Excitons and trions in spherical semiconductor quantum dots
An original iterative procedure has been developed to obtain energy spectrum of neutral and charged excitons (positive and negative trions) in spherical semiconductor quantum dots (QD) imbedded into a dielectric material. Numerical calculations are made using the effective mass approximation and Har...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2006
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irk-123456789-1215782017-06-15T03:04:25Z Excitons and trions in spherical semiconductor quantum dots Kupchak, I.M. Kryuchenko, Yu.V. Korbutyak, D.V. An original iterative procedure has been developed to obtain energy spectrum of neutral and charged excitons (positive and negative trions) in spherical semiconductor quantum dots (QD) imbedded into a dielectric material. Numerical calculations are made using the effective mass approximation and Hartree-Fock method. A combined effect of heterointerface polarization (image force potential) and finite band-off-sets on the energy spectrum of excitons and trions in QDs is considered for the first time. It is shown that binding energies of excitons and trions in such QDs can be substantially larger than those in bulk semiconductors due to spatial and “dielectric” confinement effects. 2006 Article Excitons and trions in spherical semiconductor quantum dots / I.M. Kupchak, Yu.V. Kryuchenko, D.V. Korbutyak // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 8, № 1. — С. 1-8. — Бібліогр.: 32 назв. — англ. 1560-8034 PACS 71.35.Pq, 71.35.-y http://dspace.nbuv.gov.ua/handle/123456789/121578 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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An original iterative procedure has been developed to obtain energy spectrum of neutral and charged excitons (positive and negative trions) in spherical semiconductor quantum dots (QD) imbedded into a dielectric material. Numerical calculations are made using the effective mass approximation and Hartree-Fock method. A combined effect of heterointerface polarization (image force potential) and finite band-off-sets on the energy spectrum of excitons and trions in QDs is considered for the first time. It is shown that binding energies of excitons and trions in such QDs can be substantially larger than those in bulk semiconductors due to spatial and “dielectric” confinement effects. |
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Kupchak, I.M. Kryuchenko, Yu.V. Korbutyak, D.V. |
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Kupchak, I.M. Kryuchenko, Yu.V. Korbutyak, D.V. Excitons and trions in spherical semiconductor quantum dots Semiconductor Physics Quantum Electronics & Optoelectronics |
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Kupchak, I.M. Kryuchenko, Yu.V. Korbutyak, D.V. |
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Kupchak, I.M. |
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Excitons and trions in spherical semiconductor quantum dots |
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Excitons and trions in spherical semiconductor quantum dots |
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Excitons and trions in spherical semiconductor quantum dots |
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Excitons and trions in spherical semiconductor quantum dots |
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Excitons and trions in spherical semiconductor quantum dots |
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excitons and trions in spherical semiconductor quantum dots |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2006 |
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http://dspace.nbuv.gov.ua/handle/123456789/121578 |
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Excitons and trions in spherical semiconductor quantum dots / I.M. Kupchak, Yu.V. Kryuchenko, D.V. Korbutyak // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 8, № 1. — С. 1-8. — Бібліогр.: 32 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT kupchakim excitonsandtrionsinsphericalsemiconductorquantumdots AT kryuchenkoyuv excitonsandtrionsinsphericalsemiconductorquantumdots AT korbutyakdv excitonsandtrionsinsphericalsemiconductorquantumdots |
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2025-07-08T20:09:17Z |
| last_indexed |
2025-07-08T20:09:17Z |
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1837110771862994944 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 1-8.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
1
PACS 71.35.Pq, 71.35.-y
Excitons and trions in spherical semiconductor quantum dots
I.M. Kupchak, Yu.V. Kryuchenko, D.V. Korbutyak
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
45, prospect Nauky, 03028 Kyiv, Ukraine
Phone: +38 (044) 525 6391, e-mail: div47@isp.kiev.ua
Abstract. An original iterative procedure has been developed to obtain energy spectrum
of neutral and charged excitons (positive and negative trions) in spherical semiconductor
quantum dots (QD) imbedded into a dielectric material. Numerical calculations are made
using the effective mass approximation and Hartree-Fock method. A combined effect of
heterointerface polarization (image force potential) and finite band-off-sets on the energy
spectrum of excitons and trions in QDs is considered for the first time. It is shown that
binding energies of excitons and trions in such QDs can be substantially larger than those
in bulk semiconductors due to spatial and “dielectric” confinement effects.
Keywords: semiconductor quantum dot, quantum size effect, image force potential,
excitons, trions, energy spectrum, binding energy.
Manuscript received 30.11.05; accepted for publication 15.12.05.
1. Introduction
The neutral or charged excitons (trions) are quasi-
particles composed of two or three interacting charge
carriers, e.g. of an electron and a hole (neutral exciton),
of two electrons and a hole (a negative trion Х-) or of
two holes and one electron (a positive trion Х+). In
semiconductors, the existence of three-particle
complexes bound by the Coulomb interaction was
predicted in 1958 by Lampert [1]. As was shown in that
work, in bulk semiconductors these charged electron-
hole states are energetically favourable as compared to a
three-particle state "neutral exciton plus free charge
carrier". However, their binding energy turned out to be
very small (usually, less than 1 meV) and for this reason
the experimental investigation of these states in bulk
semiconductors is substantially complicated. Never-
theless, the trions were observed for example in bulk
silicon [2] and germanium [3]. Very interesting is the
question about exciton and trion states in semiconductor
systems with spatial confinement when partial or total
quantization of electron and/or hole energy spectra
occurs. As was shown in a number of experimental and
theoretical works on quantum wells and quantum wires
(e.g., see [4-19], binding energies of exciton and trion
states increase substantially in such structures. For
example, in accordance with theoretical calculations in
the case of semiconductor quantum wells [20] the trion
binding energy should be an order of magnitude greater
than that in bulk semiconductors. In reality, this fact has
been confirmed experimentally [21]. Spatially confined
trions were observed in the structures with QDs as well.
Of special interest are the investigations of photo-
luminescence spectra of single QDs [22, 23] where lines
caused by trion transitions are clearly revealed.
In addition to spatial confinement in many
semiconductor nanostructures the effect of dielectric
confinement due to large dielectric mismatch with
environment plays very important role, too [see, e.g. 24-
27]. Unfortunately, in a majority of previous works this
effect has been considered theoretically only in the
approximation of infinitely high barriers, when wave
functions of electron and hole confined states turn to
zero at the interfaces. It is clear that with a decrease of
characteristic size of nanocrystals such approximation
becomes too bad for the finite band-off-sets in real
semiconductor-dielectric nanostructures.
In this work, we consider theoretically the exciton
and trion states that can be formed and radiate in
semiconductor QDs located in a dielectric matrix. Both
finite band-off-sets and dielectric confinement effects
are taken into account as in our previous works [9, 28]
on quantum wells and quantum wires. We exploit here a
simplified solid state model based on the effective mass
approximation. Moreover, only heavy-hole subband
forming the lowest in energy confined hole states in QDs
has been taken into account in our calculations.
Additional effects which can arise due to subband
mixing, anisotropy, magnetic and electric fields, etc. will
be considered in future.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 1-8.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
2
,
,
If an electron and a hole move simultaneously in QD,
then the energy of the ground radiative electron-hole
(exciton) transition in such a QD can be written as
follows:
X g e h XE E E E S= + + − , (1)
where Eg is the bandgap width of bulk material
containing QDs, Ee and Eh are the lowest single-particle
confinement energies in these QDs, respectively, SX is
the exciton binding energy caused by the Coulomb
interaction between an electron and a hole in QDs.
If three carriers (two electrons and a hole or two
holes and an electron) move simultaneously in a QD,
then the energy of the ground radiative electron-hole
transition in such a system may be written as
XX X
E E S− −= − (2)
for a negative trion case and
XX X
E E S+ += − (3)
for a positive trion case. In Eqs (2) and (3) the values
−
XS − and − XS + are the energy shifts of the ground
radiative trion transitions with respect to the energy of
the ground exciton transition. If these values are
negative, the corresponding trion states are energetically
favourable as compared to the states of a three-particle
system in the form of “free” exciton and “free” charge
carrier. In other words, the values − −X
S and − +X
S are
the trion “binding” energy with respect to the virtual
state of non-interacting exciton and charge carrier.
2. Model and results
The trion states, where the pair Coulomb interaction
between all the quasi-particles of which the trion is
composed plays a substantial role, in a first appro-
ximation may be built of single-particle states of a model
Hamiltonian for a spherically-symmetric semiconductor
QD in a dielectric matrix. In this case, the conduction
and valence band-off-sets at the "semiconductor QD –
dielectric matrix" heterointerface form rectangular
potential energy gaps both for electrons and holes inside
the QD region. The corresponding coordinate
dependences of the electron and hole potential energies
Uc and Uv caused by the band-off-sets can be written as
( )
( )
0, if
( )
, ifc v
e h
r R
U r
U r R
<= >
, (4)
where R is the QD radius, Ue(h) are the band-off-set
values.
In the case when the QD material dielectric constant
ε1 differs from that of the ambient ε2, the polarization of
the heterointerface by a charged point particle forms an
additional image forces potential field. The potential
energy of a particle in this field (self-action energy) can
be expressed using the Green function G (x, x') of the
following Poisson equation for the point positive unit
charge:
)(
π4
),( xxxx ′−−=′∆ δ
ε
Gx , (5)
where ε = ε1 if r < R and ε = ε2 if r > R, х' is the
coordinate of a point unit charge that forms the
electrostatic potential G (x, x') at the point x. The
solution of this equation can be conveniently written in a
form of the following expansion in spherical harmonics
Ylm [29]:
)()(),( *
0
Ω′Ω=′ ∑ ∑
∞
= −=
lmlm
l
l
lm
l YYgG xx . (6)
In the case, when both the charge and the observation
point are located inside the QD, i.e., when both Rr <′
and Rr < then
( ) ( )
( )
( ) ( )
( )
( )
1
1
1 2
2 1
1 1 2
4,
2 1
14
2 1 1
l
l l
l
l
r
g r r
l r
l r r
l l l R
π
ε
ε επ
ε ε ε
<
+
>
> <
+
= +′
+
− +
+
+ + +
(7)
where the radial coordinate r> is the greatest of the r and
'r coordinates, while r< is the least one. In the case
when both the charge and the observation point are
located beyond the QD in the barrier region, i.e., when
both Rr >' and Rr > , the corresponding expression has
the form of
.
)()1(
)(
)12(
π4
)12(
π4
),(
1
12
21
12
2
1
2
+
<>
+
+
>
<
++
−
+
+
+
+
=′
l
l
l
l
l
rr
R
ll
l
l
r
r
l
rrg
εε
εε
ε
ε
(8)
Finally, when the charge and the observation point
are at different sides of the heterointerface
( ) ( )
( )
( )
( ) ( )
1
1 2
1
1 2 1 2
4π,
2 1
4π 1
2 1 1
l
l l
l
l
r
g r r
l r
r
l l l r
ε
ε ε
ε ε ε ε
<
+
>
<
+
>
= +′
+
−
+
+ + + +
(9)
where 1 2( ) / 2ε ε ε= + . Taking into account the well-
known expression
),(),(
12
1
π4
1 *
0
1
ϕθϕθ lmlm
l
l
lm
l
l
YY
r
r
l
′′
+
=
′− ∑ ∑
∞
= −=
+
>
<
xx
,
(10)
it is clear that the first term in the right-hand sides of Eqs
(7) - (9) corresponds to the usual Coulomb field of a
point positive unit charge in the medium with the
effective dielectric constant ε1, ε2, and 1 2( ) / 2ε ε ε= + ,
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 1-8.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
3
respectively. The second term in the right-hand sides of
Eqs (7)-(9) is the image forces field component that
arises due to the QD heterointerface polarization by the
point charge.
The self-action potential energy of the charge e
located at the point x is equal to 2/),()( 2 xxGerU s ′= ,
where the prime at the Green function indicates only the
part of this function that is associated with the second
term in Eqs (7) and (8), i.e. with the self-image field.
Thus, the potential energy of a charged particle in the
image force field induced by this particle itself (the self-
action potential energy) can be expressed by the
following formulas:
( ) ( )
22
1 2
1 1 20
1
2 1
l
s
l
e l rU r
R l l R
ε ε
ε ε ε
∞
=
− + = + +∑ , if r < R
(11)
and
( ) ( )
( )2 12
1 2
2 1 20
2 1
l
s
l
e l RU r
R l l r
ε ε
ε ε ε
∞ +
=
− = − + +∑ , if r > R.
(12)
The classical point charge electrostatic potential and,
hence, the self-action potential energy has a nonphysi-
cal diverging behavior near the heterointerface
2
1 2 1,2 1 2( ) ~ ( ) /(4 ( ))sU r e zε ε ε ε ε− + , where | |z R r= −
is the distance of the charged particle from the
heterointerface. However, in real systems a transient
layer between the QD and the barrier region always
exists, therefore, the electrostatic potential is in fact
continuous [30]. The continuity of the electrostatic
potential at the heterointerface and in the transient layer
also follows from the spatial dispersion account (i.e.
account of a nonlocal character of interaction between
the charge and the induced field). Rigorous calculations
of the self-action potential energy in the transient layer is
a rather difficult problem in this case. However, if we
are interested only in one-particle self-energy shifts, then
it is not so necessary to know the exact solution of this
difficult problem in the transient region due to the fact
that contributions to self-energy shifts from the internal
part of the thin transient layer (i.e., that belonging to the
QD region) and external part (belonging to the barrier
region) will practically compensate each other and will
give, as a result, only a small correction to the total
single-particle energy. Therefore, to a first
approximation, the exact total single-particle potential
energies Uc(v)+Us within the thin transient layer (of the
order of interatomic distance) around the heterointerface
can be simply approximated by a polynomial over the
distance z = (R− r). In particular, we used in our
calculations a cubic polynomial.
For example, Fig. 1 illustrates the energy diagram for
electrons and holes in the case of a Si-SiO2 QD of
1.5 nm in diameter. Thin solid lines correspond to the
ordinary potential energy behaviour in the absence of the
image forces field, while thick solid lines show the
potential energy gaps, when self-action is taken into
account. The self-action potential energy shift Us(r) at
the center of the QD takes the value
( )
2
1 2
1 2
0
2s
eU
R
ε ε
ε ε
−
= . (13)
The smaller is the QD, the greater is the shift of the
potential gap bottom caused by the QD heterointerface
polarization. Therefore, by finding the energy levels of
single-particle states in a semiconductor QD within a
dielectric matrix, it makes sense to account for this shift
already from the very beginning, i.e., calculate electron
(hole) confinement energies in rectangular spherical
potential gaps of the depth ( ) ( ) (0)e h e h sU U U= − rather
than Ue(h). Respectively, in the final expression for the
total energy of the electron-hole transition, these shifts
of the bottoms of the potential gaps for an electron and a
hole can be easily taken into account by adding simply
2Us(0). Thus, the total single-particle potential energies
in the Schroedinger equation can be written in the form
of ( ) ( ) ( )c v s U r U r+ , where ( ) ( ) 0c vU r = for r < R and
( ) ( )( )c v e hU r U= for r > R, while ( ) ( )s sU r U r= for
r > R and ( ) ( ) ( )0s s sU r U r U= − for r < R. With
respect to the bottoms of potential gaps changed by self-
action, the energy spectrum of single-particle states ,iE α
is determined by the single-particle Schroedinger
equations ,
ˆ ( ) ( )i i
i i i iH Eα α αΨ = Ψr r , where
2
( )
ˆ ∆ ( ) ( ) .
2 ( )i i c v i s i
i i
H U r U r
m r
= − + + (14)
Here, the index i serves to denote an electron (e) or a
hole (h), mi(ri) is the effective mass that takes the values
mi1 in the QD region and mi2 in the barrier region. The
wave functions i
αΨ of single-particle states localized in
Fig. 1. Schematic potential energy diagram for electrons
and holes in the case of Si-SiO2 quantum dot of 1.5 nm in
diameter. Dotted lines show the energies of confined
electron and hole s-states in the rectangular spherically-
symmetrical potential gaps with the bottoms shown by
dashed lines.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 1-8.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
4
such potential gaps can be found in the form of an
expansion in wave functions ( ),i k irϕ of a model
Hamiltonian describing a confined movement of the
particles in rectangular spherically-symmetrical potential
gaps ( ) ( )c vU r (the bottoms of these gaps are shown in
Fig. 1 by dashed lines):
( ) ( ), ,
i
i i k i k i
k
C
α
α ϕΨ = ∑r r . (15)
To abbreviate the notation, the index k is used here to
denote the standard set of quantum numbers {n, l, m} for
a spherically-symmetrical system, the quantum numbers
α = 0, 1, 2 ... characterize energy levels of single-particle
states in a QD when self-action potential energy (third
term in the right-hand side of expression (14)) is taken
into account, too.
The wave functions )(, iki rϕ have the following
form:
( )
, , ,
,
,
,
( ) ( ) ( )
( )
( ) ( ) ,
( )
i nlm i i nl i l i nl i
l i nl
i l i nl i lm i
l i nl
N R r j k r
j k R
r R k r Y
k R
ϕ θ
θ η
η
= − +
+ − Ω
r
(16)
where θ(x) is a step-function, Ni,nl is the normalization
coefficient, 1/ 2
, 1 ,[2 ] /i nl i i nlk m E= , , 2[2 (i nl i im Uη = −
1/ 2
, )] /i nlE− , jl and kl are the usual and modified
spherical Bessel functions. The corresponding single-
particle confinement energies ,i nlE in such rectangular
spherically-symmetrical potential gaps can be found
from the equation
( )
( )
( )
( )
1 1 2 1
1 2 2 1
0l i l ii i i i
i l i i l i i i
j k R k Rk R R m m
l
m j k R m k R m m
ηη
η
+ + −
− − = . (17)
The energy levels of electron and hole confined s-
states (i.e., with the quantum number l = 0) in Si-SiO2
QD of 1.5 nm in diameter are shown in Fig. 1 by dotted
lines.
By determining the coefficients ,i kCα and energies
,iE α as a result of expansion (15) substitution into Eq.
(14) and fulfilment of corresponding integrations, we
completely solve the problem of finding the basic set of
single-particle states i
αΨ in semiconductor QD inside
this dielectric matrix. These states will be used below for
a description of exciton and trion states in spherical QDs
in the presence of the image forces field. The energies Ei
(I = e,h) in the right-hand part of the expression (1) are
expressed therewith as
,0(0)i s iE U E= + . (18)
The Hamiltonian of a two-particle system, which
determines the energies of exciton transitions in such
QD, has the following form:
( )
( )
2 2
ˆ 2 0
2 ( ) 2 ( )
( ) ( ) ( ) ( ) ,
ˆ ˆ2 0 , ,
X g s e h
e e h h
s e s h c e v h eh e h
g s e h eh e h
H E U
m r m r
U r U r U r U r U
E U H H U
= + − ∆ − ∆ +
+ + + + + =
= + + + +
( )
r r
( ) r r
(19)
where the term Ueh(re,rh) = − e2 G (re,rh) allows for both
direct Coulomb interaction between an electron and a
hole and indirect (i.e., interaction of an electron with the
image of a hole and a hole with the image of an
electron).
We will seek the wave function of the ground state of
exciton in the form of a product of the "exciton" single-
particle electron and hole wave functions ( )X
e eΨ r and
( )X
h hΨ r :
( ) ( ) ( ), X X
X e h e e h hΦ = Ψ Ψr r r r , (20)
where ( )X
e eΨ r and ( )X
h hΨ r , in turn, can be
conveniently presented in the form of an expansion in
basal single-particle wave functions i
αΨ (15):
( ) ( )X i
i i i iBα
α
α
Ψ = Ψ∑r r . (21)
In this case, the Schroedinger equation for the ground
state of exciton in a QD can be written in the form of a
coupled equations for the electron and hole wave
functions X
iΨ :
( )
( )
21ˆ , ( ) ( )
2
e
X X
e eh e h h h h e e
X X
e e
H U d
E
+ Ψ Ψ =
= Ψ
∫ r r r r r
r
and (22)
( )
( )
21ˆ , ( ) ( )
2
h
X X
h eh e h e e e h e
X X
h
H U d
E
+ Ψ Ψ =
= Ψ
∫ r r r r r
re
The system of equations (22) is solved self-
consistently in the way analogous to Hartree-Fock
technique. As a first approximation by integrating in Eqs
(22), the wave functions (15) of electron and hole
ground states are taken: ( ) ( ),0
0
X i
i i iα=Ψ = Ψr r . Further,
in the basis of the functions ( )i
iαΨ r (i.e., using the
expansion (21)), we write Eq. (22) in a matrix form,
diagonalize the matrix obtained, and find new wave
functions of an electron and a hole ( ),1
,0
X
i iΨ r of the
type (21). They correspond to the lowest one-particle
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 1-8.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
5
energy states ,1
,0
X
iE at a first iteration (the upper index 1
means that these functions and energies are obtained as a
result of a first iteration, the lower index 0 that they
correspond to the smallest eigenvalue of the
diagonalized matrix, i.e., to the ground state). With these
new wave-functions, we repeat integrations in (22),
build new matrices and so on. The iterative procedure is
continued until the relation of the energy increment
obtained in the course of the last iteration to the total
value of the energy obtained up to this moment becomes
smaller than a preset accuracy. As a rule, to have the
accuracy of the order of 10−6, three or four iterations are
enough. As a result, we obtain "exciton" single-particle
energies
e
XE and
h
XE in the exciton ground state. The
energy of the system transition from a non-excited state
into the exciton ground state (or vice versa) is expressed
in this case as EX 2 (0)
e h
X X
g sE U E E= + + + (or
e h
X X
X gE E E E= + + , if similarly to (18) we set
(0)
i
X X
s iE U E= + ). By calculating the "exciton" single-
particle energies
i
XE and real single-particle energies Ei
(18) as well as comparing the above expression for the
exciton transition energy with the expression (1), we are
able thus to determine the binding energy SX of an
exciton in a QD with finite band-off-sets and a polarized
heterointerface:
( )X X
X e h e hS E E E E= + − + . (23)
Fig. 2 shows calculated dependences of the exciton
transition energy (a) and exciton binding energy (b) on
the QD diameter for the cases of silicon QDs and those
of some AIIBVI materials located in the SiO2 matrix.
Within the considered range of QD diameters, the
behaviour of the curves in Fig. 2a can be described as
n
X gE E Ad −= + , where d is the QD diameter, and the
parameter n takes the following values: 1.461 (Si), 1.582
(CdS), 1.271 (CdSe), and 1.273 (CdTe). Noteworthy is
that in the infinite barrier height approximation these
dependences should behave practically like d−2. As seen
from Fig. 2b, due to the effects of quantum confinement
and dielectric enhancement, the binding energy of
excitons in QD of small size (~1 – 2 nm) can reach very
large values (from several tenths to several electron-
volts). A small difference between the exciton binding
energies in Si, CdSe and CdTe QDs (in Fig. 2b
corresponding curves are shown for simplicity by one
solid line) is explained by the fact that dielectric
constants of these materials are close to each other (ε
~10–12). The curve for the CdS QD in Fig. 2b (dashed
line) lie much lower because of the smaller value of the
dielectric constant in CdS material (ε ~ 5.4).
Fig. 2. Energies of the ground exciton transition in
semiconductor QDs located in SiO2 matrix as functions of QD
diameter (a). Exciton energy shifts in the exciton ground state
(exciton binding energies taken with the negative sign) in
semiconductor QDs located in SiO2 matrix as functions of QD
diameter (b). Solid curve corresponds to the cases of Si, CdSe
and CdTe QDs, dashed one to CdS QD.
The described above technique may be also
generalized to describe the trion states. As an example,
let us consider the ground state of a negative trion in
QD. Since in this case the electron subsystem should be
a spin singlet, the wave function of a negative trion in a
QD can be found in the form of a symmetrized product
of "trion" single-particle functions X
i
−
Ψ :
( ) ( ) ( )
( ) ( ) ( )
1 2 , 1 , 2
, 2 , 1
1, ,
2
,
X
X X
h e e
X X X
e e h h
α β
α β
− −
−
− − −
Φ = Ψ Ψ +
+Ψ Ψ Ψ
r r r r r
r r r
(24)
where, analogously to the case of an exciton, "trion"
single-particle functions will be presented in the form of
an expansion in basic single-particle functions )( i
i rαΨ
(15):
( ) ( )
( ) ( )
, ( ) ,
.
X e
e i e i
X h
h h h h
A
A
λ
γ λ
λ
λ
λ
λ
χ γ
−
−
Ψ = Ψ
Ψ = Ψ
∑
∑
r r
r r
. (25)
In the expressions (24) and (25), the indices α, β and
γ are the spin variables taking the values +1/2 and −1/2,
( )χ γ is an electron spin function. Although the spin
coordinate will not be taken into account further in an
explicit form, for a more accurate and rapid iteration it is
worth distinguishing functions of electrons with different
spins as it is made in the spin-unrestricted Hartree-Fock
method [31, 32]. For such a form of the wave
function ( )1 2, , hX −Φ r r r , the Schroedinger equations
for a negative trion in a QD may be written in the form
of the following system of equations:
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 1-8.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
6
Fig. 3. Trion energy shifts in the trion ground state (trion
binding energy taken with the negative sign) with respect
to the energy of the state formed by non-interacting exciton
and charge carrier in Si-SiO2 QD as a function of the QD
diameter. Solid curves correspond to the case when the
effect of interface polarization is not taken into account,
while dashed ones are obtained with such an account.
( ) ( )
( ) ( )
( ) ( )
2
1 1
2
1 2 , 2 2
, 1 , , 1
1ˆ ( ) ,
2
1 ,
2
,
X
e eh h h h h
X
ee e
X X X
e e e
H U d
U d
E
β
α α α
−
−
− − −
+ Ψ +
+ Ψ ×
×Ψ = Ψ
∫
∫
r r r r r
r r r r
r r
( ) ( )
( ) ( )
( ) ( )
2
2 2
2
1 2 , 1 1
, 2 , , 2
1ˆ ( ) ,
2
1
,
2
,
X
e eh h h h h
X
ee e
X X X
e e e
H U d
U d
E
α
β β β
−
−
− − −
+ Ψ +
+ Ψ ×
×Ψ = Ψ
∫
∫
r
r r r r
r r
r r r r
(26)
( ) ( )
( ) ( )
( ) ( )
2
1 , 1 1
2
2 , 2 2
1ˆ ( ) ,
2
1 ,
2
,
X
h h eh h e
X
eh h e
X X X
h h h h h
H U d
U d
E
α
β
−
−
− − −
+ Ψ +
+ Ψ ×
×Ψ = Ψ
∫
∫
r
r r r r
r r
r r r r
where Uee(r1,r2) = e2 G (r1,r2) includes both direct and
indirect Coulomb interaction between two electrons in a
trion. The single-particle wave functions ,
X
e γ
−
Ψ and
X
h
−
Ψ in the ground state of trion and the corresponding
energies
i
XE
−
can be determined using the iteration
procedure analogous to that described in the case of an
exciton. The energy of the system transition from a
non-excited state, when only one charge carrier is
present in the QD (electron in the case of a negative
trion) into the ground state of trion (or vice versa) is
expressed as 2 (0) (2 ) X
e h
X
g s eX
E E U E E E
−
−
−= + + − +
(or (2 ) X
e h
X
g eX
E E E E E
−
−
−= + − + , if we introduce a
designation (0)
i
X X
s iE U E
− −
= + ). Then, it follows from
the definition (2) that the energy shift of such a trion
transition with respect to the exciton transition energy is
expressed by the formula
(2 )X X X X
e e h e hX
S E E E E E
− −
− = + + − + . (27)
Similar expressions can be also obtained for the case
of a positively charged trion.
Fig. 3 shows the dependence on the silicon QD
diameter of the trion transition energy shift ( )XS − +−
with respect to the exciton transition energy. As seen
from this figure, if the effect of the heterointerface
polarization is not taken into account, then the trion
states in QD are more energetically favourable than that
of an exciton, and with the QD size reducing their
binding energy with respect to the exciton state increases
and can reach the values of 10 – 20 meV in QD of a very
small size (~1 – 2 nm). These values by factors of tens
exceed trion binding energy in the case of bulk silicon
[2]. The effect of the heterointerface polarization in Si-
SiO2 QDs considerably decreases the binding energy of
trion states with respect to the exciton state in a QD of
several nanometers in size. When the QD size decreases,
the trion transition energy becomes even greater than
that of the exciton, i.e., the trion state in principle is less
energetically favourable in a small Si-SiO2 QD than the
state in the form of a separate exciton and a charge
carrier (e.g., those separated by tunneling into adjacent
QDs).
3. Conclusion
Thus, in the present work by using the Hartree-Fock
technique we have calculated the energy of the ground
radiative exciton and trion transitions in semiconductor
QDs located in a dielectric matrix. It has been shown
that the trion transition energy can be either less or
greater than the exciton transition energy depending on
the physical parameters of the system under
consideration.
For example, in the case of a Si-SiO2 QD the barrier
for holes due to the valence band-off-set is higher than
the barrier for electrons (~ 4.6 and ~ 3.2 еV,
respectively). Besides, the effective mass of a heavy hole
in silicon (~0.5 m0) is greater than that of an electron
(~ 0.25 m0). This makes hole states to be more localized
in a QD than those of electrons. Therefore, the more
effective repulsion of two holes in the case of a
positively charged trion results in greater energies of the
trion transitions in Si-SiO2 QDs than in the case of a
negatively charged trion.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 1. P. 1-8.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
7
It has been shown that if we do not take into account
the image forces field that arises due to a substantial
mismatch in the low-frequency dielectric constants of
silicon (ε1 ~ 11.7) and silicon dioxide (ε2 ~ 2.1), the
energy of the radiative transition in the case of a
negatively charged trion is less than that of the exciton
transition. This means that the effective attractive
electron-hole interaction in such a trion exceeds the
effective repulsive electron-electron interaction, and
with the QD size decreasing, the trion state is becoming
energetically ever more favourable as compared to the
state of non-interacting exciton and electron. As far as a
positively charged trion concerned, in the considered QD
size range the trion transition energy is greater than that
of the exciton, and with size reduction, the positive
difference between the trion and exciton transition
energies increases. Taking into account the polarization
of the heterointerfaces causes a shift of the trion
transition energies towards the short-wave region.
Moreover, for small QD sizes (< 2 nm), the transition
energies of negatively charged trions become greater
than the exciton transition energies (like to the case of
positively charged trions within the whole range of
considered QD sizes).
Acknowledgements
This work was supported in part by State Fundamental
Researches Foundation of Ukraine, Russian-Ukrainian
Program “Nanophysics and Nanoelectronics” and
Program “Nanostructured Systems, Nanomaterials and
Nanotechnologies” NAS of Ukraine.
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