Download Magnetic, optical, and magneto-optical properties of MnX „X As, Sb

PHYSICAL REVIEW B
VOLUME 59, NUMBER 24
15 JUNE 1999-II
Magnetic, optical, and magneto-optical properties of MnX „X5As, Sb, or Bi…
from full-potential calculations
P. Ravindran,* A. Delin, P. James, and B. Johansson
Condensed Matter Theory Group, Department of Physics, Uppsala University, Box 530, 75121 Uppsala, Sweden
J. M. Wills
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545
R. Ahuja and O. Eriksson
Condensed Matter Theory Group, Department of Physics, Uppsala University, Box 530, 75121 Uppsala, Sweden
~Received 21 August 1998!
The magneto-optic ~MO! Kerr and Faraday spectra for manganese pnictides are calculated using the all
electron, relativistic, full-potential linear muffin-tin orbital method. The amplitude of our calculated spectra are
found to be in good agreement with corresponding experimental spectra. Although the MO property is a rather
complicated function of the diagonal and off-diagonal elements of the optical conductivity tensor, present
theory nevertheless provides very practical insight about its origin in these compounds. The largest Kerr effect
observed in MnBi can be understood as a combined effect of maximal exchange splitting of Mn 3d states and
the nearly maximal spin-orbit ~s-o! coupling of Bi. The frequency-dependent optical properties, namely reflectivity, absorption coefficient, electron-energy-loss spectra, refractive index, extinction coefficient are given.
From our calculations ~including spin-orbit coupling and orbital polarization! the site-projected spin and orbital
moments are also obtained and compared to the available experimental values and a good agreement is found.
The magnetic anisotropy energy is calculated with a minimal number of approximations for the three systems.
A disagreement between theory and experiment is found. Using the generalized gradient corrected fullpotential linear augmented plane-wave method we have calculated the unscreened plasma frequencies and the
hyperfine parameters such as electric-field gradient as well as the hyperfine field. @S0163-1829~99!01419-8#
I. INTRODUCTION
Although magneto-optical properties of magnetic metals
have been known for over a hundred years,1 it is only in the
past couple of decades that vigorous interest has been focused on this subject.2 One reason is their potential for application in the technology of high-density data storage.3 The
recent discovery of the huge polar Kerr rotation u K in MnPt3
( u K 521.3° at 1.24 eV!4 as well as previous discoveries of
very large Kerr rotations in MnPtSb ~Ref. 5! and MnBi ~Ref.
6! have created considerable interest in manganese-based intermetallic compounds.
In addition to its excellent MO properties, MnBi possesses large perpendicular magneto-crystalline anisotropy at
room temperature, and hence it is extensively investigated as
a candidate for MO recording media. However, MnBi has
rather large grain size, which brings out a high level of media noise when the signal is read out. Also, after many cycles
of write/erase, a magnetic disorder/antiferromagnetic phase
appears in the recorded spots at room temperature so that its
MO characters degrade gradually. Moreover, its high Curie
temperature (T C ) is not favorable for thermo-magnetic optical recording. Thus, MnBi does not appear to be the ideal
system for MO recording. The isostructural, isoelectronic
compound MnSb has appropriate MO properties and furthermore its T C is not as large as that of MnBi. However, its easy
axis of magnetization is in the plane below 300 K ~Ref. 7!.
This drawback may be overcome, since studies of Mn/Sb
multilayers8 and sublayers9 indicate that perpendicular an0163-1829/99/59~24!/15680~14!/$15.00
PRB 59
isotropy can be achieved. MnAs, the homologous and lightest compound in the family, possesses perpendicular anisotropy and its T C is just above room temperature. However, its
Kerr angle is considerably smaller than that of MnBi and
MnSb.
The MO effects can be studied either in transmission
~Faraday effect! or in reflection ~Kerr effect!. The Faraday
effect can only be studied in sufficiently thin films. Further,
the occurrence of multiple reflections due to Faraday rotation
in the substrate, and discontinuous polarization changes at
the various interfaces, complicates the investigation of the
Faraday effect. Consequently, rather little work, experimental as well as theoretical, has been devoted to studying the
Faraday effect compared to the Kerr effect, where such complications are largely absent. The experimental results for the
Faraday effect that do exist are greatly different from each
other, and appear to depend on the preparation and composition of the films.10 Therefore, it is of general interest to
calculate Faraday spectra for the ideal crystal.
Despite the great potential for applications in erasable
high-density recording technology for MnBi and its homologues, very few attempts have been made to understand the
systematic changes of the MO properties in these systems.
Among the MnX compounds, MnBi has a maximum rotation
of ;1.6°, which is decreasing to less than 0.3° for MnAs. A
systematic study of these materials will give more insight
into the physical origins of large MO effects, and aid in
identifying which kind of interactions are the most crucial
ones in the design of materials with desirable MO properties.
15 680
©1999 The American Physical Society
PRB 59
MAGNETIC, OPTICAL, AND MAGNETO-OPTICAL . . .
15 681
633 K in MnBi. In our calculations, the experimental lattice
parameters have been used throughout, i.e., a53.722 Å,
c55.702 Å for MnAs, a54.122 Å and c55.7549 Å for
MnSb, and a54.170 Å and c55.755 Å for MnBi.
A. FLAPW calculations
FIG. 1. Crystal structure of manganese pnictides. The white
balls represent the Mn atoms and the black balls represent the pnictide atoms.
Recent studies by Mohn, Blaha, and Schwarz11 show that if
the material has strongly aspherical spin density, the fullpotential treatment is important in order to achieve the correct magnetic ground state. Since the MO properties arise
from the exchange splitting in combination with spin-orbit
coupling, their result suggests that the full-potential treatment may be important in order to describe the MO properties properly. In addition, the calculation of MO properties
requires an accurate description of the Kohn-Sham eigenvalues and eigenvectors, which motivates use of a full potential
method. In this paper, we present systematic studies of the
electronic structure, the magnetic moments, and the optical
and MO properties of these compounds using the fullpotential linear muffin-tin orbital ~FLMTO! method.
The rest of this paper is organized as follows. The computational details regarding the calculations of the magnetic,
optical, and magneto-optical properties, as well as the hyperfine parameters are given in Sec. II. In Sec. III, the results
from our calculations are presented and discussed. In Sec.
IV, we describe the calculated electronic structure, whereas
Secs. V and VI describes the optical and magneto optical
properties, respectively. The most important conclusions
drawn from our FLMTO and full-potential linear augmented
plane-wave ~FLAPW! calculations are given in Sec. VIII.
II. COMPUTATIONAL DETAILS
The MnX compounds crystallize in the NiAs structure
~D46h , space group P63 /mmc!. The unit cell is shown in Fig.
1. It consists of two Mn atoms at the Wyckoff 2c sites ~ 31 23
2 1 3
1
4 ! and ~ 3 3 4 ! and two pnictogens at the 2a sites ~0 0 0! and
~0 0 1/2!. Each Mn atom is surrounded by a trigonally distorted octahedron of pnictogen atoms. The manganese pnictides studied here are all ferromagnets and the Curie temperature increases from 318 K in MnAs to 555 K in MnSb to
In the full-potential linearized augmented plane wave
~FLAPW! method,12 space is divided into an interstitial region ~IR! and nonoverlapping muffin-tin ~MT! spheres centered at the atomic sites. In the IR, the basis set consists of
plane waves. Inside the MT spheres, the basis set is described by radial solutions of the one-particle Schrödinger
equation, at fixed energies, and their energy derivatives times
spherical harmonics. Valence states and semicore states ~Mn
3 p, As3d, Sb4d, Bi5d) were treated in the same energy
window by using local orbitals13 as an extension to the
LAPW basis set. The charge densities and potentials in the
atomic spheres were represented by spherical harmonics up
to l56, whereas in the interstitial region these quantities
were expanded in a Fourier series with 745 stars of reciprocal lattice vectors G. The radial basis functions of each
LAPW were calculated up to l510 and the nonspherical
potential contribution to the Hamiltonian matrix had an upper limit of l54. The Brillouin-zone integration was done
with a modified tetrahedron method14 and we used 120 k
points in the irreducible wedge of the Brillouin zone ~IBZ! of
the hexagonal lattice for our self-consistent calculation and
240 as well as 420 k points in the IBZ for our unscreened
plasma frequency calculations. Though the optical properties
are believed not to be sensitive to generalized gradient corrections ~GGA!, earlier calculations15 on electric-field gradient ~EFG! show that using GGA instead of the standard local
spin-density approximation ~LSDA! can improve on the theoretical EFG values significantly. For this reason, we have
used GGA ~Ref. 16! in all our FLAPW calculations. Spinorbit coupling was not included in the FLAPW calculations.
B. FLMTO calculations
The full-potential LMTO calculations17 presented in this
paper are all electron and no shape approximation to the
charge density or potential has been used. The base geometry
in this computational method consists of a muffin-tin part
and an interstitial part. The basis set is comprised of augmented linear muffin-tin orbitals.18 Inside the muffin-tin
spheres the basis functions, charge density, and potential are
expanded in symmetry adapted spherical harmonic functions
together with a radial function and a Fourier series in the
interstitial. In the present calculation the spherical-harmonic
expansion of the charge density, potential, and basis functions were carried out up to l max56. The tails of the basis
functions outside their parent spheres are linear combinations
of Hankel or Neuman functions depending on the sign of the
kinetic energy of the basis function in the interstitial region.
For the core charge density, the Dirac equation is solved self
consistently, i.e., no frozen core approximation is used. The
calculations are based on the local density approximation to
the density functional with the exchange-correlation potential parametrized according to Hedin and Lundquist.19
In the calculation of the radial functions inside the
muffin-tin spheres, all scalar-relativistic terms are included,
15 682
P. RAVINDRAN et al.
with the small component vanishing at the sphere boundary.
The spin-orbit ~s-o! term is included directly in the Hamiltonian matrix elements for the part inside the muffin-tin
spheres, thus doubling the size of the secular matrix. Moreover, the present calculations make use of a so-called multibasis, to ensure a well-converged wave function. This means
that we use different Hankel or Neuman functions each attaching to its own radial function. We, thus, have three 4s,
two 4p, three 3d, and two 4 f orbitals for Mn, three 4s, two
5s, three 4p, two 4d for As, three 5s, two 6s, three 5p, two
5d for Sb, three 6s, and two 7s, three 6p, two 6d for Bi in
our expansion of the wave function. The direction of the
moment is chosen to be perpendicular to the basal plane
~@0001# direction!. The k-space integration was performed
using the special point method with 1152 irreducible k points
in the whole Brillouin zone for the self-consistent groundstate calculations and 9216 k points for our optical as well as
MO calculations.
Many magnetic metals possess a considerable orbital
magnetic moment in addition to the spin moment. In LSDA,
the exchange-correlation potential only depends on the spin
density and an induced spin moment corresponds to a gain in
exchange energy, as implied by Hund’s first rule for atoms.
To obtain an orbital moment the spin-orbit interaction must
be included in the Hamiltonian. However, the so-calculated
orbital moment is found to be too small to account for the
experimentally observed orbital moments. A correction to
LSDA was suggested by Eriksson et al.20 in order to restore
Hund’s second rule. This correction, orbital polarization
~OP! is added to the diagonal of the Hamiltonian matrix and
has the form DE l, j ,m l 52R l, j L j m l d l,l 8 d m l m l , where R l, j is
8
the so-called Racah parameter, which is calculated using the
radial part of the wave functions for each spin channel in a
self-consistent way. L j is the orbital moment for spin channel j and m l is the magnetic quantum number.
C. Calculation of optical properties
For metals, the components of the optical conductivity
tensor are given by a sum of interband and intraband contributions. We adopted the dipole approximation in our interband transition-optical calculations, i.e., the momentum
transfer from the initial state to the final state was neglected.
Spin-flip transitions, which are allowed within the electric
dipole approximation when spin-orbit coupling is included,
were also omitted in the present calculations, since their effect is negligible.21 The interband contribution to the absorp(abs)
tive part of the optical conductivity s ab
( v ), as a function
of frequency v of the incoming/outgoing electromagnetic
radiation, in the random phase approximation, without allowance for local field effects, was calculated by summing transitions from occupied to unoccupied states ~with fixed k vector! over the Brillouin zone, weighted with the appropriate
matrix element giving the probability for the transition by22
abs!
s ~ab
~ v !5
Ve 2
8 p 2 \m 2 v
(
nn
8
E
d 3 k ^ kn u p a u kn 8 &
3^ kn 8 u p b u kn & f kn ~ 12 f kn 8 ! d ~ e kn 8 2 e kn 2\ v ! .
~1!
PRB 59
(abs)
(1)
Note that s aa
( v )5 s aa
( v ) ~i.e., the real part!, whereas
(abs)
(2)
s ab ( v )5 s ab ( v ) ~i.e., the imaginary part! when a Þ b . In
the expression above, (p x ,p y ,p z )5p52i\¹ is the momentum operator, f kn is the Fermi-distribution function ensuring
that only transitions from occupied to unoccupied states are
counted, and u kn & the crystal wave function, corresponding
to eigenvalue e kn with crystal momentum k, and d ( e kn 8
2 e kn 2\ v ) is the condition for total energy conservation.
The evaluation of matrix elements in Eq. ~1! is done over the
muffin-tin and interstitial regions separately. Further details
about the evaluation of matrix elements are given
elsewhere.23
Because of the metallic nature of MnX, the optical spectra
in the lower energy range will have a dominant contribution
from intraband transitions. The intraband contribution to the
diagonal components of the conductivity is normally described by the Drude formula,24
v 2P
s D~ v ! 5
.
4 p @~ 1/t ! 2i v #
~2!
The relaxation time t , characterizing the scattering of charge
carriers, is dependent on the amount of vacancies and other
defects, and will therefore vary from sample to sample. Here
we have chosen, by comparing the calculated intraband contribution with experimental results, \/ t 50.2 eV for all three
compounds considered. Note that the variation of this parameter within reasonable limits has very little effect on the optical conductivity for energies larger than 1 eV. The unscreened plasma frequency v P depends on the concentration
of the charge carriers. It is worth noting that the extraction of
the Drude parameters from experimental data requires a freeelectronlike region in the optical spectra. Such a region exists for metals such as aluminum or silver, but not of the
compounds considered here. Experimental determination of
the Drude parameters for these compounds is therefore intrinsically difficult, and instead of taking the Drude parameters from experiment, we have chosen an alternative path.
Hence, we have calculated the spin-resolved unscreened
plasma frequency by integrating over the Fermi surface using the relation
v 2Pii 5
8pe2
V
^ kn j u p i u kn j &^ kn j u p i u kn j & d ~ e kn j 2 e F ! ,
(
kn
~3!
where V is the volume of the primitive cell, e F is the Fermi
energy, e is the electron charge and j is the spin. The v P
were calculated using the FLAPW method from the above
relation with 240 k points as well as 420 k points in the
irreducible wedge of the Brillouin zone. The increase in k
points from 240 to 420 changes the calculated plasma frequency in the second decimals only. Our calculated v Pii are
given in Table I.
The calculations described above yield unbroadened functions. The interband transitions are affected by scattering
events that are phenomenologically described by using a finite lifetime assumed to be independent of the detailed kind
of interband excitation. Since the lifetime of an excited state
generally decreases with increased excitation energy, it is
relevant and potentially more appropriate to broaden with a
PRB 59
MAGNETIC, OPTICAL, AND MAGNETO-OPTICAL . . .
TABLE I. Unscreened plasma frequency v Pii in eV for the MnX
compounds calculated using the FLAPW method.
Compound
direction
MnAs
xx
zz
MnSb
xx
zz
MnBi
xx
zz
Spin up
Spin down
3.612
1.644
5.117
2.462
3.409
1.237
4.596
1.556
3.175
0.740
4.440
1.004
function whose width increases with excitation energy. Thus,
broadening the calculated optical spectra was performed by
convoluting the absorptive optical conductivity with a
Lorentzian. The full width at half maximum ~FWHM! of the
Lorentzian is taken to increase linearly with energy, being
0.01 eV at photon energy 1 eV. The experimental resolution
was simulated by broadening the final spectra with a Gaussian of constant FWHM equal to 0.02 eV. The dispersive
parts of the components of the optical conductivity were calculated with a Kramer-Kronig transformation. The calculated
real and imaginary parts of the optical conductivity allow the
calculation of important optical constants. The reflectivity
spectra were derived from the Fresnel’s formula for normal
incidence assuming an orientation of the crystal surface parallel to the optical axes. The relevant relations used in our
optical constants n( v ) and k( v ), absorption coefficient
I( v ) and electron-energy-loss spectra ~EELS! can be found
elsewhere.25
15 683
rotation of the axes of the polarization ellipse, as viewed by
an observer who looks in 1z direction when the incoming
linearly polarized light from a source is traveling along the
1z direction. For the polar geometry, the Kerr rotation and
ellipticity are related to the optical conductivity through the
following relation,
11tan~ h K ! 2i u
~ 11n 1 !~ 12n 2 !
e K5
,
12tan~ h K !
~ 12n 1 !~ 11n 2 !
where n 26 are eigenvalues of the dielectric tensor corresponding to Eq. ~4!. In terms of conductivities, they are
n 26 511
4pi
~ s xx 6i s xy ! .
v
S
s xx
s5 2 s xy
0
s xy
0
s xx
0
0
s zz
D
.
~4!
Consider a plane-polarized light beam traveling in the z direction, which is parallel to the magnetization direction M.
This light beam can be resolved into two circularly polarized
beams with the spinning direction of the corresponding electric field parallel and antiparallel to M. In this paper, we use
the sign convention in which right-circular polarized ~RCP!
light has its electric-field vector E rotating in a clockwise
sense at a given point in space and the signs of Faraday
rotation u F and Kerr rotation u K are positive for a clockwise
~6!
For small Kerr angles, Eq. ~5! can be simplified to28
u K 1i h K 5
2 s xy
s xx A11 ~ 4 p i/ v ! s xx
.
~7!
The magnetic circular birefringence, also called the Faraday
rotation u F , describes the rotation of the polarized plane of
linearly polarized light on transmission through matter magnetized in the direction of light propagation. Similarly, the
Faraday ellipticity h F , which is also known as the magnetic
circular dichroism, is proportional to the difference of the
absorption for right- and left-handed circularly polarized
light.27 Thus, these quantities are simply given by29
D. Calculation of magneto-optical properties
Plane-polarized light, when reflected from a metal surface
or transmitted through a thin film with nonzero magnetization, will become elliptically polarized, with its major axis
slightly rotated with respect to the original direction. The
effect due to transmission is called the MO Faraday effect,
and the reflection effect is the MO Kerr effect.26,27 The Kerr
effect exists in several different geometries. Of these, the
polar Kerr effect for which the direction of the macroscopic
magnetization of the ferromagnetic material and the propagation direction of the linearly polarized incident light beam
are perpendicular to the plane of the surface, is by far the
largest one, and therefore this geometry is the most interesting one in connection with technological applications.
Hence, we performed our theoretical investigations for the
polar Kerr effect only. With the magnetic moment in the
@0001# direction of the hexagonal crystal, the form of the
optical conductivity tensor is
~5!
u F 1i h F 5
vd
~ n 2n 2 ! ,
2c 1
~8!
where c is the velocity of light in vacuum, and d is the
thickness of the film.
E. Calculation of Hyperfine Parameters
Hyperfine parameters such as the hyperfine field ~HFF!
and the electric-field gradient ~EFG! provide information
about the interaction of a nucleus with the surrounding
charge distribution. They can be measured by various techniques like Mössbauer spectroscopy, nuclear magnetic resonance or perturbed angular correlation. For the hyperfine parameters calculation we have used the FLAPW method as
embodied in the WIEN97 code.12 The total hyperfine field
~HFF! can be decomposed into three terms such as a dominant Fermi contact term, a dipolar term and an orbital contribution. We consider only the contact term, which in the
scalar-relativistic limit is derived from the spin densities at
the nuclear site,
8
H c 5 p m 2B @ r ↑ ~ 0 ! 2 r ↓ ~ 0 !# ,
3
~9!
while in the fully relativistic case this spin density at the
nucleus is replaced by its average over the Thomson radius
r T 5Ze 2 /mc 2 . 30
The EFG is defined as the second derivative of the electrostatic potential at the nucleus, written as a traceless tensor.
This tensor can be obtained from an integral over the nonspherical charge density r (r). For instance the principal
component V zz is given by
P. RAVINDRAN et al.
15 684
TABLE II. Calculated site-projected spin and orbital moments
for the MnX systems. Spin moments are denoted by the subscript s
and the total moment by t. X denotes the pnictogen site. All moments are in units of m B /formula unit.
Compound
MnAs
Present theory
Exp.a
Theoryb
Theoryc
MnSb
Present theory
Theoryc
Exp.d
Exp.e
Theoryf
Theoryg
Theoryh
MnBi
Present theory
Exp.i
Exp.j
Exp.k
Theoryc
Theoryl
Mns
Xs
MnX t
3.45
20.052
3.62
3.40
3.10
3.06
3.14
20.08
3.68
3.34
20.040
20.07
3.35
3.30
3.5
20.032
20.06
20.17
3.78
20.041
3.71
3.50
20.10
20.02
4.01
3.82
3.843
3.95
3.61
3.56
Yamaguchi et al. ~Ref. 35!.
Katoh et al. ~Ref. 36!
c
Oppeneer et al. ~Ref. 37!.
d
Bouwma et al. ~Ref. 38!.
e
Chen et al. ~Ref. 39!.
f
Vast et al. ~Ref. 40!.
g
Coehoorn et al. ~Ref. 41!.
h
Podoloucky et al. ~Ref. 42!.
i
Zhiquang et al. ~Ref. 43!.
j
T. Chen and W. E. Stutius ~Ref. 44!.
k
R. R. Heikes ~Ref. 45!.
l
J. Köhler and J. Kübler ~Ref. 46!.
a
b
V zz 5
E
d 3 r r ~ r!
2 P 2 ~ cos u !
r3
,
TABLE III. Calculated orbital moments for two different quantizations axes and magneto crystalline anisotropy for the MnX systems. SO indicates spin-orbit coupling and OP indicates the inclusion of orbital polarization ~Ref. 20!.
Orbital moments @ m B #
@0001#
@ 1̄100#
MnAs
MnSb
MnBi
3.85
3.27
3.55
3.50
3.32
3.24
3.33
~10!
where P 2 is the second-order Legendre polynomial. A more
detailed description of the calculation of EFG can be found
elsewhere.31
III. CALCULATED MAGNETIC PROPERTIES
AND MAGNETIC ANISOTROPY ENERGIES
Both magneto-optical effects and orbital magnetic moments have a common origin in the spin-orbit coupling and
exchange splitting. Thus, a close connection between the two
phenomena seems plausible. For this reason, as a precursor
to the discussion of magneto optical properties, we list in
Table II the spin moments as well as the total ~spin and
orbital! moments ~ferromagnetically oriented along the
@0001# direction of the crystal! along with available experimental results and magnetic moments obtained from atomic
sphere approximation– ~ASA! type calculations. In our cal-
PRB 59
MAE @meV#
SO
OP
SO
OP
SO
OP
0.055
0.106
0.222
0.073
0.134
0.264
0.040
0.080
0.174
0.051
0.098
0.198
0.54
0.53
2.0
0.73
0.77
2.6
culations we find a small moment in the interstitial region,
which is not explicitly given in the table. The total magnetic
moments in the MnX compounds obtained from our calculations are found to be in good agreement with the corresponding experimental values. The ASA calculations give 0–13 %
smaller32,33 magnetic moments than the experimental values.
In all three compounds, Mn induces a small spin moment on
the pnictogen atoms through hybridization. From spinpolarized neutron diffraction studies, Yamaguchi et al.34
found a spherically symmetric negative magnetic moment of
(0.260.13) m B at the Sb sites in MnSb. The present observation of a small negative moment in the pnictogen site in
these compounds is consistent with the above studies. Our
calculated orbital contribution to the magnetic moment is
small for all the compounds listed in Table III. The magnetic
g factor was estimated47 from g5(M o 1M s )/(M o 1M s /2),
where M s is the spin contribution to the total magnetic moment and M o is the orbital moment induced by s-o coupling
and orbital polarization. Our calculated values of g for
MnAs, MnSb, and MnBi are 1.972, 1.957, and 1.925, respectively. The magnetic g factor for MnSb obtained from our
calculation is in good agreement with the average experimental value of 1.97860.002 obtained from Einstein-de
Haas measurements.48
Additionally, the magnetic anisotropy energy ~MAE! is
calculated from the FLMTO method. The MAE is, in this
paper, defined as the difference between two selfconsistently calculated fully relativistic total energies for two
different crystal directions, E [0001] 2E [1¯ 100] . To attain good
convergence a large number of k points have been sampled
in the two different irreducible zones, which corresponds to
6000–9000 k points in the full zone. Since the number and
types of symmetry operations depend on the magnetization
direction ~quantization axis! some special care has to taken
while performing the Fourier series expansion of the density
as well as the potential and the wave function in the interstitial region. The calculations have been done both with and
without the so-called orbital polarization correction ~OP! explained in Sec. B and the results are shown in Table III.
There are a few things worth noting. The orbital moment as
well as the MAE is increased when the OP term is included
into the calculations. This is expected and in agreement with
earlier experience.49 The next observation is that the spin
direction giving the biggest orbital moment is consequently
the most stable one. This is in agreement with first-order
perturbation theory by Bruno50 as well as earlier
experience.49 A final remark is that the orbital moments, as
well as the orbital moment anisotropies, increases along the
series As-Sb-Bi ~progression towards heavier atoms!. This
trend is also found for the MAE with the exception that
PRB 59
MAGNETIC, OPTICAL, AND MAGNETO-OPTICAL . . .
TABLE IV. Principal component of the electric field gradient
V zz ~in units of 1021 V/m2 ), calculated using the FLAPW method.
Compound
V zz ~Mn!
V zz (X)
MnAs
MnSb
MnBi
0.37
0.28
0.13
12.08
18.96
31.81
MnAs and MnSb have almost the same value of the anisotropy.
The theoretical value of the MAE for MnSb ~without OP!
can be compared with a previous ASA calculation by Vast
et al.40 and experiments done by Coehoorn et al.32 One can
notice that our value is larger by approximately a factor of
two compared to both the experiments ~210 m eV/unit cell40!
and the ASA calculations. The disagreement between our
value and experiment becomes worse when the OP term is
included in the theoretical treatment. There are two major
differences between the two theoretical treatments. In the
present paper a full potential approach is adopted while Vast
et al.81 has been using an ASA approximation. Using ASA
for relatively open structures, like the MnX structures may be
questionable, especially when a very small energy difference
is to be calculated. In addition, Vast et al. has been using the
so called force theorem to approximate the MAE while, in
the present work the total energy has been calculated with an
accuracy high enough to be able to resolve the difference in
total energies for the two different spin directions. The disagreement between our theory and experiment is most notable for MnBi where our value is an order of magnitude
larger than experiment ~130 m eV/unit cell40!. This is much
larger than what one usually finds in first-principles theory.50
A disturbing experimental finding is that the MAE for these
systems deviates from the expected trend, that a heavier
ligand ~with larger spin-orbit coupling! gives larger MAE.
This is not supported by the present theory. A problem with
these systems is the temperature dependence of the MAE.
One should also note that experiments contain a dipole contribution, which is absent in the calculation.
IV. CALCULATED ELECTRIC FIELD GRADIENT
The calculated EFG from the FLAPW method are given
in Table IV. It should be noted that the EFG gradually increases in the pnictogen site when we go from MnAs to
MnBi. The large value of the EFG in Bi compared to As is
due to the more delocalized nature of 6p states of Bi, which
gives rise to a more aspherical charge density around the Bi
site. The systematic decrease in the EFG at the Mn site as
one goes from MnAs to MnBi is mainly a volume effect.
Because of the increasing interatomic distance between the
Mn atoms and the pnictogen, the Mn 3d states become more
localized in MnBi than in MnAs. As a result, the hybridization between the Mn-3d and Bi-6p states is much smaller
than the corresponding hybridization in MnAs. Due to this
weaker covalent interaction, resulting in a more spherical
charge-density distribution around Mn in MnBi, a smaller
EFG is observed. Because of the more localized nature of the
Mn-3d states in MnBi, the calculated spin and orbital mo-
15 685
TABLE V. Calculated valence, core and total contribution to the
Fermi contact hyperfine field H in kGauss at the Mn and pnictogen
sites in MnX compounds. The calculations were performed with the
FLAPW method.
Compound
MnAs
MnSb
MnBi
H~Mn!
Val.
569
593
633
Core
2432
2446
2463
H(X)
Total
137
147
169
Val.
563
718
1338
Core
5
1
28
Total
568
719
1330
ments, given in Table II, are larger in this system than in
MnAs or MnSb.
The calculated hyperfine field, arising from the Fermi
contact interaction at zero field, are given in Table V. In
order to understand the origin of the hyperfine field the site
projected valence electron contribution as well as the core
contributions are also given in this table.
Because of the opposite signs of the contributions from
the core and valence electrons, the net field at the Mn site is
much lower than the net field at the pnictogen site. We see
that the HFF systematically increases when one goes from
MnAs to MnBi. The systematic increase in the HFF at the
Mn site with increased pnictogen atomic number can be explained by increased spin moment on the Mn site. In contrast, the large increase of the HFF on the pnictogen site
when going from MnAs to MnBi, see Table V, originates
from several sources. First, the valence electrons in Bi are
more delocalized compared to the situation in As, since the
valence orbitals in Bi have a higher n and consequently more
nodes. This results in a larger overlap with the Mn 3d orbitals, which increases the induced moment at the pnictogen
site. Secondly, the magnetic moment in the neighboring Mn,
which polarizes the valence electrons, is larger in MnBi than
in MnAs. If the local moment of the particular site increases,
the core polarization will be large due to the exchange interaction of the polarized electrons with the s orbitals of the
core. Hence, the systematic increase of the core contribution
to the HFF at the Mn site as one goes from MnAs to MnBi
can be understood in terms of the increasing trend in local
moment on the Mn site. Since the local moments on the
pnictogen site are negligible, the core contribution to the
HFF at this site is small. As discussed in earlier studies51 the
large negative value of core contribution to the HFF at the
Mn site can be qualitatively explained as follows. The
majority-spin s electrons in the core will be pulled into the
region of the spin-polarized 3d shell, since the exchange
interaction is attractive, whereas the minority electrons will
be repelled from the d shell. At the Mn nuclear position an
excess of minority electrons, i.e., a negative polarization, is
therefore created.
V. BAND STRUCTURE AND DENSITY OF STATES
Several models have been proposed for the electronic
structure of the manganese pnictides. The earliest ones are
the ionic models of Goodenough,52 Albers and Hass,53 and
Bärner,54 in which Mn is in the 3 1 state, and essentially has
four d electrons. Results from measurements of the zero-field
nuclear magnetic resonance,55 XPS spectra,56 and optical and
15 686
P. RAVINDRAN et al.
transport properties57 favor a band model for the electronic
structure. In the band model, Mn has a d 6.5 configuration,
which explains the observed magnetic moment of 3.5m B per
Mn atom.
As mentioned earlier, there is no full potential calculation
on the electronic structure of these materials available. Further, the published band structures of the MnX compounds
seem inconsistent with each other. For example, our calculated eigenvalues at the A point in the band structure is different from that of Sabiyanov and Jaswal,58 but consistent
with that of Köhler and Kübler.46 Furthermore, in the
G-M -K-G symmetry directions our band structure is similar
to that of Sabiyanov and Jaswal.58 However, in these directions, particularly at K, our band structure is different from
that of Köhler and Kübler.46 MnAs, MnSb, and MnBi are
isoelectronic and isostructural, and our calculations show
that the band structures of all three compounds resemble
each other. For these reasons we have not shown the band
structure of MnAs and MnSb. In the calculated band structures of MnAs by Motizuki and Katoh59 and of MnSb by
Coehoorn, Haas, and de Groot,41, the eigenvalues at the K
point and in the M -K-G symmetry directions are rather different from the present calculation, as well as from the one
calculated by Sabiryanov and Jaswal.58
The optical and MO properties originate mainly from the
interband transitions. Therefore, it is of interest to display the
spin projected band structures. The majority-spin and
minority-spin band structures of MnBi calculated using the
scalar-relativistic FLMTO method are shown in Figs. 2~a!
and 2~b!, respectively. If we take into account the arrangement of the anions around each Mn atom in the form of a
trigonally distorted octahedral coordination, thus going from
O h to D 3d symmetry, the Mn 3d levels transform as T 2g
→A T1g 1E Tg whereas the E g level remains unaltered. The Mn
3d bands in the majority-spin band structure arise partly
from A T1g states. The narrow bands above E F in the minorityspin band structure are dominated by both E Tg and E g symmetry. Due to exchange splitting, the narrow 3d bands move
away from the Fermi level, and as a result the density of
states ~DOS! at the Fermi level becomes smaller than would
otherwise have been the case.
The angular momentum-, site-, and spin-projected density
of states for MnBi are shown in Fig. 3. From this figure, it is
clear that the valence bands of the manganese pnictides are
derived from the 3d and 4s electrons of the Mn and the
pnictogen s,p electrons. Further, the Mn 3d states are energetically degenerate with the Bi 6p states in the valence
band. This indicates the presence of covalent bonding between the manganese and pnictogen atoms in these compounds. Because of this hybridization, the calculated magnetic moment per Mn atom becomes less than 4m B ~Table
II!, which is a substantial reduction of the free-atom moment
of 5m B . It should be noted that, due to this hybridization
effect, a small net negative moment develops on the pnictogen site in all the three compounds. For MnBi, Coehoorn and
de Groot32 computed the site-, spin- and angular momentum
decomposed densities of states of the ferromagnetic state.
They found a peak for Mn 3d spin-up ~majority! states 2.5
eV below the Fermi energy, and for the Mn 3d spin down
~minority! states, 0.5 eV above E F . Our results are consistent with their studies.
PRB 59
FIG. 2. The electronic band structure of the ~a! majority, ~b!
minority spin electrons of MnBi along symmetry lines in the first
BZ from FLMTO method without s-o interactions. Labels used are
according to Bradley and Cracknell ~Ref. 80!.
Our calculated spin projected total density of states of all
three compounds are given in Fig. 4. It is interesting to note
that the Mn 3d partial density of states ~PDOS! is relatively
low at the Fermi level, which agrees with the small electronic contribution to the specific heat observed
experimentally.57 Our calculated density of states at the
Fermi level N F for MnSb is 2.78 states/~eV unit cell!, which
agrees well with the experimental value56 of 2.460.4 states/
~eV unit cell! estimated from specific heat measurements,
neglecting mass-enhancement effects by electron-phonon
and electron-magnon interaction. Our calculated value of N F
for MnAs and MnBi are 3.46 states/~eV unit cell! and 2.05
states/~eV unit cell!, respectively.
VI. OPTICAL PROPERTIES
As the reflectivity is one of the parameters which decide
the MO figure of merit, it is of relevance to study the reflectivity as a function of energy. Our calculated reflectivity
spectra for the manganese pnictides are given in Fig. 5. Because of the isoelectronic and isostructural nature of these
compounds their reflectivity spectra resemble each other. In
order to understand the optical anisotropy of these materials
PRB 59
MAGNETIC, OPTICAL, AND MAGNETO-OPTICAL . . .
15 687
FIG. 3. The spin projected angular momentum and site decomposed density of states of MnBi obtained from the s-o included
FLMTO method.
we have given the reflectivity spectra for Ei c as well as
E'c. It is interesting to note that although the crystal structure is anisotropic in nature, the calculated reflectivity spectra are relatively isotropic. A similarity in the reflectivity
between MnX compounds and transition metals60 is evident
FIG. 5. The reflectivity spectra for MnX compounds obtained
from FLMTO calculations. The experimental spectra are for MnAs
from Ref. 54, MnSb from Ref. 54, and MnBi from Ref. 73.
FIG. 4. The spin-projected total density of states of manganese
pnictides.
from Fig. 5. That is, the reflectivity falls from '100% immediately as the energy increases from zero to the infrared
region, and then decreases more slowly with very little structure through the visible to the ultraviolet region. Bärner,
Braunstein and Chock54 have measured the reflectivity spectrum of MnAs in the interval 0.3–14 eV in the NiAs structure at room temperature. For this compound, our calculation
is able to reproduce the overall trend of the experimental
reflectivity spectra. Since MnAs is easily corroded even in
weak acids, no etching, which might improve the sample
surfaces in the experimental study, was performed.54 Hence,
one can expect a lower amplitude in the experimental reflectivity spectrum than in the calculated one. The discrepancy
between experiment and theory regarding the amplitude of
reflectivity in MnAs may be due to this. Allen and
Mikkelsen61 measured the optical reflectivity spectrum of
MnSb up to 5 eV at room temperature. The reflectivity spectrum for MnBi was measured at room temperature by Di
et al.6 The good agreement between our theoretical reflectivity spectra for MnSb and MnBi, and the corresponding experimental spectra in Fig. 5, indicate that temperature does
not play a significant role for the reflectivity in these systems.
There is very little published data on the optical constants
of the manganese pnictides. The read-out figure of merit is
defined as 2 u F /I, where I is the optical-absorption coefficient. Thus, it is interesting to calculate I as a function of
wavelength. Experimental studies62 show that the optical-
15 688
P. RAVINDRAN et al.
FIG. 6. The electron energy loss function @ L( v ) # , absorption
coefficient @ I( v ) in 105 cm21 ], refractive index @ n( v ) # and extinction coefficient @ k( v ) # for MnX from FLMTO calculations.
absorption coefficients of MnSb and MnBi are rather similar
in magnitude. This is consistent with our results in Fig. 6.
The characteristic energy-loss function 2Im( e 21 ) is proportional to the probability that a fast electron moving across a
medium loses an energy E per unit length.63 The loss maximum in the EELS spectrum is associated with plasma oscillations. In the lowest panel of Fig. 6, it can be seen that the
calculated screened plasma frequencies for MnAs, MnSb,
and MnBi are 22.2, 19.7, and 19.6 eV, respectively.
In general, optical constants depend on the growth conditions of the films,64 weight ratio,65 structure of the film,66 and
different assumptions made in the analysis of the data. The
measurements by Angadi and Thanikaimani65 on MnBi films
and by Stoffel and Schneider67 on MnAs films reveal a considerable aging effect of the optical constants. So far no
single-crystal and low-temperature measurements of the optical properties of these materials are available. We hope that
our calculated optical constants given in Fig. 6 will motivate
experimental studies in the future.
VII. MAGNETO-OPTICAL PROPERTIES
Before comparing our theoretical MO spectra with the
experimental results, one has to keep the following two
points in mind. ~1! The experimental studies on MnBi show
that the Kerr effect at low temperature is much different from
that at room temperature.68 ~2! The optical constants for
PRB 59
FIG. 7. The calculated polar Kerr rotation spectra for MnX compounds. For MnBi, the experimental spectra are taken from Di et al.
~Ref. 6!, Huang et al. ~Ref. 81!, and Fumagalli et al. ~Ref. 82! and
the theoretical spectra from Sabiryanov and Jaswal ~Ref. 58!,
Köhler and Kübler ~Ref. 46!, Oppeneer et al. ~Ref. 37!, and Kulatov et al. ~Ref. 75!, for MnSb the experimental spectra from Buschow and van Engen ~Ref. 75! and the theoretical spectra from
Oppeneer et al. ~Ref. 37! and Kulatov et al. ~Ref. 76!. For MnAs,
the Kerr spectra were derived ~Ref. 77! from the complex Faraday
rotation spectra reported by Stoffel and Schneidner ~Ref. 67!, and
the theoretical spectra from Oppeneer et al. ~Ref. 37! and Kulatov
et al. ~Ref. 76!.
MnBi are sensitive not only to the film composition ratio of
Mn/Bi,69,70 but also the surface states of the films since there
is always excess Mn or Bi on the film surface.71 Further, the
reported experimental measurements are made on thin films,
whereas our theoretical spectra are valid for ideal single
crystals.
The calculated Kerr rotation spectra for the MnX compounds are given in Fig. 7 along with the available experimental spectra and theoretical results obtained from ASA
calculations. Overall, our calculated spectra are found to be
in good agreement with the experimental values. It should be
noted that the Kerr spectra obtained from the ASA calculations give higher Kerr rotation than experiment as well as
our full-potential calculation. This difference between the
calculated results may indicate that a full-potential treatment
is of importance for these materials. The experimentally
measured Kerr rotation spectra of MnBi by Di et al. and
Fumagalli et al. show two peaks, one around 1.8 eV and the
other around 3.3 eV. Our calculation is able to reproduce
PRB 59
MAGNETIC, OPTICAL, AND MAGNETO-OPTICAL . . .
both peaks. However the second peak in our spectra in Fig. 7
is shifted 0.5 eV to lower energy compared to experiment. A
tentative explanation of the difference in the peak energies
might be the sample composition. In support of our viewpoint the calculated Kerr rotation spectra of Oppeneer et al.37
for Mn2 Bi show a peak at about 4.3 eV. Interestingly, this
second peak is absent in all other calculations, except in the
one by Sabiryanov and Jaswal.58 From the studies of the
magneto-optical properties of MnBi by ternary alloying
Köhler and Kübler46 concluded that the second peak in the
experimental spectra is due to extrinsic effects.
MacDonald, Pickett, and Koelling72 show that including
the ls term as a perturbation to the energy eigenvalues of the
Hamiltonian will introduce large error for p states in heavy
metals like Pb. However, it is clear from Fig. 7 that our
calculated magneto-optical Kerr spectra for MnBi is found to
be good agreement with the experimental results at least up
to 3 eV. As the magneto-optical effect is originating from the
spin-orbit splitting, it indicates no significant error introduced in our calculation due to the above fact. The possible
reasons for the nonintroduction of errors in the present calculation is that the p states of Bi are strongly hybridized with
the d states of the Mn. Also, the Bi-p states are polarized by
the induced magnetic field arising from the neighboring Mn
atoms.
From Eq. ~7!, it can be seen that the Kerr rotation can be
enhanced by a larger off-diagonal conductivity and a smaller
diagonal part. In order to gain a more detailed insight into
the origin of the features in the magneto-optical spectra in
terms of electronic structure, the computed diagonal and offdiagonal optical conductivities are given in Figs. 8 and 9,
respectively. The frequency dependence of the absorptive
and dispersive part of the diagonal component ( s xx ) for all
the three compounds are given in Fig. 8. Di and Uchiyama73
measured the complex conductivity at 300 K in MnBi films
below 5 eV. Their measurement show that the real part of
s xx is larger than the imaginary part in the whole frequency
range considered, which is consistent with our spectra. MO
effects arise generally from both interband transitions and
intraband transitions. To understand the effect of interband
transitions on the MO properties, experimentalists often display vs xy rather than s xy . As the intraband contribution to
s xy is proportional to ( v t ) 21 , the free-electron-like intraband contribution will be a constant proportional to 1/t in
the experimental vs xx spectra. So, even though our offdiagonal conductivity does not contain the Drude term we
show vs xy in Fig. 9. It is interesting to note that the overall
topology of our imaginary part of vs xy spectra for MnBi is
similar to the Kerr rotation spectra in Fig. 7. This shows the
dominant contribution from the off-diagonal conductivity to
the MO properties.
The absorptive part of the off-diagonal optical conductivity Im( s xy ) has a direct physical interpretation. It is
proportional74 to the difference in absorption rate of left and
right circularly polarized ~LCP and RCP! light, and its sign is
directly related to the spin polarization of the states responsible for the interband transitions producing the structures in
the spectrum. Between 1.2 to 3.8 eV, Im( vs xy ) has a large
value for MnBi as shown in Fig. 9, suggesting that interband
transitions related to LCP light might be stronger in this
region. Above 3.8 eV, Im( vs xy ) decreases, indicating the
15 689
FIG. 8. Calculated complex diagonal optical conductivity spectra from MnX from FLMTO method.
dominance of interband transitions related to RCP. The
Im( vs xy ) related to interband transition will be zero where
the absorption coefficient for RCP equals to that of LCP. We
observe the first zero around 4 eV, which is consistent with
the experimental measurements.6
It is of importance to find the particular transitions responsible for the occurrence of the peak structures in the
Kerr rotation spectrum of MnBi. From Fig. 3, it is evident
that the low-energy peaks in the Kerr spectra are mainly due
to Mn3d ↓ →Bi6 p ↓ interband transition. However, a detailed
analysis of the band structure as well as the interband transitions show that a considerable Bi6 p ↑ →Mn3d ↑ interband
contribution is also present on the low-energy side of the first
peak. The second peak in the Kerr rotation spectrum of
MnBi around 2.8 eV originates from Mn3d ↓ →Bi6 p ↓ interband transition. The majority-spin interband transitions in
this energy region are very small.
The room-temperature polar Kerr rotation of MnSb has
been studied by Buschow et al.75 The middle panel of Fig. 7
shows our theoretical Kerr rotation spectrum along with their
experimental result. The experimental74 as well as the
present theoretical spectrum show a rather broad feature in
the Kerr rotation of MnSb compared to that of MnBi. For
MnSb, both ASA calculations37,76 again show larger Kerr
rotation than experiment as well the present full-potential
results. Our theoretical spectra not only reproduce the experimental trend but also the absolute values. This indicates that
our method is reliable for predicting Kerr rotation spectra.
The optical and magneto-optical properties of MnAs films
were investigated by Stoffel and Schneider67 in view of ap-
15 690
P. RAVINDRAN et al.
PRB 59
FIG. 9. Calculated complex off-diagonal optical conductivity
spectra from MnX from FLMTO method.
plications in beam-addressable memories. Our calculated
Kerr rotation spectrum for MnAs is not in as good agreement
with the experimental spectrum as was the case for MnSb
and MnBi. One possible reason is that the experimental Kerr
rotation spectrum for MnAs given in Fig. 7 is not directly
measured but is derived77 from the reported67 complex Faraday spectra of MnAs films. Further, discrepancies between
experimental and theoretical Kerr rotation spectra may also
occur due to temperature effects.
The calculated Kerr ellipticity spectra for the MnX compounds are shown in Fig. 10. Overall, our Kerr ellipticity
spectra are able to reproduce the experimental trends.
Our calculated specific Faraday rotation and ellipticity
spectra for the MnX compounds are shown in Figs. 11 and
12, respectively. For MnBi, experimental spectra are also
plotted. Older low-temperature measurements78,79 of Faraday
rotation spectra for MnBi differ substantially from the spectra in Ref. 72, which are the ones we compare our calculations to. The differences are due to different film-preparation
techniques, and effects of multiple scattering. Di and
Uchiyama calculated the intrinsic values of Faraday rotation
and ellipticity for MnBi from the measured Kerr effect at 85
K and the optical constants measured at room temperature.
Sawatzky and Street62 have reported Faraday rotation measurements for MnSb. They found that the Faraday rotation at
room temperature varies from 2.83105 deg. cm21 at 1 eV to
2.13105 deg cm21 at 6 eV with maxima around 1.3 and 2.5
eV. Unfortunately, no other experimental Faraday spectra are
available, with which we can compare our results.
The systematic increase in the MO effect from MnAs to
FIG. 10. Kerr ellipticity spectra for the MnX compounds. For
MnBi, the experimental spectra are taken from Di et al. ~Ref. 73!
and the theoretical spectra from Oppeneer et al. ~Ref. 37! and Kulatov et al. ~Ref. 76!. For MnSb, the experimental spectra are from
Buschow and van Engen ~Ref. 75! and the theoretical spectra from
Oppeneer et al. ~Ref. 37! and Kulatov et al. ~Ref. 76!. For MnAs,
the ellipticity spectra were derived ~Ref. 77! from the complex Faraday rotation spectra reported by Stoffel and Schneidner ~Ref. 67!
and the theoretical spectra are calculated by Oppeneer et al. ~Ref.
37! and Kulatov et al. ~Ref. 76!.
MnBi indicate the important role played by the pnictogen.
Two reasons can be singled out causing the increase in the
Kerr effect as the pnictogen becomes heavier. The first is the
increased volume. From the experimentally reported equilibrium lattice parameters one can see that the equilibrium volume increases from 68.4 Å3 for MnAs to 89.67 Å3 for MnBi.
Correspondingly, the Mn-X distance increases from 2.578 Å
for MnAs to 2.83 Å for MnBi. Because of the larger interatomic distance in the Mn-Bi bonds compared to the Mn-As
and Mn-Sb bonds, the Mn3d-Bi6p covalent interaction is
smaller in MnBi compared to other two compounds. As a
consequence of this, the Mn-3d band becomes narrower in
MnBi. This is the main reason for the magnetic moment
enhancement from 3.57 m B for MnAs to 3.86 m B for MnBi.
In general, narrow bands give sharper features in the optical
conductivities, which reasonably should result in an enhanced Kerr effect. The second reason is the larger spin-orbit
coupling strength in Bi than in As and Sb.
PRB 59
MAGNETIC, OPTICAL, AND MAGNETO-OPTICAL . . .
FIG. 11. Specific Faraday rotation spectra for MnX ~X5As, Sb
and Bi!. The experimental values for MnBi are from thin film measurements at 85 K in Ref. 73.
VIII. SUMMARY
Using a spin-polarized relativistic full-potential LMTO
method, we have calculated the frequency dependent MO
Kerr and Faraday effects for MnX ~X5As, Sb, or Bi!. The
calculated spectra are found to be in good agreement with
the experimental results. We are able to reproduce the experimentally observed second peak in our Kerr rotation spectra of MnBi, albeit at a lower energy than experimentally
observed. Specific Faraday rotation and ellipticity for all the
three compounds are also calculated and compared to available experimental results. Features of the MO spectra are
seen to be primarily defined by the absorptive part of the
off-diagonal conductivity s 2xy ( v ). The systematic increase
in the MO effect as one goes from MnAs to MnBi is due to
the enhancement of magnetic moment on the Mn site as well
as the increasing spin-orbit coupling strength of pnictogen.
Linear-optical properties such as reflectivity, absorption coefficient, refractive index, extinction coefficient, and
electron-energy-loss spectra are also reported.
15 691
FIG. 12. Calculated specific Faraday ellipticity spectra for MnX.
The experimental spectra for MnBi take from Ref. 73.
We have studied the trends in electronic structure, spin
moments, orbital moments, and magnetic anisotropy energies, and also calculated the hyperfine field, electric-field
gradient, and unscreened plasma frequencies using the
FLAPW method. The systematic enhancement in the magnetic moment as well as the calculated hyperfine parameters
show that the covalent interaction between the Mn and pnictogen atoms decreases systematically when we go from
MnAs to MnBi. This may be the possible reason for the
experimentally observed crystal-structure metastability of
MnBi.83–85
ACKNOWLEDGMENTS
B.J., O.E., and P.R. are thankful for financial support
from the Swedish Natural Science Research Council and for
support from the Materials Science Consortium No. 9. A.D.
acknowledges financial support from the Swedish Research
Council for Engineering Sciences. P.R. is grateful to Professor K. Schwarz for his encouragement, P. Blaha for his generous help, and Lars Nordström, Claudia Ambrosch-Draxl,
Peter Oppeneer, and Mike Brooks for useful discussions.
15 692
P. RAVINDRAN et al.
*Electronic address: [email protected]
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