Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory
A theory of laser generation on size-quantization levels in semiconductor quantum dots put in a semiconductor matrix is developed. The size of quantum dots mass of which comprises the active sphere of an injection laser is determined by the new optical method consisting of the comparison of theoreti...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2004
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| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Цитувати: | Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory / S.I. Pokutnyi // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 247-250. — Бібліогр.: 28 назв. — англ. |
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irk-123456789-1191192017-06-05T03:02:47Z Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory Pokutnyi, S.I. A theory of laser generation on size-quantization levels in semiconductor quantum dots put in a semiconductor matrix is developed. The size of quantum dots mass of which comprises the active sphere of an injection laser is determined by the new optical method consisting of the comparison of theoretical and experimental dependence of the spectrum of an electron-hole pair upon the radius of a quantum dot. 2004 Article Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory / S.I. Pokutnyi // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 247-250. — Бібліогр.: 28 назв. — англ. 1560-8034 PACS: 73.20; 78.66 http://dspace.nbuv.gov.ua/handle/123456789/119119 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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A theory of laser generation on size-quantization levels in semiconductor quantum dots put in a semiconductor matrix is developed. The size of quantum dots mass of which comprises the active sphere of an injection laser is determined by the new optical method consisting of the comparison of theoretical and experimental dependence of the spectrum of an electron-hole pair upon the radius of a quantum dot. |
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Article |
| author |
Pokutnyi, S.I. |
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Pokutnyi, S.I. Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Pokutnyi, S.I. |
| author_sort |
Pokutnyi, S.I. |
| title |
Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory |
| title_short |
Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory |
| title_full |
Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory |
| title_fullStr |
Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory |
| title_full_unstemmed |
Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory |
| title_sort |
spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/119119 |
| citation_txt |
Spectrum of electron-hole pair in quantum dots of a semiconductor nanolaser: theory / S.I. Pokutnyi // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 247-250. — Бібліогр.: 28 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT pokutnyisi spectrumofelectronholepairinquantumdotsofasemiconductornanolasertheory |
| first_indexed |
2025-07-08T15:15:22Z |
| last_indexed |
2025-07-08T15:15:22Z |
| _version_ |
1837092280031248384 |
| fulltext |
247© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 3. P. 247-250.
PACS: 73.20; 78.66
Spectrum of electron-hole pair in quantum dots
of a semiconductor nanolaser: theory
S.I. Pokutnyi
Illichivsk Educational Research Center of the Odessa National University:
17 A Danchenko str., Illichivsk, Odessa red., 68001, Ukraine;
E-mail: univer@ivt.ilyichevsk.odessa.ua, Phone / fax: + 380 (4868) 4 30 76
Abstract. A theory of laser generation on size-quantization levels in semiconductor quantum
dots put in a semiconductor matrix is developed. The size of quantum dots mass of which
comprises the active sphere of an injection laser is determined by the new optical method
consisting of the comparison of theoretical and experimental dependence of the spectrum of
an electron-hole pair upon the radius of a quantum dot.
Keywords: energy spectrum, electron-hole pair, quantum dots; semiconductor nanolaser.
Paper received 26.03.04; accepted for publication 21.10.04.
1. Introduction
Quasi-zero-dimensional structures consisting of spherical
semiconductor microcrystals with a radius of à ~ 1�102 nm
(the so-called quantum dots (QDs)), grown in transparent
semiconductor (dielectric) matrices [1�12], draw atten-
tion due to their non-linear optical properties and appli-
cations in optoelectronics (in particular, as an active range
of injection semiconductor lasers [10, 13�17]).
Optical and non-linear optical properties of such
multiphase systems are determined by the energy spec-
trum of space-limited electron-hole pairs (excitons) [1�
12, 18�21]. The energy spectrum of charge carriers in
QD of sizes à ~ 1�102 nm will be absolutely discrete [22,
23]. This property of QD is used for creating optical
nanolasers and other devices with high temperature sta-
bility of frequency generation [10, 14, 15]. The QDs di-
mensions a must be in the range of several nm for appea-
ring energy gaps between the levels of electrons and ho-
les Åe(h) to be of the order of several kT0(∆Åe(h) ~ kT0,
where k is the Bolzmann constant) and T0 is the room
temperature.
In experimental paper [14], optical properties of ver-
tically coupled InAs QDs in GaAs matrix and connected
with them instrumental characteristics of injection laser
with an active range in terms of the mass QD were stud-
ied. A strong short-wave shift of the laser generation line
of the mass QD was observed at that time.
In this present paper, the theory of size-quantization
of laser generation levels in semiconductor QDs placed
into a semiconductor matrix is developed. The expres-
sion for the Hamiltonian of an electron-hole pair in QD,
which contains both the energy of the Coulomb interac-
tion between the electron and hole as well as the energy
of interaction between electrons and holes with the field
of induced polarization on a spherical interface between
two media has been obtained.
The value of the quantum dots mass of which com-
prises the active sphere of the injection laser is determined
by the new optical method [19-23] consisting of compari-
son of theoretical and experimental [14, 15] dependence
of the spectrum of electron-hole pair on QD radius a.
2. Hamiltonian of an electron-hole pair
in a quantum dot
We shall consider a simple model of a quasi-zero-dimen-
sional system: of a neutral spherical semiconductor QD
of radius a with a dielectric function ε2 surrounded by a
medium dielectric function of which is ε1. An electron e
and a hole h with the effective masses me and mh, respec-
tively, move in this QD (re and rh are the distances of the
electron and hole from the center of QD). It is also as-
sumed that the electron and hole energy bands are para-
bolic.
248
SQO, 7(3), 2004
S.I. Pokutnyi: Spectrum of electron-hole pair in quantum dots of ...
The characteristic dimensions of the problem are à,
àå, àh, àåõ, where
ae = ε2 h
2 / mee2 , ah = ε2 h
2 / mhe2,
aex = ε2 h
2 / µe2, (1)
are the Bohr radii of an electron, a hole and an exciton in
a semiconductor with the dielectric function ε2 (e is the
electron charge, µ = memh / (me + mh) is the reduced effec-
tive mass of the exciton). The fact that all the character-
istic dimensions of the problem
à, àå, àh, àåõ » à0 (2)
are much greater than the interatomic distance à0
1), al-
lows us to consider the motion of an electron and hole in
QD in the effective mass approximation.
The conditions of charge-carrier localization near a
spherical dielectric particle were analyzed in Ref. [18,
23, 25], where the problem concerning the field induced
by the charge near the dielectric particle, put in a dielec-
tric medium, is solved and the analytic expressions for
potential energy of interaction of charge carriers with
the spherical interface between these two media are given.
In particular, in Refs [18, 23, 25] shown is that the
potential at the observation point r′ in the medium ε¼ in-
duced by a charge å at the point r in the medium εi can be
represented as the sum of potentials provided by the point
charge of the image å′(rij /r) at the point (rij = (a / r)2rδij +
+ r (1 � δij)) and the linear distribution, with the density
ρij (y, r), of the image charge along the straight line
which goes through the sphere center and the charge at
the point r:
.
))/((
),(1
)(
)/(
0)(
),/,(
0
∫
∞
′−′
+
+
−′
′
+
−′
=′
rryr
rydy
rr
rre
rr
e
irjr
ij
j
ijj
ij
j
ρ
ε
εε
ϕ
(3)
The first term in (3) determines the field, created by
the charge.
In case, when the charge e is situated in the point r in
the bulk QD with ε2, and the observation point r′ is also
situated in the bulk QD (i = j = 2), quantities ε′ (r22 | r), r22
and ρ22(y, r) are determined in the following way:
e
r
a
rre β=′ )/( 22 , r
r
a
r
=
2
2
22 ,
−Θ
−=
r
a
y
a
e
ry
a
ry
22
22 )1(),(
α
αβρ (4)
12
12
εε
εε
β
+
−
= ,
12
1
εε
ε
α
+
=
where Θ(õ) is the conventional step function.
Using the adopted model of quasi-zero-dimensional
system, we can write down the potential energy of inter-
action of the electron e and hole h (with the charge (�å)),
in the bulk QD at the points re and rh, with the field in-
duced by these quasiparticles polarization in the follow-
ing way:
)2,|2,(
2
)2,|2,(
2
),,( ehhehe rr
e
rr
e
arrV ϕϕ −= (5)
where potentials ϕ(re, 2 | rh, 2) and ϕ(rh, 2 | re, 2) created
by the hole being at the point rh, at the point re of the
position of the electron and by the electron being at the
point re, at the point rh of the position of the hole due to
Eqs (3) and (4) becomes:
,
)/(
))/(()/(
]1cos)/(2)/[(
1
)(
)2,|2,(
0
22
12
2/1222
2
2
∫
∞
−
−Θ
+
−
−
+−
−
−
−
−=
hhe
h
a
h
hehe
eh
he
rryr
rayyrady
a
e
arrarra
e
rr
e
rr
εε
β
θε
β
ε
ϕ
(6)
,
)/(
))/(()/(
]1cos)/(2)/[(
1
)(
)2,|2,(
0
22
12
2/1222
2
2
∫
∞
−
−Θ
+
−
−
+−
−
−
−
−=
eeh
e
a
e
hehe
eh
eh
rryr
rayyrady
a
e
arrarra
e
rr
e
rr
εε
β
θε
β
ε
ϕ
where the corner Θ = re rh.
Using Eqs (6), the energy potential interaction V (re,
rh, a) (5), becomes:
V(re, rh, a) = Veh(re, rh) + U(re, rh, a) , (7)
where
||
),(
2
2
eh
heeh
rr
e
rrV
−
−=
ε
� (8)
is the energy of Coulomb interaction being an electron
and a hole, and
.
)/(
))/(()/(
)(2
)/(
))/(()/(
)(2
]1cos)/(2)/[(
1
),,(
0
22
12
2
0
22
12
2
2/1222
2
2
∫
∫
∞
∞
−
−Θ
+
−
−
−
−Θ
+
−
−
+−
−=
eeh
e
a
e
hhe
h
a
h
hehe
he
rryr
rayyrady
a
e
rryr
rayyrady
a
e
arrarra
e
arrU
εε
β
εε
β
θε
β
Is the energy of interaction of an electron and a hole
with the induced polarization field at the spherical inter-
face between these two media.
(9)
S.I. Pokutnyi: Spectrum of electron-hole pair in quantum dots of ...
249SQO, 7(3), 2004
The Hamiltonian of an exciton obtained subject to
the above assumption has the following form in the
adopted model of this quasi-zero-dimensional system
,),,(),(
22
22
gheheeh
h
h
e
e
EarrUrrV
mm
H
+++
+∆−∆−=
hh
(10)
where the first two terms describe the kinetic energy of an
electron and a hole, Eg is the band gap in an infinite
semiconductor with the dielectric function ε2, and en-
ergy potential interaction Veh(re, rh), U(re, rh, a) are de-
scribed by Eqs (8) and (9).
3. Short-wave length shift of the line of laser
generation in the semiconductor quantum dots
Let's consider the results of the experiment [14]. In this
paper, optical properties of the masses of vertically
aligned pyramidal QD of GaAs (with the dielectric func-
tion ε2 = 12,5) with an average dimension à ≅ 5 nm in the
matrix GaAs (ε1 = 12) are studied. By an average dimen-
sion of a of this QD we mean an average radius a between
the sphere described around QD and the sphere inscribed
in the bulk of QD. A strong short-wave length shift of the
line of laser generation ∆E = 106 meV of the mass QD by
the temperature T = 77 K was observed.
The mass QD under study in [14] was fired by the T =
= 973 K during the period not exceeding 60 min. Short-
wave shift of the laser generation line the authors [14]
related with the reduction in the energy of localization of
charge carriers in the volume QD which was caused by
their annealing.
Since in this [14] quazi-zero-dimensional structure the
dielectric function of QD and the surrounding matrix
are assumed to differ slightly (the parameter β = 2⋅10�2
(4)), in the Hamiltonian (10) in a first approximation the
quantity of the energy of the polarization interaction of
an electron and a hole U(re, rh, a) (9) may be neglected.
As a result, the Hamiltonian (10) becomes:
,),(
22
22
gheehh
h
e
e
ErrV
mm
H ++∆−∆−=
hh
(11)
Effective masses of an electron and a hole in QD of
GaAs had respectively equals (me / m0) = 2.8⋅10�2 and
(mh / m0) = 3.3⋅10�1 [26] (me / mh) = 8.5⋅10�2, i.e.
me << mh).
It's assumed that as in [27, 28] during the annealing
of the volume QD by the T = 973 K heat ejection of a
light-weight electron occurs and the bulk of QD contains
only a hole. In the Hamiltonian (11), we shall neglect the
energy of the Coulomb interaction Veh (8) of the hole with
the electron that can be localized at a deep trap in the
matrix of GaAs. This is justified provided the distance d
between the trap and the center of the QD is large as
compared with the radius of QD a, i.e. if d >> à.
As a result, the effect of the spherical boundary QD of
the radius a on the spectrum of the hole Ånl (a) (n, l � the
principal and orbital quantum numbers) cause the shift
of all levels of size quantization of the hole:
2
2
2
2
)( nl
h
nl
am
aE ϕh
∆ (12)
where ∆ϕnl are the roots of the equation for Bessel func-
tions Jl + 1/2(ϕnl) = 0. Eq. (12) was obtained assuming that
the minimal value of hole energy is located at the center
of QD Brillouin zone, the top of the valence band is taken
as a zero.
QD dispersion to dimensions being neglected, let's
assume that short-wave shift of the line of laser genera-
tion ∆E = 106 meV of the mass QD, which is due to the
size quantization of the hole, can be written as follows
(12):
2
102
2
2
ϕ
am
E
h
h
=∆ (13)
where ϕ10 = π. Eq. (13) allows to determine an average
dimension of QD à ~ 3.3 nm studied under the conditions
of the experiments [14].
4. Conclusions
The Hamiltonian of an electron-hole pair (11) in QD of
GaAs and quantities ∆E (12), (13) were obtained assum-
ing that electron and hole bands are parabolic. Moreo-
ver, Eq. (12), (13) were found assuming that QD of GaAs
is spherical, symmetric potential well of an infinite depth
for the hole. Such approximations are justified only for
the lowest level of the hole (n, l) in QD for which the
condition can be satisfied ∆En, l (à) << V0 (where V0 is
the depth of the potential well for holes in QD).
Thus, using the new optical method based on the com-
parison of theoretical and experimental dependencies of
the energy spectrum ∆En, l (à) of charge carriers on QD
radius a gives the possibility to determine the size of quan-
tum dot, mass of which comprises the active sphere of
injection laser.
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