Annihilation of topological solitons in magnetism with spin wave burst finale: The role of nonequilibrium electrons causing nonlocal damping and spin pumping over ultrabroadband frequency range

TL;DR

This paper demonstrates that magnetic domain wall annihilation generates ultrabroadband spin waves and spin currents, revealing significant nonlocal damping effects caused by nonequilibrium electrons, which influence the emitted spin wave spectrum.

Contribution

It introduces a microscopic Hamiltonian-based model that self-consistently captures ultrabroadband spin pumping and nonlocal damping effects overlooked by previous phenomenological approaches.

Findings

Reproduces experimentally observed spin wave bursts during domain wall annihilation.

Predicts ultrabroadband spin current generation without bias voltage.

Shows nonlocal damping exceeds traditional Gilbert damping and alters spin wave spectra.

Abstract

We not only reproduce burst of short-wavelength spin waves (SWs) observed in recent experiment [S. Woo et al., Nat. Phys. 13, 448 (2017)] on magnetic-field-driven annihilation of two magnetic domain walls (DWs) but, furthermore, we predict that this setup additionally generates highly unusual} pumping of electronic spin currents in the absence of any bias voltage. Prior to the instant of annihilation, their power spectrum is ultrabroadband, so they can be converted into rapidly changing in time charge currents, via the inverse spin Hall effect, as a source of THz radiation of bandwidth $\simeq 27$ THz where the lowest frequency is controlled by the applied magnetic field. The spin pumping stems from time-dependent fields introduced into the quantum Hamiltonian of electrons by the classical dynamics of localized magnetic moments (LMMs) comprising the domains. The pumped currents carry spin-polarized electrons which, in turn, exert backaction on LMMs in the form of nonlocal damping which is more than twice as large as conventional local Gilbert damping. The nonlocal damping can substantially modify the spectrum of emitted SWs when compared to widely-used micromagnetic simulations where conduction electrons are completely absent. Since we use fully microscopic (i.e., Hamiltonian-based) framework, self-consistently combining time-dependent electronic nonequilibrium Green functions with the Landau-Lifshitz-Gilbert equation, we also demonstrate that previously derived phenomenological formulas miss ultrabroadband spin pumping while underestimating the magnitude of nonlocal damping due to nonequilibrium electrons.

Equations

$$\mathcal{H}$$

$${\partial }_t{\mathbf{M}}_i$$

$${\mathbf{T}}_i\left[{\mathbf{M}}_i(t)\right]=J_{\mathrm{sd}}(⟨{\hat{\mathbf{s}}}_i{⟩}^{\mathrm{neq}}(t)-⟨{\hat{\mathbf{s}}}_i{⟩}_t^{\mathrm{eq}})\times {\mathbf{M}}_i(t),$$

$$\hat{H}(t)=-\gamma \sum _{⟨ij⟩}{\hat{c}}_i^{†}{\hat{c}}_i-J_{\mathrm{sd}}\sum _i{\hat{c}}_i^{†}\mathbf{\sigma }\cdot {\mathbb{M}}_i(t){\hat{c}}_i.$$

$${\mathbf{\rho }}_t^{\mathrm{eq}}=-\frac{1}{\pi }\int dE\mathrm{Im}{\mathbf{G}}_t^{r}f(E).$$

Full text

Introduction.—The control of the domain wall (DW) motion Tatara2008 ; Tatara2019 ; Kim2017a within magnetic nanowires by magnetic field or current pulses is both a fundamental problem for nonequilibrium quantum many-body physics and a building block of envisaged applications in digital memories. Parkin2015 logic Allwood2002 and artificial neural networks. Grollier2016 Since DWs will be closely packed in such devices, understanding interaction between them is a problem of great interest. Thomas2012 For example, head-to-head or tail-to-tail DWs—illustrated as the left (L) or right (R) noncollinear texture of localized magnetic moments (LMMs), respectively, in Fig. 1—behave as free magnetic monopoles carrying topological charge. Braun2012 The topological charge (or the winding number) $Q\equiv-\frac{1}{\pi}\int dx\,\partial_{x}\phi$, associated with winding of LMMs as they interpolate between two uniform degenerate ground states with $\phi=0$ or $\phi=\pi$, is opposite for adjacent DWs, such as $Q_{L}=-1$ and $Q_{R}=+1$ for DWs in Fig. 1. Thus, long-range attractive interaction between DWs can lead to their annihilation, resulting in the ground state without any DWs. Kunz2009 ; Kunz2010 ; Ghosh2017 ; Kim2015 This is possible because total topological charge remains conserved, $Q_{L}+Q_{R}=0$. The nonequilibrium dynamics Manton2004 triggered by annihilation of topological solitons is also of great interest in many other fields of physics, such as cosmology, Bradley2008 gravitational waves, Nakayama2017 quantum Manton2004 and string field Dvali2003 theories, liquid crystals Shen2019 and Bose-Einstein condensates. Takeuchi2012 ; Nitta2012

The recent experiment Woo2017 has monitored annihilation of two DWs within a metallic ferromagnetic nanowire by observing intense burst of spin waves (SWs) at the moment of annihilation. Thus generated large-amplitude SWs are dominated by exchange, rather than dipolar, interaction between LMMs and are, therefore, of short wavelength. The SWs of $\sim 10$ nm wavelength are crucial for scalability of magnonics-based technologies, Chumak2015 ; Kim2010 like signal transmission or memory-in-logic and logic-in-memory low-power digital computing architectures. However, they are difficult to excite by other methods due to the requirement for high magnetic fields. Navabi2017 ; Liu2018

The computational simulations of DW annihilation, Woo2017 ; Kunz2009 ; Kunz2010 together with theoretical analysis of generic features of such a phenomenon, Ghosh2017 have been based exclusively on classical micromagnetics where one solves coupled Landau-Lifshitz-Gilbert (LLG) equations Evans2014 for the dynamics of LMMs viewed as rotating classical vectors of fixed length. On the other hand, the dynamics of LMMs comprising two DWs also generates time-dependent fields which will push the surrounding conduction electrons out of equilibrium. The nonequilibrium electrons comprise pumped spin current Tserkovnyak2005 ; Petrovic2018 ; Chen2009 (as well as charge currents if the left-right symmetry of the device is broken Chen2009 ; Bajpai2019 ) in the absence of any externally applied bias voltage. The pumped spin currents flow out of the DW region into the external circuit, and since they carry away excess angular momentum of precessing LMMs, the backaction of nonequilibrium electrons on LMMs emerges Tserkovnyak2005 as an additional damping-like (DL) spin-transfer torque (STT).

The STT, as a phenomenon in which spin angular momentum of conduction electrons is transferred to LMMs when they are not aligned with electronic spin-polarization, is usually discussed for externally injected spin current. Ralph2008 But here it is the result of complicated many-body nonequilibrium state in which LMMs drive electrons out of equilibrium which, in turn, exert backaction in the form of STT onto LMMs to modify their dynamics in a self-consistent fashion. Petrovic2018 ; Sayad2015 Such effects are absent from classical micromagnetics or atomistic spin dynamics Evans2014 because they do not include conduction electrons. This has prompted derivation of a multitude of phenomenological expressions Zhang2009 ; Kim2012 ; Foros2008 ; Tserkovnyak2009 ; Hankiewicz2008 ; Yuan2014 ; Yuan2016 ; Thonig2018 for the so-called nonlocal (i.e., magnetization-texture-dependent) and spatially nonuniform (i.e., position-dependent) Gilbert damping that could be added into the LLG equation and micromagnetics codes Weindler2014 ; Wang2015 ; Verba2018 to capture the backaction of nonequilibrium electrons while not simulating them explicitly. Such expressions do not require spin-orbit (SO) or magnetic disorder scattering, which are necessary for conventional local Gilbert damping, Kambersky2007 ; Gilmore2007 ; Starikov2010 but they were estimated Kim2012 ; Hankiewicz2008 to be usually a small effect unless additional conditions (such as narrow DWs or intrinsic SO coupling splitting the band structure Kim2012 ) are present. On the other hand, a surprising result Weindler2014 of Gilbert damping extracted from experiments on magnetic-field-driven DW being several times larger than the value obtained from standard ferromagnetic resonance measurements can only be accounted by including additional nonlocal damping.

In this Letter, we unravel complicated many-body nonequilibrium state of LMMs and conduction electrons created by DW annihilation using recently developed Petrovic2018 ; Bajpai2019a ; Suresh2021 ; Suresh2020 ; Bostrom2019 quantum-classical formalism which combines time-dependent nonequilibrium Green function (TDNEGF) Stefanucci2013 ; Gaury2014 description of quantum dynamics of conduction electrons with the LLG equation description of classical dynamics of LMMs on each atom. Evans2014 Such TDNEGF+LLG formalism is fully microscopic, since it requires only the quantum Hamiltonian of electrons and the classical Hamiltonian of LMMs as input, and numerically exact. We apply it to a setup depicted in Fig. 1 where two DWs reside at time $t=0$ within a one-dimensional (1D) magnetic nanowire attached to two normal metal (NM) leads, terminating into the macroscopic reservoirs without any bias voltage.

Our principal results are: (i) annihilation of two DWs [Fig. 2] pumps highly unusual electronic spin currents whose power spectrum is ultrabroadband prior to the instant of annihilation [Fig. 3(d)], unlike the narrow peak around a single frequency for standard spin pumping; Tserkovnyak2005 (ii) because pumped spin currents carry away excess angular momentum of precessing LMMs, this acts as DL STT on LMMs which is spatially [Figs. 2(e) and  4(b)] and time [Fig. 2(g)] dependent, as well as $\simeq 2.4$ times larger [Fig. 2(f)] than conventional local Gilbert damping [Eq. (2)]. This turns out to be remarkably similar to $\simeq 2.3$ ratio of nonlocal and local Gilbert damping measured experimentally in permalloy, Weindler2014 but it is severely underestimated by phenomenological theories Zhang2009 ; Kim2012 [Fig. 4(a),(b)].

Models and methods.—The classical Hamiltonian for LMMs, described by unit vectors $\mathbf{M}_{i}(t)$ at each site $i$ of 1D lattice, is chosen as

[TABLE]

where $J=0.1\ {\rm eV}$ is the Heisenberg exchange coupling between the nearest-neighbor LMMs; $K=0.05\ {\rm eV}$ is the magnetic anisotropy along the $x$-axis; and $D=0.007\ {\rm eV}$ is the demagnetizing field along the $y$-axis. The last term in Eq. (1) is the Zeeman energy ($\mu_{B}$ is the Bohr magneton) describing the interaction of LMMs with an external magnetic field $\mathbf{B}_{\rm ext}$ parallel to the nanowire in Fig. 1 driving the DW dynamics, as employed in the experiment. Woo2017 The classical dynamics of LMMs is described by a system of coupled LLG equations Evans2014 (using notation $\partial_{t}\equiv\partial/\partial t$)

[TABLE]

where $\mathbf{B}^{\rm eff}_{i}=-\frac{1}{\mu_{M}}\partial\mathcal{H}/\partial\mathbf{M}_{i}$ is the effective magnetic field ($\mu_{M}$ is the magnitude of LMMs); $g$ is the gyromagnetic ratio; and the magnitude of conventional local Gilbert damping is specified by spatially- and time-independent $\lambda$, set as $\lambda=0.01$ as the typical value measured Weindler2014 in metallic ferromagnets. The spatial profile of a single DW in equilibrium, i.e., at time $t=0$ as the initial condition, is given by {\bf M}_{i}(Q,X_{\rm DW})=\big{(}\cos{\phi_{i}(Q,X_{\rm DW})},\ 0,\ \sin{\phi_{i}(Q,X_{\rm DW})}\big{)}, where ${\phi_{i}}(Q,X_{\rm DW})=Q\arccos\left[\tanh\left(x_{i}-X_{\rm DW}\right)\right]$; $Q$ is the topological charge; and $X_{\rm DW}$ is the position of the DW. The initial configuration of two DWs, $\mathbf{M}_{i}(t=0)=\mathbf{M}_{i}(Q_{\rm L},X_{\rm L})+\mathbf{M}_{i}(Q_{\rm R},X_{\rm R})$, positioned at sites $X_{\rm L}=15$ and $X_{\rm R}=30$ harbors opposite topological charges $Q_{\rm R}=-Q_{\rm L}=1$ around these sites.

In general, two additional terms Zhang2004 ; Zhang2009 ; Kim2012 in Eq. (2) extend the original LLG equation—STT due to externally injected electronic spin current, Ralph2008 which is actually absent $\mathbf{T}_{i}\left[I^{S_{\alpha}}_{\mathrm{ext}}\right]\equiv 0$ in the setup of Fig. 1; and STT due to backaction of electrons

[TABLE]

driven out of equilibrium by $\mathbf{M}_{i}(t)$. Here $J_{\rm sd}=0.1$ eV is chosen as the $s$-$d$ exchange coupling (as measured in permalloy Cooper1967 ) between LMMs and electron spin. We obtain “adiabatic” Stahl2017 ; Bajpai2020 electronic spin density, $\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{eq}}_{t}=\mathrm{Tr}\,[{\bm{\rho}}^{\mathrm{eq}}_{t}|i\rangle\langle i|\otimes{\bm{\sigma}}]$, from grand canonical equilibrium density matrix (DM) for instantaneous configuration of $\mathbf{M}_{i}(t)$ at time $t$ [see Eq. (5)]. Here ${\bm{\sigma}}=(\hat{\sigma}_{x},\hat{\sigma}_{y},\hat{\sigma}_{z})$ is the vector of the Pauli matrices. The nonequilibrium electronic spin density, $\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{neq}}(t)=\mathrm{Tr}\,[{\bm{\rho}}_{\mathrm{neq}}(t)|i\rangle\langle i|\otimes{\bm{\sigma}}]$, requires to compute time-dependent nonequilibrium DM, ${\bm{\rho}}_{\mathrm{neq}}(t)=\hbar\mathbf{G}^{<}(t,t)/i$, which we construct using TDNEGF algorithms explained in Refs. Croy2009, ; Popescu2016, and combine Petrovic2018 with the classical LLG equations [Eq. (2)] using time step $\delta t=0.1$ fs. The TDNEGF calculations require as an input a quantum Hamiltonian for electrons, which is chosen as the tight-binding one

[TABLE]

Here $\hat{c}_{i}^{\dagger}=(\hat{c}_{i\uparrow}^{\dagger},\hat{c}_{i\downarrow}^{\dagger})$ is a row vector containing operators $\hat{c}_{i\sigma}^{\dagger}$ which create an electron of spin $\sigma=\uparrow,\downarrow$ at the site $i$, and $\hat{c}_{i}$ is a column vector that contains the corresponding annihilation operators; and $\gamma=1$ eV is the nearest-neighbor hopping. The magnetic nanowire in the setup in Fig. 1 consists of 45 sites and it is attached to semi-infinite NM leads modeled by the first term in $\hat{H}$. The Fermi energy of the reservoirs is set at $E_{F}=0$ eV. Due to the computational complexity of TDNEGF calculations, Gaury2014 we use magnetic field $|\mathbf{B}_{\rm ext}|=100\ {\rm T}$ to complete DW annihilation on $\sim$ ps time scale (in the experiment Woo2017 this happens within $\sim 2$ ns).

Results.—Figure 2(a) demonstrates that TDNEGF+LLG-computed snapshots of $\mathbf{M}_{i}(t)$ fully reproduce annihilation in the experiment, Woo2017 including finale when SW burst is emitted at $t\simeq 7.2$ ps in Fig. 2(b). The corresponding complete spatio-temporal profiles are animated as a movie provided in the Supplemental Material (SM). sm However, in contrast to micromagnetic simulations of Ref. Woo2017, where electrons are absent, Fig. 2(d) shows that they generate spin density $\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{neq}}(t)$ which is noncollinear with either $\mathbf{M}_{i}(t)$ or $\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{eq}}_{t}$. This leads to “nonadiabaticity” angle $(\delta_{i}^{\mathrm{neq}}-\delta_{i}^{\mathrm{eq}})\neq 0$ in Fig. 2(d) and nonzero STT [Eq. (3) and Fig. 2(e)] as self-consistent backaction of conduction electrons onto LMMs driven out of equilibrium by the dynamics of LMMs themselves. The STT vector, $\mathbf{T}_{i}=\mathbf{T}^{\mathrm{FL}}_{i}+\mathbf{T}^{\mathrm{DL}}_{i}$, can be decomposed [see inset above Fig. 2(e)] into: (i) even under time-reversal or field-like (FL) torque, which affects precession of LMM around $\mathbf{B}^{\rm eff}_{i}$; and (ii) odd under time-reversal or DL torque, which either enhances Gilbert term [Eq. (2)] by pushing LMM toward $\mathbf{B}^{\rm eff}_{i}$ or competes with it as antidamping. Figure 2(f) shows that $\mathbf{T}_{i}^{\mathrm{DL}}\left[\mathbf{M}_{i}(t)\right]$ acts like an additional nonlocal damping while being $\simeq 2.4$ times larger than conventional local Gilbert damping $\lambda\mathbf{M}_{i}\times\partial_{t}\mathbf{M}_{i}$ [Eq. (2)].

The quantum transport signature of DW vanishing within the time interval $t=6.9$–$7.2$ ps in Fig. 2(a) is the reduction in the magnitude of pumped electronic spin currents [Fig. 3(a)–(c)]. In fact, $I^{S_{x}}_{R}(t)\rightarrow 0$ becomes zero [Fig. 3(a)] at $t=7.2$ ps at which LMMs in Fig. 2(a) turn nearly parallel to the $x$-axis while precessing around it. The frequency power spectrum [red curve in Fig. 3(d)] obtained from fast Fourier transform (FFT) of $I^{S_{z}}_{\mathrm{R}}(t)$, for times prior to completed annihilation and SW burst at $t=7.2$ ps, reveal highly unusual spin pumping over an ultrabroadband frequency range. This can be contrasted with the usual spin pumping Tserkovnyak2005 whose power spectrum is just a peak around a single frequency, Bocklage2017 as also obtained [brown curve in Fig. 3(d)] by FFT of $I^{S_{z}}_{R}(t)$ at post-annihilation times $t>7.2$ ps.

The spin current in Fig. 3(a)–(c) has contributions from both electrons moved by time-dependent $\mathbf{M}_{i}(t)$ and SW hitting the magnetic-nanowire/NM-lead interface. At this interface, SW spin current is stopped and transmuted Bauer2011 ; Suresh2021 ; Suresh2020 into an electronic spin current flowing into the NM lead. The transmutation is often employed experimentally for direct electrical detection of SWs, where an electronic spin current on the NM side is converted into a voltage signal via the inverse spin Hall effect. Woo2017 ; Chumak2012 Within the TDNEGF+LLG picture, SW reaching the last LMM of the magnetic nanowire, at the sites $i=1$ or $i=45$ in our setup, initiates their dynamics whose coupling to conduction electrons in the neighboring left and right NM leads, respectively, leads to pumping Suresh2021 of the electronic spin current into the NM leads. The properly isolated electronic spin current due to transmutation of SW burst, which we denote by $I^{S_{\alpha},\mathrm{SW}}_{p}$, is either zero or very small until the burst is generated in Fig. 3(e)–(g), as expected. We note that detected spin current in the NM leads was attributed in the experiment Woo2017 solely to SWs, which corresponds in our picture to considering only $I^{S_{\alpha},\mathrm{SW}}_{p}$ while disregarding $I^{S_{\alpha}}_{p}-I^{S_{\alpha},\mathrm{SW}}_{p}$.

Discussion.—A computationally simpler alternative to our numerical self-consistent TDNEGF+LLG is to “integrate out electrons” Sayad2015 ; Onoda2006 ; Nunez2008 ; Fransson2008 ; Hurst2020 and derive effective expressions solely in terms of $\mathbf{M}_{i}(t)$, which can then be added into the LLG Eq. (2) and micromagnetics codes. Weindler2014 ; Wang2015 ; Verba2018 For example, spin motive force (SMF) theory Yamane2011 gives $I_{\mathrm{SMF}}^{S_{x}}(x)=\frac{g\mu_{B}\hbar G_{0}}{4e^{2}}[\partial{\bf M}(x,t)/\partial t\times\partial{\bf M}(x,t)/\partial x]_{x}$ for the spin current pumped by dynamical magnetic texture, so that $\mathbf{M}\times\mathcal{D}\cdot\partial_{t}\mathbf{M}$ is the corresponding nonlocal Gilbert damping. Zhang2009 ; Kim2012 Here $\mathbf{M}(x,t)$ is local magnetization (assuming our 1D system); $\mathcal{D}_{\alpha\beta}=\eta\sum_{\nu}(\mathbf{M}\times\partial_{\nu}\mathbf{M})_{\alpha}(\mathbf{M}\times\partial_{\nu}\mathbf{M})_{\beta}$ (using notation $\alpha,\beta,\nu\in\{x,y,z\}$) is $3\times 3$ spatially-dependent damping tensor; and $\eta=\frac{g\mu_{B}\hbar G_{0}}{4e^{2}}$ with $G_{0}=G^{\uparrow}+G^{\downarrow}$ being the total conductivity. We compare in Fig. 4: (i) spatial profile of $I_{\mathrm{SMF}}^{S_{x}}(x)$ to locally pumped spin current $I_{i\rightarrow j}^{S_{x}}$ Suresh2021 from TDNEGF+LLG calculations [Fig. 4(a)] to find that the former predicts negligible spin current flowing into the leads, thereby missing ultrabroadband spin pumping predicted in Fig. 3(d); (ii) spatial profile of $\mathbf{M}\times\mathcal{D}\cdot\partial_{t}\mathbf{M}$ to DL STT $T_{i}^{\mathrm{DL}}$ from TDNEGF+LLG calculations, to find that the former has comparable magnitude only within the DW region but with substantially differing profiles. Note also that Suresh2021 $\left[\sum_{i}\mathbb{T}_{i}(t)\right]_{\alpha}=\frac{\hbar}{2e}\left[I^{S_{\alpha}}_{L}(t)+I^{S_{\alpha}}_{R}(t)\right]+\sum_{i}\frac{\hbar}{2}\frac{\partial\langle\hat{\mathrm{s}}^{\alpha}_{i}\rangle^{\mathrm{neq}}}{\partial t}$, which makes the sum of DL STT plotted in Fig. 2(g) time-dependent during collision, in contrast to the sum of local Gilbert damping shown in Fig. 2(g). The backaction of nonequilibrium electrons via $\mathbf{T}_{i}\left[\mathbf{M}_{i}(t)\right]$ can strongly affect the dynamics of LMMs, especially for the case of short wavelength SWs and narrow DWs, Zhang2009 ; Kim2012 ; Wang2015 ; Verba2018 as confirmed by comparing FFT power spectra of $M_{i}^{z}(t)$ computed by TDNEGF+LLG [Fig. 4(c),(d)] with those from LLG calculations [Fig. 4(e)] without any backaction.

We note that SMF theory Yamane2011 is derived in the “adiabatic” limit, Tatara2019 ; Stahl2017 which assumes that electron spin remains in the the lowest energy state at each time. “Adiabaticity” is used in two different contexts in spintronics with noncollinear magnetic textures—temporal and spatial. Tatara2019 In the former case, such as when electrons interact with classical macrospin due to collinear LMMs, one assumes that classical spins are slow and $\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{neq}}(t)$ can “perfectly lock” Tatara2019 to the direction $\mathbf{M}_{i}(t)$ of LMMs. In the latter case, such as for electrons traversing thick DW, one assumes that electron spin keeps the lowest energy state by rotating according to the orientation of $\mathbf{M}_{i}(t)$ at each spatial point, thereby evading reflection from the texture. Tatara2019 The concept of “adiabatic” limit is made a bit more quantitative by considering Tatara2019 ratio of relevant energy scales, $J_{\mathrm{sd}}/\hbar\omega\gg 1$ or $J_{\mathrm{sd}}/\mu_{B}|\mathbf{B}_{\rm ext}|\gg 1$, in the former case; or combination of energy and spatial scales, $J_{\mathrm{sd}}d_{\mathrm{DW}}/\hbar v_{F}=J_{\mathrm{sd}}d_{\mathrm{DW}}/\gamma a\gg 1$, in the latter case (where $v_{F}$ is the Fermi velocity, $a$ is the lattice spacing and $d_{\mathrm{DW}}$ is the DW thickness). In our simulations, $J_{\mathrm{sd}}/\mu_{B}|\mathbf{B}_{\rm ext}|\approx 10$ and $J_{\mathrm{sd}}d_{\mathrm{DW}}/\gamma a\approx 1$ for $d_{\mathrm{DW}}\approx 10a$ in Fig. 2(a). Thus, it seems that fair comparison of our results to SMF theory requires to substantially increase $J_{\mathrm{sd}}$. However, $J_{\mathrm{sd}}=0.1$ eV (i.e., $\gamma/J_{\mathrm{sd}}/\sim 10$, for typical $\gamma\sim 1$ eV which controls how fast is quantum dynamics of electrons) in our simulations is fixed by measured properties of permalloy. Cooper1967

Let us recall that rigorous definition of “adiabaticity” assumes that conduction electrons within a closed quantum system Stahl2017 at time $t$ are in the ground state $|\Psi_{0}\rangle$ for the given configuration of LMMs $\mathbf{M}_{i}(t)$, $|\Psi(t)\rangle=|\Psi_{0}[\mathbf{M}_{i}(t)]\rangle$; or in open system Bajpai2020 their quantum state is specified by grand canonical DM

[TABLE]

where the retarded GF, \mathbf{G}^{r}_{t}=\big{[}E-\mathbf{H}[\mathbf{M}_{i}(t)]-{\bm{\Sigma}}_{L}-{\bm{\Sigma}}_{R}\big{]}^{-1}, and ${\bm{\rho}}^{\mathrm{eq}}_{t}$ depend parametrically Bode2011 ; Thomas2012a ; Mahfouzi2016 (or implicitly, so we put $t$ in the subscript) on time via instantaneous configuration of $\mathbf{M}_{i}(t)$, thereby effectively assuming $\partial_{t}\mathbf{M}_{i}(t)=0$. Here $\mathrm{Im}\mathbf{G}^{r}_{t}=(\mathbf{G}^{r}_{t}-[\mathbf{G}^{r}_{t}]^{\dagger})/2i$; ${\bm{\Sigma}}_{L,R}$ are self-energies due to the leads; and $f(E)$ is the Fermi function. For example, the analysis of Ref. Yamane2011, assumes $\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{neq}}(t)\parallel\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{eq}}_{t}$ to reveal the origin of spin and charge pumping in SMF theory—nonzero angle $\delta_{i}^{\mathrm{eq}}$ between $\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{eq}}_{t}$ and $\mathbf{M}_{i}(t)$ with the transverse component scaling |\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{eq}}_{t}\times\mathbf{M}_{i}(t)|/\big{(}\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{eq}}_{t}\cdot\mathbf{M}_{i}(t)\big{)}\propto 1/J_{\mathrm{sd}} as the signature of “adiabatic” limit. Note that our $\delta_{i}^{\mathrm{eq}}\lesssim 4^{\circ}$ [Fig. 2(c)] in the region of two DWs (and $\delta_{i}^{\mathrm{eq}}\rightarrow 0$ elsewhere). Additional Figs. S1–S3 in the SM, sm where we isolate two neighboring LMMs from the right DW in Fig. 1 and put them in steady precession with frequency $\omega$ for simplicity of analysis, demonstrate that entering such “adiabatic” limit requires unrealistically large $J_{\mathrm{sd}}\gtrsim 2$ eV. Also, our exact Bajpai2020 result [Figs. S1(b), S2(b) and S3(b) in the SM sm ] shows |\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{eq}}_{t}\times\mathbf{M}_{i}(t)|/\big{(}\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{eq}}_{t}\cdot\mathbf{M}_{i}(t)\big{)}\propto 1/J_{\mathrm{sd}}^{2} (instead of $\propto 1/J_{\mathrm{sd}}$ of Ref. Yamane2011, ). Changing $\hbar\omega$—which, according to Fig. 3(c), is effectively increased by the dynamics of annihilation from $\hbar\omega\simeq 0.01$ eV, set initially by $\mathbf{B}_{\rm ext}$, toward $\hbar\omega\simeq 0.1$ eV—only modifies scaling of the transverse component of $\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{neq}}(t)$ with $J_{\mathrm{sd}}$ [Figs. S1(a),  S2(a), S3(a), S4(b) and S4(d) in the SM sm ]. The nonadiabatic corrections Bajpai2020 ; Bode2011 ; Thomas2012a ; Mahfouzi2016 take into account $\partial_{t}\mathbf{M}_{i}(t)\neq 0$. We note that only in the limit $J_{\mathrm{sd}}\rightarrow\infty$, \big{(}\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{neq}}(t)-\langle\hat{\mathbf{s}}_{i}\rangle^{\mathrm{eq}}_{t}\big{)}\rightarrow 0. Nevertheless, multiplication of these two limits within Eq. (3) yields nonzero geometric STT, Stahl2017 ; Bajpai2020 as signified by $J_{\mathrm{sd}}$-independent STT [Figs. S1(c), S2(c) and S3(c) in the SM sm ]. Otherwise, “nonadiabaticity” angle is always present $(\delta_{i}^{\mathrm{neq}}-\delta_{i}^{\mathrm{eq}})\neq 0$ [Fig. 2(d)], even in the absence of spin relaxation due to magnetic impurities or SO coupling, Evelt2017 and it can be directly related to additional spin and charge pumping Suresh2020 ; Evelt2017 [see also Figs. S1(f), S2(f) and S3(f) in the SM sm ].

Conclusions and outlook.—The pumped spin current over ultrabroadband frequency range [Fig. 3(d)], as our central prediction, can be converted into rapidly changing transient charge current via the inverse spin Hall effect. Wei2014 ; Seifert2016 ; Chen2019 Such charge current will, in turn, emit electromagnetic radiation covering $\sim 0.03$–$27$ THz range (for $|\mathbf{B}_{\mathrm{ext}}|\sim 1$ T) or $\sim 0.3$–$27.3$ THz range (for $|\mathbf{B}_{\mathrm{ext}}|\sim 10$ T), which is highly sought range of frequencies for variety of applications. Seifert2016 ; Chen2019