# Development in the Scattering Matrix Theory: From Spin-Orbit-Coupling
Affected Shot Noise to Quantum Pumping

Rui Zhu***Corresponding author. Electronic address<EMAIL_ADDRESS>Department of Physics, South China University of Technology, Guangzhou 510641,
People’s Republic of China

###### Abstract

The review chapter starts by a pedagogical introduction to the general concept
of the scattering theory: from the fundamental wave-function picture to the
second-quantization language, with the aim to clear possible ambiguity in
conventional textbooks. Recent progress in applying the method to current
fluctuations and oscillating-parameter driven quantum pumping processes is
presented with inclusion of contributions by Büttiker, Brouwer, Moskalets,
Zhu, etc. In particular, the spin-orbit-coupling affected shot noise can be
dealt with by taking into account the spin-dependent scattering processes. A
large shot noise suppression with the Fano factor below 0.5 observed
experimentally can be illustrated by effective repulsion between electrons
with antiparallel spin induced by the Dresselhaus spin-orbit coupling effect.
A Floquet scattering theory for quantum-mechanical pumping in mesoscopic
conductors is developed by Moskalets et al., which gives a general picture of
quantum pumping phenomenon, from adiabatic to non-adiabatic and from weak
pumping to strong pumping.

###### pacs:

72.10.-d, 73.23.-b, 05.60.Gg, 73.50.Td, 71.70.Ej, 75.60.Ch

###### Contents

  1. I Pedagogical introduction to the general concept of the scattering theory
    1. I.1 Landauer-Büttiker conductance
    2. I.2 Further illustration of the scattering scheme in a toy system with correlated reservoirs
    3. I.3 Time-dependent scattering-matrix theory
  2. II Spin-orbit coupling affected shot noise
    1. II.1 Background
    2. II.2 Theoretical approach
    3. II.3 Numerical results in comparison with experiment
  3. III Quantum pumping beyond linear response
    1. III.1 Introduction to quantum pumping
    2. III.2 Theoretical formulation
    3. III.3 Numerical results and interpretations
  4. IV Summary and future directions
  5. V Acknowledgements

##  I Pedagogical introduction to the general concept of the scattering theory

###  I.1 Landauer-Büttiker conductance

The discussion is based on the scattering approach to electrical conductance.
This approach, as we will show, is conceptually simple and transparent.
Nevertheless, the generality of the scattering approach and its conceptual
clarity, make it the desired starting point of a discussion of noise in
electrical conductors. By expanding the time-dependent scattering matrix into
Fourier series, a description of the quantum pumping phenomenon can be given.

We start with the wave function picture and consider an electron tunneling
through a one-dimensional single barrierRef29 , which can be realized in a
semiconductor heterostructure with a layer of
${\rm{Al}}_{x}{\rm{Ga}}_{1-x}{\rm{As}}$ of width $2L$ imbedded in $\rm{GaAs}$
as shown in Fig. 1. In the effective mass approximation, the electron motion
in each layer of the structure is described by the stationary solution of the
envelop equation in the $x$-direction.

$\left[{\frac{{-\hbar^{2}}}{2}\frac{\partial}{{\partial
x}}\frac{1}{{m^{*}\left(x\right)}}\frac{\partial}{{\partial
x}}+V_{eff}\left(x\right)}\right]\psi\left(x\right)=E\psi\left(x\right).$ (1)

Here, $m^{*}$ and $V_{eff}$ are the effective mass and potential in different
regions with $E$ the energy of the transporting electron. The electron’s wave
functions are expressible as

$\Psi\left({x,t}\right)=\psi\left(x\right)e^{-iEt/\hbar}=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}c}{\left({Ae^{ikx}+Be^{-ikx}}\right)e^{-iEt/\hbar},}&{x\leq-L,}\\\
\end{array}\\\ \begin{array}[]{*{20}c}{\left({Ce^{\kappa x}+De^{-\kappa
x}}\right)e^{-iEt/\hbar},}&{-L\leq x\leq L}\\\ \end{array},\\\
\begin{array}[]{*{20}c}{\left({Ee^{ikx}+Fe^{-ikx}}\right)e^{-iEt/\hbar},}&{x\geq
L,}\\\ \end{array}\\\ \end{array}\right.$ (2)

with
$k={{\sqrt{2m^{*}E}}\mathord{\left/{\vphantom{{\sqrt{2m^{*}E}}\hbar}}\right.\kern-1.2pt}\hbar}$
and
$\kappa={{\sqrt{2m^{*}\left({V_{0}-E}\right)}}\mathord{\left/{\vphantom{{\sqrt{2m^{*}\left({V_{0}-E}\right)}}\hbar}}\right.\kern-1.2pt}\hbar}$.
The coefficients $A$ and $B$ are associated respectively with incoming and
outgoing waves on the left side relative to the barrier. Likewise, the
coefficients $E$ and $F$ are respectively outgoing and incoming waves on the
right. The scattering matrix connects the incoming and outgoing fluxes as

$\left[{\begin{array}[]{*{20}c}B\\\ E\\\
\end{array}}\right]=\left[{\begin{array}[]{*{20}c}{S_{11}}&{S_{12}}\\\
{S_{21}}&{S_{22}}\\\ \end{array}}\right]\left[{\begin{array}[]{*{20}c}A\\\
F\\\ \end{array}}\right].$ (3)

Ideal (i.e., without scattering) conducting leads connect the scattering
region to reservoirs on the left and right characterized by quasi-Fermi
energies $\mu_{1}$ and $\mu_{2}$, respectively, corresponding to the electron
densities there. These reservoirs or contacts randomize the phase of the
injected and absorbed electrons through inelastic processes such that there is
no phase relation between particles. For such an ideal 1D system, the current
injected from the left and right may be written as an integral over the
fluxRef29 ; Ref33

$I_{L}=\frac{{2e}}{{2\pi}}\left[{\int_{0}^{\infty}{dkv\left(k\right)f_{1}\left(k\right)T\left(E\right)}-\int_{0}^{\infty}{dk^{\prime}v\left({k^{\prime}}\right)f_{2}\left({k^{\prime}}\right)T\left({E^{\prime}}\right)}}\right],$
(4)

where $v(k)$ is the velocity, $T(E)$ is the transmission coefficient, which
can be obtained from the wave function Eq. (2), and $f_{1}$ and $f_{2}$ are
the reservoir distribution functions characterized by $\mu_{1}$ and $\mu_{2}$,
respectively. The integrations are only over positive $k$ and $k^{\prime}$
relative to the direction of the injected charge as positive $k$ is in
$+x$-direction and positive $k^{\prime}$ is in $-x$-direction. If we now
assume low temperatures, electrons are injected up to an energy $\mu_{1}$ from
the left lead and injected up to $\mu_{2}$ from the right one. Converting to
integrals over energy, the current becomes

$\begin{array}[]{l}I_{L}=\frac{{2e}}{{2\pi}}\left[{\int_{0}^{\mu_{1}}{dE\left({\frac{{dk}}{{dE}}}\right)v\left(k\right)T\left(E\right)}-\int_{0}^{\mu_{2}}{dE\left({\frac{{dk^{\prime}}}{{dE}}}\right)v\left({k^{\prime}}\right)T\left(E\right)}}\right]\\\
=\frac{{2e}}{{2\pi\hbar}}\int_{\mu_{2}}^{\mu_{1}}{dET\left(E\right)}.\\\
\end{array}$ (5)

It can be seen that the first term of Eq. (5) is the flux generated by
electrons injected from the left reservoir and the second term is that from
the right reservoir. The integration is done in two independent ensembles.

The Landauer-Büttiker conductance introduced above can be reproduced in the
second-quantization language. Without loss of generality, we assume the input
amplitudes from the two reservoirs in Eq. (2) to be unity. For simplicity, as
a single channel is considered, the scattering matrix can be described in the
relation

$\hat{b}_{\alpha}=S_{\alpha\beta}\hat{a}_{\beta}$ (6)

or elaborately

$\left[{\begin{array}[]{*{20}c}{\hat{b}_{L}}\\\ {\hat{b}_{R}}\\\
\end{array}}\right]=\left[{\begin{array}[]{*{20}c}{S_{11}}&{S_{12}}\\\
{S_{21}}&{S_{22}}\\\
\end{array}}\right]\left[{\begin{array}[]{*{20}c}{\hat{a}_{L}}\\\
{\hat{a}_{R}}\\\
\end{array}}\right]=\left[{\begin{array}[]{*{20}c}r&{t^{\prime}}\\\
t&{r^{\prime}}\\\
\end{array}}\right]\left[{\begin{array}[]{*{20}c}{\hat{a}_{L}}\\\
{\hat{a}_{R}}\\\ \end{array}}\right],$ (7)

where operators $a_{L/R}$ annihilate electrons incident upon the sample from
the left/right reservoir and operators $b_{L/R}$ describe electrons in the
outgoing states. $t$ and $t^{\prime}$ are the transmission amplitudes of
electrons incident rightward and rightward, respectively, and $r$ and
$r^{\prime}$ are the corresponding reflection amplitudes, as defined
conventionally. Hence, the Landauer-Büttiker formula of the current can be
expressed as

$I_{L}=\frac{e}{{2\pi\hbar}}\int{dE\left\langle{\left[{a_{L}^{\dagger}\left(E\right)a_{L}\left(E\right)-b_{L}^{\dagger}\left(E\right)b_{L}\left(E\right)}\right]}\right\rangle},$
(8)

where $\left\langle\cdots\right\rangle$ calculates the quantum statistical
average of the product of an electron creation operator and annihilation
operator of a Fermi gas. As a conventional electron reservoir is considered,
we have

$\begin{array}[]{l}\left\langle{a_{L}^{\dagger}\left(E\right)a_{L}\left({E^{\prime}}\right)}\right\rangle=f_{L}\left(E\right)\delta\left({E-E^{\prime}}\right),\\\
\left\langle{a_{R}^{\dagger}\left(E\right)a_{R}\left({E^{\prime}}\right)}\right\rangle=f_{R}\left(E\right)\delta\left({E-E^{\prime}}\right),\\\
\left\langle{a_{L}^{\dagger}\left(E\right)a_{R}\left({E^{\prime}}\right)}\right\rangle=\left\langle{a_{R}^{\dagger}\left(E\right)a_{L}\left({E^{\prime}}\right)}\right\rangle=0.\\\
\end{array}$ (9)

It should be noted from the third formula of Eq. (9) that electrons incident
from the left and right reservoirs are completely incoherent and the phases of
the two reservoirs are randomized and completely unrelated. Substituting Eq.
(9) to Eq. (8), the formula of the current expressed in Eq. (5) can be
reproduced.

It is interesting to see that if we could build up a system with the conductor
connected to two correlated reservoirs, in which the quantum statistical
average of the cross product reads

$\begin{array}[]{l}\left\langle{a_{L}^{\dagger}\left(E\right)a_{R}\left({E^{\prime}}\right)}\right\rangle=\left\langle{a_{R}^{\dagger}\left(E\right)a_{L}\left({E^{\prime}}\right)}\right\rangle=f_{3}\left(E\right)\delta\left({E-E^{\prime}}\right),\\\
f_{3}\left(E\right)=\left\\{\begin{array}[]{l}1,\begin{array}[]{*{20}c}{}\hfil&{\mu<\min\left({\mu_{1},\mu_{2}}\right),}\\\
\end{array}\\\ 0,\begin{array}[]{*{20}c}{}\hfil&{{\rm{others}},}\\\
\end{array}\\\ \end{array}\right.\\\ \end{array}$ (10)

the system can generate a current demonstrating the interference between the
electron states incident from the left reservoir and the right one.

$I_{L}=-\frac{e}{{2\pi\hbar}}\int_{0}^{\mu_{1}}{dE}\left[{\int_{0}^{\mu_{1}}{\left|r\right|^{2}dE}+\int_{0}^{\mu_{2}}{\left|t\right|^{2}dE}+2\int_{0}^{\mu\left({\min}\right)}{{\mathop{\rm
Re}\nolimits}\left({r^{*}t^{\prime}}\right)dE}}\right].$ (11)

In the next subsection, we would use a toy system based on correlated
reservoirs to further illustrate the scattering scheme. Significant difference
between uncorrelated and correlated reservoirs is illuminated. It is also
elaborated that in scattering problems the transmission and reflection process
demonstrates the quantum state spanned throughout the space.

###  I.2 Further illustration of the scattering scheme in a toy system with
correlated reservoirs

We consider incidence only from the left in Eq. (2) and the input flux is
assumed to be unity. The wave function in different scattering regions can be
expressed as

$\psi\left(x\right)=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}c}{e^{ikx}+re^{-ikx},}&{x\leq-L,}\\\
\end{array}\\\ \begin{array}[]{*{20}c}{Ce^{\kappa x}+De^{-\kappa x},}&{-L\leq
x\leq L}\\\ \end{array},\\\ \begin{array}[]{*{20}c}{te^{ikx},}&{x\geq L.}\\\
\end{array}\\\ \end{array}\right.$ (12)

An incident particle is transmitted with probability $\left|t\right|^{2}$ and
reflected with probability $\left|r\right|^{2}$. It is determined by the wave
function that the particle momentum has value of $\hbar k$ with probability
${{\left({1+\left|t\right|^{2}}\right)}\mathord{\left/{\vphantom{{\left({1+\left|t\right|^{2}}\right)}2}}\right.\kern-1.2pt}2}$
and $-\hbar k$ with probability
${{\left|r\right|^{2}}\mathord{\left/{\vphantom{{\left|r\right|^{2}}2}}\right.\kern-1.2pt}2}$.
The mean value of the momentum $\left|t\right|^{2}\hbar k$ characterizes the
density of the probability current $\bf{J}$, which can also be obtained from
the continuity equation
${{\partial\rho}\mathord{\left/{\vphantom{{\partial\rho}{\partial{\rm{t}}}}}\right.\kern-1.2pt}{\partial{\rm{t}}}}+\nabla\cdot{\bf{J}}=0$
with $\rho$ the probability density.

An interesting interference pattern can be observed if we consider the
incidence from the left and the right is correlated. We consider a toy system
of a one-dimensional electron gas subject to two oscillating gate volatages
(see Fig. 2). The inspiration comes from the quantum pumping phenomenon. The
single-particle Hamiltonian reads

$H=-\frac{\hbar}{{2m^{*}}}\frac{{\partial^{2}}}{{\partial
x^{2}}}+U\left({x,t}\right),$ (13)

with
$U\left({x,t}\right)=\Theta\left({x+2L}\right)\Theta\left({-L-x}\right)U_{1}\left(t\right)+\Theta\left({x-L}\right)\Theta\left({2L-x}\right)U_{2}\left(t\right).$
The two barriers $U_{1}$ and $U_{2}$ are adiabatically modulated at the
frequency $\omega$ with a phase difference $\phi$.

$\begin{array}[]{l}U_{1}\left(t\right)=U_{10}+U_{1\omega}\sin\omega t,\\\
U_{2}\left(t\right)=U_{20}+U_{2\omega}\sin\left({\omega t-\phi}\right).\\\
\end{array}$ (14)

We assume $\omega$ is extremely small so that a static treatment is valid.
Therefore, it is tolerable to consider only the zero order of the Fourier
component of the time-dependent scattering matrix without taking into account
the photon-absorption/emmission processes, which is done in adiabatic quantum
pumping theory.

As shown in Fig. 2, the electrons are incident from the left and right
reservoirs with identical amplitudes at zero bias, which is set to be unity
without impairing generality. As assumed, incidence from the two correlated
reservoirs characterize a coherent single-particle state. The single-particle
wave function at a certain time has the following form.

$\Psi\left({x,t}\right)=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}c}{\left[{e^{ikx}+\left({r+t^{\prime}e^{i\theta}}\right)e^{-ikx}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{x\leq-2L,}\\\
\end{array}\\\
\begin{array}[]{*{20}c}{\left[{A_{2}e^{\kappa_{2}x}+B_{2}e^{-\kappa_{2}x}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{-2L\leq
x\leq-L}\\\ \end{array},\\\
\begin{array}[]{*{20}c}{\left[{A_{3}e^{ikx}+B_{3}e^{-ikx}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{-L\leq
x\leq L}\\\ \end{array},\\\
\begin{array}[]{*{20}c}{\left[{A_{4}e^{\kappa_{4}x}+B_{4}e^{-\kappa_{4}x}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{L\leq
x\leq 2L}\\\ \end{array},\\\
\begin{array}[]{*{20}c}{\left[{\left({t+r^{\prime}e^{i\theta}}\right)e^{ikx}+e^{-ikx}}\right]e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}},}&{x\geq
2L.}\\\ \end{array}\\\ \end{array}\right.$ (15)

$k={{\sqrt{2m^{*}E}}\mathord{\left/{\vphantom{{\sqrt{2m^{*}E}}\hbar}}\right.\kern-1.2pt}\hbar}$
and
$\kappa_{2/4}={{\sqrt{2m^{*}\left({U_{1/2}-E}\right)}}\mathord{\left/{\vphantom{{\sqrt{2m^{*}\left({U_{1/2}-E}\right)}}\hbar}}\right.\kern-1.2pt}\hbar}$.
$t$ and $r$ quantify the transmission and reflection amplitudes of the
electrons incident from the left reservoir while $t^{\prime}$ and $r^{\prime}$
quantify those incident from the right with $t^{\prime}=t$ and
$r^{\prime}={{-tr^{*}}\mathord{\left/{\vphantom{{-tr^{*}}{t^{*}}}}\right.\kern-1.2pt}{t^{*}}}$
. We introduce a geometric phase $\theta$ to describe the unavoidable phase
difference between the electrons injected from the two reservoirs. In the toy
approach, correlated reservoirs are assumed, which justifies a particular
value of $\theta$. For simplicity and without violation of the physics we take
$\theta=0$. It is noted that in conventional real reservoirs mixed-ensemble
integral should be applied, i.e., the probability flow of the electron
incident from the left reservoir absolutely cancels out that of the incidence
backward at zero bias as a result of the randomized phase distribution, which
absolutely differs from our toy consideration.

From the continuity equation

$\frac{{\partial\rho}}{{\partial t}}+\nabla\cdot{\bf{J}}=0,$ (16)

we could derive the probability current flow as functions of the transmission
and reflection amplitudes as

$j_{x}=-\frac{{i\hbar}}{{2m}}\left[{\psi^{\dagger}\frac{\partial}{{\partial
x}}\psi-\left({\frac{\partial}{{\partial
x}}\psi^{\dagger}}\right)\psi}\right]=-\frac{{4\hbar k}}{m}{\mathop{\rm
Re}\nolimits}\left({r^{*}t^{\prime}}\right).$ (17)

We can also see from the wave function Eq. (15) that the particle momentum has
value of $\hbar k$ with probability ${{\left[{1-{\mathop{\rm
Re}\nolimits}\left({tr^{*}}\right)}\right]}\mathord{\left/{\vphantom{{\left[{1-{\mathop{\rm
Re}\nolimits}\left({tr^{*}}\right)}\right]}2}}\right.\kern-1.2pt}2}$ and
$-\hbar k$ with probability ${{\left[{1+{\mathop{\rm
Re}\nolimits}\left({tr^{*}}\right)}\right]}\mathord{\left/{\vphantom{{\left[{1+{\mathop{\rm
Re}\nolimits}\left({tr^{*}}\right)}\right]}2}}\right.\kern-1.2pt}2}$. The mean
value of the momentum $-{\mathop{\rm Re}\nolimits}\left({tr^{*}}\right)\hbar
k$ characterizes the density of the probability current of Eq. (17). The net
current density can be described by the period-average of the probability
current density multiplied by the carrier charge and density. The accumulated
contribution by electrons within the $\pm\hbar\omega$ sidebands is taken into
account by an integral. Without dynamic modulation, no current occurs as the
$\pm\hbar\omega$ energy channel is closed even when the probability current
density is nonzero for asymmetric barrier configuration. The current density
as a function of the Fermi level thus becomes

$I_{L}\left(E_{F}\right)=-\frac{{4\hbar}}{{m^{*}}}\int_{E_{F}-\hbar\omega}^{E_{F}+\hbar\omega}{\frac{{ek\rho_{e}N\left(E\right)f\left(E\right)}}{{\left({{{2\pi}\mathord{\left/{\vphantom{{2\pi}\omega}}\right.\kern-1.2pt}\omega}}\right)}}\int_{0}^{\frac{{2\pi}}{\omega}}{{\mathop{\rm
Re}\nolimits}\left[{r^{*}\left(t\right)t^{\prime}\left(t\right)}\right]dt}dE},$
(18)

where the density of states of a one-dimensional electron gas is

$N\left(E\right)=\frac{V}{{\pi\hbar}}\sqrt{\frac{{m_{e}}}{{2E}}}.$ (19)

Here, $k$ is the wave vector of the electron. $e$ is the electron charge.
$\rho_{e}$ is the carrier density of the two-dimensional electron gas in which
the quantum wire is confined. $f(E)$ is the Fermi-Dirac distribution function
of the leads. $V$ quantifies the volume of the one-dimensional wire and
$m_{e}$ is the mass of a free electron.

Here, in the toy configuration, quantum interference is remarkably
demonstrated. The single-electron state Eq. (15) interferes with itself and
carries the probability flow and hence the net current through oscillating
cycles.

We numerically calculated the current in a one-dimensional electron gas (2DEG)
based on the GaAs/AlGaAs heterostructures with the average carrier density
$\rho_{e}\sim 10^{11}$ cm-2 and the effective mass of the electron $m^{*}\sim
0.067m_{e}$Ref73 . The width of the two gate potential barriers $L=20$ Å
equally separated by a $2L=40$ Å width well. The amplitudes of the modulations
$U_{1\omega}=U_{2\omega}=1$ meV. All of the above setups are not essential as
we are dealing with an assumed toy structure.

In Fig. 3, it is shown that the time-integrated current demonstrates a
sinusoidal pattern as a function of the phase difference between two
oscillating parameters, which is a result of quantum interference. The
probability density flow formulated in Eq. (17) is a result of phase
difference between transmission forward and backward. Quantum phase
interference gives rise to nonzero probability density flow for asymmetric
barrier configurations. The absolute strength of the probability flow
increases as the height difference between the two potential barriers
increases determined by the phase difference $\phi$. Fig. 4 presented the time
variation of the net current within an oscillating cycle. Considering the
time-averaged effect, the integrated asymmetry is different. When the phase
difference between the two modulations approaches $\pi$, time-reversal
symmetry destroys the time-integrated current to zero although the probability
density flow maximizes at a certain time. i.e. the probability flow in half a
pumping cycle completely offsets that of the other. Therefore a sinusoidal
dependence on $\phi$ occurs in the time-integrated current.

In real reservoirs, the single-particle state between reservoirs has a
definite momentum direction determined by its source. When coherent reservoirs
can be realized in any form, however, quantum states within the mesoscopic
conductor can be expressed as Eq. (15) and double-slit interference pattern is
observable in an electron device.

###  I.3 Time-dependent scattering-matrix theory

The scattering-matrix equation
$\hat{b}_{\alpha}=S_{\alpha\beta}\hat{a}_{\beta}$ with $\alpha$ and $\beta$
indexes of lead, channel, and spin introduced in Sec. I.A. for the Landauer-
Büttiker conductance characterizes the transport properties through a
conductor at a certain bias.

In a more general situation with dynamic processes, e.g. in quantum pumping, a
time-dependent scattering matrix can be introduced as follows.

$\hat{b}_{\alpha}\left(t\right)=\int_{-\infty}^{\infty}{S_{\alpha\beta}\left({t,t^{\prime}}\right)\hat{a}_{\beta}\left({t^{\prime}}\right)dt^{\prime}},\begin{array}[]{*{20}c}{}\hfil&{t\geq
t^{\prime},}\\\ \end{array}$ (20)

with $\alpha$ and $\beta$ general indexes denoting the lead, channel, and
spin. An incident state $\hat{a}_{\beta}$ at time $t^{\prime}$ is scattered
into the outgoing state $\hat{b}_{\alpha}$ at time $t$ with the amplitude
$S_{\alpha\beta}\left({t,t^{\prime}}\right)$.

The time-dependent scattering-matrix picture described by Eq. (20) is exactly
equivalent to the time-dependent Schrödinger equation with the elements of the
scatting matrix amplitudes of the wave function. In usual cases, the time-
dependent Schrödinger equation cannot be solved exactly, similarly to the
time-dependent scattering matrix. In the static or adiabatic cases, it is
advantageous to use an analog of the Wigner transform for the matrix
$S_{\alpha\beta}\left({t,t^{\prime}}\right)$trs ,

$S_{\alpha\beta}\left({E,t}\right)=\int_{-\infty}^{\infty}{e^{iE\left({t-t^{\prime}}\right)}S_{\alpha\beta}\left({t,t^{\prime}}\right)dt^{\prime}}.$
(21)

An on-time scattering process $S_{\alpha\beta}\left({t,t^{\prime}}\right)$
with $t=t^{\prime}$ is sufficient to describe the bias-driven conductance.
Namely, from Eq. (21), we can use $S_{\alpha\beta}\left({E}\right)$ with $E$
labeling the energy channel to fully capture the transport physics.

When the scattering time $t-t^{\prime}$ is small (i.e., the dynamic
characteristic frequency is much smaller than the inverse Wigner time delay),
the dynamics can be approximated into the instant-scattering picture.
Physically this means that the scattering matrix changes only a little while
an electron is scattered by the mesoscopic sample under dynamic modulation, in
which we use the term “adiabatic”. In adiabatic dynamics, we can use low-order
Fourier components of $S_{\alpha\beta}\left({E,t}\right)$ to characterize
transport physics. Therefore, small $t-t^{\prime}$ is transformed into
variation of the particle energy by side-band broadening around the Fermi
level.

In Sec. III, we would illustrate the time-dependent scattering approach in
adiabatic quantum pumping beyond the linear-response approximation.

##  II Spin-orbit coupling affected shot noise

###  II.1 Background

Current fluctuations are present in almost all kinds of conductors and have
been developed into a very active and fascinating subfield of mesoscopic
physics (for review see Ref. Ref20, ). At low temperatures, thermal
fluctuations are extremely small, the current fluctuation properties are
governed by the so-called shot noise, which is a consequence of the
quantization of charge. Shot noise is useful to obtain information on a system
which is not available through conductance measurements. In particular, shot
noise experiments can determine the quantum correlation of electrons, the
charge and statistics of the quasi-particles relevant for transport, and
reveal information on the potential profile as well as internal energy scales
of mesoscopic systemsRef34 ; Ref35 ; Ref36 ; Ref37 ; Ref38 ; Ref39 ; Ref40 ;
Ref41 ; Ref42 ; Ref43 ; Ref44 ; Ref45 ; Ref22 ; Ref60 . Shot noise is
generally more sensitive to the effects of electron-electron interactions than
the average conductance.

A convenient measure of shot noise is the Fano factor $F$, which is the ratio
of the actual shot noise and the Poisson noise. The Poisson noise would be
achieved in measurement if the transport is carried by single independent
electrons. Four typical values of the Fano factor characterize the shot noise
properties of different mesoscopic conductors.

1\. $F=1$ characterizes Poissonian processes. Particles are completely
independent during transport corresponding to channels through which
transmission is exponentially small. In diffusive transport, they are the so-
called approximately-closed channels. Typical conductors featuring $F=1$
include tunneling junction, Schottky-barrier diode, and asymmetric double-
barrier diode.

2\. $F=0$ characterizes ballistic transport. In ballistic transport,
transmission approaches the maximum of unity. Free particles wave function
extends throughout the space, i.e. particle beams exhibit full coherence. In
diffusive transport, they are the so-called open channels. Typical conductors
featuring $F=0$ include pure metal and free two-dimensional electron gas.

3\. $F=1/3$ characterizes diffusive transport. Open channels and closed
channels are distributed randomly. As a result of ensemble average of
channels, the strength shot noise falls between $F=1$ and $F=0$. Typical
conductors featuring $F=1/3$ include diffusive metals, graphene, and two-
dimensional electron gas modulated by magnetic barriers.

4\. $F=1/2$ characterizes ballistic transport constrained by Pauli principle.
The Pauli principle forbids two electrons to be in the same channel
simultaneously. As a result, the shot noise is suppressed. All kinds of
conductors are subject to the Pauli principle. In the symmetric-double-barrier
diode, where Pauli exclusion is the only correlation between particles, the
shot noise features $F=1/2$.

With increased attentionRef46 ; Ref47 ; Ref48 ; Ref49 ; Ref50 ; Ref51  to
semiconductor spintronics, materials such as ${\rm{GaSb}}$, ${\rm{InAs}}$, and
${\rm{InSb}}$ with considerably strong spin-orbit coupling (SOC) constantRef51
; Ref52  are becoming widely used in mesoscopic conductors, observationsRef34
; Ref37 ; Ref38  beyond the formalism of earlier theory on current shot noise
are reported. The scattering approach is developed to derive a general formula
for the shot noise in the presence of the SOC effect and apply it to the
double-barrier resonant diode (DBRD) systems. It is demonstrated that the
microscopic origin of the super-suppression of the shot noise observed in
experimentRef34 ; Ref37 ; Ref38  is the bunching interaction between electrons
with opposite spins resulting from the Dresselhaus $k^{3}$ termsRef53 ; Ref54
in the effective Hamiltonian of the bulk semiconductor of the barriers.

###  II.2 Theoretical approach

In this part, the effect of the Dresselhaus SOC to the shot noise properties
in the DBRD structure connected to ferromagnetic or normal metal leads is
considered. The theory can be generalized to other coherent mesoscopic
conductors subject to Dresselhaus and/or Rashba SOC.

The Dresselhaus SOC is caused by the bulk inversion asymmetry and exists
broadly in III-V compound semiconductors with zinc-blende crystal
structuresRef53 . Consider the transmission of electrons with identical wave
vector ${\bf{k}}=({\bf{k}}_{\parallel},k_{z})$ through a certain potential
barrier grown along $z\parallel[001]$ direction. ${\bf{k}}_{\parallel}$ is the
wave vector in the plane of the barrier and $k_{z}$ is the wave vector
component normal to the barrier. The electron Hamiltonian of the barrier in
the effective-mass approximation contains the spin-dependent $k^{3}$
Dresselhaus term

$\hat{H}_{D}=\gamma(\hat{\sigma}_{x}k_{x}-\hat{\sigma}_{y}k_{y})\frac{{\partial^{2}}}{{\partial
z^{2}}},$ (22)

where $\gamma$ is the material constant denoting the strength of the
Dresselhaus SOC, $\hat{\sigma}_{x}$ and $\hat{\sigma}_{y}$ are the Pauli
matrices. The Dresselhaus term can be diagonalized by the spinors

$\chi_{\pm}=\frac{1}{{\sqrt{2}}}\left({\begin{array}[]{*{20}c}1\\\ {\mp
e^{-i\varphi}}\\\ \end{array}}\right),$ (23)

which describe the spin-up (“$+$”) and spin-down (“$-$”) electron eigenstates.

Suppose the system is a layered mesoscopic conductor with its potential
profile described by $V_{0}(z)$ (see Fig. 5), the electron motion in each
layer of the structure is described by the Hamiltonian

$\hat{H}=-\frac{{\hbar^{2}}}{{2m^{*}}}\nabla^{2}+V(z)+\hat{H}_{D},$ (24)

where $V(z)=V_{0}(z)-eF(z+b)\Theta(z+b)\Theta(a+c-z)$ with $F$ the magnitude
of the electric field, $\Theta(z)$ the step function, and $-b$ and $a+c$ the
longitudinal coordinates of surfaces in $z$ direction. Our discussion is
within the framework of single electron approximation and coherent
tunnelingRef55 ; Ref56 , and only zero-frequency noise at zero temperature is
considered. Under the assumption that $k_{\parallel}$ is conserved during the
tunneling, the wave functions for the electrons with definite longitudinal
electron energy ($E_{z}$) can be obtained from Schrödinger equation based on
the Hamiltonian given in Eq. (24), which can be diagonalized by spinors
$\chi_{\pm}$. So, the wave functions become

$\psi(\mathord{\buildrel{\lower
3.0pt\hbox{$\scriptscriptstyle\rightharpoonup$}}\over{r}})=\left\\{{\begin{array}[]{*{20}c}{\exp(i{\bf{k}}_{\parallel}\cdot{\bf{\rho}})\left({\sum\limits_{j=\pm}{\sqrt{\frac{{m_{1}}}{{\hbar
k_{1j}}}}\exp(ik_{1j}z)\chi_{j}}+\sum\limits_{j=\pm}{r_{j}\sqrt{\frac{{m_{1}}}{{\hbar
k_{1j}}}}\exp(-ik_{1j}z)\chi_{j}}}\right),}&{z<b,}\\\
{\exp(i{\bf{k}}_{\parallel}\cdot{\bf{\rho}})\sum\limits_{j=\pm}{\Phi_{\xi
j}(z)},}&{-b\leq z<a+c,}\\\
{\exp(i{\bf{k}}_{\parallel}\cdot{\bf{\rho}})\sum\limits_{j=\pm}{t_{j}\sqrt{\frac{{m_{5}}}{{\hbar
k_{5j}}}}\exp(ik_{5j}z)\chi_{j}},}&{z\geq a+c,}\\\ \end{array}}\right.$ (25)

where $\Phi_{\xi j}$ denotes the wave function in the conductor region, and
$t_{j}$ and $r_{j}$ are the transmission and reflection amplitudes, which can
be calculated using the transfer-matrix methodRef57 . The spin “$+$” and spin
“$-$” components of the electron wave functions transport separately without
correlation. As standard scattering method is applied, we introduce creation
and annihilation operators of electrons in the scattering states. Schematics
of the scattering states are shown in Fig. 6. Operators
$\hat{a}_{Ln\sigma}^{\dagger}(E)$ and $\hat{a}_{Ln\sigma}(E)$ create and
annihilate electrons with total energy $E$ and spin polarization $\sigma$ in
the transverse channel $n$ in the left lead, which are incident upon the
sample. In the same way, the creation $\hat{b}_{Ln\sigma}^{\dagger}(E)$ and
annihilation $\hat{b}_{Ln\sigma}(E)$ operators describe electrons in the
outgoing states. They obey anti-commutation relations. Therefore, we can write
the scattering matrix of the sample as

$\begin{array}[]{c}\left({\begin{array}[]{*{20}c}{\hat{b}_{Ln\uparrow}}\\\
{\hat{b}_{Ln\downarrow}}\\\ {\hat{b}_{Rn\uparrow}}\\\
{\hat{b}_{Rn\downarrow}}\\\
\end{array}}\right)=\frac{1}{2}\underbrace{\left({\begin{array}[]{*{20}c}1&1&0&0\\\
{e^{-i\varphi}}&{-e^{-i\varphi}}&0&0\\\ 0&0&1&1\\\
0&0&{e^{-i\varphi}}&{-e^{-i\varphi}}\\\
\end{array}}\right)}_{M_{1}}\times\underbrace{\left({\begin{array}[]{*{20}c}{r_{+}}&0&{t_{+}^{\prime}}&0\\\
0&{r_{-}}&0&{t_{-}^{\prime}}\\\ {t_{+}}&0&{r_{+}^{\prime}}&0\\\
0&{t_{-}}&0&{r_{-}^{\prime}}\\\ \end{array}}\right)}_{s}\\\
\times\underbrace{\left({\begin{array}[]{*{20}c}1&{-e^{i\varphi}}&0&0\\\
1&{e^{i\varphi}}&0&0\\\ 0&0&1&{-e^{i\varphi}}\\\ 0&0&1&{e^{i\varphi}}\\\
\end{array}}\right)}_{M_{2}}\times\left({\begin{array}[]{*{20}c}{\hat{a}_{Ln\uparrow}}\\\
{\hat{a}_{Ln\downarrow}}\\\ {\hat{a}_{Rn\uparrow}}\\\
{\hat{a}_{Rn\downarrow}}\\\ \end{array}}\right),\\\ \end{array}$ (26)

with $t^{\prime}=t,r^{\prime}=-\frac{t}{{t^{*}}}r^{*}$,
$T_{j}(E)=\left|{t_{j}}\right|^{2},R_{j}(E)=\left|{r_{j}}\right|^{2}$, and
$j=\pm$. The matrices $M_{1}$ and $M_{2}$ are unitary transformations between
spin “$\uparrow\downarrow$” states and spin “$\pm$” states, $s$ is the
scattering matrix connecting the incoming and outgoing spin “$\pm$” states of
the $n$th channel. The current of the system can be derived as follows

$\hat{I}_{L}(t)=\frac{e}{{2\pi\hbar}}\sum\limits_{\alpha\beta}{\sum\limits_{mn}{\sum\limits_{\sigma^{\prime}\sigma^{\prime\prime}}{\sum\limits_{\sigma}{\int{dE}\int{dE^{\prime}}e^{i(E-E^{\prime})t/\hbar}\hat{a}_{\alpha
m\sigma^{\prime}}^{\dagger}}}}}(E)A_{\alpha,\beta,\sigma^{\prime},\sigma^{\prime\prime}}^{m,n,\sigma}(L;E,E^{\prime})\hat{a}_{\beta
n\sigma^{\prime\prime}}(E^{\prime}),$ (27)

where

$\begin{array}[]{l}A_{\alpha,\beta,\sigma^{\prime},\sigma^{\prime\prime}}^{m,n,\sigma}(L;E,E^{\prime})=\\\
\delta_{\alpha L}\delta_{mm}\delta_{\beta
L}\delta_{\sigma^{\prime}\sigma}\delta_{\sigma^{\prime\prime}\sigma}-\sum\limits_{\xi^{\prime}\xi^{\prime\prime}}{\sum\limits_{k}{M_{2\xi^{\prime\prime}\sigma^{\prime}}^{\dagger}s_{L,\alpha;m,k;\xi^{\prime\prime}}^{\dagger}}(E)M_{1\sigma\xi^{\prime\prime}}^{\dagger}M_{1\sigma\xi^{\prime}}s_{L,\beta;k,n;\xi^{\prime}}(E^{\prime})M_{2\xi^{\prime}\sigma^{\prime\prime}}}.\\\
\end{array}$ (28)

For a system at thermal equilibrium, the quantum statistical average of the
product of an electron creation operator and annihilation operator of a Fermi
gas with spin polarization is

$\begin{array}[]{*{20}c}{\left\langle{a_{Ln\uparrow}^{\dagger}a_{Ln\uparrow}}\right\rangle=f_{Lp},}&{\left\langle{a_{Ln\downarrow}^{\dagger}a_{Ln\downarrow}}\right\rangle=f_{La},}&{\left\langle{a_{Ln\uparrow}^{\dagger}a_{Ln\downarrow}}\right\rangle=\left\langle{a_{Ln\downarrow}^{\dagger}a_{Ln\uparrow}}\right\rangle=\frac{1}{2}(f_{Lp}-f_{La}).}\\\
\end{array}$ (29)

Without loss of generality, we set the unit vector directed along the spin
orientation ${\bf{n}}_{s}=(1,0,0)$. Making use of Eqs. (24)-(26), after some
algebra, we can obtain the expression for the zero-frequency noise power

$\begin{array}[]{l}S_{\alpha\beta}\equiv
S_{\alpha\beta}(0)=\frac{{e^{2}}}{{4\pi\hbar}}\sum\limits_{\delta\delta_{2}}{\sum\limits_{mn}{\sum\limits_{\scriptstyle\sigma^{\prime}\sigma^{\prime\prime}\hfill\atop\scriptstyle\sigma_{2}^{\prime}\sigma_{2}^{\prime\prime}\hfill}{\sum\limits_{\scriptstyle\sigma\hfill\atop\scriptstyle\sigma_{2}\hfill}{\int{dE}}}}}\\\
\times\left\\{{A_{\delta,\delta_{2},\sigma^{\prime},\sigma^{\prime\prime}}^{m,n,\sigma}(\alpha;E,E)A_{\delta_{2},\delta,\sigma_{2}^{\prime},\sigma_{2}^{\prime\prime}}^{n,m,\sigma_{2}}(\beta;E,E)+A_{\delta,\delta_{2},\sigma^{\prime},\sigma^{\prime\prime}}^{m,n,\sigma}(\beta;E,E)A_{\delta_{2},\delta,\sigma_{2}^{\prime},\sigma_{2}^{\prime\prime}}^{n,m,\sigma_{2}}(\alpha;E,E)}\right\\}\\\
\times\left\\{{\left[{\delta_{\sigma^{\prime\prime}\sigma_{2}^{\prime}}\delta_{\sigma^{\prime}\sigma_{2}^{\prime\prime}}f_{\delta}(E)+\delta_{\sigma^{\prime\prime}\sigma_{2}^{\prime}}\delta_{\bar{\sigma}^{\prime}\sigma_{2}^{\prime\prime}}\frac{1}{2}\left({f_{p\delta}(E)-f_{a\delta}(E)}\right)}\right]}\right.\\\
\left.{-\left[{\delta_{\sigma^{\prime}\sigma_{2}^{\prime\prime}}f_{\delta}(E)+\delta_{\bar{\sigma}^{\prime}\sigma_{2}^{\prime\prime}}\frac{1}{2}\left({f_{p\delta}(E)-f_{a\delta}(E)}\right)}\right]\times\left[{\delta_{\sigma^{\prime\prime}\sigma_{2}^{\prime}}f_{\delta_{2}}(E)+\delta_{\bar{\sigma}^{\prime\prime}\sigma_{2}^{\prime}}\frac{1}{2}\left({f_{p\delta_{2}}(E)-f_{a\delta_{2}}(E)}\right)}\right]}\right\\}.\\\
\end{array}$ (30)

Eq. (30) gives a general formula to calculate the shot noise in mesoscopic
conductors subject to the SOC effect. It can be used to predict the low-
frequency noise properties of arbitrary multi-channel, multi-probe phase-
coherent conductors in the presence of Dresselhaus SOC. It can be naturally
extended to the system with Rashba SOC and the system with both Dresselhaus
SOC and Rashba SOC. In the limit of zero SOC and when the conductor is
connected to normal metal leads, there is $f_{\alpha a}(E)=f_{\alpha p}(E)$
and $T_{n+}(E)=T_{n-}(E)=T(E)$, thus Eq. (30) reconverts to the formula
provided by BüttikerRef58  concerning scalar electron systems without the spin
degree of freedom inducing observable difference.

###  II.3 Numerical results in comparison with experiment

To further demonstrate our theory, we provide numerical results based on real
heterostructures in comparison with experiment.

The double-barrier structure considered here is constructed of layers of
${\rm{Ga}}_{\rm{x}}{\rm{Al}}_{{\rm{1-x}}}{\rm{Sb}}$ with
$x=0.15/0.3/0/0.3/0.15$ ($x=0.3$ for the barriers and $x=0$ for the well),
which are known to be semiconductors with relatively strong Dresselhaus
SOCRef55 ; Ref56 . Our target setup is a symmetric double-barrier structure
with the thickness of the well $a=30$ ${\rm{{\AA}}}$ and the thickness of the
two barriers $b=c=50$ ${\rm{{\AA}}}$. The height of the barrier $V_{b}=230$
${\rm{meV}}$ and the depth of the well $V_{w}=200$ ${\rm{meV}}$ are given by
the heterostructure propertiesRef57  (cf. Fig. 5). We assume that in the whole
region the effective mass $m^{*}=0.053m_{e}$Ref55 ; Ref56 . The chemical
potential of the two electrodes is set to be ${\rm{12}}$ ${\rm{meV}}$. For
comparison, we choose the Dresselhaus constant $\gamma=0,40,80,120$
${\rm{eV}}$ ${\rm{{\AA}^{3}}}$.

We see the current fluctuation of the system caused by Dresselhaus SOC. Fig. 7
presents results of the electric current $I$ and the shot noise $S$ versus the
external bias. It is demonstrated that the peaks of the current and of the
shot noise are lowered as the Dresselhaus constant $\gamma$ increases, and a
concave down in the ascending side of the noise curve is obvious for non-zero
$\gamma$. The concaveness gives rise to nadirs far below 0.5 in the Fano
factor (see Fig. 8 (a)). To compare with experiment, $S$ vs $I$ curves
normalized to unity are displayed in Fig. 8 (b). In the positive differential
conductance region, the shot noise follows the value of uncorrelated electrons
($2eI$) for small tunnel currents and is significantly suppressed for larger
currents and eventually increases. The suppression is above one-half for
$\gamma=0$, near one-half for $\gamma=4.0\times 10^{-29}$ ${\rm{eV}}$
${{\rm{m}}^{\rm{3}}}$, and below one-half for $\gamma$ larger than $8.0\times
10^{-29}$ ${\rm{eV}}$ ${{\rm{m}}^{\rm{3}}}$. In the negative differential
conductance (NDC) region, the Coulomb interaction or charging effect in the
well enhances the shot noise and overweighs the effect of SOCRef59 .
Iannaccone et al. have focused on the NDC region and obtained enhanced shot
noiseRef36 .

The results shown in Figs. 7-8 can be understood from the following. The SOC
interaction behaves like a pseudo magnetic field and induces split of
different spin components of the resonant level in the barrier structureRef55
, which contribute collectively to the electric current and shot noise. Thus,
the current is lowered at the peak and simultaneously lifted in both sides
around the peak in the current-bias spectra. When the SOC is present, spin
“$\uparrow$” electrons exclude spin “$\downarrow$” electrons as well as spin
“$\uparrow$” ones. Therefore, the large noise suppressions are a consequence
of the repulsion between current pulses of different spin states in addition
to the consequence of the Pauli blockade and Coulomb repulsion.

##  III Quantum pumping beyond linear response

###  III.1 Introduction to quantum pumping

Generally speaking, the transport of matter from low potential to high
potential excited by absorbing energy from the environment can be described as
a pump process. The driving mechanics of classic pumps is straightforward and
well understoodRef61 . The concept of a quantum pump is initiated several
decades agoRef62  with its mechanism involving coherent tunneling and quantum
interference. Research on quantum pumping has attracted heated interest since
its experimental realization in an open quantum dotRef1 ; Ref2 ; Ref3 ; Ref4 ;
Ref5 ; Ref6 ; Ref7 ; Ref8 ; Ref9 ; Ref10 ; Ref11 ; Ref12 ; Ref13 ; Ref14 ;
Ref15 ; Ref16 ; Ref17 ; Ref18 ; Ref19 ; Ref63 ; Ref64 ; Ref65 ; Ref66 ; Ref67
; Ref68 ; Ref69 ; Ref70 ; Ref71 ; Ref72 .

The mechanisms of an adiabatic quantum pump can be demonstrated in a
mesoscopic system modulated by two oscillating barriers (see Fig. 9). To
prominently picture the charge flow driven process within a cyclic period, the
two potential barriers are modulated with a phase difference of $\pi/2$ in the
manner of $U_{1}=U_{0}+U_{1\omega}\sin t$ and
$U_{2}=U_{0}+U_{2\omega}\sin(t+\pi/2)$. Our discussion is within the framework
of the single electron approximation and coherent tunneling. The Pauli
principle is taken into account throughout the pumping process. The Fermi
energy of the two reservoirs and the inner single-particle state energy are
equalized to eliminate the external bias and secure energy-conserved tunneling
[The kinetic properties (charge current, heat current, etc.) depend on the
values of the scattering matrix within the energy interval of the order of
${\rm{max}}(k_{B}T,\hbar\omega)$ near the Fermi energy. In the low-frequency
($\omega\rightarrow 0$) and low-temperature ($T\rightarrow 0$) limit we assume
the scattering matrix to be energy independent]. As shown in Fig. 9, the
transmission strengths between one of the reservoirs and the inner single-
particle state are denoted by $t_{1}$-$t_{4}$. When $t\in[0,\pi/2]$, $\sin t$
changes from 0 to 1 and $\sin(t+\pi/2)$ changes from 1 to 0. Considering the
time-averaged effect, the chance of $U_{1}>U_{2}$ and $U_{1}<U_{2}$ is equal.
Therefore, the probability of $t_{1}$ and $t_{3}$ balance out. The tunneling
quantified by $t_{2}$ and $t_{4}$ do not occur since the inner particle state
is not occupied. When $t\in[\pi/2,\pi]$, $\sin t$ changes from 1 to 0 and
$\sin(t+\pi/2)$ changes from 0 to -1. $U_{1}>U_{2}$ invariably holds in this
time regime. The probability of $t_{3}$ prevails and a net particle flow is
driven from the right reservoir to the middle state. When $t\in[\pi,3\pi/2]$,
$\sin t$ changes from 0 to -1 and $\sin(t+\pi/2)$ changes from -1 to 0. The
probability of $t_{2}$ and $t_{4}$ balance out and the tunneling quantified by
$t_{1}$ and $t_{3}$ are excluded from the Pauli principle. No net time-
averaged tunneling occurs. When $t\in[3\pi/2,2\pi]$, $\sin t$ changes from -1
to 0 and $\sin(t+\pi/2)$ changes from 0 to 1. $U_{1}$ maintains a lower height
than $U_{2}$, which drives the particle in the inner state to the left
reservoir. Through one whole pumping cycle, electrons are pumped from the
right reservoir to the left by absorbing energy from the two oscillating
sources. The tunneling is governed by quantum coherence. In each period, the
pumping process repeats and the particles are driven continuously in the same
direction as time accumulates. Direction-reversed pumped current can be
obtained with reversed phase difference of the two oscillating gates. The
direction of the pumped current is from the phase-leading gate to the phase-
lagged one without exception when we assume that higher barriers admit smaller
transmission probability. It can find resemblance in its classical turnstile
counterpartRef61  in which the fore-opened gate admits transmission ahead of
the later-opened one driving currents in corresponding manner.

The current and noise properties in various quantum pump structures and
devices were investigated such as the magnetic-barrier-modulated two
dimensional electron gasRef4 , mesoscopic one-dimensional wireRef6 ; Ref65 ,
quantum-dot structuresRef61 ; Ref5 ; Ref11 ; Ref12 ; Ref71 , mesoscopic rings
with Aharonov-Casher and Aharonov-Bohm effectRef7 , magnetic tunnel
junctionsRef10 , chains of tunnel-coupled metallic islandsRef68 , the
nanoscale helical wireRef69 ,the Tomonaga-Luttinger liquidRef67 , and
garphene-based devicesRef63 ; Ref64 . Theory also predicts that charge can be
pumped by oscillating one parameter in particular quantum configurationsRef66
. A recent experimentRef70  based on two parallel quantized charge pumps
offers a way forward to the potential application of quantum pumping in
quantum information processing, the generation of single photons in pairs and
bunches, neural networking, and the development of a quantum standard for
electrical current. Correspondingly, theoretical techniques have been put
forward for the treatment of the quantum pumpsRef2 ; Ref3 ; Ref18 ; Ref65 ;
Ref68 ; Ref72 . One of the most prominent is the scattering approach proposed
by Brouwer who presented a formula that relates the pumped current to the
parametric derivatives of the scattering matrix of the system. Driven by
adiabatic and weak modulation (the ac driving amplitude is small compared to
the static potential), the pumped current was found to vary in a sinusoidal
manner as a function of the phase difference between the two oscillating
potentials. It increases linearly with the frequency in line with experimental
finding. The Floquet scattering theory is developedRef72  for quantum-
mechanical pumping in mesoscopic conductors. It can be used to investigate
quantum pumping behavior at arbitrary pumping amplitude and frequency.

As an example to demonstrate the Floquet scattering theory, we focus on the
experimentally observed deviation from the weak-pumping theory with only the
first-order parametric derivative of the scattering matrix considered. By
expanding the scattering matrix to higher orders of the time and modulation
amplitude, experimental observation can be interpreted by multi-energy-
quantum-related processes.

###  III.2 Theoretical formulation

We use the scattering matrix approach to describe the response of a mesoscopic
phase-coherent sample to two slowly oscillating (with a frequency $\omega$)
external real parameters $X_{j}(t)$ (gate potential, magnetic flux, etc.),

$\begin{array}[]{*{20}c}{X_{j}\left(t\right)=X_{0,j}+X_{\omega,j}e^{i\left({\omega
t-\varphi_{j}}\right)}+X_{\omega,j}e^{-i\left({\omega
t-\varphi_{j}}\right)},}&{j=1,2.}\\\ \end{array}$ (31)

$X_{0,j}$ and $X_{\omega,j}$ measure the static magnitude and ac driving
amplitude of the two parameters, respectively. The phase difference between
the two drivers is defined as $\phi=\varphi_{1}-\varphi_{2}$. The mesoscopic
conductor is connected to two reservoirs at zero bias. The scattering matrix
$\hat{s}$ being a function of parameters $X_{j}(t)$ depends on time.

A time-dependent scattering matrix can be introduced as follows.

$\hat{b}_{\alpha}\left(t\right)=\int_{-\infty}^{\infty}{S_{\alpha\beta}\left({t,t^{\prime}}\right)\hat{a}_{\beta}\left({t^{\prime}}\right)dt^{\prime}},\begin{array}[]{*{20}c}{}\hfil&{t\geq
t^{\prime}.}\\\ \end{array}$ (32)

Its Wigner transform reads

$S_{\alpha\beta}\left({E,t}\right)=\int_{-\infty}^{\infty}{e^{iE\left({t-t^{\prime}}\right)}S_{\alpha\beta}\left({t,t^{\prime}}\right)dt^{\prime}}.$
(33)

We assume the scattering time $t-t^{\prime}$ is small. Up to corrections of
order $\hbar\omega/\gamma$ ($\gamma$ measures the escape rate), the matrix
$S_{\alpha\beta}\left({E,t}\right)$ is equal to the “instantaneous” scattering
matrix $S_{X}(E)$, which is obtained by “freezing” all parameters $X_{j}$ to
their values at time $t$. Below, we use the instant scattering matrix
$\hat{s}\left(t\right)$ in place of $S_{\alpha\beta}\left({E,t}\right)$ to
describe the physics for simplicity. The kinetic properties (charge current,
heat current, etc.) depend on the values of the scattering matrix within the
energy interval of the order of ${\rm{max}}(k_{B}T,\hbar\omega)$ near the
Fermi energy. In the low-frequency ($\omega\rightarrow 0$) and low-temperature
($T\rightarrow 0$) limit we assume the scattering matrix to be energy
independent. To investigate the deviation from the small amplitude
$X_{\omega,j}$ limit, we expand the scattering matrix $\hat{s}(t)$ into Taylor
series of $X_{j}(t)$ to second order at $X_{0,j}$ with the terms linear and
quadratic of $X_{\omega,j}$ present in the expansion,

$\hat{s}\left(t\right)\approx\hat{s}_{0}\left({X_{0,j}}\right)+\hat{s}_{-\omega}e^{i\omega
t}+\hat{s}_{+\omega}e^{-i\omega t}+\hat{s}_{2}+\hat{s}_{-2\omega}e^{2i\omega
t}+\hat{s}_{+2\omega}e^{-2i\omega t},$ (34)

with

$\left\\{\begin{array}[]{l}\hat{s}_{\pm\omega}=\sum\limits_{j=1,2}{X_{\omega,j}e^{\pm
i\varphi_{j}}{{\partial\hat{s}}\mathord{\left/{\vphantom{{\partial\hat{s}}{\partial
X_{j}}}}\right.\kern-1.2pt}{\partial X_{j}}}},\\\
\hat{s}_{2}=\sum\limits_{j=1,2}{X_{\omega,j}^{2}{{\partial^{2}\hat{s}}\mathord{\left/{\vphantom{{\partial^{2}\hat{s}}{\partial
X_{j}^{2}}}}\right.\kern-1.2pt}{\partial X_{j}^{2}}}},\\\ \hat{s}_{\pm
2\omega}=\frac{1}{2}\sum\limits_{j=1,2}{X_{\omega,j}^{2}e^{\pm
2i\varphi_{j}}{{\partial^{2}\hat{s}}\mathord{\left/{\vphantom{{\partial^{2}\hat{s}}{\partial
X_{j}^{2}}}}\right.\kern-1.2pt}{\partial X_{j}^{2}}}}.\\\ \end{array}\right.$
(35)

It can be seen from the equations that higher orders of the Fourier spectra
enter into the scattering matrix. As a result, both the nearest and next
nearest sidebands are taken into account, which implies that a scattered
electron can absorb or emit an energy quantum of $\hbar\omega$ or
$2\hbar\omega$ before it leaves the scattering region. In principle, third or
higher orders in the Taylor series can be obtained accordingly. However, the
higher-order parametric derivatives of the scatter matrix diminish
dramatically and approximate zero. Numerical calculation demonstrates that
even in relatively large amplitude modulation, their contribution is
negligible.

The pumped current depends on the values of the scattering matrix within the
energy interval of the order of $\max\left({k_{B}T,2\hbar\omega}\right)$ near
the Fermi energy. In the low-temperature limit ($T\to 0$), an energy interval
of $2\hbar\omega$ is opened during the scattering process.

The mesoscopic scatterer is coupled to two reservoirs with the same
temperatures $T$ and electrochemical potentials $\mu$. Electrons with the
energy $E$ entering the scatterer are described by the Fermi distribution
function $f_{0}(E)$, which approximates a step function at a low temperature.
Due to the interaction with an oscillating scatterer, an electron can absorb
or emit energy quanta that changes the distribution function. A single
transverse channel in one of the leads is considered. Applying the hypothesis
of an instant scattering, the scattering matrix connecting the incoming and
outgoing states can be written as

$\hat{b}_{\alpha}\left(t\right)=\sum\limits_{\beta}{s_{\alpha\beta}\left(t\right)\hat{a}_{\beta}\left(t\right)}.$
(36)

Here $s_{\alpha\beta}$ is an element of the scattering matrix $\hat{s}$; the
time-dependent operator is
$\hat{a}_{\alpha}\left(t\right)=\int{dE\hat{a}_{\alpha}\left(E\right)e^{{{-iEt}\mathord{\left/{\vphantom{{-iEt}\hbar}}\right.\kern-1.2pt}\hbar}}}$,
and the energy-dependent operator ${\hat{a}_{\alpha}\left(E\right)}$
annihilates particles with total energy E incident from the $\alpha$ lead into
the scatter and obey the following anticommutation relations

$\left[{\hat{a}_{\alpha}^{\dagger}\left(E\right),\hat{a}_{\beta}\left({E^{\prime}}\right)}\right]=\delta_{\alpha\beta}\delta\left({E-E^{\prime}}\right).$
(37)

Note that above expressions correspond to single- (transverse) channel leads
and spinless electrons. For the case of many-channel leads each lead index
($\alpha$, $\beta$, etc.) includes a transverse channel index and any
repeating lead index implies implicitly a summation over all the transverse
channels in the lead. Similarly an electron spin can be taken into account.

Using Eqs. (34) and (36) and after a Fourier transformation we obtain

$\begin{array}[]{l}\hat{b}_{\alpha}\left(E\right)=\sum\limits_{\beta}{\left[{\hat{s}_{0,\alpha\beta}\hat{a}_{\beta}\left(E\right)+\hat{s}_{2,\alpha\beta}\hat{a}_{\beta}\left(E\right)+\hat{s}_{-\omega,\alpha\beta}\hat{a}_{\beta}\left({E+\hbar\omega}\right)}\right.}\\\
\left.{+\hat{s}_{+\omega,\alpha\beta}\hat{a}_{\beta}\left({E-\hbar\omega}\right)+\hat{s}_{-2\omega,\alpha\beta}\hat{a}_{\beta}\left({E+2\hbar\omega}\right)+\hat{s}_{+2\omega,\alpha\beta}\hat{a}_{\beta}\left({E-2\hbar\omega}\right)}\right].\\\
\end{array}$ (38)

The distribution function for electrons leaving the scatterer through the lead
$\alpha$ is
$f_{\alpha}^{\left({out}\right)}\left(E\right)=\left\langle{\hat{b}_{\alpha}^{\dagger}\left(E\right)\hat{b}_{\alpha}\left(E\right)}\right\rangle$,
where $\left\langle\cdots\right\rangle$ means quantum-mechanical averaging.
Substituting Eq. (38) we find

$\begin{array}[]{l}f_{\alpha}^{\left({out}\right)}\left(E\right)=\sum\limits_{\beta}{\left[{\left|{\hat{s}_{0,\alpha\beta}+\hat{s}_{2,\alpha\beta}}\right|^{2}f_{0}\left(E\right)+\left|{\hat{s}_{-\omega,\alpha\beta}}\right|^{2}f_{0}\left({E+\hbar\omega}\right)}\right.}\\\
\left.{\left|{\hat{s}_{+\omega,\alpha\beta}}\right|^{2}f_{0}\left({E-\hbar\omega}\right)+\left|{\hat{s}_{-2\omega,\alpha\beta}}\right|^{2}f_{0}\left({E+2\hbar\omega}\right)+\left|{\hat{s}_{+2\omega,\alpha\beta}}\right|^{2}f_{0}\left({E-2\hbar\omega}\right)}\right].\\\
\end{array}$ (39)

The distribution function for outgoing carriers is a nonequilibrium
distribution function generated by the nonstationary scatterer. The Fourier
amplitudes of the scattering matrix
${\left|{\hat{s}_{-\omega,\alpha\beta}}\right|^{2}}$
(${\left|{\hat{s}_{+\omega,\alpha\beta}}\right|^{2}}$) is the probability for
an electron entering the scatterer through the lead $\beta$ and leaving the
scatterer through the lead $\alpha$ to emit (to absorb) an energy quantum
$\hbar\omega$ and ${\left|{\hat{s}_{-2\omega,\alpha\beta}}\right|^{2}}$
(${\left|{\hat{s}_{+2\omega,\alpha\beta}}\right|^{2}}$) is that of the energy
quantum $2\hbar\omega$ process.
${\left|{\hat{s}_{0,\alpha\beta}+\hat{s}_{2,\alpha\beta}}\right|^{2}}$ is the
probability for the same scattering without the change of an energy with the
second-order term $\hat{s}_{2,\alpha\beta}$ much smaller than the zero-order
term $\hat{s}_{0,\alpha\beta}$ in weak-modulation limit ($X_{\omega,j}\ll
X_{0,j}$) and can be omitted therein.

Using the distribution functions $f_{0}(E)$ for incoming electrons and
$f_{\alpha}^{out}(E)$ for outgoing electrons, the pumped current measured at
lead $\alpha$ reads

$I_{p}=\frac{e}{{2\pi\hbar}}\int_{0}^{\infty}{\left\langle{\hat{b}_{\alpha}^{\dagger}\left(E\right)\hat{b}_{\alpha}\left(E\right)}\right\rangle-\left\langle{\hat{a}_{\alpha}^{\dagger}\left(E\right)\hat{a}_{\alpha}\left(E\right)}\right\rangle
dE}.$ (40)

Substituting Eqs. (34) and (30) we get

$\begin{array}[]{c}I_{p}=\frac{{e\omega}}{{2\pi}}\sum\limits_{\beta,j_{1},j_{2}}{X_{\omega,j_{1}}X_{\omega,j_{2}}\frac{{\partial
s_{\alpha\beta}}}{{\partial X_{j_{1}}}}\frac{{\partial
s_{\alpha\beta}^{*}}}{{\partial
X_{j_{2}}}}2i\sin\left({\varphi_{j_{1}}-\varphi_{j_{2}}}\right)}\\\
+\frac{{e\omega}}{{2\pi}}\sum\limits_{\beta,j_{1},j_{2}}{X_{\omega,j_{1}}^{2}X_{\omega,j_{2}}^{2}\frac{{\partial^{2}s_{\alpha\beta}}}{{\partial
X_{j_{1}}^{2}}}\frac{{\partial^{2}s_{\alpha\beta}^{*}}}{{\partial
X_{j_{2}}^{2}}}i\sin\left[{2\left({\varphi_{j_{1}}-\varphi_{j_{2}}}\right)}\right]}.\\\
\end{array}$ (41)

Quantum pumping properties beyond the theory based on first-order parametric
derivative of the scattering matrix are demonstrated in Eq. (41). By taking
higher orders of the Fourier spectrum of the scattering matrix into
consideration, double $\hbar\omega$ energy quantum (or a $2\hbar\omega$ energy
quantum) emission (absorption) processes coact with single $\hbar\omega$
quantum processes. In the weak-modulation limit, the second term in the right-
hand side of Eq. (41) is small, which implies that double $\hbar\omega$
quantum processes are weak and therefore not observable. As the ac driving
amplitude is enlarged, this term increases markedly and contribution from
double $\hbar\omega$ quantum processes takes effect. As a result, the
dependence of the pumped current on the phase difference between two driving
oscillations deviates from sinusoidal and changes from $\sin\phi$ to $\sin
2\phi$, which is observed in experimentRef1 . Moreover, the relation between
the pumped current and the ac driving amplitude $X_{\omega,j}$ is reshaped. It
is also seen that the linear dependence of the pumped current on the
oscillation frequency holds for multi-quanta-related processes.

###  III.3 Numerical results and interpretations

Here, numerical results of the pumped current in a two-oscillating-potential-
barrier modulated nanowire are presented and comparison with experiment is
given. We consider a nanowire modulated by two gate potential barriers with
equal width $L=20$ Å separated by a $2L=40$ Å width well (see Fig. 10). The
electrochemical potential of the two reservoirs $\mu$ is set to be $60$ meV
according to the resonant level within the double-barrier structure. The two
oscillating parameters in Eq. (31) correspond to the two ac driven potential
gates $X_{1,2}\left(t\right)\to U_{1,2}\left(t\right)$ with all the other
notations correspond accordingly. We set the static magnitude of the two gate
potentials $U_{0,1}=U_{0,2}=U_{0}=100$ meV and the ac driving amplitude of the
modulations equal $U_{\omega,1}=U_{\omega,2}=U_{\omega}$.

In Fig. 11, the dependence of the pumped current on the phase difference
between the two ac oscillations is presented. In weak-modulation regime
(namely $U_{\omega}$ is small), sinusoidal behavior dominates. Here, three
relatively large $U_{\omega}$ is selected to reveal the deviation from the
sinusoidal dependence. (The magnitude of the pumped current mounts up in
power-law relation as a function of $U_{\omega}$ as shown in Fig. 12. The
sinusoidal curve for small $U_{\omega}$ would be flat and invisible in the
same coordinate range.) It can be seen from the figure that the $I_{p}$-$\phi$
relation varies from sinusoidal ($\sin\phi$) to double-sinusoidal ($\sin
2\phi$) as the ac oscillation amplitude is increased. The interpretation
follows from Eq. (41). The single $\hbar\omega$ quantum emission (absortion)
processes feature a sinusoidal behavior while the $2\hbar\omega$ quantum
emission (absortion) processes feature a double-sinusoidal behavior when the
Fourier index is doubled. As $U_{\omega}$ is increased, double $\hbar\omega$
quantum processes gradually parallel and outweigh the single $\hbar\omega$
quantum ones. It is also demonstrated that when the single $\hbar\omega$
quantum processes have the effect of $\sin\phi$ dependence, the double
$\hbar\omega$ quantum processes induce a $-\sin 2\phi$ contribution with a
sign flip, which can be understood from the sign change of the derivative of
the scattering matrix. The effect of three- and higher $\hbar\omega$ quantum
processes is small even for large $U_{\omega}$ comparable to $U_{0}$. The
experimental observationsRef1  as a deviation from the weak-modulation limit
are revealed by our theory.

ExperimentRef1  also discovered that for weak pumping the dependence of the
pumped current on the pumping strength obeys a power of 2 relation, as
expected from the simple loop-area argumentRef2 ; for strong pumping, power of
1 and 1/2 relation is observed. We presented in Fig. 12 the numerical results
based on our theory of the $I_{p}$-$U_{\omega}$ relation at a fixed $\phi$. To
demonstrate its power-law dependence, natural logarithm of the variables is
applied. From Eq. (41), it can be seen that for large ac driving amplitude
$U_{\omega}$, contribution of double $\hbar\omega$ quantum processes
(formulated in the second term on the right hand side of the equation) causes
the $I_{p}$-$U_{\omega}$ relation to deviate from its weak-modulation limit,
the latter of which is $I_{p}\propto U_{\omega}^{2}$. For different phase
difference between the two ac drivers, the deviation is different. At
$\phi=\pi$ the pumped current is invariably zero regardless of the order of
approximation determined by time-reversal symmetry. At $\phi=\pi/2$, $\sin
2\phi$ is exact zero, and no difference is incurred by introducing higher
order effect. If we shift the value of $\phi$ to $0.49\pi$, the abating effect
of the double $\hbar\omega$ quantum processes has the order of
$U_{\omega}^{4}$ with the small second-order parametric derivative of the
scattering matrix smoothing that effect a bit. Consequently, a power of $2\to
1\to 1/2$ relation is obtained and visualized by the curve fit, which is
analogous to experimental findings. For different values of $\phi$, sharper
abating and augmental effect occurs with analogous mechanisms. It is possible
that the experimentRef1  was done at the phase difference close to $\pi/2$
while trying to approach maximal pumped current in the adiabatic and weak-
pumping limit.

##  IV Summary and future directions

The scattering matrix method is initiated by Landauer and Büttiker to
investigate the conductance of multi-terminal and multi-channel mesoscopic
conductors. The spin degree of freedom can be included in the formalism by
enlargement of the dimension of the scattering matrix. The current-current
correlation and spin-spin correlation, such as the shot noise, can be
calculated from the cross products of the scattering matrix. Along this
direction, higher-order correlation function can also be considered. Dynamic
transport processes including the quantum pumping behavior can be dealt with
by the time-dependent scattering approach.

The development of the scattering theory enables its potential applications in
currently open issues. The interaction can be included to the scattering
matrix by the renormalization factors. The shot noise properties in various
conductors with active spin degree of freedom can be considered. The time-
dependent scattering theory provides a way to deal with dynamic quantum
issues. Some particular problems include non-harmonically driven quantum
pumping, spin pumping in racetrack memory applications, and multiferroic
transport dynamics, etc..

##  V Acknowledgements

This project was supported by the Nature Science Foundation of SCUT (No.
x2lxE5090410) and the Graduate Course Construction Project of SCUT (No.
yjzk2009001). The author would like to express sincere appreciation to
Professor Wenji Deng, Dr. Brian M. Walsh, Professor Jamal Berakdar, Professor
Michael Moskalets, and Professor Liliana Arrachea for valuable enlightenment
to the topic from discussions with them.

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Figure 1: Schematic demonstration of a single-barrier tunneling problem. The
quantum states between two reservoirs at zero bias is indicated.  Figure 2:
Schematic demonstration of a quantum wire modulated by two potential barriers.
The quantum state between two reservoirs at zero bias is indicated. Figure 3:
Time-integrated current density as a function of the phase difference between
the two modulations for different modulation frequencies. The Fermi level of
the two reservoirs $E_{F}=60$ meV counting from the conduction band edge of
the electron gas structure. Figure 4: Time variation of the current density
within an oscillating cycle. Different phase difference between the two
modulations is considered. The modulation frequency is set to be $\omega=10$
MHz and the Fermi level $E_{F}$ to be 60 meV. Figure 5: Schematics of the
double-barrier resonant diode. The resonant level is sketched between the two
barriers. The upper panel demonstrates the resonant level in conventional
diode without the spin-orbit coupling (SOC). In the lower panel, the SOC
behaves like a pseudomagnetic field and induces a split of different spin
components of the resonant level in the barrier structure, which contribute
collectively to the electric current and shot noise. Figure 6: Schematics of
the scattering approach. Spin components of the incoming and outgoing states
are indicated. Figure 7: Current $I$ (a) and shot noise $S$ (b) as functions
of the applied bias $EV$ of electrons traversing a symmetric DBRD structure
with different Dresselhaus constants $\gamma$. Figure 8: Fano factor as a
function of the applied bias $EV$ (a) and $S/2eI_{norm}$ vs $I/I_{norm}$ (b)
of electrons traversing a symmetric DBRD structure with different Dresselhaus
constants $\gamma$. Two straight lines in (b) show the full shot noise value
($2eI$) and half of its value for comparison. Figure 9: The tunneling scenario
of an adiabatic quantum pump. The two shadowed blocks represent the left and
right electron reservoirs respectively. The two barriers oscillate
adiabatically in time. The middle bar indicates the single-particle state
between the two barriers. The Fermi levels of the two reservoirs are the same
and are leveled to the single-particle state within the conductor.
$t_{1}$-$t_{4}$ indicate the transmission amplitudes between one of the two
reservoirs and the middle single-particle state. Figure 10: Schematics of the
quantum pump: a nanowire modulated by two ac driven potential barriers. Figure
11: Pumped current as a function of the phase difference between the two
modulations for different ac driving amplitudes. Figure 12: Pumped current as
a function of the ac driving amplitude $U_{\omega}$ along with fits to
$I_{p}\propto U_{\omega}^{2}$ (red solid circle) below 35 meV, $I_{p}\propto
U_{\omega}$ (green upward triangle) below 41 meV, and $I_{p}\propto
U_{\omega}^{1/2}$ above 41 meV (blue downward triangle). To demonstrate its
power-law dependence, natural logarithm of the variables is applied. The phase
difference between the two ac driver $\phi=0.49\pi$. Inset is the zoom-in of
the circled region.