Nanoscale physics

Size Effects of a Nanoobject in Magnetic Field
Lukiyanets B.A., Matulka D.V.
Department of Applied Physics and Nanomaterial Science, Lviv Polytechnic National University, S. Bandera str., 12, Lviv, Ukraine, 79000
E-mail: dariya2009@gmail.com

The influence of the shape and size of nanoobjects on their physical properties (electronic, optical, optoelectric, and others) is an important problem of modern nanotechnology.
The electronic spectrum of crystals is the basis of microelectronics. It is known that the spectrum of an infinite crystal is quasi-continuous, while the  spectrum of a nanosized crystal  is noticeably discrete. The degree of such discreteness depends on the size and shape of the nanoobject and because the recognition of the physics of such dependence and its manifestations in the phenomena of microelectronics remains urgent.
Below we will analyze, within the perturbation theory, the change of the electronic spectrum of a nanoobject by an applied magnetic field, depending on the size of the nanoobject of the shape of a rectangular parallelepiped with the sides  located along   axies, respectively. Let the potential energy of an electron in it is the 3-dimensional infinite deep square well. The wavelength function of an electron in this case is 
.
Let us place the nanoobject into a uniform magnetic field whose vector potential is. The Hamiltonian of the electronic subsystem in the magnetic field  without taking into account of the spin-orbital interaction has the following form:
.
Here  are Pauli matrix.
After transformations, the Hamiltonian can be rewritten as follows:
,
where is the Hamiltonian of the nanoobject without magnetic field and

is the perturbation Hamiltonian which describes the interaction of an electron with magnetic field.
We analyze two cases of the magnetic field  orientation: a) The Landau gauge case, . Taking into account the relationship , the field is oriented along , and b) The gauge – magnetic field  is in XOY plane with different orientation, depending on parameter , in particular, at  = 1,  is oriented along the diagonal in  XOY plane, and at = 0 along OY axis.
In the case a) the change by the perturbation Hamiltonian  of the electron spectrum with the Hamiltonian  and the displaced by the magnetic field by the value for the spin state (and by -  for the spin state ) within the framework of the perturbation theory will be
In this case, the 1st correction is zero, and the dependence as a result of the 2nd correction on the geometric sizes of the nanoobject to the ground state  in terms of the units  is shown in Fig. 1 (here and below a, b, c in nm). Note that the obtained corrections do not depend on the size of the nanoobject along the direction of the applied magnetic field.
In the gauge . the correction  with   = 1 ( is oriented along the diagonal in  XOY plane), in contrast to the analogous one in the case a), contains the dependence of the length of c, as well as of its ratios to a and b, namely , . Fig.2 shows the analogous to Fig. 1 dependences of the 2nd correction for the size family  ( nm).  
		
Fig.1.	Fig.2.	
Fig. 3 shows the dependencies  for the family  and (a) and and (b). There is clearly seen the intersection of two pairs of planes. The points  on the line of intersection are those in which the corrections corresponding to ,  are of the same values.
		
Fig, 3a)	Fig, 3 b)	
The analysis of the electron spectrum of a nanoobject the shape of rectangular parallelepiped placed in an external magnetic field, depending on the size of the object, indicates that in the framework of perturbation theory the magnitudes of the corrections depend both on the magnitude of the magnetic field  (it is greater for greater fields) and on its orientation relative Thus, in order to purposefully change the electronic spectrum of the nanoobject by magnetic field , one should take into account not only  the orientation of  in the nanoobject, but also its shape and relationship between its geometrical characteristics.