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Magnetic properties of electrons confined in an anisotropic cylindrical potential
Physica B 452 (2014) 113–118
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Magnetic properties of electrons confined in an anisotropic cylindrical potential Zlatko Nedelkoski a, Irina Petreskaa,b,n a b
Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, P.O. Box 162, 1001 Skopje, Macedonia Institut für Mathematische Physik, Technische Universität Braunschweig, Mendelssohnstrasse 3, 38106 Braunschweig, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 22 January 2014 Received in revised form 23 May 2014 Accepted 5 July 2014 Available online 14 July 2014
In the present paper a theoretical model, describing the effects of external electric and magnetic fields on an electron confined in an anisotropic parabolic potential, is considered. The exact wave functions are used 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 external magnetic field, expressed via the Larmor frequency are analyzed. We also include the dependencies of the magnetic dipole momentum and its fluctuations on the effective mass of the electron, considering some experimental values for low-dimensional systems, that are extensively studied for various applications in electronics. Our results suggest that the average momentum or its fluctuations are strongly related to the effective mass of the electron. Having on mind that parabolically shaped potentials have very wide area of application in the low-dimensional systems, such as quantum dots and rings, carbon nanotubes, we believe that the proposed model and the consequent analysis is of general importance, since it offers exact analytical approach. & 2014 Elsevier B.V. All rights reserved.
Keywords: Anisotropic parabolic potential Effective mass Magnetic dipole momentum Low-dimensional systems
1. Introduction Modeling of real physical and chemical systems using various types of parabolic potentials is widely exploited in treating a broad class of phenomena in condensed matter physics, such as optical transitions in solids, molecular vibrations, phononic vibrations up to vibronic transitions, and excitonic transitions. On the other hand, the advancement of spectroscopic techniques for studying molecular vibronic transitions and excitonic transitions in solid state produces a large amount of experimental data (energy spectrum, Fermi surfaces, effective mass of confined electrons), providing a ground to develop reliable theoretical models and determine the limits of applicability of the known models. So far, it has been shown in a number of excellent papers that electronic structure, optical transition, absorption coefficients of newly fabricated low-dimensional quantum systems, such as quantum dots, quantum wires, quantum rings, where one deals with N-electrons, confined in one or three dimensions under various potential shapes are successfully modeled by parabolic potentials
n
Corresponding author. Tel.: þ 389 71 681 997; fax: þ 389 23 228 141. E-mail address: [email protected] (I. Petreska).
http://dx.doi.org/10.1016/j.physb.2014.07.008 0921-4526/& 2014 Elsevier B.V. All rights reserved.
[1-14], to name but a few. These systems resemble many interesting electronic, optical and magnetic properties, thus development of theoretical models to rationalize and understand experimentally detected features is of crucial importance. In Ref. [14] electronic properties of anisotropic quantum dots are studied analytically, including the effects of the magnetic field magnitude and anisotropy on the energy levels. The theory and the modeling of anisotropic quantum systems have attracted much attention recently, because a series of interesting properties of anisotropic quantum dots have been found. For example, resonance Raman scattering in the anisotropic quantum dots subjected to magnetic field suggests that such a quantum dot could be used as a phonon modes detector [13,15]. In Ref. [11] N-electron quantum dots with several shapes of confining potentials at high magnetic fields are investigated in the frameworks of configurations interaction scheme with a multi-centered single-electron wave functions in Cartesian coordinates. In the paper, among the other shapes, the authors also consider anisotropic two-dimensional parabolic potential with Landau gauge and in order to verify the validity of the proposed method, comparison with isotropic three-dimensional parabolic potential is provided. Undoubtedly, the model of linear harmonic oscillator (parabolic confining potential) with its simplicity is still an important
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reference for theoretical description of emerging quantum systems. Stating the Schrödinger equation for a class of systems, manifesting some common properties, such as harmonicity and anisotropy, and even more finding analytical solutions enable a general and systematic approach in treating, predicting and analyzing a broader class of problems. It is worth to emphasize here that the aforementioned properties could not be extrapolated from the bulk properties, thus finding exact forms of singleelectron wave functions is of great importance, since it provides a basis set for expanding N-electron wave functions describing the aforementioned few-electron nanostructures. In our previous work we have stated an analytically solvable model for an axially symmetrical anisotropic quantum oscillator in the presence of electric and magnetic fields, obtaining the nondegenerate energy spectrum and normalized wave functions, as well as selection rules in dipole approximation for the considered system [16]. A perturbation theory approach, utilizing the derived basis set, was also applied to inspect the effects of symmetry removal in the presence of external fields. In the present work we extend this model, by investigating also magnetic properties of a system that could be described as anisotropic cylindrical oscillator. Assuming the applicability of this theoretical model in investigation of the low-dimensional structures, such as quantum wires, dots and rings, we here adopt effective mass approximation and consider 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 in electric and magnetic fields that are further utilized to obtain orbital magnetic dipole momentum. It is worth to be mentioned here that the orbital magnetic momentum is predominant over a spin one in some of the emerging novel materials, such as carbon nanotubes for example [17]. Obtaining normalized basis set and calculating the current density are also very important to analyze the edge states in quantum dots [12]. We further consider statistical mean values of the magnetic dipole momentum and its fluctuations within a canonical ensemble approach. An extensive analysis of the dependence of the magnetic fluctuations on the temperature and the external magnetic field is provided. We have also obtained exact expression for the magnetic dipole– dipole interaction of the confined electrons that could be further used to perturbatively analyze the effects of long-range interactions of electrons. As we mentioned above, the potential area of application of such a model is wide, considering that the effective model Hamiltonians of electrons in low-dimensional structures often contain a parabolic potential.
2. Methodology and calculations Let us first give the statement of the model. We consider an anisotropic parabolic potential energy function of the form [18,20,21] U ðHÞ ¼
mn 2 2 ðω0 ρ þ ω2z z2 Þ; 2
ð1Þ
where for the mass of the oscillator we use the effective mass mn of the electron, and ω0 and ωz are the classical angular frequencies in the aforementioned potential. The effective mass approximation with such potentials has been used in many papers treating various shapes of semiconducting low-dimensional structures. For example, parabolically shaped confining potential is used in Ref. [1] to investigate the linear and the nonlinear optical absorption of quantum dots and rings made of GaAs. Intersubband transitions in semiconducting materials and the optical properties with parabolically shaped potential plus some additional terms are also studied in Ref. [2,3]. Similar but isotropic case is considered in Ref. [4]. Third harmonic generation in GaAs/AlAs cylindrical quantum dots within
the frameworks of such models is investigated in Ref. [6–8]. In Ref. [9,10] the electronic states of narrow band gap semiconductor microcrystal, as well as interband transitions and absorption coefficients in cylindrical quantum dots made of GaAs, are studied. It is worth to mention that the effective mass of electrons and holes in solids is usually ð0:01–10Þm0 , where m0 stands for the free electron mass, e.g. in GaAs it is 0:067m0 [22]. Further, the considered oscillator exhibits influence from external electric and magnetic fields and their explicit forms are provided below: E ¼ Ez ez ;
B ¼ B0 ez :
ð2Þ
The Hamiltonian of this system, taking into account the influences of external fields, is given by 1 H^ 0 ¼ ð iℏ∇ qAÞ2 þU ðHÞ q z Ez ; 2mn
ð3Þ
where q represents the charge of the electron. It is worth to mention here that the first statements of such Hamiltonians and valuable results, widely applicable also to emerging low-dimensional quantum systems, date back to seminal works of Fock and Darwin [18,19]. The corresponding Schrödinger equation has the following form: H^ 0 Ψ ðρ; ϕ; zÞ ¼ Eð0Þ Ψ ðρ; ϕ; zÞ:
ð4Þ
The potential function (1) is invariant by rotation around z-axis. Likewise, both external fields are of form which does not destroy the initial cylindrical symmetry of the oscillator. This fact naturally imposes to solve Schrödinger equation in cylindrical coordinates. A suitable choice for the vector potential A which enables analytical solution of the Schrödinger equation is the following: A ¼ ðB0 =2Þρeϕ , here eϕ is an ort vector in azimuthal direction. This vector potential meets both required conditions div A ¼ 0; rot A ¼ B. As it is shown in Ref. [16] the exact wave function is given by the following expression: " !# 1 ρ2 β 2 ψ ðρ; φ; zÞ ¼ C nρ ;jml j;nz eıml φ ρjml j exp þ α z z 2 ρ20 2α2z ! pffiffiffiffiffiffi ρ2 β l jÞ αz z 2 ; ð5Þ H nz Lnðjm ρ 2α z ρ20 with normalization constant: C nρ ;jml j;nz ¼
αz
π 3 22nz
1=4
1
jm j þ 1
ρ0
l
"
#1=2 nρ ! : nz !Γ nρ þ jml jþ 1
ð6Þ
Regarding notation, we have introduced the following labels: λ ¼ ð2mn Eð0Þ =ℏ2 ; α0 ¼ ðmn ω0 Þ=ℏ; αz ¼ ðmn ωz Þ=ℏ; β ¼ ð2mn qEz Þ=ℏ2 ; ρ40 ¼ ½α20 þ ðq2 B20 Þ=ð4ℏ2 Þ 1 , where ml is the magnetic quantum number with allowed values 0; 71; 7 2‥. Both quantum numbers nz and nρ are allowed to values 0; 1; 2; ‥. Eigenenergies, calculated analytically as well, are of the following form [16]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Eð0Þ nρ ;nz ;ml ¼ ℏ½ ω0 þ ω ð2nρ þ jml j þ 1Þ ml ω þ ωz ð1=2 þ nz Þ ðq2 E2z Þ=ð2mn ω2z Þ;
ð7Þ
where we have introduced the Larmor frequency ω ¼ qB0 =2mn . 2.1. Current density in electric and magnetic fields. Magnetic dipole momentum calculation Knowing the exact quantum states in this potential as well as eigenenergies we are able to proceed finding analytical expressions 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) 113–118
Similarly, for the mean value of the angular quantum number we get
density which is given by [23] j¼
115
2
iqℏ q n n ½Ψ ∇Ψ Ψ ∇Ψ n AjΨ j2 : 2mn m
ð8Þ
The values of magnetic and electric dipole momenta can only be different than zero in z-direction. This component of the electric dipole momentum is given by [24] q2 Ez 〈d〉 ¼ q〈z〉 ¼ qz0 ¼ n 2 ; m ωz
ð9Þ
with z0 we denote z-coordinate of the classical equilibrium position of the considered system. Notice that the mean value of the electric dipole momentum depends only on the electric field without any influence from the external magnetic field. The magnetic dipole momentum induced by the electron current flowing through the element surface dS in the plane ρ–z is given by
〈ml 〉 ¼
βℏðjml jω0 ωml Þ 1 ml e βℏðjml jω0 ωml Þ ∑1 ml ¼ 1 e
∑1 ml ¼
¼
shðβℏωÞ ; chðβℏω0 Þ
ð17Þ
and for the absolute value 〈jml j〉 ¼
βℏðjml jω0 ωml Þ 1 jml je βℏðjml jω0 ωml Þ ∑1 ml ¼ 1 e
∑1 ml ¼
¼
chðβ ℏω0 Þ chðβℏωÞ 1 : shðβ ℏω0 Þ chðβℏω0 Þ
ð18Þ
Finally, for the expectations value of the magnetic dipole moment 〈μ〉 holds shðβ ℏωÞ ω chðβℏω0 Þ chðβ ℏωÞ 1 0 : ð19Þ cthð β ℏ ω Þ þ2 〈μ〉 ¼ μ~ chðβ ℏω0 Þ ω0 shð2βℏω0 Þ
dμ ¼ πρ2 jϕ dS:
ð10Þ
For the total magnetic dipole momentum holds Z μ ¼ πρ2 jϕ dρ dz:
Labeling γ ¼ βℏω as well as γ 0 ¼ β ℏω0 , after simple transformations the mean value of the magnetic dipole momentum can be written as sh γ γ ch γ 0 th γ þ 〈μ〉 ¼ μ~ : ð20Þ ch γ 0 γ 0 sh γ 0
ð11Þ
Following the same procedure, for the mean square magnetic moment 〈μ2 〉, we obtain 〈μ2 〉 ¼ μ~ 2 ½〈m2l 〉 þ τ2 ðωÞf4〈n2ρ 〉 þ 〈m2l 〉þ 1 þ 4〈nρ 〉〈jml j〉
The angular component of j can be written as " # n iqℏ ∂Ψ q2 B0 ρ n ∂Ψ Ψ jΨ j2 : Ψ jϕ ¼ 2mn ρ 2mn ∂ϕ ∂ϕ
þ 4〈nρ 〉þ 2〈jml j〉g
ð12Þ
2τðωÞf2〈nρ 〉〈ml 〉 þ 〈ml 〉 þ 〈ml jml j〉g;
ð21Þ
Knowing the explicit form of the wave function we can carry out the integrals and the magnetic dipole momentum equals
where 〈nρ 〉, 〈ml 〉 and 〈jml j〉 are given by (16), (17) and (18), respectively, while 〈n2ρ 〉, 〈m2l 〉 and 〈ml jml j〉 are calculated according to
ℏq q2 B μ ¼ n ml n0 ρ20 ð2nρ þ ml þ 1Þ: 2m 4m
ð13Þ
〈n2ρ 〉 ¼
If we employ the introduced label ω ¼ qB0 =ð2mn Þ and define qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω0 ¼ ω20 þ ω2 , the magnetic dipole momentum along z-axis
〈m2l 〉 ¼
0
equals ð14Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where τðωÞ ¼ ω= ω20 þ ω2 ¼ ω=ω0 and μ~ ¼ ℏq=2mn is equivalent of
μ and d can be derived
ð0Þ using μ ¼ ∂Eð0Þ ðnρ ;nz ;ml Þ =∂B0 ; d ¼ ∂E ðnρ ;nz ;ml Þ =∂E z , as well.
2.2. Mean value and fluctuations of the magnetic dipole momentum Considering a system in equilibrium with a heat bath, we further calculate the statistical mean value (expectation value) of the magnetic dipole momentum in the canonical ensemble frameworks. To revise, the expectation value of a quantity a for a system in equilibrium with a heat bath at temperature T is given by 〈a〉 ¼
∑n aðnÞe βHn ; Z
ð15Þ
where the term β is given by β ¼ 1=kT (k is Boltzmann's constant), Hn is the nth eigenvalue of the Hamiltonian and Z is the canonical partition function, defined as Z ¼ ∑n e βHn . Since we have already obtained the eigenenergies (7) for our system, the evaluation of the statistical mean of the magnetic dipole moment is straightforward. From the expression (14) it follows that we first need to find the mean values of the quantum numbers 〈nρ 〉, 〈ml 〉, and 〈jml j〉. Utilizing the definition of the canonical ensemble mean values (15), the following results are obtained. For the radial quantum number mean value 〈nρ 〉, one reads 〈nρ 〉 ¼
∑nρ ;ml ;nz nρ e ∑nρ ;ml ;nz e
βEð0Þ nρ ;m ;nz l
βEð0Þ nρ ;m ;nz l
2βℏω nρ ∑1 nρ ¼ 0 nρ e 0
¼
2βℏω nρ ∑1 nρ ¼ 0 e 0
;
2 βℏðjml jω0 ωml Þ 1 ml e ; βℏðjml jω0 ωml Þ ∑1 ml ¼ 1 e
∑1 ml ¼
ð22Þ
ð23Þ
and
μ ¼ ℏq=2mn ½ml τðωÞð2nρ þjml j þ1Þ ¼ μ~ ½ml τðωÞð2nρ þ jml j þ 1Þ;
the Bohr magneton. The expressions for
2 2βℏω nρ ∑1 nρ ¼ 0 nρ e
2βℏω nρ ∑1 nρ ¼ 0 e 0
¼
1 e2βℏω0 1
:
ð16Þ
〈ml jml j〉 ¼
β ℏðjml jω0 ωml Þ 1 ml jml je : 0 1 β ℏðjm l jω ωml Þ ∑ml ¼ 1 e
∑1 ml ¼
ð24Þ
We do not give the final form of the expression for 〈μ2 〉 here, because of its robustness. 2.3. Magnetic dipolar interaction So far we have obtained all the prerequisites to derive magnetic dipolar interaction that is important for many various aspects [25,26]. It is not our intention to go into detailed analysis regarding this issue, but considering the broad area of applicability of this general model we find it useful to present these results, as well. We consider the following configuration of magnetic dipoles: the position of the magnetic dipoles is given by the position vectors r1 ð0; 0; 0Þ and r2 ð0; r cos α; r sin αÞ. The magnetic field as well as magnetic dipoles is directed along z-axis. The expression for potential energy which belongs to the interaction of these dipoles is given by [24] U¼
μ0
ð4π r 3 Þ
½3ðμ1 ; r0 Þðμ2 ; r0 Þ ðμ2 ; μ1 Þ;
ð25Þ
where r ¼ r2 r1 and r0 ¼ r=r the corresponding ort. From where it follows U¼
μ0
4π r 3
μ1 μ2 ð3 sin 2 α 1Þ:
ð26Þ
When the interaction between these dipoles is small, the average dipolar interaction will simply depend on the product 〈μ1 〉〈μ2 〉, so in (26) we can straightforwardly substitute for the independent mean values of the magnetic momenta μ1 and μ2 from (20), which
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Z. Nedelkoski, I. Petreska / Physica B 452 (2014) 113–118
leads to 2
U¼
μ0 μ~ 4π r 3
2 sh γ γ ch γ 0 th γ 0 þ ð3 sin 2 α 1Þ: 0 0 ch γ γ sh γ
ð27Þ
Likewise, employing the expression 3μ0 ðr μ1 Þ μ2 þ ðr μ2 Þ μ1 2rðμ2 ; μ1 Þ F1;2 ¼ 4π r 5 5rððμ1 rÞ; ðμ2 rÞÞ ; þ 2 r
ð28Þ
for the force between these dipoles, we can explicitly calculate it and the result is given by
where T is the temperature and T 0 ¼ ℏω0 =k. The rest of the quantities are defined in Sections 2.1 and 2.2. Canonical mean value manifests the behavior that would be an interplay between the temperature and magnetic field effects at the same time. Due to this, at lower magnetic fields the mean orbital magnetic momentum decreases with temperature, while at higher magnetic fields the thermal fluctuations have less significant influence on the relatively stable value of the magnetic momentum (see for example graph corresponding to t¼ 10 in Fig. 1 or graph corresponding to w¼ 4, Fig. 2). Reasonably this is reflected in the relative fluctuations (Figs. 3 and 4), where 0.5
F1;2 ¼ ð3μ0 Þ=ð4π r 4 Þμ1 μ2 ½ð cos α 5 cos α sin 2 αÞj þ ð3 sin α 5 sin 3 αÞk;
ð29Þ 0.1
ð30Þ 0.3 −0.5 1.0 2.0
−1.5
In this section we represent the obtained results graphically, analyzing the temperature and magnetic field influence on the average magnetic dipole momentum and its fluctuations. In all the plots we use relative dimensionless quantities defined by w¼
ω ; ω0
t¼
T ; T0
R¼
4.0
−1
2.4. Graphical representation and discussion
〈μ〉 ; μ~
0.5
μrel
As in the above-mentioned case i.e. if the interaction between them is small, using the magnetic momenta means from (20), one obtains 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3μ μ~ 2 sh γ γ ch γ F 1;2 ¼ 0 4 0 th γ 0 þ ð5 sin 4 α 2 sin 2 α þ 1Þ: 0 0 sh γ 4π r ch γ γ ð31Þ
μrel ¼
0.0
0
while its magnitude is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 F 1;2 ¼ ð3μ0 Þ=ð4π r 4 Þμ1 μ2 5 sin α 2 sin α þ 1:
〈μ2 〉 〈μ〉2 ; μ~ 2
ð32Þ
−2
0
0.5
1
1.5
2
2.5
3
3.5
4
T/T0
Fig. 2. Dependence of the relative magnetic dipole momentum μrel on the temperature, at various values of frequency parameter w, shown next to each curve.
0
7
1
−1
0.01
0.5 6
2 3
5
−2
1
4
−3
R
μrel
5
3 −4
1.5
8
2 −5
2
10
1 −6
0
1
2
3
4
5
6
7
8
9
10
ω/ω0
Fig. 1. Dependence of the relative magnetic dipole momentum μrel on the magnetic field magnitude, expressed via the relative Larmor frequency parameter w, defined in (32). The plots are represented for different values of temperature parameter t, shown next to each curve.
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
T/T0
Fig. 3. Dependence of the relative fluctuation R of magnetic dipole momentum on the 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) 113–118
the stabilization of the fluctuations at higher magnetic fields is obvious. Naturally, the higher the temperature, the higher the threshold magnetic field value needed to stabilize the fluctuations. Observing these plots one can conclude that the fluctuations manifest Gaussian alike form. Observing Figs. 1 and 2, one can notice that the average magnetic moment is negative, which implies the expected Landau-like diamagnetism of the studied system. Also, analyzing the analytical expressions for the magnetic momentum Eq. (13), it
0 2 −2
3
16
0.5
−6
−8
0.1
−10
14
10
−4
μrel,0
18
117
−12
12 −14
0.067
R
10 −16
8
2
1 0.5 0
1
2 ω/ω0
3
4
Fig. 4. Dependence of the relative fluctuation R of magnetic dipole momentum on the magnetic field magnitude, expressed via the relative Larmor frequency parameter w, at various values of the temperature parameter t, shown next to each curve. The parameters are defined in (32).
80
70
2
3
4
5
6
Fig. 6. Effects of the effective mass on relative magnetic momentum μrel;0 , defined in units of μ~0 ¼ ℏq=2m0 . Dependence of R0 on the relative Larmor frequency w is presented 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. The plots are given for the value of the temperature parameter t¼ 0.5.
4
0
1
ω/ω0
6
2
0
can be noticed that it leads to correct results in the limiting cases, thus in the absence of the magnetic field, the magnetic momentum is simply a multiple of Bohr magnetons μ ¼ ðℏq=2mn Þml , while the magnetic momentum mean value is consequently equal to zero. Finally, influence of the effective mass on the analyzed quantities is also considered. To show the curve for one realistic example, we insert the plots for effective mass of an electron in GaAs-based quantum dots and rings (mn ¼ 0:067m0 ). Sensitivity of the magnetic dipole momentum and its fluctuations on the effective mass could be useful for the estimation of effective mass in newly fabricated quantum structures (Figs. 5 and 6).
0.067
3. Conclusions 60
R0
50
40
30
0.1
20
10
0
0.2 1 0
0.5
1
1.5
2
2.5
3
3.5
4
ω/ω0
Fig. 5. Effects of the effective mass on the relative fluctuation R0, defined in units of μ~20 ¼ ðℏq=2m0 Þ2 . Dependence of R0 on the relative Larmor frequency w is presented 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. The plots are given for the value of the temperature parameter t¼ 0.5.
In the present paper a theoretical model, describing an electron confined in an anisotropic parabolic potential in the presence of mutually parallel electric and magnetic fields, was considered. The work offers exact analytical treatment of the stated problem, that is applicable to a wide class of phenomena in condensed matter physics. Wave functions derived in our previous works, adjusted to effective mass approach, were used to find exact expressions of the electron current density. It was further used to obtain orbital magnetic dipole momentum. Having the explicit expressions for the magnetic dipole momentum, we have calculated the mean values in canonical ensemble approach, assuming thermal equilibrium with a heat bath. Without going into so many details, we have also presented the exact expressions of the dipole–dipole interactions, which is also of relevance for model studies. Analyzing the graphical representations of the dependencies of the magnetic momentum mean values and its fluctuations on the magnetic field and temperature, it was concluded that the proposed model leads to expected behaviors. Namely, the thermal fluctuations of the orbital magnetic momentum decrease at high magnetic fields, while at low fields temperature has significant effects on these fluctuations. Negative values of the magnetic
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Z. Nedelkoski, I. Petreska / Physica B 452 (2014) 113–118
momentum just confirm the Landau-like diamagnetism of the studied system. Moreover, the influence of the effective mass enables one to at least approximately estimate it, analyzing statistical mean values of the magnetic dipole momentum, as well as its fluctuations. We find this very important because determination of the effective mass is never a trivial problem, while the experimental measurement of statistical averages of the magnetic dipole moments is more common. As a concluding remark, it is important to emphasize that the advantage of the proposed model is the general and exact analytical treatment it offers, applicable to a wide class of systems described by anisotropic parabolically shaped potential. Quantum dots, quantum rings, carbon nanotubes, magnetic nanoparticles etc. are some of the systems that fall within the aforementioned class. Even though the models like one we propose are a simplified picture to rather complex phenomena, they provide the necessary reference for a comprehensive theoretical approach, enabling at the same time rationalization of experimentally observed results, as well as prediction of the behavior of novel prospective quantum structures. For example, introduction of complete single particle basis set that includes the effect of anisotropy and external magnetic fields could further serve as a ground to study more subtle perturbational effects. Even more, it is shown that many other relevant quantities, such as current density, magnetic momentum and its statistical treatment, could be also derived to a compact analytical form within the frameworks of this model. Acknowledgement The authors are grateful to Prof. Gertrud Zwicknagl for the helpful discussions on the subject matter and for providing some useful references.
References [1] [2] [3] [4] [5] [6]
C.M. Duque, A.L. Morales, M.E. Mora-Ramos, J. Lumin. 143 (2013) 81. G. Liu, K. Guo, C. Wang, Physica B 407 (2012) 2334. G. Liu, K. Guo, H. Hassanabadi, Physica B 407 (2012) 3676. S.M. Ikhdair, M. Hamzavi, R. Sever, Physica B 407 (2012) 4523. T. Chenn, W. Xie, S. Liang, Physica B 407 (2012) 263. S. Shao, K.-X. Guo, Z.-H. Zhang, N. Li, Chao Peng, Solid State Commun. 151 (2011) 289. [7] Shuai Shao, Kang-Xian Guo, Zhi-Hai Zhang, Ning Li, Chao Peng, Superlattices Microstruct. 48 (2010) 541. [8] Guanghui Wang, Qi Guo, Physica B 403 (2008) 37. [9] M.S. Atoyan, E.M. Kazaryan, H.A. Sarkisyan, Physica E 22 (2004) 860. [10] E.M. Kazaryan, L.S. Petrosyan, H.A. Sarkisyan, Physica E 11 (2001) 362. [11] B. Szafran, F.M. Peeters, S. Bednarek, J. Adamowski, Phys. Rev. B 69 (2004) 125344. [12] C.S. Lent, Phys. Rev. B 43 (5) (1991) 4179. [13] T. Tanaka, Y. Arakawa, Phys. Rev. B 50 (11) (1994) 7719. [14] A.V. Madhav, T. Chakraborty, Phys. Rev. B 49 (12) (1994) 8163. [15] A.V. Shorokhov, V.A. Margulis, arXiv:1001.1037v1, 2010. [16] I. Petreska, T. Sandev, Z. Nedelkoski, Lj. Pejov, Cent. Eur. J. Phys. 11 (4) (2013) 412. [17] E.D. Minot, Y. Yaish, V. Sazonova, P.L. McEuen, Nature 428 (2004) 536. [18] V. Fock, Z. Phys. 47 (1928) 446. [19] C.G. Darwin, Proc. Camb. Philos. Soc. 27 (1930) 86. [20] A. Sommerfeld, Wave Mechanics, E.P. Dutton and Co. Inc., New York, 1930. [21] L. Pauling, E. Bright Wilson Jr., Introduction to Quantum Mechanics with Applications to Chemistry, The McGraw-Hill Book Co., New York, 1935. [22] S.M. Sze, K.K. Ng, Physics of Semiconductor Devices, Third ed., John Willey and Sons, Inc., Hoboken, New Jersey, 2007. [23] Leslie E. Balentine, Quantum Mechanics—A Modern Development, World Scientific Publishing Co. Pte. Ltd., Singapore, 1998. [24] D.J. Griffiths, Introduction to Electrodynamics, Prentice Hall Inc., Upper Saddle River, New Jersey, 1981. [25] J. García-Otero, M. Porto, J. Rivas, A. Bunde, Phys. Rev. Lett. 84 (1) (1999) 167. [26] S. Giovanazzi, A. Görlitz, T. Pfau, Phys. Rev. Lett. 89 (13) (2002) 130401-1.
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Physica B journal homepage: www.elsevier.com/locate/physb
Magnetic properties of electrons confined in an anisotropic cylindrical potential Zlatko Nedelkoski a, Irina Petreskaa,b,n a b
Institute of Physics, Faculty of Natural Sciences and Mathematics, Ss. Cyril and Methodius University, P.O. Box 162, 1001 Skopje, Macedonia Institut für Mathematische Physik, Technische Universität Braunschweig, Mendelssohnstrasse 3, 38106 Braunschweig, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 22 January 2014 Received in revised form 23 May 2014 Accepted 5 July 2014 Available online 14 July 2014
In the present paper a theoretical model, describing the effects of external electric and magnetic fields on an electron confined in an anisotropic parabolic potential, is considered. The exact wave functions are used 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 external magnetic field, expressed via the Larmor frequency are analyzed. We also include the dependencies of the magnetic dipole momentum and its fluctuations on the effective mass of the electron, considering some experimental values for low-dimensional systems, that are extensively studied for various applications in electronics. Our results suggest that the average momentum or its fluctuations are strongly related to the effective mass of the electron. Having on mind that parabolically shaped potentials have very wide area of application in the low-dimensional systems, such as quantum dots and rings, carbon nanotubes, we believe that the proposed model and the consequent analysis is of general importance, since it offers exact analytical approach. & 2014 Elsevier B.V. All rights reserved.
Keywords: Anisotropic parabolic potential Effective mass Magnetic dipole momentum Low-dimensional systems
1. Introduction Modeling of real physical and chemical systems using various types of parabolic potentials is widely exploited in treating a broad class of phenomena in condensed matter physics, such as optical transitions in solids, molecular vibrations, phononic vibrations up to vibronic transitions, and excitonic transitions. On the other hand, the advancement of spectroscopic techniques for studying molecular vibronic transitions and excitonic transitions in solid state produces a large amount of experimental data (energy spectrum, Fermi surfaces, effective mass of confined electrons), providing a ground to develop reliable theoretical models and determine the limits of applicability of the known models. So far, it has been shown in a number of excellent papers that electronic structure, optical transition, absorption coefficients of newly fabricated low-dimensional quantum systems, such as quantum dots, quantum wires, quantum rings, where one deals with N-electrons, confined in one or three dimensions under various potential shapes are successfully modeled by parabolic potentials
n
Corresponding author. Tel.: þ 389 71 681 997; fax: þ 389 23 228 141. E-mail address: [email protected] (I. Petreska).
http://dx.doi.org/10.1016/j.physb.2014.07.008 0921-4526/& 2014 Elsevier B.V. All rights reserved.
[1-14], to name but a few. These systems resemble many interesting electronic, optical and magnetic properties, thus development of theoretical models to rationalize and understand experimentally detected features is of crucial importance. In Ref. [14] electronic properties of anisotropic quantum dots are studied analytically, including the effects of the magnetic field magnitude and anisotropy on the energy levels. The theory and the modeling of anisotropic quantum systems have attracted much attention recently, because a series of interesting properties of anisotropic quantum dots have been found. For example, resonance Raman scattering in the anisotropic quantum dots subjected to magnetic field suggests that such a quantum dot could be used as a phonon modes detector [13,15]. In Ref. [11] N-electron quantum dots with several shapes of confining potentials at high magnetic fields are investigated in the frameworks of configurations interaction scheme with a multi-centered single-electron wave functions in Cartesian coordinates. In the paper, among the other shapes, the authors also consider anisotropic two-dimensional parabolic potential with Landau gauge and in order to verify the validity of the proposed method, comparison with isotropic three-dimensional parabolic potential is provided. Undoubtedly, the model of linear harmonic oscillator (parabolic confining potential) with its simplicity is still an important
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reference for theoretical description of emerging quantum systems. Stating the Schrödinger equation for a class of systems, manifesting some common properties, such as harmonicity and anisotropy, and even more finding analytical solutions enable a general and systematic approach in treating, predicting and analyzing a broader class of problems. It is worth to emphasize here that the aforementioned properties could not be extrapolated from the bulk properties, thus finding exact forms of singleelectron wave functions is of great importance, since it provides a basis set for expanding N-electron wave functions describing the aforementioned few-electron nanostructures. In our previous work we have stated an analytically solvable model for an axially symmetrical anisotropic quantum oscillator in the presence of electric and magnetic fields, obtaining the nondegenerate energy spectrum and normalized wave functions, as well as selection rules in dipole approximation for the considered system [16]. A perturbation theory approach, utilizing the derived basis set, was also applied to inspect the effects of symmetry removal in the presence of external fields. In the present work we extend this model, by investigating also magnetic properties of a system that could be described as anisotropic cylindrical oscillator. Assuming the applicability of this theoretical model in investigation of the low-dimensional structures, such as quantum wires, dots and rings, we here adopt effective mass approximation and consider 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 in electric and magnetic fields that are further utilized to obtain orbital magnetic dipole momentum. It is worth to be mentioned here that the orbital magnetic momentum is predominant over a spin one in some of the emerging novel materials, such as carbon nanotubes for example [17]. Obtaining normalized basis set and calculating the current density are also very important to analyze the edge states in quantum dots [12]. We further consider statistical mean values of the magnetic dipole momentum and its fluctuations within a canonical ensemble approach. An extensive analysis of the dependence of the magnetic fluctuations on the temperature and the external magnetic field is provided. We have also obtained exact expression for the magnetic dipole– dipole interaction of the confined electrons that could be further used to perturbatively analyze the effects of long-range interactions of electrons. As we mentioned above, the potential area of application of such a model is wide, considering that the effective model Hamiltonians of electrons in low-dimensional structures often contain a parabolic potential.
2. Methodology and calculations Let us first give the statement of the model. We consider an anisotropic parabolic potential energy function of the form [18,20,21] U ðHÞ ¼
mn 2 2 ðω0 ρ þ ω2z z2 Þ; 2
ð1Þ
where for the mass of the oscillator we use the effective mass mn of the electron, and ω0 and ωz are the classical angular frequencies in the aforementioned potential. The effective mass approximation with such potentials has been used in many papers treating various shapes of semiconducting low-dimensional structures. For example, parabolically shaped confining potential is used in Ref. [1] to investigate the linear and the nonlinear optical absorption of quantum dots and rings made of GaAs. Intersubband transitions in semiconducting materials and the optical properties with parabolically shaped potential plus some additional terms are also studied in Ref. [2,3]. Similar but isotropic case is considered in Ref. [4]. Third harmonic generation in GaAs/AlAs cylindrical quantum dots within
the frameworks of such models is investigated in Ref. [6–8]. In Ref. [9,10] the electronic states of narrow band gap semiconductor microcrystal, as well as interband transitions and absorption coefficients in cylindrical quantum dots made of GaAs, are studied. It is worth to mention that the effective mass of electrons and holes in solids is usually ð0:01–10Þm0 , where m0 stands for the free electron mass, e.g. in GaAs it is 0:067m0 [22]. Further, the considered oscillator exhibits influence from external electric and magnetic fields and their explicit forms are provided below: E ¼ Ez ez ;
B ¼ B0 ez :
ð2Þ
The Hamiltonian of this system, taking into account the influences of external fields, is given by 1 H^ 0 ¼ ð iℏ∇ qAÞ2 þU ðHÞ q z Ez ; 2mn
ð3Þ
where q represents the charge of the electron. It is worth to mention here that the first statements of such Hamiltonians and valuable results, widely applicable also to emerging low-dimensional quantum systems, date back to seminal works of Fock and Darwin [18,19]. The corresponding Schrödinger equation has the following form: H^ 0 Ψ ðρ; ϕ; zÞ ¼ Eð0Þ Ψ ðρ; ϕ; zÞ:
ð4Þ
The potential function (1) is invariant by rotation around z-axis. Likewise, both external fields are of form which does not destroy the initial cylindrical symmetry of the oscillator. This fact naturally imposes to solve Schrödinger equation in cylindrical coordinates. A suitable choice for the vector potential A which enables analytical solution of the Schrödinger equation is the following: A ¼ ðB0 =2Þρeϕ , here eϕ is an ort vector in azimuthal direction. This vector potential meets both required conditions div A ¼ 0; rot A ¼ B. As it is shown in Ref. [16] the exact wave function is given by the following expression: " !# 1 ρ2 β 2 ψ ðρ; φ; zÞ ¼ C nρ ;jml j;nz eıml φ ρjml j exp þ α z z 2 ρ20 2α2z ! pffiffiffiffiffiffi ρ2 β l jÞ αz z 2 ; ð5Þ H nz Lnðjm ρ 2α z ρ20 with normalization constant: C nρ ;jml j;nz ¼
αz
π 3 22nz
1=4
1
jm j þ 1
ρ0
l
"
#1=2 nρ ! : nz !Γ nρ þ jml jþ 1
ð6Þ
Regarding notation, we have introduced the following labels: λ ¼ ð2mn Eð0Þ =ℏ2 ; α0 ¼ ðmn ω0 Þ=ℏ; αz ¼ ðmn ωz Þ=ℏ; β ¼ ð2mn qEz Þ=ℏ2 ; ρ40 ¼ ½α20 þ ðq2 B20 Þ=ð4ℏ2 Þ 1 , where ml is the magnetic quantum number with allowed values 0; 71; 7 2‥. Both quantum numbers nz and nρ are allowed to values 0; 1; 2; ‥. Eigenenergies, calculated analytically as well, are of the following form [16]: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Eð0Þ nρ ;nz ;ml ¼ ℏ½ ω0 þ ω ð2nρ þ jml j þ 1Þ ml ω þ ωz ð1=2 þ nz Þ ðq2 E2z Þ=ð2mn ω2z Þ;
ð7Þ
where we have introduced the Larmor frequency ω ¼ qB0 =2mn . 2.1. Current density in electric and magnetic fields. Magnetic dipole momentum calculation Knowing the exact quantum states in this potential as well as eigenenergies we are able to proceed finding analytical expressions 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) 113–118
Similarly, for the mean value of the angular quantum number we get
density which is given by [23] j¼
115
2
iqℏ q n n ½Ψ ∇Ψ Ψ ∇Ψ n AjΨ j2 : 2mn m
ð8Þ
The values of magnetic and electric dipole momenta can only be different than zero in z-direction. This component of the electric dipole momentum is given by [24] q2 Ez 〈d〉 ¼ q〈z〉 ¼ qz0 ¼ n 2 ; m ωz
ð9Þ
with z0 we denote z-coordinate of the classical equilibrium position of the considered system. Notice that the mean value of the electric dipole momentum depends only on the electric field without any influence from the external magnetic field. The magnetic dipole momentum induced by the electron current flowing through the element surface dS in the plane ρ–z is given by
〈ml 〉 ¼
βℏðjml jω0 ωml Þ 1 ml e βℏðjml jω0 ωml Þ ∑1 ml ¼ 1 e
∑1 ml ¼
¼
shðβℏωÞ ; chðβℏω0 Þ
ð17Þ
and for the absolute value 〈jml j〉 ¼
βℏðjml jω0 ωml Þ 1 jml je βℏðjml jω0 ωml Þ ∑1 ml ¼ 1 e
∑1 ml ¼
¼
chðβ ℏω0 Þ chðβℏωÞ 1 : shðβ ℏω0 Þ chðβℏω0 Þ
ð18Þ
Finally, for the expectations value of the magnetic dipole moment 〈μ〉 holds shðβ ℏωÞ ω chðβℏω0 Þ chðβ ℏωÞ 1 0 : ð19Þ cthð β ℏ ω Þ þ2 〈μ〉 ¼ μ~ chðβ ℏω0 Þ ω0 shð2βℏω0 Þ
dμ ¼ πρ2 jϕ dS:
ð10Þ
For the total magnetic dipole momentum holds Z μ ¼ πρ2 jϕ dρ dz:
Labeling γ ¼ βℏω as well as γ 0 ¼ β ℏω0 , after simple transformations the mean value of the magnetic dipole momentum can be written as sh γ γ ch γ 0 th γ þ 〈μ〉 ¼ μ~ : ð20Þ ch γ 0 γ 0 sh γ 0
ð11Þ
Following the same procedure, for the mean square magnetic moment 〈μ2 〉, we obtain 〈μ2 〉 ¼ μ~ 2 ½〈m2l 〉 þ τ2 ðωÞf4〈n2ρ 〉 þ 〈m2l 〉þ 1 þ 4〈nρ 〉〈jml j〉
The angular component of j can be written as " # n iqℏ ∂Ψ q2 B0 ρ n ∂Ψ Ψ jΨ j2 : Ψ jϕ ¼ 2mn ρ 2mn ∂ϕ ∂ϕ
þ 4〈nρ 〉þ 2〈jml j〉g
ð12Þ
2τðωÞf2〈nρ 〉〈ml 〉 þ 〈ml 〉 þ 〈ml jml j〉g;
ð21Þ
Knowing the explicit form of the wave function we can carry out the integrals and the magnetic dipole momentum equals
where 〈nρ 〉, 〈ml 〉 and 〈jml j〉 are given by (16), (17) and (18), respectively, while 〈n2ρ 〉, 〈m2l 〉 and 〈ml jml j〉 are calculated according to
ℏq q2 B μ ¼ n ml n0 ρ20 ð2nρ þ ml þ 1Þ: 2m 4m
ð13Þ
〈n2ρ 〉 ¼
If we employ the introduced label ω ¼ qB0 =ð2mn Þ and define qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω0 ¼ ω20 þ ω2 , the magnetic dipole momentum along z-axis
〈m2l 〉 ¼
0
equals ð14Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where τðωÞ ¼ ω= ω20 þ ω2 ¼ ω=ω0 and μ~ ¼ ℏq=2mn is equivalent of
μ and d can be derived
ð0Þ using μ ¼ ∂Eð0Þ ðnρ ;nz ;ml Þ =∂B0 ; d ¼ ∂E ðnρ ;nz ;ml Þ =∂E z , as well.
2.2. Mean value and fluctuations of the magnetic dipole momentum Considering a system in equilibrium with a heat bath, we further calculate the statistical mean value (expectation value) of the magnetic dipole momentum in the canonical ensemble frameworks. To revise, the expectation value of a quantity a for a system in equilibrium with a heat bath at temperature T is given by 〈a〉 ¼
∑n aðnÞe βHn ; Z
ð15Þ
where the term β is given by β ¼ 1=kT (k is Boltzmann's constant), Hn is the nth eigenvalue of the Hamiltonian and Z is the canonical partition function, defined as Z ¼ ∑n e βHn . Since we have already obtained the eigenenergies (7) for our system, the evaluation of the statistical mean of the magnetic dipole moment is straightforward. From the expression (14) it follows that we first need to find the mean values of the quantum numbers 〈nρ 〉, 〈ml 〉, and 〈jml j〉. Utilizing the definition of the canonical ensemble mean values (15), the following results are obtained. For the radial quantum number mean value 〈nρ 〉, one reads 〈nρ 〉 ¼
∑nρ ;ml ;nz nρ e ∑nρ ;ml ;nz e
βEð0Þ nρ ;m ;nz l
βEð0Þ nρ ;m ;nz l
2βℏω nρ ∑1 nρ ¼ 0 nρ e 0
¼
2βℏω nρ ∑1 nρ ¼ 0 e 0
;
2 βℏðjml jω0 ωml Þ 1 ml e ; βℏðjml jω0 ωml Þ ∑1 ml ¼ 1 e
∑1 ml ¼
ð22Þ
ð23Þ
and
μ ¼ ℏq=2mn ½ml τðωÞð2nρ þjml j þ1Þ ¼ μ~ ½ml τðωÞð2nρ þ jml j þ 1Þ;
the Bohr magneton. The expressions for
2 2βℏω nρ ∑1 nρ ¼ 0 nρ e
2βℏω nρ ∑1 nρ ¼ 0 e 0
¼
1 e2βℏω0 1
:
ð16Þ
〈ml jml j〉 ¼
β ℏðjml jω0 ωml Þ 1 ml jml je : 0 1 β ℏðjm l jω ωml Þ ∑ml ¼ 1 e
∑1 ml ¼
ð24Þ
We do not give the final form of the expression for 〈μ2 〉 here, because of its robustness. 2.3. Magnetic dipolar interaction So far we have obtained all the prerequisites to derive magnetic dipolar interaction that is important for many various aspects [25,26]. It is not our intention to go into detailed analysis regarding this issue, but considering the broad area of applicability of this general model we find it useful to present these results, as well. We consider the following configuration of magnetic dipoles: the position of the magnetic dipoles is given by the position vectors r1 ð0; 0; 0Þ and r2 ð0; r cos α; r sin αÞ. The magnetic field as well as magnetic dipoles is directed along z-axis. The expression for potential energy which belongs to the interaction of these dipoles is given by [24] U¼
μ0
ð4π r 3 Þ
½3ðμ1 ; r0 Þðμ2 ; r0 Þ ðμ2 ; μ1 Þ;
ð25Þ
where r ¼ r2 r1 and r0 ¼ r=r the corresponding ort. From where it follows U¼
μ0
4π r 3
μ1 μ2 ð3 sin 2 α 1Þ:
ð26Þ
When the interaction between these dipoles is small, the average dipolar interaction will simply depend on the product 〈μ1 〉〈μ2 〉, so in (26) we can straightforwardly substitute for the independent mean values of the magnetic momenta μ1 and μ2 from (20), which
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Z. Nedelkoski, I. Petreska / Physica B 452 (2014) 113–118
leads to 2
U¼
μ0 μ~ 4π r 3
2 sh γ γ ch γ 0 th γ 0 þ ð3 sin 2 α 1Þ: 0 0 ch γ γ sh γ
ð27Þ
Likewise, employing the expression 3μ0 ðr μ1 Þ μ2 þ ðr μ2 Þ μ1 2rðμ2 ; μ1 Þ F1;2 ¼ 4π r 5 5rððμ1 rÞ; ðμ2 rÞÞ ; þ 2 r
ð28Þ
for the force between these dipoles, we can explicitly calculate it and the result is given by
where T is the temperature and T 0 ¼ ℏω0 =k. The rest of the quantities are defined in Sections 2.1 and 2.2. Canonical mean value manifests the behavior that would be an interplay between the temperature and magnetic field effects at the same time. Due to this, at lower magnetic fields the mean orbital magnetic momentum decreases with temperature, while at higher magnetic fields the thermal fluctuations have less significant influence on the relatively stable value of the magnetic momentum (see for example graph corresponding to t¼ 10 in Fig. 1 or graph corresponding to w¼ 4, Fig. 2). Reasonably this is reflected in the relative fluctuations (Figs. 3 and 4), where 0.5
F1;2 ¼ ð3μ0 Þ=ð4π r 4 Þμ1 μ2 ½ð cos α 5 cos α sin 2 αÞj þ ð3 sin α 5 sin 3 αÞk;
ð29Þ 0.1
ð30Þ 0.3 −0.5 1.0 2.0
−1.5
In this section we represent the obtained results graphically, analyzing the temperature and magnetic field influence on the average magnetic dipole momentum and its fluctuations. In all the plots we use relative dimensionless quantities defined by w¼
ω ; ω0
t¼
T ; T0
R¼
4.0
−1
2.4. Graphical representation and discussion
〈μ〉 ; μ~
0.5
μrel
As in the above-mentioned case i.e. if the interaction between them is small, using the magnetic momenta means from (20), one obtains 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3μ μ~ 2 sh γ γ ch γ F 1;2 ¼ 0 4 0 th γ 0 þ ð5 sin 4 α 2 sin 2 α þ 1Þ: 0 0 sh γ 4π r ch γ γ ð31Þ
μrel ¼
0.0
0
while its magnitude is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 2 F 1;2 ¼ ð3μ0 Þ=ð4π r 4 Þμ1 μ2 5 sin α 2 sin α þ 1:
〈μ2 〉 〈μ〉2 ; μ~ 2
ð32Þ
−2
0
0.5
1
1.5
2
2.5
3
3.5
4
T/T0
Fig. 2. Dependence of the relative magnetic dipole momentum μrel on the temperature, at various values of frequency parameter w, shown next to each curve.
0
7
1
−1
0.01
0.5 6
2 3
5
−2
1
4
−3
R
μrel
5
3 −4
1.5
8
2 −5
2
10
1 −6
0
1
2
3
4
5
6
7
8
9
10
ω/ω0
Fig. 1. Dependence of the relative magnetic dipole momentum μrel on the magnetic field magnitude, expressed via the relative Larmor frequency parameter w, defined in (32). The plots are represented for different values of temperature parameter t, shown next to each curve.
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
T/T0
Fig. 3. Dependence of the relative fluctuation R of magnetic dipole momentum on the 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) 113–118
the stabilization of the fluctuations at higher magnetic fields is obvious. Naturally, the higher the temperature, the higher the threshold magnetic field value needed to stabilize the fluctuations. Observing these plots one can conclude that the fluctuations manifest Gaussian alike form. Observing Figs. 1 and 2, one can notice that the average magnetic moment is negative, which implies the expected Landau-like diamagnetism of the studied system. Also, analyzing the analytical expressions for the magnetic momentum Eq. (13), it
0 2 −2
3
16
0.5
−6
−8
0.1
−10
14
10
−4
μrel,0
18
117
−12
12 −14
0.067
R
10 −16
8
2
1 0.5 0
1
2 ω/ω0
3
4
Fig. 4. Dependence of the relative fluctuation R of magnetic dipole momentum on the magnetic field magnitude, expressed via the relative Larmor frequency parameter w, at various values of the temperature parameter t, shown next to each curve. The parameters are defined in (32).
80
70
2
3
4
5
6
Fig. 6. Effects of the effective mass on relative magnetic momentum μrel;0 , defined in units of μ~0 ¼ ℏq=2m0 . Dependence of R0 on the relative Larmor frequency w is presented 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. The plots are given for the value of the temperature parameter t¼ 0.5.
4
0
1
ω/ω0
6
2
0
can be noticed that it leads to correct results in the limiting cases, thus in the absence of the magnetic field, the magnetic momentum is simply a multiple of Bohr magnetons μ ¼ ðℏq=2mn Þml , while the magnetic momentum mean value is consequently equal to zero. Finally, influence of the effective mass on the analyzed quantities is also considered. To show the curve for one realistic example, we insert the plots for effective mass of an electron in GaAs-based quantum dots and rings (mn ¼ 0:067m0 ). Sensitivity of the magnetic dipole momentum and its fluctuations on the effective mass could be useful for the estimation of effective mass in newly fabricated quantum structures (Figs. 5 and 6).
0.067
3. Conclusions 60
R0
50
40
30
0.1
20
10
0
0.2 1 0
0.5
1
1.5
2
2.5
3
3.5
4
ω/ω0
Fig. 5. Effects of the effective mass on the relative fluctuation R0, defined in units of μ~20 ¼ ðℏq=2m0 Þ2 . Dependence of R0 on the relative Larmor frequency w is presented 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. The plots are given for the value of the temperature parameter t¼ 0.5.
In the present paper a theoretical model, describing an electron confined in an anisotropic parabolic potential in the presence of mutually parallel electric and magnetic fields, was considered. The work offers exact analytical treatment of the stated problem, that is applicable to a wide class of phenomena in condensed matter physics. Wave functions derived in our previous works, adjusted to effective mass approach, were used to find exact expressions of the electron current density. It was further used to obtain orbital magnetic dipole momentum. Having the explicit expressions for the magnetic dipole momentum, we have calculated the mean values in canonical ensemble approach, assuming thermal equilibrium with a heat bath. Without going into so many details, we have also presented the exact expressions of the dipole–dipole interactions, which is also of relevance for model studies. Analyzing the graphical representations of the dependencies of the magnetic momentum mean values and its fluctuations on the magnetic field and temperature, it was concluded that the proposed model leads to expected behaviors. Namely, the thermal fluctuations of the orbital magnetic momentum decrease at high magnetic fields, while at low fields temperature has significant effects on these fluctuations. Negative values of the magnetic
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momentum just confirm the Landau-like diamagnetism of the studied system. Moreover, the influence of the effective mass enables one to at least approximately estimate it, analyzing statistical mean values of the magnetic dipole momentum, as well as its fluctuations. We find this very important because determination of the effective mass is never a trivial problem, while the experimental measurement of statistical averages of the magnetic dipole moments is more common. As a concluding remark, it is important to emphasize that the advantage of the proposed model is the general and exact analytical treatment it offers, applicable to a wide class of systems described by anisotropic parabolically shaped potential. Quantum dots, quantum rings, carbon nanotubes, magnetic nanoparticles etc. are some of the systems that fall within the aforementioned class. Even though the models like one we propose are a simplified picture to rather complex phenomena, they provide the necessary reference for a comprehensive theoretical approach, enabling at the same time rationalization of experimentally observed results, as well as prediction of the behavior of novel prospective quantum structures. For example, introduction of complete single particle basis set that includes the effect of anisotropy and external magnetic fields could further serve as a ground to study more subtle perturbational effects. Even more, it is shown that many other relevant quantities, such as current density, magnetic momentum and its statistical treatment, could be also derived to a compact analytical form within the frameworks of this model. Acknowledgement The authors are grateful to Prof. Gertrud Zwicknagl for the helpful discussions on the subject matter and for providing some useful references.
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