# Spin-orbit driven terahertz optical response in ferromagnetic Fe-Co-Al
alloys

Ming Lei<EMAIL_ADDRESS>Chemical and Environmental Engineering, University
of California Riverside, California 92521, USA  Sinisa Coh  Materials Science
and Mechanical Engineering, University of California Riverside, California
92521, USA

###### Abstract

We study the magneto-optical properties of Fe-Co-Al ordered alloys in the
terahertz range of frequencies. We find that magnetism can modify the
reflection of light from these magnets and that this modification strongly
depends on the frequency of incoming light in the terahertz range. For
example, we find that below 10 THz Co3Al has nearly constant $\sigma_{xy}$ and
that above 10 THz it is reduced by about 50 times. Furthermore, we find a
strong dependence of $\sigma_{xy}$ on chemical composition. For example, we
find that the addition of Al to Fe changes the sign of $\sigma_{xy}$, while
the addition of Co to Fe leads to non-monotonic dependence of $\sigma_{xy}$ on
Co-concentration.

##  I Introduction

A magnet with a cubic crystal structure and magnetization pointing along the
z-axis, has a non-zero off-diagonal component of the optical conductivity,
$\sigma_{xy}(\omega)$. This contribution to the conductivity occurs because
magnetic order in such a system breaks time reversal symmetry. The breaking of
the time-reversal symmetry is propagated to the electronic degrees of freedom
by the spin-orbit interaction. Therefore, the magnitude of
$\sigma_{xy}(\omega)$ is dictated by the spin-orbit interaction
strength.Nagaosa  _et al._ (2010) Presence of off-diagonal conductivity
$\sigma_{xy}$, and thus in-directly the presence of magnetic order, can be
detected optically by comparing the polarization of light incident to, and
reflected from, the surface of a magnet. This is the so-called magneto-optical
Kerr effect, MOKE.LL.D. (1877) Opposite is also true. If one changes the
magnetization direction along the z-axis,

$M_{z}\rightarrow-M_{z},$

this will result in,

$\sigma_{xy}\rightarrow-\sigma_{xy}.$

Therefore, one can use direction of magnetization to control the way in which
light reflects from a surface of a magnet. This effect has been used in memory
storage devicesKryder (1985); Meiklejohn (1986) and creation of tunable
photonic materials.Atmatzakis  _et al._ (2014); Wang  _et al._ (2020)

At zero frequency, the static conductivity $\sigma_{xy}(\omega=0)$ is also
referred to as the anomalous Hall effect (AHE). In materials with very few
impurities, so that the diagonal conductivity $\sigma_{xx}$ is above $10^{6}$
$(\Omega~{}\textrm{cm})^{-1}$, the dominant contribution to AHE originates
from scattering off of impurities. On the other hand, somewhat paradoxically,
in materials with moderate amount of impurities, with diagonal conductivity
around $10^{4}$–$10^{6}$ $(\Omega~{}\textrm{cm})^{-1}$, the dominant
contribution to AHE is intrinsic, independent of the number of
impurities.Nagaosa  _et al._ (2010) The part of the intrinsic contribution to
AHE is given by the sum of the Berry curvature of the Bloch wave function over
the occupied states.Haldane (2004) However, in the dynamic case, the
$\sigma_{xy}(\omega\neq 0)$ can no longer be written as a sum over the Berry
curvature. Nevertheless, we expect that for small enough $\omega/(2\pi)$, in
the THz range, the intrinsic contribution still dominates
$\sigma_{xy}(\omega)$, which is consistent with findings in Ref. Matsuda  _et
al._ , 2020.

There are many calculations of static intrinsic AHC in the literature. For
example, YaoYao  _et al._ (2004) calculated the AHC of ferromagnetic bcc Fe,
and the calculated $\sigma_{xy}$ at zero frequency is 751 $(\Omega~{}{\rm
cm})^{-1}$. WangWang  _et al._ (2007) found that at zero frequency, AHC in bcc
Fe, fcc Ni and hcp Co, and $\sigma_{xy}$ is 753, $-$2203 and 477
$(\Omega~{}{\rm cm})^{-1}$, respectively. BiancoBianco  _et al._ (2014)
calculated the AHC in Fe3Co, and the $\sigma_{xy}$ value at zero frequency is
452 $(\Omega~{}{\rm cm})^{-1}$. HuangHuang  _et al._ (2015) calculated the AHC
of Heusler compound such as Co2FeAl and found it to be 39 $(\Omega~{}{\rm
cm})^{-1}$.

While $\sigma_{xy}$ originates from the magnetism, the coefficient of
proportionality between $\sigma_{xy}$ and magnetic moment is difficult to
predict without performing an explicit first-principles calculation. As
discussed in Refs. Yao  _et al._ , 2004; Bianco  _et al._ , 2014; Wang  _et
al._ , 2006, 2007, different parts of the reciprocal space can have either
positive or negative contribution to $\sigma_{xy}$, as sign will in general
depend on position of the Fermi level relative to the subtle spin-orbit
induced band gaps in the band-structure. Therefore, we might expect that
generally $\sigma_{xy}$ will be a very sensitive function of the electronic
band structure. As a consequence, we expect a rich dependence of dynamic
$\sigma_{xy}(\omega)$ in the low-frequency range. In particular, we expect
that there will be a strong frequency dependence of $\sigma_{xy}(\omega)$ when
$\hbar\omega$ is close to the energy of the spin-orbit split bands in the band
structure. The energy of the spin-orbit split bands is on the order of tens of
meV in ferromagnetic metals such as Fe, Co, or Ni. These energies lie in the
range of frequencies $\sim$10 THz, which is within the so-called terahertz-
gap: the range of frequencies in the electromagnetic spectrum that are
currently unexplored compared to larger frequencies.

Moreover, we expect that $\sigma_{xy}$ will be very sensitive on alloying, as
even subtle changes in the electronic band structure will change the position
of spin-orbit split bands relative to the Fermi level. Therefore, in this
study we use first-principles techniques to calculate the
$\sigma_{xy}(\omega)$ of Fe-Co-Al alloys in the THz regime. Our calculations
show that alloying is a feasible way to change the THz optical response of
ferromagnetic metals such as Fe or Co. As expected, we find unusually strong
dependence of the $\sigma_{xy}$ in these alloys on both frequency and alloy
composition.

Most other studies of frequency dependent $\sigma_{xy}$ are done in the
optical regime, at much higher energies. For example, Ref. Kim  _et al._ ,
2002 computed $\sigma_{xy}$ for Fe3Co and FeCo3 in the 0.5–5.2 eV energy
range. Next, Ref. Kumar  _et al._ , 2010 computed the optical response of
Fe4-xCox (with $x=1$–$3$) up to the 13 eV range, while Ref. Rhee, 2004;
Adebambo  _et al._ , 2010 reported $\sigma_{xy}$ for FeAl in the 0.5–6 eV
range. Recently, some studies have focused on the off-diagonal optical
response in the terahertz (THz) regime. SeifertSeifert  _et al._ (2021)
studied the off-diagonal optical response in DyCo5, Co0.32Fe0.68 and
Gd0.27Fe0.73, and they measured somewhat stronger dependence of $\sigma_{xy}$
on $\omega$ in the range of frequencies below 10 THz, than above 10 THz.
MatsudaMatsuda  _et al._ (2020) studied the off-diagonal optical response in
Weyl antiferromagnet Mn3Sn at very low energy (10 meV) and they find weak
dependence of $\sigma_{xy}$ on the frequency. On the other hand, calculated
and measured $\sigma_{xy}$ in the THz regime is very frequency dependent in
SrRuO3, as shown in Refs. Kim  _et al._ , 2010; Fang  _et al._ , 2003; Shimano
_et al._ , 2011.

We organize the paper as follows. In Sec. II we show the calculation methods.
In Sec. III we present and analyze our results. We conclude in Sec. IV.

##  II Methods

We use Quantum Espresso packageGiannozzi  _et al._ (2009) to calculate
$\sigma_{xy}(\omega)$ of ordered alloys of Fe, Co, and Al. We use the
Generalized Gradient Approximation (GGA) of Perdew, Burke and Ernzerhof
(PBE)Perdew  _et al._ (1996) along with the optimized norm-conserving
Vanderbilt (ONCV) pseudopotentials which include spin-orbit
interaction.Schlipf and Gygi (2015); Corso and Conte (2005) We choose 120 Ha
kinetic-energy cutoff for the plane-wave expansion of the valence wave
functions. A 16$\times$16$\times$16 Monkhorst-Pack grid and a smearingMarzari
_et al._ (1999) of 0.01 Ry are used to sample the electron’s Brillouin zone.
We computed $\sigma_{xy}(\omega)$ using the standard Kubo formula,Kubo (1957)

$\displaystyle\sigma_{\alpha\beta}(\omega)=$ (1)
$\displaystyle\frac{ie^{2}\hbar}{VN_{k}}\lim_{\delta\to
0}\sum_{\textbf{k}}\sum_{nm}\frac{f_{m\textbf{k}}-f_{n\textbf{k}}}{\varepsilon_{m\textbf{k}}-\varepsilon_{n\textbf{k}}}\frac{\mathinner{\langle\phi_{n\textbf{k}}|}v_{\alpha}\mathinner{|\phi_{m\textbf{k}}\rangle\mathinner{\langle\phi_{m\textbf{k}}|}v_{\beta}\mathinner{|\phi_{n\textbf{k}}\rangle}}}{\varepsilon_{m\textbf{k}}-\varepsilon_{n\textbf{k}}-\hbar\omega-i\delta/2}.$

Here, $\alpha$ and $\beta$ denote Cartesian directions. $V$ is the cell
volume, $N_{k}$ is the number of k-points, and $f_{n\textbf{k}}$ is the Fermi-
Dirac distribution function. $\omega$ is the optical frequency. We use Wannier
interpolationMarzari  _et al._ (2012); Mostofi  _et al._ (2008) in evaluating
Eq. 1, as these calculations require a very dense sampling of the Brillouin
zone. We tested the convergence of $\sigma_{xy}$ to the choice of k mesh. We
use adaptive k-mesh refinement to accelerate the convergenceYao  _et al._
(2004) by adding a 5$\times$5$\times$5 fine mesh around regions with large
contribution to $\sigma_{xy}(\omega)$. For pure metals (Fe and Co), a
250$\times$250$\times$250 primary k-mesh is enough to achieve the convergence
of $\sigma_{xy}$. For calculations with two atoms per unit cell and four atoms
per unit cell, a 200$\times$200$\times$200 and 150$\times$150$\times$150
k-mesh, respectively, was enough to achieve convergence.

To validate the reliability of our calculation approach, we compare our
calculation results with previous calculations. Our calculated $\sigma_{xy}$
of Fe and Co at zero frequency are 758 and 471 $(\Omega~{}{\rm cm})^{-1}$,
respectively, which agrees very well with the previous calculation result of
753 $(\Omega~{}{\rm cm})^{-1}$ in Fe and 477 $(\Omega~{}{\rm cm})^{-1}$ in
Co.Wang  _et al._ (2007) Furthermore, the calculated $\sigma_{xy}$ of FeCo and
Fe3Co at zero frequency is 226 and 416 $(\Omega~{}{\rm cm})^{-1}$,
respectively. The measured $\sigma_{xy}$ of Fe0.68Co0.32 at zero frequency is
about 350 $(\Omega~{}{\rm cm})^{-1}$Seifert  _et al._ (2021), which lies
between the theoretical values of FeCo and Fe3Co. In addition, we calculated
the Kerr angle of Fe in the range of 0–1.2 eV by using our calculated
$\sigma_{xy}$ values and experimental $\sigma_{xx}$ values from Ref. Krinchik,
1964; Ordal  _et al._ , 1983. Our calculated Kerr angle matches very well with
the results in Ref. Krinchik and Artem’ev, 1968.

In all of our calculations we assumed that the magnetization points along the
$[001]$ direction. We did not explore what happens to the off-diagonal
conductivity when magnetization points in any other crystallographic
direction, such as $[011]$ or $[111]$. We decided to restrict our calculations
to those with the magnetization axis pointing along $[001]$ direction as our
goal here is to compare $\sigma_{xy}$ only as a function of chemical
composition of the alloy and frequency. Furthermore, for consistency, in each
calculation we chose the same sign of magnetization along the $[001]$
direction.

##  III Results and discussion

Now we present our results for ordered Fe-Co-Al alloys. We start by discussing
the computed lattice constants and crystal structure of these alloys. Our
results are shown in Table 1. Most of the ordered alloys we studied order in
the bcc-derived structures B2 and D03.Popova (2012); Ikeda  _et al._ (2002);
Okamoto (2016) The B2 structure is a bcc-derived structure with two atoms in
the primitive unit cell. This structure is therefore present in alloys with
1–1 ratio of two elements. On the other hand, D03 structure contains four
atoms in the primitive unit cell, so it is present in ordered binary alloys
with 1–3 ratio of constituent elements, or 1–1–2 ratio in the case of ternary
alloys. Among the ordered alloys we studied, the only ones that are not in the
bcc-derived structure are: Al, FeAl3, CoAl3, Co, and Co3Al. However three of
these (Al, FeAl3, and CoAl3) are non-magnetic, regardless of their crystal
structure, so their $\sigma_{xy}$ is identically zero. Therefore, we don’t
discuss these cases in more detail. Next, in the case of Co, we explicitly
showed that $\sigma_{xy}(\omega)$ is very similar in bcc and hcp structures.
Therefore, for more consistent comparison with other members of the Fe-Co-Al
family of compounds, we will show results for pure Co in its bcc structure.
Finally, we expect that the remaining exception, Co-rich Co3Al, will also have
$\sigma_{xy}(\omega)$ that doesn’t strongly depend on the structure, so that
we are justified in studying Co3Al in the bcc-derived structure (and not in
the lowest energy structure, the fcc-derived L12).

We find that the computed lattice constants are close to the experimentally
measured values. Small deviations, on the order of 1% are due to approximate
exchange-correlation functional in GGA, as well as due to thermal expansion,
as the experimental data in Table 1 are taken at room temperature.

Calculated magnetic moments per atom are also given in Tab. 1. Most of these
ordered alloys are magnetic in our calculations, with the exception of Al-rich
compounds, such as FeAl3, CoAl3 and CoAl. These findings are in agreement with
previous studies.Gonzales-Ormeño  _et al._ (2002); Parsons  _et al._ (1958)

Table 1: Calculated magnetic moment and lattice constant of ordered Fe-Co-Al alloys. The magnetic moment is on a per atom basis, and this includes either nominally magnetic atom (Fe, Co), or non-magnetic atom (Al).  |  | $M$ | $a^{\rm calc}$ | $a^{\rm exp}$
---|---|---|---|---
|  | ($\mu_{\rm B}/{\rm atom}$) | (Å) | (Å)
Fe | bcc | $2.26$ | 2.84 | 2.84
Fe3Co | D03 | $2.36$ | 5.71 | 5.66
FeCo | B2 | $2.30$ | 2.85 | 2.85
FeCo3 | D03 | $2.02$ | 5.66 | 5.66
Co | bcc | $1.80$ | 2.82 | 2.82
Co | hcp | $1.62$ | $a=2.50$ | $a=2.50$
$c=4.04$ | $c=4.09$
Fe3Al | D03 | $1.55$ | 5.76 | 5.79
FeAl | B2 | $0.36$ | 2.88 | 2.91
FeAl3 | D03 | . | 5.98 |
Co3Al | D03 | $1.05$ | 5.69 |
CoAl | B2 | . | 2.86 | 2.86
CoAl3 | D03 | . | 6.01 |
Fe2CoAl | D03 | $1.54$ | 5.76 | $5.73$
FeCo2Al | D03 | $1.25$ | 5.71 | $5.73$

###  III.1 Fe and Co

Figure 1 shows calculated $\sigma_{xy}(\omega)$ of pure Fe and Co metal in the
range of energies from $\hbar\omega\sim 0$–$0.1$ eV. This corresponds to the
range of frequencies $\omega/(2\pi)\sim 0$–$25$ THz. As expected, we find a
strong variation of $\sigma_{xy}(\omega)$ as a function of frequency, in both
Fe and Co. These variations are the strongest around $10$–$15$ THz. We
attribute these modifications to the fact that the characteristic spin-orbit
gaps in these metals occur in the same range of energies. For Fe, we find that
the minimal value of $\sigma_{xy}$, in the studied frequency range, is 634
$(\Omega\ {\rm cm})^{-1}$ at $14.5$ THz, while the maximal value is 1.5 times
larger, 950 $(\Omega\ {\rm cm})^{-1}$ at a nearby frequency of $16$ THz. In
the case of Co, the minimal value is 233 $(\Omega\ {\rm cm})^{-1}$ at $17$
THz, and maximal value is 2.5 times larger, 582 $(\Omega\ {\rm cm})^{-1}$ at
$10$ THz. We compared $\sigma_{xy}(\omega)$ in bcc and hcp structure of Co and
we found qualitatively similar responses, as shown in Fig. 1. Therefore, at
least in the case of Co, the crystalline structure doesn’t have a strong
effect on $\sigma_{xy}(\omega)$.

Figure 1:  Calculated real part of $\sigma_{xy}$ for bcc Fe, bcc Co, and hcp
Co as a function of $\omega$.

###  III.2 Fe-Co alloys

Figure 2:  Calculated real part of $\sigma_{xy}$ for bcc Fe, D03 Fe3Co, B2
FeCo, D03 FeCo3 and bcc Co as a function of $\omega$.

After analyzing $\sigma_{xy}$ in pure Fe and Co, we now turn to the ordered
Fe-Co alloys. We considered three ordered alloys of Fe and Co, these are
Fe3Co, FeCo, and FeCo3. The crystalline ordered structures of these alloys are
D03, B2 and D03, respectively. In what follows we show the data for pure Co in
its bcc phase because all three ordered alloys are derivatives of the bcc
structure. However, as we just discussed, bcc and hcp Co have a very similar
value of $\sigma_{xy}$. Our results for Fe-Co alloys are shown in Fig. 2.
Starting from pure Fe, we find that adding Co at first significantly reduces
the value of $\sigma_{xy}$. In particular, $\sigma_{xy}$ in Fe3Co is on
average about 2 times lower than that of pure Fe. Furthermore, spectral
features of alloy Fe3Co are distinct from that of pure Fe. While both Fe and
Co have nearly constant $\sigma_{xy}$ up to 10 THz, we find that Fe3Co shows a
strong dependence on frequency starting already around 3 THz. This finding of
a large sensitivity of $\sigma_{xy}$ on chemical composition is in agreement
with our expectation that $\sigma_{xy}$ is very sensitive on details of the
band structure, as spin-orbit split bands give either positive or negative
contribution to $\sigma_{xy}$, depending on the location of the Fermi level.
Adding even more Co, as in FeCo, we find that the value of $\sigma_{xy}$ is
reduced even further. In particular, $\sigma_{xy}$ is about 4 times smaller in
FeCo than in pure Fe. Moreover, the spectral features of FeCo are surprisingly
constant in the entire range of frequencies we studied. Further addition of Co
increases the value of $\sigma_{xy}$. For example, in the case of FeCo3,
$\sigma_{xy}$ is similar in magnitude to that of pure Co. Spectral features of
FeCo3 are strongest among the Fe-Co alloys we studied.

The non-monotonic dependence of $\sigma_{xy}$ on Co concentration is
reminiscent of a non-monotonic dependence of magnetic damping observed in Fe-
Co alloys.Schoen  _et al._ (2016) While magnetic damping is not directly
related to $\sigma_{xy}$, we expect that both can be related to the changes in
the nature of the electronic band structure near the Fermi level in Fe-Co
alloys.

We note that variation in $\sigma_{xy}$ on Co concentration can’t be
rationalized with variation of the magnetic moment, as we find that all
members of the Fe-Co family of alloys have nearly constant magnetic moment per
atom, on the order of $\sim 2\mu_{\rm B}$ (see Table 1). Therefore, even
though $\sigma_{xy}$ can be used as a signature of magnetic order, one is not
necessarily proportional to the other. As discussed earlier, the role of the
magnetic order here is only to break the time-reversal symmetry, while the
magnitude and spectral properties of $\sigma_{xy}$ are driven by the spin-
orbit interaction of electron bands close to the Fermi level.

Comparing the band structure of all Fe–Co ordered alloys we studied, we find
that the overall band structure is not changed much, only the position of the
Fermi level, relative to the rigid band structure is increasing with the
addition of Co. Therefore, we attribute changes in the $\sigma_{xy}$ of Fe–Co
alloys to the changes in the relative position of the Fermi level, and not to
changes in the band structure itself.

###  III.3 Fe-Al alloys

Figure 3:  Calculated real part of $\sigma_{xy}$ for bcc Fe, D03 Fe3Al, B2
Fe0.50Al0.50 and bcc Al as a function of $\omega$.

Next we discuss calculated $\sigma_{xy}$ in Fe-Al alloys. Our results are
shown in Fig. 3. Since Al is non-magnetic, the $\sigma_{xy}$ response is, by
symmetry, zero at all frequencies. By introducing Al, the $\sigma_{xy}$
response of Fe3Al lies between that of pure Fe and Al, as one might expect.
However, unexpectedly, when the Al concentration reaches 50%, $\sigma_{xy}$
response of the FeAl changes sign in the entire frequency range from 0 to 20
THz. We ensured that in all of these cases, Fe and Fe-Al alloys have
magnetization pointing in the same direction. A similar result is observed in
Co/Pd multi-layers in Ref Keskin  _et al._ , 2013, where the sign of
$\sigma_{xy}$ changes depending on the relative concentration of Co to Pd.

###  III.4 Co-Al alloys

Figure 4:  Calculated real part of $\sigma_{xy}$ for bcc Co, D03 Co3Al and bcc
Al as a function of $\omega$.

The next binary alloys we discuss are Co-Al alloys. Results for these alloys
are shown in Fig. 4. While CoAl and CoAl3 are not magnetic, the Co-rich
compound Co3Al is magnetic in our calculations. At low frequencies, below 10
THz, we find that $\sigma_{xy}$ for Co3Al is quite large, and somewhat
constant, with the value of $\sim 500$ $(\Omega~{}{\rm cm})^{-1}$. However,
above 10 THz, $\sigma_{xy}$ is reduced to only $\sim 10$ $(\Omega~{}{\rm
cm})^{-1}$. Therefore, in the case of Co3Al, we find that $\sigma_{xy}$ is
nearly constant above and below $\sim 10$ THz, but that magnitude above $\sim
10$ THz is reduced 50-fold relative to the magnitude below $\sim 10$ THz.

###  III.5 Fe-Co-Al ternary alloys

Figure 5:  Calculated real part of $\sigma_{xy}$ for D03 Fe3Al, D03 Fe2CoAl,
D03 FeCo2Al and D03 Co3Al as a function of $\omega$.

So far, we discussed the $\sigma_{xy}$ response of binary alloys. In what
follows, we consider several Fe-Co-Al ternary alloys. We kept the Al
concentration at 25%, and we varied the relative concentration of Fe and Co.
The results for these alloys are shown in Fig. 5. The behavior of Fe2CoAl is
qualitatively similar to that of Fe3Al, so the replacement of Fe with Co did
not change qualitatively $\sigma_{xy}(\omega)$. Quantitatively, we find that
$\sigma_{xy}$ is reduced about $\sim 1.5$ times lower in Fe2CoAl relative to
Fe3Al. However, with even larger concentration of Co, as in FeCo2Al, we find
$\sigma_{xy}(\omega)$ that is both qualitatively and quantitatively different
from other members of the Fe-Co-Al family of compounds. In particular, we find
an unusually small $\sigma_{xy}$ in FeCo2Al, around 80 $(\Omega~{}{\rm
cm})^{-1}$, that is almost insensitive on the frequency $\omega$ in the entire
range between 0 and 25 THz.

###  III.6 Optical conductivity and magnetic moment

As we discussed earlier, $\sigma_{xy}$ is often taken as an optical signature
of magnetic order. Therefore, it is natural to ask whether a material with
large magnetization will also have a large $\sigma_{xy}$. Figure 6 shows the
relationship between $\sigma_{xy}$ and the magnetic moment per atom, for all
of the compounds we studied. For each compound we show on the vertical scale
of Fig. 6 the range of maximal and minimal values of the calculated
$\sigma_{xy}$ in the range of frequencies from 0 to 25 THz. As can be seen
from the figure, we find that in the case of Fe–Co alloys, the magnetic
moments per atom are nearly the same, on the order of $\sim 2$ $\mu_{\rm B}$,
but the $\sigma_{xy}$ ranges from 150, all the way to 950 $(\Omega~{}{\rm
cm})^{-1}$. Same is true for the other compounds we studied. The most drastic
example is FeAl in which $\sigma_{xy}$ even changes sign relative to that in
Fe. Therefore we can conclude that materials with larger magnetic moment do
not necessarily have larger $\sigma_{xy}$. This is not too surprising, as the
large magnetization arises from the large difference in population of
dominantly spin-up and spin-down bands. However, large $\sigma_{xy}$ in the
low-frequency regime relies on detailed information about spin-orbit split
bands near the Fermi level.

###  III.7 Remaining components of the $\sigma$ tensor

So far we focused on the real part of $\sigma_{xy}$, as this is the most
studied part of the optical conductivity tensor that is dependent on the
magnetic order. However, by symmetry, there are two other components of the
conductivity tensor that can be controlled by the magnetic order. These are
the imaginary part of the off-diagonal component $\sigma_{xy}$, as well as the
difference between the diagonal components of the conductivity tensor along
and perpendicular to the direction of the magnetic moment.

We first analyze the imaginary part of $\sigma_{xy}$. While the real part of
$\sigma_{xy}$ is related to the change in the polarization direction in the
polar-MOKE geometry, the imaginary part is related to the change in the
ellipticity of the reflected light. The imaginary part of $\sigma_{xy}$ is
shown in the supplement for all compounds we studied. As would be expected
from the Kramers-Kronig relationship, imaginary part of $\sigma_{xy}$ is also
strongly dependent on both chemical composition and frequency. For example, we
find that the step-like spectral feature we found for the real part of
$\sigma_{xy}$ in Co3Al is accompanied with a single sharp feature in the
imaginary part of $\sigma_{xy}$, at nearly the same frequency ($\sim 10$ THz),
as one might expect. Similarly sharp features in the imaginary part of
$\sigma_{xy}$ we find in the case of Co and Fe3Al. The spectral features of
the imaginary part of $\sigma_{xy}$ are much more complex for the other
compounds we studied. This is particularly true for alloys containing Fe
magnetic atoms.

Now we analyze the other component of the conductivity tensor that depends on
the presence of the magnetic order. By symmetry, the magnetization along the
$z$-axis introduces a spin-orbit driven difference between the $\sigma_{zz}$
(along the magnetization axis) and $\sigma_{xx}=\sigma_{yy}$ (perpendicular to
the magnetization axis). While $\sigma_{xy}$ is zero without magnetic order,
the diagonal components $\sigma_{xx}$, $\sigma_{yy}$, and $\sigma_{zz}$ are
not. Therefore, here we don’t focus on these diagonal components individually,
but instead we focus on their difference, $\sigma_{zz}-\sigma_{xx}$. The
calculated values of $\sigma_{zz}-\sigma_{xx}$ for all compounds we studied
can be found in the supplement. We again find strong variations of
$\sigma_{zz}-\sigma_{xx}$ both as a function of frequency and as a function of
chemical composition. In the case of Fe-Co family of compounds, we find that
$\sigma_{zz}-\sigma_{xx}$ in Fe and FeCo have a sharp peak around 15 THz, and
are nearly zero below 14 THz. Co shows two sharp features, with opposite
signs, one at around 9 and another around 20 THz. Addition of 25% of Al to Fe
strongly changes the value of $\sigma_{zz}-\sigma_{xx}$. While Fe has a sharp
feature around 15 THz, Fe3Al has a sharp feature around 3 THz. Addition of
even more Al, as in FeAl, introduces two sharp features, at around 10 and 15
THz. Addition of Al to Co also introduces significant qualitative and
quantitative changes to $\sigma_{zz}-\sigma_{xx}$.

Figure 6:  The summary of $\sigma_{xy}$ in the range of 0$\sim$0.1 eV and
magnetic moment per atom of each metal and alloy. The bar shows the maximum
and minimum of the $\sigma_{xy}$ in the range of 0$\sim$0.1 eV.

##  IV Conclusion

We find a strong dependence of $\sigma_{xy}(\omega)$ in the terahertz
frequency range for Fe-Co-Al ordered alloys. For example, in the case of Co3Al
we find nearly 50-fold reduction in $\sigma_{xy}(\omega)$ above 10 THz as
compared to $\sigma_{xy}(\omega)$ below 10 THz. On the other hand, in the case
of FeCo2Al we find nearly constant $\sigma_{xy}(\omega)$ in the entire range
from 0 to 25 THz. Furthermore, we also find a strong dependence of
$\sigma_{xy}(\omega)$ on the chemical composition. For example, the addition
of Al to Fe can change the sign of $\sigma_{xy}(\omega)$, so that
$\sigma_{xy}(\omega)$ is positive in Fe and Fe3Al but negative in the case of
FeAl. Similarly, addition of Co to Fe produces a non-monotonic dependence of
$\sigma_{xy}(\omega)$ on Co concentration. As an example,
$\sigma_{xy}(\omega)$ in FeCo is about 4 times smaller than that in Fe, and 3
times smaller than that in Co. We attribute both of these strong variations,
with frequency and composition, to the subtle changes in the electronic
structure induced by the presence of the spin-orbit interaction. However,
unlike in the case of AHE, the analysis of the origin of $\sigma_{xy}(\omega)$
is additionally complicated by the fact that Kubo expression for
$\sigma_{xy}(\omega)$ involves transition between empty and occupied states,
while in the case of AHE $\sigma_{xy}(\omega=0)$ can be written as integral of
the Berry curvature over occupied set of bands. Therefore, we leave a more
detailed analysis of the origin of these strong variations of
$\sigma_{xy}(\omega)$ to future studies.

Our calculations are performed in the limit of infinite electron lifetime.
While this is justified for materials with moderate amount of disorder, with
diagonal conductivity around $10^{4}$–$10^{6}$ $(\Omega~{}\textrm{cm})^{-1}$,
we leave for future work discussion of role of disorder on
$\sigma_{xy}(\omega)$ in the terahertz range for materials that are not within
the moderate range of disorder. We expect that at low enough frequencies the
phenomenology of $\sigma_{xy}(\omega)$ will be the same as that of
$\sigma_{xy}(\omega=0)$, so that with a moderate amount of disorder the
dominant contribution to $\sigma_{xy}(\omega)$ is intrinsic, but with less
disorder scattering from impurities start to dominate.Nagaosa  _et al._ (2010)
We show in supplement calculated $\sigma_{xy}(\omega)$ in which we
approximately incorporate the effect of finite carrier lifetime. While finite
carrier lifetime approximation washes out some of the spectral features in
$\sigma_{xy}(\omega)$, we still find that lot of qualitative features remain,
such as change in sign of $\sigma_{xy}$ near $20$ THz, or sharp decline in
$\sigma_{xy}$ in Co3Al around 10 THz, or non-monotonic dependence of
$\sigma_{xy}$ of Co-concentration in Fe-Co alloys.

Our findings indicate that alloys such as Fe-Co-Al would be of interest in
creation of magnetic optical metamaterials in which direction of magnetic
moment and chemical composition are used to control its interaction with
light. While our study focused mostly on real part of $\sigma_{xy}$, we note
that there are additional components of the conductivity tensor which can be
controlled with the magnetic field, such as imaginary part of $\sigma_{xy}$,
as well as difference between the diagonal components
$\sigma_{zz}-\sigma_{xx}$. We also note that change in the polarization of
light is dependent not only on $\sigma_{xy}$, but also on diagonal components
of $\sigma$ tensor. Therefore, any additional spectral features present in the
diagonal components of $\sigma$ will additionally modify the MOKE signal in
these alloys.

###### Acknowledgements.

This work was supported by grant NSF DMR-1848074. Computations were performed
using the HPCC computer cluster at UCR. We acknowledge discussion with Igor
Barsukov and Richard B. Wilson.

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in 1,2,3 See pages \x, of si.pdf


11institutetext: 1École de Technologie Supérieure, Montreal, Canada
11email<EMAIL_ADDRESS>
# Curating Subject ID Labels using Keypoint Signatures

Laurent Chauvin 11 Matthew Toews 11

Subject ID labels are unique, anonymized codes that can be used to group all
images of a subject while maintaining anonymity. ID errors may be
inadvertently introduced manually error during enrollment and may lead to
systematic error into machine learning evaluation (e.g. due to double-dipping)
or potential patient misdiagnosis in clinical contexts. Here we describe a
highly efficient system for curating subject ID labels in large generic
medical image datasets, based on the 3D image keypoint representation, which
recently led to the discovery of previously unknown labeling errors in widely-
used public brain MRI datasets [1].

Method: ID errors are identified using a pair-wise (dis)similarity measure
$d(A,B)$ to evaluate and flag all image pairs $(A,B)$ that are unexpectedly
similar or dissimilar. The primary computational challenge is the $O(N^{2})$
algorithmic complexity of pairwise comparisons for a set of $N$ images, which
becomes intractable as $N\rightarrow\infty$. $d(A,B)$ is defined by the log of
the Jaccard distance [3] between keypoint sets $A=\\{f_{i}\\}$ and
$B=\\{f_{j}\\}$, as illustrated in Figure 1. Salient 3D SIFT-Rank keypoints
[7] are detected in individual images, where each keypoint is a generic local
anatomical pattern and is represented as a compact 64-element descriptor
derived from rank-ordered gradient histogram elements [6]. Enumerating nearest
neighbor (NN) descriptor matches, i.e. equivalent anatomical patterns between
$N^{2}$ image pairs is achieved in $O(N~{}log~{}N)$ complexity via approximate
K-NN search using the KD-tree structure [5].

Figure 1: Computing the log Jaccard distance $d(A,B)=-\log J(A,B)$ between a
pair of generic medical image volumes $(A,B)$ involves (left to right) SIFT-
Rank keypoint detection [7], K-NN keypoint matching and Jaccard overlap
evaluation.

Results: A total of $N=7536$ MRIs of $3334$ unique subjects are pooled from
four large public neuroimaging databases and used in evaluation: the Human
Connectome Project (HCP) [8], the Open Access Series of Imaging Studies
(OASIS) 1 and 3 [4], and the Alzheimer’s Disease Neuroimaging Initiative
(ADNI) [2]. Each of $N(N-1)/2=28,391,880$ image pairs $(A,B)$, is associated
with a unique pair-wise relationship label: same subjects (SM), monozygotic
twins (MZ), dizygotic twins (DZ), full-sibling (FS), and unrelated (UR),
images in different databases are assumed to be unrelated. As expected,
pairwise distance distributions in Figure 2 are lowest and highest for the
same (SM, pink) and unrelated subjects (UR, blue), respectively, with twin and
non-twin sibling distances falling in between. Obvious outliers (blue, pink
points) reveal labeling errors (see Figure 2), which were visually confirmed
to be either images of the same subject mislabeled as different (blue dots) or
images of different subjects labeled as the same (pink).

Figure 2: Distributions of the pairwise log Jaccard distance $d(A,B)$
conditional on relationship labels, dots indicate obvious label
inconsistencies, with visual examples of same subject images mislabelled as
different (a,b) and different subject images labeled as the same (c).

Discussion: For a typical $256^{3}$-voxel image, keypoint detection requires
1-2 seconds per image via a GPU implementation 111https://github.com/3dsift-
rank/3DSIFT-Rank and produces 1000-5000 keypoints per image depending on the
content. Enumerating all NN images requires 100-350 milliseconds per image.
The memory footprint for keypoints is 100x smaller than image data and could
be incorporated as a DICOM image header. The HCP has since removed duplicate
subjects.

## References

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  * [2] Jack, C.R.C., Bernstein, M.A., Fox, N.C.N., Thompson, P., Alexander, G., Harvey, D., Borowski, B., Britson, P.J., L. Whitwell, J., Ward, C., Dale, A.M., Felmlee, J.P., Gunter, J.L., Hill, D.L., Killiany, R., Schuff, N., Fox-Bosetti, S., Lin, C., Studholme, C., DeCarli, C.S., Gunnar Krueger, G., Ward, H.A., Metzger, G.J., Scott, K.T., Mallozzi, R., Blezek, D., Levy, J., Debbins, J.P., Fleisher, A.S., Albert, M., Green, R., Bartzokis, G., Glover, G., Mugler, J., Weiner, M.W., Gunnar Krueger, H.A., Ward, H.A., Metzger, G.J., Scott, K.T., Mallozzi, R., Blezek, D., Levy, J., Debbins, J.P., Fleisher, A.S., Albert, M., Green, R., Bartzokis, G., Glover, G., Mugler, J., Weiner, M.W.: The Alzheimer’s disease neuroimaging initiative (ADNI): MRI methods. Journal of Magnetic Resonance Imaging 27(4), 685–691 (4 2008). https://doi.org/10.1002/jmri.21049
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  * [4] Marcus, D.S., Wang, T.H., Parker, J., Csernansky, J.G., Morris, J.C., Buckner, R.L.: Open Access Series of Imaging Studies (OASIS): Cross-sectional MRI Data in Young, Middle Aged, Nondemented, and Demented Older Adults. Journal of Cognitive Neuroscience 19(9), 1498–1507 (9 2007). https://doi.org/10.1162/jocn.2007.19.9.1498
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