# Elasto-Dynamical Induced Spin and Charge Pumping in Bulk Heavy Metals

Farzad Mahfouzi<EMAIL_ADDRESS>Department of Physics and
Astronomy, California State University, Northridge, CA, USA  Nicholas Kioussis
<EMAIL_ADDRESS>Department of Physics and Astronomy, California State
University, Northridge, CA, USA

###### Abstract

Analogous to the Spin-Hall Effect (SHE), ab initio electronic structure
calculations reveal that acoustic phonons can induce charge (spin) current
flowing along (normal to) its propagation direction. Using Floquet approach we
have calculated the elastodynamical-induced charge and spin pumping in bulk Pt
and demonstrate that: (i) While the longitudinal charge pumping is an
intrinsic observable, the transverse pumped spin-current has an extrinsic
origin that depends strongly on the electronic relaxation time; (ii) The
longitudinal charge current is of nonrelativstic origin, while the transverse
spin current is a relativistic effect that to lowest order scales linearly
with the spin-orbit coupling strength; (iii) both charge and spin pumped
currents have parabolic dependence on the amplitude of the elastic wave.

###### pacs:

72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg

Introduction: One of the primary objectives in the field of spintronics is the
development of efficient means to generate pure spin current, which can be in
turn used to manipulate the magnetization configuration and damping rate in
magnetic based memory bits[1], spin transistors[2, 3], antennas[4], and
sensors[5]. Generating spin current in materials with strong spin-orbit
coupling (SOC) through the spin Hall effect[6] (SHE), i.e., the transverse
spin current generation by electric field, has recently been the focus of
intensive research both theoretically and experimentally[7, 8]. The SHE is
often parameterized by the charge-to-spin current conversion efficiency,
referred to as the spin-Hall angle, $\Theta_{SH}$, whereby the spin current is
given by
$\vec{I}_{\vec{S}}=\frac{\hbar}{2e}\Theta_{SH}\vec{I}_{C}\times\vec{e}_{\vec{S}}$.
Here, $\vec{e}_{\vec{S}}$ is the spin polarization unit vector and
$\vec{I}_{C}$ is the charge current. Experimentally, the conventional approach
to quantify $\Theta_{SH}$ is through spin-orbit torque measurements in
ferromagnetic/normal metal bilayer devices.[9, 10, 11, 12, 13, 14, 15]

Another commonly used mechanism to generate dc spin current is the spin
pumping from a precessing ferromagnet (FM) into an adjacent normal metal (NM)
[16, 17, 18, 19, 20]. The generated spin current flowing through the FM/NM
interface is
${I}_{\vec{S}}=\frac{\hbar}{4\pi}g_{\uparrow\downarrow}\vec{e}_{\vec{S}}\cdot\vec{m}\times\partial\vec{m}/\partial
t$, where $\vec{m}$ and $g_{\uparrow\downarrow}$ denote the direction of the
magnetization and the spin-mixing conductance, respectively.
$g_{\uparrow\downarrow}$ is often determined experimentally through the change
of Gilbert damping in the presence and absence of the normal metal adjacent to
the FM.[21] In addition, other approaches, such as, the spin polarization
effect in magnetic tunnel junctions,[22, 23] the spin Seebeck effect[25, 26,
27, 24], spin pumping from magnons excited in response to elastic waves,[28,
29] and spin current pulses produced by ultrafast laser induced
demagnetization process [30, 31] have also been proven to be promising for the
generation of spin currents in spintronic devices.

Recently, a different approach to generate spin current was demonstrated
experimentally in X/CoFeB/MgO (X=W,Pt,W) heterostructures where the spin
current emerges from the lattice dynamics in strong spin-orbit nonmagnetic
metals (Pt, W), and flows transverse to the propagation direction of the
surface acoustic wave[32]. This is similar to the conventional SHE where the
spin current propagates orthogonal to the electrical current. To account for
the experimental results Kawada et al. suggested [32] that the spin current
must scale with the SOC and the time derivative of the lattice displacement
along the wave propagation direction. The suggested plausible mechanisms of
the generation of acoustic spin Hall effect include a dynamic change in the
Berry curvature of electron wavefunction induced by the time-dependent lattice
displacement giving rise to a Berry electric field,[33, 34] and/or SOC-
mediated spin-lattice coupling resembling the Rashba Hamiltonian[35].
Nevertheless, the underlying atomistic mechanism remains unresolved.

In this Letter, using ab initio based electronic structure calculations we
reveal the emergence of a dc charge (spin) current in response to acoustic
phonons in heavy metals (Pt), shown schematically in Fig. 1(a), where the spin
current flows transverse to the phonon propagation direction. We demonstrate
that the phonon-induced spin current is a relativistic effect arising from the
spin-orbit coupling (SOC) where, to the lowest order depends linearly on the
SOC strength. Analogous to the SHE, the spin polarization orientation is
orthogonal to both the spin current and phonon propagation directions. We show
that the phonon-induced charge (spin) current saturates (diverges) in the
limit of ballistic transport regime, signifying its intrinsic (extrinsic)
origin. This is in sharp contrast to the electric field-induced charge (spin)
current where the corresponding longitudinal (transverse) conductivity is
extrinsic (intrinsic).

Theoretical Formalism: In the linear response regime the electronic
Hamiltonian of a bulk system under a time- and position-dependent strain,
$\epsilon_{ij}(\vec{R},t)$ ($i,j=x,y,z$), is given by[36, 37]

$\displaystyle\hat{\bf H}=\hat{\bf H}^{0}+\frac{1}{2}\sum_{ij}(\hat{\bf
H}^{ij}\hat{\bm{\epsilon}}_{ij}(t)+\hat{\bm{\epsilon}}_{ij}(t)\hat{\bf
H}^{ij}).$ (1)

Here, bold symbols denote matrices in real space, hat symbols denote matrices
in atomic orbital Hilbert space, $\hat{\bf H}^{0}$ is the Hamiltonian in the
absence of strain, ${\hat{\bf H}^{ij}}=\partial\hat{\bf
H}/\partial\epsilon_{ij}|_{\epsilon_{ij}\rightarrow 0}$ is the deformation
Hamiltonian term associated with the coupling between electrons and local
strain, $[\hat{\bf
H}]_{\vec{R},\vec{R}^{\prime}}=\hat{H}_{\vec{R}-\vec{R}^{\prime}}$,
$\vec{R},\vec{R}^{\prime}$ are the positions of the unit cells, and the time-
and position-dependent local strain is a diagonal matrix with elements given
by,
$[{\hat{\bm{\epsilon}}}_{ij}]_{\vec{R},\vec{R}^{\prime}}=\hat{1}{\epsilon}_{ij}(\vec{R},t)\delta_{\vec{R},\vec{R}^{\prime}}$.
Eq. (1) assumes that spatial variation of strain is adiabatic and hence
ignores the dependence of the Hamiltonian on the gradience of strain. For a
single phonon mode of wave vector, $\vec{q}$, and frequency, $\omega$, we
have, $\epsilon_{ij}(\vec{R},t)=Re(\epsilon^{\vec{q}}_{ij}e^{i\omega
t+i\vec{q}\cdot\vec{R}})$. In this case, the time and position dependence of
the Hamiltonian can be removed by applying a gauge transformation, $[\hat{\bf
U}]_{NM}=\delta_{NM}\hat{1}e^{iN\omega t+iN\vec{q}\cdot\vec{R}}$, where the
capital letters, N and M refer to phonon states. The resulting time-
independent system is referred to as the Floquet space, where the
corresponding Hilbert space is extended to include the phononic degrees of
freedom, with the total Hamiltonian given by

$\displaystyle[\hat{\mathcal{H}}_{\vec{k}}]_{NM}$
$\displaystyle=\hat{H}^{0}_{\vec{k}+N\vec{q}}\delta_{NM}$
$\displaystyle+\frac{1}{2}\sum_{ij}\hat{T}^{ij}_{\vec{k},N}\epsilon^{\vec{q}}_{ij}\Big{(}\delta_{N,M+1}+\delta_{N,M-1}\Big{)},$
(2)

where
$\hat{T}^{ij}_{\vec{k},N}=\frac{1}{2}\big{(}\hat{H}^{ij}_{\vec{k}+N\vec{q}}+\hat{H}^{ij}_{\vec{k}+(N+1)\vec{q}}\big{)}$,
$N,M=0,\ldots,N_{ph}$, and $N_{ph}$ is the cutoff for the number of phonons.
The hybrid electron-phonon Hamiltonian is shown schematically in Fig. 1(b),
where the additional dimension is introduced to account for the single mode
phononic degree of freedom. The dependence of the quasi-particle chemical
potential on the number of phonons results in a quasiparticle transport along
the auxiliary direction which describes the attenuation of the single mode
elastic wave through electron-phonon scattering.[38] In this case, due to the
conservation of momentum, both electrons and holes are dragged along the
phonon propagation direction, whereby systems with electron-hole asymmetry
experience a net pumped charge current. This mechanism of pumping is, in
essence, similar to the phonon drag effect that gives rise to an enhancement
of Seebeck coefficient which is often treated theoretically using coupled
electron-phonon Boltzmann transport equations.[39]

Figure 1:  (a) Schematic of the elastodynamical-induced charge and spin
current in bulk heavy metal under a time- and position-dependent strain
$\epsilon_{xx}(\vec{R},t)=Re(\epsilon_{xx}^{\vec{q}}e^{i\omega
t+i\vec{q}\cdot\vec{R}})$, of wavevector $\vec{q}$ along the x direction. The
charge current propagates along $\vec{q}$, while the spin current flows (along
$z$) orthogonal to the phonon and spin-polarization (y) directions. (b)
Floquet space representation of the combined electron-phonon system. The
phononic part consists of a single mode with wave vector $\vec{q}$ and a
cutoff number, $N_{ph}$, of phonons. The $N$th layer represents the electronic
system with Hamiltonian $H^{0}_{\vec{k}+N\vec{q}}$ and chemical potential
$\mu+N\hbar\omega$ that is coupled to the other electronic systems with
different number of phonons though the electron-phonon coupling matrix,
$T^{ij}_{\vec{k},N}$, $i,j=x,y,z$.

For an isolated system, the wave function of the coupled electron-phonon
system is of the form,
$|N\alpha\vec{k}\rangle=|N\rangle\otimes|\alpha\vec{k}\rangle$, where,
$\otimes$ refers to the Kronecker product, and $\alpha$ denotes the atomic
orbitals and spin of the electron Bloch states. The single-quasi-particle
retarded Green function and the corresponding density matrix can be obtained
from,[38, 40]

$\displaystyle\hat{\bm{\rho}}_{\vec{k}}=\frac{\eta}{\pi}\int
dE\hat{\bm{G}}_{\vec{k}}(E){f}(E\hat{\bf
1}-\hat{\bm{\Omega}})\hat{\bm{G}}_{\vec{k}}^{\dagger}(E),$ (3)

where the Green’s function is calculated from,

$\displaystyle\left(E-i\eta-\hat{\bf\mathcal{H}}_{\vec{k}}-\hat{\bm{\Omega}}\right)\hat{\bm{G}}_{\vec{k}}(E)=\hat{\bm{1}}.$
(4)

Here, $[\hat{\bm{\Omega}}]_{NM}=N\hbar\omega\delta_{NM}\hat{1}$, is the single
mode phononic Hamiltonian, $f(E)$ is the Fermi-Dirac distribution function and
$\eta=\hbar/2\tau$ is the energy broadening parameter which is inversely
proportional to the electronic relaxation time, $\tau$. Using density matrix
given by Eq. (4), the charge and spin currents are determined from,

$\displaystyle\vec{I}$
$\displaystyle=e\langle\vec{\hat{\bm{v}}}_{\vec{k}}\rangle,$ (5a)
$\displaystyle\vec{I}^{S_{i}}$
$\displaystyle=\frac{\hbar}{2}Re\langle\hat{\sigma}_{i}\vec{\hat{\bm{v}}}_{\vec{k}}\rangle,$
(5b)

where,
$\langle...\rangle=\sum_{\vec{k}}Tr[...\hat{\bm{\rho}}_{\vec{k}}]/(VN_{k}N_{ph})$
is the expectation value, $V$ is the unit of the unit cell, $N_{k}$ is the
number of k-points in the summation and
$\vec{\hat{\bm{v}}}_{\vec{k}}=\partial\hat{\mathcal{H}}_{\vec{k}}/\partial\vec{k}$
is the electronic group velocity operator. In the ballistic regime and in
linear response to the phonon frequency, $\omega$, the density matrix is given
by,

$\displaystyle\hat{\bm{\rho}}_{\vec{k}}\approx
f(\mathcal{\hat{H}}_{\vec{k}}+\hat{\bm{\Omega}})+\frac{\eta}{\pi}\hat{\bm{G}}_{\vec{k}}(E_{F})\hat{\bm{\Omega}}\hat{\bm{G}}_{\vec{k}}^{\dagger}(E_{F}).$
(6)

The first and second terms in Eq. (6) are referred to as the Fermi sea and
Fermi surface contributions, respectively. The Fermi surface contribution to
the density matrix and the resulting pumped charge and spin currents can be
separated into even and odd components with respect to $\eta$ (or relaxation
time),
$\hat{\bm{\rho}}^{in/ex}_{\vec{k}}=(\hat{\bm{\rho}}_{\vec{k}}(\eta)\pm\hat{\bm{\rho}}_{\vec{k}}(-\eta))/2$.
To lowest order, the even (odd) component is expected to be independent of
(inversely proportional to) $\eta$, demonstrating its intrinsic (extrinsic)
character. The term “extrinsic” is defined as the odd component of the density
matrix with respect to the broadening parameter $\eta$, which, to lowest
order, is inversely proportional to $\eta$ and originates from the intraband
contributions (See Supplemental Material [52]).

In the ballistic regime, $\eta\rightarrow 0$, the intrinsic (even) component
of the charge current can be written in terms of the Berry curvature,

$\displaystyle\vec{I}^{in}$
$\displaystyle=\frac{2e}{VN_{k}N_{ph}}\sum_{n\vec{k}}f(\varepsilon_{n\vec{k}})Im\bra{\frac{\partial\psi_{n\vec{k}}}{\partial\vec{k}}}\hat{\bm{U}}^{\dagger}\frac{\partial}{\partial
t}\bm{\hat{U}}\ket{\psi_{n\vec{k}}}$
$\displaystyle=\frac{2e}{VN_{k}N_{ph}}\sum_{n\vec{k}}f(\varepsilon_{n\vec{k}})Re\bra{\frac{\partial\psi_{n\vec{k}}}{\partial\vec{k}}}\bm{\hat{\Omega}}\ket{\psi_{n\vec{k}}},$
(7)

where, we use perturbation theory to express,
$\ket{\frac{\partial\psi_{n\vec{k}}}{\partial\vec{k}}}=\sum_{m}\frac{(\vec{\bm{\hat{v}}})_{nm}\ket{\psi_{m\vec{k}}}}{\varepsilon_{n\vec{k}}-\varepsilon_{m\vec{k}}-i\eta}$,
with
$\vec{\bm{\hat{v}}}_{nm,\vec{k}}=\bra{\psi_{n\vec{k}}}\vec{\bm{\hat{v}}}\ket{\psi_{m\vec{k}}}$
being the group velocity matrix elements, and the energy broadening parameter,
$\eta$, is introduced to avoid divergences at the degenerate points. The
wavefunction, $\ket{\psi_{n\vec{k}}}$ and energy dispersion,
$\varepsilon_{n\vec{k}}$ are the eigenvectors and eigenvalues of the Floquet
Hamiltonian, $\hat{\bf\mathcal{H}}_{\vec{k}}$, respectively. The intrinsic
component of the pumped spin current can be calculated from an expression
similar to Eq. (7), by replacing the charge current operator,
$e\vec{\bm{\hat{v}}}$, with the spin current operator given by the Hermitian
part of $\hbar\hat{\sigma}_{i}\vec{\bm{\hat{v}}}/2$. Furthermore, it should be
noted that in systems with small bandgap at the Fermi surface, topological
nature of Eq. (7) can yield quantized values (i.e. independent of elastic wave
amplitude) where electrons and holes are trapped and dragged by the elastic
wave, resulting in the so-called Thouless pumping.[41, 42]

Table 1: Values of the intrinsic and extrinsic contributions to the elastodynamical-induced components of the pumped charge, $\vec{I}$, and spin current, $\vec{I}^{S_{i}}$, divided by the acoustic phonon frequency, $\omega$, (in units of $10^{10}/e\Omega m^{2}$), for phonon wavevector, $\vec{q}=(\pm 0.01,0,0)$(Å-1) and strain amplitude $\epsilon^{\vec{q}}_{xx}=10^{-3}$. |  | Intrinsic (Even) |  |  | Extrinsic (Odd) |
---|---|---|---|---|---|---
$i=$ | $x$ | $y$ | $z$ | x | y | z
$I_{i}(\vec{q})$/$\omega$ |  $\pm$80 | 0 | 0 | 0 | 0 | 0
$I^{S_{x}}_{i}(\vec{q})$/$\omega$ | 0 | 0 | 0 | 0 | 0 | 0
$I^{S_{y}}_{i}(\vec{q})$/$\omega$ | 0 | 0 | 0 | 0 | 0 |  $\mp$ 0.7
$I^{S_{z}}_{i}(\vec{q})$/$\omega$ | 0 | 0 | 0 | 0 |  $\pm$0.7 | 0

Computational Approach: The tight-binding Hamiltonian, $\hat{H}_{\vec{k}}$
matrices for bulk Pt under different values of strain, $\epsilon_{ij}$, are
calculated using the Linear Combination of Atomic Orbitals OpenMX package[43,
44, 45]. For a nonmagnetic material, the one-electron Kohn-Sham Hamiltonian
can be expressed by[38, 47, 46],

$\displaystyle\hat{H}_{\vec{k}}(\\{\epsilon_{ij}\\})$
$\displaystyle=\hat{H}_{\vec{k}}^{K}(\\{\epsilon_{ij}\\})\hat{1}_{2\times
2}+\xi\hat{H}^{soc}_{\vec{k}}(\\{\epsilon_{ij}\\}),$ (8)

where, the first term represents the kinetic component of the Hamiltonian, the
second term is the SOC contribution, and $\xi$ is the scaling factor. The
effect of strain, ${\epsilon_{ij}}$, is to modify the primitive lattice
vectors, $\vec{a}^{\prime}_{i}$, such that,
$(\vec{a}^{\prime}_{i}-\vec{a}_{i})\cdot\vec{e}_{j}=\sum_{k}\vec{a}_{i}\cdot\vec{e}_{k}\epsilon_{kj}$,
where the $\vec{e}_{j}$’s denote unit vectors in Cartesian coordinates. The
electron-phonon coupling terms in Eq. (1) are calculated by fitting Eq. (8) to
a polynomial function, and in turn calculate,
$\hat{H}^{ij}_{\vec{k}}=\partial\hat{H}_{\vec{k}}/\partial\epsilon_{ij}|_{\epsilon_{ij}\rightarrow
0}$. The self-consistent (SCF) calculations employ the Troullier-Martins type
norm-conserving pseudopotentials[48] with partial core correction. The
equilibrium lattice constant of bulk Pt was set to $a=3.96$ Å. In the SCF
calculations we used a $24^{3}$ k-point mesh in the first Brillouin zone, and
an energy cutoff of 350 Ry for numerical integrations in the real space grid.
For the exchange correlation functional the LSDA[49] parameterized by Perdew
and Wang[50] was used.

Results and Discussion: Table I lists the values of the intrinsic and
extrinsic contributions to the elastodynamical-induced components of the
pumped charge, $\vec{I}$, and spin current, $\vec{I}^{S_{i}}$, divided by the
acoustic phonon frequency, $\omega$, (in units of $10^{10}/e\Omega m^{2}$),
for $\vec{q}=(\pm 0.01,0,0)$(Å-1), strain amplitude
$\epsilon^{\vec{q}}_{xx}=10^{-3}$ and $\eta$ = 0.01eV. The charge current is
dominated by the intrinsic component (even function of $\eta$) and is parallel
to the elastic wave propagation direction, $\vec{I}\parallel\vec{q}$, while
the spin current is extrinsic (odd function of $\eta$) and propagates normal
to $\vec{q}$ and the spin polarization direction,
$\vec{I}^{\vec{S}}\parallel\vec{e}_{\vec{S}}\times\vec{q}$. Fig. 2(a) displays
the calculated pumped charge (dashed curve) and spin (solid curve) currents
versus the wavevector of the elastic wave propagating along $x$, respectively,
with $\epsilon^{\vec{q}}_{xx}$ = $10^{-3}$ and $\eta$= 0.01eV. Both charge and
spin currents are odd functions of the wavevector with a linear dependence on
$q_{x}$ close to the $\Gamma$-point. This is analogous to the electric field-
induced charge and spin current, where the external electric field is replaced
with $\hbar\omega\vec{q}/e$. The difference, however, is that in contrast to
the extrinsic (intrinsic) nature of the Ohmic (spin-Hall) current, the phonon-
induced charge (spin) current exhibits intrinsic (extrinsic) character.

Figure 2: (a) Elastodynamical pumped spin (left-hand ordinate) and charge
(right-hand ordinate) current divided by the phonon frequency, $\omega$,
versus the phonon wavevector $\vec{q}$ along $x$-axis with strain amplitude
$\epsilon^{\vec{q}}_{xx}=10^{-3}$ and energy broadening parameter $\eta$= 10
meV. (b) Efficiency of the extrinsic elastodynamical-induced spin current,
$I^{S_{y};ex}_{z}/q_{x}\omega$ , (left-hand ordinate) and intrinsic charge
current, $I^{in}_{x}/q_{x}\omega$, (right-hand ordinate) in the limit
$q_{x}\rightarrow$ 0, versus the square of the strain amplitude,
$(\epsilon^{\vec{q}})^{2}_{xx}$, where $\epsilon^{\vec{q}}_{xx}$ ranges
between 0 and 1$\%$. (c) Total (stars) and Fermi surface (circles)
contributions to the Elastodynamical Longitudinal Charge Conductivity (ELCC)
versus $\eta$ for $N_{ph}$= 5 and 10 shown with solid and dashed curves,
respectively. (d) Fermi surface contribution to the Elastodynamical Transverse
Spin Conductivity (ETSC) versus $\eta$ for different values of $N_{ph}$.
Insets in (c) and (d): ELCC and ETSC versus SOC scaling factor or $\eta=0.1$
eV.

The slope of the pumped charge and spin currents with respect to $q_{x}$ in
the limit of $q_{x}\rightarrow 0$, describes the efficiency of the phonon-
induced electronic transport in the material. Hence, in Fig. 2(b) we show the
variation of the elastodynamical-induced even (intrinsic) component of the
charge current, $I^{in}_{x}/q_{x}\omega$, and odd (extrinsic) component of the
spin current, $I^{S_{y};ex}_{z}/q_{x}\omega$ , versus
$(\epsilon^{\vec{q}}_{xx})^{2}$. The calculations reveal that both the charge
(dashed curve) and the spin current (solid curve) vary quadrtiaclly with the
amplitude of the elastic wave for a wide range of strain. These results
suggest that the strain dependence of the elastodynamical-induced pumped
charge and spin currents can be written in the general form,

$\displaystyle\vec{I}({\vec{q}})/\omega$ $\displaystyle=\vec{q}\sum_{i\leq
j,k\leq l}\gamma^{ch}_{ij,kl}\epsilon^{\vec{q}}_{ij}\epsilon^{\vec{q}}_{kl}$
(9a) $\displaystyle\vec{I}^{\vec{S}}({\vec{q}})/\omega$
$\displaystyle=\frac{\hbar}{2e}\vec{e}_{\vec{S}}\times\vec{q}\sum_{i\leq
j,k\leq l}\gamma^{sp}_{ij,kl}\epsilon^{\vec{q}}_{ij}\epsilon^{\vec{q}}_{kl},$
(9b)

where, $\gamma^{ch}_{ij,kl}$ and $\gamma^{sp}_{ij,kl}$ are the Elastodynamical
Longitudinal Charge Conductivity (ELCC) and Elastodynamical Transverse Spin
Conductivity (ETSC), respectively. Given, $\epsilon_{ij}=\epsilon_{ji}$, in
order to avoid double counting we consider only strains with, $i\leq j$. One
can then use the Voigt notation ($[1,2,3,4,5,6]\equiv[xx,yy,zz,yz,xz,xy]$) to
represent the $\gamma_{ij,kl}^{ch/sp}$ tensor elements as a 6$\times$6
symmetric matrix.

The variation of the ELCC versus the broadening energy, $\eta$, is shown in
Fig. 2(c) for phonon cutoff number, $N_{ph}=$ 5 and 10. We display the
variation of the total ELCC and its Fermi surface contribution, where the
former is determined from Eq. (7), while the latter is calculated from Eq.
(5a) where the density matrix is given only by the second term in Eq. (6). We
find that in the limit of ballistic regime ($\eta\rightarrow$ 0), the ELCC
saturates to a finite value consistent with the expected behavior of an
intrinsic observable. On the other hand, in the limit of large $\eta$
(diffusive regime), while the ELCC becomes independent of $N_{ph}$, the
deviation between the total and Fermi surface contribution becomes more
significant, due to the fact that Eq. (7) is derived and hence valid in the
small $\eta$ limit. The inset in Fig. 2(c) shows the dependence of ELCC on the
SOC scaling factor with a finite value in the absence of SOC, demonstrating
its non-relativistic nature.

Fig. 2(d) shows the variation of the extrinsic component of ETSC (only the
Fermi surface contribution) versus $\eta$ for $N_{ph}=$ 5 and 10. In contrast
to the expected $1/\eta$ dependence, we find that in the limit of
$\eta\rightarrow 0$ the ETSC is proportional to $\eta$ and reaches a peak with
increasing $\eta$. The peak ETSC value increases with $N_{ph}$, and the
corresponding $\eta_{max}$ decreases with increasing $N_{ph}$, suggesting that
ETSC$\propto\eta/(N^{-1}_{ph}+c\eta^{2})$, where $c$ is a constant. This
means, a larger phonon cutoff number is required to converge the ETSC value in
the clean limit (small $\eta$ limit). The inset in Fig. 2(d) shows the ETSC
versus the SOC scaling factor, $\xi$, demonstrating its relativistic nature,
which, analogous to the SHE, to lowest order in $\xi$, is proportional to the
SOC strength. It is worth noting that the non-monotonic dependence of ETSC and
its sign reversal with SOC for small $\eta$ suggests that in heavy metals
(with large SOC) such as Pt a perturbative treatment of ETSC with respect to
SOC may fail. Interestingly, the results of ETSC versus SOC shown in Fig. S1
in the supplemental material[52] for larger $\eta$ of 0.5 eV exhibits a
monotonic behavior.

Thus far, we have focused only on the elastodynamical-induced pumping in
response to longitudinal elastic wave, $\epsilon_{xx}(\vec{R},t)$. In the
following we present the results for the non-zero matrix elements of the
6$\times$6 elastodynamical charge/spin conductivity matrix. The values of the
diagonal matrix elements for the even (intrinsic) component of the ELCC matrix
(in units of $10^{9}/e\Omega m$) for $\vec{q}\parallel x$ and $\eta=25\ meV$,
are $\gamma^{ch}_{ii}=[1.5,2.5,2.5,0.8,1,1]$, ($i$=1, $\ldots$, 6), while
those of the off-diagonal matrix elements are
$\gamma^{ch}_{12}=\gamma^{ch}_{13}=-0.6$, and $\gamma^{ch}_{23}=-2.4$, where
$\gamma^{ch/sp}_{ij}=\gamma^{ch/sp}_{ji}$. Note that similar to the elastic
stiffness and magneto-elastic tensor elements[51], the symmetry of the crystal
structure for the elastodynamical electronic transport reduces the number of
independent tensor elements. However, the difference is that the elastic wave
propagation along $x$-axis breaks the cubic symmetry and the ELCC matrix
elements resemble those of a tetragonal system instead.

Similar calculations for the extrinsic component of the ETSC (in units of
$10^{7}/e\Omega m$) with spin current along the $z$-axis and spin-polarization
along $y$-axis, yields that the nonzero diagonal elements are,
$\gamma^{sp}_{ii}=[3.5,2.6,-0.6,2,0.5,-0.4]$ and the off diagonal elements
are, $\gamma^{sp}_{12}=-3.3$, $\gamma^{sp}_{13}=-1.6$ and
$\gamma^{sp}_{23}=1.6$. Note that all of the nonzero elements of ETSC are
independent, similar to the elastic stiffness matrix elements of an
orthorhombic crystal structure. This is due to the fact that the choice of the
spin current direction along $z$ renders the $yz$-plane (i.e., normal to
$\vec{q}\parallel\vec{e}_{x}$) anisotropic.

As demonstrated in the Supplemental Material[52], the underlying mechanism of
the phonon-induced pumping can be attributed to the electron-hole asymmetry,
which results in a negative (positive) sign for ELCC when the transport is
electron-(hole-)like. The derivative of density of states (DOS) with respect
to the energy can be employed to qualitatively determine the sign of the
phonon-induced charge pumping. We find a correlation between ELCC, ETSC and
the derivative of DOS as a function of chemical potential shift [see Fig. S2
in [52]], suggesting that the ETSC may be explained by the phonon drag charge
current combined with the spin-Hall effect.

An approximate value of the experimental ELCC can be determined from
$\vec{I}\approx\omega\vec{q}\gamma^{ch}_{eff}\epsilon_{eff}^{2}$, using the
measured values[32] of pumped charge current, $I=7.5\times 10^{3}(A/m^{2})$,
in response to an elastic wave with frequency $\nu=$193 MHz, wavelength,
$\lambda$= 20 $\mu m$, and strain amplitude $\epsilon_{eff}\approx
0.01\%$.[53] An effective value of ELCC, $\gamma^{ch,exp}_{eff}\approx
10^{12}(1/e\Omega m)$ is then estimated, which is near the upper limit of the
theoretical value in Fig. 2(c), corresponding to the clean limit,
$\eta\rightarrow 0$. A more accurate comparison with experiment requires
taking into account all ELCC tensor elements. Moreover, the spin current
calculated in this work accounts only for the conventional bulk component and
neglects the spin-torque [54, 55] and the FM/NM interfacial [46]
contributions. Therefore, a proper comparison with experiment requires
calculations of the elastodynamical spin orbit torque in bilayer systems. It
should also be noted that the relaxation time approximation is reliable only
in the small $\eta$ limit, since the limit of large $\eta$ violates
conservation laws, resulting in a smaller pumped spin-current.[56, 38]

In conclusion, we have developed a Floquet-based approach using density
functional theory and demonstrated the emergence of charge and spin currents
induced by an acoustic phonon in bulk Pt. The calculations unveil the
underlying atomistic mechanism of the recently discovered acoustic spin Hall
effect in strong spin-orbit metals.[32] We find that the pumped charge (spin)
current is intrinsic (extrinsic) property, flows along (normal to) to the
phonon wavevector, and is of nonrelativistic (relativistic) origin. It is
worth mentioning that the phonon-induced pumped charge and spin current effect
presented in this Letter can be generalized to other bosonic excitations (e.g.
magnons, photons, etc) and their combinations.

The authors are grateful for fruitful discussions with Masamitsu Hayashi and
Takuya Kawada. The work is supported by NSF ERC-Translational Applications of
Nanoscale Multiferroic Systems (TANMS)- Grant No. 1160504, NSF PFI-RP Grant
No. 1919109, and by NSF-Partnership in Research and Education in Materials
(PREM) Grant No. DMR-1828019.

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