We start with the simplest case that could be easily treated on the basis of traditional approach - a NS shaped as a rectangular prism with a square base (with the sides a
oriented along the axes x
; the side c
is set along the z
direction). Assuming, as it is usually done in the literature, the absence of a potential inside the NS and separating the variables, we look for the solution of the stationary Schrödinger equation ΔΨ
0 (where k2
being the particle's effective mass) as the product of plain waves propagating in both directions along the coordinate axes:
For this case, the even mirror boundary conditions are as follows [10
That renders the following solution (Equation
1) of the Schrödinger equation:
with wave vector components
It gives the following energy spectrum:
The odd mirror boundary conditions are obtained from Equation
2 by inverting the sign of the left-hand-side function. The solution will then be as follows:
The wave vector components will be the same as that presented in Equation
4, yielding the same energy spectrum (Equation
5). Using the traditional impenetrable wall boundaries, one will also obtain the solution in the form (Equation
6) that coincides with the OMBC solution that has a vanishing Ψ function at the boundary. Therefore, the energy spectrum is the same for both types of mirror boundary conditions and impenetrable wall boundary, although the solutions themselves are not equal. In [7
], we demonstrated that for NS of spherical shape, the energy spectrum found with EMBC (weak confinement) is different from that corresponding to impenetrable walls conditions.
5, it is evident that the energy spectrum of prismatic (cylindrical) NS is a sum of the spectra corresponding to the two-dimensional cross-section NS (a square with side length a
) and the one-dimensional wire of length c
. In a similar manner, the spectrum for cylinders with other cross-section shapes can be constructed using the solutions for two-dimensional triangular or hexagonal structures analyzed previously [8
]. Below, we present the analysis of cylindrical NS.
Let us consider a nanostructure with a circular cross section of diameter a
and cylinder height c
. The solution of the problem using a traditional approach can be found in [12
]. In our case, we make variable separation in cylindrical coordinates:
We note that the value of p
defines the angular momentum: L
In the case of EMBC, one can apply mirror reflection from the base, which gives B
, resulting in the following wave function:
Strong confinement (OMBC) gives B
, which introduces sinkz
instead of coskz
The radial function F(r) is the solution of the following radial equation:
It is Bessel's differential equation regarding the variables kr
, the solution of which is given by the cylindrical Bessel function of integer order |p
); with, k
. Here, m
is the effective mass of the particle, and En
is the quantized kinetic energy corresponding to the motion in two-dimensional circular quantum well. The total energy consists of energy contribution for the motion within cross-section plane and along the vertical axis z
The energy En
depends on the values of k
and is obtained using boundary conditions. In the traditional case of impenetrable walls, the Ψ function vanishes at the boundary so that the energy values are determined by the roots (nodes) of the cylindrical Bessel function (see Figure
for different order numbers n
, and also Table
). The same situation will take place for OMBC, yielding zero wave function at the boundary so that the nodes q|p|i
of the Bessel function will define the energy values.
Cylindrical Bessel functionsJn(x). Curve numbers correspond to order n.
Argument values at nodes and extremes of cylindrical Bessel function
If the EMBC are used, the situation becomes different since the function values in the points approaching the boundary of the nanostructure should match those in the image points, making the boundary to correspond to the extremes of the Bessel function (which was strictly proved for the spherical quantum dots (QDs) [10
gives several values of the Bessel function argument kr
corresponding to the function nodes (q|p|i
) and extremes (t|p|i
) calculated for function orders 0, 1, 2, and 3.
At the boundary, r
/2; therefore, the corresponding value of k
for OMBC and 2t|p|i
for EMBC. The energy spectrum for a particle confined in a circular-shaped quantum well is as follows:
Here, the parameter s|p|i takes the values of q|p|i for OMBC (strong confinement) and t|p|i for EMBC (weak confinement).
The quantization along the z
axis for both the boundary condition types will be
, yielding the total energy
In the case of EMBC, the ground state (GS) energy will be obtained with t11
In the OMBC case, the GS will be determined by the smallest q value of 2.4:
10, 11, and 11A can be used for the analysis of optical processes in the NS discussed. In particular, blueshift in exciton ground state can be found from Equations
11 and 11A if one substitutes a reduced exciton mass in place of particle mass m
. Using Equation
10, it is possible to obtain in a similar way the energies corresponding to the higher excited states.
For long NS with sufficiently large c
, the second term in energy does not affect the GS. Thus, the solution for cylindrical NS based on even mirror boundary conditions EMBC (weak confinement) gives the GS shift due to quantum confinement that is (2.4/1.625)2
2.18 times smaller than the value obtained for the strong confinement case. In the case of spherical QD [10
], the difference was four times. It is reasonable that for strong confinement, the blue shift value exceeds that obtained for the weak confinement case. To illustrate this, we present in Figure
the comparison of ground state energy obtained with OMBC and EMBC (using Equations
11 and 11A) on NS diameter for a cylindrical quantum well with parameters of silicon (effective mass for electron 0.26 and 0.49 for a hole, which corresponds to reduced exciton mass of 0.17; bandgap is 1.1
eV for 300
K). As one can see from the figure, the difference of the exciton bandgap scales down with increase of the NS diameter, with invariably higher values observable for the strong confinement case described by OMBC.
Dependence of ground state energy on diameter of a cylindrical nanostructure. The plot shows the data obtained with odd and even mirror boundary conditions for an NS with parameters of silicon.
The choice of OMBC or EMBC has to be made taking into account the probability of electron tunneling through the walls forming the nanostructure. One can expect that in the case of isolated NS strong confinement (OMBC), approximation will be more appropriate, whereas for NS surrounded by other solid or liquid media (core-shell QDs [10
] and pores in semiconductor media), weak confinement with EMBC should be used.