# Charge pumping with strong spin-orbit coupling:
Fermi surface breathing and higher harmonic generation

Aurélien Manchon  email<EMAIL_ADDRESS>Armando Pezo  Aix-
Marseille Université, CNRS, CINaM, Marseille, France

###### Abstract

Spin and charge pumping induced by a precessing magnetization has been
instrumental to the development of spintronics. Nonetheless, most theoretical
studies so far treat the spin-orbit coupling as a perturbation, which
disregards the dynamical competition between exchange and spin-orbit fields.
In this work, based on Keldysh formalism and Wigner expansion, we develop an
adiabatic theory of spin and charge pumping adapted to multiorbital systems
with arbitrary spin-orbit coupling. We apply this theory to the magnetic
Rashba gas and magnetic graphene cases and discuss the pumped ac and dc
current. We show that the pumped current possesses both intrinsic and
extrinsic contributions, akin to the magnetic damping. In addition, we find
that higher harmonics can be generated under large angle precession and we
propose a couple of experimental setups where such an effect could be
experimentally observed.

##  I Introduction

Adiabatic spin pumping [1, 2] has been instrumental to the development of
spintronics over the past two decades. It is now routinely used to inject pure
spin currents from a magnetic spin source into an adjacent metal, enabling the
investigation of spin-to-charge interconversion processes in a wide range of
materials, from transition metal compounds [3] to two dimensional gases [4],
oxide heterostructures [5], topological surface states [6, 7], van der Waals
heterostructures (see for instance Ref. [8]) but also other magnetic materials
such as antiferromagnets [9] and spin liquids [10]. Although the magnetic spin
source is usually a ferromagnet excited at magnetic resonance, the recent
demonstration of spin-to-charge interconversion using antiferromagnetic
resonance [11, 12] opens appealing avenues for the generation of very high
frequency currents via spin pumping. In the standard theory of spin pumping
[1, 2], the spin current induced by the precessing magnetization ${\bf m}$ and
injected in the adjacent metal reads

$\displaystyle{\cal J}_{s}=\eta_{r}{\bf m}\times\partial_{t}{\bf
m}+\eta_{i}\partial_{t}{\bf m},$ (1)

where $\eta_{r,i}$ are coefficients related to the spin mixing conductance at
the interface between the magnet and the nonmagnetic metal, and to the spin
relaxation in the metal [13, 14]. When spin-orbit coupling is present in the
metal, the spin current is converted into a charge current that takes the
general form

$\displaystyle{\bf J}_{c}=\alpha_{H}\eta_{r}{\bf z}\times({\bf
m}\times\partial_{t}{\bf m})+\alpha_{H}\eta_{i}{\bf z}\times\partial_{t}{\bf
m},$ (2)

where ${\bf z}$ is normal to the interface and $\alpha_{H}$ is the spin-to-
charge conversion efficiency, proportional to the spin-orbit coupling
strength, and whose specific structure depends on the involved mechanism (spin
Hall effect, Rashba-Edelstein effect, possibly spin swapping etc.). Equation
(2) is widely used to interpret experimental data and quantify the physical
parameters such as the spin-mixing conductance itself, the spin-to-charge
conversion efficiency and the spin relaxation length [3, 15, 16, 17, 18].
Notice that, to date, the vast majority of experiments have focused on the
time-averaged, rectified part of the pumped charge current ${\bf J}_{c}|_{\rm
dc}=\alpha_{H}\eta_{r}\langle{\bf z}\times({\bf m}\times\partial_{t}{\bf
m})\rangle$, and only a handful of them have achieved to measure the ac
contribution [19, 20, 21].

Figure 1: (Color online) (a) Sketch of the nonlinear spin dynamics expected
when a strong Rashba spin-orbit coupling coexists with s-d exchange. (b) Band
structure of the magnetic Rashba gas for ${\bf m}={\bf y}$ (red) and $-{\bf
y}$ (black) for (top) $\Delta=0.1t$, $t_{R}=0.1t$ and (bottom) $\Delta=1t$,
$t_{R}=0.3t$. (c) Corresponding Fermi surface computed at $E_{F}=-1t$, showing
strong ”breathing” effect upon magnetization reorientation. In these
calculations, $t=0.2$.

An important shortcoming of the standard theory of spin pumping is that it
formally applies in the presence of vanishingly small spin-orbit coupling
compared to the s-d exchange between conduction and localized electrons (about
1-2 eV in Fe, Co, Ni compounds). In fact, in most experiments the adjacent
metal rather possesses a large spin-orbit coupling, i.e., a few 100 meV (heavy
metals, topological insulators and Weyl semimetals, to name a few). In other
words, the spin mixing conductance approach is not adapted to treat these
systems and overlooks the competition between exchange and spin-orbit
interactions. As a matter of fact, in noncentrosymmetric multilayers
interfacial spin-orbit splitting adopts the form of Rashba spin-orbit coupling
[22], which substantially modifies the spin dynamics at the interface. As
sketched in Fig. 1(a), the time-dependent current ${\bf j}_{c}(t)$ pumped by
the precessing magnetization is accompanied by the so-called Rashba field,
${\bf B}_{R}(t)\propto{\bf z}\times{\bf j}_{c}(t)$, which competes with the
s-d exchange to drag the itinerant spin away from the magnetization. Since
this Rashba field is itself proportional to the charge current, one naturally
expects a massive modification of the spin dynamics in the limit of strong
spin-orbit coupling. In a recent paper, Chen and Zhang [23] proposed a Green’s
function approach to compute the induced spin current in the presence of spin-
orbit coupling. Unfortunately, this theory is limited to small angle
precession such that the spin-orbit dynamics is overlooked. In a recent work
[24], we recently performed time-dependent quantum transport simulations and
reported the progressive emergence of higher harmonics upon increasing the
strength of Rashba spin-orbit coupling. This result, confirmed by another work
[25], clearly advocates for the presence of nonlinear itinerant spin dynamics
but its lack of transparency hinders a precise understanding of the underlying
physics.

In the present work, we address this problem by adopting a different
theoretical approach. Based on Keldysh formalism we first derive a formula for
the charge pumping that is valid in the slow dynamics regime and, most
importantly, valid for the full range of spin-orbit coupling and exchange
interaction. We then apply this formalism to the paradigmatic magnetic Rashba
gas and magnetic graphene, and demonstrate that the harmonic generation is a
direct consequence of the Fermi surface breathing. We then suggest several
materials systems and configurations in which the harmonic generation could be
observed.

##  II Adiabatic pumping and Fermi surface breathing

###  II.1 Keldysh theory of adiabatic pumping

Let us start from Keldysh-Dyson equation [26] in Wigner representation, i.e.,
only the macroscopic coordinates (time and position) of the center-of-mass of
the wave packet are treated explicitly while its microscopic internal degrees
of freedom are Fourier transformed (see, e.g., Ref. [27]). In the present
theory, we are interested in deriving the response of an observable ${\cal
O}$, expressed through the lesser Green’s function $G^{<}_{\bf k}$, to the
first order in magnetization dynamics, $\partial_{t}{\cal H}_{\bf k}$.
Keldysh-Dyson equations can be rewritten

$\displaystyle(\varepsilon-{\cal H}_{\bf k}-\Sigma^{R})\otimes G^{R}_{\bf
k}=1,$ (3) $\displaystyle G^{<}_{\bf k}=G^{R}_{\bf k}\otimes\Sigma^{<}\otimes
G^{A}_{\bf k},$ (4)

where ${\cal H}_{\bf k}$ is the unperturbed Hamiltonian, $\otimes={\rm
exp}[i\hbar(\partial_{t}\partial_{\varepsilon}-\partial_{\varepsilon}\partial_{t})]$
is Moyal’s product that emerges from Wigner transform, and
$\Sigma^{R,<}=n_{i}V_{0}^{2}\int d^{3}{\bf k}/(2\pi)^{3}G_{\bf k}^{R,<}$ is
the (retarded, lesser) self-energy in the presence of delta-like impurities
with potential $V_{0}$ and density $n_{i}$. We now expand these two equations
to the first order in $\partial_{t}{\cal H}_{\bf k}$ and after some algebra,
one obtains

$\displaystyle G^{<}_{\bf k}$ $\displaystyle=$ $\displaystyle G^{R}_{\bf
k}\Sigma^{<}G^{A}_{\bf k}-\frac{i\hbar}{2}\left(G^{R}_{{\bf
k}0}\partial_{t}{\cal H}_{\bf k}\partial_{\varepsilon}G^{A}_{{\bf
k}0}-\partial_{\varepsilon}G^{R}_{{\bf k}0}\partial_{t}{\cal H}_{{\bf
k}}G^{A}_{{\bf k}0}\right),$ $\displaystyle G^{R}_{\bf k}$ $\displaystyle=$
$\displaystyle G^{R}_{{\bf k}0}-\frac{i\hbar}{2}\left(G^{R}_{{\bf
k}0}\partial_{t}{\cal H}_{{\bf k}}\partial_{\varepsilon}G^{R}_{{\bf
k}0}-\partial_{\varepsilon}G^{R}_{{\bf k}0}\partial_{t}{\cal H}_{{\bf
k}}G^{R}_{{\bf k}0}\right)$ (6)

We then simply insert Eq. (6) into Eq. (LABEL:eq:keldysh2), and compute the
response of an observable ${\cal O}$ as

$\displaystyle{\cal O}=\int\frac{d\varepsilon}{2i\pi}{\rm Tr}_{\bf
k}[\hat{O}G^{<}_{\bf k}].$ (7)

Here, ${\rm Tr}_{\bf k}[...]=\int\frac{d^{3}{\bf k}}{(2\pi)^{3}}{\rm
Tr}[...]$. By posing ${\cal O}=\delta{\cal O}_{j}\partial_{t}m_{j}$, and
$\partial_{t}{\cal H}_{\bf k}=-{\bf T}_{\bf k}\cdot\partial_{t}{\bf m}$, where
${\bf T}_{\bf k}=-\partial_{\bf m}{\cal H}_{\bf k}$ is the torque operator, we
obtain two contributions

$\displaystyle\delta{\cal O}^{\rm
surf}_{j}=\hbar\int\frac{d\varepsilon}{2\pi}{\rm ReTr}_{\bf
k}[\hat{O}(G^{R}_{{\bf k}0}-G^{A}_{{\bf k}0})T_{\bf k}^{j}G^{A}_{{\bf
k}0}]\partial_{\varepsilon}f(\varepsilon),$ (8) $\displaystyle\delta{\cal
O}^{\rm sea}_{j}=\hbar\int\frac{d\varepsilon}{2\pi}{\rm ReTr}_{\bf
k}[\hat{O}G^{A}_{{\bf k}0}[T_{\bf k}^{j},G^{A}_{{\bf k}0}]G^{A}_{{\bf
k}0}]f(\varepsilon).$ (9)

These two expressions represent the Fermi surface and Fermi sea contributions
to the adiabatic pumping. It is noteworthy to point out that this theory
generalizes the theory of the magnetic damping proposed by Gilmore et al.
[28], in which intrinsic and extrinsic electronic contributions to the
magnetic damping are discussed. In our formalism, the magnetic damping can be
computed simply by replacing $\hat{O}$ by the torque operator, resulting in
the torque-torque correlation introduced by Kambersky [29]. These formulas are
valid for slow dynamics but, most importantly, remain exact for all values of
spin-orbit coupling and exchange, as well as for any direction of the
magnetization. It is also well adapted to multiband systems and
heterostructures and can be used to compute spin, charge and orbital pumping
in realistic materials.

###  II.2 Pumping and Fermi surface breathing

Let us now evaluate the charge current pumped by the slowly precessing
magnetization. To do so, we consider a magnetic Rashba gas on a hexagonal
lattice, defined by the Hamiltonian [30]

$\displaystyle{\cal H}_{0}=\varepsilon_{\bf
k}+\Delta\hat{\bm{\sigma}}\cdot{\bf m}+\frac{t_{\rm R}}{a}{\bm{\eta}}_{\bf
k}\cdot(\hat{\bm{\sigma}}\times{\bf z}).$ (10)

The sum is taken over the nearest neighbors, i.e., ${\bf u}={\bf a},{\bf
b},-{\bf a}-{\bf b}$, ${\bf a}$ and ${\bf b}$ being the lattice vectors and
$a$ the lattice parameter. Details can be found in Ref. [30]. Explicitly,
$\varepsilon_{\bf k}=-2t(\cos{\bf k}\cdot{\bf a}+\cos{\bf k}\cdot{\bf
b}+\cos{\bf k}\cdot{\bf c})$, and ${\bm{\eta}}_{\bf k}=2({\bf a}\sin{\bf
k}\cdot{\bf a}+{\bf b}\sin{\bf k}\cdot{\bf b}+{\bf c}\sin{\bf k}\cdot{\bf
c})$. Here, $t$ is the nearest-neighbor hopping parameter (fixed to 0.2 in the
present work), $\Delta$ is the s-d exchange between the conduction electrons
and the localized ones, $t_{R}$ is the linear Rashba spin-orbit coupling
coming from inversion symmetry breaking normal to the (a, b) plane. Because of
the coexistence of s-d exchange and spin-orbit coupling, the band structure
and Fermi surface of the gas are highly sensitive to the magnetization
direction, as shown in Fig. 1(b) and (c), respectively. These figures show the
band structure and Fermi surface when the magnetization lies along ${\bf y}$
(red) and $-{\bf y}$ (black) for two different situations, $\Delta=0.1$ and
$t_{R}=0.1$ (top) and $\Delta=1$ and $t_{R}=0.3$ (bottom). One immediately
sees that in the adiabatic regime, the high sensitivity of the Fermi surface
to the magnetization direction results in a so-called ”breathing”, i.e., the
periodic modulation of the Fermi surface driven by the precessing
magnetization [31] (short movies of the breathing can be found in Ref. [32]).
During the breathing, states of opposite spin chirality are pumped from one
side of the Fermi surface to the other, resulting in a periodic charge
current. Notice that this Fermi surface breathing is also at the origin of
electron-mediated magnetic damping [33, 34, 28]. Interesting, whereas the
Fermi surface remains mostly circular for the first set of parameters [top
panels in Fig. 1(c)], there is a region in parameter space when the distortion
is much more dramatic [bottom panels in Fig. 1(c)], manifesting the drastic
competition between spin-orbit coupling and exchange. As discussed in the next
sections, in this particular regime, current harmonics can emerge.

##  III dc and ac charge pumping

###  III.1 Preliminary insights

Without loss of generality, we consider a magnetization precessing around the
${\bf x}$ axis (see Fig. 2), ${\bf
m}=(\cos\theta,\sin\theta\sin\phi,-\sin\theta\cos\phi)$, with $\phi=\omega t$.
In the standard case of a ferromagnet/nonmagnetic metal bilayer, studied in
most of the literature [3, 15, 16, 17, 18], the charge pumping is due to the
spin Hall effect in the nonmagnetic metal and computed by combining
magnetoelectronic circuit theory with drift-diffusion equation [1, 2]. In this
case, the charge current reads

$\displaystyle{\bf
J}_{c}\approx{\tilde{\alpha}}_{N}\frac{{\tilde{\lambda}}_{N}}{d_{N}}(2{\tilde{G}}^{r}_{\uparrow\downarrow}{\bf
z}\times({\bf m}\times\partial_{t}{\bf
m})+2{\tilde{G}}^{i}_{\uparrow\downarrow}{\bf z}\times\partial_{t}{\bf m}),$
(11)

where

$\displaystyle 2{\tilde{G}}^{r,i}_{\uparrow\downarrow}$ $\displaystyle=$
$\displaystyle\frac{2G^{r,i}_{\uparrow\downarrow}}{1+\frac{2G_{\uparrow\downarrow}\tilde{\lambda}_{N}}{\sigma_{N}}},\;{\tilde{\lambda}}_{N}=\frac{\lambda_{N}}{\tanh\frac{d_{N}}{\lambda_{N}}},$
(12) $\displaystyle{\tilde{\alpha}}_{N}$ $\displaystyle=$
$\displaystyle\alpha_{N}\left(1-\cosh^{-1}\frac{d_{N}}{\lambda_{N}}\right).$
(13)

$G^{r,i}_{\uparrow\downarrow}$ are the real and imaginary parts of the spin
mixing conductance, and $\alpha_{N}$, $\lambda_{N}$, $d_{N}$ and $\sigma_{N}$
are the spin Hall angle, spin relaxation length, thickness and conductivity of
the nonmagnetic metal. For the precessing magnetization adopted in our
calculations and considering
${G}^{r}_{\uparrow\downarrow}\gg{G}^{i}_{\uparrow\downarrow}$ [35],

$\displaystyle{\bf J}_{c}^{dc}$ $\displaystyle=$
$\displaystyle\omega{\tilde{\alpha}}_{N}\frac{{\tilde{\lambda}}_{N}}{d_{N}}2{\tilde{G}}^{r}_{\uparrow\downarrow}\sin^{2}\theta{\bf
y},$ (14) $\displaystyle{\bf J}_{c}^{ac}$ $\displaystyle=$
$\displaystyle\omega{\tilde{\alpha}}_{N}\frac{{\tilde{\lambda}}_{N}}{d_{N}}\left(2{\tilde{G}}^{r}_{\uparrow\downarrow}\frac{\sin
2\theta}{2}\sin\omega
t+2{\tilde{G}}^{i}_{\uparrow\downarrow}\sin\theta\cos\omega t\right){\bf x}.$

As mentioned in the introduction, this theory gives the well-known result that
a dc current is injected transverse to the precession axis, Eq. (14), and an
ac current is generated along it, Eq. (LABEL:eq:spt2). Let us now turn our
attention to the case of a magnetic Rashba gas.

Although the scattering matrix formalism used in Refs. [1, 2] is well adapted
to current-perpendicular-to-plane geometries, it is not suited to current-in-
plane geometries. The charge pumping can be readily computed using the
semiclassical wave packet formalism developed by Sundaram and Niu [36]. In
this approach, the group velocity of state $n$, ${\bf v}_{n}$, is associated
with the time-momentum Berry curvature ${\bm{\Omega}}^{n}_{t{\bf k}}$ and
reads

$\displaystyle{\bf v}_{n}={\bm{\Omega}}^{n}_{t{\bf k}}=2i{\rm
Im}\left[\langle\partial_{t}u_{n}|\partial_{\bf k}u_{n}\rangle\right],$ (16)

where $|u_{n}\rangle$ is the periodic part of the Bloch state. In the limit of
slow dynamics, the spin remains aligned on the effective field due to exchange
and spin-orbit coupling and because of spin-momentum locking, the wave
function acquires a geometrical phase which results in adiabatic charge
pumping. The induced intrinsic current is therefore ${\bf J}_{c}^{\rm
int}=-e\sum_{n}\int d^{2}{\bf k}/(2\pi)^{2}{\bm{\Omega}}^{n}_{t{\bf
k}}f(\varepsilon_{\bf k}^{n})$. Let us now compute this current in the case of
the free magnetic Rashba electron gas, defined by the Hamiltonian

$\displaystyle{\cal
H}=\frac{\hbar^{2}k^{2}}{2m}+\Delta\hat{\bm{\sigma}}\cdot{\bf m}+\alpha_{\rm
R}\hat{\bm{\sigma}}\cdot({\bf p}\times{\bf z}),$ (17)

where $\alpha_{\rm R}$ is the Rashba strength. Equation (17) can be derived
directly from Eq. (10) taking $|{\bf k}|\ll\pi/a$. The eigenstates are

$\displaystyle|+\rangle=\left(\begin{matrix}-e^{-i\phi_{k}}\cos\frac{\theta_{k}}{2}\\\
\sin\frac{\theta_{k}}{2}\end{matrix}\right),\;|-\rangle=\left(\begin{matrix}e^{-i\phi_{k}}\sin\frac{\theta_{k}}{2}\\\
\cos\frac{\theta_{k}}{2}\end{matrix}\right),$ (18)
$\displaystyle\varepsilon_{\bf
k}^{n}=\frac{\hbar^{2}k^{2}}{2m}+n\lambda_{k},\;n=\pm 1,$ (19)

with

$\displaystyle\lambda_{k}$ $\displaystyle=$
$\displaystyle\sqrt{\Delta^{2}+\alpha_{\rm R}^{2}k^{2}+2\Delta\alpha_{\rm
R}({\bf p}\times{\bf z})\cdot{\bf m}}$ (20) $\displaystyle\cos\theta_{k}$
$\displaystyle=$ $\displaystyle-\frac{\Delta}{\lambda_{k}}\sin\theta\cos\phi,$
(21) $\displaystyle\tan\phi_{k}$ $\displaystyle=$
$\displaystyle\frac{\Delta\sin\theta\sin\phi-\alpha_{\rm
R}k_{x}}{\Delta\cos\theta+\alpha_{\rm R}k_{y}}.$ (22)

With these definitions, the Berry curvature for band $n$ reads

$\displaystyle{\bm{\Omega}}_{t{\bf k}}^{n}=n(\partial_{\bf
k}\theta_{k}\partial_{t}\phi_{k}-\partial_{\bf
k}\phi_{k}\partial_{t}\theta_{k})\frac{\sin\theta_{k}}{2}.$ (23)

After some algebra, we find that the intrinsic pumped current density reads,
to the lowest order in Rashba strength $\alpha_{\rm R}$,

$\displaystyle{\bf j}_{p}=-\frac{e\omega}{\lambda_{\rm R}}\left(\frac{\sin
2\theta}{2}\sin\omega t{\bf x}+\sin^{2}\theta{\bf y}\right),$ (24)

where $\lambda_{\rm R}=\hbar/(\alpha_{\rm R}m)$ is the Rashba precession
length. Equation (24) shows that a dc current is injected transverse to the
precession axis and an ac current is generated along it, which is expected
from the standard theory, Eqs. (14)-(LABEL:eq:spt2). The very same angular
dependence is obtained. We emphasize that this expression is correct in the
limit of small spin-orbit coupling and only accounts for the intrinsic
contribution to the charge pumping, disregarding the Fermi surface breathing.

###  III.2 Charge pumping in the magnetic Rashba gas

We now compute the charge current pumped by the precessing magnetization using
our theory. With the definition of ${\bf m}$ given in the previous section,
the torque operator is ${\bf T}\cdot\partial_{t}{\bf
m}=\omega\Delta\sin\theta\alpha(t)$, with $\alpha(t)=\sigma_{y}\sin\omega
t+\sigma_{z}\cos\omega t$. Therefore, the charge current reads

$\displaystyle{\bf J}_{c}^{\rm surf}=-\omega\sin\theta\frac{\Delta}{2\pi}{\rm
ReTr}_{\bf k}[{\bf
v}(G^{R}_{0}-G^{A}_{0})\alpha(t)G^{A}_{0}]\partial_{\varepsilon}f(\varepsilon),$
(25) $\displaystyle{\bf J}_{c}^{\rm
sea}=\omega\frac{\Delta}{2\pi}\sin\theta\int d\varepsilon{\rm ReTr}_{\bf
k}[{\bf v}G^{A}_{0}[\alpha(t),G^{A}_{0}]G^{A}_{0}]f(\varepsilon).$ (26)

Here we defined ${\rm Tr}_{\bf k}=\int\frac{d^{2}{\bf k}}{(2\pi)^{2}}{\rm
Tr}$. In the limit of vanishing disorder, Eq. (26) is simply the intrinsic
current induced by the time-momentum Berry curvature, Eq. (16). Equation (25)
has not been derived before though, and is associated with the Fermi surface
breathing, as already mentioned.

We start our investigation by adopting the set of parameters $\Delta=0.1$,
$t_{R}=0.1$, for which the Fermi surface remains mostly circular (see Fig. 1).
The pumped current components computed for the Rashba gas are displayed in
Fig. 2(b-e) and several remarks are in order. First, the signal of all four
current components increases steadily with the cone angle, which is expected.
Second, whereas Eqs. (14)-(LABEL:eq:spt2) predict an oscillatory current only
along ${\bf x}$, our calculations predict that in the case of the Rashba gas
an oscillatory current develops along ${\bf y}$, dominated by the extrinsic
contribution, i.e., the Fermi surface breathing [Fig. 2(e)]. This distinct
feature can be traced back to the competition between the exchange and the
Rashba field, depicted in Fig. 1, and which results in a time-dependent
modulation of the effective field ${\bf B}_{\rm eff}$ along ${\bf x}$ that
produces the oscillatory current along ${\bf y}$. Notice also that the
intrinsic current displays a cosine dependence whereas the extrinsic current
displays a sine dependence. This phase shift between intrinsic and extrinsic
currents is also present in the conventional theory, Eq. (LABEL:eq:spt2),
where $J_{c,x}^{ac}\propto{\tilde{G}}^{r}_{\uparrow\downarrow}\sin\omega
t+{\tilde{G}}^{i}_{\uparrow\downarrow}\cos\omega t$. Finally, we find that the
extrinsic contribution to the pumped current is much larger than the intrinsic
one, typically two orders of magnitudes in Fig. 2(b-e). The extrinsic
contribution is, by definition, inversely proportional to the disorder
broadening $\Gamma$, which in our model is a tunable parameter, fixed to
$\Gamma=0.1$ eV. Decreasing this parameter would lead to an enhancement of the
extrinsic contribution. The intrinsic contribution, on the other hand, is
related to the time-momentum Berry curvature, and is therefore expected to be
very sensitive to avoided crossing points in the band structure, akin to
anomalous Hall effect (see, for instance discussions in Ref. [37]). Therefore,
the relative magnitude of the intrinsic and extrinsic contributions is
therefore not only band structure-dependent but also disorder-dependent, which
opens particularly appealing avenues for charge pumping engineering in quantum
materials, such as magnetic Weyl semimetals.

Figure 2: (Color online) (a) Sketch of the charge pumping configuration. Time-
dependence of the intrinsic (b,c) and extrinsic (d,e) currents as a function
of the cone angle, $\theta\in[5^{\circ},65^{\circ}]$. Both $J_{x}$ (d,e) and
$J_{y}$ (b,c) are displayed and the parameters are $\Delta=0.1$, $t_{R}=0.1$,
$\Gamma=0.1$ and $E_{F}=-1$.

We now investigate the influence of the relative strength of the Rashba and
exchange interactions on the dc (Fig. 3) and ac (Fig. 4) currents. The dc
current displays two interesting features. First, the overall magnitude of the
pumped current increases with the exchange $\Delta$, which is expected from
the linear response theory. Second, and more interestingly, the pumped dc
current depends on the Rashba strength $t_{R}$ in a nontrivial manner. At
small Rashba strength, both intrinsic and extrinsic contributions first
increase steadily before reaching a maximum. Once the maximum is reached, the
magnitude of the pumped dc current decreases, switches sign and increases
again. The Rashba strength at which the sign reversal occurs decreases when
increasing the exchange, which is particularly clear for the intrinsic
contribution [Fig. 3(a)]. Finally, at large Rashba strength, the pumped
current saturates. This behavior is obviously in stark contrast with the
conventional theory of spin pumping and illustrates the complex interplay
between exchange, Rashba field and spin-to-charge conversion.

Figure 3: (Color online) Dependence of the (a) intrinsic and (b) extrinsic dc
current along ${\bf y}$ as a function of the Rashba and exchange interactions.
The parameters are $\Delta\in[0.1,1]$, $t_{R}\in[0,1.3]$, $\Gamma=0.1$,
$E_{F}=-1$ and $\theta=65^{\circ}$.

To complete the physical picture, Fig. 4 displays the magnitude of the ac
homodyne ($\sim\omega$) charge current as a function of the strength of
exchange and Rashba interactions. We emhasize that we show the absolute value
of the ac components. The extrinsic contributions [Fig. 4(c,d)] increase
steadily with the Rashba strength and saturate at large spin-orbit
interaction, similarly to the dc case. In contrast, the intrinsic
contributions [Fig. 4(a,b)] reach a maximum, collapse and saturate at large
Rashba strength. Since we only account for the absolute value of the ac
current, the collapse observed in Fig. 4(a,b) is associated with a $\pi$ shift
(i.e., a sign change) akin to the one observed for the intrinsic dc current
[Fig. 3(a)].

Figure 4: (Color online) Dependence of the ac current along (a,c) ${\bf x}$
and (b,d) ${\bf y}$ as a function of the Rashba and exchange interactions.
Both (a,b) intrinsic and (c,d) extrinsic contributions are shown. The
parameters are the same as in Fig. 3.

As discussed above, the competition between exchange and spin-orbit coupling
leads to a regime of parameters when the Fermi surface breathing induces
higher harmonics due to the dramatic distortion of the Fermi surface.
Importantly, we find that the harmonics are particularly strong for the
intrinsic current contributions and rather negligible in the extrinsic
contribution [compare Fig. 2(c) with Fig. 2(e), for instance]. In the
following, we focus on the intrinsic currents. Figure 5 shows such a
situation, obtained when $\Delta=1$, $E_{F}=-1$, and $\theta=65^{\circ}$. The
current along ${\bf x}$ possesses odd harmonics ($(2n+1)\omega$) whereas the
current along ${\bf y}$ possesses even harmonics ($2n\omega$), due to symmetry
(only $J_{y}$ produces a rectified, dc current). Although their magnitude
decreases with the harmonics number (under the adiabatic assumption, our
theory is based on a perturbative expansion of the magnetization dynamics), we
find that in a certain region of Rashba strength, close to the maximum dc
current obtained in Fig. 3(a), i.e., $t_{R}\in[0.3,0.6]$, the first few
harmonics are comparable in magnitude. This results in a rather complex time-
dependence of the intrinsic signal, as shown in Figure 5(b,d) for selected
cases.

Figure 5: (Color online) Dependence of the current harmonics as a function of
the Rashba strength for $\Delta=1$ for the intrinsic contribution of the
current along (a) ${\bf x}$ and (c) ${\bf y}$. The shaded area emphasizes the
parameter region when harmonics are sizable. Time-dependence of the intrinsic
current along (b) ${\bf x}$ and (d) ${\bf y}$ for $t_{R}=0.2,\;0.3,\;0.4$ and
0.5.

###  III.3 Magnetic graphene

For the sake of completeness, we now consider the case of magnetic graphene,
Fig. 6(a), whose band structure presents the peculiarity to be highly
sensitive to the magnetization direction. The Hamiltonian is obtained by
regularizing Hamiltonian (10) on a honeycomb lattice. Setting
$\theta=45^{\circ}$, $t=1$ eV, and $t_{R}=0.15$ eV, we obtain the band
structures reported in Fig. 6(c,e) for $\Delta=0.1$ eV and $\Delta=0.75$ eV,
respectively. For a better comparison with the Rashba gas studied above, we
set the lattice parameter to $a=1$ nm. Figures 6(b,d) show the Fermi surface
when setting the magnetization along $\pm{\bf x}$ for two different
illustrative cases. Because the breathing is governed by the Rashba spin-orbit
coupling, we obtain a distortion that is qualitatively similar to the one in
Fig. 1, suggesting strong charge pumping.

Figure 6: (Color online) (a) Schematics of spin pumping in magnetic graphene
with Rashba spin-orbit coupling. (b,d) Fermi surface for $\Delta=0.1$ eV
($\Delta=0.75$ eV) and $t_{R}=0.15$ eV when setting the magnetization along
$\pm{\bf x}$. (c,e) Corresponding band structure when setting the
magnetization along ${\bf z}$. The dashed lines correspond to the energy at
which the surfaces in (b,d) are taken.

The time-dependence of the intrinsic and extrinsic currents are reported in
Fig. 7 for parameters comparable to that of the Rashba gas, i.e., $t=0.2$ eV,
$\Delta=1.0$ eV, $E_{F}=-0.5$ eV and $t_{R}=0.3$ (black), 0.5 (red) and 0.7 eV
(blue). For this set of parameters, the time-dependence of the pumped current
is radically different from the conventional one reported in Fig. 2, and
substantially deviates from an harmonic response (i.e., $\cos\omega t$,
$\sin\omega t$). This behavior is directly associated with the high
sensitivity of the band structure on the magnetization, as seen in Fig. 6, and
results in high harmonics.

Figure 7: (Color online) Time-dependence of the intrinsic (a,b) and extrinsic
(c,d) currents for $t=0.2$ eV, $\Delta=1.0$ eV, $E_{F}=-0.5$ eV and
$t_{R}=0.3$ (black), 0.5 (red) and 0.7 eV (blue).

Finally, taking a hopping parameter closer to that of graphene, $t=1$ eV, Fig.
8 shows the dependence of the harmonics of the intrinsic current, obtained
when $\Delta=0.75$ eV, $E_{F}=-1.15$ eV, and $\theta=45^{\circ}$. As expected,
the current along ${\bf x}$ possesses odd harmonics ($(2n+1)\omega$) whereas
the current along ${\bf y}$ possesses even harmonics ($2n\omega$). Upon
increasing the Rashba strength, the homodyne component along ${\bf x}$
increases steadily up to a maximum that lies out of the range studied here.
The other harmonic components all reach a maximum within the range
$t_{R}\in[0,1]$ eV suggesting that second and third harmonic are quite sizable
for a reasonable Rashba strength, resulting in the complex time-dependence of
the intrinsic signal, see Fig. 8(b,d).

Figure 8: (Color online) Dependence of the current harmonics as a function of
the Rashba strength for $t=1$ eV, $\Delta=0.75$ eV and $E_{F}=-1.15$ eV for
the intrinsic contribution of the current along (a) ${\bf x}$ and (c) ${\bf
y}$. Time-dependence of the intrinsic current along (b) ${\bf x}$ and (d)
${\bf y}$ for $t_{R}\in[0.1,1]$ eV.

##  IV Discussion and Conclusion

The formalism described in the present Article extends the traditional theory
of spin and charge pumping [1, 2] by covering the full range of exchange and
spin-orbit coupling. It can be readily adapted to address spin, charge, and
orbital pumping in multiband heterostructures. It is particularly well adapted
to compute adiabatic pumping in realistic heterostructures obtained from
density functional theory. Among the important features uncovered by the
present theory, we point out the importance of both intrinsic and extrinsic
contributions to both the dc and ac currents, akin to magnetic damping [28],
an aspect that is overlooked by the traditional theory of spin pumping. This
theoretical framework is instrumental to investigate spin-charge
interconversion in strongly spin-orbit coupled systems such as the surface of
topological heterostructures, involving topological insulators and Weyl
semimetals, for instance.

Figure 9: (Color online) Two devices implementing charge pumping induced by
large angular precession. In device (a), the magnetization autooscillation is
induced by spin transfer torque, whereas in device (b) it is induced by spin-
orbit torque. Notice that in both cases, the pumped current is collected along
the magnetization direction (black connectors), decoupled from the driving
current (red connectors).

Finally, we would like to comment on the adiabatic harmonic generation
displayed by Figs. 5 and 8. In contrast with Ref. 24 that displays tens of
harmonics, the adiabatic theory only shows a few of them. The present theory
differs from Ref. 24 in several aspects: first Ref. 24 computes the time-
dependent Schrödinger equation in a finite size ribbon in the Landauer-
Büttiker configuration (a conductor connected to two leads). As such, the
internal dynamics of the electron spin is computed numerically and the quantum
interference between the electronic modes contributes to the complex time-
dependent current. In the present theory, it is assumed that the spin is
aligned on the effective field (exchange field+Rashba field, ${\bf B}_{\rm
eff}$ in Fig. 1) so that the internal spin dynamics is neglected. In addition,
our model Hamiltonian is not defined in real space but in momentum space. It
is therefore unsurprising that several features associated with quantum
interference and real-time spin evolution are absent in the present work.

Our calculations suggest that these harmonics are mostly present in the
intrinsic contribution, associated with the time-momentum Berry curvature. A
natural research direction is therefore to identify materials systems where
such a Berry curvature is enhanced, e.g. in magnetic Weyl semimetals such as
Co3Sn2S2 [38, 39] or Mn3Sn [40]. In addition, the harmonics computed here are
obtained for large precession angles, which are not achievable using
ferromagnetic resonance techniques. Nonetheless, such large angles can be
obtained via current-driven autooscillations [41, 42]. This inspires us to
propose two devices in Fig. 9 based on (a) spin transfer torque and (b) spin-
orbit torque that can excite such large-angle autooscillations. In the
presence of strong spin-orbit coupling, large Fermi surface breathing is
expected, which could trigger harmonic currents.

We acknowledge that the generation of harmonic currents via spin pumping is a
daring technical challenge. In fact, to the best of our knowledge, only a
handful of experiments have achieved the detection of the homodyne ac current
[19, 20, 21]. The difficulty is that not only one needs to get rid of the
frequency response of the radiofrequency setup itself, but in addition, in
most experiments the magnetization precession is not circular but rather
ellipsoidal, resulting in harmonics of purely magnetic origin [19]. Hence, one
needs to ensure that the observed harmonics are of purely electronic nature,
i.e., from the competition of exchange and spin-orbit coupling, which is
certainly an interesting challenge for experiments.

###### Acknowledgements.

A.P. acknowledges support from the ANR ORION project, grant
ANR-20-CE30-0022-01 of the French Agence Nationale de la Recherche. A. M.
acknowledges support from the Excellence Initiative of Aix-Marseille
Université - A*Midex, a French ”Investissements d’Avenir” program.

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