# Photoinduced Nonperturbative Valley Polarization in Graphene

Hamed Koochaki Kelardeh<EMAIL_ADDRESS>Max Planck Institute for the
Physics of Complex Systems, Nöthnitzer Straße 38, D-01187 Dresden, Germany
Mohammadreza Eidi  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Straße 38, D-01187 Dresden, Germany  Takashi Oka  The Institute for
Solid State Physics, The University of Tokyo, Japan  Jan Michael Rost  Max
Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38,
D-01187 Dresden, Germany

###### Abstract

We investigate a valleytronic device based on graphene with charge separation
at different sublattices and correspondingly at nonequivalent valleys. We
characterize the maximality condition of valley polarization and investigate
the parameters and conditions upon which we can coherently control the
carriers and store data via valley degree of freedom. The valley polarization
is controlled by the amplitude as well as the carrier-envelope phase of the
pulse – one cycle optical field - and the curvature of the electron trajectory
in the reciprocal space. When strong-field excitation is taken into account,
the optical selection rule in perturbative optics is replaced by the
nonadiabatic geometric effects. As a result, a nonperturbative valley
polarization in two dimensional Dirac materials is induced regardless of
having an intrinsic bandgap. Microscopically, such a nonreciprocal response of
graphene in the chiral electric field is encoded by the quantum Berry phase,
as a (pseudo) magnetoelectric monopole.

##  I  Introduction

Today, geometric effects and controlling various phases of matter from
superconducting to semiconducting states to topological insulators with
conducting edge or surface states is a central topic in condensed matter
physics with potential applications in quantum computing and room temperature
superconductivity Xiao  _et al._ (2010); Qi and Zhang (2011).

Among different quantum states, valley degree of freedom and manifestation of
nontrivial quantum phases together with the generation of topological states
of light has lately drawn significant attentions Schaibley  _et al._ (2016);
Kelardeh  _et al._ (2016); Khanikaev and Shvets (2017); Vitale  _et al._
(2018); Jiménez-Galán  _et al._ (2020); Ozawa  _et al._ (2019). The
possibility of incorporating valley degree of freedom to store and carry
information has drawn numerous attention and led to substantial electronic
applications labeled as valleytronics. The valley pseudospin has potential to
serve as a robust quantum signal processor and data storage with a petahertz
bandwidth.

Theoretical and experimental investigation of Valley polarization have
predominantly been performed on insulating state such as monolayer hexagonal
Boron Nitride Song  _et al._ (2017), and Diamond Isberg  _et al._ (2013), as
well as semiconductors such as transition metal dichalcogenides (TMDC’s) Cao
_et al._ (2012); Mak  _et al._ (2012); Jones  _et al._ (2013); Yoshikawa  _et
al._ (2019); Zeng  _et al._ (2012); Ye  _et al._ (2017); Langer  _et al._
(2018), and Silicon Salfi  _et al._ (2014). Besides, the semimetallic systems
with broken inversion symmetry such as Bismuth Zhu  _et al._ (2012, 2017),
gapped graphene Azar Oliaei Motlagh  _et al._ (2019), bilayer graphene
Shimazaki  _et al._ (2015); Sui  _et al._ (2015); Kumar  _et al._ (2020), and
graphene superlattice Yankowitz  _et al._ (2012); Gorbachev  _et al._ (2014)
have been suggested as potential candidates for valley-contrasting
optoelectronic devices. It has been frequently reported that the presence of a
gap is a precondition for having valley polarization Yao  _et al._ (2008);
Lensky  _et al._ (2015); Azar Oliaei Motlagh  _et al._ (2019). In this report,
we are going to reexamine such a widely accepted notion and address the
following important question: can we expect a sizable electric field-induced
VP in gapless Dirac semimetals? Answering this question is imperative
considering the fact that introducing an eV-scale staggered bandgap ( also
called Semenoff mass Semenoff (1984)) in graphene, which is required for
having a measurable VP in noncentrosymmetric graphene systems is
technologically challenging.

We report VP in gapless graphene with efficiency as high as 35$\%$ by
utilizing a chiral optical pulse with few-cycle (sub ten femtosecond) and
strong-field (volt per angstrom scale) with controlled polarity, amplitude,
and carrier-envelope phase (CEP). Such a noticeable VP in monolayer graphene
is attributed to the nonperturbative dynamics of electrons in the strong
electric field of the laser pulse, global topology of the honeycomb band
structure and the nonequivalent Berry phase of $\pm\pi$ at the K and K′
valleys.

In a nonperturbative nonlinear light-matter interaction where the light pulse
has a strong amplitude (comparable with the interatomic field) and contains
only one or two cycles, the incident electric field breaks the conservation
law and gives rise to nonadiabatic topological effects and generation of the
valley-contrasting population in the reciprocal space.

In a honeycomb crystal, the two degenerated valleys can be distinguished by
pseudovector quantities, namely the Berry phase, connection, and curvature.
Berry phase describes the phase of quantum mechanical wave functions
accumulated along a closed loop in the reciprocal space (or any parametric
space). Such a topological phase defines the topology of electronic states and
plays a fundamental role in many emerging phenomena in condensed-matter
systems, such as High-order harmonic generation Silva  _et al._ (2019); Chacón
_et al._ (2020); Yue and Gaarde (2020); Moos  _et al._ (2020), quantum spin
Hall effect Wu  _et al._ (2018), anomalous quantum Hall effect Nagaosa  _et
al._ (2010); Sato  _et al._ (2019); McIver  _et al._ (2020). Moreover, Berry
connection which is defined as ${\mathbfcal
A}^{{\rm{nn}}}=\left\langle{{\phi_{n}}|i\gradient|{\phi_{n}}|}{g({\bf{q}})}\right|{e^{i{\phi_{\bf{q}}}}}$,
with ${\phi_{\bf{q}}}={\tan^{-1}}\left({\frac{{{\mathop{\rm
Im}\nolimits}(g({\bf{q}}))}}{{{\mathop{\rm
Re}\nolimits}(g({\bf{q}}))}}}\right)$.

Accordingly, the eigenstates and eigenenergies of the conduction and valence
bands can be found from the above Hamiltonian, ${\cal H}_{0}$, as follows

$\phi_{\bf{q}}^{c/v}({\bf{r}})=\frac{{e^{i{\bf{q}}\cdot{\bf{r}}}}}{{\sqrt{2{E_{c/v}}(\delta+{E_{c/v}})}}}\left({\begin{array}[]{*{20}{c}}{\delta+{E_{c/v}}}\\\
{\left|g\right|{e^{-i{\phi_{\bf{q}}}}}}\end{array}}\right)$ (5)

${E_{c/v}}({\bf{q}})=\pm\sqrt{{\delta^{2}}+{{\left|{g({\bf{q}})}\right|}^{2}}}$
(6)

where the +/- signs correspond to the conduction (c)/ valence (v) band,
respectively.

An applied electric field generates both the intraband (adiabatic) and
interband (nonadiabatic) electron dynamics. The intraband dynamics is
determined by the Bloch acceleration theorem in the reciprocal space,
$\hbar{\bf{\dot{k}}}=e{\bf{F}}({\bf{t}})$. For an electron with initial
momentum $\bf{q}$ the electron dynamics is described by the time dependent
wave vector given by ${\bf{k}}(t)={\bf{q}}-e{\hbar^{-1}}{\bf{A}}(t)$ with
${\bf{A}}(t)=-\int_{-\infty}^{t}{\bf{F}}(t^{\prime})dt^{\prime}$ as the vector
potential of the laser field. In fact, the electron wave packet with initial
crystal wave vector $\bf{q}$ transforms to a trajectory-guided
vector,${\bf{q}}\mapsto{\bf{k}}(t)$, where its time-dependent trajectory is
governed by the vector potential of the incident pulse.

The most general solution to the time-dependent Schrödinger equation (1) can
be written in the interaction representation as

${\Psi_{\bf{q}}}({\bf{r}},t)=\sum\limits_{n=v,c}{\int_{BZ}{\beta_{{}_{{\bf{k}}(t)}}^{n}\Phi_{n,{\bf{q}}}^{({\rm{H}})}({\bf{r}},t)}}$
(7)

where
$\Phi_{n,{\bf{q}}}^{({\rm{H}})}({\bf{r}},t)=\phi_{{\bf{k}}({\bf{t}})}^{n}({\bf{r}}){e^{-\frac{i}{\hbar}\int_{-\infty}^{t}{E_{n}^{T}[{\bf{k}}(t^{\prime})]dt^{\prime}}}}$
are the time-dependent adiabatic basis set, i.e., the Houston functions and
are the solutions of Schrödinger equation within a single band without an
interband coupling.

$E_{n}^{T}[{\bf{k}}(t^{\prime})]={E_{\rm{n}}}[{\bf{k}}(t^{\prime})]+e{\bf{F}}(t^{\prime})\cdot{\mathbfcal{A}^{({\rm{nn}})}}[{\bf{k}}(t^{\prime})]$
(8)

is called the modified band energy, accounting the dynamic phase [first term],
as well as the geometric (Berry) phase [second term]. Here ${{\mathbfcal
A}^{(nn)}}=i\left\langle{\phi_{{\bf{k}}({\bf{t}})}^{n}|{\gradient_{\bf{q}}}|\phi_{{\bf{k}}({\bf{t}})}^{n}}\right\rangle$
is the global energy of ${n^{\rm{th}}}$-band including the Bloch energy
dispersion [First term], as well as the field-induced geometrical counterpart
[second term]. The later term is critical to detect the topological
information of solid state systems including the peculiarities observed in the
quantum Hall effect regime and pseudospin-related Berry’s phase Kelardeh  _et
al._ (2017).

$\phi_{{\bf{k}}({\bf{t}})}^{n}({\bf{r}})$ are the Bloch eigenstates of
graphene (Eq. 5), and $n=v,c$ stands for the valence and conduction bands,
respectively. The expansion coefficients ${\beta_{c}}(t)$ and ${\beta_{v}}(t)$
in Eq. 7 interpret as the probability amplitudes that at time $t$ the electron
(or hole) is in the conduction or valence band. They satisfy the following
system of integro-differential equations

$\begin{array}[]{*{20}{l}}{i\hbar\dot{\beta}_{{\bf{k}}(t)}^{c}=e{\bf{F}}(t)\cdot{\bf{Q}}_{{\bf{k}}(t)}^{cv}\beta_{{\bf{k}}(t)}^{v}}\\\
{i\hbar\dot{\beta}_{{\bf{k}}(t)}^{v}=e{\bf{F}}(t)\cdot{\bf{Q}}_{{\bf{k}}(t)}^{cv*}\beta_{{\bf{k}}(t)}^{c}}\end{array},$
(9)

where

${\bf{Q}}_{{\bf{k}}(t)}^{cv}={\mathbfcal
A}_{{\bf{k}}(t)}^{cv}{e^{\frac{i}{\hbar}\int_{-\infty}^{t}{\left({E_{c}^{T}[{\bf{k}}(t^{\prime})]-E_{v}^{T}[{\bf{k}}(t^{\prime})]}\right)dt^{\prime}}}}$
(10)

determines the matrix element of interband interaction where ${\mathbfcal
A}_{{\bf{k}}(t)}^{cv}$ is the interband Berry connection. The interband Berry
connection can be defined in terms of the transition dipole moments (TDM)
${\bf{D}}_{{\bf{k}}(t)}^{cv}$ Kelardeh  _et al._ (2015) as
${\bf{D}}_{{\bf{k}}(t)}^{cv}=e{\mathbfcal A}_{{\bf{k}}(t)}^{cv}$. TDM
determines optical transitions between the VB and CB at a crystal momentum
$\mathbf{q}$. The exponential factor in Eq. 10 which is calculated from Eq. 8,
defines the global phase difference between the states of the CB and VB (i.e.,
the generalized band offset).

We can write the intraband and interband components of the Berry connection
operator In the two-band graphene system, the Berry connection has the
following matrix form

$\mathbfcal{\hat{A}}(\mathbf{q})=\left({\begin{array}[]{*{20}{c}}{{\mathbfcal{A}^{cc}}}&{{\mathbfcal{A}^{cv}}}\\\
{\mathbfcal{A}^{cv*}}&{{\mathbfcal{A}^{vv}}}\end{array}}\right)$ (11)

with

${{\mathbfcal
A}^{nm}}=i\left\langle{\phi_{\bf{q}}^{(n)}|{\gradient}|\phi_{\bf{q}}^{(m)}}\right\rangle$
(12)

Substituting Eq. 5 into Eq. 12, one finds the an analytical expression for the
matrix elements of the Berry connection in the tight-binding approximation.

The intraband components read

$\displaystyle\begin{array}[]{l}A_{x}^{cc/vv}=\frac{{a{\gamma^{2}}}}{{\sqrt{3}}}\frac{{1+{c_{0}}({c_{3}}-2{c_{0}})}}{{{u_{c/v}}}}\\\
A_{y}^{cc/vv}=a{\gamma^{2}}\frac{{{s_{0}}{s_{3}}}}{{{u_{c/v}}}}\end{array}$
(15)

and interband Berry connection:

$\displaystyle\begin{array}[]{l}{\cal{A}}_{x}^{cv}=\frac{{a{\gamma^{2}}}}{{2\sqrt{3}{E_{c}}\left|g\right|}}\left[{1+{c_{0}}({c_{3}}-2{c_{0}})}\right]+i\frac{{\sqrt{3}\delta
a{\gamma^{2}}}}{{2E_{c}^{2}\left|g\right|}}{c_{0}}{s_{3}}\\\
{\cal{A}}_{y}^{cv}=\frac{{a{\gamma^{2}}}}{{2{E_{c}}\left|g\right|}}{s_{0}}{s_{3}}+i\frac{{\delta
a{\gamma^{2}}}}{{2E_{c}^{2}\left|g\right|}}{s_{0}}\left({{c_{3}}+2{c_{0}}}\right)\end{array}$
(18)

where ${u_{c/v}}=\left|{{g^{2}}}\right|+{(\delta+{E_{c/v}})^{2}}$, and
${c_{0}}=\cos\left({a{k_{y}}/2}\right)$,
${s_{0}}=\sin\left({a{k_{y}}/2}\right)$,
${c_{3}}=\cos\left({\sqrt{3}a{k_{x}}/2}\right)$,
${s_{3}}=\sin\left({\sqrt{3}a{k_{x}}/2}\right)$.

So far we treated the graphene system as a closed two-level system; now, we
extend our model to incorporate dephasing and decoherence into the driven
solid-state electron dynamics. To this end, we employ the Liouville von
Neumann equation and propagate the reduced density matrix in the length gauge
interaction picture to describe quantum dynamics in the presence of the
relaxation process as below:

$\left\\{{\begin{array}[]{*{20}{l}}{\dot{\rho}_{{\bf{k}}(t)}^{cv}=\frac{i}{\hbar}{\bf{F}}(t)\cdot{\bf{Q}}_{{\bf{k}}(t)}^{cv}\left[{\rho_{{\bf{k}}(t)}^{cc}-\rho_{{\bf{k}}(t)}^{vv}}\right]-\gamma\rho_{{\bf{k}}(t)}^{cv}}\\\
{\dot{\rho}_{{\bf{k}}(t)}^{cc}=2{\rm{Re}}\left[{\frac{i}{\hbar}{\bf{F}}(t)\cdot{\bf{Q}}_{{\bf{k}}(t)}^{cv*}\rho_{{\bf{k}}(t)}^{cv}}\right]-\gamma\rho_{{\bf{k}}(t)}^{cc}}\end{array}}\right.$
(19)

${\rho^{ij}}$ are the solution of the Born-Markov Master equation by averaging
over the thermal bath degrees of freedom:

$\rho(t)={\rm{T}}{{\rm{r}}_{B}}\left({\left|\Psi\right\rangle\left\langle\Psi\right|}\right)=\left({\begin{array}[]{*{20}{c}}{\rho_{{\bf{k}}(t)}^{cc}}&{\rho_{{\bf{k}}(t)}^{cv}}\\\
{\rho_{{\bf{k}}(t)}^{cv*}}&{\rho_{{\bf{k}}(t)}^{vv}}\end{array}}\right)$ (20)

The diagonal terms in Eq. 20 represent the band population
$\rho_{{\bf{k}}(t)}^{cc}={\left|{\beta_{{\bf{k}}(t)}^{c}}\right|^{2}}$ and the
nondiagonal terms define the polarization function. Also note that
${\rho^{vv}}=1-{\rho^{cc}}$.

The relaxation rate $\gamma$ in Eq. 19 has an inverse proportion of the
dephasing time, ${\gamma_{({\rm{PHz}})}}=\frac{1}{{{T_{({\rm{fs}})}}}}$.

The Liouville-Von Neumann equation (Eq.19) in our settings goes beyond the
Boltzman transport theory where only the band dispersion and the consequent
group velocity appears in the equation.

We assume that the VB is fully occupied and the CB is empty. The applied pulse
is an intense single optical oscillation. In the numerical simulation, the
important precondition for the chosen electric field waveform impose that
$\int_{-\infty}^{\infty}{{\bf{F}}(t^{\prime})dt^{\prime}}=0$. We characterize
the laser waveform such that it satisfies this condition, by defining
${\bf{A}}(t)$ and obtaining ${\bf{F}}(t)$ as the temporal derivative of it. We
employ the following vector potential waveform for the elliptically-polarized
pulse.

$\begin{array}[]{l}{A_{x}}(t)={F_{0}}{\omega^{-1}}{e^{-{{(t/\tau)}^{2}}}}\sin(\omega
t+{\phi_{\rm{CEP}}}),\\\
{A_{y}}(t)=\pm{\varepsilon}{F_{0}}{\omega^{-1}}{e^{-{{(t/\tau)}^{2}}}}\cos(\omega
t+{\phi_{\rm{CEP}}}),\end{array}$ (21)

${F_{0}}$ is the field amplitude, $\tau$ is the pulse length corresponding to
carrier frequency $\omega$ = 1.5 eV/$\hbar$, and ${\phi_{\rm{CEP}}}$ is the
CEP. Sign $\pm$ determines the right and left-handedness of the pulse.
$\varepsilon\in[0,1]$ controls the degree of ellipticity and hence the
curvature of electron trajectory.

System of equations 19 determines the laser-induced electron dynamics; solving
these coupled diffrentio-integral equations, we obtain reciprocal space
distribution of electrons in the conduction and valence bands.
Correspondingly, the time-dependent CB population is defined by
$\rho_{{\bf{k}}(t)}^{cc}={\left|{\beta_{{\bf{k}}(t)}^{c}}\right|^{2}}$. The
residual value of the CB population, ${\rho_{{\bf{k}}(t\to{t_{f}})}^{cc}}$, is
defined as the population after the pulse.
We define the VP as

${\rm{VP}}=\frac{{\left|{{n_{\rm{K}}}-n{}_{{\rm{K^{\prime}}}}}\right|}}{{{n_{\rm{K}}}+n{}_{{\rm{K^{\prime}}}}}},$
(22)

with

$\begin{array}[]{*{20}{l}}{{n_{\rm{K}}}=\int_{-{q_{y}}/\sqrt{3}}^{{q_{y}}/\sqrt{3}}{\int_{0}^{3\kappa/2}{\rho_{{\bf{k}}(t\to{t_{f}})}^{cc}{\rm{d}}{q_{y}}{\rm{d}}{q_{x}}}},}\\\
{{n_{{\rm{K^{\prime}}}}}=\int_{{q_{y}}/\sqrt{3}}^{-{q_{y}}/\sqrt{3}}{\int_{-3\kappa/2}^{0}{\rho_{{\bf{k}}(t\to{t_{f}})}^{cc}{\rm{d}}{q_{y}}{\rm{d}}{q_{x}}}},}\end{array}$
(23)

as the total population of the K- and $\rm{K^{\prime}}$ valleys inside their
corresponding triangle and $\kappa=4\pi/3{a_{0}}$ is a constant. Due to the
honeycomb structure of graphene, the triangular meshgrid is chosen to fully
take in the population proportions throughout the BZ, as illustrated in Fig.
1.
The fine (eye-drop shaped) line within the bright colored CB momentum
distribution represent the separatrix, which is the superposition of initial
conditions where their trajectories pass through the Dirac point. In other
words, the separatrix is the mirror symmetry of the vector potential
(polarization state), that steer the trajectory of electron in the momentum
space.

Figure 1: Schematic illustration of the CB population excitation of graphene
at the end of a single-cycle circular field with $F_{0}=0.9{\rm{V/\AA}}$,
CEP=0 and dephasing time T=100 fs. ${\rho_{{\bf{k}}(t\to{t_{f}})}^{cc}}$,
which is calculated from Eq. 19 is color mapped between 0 (no excitation
states) to 1 (full occupation states) in the reciprocal space. The first BZ
and the inequivalent $\rm{K}$ and $\rm{K^{\prime}}$ points are shown. The
triangular borderline in-between the $\rm{K}$ and $\rm{K^{\prime}}$ valleys
are also indicated.

##  III Result and discussion

Valley degeneracy in the momentum space of the conduction and valence bands
presents an additional degree of freedom for charge carrier manipulation. As
its counterpart spin states in spintronics, controlling the population of
valley states is essential to the development of valley-based electronics.

We show that the single-cycle pulse results in a significantly large VP. The
controlling parameters of the optical pulse are the field amplitude $F_{0}$,
ellipticity (tuned by $\varepsilon$), and the carrier envelope phase
${\phi_{\rm{CEP}}}$. polarization state

###  III.1  Field Amplitude and Ellipticity

Firstly, we look into the simultaneous roles of ellipticity and field
amplitude on VP and find the optimal $\varepsilon$ and $F_{0}$ for which the
VP is maximized. We set CEP=0 and exclude the relaxation processes in this
section. Fig. 2(a) plots the VP versus $F_{0}$ for a range of polarization
states from linear ($\varepsilon$=0) to circular pulse ($\varepsilon$=1).
Evidently, since the linear pulse preserves the time-reversal symmetry, its
corresponding VP is zero.

We span through a wide range of $F_{0}$ up until 1.4 ${\rm{V/\AA}}$. In this
field region, VP increases monotonically for $\varepsilon$=0.25 and 0.5,
whereas for $\varepsilon$=0.75 and 1, VP after a slow and oscillatory increase
at small fields, grows exponentially to its highest amount then falls at
higher fields. The highest VP corresponds to the circular pulse
($\varepsilon=1$) with field amplitude ${F_{0}}\simeq 0.9\,{\rm{V/\AA}}$.

Fig. 2(b), on the other hand, plots the VP as a function of polarization state
for different laser field amplitudes. For $F_{0}<0.5{\rm{V/\AA}}$, VP is
approximately zero with practically no influence by the ellipticity
modulation. However, at moderate fields $\sim$ 0.5 to 0.9 ${\rm{V/\AA}}$, VP
is abruptly enhanced for $\varepsilon\geq 0.5$.

Figure 2: (a) depicts the VP as a function of field amplitude, $F_{0}$, for
different ellipticities $\varepsilon$. the purple color corresponds to the
circularly polarized pulse, in which the VP increases monotonically with field
and reaches to a maximum of 35 percent at ${F_{0}}=0.9\,{\rm{V/\AA}}$, before
sharply falling at higher fields. (b) VP is plotted versus ellipticity
$\varepsilon$ for different field amplitude $F_{0}$. For linearly polarized
pulse VP=0 since such pulse preserve the time-reversal symmetry. For
$F_{0}>0.5{\rm{V/\AA}}$, a threshold $\varepsilon\simeq 0.5$ is observed above
which the VP steeply increases.

###  III.2  Induced Current and Net Charge Transport

To understand the behavior of VP versus field amplitude and ellipticity, we
calculate the generated photocurrent and net charge transfer through the
graphene system. The time-dependent electric field of the optical pulse causes
the polarization of the system which generates an electric current
${\bf{J}}(t)=\left\\{{{J_{x}}(t),{J_{y}}(t)}\right\\}$. Both intraband
($\mathbf{J}^{\rm{intra}}(t)$) and interband ($\mathbf{J}^{\rm{inter}}(t)$)
currents contribute to the total current,
$\mathbf{J}(t)=\mathbf{J}^{\rm{intra}}(t)+\mathbf{J}^{\rm{inter}}(t)$, in the
system. the two-band density matrix formalism we incorporated in the main text
results in the following equations for the interaband and interband currents:

$\begin{array}[]{*{20}{l}}{{{\bf{J}}_{{\mathop{\rm
int}}ra}}(t)=2e\sum\limits_{\alpha=c,v}{\int_{\overline{{\rm{BZ}}}}{{{\bf{V}}^{\alpha}}}}\left({{\bf{k}}({\bf{q}},t)}\right){\rho^{\alpha\alpha}}({\bf{q}},t)d{\bf{q}},}\\\
{{{\bf{J}}_{{\mathop{\rm
int}}er}}(t)=2e\int_{\overline{{\rm{BZ}}}}{{{\bf{V}}^{vc}}}\left({{\bf{k}}({\bf{q}},t)}\right){\rho^{cv}}({\bf{q}},t)d{\bf{q}}+{\rm{c}}.{\rm{c}}.}\end{array}$
(24)

${{\rho^{\alpha\alpha}}({\bf{q}},t)}$ is the $\alpha$th-band occupation (with
the index $m$ running over the valence and conduction bands, i.e. $\alpha=v$
and $c$, respectively) and ${{\rho^{cv}}({\bf{q}},t)}$ is the interband
coherence. The factor of 2 in Eq. 24 is due to the spin degeneracy.

${{\bf{V}}^{\alpha}}$ and ${{\bf{V}}^{\alpha\alpha^{\prime}}}$, ($\alpha$ and
$\alpha^{\prime}$ interchange between $c$ and $v$) are the matrix elements of
the velocity operator where in the two-band picture has the following form:

${\bf{V}}=\left({\begin{array}[]{*{20}{c}}{{{\bf{V}}^{cc}}}&{{{\bf{V}}^{cv}}}\\\
{{{\bf{V}}^{cv*}}}&{{{\bf{V}}^{vv}}}\end{array}}\right).$

The intraband velocity, within the quantum kinetic theory, is defined as

$\begin{array}[]{l}{{\bf{V}}^{\alpha}}=\frac{1}{\hbar}{\gradient_{\bf{k}}}E_{\alpha}^{T}[{\bf{k}}(t)]=\\\
\,\,\,\,\,\,\,\,\,\,\frac{1}{\hbar}\left[{{\gradient_{\bf{k}}}{E_{\alpha}}[{\bf{k}}(t)]+e{\gradient_{\bf{k}}}\left({{\bf{F}}(t)\cdot{{\bf{A}}^{(\alpha\alpha)}}[{\bf{k}}(t)]}\right)}\right]\end{array}$
(25)

and interband velocity matrix element as

${{\bf{V}}^{\alpha\alpha^{\prime}}}({\bf{k}})=\frac{i}{\hbar}{\bf{Q}}_{{\bf{k}}({\bf{q}},t)}^{\alpha\alpha^{\prime}}\left[{E_{\alpha}^{T}({\bf{k}}({\bf{q}},t))-E_{\alpha^{\prime}}^{T}({\bf{k}}({\bf{q}},t))}\right]$
(26)

${\bf{Q}}_{{\bf{k}}({\bf{q}},t)}^{\alpha\alpha^{\prime}}$ in the above
equation is obtained from Eq. 10 and is related to the interband Berry
connection.

It is important to note that the Berry connection and curvature effects come
into play in the matrix elements of both inetra- and interband velocities.

Intraband velocity contains two terms:
${\bf{V}}_{\rm{gr}}^{\alpha}({\bf{k}})={\gradient_{{\bf{k}}{\kern
1.0pt}}}{E_{\alpha}}({\bf{k}})$ is the particle (i.e. electron or hole) group
velocity with $E_{\alpha}({\bf k})$ the bands’ energy dispersion, and
${\bf{V}}_{{\rm{anom}}}^{\alpha}({\bf{k}})=e{\gradient_{\bf{k}}}\left({{\bf{F}}(t)\cdot{{\bf{A}}^{(\alpha\alpha)}}[{\bf{k}}({\bf{q}},t)]}\right)$
is the anomalous velocity that captures the quantum geometry of the Bloch
wavefunction and Berry phase Stephanov and Yin (2012). ${\bf k}$ is the quasi
momentum defined in terms of the crystal wave vector ${\bf q}$ and the vector
potential ${\bf A}(t)$ of the laser’s electric field as
${\bf{k}}({\bf{q}},t)={\bf{q}}-e/\hbar{\bf{A}}(t)$.

The current induced by the single-cycle optical pulse, respectively, results
in the charge transfer across the system, which can be calculated from the
following expression

${\bf{Q}}=\int_{-\infty}^{\infty}{{\bf{J}}(t){\rm{d}}t}$ (27)

Fig. 3(a) plots the time-dependent current in graphene for the circularly
polarized pulse with different Field amplitudes. From the numerical
calculations, we observe that the current changes drastically, and this occurs
in the same situation where the VP shows the switching. The reason that the
photocurrent and VP show the common behavior can be explained by the
trajectory of e in the momentum space. The dominant excitation taking place at
$\rm{K}$-point will move as ${\bf{q}}-e/\hbar{\bf{A}}(t)$, and when they move
to the $\rm{K^{\prime}}$-valley, the group velocity, as well as the Anomalous
velocity, will show the sign flip. Such a sign change in the current density,
occurs roughly at $F_{0}\sim 0.9{\rm{V/\AA}}$, where the maximum VP appears
(c.f. Sec. III.1).

The residual population and current, in turn, translate to a transferred
charge density and is plotted in Fig. 3(b) as a function of the peak laser
field. Respective to the sign change of current in Fig. (a), Q also reacts on
the critical field and changes its slope at $F_{0}\sim 0.9{\rm{V/\AA}}$. Such
observable substantiate the VP field dependence as represented in Fig. 2(a).

Figure 3:  (a) photoinduced current in graphene is plotted within the
incidence of the single-cycle circular field. The corresponding field
amplitudes, $F_{0}$ associated with different colors are indicated. The
current density oscillates at high field and changes sign at
${F_{0}}>0.9{\rm{V/\AA}}$ associated with the maximum valley polarization. (b)
Transferred charge density through graphene monolayer as a function of field
amplitude $F_{0}$, also manifests the resulting field dependence of the
valley-contrasting excitation.

###  III.3  Relaxation Dynamics

In the previous section, we explored the variation of VP as a function of
field amplitude and laser waveform, without taking the role of relaxation into
account (i.e., $T\to\infty$ in Eq. 19). In this and following sections, we
study in what manner the relaxation mechanisms alter functionality and
amplitude of the VP.

In previous section we showd that, in the absence of relaxation, the circular
pulse with $F_{0}=0.9{\rm{V/\AA}}$ gives rise to a maximum VP for a single-
cycle optical laser. In Fig. 4 we plot the VP as a function of dephasing time
T, with field parameter corresponding to the maximum VP. As depicted, the VP
falls suddenly for ultrafast dephasing time $T<3$ fs. Notably, for the fastest
relaxation time T=1 fs, we expect to have 22$\%$ of the VP.

Figure 4:  Valley polarization (VP) plotted as a function of the dephasing
time. VP is practically unaffected by the relaxation up until $T\simeq 3$ fs
where it exponentially decays to a magnitude of $\sim$0.22 at $T=1$ fs. The
laser in this case is a single cycle circular pulse with
${F_{0}}=0.9\,{\rm{V/\AA}}$ and CEP=0.

We further look into the influence of the dephasing time, T, on the field
amplitude ($F_{0}$) and ellipticity ($\varepsilon$) dependence of the VP. Fig.
5 plots VP as a function of amplitude of the pulse $F_{0}$ (a), and versus
curvature of the pulse $\varepsilon$ (b). Two cases of relaxation time have
been examined: The black line correspond to a superfast decaying time (T=1
fs), and the red line correspond to the relatively slow relaxation (T=100 fs).
the pulse length is 5 fs. the fast decay rate suppress the maximum VP from 35
percent to approximately 20 percent, while the general behavior of the VP
remains roughly the same as the case with no or slow relaxation.

Figure 5:  (a) VP as a function of the field amplitude for a circular pulse.
Two relaxation times T=1 fs (red line) and T=100 fs (black line). VP increases
monotonically with field and reaches to a maximum of 35 percent (red), and 22
percent (black, fast dephasing and decoherence) at
${F_{0}}=0.9\,{\rm{V/\AA}}$, before sharply falling at higher fields. (b) VP
versus pulse ellipticity with field amplitude ${F_{0}}=0.9\,{\rm{V/\AA}}$.
Similarly, red and black correspond to T=1 and 100 fs, respectively.

In Fig 6(a) the impact of relaxation processes on the excitation distribution
of graphene is illustrated at the end of the single-cycle circular pulse for
decoherence time T=2 fs. The triangular borderline seperating the proportion
of $\rm{K}$ and $\rm{K^{\prime}}$ valleys for calculating ${n_{\rm{K}}}$ and
${n_{\rm{K^{\prime}}}}$ in Eq. 23 are indicated in the density plot of CB
population distribution. The VP, which is calculated by Eq. 22 is the total
population over the triangular surface of K subtracted from the
$\rm{K^{\prime}}$-triangle, normalized to 1.

Figure 6: [Color online] (a) The influence of fast relaxation (T=2 fs) on the
CB population distribution at the end of the pulse for circular waveform (to
be compared with Fig. 1 with T=100 fs). Compared to Fig. 1 the maximum
population reduces from 1 to 0.6. Also, the CB population deforms along the
electron trajectory as time progresses arising to an asymmetric population
distribution (part b).  Figure 7:  Exhibits VP versus the $\phi_{\rm{CEP}}$
for three different relaxation times. For all case scenarios, the VP
periodically modulates with on and off periodicity of 60 degree.

The fast electron scattering impacts on the magnitude of electron wave packet
as well as its phase information, giving rise to an asymmetric CB distribution
with maximum amplitude of 0.6. In Fig. 6(b) we compare the time evolution of
the population formation close to the Dirac point for T=2 fs (lower panel)
with the case of long relaxation time T=100 fs (upper panel). The
corresponding extended BZ distribution of T=100 is depicted in Fig. 1. The
maximum population for T=2 fs reduces to 0.6, in contrast to the case for
T=100 fs where population maxes out to 1. Besides, the fast relaxation time
partially smears the excitation distribution along the cyclic path of
electron.

###  III.4  Carrier Envelope Phase

We have seen in section III.1 that the VP can be optimally controlled by the
amplitude of the laser field. In fact, a threshold of $~{}0.5{\rm{V/\AA}}$ is
observed in Fig. 2(a), below which there is no VP irrespective of the
polarization state and laser waveform. Above this field threshold, the single
cycle pulse exhibits dissimilar behavior for different ellipticities.
Likewise, VP behaves in a different way as a function of ellipticity above and
below the $\varepsilon\simeq 0.5$ threshold (see Fig. 2b).

In Fig. 7 the carrier envelope phase (${\phi_{{\rm{CEP}}}}$ in Eq. 21)
dependence of VP is plotted for three different relaxation times: T = 1, 2,
and 100 fs. The case of a circularly polarized pulse is considered with
amplitude ${F_{0}}=0.9\,{\rm{V/\AA}}$ of the electric field corresponding to
the optimum VP observed in Sec. III.1, where ${\phi_{{\rm{CEP}}}}$ was set to
zero. We extend over the full range of CEP from 0 to $2\pi$.

Fig. 7 reveals that the VP is highly sensitive to the pulse orientation with
respect to the graphene sheet which is controlled by the CEP,
${\phi_{{\rm{CEP}}}}$ in Eq. 21. The VP switches on and off depending on the
CEP angle with periodicity of ${60^{\circ}}$. Such a periodic behavior is due
to the fact that graphene belongs to the Symmorphic ${\rm{C}}_{6v}$ point
group with ${60^{\circ}}$ rotational symmetry in real and reciprocal space.

Another important physical point which is drawn from Fig. 7 is the shift of VP
with respect to the relaxation time. Such a shift can be understood by
operating a unitary transformation on elements of the density matrix in Eq. 19
as ${\rho^{ij}}\mapsto{{\tilde{\rho}}^{ij}}{e^{-\gamma t}}$. Subsequently, we
find

${\begin{array}[]{*{20}{l}}{\dot{\tilde{\rho}}_{{\bf{k}}(t)}^{cv}=\frac{ie}{\hbar}{\bf{F}}(t)\cdot{\bf{Q}}_{{\bf{k}}(t)}^{cv}\left[{\tilde{\rho}_{{\bf{k}}(t)}^{cc}-\tilde{\rho}_{{\bf{k}}(t)}^{vv}}\right]}\\\
{\dot{\tilde{\rho}}_{{\bf{k}}(t)}^{cc}=2{\rm{Re}}\left[{\frac{ie}{\hbar}{\bf{F}}(t)\cdot{\bf{Q}}_{{\bf{k}}(t)}^{cv*}\tilde{\rho}_{{\bf{k}}(t)}^{cv}}\right]}\end{array}}$
(28)

Thus in Eq. 23 we can transform the relaxation rate to the exponent. Since the
carrier envelope phase acting as a rotation operator on the electron
trajectory in reciprocal space, under the influence of the relaxation
processes, acts like a shift vector on the density matrix elements.

##  IV Conclusions

The principal challenge in the development of valleytronics is to lift the
valley degeneracy of charge carriers in a controlled way. The ability to
exploit valley polarization (VP) has been rather limited until the recent
emergence of the 2D materials. 2D materials with honeycomb structures
preeminent by graphene offer a combination of properties not obtainable from
conventional thin-film materials.

Since the advent of graphene, various possibilities have been envisioned and
explored in it for photonic, plasmonic and optoelectronic devices. However,
the generation and detection of valley polarization and feasibility of
valleytronics in pristine graphene has been elusive due to the presence of
inversion symmetry. In this article, we have proposed a potential prototype of
valleytronic device based on monolayer graphene. we have characterized the
optimality condition of valley polarization and investigated the parameters
upon which we can control the carriers in graphene and store data via valley
degree of freedom.

We have shown the the valley dependent Berry phase results in a valley-
contrasting population and carrier transport in pristine graphene. Exciting
graphene with sub- 10 femtosecond light pulse creates nonequilibrium charge
states in a highly nonlinear fashion. Nonlinear properties are the basis of
functional optical devices, as they enable functions such as ultrafast
modulation and control, and optical gain.

In fact, driving a system by strong coherent fields break energy-momentum
conservation law, and nonadiabatic geometric effects dominates the dynamical
processes. Hence, the perturbative optical phenomena which is induced by the
absorption and emission of photons is replaced by the nonadiabatic geometric
effects. Respectively, a nonperturbative replica for the common selection rule
is set off with the elemental of symmetry, chirality and topology in the
optically allowed and forbidden transitions.

According to our results, a circularly polarized pulse with a single-cycle
carrier generates a Valley population with efficiency as high as 35$\%$ which
is substantial and disprove the general conception that the valley degeneracy
can hardly be lifted in gapless materials. Moreover, the VP is susceptible to
the orientation of the laser waveform with respect to the graphene sheet. VP
modulates periodically from its maximum value to almost zero depending on the
CEP angle with a periodicity of ${60^{\circ}}$, indicating a clear-cut and
robust modulation of the VP. This periodic switching of VP with respect to CEP
is understandable considering the fact that graphene belongs to the Symmorphic
${\rm{C}}_{6v}$ point group with ${60^{\circ}}$ rotational symmetry in real
and reciprocal space. We further investigate the effect of relaxation on the
resultant VP. Notably, even for an ultrafast decoherence and dephasing time of
1 fs, we expect to have more than twenty percent of the valley-contrasting
excitation.

The predicted induction of the lightwave valley polarization with a few-cycles
of optical lasers in graphene could be useful for valleytronics applications,
and development of electronic devices such as valley-polarized optoelectronic
emitters, valley optical interconnects, and ultrafast data storage with utmost
reliability and robustness.

###### Acknowledgements.

We would like to thank Alexandra Landsman, Mark Stockman, Vadym Apalkov and
Lisa Ortmann for fruitful discussions.

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