Polaron in cylindrical and spherical quantum dots
Polaron states in cylindrical and spherical quantum dots with parabolic confinement potentials are investigated applying the Feynman variational principle. It is observed that for both kinds of quantum dots the polaron energy and mass increase with the increase of Frohlich electron-phonon coupling c...
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| Цитувати: | Polaron in cylindrical and spherical quantum dots / L.C. Fai, V. Teboul, A. Monteil, I. Nsangou, S. Maabou // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 157-166. — Бібліогр.: 38 назв. — англ. |
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irk-123456789-1188862017-06-02T03:03:02Z Polaron in cylindrical and spherical quantum dots Fai, L.C. Teboul, V. Monteil, A. Nsangou, I. Maabou, S. Polaron states in cylindrical and spherical quantum dots with parabolic confinement potentials are investigated applying the Feynman variational principle. It is observed that for both kinds of quantum dots the polaron energy and mass increase with the increase of Frohlich electron-phonon coupling constant and confinement frequency. In the case of a spherical quantum dot, the polaron energy for the strong coupling is found to be greater than that of a cylindrical quantum dot. The energy and mass are found to be monotonically increasing functions of the coupling constant and the confinement frequency. Досліджуються стани полярона в циліндричних і квантових крапках з параболічними обмежуючими потенціалами, використовуючи варіаційний принцип Фейнмана. Знайдено, що для обох типів квантових крапок енергія і маса полярона зростає з ростом постійної Фрьоліха електрон-фононного зв’язку і обмежуючої частоти. Показано, що у випадку сферичної квантової крапки енергія полярона для сильного зв’язку є більшою, ніж у випадку циліндричної квантової крапки. Знайдено, що енергія і маса є монотонно зростаючими функціями постійної зв’язку і обмежуючої частоти. 2004 Article Polaron in cylindrical and spherical quantum dots / L.C. Fai, V. Teboul, A. Monteil, I. Nsangou, S. Maabou // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 157-166. — Бібліогр.: 38 назв. — англ. 1607-324X PACS: 78.67.-n, 78.67.Hc, 71.38.-k DOI:10.5488/CMP.7.1.157 http://dspace.nbuv.gov.ua/handle/123456789/118886 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Polaron states in cylindrical and spherical quantum dots with parabolic confinement potentials are investigated applying the Feynman variational principle. It is observed that for both kinds of quantum dots the polaron energy and mass increase with the increase of Frohlich electron-phonon coupling constant and confinement frequency. In the case of a spherical quantum dot, the polaron energy for the strong coupling is found to be greater than that of a cylindrical quantum dot. The energy and mass are found to be monotonically increasing functions of the coupling constant and the confinement frequency. |
| format |
Article |
| author |
Fai, L.C. Teboul, V. Monteil, A. Nsangou, I. Maabou, S. |
| spellingShingle |
Fai, L.C. Teboul, V. Monteil, A. Nsangou, I. Maabou, S. Polaron in cylindrical and spherical quantum dots Condensed Matter Physics |
| author_facet |
Fai, L.C. Teboul, V. Monteil, A. Nsangou, I. Maabou, S. |
| author_sort |
Fai, L.C. |
| title |
Polaron in cylindrical and spherical quantum dots |
| title_short |
Polaron in cylindrical and spherical quantum dots |
| title_full |
Polaron in cylindrical and spherical quantum dots |
| title_fullStr |
Polaron in cylindrical and spherical quantum dots |
| title_full_unstemmed |
Polaron in cylindrical and spherical quantum dots |
| title_sort |
polaron in cylindrical and spherical quantum dots |
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Інститут фізики конденсованих систем НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/118886 |
| citation_txt |
Polaron in cylindrical and spherical quantum dots / L.C. Fai, V. Teboul, A. Monteil, I. Nsangou, S. Maabou // Condensed Matter Physics. — 2004. — Т. 7, № 1(37). — С. 157-166. — Бібліогр.: 38 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT failc polaronincylindricalandsphericalquantumdots AT teboulv polaronincylindricalandsphericalquantumdots AT monteila polaronincylindricalandsphericalquantumdots AT nsangoui polaronincylindricalandsphericalquantumdots AT maabous polaronincylindricalandsphericalquantumdots |
| first_indexed |
2025-07-08T14:50:16Z |
| last_indexed |
2025-07-08T14:50:16Z |
| _version_ |
1837090699741233152 |
| fulltext |
Condensed Matter Physics, 2004, Vol. 7, No. 1(37), pp. 157–166
Polaron in cylindrical and spherical
quantum dots
L.C.Fai 1 , V.Teboul 2 , A.Monteil 2 , I.Nsangou 1 , S.Maabou 2
1 Department of Physics, Faculty of Science, University of Dschang,
Cameroon
2 Laboratoire POMA, UMR CNRS 6136,Université d’Angers, 2 Bd Lavoisier,
49045 Angers, France
Received November 30, 2003, in final form January 21, 2004
Polaron states in cylindrical and spherical quantum dots with parabolic con-
finement potentials are investigated applying the Feynman variational prin-
ciple. It is observed that for both kinds of quantum dots the polaron energy
and mass increase with the increase of Fröhlich electron-phonon coupling
constant and confinement frequency. In the case of a spherical quantum
dot, the polaron energy for the strong coupling is found to be greater than
that of a cylindrical quantum dot. The energy and mass are found to be
monotonically increasing functions of the coupling constant and the con-
finement frequency.
Key words: polaron, polaron energy, polaron mass, parabolic
confinement, Fröhlich electron-phonon coupling constant, quantum dot
PACS: 78.67.-n, 78.67.Hc, 71.38.-k
1. Introduction
Recent advances in the technology of fabrication of quasi-2D, -1D, -0D nanocrys-
tals have stimulated the theoreticians’ interest in formulating models describing
physical phenomena associated with nanocrystals [1–12]. These structures are at-
tractive both for scientific investigation and for the development of a new generation
of electronic devices.
Nowadays not only perfect planar multilayer structures but also cylindrical and
spherical structures are being intensively investigated [7–13]. Polaron energy is eval-
uated in [13–15] using perturbation theory, in [16] using the weak coupling approxi-
mation, in [17] using the dielectric continuum model and in [18] using the Feynman
variational principle.
The polaron concept was introduced by Landau [19] as the autolocalised state
of a charge carrier in a homogeneous polar medium. The quantum dot is one of the
simplest quantum confined systems. For the strong electron-phonon interaction an
c© L.C.Fai, V.Teboul, A.Monteil, I.Nsangou, S.Maabou 157
L.C.Fai et al.
electron localises at a small region. A polaron is a quasi particle that arises due to the
conduction electron (or hole) together with its self-induced polarization in an ionic
crystal or in a polar semiconductor [20]. To classify polarons, the Fröhlich electron-
phonon coupling constant value, α is weak-coupling if α < 1, strong-coupling if α > 7
and intermediate-coupling between these ranges. The majority of crystals are weak
or intermediate-coupling polarons. Strong coupling is not attained even in strong
ionic crystals such as alkaline halides. The polaron character is well pronounced
only for strong coupling [21].
It is possible to reduce the lower bound of the electron-phonon coupling con-
stant’s threshold value in nanocrystals to within weak or intermediate-coupling
range. When investigating the polaron in nanocrystals, we should consider both
the electron and the phonon confinements. The electron confinement is described in
[1,22–27].
The present investigation also arises from the recent advances in fabrication of
nanocrystal with strong ionic coupling [28]. The spectra of polar optical vibrations
in 0D and 1D structures are investigated in [29]. [30] investigated the bulk and
interface polarons in quantum wires and dots. Polaron in a spherical quantum dot is
studied in [7,31–33] and in a cylindrical quantum dot in [34,35]. It is known that for
decreasing dimensionality of a structure (3D → 2D → 1D → 0D) polaron effects are
enhanced, but the number of independent directions of charge transport decreases
[36]. Thus, quasi 1D present certain interest as structures with maximal polaron
effect under the condition of the existence of at least one charge transport direction.
Quantum dot systems attract much attention in electronics and optics [8,37].
In this paper the polaron states in spherical and cylindrical quantum dots are
investigated applying the Feynman variational principle. This results in the up-
per bound polaron ground state energy for an arbitrary Fröhlich electron-phonon-
coupling constant. The electronic confinement is selected for the spherical and cylin-
drical quantum dots in the form of a parabolic confinement potential. For the
parabolic confinement, rigid interface boundaries are absent and interface-like pho-
non modes are smoothly distributed in space rather than localized near a sharp
boundary. Then we examine the electron interaction only with 3D longitudinal po-
lar optical phonons (3D-phonon approximation). Interaction between electrons is
described by 3D coulomb potentials of corresponding symmetries and the Fröhlich
3D Hamiltonian is chosen to describe electron-phonon interaction. Consequently,
interface phonons may not be considered. This approach seems to be adequate since
the integral polaron effects resulting from the summation over all phonon modes ap-
pear to be weakly dependent on the details of the phonon spectra. In the phenomena
with integral phonon effects we resort to numerical results.
In [33,38] interface-type longitudinal polar optical phonons have no contribution
to polaron effects. In [7,30] bulk-type phonons play the dominant role in the polaron
energy shift.
158
Polaron in cylindrical and spherical quantum dots
2. Feynman variational principle
The Feynman variational principle is one of the most effective methods when in-
vestigating the polaron problem for arbitrary values of the electron-phonon-coupling
constant α. To evaluate the polaron energy and the effective mass, the Feynman
variational principle [27] is used:
lnZ > lnZF = lnZ0 − 〈S [~r ] − S0 [~r ]〉 , (2.1)
where the angle brackets in 2.1 denote averaging over electron paths and are defined
as follows:
〈F [~r ]〉 =
Sp
∫
D~r F [~r ] exp {S0 [~r ]}
Sp
∫
D~r exp {S0 [~r ]} ,
where D~r denotes path integration, Sp is the spur, S0[~r ] is the model action func-
tional with:
S0 [~r ] =
∫
L0dt;
then S [~r ] is the exact action functional of the exact system:
S [~r ] =
∫
Ldt,
where ~r is the radius vector, L and L0 are the Lagrangians of the exact and model
systems respectively. In 2.1, the quantity Z0 is the statistical sum of the model
system andZF is the Feynman statistical sum defined respectively as follows:
Z0 = Sp
∫
D~r exp {S0 [~r ]},
Z = Sp
∫
D~r exp {S [~r ]}. (2.2)
The functions S and S0 in 2.1 are obtained after the path integration over phonon
variables and coordinates of the “fictitious” particles, respectively.
The thermodynamic energy E of a system is expressed through Z by the formula:
E = − d
dλ
lnZ, λ =
1
T
, (2.3)
from where
E 6 EF , (2.3’)
where
EF = − d
dλ
lnZF .
Here T is the absolute temperature (in units of the Bolztman’s constant) and
EF is the Feynman energy. The lowest energy level of the system is obtained as a
limiting case of 2.3 as T → 0, i.e. λ → ∞. Thus, the upper bound ground state
energy may be obtained without solving any wave equation.
159
L.C.Fai et al.
3. Feynman polaron in a cylindrical quantum dot
Consider the motion of an electron in a cylindrical quantum dot with a symmetric
parabolic confinement potential. The Hamiltonian that describes the electron and
phonon subsystem including their interaction with lattice vibrations is:
Ĥ = Ĥe + Ĥph + Ĥe−ph , (3.1)
where Ĥe is the electron Hamiltonian in cylindrical coordinates:
Ĥe =
P̂ 2
⊥
2m⊥
+
P̂ 2
||
2m||
+
m⊥Ω2
⊥
2
ρ2 +
m||Ω
2
||
2
z2.
Here ρ2 = x2 + y2; P̂⊥ , m⊥ and P̂|| , m|| are components of the operator of the
momentum and electron band mass in the transversal and longitudinal directions
respectively; Ω is the frequency characterizing the parabolic confinement potential;
and
Ĥph =
∑
~q
~ ω~q b̂~q + b̂~q
Ĥe−ph =
∑
~q
[
γ~q b̂~q ei~q ~r + γ~q b̂~q e−i~q ~r
]
(3.2)
are respectively the phonon contribution and electron-phonon interaction Hamilto-
nians. Here γ~q is the amplitude of the electron-phonon interaction numbered by the
wave vector ~q :
γ~q =
[
απ
V
(
~ ωo
q
)2
Rp
]
1
2
, Rp =
(
~
2meωo
)
1
2
, q2 = q2
⊥ + q2
||
and ω0 is the non-dispersional phonon frequency.
Considering the Hamiltonian 3.1, the model lagrangian L0 for the system (con-
sidering the ground state) is selected in the one-oscillatory approximation:
L0 = −m⊥ρ̇
2
2~ 2
− m||ż
2
2~ 2
− m⊥Ω2
⊥
2
ρ2 −
m||Ω
2
||
2
z2 − M⊥Ṙ
2
2~ 2
− M||Ż
2
2~ 2
− κ⊥
2
(R− ρ)2 − κ||
2
(z − Z)2 ; κj = Mjωjf , j = ⊥, (3.3)
where Rand Z are the coordinates of the model particle and ρ with z are the co-
ordinates of the electron, respectively. The quantities M⊥,M||, κ⊥, and κ|| serve as
variational parameters. ωf is the elastic coupling frequency.
From the Lagrangian in 3.3, the transverse and longitudinal equations of motion
for the model system are independent. In 3.3 the time t = −i~ τ and ψ̇ = dψ/dτ .
The model Lagrangian in 3.3 simulates the interacting electron-phonon system.
From the equation of motion the eigen frequencies are obtained:
ω2
ij =
1
2
(
κj
mj
+ Ω2
j
)
− (−1)i 1
2
[
(
κj
mj
+ Ω2
j
)2
−
4κjΩ
2
j
mj
]
1
2
,
i = 1, 2, j = ⊥, || . (3.4)
160
Polaron in cylindrical and spherical quantum dots
Considering 2.1 and 2.3, the polaron dimensionless energy E is evaluated:
E =
2
∑
j=1
(
1
2
)j
[
Ωj
(
1
2
)−1
+
(Ωj − ω1j)
2 (Ωj − ω2j)
2
Ω2
j (ω1j + ω2j)
]
− α
∞
∫
0
F (A⊥, A||)e
−τdτ ,
j = 1 → ⊥, j = 2 → || . (3.5)
The dimensionless effective mass M is conveniently obtained:
M =
∑
j
(
Ω2
j − ω2
2j
) (
ω2
1j − Ω2
j
)
ω2
1jω
2
2j
, (3.6)
where
F (A⊥, A||) =
√
2
π
1
(
|A|| − A⊥|
)1/2
Ar sinh
(
A||
A⊥
− 1
)1/2
, A|| > A⊥ ,
arcsin
(
1 − A||
A⊥
)1/2
, A|| > A⊥ ,
Aj =
∑ aij
ωij
(
1 − e−ωijτ
)
, j = ⊥, || , a1 =
ω2
1 − ω2
f
ω2
1 − ω2
2
, a2 =
ω2
f − ω2
2
ω2
1 − ω2
2
. (3.7)
4. Feynman polaron in a spherical quantum dot
Consider the motion of an electron in a spherical quantum dot with a spherical
symmetric parabolic confinement potential. The model Lagrangian L0 of the system
is selected:
L0 = −m~̇r
2
2~ 2
− Mf
~̇R2
2~ 2
− mΩ2~r 2
2
−
Mfω
2
f
(
~R− ~r
)2
2
, (4.1)
where ~R is the coordinate of the fictitious particle. The quantities Mf and ωf are
the mass and the elastic coupling frequencies of the fictitious particle, respectively.
Both of them serve as variational parameters. From the equation of motion of the
model system there follow the eigen frequencies:
ω2
j =
1
2
κ
µ
+ Ω2 − (−1)i
√
(
κ
µ
+ Ω2
)2
− 4Ω2κ
µ
,
j = 1, 2, µ =
mM
m+M
, κ = Mfω
2
f . (4.2)
From 2.1 and 2.3, the Feynman polaron variational energy E is given by:
E =
3
2
Ω +
3
4
(ω1 − Ω)2 (Ω − ω2)
2
Ω2 (ω1 + ω2)
− α√
π
∫
e−τdτ
A1/2
, (4.3)
161
L.C.Fai et al.
where
A =
2
∑
j=1
aj
ωj
(
1 − e−ωjτ
)
.
The polaron effective mass M may be conveniently evaluated from the equation
of the eigen frequencies 4.2:
M =
(ω2
1 − Ω2) (Ω2 − ω2
2)
ω2
1ω
2
2
.
5. Conclusions
Figures 1 and 2 show the plot of the polaron energy (absolute value) versus
Fröhlich electron-phonon-coupling constant for the cylindrical and spherical quan-
tum dots, respectively. The plots slightly deviate from a linear relation. Here, the
polaron energy increases with the increase of the electron-phonon-coupling constant.
From the plots, the confinement strengthening enhances the electron-phonon inter-
action. The Polaron energy for the case of the spherical quantum dot is greater
than that of the cylindrical quantum dot. For example, if we consider α = 8.25
then for the cylindrical quantum dot we have the following values for the energy
Ec = 25.1072; 20.50333; 15.711. These values of the energy correspond, respective-
ly, to the confinement frequencies Ω = 7; 5; 3. For the case of the spherical quan-
tum dot for α = 8.25 we haveEs = 25.29462; 20.66347; 15.84761 that correspond,
respectively, to the following confinement frequencies Ω = 7; 5; 3. We have also ex-
amined the polaron energies at the confinement frequency Ω = 10.3. For the case
of the cylindrical quantum dot we have Ec = 29.75767; 25.58428; 21.48544 that cor-
respond, respectively, to α = 7; 5; 3. For the case of the spherical quantum dot we
have Es = 29.90701; 25.64913; 21.50619 that correspond, respectively, to α = 7; 5; 3.
This also shows that the polaron energy for the case of the spherical quantum dot is
greater than that of the cylindrical quantum dot. This confirms the results in [36].
In figures 3 and 4 the mass increases with the increase of the electron phonon
coupling constant. This coincides with the results obtained in [30]. Figures 3 and
4 also show that though the graphs of the mass are lifted in the same fashion, the
one for the cylindrical dot is greater in numerical value than that of the spherical
quantum dot. From figures 1 to 4 it is observed that the polaron energy and mass
are monotonically increasing functions of the coupling constant. It is observed that
regions of weak and intermediate polarons overlap with those of strong coupling
polarons (i.e., the regions of strong coupling polarons are shifted to those of weak
and intermediate polarons).
162
Polaron in cylindrical and spherical quantum dots
Figure - 1 Plot of the polaron energy versus Fröhlich
electron-phonon constant (cylindical dot)
0
5
10
15
20
25
30
0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25
αααα
Ec
Ω=7
Ω=5
Ω=3
Ω=1.5
Ω=0.5
Figure 1. Plot of the polaron energy versus Fröhlich electron-phonon constant
(cylindrical dot)
Figure - 2 Plot of the polaron energy versus Fröhlich electron-
phonon constant (spherical dot)
0
5
10
15
20
25
30
0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25
αααα
Es Ω=7
Ω=5
Ω=3
Ω=1.5
Ω=0.5
Figure 2. Plot of the polaron energy versus Fröhlich electron-phonon constant
(spherical dot)
163
L.C.Fai et al.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25
aaaa
Mc
W=7
W=5
W=3
W=1.5
W=0.5
Figure 3. Plot of the polaron mass versus Fröhlich electron-phonon constant
(cylindrical dot)
Figure - 4 Plot of the polaron mass versus Fröhlich electron-
phonon constant (spherical dot)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25
αααα
Ms
Ω=7
Ω=5
Ω=3
Ω=1.5
Ω=0.5
Figure 4. Plot of the polaron mass versus Fröhlich electron-phonon constant
(spherical dot)
164
Polaron in cylindrical and spherical quantum dots
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Полярон в циліндричних і сферичних квантових
крапках
Л.С.Фаі 1 , В.Тебул 2 , А Монтейл 2 , І.Нсангу 1 , С.Маабу 2
1 Кафедра фізики, факультет природничих наук, університет
м. Джанг, Камерун
2 Університет м. Анже, вул.Лавуаз’є, 2, 4945 Анже, Франція
Отримано 30 листопада 2003 р., в остаточному вигляді –
21 січня 2004 р.
Досліджуються стани полярона в циліндричних і квантових крапках з
параболічними обмежуючими потенціалами, використовуючи варіа-
ційний принцип Фейнмана. Знайдено, що для обох типів квантових
крапок енергія і маса полярона зростає з ростом постійної Фрьолі-
ха електрон-фононного зв’язку і обмежуючої частоти. Показано, що
у випадку сферичної квантової крапки енергія полярона для сильно-
го зв’язку є більшою, ніж у випадку циліндричної квантової крапки.
Знайдено, що енергія і маса є монотонно зростаючими функціями
постійної зв’язку і обмежуючої частоти.
Ключові слова: полярон, енергія полярона, параболічне
обмеження, постійна Фрьоліха електрон-фононної взаємодії
PACS: 78.67.-n, 78.67.Hc, 71.38.-k
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