Typeset with jpsj2.cls <ver.1.2> Full Paper
Fully Relativistic Full-Potential Calculations of Magnetic Moments in
Uranium Monochalcogenides with the Dirac Current
Shugo Suzuki and Hidehisa Ohta
Institute of Materials Science, University of Tsukuba, Tsukuba 305-8573
(Received April 26, 2010)
We study the orbital, spin, and total magnetic moments in uranium mono chalcogenides,
UX where X=S, Se, and Te, using the fully relativistic full-potential calculations based on
the spin density functional theory. In particular, the orbital magnetic moments are calculated
with the Dirac current. We employ two methods which adopt distinctly different basis sets;
one is the fully relativistic full-potential linear-combination-of-atomic-orbitals (FFLCAO)
method and the other is the fully relativistic full-potential mixed-basis (FFMB) method.
Showing that the orbital magnetic moments calculated using the FFLCAO method and
those calculated using the FFMB method agree very well with each other, we demonstrate
that, in contrast to the conventional method, the method with the Dirac current enables us
to calculate the orbital magnetic moments even if the basis set includes basis functions with
no definite angular momenta, e.g., the plane waves in the FFMB method. Furthermore, it
is found that the orbital magnetic moments obtained in this work are larger by nearly 0.4
µ
B
than those obtained using the conventional method. This is crucial because the resultant
differences in the total magnetic moments are about 30 %. We compare the results of this
work with those of previous theoretical and experimental studies.
KEYWORDS: Dirac current, orbital magnetic moment, fully relativistic calculations, full-
potential calculations, uranium monochalcogenide, spin density functional theory
1. Introduction
For more than a half century, the properties of actinide compounds have been studied both
experimentally and theoretically.
1)
Among them, uranium monochalcogenides, UX where X =
S, Se, and Te, have been studied extensively as a typical material. In particular, their magnetic
properties have attracted much attention.
2–11)
The experimental studies have revealed that
UX are ferromagnetic at low temperatures. One remarkable feature of the ferromagnetism in
UX is that it is the orbital magnetic moments, M
orb
, that dominate in the total magnetic
moments, M
tot
, overcoming the spin magnetic moments, M
spin
.
12)
The calculation of M
orb
is subtle in contrast to the calculation of M
spin
; for the latter
quantity, we need only to integrate the spin density. So far, for calculating M
orb
, a conventional
method has been widely used.
13)
In this method, the magnetic-moment operator is defined
in an appropriate way, and M
orb
are then calculated using the expectation values of the
magnetic-moment operator in the Bloch states. In actual calculations, the evaluation of the
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J. Phys. Soc. Jpn. Full Paper
elements of the magnetic-moment operator requires little more than a rearrangement of the
overlap matrix. The magnetic moments of the unit cell calculated using this method can be
partitioned into the magnetic moments of the constituent atoms or atomic orbitals attributing
each basis function to the atom or atomic orbital to which the basis function belongs. A
disadvantage of this method is that one cannot use basis functions with no definite angular
momenta, e.g., plane waves.
Another method for calculating the magnetic moments is to use the current density as
described in the textbooks of electrodynamics.
14)
That is, if the current density is obtained,
one can calculate the magnetic moments by integrating the cross product between the posi-
tion vector and the current density. For UX as well as other actinide compounds, since the
relativistic effects are significant, the calculation of the current density should be p erformed
using the Dirac current because this includes all the relativistic effects. In the fully relativis-
tic calculations based on the density functional theory adopting a single-particle equation of
the Kohn-Sham-Dirac type, the Dirac current is calculated simply using the Dirac matrices.
Furthermore, in contrast to the conventional method for calculating M
orb
, this method has
an advantage that the procedure can be applied even if the basis set includes basis functions
with no definite angular momenta. This is favorable because the physical quantities should be
calculated whatever the basis set is if its quality is good. However, to our knowledge, the cal-
culation of the magnetic moments in UX as well as other actinide compounds with the Dirac
current has not been reported so far. Thus, it seems interesting to compare M
orb
calculated
using the conventional method and those calculated using the method with the Dirac current.
When applying the method with the Dirac current, the following point should be noted.
Since the integral of the cross product between the position vector and the Dirac current does
not converge if the integral is performed over an infinitely extended system, as is the case for
a crystalline solid, because of the position vector in the integrand. For this reason, an appro-
priate atomic partitioning scheme is needed. One natural choice is to use the Voronoi cells. In
actual calculations, since the Voronoi cells with sharp boundaries are not suitable for accurate
numerical calculations, the Voronoi cells with smooth boundaries are useful instead.
15–17)
It is worth pointing out that, strictly speaking, there is no guarantee of reproducing M
orb
if one employs the spin density functional theory (SDFT), the framework used widely so far,
in which only the electron density and the spin density are taken as basic variables. The theory
that takes the current density as an additional basic variable has been developed, known as
the current density functional theory (CDFT).
18–23)
Although, even within SDFT, M
orb
in
UX induced by spin-orbit coupling largely contribute to M
tot
, M
orb
are most likely enhanced
considerably when taking account of the exchange-correlation effects due to the current density
as shown for 3d ferromagnetic metals.
24)
On the other hand, even if we restrict ourselves within
SDFT, it seems important to compare M
orb
calculated using the two different methods, i.e.,
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the conventional method and the method with the Dirac current; since all the calculations of
M
orb
in UX reported so far have been performed within SDFT, the restriction within SDFT
at this stage may be useful for unambiguous comparison and also for step-by-step progress.
In this work, we study M
orb
, M
spin
, and M
tot
in UX using the fully relativistic full-
potential calculations. In particular, M
orb
are calculated with the integral of the cross product
between the position vector and the Dirac current. The calculations are performed with the
fully relativistic full-potential linear-combination-of-atomic-orbitals (FFLCAO) method and
the fully relativistic full-potential mixed-basis (FFMB) method, both based on SDFT within
the local spin density approximation (LSDA).
25)
With the two methods, whose basis sets are
distinctly different from each other, we can examine the reliability of the results with respect
to the quality of basis sets. In §2, we describe the method of calculations. The results and
discussion are given in §3. Here, we compare the results of the FFLCAO calculations and those
of the FFMB calculations. We also compare the results of this work with those of previous
theoretical and experimental studies. Finally, we give the conclusions of this work in §4.
2. Method of Calculations
We begin with the following self-consistent equations:
[
cα · p + (β − I)mc
2
+ V
es
(r) + V
xc
(r) + βΣ · B
xc
(r)
]
ψ
ν
(r) = ε
ν
ψ
ν
(r) ,
(1)
ρ(r) = −e
∑
ν
f
ν
ψ
ν
(r)
†
ψ
ν
(r) ,
(2)
and
m(r) = −e
∑
ν
f
ν
ψ
ν
(r)
†
βΣψ
ν
(r) .
(3)
In the Dirac Hamiltonian in the left-hand side of eq. (1), c and m denote the speed of light and
the rest mass of an electron, respectively, and the rest energy of an electron, mc
2
, is subtracted.
Also, α and β are the Dirac matrices in the usual representation.
26)
In the self-consistent equa-
tions, the four-component spinor ψ
ν
(r) is the one-electron wave function of the νth level with
the energy eigenvalue ε
ν
and the occupation number f
ν
; for a crystalline solid, ν represents
the band index n and the wave vector k. In eq. (1), V
es
(r) is the electrostatic potential orig-
inated in the nuclear charges and the electron charge density, where the latter is denoted by
ρ(r) in eq. (2) with e being the electron charge. Also, V
xc
(r) = [V
up
xc
(r) + V
down
xc
(r)]/2 is the
effective scalar potential that describes the spin-independent part of the exchange-correlation
potential and B
xc
(r) = [V
up
xc
(r) − V
down
xc
(r)]/2 e
z
is the effective magnetic field that describes
the spin-dependent part of the exchange-correlation potential, where V
up
xc
(r) and V
down
xc
(r)
represent the exchange-correlation potentials for up- and down-spin electrons, respectively,
and e
z
represents the unit vector along the z axis; B
xc
(r) is originated in the spin magnetiza-
tion density, m(r), which is calculated with Σ = I
2
⊗ σ where I
2
is the 2×2 unit matrix and
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σ are the usual 2×2 Pauli spin matrices. The electron charge density ρ(r) and the spin mag-
netization density m(r) are calculated with ψ
ν
(r) and f
ν
. The Dirac current is then obtained
with the following equation:
j(r) = −e
∑
ν
f
ν
ψ
ν
(r)
†
cαψ
ν
(r) .
(4)
It is crucial to note that j(r) consists of not only the orbital contribution but also the spin
contribution according to the Gordon decomposition.
27)
To divide m(r) and j(r) into atomic components, we use the atomic partitioning scheme
adopting the Voronoi cells with smo oth boundaries.
15–17)
In this scheme, the weight function
associated with the ath atom, w
a
(r), is introduced as follows:
∑
a
w
a
(r) = 1 ,
(5)
where
w
a
(r) = p(r − r
a
)/
∑
b
p(r − r
b
)
(6)
with p(r) being a function which typically is large for small arguments and small for large
arguments. In this work, we use p(r) = [exp(1/nr) − 1 − 1/nr]
n
with n = 5; instead of taking
the limit n → ∞, we take n = 5 for performing the numerical integration accurately. Using
w
a
(r), a function of space variables, f(r), is divided into atomic components, f
a
(r), as follows:
f(r) =
∑
a
f
a
(r) ,
(7)
where f
a
(r) = w
a
(r)f(r). The integral of f(r) over the whole solid, I, is also divided into
atomic components, I
a
, as follows:
I =
∑
a
I
a
,
(8)
where
I =
∫
f(r) dr (9)
and
I
a
=
∫
f
a
(r) dr . (10)
Thus, using the spin magnetization associated with the ath atom, m
a
(r) = w
a
(r)m(r), we
calculate the atomic spin magnetic moment, M
spin
a
:
M
spin
a
=
∫
m
a
(r) dr . (11)
Also, using the Dirac current associated with the ath atom, j
a
(r) = w
a
(r)j(r), we calculate
the atomic total magnetic moment, M
tot
a
:
M
tot
a
=
1
2c
∫
r × j
a
(r) dr . (12)
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Here, it may be worth mentioning again that j(r) consists of not only the orbital contribu-
tion but also the spin contribution. Accordingly, M
tot
a
consists of both the orbital and spin
contributions. Finally, we calculate the atomic orbital magnetic moment, M
orb
a
:
M
orb
a
= M
tot
a
− M
spin
a
.
(13)
An important point to be noted is that M
tot
a
calculated with eq. (12) is independent of the
choice of the origin of the position vector only if
∫
j
a
(r) dr = 0 . (14)
We have checked that this condition is always satisfied for the calculated results given in the
next section.
We here remark that the conventional method used in previous theoretical studies for
calculating M
orb
is different from that used in this work although the method for calculating
M
spin
is the same. The formula used previously for calculating M
orb
is the following:
13)
M
orb
a
= −e Re
(
∑
p∈a
∑
q
∑
ν
f
ν
C
∗
pν
C
qν
∫
χ
p
(r)
†
βlχ
q
(r)dr
)
.
(15)
Here χ
p
(r) are the basis functions employed in the calculations and C
pν
are the coefficients in
the expansion of ψ
ν
(r) with χ
p
(r). Also, l represent the angular momentum operator, r × p.
It is important to note that eq. (15) is applicable only if we can evaluate lχ
q
(r) definitely;
for example, it is impossible to calculate M
orb
with eq. (15) if the basis set includes plane
waves. On the contrary, we can use eqs. (11)-(13) for calculating M
orb
with any type of basis
function if the quality of the basis set is good.
UX crystallize in the NaCl structure exhibiting a strong magnetic anisotropy with an
easy axis in the [111] direction.
1)
The experimental lattice constants of US, USe, and UTe
are 5.489, 5.740, and 6.155
˚
A, respectively. These exp erimental lattice constants were used in
our calculations. We assumed that the magnetization axis is in the [111] direction, which was
taken as the z axis in our calculations. The basis functions adopted in the FFLCAO method
consist of the following four-component atomic orbitals: 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f,
5s, 5p, 5d, 5f, 6s, 6p, 6d, and 7s orbitals of the neutral U atom, 5f, 7s, and 7p orbitals of
the U
2+
atom, 1s, 2s, 2p, 3s, and 3p atomic orbitals of the neutral S atom, and 3s, 3p, and
3d orbitals of the S
2+
atom, 1s, 2s, 2p, 3s, 3p, 3d, 4s, and 4p atomic orbitals of the neutral
Se atom, and 4s, 4p, and 4d orbitals of the Se
2+
atom, 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 5s,
and 5p atomic orbitals of the neutral Te atom, and 5s, 5p, and 5d orbitals of the Te
2+
atom.
Also, the basis functions adopted in the FFMB method consist of the four-component atomic
orbitals of neutral U and X atoms used in the FFLCAO method and four-component plane
waves, which are positive-energy solutions of the Dirac equation for a free electron. In this
work, we chose the cut-off energy of the four-component plane waves to be 50 eV. This cut-off
energy corresponds to about 40, 50, and 60 four-component plane waves per each k point
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