Size effects of a nanoobject in magnetic field

B.A. Lukiyanets; D.V. Matulka

arxiv: 1906.11581 · v1 · pith:2W3MN3PGnew · submitted 2019-06-27 · ❄️ cond-mat.mes-hall

Size effects of a nanoobject in magnetic field

B.A. Lukiyanets , D.V. Matulka This is my paper

Pith reviewed 2026-05-25 14:56 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall

keywords nanoobjectrectangular parallelepipedmagnetic fieldperturbation theoryelectronic spectrumsize effectsvector potential

0 comments

The pith

The first correction to the energy spectrum of electrons in a rectangular nano-parallelepiped vanishes for any magnetic-field orientation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models a nanoobject as a rectangular parallelepiped with sides a, b, c and uses perturbation theory to compute how an external magnetic field shifts its electronic energy levels. It establishes that the first-order correction in the field strength is identically zero no matter how the field is oriented. In one gauge choice the second-order correction is independent of the length c, while in the other gauge it depends on c and on the ratios a/c and b/c. The analysis further identifies particular sizes in the base plane for which the second-order shifts become the same even when the height c is varied.

Core claim

Within first-order perturbation theory the linear-in-B correction to every energy eigenvalue is zero for both the Landau gauge with B along c and the gauge with A = (B z, 0, α B y). The quadratic correction in the second gauge depends explicitly on the length c and on the aspect ratios a/c, b/c; for certain discrete values of a and b the quadratic shift is independent of c.

What carries the argument

Perturbative expansion of the kinetic term that contains the vector potential, evaluated between the exact zero-field eigenfunctions of the three-dimensional infinite rectangular well.

If this is right

The spectrum receives no shift linear in B regardless of field direction.
When the vector potential has components that couple the x and y directions, the quadratic shift depends on the out-of-plane length c.
Special base-plane dimensions exist that make the quadratic shift identical for different values of c.
The dependence on c disappears in the gauge where B lies strictly along the c axis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Device designers could choose base dimensions to render the magnetic response insensitive to small variations in height.
The result supplies a concrete test for whether real nanoobjects behave as ideal infinite wells before higher-order or boundary effects intervene.
The same perturbative framework could be applied to other confining potentials whose zero-field states are known exactly.

Load-bearing premise

The nanoobject is an ideal rectangular box whose unperturbed states are exactly those of a particle in a three-dimensional infinite well, and the magnetic field is weak enough that perturbation theory applies.

What would settle it

Measurement of a nonzero linear term in B in the energy levels of a rectangular quantum dot or wire would contradict the claim that the first-order correction vanishes for any orientation.

Figures

Figures reproduced from arXiv: 1906.11581 by B.A. Lukiyanets, D.V. Matulka.

**Figure 1.** Figure 1: (Colour online) Dependence of the 2nd correction in terms of the units[−( eB m ) 2 ·( 32 π2 ) 2 · 2·10−18 ·m π2 ] on the sizes in XOY plane in the Landau gauge case, A® = (0, Bx, 0) (magnetic field oriented along the OZ) (a, b in nm) [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Figure 2: (Colour online) Dependence of the 2nd correction in terms of the units[−( eB m ) 2 ·( 32 π2 ) 2 · 2·10−18 ·m π2 ] on the sizes in XOY plane in the gauge case, A® = (Bz, 0, By) (the direction of the magnetic field coincides with the bisection of the XOY), c1 = 3, c2 = 4, c3 = 5, c4 = 6 (a, b, c in nm). +α · b c 2 · Õ0 k2k3    2k3 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: shows the dependencies ∆ 0 2 E (111) = f (a, b) for the family c1 = 1 and c2 = 3 (a) and c1 = 1 and c2 = 4 (b). There is clearly seen an intersection of two pairs of planes. The points ai , bi on the line of intersection are those in which the corrections corresponding to c1, c2 are of the same values. From the a) b) [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Standard perturbation theory on the 3D box in two gauges; first-order term vanishes as expected and second-order size dependence follows directly from the matrix elements.

read the letter

The colleague should know that this paper applies non-degenerate perturbation theory to the particle-in-a-3D-rectangular-box states with two standard vector potentials. The first-order correction is zero in both cases, and the second-order term is independent of height c in the Landau gauge but depends on c and the a/b ratios in the second gauge, with particular aspect ratios that remove the c dependence again. That is the entire content. The calculation itself follows the usual route: the unperturbed infinite-well wavefunctions are known exactly, the perturbation is the A·p + A² term, and the sums are evaluated for the given geometry. The stress-test note is correct that nothing in the claims requires extra assumptions beyond the standard setup. The vanishing first-order piece is immediate for real wavefunctions, and the gauge difference is immediate from whether A carries z dependence. The special sizes are just the algebraic condition that makes the c-dependent pieces cancel in the sum. What the paper does well is to write down the explicit dependence for this geometry. The soft spot is the complete absence of novelty. These results are the direct, unsurprising output of the matrix-element structure for this problem; they do not introduce a new method, resolve an inconsistency in the literature, or supply a quantity that was previously only known numerically or empirically. The abstract gives no evidence of numerical verification, comparison to exact diagonalization, or device-relevant consequences. This is the sort of calculation a reader could reproduce in a few hours if needed for modeling. It is not the kind of work that advances mesoscopic physics or supplies a result worth citing. A serious editor should desk-reject rather than send it to referees.

Referee Report

0 major / 3 minor

Summary. The manuscript analyzes the electronic spectrum of a rectangular parallelepiped nanoobject modeled as a particle in a 3D infinite well, subjected to a uniform magnetic field via two vector-potential choices. Using non-degenerate perturbation theory, it shows that the first-order correction vanishes for any orientation. The second-order correction is independent of the length c when B is along z (Landau gauge A=(0,Bx,0)), but depends on c and the aspect ratios a/b/c for the second gauge A=(Bz,0,α B y). Special (a,b) values are reported at which the second-order shift becomes independent of c.

Significance. If the explicit sums are correct, the work supplies parameter-free analytic expressions for the leading diamagnetic shift in a solvable confined geometry. The symmetry argument for the vanishing linear term and the identification of aspect ratios that decouple the quadratic shift from one dimension are directly usable for interpreting size-dependent magnetospectroscopy in lithographic nanostructures.

minor comments (3)

Abstract: the sentence 'There was found the existence of such nanoobject sizes...' is grammatically awkward and should be rephrased for clarity (e.g., 'We identify nanoobject sizes in the XOY plane...').
The manuscript should state the regime of validity of the perturbation expansion (i.e., ħω_c ≪ level spacing set by a,b,c) and give a numerical example of the maximum B for which the reported O(B²) term remains the leading correction.
Notation for the quantum numbers (n_x, n_y, n_z) and the explicit form of the unperturbed wavefunctions should be introduced once in the main text rather than assumed from the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript on size effects of a rectangular parallelepiped nanoobject in a magnetic field. The recommendation for minor revision is noted. No explicit major comments were provided in the report, so we have no points requiring rebuttal or clarification at this time.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation applies non-degenerate perturbation theory directly to the known eigenstates of the 3D infinite well. The first-order term vanishes by symmetry of real wavefunctions under the given vector potentials; this is an immediate consequence of the matrix-element definition and is not imposed by any self-definition or prior result. Second-order corrections are explicit sums over overlap integrals whose dependence on a, b, c follows algebraically from the unperturbed wavefunctions and the chosen gauges; no parameter is fitted and then relabeled as a prediction. No self-citations appear as load-bearing premises, and the special (a,b) ratios that make the shift independent of c are straightforward numerical consequences of those integrals. The calculation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard quantum-mechanical model of a particle confined in a rectangular box plus the minimal-coupling substitution for the magnetic field. No new entities are postulated and no parameters are fitted to data.

axioms (2)

domain assumption Electrons inside the nanoobject obey the Schrödinger equation for a particle in a three-dimensional rectangular infinite well whose eigenfunctions and eigenvalues are known exactly.
This is the unperturbed Hamiltonian implicitly used when applying perturbation theory to the magnetic term.
domain assumption The magnetic field is weak enough that first-order perturbation theory is valid and the vector potential can be introduced via minimal substitution without altering the boundary conditions.
Standard assumption for orbital magnetic effects in mesoscopic systems.

pith-pipeline@v0.9.0 · 5833 in / 1586 out tokens · 31028 ms · 2026-05-25T14:56:14.571168+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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InTRoduCTIon The inﬂuence of the shape and size of nanoobjects on their physical properties (electronic, optical, optoelectric, and others) is an important problem in modern nanotechnology. Theelectronicspectrumofcrystalsisthebasisofmicroelectronics.Itisknownthatthespectrumofan inﬁnitecrystalisquasi-continuous,whilethespectrumofananosizedcrystalisnoticeab...

work page internal anchor Pith review Pith/arXiv arXiv 1906
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Model.CAlCulATIonS Letusconsiderananoobjectoftheshapeofarectangularparallelepipedwiththesides a, b, clocated along OX, OY, OZ axes, respectively. Let the potential energy of an electron in it be the sum of three terms U (®r)= U(x) + U(y) + U(z) , (2.1) where each of them is an inﬁnite deep square well, i.e., U(s) = { 0, if s = x∈[ 0, a]or y∈[ 0, b]or z∈[ ...

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(3.1) Let us analyze two cases of the magnetic ﬁeld®B orientation: a) the Landau gauge case,®A(®r) = (0, Bx, 0)

3hIFToFlevelSbyAMAgneTIC 1eld Using (2.9), we consider the shift of stationary states caused by the magnetic ﬁeld: ∆E(nanbnc)≡⟨nanbnc|ˆH|nanbnc⟩− ⣨ nanbnc ˆH0 + eℏ 2m ®B®σ nanbnc ⟩ =⟨nanbnc|ˆH1|nanbnc⟩ + Õ′ nx ny nz ⟨nanbnc|ˆH1|nxnynz⟩⟨nxnynz|ˆH1|nanbnc⟩ Ena nb nc− Enx ny nz . (3.1) Let us analyze two cases of the magnetic ﬁeld®B orientation: a) the Landa...

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ConCluSIonS The analysis of the electron spectrum of a nanoobject and of the shape of rectangular parallelepiped placed in an external magnetic ﬁeld, depending on the size of the object, indicates that (1) in the framework of perturbation theory, the correction to the spectrum appears only from the second term onward; (2) the magnitudes of the corrections...

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InTRoduCTIon The inﬂuence of the shape and size of nanoobjects on their physical properties (electronic, optical, optoelectric, and others) is an important problem in modern nanotechnology. Theelectronicspectrumofcrystalsisthebasisofmicroelectronics.Itisknownthatthespectrumofan inﬁnitecrystalisquasi-continuous,whilethespectrumofananosizedcrystalisnoticeab...

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Model.CAlCulATIonS Letusconsiderananoobjectoftheshapeofarectangularparallelepipedwiththesides a, b, clocated along OX, OY, OZ axes, respectively. Let the potential energy of an electron in it be the sum of three terms U (®r)= U(x) + U(y) + U(z) , (2.1) where each of them is an inﬁnite deep square well, i.e., U(s) = { 0, if s = x∈[ 0, a]or y∈[ 0, b]or z∈[ ...

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(3.1) Let us analyze two cases of the magnetic ﬁeld®B orientation: a) the Landau gauge case,®A(®r) = (0, Bx, 0)

3hIFToFlevelSbyAMAgneTIC 1eld Using (2.9), we consider the shift of stationary states caused by the magnetic ﬁeld: ∆E(nanbnc)≡⟨nanbnc|ˆH|nanbnc⟩− ⣨ nanbnc ˆH0 + eℏ 2m ®B®σ nanbnc ⟩ =⟨nanbnc|ˆH1|nanbnc⟩ + Õ′ nx ny nz ⟨nanbnc|ˆH1|nxnynz⟩⟨nxnynz|ˆH1|nanbnc⟩ Ena nb nc− Enx ny nz . (3.1) Let us analyze two cases of the magnetic ﬁeld®B orientation: a) the Landa...

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ConCluSIonS The analysis of the electron spectrum of a nanoobject and of the shape of rectangular parallelepiped placed in an external magnetic ﬁeld, depending on the size of the object, indicates that (1) in the framework of perturbation theory, the correction to the spectrum appears only from the second term onward; (2) the magnitudes of the corrections...

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Tholkappiyan, A.N

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work page doi:10.1080/16583655.2018.1473946 2018

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Abdulsattar M.A., Hussein M.T., Jamal R.K., Kasim T., Adv. Mater. Phys. Chem., 2012,2, No. 4, 267–274, doi:10.4236/ampc.2012.24039

work page doi:10.4236/ampc.2012.24039 2012

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LifshitzE.,SiebbelesL.,Phys.Chem.Chem.Phys.,2014, 16,No.47,25677–25678,doi:10.1039/C4CP90174F

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Helmlinger J., Sengstock C., Groß-Heitfeld C., Mayer C., Schildhauer T.A., Köller M., Epple M., RSC Adv., 2016, 6, 18490–18501, doi:10.1039/C5RA27836H. 23704-7 B.A.,UkIyANeTS,D.6.-ATULkA

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Holovatsky V.A., Voitsekhivska O.M., Yakhnevych M.Ya., Physica E, 2017,93, 295–300, doi:10.1016/j.physe.2017.06.019

work page doi:10.1016/j.physe.2017.06.019 2017

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Holovatsky V.A., Holovatskyi I.V., Yakhnevych M.Ya., Physica E, 2018,104, 58–63, doi:10.1016/j.physe.2018.07.020

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Grundmann M., The Physics of Semiconducnors, Springer, Cham, 2006

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ЛьвIвська полIтехнIка

Prudnikov A.B., Brychkov Yu.A., Marichev O.I., Integrals and Series, CRC Press, Amsterdam, 1992. РозмIрнI ефекти в нанооб’єктах у магнIтному полI Б.А.ЛукIянець,Д.В.Матулка НацIональний унIверситет“ЛьвIвська полIтехнIка”,вул.С.Бандери,12,79000ЛьвIв,Україна Проведений теоретичний аналIз впливу розмIрIв на фIзичнI властивостI нанооб’єкта у виглядI прямоку- т...

work page 1992