# Spin elastodynamic motive force

Takumi Funato1,2 and Mamoru Matsuo2,3,4,5 1Center for Spintronics Research
Network, Keio University, Yokohama 223-8522, Japan  2Kavli Institute for
Theoretical Sciences, University of Chinese Academy of Sciences, Beijing,
100190, China.  3CAS Center for Excellence in Topological Quantum Computation,
University of Chinese Academy of Sciences, Beijing 100190, China  4RIKEN
Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai,
319-1195, Japan

###### Abstract

The spin-motive force (SMF) in a simple ferromagnetic monolayer caused by a
surface acoustic wave is studied theoretically via spin-vorticity coupling
(SVC). The SMF has two mechanisms. The first is the SVC-driven SMF, which
produces the first harmonic electromotive force, and the second is the
interplay between the SVC and the magentoelastic coupling, which produces the
d.c. and second harmonic electromotive forces. We show that these electric
voltages induced by a Rayleigh-type surface acoustic wave can be detected in
polycrystalline nickel. No sophisticated device structures, non-collinear
magnetic structures, or strong spin-orbit materials are used in our approach.
Consequently, it is intended to broaden the spectrum of SMF applications
considerably.

###### pacs:

Valid PACS appear here

Introduction.— Spintronics is concerned with the interconversion of charge
transport and spin dynamics. The use of charge current to control
magnetization structures, such as magnetic domain wall, has been extensively
investigated for spin-torque oscillatorsKiselev  _et al._ (2003); Rippard  _et
al._ (2004); Katine and Fullerton (2008) and magnetoresistive random access
memory applicationshosomiNovelNonvolatileMemory2005 ; Ikeda  _et al._ (2007);
Matsunaga  _et al._ (2008). The spin-transfer torqueSlonczewski (1996); Berger
(1996); Katine  _et al._ (2000) associated with adiabatically following
conduction electron spins to the magnetization through the s-d coupling plays
a significant role. The spin motive force (SMF), a current-driven phenomenon
related to the spin-dependent force, was theoretically hypothesized as an
inverse spin-transfer torque effectBerger (1986); Volovik (1987); Stern
(1992); Barnes and Maekawa (2007). The first experiment involves observing the
SMF in a nanowire due to magnetic domain wall motionYang  _et al._ (2009,
2010); Hayashi  _et al._ (2012).

In nonuniform magnetic textures formed by a comb-shaped deviceYamane  _et al._
(2011), the magnetic vortex on a gyrating diskTanabe  _et al._ (2012), a
wedged-shaped deviceNagata  _et al._ (2015), exchange-coupled ferromagnetic
bilayersZhou  _et al._ (2019), and helical magnetismNagaosa (2019); Yokouchi
_et al._ (2020), the SMF induced by excited magnetization dynamics was seen.
The SMF induced by the oscillating magnetic field in systems with Rashba spin-
orbit interaction (SOI)Kim  _et al._ (2012) and temporarily varying gate
voltage in systems with strong SOIHo  _et al._ (2012, 2014, 2015); Yamane  _et
al._ (2013) have not been observed experimentally. There is little variety in
experimental reports on the SMFs because of experimental restrictions such as
sophisticated device structures, non-collinear magnetic structures, and strong
SOI systems.

Figure 1: Schematics of the mechanism of the spin elastodynamic motive force.
In a ferromagnetic metal, the symmetric part of the deformation couples to
magnetization (the magneto-elastic coupling) while the anti-symmetric part
couples to electron spin (the spin-vorticity coupling). When a surface
acoustic wave is excited in the ferromagnet, these couplings generate two
different spatially nonuniform spin dynamics of the magnetization and electron
spins. With the s-d coupling between magnetization and spins, the
unconventional spin motive force is induced. In particular, the spin-vorticity
coupling allows the spin motive force that does not rely on complicated
structures and strong spin-orbit interacting materials.

We propose the SMF induced by surface acoustic waves (SAWs) via spin-
elastodynamics, which we call “spin elastodynamic motive force (SEMF)”, to
overcome conventional limitations. Three couplings drive the spin-
elastodynamics in a ferromagnetic metal: magneto-elastic coupling (MEC), spin-
vorticity coupling (SVC), and s-d coupling (Fig. 1). The MEC has been
exploited for driving non-equilibrium spin dynamics with a SAWDreher  _et al._
(2012); weiler2012 . In the presence of a SAW, the SVC, the coupling of
electron spins and the vorticity of the lattice of the moving materials, play
a significant role for spin manipulationsMatsuo  _et al._ (2011a, b, c, 2013a,
2013b, 2014, 2017a). SVC has attracted much attention because it generates
spin current without SOI. The spin current generated via SVC has been
experimentally observed in spin-current generation by vortices associated with
liquid metal flowTakahashi  _et al._ (2016, 2020); Tabaei Kazerooni  _et al._
(2020, 2021); Matsuo  _et al._ (2017b) and by vortex motion of lattice
associated with SAWsKobayashi  _et al._ (2017); Kurimune  _et al._ (2020a, b);
Tateno  _et al._ (2020, 2021). The effective magnetic field owing to the
rotational motion has been directly detected using NMR and NQRChudo  _et al._
(2014, 2015); Harii  _et al._ (2015); Ono  _et al._ (2015); Arabgol and
Sleator (2019); Chudo  _et al._ (2021a, b).

Due to the rotational motion of the lattice associated with the SAWs, the
conduction electrons are subjected to SVC when they are applied to
ferromagnetic metals. Through the s-d coupling, the conduction electron spins
adiabatically follow the direction of the magnetization, and the SVC is
modulated by the magnetization’s precession motion, which causes the SEMF.
Simultaneously, the MEC operates on the magnetization to excite the
magnetization precession owing to lattice strain dynamics.

The current study finds a way to get over the SMF’s conventional limitations.
Thus, we concentrate on the SEMF induced by a SAW via SVC, a material-
independent interaction. We calculate the SEMF up to second-order in lattice
displacement via treating the influence of magnetization dynamics as a unitary
transformation of the spin space. The SEMF has two mechanisms. The first
produces an a.c. electromotive force oscillating the SAW’s frequency $\omega$,
which is induced by the gradient spin accumulation via the SVC (SV-SMF), and
the second produces d.c. electromotive force and a.c. one oscillating
$2\omega$ due to the interplay between the SVC acting on conduction electron
spins and the MEC acting on magnetization (SVME-SMF). The SVME-SMF generates
d.c. electromotive force in the film thickness direction when an out-of-plane
magnetic field is applied. These findings show that the SVME-SMF causes non-
reciprocity of the electromotive force.

Model.— We consider the free-electron system and coupled to the magnetization
through the s-d coupling. The conduction electron’s Lagrangian is expressed by

$\displaystyle\mathcal{L}_{\text{el}}=\int_{\bm{x}^{\prime}}c^{\prime\dagger}_{\bm{x}^{\prime}t}\left[i\hbar\partial_{t}+\frac{\hbar^{2}}{2m}\nabla^{\prime
2}-V(\bm{x}^{\prime})-J_{\text{sd}}\bm{m}^{\prime}\cdot\hat{\bm{\sigma}}\right]c^{\prime}_{\bm{x}^{\prime}t},$
(1)

where $c^{\prime\dagger}_{\bm{x}^{\prime}t}$ and
$c^{\prime}_{\bm{x}^{\prime}t}$ are electron creation and annihilation
operators, respectively,
$\hat{\bm{\sigma}}=(\hat{\sigma}^{x},\hat{\sigma}^{y},\hat{\sigma}^{z})$ are
the Pauli matrices, $\bm{m}^{\prime}$ is unit vector of the magnetization,
$J_{\text{sd}}$ is exchange splitting constant, and $V$ is potential due to
the lattice and impurities. When the lattice distortion dynamics are induced,
the potentials are modulated as $V(\bm{x}^{\prime}-\bm{u})$, where $\bm{u}$ is
the displacement vector of the lattice. Treating the lattice distortion
effect, we perform the local coordinate transformation from the laboratory
frame $\bm{x}^{\prime}$ to the “rotate frame” $\bm{x}=\bm{x}^{\prime}-\bm{u}$.
The conduction electron’s Lagrangian is expressed as
$\mathcal{L}=\mathcal{L}_{\text{el}}+\mathcal{L}_{\text{sv}}$, where
$\mathcal{L}_{\text{el}}$ is given by

$\displaystyle\mathcal{L}_{\text{el}}=\int_{\bm{x}}c^{\dagger}_{\bm{x}t}\left[i\hbar\partial_{t}+\frac{\hbar^{2}}{2m}\nabla^{2}-V(\bm{x})-J_{\text{sd}}\bm{m}\cdot\hat{\bm{\sigma}}\right]c_{\bm{x}t},$
(2)

where $c_{\bm{x}t}=\sqrt{1+\nabla\cdot\bm{u}}c^{\prime}_{\bm{x}^{\prime}t}$
and $\bm{m}^{\prime}=\mathcal{R}(r_{ij})\bm{m}$ is the unit vector of the
magnetization in the rotate frame with $\mathcal{R}(r_{ij})$ being the $SO(3)$
rotation matrix and $r_{ij}=\frac{1}{2}(\partial_{i}u^{j}-\partial_{j}u^{i})$
being the rotation tensor associated with the coordinate transformation. The
$\mathcal{L}_{\text{sv}}$ is the SVC:

$\displaystyle\mathcal{L}_{\text{sv}}=\frac{\hbar}{4}\int_{\bm{x}}c^{\dagger}_{\bm{x}t}\hat{\bm{\sigma}}\cdot\bm{\Omega}c_{\bm{x}t},$
(3)

where $\Omega_{k}=2\epsilon_{ijk}\partial_{t}r_{ij}$ is the vorticity of the
lattice motion. Note that the modulation of the kinetic energy and the time
derivative appear; however, they are not dominant in this study and are
neglected. We perform rotational transformation in the spin space
$c_{\bm{x}t}=U(\bm{x},t)a_{\bm{x}t}$ with the direction of magnetization
$\bm{m}$ in the $z$-direction, where $a^{\dagger}_{\bm{x}t}$ and $a_{\bm{x}t}$
are the electron creation and annihilation operators, respectively, after the
rotational transformation. Here the unitary operator $U$ is determined to
satisfy $c^{\dagger}\bm{m}\cdot\hat{\bm{\sigma}}c=a^{\dagger}\sigma^{z}a$. The
Lagrangian including the SVC after the transformation is given by

$\displaystyle\tilde{\mathcal{L}}_{\text{el}}=\int_{\bm{x}}a^{\dagger}_{\bm{x}t}\left[i\hbar\partial_{t}-\frac{(\bm{p}+e\hat{\bm{A}}^{\text{s}})^{2}}{2m}-\tilde{V}+e\hat{A}^{\text{s}}_{0}-J_{\text{ex}}\sigma^{z}\right]a_{\bm{x}t},$
(4)

where $\bm{p}=-i\hbar\nabla$ is the momentum operator and
$\tilde{V}=U^{\dagger}VU$. Here, $\hat{A}^{\text{s}}_{0}$ and
$\hat{\bm{A}}^{\text{s}}$ are the effective spin scalar and vector potentials,
given by
$\hat{A}_{0}^{\text{s}}=\frac{i\hbar}{e}U^{\dagger}\partial_{t}U+\frac{\hbar}{4e}U^{\dagger}(\hat{\bm{\sigma}}\cdot\bm{\Omega})U$,
and $\hat{\bm{A}}^{\text{s}}=-\frac{i\hbar}{e}U^{\dagger}\nabla U$.
Particularly, the second term of $\hat{A}^{\text{s}}_{0}$ is spin scalar
potential induced by the SVC. The effective electric field is obtained by
$\hat{\bm{E}}^{\text{s}}=-\partial_{t}\hat{\bm{A}}^{\text{s}}-\nabla\hat{A}^{\text{s}}_{0}$.
The total SMF is given by
$\bm{E}_{\text{smf}}=P\text{tr}(\hat{\sigma}^{z}\hat{\bm{E}}^{\text{s}}),$
where $P$ is the spin polarization of conduction electrons. The total SMF
contains conventional SMF and the SEMFs.

$\displaystyle\bm{E}_{\text{smf}}=\bm{E}_{\text{con}}-\frac{P\hbar}{2e}\nabla(\bm{m}\cdot\bm{\Omega}),$
(5)

where the first term
$E_{\text{con},i}=\frac{P\hbar}{e}\bm{m}\cdot\left(\partial_{t}\bm{m}\times\partial_{i}\bm{m}\right)$
is the conventional SMF, and the second term is the SEMFs, which is dominant
in this setup.

The spin dynamics generated by SAWs under a static magnetic field is
determined by

$\displaystyle\partial_{t}\bm{m}^{\prime}=\gamma(\bm{h}_{\text{eff}}+\bm{h}_{\text{me}})\times\bm{m}^{\prime}-\alpha(\partial_{t}\bm{m}^{\prime})\times\bm{m}^{\prime},$
(6)

where $\gamma(>0)$ is the gyromagnetic constant, $\alpha$ is the Gilbert
damping constant, and $h_{\text{eff}}$ is a static effective magnetic field
containing the static magnetic field and an anisotropy field. We assume that
the Barnett field is negligible compared to the effective magnetic field due
to the MEC, which is given bychikazumi

$\displaystyle
h_{\text{me},i}=-\frac{2}{M_{\text{S}}}\sum_{j}e_{ij}m_{j}^{\prime}\Bigl{[}b_{1}\delta_{ij}+b_{2}(1-\delta_{ij})\Bigr{]},$
(7)

where $e_{ij}=\frac{1}{2}(\partial_{i}u_{j}+\partial_{j}u_{i})$ is the strain
tensor, $M_{\text{S}}$ is the saturation magnetization, and $b_{1}$ and
$b_{2}$ are the MEC constants. The magnetization processes around
$h_{\text{eff}}$ with the resonance frequency of the Kittel modes
$\omega_{\text{K}}$ in the spin-wave approximation. The magnetization vector
in the rotate frame is given by
$\bm{m}=\bm{m}_{0}+\delta\bm{m}+\mathcal{O}(u^{2})$, where
$\bm{m}_{0}=(\sin\theta_{0}\cos\phi_{0},\sin\theta_{0}\sin\phi_{0},\cos\theta_{0})$
is the static component and $\delta\bm{m}$ is the dynamical component. Note
that the magnetization modulation associated with the rotational
transformation is negligible because of a higher order of Gilbert damping.

The lattice displacement $\bm{u}$ with the Rayleigh-type SAW (R-SAW)
proceeding in the $x$-direction is given bycleland

$\displaystyle\bm{u}=u_{0}e^{iq_{\mu}x^{\mu}}\left(\begin{matrix}i\mathfrak{s}_{q}\left(e^{\kappa_{\text{t}}z}-\frac{2\kappa_{\text{t}}\kappa_{\text{l}}}{\kappa_{\text{l}}^{2}+q^{2}}e^{\kappa_{\text{t}}z}\right)\\\
0\\\
\frac{\kappa_{\text{l}}}{|q|}\left(e^{\kappa_{\text{l}}z}-\frac{2q^{2}}{\kappa_{\text{t}}^{2}+q^{2}}e^{\kappa_{\text{t}}z}\right)\end{matrix}\right),$
(8)

where $u_{0}$ is the deformation amplitude of R-SAW, $\mathfrak{s}_{q}$ is the
sign function of $q$, and $\kappa_{\text{l}}^{-1}$ and
$\kappa_{\text{t}}^{-1}$ are decay lengths of longitudinal and transverse
waves, respectively. Here, $q_{\mu}=(-\omega,\bm{q})$ is the four-vector with
$\omega$ being a frequency of R-SAW and $\bm{q}=(q,0,0)$ being a wavenumber of
a longitudinal wave, and $x^{\mu}=(t,\bm{x})$ is the four-vector of time and
coordinate. The vorticity $\bm{\Omega}$ is given by

$\displaystyle\Omega_{y}=\frac{2r_{0}\omega^{2}}{c_{\text{R}}}\mathfrak{s}_{q}e^{\kappa_{\text{t}}z}e^{iq_{\mu}x^{\mu}},$
(9)

where $c_{\text{R}}=\omega/|q|$ is a R-SAW velocity and $r_{0}=|u_{z}(z=0)|$.

Figure 2:  The dependence of the SVME-SMF on the depth from the surface (a)(b) and the direction of the applied magnetic field (c)(d). The red and blue lines represent the second harmonic component in the $x$-direction and $z$-direction, and the green lines represent the d.c. component in the $z$-direction. The depth from the surface $\tilde{z}$ is normalized by the decay constant of the longitudinal wave $\kappa_{\text{l}}$. Here, we assume the isotropic system, and $b_{1}=b_{2}$ is satisfied. (a) and (c) show the case of an in-plane magnetic field applied ($\theta_{0}=\frac{\pi}{2}$), and the d.c. component vanishes. (b) and (d) show the case of an out-of-plane magnetic field applied ($\phi_{0}=0$), and both the second harmonic and d.c. components are induced. According to (c), the characteristic non-reciprocity is found in the $z$-direction SMF.  Table 1:  The characteristics of the SEMFs containing the SV-SMF and SVME-SMF. The SEMFs do not require any device and material restrictions such as nonuniform device structure, non-collinear magnetization, and strong SOI systems if only the SAW devices are available. IP and OP represent the cases where the magnetic field is in the in-plane and out-of-plane directions, respectively. The SEMFs produce various types of electromotive forces such as d.c. and a.c. oscillating at $\omega$ and $2\omega$.  Mechanisms | Interaction | Device | IP ($\theta_{0}=0$) | OP ($\theta_{0}\neq 0$)
---|---|---|---|---
SVME-SMF | SVC, MEC | SAW device | $2\omega$ | d.c., $2\omega$
SV-SMF | SVC | $\omega$ |  $\omega$

Results.— The SEMFs contribute to the SMF because the conventional SMF, as is
well-known, turns out to vanish in the non-collinear magnetization structure.
The SMF is given by up to second-order in $u$.

$\displaystyle\bm{E}_{\text{smf}}=\bm{E}_{\text{sv}}+\bm{E}_{\text{s-m}}.$
(10)

The first term
$\bm{E}_{\text{sv}}=-\frac{\hbar}{2e}\nabla(\bm{m}_{0}\cdot\bm{\Omega})$ is
the a.c. component oscillating at $\omega$ due to the gradient spin
accumulation induced by the SVC, which we call “SV-SMF”, given by

$\displaystyle\left(\begin{matrix}E_{\text{sv},x}\\\
E_{\text{sv},z}\end{matrix}\right)_{\omega}=\mathcal{E}_{\text{sv}}\sin\theta_{0}\sin\phi_{0}e^{\kappa_{\text{t}}z}\left(\begin{matrix}\sin(q_{\mu}x^{\mu})\\\
-\frac{\kappa_{\text{t}}}{|q|}\mathfrak{s}_{q}\cos(q_{\mu}x^{\mu})\end{matrix}\right),$
(11)

where $\mathcal{E}_{\text{sv}}=\frac{P\hbar
r_{0}}{ec_{\text{R}}^{2}}\omega^{3}$ a function of $\omega$ with the dimension
of an electric field. The second term in Eq. (10) is given by
$\bm{E}_{\text{s-m}}=-\frac{\hbar}{2e}\nabla(\delta\bm{m}\cdot\bm{\Omega})$
from the combination of SVC and MEC, which we call “SVME-SMF”. The SVME-SMF
contains the second harmonic component oscillating at $2\omega$ and d.c.
component. The second harmonic component is given by

$\displaystyle\left(\begin{matrix}E_{\text{s-m},x}\\\
E_{\text{s-m},z}\end{matrix}\right)_{2\omega}=\mathcal{E}_{\text{s-m}}\mathfrak{s}_{q}\left(\begin{matrix}g_{x}\sin(2q_{\mu}x^{\mu}+\mathfrak{s}_{q}\Theta_{x})\\\
g_{z}\cos(2q_{\mu}x^{\mu}+\mathfrak{s}_{q}\Theta_{z})\end{matrix}\right),$
(12)

and the d.c. component is given by

$\displaystyle\left(\begin{matrix}E_{\text{s-m},x}\\\
E_{\text{s-m},z}\end{matrix}\right)_{\text{dc}}=\mathcal{E}_{\text{s-m}}\mathfrak{s}_{q}\left(\begin{matrix}0\\\
g_{\text{dc}}\end{matrix}\right),$ (13)

where $\mathcal{E}_{\text{s-m}}=\frac{P\hbar\gamma
b_{1}}{2e\omega_{\text{K}}\alpha
M_{\text{s}}}\frac{u_{0}r_{0}}{c_{\text{R}}^{3}}\omega^{4}$ is the function of
$\omega$, and $g_{x/z}$, $g_{\text{dc}}$ are dimensionless functions of $z$,
$\theta_{0}$, $\phi_{0}$, and $\mathfrak{s}_{q}$. The dependence of the SVME-
SMF on the depth from the surface and direction of the applied magnetic field
are shown in Fig. 2. According to Fig. 2(c), the second harmonic component on
the $z$-direction has characteristic non-reciprocity. The interplay causes
this among the three effects: the rotation acting on the electrons via the
SVC, the strain acting on the magnetization via MEC, and the s-d
couplingsasaki2017 ; mingran2018 ; tateno2020 ; puebla2020 ; mingran2020 .
Note that $\Theta_{x/z}$ are dimensionless functions representing phase shift
and turn out to vanish in $\theta_{0}=\frac{\pi}{2}$.

Discussion.— Let us estimate the magnitude of the SV-SMF
$\mathcal{E}_{\text{sv}}$ and the SVME-SMF $\mathcal{E}_{\text{s-m}}$ in
polycrystalline nickel with strong ME coupling. When the R-SAW with frequency
$f=10$GHz is applied, SV-SMF is estimated as $\mathcal{E}_{\text{sv}}\sim
5.01\times 10^{-2}$V/m, and SVME-SMF is estimated as
$\mathcal{E}_{\text{s-m}}\sim 5.25\times 10^{-3}$V/m. This estimation suggests
that an observable voltage, about nV order, can be induced for a distance of
about $\mu$m. Here, longitudinal wave velocity is $c_{\text{l}}=6.04\times
10^{3}$m/s, transverse wave velocity is $c_{\text{t}}=3.00\times 10^{3}$m/s
rikanenpyo , R-SAW velocity is $c_{\text{R}}=2.80\times 10^{3}$m/s landau ,
saturation magnetization is $M_{\text{s}}=0.61$ T Seberino and Bertram (1997),
damping constant is $\alpha=4.5\times 10^{-2}$ Walowski  _et al._ (2008),
gyromagnetic ratio is $\gamma=2.41\times 10^{5}$ mA-1s-1 Meyer and Asch
(1961), MEC constant is $b_{1}=b_{2}=9.5$MJ/$\text{m}^{3}$, and lattice
displacement is $u_{0}=3.5\times 10^{-12}$m, which is 1% of the nickel lattice
constant.

Surprisingly, the current processes in typical ferromagnetic materials with
simple structures, such as nickel, can induce the SMF. Complex device
structures or non-collinear magnetization structures are required in
conventional SMFs. A Rashba device with spatial inversion symmetry breaking, a
gate voltage modulation, and a strong SOI material are all material
constraints for SOI-induced SMFs. Because there are no restrictions on device
structures or materials, the SEMFs enable technical applications, such as
microfabrication on magnetic materials. The SEMFs simultaneously generate the
d.c. and a.c. components oscillating at $\omega$ and $2\omega$, and the
current mechanisms are summarized in Table 1.

Conclusion.— To address the limits of typical SMF processes, we theoretically
analyzed the SEMF, which is the SMF induced by the R-SAW via SVC. We
discovered that the SEMF has two mechanisms. The first is SV-SMF generates the
first harmonic electromotive force due to gradient spin accumulation generated
by SVC, and the second is SVME-SMF generates d.c. and second harmonic
electromotive force due to the combination of SVC and MEC. Particularly, the
d.c. SMF is induced under an applied out-of-plane magnetic field. The results
suggest that the second harmonic component on the $z$-direction has
characteristic non-reciprocity. Our estimation suggests that the electric
voltage induced by the SEMFs is detectable in polycrystalline nickel. The
SEMFs can be generated in ferromagnet without restrictions on device structure
or materials if only the SAW device is available. Therefore, the SEMFs are
expected to expand the range of the SMF applications greatly.

We would like to thank Y. Nozaki, K. Yamanoi, T. Horaguchi, J. Fujimoto, and
D. Oue for enlightening discussions. This work was partially supported by JST
CREST Grant No. JPMJCR19J4, Japan. TF is supported by JSPS KAKENHI for Grants
No. 21K20356. MM is supported by JSPS KAKENHI for Grants (Nos. 20H01863 and
21H04565) and the Priority Program of the Chinese Academy of Sciences, Grant
No. XDB28000000.

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# Harmonic generation predominantly from a single spin channel in a half metal

G. P. Zhang∗ Department of Physics, Indiana State University, Terre Haute,
Indiana 47809, USA  Y. H. Bai  Office of Information Technology, Indiana State
University, Terre Haute, Indiana 47809, USA

###### Abstract

Harmonic generation in atoms and molecules has reshaped our understanding of
ultrafast phenomena beyond the traditional nonlinear optics and has launched
attosecond physics. Harmonics from solids represents a new frontier, where
both majority and minority spin channels contribute to harmonics. This is true
even in a ferromagnet whose electronic states are equally available to optical
excitation. Here, we demonstrate that harmonics can be generated mostly from a
single spin channel in half metallic chromium dioxide. An energy gap in the
minority channel greatly reduces the harmonic generation, so harmonics
predominantly emit from the majority channel, with a small contribution from
the minority channel. However, this is only possible when the incident photon
energy is well below the energy gap in the minority channel, so all the
transitions in the minority channel are virtual. The onset of the photon
energy is determined by the transition energy between the dipole-allowed
transition between the O-$2p$ and Cr-$3d$ states. Harmonics mainly from a
single spin channel can be detected, regardless of laser field strength, as
far as the photon energy is below the minority band energy gap. This
prediction should be tested experimentally.

###### pacs:

42.65.Ky, 78.66.Tr

##  I Introduction

Rapid developments of high harmonic generation (HHG) in atoms and molecules
mcpherson1987 ; ferray1988  have opened new frontiers in material research and
ultrafast science. Starting from 1990s, interest has shifted from atoms,
molecules to nanostructure and actual solids farkas1992 ; vonderlinde1995 ;
prl05 ; ganeev2009a ; ghimire2011 ; luu2015 ; garg2016 ; nd . Solids are much
more complex, and they demand more from HHG. For instance, can HHG directly
probe band structures in a solid vampa2015a ; vampa2015b ; prb20 ? Developing
harmonics into a tool to explore properties in topological materials such as
Bi2Se3 jia2019 ; baykusheva2021 , MnBi2Te4 jia2020b , MoS2 jia2020a  and spin-
orbit coupled systems lysne2020  is particularly important, given that vast
majority of devices are solids. Up to now, most of materials investigated so
far are nonmagnetic and most of them are insulators and semiconductors
ghimire2011 ; garg2016 ; you2017 . In ferromagnets, both majority and minority
spin channels contribute to harmonic signals nc18 , so it is difficult to
disentangle their respective contributions cleanly takayoshi2019 . In
antiferromagnets, one encounters the same difficulty where two sublattices
with opposite spin orientations both contribute to harmonic signals Tancogne-
Dejean2018 . So far, pure spin polarized harmonics in solids have not
materialized, but are in huge demand in materials where spin dynamics is
important. In laser-induced ultrafast demagnetization eric ; ourreview , one
needs to know how differently two spin channels evolve with time. THz emission
was reported in Cr(30 $\rm\AA$)/Ni(42 $\rm\AA$)/Cr(70 $\rm\AA$) films eric2004
, and more recently in Co/ZnO/Pt and Co/Cu/Pt multilayers li2018  and
MgO/Fe/MgO thin films zhang2020 . THz emission only allows one to access
magnetization on a picosecond time scale. Being able to detect the spin signal
from harmonics beyond THz emission potentially provides a new way to
investigate spin wave dynamics zhang2002 ; muller2009  on a few hundred
femtoseconds.

In this paper, we demonstrate harmonic generation mostly from a single spin
channel in half metallic $\rm CrO_{2}$ schwarz1986 ; coey . $\rm CrO_{2}$ has
no energy gap in the majority spin channel, but has a gap in the minority one.
The minority state is away from the Fermi level and the spin-orbit coupling is
weak, leading to nearly 100% spin polarization soulen1998 . These interesting
properties manifest themselves in harmonic generation. The energy gap in the
minority band limits the efficiency of harmonic generation, but the effect on
each order of harmonics is different. The fundamental harmonic, the first
harmonic, is weakly affected, and has no strong dependence on the incident
laser photon energy. A much stronger effect is found in higher orders of
harmonics. With the laser parameters used, we find that upon excitation of a
0.4-eV pulse, the fifth harmonic from the majority channel is 60 times
stronger than that from the minority signal, making it a nearly pure single-
spin channel harmonic. This difference disappears when a higher photon energy
is used. Such a dramatic effect is closely related to the dipole-allowed
transitions between the oxygen $2p$ states and chromium $3d$ state, where the
strongest transition is slightly above 2 eV. It is generally believed that
harmonic generation does not have the spin selectivity. Our finding here
demonstrates that optical harmonics can be a powerful tool to access spin
information in half-metals, where harmonics are largely emitted from a single
spin channel.

The rest of the paper is arranged as follows. In Sec. II, we present our
theoretical formalism. The results and discussions are in Sec. III. We
conclude this paper in Sec. IV.

##  II Theoretical formalism

We employ the first-principles density functional theory and couple it to the
time-dependent Liouville equation. This formalism represents an important
alternative to the time-dependent density functional theory and rigorously
respects the Pauli exclusion principles jpcm16 . Here in brief, we solve the
the spin-polarized Kohn-Sham equation wien2k ; blaha2020 ; np09 ; prb09 ,

$\left[-\frac{\hbar^{2}\nabla^{2}}{2m_{e}}+V_{ne}+V_{ee}+V_{xc}\right]\psi_{i{\bf
k}}^{\sigma}({\bf r})=E_{i{\bf k}}^{\sigma}\psi_{i{\bf k}}^{\sigma}({\bf r}),$
(1)

where $\sigma$ is the spin index, $m_{e}$ is the electron mass, the terms on
the left-hand side represent the kinetic energy, nuclear-electron attraction,
electron-electron Coulomb repulsion and exchange correlation (generalized
gradient approximation pbe ), respectively. $\psi_{i{\bf k}}^{\sigma}({\bf
r})$ is the Bloch wavefunction of band $i$ for spin $\sigma$ at crystal
momentum ${\bf k}$, and $E_{i{\bf k}}^{\sigma}$ is the band energy. Our two
spin channels are not mixed because of weak spin-orbit coupling soulen1998 .
This allows us to cleanly investigate the contributions from these two spin
channels. We use the full-potential augmented plane wave method as implemented
in the Wien2k code wien2k , where within the Muffin-tin sphere atomic basis
functions are used and in the interstitial region a plane wave is used. These
two basis functions are matched both in value and slope at the Muffin-tin
boundary.

To investigate high harmonic generation, we solve the Liouville equation in
real time. Different from prior studies prb19 , we have to solve the equation
for each spin channel $\sigma$,

$\frac{\partial\rho_{\bf k}^{\sigma}}{\partial
t}=\frac{1}{i\hbar}[H_{0}^{\sigma}+H_{I}^{\sigma},\rho_{\bf
k}^{\sigma}]-\frac{\rho_{\bf k}^{\sigma}-\rho_{\bf k}^{\sigma}(0)}{T_{1(2)}},$
(2)

so the computational cost doubles. Here $\rho_{\bf k}^{\sigma}$ is the time-
dependent density matrix with spin $\sigma$ at the ${\bf k}$ point and
$\rho_{\bf k}^{\sigma}(0)$ is its initial value. The second term on the right
side describes the decay of the density matrix to its initial value shen . For
a diagonal density matrix element $\rho_{{\bf k};ii}^{\sigma}$, the second
term becomes $-[\rho_{{\bf k};ii}^{\sigma}-\rho_{{\bf
k};ii}^{\sigma}(0)]/T_{1}$, where $i$ is the band index and $T_{1}$ is called
the longitudinal relaxation time. For an off-diagonal density matrix element
$\rho_{{\bf k};i\neq j}^{\sigma}$, the second term is $-\rho_{{\bf k};i\neq
j}^{\sigma}/T_{2}$, where $i$ and $j$ are the band indices and $T_{2}$ is
called the transverse relaxation time. Note that the initial off-diagonal
matrix elements are zero. $T_{1}$ and $T_{2}$ are different from those in NMR
where $T_{1}$ is used for the $z$ component of magnetization $M_{z}$ and
$T_{2}$ is used for the $x$ and $y$ components kittel . $H_{0}$ is the field-
free system Hamiltonian. $H_{I}^{\sigma}$ represents the interaction between
the laser field and system, $H_{I}^{\sigma}=-\sum_{{\bf k};i,j}\rho_{{\bf
k};i,j}^{\sigma}{\bf P}_{{\bf k};j,i}^{\sigma}\cdot{\bf A}(t)$. ${\bf P}_{{\bf
k};ij}^{\sigma}$ is the momentum matrix element between bands $i$ and $j$ at
the ${\bf k}$ point, and is calculated within the Wien2k code, but these
matrix elements must be converted properly before use (see the details in Ref.
ourbook ). ${\bf A}(t)$ is the vector potential, ${\bf
A}(t)=A_{0}\sin^{2}(t/\tau)(\cos(\omega t)\hat{x}\pm\sin(\omega t)\hat{y}),$
where $A_{0}$ is the amplitude of the vector potential in units of
Vfs$/\rm\AA$ nc18 , $\hat{x}$ and $\hat{y}$ are the unit vectors along the $x$
and $y$ axes, respectively, $t$ is the time, $\omega$ is the carrier
frequency, \+ and $-$ refer to the left ($\sigma^{-}$) and right
($\sigma^{+}$) circularly polarized light within the $xy$ plane, respectively.
$\tau$ is the laser pulse duration. We choose the duration $\tau$ to be 64
cycles of laser period prl05 . This is sufficient to resolve harmonics at
different orders. If the laser polarization is within the $yz$ plane, those
unit vectors must be changed properly.

##  III Results and discussions

$\rm CrO_{2}$ crystallizes in the rutile structure, with the space group $\rm
P4_{2}/mnm$. The Cr atoms form a body-centered tetragonal lattice. If we
orient the structure in such a way that the Cr atom is at the center of the
oxygen distorted octahedron, then two apex oxygen atoms $O_{1}$ and $O_{2}$
are placed along the vertical direction. Four equatorial oxygen atoms
$O_{3}$,…, and $O_{6}$, whose Wyckoff positions are $(u,u,0)$, $(u,u,1)$,
$(1-u,1-u,0)$, and $(1-u,1-u,1)$, are placed in the horizontal plane. The
distance between the body center Cr atom and $O_{1}$ and $O_{2}$ is shorter,
$d_{a}=1.8962\rm\AA$, and that between the body center Cr atom and $O_{3}$,…,
and $O_{6}$ is longer, $d_{e}=1.9087\rm\AA$, both of which are computed from
$d_{a}=\sqrt{2}ua$ and $d_{e}=\sqrt{2(\frac{1}{2}-u)^{2}a^{2}+(c/2)^{2}}$
sorantin1992 . There are two distances between neighboring equatorial O atoms,
$c$ and $\sqrt{2}(2u-1)a$, and normally they differ, so one does not have an
ideal octahedron.

Our ${\bf k}$ mesh is $29\times 29\times 45$, with 2760 points in the
irreducible Brillouin zone. The product of planewave cutoff $K_{\rm max}$ and
Muffin-tin radius $R_{\rm MT}$, $R_{\rm MT}K_{\rm max}$, determines the
accuracy of our calculation. Our $R_{\rm MT}K_{\rm max}$ is 9, which is more
than enough for our purpose. $R_{\rm MT}(Cr)$ is 1.89 a.u. and $R_{\rm MT}(O)$
is 1.67 a.u. We optimize the internal coordinates of atoms, and we compare our
results with the experimental ones in Table 1. We see that the agreement with
the experiment is excellent. Our theoretical spin moment for formula unit is
2.0 $\mu_{B}$, which agrees with the experimental result of 2.0 $\mu_{B}$
nearly perfectly and prior theoretical calculations schwarz1986 ; sorantin1992
.

Figure 1(a) displays the partial density of states (PDOS) for the majority
(positive value) and minority (negative value) spins. The solid line and
dotted line denote the Cr-$3d$ and O-$2p$ majority states, respectively. The
vertical dashed line represents the Fermi energy which is set at 0. One sees
that both Cr-$3d$ and O-$2p$ states cross the Fermi level, which is why $\rm
CrO_{2}$ is a metal. Each cell has 28 valence majority electrons and 24
minority electrons. The Fermi energy nearly cuts at the local minimum of
Cr-$3d$ states, stabilizing $\rm CrO_{2}$ sorantin1992 . The minority PDOS is
quite different (see the solid and dashed lines on the negative axis). At the
Fermi level, there is no DOS, with a band gap of 1.43 eV schwarz1986 . These
features are fully consistent with prior studies schwarz1986 ; sorantin1992 ;
mazin1999 . What is important for harmonic generation is that the valence band
is dominated by the O-$2p$ state, while the conduction band is dominated by
the Cr-$3d$ state. This constitutes an ideal case for dipole-allowed
transitions.

We find there is no major difference between left and right circularly
polarized light for our system, so in the following we use $\sigma^{+}$, with
the photon energy 0.4 eV and field amplitude $A_{0}=0.03$ Vfs/$\rm\AA$. Our
interested observable is the expectation value of the momentum operator,
$\langle{\bf P}^{\sigma}(t)\rangle=\langle\sum_{{\bf k};i,j}\rho_{{\bf
k};i,j}^{\sigma}{\bf P}_{{\bf k};j,i}^{\sigma}\rangle,$ where the summation is
over the Brillouin zone and band states. In our calculation, we include all
the states from Cr-$3s$ states (energy at $-4.89$ Ry for spin up and $-4.73$
Ry for spin down), Cr-$3p$ state (energy at $-2.79$ Ry for spin up and $-2.64$
Ry for spin down), O-$2s$ (-0.95 Ry), all the way up to 4.95 Ry, with 10 Ry
energy window width covering all the possible channels for excitation. To
compute harmonic spectra, we Fourier transform $\langle{\bf
P}^{\sigma}(t)\rangle$ into the frequency domain nc18 ; prl05 ; jia2020a ,

${\bf P}(\Omega)=\int_{-\infty}^{\infty}{\bf P}(t){\rm e}^{i\Omega t}{\cal
W}(t)dt,$ (3)

where ${\cal W}(t)$ is the window function nc18 .

Figure 1(b) shows harmonic signals from the majority (top) and minority
(bottom) spin channels, respectively. We see that all the harmonics appear at
odd orders because our system has an inversion symmetry. The majority spin
channel has a stronger harmonic signal than the minority spin channel, and it
has a broader peak. This difference is expected from the PDOS. Recall that the
minority band has an energy gap, so with the photon energy of 0.4 eV, the
excitation is virtual. The majority band has no such gap, so electrons in the
valence band can successively excite into conduction bands and then emit
harmonics.

We notice a crucial difference when we compare the 3rd harmonic signal with
the 5th harmonic signal from the majority and minority spin channels. The
difference in the 5th harmonic between the majority and minority spin channels
is much larger. A very weak 5th harmonic in the minority channel is
interesting because it potentially provides an opportunity to generate
harmonics from a single spin channel. We note that in a nonmagnetic material
jia2019 , there is no difference in harmonics between spin up and spin down
channels. In ferromagnets, this difference is also small because each channel
is able to generate harmonics with a small difference in strength nc18 ;
jia2020a ; jia2020b . A challenge in ferromagnetic metals is that the electron
deexcitation is extremely fast nc18 , so it is difficult to detect those
harmonics. CrO2 is very unique. It is known that laser-induced ultrafast
demagnetization lasts very long zhang2006 , partly because its spin-orbit
coupling is very weak. This prolonged process is advantageous to detect
harmonic signals.

Next we investigate how laser parameters affect harmonic generation in CrO2.
We increase the photon energy to 0.8 eV, while keeping the rest of laser
parameters unchanged. Figure 2(b) shows that the 5th harmonic in the minority
channel at 0.8 eV is slightly smaller than that in the majority channel, very
different from that at 0.4 eV (see Fig. 2(a)). Energetically, when the photon
energy $\hbar\omega$ is 0.4 eV, $5\hbar\omega=2$ eV is slightly above the gap
energy of 1.43 eV. Because a transition from the O-$2p$ to Cr-$3d$ does not
occur immediately at this band gap energy, one is still in the virtual
excitation regime. This situation is important because not all half metals
have this property. For instance, in another half-metal, Mn2RuGa does not have
this unique feature, so its harmonic generation from the majority channel is
similar to that in the minority channel. With $\hbar\omega=0.8$ eV,
$5\hbar\omega=4$ eV is well into the real excitation regime, so the band gap
suppression is no longer at work. This is further verified when we increase
the incident photon energy to 1.6 eV. Figure 2(c) shows that harmonic strength
in the minority and majority channels is almost exactly the same, so harmonics
are generated from both spin channels. Figure 2(d) compares the harmonic
signal ratio of the majority harmonic signal to the minority harmonic signal
as a function of incident photon energy. We see that in general higher order
harmonics have a much stronger variation. The ratio is larger when the
incident photon energy is small. At 0.4 eV, the fifth harmonic has a ratio of
67 with $A_{0}=\rm 0.03Vfs/\AA$. This photon energy window predicted here is
critical to future experiments.

To show that our prediction is not specific to a particular laser field
strength, we choose several vector potential amplitudes. We increase the
amplitude $A_{0}$ from 0.01 to 0.05 $\rm Vfs/\AA$ and include both the third
and fifth harmonics. Figure 3(a) displays the third harmonic signal as a
function of $A_{0}$ for spin up (filled circles) and spin down (filled boxes)
channels. The photon energy is 0.4 eV and the duration is also 64 cycles of
the laser period as above. We see that both signals increase superlinearly. We
fit the signal to a power function and find that the signal scales with
$A_{0}$ as $A_{0}^{2.37}$. According to the traditional nonlinear optics shen
, the polarization ${\cal P}$ is proportional to the response function $R(t)$
multiplied by the external field $E(t)$ mukamel , and for the third harmonic
${\cal P}^{(3)}\propto R^{(3)}(t)E(t)^{3}$. If $R^{(3)}(t)$ is significantly
modulated by band states, the scaling exponent differs from 3. This shows that
real excitations among band states contribute to the harmonic signal, which is
exactly what one expects from the density of states in the spin majority
channel (Fig. 1(a)). The spin minority channel is different. We see that the
signal scales as $A_{0}^{2.99}$ (filled boxes in Fig. 3(a)), with the exponent
close to 3. This again shows that the excitation is virtual, as expected.

Different excitations seen above also leave a crucial hallmark on the power
spectrum as well. Figure 3(b) shows the third harmonic signal as a function of
$A_{0}$, from 0.01 to 0.05 $\rm Vfs/\AA$, denoted respectively by the circles,
boxes, diamonds, up triangles and left triangles. We see that the peaks for
the spin up channel are not entirely symmetric with respect to the 3rd order.
This is an indication that harmonics are generated through real states (Fig.
1). In the spin down channel, the harmonic peak is highly symmetric (see Fig.
3(c)). In other words, the harmonic asymmetry is a hallmark for transitions
that involve real band states, which is crucial for electronic structure
detection in solids.

We can further demonstrate this in the fifth harmonic. Figure 3(d) shows the
5th harmonic signal as a function of $A_{0}$ for the majority spin (circles)
and minority spin (boxes) channels. As we increase $A_{0}$, the signal scales
as $A_{0}^{4.32}$, indicative of real transitions among band states. To reveal
further details in harmonics, we also plot the harmonic spectrum. Figure 3(e)
shows that the harmonic peak is not symmetric with respect to 5. All the
harmonic signals are multiplied by 100 for an easy view. In the spin down
channel, we note that the nominal 5th peak does not appear at 5 (Fig. 3(f)),
where harmonic signals are multiplied by 1000 for an easy view. This is
because the main excitation occurs above 2 eV, which is fully consistent with
our observation in the photon energy dependence in Fig. 2. In comparison with
the majority signal, the minority signal is extremely weak even at 0.05 $\rm
Vfs/\AA$. Quantitatively, we find the ratio between the majority and minority
5th order signals is 46, or about 2%.

To establish our results on a solid ground, we carry out several additional
tests. We first check our harmonic signal convergence with the Brillouin zone
sampling by increasing the number of $k$ points from $18\times 18\times 28$,
$19\times 19\times 30$, to $29\times 29\times 45$, and we find the results are
well converged. For all the results in this paper, we use a ${\bf k}$ mesh of
$29\times 29\times 45$, with 2760 points in the irreducible Brillouin zone, or
10830 in the full Brillouin zone. Secondly, we examine the influence of
relaxation times $T_{1}$ and $T_{2}$ on the harmonic signal. $T_{2}$ is
normally shorter than $T_{1}$, but their precise values are unknown, so to
this end, we take $T_{1}=200$ fs and $T_{2}=100$ fs. Extremely smaller $T_{1}$
and $T_{2}$ were used before in ZnO vampa2014 . Figures 4(a)-(d) show the
results for the same laser parameters used in Fig. 1(a), but with $T_{1}=400$
fs and $T_{2}=200$ fs. Their influence on our harmonic signal is very weak
since our laser pulse duration is on the order of 200 fs. When we increase
$T_{1}$ and $T_{2}$ from 200 and 100 fs to 400 and 200 fs, the fifth order
harmonic changes from $0.36\times 10^{-2}$ to $0.50\times 10^{-2}$ for the
majority channel. The change is even smaller for the minority band, and it
only changes from $0.55\times 10^{-4}$ to $0.52\times 10^{-4}$ for the
minority channel in the same unit. This huge difference between the spin
majority and minority channels remains, regardless of the $x$ (Figs. 4(a) and
4 (c)) or $y$ (Figs. 4(b) and (d) or Figs. 4(e) and 4(f)) or $z$ component
(Figs. 4(f) and 4(h)). Since $\rm CrO_{2}$ is tetragonal, we wonder whether
the laser polarization in the $yz$ plane ($\sigma_{yz}$), instead of the $xy$
plane ($\sigma_{xy}$), makes a difference. Figures 4(e)-(h) show our results
with the same relaxation times as Figs. 4(a)-(d). We find that for the
majority band, the $y$ component maximum of the 5th harmonic signal is
similar, but the line shape is slightly different. Under $\sigma_{yz}$
excitation, the peak appears more symmetric (see Fig. 4(e)) and has no
structure in it, while under $\sigma_{xy}$ excitation, the peak has more
structures. This difference is due to the different band states involved
during harmonic generation. This potentially provides another interesting
direction for experimental investigation as a function of laser polarization
angle, in a similar way as done in MgO you2017 . For the minority band, the
difference is much smaller (see Figs. 4(g) and 4(f)). Importantly, the ratio
of the majority 5th harmonic to that of the minority remains the same. This
concludes that the fifth order harmonic emission is mainly from the majority
channel. This paves the way to future experimental detection of harmonics from
a single spin channel. We believe that the results found here are interesting
and should motivate future experimental and theoretical investigations.

##  IV conclusion

We have investigated harmonic generation from half-metallic $\rm CrO_{2}$.
This nearly 100% spin polarized material shows peculiar features that are not
found in other materials. The majority spin channel has no energy gap just
like a metal, but in the minority channel has 1.4-eV energy gap. Upon laser
excitation, this leads to a significant difference in harmonic generation in
these two channels. This difference depends on the harmonic order and photon
energy $\hbar\omega$ used. The higher the order is, the larger the effect is.
At $\hbar\omega=0.4$ eV and the fifth harmonic, we find that the ratio of the
majority channel signal to the minority signal exceeds 60. This makes the 5th
harmonic as if it is emitted nearly purely from a single spin channel.
Microscopically, in the minority channel, transitions are mainly from the
O-$2p$ states to Cr-$3d$ states, so the onset of single spin channel harmonics
starts at 2 eV. This explains why the fifth order shows such a strong effect.
In general, it is known that harmonics are less sensitive to material
properties. Here, we show that when a material has a peculiar difference in
spin channels, harmonics can be very powerful. Our prediction, if confirmed
experimentally, potentially represents a new direction for high harmonic
generation for solids ghimire2011 ; luu2015 ; garg2016 ; nd ; neufeld2019 .

###### Acknowledgements.

This work was solely supported by the U.S. Department of Energy under Contract
No. DE-FG02-06ER46304. Part of the work was done on Indiana State University’s
high performance Quantum and Obsidian clusters. The research used resources of
the National Energy Research Scientific Computing Center, which is supported
by the Office of Science of the U.S. Department of Energy under Contract No.
DE-AC02-05CH11231<EMAIL_ADDRESS>https://orcid.org/0000-0002-1792-2701

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Table 1: Optimized structural parameters of $\rm CrO_{2}$. Cr atoms are at $(2a)$ positions, (0,0,0) and $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$, and O atoms at $(4f)$ positions, ($u$,$u$,0), ($\bar{u}$,$\bar{u}$,0), ($\frac{1}{2}+u,\frac{1}{2}-u,\frac{1}{2}$), and $(\frac{1}{2}-u,\frac{1}{2}+u,\frac{1}{2})$. Our theoretical values are listed under “Theory”, with the experimental ones in parenthesis. Our theoretical $u$ is 0.303, in good agreement with the experimental $u=0.301\pm 0.004$ cloud1962 . The experiment lattice constants are $a=4.4218~{}\rm\AA$ and $c=2.9182~{}\rm\AA$ cloud1962 , $a=4.423~{}\rm\AA$ and $c=2.918~{}\rm\AA$ singh2009 , $a=4.421~{}\rm\AA$ and $c=2.916~{}\rm\AA$ pathak2009 , and $a=4.421~{}\rm\AA$ and $c=2.916~{}\rm\AA$ thamer1956 . We adopt the experimental lattice constant from Cloud et al. cloud1962 .  Atom | $x$ | $y$ | $z$
---|---|---|---
|  Theory | Experiment |  Theory | Experiment |  Theory | Experiment
Cr1 | 0 | (0) | 0 | (0) | 0 | (0)
Cr2 | $\frac{1}{2}$ | ($\frac{1}{2}$) | $\frac{1}{2}$ | ($\frac{1}{2}$) | $\frac{1}{2}$ | ($\frac{1}{2}$)
O1 | 0.303 | (0.301) | 0.303 | (0.301) | 0 | (0)
Figure 1:  (a) The partial density of states (PDOS) for Cr-$3d$ and O-$2p$
states for the majority (thick solid and dotted lines on the positive axis)
and minority spins (solid and dashed lines on the negative axis). The vertical
dashed line denotes the Fermi energy. (b) Logarithmic scale of harmonics from
the majority and minority spin channels. Here we use the laser duration of 64
cycles of laser period. The photon energy is 0.4 eV and the laser field
amplitude $A$ is 0.03 $\rm Vfs/\AA$. A circularly polarized pulse is used.
Figure 2:  Dependence of harmonics on photon energies $\hbar\omega$ (a) 0.4
eV, (b) 0.8 eV and (c) 1.6 eV. We choose the vector potential $A_{0}=0.03\rm
Vfs/\AA$ and the duration of the laser pulse is 64 cycles of laser period.
Harmonics from the minority channel are plotted on the negative axis. (a) At
0.4 eV, the majority and minority spin channels produce a similar signal
strength at the 1st and 3rd orders, but differ a lot at the 5th order, where
the majority dominates over the minority spin channel. This presents an
opportunity to generate harmonic mainly from a single spin channel. As we
increase photon energy to 0.8 eV (b), the minority 5th harmonic increases its
signal strength, so it starts to compete with that from the majority channel.
This trend continues as we increase the photon energy to 1.6 eV (c), where the
majority and minority channels have a similar strength. (d) Ratio of the
majority harmonic signal to the minority harmonic signal as a function of
photon energy for the first (circles), third (squares) and fifth (diamonds)
harmonics. At 0.4 eV, the ratio for the fifth harmonic reaches 67.  Figure 3:
Dependence of harmonic signals on the laser field amplitude $A_{0}$. The
photon energy is 0.4 eV, and pulse duration is 64 cycles of the laser period.
Here we use a dense $k$ mesh of $29\times 29\times 45$. (a) Third-order
harmonics as a function of $A_{0}$ for spin up (filled circles) and spin down
(filled boxes), which are fitted to a power of $A_{0}^{2.37}$ and
$A_{0}^{2.99}$. (b) Third harmonic from the spin up channel. The circles,
boxes, diamonds, upper triangles and left triangles denote $A_{0}=0.01$, 0.02,
0.03, 0.04 and 0.05 $\rm Vfs/\AA$, respectively. (c) Third harmonic from the
spin down channel. All the legends have the same meaning as those in (b). (d)
Fifth-order harmonics as a function of $A_{0}$ for spin up (filled circles)
and spin down (filled boxes). The spin up signal can be fitted to a power of
$A_{0}^{4.32}$. (e) The fifth harmonic from the spin up channel. Here all the
signals are multiplied by 100 for an easy view. All the legends have the same
meaning as those in (b). (f) Fifth harmonic from the spin down channel. Here
all the signals are multiplied by 1000. All the legends have the same meaning
as those in (b).  Figure 4: Influence of relaxation times and laser
polarization on harmonics. (a)-(d) use a different set of relaxation times,
$T_{1}=400$ and $T_{2}=200$ fs with laser polarization within the $xy$ plane.
(a) and (c) show the majority and minority harmonic signals along the $x$
direction respectively, which should be compared with those in Fig. 1(b) with
$T_{1}=200$ and $T_{2}=100$ fs. (b) and (d) show the majority and minority
harmonic signals along the $y$ direction, respectively. (e)-(h) use the laser
polarization in the $yz$ plane with $T_{1}=400$ and $T_{2}=200$ fs. (e) and
(g) show the majority and minority harmonic signals along the $y$ direction,
respectively. (f) and (h) show the majority and minority harmonic signals
along the $z$ direction, respectively. For all the figures, the photon energy
is 0.4 eV, the pulse duration is 64 cycles of the laser period and the laser
field amplitude $A_{0}$ is 0.03 $\rm Vfs/\AA$.