# Size effects of a nanoobject in magnetic field

## TL;DR

This paper theoretically investigates how the size and magnetic field orientation affect the electronic spectrum of a nanoobject, revealing size-dependent spectral corrections and specific nanoobject dimensions with invariant spectral corrections.

## Contribution

It provides a detailed perturbation theory analysis of size effects on nanoobject spectra under magnetic fields, considering different field orientations and identifying size conditions for spectral invariance.

## Key findings

- First-order spectral correction is zero regardless of magnetic field orientation.
- Spectral correction depends on the nanoobject's size and aspect ratios in certain field configurations.
- Existence of nanoobject sizes where spectral corrections are invariant to the length of side c.

## Abstract

A theoretical analysis of physical properties of the effect of size of a nanoobject in the form of a rectangular parallelepiped whose sides $a$, $b$, $c$, are oriented along the $OX$, $OY$, $OZ$, respectively, is carried out. In the framework of the perturbation theory, changes in the electronic spectrum of the nanoobject caused by an external magnetic field $\vec{B}$, depending on its size, are analyzed. We consider two cases of the fields which are described 1) by the Landau gauge, $\vec{A}(\vec{r})=\left(0,Bx,0\right)$ ($\vec{B}$ is oriented along the side $c$) and 2) by $\vec{A}(\vec{r})=\left(Bz,0,\alpha By\right)$ ($\alpha$ is a parameter; at $\alpha = 0$, $\vec{B}$ is directed along $OX$ axis, and at $\alpha = 1$, $\vec{B}$ is directed along the diagonal in $XOY$ plane). Firstly, it is shown that the first correction to the spectrum is zero, regardless of $\vec{B}$ orientation. Secondly, it is established that, in contrast to the case of the field orientation 1), where the correction does not depend on the length of $c$, in the case 2) such correction depends both on $c$ and on its ratios to the lengths of $a$ and $b$. There was found the existence of such nanoobject sizes in $XOY$ plane at which the corrections to the spectrum are the same for different lengths of $c$ of the nanoobject.