Magnetic properties of electrons confined in an anisotropiccylindrical potential

Zlatko Nedelkoski a, Irina Petreskaa,b,n

a Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, P.O. Box 162, 1001 Skopje, Macedoniab Institut für Mathematische Physik, Technische Universität Braunschweig, Mendelssohnstrasse 3, 38106 Braunschweig, Germany

a r t i c l e i n f o

Article history:Received 22 January 2014Received in revised form23 May 2014Accepted 5 July 2014Available online 14 July 2014

Keywords:Anisotropic parabolic potentialEffective massMagnetic dipole momentumLow-dimensional systems

a b s t r a c t

In the present paper a theoretical model, describing the effects of external electric and magnetic fieldson an electron confined in an anisotropic parabolic potential, is considered. The exact wave functions areused to calculate electron current and orbital magnetic dipole momentum for the single electron.Exact expressions, giving the force and energy of the dipole–dipole interaction, are also determined. Further,the system is coupled to a heat bath, and mean values and fluctuations of the magnetic dipole momentum,utilizing the canonical ensemble are calculated. Influences of the temperature, as well as the externalmagnetic field, expressed via the Larmor frequency are analyzed. We also include the dependencies of themagnetic dipole momentum and its fluctuations on the effective mass of the electron, considering someexperimental values for low-dimensional systems, that are extensively studied for various applications inelectronics. Our results suggest that the average momentum or its fluctuations are strongly related to theeffective mass of the electron. Having on mind that parabolically shaped potentials have very wide area ofapplication in the low-dimensional systems, such as quantum dots and rings, carbon nanotubes, we believethat the proposed model and the consequent analysis is of general importance, since it offers exact analyticalapproach.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Modeling of real physical and chemical systems using varioustypes of parabolic potentials is widely exploited in treating a broadclass of phenomena in condensed matter physics, such as opticaltransitions in solids, molecular vibrations, phononic vibrations upto vibronic transitions, and excitonic transitions. On the otherhand, the advancement of spectroscopic techniques for studyingmolecular vibronic transitions and excitonic transitions in solidstate produces a large amount of experimental data (energyspectrum, Fermi surfaces, effective mass of confined electrons),providing a ground to develop reliable theoretical models anddetermine the limits of applicability of the known models. So far, ithas been shown in a number of excellent papers that electronicstructure, optical transition, absorption coefficients of newlyfabricated low-dimensional quantum systems, such as quantumdots, quantum wires, quantum rings, where one deals withN-electrons, confined in one or three dimensions under variouspotential shapes are successfully modeled by parabolic potentials

[1-14], to name but a few. These systems resemble many interest-ing electronic, optical and magnetic properties, thus developmentof theoretical models to rationalize and understand experimentallydetected features is of crucial importance. In Ref. [14] electronicproperties of anisotropic quantum dots are studied analytically,including the effects of the magnetic field magnitude and aniso-tropy on the energy levels. The theory and the modelingof anisotropic quantum systems have attracted much attentionrecently, because a series of interesting properties of anisotropicquantum dots have been found. For example, resonance Ramanscattering in the anisotropic quantum dots subjected to magneticfield suggests that such a quantum dot could be used as a phononmodes detector [13,15]. In Ref. [11] N-electron quantum dots withseveral shapes of confining potentials at high magnetic fields areinvestigated in the frameworks of configurations interactionscheme with a multi-centered single-electron wave functions inCartesian coordinates. In the paper, among the other shapes, theauthors also consider anisotropic two-dimensional parabolic poten-tial with Landau gauge and in order to verify the validity of theproposed method, comparison with isotropic three-dimensionalparabolic potential is provided.

Undoubtedly, the model of linear harmonic oscillator (parabolicconfining potential) with its simplicity is still an important

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Physica B

http://dx.doi.org/10.1016/j.physb.2014.07.0080921-4526/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ389 71 681 997; fax: þ389 23 228 141.E-mail address: irina.petreska@pmf.ukim.mk (I. Petreska).

Physica B 452 (2014) 113–118

reference for theoretical description of emerging quantum sys-tems. Stating the Schrödinger equation for a class of systems,manifesting some common properties, such as harmonicity andanisotropy, and even more finding analytical solutions enable ageneral and systematic approach in treating, predicting andanalyzing a broader class of problems. It is worth to emphasizehere that the aforementioned properties could not be extrapolatedfrom the bulk properties, thus finding exact forms of single-electron wave functions is of great importance, since it providesa basis set for expanding N-electron wave functions describing theaforementioned few-electron nanostructures.

In our previous work we have stated an analytically solvablemodel for an axially symmetrical anisotropic quantum oscillator inthe presence of electric and magnetic fields, obtaining the non-degenerate energy spectrum and normalized wave functions, aswell as selection rules in dipole approximation for the consideredsystem [16]. A perturbation theory approach, utilizing the derivedbasis set, was also applied to inspect the effects of symmetryremoval in the presence of external fields. In the present work weextend this model, by investigating also magnetic properties of asystem that could be described as anisotropic cylindrical oscillator.Assuming the applicability of this theoretical model in investiga-tion of the low-dimensional structures, such as quantum wires,dots and rings, we here adopt effective mass approximation andconsider the motion of electron in anisotropic parabolic potential.We use the previously derived analytical solutions by our group[16] to carry out calculations of the electron current density inelectric and magnetic fields that are further utilized to obtainorbital magnetic dipole momentum. It is worth to be mentionedhere that the orbital magnetic momentum is predominant over aspin one in some of the emerging novel materials, such as carbonnanotubes for example [17]. Obtaining normalized basis set andcalculating the current density are also very important to analyzethe edge states in quantum dots [12]. We further considerstatistical mean values of the magnetic dipole momentum andits fluctuations within a canonical ensemble approach. An exten-sive analysis of the dependence of the magnetic fluctuations onthe temperature and the external magnetic field is provided. Wehave also obtained exact expression for the magnetic dipole–dipole interaction of the confined electrons that could be furtherused to perturbatively analyze the effects of long-range interac-tions of electrons. As we mentioned above, the potential area ofapplication of such a model is wide, considering that the effectivemodel Hamiltonians of electrons in low-dimensional structuresoften contain a parabolic potential.

2. Methodology and calculations

Let us first give the statement of the model. We consider ananisotropic parabolic potential energy function of the form [18,20,21]

UðHÞ ¼mn

2ðω2

0ρ2þω2

z z2Þ; ð1Þ

where for the mass of the oscillator we use the effective mass mn ofthe electron, and ω0 and ωz are the classical angular frequencies inthe aforementioned potential. The effective mass approximationwithsuch potentials has been used in many papers treating variousshapes of semiconducting low-dimensional structures. For example,parabolically shaped confining potential is used in Ref. [1] toinvestigate the linear and the nonlinear optical absorption ofquantum dots and rings made of GaAs. Intersubband transitions insemiconducting materials and the optical properties with paraboli-cally shaped potential plus some additional terms are also studied inRef. [2,3]. Similar but isotropic case is considered in Ref. [4]. Thirdharmonic generation in GaAs/AlAs cylindrical quantum dots within

the frameworks of such models is investigated in Ref. [6–8]. InRef. [9,10] the electronic states of narrow band gap semiconductormicrocrystal, as well as interband transitions and absorption coeffi-cients in cylindrical quantum dots made of GaAs, are studied. It isworth to mention that the effective mass of electrons and holes insolids is usually ð0:01–10Þm0, where m0 stands for the free electronmass, e.g. in GaAs it is 0:067m0 [22].

Further, the considered oscillator exhibits influence from externalelectric and magnetic fields and their explicit forms are providedbelow:

E¼ Ezez; B¼ B0ez: ð2Þ

The Hamiltonian of this system, taking into account the influences ofexternal fields, is given by

H0 ¼1

2mnð� iℏ∇�qAÞ2þUðHÞ �q � z � Ez; ð3Þ

where q represents the charge of the electron. It is worth to mentionhere that the first statements of such Hamiltonians and valuableresults, widely applicable also to emerging low-dimensional quan-tum systems, date back to seminal works of Fock and Darwin[18,19]. The corresponding Schrödinger equation has the followingform:

H0Ψ ðρ;ϕ; zÞ ¼ Eð0ÞΨ ðρ;ϕ; zÞ: ð4Þ

The potential function (1) is invariant by rotation around z-axis.Likewise, both external fields are of formwhich does not destroy theinitial cylindrical symmetry of the oscillator. This fact naturallyimposes to solve Schrödinger equation in cylindrical coordinates.A suitable choice for the vector potential A which enables analyticalsolution of the Schrödinger equation is the following: A¼ ðB0=2Þρeϕ,here eϕ is an ort vector in azimuthal direction. This vector potentialmeets both required conditions div A¼ 0; rot A¼ B. As it is shownin Ref. [16] the exact wave function is given by the followingexpression:

ψ ðρ;φ; zÞ ¼ Cnρ ;jmlj;nz � eımlφ � ρjmlj � exp �12

ρ2

ρ20

þαz z� β2α2

z

� �2 !" #

�LðjmljÞnρ

ρ2

ρ20

!� Hnz

ffiffiffiffiffiffiαz

pz� β

2α2z

� �� �; ð5Þ

with normalization constant:

Cnρ ;jml j;nz ¼αz

π322nz

� �1=4

� 1

ρjmljþ10

� nρ!nz!Γ nρþjmljþ1

� �" #1=2

: ð6Þ

Regarding notation, we have introduced the following labels:λ¼ ð2mnEð0Þ=ℏ2; α0 ¼ ðmnω0Þ=ℏ; αz ¼ ðmnωzÞ=ℏ; β¼ ð2mnqEzÞ=ℏ2;ρ40 ¼ ½α2

0þðq2B20Þ=ð4ℏ2Þ��1, where ml is the magnetic quantum

number with allowed values 0; 71; 72‥. Both quantum numbersnz and nρ are allowed to values 0;1;2;‥. Eigenenergies, calculatedanalytically as well, are of the following form [16]:

Eð0Þnρ ;nz ;ml¼ ℏ½

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

0þω2q

ð2nρþjmljþ1Þ�mlωþωzð1=2þnzÞ�

�ðq2E2z Þ=ð2mnω2z Þ; ð7Þ

where we have introduced the Larmor frequency ω¼ qB0=2mn .

2.1. Current density in electric and magnetic fields. Magnetic dipolemomentum calculation

Knowing the exact quantum states in this potential as well aseigenenergies we are able to proceed finding analytical expres-sions of quantities related to the magnetic dipole momentum.A particle with charge q and effective mass mn creates current

Z. Nedelkoski, I. Petreska / Physica B 452 (2014) 113–118114

density which is given by [23]

j¼ iqℏ2mn

½Ψ∇Ψ n�Ψ n∇Ψ �� q2

mnAjΨ j2: ð8Þ

The values of magnetic and electric dipole momenta can only bedifferent than zero in z-direction. This component of the electricdipole momentum is given by [24]

⟨d⟩¼ q⟨z⟩¼ qz0 ¼q2Ezmnω2

z; ð9Þ

with z0 we denote z-coordinate of the classical equilibrium positionof the considered system. Notice that the mean value of the electricdipole momentum depends only on the electric field without anyinfluence from the external magnetic field. The magnetic dipolemomentum induced by the electron current flowing through theelement surface dS in the plane ρ–z is given by

dμ¼ πρ2jϕ dS: ð10Þ

For the total magnetic dipole momentum holds

μ¼Z

πρ2jϕ dρ dz: ð11Þ

The angular component of j can be written as

jϕ ¼ iqℏ2mnρ

Ψ∂Ψ n

∂ϕ�Ψ n∂Ψ

∂ϕ

" #�q2B0ρ

2mnjΨ j2: ð12Þ

Knowing the explicit form of the wave function we can carry outthe integrals and the magnetic dipole momentum equals

μ¼ ℏq2mn

ml�q2B0

4mnρ20ð2nρþ ml þ1Þ:

ð13Þ

If we employ the introduced label ω¼ qB0=ð2mnÞ and define

ω0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

0þω2q

, the magnetic dipole momentum along z-axis

equals

μ¼ ℏq=2mn½ml�τðωÞð2nρþjmljþ1Þ�¼ ~μ½ml�τðωÞð2nρþjmljþ1Þ�; ð14Þ

where τðωÞ ¼ω=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiω2

0þω2q

¼ω=ω0 and ~μ ¼ ℏq=2mn is equivalent of

the Bohr magneton. The expressions for μ and d can be derived

using μ¼ �∂Eð0Þðnρ ;nz ;mlÞ=∂B0; d¼ �∂Eð0Þðnρ ;nz ;mlÞ=∂Ez , as well.

2.2. Mean value and fluctuations of the magnetic dipole momentum

Considering a system in equilibrium with a heat bath, wefurther calculate the statistical mean value (expectation value) ofthe magnetic dipole momentum in the canonical ensemble frame-works. To revise, the expectation value of a quantity a for a systemin equilibrium with a heat bath at temperature T is given by

⟨a⟩¼∑naðnÞe�βHn

Z; ð15Þ

where the term β is given by β¼ 1=kT (k is Boltzmann's constant),Hn is the nth eigenvalue of the Hamiltonian and Z is the canonicalpartition function, defined as Z ¼∑ne�βHn . Since we have alreadyobtained the eigenenergies (7) for our system, the evaluation ofthe statistical mean of the magnetic dipole moment is straightfor-ward. From the expression (14) it follows that we first need to findthe mean values of the quantum numbers ⟨nρ⟩, ⟨ml⟩, and ⟨jmlj⟩.Utilizing the definition of the canonical ensemble mean values(15), the following results are obtained. For the radial quantumnumber mean value ⟨nρ⟩, one reads

⟨nρ⟩¼∑nρ ;ml ;nznρe

�βEð0Þnρ ;ml ;nz

∑nρ ;ml ;nz e�βEð0Þnρ ;ml ;nz

¼∑1

nρ ¼ 0nρe�2βℏω0nρ

∑1nρ ¼ 0e

�2βℏω0nρ¼ 1e2βℏω0 �1

: ð16Þ

Similarly, for the mean value of the angular quantum numberwe get

⟨ml⟩¼∑1

ml ¼ �1mle�βℏðjml jω0 �ωmlÞ

∑1ml ¼ �1e�βℏðjmljω0 �ωmlÞ

¼ shðβℏωÞchðβℏω0Þ; ð17Þ

and for the absolute value

⟨jmlj⟩¼∑1

ml ¼ �1jmlje�βℏðjmljω0 �ωmlÞ

∑1ml ¼ �1e�βℏðjmljω0 �ωmlÞ

¼ chðβℏω0Þ chðβℏωÞ�1shðβℏω0Þ chðβℏω0Þ : ð18Þ

Finally, for the expectations value of the magnetic dipole moment⟨μ⟩ holds

⟨μ⟩¼ ~μshðβℏωÞchðβℏω0Þ�

ωω0 cthðβℏω0Þþ2

chðβℏω0Þ chðβℏωÞ�1shð2βℏω0Þ

� �� �: ð19Þ

Labeling γ ¼ βℏω as well as γ0 ¼ βℏω0, after simple transformationsthe mean value of the magnetic dipole momentum can be written as

⟨μ⟩¼ ~μsh γch γ0

� γγ0

th γ0 þ ch γsh γ0

� �� �: ð20Þ

Following the same procedure, for the mean square magneticmoment ⟨μ2⟩, we obtain

⟨μ2⟩¼ ~μ2½⟨m2l ⟩þτ2ðωÞf4⟨n2

ρ⟩þ ⟨m2l ⟩þ1þ4⟨nρ⟩⟨jmlj⟩

þ4⟨nρ⟩þ2⟨jmlj⟩g�2τðωÞf2⟨nρ⟩⟨ml⟩þ ⟨ml⟩þ ⟨mljmlj⟩g�; ð21Þ

where ⟨nρ⟩, ⟨ml⟩ and ⟨jmlj⟩ are given by (16), (17) and (18), respec-tively, while ⟨n2

ρ⟩, ⟨m2l ⟩ and ⟨mljmlj⟩ are calculated according to

⟨n2ρ⟩¼

∑1nρ ¼ 0n

2ρe

�2βℏω0nρ

∑1nρ ¼ 0e

�2βℏω0nρ; ð22Þ

⟨m2l ⟩¼

∑1ml ¼ �1m2

l e�βℏðjml jω0 �ωmlÞ

∑1ml ¼ �1e�βℏðjmljω0 �ωmlÞ

; ð23Þ

and

⟨mljmlj⟩¼∑1

ml ¼ �1mljmlje�βℏðjml jω0 �ωmlÞ

∑1ml ¼ �1e�βℏðjml jω0 �ωmlÞ

: ð24Þ

We do not give the final form of the expression for ⟨μ2⟩ here, becauseof its robustness.

2.3. Magnetic dipolar interaction

So far we have obtained all the prerequisites to derive magneticdipolar interaction that is important for many various aspects[25,26]. It is not our intention to go into detailed analysis regardingthis issue, but considering the broad area of applicability of thisgeneral model we find it useful to present these results, as well.We consider the following configuration of magnetic dipoles: theposition of the magnetic dipoles is given by the position vectorsr1 ð0;0;0Þ and r2 ð0; r � cos α; r � sin αÞ. The magnetic field as wellas magnetic dipoles is directed along z-axis. The expression forpotential energy which belongs to the interaction of these dipolesis given by [24]

U ¼ � μ0

ð4πr3Þ½3ðμ1; r0Þðμ2; r0Þ�ðμ2;μ1Þ�; ð25Þ

where r¼ r2�r1 and r0 ¼ r=r the corresponding ort. Fromwhere itfollows

U ¼ � μ0

4πr3μ1μ2ð3 sin 2 α�1Þ: ð26Þ

When the interaction between these dipoles is small, the averagedipolar interaction will simply depend on the product ⟨μ1⟩⟨μ2⟩, soin (26) we can straightforwardly substitute for the independentmean values of the magnetic momenta μ1 and μ2 from (20), which

Z. Nedelkoski, I. Petreska / Physica B 452 (2014) 113–118 115

leads to

U ¼ �μ0 ~μ2

4πr3sh γch γ0

� γγ0

th γ0 þ ch γsh γ0

� �� �2ð3 sin 2 α�1Þ: ð27Þ

Likewise, employing the expression

F1;2 ¼3μ0

4πr5

�ðr � μ1Þ � μ2þðr � μ2Þ � μ1�2rðμ2;μ1Þ

þ5rððμ1 � rÞ; ðμ2 � rÞÞr2

�; ð28Þ

for the force between these dipoles, we can explicitly calculate itand the result is given by

F1;2 ¼ ð3μ0Þ=ð4πr4Þμ1μ2½ð cos α�5 cos α sin 2 αÞjþð3 sin α�5 sin 3 αÞk�; ð29Þ

while its magnitude is given by

F1;2 ¼ ð3μ0Þ=ð4πr4Þμ1μ2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 sin 4 α�2 sin 2 αþ1

q: ð30Þ

As in the above-mentioned case i.e. if the interaction betweenthem is small, using the magnetic momenta means from (20), oneobtains

F1;2 ¼3μ0 ~μ

2

4πr4sh γch γ0

� γγ0

th γ0 þ ch γsh γ0

� �� �2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið5 sin 4 α�2 sin 2 αþ1Þ

q:

ð31Þ

2.4. Graphical representation and discussion

In this section we represent the obtained results graphically,analyzing the temperature and magnetic field influence on theaverage magnetic dipole momentum and its fluctuations. In all theplots we use relative dimensionless quantities defined by

μrel ¼⟨μ⟩~μ; w¼ ω

ω0; t ¼ T

T0; R¼ ⟨μ2⟩�⟨μ⟩2

~μ2 ; ð32Þ

where T is the temperature and T0 ¼ ℏω0=k. The rest of thequantities are defined in Sections 2.1 and 2.2.

Canonical mean value manifests the behavior that would be aninterplay between the temperature and magnetic field effectsat the same time. Due to this, at lower magnetic fields the meanorbital magnetic momentum decreases with temperature, while athigher magnetic fields the thermal fluctuations have less signifi-cant influence on the relatively stable value of the magneticmomentum (see for example graph corresponding to t¼10in Fig. 1 or graph corresponding to w¼4, Fig. 2). Reasonablythis is reflected in the relative fluctuations (Figs. 3 and 4), where

0 1 2 3 4 5 6 7 8 9 10−6

−5

−4

−3

−2

−1

0

ω/ω0

μ rel

10

8

5

3

2

1 0.01

Fig. 1. Dependence of the relative magnetic dipole momentum μrel on the magneticfield magnitude, expressed via the relative Larmor frequency parameter w, definedin (32). The plots are represented for different values of temperature parameter t,shown next to each curve.

0 0.5 1 1.5 2 2.5 3 3.5 4−2

−1.5

−1

−0.5

0

0.5

T/T0

μ rel

0.0

0.1

0.3

0.5

1.02.0

4.0

Fig. 2. Dependence of the relative magnetic dipole momentum μrel on thetemperature, at various values of frequency parameter w, shown next toeach curve.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

T/T0

R

0.5

1

1.5

2

Fig. 3. Dependence of the relative fluctuation R of magnetic dipole momentum onthe temperature parameter t, at various values of the frequency parameter w,shown next to each curve. The parameters are defined in (32).

Z. Nedelkoski, I. Petreska / Physica B 452 (2014) 113–118116

the stabilization of the fluctuations at higher magnetic fields isobvious. Naturally, the higher the temperature, the higher thethreshold magnetic field value needed to stabilize the fluctuations.Observing these plots one can conclude that the fluctuationsmanifest Gaussian alike form.

Observing Figs. 1 and 2, one can notice that the averagemagnetic moment is negative, which implies the expectedLandau-like diamagnetism of the studied system. Also, analyzingthe analytical expressions for the magnetic momentum Eq. (13), it

can be noticed that it leads to correct results in the limiting cases,thus in the absence of the magnetic field, the magnetic momen-tum is simply a multiple of Bohr magnetons μ¼ ðℏq=2mnÞml, whilethe magnetic momentum mean value is consequently equalto zero.

Finally, influence of the effective mass on the analyzed quan-tities is also considered. To show the curve for one realisticexample, we insert the plots for effective mass of an electron inGaAs-based quantum dots and rings (mn ¼ 0:067m0). Sensitivity ofthe magnetic dipole momentum and its fluctuations on theeffective mass could be useful for the estimation of effective massin newly fabricated quantum structures (Figs. 5 and 6).

3. Conclusions

In the present paper a theoretical model, describing an electronconfined in an anisotropic parabolic potential in the presence ofmutually parallel electric and magnetic fields, was considered. Thework offers exact analytical treatment of the stated problem, thatis applicable to a wide class of phenomena in condensed matterphysics. Wave functions derived in our previous works, adjusted toeffective mass approach, were used to find exact expressions ofthe electron current density. It was further used to obtain orbitalmagnetic dipole momentum. Having the explicit expressions forthe magnetic dipole momentum, we have calculated the meanvalues in canonical ensemble approach, assuming thermal equili-brium with a heat bath. Without going into so many details, wehave also presented the exact expressions of the dipole–dipoleinteractions, which is also of relevance for model studies.

Analyzing the graphical representations of the dependencies ofthe magnetic momentum mean values and its fluctuations on themagnetic field and temperature, it was concluded that the pro-posed model leads to expected behaviors. Namely, the thermalfluctuations of the orbital magnetic momentum decrease at highmagnetic fields, while at low fields temperature has significanteffects on these fluctuations. Negative values of the magnetic

0 1 2 3 40

2

4

6

8

10

12

14

16

18

ω/ω0

R

3

2

1

0.5

Fig. 4. Dependence of the relative fluctuation R of magnetic dipole momentumon the magnetic field magnitude, expressed via the relative Larmor frequencyparameter w, at various values of the temperature parameter t, shown next to eachcurve. The parameters are defined in (32).

0 0.5 1 1.5 2 2.5 3 3.5 40

10

20

30

40

50

60

70

80

ω/ω0

R0

0.067

0.1

0.2

1

Fig. 5. Effects of the effective mass on the relative fluctuation R0, defined in units of~μ20 ¼ ðℏq=2m0Þ2. Dependence of R0 on the relative Larmor frequency w is presentedon this graph. The effective mass values are shown next to each curve. Valuemn ¼ 0:067m0 corresponds to the effective mass of electrons in GaAs. The plots aregiven for the value of the temperature parameter t¼0.5.

0 1 2 3 4 5 6−16

−14

−12

−10

−8

−6

−4

−2

0

ω/ω0

μ rel,0

102

0.5

0.1

0.067

Fig. 6. Effects of the effective mass on relative magnetic momentum μrel;0, definedin units of ~μ0 ¼ ℏq=2m0. Dependence of R0 on the relative Larmor frequency w ispresented on this graph. The effective mass values are shown next to each curve.Value mn ¼ 0:067m0 corresponds to the effective mass of electrons in GaAs. Theplots are given for the value of the temperature parameter t¼0.5.

Z. Nedelkoski, I. Petreska / Physica B 452 (2014) 113–118 117

momentum just confirm the Landau-like diamagnetism of thestudied system. Moreover, the influence of the effective massenables one to at least approximately estimate it, analyzingstatistical mean values of the magnetic dipole momentum, aswell as its fluctuations. We find this very important becausedetermination of the effective mass is never a trivial problem,while the experimental measurement of statistical averages of themagnetic dipole moments is more common.

As a concluding remark, it is important to emphasize that theadvantage of the proposed model is the general and exactanalytical treatment it offers, applicable to a wide class of systemsdescribed by anisotropic parabolically shaped potential. Quantumdots, quantum rings, carbon nanotubes, magnetic nanoparticlesetc. are some of the systems that fall within the aforementionedclass. Even though the models like one we propose are a simplifiedpicture to rather complex phenomena, they provide the necessaryreference for a comprehensive theoretical approach, enabling atthe same time rationalization of experimentally observed results,as well as prediction of the behavior of novel prospective quantumstructures. For example, introduction of complete single particlebasis set that includes the effect of anisotropy and external magneticfields could further serve as a ground to study more subtle perturba-tional effects. Even more, it is shown that many other relevantquantities, such as current density, magnetic momentum and itsstatistical treatment, could be also derived to a compact analyticalform within the frameworks of this model.

Acknowledgement

The authors are grateful to Prof. Gertrud Zwicknagl for thehelpful discussions on the subject matter and for providing someuseful references.

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