Journal of Physics and Chemistry of Solids 75 (2014) 391–396
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Journal of Physics and Chemistry of Solids
journal homepage: www.elsevier.com/locate/jpcs
Pressure and disorder effects on the half-metallic character
and magnetic properties of the full-Heusler alloy Co2FeSi
Xingfeng Zhu a,n, Yaxu Wang a, Lu Wang b, Yafei Dai a, Chenglin Luo a
a
b
Department of Physics, Nanjing Normal University, Nanjing 210097, China
Department of Physics, Nanjing University of Information Science and Technology, Nanjing 210044, China
art ic l e i nf o
a b s t r a c t
Article history:
Received 3 June 2013
Received in revised form
25 October 2013
Accepted 18 November 2013
Available online 23 November 2013
We investigate the pressure and site disorder effects on the half-metallicity and magnetic properties of
the full-Heusler alloy Co2FeSi using first-principles density functional theory within the GGA and
GGA þU schemes. The calculated lattice constant, bulk modulus and total magnetic moments are in
excellent agreement with recent experiments. The volume compression leads to a slight increase of the
minority band gap, i.e., the half-metallic properties of Co2FeSi can maintain under pressure. The disorder
calculations reveal that Fe–Co type disorder significantly destroys the half-metallic character and reduces
the spin polarization of Co2FeSi while disorder between Fe and Si can maintain half-metallic properties.
Our results also show that the Fe–Co type disorder leads to degradation of the magnetism while the Fe–Si
type disorder affects hardly the magnetism as observed in Co2FeSi.
& 2013 Elsevier Ltd. All rights reserved.
Keywords:
Half-metal
Atomic disorder
Electronic structure
Magnetic properties
1. Introduction
Heusler alloys that are considered to be good candidates for
spintronic applications are characterized by a metallic behavior in
the majority spin channel and a semiconducting behavior with a
gap in the minority channel. Such a behavior provides a 100%
spin polarization at the Fermi level, which is necessary for spin
injection in spintronic devices [1,2]. The ternary intermetallics
with composition XYZ and X 2 YZ, the half and full Heusler alloys,
respectively, contain two different transition metal atoms X and
Y and a third element Z which is a nonmagnetic metal or
nonmetallic element from IIIb or IVb column. The full Heusler
compounds are well ordered alloy with face centered cubic L21
(SG: Fm3m) structure [3–5]. Among the Heusler compounds,
Co2FeSi plays an important role since it has unique half-metallic
band structure with highest Curie temperature of 1100 K reported
so far and exhibits the highest magnetic moment of 6 μB per
formula unit, making it ideal candidates for applications.
Application of high pressure is a unique method for obtaining
important knowledge about the electronic properties of materials
because the density of state (DOS), Fermi surface, bandwidth, as
well as the volume and structure of the materials, etc., can be
continuously controlled by applying high pressure. Picozzi et al.
theoretically investigated the effect of hydrostatic pressure on the
electronic structure in Co2MnX (X ¼Si, Ge, and Sn) and found an
n
Corresponding author. Tel.: þ 86 25 83598305.
E-mail address: [email protected] (X. Zhu).
0022-3697/$ - see front matter & 2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jpcs.2013.11.008
increase of the half-metallic band gap and progressive shifts of
Fermi level (EF) from the valence band (VB) into the band gap
by volume compression [6]. Alka et al. published a study on phase
stability of Heusler compound Co2FeSi under pressure based
in-situ x-ray powder diffraction measurements and the compound
is structurally stable up to 24.8 GPa [7]. It has been predicted that
spin polarization of Co-based Heusler alloys is sensitive to this site
disorder [6,9,10]. Although in the L21 structure the Co atoms are
placed on the second neighbor positions, their interaction plays an
important role in magnetic properties of these compounds [11,12].
The L21structure of Co2YZ lowers to B2 or further to A2 structure,
depending on a degree of the atomic site disorder.
In the present study, we focus our attention on the effects of
pressure and atomic disorder on the half-metallicity of the fullHeusler alloy Co2FeSi, using first-principles calculations. First, we
investigate the effect of the applied pressure on the electronic and
magnetic properties of the alloy Co2FeSi. Then, we investigate the
effects of atomic disorder and discuss how the half-metallicity of
the alloy changes by the pressure and the atomic disorder among
Co, Fe, and Si.
2. Computational details
First-principles calculations of Co2FeSi are performed in a
plane-wave basis set using the projector augmented wave (PAW)
[13,14] method in the generalized gradient approximation (GGA)
as it is implemented in the Vienna ab initio simulation program
(VASP) [15,16]. To treat exchange and correlation effects, we use
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X. Zhu et al. / Journal of Physics and Chemistry of Solids 75 (2014) 391–396
both GGA and the semiempirical GGA þU [17–20] method within a
rotationally invariant formalism for a better description of the
localized transition metal d electrons. We use here the simple
formulation by Liechtenstein et al. [19] and Dudarev et al. [20],
where a single parameter Ueff determines an orbital-dependent
correction to the density functional theory (DFT) energy. Ueff is
generally expressed as the difference between two parameters,
the Hubbard U, which is the Coulomb-energetic cost to place two
electrons at the same site, and an approximation of the Stoner
exchange parameter J, which is almost constant at about 1 eV.
We observe that the total magnetic moment is fixed at the experimental value of 6 μB in the range of U ¼2–5.5 eV. Therefore, in this
range of effective Coulomb exchange interaction, GGA þU predicts
a halfmetallic character for Co2FeSi. Here, we choose U¼ 3.5 eV
J ¼0.9 eV for the 3d orbitals of Co atoms and U¼3.4 eV J¼ 0.9 eV
for the 3d orbitals of Fe atoms which are sufficient to produce
measured magnetic moment and, moreover, are consistent with
the first-principles Hartree–Fock-calculated values of effective
U parameter for transition-metal elements. A plane-wave basis
and projector augmented-wave pseudopotentials are used, with
Co d8s1, Fe d7s1 and Si s2p2 electrons treated self-consistently.
A 350 eV plane-wave cutoff results in good convergence of the
total energies. Forces on atoms were calculated, and atoms were
allowed to relax using a conjugate gradient technique until their
residual forces had converged to less than 0.01 eV/Å. The Brillouin
zone integration is performed using Monkhorst-Pack grids of
10 10 10 during the iterations; but, to obtain higher quality
state densities and to check the stability of the results, this number
of k-points was increased to 12 12 12 after convergence was
reached.
The structural optimizations were performed for the 16-atom
cell, which corresponds to a cubic unit cell consisting of four
formula units. The L21 structure may conveniently be described as
four interpenetrating fcc sub-lattices with Co atoms placed on the
Wycoff position 8c (1/4, 1/4, 1/4), Fe atoms on 4a (0, 0, 0) and Si
atoms on (0.5, 0.5, 0.5) position, as shown in Fig. 1. Each Fe or Si
has eight Co atoms as first neighbors, placed in octahedral
symmetry position, while each Co has four Fe and Si atoms as
first neighbors, and the symmetry of the crystal is reduced to the
tetragonal one due to displaying the disorder in Co2FeSi. In order
to understand the magnetic properties in disordered Co2FeSi, the
Co and Fe are designated as Co1, Co2, Co3, Co4, Co5, Co6, Co7, Co8,
Fe1, Fe2, Fe3 and Fe4 (labeled in Fig. 1). We consider two types of
disorder of the full-Heusler alloy, one is a Fe–Co type disorder
where interchanging the Fe and Co sites corresponding to Co2FeSi,
and the other is a Fe–Si type disorder where interchanging the Fe
and Si sites corresponding to Co2FeSi.
3. Results and discussion
The lattice constant of this alloy is obtained by optimizing the
structure using Murnaghan equation of state
" #
B0 V
V 0 B′
V0
1 þ E0
þ B′ 1 ð1Þ
Etot ðVÞ ¼
B′ðB′ 1Þ
V
V
where B0 is the bulk modulus, B′ is the bulk modulus derivative,
and V0 the equilibrium volume. The equilibrium lattice constant
and bulk modulus are calculated using both GGA and GGA þ U;
the results are compared with available experimental data in
Table 1. A comparison between the equilibrium lattice constants
predicted using GGA and GGA þU for the exchange and correlation
functional shows that GGA þ U is essential to accurately reproduce
the equilibrium structural properties of these Heusler alloys. GGA
underestimates the equilibrium volume with respect to experiment, whereas the GGA þU gives larger lattice constants and this
results in a better agreement with experiment. We see from
Table 1 that the GGA bulk modulus are also smaller than GGA þU,
as already observed for the 3d ferromagnetic metals. It therefore
seems that the GGA þ U results are in better agreement with
experiment, and so results in an overall more accurate description
of the structural properties of these Heusler compounds. In order
to investigate the stability of the alloys, tetragonal distortions
using GGA and GGA þU methods are applied (changing the c/a
ratio), see Fig. 2. From this figure, one can see that Co2FeSi has a
minimum energy at c/a ¼1 and a second minimum occurs at
higher energies for c/a close to 1.2 under GGA while there is only
one energy minimum at c/a ¼1 under GGA þU which means that
Co2FeSi is stable in L21 structure.
Fig. 3 shows the total DOS of Co2FeSi for V/V0 ¼ 1.00–0.82 in
both GGA and GGA þU methods. The majority spin band structure
is strongly metallic, while the minority spin band structure shows
a semiconducting gap around the Fermi level (EF) in Co2FeSi. The
volume compression leads to a slight increase of the minority
band gap, where the valence band edges move away from the EF
although the conduction band edges move toward the EF in the
minority-spin state. Moreover, at fixed lattice constant, GGA þU
predicts a larger band gap compared to GGA; the valence bands
are shifted downwards, while EF is more or less fixed with respect
to the conduction band edges. The GGA and GGA þU methods
reveal somewhat different values of the magnetic moment of
the alloy. The GGA þU result for the magnetic moment is almost
exactly that expected from Slater–Pauling rule, μT ¼ 6, μCo ¼ 1:476,
Table 1
The calculated lattice constant (a) in Å, total magnetic moments (M T ) in μB /f.u.,
local magnetic moments, μX (X ¼Co, Fe and Si), in μB =atom, bulk modulus (B0) in
GPa and its derivative (B′) in comparison with other theoretical studies and
experimental data for Co2FeSi alloy.
a
Mt
μCo
μFe
μSi
B0
B′
a
GGA þ U
Expt
5.618
5.08
1.200
2.679
0.003
207.1
4.62
5.639
6
1.476
3.143
0.001
209.3
4.67
5.64ab 5.65cd
6ab 6.17 0.2d
1.2 7 0.1a 1.38 7 0.2d
2.6 7 0.1a 3.017 0.4d
–
240c
4c
Reference [21].
Reference [22].
c
Reference [7].
d
Reference [8].
b
Fig. 1. Co2FeSi Heusler (L21)-type ordered structure where the Co atoms sit at the
A, C sites and the Fe, Si atoms sit at B, D sites, respectively.
GGA
X. Zhu et al. / Journal of Physics and Chemistry of Solids 75 (2014) 391–396
μFe ¼ 3:143, while GGA yield μT ¼ 5:08. The GGA þ U results of local
magnetic moments of Co and Fe are larger than experimental data
while the GGA results are in better agreement with experiment.
The magnetic moments in experiment derive from a sum rule
analysis at T ¼300 K and μ0H¼0.4 T which have a rather high
degree of disorder [21]. In Fig. 4, we show the total magnetic
moments and the local magnetic moments of Co and Fe as a
function of applied lattice constant. The total magnetic moment of
GGA þU is saturated at a ¼ aeq while the total magnetic moment of
GGA straightly increases with the lattice constant. Note that the
Fig. 2. The total energy as a function of tetragonal deformations (c/a) for the alloys
Co2FeSi. The energies are plotted with reference to the energy of the cubic phase
(c/a ¼1).
393
Co and Fe magnetic moments increase with increasing the lattice
constant; this general behavior can be explained considering that
as the Co and Mn atoms are pulled apart the increased 3d
localization leads to larger magnetic moments.
The effects of atomic disorder are simulated by interchanging
the Fe and Co/Si sites in Co2FeSi, so as to generate different
chemical environments surrounding each inequivalent Fe and Co
site. In the perfect L21 the lattice consists of four interpenetrating
fcc sublattices which are occupied in the sequence Co–Fe–Co–Si.
The Co1–Co4 atoms sit at the A site and the Co5–Co8 atoms sit at
the C site (see Fig. 1). Disorder concentration c is defined as the
ratio of the number Fe antisites on Fe/Si sublattice to the total
number of Fe/Si atoms. With this definition, the concentration
c¼ 0.5 corresponds to the perfect B2 order, i.e., a complete mixing
of Fe and Si elements on Y and Z sublattices. For c ¼0.25 in Fe–Co
type disorder, we consider tow kinds of disorder configurations:
interchanging (I) the Fe and Co(A) sites and (II) the Fe and Co
(C) sites. For c ¼0.5 in Fe–Co type disorder, we consider three kinds
of disorder configurations: interchanging (I) 2Fe atoms and 2Co
(A) sites, (II) 2Fe atoms and 2Co(C) sites and (III) 2Fe atoms and Co
(A) Co(C) sites. In Table 2, we show the calculated lattice constant,
the total energy relative to the ordered Co2FeSi, and the total
magnetic moment as the disorder level c. It is found that the cubic
structure in Co2FeSi cannot be maintained in the disordered
structure which have slight distortion except configurations (III)
of c¼0.5. For configurations (III) of c ¼0.5, the structure of Co2FeSi
has large deformation and the total energy relative to that of the
ordered structure is extremely large which means that this type of
Fig. 3. The total DOS of Co2FeSi for the distortion V/V0 ¼1.00 0.82 using GGA and GGA þU approximations.
394
X. Zhu et al. / Journal of Physics and Chemistry of Solids 75 (2014) 391–396
Fig. 4. The total magnetic moments and the local magnetic moments of Co and Fe in μB as a function of applied lattice constant using GGA and GGA þ U approximations.
Table 2
The calculated lattice constant (a,b,c) in Å, total energy relative to that of the ordered structure (ΔE) in meV/f.u, and total magnetic moment (M T ) in μB =f :u: as disorder level c
of Co2FeSi.
Fe–Co disorder
Fe–Si disorder
c ¼0.5
c¼ 0.25
I
a
b
c
ΔE
MT
GGA
5.600
5.600
5.600
27
4.73
II
GGA þ U
5.603
5.603
5.603
250
5.36
GGA
5.601
5.601
5.601
28
4.73
c ¼0.25
I
GGA þ U
5.602
5.602
5.602
253
5.36
GGA
5.582
5.582
5.604
100
4.38
II
GGA þ U
5.616
5.616
5.623
502
5.12
GGA
5.581
5.594
5.594
105
4.38
disorder (replacing both the nearest neighbor Co atoms) is difficult
to exist. For c¼ 0.25 and c ¼0.5 in Fe–Co type disorder, the total
energy of the configurations (I) relative to the ordered Co2FeSi is
lowest. The total energy of the Fe–Co type disorder relative to the
ordered Co2FeSi rapidly increases compared with the Fe–Si type
disorder. Thus, it is expected that the concentration of Fe–Co type
disorder is unlikely larger compared with the Fe–Si type disorder.
Our calculated disordered models using GGA and GGA þU methods yield the ground state to be magnetic with a reduced total
magnetic moment for Fe–Co and Fe–Si type disorder compared to
that of the fully ordered sample in Table 1. The calculated results
are in agreement with the experimental observation of a lower
total magnetic moment resulting from disorder defects [21].
Here we discuss the magnetic properties and half-metallic
character using the configuration (I) of Fe–Co type disorder for
c¼ 0.25 and c ¼0.5 which have the lowest total energy. The
magnetic moments at various Fe and Co inequivalent sites of
Fe–Co and Fe–Si type disorder are given in Table 3. We discuss our
magnetic property based on GGA methods because there is not
difference between the results of GGA and GGA þU except the
larger value with GGA þU than GGA. In the Fe–Co type disorder
case: the magnetic moments of the Co at Fe site (CoðFe′Þ) increases
from 1:2 μB =f:u: to 1:397 μB =f:u: at c¼0.25, then to 1:445 μB =f:u: at
c¼ 0.5 as the Co nearest neighbors are replaced by magnetic
c ¼0.5
III
GGA þU
5.601
5.599
5.605
513
5.15
GGA
–
–
–
302
3.32
GGA þ U
–
–
–
1678
4.23
GGA
5.551
5.551
5.732
153
4.85
GGA þ U
5.661
5.661
5.603
188
5.97
GGA
5.497
5.497
5.872
209
4.74
GGA þU
5.609
5.609
5.707
233
5.93
Fe and Co elements while the magnetic moments of the Fe at
Co site (FeðCo′Þ) decreases sharply from 2:679 μB =f:u: to 1:728 μB =f:u:
at c¼ 0.25, then to 1:636 μB =f:u: at c¼ 0.5 as the Co nearest
neighbors are replaced by nonmagnetic Si elements; the magnetic
moments of the Co8 and Co2 are slightly different in spite of that
they have the same nearest and next-nearest chemical environments which may due to the difference of the atom sites; the
magnetic moments of the Fe are basically not affected by the
Fe–Co type disorder while the magnetic moments of Co decrease
with the nearest number of Fe; the next-nearest chemical environments have little effect on magnetic moments of Fe and Co.
In Fe–Si type disorder case: note here that the Fe and Si sites have
as nearest neighbors eight Co atoms and thus in this case both
Co and Fe atoms have similar spin magnetic moments as Co and
Fe in ordered L21 cases; the next-nearest chemical environments
have little effect on magnetic moments of Fe and Co.
The calculated density of the electronic states (DOS) for Co2FeSi
alloy with the L21-type crystal structure obtained using the GGA
and GGA þU approaches, are shown in Fig. 5 (c¼ 0.0). The minority
DOS exhibits a clear gap around EF, confirming the half-metallic
character of the material. It is seen that the main contributions to
the resulting DOS of Co2FeSi alloy obtained by either GGA or
GGA þU methods are due to the Co and Fe atoms. The most intense
peaks of the DOS are formed by the coincident in energy Co and Fe
X. Zhu et al. / Journal of Physics and Chemistry of Solids 75 (2014) 391–396
Table 3
The atoms in parenthesis show the atoms which were replaced, e.g., Co (Fe1) in the
table means the Co replace the Fe1 site. The third and fourth columns give the
nearest and the next-nearest number of Co, Fe and Si neighbors, respectively. The
last column gives the magnetic moment at various inequivalent sites in μB .
Concentration
Sites
nn
nnn
(Fe–Co disorder)
c ¼ 0.25
c ¼ 0.5
c ¼ 0.25
c ¼ 0.5
Co2
4Siþ 3Feþ Co
Co3
4Siþ 3Feþ Co
Co4
4Siþ 3Feþ Co
Co5
4Siþ 3Feþ Co
Co6
4Siþ 3Feþ Co
Co7
4Siþ 3Feþ Co
Co8
4Siþ 3Feþ Co
CoðFe1′Þ
5Co þ Fe
FeðCo1′Þ
4Siþ 3Feþ Co
Fe2
6Co
Fe3
6Co
Fe4
6Co
Co3
4Siþ 2Feþ 2Co
Co4
4Siþ 2Feþ 2Co
Co5
4Siþ 2Feþ 2Co
Co6
4Siþ 2Feþ 2Co
Co7
4Siþ 2Feþ 2Co
Co8
4Siþ 2Feþ 2Co
CoðFe1′Þ
2Feþ 4Co
CoðFe2′Þ
2Feþ 4Co
FeðCo1′Þ
4Siþ 2Feþ 2Co
FeðCo2′Þ
4Siþ 2Feþ 2Co
Fe3
6Co
Fe4
6Co
(Fe–Si disorder)
Fe(Si)
8Co
Fe2
8Co
Fe3
8Co
Fe4
8Co
Fe(Si)
8Co
Fe3
8Co
Fe4
8Co
6Co
6Co
6Co
5Co þ Fe
5Co þ Fe
5Co þ Fe
6Co
6Si
6Co
6Si
6Si
6Si
6Co
6Co
1Feþ 5Co
2Feþ 4Co
2Feþ 4Co
6Co
6Si
6Si
6Co
6Co
6Si
6Si
6Si
2Feþ 4Si
2Feþ 4Si
4Feþ 2Si
6Si
2Feþ 4Si
2Feþ 4Si
Magnetic moments
GGA
GGAþ U
1.130
1.137
1.127
1.166
1.168
1.161
1.072
1.397
1.728
2.660
2.659
2.658
1.052
1.051
1.018
1.091
1.091
1.017
1.445
1.445
1.636
1.636
2.626
2.630
GGA
2.616
2.719
2.719
2.647
2.626
2.624
2.626
1.252
1.252
1.252
1.263
1.263
1.263
1.199
1.628
1.969
2.982
2.982
2.982
1.308
1.307
1.266
1.359
1.359
1.263
1.997
1.838
2.245
2.241
2.985
2.985
GGAþ U
3.060
3.159
3.159
3.159
3.101
3.100
3.100
395
states. As expected, there exists a splitting between the valence eu
and t1u states owing to the octahedral crystal-field effect. A
detailed analysis shows that the contribution to the hybridized
states is due mainly to the Co and Fe 3d states, which indicates a
covalent character of their interaction. The contribution to the
total DOS from the Si states is small. This means that the Si atoms
form essentially ionic bonds with the surrounding atoms. The high
density below EF is dominated by d-states being located at Co and
Fe sites. Inspecting the majority DOS one finds a small density of
states near EF. This density is mainly derived from states located at
Co and Si sites. In the structures, where Co atoms are nearestneighbors, there is a region of low density of states around the
Fermi level in the majority-spin band and a high degree of spinpolarization is exhibited. Fig. 5 also shows the total DOS and the
partial DOS of Mn and Co 3d for the Fe–Co disorder using the spinpolarized GGA and GGA þU calculations. The John–Teller (JT)
distortion and the tilting of the octahedra result in further splitting
of eu states. For GGA, the occurrence of Fe–Co type disorder
destroys the half-metallic character due to Co and Fe d DOS
located just in proximity to EF and the projected density of states
reveals that this is almost entirely due to the disordered Co d
states. For GGA þU, at c¼0.25 the Fe–Co disorder does not
significantly degrade the half-metallicity of the alloy while at
c¼ 0.5 half metallicity is lost, due to the antisite Co d states and the
spin polarization at the Fermi level is strongly reduced.
The total DOS and partial DOS of Co 3d and Fe 3d in Fe–Si type
disordered Co2FeSi are shown in Fig. 6. The Co atoms remain at
their X site and still form a lattice with octahedral symmetry. Thus
the gap created in the ordered structure between the eu and t1u
states, which obey the octahedral group symmetry and are
exclusively located at the Co sites, is not affected by the Fe–Si
disorder. Thus, the Fe–Si type disorder can maintain half-metallic
properties where a gap similar to the one in the ordered case
exists although partially filled by conduction band. The partial DOS
of Co 3d at c ¼0.25 and c¼ 0.5 in Fe–Si type disordered case
Fig. 5. Total densities of states and partial densities of states of Co 3d and Fe 3d in ordered and Fe–Co type disordered Co2FeSi using the GGA and GGA þ U approximations.
396
X. Zhu et al. / Journal of Physics and Chemistry of Solids 75 (2014) 391–396
Fig. 6. Total densities of states and partial densities of states of Co 3d and Fe 3d in Fe–Si type disordered Co2FeSi using the GGA and GGA þ U approximations.
remains almost unaltered. The partial DOS of Fe and FeðSi′Þ 3d at
c¼ 0.5 is exactly same where we show them together as Fe 3d in
Fig. 6. Since Fe–Si type disorder do not destroy the half-metallic
character of the ordered compounds we expect that the Slater–
Pauling rule still holds for them and this is confirmed from the
calculated total spin magnetic moments which have the similar
values as in the ordered cases (see Table 2).
4. Conclusions
Employing first-principles electronic structure calculations, we
have investigated the electronic and magnetic properties of
Co2FeSi under pressure and disorder. The calculated total magnetization of 6 μB in Co2FeSi is in excellent agreement with recent
experiments. The volume compression leads to a slight increase of
the minority band gap, where the valence band edges move away
from the EF although the conduction band edges move toward the
EF in the minority-spin state. The magnetic moments of Co and Fe
are determined mainly by nearest neighbors atoms and the nextnearest chemical environments have little effect on magnetic
moments of Fe and Co. The Co2FeSi with Fe–Si type disorder can
maintain half-metallic properties but the Fe–Co type disorder
model is prohibitive for half-metallic properties.
Acknowledgments
Projects 11147185 and 21203097 are supported by Natural
Science Foundation of China. Numerical calculation was carried
out using the facilities of the Department of Physics in Nanjing
Normal University.
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