Wave Functions and Energy Levels

An electron in a semiconductor is characterized by the effective mass m^* generally smaller than the free electron mass m₀. Then, the de Broglie wavelength of an electron in a semiconductor λ is greater than that of a free electron λ⁰:

0

0 *

2 *

m

h h

p m E m

λ= = =λ (2.1)

Where h is Planck constant and p is electron momentum. If geometrical size of semiconductor sample with dimensions X, Y, and Z introduced: Since only an integer number of half-waves of the electron or hole can be put in any finite size system, instead

of continuous energy spectrum and a continuous number of the electron states, a set of discrete electron states and energy levels are obtained, which are characterized by the corresponding number of half wavelength. And this phenomenon generally referred as quantization of electron motion. One can distinguish four different cases, depending on the system dimensions:

Three dimensional or bulk situation, in which quantization of electron is not significant at all, and an electron behaves like free particle in the crystal and characterized with the effective mass m^*:

λ<< X, Y, Z,

In quantum well or two-dimensional system, the quantization of electron occurs in one dimension in the growth direction while in the other two directions electron motion is free:

λ~ X << Y, Z

In quantum wire or one dimensional system case, quantization occurs in two dimensions and electron moves freely along the wire:

λ ~ X~Y<<Z

In zero dimensional or nanocrystal (quantum dot) case, the quantization occurs in all directions and the electron cannot move freely in any directions:

λ~ X ~ Y ~ Z

Nanostructures are quantum mechanical systems inasmuch as their sizes are comparable with the typical de Broglie wavelength of electrons in solids, so that a quantum mechanical treatment of the problem is strictly needed to determining the wave function of a single electron or of the whole system. The wave function Ψ of an electron or electron system satisfies the principal equation of quantum mechanics, the Schrödinger equation,

0 i ∂Ψ −ΗΨ =t

ℏ ∂ (2.2)

Where Ηis the Hamiltonian of the system,

is the potential energy and the first term is the kinetic energy operator. If V(r
) is assumed to time independent, Ψ can be separated to its time and spatial coordinates:

) Substituting this in to the Eq. (1.2) we get the time independent Schrödinger equation:

( ) ( ) ( )

The major aim of solving this stationary Schrödinger equation in quantum dot system is related with electron (hole) bound states in dot. In this case, one can get discrete energies of bound state, can calculate relaxation of excited states due to the interactions with free electrons, phonons, and defects, and lastly results of interaction with electromagnetic field.

To solve the equation above two important simplifications can be imposed; isotropic effective mass m^* i.e. independent of both position and the energy of the electron and the other is idealized step like potential profile, which can be analyzed easily. The simplest potential V(x, y, z) of this type is For this potential profile the solution of Eq. (1.5) can be written down as

) dot (box) can be obtained. Generally, all energies are different, that is not degenerated,

but if two or all dimensions are equal or the dimensions ratios are integers, some levels will coincide with different quantum numbers and this coincidence results in degeneracy. This discrete spectrum in quantum dot and the lack of free electron propagation are the main distinguishing features of quantum dots or boxes from other systems (3D, 2D or 1D). As it is well known, these features are typical for atomic systems.

Due to the similarity with atoms, QD generally studied with the shape of spherical dots. In this case, the potential is

R Where R is the radius of the dot and r is magnitude of radius vector. It is known from quantum mechanics that for this situation the solution of Schrödinger equation can expressed by separating it into its angular and radial parts.

)

Writing the Schrödinger equation for the radial function R(r) ),

equation into one-dimensional coordinate due to the spherical symmetry. It is clear from Eq. (2.11) that, the effective potential V_eff(r) depends on quantum number l, but does not depend on m and the energy is function of principal quantum number n from and the angular momentum l.

In the simple cases for l=0 the solution of Eq. (2.11) can be easily obtained with the potential V(r) as;

R

Using the boundary conditions the equation for the energy;

w Being the potential well deep enough, the solution to Eq. (2.14) is;

2 Therefore taking into account the dot radius, potential well must be large enough to

confine the electron. That is the behavior of energies in confined system is expected as a function of the spatial dimension, namely the confinement energy decreases as the size of the system (nanocrystal size) increases.

In the practical application the confinement potential can be supplied by growing the nanocrystal in a higher band gap matrix than the dot material. The discontinuities at the conduction band and the valance band allow the confinement of electrons and holes. In the case of type-I heterostructures both electrons and holes are confined within the nanocrystal itself, for type-II heterostructures electrons and holes confined in either nanocrystal or matrix separately. This situation can be very useful for development of quantum dot solar cell and would increase the responsivity of the QDIP (quantum dot infrared photo detector) as a result of the different transport media for the excited carriers in the devices; the unwanted recombination of electrons and holes can be minimized. There is a terminology, scarcely mentioned, known as antidot structure which is the situation that the nanocrystal rejects the created carrier pairs in itself or near its surface. Now the band gap of confining matrix potential is lower in the potential value than the nanocrystal; therefore carriers tend to settle in the matrix material and

Actually, the mentioned theory above cannot exhibit full structure of real nanocrystals in practice but only gives some hints to the real case; in reality the confining potential can not be infinite. There are a lot of parameters affect energy band structure of nanocrystal that have to be accounted: strain in the bonds due to lattice mismatch, polarization effect due to differences in dielectric constants, surface states between the core and matrix and variations in the shapes, etc.