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Physica B 421 (2013) 127–131 Contents lists available at SciVerse ScienceDirect Physica B journal homepage: www.elsevier.com/locate/physb Two electrons in a cylindrical quantum dot under constant magnetic field Agile Mathew a,n, Malay K. Nandy b a b Center for Nanotechnology, Indian Institute of Technology Guwahati 781039, India Department of Physics, Indian Institute of Technology Guwahati 781039, India art ic l e i nf o a b s t r a c t Article history: Received 28 November 2012 Received in revised form 13 March 2013 Accepted 13 April 2013 Available online 20 April 2013 We present the results obtained for the problem of two electrons in a cylindrical quantum dot with finite step potential in the presence of orthogonal magnetic field. The method we adopted is linear variational theory, where the basis states are constructed from single electron eigenfunctions of the harmonic oscillator potential. We show how the two electron energy levels vary with the magnetic field for various quantum numbers. Magnetization of the system is then calculated after determining its free energy at a non-zero temperature. Finally, we also plot the electron density and pair correlation function for various quantum numbers and field strengths. & 2013 Published by Elsevier B.V. Keywords: Cylindrical quantum dot Step potential Linear variational theory Energy spectrum Magnetization Pair correlation 1. Introduction Quantum dots in magnetic fields are extensively studied in the context of nanotechnology, both theoretically as well as experimentally [1,2]. The confinement potential that makes up these dots can be created either through semiconductor band-gap engineering alone or them along with electrical potential applied through external metallic electrodes. By controlling the fabrication process we can tune the properties of these zero-dimensional nanostructures. Technology is now matured enough to have precise control over the number of electrons inside these dots [3]. To theoretically study these dots various model potentials are employed, such as rectangular, harmonic oscillator, cylindrical, inverted-Gaussian, spherical ,…etc [4–8]. Though for majority of cases harmonic oscillator model potential is a good approximation, for certain optically active self-assembled quantum dots, cylindrical potential is a better approximation [9]. In the present paper, we focus on a two electron cylindrical quantum dot with a finiteheight discontinuous confinement potential in the presence of a magnetic field in its axial direction. The one-electron problem for such a GaAs=Ga1−x Alx As dot was studied earlier [10]. Two electron quantum dots are interesting because it is the simplest case for studying the effect of electron–electron interaction [11]. We can anticipate that the energy spectrum of a multielectron quantum dot is going to depend on various factors such as n Corresponding author. Tel: +91 9957 817471. E-mail addresses: [email protected] (A. Mathew), [email protected] (M.K. Nandy). 0921-4526/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.physb.2013.04.022 confinement due to external magnetic field, confinement due to dot-potential, and finally the electron-electron interaction and correlation effects. For a dot with very small height to diameter ratio, we can safely assume that electrons are restricted to the ground state sub-band in the z-direction and thus the problem reduces to the case of a two-dimensional circular disc. The energy levels for two electron quantum dot with harmonic potential was studied extensively by many researchers [12–17]. A similar study for discontinuous model potential was done by Peeters and Schweigert wherein they solved the one electron problem using finite difference scheme, and then used those results to do variational analysis for the two electron problem [18]. In the present work, we employ linear variational analysis where the trial wave function is constructed out of single electron harmonic oscillator wavefunctions. The paper is organized as follows: Section 2 discusses the theoretical model and the basic analytical steps involved, the results are presented and discussed in Section 3, and finally the paper is concluded in Section 4. 2. Theory and procedure In the following analysis we consider a quantum dot based on GaAs=Ga1−x Alx As heterostructure, but assume that its effective mass remain constant across the heterogeneous boundary. This is a reasonable approximation for a potential step of size 100 meV, as it amounts to only a 15% difference in the effective mass for the Ga1−x Alx As outer region in comparison to that of the inner GaAs region [10]. As the electron densities for the lower energy states 128 A. Mathew, M.K. Nandy / Physica B 421 (2013) 127–131 are negligible in the outer region, this difference does not contribute much to our problem. Further, we assume a much stronger confinement in the axial direction of the dot relative to that in its radial direction, thus justifying our two dimensional model for the analysis [18]. Under these assumptions, the Hamiltonian of a two electron cylindrical quantum dot system in constant magnetic field B can be written as !2 1 ! gμB ! ! q2 H¼ ∑ ð1Þ S þ qA Þ ðp þ Vðr Þ þ B þ ! ! i i i i n 2m ℏ κjr 1 −r 2 j i ¼ 1;2 e 0; ! where, Ai ¼ B=2ð0; r i ; 0Þ and Vðr i Þ ¼ V 0; r i ≤r 0 r i 4r 0 . Here ri and pi are the conjugate position and momentum coordinates of the ith electron, mne is the electron effective mass, q is the electron charge, g is the Landé g-factor of an electron ! inside the quantum dot, μB is the Bohr magneton, Si is the spin angular momentum of the ith electron, and κ ¼ 4πϵ, the strength of Coulomb interaction in SI units. Here, the value of quantum dot permittivity, ϵ, is given an average value of 13:1ϵ0 throughout the material. In the operator form Eq. (1) becomes " ! 2 ℏ2 ∂2 1 ∂ 1 ∂2 ^ þ þ H¼ ∑ − r i ∂r i r 2i ∂ϕ2i 2mne ∂r 2i i¼1 # 2 ωC ^ 1 n ωC ℏωL ðiÞ q2 2 ð2Þ s^z r i þ V ðr i Þ þ þ þ ! ! Lz þ m 2 i 2 e 2 2 κjr 1 −r 2 j where, L^ zi ¼ −iℏð∂=∂ϕi Þ, the z-component of the angular momentum operator of the ith electron, ωC ¼ qB=mne , the cyclotron frequency, and ωL ¼ gðqB=2mne Þ, the Larmor precession frequency. By adding and subtracting a harmonic oscillator potential 1 n 2 2 2me ω0 r i for each electron, we can re-write the above Hamiltonian as ^ ^ 2Þ H^ ¼ H 0 ð1; 2Þ þ H′ð1; ð3Þ ^ ^ 2Þ are as given below. The definitions of H 0 ð1; 2Þ and H′ð1; ^ H 0 ð1; 2Þ ¼ H^ ho ð1Þ þ H^ ho ð2Þ where, ℏ2 H^ ho ðiÞ ¼ − 2mne with ω ¼ ∂2 1 ∂ 1 ∂2 þ þ 2 2 2 r i ∂r i r i ∂ϕi ∂r i ð4Þ ! þ ωC ^ 1 L z þ mn ω2 r 2i 2 i 2 e jψ〉 ¼ c1 jχ 1 〉 þ c2 jχ 2 〉 þ ⋯ þ ci jχ i 〉 þ ⋯ þ cd jχ d 〉 ð10Þ where, d is the dimension of the basis set, coefficients c1 ; c2 ; …; cd are variational parameters and jχ i 〉 s are the orthonormalized twoelectron states. If s1 and s2 are the spin quantum numbers of individual electrons (with each having a value 12), and if s is the total spin quantum number of the two-electron system, then by angular momentum addition rule s can take only two values viz. s¼0 corresponding to js1 −s2 j, and s ¼1 corresponding to js1 þ s2 j. For s ¼0 state, the z-component of the total spin operator, S^ z , can have only one value for its quantum number viz. ms ¼0 (therefore the name, singlet). Whereas for the s¼1 state, it can take three values viz. ms ¼ 0; 7 1 (therefore the name, triplet). The orbital part of a singlet states must be symmetric and that of the triplet states must be antisymmetric. These can be easily constructed out of one-electron wavefunctions by taking Slater permanent and determinant respectively. For example, if jφn1 ;m1 〉 and jφn2 ;m2 〉 are any two distinct one-electron eigen states, then jχ n71 ;m1 ;n2 ;m2 ð1; 2Þ〉 ¼ jφn1 ;m1 ð1Þ〉jφn2 ;m2 ð2Þ〉 7 jφn2 ;m2 ð1Þ〉jφn1 ;m1 ð2Þ〉 pffiffiffi 2 ð11Þ is a valid orthonormalized symmetric (antisymmetric) twoelectron orbital wavefunction constructed out of them, except pffiffiffi for the case ðn1 ; m1 Þ ¼ ðn2 ; m2 Þ where an additional factor of 2 must be taken care of. This, multiplied by their appropriate spinwavefunction counterpart ðjs〉Þ, forms the required basis wavefunctions in Eq. (10) jχ i 〉 ¼ jχ i7 〉js ¼ 0ð1Þ〉 ð12Þ ^ L^ z ¼ 0, the total z-component of the angular momenSince ½H; tum, M ¼ m1 þ m2 , must be same for all terms in the trial ^ S^2 ¼ ½H; ^ S^ z ¼ 0, every term in wavefuntion. Similarly, since ½H; the trial wavefunction must have the same value for the s and ms quantum numbers. Thus, we can do variational analysis for each combinations of M, s and ms values separately. For each case, the linear variational analysis reduces to solving an eigen value problem of the type given below ð5Þ ½HfCg ¼ EfCg ð6Þ Here, ½H is a square matrix of size d d with elements ^ 〉, fCg is a column matrix of size d 1 with unknown H ij ¼ 〈χ i jHjχ j coefficients ci in Eq. (10) as elements, and E is their corresponding eigen energy. The matrix ½H can be written as a sum of matrices qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω20 þ ðωC =2Þ2 . Similarly, ^ ^ ð1; 2Þ þ Z^ ð1; 2Þ þ C^ ð1; 2Þ H′ð1; 2Þ ¼ W Now, in linear variational theory [19], we consider a trial wavefunction as given below ð13Þ 0 ^ which we ½H 0 and ½H′ corresponding to the operators H^ and H′ ^ ð1; 2Þ ¼ ∑2 ½Vðr i Þ−1mn ω2 r 2 , the residue potential term, where, W i¼1 2 e 0 i Z^ ð1; 2Þ ¼ ððℏωL Þ=2Þ∑2i ¼ 1 s^ ðiÞ the Zeeman term and z , ! ! 2 ^ C ð1; 2Þ ¼ q =ðκj r 1 − r 2 j), the Coulomb term. The eigenvalues and eigenfunctions of H^ ho are well known [1] and are as given below of one-electron wavefunctions jφn;m 〉, it may be noticed that ½H 0 has non-zero elements only along its diagonal. H^ ho φn;m ðr; ϕÞ ¼ E n;m φn;m ðr; ϕÞ H 0ij ¼ 〈χ n71 ;m1 ;n2 ;m2 jH^ ho ð1Þ þ H^ ho ð2Þjχ n73 ;m3 ;n4 ;m4 〉 ð7Þ where, n ¼ 0; 1; 2… and m ¼ 0; 71; 72… are the radial and azimuthal quantum numbers, respectively, and m ð8Þ E n;m ¼ ℏωð2n þ mþ1Þ þ ℏωC 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðjmj þ nÞ! exp φn;m ðr; ϕÞ ¼ 1þjmj πn! jmj! aB " # " # −r 2 r2 jmj þ imϕ r 1 F 1 −n; mþ1; 2 ð9Þ 2a2B aB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where aB ¼ ℏ=mne ω, the effective Bohr radius of the quantum dot. have defined earlier. The elements of ½H 0 for the case of singlet (+) and triplet (−) states are given by Eq. (14). Due to the orthogonality ¼ δn1 n3 δm1 m3 δn2 n4 δm2 m4 ðE n3 ;m3 þ E n4 ;m4 Þ ð14Þ Similarly, the elements of ½H′ are also evaluated. In the case of triplet states, the result after substituting for jχ − 〉 from Eq. (11) becomes ^ H′ij ¼ 〈φn1 ;m1 ð1Þφn2 ;m2 ð2ÞjH′ð1; 2Þjφn3 ;m3 ð1Þφn4 ;m4 ð2Þ〉 ^ −〈φn1 ;m1 ð1Þφn2 ;m2 ð2ÞjH′ð1; 2Þjφn4 ;m4 ð1Þφn3 ;m3 ð2Þ〉 ð15Þ A little care must be given while evaluating the elements of ½H′ for singlet states. This is because, we can build symmetric orbital states out one-electron eigenstates even when n1 ¼ n2 and m1 ¼ m2 , and those states are little different from the general A. Mathew, M.K. Nandy / Physica B 421 (2013) 127–131 symmetric states given by Eq. (11). For the general states ^ H′ij ¼ 〈φn1 ;m1 ð1Þφn2 ;m2 ð2ÞjH′ð1; 2Þjφn3 ;m3 ð1Þφn4 ;m4 ð2Þ〉 ^ ð1Þφ ð2ÞjH′ð1; 2Þjφ ð1Þφ ð2Þ〉 þ〈φ n1 ;m1 n2 ;m2 n4 ;m4 n3 ;m3 ð16Þ When n1 ¼ n2 and m1 ¼ m2 and n3 ¼ n4 and m3 ¼ m4 , the elements of ½H′ for singlet states becomes ^ 2Þjφn3 ;m3 ð1Þφn3 ;m3 ð2Þ〉 H′ij ¼ 〈φn1 ;m1 ð1Þφn1 ;m1 ð2ÞjH′ð1; ð17Þ The final case is when quantum numbers are equal only on one side, say n1 ¼ n2 and m1 ¼ m2 , the elements of ½H′ for singlet states is given by pffiffiffi ^ 2Þjφn3 ;m3 ð1Þφn4 ;m4 ð2Þ〉 ð18Þ H′ij ¼ 2〈φn1 ;m1 ð1Þφn1 ;m1 ð2ÞjH′ð1; All the elements of ½H′ were found out through numerical integration in MATHEMATICA software. To simplify the case, we can split and write ½H′ as a sum of three matrices corresponding to the terms defined in Eq. (6) as ½H′ ¼ ½W þ ½Z þ ½C ð19Þ Now, the matrix [Z] is diagonal due to the orthonormality of orbital part of the basis wavefunctions. Furthermore, these diagonal elements are also zero for ms ¼0 states ðj↑↓〉p7ffiffi2j↓↑〉Þ. But for ms ¼ 71 states ðj↑↑〉; j↓↓〉Þ, it give rises to a constant value, 7 ℏωL for all the diagonal elements. Similarly, many of the elements in [W] are zeros due to the orthogonality of m1 ≠m3 or m2 ≠m4 condition. Again, because of the azimuthal and exchange symme^ ð1; 2Þ term, all the non-zero [W] elements require tries of the W only one-variable numerical integrations. The evaluation of elements in the [C] matrix involves calculation of terms like C 12;34 ¼ 〈φn1 ;m1 ð1Þφn2 ;m2 ð2ÞjC^ ð1; 2Þjφn3 ;m3 ð1Þφn4 ;m4 ð2Þ〉: ð20Þ pffiffiffiffiffiffi imϕ If we substitute φn;m ðr; ϕÞ ¼ ρn;m ðrÞe = 2π in Eq. (20) we get Z ∞ q2 r 1 dr 1 ρnn1 ;m1 ðr 1 Þρn3 ;m3 ðr 1 Þ C 12;34 ¼ 2πκ r1 ¼ 0 Z ∞ r 2 dr 2 ρnn2 ;m2 ðr 2 Þρn4 ;m4 ðr 2 ÞIðr 1 ; r 2 Þ ð21Þ 129 spectrum and corresponding electron wavefunctions. We have done this by using the subroutine Eigensystem available in MATHEMATICA software. This process is done separately for each value of quantum numbers (M,s) and it is then repeated for values of B ranging from 0 to 30 Tesla. The electron density, ηðrÞ and pair correlation function, ! f pc ð r Þ are then evaluated using these resultant wavefunctions, following the definitions mentioned in Ref. [18] as given below 2 ! ! ! ηð r Þ ¼ ∑ 〈δð r − r i Þ〉 i¼1 ! ! ! ! f pc ð r Þ ¼ 〈δð r −r 1 þ r 2 Þ〉 ð24Þ 3. Results and discussion The results obtained here are classified in terms of quantum numbers (M,s), which form separate infinite-dimensional subspaces within the Hilbert space of the problem. The accuracy of results obtained through linear variational analysis for each such case, depends on how well the basis set span over the subspace of interest. Though the lower energy states are mostly spanned by the lower energy basis functions, the accuracy of results gets better as we increase the size of the basis set. This size, in turn depends on the number of one-electron eigenfunctions we begin with, from which we construct those basis functions. In the present paper, we have used single electron eigenfunctions jφn;m 〉 s with quantum numbers n ranging from 0 to ν, and m ranging from −μ to μ. This will result in a total number of N 1 ¼ ðν þ 1Þð2μ þ 1Þ single electron wavefunctions. From N1 one-electron wavefunctions, we can construct N 1 ðN 1 þ 1Þ=2 symmetric wavefunctions for singlet states, and N 1 ðN 1 −1Þ=2 antisymmetric wavefunctions for triplet states. The number of two-electron basis functions that can be constructed for a given ν, μ, M and s values is given by h i ν ðν þ 1Þ ðν þ 1Þ μ þ 1− jMj 2 − 2 −s ; M is even N2 ¼ ð25Þ ; M is odd ðν þ 1Þ2 μ− jMj−1 2 r2 ¼ 0 R 2π R 2π where, Iðr 1 ; r 2 Þ ¼ ϕ ¼ 0 dϕ1 ϕ ¼ 0 dϕ2 exp iðð−m1 þ m3 Þϕ1 þ ð−m2 þ ! ! ! ! m4 Þϕ2 Þ=j r 1 − r 2 j. Using the multipole expansion of 1=j r 1 − r 2 j and simplifying further, Iðr 1 ; r 2 Þ becomes 1 ∞ r o l ðl−m1 þ m3 Þ! m1 −m3 ðP ∑ ð0ÞÞ2 ð22Þ Iðr 1 ; r 2 Þ ¼ 4π 2 r 4 l ¼ 0 r 4 ðl þ m1 −m3 Þ! l where, r 4 ¼ r 1 and r o ¼ r 2 when r 1 4 r 2 and viceversa, and Pm l is the associated Legendre polynomial. Substituting this expression for Iðr 1 ; r 2 Þ in Eq. (21), we get C 12;34 ¼ Z ∞ r1 ¼ 0 ∞ 2πq2 ðl−m1 þ m3 Þ! m1 −m3 ∑ ðP ð0ÞÞ2 κ l ¼ jm1 −m3 j ðl þ m1 −m3 Þ! l r 1 dr 1 ρnn1 ;m1 ðr 1 Þρn3 ;m3 ðr 1 Þ "Z lþ1 r2 r1 r2 ¼ 0 l # Z ∞ r1 dr 2 ρnn2 ;m2 ðr 2 Þρn4 ;m4 ðr 2 Þ þ r2 r2 ¼ r1 r1 dr 2 ρnn2 ;m2 ðr 2 Þρn4 ;m4 ðr 2 Þ ð23Þ Thus we have replaced the original four-variable integration given in Eq. (20) with an infinite sum of two-variable integrations. But since the result is pretty accurate with only first few (say 30) terms of Eq. (23), this new formula helps us to achieve fast computation of the elements in [C]. Once all the elements of [H] is evaluated, we proceed to solve the eigen value problem in Eq. (13) which will then give us the energy where, μ þ 1−jMj=2 and μ−ðjMj−1Þ=2 respectively are, for even and odd cases of M, the number of possible combinations of m1 and m2 such that m1 þ m2 ¼ M. Now, ðν þ 1Þ2 is the number of possible combinations for n1 and n2 for each allowed combination of m1 and m2 values. The νðν þ 1Þ=2 in Eq. (25) is the number of repeated cases of two-electron wavefunctions for the m1 ¼ m2 case, due to the permutation of n1 and n2 values. Finally the subtraction of ðν þ 1Þs term removes the prohibited case of ðn1 ; m1 Þ ¼ ðn2 ; m2 Þ from the antisymmetric orbital wavefunctions of triplet states. The results plotted in this section are for the case, ν ¼ 2 and μ ¼ 3, giving N1 ¼ 21, in which case we obtain 231 singlet basis states and 210 triplet basis states. These states are then classified according to their M values, giving rise to different basis dimensions for each case, as listed in Table 1. It may be noticed from the Table that the largest basis set is obtained for M¼ 0 and the smallest basis set is for M ¼ 7 2μ. Therefore, for a given analysis described here, we expect more accurate results for terms with smaller jMj values. An important parameter we must fix beforehand is the value of ω0 in Eq. (5). We expect best results when the harmonic oscillator potential V ho ðω0 ; rÞ ¼ 12mne ω20 r 2 effectively mimics the most of the cylindrical step potential of the original problem. To find this value of ω0 , we have expressed V ho ðω0 ; r 0 Þ ¼ kV 0 , where V0 is the stepsize and k is a dimensionless number whose value we need to determine. Then ω0 becomes sffiffiffiffiffiffiffiffiffiffiffi 2kV 0 ð26Þ ω0 ðkÞ ¼ mne r 20 130 A. Mathew, M.K. Nandy / Physica B 421 (2013) 127–131 E (meV) Table 1 Dimensions of two-electron basis set for different M values. 100 M value Dimension of singlet basis Dimension of triplet basis −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 6 9 15 18 24 27 33 27 24 18 15 9 6 3 9 12 18 21 27 30 27 21 18 12 9 3 90 80 70 60 50 40 5 10 15 20 25 30 20 25 30 B (Tesla) µmag / µB T=4K 5 E (meV) 100 5 90 10 15 B (Tesla) −5 80 70 −10 60 −15 50 T=0.2K −20 40 5 10 15 20 25 30 B (Tesla) Fig. 1. Two electron energy spectrum for a cylindrical quantum dot. Here, the dotted (continuous) curves stands for s ¼1 (s ¼ 0) states. The colors red, green, blue, orange and brown stand for M ¼ −2; −1; 0; 1; 2 respectively. The results plotted are for an r 0 ¼ 15 nm and V 0 ¼ 100 meV. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.) By tuning the value of k, we found that there exists a minimum for the ground state energy when k ¼0.25. This corresponds to ℏω0 ¼ 15:85 meV for a quantum dot with r 0 ¼ 15 nm and V 0 ¼ 100 meV. The energy spectrum obtained for various quantum numbers are plotted as a function of external magnetic field strength in Fig. 1. We have chosen r 0 ¼ 15 nm, to avoid the piling up of many energy levels in a window of energy between 0 and 100 meV, but at the same time granting us decent number of curves for observation. In the figure, we have shown curves for both singlet (continuous) and triplet (dashed) states, and in each case M varying from −2 to +2 which are represented by various color codes, as mentioned in the figure caption. Curves with higher energies for the same color code corresponds to excited states, which may be labeled by a new quantum number N ¼ 1; 2; 3… etc. Notice from the figure that at 0 T, the states corresponding to 7M are degenerate and it is lifted off as the magnetic field is switched on. The curves for different values of ms quantum number are suppressed for the sake of clarity. If we add them, the only difference it will make is to introduce two more levels for each triplet states, which differ from the present one by 7ℏωL . The results we obtained show considerable similarity with those in Ref. [18], where they have plotted for cylindrical quantum dot with V 0 ¼ 500 meV and r 0 ¼ 10 nm and r 0 ¼ 20 nm, respectively. Similar to their cases, grouping of energy levels are noticed at higher magnetic field strengths as shown in Fig. 1. Similarly, singlet– triplet crossing is observed for the ground state, with the increase of external magnetic field. Our results for 15 nm dot nicely interpolates with the two cases in Ref. [18], in terms of the scales of energy and magnetic field strength as can be noticed from the graphs. The first singlet–triplet crossing here takes place at around Fig. 2. (a) Two electron energy spectrum including the multiplicity of triplet states due to Zeeman interaction, (b) Magnetization at T ¼ 0.2 K (blue) and 4 K (red) are shown as a function of magnetic field strength. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.) 12 T and the second one happens at around 29 T, which were 10 T and 19 T in their case of 20 nm quantum dot. The discrepancies could be attributed to the differences in confinement energies of both cases due to the differences in r0 and V0 values. In Fig. 2, we have plotted the magnetization of the present twoelectron quantum dot as a function of magnetic field strength. This is evaluated by taking the negative derivative of free energy with respect to magnetic field as shown below [20] μmag ¼ − ∂F ∂B ð27Þ where, F ¼ −kT lnZ(B) is the free energy of the system, and ZðBÞ ¼ ∑i e−Ei ðBÞ=kT is the system partition function. Here, B is the magnetic field strength and Eis are the energy levels plotted in Fig. 2(a). The results shown are for temperatures T¼ 4 K and T¼ 0.2 K, where the magnitude of magnetization are expressed in units of Bohr magneton ðμB ¼ 5:79 10−2 meV=TÞ. It may be noticed that, we now have lesser number of singlet–triplet transitions for the ground state as compared to that in Fig. 1 and the initial one takes place early at around 8 T. This is because we now have taken into account the Zeeman effect, whose contribution is important for the calculation of free energy. Similar results for two electrons in parabolic quantum dot potential were shown much earlier in Ref. [21]. In comparison, we have noticed only one oscillation in the magnetization curve as opposed to many in their case for a given range of magnetic field strength. This can be attributed to the large effective confinement radius (almost double in comparison) in their case. We also noticed larger than paramagnetic contribution in the step size of magnetization at the point of transition, which definitely points to the contribution from exchange interaction and orbital motion. The step is found to be abrupt for very low values of T but smoothens more with the increase of temperature, as the value of kT becomes comparable to that of the spacing between energy levels. Since magnetization can A. Mathew, M.K. Nandy / Physica B 421 (2013) 127–131 ED (nm−2) 131 CF (nm−2) 0.0020 0.012 0.010 0.008 0.006 0.004 0.002 (0, 0) 0.0015 (0, 0) 0.0010 0.0005 5 10 15 20 r (nm) ED (nm−2) 10 20 30 40 r (nm) CF (nm−2) 0.005 0.0010 0.004 (-1, 1) (-1, 1) 0.0008 0.003 0.0006 0.002 0.0004 0.001 0.0002 5 10 15 20 r (nm) 10 20 30 40 r (nm) CF (nm−2) ED (nm−2) 0.005 0.004 0.0008 (-2, 0) 0.003 0.0006 0.002 0.0004 0.001 0.0002 5 10 15 20 r (nm) (-2, 0) 10 20 30 40 r (nm) Fig. 3. Left column: radial electron density (ED), ηðr; ϕ ¼ 0Þ. Right column: pair correlation function (CF), f pc ðr; ϕ ¼ 0Þ. The continuous, dashed and dotted curves corresponds to B¼0, 10 and 20 Tesla, respectively. The rows 1, 2 and 3 corresponds to quantum numbers (0,0), (−1,1) and (−2,0), respectively. All results are for an r 0 ¼ 15 nm. be measured experimentally, these results are useful for the determination of spin states in spintronic applications as reported in Ref. [22]. We have also plotted the electron radial densities and pair correlation functions for various quantum numbers (M,s) and magnetic field strengths. These are shown in Fig. 3, where we again notice a very compliant results with that in Ref. [18]. The qualitative nature and quantitative values of both the radial electron densities and pair correlation functions look very appealing by comparison of these results. At higher magnetic fields, these functions are seen to be pressed toward the center of the dot, due to the magnetic confinement. But for low magnetic fields, the electron–electron interaction pushes them apart, while the dot confinement potential limits the extent to which they can be separated. Due to the inherent anti-symmetry in the orbital wavefunction of the triplet states, f pc ð0Þ ¼ 0 as two electrons cannot be in the same place in such states. Where as, for all singlet states, there exist a non-zero value for f pc ð0Þ. 4. Conclusion We have conducted a linear variational analysis for a two-electron cylindrical quantum dot with step potential, with a trial wavefunction constructed out of harmonic oscillator eigen functions. We fixed the strength of this oscillator potential by tuning the oscillator frequency for minimum ground state energy. We plotted the two-electron energy spectrum as a function of external magnetic field strength for various values of quantum numbers (M,s). From these, we evaluated the free energy and then determined the dependency of magnetization with respect to the magnetic field strength. Similarly, we have also plotted radial electron density and pair correlation functions for different quantum numbers and different magnetic field strengths. We could convincingly obtain all the qualitative and even most of the quantitative features of the exact results obtained by Peeters and Schweigert [18], where they have considered around 1000 basis states for each combination of (M,s) quantum numbers. In comparison, we considered only less than 50 in size basis set to obtain the results displayed. This shows that, a very few low-energy harmonic oscillator eigenfunctions are able to effectively capture the quantum dot cylindrical step-potential. Our method also eliminates the requirement of finite element analysis for obtaining one electron wavefunctions. 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