe variational method. Herein, the cylin-
drical potential well is approximated by a spherical well with the radius assumed as
variational parameters. The numerical calculations are performed for the HgS/ZnS
heterosystem. The energy levels dependences on cylinder sizes are analysed.
1. Hamiltonian of the system and solution of Schr ödinger equa-
tion in zeroth approximation
The nanoheterosystem consisting of HgS semiconductor nanocrystal having the
cylindrical shape with ρ0 radius and 2h height embedded into the ZnS semiconductor
medium is under study. The beginning of the energy starts from the energy of the
bottom of ZnS conductive band since an electron is moving inside the the potential
well
U(~r) =
{
−V0; 0 6 ρ 6 ρ0, −h 6 z 6 h,
0; other region.
(1)
864
Electron spectrum in confined cylindrical nanoheterosystem. . .
Thus, one has to solve the problem of quasiparticle energy spectrum and wave
functions in rectangular potential well which has the shape of cylinder with finite
height.
The Schrödinger equation has the form
(
−~
2
2
~∇
1
m(~r)
~∇+ U(~r)
)
ψ(~r) = Eψ(~r), (2)
where
m(~r) =
{
m0; 0 6 ρ 6 ρ0, −h 6 z 6 h,
m1; other region,
(3)
Equation (2) with the potential (1) and masses (3) cannot be solved exactly. There-
fore, using the Bethe variational method, cylinder is approximated by a sphere with
r0 radius, playing the role of the variational parameter. The approximated potential
is written as
Vs(r) =
{
−V0; 0 6 r 6 r0,
0; r0 < r <∞
(4)
and approximated mass
ms(r) =
{
m0; 0 6 r 6 r0,
m1; r0 < r <∞.
(5)
The Hamiltonian of the system is given as
H = H0 +∆H, (6)
where
H0 = −
~
2
2
~∇
1
ms(r)
~∇+ Vs(r) (7)
is the electron Hamiltonian in approximated spherical heterosystem,
∆H = −
~
2
2
~∇
1
m(~r)
~∇+
~
2
2
~∇
1
ms(r)
~∇+ U(~r)− Vs(r) (8)
is the Hamiltonian of perturbation.
It is clear that the energy states in cylindrical quantum well must be located in
the interval between the corresponding states in inscribed and described spherical
quantum wells. Thus, the Hamiltonian of quasiparticle is assumed as a basic part
of H0 Hamiltonian in spherical potential well with certain r 0 radius the magnitude
of which would be fixed, according to the Bethe variational method, from the func-
tional of the total energy. The Schrödinger equation with H0 is solved exactly in the
spherical coordinate system and has the form
H0(r, ϑ, φ)Ψ(r, ϑ, φ) = E0Ψ(r, ϑ, φ). (9)
Its solution is written as
Ψnlm(r, ϑ, φ) = Rnl(r)Ylm(ϑ, φ). (10)
865
V.A.Holovatsky et al.
The radial functions are given by
Rnl(r) =
{
R
(0)
nl (r) = A0jl(knr); r 6 r0,
R
(1)
nl (r) = A1h
(+)
l (iχnr); r > r0,
(11)
where
k =
√
2m0(V0 − |E|)
~2
, χ =
√
2m1|E|
~2
. (12)
In the media interface (r = r0) the following conditions should be satisfied
R0
nl(r0)|r=r0 = R1
nl(r0)|r=r0 ,
1
m0
dR0
nl
dr
∣
∣
∣
∣
r=r0
=
1
m1
dR1
nl
dr
∣
∣
∣
∣
r=r0
.
(13)
From (13) one can get the dispersion equation for defining the quasiparticle
energy spectrum
1
m0
(
l − k0nr0
jl+1(k
0
nr0)
jl(k0nr0)
)
=
1
m1
(
l − iχ1
nr0
h+l+1(iχ
1
nr0)
h+l (iχ
1
nr0)
)
. (14)
Coefficients A1 and A0 have the relationship
A1 = A0
jl(knr0)
h+l (iχnr0)
. (15)
In the case l = 0, equation (14) has the form
1 + χ1
nr0 −
m1
m0
(1− k0nr0 cot(k
0
nr0)) = 0. (16)
The r0 variational parameter is found in the framework of variational method
from the minimum condition of the functional built from the basic Hamiltonian H 0
and perturbation ∆H averaged on the wave functions (11).
2. Calculation of non-spherical correction
The electron energy spectrum in the first approximation is defined as a solution
of zeroth problem obtained from dispersion equation (14) plus the correction found
on the wave functions of the previous section. Thus
Enlm(r0) = ǫnl(r0)+ < nlm|∆Ĥ|nlm > . (17)
The correction to the basic Hamiltonian is convenient to write as
∆Ĥ = ∆Ĥ0 +∆Ĥ1, (18)
866
Electron spectrum in confined cylindrical nanoheterosystem. . .
where ∆H0 = T0 + U0 is the Hamiltonian operating to the wave function in space
inside the sphere with r0 radius but outside the cylinder (Ω0); ∆H1 = T1+U1 is the
Hamiltonian operating to the wave function in space outside the sphere but inside
the cylinder (Ω1). From equation (8) there are
∆Ĥ0 = Θ0(~r)
{
~
2
2
(
1
m1
−
1
m0
)[
d2
dr2
−
l(l + 1)
r2
]
+ V0
}
, (19)
∆Ĥ1 = Θ1(~r)
{
~
2
2
(
1
m0
−
1
m1
)[
d2
dr2
−
l(l + 1)
r2
]
− V0
}
, (20)
where Θ0(~r) and Θ1(~r) functions define the region of operation of corresponding
operators and have the form
Θ0(1)(~r) =
{
1; ~r ∈ Ω0(1),
0; ~r /∈ Ω0(1).
(21)
The wave functions (11) are the eigenfunctions of ∆H operator, then the functional
(17) is transformed to the equation
Enlm(r0) = ǫnl(r0) +
{
(1−
m0
m1
)ǫnl +
m0
m1
V0
}
I
(0)
nlm +
{
(1−
m1
m0
)ǫnl − V0
}
I
(1)
nlm, (22)
with the following notations
I
(0)
nlm =
∫∫
Ω0
∫
|R
(0)
nl (r)|
2|Ylm(θ, φ)|
2r2dr sin θdθdφ, (23)
I
(1)
nlm =
∫∫
Ω1
∫
|R
(1)
nl (r)|
2|Ylm(θ, φ)|
2r2dr sin θdθdφ. (24)
Taking into account that
Ylm(θ, φ) = NlmP
(m)
l (cos θ)eimφ, (25)
where
Nlm =
√
(l − |m|)!(2l + 1)
4π(l + |m|)!
(26)
integrals (23) and (24) can be rewritten as
I
(0)
nlm = 2
(l − |m|)!(2l + 1)
(l + |m|)!
∫∫
S0
|R
(0)
nl (r)|
2r2dr|P
(m)
l |2 sin θdθ, (27)
I
(1)
nlm = 2
(l − |m|)!(2l + 1)
(l + |m|)!
∫∫
S1
|R
(1)
nl (r)|
2r2dr|P
(m)
l |2 sin θdθ, (28)
where the integrating regions S0 and S1 for the case ρ < r0 < h are shown in figure 2.
From (27) and (28) one can see that the degeneration over the quantum number
m is taken off and the difference of energy levels with different m characterizes the
magnitude of nanostructure “non-spherical” shape.
867
V.A.Holovatsky et al.
Figure 2. Scheme of integration regions S0 and S1 for the case ρ0 < r0 < h.
3. Analysis of the results
The calculation of electron spectrum in cylindrical nanoheterosystem HgS/ZnS
is performed using the formulas obtained in the previous sections. The crystals pa-
rameters are given in table 1. The electron spectrum of the same system is calculated
in the model of the infinitely deep potential well for comparison. It is well known
that in this model the electron wave function is
Ψnnρm(ρ, φ, z) = AJm(χnρmρ/ρ0)f(z), (29)
where
f(z) =
√
1
h
cos
(nπz
2h
)
; n = 1, 3, 5,
√
1
h
sin
(nπz
2h
)
; n = 2, 4, 6.
(30)
Table 1. Material parameters
Material me a (Å) Ve (eV)
ZnS 0.28 5.41 –5.0
HgS 0.036 5.85 –3.1
868
Electron spectrum in confined cylindrical nanoheterosystem. . .
10 15 20 25
0,0
0,1
0,2
0,3
2h
ρ
0
h=ρ
0
4
3
2
1
ρ
0
/a
HgS
E
, e
V
Figure 3. Dependence of electron ground state energy on cylinder radius: curve 1
– electron in cylindrical potential well with finite depth, 2 – electron in sphere de-
scriben around the cylinder, 3 – electron in the sphere inscriben into the cylinder,
4 – electron in infinitely deep cylinder.
The energy spectrum is defined from the equation
Enρm =
~
2
2m0
(
π2n2
4h2
+
ξ2nρm
ρ20
)
, (31)
where ξnρm - the zeros of Bessels functions Jm(ξnρm).
From physical considerations it is clear that the energy levels from the electron
spectrum in a cylinder of finite sizes must be located between the corresponding
energy levels of the electron in the sphere inscribed into the cylinder and described
around it. Thus, the calculation and analysis of electron spectrum is performed for
both spheres.
The dependence of energy levels on cylinder sizes was obtained at the condition
ρ0 = h. The results are shown in figure 3 and figure 4.
From the figures one can see the following features. All energy levels in all models
are shifted to the bottom of the well when the quantum well is widening. At any well
sizes, all energy levels in the model under research are really located between the
respective levels of inscriben and describen spheres models. As far as the model of
infinitely deep potential well is concerned, figure 3 proves that it gives satisfactory
results for the ground level in the wells with a rather big size (ρ0 > 15aHgS). When
ρ0 < 15aHgS the unexactness of this model sharply increases. It is clear, because
869
V.A.Holovatsky et al.
10 15 20 25
0,0
0,1
0,2
0,3
0,4
0,5
m=1
m=0
ρ
0
/a
HgS
E
, e
V
Figure 4. Energy of the electron in state n = 1, l = 1 as a function of cylinder
radius: 1,2 – electron in cylindrical potential well with finite depth and quantum
number m = 0, 1, respectively; 3(4) – electron in the sphere describen around (in-
scriben into) the cylinder; 5,6 – electron in infinitely deep cylinder with quantum
number m = 0, 1, respectively.
when the height of the potential barrier is rather big (V0 = V ZnS
e − V HgS
e = 1, 9 eV,
see table 1) the electron has a small probability to tunnel into ZnS medium and,
thus, such states are well described by the model of infinitely deep potential well.
When cylinder sizes become smaller, the energy levels shift into the region of higher
energies and, thus, the probability of tunnelling into the external medium increases
because the barrier effective height decreases. Due to these, the model of infinitely
deep well is rough for the description of such states. Figure 4 proves that the excited
states with l = 1 are degenerated in both models with spheres. The degeneration
is partly taken off for the model under research and for infinitely deep potential
model. Herein it is clear that the latter one is extremely rough in the region of
quantum dots characteristic sizes (100 ÷ 150 Å). The reason for such a situation
was explained before. In the model under research both levels (n = 1, l = 1, m = 0
and n = 1, l = 1, m = ±1) are located between the corresponding degenerated
levels of inscriben and describen spheres and the magnitude of energy difference
between states with m = ±1 and m=0 characterizes the “non-spherical” shape of
the quantum dot.
870
Electron spectrum in confined cylindrical nanoheterosystem. . .
Finally, we must note that the presented method can be used for the approxi-
mated spectrum calculation for the quantum dots the geometric shape of which does
not permit to obtain the exact solutions.
References
1. Alferov Zh.I. History and future of semiconductor heterostructures. // Fiz. Tehn. Pol.,
1998, vol. 32, No. 1, p. 3–11 (in Russian).
2. Bimberg D. Quantum dots: paradigm changes in semiconductor physics. // Fiz. Tehn.
Pol., 1998, vol. 33, No. 9, p. 1044–1048 (in Russian).
3. Schoos D., Mews A., Weller H. Quantum dot quantum well CdS/HgS/CdS. // Phys.
Rev. B., 1994, vol. 49, No. 24, p. 17072–17078.
4. Ledentsov N.N., Ustinov V.M., Shchukin V.A., Kop’ev P.S., Alferov Zh.I., Bimberg D.
Quantum dot heterostructures: fabrication, properties, lasers. // Fiz. Tehn. Pol., 1998,
vol. 32, No. 4, p. 385–410 (in Russian).
Спектр електрона в обмеженій циліндричній
наногетеросистемі зi скінченною глибиною
потенціальної ями
В.А.Головацький, О.М.Войцехівська, М.Я.Міхальова,
М.М.Ткач
Чернівецький національний університет,
274012 Чернiвцi, вул. Коцюбинського, 2
Отримано 9 листопада 2000 р.
На основі варіаційного методу з варіаційним переметром у гаміль-
тоніані розраховано електронний спектр у циліндричній квантовій
точці HgS, розміщеній у середовищі ZnS. Встановлена залежність
енергетичного спектру, розрахованого в рамках моделі безмежно
глибокої потенціальної ями, задовільно працює лише для основного
рівня і при досить великих розмірах квантової ями.
Ключові слова: електрон, квантова точка, нанокристал
PACS: 79.60.jv
871
872
|