# The Giant Spin Hall Effect at Optical Frequencies

P. Elliott  Max-Born-Institute for Non-linear Optics and Short Pulse
Spectroscopy, Max-Born Strasse 2A, 12489 Berlin, Germany  S. Shallcross  Max-
Born-Institute for Non-linear Optics and Short Pulse Spectroscopy, Max-Born
Strasse 2A, 12489 Berlin, Germany  S. Sharma  Max-Born-Institute for Non-
linear Optics and Short Pulse Spectroscopy, Max-Born Strasse 2A, 12489 Berlin,
Germany

###### Abstract

We generalize the spin Hall angle to laser pulses of finite frequencies in the
linear response regime and predict a giant optical spin Hall effect. Namely,
for certain transition metal elements, at particular frequencies, the spin
current can be a significant fraction of the charge current, and even exceed
it for XUV frequencies. By maximizing spin current while minimizing the charge
current, we thus minimize a major source of heating in spintronic devices. We
employ ab-initio time-dependent density functional theory (TDDFT), and with
real-time simulations calculate the conductivity and transverse spin
conductivity for all $3$d, $4$d, and $5$d transition metals for frequencies up
to $50$ eV. In the XUV frequency range we find values greater than $1$ for the
spin Hall angle, indicating spin currents larger than the charge current can
be generated.

In the field of spintronics, the electronic spin degree of freedom is utilized
instead of, or in conjunction with, the charge degree of freedom to design
more efficient technologies. While it is a relatively young research field, it
has already had a significant impact on practical electronics thanks to the
integration of the giant magnetoresistanceBaibich  _et al._ (1988); Binasch
_et al._ (1989) (GMR) effect into commercially available devices. Similarly,
magnetoresistive RAM (MRAM) in which, unlike conventional RAM, bits are stored
via the magnetic moment, is another highly promising area of spintronics. An
emerging form of MRAM devices are based on the spin-orbit torque (SOT)Manchon
_et al._ (2019), where the magnetic moment is manipulated via a charge current
flowing in the substrate, and not in the magnetic material itselfMiron  _et
al._ (2011). This is possible when the substrate has strong spin-orbit
coupling which can convert a charge current into a perpendicular spin current
via the spin Hall effect. Thus, a key component of SOT-MRAM is the spin Hall
effect (SHE).

In the ordinary Hall effect, discovered in $1879$, an external magnetic field
perpendicular to a charge current generates an additional charge current
perpendicular to both. The spin Hall effectDyakonov and Perel (1971), in
contrast, produces a spin current perpendicular to both the charge current
direction and the spin quantization axis in the absence of an external
magnetic field. In $2004$ the accumulation of opposite spin polarizations on
the two sides of a Hall bar was observed, confirming the existence of the spin
Hall effect experimentallyKato  _et al._ (2004); Wunderlich  _et al._ (2005)
and beginning the era of SHE physics.

Typically the SHE has been studiedOppeneer  _et al._ (1992); Sinova  _et al._
(2003); Guo  _et al._ (2005); Yao and Fang (2005); Freimuth  _et al._ (2010);
Guo  _et al._ (2008); Freimuth  _et al._ (2014); Wang  _et al._ (2016); Dolui
and Nikolić (2017); Turek  _et al._ (2019); Ryoo  _et al._ (2019); Haney  _et
al._ (2013); Tanaka  _et al._ (2008) and utilized in the static, zero-
frequency, or dc, limit of electric fields. However, recently, the spin-Hall
effect was studied for bilayers in the THz frequency regimeGuimarães  _et al._
(2017) and the inverse spin-Hall effect was used to generate Terahertz
pulsesSeifert  _et al._ (2016) from current pulses themselves generated by
femtosecond laser pulses. Extending the SHE into the domain of optical-XUV
frequency electric fields, such as those in femtosecond laser pulses, will be
of crucial importance in uniting the fields of spintronics and ultrafast
magnetismBovensiepen (2009), and expected to drastically improve the speed and
efficiency of spintronic devices. In this work, we calculate the spin-current
tensor for non-collinear systems and extend the definition of the spin-Hall
angle to the finite frequency domain. This allows us to predict a remarkable
behavior for the frequency dependent spin current conductivities finding for
several transition metals that the frequency dependent spin Hall angle greatly
exceeds that found in the dc limit, and can even exceed unity, implying that
spin currents of greater magnitude than the driving charge current can be
generated. Such a situation represents the ideal physics for spintronics
devices as the important spin current is maximised while the heat deposited
due to Joule heating by the charge current is minimized.

The underlying physics of the spin Hall effect is typically due to the
relativistic effect of spin-orbit couplingEngel  _et al._ (2007); Sinova  _et
al._ (2015); Reiss  _et al._ (2021). In the intrinsic form of the SHE, spin-
orbit coupling induces a spin texture (or Berry curvature) in the bands of the
material, which then acts like a spin-dependent magnetic field deflecting
moving electrons. In the extrinsic case, the additional spin-orbit interaction
due to impurities cause difference scattering events that depend on the spin
of an electron. In this work we will only consider the intrinsic effect.

Our method of choice is time-dependent density functional theory (TDDFT) which
is an _ab-initio_ method for calculating electron dynamicsRunge and Gross
(1984). This method has had enormous success in both, explaining the
underlying physics of ultrafast processesKrieger  _et al._ (2015); Falke  _et
al._ (2014), and also predicting new phenomena, such as OISTRElliott  _et al._
(2016); Dewhurst  _et al._ (2018), which was subsequently experimentally
verifiedSteil  _et al._ (2020); Chen  _et al._ (2019); Willems  _et al._
(2020); Siegrist  _et al._ (2019); Hofherr  _et al._ (2020); Golias  _et al._
(2021). Thus, TDDFT is the ideal method to explore the behavior of the SHE at
finite frequencies. For the choice of materials, we will explore all $3d$,
$4d$, and $5d$ transition metals (except Mn due to the large number of atoms
in the unit cellHafner and Hobbs (2003)), as these have strong SOC and are
known to display a strong SHE in the dc limit.

Figure 1: The frequency-dependent transverse spin conductivity for several
transition metals calculated by real-time TDDFT in the linear response regime,
shown for optical frequencies.

In TDDFT, the charge current and spin currents are calculated from the
expectation value of the time-dependent Kohn-Sham wavefunction at each
timestep. For more information on the propagation scheme, TDDFTRunge and Gross
(1984), and the computational parametersHaas  _et al._ (2009); Janthon  _et
al._ (2014); Li  _et al._ (2015), please see the supplementary material
available at [URL will be inserted by publisher]. The charge current operator,
$\hat{{\bf j}}$ contains the usual paramagnetic and diamagnetic terms (as well
as a small correction due to SOC). On the other hand the spin current operator
(which measures spin flow in direction $a=x,y,z$ of spin along quantization
axis $\mu=x,y,z$) is not uniquely definedVernes  _et al._ (2007). A fact that
follows from the absence of a continuity equation for the spin current and
magnetization, due to non-conservation of the magnetic moment in the presence
of SOC. The simplest choice is:

$\hat{\mathbf{j}}^{S}_{a\mu}=\mathbf{\hat{j}}_{a}\bm{\hat{\sigma}}_{\mu}$ (1)

which satisfies the continuity equation in the absence of SOC.

To calculate the charge conductivity tensor, ($\sigma_{ab}$), and the spin
conductivity tensor, ($\sigma^{S}_{a\mu b}(\omega)$), in the linear response
regime, a real time TDDFT calculation is performed using the ELK electronic
structure codeelk  for a weak perturbation and the resulting currents Fourier
transformed to frequency space:

$\displaystyle\sigma_{ab}(\omega)$ $\displaystyle=\frac{{\bf
j}_{a}(\omega)}{{\bf E}_{b}(\omega)}$ (2) $\displaystyle\sigma^{S}_{a\mu
b}(\omega)$ $\displaystyle=\frac{{\bf j}^{S}_{a\mu}(\omega)}{{\bf
E}_{b}(\omega)}$ (3)

where ${\bf E}$ is a weak perturbing electric field, related to the vector
potential by ${\bf E}(t)=-1/c\ \textrm{d}{\bf A}_{\rm ext}(t)/\textrm{d}t$
with $c$ the speed of light. The spin Hall effect is measured by the term
$\sigma^{S}_{xyz}(\omega)$, which we denote as the spin Hall conductivity or
the transverse spin conductivity. It measures the spin current flowing in
direction $x$ of the spin oriented along $y$ due to an applied electric field
in the $z$ direction (which also drives a charge current in the
$z$-direction).

Figure 2: The spin Hall angle, defined as the ratio of the transverse spin
conductivity and the charge conductivity, or equivalently the Fourier
transformed perpendicular spin current divided by the charge current, for
several transition metals. The giant optical spin Hall effect can be seen for
the large values of the spin Hall angle at optical frequencies.

In Fig. 1, we show the transverse spin conductivity for those transition
metals with the largest SHE (for others see SI). As can be seen, there is
significant frequency dependence. At low frequencies (from $0$ to $1$ eV), W,
Pd, and Pt have the largest spin Hall conductivities, unsurprising given these
elements display a giant SHE in the dc limitMorota  _et al._ (2011); Sagasta
_et al._ (2016); Pai  _et al._ (2012); Hao and Xiao (2015); Qu  _et al._
(2018). It is interesting to note that the spin conductivity is finite at
$\omega=0$, in contrast to the charge current where electron-phonon scattering
is required to prevent the current diverging in this limit. This highlights
the difference between the physics of charge current and spin currents. Moving
to higher frequencies, we see that other materials such as Au, Re, and Os show
the best spin current response to an electric field, although W remains strong
for a wide range of frequencies (even after it switches sign). For the optical
range of $1.5-3.0$ eV, Au and W exhibit the largest spin currents. We note
that the Kubo-Greenwood independent-particle formalism results for the W and
Pt spin conductivities calculated in Ref. Stamm  _et al._ (2017) agree with
the TDDFT results of Fig. 1, indicating that exchange-correlation corrections
are small, at least in this case.

In the dc case, the ratio between the spin current and the charge current is
known as the spin Hall angle (SHA) and is commonly used to characterize the
SHE. It may be generalizedGuimarães  _et al._ (2017) to finite frequencies:

$\theta^{\rm SH}(\omega)=\frac{{\bf j}^{S}_{xy}(\omega)}{{\bf
j}_{z}(\omega)}=\frac{\sigma^{S}_{xyz}(\omega)}{\sigma_{zz}(\omega)}$ (4)

Figure 3: The charge and transverse spin currents in Au induced by laser
pulses of frequency (a) $2.1$ eV and (b) $4.2$ eV. This demonstrates the giant
optical spin Hall effect in Au for laser pulses with frequency $2.1$ eV.

In Fig. 2, we plot the transition metal elements with the largest SHAs (for
other metals see SI). Immediately we see a much richer structure than the spin
conductivity alone, and find the interesting result that the SHA at certain
frequencies is very large and can even exceed the values of W and Pt at zero
frequency. Indeed, even W and Pt themselves have higher values at finite
frequency than at zero frequency. We name this phenomenon as the giant optical
spin Hall angle (this term has also been used in the context of a polariton
spin Hall effectKavokin  _et al._ (2005), however it is the natural name for
the effect described in this work). An enhancement of the dynamical spin Hall
angle was also observed in Ref. Guimarães  _et al._ , 2017 in the THZ
frequency regime due to changes in the band structure induced by ferromagnetic
layers.

To demonstrate how this giant optical spin Hall would behave in an
experimental setup with a finite duration pulse, we simulate the response of
Au to two different frequency pulses. From Fig. 2, we see that Au has the
largest SHA at $2.1$ eV and a value roughly half at $4.2$ eV. Thus in Fig. 3,
we compare the charge current and transverse spin current induced by lasers
oscillating at these frequencies. Both pulses have a duration full width at
half maximum (FWHM) of $12.1$ fs and a peak intensity of $10^{9}$ W/cm2.

In Fig. 3 (a), we see that at $\omega=2.1$ eV, the charge and spin currents
are of similar magnitude, while in Fig. 3 (b), the charge current is much
larger than the spin current. This is exactly as expected from Fig. 2, which
says that the ratio of the spin current to the charge current is larger at
$\omega=2.1$ eV than at $\omega=4.2$ eV. In this case the amplitude of the
charge current at both frequencies is similar, meaning the spin current at
$\omega=2.1$ eV is much larger. We can also see that the charge and spin
currents are $180^{\circ}$ out of phase with each other, as the SHA is
negative. In both cases, the current oscillations will continue after the
laser pulse, however decoherence and relaxation to the ground-state will damp
these out on the $100$ fs to $1$ ps timescale.

Figure 4: Tuning the spin Hall properties via ordered alloying of Pt and Au
showing the frequency dependence of the (a) spin Hall angle and (b) the
transverse spin Hall conductivity. For comparison, we show the elemental
values for Au and Pt, as well as their their average, in each case.

The giant SHA can be interpreted as the appropriate frequency to maximize the
spin current for a given charge current amplitude, or equivalently minimize
the charge current for a desired amplitude of spin current. This has great
significance as the smaller the charge current, the less energy deposited into
the system due to Joule heating. As dissipation of heat is one of the limiting
factors currently hindering technological development, the giant SHA can be
utilized in the quest for more efficient devices. One such device where the
giant SHA can be exploited would be ultrafast SOT devices, where a magnetic
moment is controlled via spin-currents generated from ultrafast laser pulses.
In a SOT geometry, a laser incident on a SHE layer could create an oscillating
spin-current that penetrates the magnetic layer. The induced dynamics of the
magnetic moment due to such a spin current is an open question, which will be
studied in future work.

A key quality required for efficient technological transfer is the ability to
tailor the material for desired properties. For example, whether transition
metal elements could be combined to enhance or extend the frequency range of
the SHE. From Fig. 1, we see that while both Pt and Au have large spin Hall
conductivities, they occur in different frequency ranges: $0$ eV to $2$ eV for
Pt, and $2$ eV to $4$ eV for Au. In Fig. 4(b) we show that mixing Pt and Au
together in an ordered alloy (L$1_{0}$ geometry with the Au lattice spacing)
produces a material with a large spin Hall conductivity over the entire NIR,
visible, and VUV range between $0$ eV to $4$eV. In Fig. 4(b), we also see that
the combined PtAu spin Hall conductivity is simply the average of the two
independent Au and Pt values. However this will not be the case for the spin
Hall angle. In Fig. 4(a), we see the remarkable result that the giant spin
Hall angle seen in Pt at $0.75$ eV can be further enhanced in the PtAu alloy.
This is evidently due to the changes in the charge conductivity caused by the
altered band structure of the alloy. This new value of this giant spin Hall
angle exceeds all those seen in the elemental cases of Fig. 2, demonstrating
how simple alloying can be used to tailor and optimize the SHE.

Figure 5: The spin Hall angle, as in Fig. 2, extended to the XUV frequency
range, where values several times larger than unity can be seen.

At near infra-red (NIR) and optical frequencies, we saw in Fig. 2 that large
spin Hall angles are possible at finite frequencies. The largest of these was
Au with a value of $-0.4$ or a spin current magnitude of about $40\%$ of the
charge current. We now show that in the x-ray ultraviolet (XUV) range, spin-
currents several times larger than the charge current are possible.

In Fig. 5, we extend the spin Hall conductivities of Fig. 2 into the XUV
frequency range up to $55$ eV. The dominant transition in this frequency range
is from the $3p/4p/5p$ orbitals to the unoccupied $3d/4d/5d$ orbitals at the
Fermi level, otherwise known as the M2,3, N2,3, and O2,3 absorption edges in
XAS. As with the optical frequency range, the charge and spin currents are
determined by the interplay between the spin texture/Berry curvature of the
occupied and unoccupied states in the excitation, with the added complication
of strong spin orbit splitting of the p states. However, as we see in Fig. 5
this combination can create scenarios where a spin current is generated that
is even larger than the charge current, i.e. a SHA above/below $\pm 1$.

Only the materials with the largest XUV SHA are shown in Fig. 5. Curiously
these materials are a mix of $4$d and 5$d$ elements, in contrast to the
results presented in Figs. 1 and 2, which are mostly $5$d elements. The
exceptions being W and Re, which appear in all $3$ figures.

We again examine the behavior of the currents when using a finite duration
pulse with a frequency centered at one of the SHA peaks. In Fig. 6 we show the
charge current and the perpendicular spin current induced by a pulse with
frequency $\omega=35$ eV, a FWHM of $12.1$ fs, and a peak intensity of
$10^{10}$ W/cm2. Additionally, for such a high frequency pulse, the timestep
in our time propagation algorithm must be decreased to $\Delta t=0.05$ a.u.,
to ensure accuracy and stability. As expected from Fig. 5, the spin-current is
more than double the charge current, thus demonstrating a giant XUV spin Hall
angle. In the supplemental material, we show the charge and spin
conductivities for each element. This giant SHA is due to the charge current
becoming small for a perturbation at this frequency while the spin-current
response is similar to that in the optical range.

Figure 6: The charge and transverse spin currents in W induced by a laser with
frequency $35$ eV. The amplitude of the spin current is several times larger
than the charge current.

In conclusion we have calculated the charge and transverse spin-Hall
conductivities in the linear response regime using real-time TDDFT for laser
pulses in the NIR, optical, and XUV frequency ranges for almost all transition
metal elements. We find the element with the highest spin response strongly
depends on this frequency. For example, at optical frequencies, it is Au and W
which show the strongest SHE unlike the dc limit where Pt is the strongest.

We have generalized the concept of the spin Hall angle to finite frequency
using the Fourier transform of the induced charge current and perpendicular
spin current. We found a giant intrinsic spin Hall angle exists for certain
materials with laser pulses at particular frequencies, and have demonstrated
this effect in Au where the ratio between the charge current and spin current
can be maximized by choosing the pulse frequency. This minimizes a major
source of heating in spintronic devices, namely Joule heating demonstrating
the promise of combining the Spin Hall physics with ultrafast laser pulses. In
future work, we will calculate the induced magnetization dynamics in SOT
devices due to these ultrafast spin-currents. We also observed that for XUV
frequencies, in the range $20-50$ eV, SHE spin currents could be generated
that even by several times the charge current in amplitude. Finally we shown
that by designing multi-component alloys we could enhance and extend the range
of frequencies showing a significant SHE beyond those found in the constituent
components. In future work we plan to study more complicated structures, in
particular how the spin Hall conductivity depends on the interface.

##  I Acknowledgments

PE thanks DFG for funding through project 438494688. Shallcross would like to
thank DFG for funding through SPP 1840 QUTIF Grant No. SH 498/3-1, and Sharma
thanks DFG for funding through project-ID 328545488 TRR227 (project A04).

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