Quantum Confinement in Nanometric Structures
Magdalena L. Ciurea*, Vladimir Iancu**

*National Institute of Materials Physics, Bucharest-Magurele 77125, Romania
(Tel: +40-21-369-0185; e-mail: ciurea@infim.ro)
** University “Politehnica” of Bucharest, Bucharest 060042, Romania
(e-mail: codas@physics.pub.ro)
Abstract: This paper discusses the quantum confinement effects in nanometric structures that form low
dimensional systems. In such systems, each surface/interface acts like a potential barrier, i.e. the wall of a
quantum well, generating new energy levels. These levels are computed in a model that uses the
approximation of the infinite rectangular quantum wells. Different applications of the model are
discussed. The errors with respect to the experimental data are proved to be of the same order of
magnitude as the differences between the infinite and the finite quantum well levels.
Keywords: Quantum confinement, quantum well, nanolayers, nanowires, quantum dots.

1. INTRODUCTION
The study of the nanometric structures, i.e. low dimensional
structures (LDS), presents a great interest for both its
fundamental aspects and its numerous applications (Brewer et
al., 2002; LaVan et al., 2003; Gaburro et al., 2004; Li et al.,
2004; McDonald et al., 2004; Arango, 2005; Walters et al.,
2005; Fert, 2008; Ihn et al., 2008; Shields, 2008).
A structure is considered as a LDS if it has nanometric size
on at least one direction. In fact, all structures have three
dimensions. However, if their size on at least one direction is
small enough (no more than one order of magnitude greater
than the interatomic distance on that direction), the structure
can be considered as quasi-low dimensional. In these
structures, the ratio between the number of atoms located at
the surface/interface N S  and the total number of atoms N
can be expressed as
N S  N  23    a d  ,
(1)
where δ is the dimensionality of the structure, a the (mean)
interatomic distance, and dδ the (minimum) LDS size. The
relation is exact for 2D plane, 1D cylindrical, and 0D
spherical symmetry, respectively. If one takes a ≈ 0.25 nm,
then N S  N  0.5 for d0 = 3 nm, d1 = 2 nm, and d2 = 1 nm.
If one takes d0 = 5 nm (20 interatomic distances),
N S  N  0.3 . One can see that the surface/interface plays a
very important role in nanometric structures.
On the other hand, each surface/interface acts like a potential
barrier. Therefore, one can consider that they represent the
walls of quantum wells, inducing quantum confinement (QC)
effects, and in particular generating new energy levels. There
are two aspects that arise from this interpretation: the depth
and the shape of the quantum well. It was proved (Iancu and
Ciurea, 1998) that the QC represents a zero order effect,
while the nature of the material represents only a first order
effect. Consequently, the infinite quantum well must be a
good first approximation. The shape of the quantum well
determines the series of ratios of the differences between the
QC levels (corresponding to the possible transitions). By
comparing the theoretical ratios (computed for rectangular,
parabolic and Woods-Saxon quantum wells) with the
experimental ones, we have reached the conclusion that the
rectangular quantum well is the best approximation (Ciurea et
al., 2006). Therefore, we will use in the following the infinite
rectangular quantum well (IRQW) model.
The present paper applies this model to the study of the
nanometric structures. Section 2 deals with the 2D systems,
Section 3 with the 1D and Section 4 with the 0D ones. The
last Section presents the conclusions.
2. 2D STRUCTURES
The 2D structures are layers of nanometric thickness. The
best ones are monoatomic (e.g. monoatomic graphene). In the
most cases, these layers are plane, parallel with the crystalline
planes. Then, the electron Hamiltonian can be exactly split as
the sum of two parts:
– a parallel part (i.e. parallel with the layer surface), which is
Bloch-type, leading to a 2D band structure;
– an orthogonal part, which can be approximated with an
IRQW, leading to QC levels.
Therefore, the electron energy has the form

 
 2   n k x , k y  2 2  2

 m d p
* 2

2
,
(2)

*
where  n k x , k y is the 2D band energy, m 
is the effective
mass on the confinement direction, and p > 0 is a natural
number.
In order to locate properly the QC levels, let us consider the
case of absolute zero temperature. Then, the highest electron
energy is at the top of the valence band and on the
fundamental QC level. This means that the fundamental QC
level is located at the top of the valence band and the other
QC levels are located in the band gap. To mark this, we will
shift the zero of the QC energy and measure it from the top of
the valence band:
     m d 
 2   m d p  1     k , k   
 2    n k x , k y  2 2  2
2 2

* 2

* 2

2
s
n
x
y
(3)
p 1 .

Here  ns  k x , k y is the shifted band energy and  p1 the
QC level energy (  0  0 ).
As an application, let us consider the contribution of the QC
levels to the functioning of quantum well solar cells (Iancu
and Fara, 2007). In order to evaluate the internal quantum
efficiency for the absorption on the QC levels we need to
evaluate the matrix element of the electric dipole interaction
Hamiltonian
 
p f z
p z
2
  e  E  r sin
sin i dz .
d 0
d
d
d
(4)
To have absorption, H fi must be different from
electric field is parallel with the layer (e.g.
incidence), the quantum selection rule is p f 
zero. If the
at normal
pi . Such a
case involves only resonant levels. If the field is orthogonal
on the layer, p f  pi  2 p  1 . This corresponds to the QC
levels. Obviously, the absorbed wavelengths are different.
Then the quantum efficiencies and the corresponding
transition energies are


(5)
E  hc   E g   2  2 2m||*d 2
(6)
||  8 2 e 2 c 0 r  ,
(where 1 m||*  1 me*||  1 m*g|| is the exciton mass) for the
transition between the first symmetric resonant levels at
normal incidence, and

  


(7)

(8)
4
   4096 e 2  2c 0 r2   p 2 4 p 2  1  ,



In the following, we will consider only cylindrical nanowires,
to facilitate the identification of the orbital magnetic quantum
number. Because of this choice, the splitting of the
Hamiltonian as the sum of a longitudinal part and a
transversal one is no longer exact. However, this splitting is a
good approximation. Then the electron energy is

 1   n k z   2 2  2
 m d  x
* 2

2
p, l ,
(9)
where   x p,l is the p-th non-null zero of the cylindrical
The modelling of the 2D structures consider the change of the
band gap from one layer to the following one as possible
quantum wells and introduce QC levels in the conduction and
valence bands. However, these are not proper QC levels, but
resonant levels. They do not contribute to the transport
properties, but to the optical ones. To have a complete
analysis of the behaviour of a LDS, one has to take both QC
and resonant levels into account. As the resonant levels are
well known (see Harrison, 2005), in this paper we are
concerned only with the proper QC levels.
H fi
3. 1D STRUCTURES


* 2
E  hc    2  2 2m
d  4 p2 1 ,
for the transitions between QC levels.
It has to be remarked that the resonant levels appear only in
multilayer structures, namely in the layers with smaller gap,
while the QC levels appear in all the layers.
Bessel function J l x  .
Once again we can analyze the case of absolute zero
temperature to find out that the QC levels are located in the
band gap, so that we can shift the zero of the QC energy and
measure it from the top of the valence band:
  m d  x 

 2   m d  x  x 
 1   n k z   2 2  2
2 2
  s  k
n
z
* 2

* 2

2
p, l
2
1, 0
2
1, 0
(10)
   p1, l ,
where  ns  k z  is the shifted band energy and  p1, l the QC
level energy (  0, 0  0 ).
If we try to analyze different excitation transitions, we have
to remember that the valence band acts like an infinite
particle reservoir, so that all excitations start from the
fundamental QC level. Then, the ratio between consecutive
transition energies (playing the role of activation energies) is

R 1   p, l  p, l  x 2p''1, l ''  x12, 0
 x
2
p'1, l '

 x12, 0 . (11)
The choice of the quantum selection rules depends on the
kind of excitation we have. In the case of thermal excitation,
the condition is that the energy variation should be minimum.
In the case of an electrical transition (i.e. under high electric
field, eU  k BT ), l  0 . In the case of an optical
transition, l  1 .
As an application, we will analyze the case of nanocrystalline
porous silicon (nc-PS). In a previous paper (Ciurea et al.,
1999), we have discussed the microstructure of nc-PS films
that present a double scale of porosity: an alveolar columnar
micropore structure (pore diameter of 1.5 – 3 μm), and a
nanoporous structure of the alveolar walls (100 – 200 nm
thickness). High resolution transmission electron microscopy
(HRTEM) images proved that these walls form a nanowire
network, with nanowire diameter of 1 – 5 nm (see Fig. 1).
The investigation of the temperature dependence of the dark
current in these nc-PS films (Ciurea et al., 1998) proved that
the characteristics were of Arrhenius type. For fresh samples,
only one activation energy, E1 = (0.52 ± 0.03) eV, was
observed (see Fig. 2a). For samples stabilized by controlled
oxidation, two activation energies, E1 = (0.55 ± 0.05) eV, E2
= (1.50 ± 0.30) eV, appeared, the change occurring rather
abruptly at T ≈ 280 K (see Fig. 2b).
Fig. 1. HRTEM detail of the alveolar wall of nc-PS, shown
by lattice fringes contrast with respect to amorphous silicon
oxide and glue (Ciurea et al., 1999). Reused with permission
from M. L. Ciurea, V. Iancu, V. S. Teodorescu, L. C. Nistor,
and M. G. Blanchin, Journal of Electrochemical Society 146,
2517, 1999. Copyright  1999, The Electrochemical Society,
Inc.
The ratio of the two energy values is E 2 E1  2.727 . From
this ratio and (11), we can identify E1   1, 0 and
E2   2, 0 . Using (10), we then find ds = (3.22 ± 0.05) nm, in
agreement with the microstructure investigations. If we use
the same identification for the fresh samples, we find df =
(3.31 ± 0.03) nm. This means that by oxidation the diameter
decreased with less than 1 Å, which is absurd. This
discrepancy arose from the fact that we have used the
effective mass approximation (EMA), which is no longer
valid at nanometric scale. In this approximation, the energy is
inversely proportional with the square of the diameter. A
thorough analysis, performed by using the linear combination
of atomic orbitals (LCAO) method (Delerue et al., 1993),
Fig. 2. I – T characteristics taken in dark on (a) fresh and
(b) stabilized nc-PS (Ciurea et al., 1998). Reprinted from
Thin Solid Films 325, M. L. Ciurea, I. Baltog, M. Lazar, V.
Iancu, S. Lazanu, and E. Pentia, “Electrical behaviour of
fresh and stored porous silicon films”, 271, Copyright 
1998, with permission from Elsevier.
proved that  ~ d  , with α = 1.02 for cylindrical
nanowires, leading to df = (3.40 ± 0.03) nm. This means that
the oxide layer at the surface of the nanowires is
monoatomic.
The phototransport (PT) in nc-PS was studied by tracing the
spectral dependence of the photocurrent (Iancu et al., 2007).
Several maxima and shoulders were identified in the I – λ
characteristics (see Fig. 3) and all but one could be identified
with transitions between QC levels (see Table 1). The
maximum No. 6, at 873 nm (1.42 eV), as attributed to surface
states. The relative errors made by the model in both cases (I
– T and I – λ characteristics) were under 3 %.
Fig. 3. I – λ characteristics taken at 20V on stabilized nc-PS
(Iancu et al., 2007). Reprinted with permission from Iancu,
V., M. L. Ciurea, I. Stavarache, and V. S. Teodorescu (2007),
Journal of Optoelectronics and Advanced Materials 9, 2638.
Copyright  1998.
Table 1. QC transitions identified in PT measurements on
nc-PS (Iancu et al., 2007). Reprinted with permission
from Iancu, V., M. L. Ciurea, I. Stavarache, and V. S.
Teodorescu (2007), Journal of Optoelectronics and
Advanced Materials 9, 2638. Copyright  1998.
No. λ (nm) Eexp (eV)
1.
506
2.45
2.
575
2.16
3.
631
1.96
4.
719
1.72
5.
825
1.50
6.
875
1.42
7.
935
1.33
8.
1025
1.21
Transition
(1, 2) → (2, 3)
(0, 0) → (1, 2)
(1, 1) → (0, 3)
(2, 0) → (1, 2)
(2, 0) → (3, 1)
–
(1, 0) → (0, 2)
(1, 0) → (2, 1)

R 0   p, l   p, l

 x 2p''1, l ''  x 2p1, l
  p,l   p, l 
 x
2
p '1, l '

 x 2p1, l .
(15)
If the dots are bigger, one has a proper band structure and
(15) takes the same form as (11).
We will apply these results to a Si – SiO2 nanocomposite,
formed by nanocrystalline silicon (nc-Si) quantum dots
embedded in an amorphous silicon dioxide (a-SiO2) matrix.
The microstructure investigations proved that, for nc-Si
volume concentration x in the interval 50 – 75 %, most of the
dots have diameters around 5 nm (Ciurea et al., 2006,
Teodorescu et al., 2008), as it can be seen from Fig. 4.
4. 0D STRUCTURES
As 0D structures, we will consider only spherical dots, for
similar reasons as for the nanowires. In the case of dots, there
appears a specific behaviour (Ciurea et al., 2006). When the
diameter is small enough (under about 20 interatomic
distances, i.e. about 5 nm), one has no longer a proper band
structure, but sets of levels forming quasibands, separated by
rather large intervals (quasigaps). More than that, the
momentum conservation law no longer applies (Heitmann et
al., 2004). Such dots are usually called “quantum dots”.
In a quantum dot, the energy is simply

 m d  x
 0  2 2  2
2
e
2
p, l
,
(12)
where   x p,l is the p-th non-null zero of the spherical
Bessel function jl x  and the effective mass is replaced by
the free electron mass (without a band structure, the concept
of effective mass becomes meaningless). Indeed, from the
LCAO computations (Delerue et al., 1993), we have
 ~ d  , with α = 1.39 for spherical dots. This means that
we can approximate the effective mass as


*
*
m*  m
 a d  me  m
,
(13)
with β ≈ 1. It is easy to see that m*  me for quantum dots.
Once again, the QC levels are located in the quasiband gap;
and once again we can measure the QC energy from the
fundamental state (  0, 0  0 ), by writing
  m d  x
 2   m d  x
 0   2 2  2
2 2
e
e
2
2
2
1, 0
2
p, l

 x12, 0  EV   p 1, l .
(14)
When one considers the transitions, one has the same
selection rules as in the case of nanowires. However, as there
is no more valence band, i.e. no more particle reservoir, the
transitions are made from one QC level to the next permitted
one (following the selection rules), and (11) becomes
Fig. 4. HRTEM image of a Si – SiO2 sample with x = 66 %
(Ciurea et al., 2006). Reprinted from Chemical Physics
Letters 423, 225, M. L. Ciurea, V. S. Teodorescu, V. Iancu,
and I. Balberg, “Electrical transport in Si-SiO2
nanocomposite films”, 225, Copyright  2006, with
permission from Elsevier.
The I – T characteristics, measured at different voltages on a
sample with the nc-Si volume concentration x = 66 % (Ciurea
et al., 2006) are presented in Fig. 5. One can see that, at low
voltages, there are three activation energies, E1 = (0.22 ±
0.02) eV, E2 = (0.32 ± 0.02) eV, and E3 = (0.44 ± 0.02) eV.
By using (11) and (15), one obtains the confirmation that the
nc-Si form quantum dots. At the same time, one can identify
the transitions between QC levels, by taking E1  1,1   0,1 ,
E2   2,1  1,1 , and E3   3,1   2,1 . From (14) one
obtains d = (5.2 ± 0.4) nm, in agreement with the
microstructure measurements. Then, the model errors are
smaller than 3 %.
It can be observed in Fig. 5 that the first activation energy
appears only at low voltages. This fact was explained by
studying the I – V characteristics. The characteristic taken at
the same concentration is presented in Fig. 6 (Ciurea et al.,
2006). From Fig. 4 one can see that the quantum dots form
chains, but these chains are not long enough to reach from
one electrode to the other one (separated by 1 mm distance).
Therefore, the carriers tunnel through the a-SiO2 regions. The
height of the potential barrier was estimated from the nc-PS
measurements to be 2.2 eV (Ciurea et al., 1998).
dots, m*  me ), U b  U N is the mean bias applied on a
barrier of height φ and width δ, and N is the number of
barriers.
One can see that there are only three fit parameters, as (16)
can be put in the form (Ciurea et al., 2006)


I  I 0 signU  1  U U 0 exp   1  U U 0
 exp   .
Fig. 5. I – T characteristics taken in dark on a Si – SiO2
sample with x = 66 % (Ciurea et al., 2006). Reprinted from
Chemical Physics Letters 423, 225, M. L. Ciurea, V. S.
Teodorescu, V. Iancu, and I. Balberg, “Electrical transport in
Si-SiO2 nanocomposite films”, 225, Copyright  2006, with
permission from Elsevier.

(17)
Here I 0  a  , U 0  N  e ,    1 2 , and q  e . By
fitting the experimental curve and using the value φ = 2.2 ±
0.1 eV, we have obtained δ = (0.97 ± 0.05) nm and N = 87 ±
4. Then eU max N  1 6 (the barrier is indeed trapezoidal)
and for U = 25 V, Ub = (0.29 ± 0.01) V > E1/e (the first level
is already excited by the applied field).
Another application consists in evaluating the internal
quantum efficiency for the quantum dot solar cells (Iancu et
al., 2008). Inside the dot, the wavefunction is


 n, l , m r, ,   N n, l R 3 2 jl zn 1, l r R Yl , m  ,  , (18)
where R = d/2 is the dot radius, Yl ,m  ,   is the spherical
harmonics, zn 1, l  0 is the (n + 1)-th non-null zero of the
spherical Bessel function jl x  ( z0, l  0 ), and
1

N n, l    jl2 z n 1, l u u 2 du
0



1 2
(19)
is the normalization constant. The light beam can be
considered as parallel with the Oz axis, due to the spherical
symmetry of the dots. Then, the absorption selection rules are
Δl = ± 1, Δm = ± 1.
Fig. 6. I – V characteristic taken in dark on a Si – SiO2
sample with x = 66 % (Ciurea et al., 2006). Reprinted from
Chemical Physics Letters 423, 225, M. L. Ciurea, V. S.
Teodorescu, V. Iancu, and I. Balberg, “Electrical transport in
Si-SiO2 nanocomposite films”, 225, Copyright  2006, with
permission from Elsevier.
The number of tunnelled barriers must be of the order of
hundreds or more, so that the barrier becomes trapezoidal
under the applied field. Consequently, the high field-assisted
tunnelling is described by the Simmons formula (Simmons,
1963)





I  a  exp      qU b  exp     qU b , (16)
where a is a constant proportional with the number of
equivalent paths for the carriers, q is the carrier charge

( q  e ),   8m*  2

12
(remember that, for quantum
If we compute now the internal quantum efficiency for the
absorption threshold, we find that, as in the case of 2D
structures, the wavelength is proportional with the square of
the size, while the internal quantum efficiency is sizeindependent. For a Si – SiO2 structure with dot diameter d =
5 nm, we have found that the threshold wavelength is λthr ≈
15.6 nm and the corresponding internal quantum efficiency is
η ≈ 8.26 %.
5. CONCLUSIONS
We have analyzed the quantum confinement effects in 2D,
1D, and 0D nanometric structures. These effects were
modelled by means of the IRQW approximation. The model
explains most of the phenomena observed in such structures.
Indeed, almost all the energies measured in electrical
transport and phototransport can be interpreted as due to
transitions between QC levels. The fact that different
phenomena lead to different energies is related to the
selection rules. The differences between the results of the
model and the experimental data are produced by the fact that
the depth of the quantum well is finite, as well as by the size
and shape distribution.
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