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Physica B 421 (2013) 127–131
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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. However, our results are valid only under the assumption that the effective mass and permittivity of the dot are almost
constant throughout the material.
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