Dynamical Spin-Orbit-Based Spin Transistor

F. N. Gürsoy1, P. Reck2, C. Gorini2,3, K. Richter 2, I. Adagideli 1,4*

1 Faculty of Engineering and Natural Sciences, Sabancı University, Orhanlı-
Tuzla, 34956, Turkey

2 Institut für Theoretische Physik, Universität Regensburg, D-93040
Regensburg, Germany

3 Université Paris Saclay, CEA, CNRS, SPEC, 91191, Gif-sur-Yvette, France

4 MESA+ Institute for Nanotechnology, University of Twente, 7500 AE Enschede,
The Netherlands *<EMAIL_ADDRESS>
## Abstract

Spin-orbit interaction (SOI) has been a key tool to steer and manipulate spin-
dependent transport properties in two-dimensional electron gases. Here we
demonstrate how spin currents can be created and efficiently read out in nano-
or mesoscale conductors with time-dependent and spatially inhomogenous Rashba
SOI. Invoking an underlying non-Abelian SU(2) gauge structure we show how
time-periodic spin-orbit fields give rise to spin-motive forces and enable the
generation of pure spin currents of the order of several hundred nano-Amperes.
In a complementary way, by combining gauge transformations with ”hidden”
Onsager relations, we exploit spatially inhomogenous Rashba SOI to convert
spin currents (back) into charge currents. In combining both concepts, we
devise a spin transistor that integrates efficient spin current generation, by
employing dynamical SOI, with its experimentally feasible detection via
conversion into charge signals. We derive general expressions for the
respective spin- and charge conductances, covering large parameter regimes of
SOI strength and driving frequencies, far beyond usual adiabatic approaches
such as the frozen scattering matrix approximation. We check our analytical
expressions and approximations with full numerical spin-dependent transport
simulations and demonstrate that the predictions hold true in a wide range
from low to high driving frequencies.

###### Contents

  1. 1 Introduction
  2. 2 Model and its non-Abelian gauge structure
    1. 2.1 General case
    2. 2.2 Spin electric field from time-dependent SOI
    3. 2.3 Spin magnetic field from spatially inhomogeneous SOI
  3. 3 Transport formalism: linear response and Floquet theory
    1. 3.1 Linear response currents
      1. 3.1.1 Basic expressions for charge and spin conductance
      2. 3.1.2 Link between spin and charge conductance via gauge transformation
    2. 3.2 Floquet scattering theory
      1. 3.2.1 General framework
      2. 3.2.2 Floquet theory in the adiabatic limit
  4. 4 Spin current generation
    1. 4.1 AC spin current in the low-frequency regime
    2. 4.2 AC spin current in the general frequency regime
    3. 4.3 AC spin current generation: comparison with numerics
    4. 4.4 Adiabatic approximation versus analytical results
  5. 5 Charge signal from a spin current
    1. 5.1 Dynamical SOI-based spin transistor setting
    2. 5.2 AC charge signal in the low-frequency regime
    3. 5.3 AC charge signal in the high-frequency regime
    4. 5.4 Simulating the dynamical spin-transistor functionality
  6. 6 Conclusions
  7. A Tight-binding Hamiltonian for 2DEG with Rashba SOI
  8. B Floquet Hamiltonian with time-dependent Rashba SOI
  9. C Comparing the Floquet scattering matrix with its adiabatic approximation
  10. D Spin and charge current calculation in the Landauer-Büttiker Formalism
  11. E Calculation of the dwell time
  12. F Role of the number of Floquet bands for the numerical results

##  1 Introduction

Following the theoretical proposal for a spin field-effect transistor by Datta
and Das in 1990 [1], much research has focused on the realization of spin-
based transistors that might eventually be more efficient and faster compared
to the present transistors based on charge transport. The central issue for
spin-based transistors is the realization of efficient generation,
manipulation and detection schemes for spin currents and spin accumulations.
While the conventional means of doing this involve ferromagnetic structures
[2], these steps could also be achieved via all-electrical means, hence
bypassing the need for magnetic components, by exploiting spin-orbit
interaction (SOI). The proposals based on exploiting the phenomena of current-
induced spin accumulation [3] and the spin Hall effect – the generation of
spin currents transverse to an applied electric field [4] – became the most
commonly employed lines of study to achieve this goal.

The SOI can be induced by the electric fields of either charge impurities or
the crystal lattice. In the latter case, one speaks of an “intrinsic” SOI.
Prominent examples are Rashba- and Dresselhaus-type SOI in semiconductor
quantum wells [5]. While the Dresselhaus SOI arises from a crystal’s lack of
inversion center, the Rashba term arises in low-dimensional systems, and it is
induced by inversion symmetry breaking generated by the confining electric
field of a quantum well. Because this electric field can be tuned by top or
back gates, as established in two-dimensional electron gases (2DEGs) over 20
years ago [6], it is possible to modulate the Rashba SOI both spatially, and
in principle, temporally. A spatial variation can generate a spin dependent
force [7], and a temporally oscillating SOI may be capable of inducing charge
and spin currents in the absence of any bias voltage [8, 9]. Such physics is
most conveniently handled by rewriting the SOI in terms of non-Abelian gauge
fields [10, 11, 12]. Among other things, such an approach allows one to
directly identify spin-electric and spin-magnetic fields [8, 14, 13, 16, 15].
The latter accelerate electrons longitudinally or sideways à la Lorentz,
respectively, but their sign depends on the electronic spin – opposite spin
species are accelerated and bent in opposite directions. While these fields
may exist in the presence of time-independent and homogeneous spin-orbit (and
Zeeman) interaction, additional components appear when the SOI is time- and
space-dependent [14, 13]. Such components were recently studied and proposed
as additional means to control spin transport at the mesoscopic level [16].

In this manuscript we will exploit the time- and space-tunability of Rashba
SOI to explore how the non-Abelian (spin) gauge fields induce spin transport.
We note from the outset that the effect of driven Rashba SOI has been studied
in different contexts ranging from diffusive metallic systems [13] to
ballistic quantum rings [17], to 1D electron gases [18, 19] and graphene [20].
However, usual Onsager reciprocal relations valid for spin transport [22, 21,
15] become more restrictive in the presence of a non-Abelian gauge structure
such as induced by the space- and time-dependent Rashba interaction. Here we
will explore the consequences of these “hidden” Onsager relations recently
established for spin and charge transport in 2DEGs [16]. Among other things,
our approach will allow us to obtain general formulae that go beyond the
frozen scattering matrix approximation [23] valid in the adiabatic limit. As a
demonstration, we will devise a multi-terminal mesoscopic spin transistor,
whose working principle builds upon the “hidden” Onsager relations [16]. While
the transitor operates on pure spin currents, the output signal is a charge
signal which can be detected by simple experimental procedures. The spin
transistor setting, combining time-dependent and spatially-inhomogenous SOI,
is shown schematically in Fig. 1: The AC-modulated Rashba SOI on the right
injects a pure spin current into the left region. The latter is then converted
into a charge signal by a spin magnetic field, induced by a static but non-
homogeneous Rashba SOI, and read out as a voltage $V_{\rm out}$ between
contacts 1 and 2.

Figure 1: Sketch of the dynamical spin transistor. A time-dependent top gate
voltage generates a spin motive force on the right. Then the generated spin
current is read out as a charge current on the left part via spatially
inhomogeneous Rashba coupling.

The paper is organized as follows. In Sec. 2, we introduce the model
Hamiltonian and the non-Abelian gauge field approach, while in Sec. 3 we
define the transport formalism, where we also discuss basics of the Floquet
theory that will be necessary for our treatment. Our gauge field analytics are
based on Ref. [16], allowing us to approximately express the desired spin-
dependent quantities (time-dependent spin currents, AC spin conductances,
etc.) in terms of conventional charge transport quantities (time-dependent
charge currents, impedances, etc.), and to make general symmetry-based
predictions concerning the expected output signals. We test such predictions
against Floquet-based numerics which do not rely on any non-Abelian gauge
manipulation, and which also allow us to go beyond analytically tractable low-
frequency regimes. Thereby we demonstrate that our analytical expressions are
applicable in a large range from low to high driving frequencies. In Sec. 4 we
discuss spin current generation in a mesoscopic sample with homogeneous but
time-dependent Rashba SOI, while in Sec. 5 we show how such spin signals can
be non-locally converted into charge ones by the non-homogeneous but static
Rashba SOI within a second mesoscopic cavity. This combined functionality is
integrated and tested for a dynamical SOI-based spin transistor. Each section
opens with low-frequency transport, before moving on to the high-frequency
regime. Finally we present our conclusions in Sec. 6. To keep the manuscript
as self-contained as possible, we gather a series of technical details in the
appendices.

##  2 Model and its non-Abelian gauge structure

###  2.1 General case

Our starting point is the standard low-energy model for a spin-orbit coupled
electron or hole gas in 2D [5]. The Hamiltonian reads

$H=\frac{p^{2}}{2m}+{\bf b}(\bf p)\cdot\sigma+W(\bf x).$ (1)

Here $W({\bf x})$ is the electrostatic potential specifying the static
environment, which might originate from gates, applied bias, impurities etc.,
$m$ is the effective electron mass, and ${\bf b}({\bf p})$ is a spin-orbit
field, coupling spin to momentum. We assume that the strength of this SOI can
be controlled externally and more generally, both in a position- and time-
dependent way: ${\bf b}({\bf p})\to{\bf b}({\bf p};{\bf x},t)$. For the rest
of the paper we specialize to electrons and light holes for which this field
is linear in momentum. Then the Hamiltonian can be rewritten introducing non-
Abelian gauge fields. 111 See e.g. Refs. [10, 11, 14, 13, 16, 15]. Note that
the representation in terms of non-Abelian gauge fields is actually exact, up
to an irrelevant constant, if the spin-orbit field is constant in time and
homogeneous in space. Following the notation of Ref. [16], the Hamiltonian
reads

$H=-\frac{D_{\mu}D_{\mu}}{2m}+V({\bf x}).$ (2)

Here $D_{\mu}=\partial_{\mu}-(ik_{so}/2)\sigma^{a}A^{a}_{\mu}$ is the $SU(2)$
covariant derivative along $\mu=x,y$, with $\sigma^{a}$ the Pauli matrices
($a=x,y,z$), $A^{a}_{\mu}({\bf x},t)$ a dimensionless $SU(2)$ vector potential
and $V=W-k_{so}^{2}(A^{a}_{\mu}A^{a}_{\mu})/(8m)$. Unless specified otherwise,
we assume that the repeated indices are summed over. The SOI strength is
controlled by the parameter $k_{so}$ which is typically much smaller than the
Fermi momentum $k_{F}$. In terms of this spin-orbit parameter, the accuracy of
Eq. (2) is ${\cal O}(k_{so}/k_{F})^{2}$. We now make the central assumption
that $L\ll l_{so}$, where $l_{so}=\pi/|k_{so}|$ is the spin-orbit length and
$L$ the system size. This is usually fulfilled for experimental realizations
using systems at nano- to mesoscales. We also assume that the time-dependence
of $A_{\mu}^{a}(t)$ is slow, then
$x_{\mu}k_{so}\partial_{t}A_{\mu}^{a}\ll\epsilon_{F}$, where $\epsilon_{F}$ is
the Fermi energy. $SU(2)$ gauge transformations are unitary transformations of
the form

$U=\exp(i\Lambda_{a}({\bf x},t)\sigma_{a}/2)\,.$ (3)

It is then straightforward to show that this transformation maps
$A^{a}_{\mu}({\bf x},t)\rightarrow(A^{\prime})^{a}_{\mu}({\bf x},t)$ and
$V({\bf x},t)\rightarrow V^{\prime}({\bf x},t)$, where

$\displaystyle(A^{\prime})^{a}_{\mu}$
$\displaystyle=A^{a}_{\mu}-\epsilon^{abc}\Lambda^{b}A^{c}_{\mu}+\frac{1}{k_{so}}\partial_{\mu}\Lambda^{a}\quad,$
(4) $\displaystyle V^{\prime}$
$\displaystyle=V-\sigma^{a}\frac{1}{2}\frac{\partial\Lambda^{a}}{\partial
t}\quad.$ (5)

These transformations allow us, among other advantages, to gauge away the
homogeneous and time-independent components of the spin-orbit field, up to
quadratic order in the coupling constant [12, 24].

For a concrete example, we now specialize to a 2D electron gas with a Rashba
spin-orbit interaction 222 We note that the essential spin transport physics
of a 2DEG with linear-in-momentum SOI is captured by Eq. (6), while details
such as the precise spin polarization direction will depend on the presence
and the strength of additional contributions, e.g. à la Dresselhaus.

$H=\frac{p^{2}}{2m}+\frac{1}{2}\\{\alpha_{R},\left(\sigma^{x}p_{y}-\sigma^{y}p_{x}\right)\\}+V({\bf
x})\,,$ (6)

where the Rashba coupling constant $\alpha_{R}$ can be a function of both
position and time. However, for the sake of simplicity, we focus on regions of
either time- or spatially dependent Rashba coupling, i.e.
$\alpha_{R}(t),\alpha_{R}({\bf x})$, and combine them later using rules for
the combination of the respective scattering matrices. We identify and study
two main effects, the generation of spin electric and spin magnetic fields to
be discussed in the following two subsections.

###  2.2 Spin electric field from time-dependent SOI

A time-dependent Rashba SOI constant $\alpha(t)=k_{so}\sin(\Omega t)$, with
$T=2\pi/\Omega$ the AC modulation period from a top gate[6], will generate the
spin-motive driving forces, see Fig. 2.

Figure 2: Sketch of the spin current source. A time-dependent top gate voltage
is applied to a (chaotic) ballistic cavity connected to two leads, which
controls the Rashba coupling and thus creates a spin motive force.

In this case $A_{\mu}^{a}(t)=\epsilon_{\mu}^{a}\sin(\Omega t)$, and the
$SU(2)$ gauge transformation (3) becomes

$U=\exp\big{(}{-ix_{\mu}{k_{so}}A^{a}_{\mu}\sigma^{a}/2}\big{)}\,.$ (7)

To order $(k_{so}L)^{2}$ one obtains a vanishing vector potential, while
$\partial_{t}A_{\mu}^{a}(t)$ generates a non-zero $SU(2)$ scalar potential

$\displaystyle(A^{\prime})_{\mu}^{a}=$ $\displaystyle\,0\,,$ (8)
$\displaystyle(V^{\prime})^{a}=$ $\displaystyle
x_{\mu}{k_{so}}\partial_{t}A_{\mu}^{a}\sigma^{a}/2\,,$ (9)

yielding the transformed Hamiltonian

$H^{\prime}=-\frac{1}{2m}\partial_{\mu}\partial_{\mu}+(x_{\mu}{k_{so}}\partial_{t}A_{\mu}^{a})\frac{\sigma^{a}}{2}+V({\bf
x}).$ (10)

A further global spin rotation[25] $\sigma^{a}\rightarrow\sigma^{z}$ then
leads to the diagonal Hamiltonian

$\displaystyle H_{d}=$ $\displaystyle\frac{p^{2}}{2m}+V({\bf
x})+\frac{V^{s}(t)}{2}\sigma^{z},$ (11) $\displaystyle V^{s}\equiv$
$\displaystyle\epsilon_{\mu}^{s}x_{\mu}k_{so}\partial_{t}\sin(\Omega t)\quad,$
(12)

i.e. opposite-spin electrons feel a different electric field

${\mathcal{E}}_{\mu}^{\uparrow\downarrow}=-\partial_{\mu}\left[V({\bf x})\pm
V^{s}(t)/2\right]\equiv E_{\mu}\pm{\mathcal{E}}^{s}_{\mu}.$ (13)

The component ${\mathcal{E}}^{s}_{\mu}$ is a spin-electric field accelerating
opposite spin species in opposite directions.

###  2.3 Spin magnetic field from spatially inhomogeneous SOI

A spatially inhomogeneous Rashba SOI constant $\alpha_{R}({\bf
x})=k_{so}\bar{\alpha}_{R}(\mathbf{x})$, with $\bar{\alpha}_{R}$ being a
dimensionless function, can either create spin currents or convert spin
currents into charge currents, see Fig. 1. This interconversion is achieved by
the spin-dependent Lorentz force due to a ”spin” magnetic field [16, 24]

$\mathcal{B}^{a}=\partial_{x}A_{y}^{a}({\bf x})-\partial_{y}A_{x}^{a}({\bf
x})$ (14)

where $A_{\mu}^{a}({\bf x})=\bar{\alpha}_{R}(\mathbf{x})\epsilon^{a}_{\mu}.$
Below we specialize to the case

$A_{\mu}^{a}=k_{so}\alpha_{0}(\mathbf{x}\cdot\mathbf{f})\epsilon^{a}_{\mu},$
(15)

with $\mathbf{f}$ a unit vector determining the fixed direction of the
gradient of $\bar{\alpha}_{R}$. We now use the decomposition

$A_{\mu}^{a}=-(\partial_{\mu}\chi^{a}+\epsilon_{\mu\nu}\partial_{\nu}\phi^{a})$
(16)

for each spin component. Then we perform an $SU(2)$ gauge transformation

$U=\exp\left({ik_{so}\,\chi^{a}\sigma^{a}/2}\right),$ (17)

and consider effects to linear order in $k_{so}$. The vector potential becomes

$\displaystyle(A^{\prime})_{\mu}^{a}=$
$\displaystyle\,\epsilon_{\mu\nu}\,\partial_{\nu}\,\phi^{a},$ (18)
$\displaystyle(V^{\prime})^{a}=$ $\displaystyle\,V.$ (19)

The transformed Hamiltonian becomes, up to linear order in the spin-orbit
coupling,

$H^{\prime}=-\frac{1}{2m}[\partial_{\mu}+i\frac{k_{so}}{2}\epsilon_{\mu\nu}\partial_{\nu}\varphi({\bf
x})\mathbf{\sigma}\cdot{\bf f}]^{2}+V({\bf x})$ (20)

where $\phi^{a}=\varphi({\bf x})f^{a}$. We now perform a global spin rotation
$\mathbb{\sigma}\cdot{\bf f}\rightarrow\sigma^{z}$ to obtain

$H_{d}=\frac{1}{2m}({\bf p}+k_{so}{\bf a})^{2}\sigma^{z}+V(\bf x),$ (21)

where ${\bf a}=\alpha_{0}(\hat{\bf z}\times{\bf x})/2$. Hence the hamiltonian
structure implies that the two spin species decouple, each subject to a
homogeneous magnetic field (as well as Lorentz force) of opposite sign. The
strength of this magnetic field is given by $\sum_{a}\mathcal{B}^{a}f^{a}$, in
agreement with Eq. (15).

##  3 Transport formalism: linear response and Floquet theory

###  3.1 Linear response currents

####  3.1.1 Basic expressions for charge and spin conductance

We consider a mesoscopic sample attached to leads labeled by $i,j$. The leads
are in contact with different metallic reservoirs used to apply spin voltages.
The linear response Landauer-Büttiker formula for the $a$-polarized spin
current flowing into/out of contact $i$ is generalized as follows [26, 27]

$I_{i}^{\alpha}=\sum_{\beta}(2N_{i}\delta_{\alpha\beta}-G^{\alpha\beta}_{ii})V^{\beta}_{i}-\sum_{\beta}\sum_{j\neq
i}G^{\alpha\beta}_{ij}V^{\beta}_{j},$ (22)

where $N_{i}$ is the number of channels in lead $i$. We use Latin letters
(a=x,y,z) to denote the spin components of the current, while Greek letters
commonly describe both electric (0) and spin components (a). Physically, the
spin voltage $V^{b}_{j}=\mu^{b}_{j}/e$ represents an imbalance of spins
polarized along a $b$-axis in reservoir $j$, with spin accumulation
$\mu^{b}_{j}=\mu^{(\uparrow)}_{j}-\mu^{(\downarrow)}_{j}$. The conductance is
a ($4\times 4$)-matrix in the combined spin-charge space:

$G_{ij}=\left(\begin{array}[]{cc}G_{ij}^{00}&G_{ij}^{0b}\\\
G_{ij}^{a0}&G_{ij}^{ab}\end{array}\right),$ (23)

where the superscript “0” indicates the charge component. Here $G\equiv
G^{00},G^{0b},G^{b0}$ and $G^{ab}$ are, respectively, the charge, charge-spin,
spin-charge and spin-spin conductances. They are defined as

$\displaystyle G^{00}_{ij}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{h}\sum_{m,n}{\rm Tr}[t_{mn}^{\dagger}t_{mn}]\,,$
(24) $\displaystyle G^{0b}_{ij}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{h}\sum_{m,n}{\rm
Tr}[t_{mn}^{\dagger}t_{mn}\sigma^{b}]\,,$ (25) $\displaystyle G^{b0}_{ij}$
$\displaystyle=$ $\displaystyle\frac{e^{2}}{h}\sum_{m,n}{\rm
Tr}[t_{mn}^{\dagger}\sigma^{b}t_{mn}]\,,$ (26) $\displaystyle G^{ab}_{ij}$
$\displaystyle=$ $\displaystyle\frac{e^{2}}{h}\sum_{m,n}{\rm
Tr}[t_{mn}^{\dagger}\sigma^{a}t_{mn}\sigma^{b}]\,.$ (27)

Here, $t_{mn}$ is the $(2\times 2)$-matrix of spin dependent transmission
amplitudes connecting channel $n$ in lead $j$ to channel $m$ in lead $i$, and
$\sigma^{a}$ are the Pauli spin matrices.

Even in the absence of any charge bias the system can feature a charge
response given by

$I^{0}_{i}=-\sum\limits_{b}G_{ii}^{0b}V^{b}_{i}-\sum_{b}\sum\limits_{j\neq
i}G_{ij}^{0b}V_{j}^{b},$ (28)

in addition to the spin response

$I^{a}_{i}=\sum\limits_{b}(2N_{i}\delta_{ab}-G^{ab}_{ii})V^{b}_{i}-\sum_{b}\sum\limits_{j\neq
i}G_{ij}^{ab}V_{j}^{b}.$ (29)

####  3.1.2 Link between spin and charge conductance via gauge transformation

The gauge transformations discussed in Sec. 2 allow us to express the
mesoscopic spin conductance $G^{a}_{ij}=G^{\uparrow}_{ij}-G^{\downarrow}_{ij}$
through charge conductances. In the case of a 2DEG with spatially
inhomogeneous Rashba SOI described in Sec. 2.3 the $a$-th component of the
spin conductance can be written in terms of the charge conductance depending
on the spin-magnetic field ${\cal B}^{a}_{z}$ in Eq. (15):

$G_{ij}^{a}\approx\left[G_{ij}({\cal B}^{a}_{z})-G_{ij}(-{\cal
B}^{a}_{z})\right]f^{a}.$ (30)

Using Onsager relations and current conservation, one can show that
$G_{ij}^{a}$ in Eq. (30) is only non-zero in the absence of time reversal
symmetry. Under the assumption $L\ll l_{so}$ – i.e. a moderate Rashba constant
$k_{so}$ – the spin precession conductance $G^{ab}$ is equivalent to the
charge conductance, $G^{ab}\propto\delta^{ab}G^{00}$. Hence all spin
conductances in Eqs. (28) and (29) can be expressed in terms of the charge
conductance to linear order in $k_{so}$. Besides, the spin voltages in Eqs.
(28) and (29) can be obtained in a 2DEG with time-dependent Rashba SOI, where
up and down spins generate a spin dependent voltage as
$V^{a}=V^{\uparrow}-V^{\downarrow}$.

In our full numerical calculations below we will compute the driven charge and
spin currents without resorting to the approximate $SU(2)$ manipulations
described here and in Sec. 2. Using these numerics as reference calculations,
we will be able to show that the description in terms of spin-electric and
spin-magnetic fields acting on effectively decoupled
$\uparrow\downarrow$-electrons is very accurate up to $L\sim l_{so}$.

###  3.2 Floquet scattering theory

####  3.2.1 General framework

When interacting with a dynamical scatterer with oscillation frequency
$\Omega$, an electron impinging at energy $E$ can absorb or emit an integer
amount of energy quanta $n\hbar\Omega$, leaving the scattering region with
energy $E_{n}=E+n\hbar\Omega$ ($n=0,\pm 1,\pm 2,\dots$). The scattering matrix
then depends on the initial and final energies $E$ and $E_{n}$, and is
referred to as the Floquet scattering matrix, $S_{F,im,jm^{\prime}}(E_{n},E)$
[28]. This matrix gives the amplitude for a process where an electron from
channel $m^{\prime}$ in lead $j$ at energy $E$ is scattered into channel $m$
within lead $i$ at energy $E_{n}$.

We now calculate the Floquet scattering matrix. To this end, one has to
numerically solve the time-dependent Schrödinger equation. As reviewed in
Appendix B, Floquet theory allows one to express the full time evolution in
terms of an effective static matrix Hamiltonian, expressed in terms of the
Floquet states $|n\rangle$,

$H_{F}=\sum_{n=-\infty}^{\infty}\big{(}(H_{0}-n\hbar\Omega)|n\rangle\langle
n|+\frac{iH_{1}}{2}(|n\rangle\langle n+1|-|n\rangle\langle n-1|)\big{)}\,.$
(31)

Here, $H_{0}=-\frac{\hbar^{2}}{2m}(\partial_{x}^{2}+\partial_{y}^{2})$ and
$H_{1}=ik_{so}(\sigma_{y}\partial_{x}-\sigma_{x}\partial_{y})\,$ are the
kinetic and the Rashba contributions – recall from Sec. 2 that the time-
modulated Rashba SOI is homogeneous, i.e. $\alpha_{R}=\alpha_{R}(t)$. The
Floquet states $|n\rangle$ are defined as the basis vectors of the periodic
eigenfunctions $|\phi_{\eta}(\vec{r},t)\rangle$ of the Floquet Hamiltonian

$|\phi_{\eta}(\vec{r},t)\rangle=\sum_{n}e^{-in\Omega t}|n\rangle\,.$ (32)

For numerical calculations one has to truncate the (infinite) sum over Floquet
bands $n$. In Appendix F we show that including up to 21 bands ($|n|\leq 10$)
provides enough accuracy for our purposes. After obtaining the elements of the
Floquet scattering matrix, we calculate the AC currents in the absence of a
bias voltage. The charge/spin currents are expressed as

$I_{i}^{\alpha}(t)=\sum_{l=-\infty}^{\infty}e^{-il\Omega
t}I_{i,l}^{\alpha}\,.$ (33)

The Fourier components are then determined in terms of the Floquet scattering
matrices $S_{F}(E_{n},E)$ as

$I_{i,l}^{\alpha}=\frac{e}{h}\frac{\hbar}{2}\int
dE\sum_{n=-\infty}^{\infty}\sum_{j=1}^{N_{r}}\sum_{m\in i,m^{\prime}\in
j}Tr[S^{\dagger}_{F,im,jm^{\prime}}(E_{n},E)\sigma^{\alpha}S_{F,im,jm^{\prime}}(E_{l+n},E)]\,.$
(34)

where $N_{r}$ denotes the number of leads, and $\sigma^{0}=1$. We refer the
readers to Appendix D for further details.

To obtain the dynamical scattering amplitude and determine the spin and charge
currents, we use the tight binding form of the Floquet Hamiltonian (31) (see
Appendix A) and calculate the scattering matrix using the software package
Kwant [29]. We consider only $l=\pm 1$ for the AC response.

We will consider a wide range of frequencies, numerically probing both the
adiabatic and high-frequency regimes. These are determined by the typical
internal energy scales of the scattering matrix. For few channel ballistic
transport, this scale is related to the inverse time of flight of an electron
between two leads, which is calculated using the Wigner-Smith time-delay
matrix [30, 31] as shown in Appendix E. If the frequency is much smaller than
the inverse of the time of flight, $\Omega\tau\ll 1$, the scattering process
is considered to be in the adiabatic limit.

####  3.2.2 Floquet theory in the adiabatic limit

In the adiabatic limit one can rely on the so-called frozen scattering matrix
$S(t,E)$, which is defined as the static scattering matrix calculated with the
time-dependent parameters at a given time $t$,

$S(t,E)=\sum_{n=-\infty}^{\infty}e^{-in\Omega t}S_{n}(E)\,.$ (35)

In the adiabatic limit the Floquet scattering matrix can be expanded in powers
of the frequency, and the Fourier transform of the frozen scattering matrix
$S_{n}(E)$ gives the zeroth order approximation of the Floquet scattering
matrix (see Appendix C),

$S_{F}(E_{n},E)=S_{n}+O(\Omega)\,.$ (36)

We will compute the adiabatic approximation (36) of the Floquet scattering
matrix and check the limits of its validity in Appendix C.

##  4 Spin current generation

We first study spin current generation from a spin-dependent voltage using Eq.
(29) in both the adiabatic and high frequency regimes. We consider a 2D
chaotic cavity with only time-dependent Rashba coupling, induced by an AC top
gate voltage, see Fig. 2. The Rashba SOI strength has a time dependence given
by $\alpha_{R}(t)=k_{so}\sin(\Omega t)$, with $\Omega$ the driving frequency.
As we discussed above, this time-dependent coupling induces a spin voltage on
the leads, and the spin voltage difference between two leads results in a spin
current polarized in the $y$-direction. Throughout this section we will be
working in the linear response regime and ignore effects nonlinear in the
spin-orbit coupling. We derive our general spin current formulae in Secs. 4.1
and 4.2. Next we present our numerical results in Sec. 4.3, also including a
comparison with our analytical results in Sec. 4.4.

###  4.1 AC spin current in the low-frequency regime

In the low-frequency regime, one can neglect the frequency dependence of the
charge conductance. The spin current in Eq. (29) can thus be computed using
the DC conductance. To leading order in the spin-orbit coupling, all spin
effects are included in the spin voltage. Hence $G^{ab}=\delta^{ab}G$. We note
that the case of a large Rashba coupling with a small time-dependent part can
also be treated with our gauge transformation method. Then the results of this
section needs to be modifed to account for spin precession. In the absence of
a bias voltage ($V^{0}=0$), the spin current at lead 1 is the current
generated by the spin voltage

$I^{a}_{1}=\sum\limits_{j}(2N_{1}\delta_{1j}-G_{1j})V_{j}^{a}.$ (37)

The spin voltage from the time-dependent Rashba SOI reads

$V^{a,\uparrow(\downarrow)}=\pm\partial_{t}\alpha_{R}(t)\epsilon^{a}_{\mu}{\it{l}_{\mu}}.$
(38)

Here $\it{l}_{\mu}$ is the system size in the direction $\mu=x,y$. For the
system shown in Fig. 2, Rashba SOI generates a spin voltage with spin
direction $y$ along the $x$ axis. Up and down spins feel opposite voltage
biases, hence total the spin voltage bias is

$V_{1(2)}^{y}=V_{1(2)}^{\uparrow}-V_{1(2)}^{\downarrow}=\pm\partial_{t}\alpha_{R}(t)L_{12}$
(39)

with $L_{ij}$ the distance between lead $i$ and lead $j$ According to Eq.
(37), this spin voltage then drives a spin current

$I^{y}_{1}(t)=(2N_{1}-G_{11}+G_{12})\,\partial_{t}\alpha_{R}(t)\,L_{12}\,.$
(40)

To summarize, we expressed the spin current in terms of the DC charge
conductance and time-dependent $\alpha_{R}$ in the small-$\Omega$ limit. This
is our first prediction for the spin current in the low-frequency regime.

###  4.2 AC spin current in the general frequency regime

In a system driven by a time-dependent voltage, both particle and displacement
currents will appear. In order to take these into account, we no longer
neglect the frequency-dependence of the AC conductance, which in Fourier space
is given by the admittance $G_{ij}(\omega)$. In linear response we have

$G_{ij}(\omega)=\langle\delta I_{i}(\omega)/\delta V_{j}(\omega)\rangle.$ (41)

In the presence of a bias voltage $V(t)=V(\omega)e^{i\omega t}$, the current
$I_{i}(\omega)$ is calculated via the Floquet formalism. Using linear response
theory, the AC conductance can then be obtained as [33, 34, 32]

$G_{ij}(\omega)=\frac{e^{2}}{h}\int dE\quad
Tr\\{\delta_{ij}\mathbb{1}_{i}-S^{\dagger}_{ij}(E)S_{ij}(E+\hbar\omega)\\}\frac{f_{j}(E)-f_{j}(E+\hbar\omega)}{\hbar\omega}.$
(42)

The AC charge current driven by $V(\omega)$ is then given by

$I_{i}(\omega)=\sum_{j}G_{ij}(\omega)V_{j}(\omega).$ (43)

In our case, a time-dependent SOI $\alpha_{R}(t)=\alpha_{R}(\omega)e^{i\omega
t}$ leads to a (spin) voltage in the frequency domain:

$V^{y}(\omega)=\pm\,i\omega\alpha_{R}(\omega)L,$ (44)

which in turn drives a spin current

$\displaystyle
I^{y}_{1}(\omega)=(2N_{1}-G(\omega)_{11}+G(\omega)_{12})i\omega\alpha_{R}(\omega)L_{12}\,.$
(45)

Equation (45) is our main result for the spin (generation) conductance. It
generalizes Eq. (40) to high frequencies and applies beyond the range of
validity of the adiabatic approximation (see Fig. 5).

For general time-dependent $\alpha_{R}$, we can obtain the time-dependent spin
current via the inverse Fourier transform of $I(\omega)$:

$I^{y}_{1}(t)=\int\frac{d\omega}{2\pi}e^{i\omega t}I^{y}_{1}(\omega)\,.$ (46)

###  4.3 AC spin current generation: comparison with numerics

We now perform numerical simulations to check our predictions summarized in
Eqs. (40) and (45) for spin current. The Rashba SOI has an AC-form
$\alpha_{R}(t)=k_{so}\sin(\Omega t)$, with $k_{so}\sim 1/L$. The system size
$L$ is taken to be $L=50a$, with $a$ the lattice constant of our discretized
model (see Appendix A). The precise system shape is depicted in the top right
inset of Fig. 3. The width of the leads is $10a$. We choose parameters
appropriate for an InAs 2DEG material with effective mass $m=0.023\,m_{0}$,
and we fix the lattice spacing at $a=2$ nm. The magnitude of the Rashba
coupling is $0.8\cdot 10^{-11}$ eVm which is in the experimental range of InAs
systems[6].

We calculate the AC spin current polarized in the $y$ direction,
$I^{y}(t)=I^{y}\cos(\Omega t)$, induced by the time-dependent Rashba SOI
$\alpha_{R}(t)=k_{so}\sin(\Omega t)$. The spin current given in Eq. (34) is
computed using the Floquet scattering matrix, and then compared with our
analytical results in Eqs. (40) and (45).

Figure 3: Comparison of the AC spin currents generated by the spin-dependent
voltage (Eq. (40), solid line) and numerically (dashed) directly for the time-
dependent Rashba SOI $\alpha_{R}(t)=k_{so}\sin(\Omega t)$ with
$\Omega\tau\approx 0.3$ and varying values $k_{so}L=0.3$ (blue), $1$ (red),
$2$ (orange), $3$ (green).

We first check the validity of our approximation (40) in the low-frequency
regime. We choose $\Omega/2\pi\approx 100$ GHz, corresponding to
$\Omega\tau\approx 0.3$. In Fig. 3 we perform the comparison for varying
values of $k_{so}$: $k_{so}L=0.3$, $1$, $2$ and $3$. We deliberately show a
small range of the energy values to make the differences clear. Even though
our approximation, in principle, requires $k_{so}L\ll\pi$, we see that up to
$k_{so}L=1$ our theory still gives quantitative agreement with the full
numerical results as can be seen in Fig. 3 (red curve).

Figure 4: AC spin currents generated by the time-dependent Rashba SOI in the
absence of any charge bias. The numerical Floquet results (green) are compared
to the spin current in $y$ direction in the presence of the time-dependent
spin voltage (orange). The spin currents (orange) are calculated using Eq.
(40) in the low-frequency regime in panels a) and b) and by Eq. (46) in the
high-frequency regime in panels c), d) and e). The system size is $L=50a$ and
the Rashba SOI is $\alpha_{R}(t)=k_{so}\sin(\Omega t)$ with $k_{so}L=1$. The
frequency values are indicated at the top left corner of each panel,
$\Omega\tau\approx 0.03$, $0.3$, $1$, $2.1$ and $3$ in panels a) and b), etc.,
respectively. The system geometry is shown in the top right inset.

We also explore the validity of our analytical predictions as we vary $\Omega$
relative to $1/\tau$, the inverse dwell time. As outlined in App. E, we
compute the dwell time using the Wigner-Smith time delay matrix for particular
Fermi energies which are determined in the range of $0.15$ and $0.27$ $eV$
when only $2$ transverse channels are open. Taking the energy average of the
dwell time gives as an approximate time of flight $\tau$ for $2$ open
channels.

We compare our analytical prediction with the numerical Floquet results (33)
and (34) for a range of frequencies. In the low-frequency regime where
$\Omega\tau<1$, we can neglect the frequency dependence of the conductance in
Eq. (42). We then use Eq. (40) to obtain our prediction and show that the
result agrees well with the Floquet result as shown in Fig. 4 in panels a) and
b). We note that the spin current is in the nano-Ampere range where the
corresponding frequencies $\Omega/2\pi$ are approximately between $10$ and
$100$ GHz.

In the high-frequency regime where $\Omega\tau>1$, the AC spin current is
calculated by incorporating the effect of the AC bias voltage and the
admittance in Eq. (42). That is, we use Eqs. (45) and (46). We find reasonable
agreement for the higher frequencies up to $\Omega\tau\approx 3$, see Fig. 4.
The frequencies $\Omega/2\pi$ corresponding to those in panels c),d) and e) of
Fig. 4 are $330$, $700$ GHz and $1$ THz, respectively. These are in
experimental reach. Notably, we find that in this high-frequency regime we can
generate spin currents up to $250$ nano-Amperes using a geometry with a length
scale of about $100$ nm.

###  4.4 Adiabatic approximation versus analytical results

In this section, we compare our analytical results with the adiabatic
approximation of the Floquet scattering matrix in Eq. (36), which is obtained
by the frozen scattering matrix in Eq. (35). In Fig. 5, we choose the Rashba
coupling constant $k_{so}L=0.01$ to obtain better agreement for our analytical
result. The driving frequency is chosen in the high-frequency regime, i.e.,
$\Omega\tau\approx 1.5$ corresponding to a driving frequency $\Omega/2\pi$ of
approximately 500 GHz.

Figure 5: AC spin currents generated by the time-dependent Rashba SOI in the
absence of a bias voltage. In the high-frequency regime, the numerical Floquet
results (green) are compared to the adiabatic approximation (blue) and the
spin current in $y$-direction calculated from the time-dependent spin voltage,
Eq. (46) (orange). The system size is $L=50a$, and the Rashba SOI is
$\alpha_{R}(t)=k_{so}\sin(\Omega t)$ with $k_{so}L=0.01$ and
$\Omega\tau\approx 1.5$.

The adiabatic approximation should fail beyond frequencies around
$\Omega\tau\approx 1$. Indeed, Fig. 5 shows that the adiabatic approximation
breaks down at high frequencies while our prediction (46) still shows good
agreement with the numerical Floquet result. We note that in addition to
having a much wider range of validity, the computation of our analytical
result is much easier and takes much less time compared to the frozen
scattering matrix approximation.

##  5 Charge signal from a spin current

###  5.1 Dynamical SOI-based spin transistor setting

In this section we discuss how to detect the AC spin current generated by the
spin potential induced through a time-dependent SOI. Usually this can be
achieved with a device with nonzero spin (detection) conductance. While the
conventional means of detecting spin currents are based on ferromagnetic
leads, here we will exploit the SU(2) gauge structure of the Rashba SOI
further, to design a device with a non-zero spin magnetic field. The latter
can then convert spin currents into charge signals.

The spin (detection) conductance in Eq. (28) is converted into the difference
of two charge magneto-conductances according to Eq. (30). Then an Onsager
relation implies that the total charge conductance vanishes in the presence of
time-reversal symmetry. On the other hand, either breaking the time-reversal
symmetry e.g. via an applied magnetic field or connecting a third terminal to
the system [16] results in a nonzero charge conductance and an AC charge
current. As a result, controlling the symmetry properties of the system, we
can obtain on/off states of this dynamical spin transistor. Below we will take
the route where we connect a third terminal.

Figure 6: Setup of the dynamical spin-orbit based spin transistor. The AC gate
voltage in the right cavity induces a dynamical Rashba SOI, which creates an
AC spin current that flows into the left cavity. There, due to a spatially
inhomogeneous Rashba SOI, the spin current is transformed again into a charge
current.

To explore how this conversion works in detail, we consider a system that
consists of two subsystems, containing a spatially inhomogeneous and a time-
dependent Rashba SOI, respectively, see Fig. 6. In Fig. 6, the right subsystem
consists of a 2D quantum wire (with length $L$ and width $W$) with a time-
dependent SOI $\alpha_{R}(t)=k_{so}\sin(\Omega t)$ engineered by an AC gate
voltage. The left subsystem contains a ballistic ring (with diameter $L$) with
a spatially inhomogeneous SOI $\alpha_{R}(\mathbb{x})=k_{so}(L-y)/L$. This way
spin currents generated in the right subsystem via a spin electric field are
converted into charge currents via a spin magnetic field in the left
subsystem.

###  5.2 AC charge signal in the low-frequency regime

In the absence of a bias voltage $V$, the charge current in lead 1 follows
from Eq. (28) as

$I_{1}=-\sum\limits_{j,b}G_{1j}^{0b}V_{j}^{b},$ (47)

with spin direction $b=x,y,z$. We express the spin conductance in terms of the
charge magneto-conductance (30). In the low-frequency regime, we again neglect
the frequency dependence of the conductance. Then the charge current in lead
$i$ reads

$I_{1}\approx-\sum_{b}\sum_{j}(G_{1j}(\mathcal{B}^{b})-G_{1j}(-\mathcal{B}^{b}))f^{b}V^{b}_{j}.$
(48)

For spatially inhomogeneous SOI $\alpha_{R}(\mathbb{x})=k_{so}(L-y)/L$ we
obtain from Eq. (15) the spin magnetic field components $\mathcal{B}^{a}_{z}$
in the $x$\- and $y$-directions as

$\displaystyle\mathcal{B}^{x}$
$\displaystyle=-\partial_{x}\alpha_{R}(\mathbb{x})f^{x}\,,$
$\displaystyle\mathcal{B}^{y}$
$\displaystyle=-\partial_{y}\alpha_{R}(\mathbb{x})f^{y}\,.$ (49)

The spin voltages generated by the time-dependent SOI
$\alpha_{R}(t)=k_{so}\sin(\Omega t)$ are obtained from Eq. (12) as

$\displaystyle
V_{1(3)}^{y}=V_{1(3)}^{\uparrow}-V_{1(3)}^{\downarrow}=\pm\partial_{t}\alpha_{R}(t)L\,.$
(50)

We now have all the ingredients to calculate the charge current in lead $1$.
We get

$I_{1}(t)=G_{s}\partial_{t}\alpha_{R}(t)L\,,$ (51)

where

$G_{s}=-G_{11}(\mathcal{B}^{y})+G_{11}(-\mathcal{B}^{y})+G_{13}(\mathcal{B}^{y})-G_{13}(-\mathcal{B}^{y})\,.$
(52)

This represents our further prediction demostrating that the spin current can
be converted into a charge current in a spatially inhomogeneous system by
means of a spin magnetic field.

###  5.3 AC charge signal in the high-frequency regime

As explained in Section 4.2, at high frequency we should retain the frequency
dependence of the AC conductance. We follow the same steps as in Section 4.2
and rewrite the AC charge current expression (48), but retaining the frequency
dependence given in Eq. (42), as

$I_{1}(\omega)\approx-\sum_{b}\sum_{j}[G_{1j}(\mathcal{\omega,B}^{b})-G_{1j}(\omega,-\mathcal{B}^{b})]f^{b}V(\omega)^{b}_{j}\,,$
(53)

where the spin-dependent magnetic field is $\mathcal{B}^{y}=k_{so}L^{-1}f^{y}$
and the spin voltage is $V^{\uparrow(\downarrow)}(\omega)=\pm
i\omega\alpha(\omega)\,L\,$. This constitutes our main result for the spin
(detection) conductance, which applies to both high- and low-frequency
regimes.

Figure 7: Dyanmical spin transistor function: AC charge current in Lead 1, see
Fig. 6, generated by the time-dependent and spatially inhomogeneous Rashba
SOI. Numerical results based on the Floquet formalism (green) are compared to
the current generated by the spin-dependent voltage and pseudo-magnetic field
(orange) in the low- and high-frequency regime. The analytical results in a)
and b) are calculated based on Eq. (52) and in panels c) to e) using Eq. (53).
The dynamical Rashba SOI in the right half system is
$\alpha_{R}(t)=k_{so}\sin(\Omega t)$ with $k_{so}L=1$ and the spatially
inhomogeneous Rashba SOI in the left half system (the ring) is
$\alpha_{R}(\mathbb{x})=k_{so}(L-y)/L$. The size of each system part is
$L=50a$. The frequencies used are $\Omega\tau\approx 0.1$, $0.3$, $1$, $1.5$
and $2.1$ from top to bottom panel.

###  5.4 Simulating the dynamical spin-transistor functionality

We performed numerical transport simulations of the spin transistor in Fig. 6
and explore the range of validity of our analytical result (53). We choose the
time-dependent Rashba SOI to be $\alpha_{R}(t)=k_{so}\sin(\Omega t)$ and the
spatially inhomogeneous Rashba SOI to be
$\alpha_{R}(\mathbb{x})=k_{so}(L-y)/L$, where the Rashba SOI constant is
selected such that $k_{so}L=1$, where $L$ is the system size along the $x$
direction for the right subsystem. The shape of the system is shown at the top
right corner of Fig. 6. The right part is a 2D wire with length $L=50a$, width
$W=30a$ and one connected lead of width $10a$. The left part is a ballistic
ring with inner radius of $10a$, outer radius of $25a$ and two connected leads
of width $10a$. The two parts are connected via a bridge of width $10a$ made
of the same material.

We consider $3$ open channels and choose the Fermi energy between $0.29$ and
$0.47$ eV. After computing the dwell time to specify the range of the
frequency, we calculate the AC charge currents $I(t)=I\cos(\Omega t)$ in the
absence of the bias voltage on the left using the Floquet scattering matrix
given in Eq. (33) and Eq. (34). Finally, we compare the Floquet result for AC
charge current with our analytical prediction both in the low and the high-
frequency regimes. Our analytical results are obtained from Eq. (52) for
$\Omega\tau\approx 0.1$, $0.3$ and shown in Fig. 6a) and b). Here, the
frequencies $\Omega/2\pi$ are approximately $40$ and $120$ GHz. We find
excellent agreement between our analytical results and the numerically
obtained AC charge current. Moreover, even in the low-frequency regime, we see
that the mechanism produces few-nA charge currents that are experimentally
observable.

For higher frequencies where $\Omega>1/\tau$, we compare our prediction based
on Eq. (53) with the numerical dynamical conductance using driving frequencies
$\Omega\tau\approx 1$, $1.5$, and $2.1$. These values correspond to
$\Omega/2\pi$ being approximately $400$, $600$, and $840$ GHz. In Fig.
6c),d),e) we also observe on the whole a fairly good match between the two
currents in the high-frequency regime up to $\Omega\tau\approx 2.1$.

##  6 Conclusions

We considered the generation and detection of pure spin currents in mesoscopic
2DEG cavities with time- and space-modulated Rashba SOI. Such modulations can
be realized by applying local gate voltages, allowing an all-electrical
architecture. Our underlying concept is based on the non-Abelian gauge
structure of spin-orbit fields. By means of a tailored $SU(2)$ gauge
transformation we converted the general Hamiltonians with time-periodic or
spatially inhomogenous Rashba SOI into corresponding approximate Hamiltonians
involving instead emergent $U(1)$ fields.

For the time-dependent Rashba SOI, the $U(1)$ fields correspond to spin-
dependent voltages with opposite associated electrical fields for spin-up and
-down electrons, respectively. Then the total spin-dependent voltage becomes
$V^{\uparrow}-V^{\downarrow}$ to linear order in the Rashba SOI strehgth. Thus
the action of the time-dependent Rashba SOI can be considered as a spin-motive
force, that enables to generate a pure AC spin current in the absence of an
applied bias voltage.

Analogously, in the presence of a spatially inhomogeneous Rashba SOI, one can
show that an appropriate gauge transformation relates the Rashba spin-orbit
fields to magnetic pseudo fields $\mathcal{B}$ with opposite directions for
the two spin species. Thereby, the spin conductance is rewritten as a charge
magneto-conductance where opposite spins experience anti-parallel spin
magnetic fields: $G^{\uparrow}-G^{\downarrow}\approx
G(\mathcal{B})-G(-\mathcal{B})$. Thereby the generated AC spin current due to
the dynamical Rashba coupling can be read out as an AC charge current by means
of an appropriately devised spatially inhomogeneous SOI. Moreover, in view of
an Onsager relation and current conservation, the AC-generated conductance
difference, $G(\mathcal{B})-G(-\mathcal{B})$, is suppressed for a two-terminal
setting in presence of time-reversal symmetry. Therefore the transistor on/off
state can be controlled via the coupling to a third lead or upon changing the
symmetry properties of the system. By combining all these aspects, we devised
and proposed a dynamical SOI-based spin-transistor which integrates a
controllable spin current generation and its detection solely by electric
means.

We checked our analytical approximations by means of numerical calculations
using Floquet theory which allows us to calculate time-periodic spin and
charge currents within a Floquet scattering matrix formalism. We presented
results and compared the different approaches both in the regimes of adiabatic
($\Omega\tau\ll 1$) and high-frequency ($\Omega\tau\geq 1$) driving. The major
assumptions behind our analytical derivations are $k_{so}L\leq\pi$ and
$(k_{so}L)(\hbar\Omega)\leq\epsilon_{F}$. The latter bound implies that our
approach is valid and facilitates quantum calculations in a frequency regime
that extends far beyond the adiabatic limit, often assumed in analytical
treatments.

The driving frequencies considered cover a broad window ranging, after
rescaling, up to frequencies of about 1 THz. Our numerical simulations show
furthermore that an AC spin current of several hundred nano-Amperes can be
obtained in a $100$ nm 2DEG InAs system for driving frequencies of about 1 THz
and a realistic Rashba coupling $0.8\cdot 10^{-11}$ eVm. This shows that an
experimental realization of the dynamic spin transistor functionality proposed
is in experimental reach.

## Acknowledgements

We thank M. Wimmer and A.M. Bozkurt for helpful discussions. IA and FNG are
grateful to the hospitality of Regensburg University where parts of this
project were carried out. FNG is also grateful to the hospitality of Utrecht
University.

##### Funding information:

This work was supported by TUBITAK under grant 110T841 and the programme
2214-A, and by the Deutsche Forschungsgemeinschaft (German Science
Foundation), Project-ID 314695032–SFB 1277 (project A07).

##  Appendix A Tight-binding Hamiltonian for 2DEG with Rashba SOI

In the case of a 2DEG with spatially inhomonegeous Rashba SOI in the
$x$-$y$-plane, the continuum Hamiltonian we use for our analytical
calculations reads

$H=\frac{\hbar^{2}}{2m^{*}}(k_{x}^{2}+k_{y}^{2})+\frac{1}{2}\\{\alpha_{R}(\mathbf{x}),\left(\sigma^{x}p_{y}-\sigma^{y}p_{x}\right)\\}.$
(54)

where $\alpha_{R}(\mathbf{x})$ is the Rashba SOI strength, $m^{*}$ is
effective mass of an electron and $\sigma$ denotes Pauli matrices. In our
numerical simulations we use the discretized version of the Hamiltonian (54)
[35, 36]. It is defined on a 2D square lattice with a lattice constant $a$,

$\displaystyle H_{tb}$ $\displaystyle=$
$\displaystyle\sum_{k,l,\sigma,\sigma{{}^{\prime}}}4t\,(c^{\dagger}_{k,l,\sigma}c_{k,l,\sigma{{}^{\prime}}})+\sum_{k,l,\sigma}t\,(c^{\dagger}_{k+1,l,\sigma}c_{k,l,\sigma}+c^{\dagger}_{k,l+1,\sigma}c_{k,l,\sigma})+$
(55)
$\displaystyle\sum_{k,l,\sigma,\sigma{{}^{\prime}}}\frac{1}{2a}\frac{1}{2}(\alpha_{R,k}+\alpha_{R,k+1})\,c^{\dagger}_{k+1,l,\sigma}(i\sigma_{y})^{\sigma\sigma{{}^{\prime}}}c_{k,l,\sigma{{}^{\prime}}}+$
$\displaystyle\sum_{k,l,\sigma,\sigma{{}^{\prime}}}-\frac{1}{2a}\frac{1}{2}(\alpha_{R,l}+\alpha_{R,l+1})\,c^{\dagger}_{k,l+1,\sigma}(i\sigma_{x})^{\sigma\sigma{{}^{\prime}}}c_{k,l,\sigma{{}^{\prime}}}$

where $c^{\dagger}_{k,l,\sigma}$ is the operator that creates an electron with
spin $\sigma$ at the lattice point ($k$,$l$), and the hopping amplitude is
$t=-{\hbar^{2}}/({2m^{*}a^{2}})$.

##  Appendix B Floquet Hamiltonian with time-dependent Rashba SOI

We review the conversion of a time-dependent Hamiltonian into a static matrix
Hamiltonian with Floquet states [37]. Floquet theory is particularly
functional for understanding the behavior of quantum mechanical systems with a
time-periodic Hamiltonian:

$H(\vec{r},t+\mathcal{T})=H(\vec{r},t),$

with period $\mathcal{T}=2\pi/\Omega$ and $\Omega$ is given by the frequency
of the periodically driven potential. The solutions of the time-dependent
Schrödinger equation are given by the Floquet wave functions.

Consider a 2D quantum mechanical system with a time-periodic Hamiltonian that
can be separated into a static and a time-dependent part as

$H(\vec{r},t)=H_{0}(\vec{r})+H_{1}(\vec{r},t)$ (56)

where the kinetic term is
$H_{0}(\vec{r})=-\frac{\hbar^{2}}{2m}(\partial_{x}^{2}+\partial_{y}^{2})$. The
time-dependent Schrödinger equation can be put into the form

$\displaystyle{H}_{F}(\vec{r},t)|\Psi(\vec{r},t)\rangle=0\,,$ (57)

where the hermitian Floquet Hamiltonian ${H}_{F}(\vec{r},t)$ is related to the
original Hamiltonian through

$\displaystyle{H}_{F}(\vec{r},t)=H(\vec{r},t)-i\hbar\frac{\partial}{\partial
t}\,.$ (58)

The solution of Eq. (57) is formally given by the Floquet states

$|\Psi_{\eta}(\vec{r},t)\rangle=e^{-iE_{\eta}t/\hbar}|\phi_{\eta}(\vec{r},t)\rangle\,,$
(59)

where $|\phi_{\eta}(\vec{r},t)\rangle$ is called a Floquet mode. Floquet modes
are eigenfunctions of the Floquet Hamiltonian ${H}_{F}(\vec{r},t)$ with
eigenvalues (quasi-energies) $E_{\eta}$:

$\Big{(}H(t)-i\hbar\frac{\partial}{\partial
t}\Big{)}|\phi_{\eta}(\vec{r},t)\rangle={H}_{F}(\vec{r},t)|\phi_{\eta}(\vec{r},t)\rangle=E_{\eta}|\phi_{\eta}(\vec{r},t)\rangle\,.$
(60)

with periodic eigenfunctions

$\displaystyle|\phi_{\eta}(\vec{r},t)\rangle=\sum_{n}e^{-in\Omega
t}|n\rangle,\quad|\phi_{\eta}(\vec{r},t)\rangle=|\phi_{\eta}(\vec{r},t+\mathcal{T})\rangle\,.$
(61)

Here $|n\rangle$ are basis vectors of the eigenfunctions of the Floquet
Hamiltonian and referto the Floquet states. In Fourier space, the time-
dependent Hamiltonian can now be expressed as a static matrix Hamiltonian

$H_{F}=\sum_{n=-\infty}^{\infty}\big{(}(H_{0}-n\hbar\Omega)|n\rangle\langle
n|+\frac{iH_{1}}{2}(|n\rangle\langle n+1|-|n\rangle\langle n-1|)\big{)}.$ (62)

Hence the original time-periodic system has been coverted into a multi-channel
stationary system expanded in the Floquet states. Using the Hamiltonian (62),
we can numerically compute the Floquet states for each Floquet channel to
obtain the elements of the Floquet scattering matrix.

In the case of the Rashba SOI $\alpha(t)=k_{so}\sin(\Omega t)$ we have
$H_{1}=ik_{so}(\sigma_{y}\partial_{x}-\sigma_{x}\partial_{y})$.

##  Appendix C Comparing the Floquet scattering matrix with its adiabatic
approximation

For a further analysis of our spin current calculations we compare results
based on the full Floquet scattering matrix $S_{F}(E_{n},E)$ with its
adiabatic approximation for different frequencies, also to see the range of
validity of an adiabatic treatment. In the adiabatic limit $S_{F}$ can be
expanded in powers of the frequency $\Omega$ [23],

$S_{F}(E_{n},E)=S_{n}+O(\Omega)\,,$ (63)

The zeroth-order terms $S_{n}$ are the Fourier coefficients of the frozen
scattering matrix, that is the stationary scattering matrix

$S(t,E)=\sum_{n=-\infty}^{\infty}e^{-in\Omega t}S_{n}(E)\,.$ (64)

calculated for a given time $t$.

In Fig. 8, we compare the adiabatic and full Floquet spin currents for the
chaotic ballistic system of Fig. 2 for low and high frequencies. The results
agree in the adiabatic limit $\Omega\tau\approx 0.1$ as expected and show
semi-quantitative agreement up to the frequency of $\Omega\tau\approx 1$.

Figure 8:  Spin currents for the chaotic ballistic system, Fig. 2, calculated
by using the full Floquet scattering matrix (orange) and by the adiabatic
approximation, Eq. (63) (blue) for $k_{so}L=1$ and $\Omega\tau\approx$ 0.1
(a), 1 (b), 1.5 (c) and 3 (d).

##  Appendix D Spin and charge current calculation in the Landauer-Büttiker
Formalism

Here we sketch the derivation of the expressions (33) and (34) for the AC
spin/charge current using the Landauer-Büttiker formalism. We start with the
charge/spin current operator. Recall that $\alpha=0,x,y,z$ indicates the
components of the charge current and the spin current, respectively:

$\hat{I}_{i}^{\alpha}(t)=\frac{e}{4\pi}\int
dEdE^{\prime}\,e^{i(E-E^{\prime})/\hbar}[\hat{a}^{\dagger}_{im,\sigma}(E)\sigma^{\alpha}\hat{a}_{im,\sigma^{\prime}}(E^{\prime})-\hat{b}^{\dagger}_{im,\sigma}(E)\sigma^{\alpha}\hat{b}_{im,\sigma^{\prime}}(E^{\prime})]$
(65)

where $\hat{a}^{\dagger}$ ($\hat{a}$) and $\hat{b}^{\dagger}$ ($\hat{b}$) are
the creation (annihilation) operators for incident and scattered electrons,
respectively, $m$ denotes the channel in lead $i$, and $\sigma$ represents the
spin operator. Note that Eq. (65) is derived under the assumption
$E-E^{\prime}\ll E_{F}$.

For a periodically driven system, the relation between the operators $\hat{a}$
and $\hat{b}$ is given by the Floquet scattering matrix

$\hat{b}_{im,\sigma}(E)=\sum_{n=-\infty}^{\infty}\sum_{j=1}^{N_{r}}\sum_{m^{\prime}\in
j}\sum_{\sigma^{\prime}=\pm
1}S^{\sigma\sigma^{\prime}}_{F,im,jm^{\prime}}(E,E_{n})\hat{a}_{jm^{\prime},\sigma^{\prime}}(E_{n})$
(66)

where $n$ denotes the Floquet states with $E_{n}=E+n\hbar\Omega$. We write the
expectation value of the current as inverse Fourier series,

$I_{i}^{\alpha}(t)\equiv\langle\hat{I}_{i}^{\alpha}(t)\rangle=\sum_{l=-\infty}^{\infty}e^{-il\Omega
t}I^{\alpha}_{i,l}\,,$ (67)

with $I^{\alpha}_{i,l}=\langle\hat{I}^{\alpha}_{i}(l\Omega)\rangle$. This
yields, in view of Eqs. (66) and (69),

$I^{\alpha}_{i,l}=\frac{e}{4\pi}\int_{-\infty}^{\infty}dE\sum_{n=-\infty}^{\infty}\sum_{i=1}^{N_{r}}\sum_{m\in
i,m^{\prime}\in
j}\\{f_{j}(E)-f_{i}(E_{n})\\}Tr[S_{F,im,jm^{\prime}}^{\dagger}(E_{n},E)\sigma^{\alpha}S_{F,im,jm^{\prime}}(E_{l+n},E)].$
(68)

Here we use that the quantum statistical average of the operator
$\langle{\hat{a}}^{\dagger}\hat{a}\rangle$ gives the Fermi function for
electrons in lead $i$,

$\langle\hat{a}^{\dagger}_{im,\sigma}(E)\hat{a}_{jm^{\prime},\sigma^{\prime}}(E^{\prime})\rangle=\,\delta_{ij}\delta_{mm^{\prime}}\delta_{\sigma\sigma^{\prime}}\delta(E-E^{\prime})f_{i}(E).$
(69)

When the chemical potential difference between the leads vanishes, the Fermi
functions are equal, $f_{i}(E)=f_{j}(E)=f_{0}(E)$. Furthermore in the limit
$\Omega\rightarrow 0$ one can expand the Fermi function with energy
$E_{n}=E+n\hbar\Omega$ and obtains

$f_{0}(E)-f_{0}(E_{n})=-\frac{df_{0}(E)}{dE}n\hbar\Omega+{\cal{O}}({\Omega}^{2})\,.$
(70)

Substituting into Eq. (68), we obtain

$I_{i,l}^{\alpha}=\frac{e}{4\pi}\int_{-\infty}^{\infty}dE\bigg{(}-\frac{\partial
f_{0}}{\partial
E}\bigg{)}\sum_{n=-\infty}^{\infty}\sum_{i=1}^{N_{r}}\sum_{m\in
i,m^{\prime}\in
j}(n\hbar\Omega)Tr[S_{F,im,jm^{\prime}}^{\dagger}(E_{n},E)\sigma^{\alpha}S_{F,im,jm^{\prime}}(E_{l+n},E)]\,.$
(71)

In the zero-temperature limit, only electrons at the Fermi energy $E_{F}$
contribute to the current [$\frac{\partial f_{0}}{\partial
E}=\delta(E-E_{F}))$], and the integral can be evaluated. This yields

$I_{i,l}^{\alpha}=\frac{e}{4\pi}\hbar\Omega\sum_{n=-\infty}^{\infty}\sum_{j=1}^{N_{r}}\sum_{m\in
i,m^{\prime}\in
j}nTr[S_{F,imjm^{\prime}}^{\dagger}(E_{F,n},E_{F})\sigma^{\alpha}S_{F,im,jm^{\prime}}(E_{F,l+n},E_{F})]\,.$
(72)

We calculate the full time-dependent current by substituting this expression
into Eq. (67).

##  Appendix E Calculation of the dwell time

Here we calculate the dwell time of an electron spent in the scattering system
with a time-dependent potential. The dwell time is given by the energy
derivative of the scattering phase shift in terms of scattering matrix [30,
31]. The diagonal elements of the Wigner-Smith time-delay matrix,

$Q(E)=-i\hbar S^{\dagger}(E)\frac{\partial S(E)}{\partial E}\;,$ (73)

are real and give the proper time delays for each transport channel. Taking
the average over all channels, one obtains the time delay

$\tau_{W}(E)=\frac{1}{N}Tr\\{Q\\}.$ (74)

For the time-periodic systems we compute the dwell time using the Floquet
scattering matrix,

$\tau_{W}(E_{F})=-\frac{i\hbar}{N^{\prime}}Tr\bigg{\\{}\sum_{n}S_{F}^{\dagger}(E_{F},E_{F,n})\frac{\partial
S_{F}(E_{F},E_{F,n})}{\partial E}\bigg{\\}},$ (75)

where $N^{\prime}=2N(2n+1)$ is the total number of scattering channels that
results from the total number of Floquet bands $n$ and spin polarization. We
also compute the energy averaged dwell time denoted as $\tau$. The energy
window is chosen such that the number of open channels is constant. This
averaged dwell time $\tau$ is a convenient measure to physically distinguish
the low-frequency and high-frequency regimes $\Omega<1/\tau$ and
$\Omega>1/\tau$, respectively.

##  Appendix F Role of the number of Floquet bands for the numerical results

Here we present our procedure to determine the number of the necessary Floquet
channel for performing numerically converged simulations. In the calculation
of the spin current using Eq. (31) we need to choose a cut-off value in the
sum over the Floquet side bands $n$. It is essential to know the number of the
bands prior to numerical calculations as the time cost of a calculation
heavily depends on this number. To determine this number, we compute the spin
current for a selected number of Floquet states $2n+1$ and change $n$ to
calculate the precision defined by $\Delta I_{n}=(I_{n}-I_{n-1})/I_{n}$. This
quantity is plotted in Fig. 9 for $k_{so}L=0.2$ and $2$ and for
$\Omega\tau\approx 0.1$ and $1$. Generally, we observe that smaller cut-off
values $n$ suffice for smaller choices of $k_{so}$, as expected. We also find
that our chosen cut-off value $n=10$ provides a sufficiently good
approximation to the time-dependent currents.

Figure 9: The convergence in $n$ of the AC spin currents generated by the
time-dependent Rashba SOI, where $\Delta I_{n}=(I_{n}-I_{n-1})/I_{n}$ and
$I_{n}$ is the current calculated with maximum Floquet states $N$. The
parameters correspond to $\Omega\tau\approx 0.1$ and $1$, and $k_{so}L=0.2$
and $2$.

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