# Edge photogalvanic effect caused by optical alignment of carrier momenta in
2D Dirac materials

M. V. Durnev  S. A. Tarasenko  Ioffe Physical-Technical Institute, 194021 St.
Petersburg, Russia

###### Abstract

We show that the inter-band absorption of radiation in a 2D Dirac material
leads to a direct electric current flowing at sample edges. The photocurrent
originates from the momentum alignment of electrons and holes and is
controlled by the radiation polarization. We develop a microscopic theory of
such an edge photogalvanic effect and calculate the photocurrent for gapped
and gapless 2D Dirac materials, also in the presence of a static magnetic
field which introduces additional imbalance between the electron and hole
currents. Further, we show that the photocurrent can be considerably
multiplied in a ratchet-like structure with an array of narrow strips.

##  I Introduction

Photoelectric phenomena in 2D materials are the topic of active research in
recent years aimed at the development of photodetectors and energy harvesting
devices Koppens2014 . Various mechanisms of photocurrent generation based on
photothermoelectric Xu2010 ; Cai2014 ; Castilla2019 , photovoltaic and
bolometric Freitag2013 , plasmonic Vicarelli2012 ; Muraviev2013 ; Bandurin2018
, photon drag Karch2010 ; Entin2010 ; Obraztsov2014 , ratchet and
photogalvanic Tarasenko2011 ; Drexler2013 ; Geller2015 ; Fateev2017 ;
Kheirabadi2018 ; Quereda2018  effects have been reported and studied. Direct
electric currents arise here due to the lack of space inversion in the 2D
structure introduced by $p$-$n$ junctions, metallic contacts, substrate,
inhomogeneity of illumination or heating, or photon wave vector. Space
inversion symmetry is also broken naturally at the edges of 2D material. This
gives rise to the edge photogalvanic effect (EPGE) observed recently in
monolayer and bilayer graphene illuminated by THz radiation Karch2011 ;
Candussio2020  and in graphene in the quantum Hall effect regime Plank2018 .
The EPGE is reminiscent of the surface photogalvanic effect studied in bulk
semiconductor crystals and metal films Magarill1979 ; Alperovich1981 ;
Alperovich1989 ; Gurevich1993 ; Schmidt2015 ; Mikheev2018 .

The EPGE in 2D materials has been studied so far for low-frequency radiation
inducing indirect intraband (Drude-like) optical transitions in doped
structures Karch2011 ; Candussio2020 ; Plank2018 . These transitions are quite
weak since they can occur only if the scattering of carriers by static defects
or phonons is involved. With increasing the frequency of radiation direct
optical transitions between the valence and conduction bands come into play
and the absorption dramatically rises. Even in 2D monolayers, such as graphene
or transition metal dichalcogenides (TMCD), the absorbance related to the
interband optical transitions is known to reach few percent Nair2008 .
Accordingly, one can expect the enhanced photoresponse in this regime.

Here, we study the edge photogalvanic effect in 2D materials caused by direct
interband optical transitions. The edge photocurrent emerges due to the
alignment of charge carrier momenta by linearly polarized electromagnetic wave
with a subsequent scattering of the carriers at the edge and consists of
electron and hole contributions. We develop a microscopic theory of the EPGE
for a large class of 2D materials with the gapped or gapless Dirac-like energy
spectrum, such as monolayer and bilayer graphene, monolayers of TMCD,
HgTe/CdHgTe quantum wells with close-to-critical thickness, etc. We show that
the current is controlled by the radiation polarization, its magnitude reaches
1 nA per W/cm2 of the radiation intensity for a single edge and can be
enhanced to 1 $\mu$A in a ratchet-like structure consisting of multiple narrow
strips. We also investigate the effect of a static magnetic field applied
normally to the 2D plane and show that the field modifies the polarization
dependence of the currents and introduces additional imbalance between the
electron and hole currents. Taking into account the important role of edge
regions in micro- and nano-scale devices, we expect that the EPGE can
determine the photoresponse of small-size devices and find applications in
detectors of terahertz and infrared radiation and radiation polarization.

The paper is organized as follows. In Sec. II we describe the model of the
edge photocurrent formation, calculate the rate of optical transitions and
momentum alignment in 2D Dirac materials, and present the microscopic theory
of the EPGE in a semi-infinite system. In Sec. III we study the effect of a
static magnetic field on the photocurrent. In Sec. IV we calculate the
photocurrent and its spatial distribution in a single strip and a strip
structure. Results of the paper are summarized in Sec. V.

##  II Model and theory

The proposed microscopic mechanism of the edge photogalvanic effect is a two-
step process sketched in Fig. 1. We consider that a semi-infinite 2D structure
is illuminated by normally-incident linearly polarized radiation which causes
direct optical transitions between the valence and conduction bands.

At the first step, the absorption of linearly polarized radiation leads to the
momentum alignment of photoelectrons (and photoholes), Fig. 1a. This
phenomenon stems from the fact that the optical transition probability depends
on the relative orientation between the quasi-momentum $\bm{p}$ and the
electric field $\bm{E}$ of the radiation. The phenomenon is well known for
bulk semiconductors Alperovich1981 ; Zemskii1976 ; Dymnikov1976 ;
Zakharchenya1982  and 2D systems based on semiconductor quantum wells
Merkulov1990 , graphene Hartmann2011 ; Golub2011 , etc. The optically aligned
electrons and holes are characterized by anisotropic but even-in-$\bm{p}$
distribution in the momentum space (shown by closed blue curve in Fig. 1b).
Since the distribution is even in $\bm{p}$, no electric current is generated
in the “bulk” of the 2D plane.

The direct electric current $\bm{J}$ emerges at the second step as a result of
the scattering of optically aligned electrons at the edge, which introduces a
local asymmetry in the electron distribution in the momentum space. Only those
carriers who were created within the mean free path from the edge may
contribute to the current, therefore the current flows in a narrow strip at
the edge, as shown in Fig. 1b. The edge current is expected to have distinct
polarization dependence: it flows in the opposite directions for the radiation
polarized at $\pm\pi/4$ angle with respect to the edge and vanishes if the
radiation is polarized perpendicular to or along the edge.

Figure 1:  (a) Optical alignment of electron and hole momenta in 2D Dirac
structure. The probability of interband transitions (shown by vertical arrows
of different thicknesses) induced by linearly polarized radiation depends on
the direction of the electron momentum $\bm{p}$. This leads to an anisotropic
distribution of photoelectrons and photoholes in the $\bm{p}$ space. (b)
Mechanism of the edge photogalvanic effect. Optical alignment of electrons
(shown by closed blue curve) followed by electron scattering from the edge of
2D structure results in a direct electric current $\bm{J}$ flowing along the
edge. The current flows within a narrow strip near the edge with the width
determined by the mean free path $l_{0}$.

Below, we develop a microscopic theory of the edge photogalvanic effect. We
calculate the excitation spectrum of the photocurrent and its dependence on
the band-structure parameters.

###  II.1 Band structure and momentum alignment

Consider first the band structure of 2D materials and interband optical
transitions. The electron and hole states in a wide class of direct gap Dirac
materials can be described by the effective Hamiltonian Bernevig2006 ;
Neto2009 ; Kormanyos2013

$H=a\,\bm{p}\cdot\bm{\sigma}+(\delta_{0}+bp^{2})\sigma_{z}+dp^{2}I\,,$ (1)

where $\bm{p}=(p_{x},p_{y})$ is the electron momentum counted from the Dirac
point, $\sigma_{j}$ ($j=x,y,z$) are the Pauli matrices, $I$ is the identity
matrix, $\delta_{0}$, $a$, $b$, and $d$ are the band structure parameters. The
Hamiltonian contains all essential ingredients: the band gap $2\delta_{0}$,
the linear-in-$\bm{p}$ coupling between the valence- and conduction-band
states, and the parabolic term $dp^{2}I$ introducing electron-hole asymmetry.
The Dirac states can have additional spin or/and valley degrees of freedom.

The energy spectrum and the wave functions of the conduction (“$c$”) and
valence (“$v$”) bands in the electron representation are given by

$E_{c/v,p}=\pm\varepsilon_{p}+dp^{2}$ (2)

and

$\Psi_{c/v,\bm{p}}=\frac{1}{\sqrt{2\varepsilon_{p}(\varepsilon_{p}\mp\delta)}}\left(\begin{array}[]{c}-ap_{-}\\\
\delta\mp\varepsilon_{p}\end{array}\right),$ (3)

respectively, where $\varepsilon_{p}=\sqrt{\delta^{2}+(ap)^{2}}$,
$\delta=\delta_{0}+bp^{2}$, and $p_{\pm}=p_{x}\pm ip_{y}$.

The Hamiltonian of electron-photon interaction has the form

$V=-(e/c)\bm{A}(t)\cdot\nabla_{\bm{p}}H\,,$ (4)

where $e$ is the electron charge, $c$ is the speed of light,
$\bm{A}(t)=\bm{A}\exp{(-i\omega t)}+{\rm c.c.}$ is the vector potential of the
field, $\omega$ is its frequency, and $\bm{A}$ is its (complex) amplitude
related to the electric field amplitude by $\bm{E}=(i\omega/c)\bm{A}$.

The electromagnetic field excites electrons from the filled valence band to
the empty conduction band. The matrix element of such optical transitions
$(v,\bm{p})\rightarrow(c,\bm{p})$ has the form

$V_{\bm{p}\bm{p}}=-\frac{e|a|}{\varepsilon_{p}pc}\left[(bp^{2}-\delta_{0})(\bm{A}\cdot\bm{p})-\mathrm{i}\varepsilon_{p}(\bm{A}\times\bm{p})_{z}\right]\>.$
(5)

The rate of optical transitions is given by Fermi’s golden rule

$g_{\bm{p}}=\frac{2\pi}{\hbar}|V_{\bm{p}\bm{p}}|^{2}\delta(2\varepsilon_{p}-\hbar\omega)\>,$
(6)

where $|V_{\bm{p}\bm{p}}|^{2}$ is found from Eq. (5) and has the form

$\displaystyle|V_{\bm{p}\bm{p}}|^{2}$ $\displaystyle=$ $\displaystyle\frac{\pi
e^{2}a^{2}I}{\varepsilon_{p}^{2}\,\omega^{2}c\,n}\\{(\delta_{0}-bp^{2})^{2}+\varepsilon_{p}^{2}+2\varepsilon_{p}(\delta_{0}-bp^{2})S_{3}$
(7) $\displaystyle-$
$\displaystyle(a^{2}+4b\delta_{0})[(p_{x}^{2}-p_{y}^{2})S_{1}+2p_{x}p_{y}S_{2}]\\}\,.$

Here, $I=cn|\bm{E}|^{2}/(2\pi)$ is the intensity of radiation, $n$ is the
refractive index of the dielectric medium surrounding the 2D Dirac material,
and $S_{1}=(|E_{x}|^{2}-|E_{y}|^{2})/|\bm{E}|^{2}$,
$S_{2}=(E_{x}E_{y}^{*}+E_{y}E_{x}^{*})/|\bm{E}|^{2}$, and
$S_{3}=i(E_{x}E_{y}^{*}-E_{y}E_{x}^{*})/|\bm{E}|^{2}$ are the Stokes
parameters of the radiation polarization.

The generation rate (6) includes the polarization-independent contribution,
the terms $\propto S_{1}$ and $\propto S_{2}$ describing the momentum
alignment of electrons by linearly polarized radiation, and the contribution
sensitive to the circular polarization. The latter has the opposite signs for
the pair of Dirac cones (e.g., spin subbands or valleys) related by time
reversal symmetry and describes the spin/valley polarization of electrons by
circularly polarized radiation Mak2012 . In contract, the optical alignment of
electron momenta by linearly polarized radiation is the same for all Dirac
cones.

###  II.2 Edge photocurrent

In this section we calculate the dc edge current emerging in a semi-infinite
sample at homogeneous illumination by linearly polarized radiation. We
consider a 2D Dirac material occupying a half-plane $x\geq 0$ with the edge
parallel to $y$-axis, see Fig. 1b.

The edge electric current consists of electron and hole contributions,
$J_{y}^{e}$ and $J_{y}^{h}$, respectively, and is given by

$J_{y}=J_{y}^{e}+J_{y}^{h}\,,\;\;\;J_{y}^{e/h}=\int\limits_{0}^{+\infty}j_{y}^{e/h}(x)dx\,,$
(8)

where $j_{y}^{e}(x)$ and $j_{y}^{h}(x)$ are the local current densities.
Below, we calculate the electron contribution $J_{y}^{e}$. The hole
contribution $J_{y}^{h}$ can be computed in a similar way.

The density of electric current in the conduction band is expressed via the
electron distribution function $f(x,\bm{p})$ as follows

$j_{y}^{e}(x)=e\nu\sum\limits_{\bm{p}}v_{y}f(x,\bm{p})\,,$ (9)

where $\nu$ is the factor of spin and valley degeneracy of Dirac states (e.g.,
$\nu=4$ for graphene) and $\bm{v}=\nabla_{\bm{p}}E_{c,\bm{p}}$ is the electron
velocity. The distribution function is found from the kinetic equation

$v_{x}\frac{\partial f}{\partial x}=g_{\bm{p}}+{\rm St}f\,,$ (10)

where $g_{\bm{p}}$ is the optical generation rate of electrons and ${\rm St}f$
is the collision integral.

The collision integral in the relaxation time approximation is given by

${\rm St}f=-\frac{f(x,\bm{p})-\langle f(x,\bm{p})\rangle}{\tau}\,,$ (11)

where $\langle f(x,\bm{p})\rangle$ is the distribution function averaged over
the directions of $\bm{p}$ and $\tau$ is the relaxation time. Note that the
collision integral above does not describe the relaxation of the zero angular
harmonic of the distribution function. The corresponding relaxation times,
governed by the processes of energy relaxation and recombination, are
typically much larger than $\tau$ and do not affect the anisotropic part of
the distribution function.

The collision integral should be supplemented with the boundary condition at
$x=0$. We consider diffuse or specular reflection of electrons from the sample
edge. In the case of diffuse scattering, the distribution of the particles
reflected from the edge is even in $p_{y}$, i.e.,
$f(0,p_{x}>0,p_{y})=f(0,p_{x}>0,-p_{y})$. For specular reflection, the
distribution satisfies $f(0,p_{x},p_{y})=f(0,-p_{x},p_{y})$. Additional
condition $\sum_{\bm{p}}v_{x}f(x,\bm{p})=0$ comes from the lack of electron
flux to the edge in the absence of spatially inhomogeneous generation and
recombination of carriers.

To calculate the electric current along $y$ we decompose the distribution
function $f(x,\bm{p})$ into the symmetric and asymmetric in $p_{y}$ parts as
follows

$f^{(s/a)}(x,p_{x},p_{y})=\frac{1}{2}[f(x,p_{x},p_{y})\pm
f(x,p_{x},-p_{y})]\,.$ (12)

The asymmetric part satisfies the equation

$v_{x}\frac{\partial f^{(a)}}{\partial
x}=-\frac{f^{(a)}}{\tau}+g_{\bm{p}}^{(a)}\,,$ (13)

where $g_{\bm{p}}^{(a)}$ is the asymmetric-in-$p_{y}$ part of the generation
term $g_{\bm{p}}$. Solution of Eq. (13) with the boundary conditions discussed
above has the form

$f^{(a)}(x,\bm{p})=\tau g_{\bm{p}}^{(a)}+\tau\left[\zeta
g_{-p_{x},p_{y}}^{(a)}-g_{\bm{p}}^{(a)}\right]\exp\left(-\frac{x}{v_{x}\tau}\right)\Theta(p_{x})\,,$
(14)

where $\zeta$ is the dimensionless parameter defined by character of edge
scattering ($\zeta=0$ for diffuse scattering and $\zeta=1$ for specular
reflection) and $\Theta(p_{x})$ is the Heaviside step function.

Multiplying $f^{(a)}(x,\bm{p})$ by $ev_{y}$, summing up the result over
$\bm{p}$ and integrating by $x$, and taking into account that the generation
rate $g_{\bm{p}}$ is an even function of $\bm{p}$, we obtain the electric
current

$J_{y}^{e}=-e\nu\frac{1+\zeta_{e}}{2}\sum\limits_{\bm{p}}\tau_{e}^{2}v_{e,x}v_{e,y}\,g_{\bm{p}}\,.$
(15)

Here, the parameter of edge scattering specularity, the relaxation time, and
the velocity related to the electrons in the conduction band are denoted as
$\zeta_{e}$, $\tau_{e}$, and $\bm{v}_{e}$, respectively. We note that the
result (15) in the case of specular reflection can be also obtained without
the explicit calculation of the distribution function (14) (see Appendix).

Similar calculations show that the electric current carried by holes in the
valence hand has the opposite sign and is given by

$J_{y}^{h}=e\nu\frac{1+\zeta_{h}}{2}\sum\limits_{\bm{p}}\tau_{h}^{2}v_{h,x}v_{h,y}\,g_{\bm{p}}\,,$
(16)

where $\zeta_{h}$, $\tau_{h}$, and $\bm{v}_{h}=-\nabla_{\bm{p}}E_{v,\bm{p}}$
are the corresponding hole parameters in the hole representation.

Equations (15) and (16) are quite general and can be applied to any 2D
material. They show that, in systems with full electron-hole symmetry, the net
electric current $J_{y}^{e}+J_{y}^{h}$ vanishes. In real systems, where the
electron-hole symmetry is lifted intrinsically (in the energy spectrum) or
extrinsically (e.g., by doping), the net photocurrent is non-zero.

For the interband optical transitions, the generation term is given by Eqs.
(6) and (7). Putting these equations into Eqs. (15) and (16), summing up over
$\bm{p}$, and assuming that $|b|\ll a^{2}/\delta_{0}$, we finally obtain the
electron and hole contributions to the edge current

$J_{y}^{e/h}=\pm\frac{e\eta
a^{2}(1+\zeta_{e/h})\tau_{e/h}^{2}v_{e/h}^{2}\,p_{*}^{2}}{2(\hbar\omega)^{3}}\Theta(\hbar\omega-2\delta_{0})IS_{2},$
(17)

where $\eta=\pi\nu e^{2}/(4\hbar cn)$ is the absorbance of the 2D Dirac
material at $\hbar\omega\gg\delta_{0}$,
$p_{*}=\sqrt{(\hbar\omega)^{2}-(2\delta_{0})^{2}}/(2a)$ is the momentum of
photoexcited electrons and holes,
$v_{e/h}=a\sqrt{1-(2\delta_{0}/\hbar\omega)^{2}}(1\pm\hbar\omega d/a^{2})$ are
the corresponding electron and hole velocities, and $\Theta(x)$ is the
Heaviside step function. The relaxation times $\tau_{e}$ and $\tau_{h}$ are
taken at the electron and hole energies
$E_{e/h}(p_{*})=\hbar\omega/2\pm(d/4a^{2})[(\hbar\omega)^{2}-(2\delta_{0})^{2}]$,
respectively.

As follows from Eq. (17), the polarization dependence of the edge current is
determined by the Stokes parameter
$S_{2}=(E_{x}E_{y}^{*}+E_{y}E_{x}^{*})/|\bm{E}|^{2}$. The current reaches
maxima of the opposite signs for the radiation polarized at $\pm 45^{\circ}$
with respect to the edge and vanishes for the radiation polarized along or
normal to the edge, in accordance with the model sketched in Fig. 1.

Figure 2 shows the excitation spectra of the electron and hole currents,
$J_{y}^{e}$ and $J_{y}^{h}$, respectively, and the total edge photocurrent
$J_{y}=J_{y}^{e}+J_{y}^{h}$ in gapped and gapless systems. The spectra are
calculated after Eq. (17) for the case when the energy dependence of the
electron/hole scattering rate $\tau_{e/h}^{-1}(\varepsilon)$ follows the
electron/hole density of states $D_{e/h}(\varepsilon)$, which is relevant to
short-range scattering. For weak electron-hole asymmetry ($|d|\ll
a^{2}/\delta_{0}$), the densities of states have the form

$D_{e/h}(\varepsilon)=\frac{\nu}{2\pi\hbar^{2}}\frac{\varepsilon}{a^{2}}\left[1\mp\frac{d}{a^{2}}\frac{(3\varepsilon^{2}-\delta_{0}^{2})}{\varepsilon}\right]\Theta(\varepsilon-\delta_{0})\>,$

and we plot the spectra for the relaxation times

$\tau_{e/h}(\varepsilon)=\tau_{0}(\tilde{\varepsilon})\frac{\tilde{\varepsilon}}{\varepsilon}\left[1\pm\frac{d}{a^{2}}\frac{(3\varepsilon^{2}-\delta_{0}^{2})}{\varepsilon}\right]\>,$

where $\tau_{0}(\tilde{\varepsilon})$ is the relaxation time at the energy
$\tilde{\varepsilon}$ in the absence of electron-hole asymmetry. The
parameters used for calculations are given in the caption of Fig. 2.

In gapped Dirac materials (Fig. 2a), the edge photocurrent is generated if the
photon energy exceeds the band gap. Above this threshold, the photocurrent
increases with the photon energy, reaches a maximum, and then decreases. In
gapless materials with the Fermi energy lying at the Dirac point (Fig. 2b),
the photocurrent monotonically decreases with the photon energy. The electron
and hole contributions to the photocurrent can be estimated as
$J_{y}^{e/h}\sim e\eta Il_{e/h}^{2}/(\hbar\omega)$, where
$l_{e/h}=v_{e/h}\tau_{e/h}$ are the mean free paths of photoexcited electrons
and holes. For the mean free path of 1 $\mu$m and $\hbar\omega=20$ meV, the
photocurrent normalized by the radiation intensity is of the order of nA
cm2/W.

Figure 2:  Excitation spectra of the electron $J_{y}^{e}$ and hole $J_{y}^{h}$
contributions to the edge photocurrent and the total edge photocurrent $J_{y}$
for gapped and gapless 2D Dirac materials. The spectra are calculated after
Eq. (17) for short-range scattering of carriers in the 2D bulk and specular
scattering from the edge ($\zeta_{e/h}=1$), the band structures parameters
$a=10^{8}$ cm/s and $\delta_{0}d/a^{2}=0.03$, the spin and valley degeneracy
$\nu=4$, the band gap $2\delta_{0}=20$ meV, the relaxation time
$\tau_{0}(10\,{\rm meV})=1$ ps, the refractive index of the surrounding medium
$n=3$, and the radiation intensity $I=1$ W/cm2. The radiation is linearly
polarized at 45∘ with respect to the structure edge.

##  III Effect of magnetic field

Now, we study the effect of a magnetic field $\bm{B}$ applied along the 2D
plane normal $z$ on the edge photocurrent. We show that the magnetic field
modifies the polarization dependence and magnitude of the photocurrent and,
more interestingly, enables the generation of a net electric current even in a
system with electron-hole symmetry. Similarly to the consideration in the
previous section, we calculate the electron contribution to the edge current
and then discuss both the electron and hole contributions.

In a classical magnetic field, the kinetic equation for the steady-state
distribution function of electrons in the conduction band has the form

$v_{x}\frac{\partial f}{\partial
x}+\frac{e}{c}(\bm{v}\times\bm{B})\cdot\frac{\partial
f}{\partial\bm{p}}=g_{\bm{p}}-\frac{f(x,\bm{p})-\langle
f(x,\bm{p})\rangle}{\tau}\,.$ (18)

Multiplying Eq. (18) by $v_{y}$, averaging the result over the direction of
$\bm{p}$, and taking into account that $\langle v_{y}g_{\bm{p}}\rangle=0$ we
obtain the equation

$\left\langle v_{x}v_{y}\frac{\partial f}{\partial
x}\right\rangle+\frac{eB_{z}}{c}\left\langle v_{x}v_{y}\frac{\partial
f}{\partial p_{y}}-v_{y}^{2}\frac{\partial f}{\partial
p_{x}}\right\rangle=-\frac{\langle v_{y}f\rangle}{\tau}\,.$ (19)

Further, multiplying Eq. (19) by $\tau$ and summing up over $\bm{p}$ we obtain
after some algebra

$j_{y}^{e}(x)=-e\nu\sum_{\bm{p}}\tau v_{x}v_{y}\frac{\partial f}{\partial
x}+\frac{eB_{z}}{c}\sum_{\bm{p}}\frac{\tau}{m_{c}}v_{x}f\,,$ (20)

where $m_{c}=p/v$ is the cyclotron mass.

In the absence of spatially inhomogeneous recombination of carriers, the
electron (and hole) flux to the edge given by the term $\left\langle
v_{x}f\right\rangle$ vanishes. Then, the second term in the right-hand side of
Eq. (20) can be neglected, and Eq. (20) yields

$J_{y}^{e}=-e\nu\sum\limits_{\bm{p}}\tau
v_{x}v_{y}[f(\infty,\bm{p})-f(0,\bm{p})]\,.$ (21)

The edge current is determined by the difference between the steady-state
distribution functions at the edge and in the 2D bulk.

For specular reflection of carriers at the edge, $f(0,\bm{p})$ is even in
$p_{x}$ and the term $\sum_{\bm{p}}\tau v_{x}v_{y}f(0,\bm{p})$ vanishes. The
rest contribution $\sum_{\bm{p}}\tau v_{x}v_{y}f(\infty,\bm{p})$ can be
readily calculated analytically from the kinetic Eq. (18) with the first term
in the left-hand side being neglected. Such a calculation with the generation
term (6) shows that the anisotropic part of the electron distribution function
far from the edge has the form

$\displaystyle\delta f(\infty,\bm{p})=-\frac{8\pi^{2}\tau
e^{2}a^{2}(a^{2}+4b\delta_{0})I}{(\hbar\omega)^{3}\omega
cn}\delta(2\varepsilon_{p}-\hbar\omega)$
$\displaystyle\times\left[(p_{x}^{2}-p_{y}^{2})\frac{S_{1}+2\omega_{c}\tau
S_{2}}{1+(2\omega_{c}\tau)^{2}}+2p_{x}p_{y}\frac{S_{2}-2\omega_{c}\tau
S_{1}}{1+(2\omega_{c}\tau)^{2}}\right],\;\;$ (22)

where $\omega_{c}=eB_{z}/{m_{c}c}$ is the cyclotron frequency. Finally, the
electron and hole contributions to the edge photocurrent at $|b|\ll
a^{2}/\delta_{0}$ are given by

$J_{y}^{e/h}=\pm\frac{e\eta
a^{2}\tau_{e/h}^{2}v_{e,h}^{2}\,p_{*}^{2}}{(\hbar\omega)^{3}}\frac{S_{2}\mp
2\omega_{e/h}\tau_{e/h}S_{1}}{1+(2\omega_{e/h}\tau_{e/h})^{2}}\\\
\times\Theta(\hbar\omega-2\delta_{0})I\,,$ (23)

where $\omega_{e/h}=eB_{z}/(m_{e/h}c)$ are the cyclotron frequencies of the
photoexcited electrons and holes and
$m_{e/h}=p_{*}/v_{e/h}=\hbar\omega/[2a^{2}(1\pm\hbar\omega d/a^{2})]$ are the
corresponding effective masses.

For diffuse scattering from the edge, the contribution to the edge current
from the sum $\sum_{\bm{p}}\tau v_{x}v_{y}f(\infty,\bm{p})$ is still given by
Eq. (23). The second contribution in Eq. (21) given by the term
$\sum_{\bm{p}}\tau v_{x}v_{y}f(0,\bm{p})$ does not vanish any more and has to
be calculated numerically.

Figure 3 shows the dependence of the edge photocurrent on magnetic field
calculated after Eq. (23) for specular reflection of carriers from the edge.
Figures 3a and 3b correspond to the radiation polarized at 45∘ to the edge and
along the edge, respectively. Since $\omega_{e/h}\tau_{e/h}\propto l_{e/h}$,
where $l_{e/h}=v_{e/h}\tau_{e/h}$ are the mean free paths of the carriers, the
difference between the hole and electron contributions to the photocurrent is
determined by the single parameter $l_{h}/l_{e}$. In the calculations we set
$l_{h}/l_{e}=0.7$.

The radiation polarized at 45∘ to the edge induces the electron $J_{y}^{e}$
and hole $J_{y}^{h}$ edge photocurrents of the opposite sign, Fig. 3a. The
photocurrents decrease in the magnetic field $B_{z}$ following the Hanle
curves, see Eq. (23). The net electric current is non-zero because of the
electron-hole asymmetry.

In the magnetic field, the edge photocurrents can be also excited by radiation
polarized along (as well as perpendicular to) the edge, Fig. 3b. For this
polarization, the electron $J_{y}^{e}$ and hole $J_{y}^{h}$ contributions have
the same sign and add up to each other. The net electric current does not
require the presence of electron-hole asymmetry in the 2D system. Therefore,
quite a strong photoelectric response can be expected even in close-to-
symmetric systems like graphene.

The diffuse edge scattering leads to a reduction of the edge photocurrent as
compared to the specular one, as shown in the insets of Figs. 3a and 3b. At
zero magnetic field, the ratio of the edge currents in the structures with
mirror ($\zeta_{e/h}=1$) and rough ($\zeta_{e/h}=0$) edges is 2:1, as also
follows from Eq. (17), whereas at $B_{z}\neq 0$ the ratio depends on the
magnetic field and the radiation polarization.

Figure 3:  Magnetic field dependence of the edge photocurrents excited by
radiation (a) linearly polarized at 45∘ with respect to the structure edge
(the Stokes parameter $S_{2}=1$) and (b) linearly polarized along the edge
(the Stokes parameter $S_{1}=1$). $J_{y}^{e}$ and $J_{y}^{h}$, and $J_{y}$ are
the electron and hole contributions to the photocurrent, and the total
photocurrent, respectively. The photocurrents are calculated for specular
scattering from the edge, the band gap $2\delta_{0}=20$ meV, the photon energy
$\hbar\omega=30$ meV, the spin and valley degeneracy $\nu=4$, the refractive
index of the surrounding medium $n=3$, the electron and hole mean free paths
$l_{e}=1$ $\mu$m and $l_{h}=0.7~{}l_{e}$, respectively, and the radiation
intensity $I=1$ W/cm2. The insets show the comparison between the total edge
photocurrents in the structures with specular ($\zeta_{e/h}=1$) and diffuse
($\zeta_{e/h}=0$) edge scattering.

Numerical solution of the kinetic Eq. (18) allows us to find the spatial
distribution of the photocurrent density $j_{y}(x)$. The results of such
calculations are shown in Fig. 4. As expected, the edge photocurrent flows
within a narrow strip near the edge. The width of this strip is of the order
of the carrier mean free path $l_{e/h}$ at zero magnetic field and close to
the cyclotron diameter $2v_{e/h}/\omega_{e/h}$ at high magnetic fields.
Interestingly, at $\omega_{e/h}\tau_{e/h}\gtrsim 1$, the edge photocurrent
flows in the opposite directions in different regions near the edge, see Fig.
(4)b.

Figure 4:  Spatial distributions of the electron current density
$j_{y}^{e}(x)$ near the sample edge at different magnetic fields: (a)
$\omega_{e}\tau_{e}=0,~{}0.5,~{}1,~{}2$ and (b) $\omega_{e}\tau_{e}=1,~{}2$ at
larger scale. The distributions are calculated for specular reflection of
electrons from the edge, the radiation with the Stokes parameter $S_{2}=1$,
and the parameters listed in the caption of Fig. 3.

In addition to the current controlled by the radiation polarization, in an
external magnetic field there can emerge a polarization-independent
photocurrent. Physically, it originates from the 2D analogue of the the
Kikoin-Noskov photo-electro-magnetic effect known for 3D materials Kikoin1934
; Chernichkin2012 . The effect is related to an enhanced recombination rate of
electrons and holes at the edge of the material, which may occur due to
increased density of defect at the edge. The edge recombination leads to a
local decrease of the electron and hole densities, which induces diffusive
fluxes of the both types of carriers. At zero magnetic field, the fluxes are
directed to the edge. The net electric current is zero because the currents
carried by electrons and holes compensate each other in the steady-state
regime. The flux density can be estimated as $i_{x}(x)\sim
a_{0}(\Delta\gamma_{\rm r}/\gamma_{\rm r})G\exp(-\sqrt{\gamma_{\rm r}/D}\,x)$
where $a_{0}$ is a width of the order of few lattice constants where the
recombination rate in enhanced, $\gamma_{\rm r}=1/\tau_{\rm r}$, $\tau_{\rm
r}$ is the lifetime of carriers far from the edge, $\Delta\gamma_{\rm r}$ is
the extra recombination probability near the edge,
$G=\eta(\omega)I/(\hbar\omega)$ is the rate of carrier generation,
$\eta(\omega)$ is the absorbance, and $D$ is the diffusion coefficient of
thermalized carriers. The external magnetic field deflects the fluxes in the
opposite directions giving rise to a net electric current along the edge. This
edge photocurrent flows within the diffusion length $\sqrt{D\tau_{\rm r}}$
near the edge and can be estimated as

$J_{y}^{\rm KN}\sim\frac{e\eta
a_{0}\sqrt{D\tau_{r}}}{\hbar\omega}\frac{\Delta\gamma_{\rm r}}{\gamma_{\rm
r}}\frac{\omega_{c}\tau}{1+(\omega_{c}\tau)^{2}}I\,.$ (24)

The ratio of the Kikoin-Noskov current (24) to the current caused by momentum
alignment (23) can be estimated as $(\Delta\gamma_{\rm r}/\gamma_{\rm
r})(a_{0}/l)\sqrt{\tau_{\rm r}/\tau}$, where $l=v\tau$ is the mean free path.
One can expect that, in high-mobility structures, the current caused by
momentum alignment predominates.

##  IV Photocurrent in a strip structure

The edge photocurrent is formed in a narrow strip of the width determined by
the carrier mean free path. The bulk of 2D system absorbs radiation but is not
involved in the current generation. Therefore, thinking of possible design of
the structure with enhanced photoresponse we consider a ratchet structure
consisting of $N$ narrow asymmetric strips, Fig 5a. Optical excitation of such
a structure leads to a photocurrent in each strip and the total current is
increased by the factor of $N$.

Figure 5:  (a) Sketch of a ratchet structure consisting of $N$ narrow
asymmetric strips. The incident radiation induces dc electric current $\bm{J}$
in each strip, and the total current flowing in the structure is $N\bm{J}$.
(b) Geometry of an individual strip.

Now, we calculate the photocurrent in an individual strip and study how it
depends on the strip width and boundary conditions. We consider that the 2D
Dirac material is located at $-w/2\leq x\leq w/2$, where $w$ is the strip
width, Fig. 5b. The distribution function of electrons in the strip is then a
solution of the kinetic Eq. (18) with boundary conditions at $x=-w/2$ and
$x=w/2$. At zero magnetic field, the asymmetric-in-$p_{y}$ part of the
electron distribution function has the form

$f^{(a)}(x,p_{x},p_{y})=\tau\left[g_{\bm{p}}^{(a)}+C_{\bm{p}}\exp\left(-\frac{x}{v_{x}\tau}\right)\right]\,,$
(25)

where

$\displaystyle C_{\bm{p}}=$ $\displaystyle-$
$\displaystyle\frac{S(w/|2v_{x}\tau|)}{S(w/|v_{x}\tau|)}g^{(a)}_{\bm{p}}$
$\displaystyle+$
$\displaystyle[\zeta_{l}\Theta(p_{x})+\zeta_{r}\Theta(-p_{x})]\frac{\sinh(w/|2v_{x}\tau|)}{S(w/|v_{x}\tau|)}g^{(a)}_{-p_{x},p_{y}}\,,$

$S(x)$ is the function defined by
$S(x)=[\exp(x)-\zeta_{l}\zeta_{r}\exp(-x)]/2$, $g^{(a)}_{\bm{p}}$ is the
asymmetric-in-$p_{y}$ part of $g_{\bm{p}}$, and $\zeta_{l/r}$ are the
parameters of scattering specularity at the left/right edges of the strip,
respectively.

Using the same method as described in Sec. III, one can show that the total
electron photocurrent in the strip $J_{y}^{e}=\int_{-w/2}^{w/2}j_{e}(x)dx$ is
given by

$J_{y}^{e}=-e\nu\sum\limits_{\bm{p}}\tau
v_{x}v_{y}[f(w/2,\bm{p})-f(-w/2,\bm{p})]\,.$ (27)

The photocurrent is proportional to the difference of the distribution
functions at the right and left edges of the strip and vanishes in a strip
possessing the $x\to-x$ mirror symmetry. This symmetry is violated if the
strip has an asymmetric shape, as shown in Fig. 5, or an asymmetric static
potential $U(x)$. The current $J_{y}^{e}$ is also nonzero in a strip with
different roughness of the left and right edges.

Substituting the distribution function Eq. (25) at $x=\pm w/2$ into Eq. (27),
we finally obtain

$J_{y}^{e}=e\nu\left(\zeta_{r}-\zeta_{l}\right)\sum\limits_{\bm{p}}\tau^{2}v_{x}v_{y}\frac{[\exp(w/v_{x}\tau)-1]^{2}g_{\bm{p}}}{\exp(2w/v_{x}\tau)-\zeta_{l}\zeta_{r}}\Theta(p_{x})\,.$
(28)

As expected, the total current in the strip is proportional to the difference
$\zeta_{r}-\zeta_{l}$.

Figure 6:  (a) Spatial distribution of the electron current density in the
strip, where electrons are scattered specularly at the left edge
($\zeta_{l}=1$) and diffusely at the right edge ($\zeta_{r}=0$). The
distributions are plotted for different ratios between the strip width $w$ and
the mean free path $l_{e}$. (b) Total electron current flowing in the strip as
a function of the strip width. The curves are calculated for the same
parameters as used in Fig. 3a.

Figure 6a shows the distribution of the electron photocurrent $j_{y}^{e}(x)$
in the cross section of the strip. The distributions are calculated by numeric
summation of Eq. (9) with the distribution function given by Eq. (25). It is
assumed that the strip edges have different scattering properties: Electrons
are reflected specularly at the left edge ($\zeta_{l}=1$) and diffusely at the
right edge ($\zeta_{r}=0$). When the strip width $w$ is much larger than the
electron mean free path $l_{e}$, the photocurrents are concentrated at the
edges and vanish in the bulk of the strip. The current densities in that case
coincide with those calculated in Sec. II.2 for a semi-infinite structure. The
currents at the opposite sides of the strip flow in the opposite directions.
The total current at the left edge is twice larger than the current at the
right edge, in agreement with Eq. (15). In narrow strips, when the strip width
approaches the mean free path, the edge currents merge and the photocurrent is
generated in the whole cross section of the strip.

The total photocurrent $J_{y}^{e}$ generated in the strip as a function of the
strip width is shown in Fig. 6b. The photocurrent increases with the strip
width $w$ in narrow strips and saturates at $w\approx 3l_{e}$. Further
increase of the strip width does not affect the current magnitude. Projecting
the results to $3\times 3$ mm2 sample with $N=10^{3}$ strips we estimate the
total current of about $1~{}\mu$A per W/cm2 in the terahertz spectral range.
Such structures based on 2D Dirac materials can be used as fast detectors of
terahertz and infrared radiation and its polarization.

##  V Summary

To summarize, we have studied theoretically the edge photogalvanic effect in
2D materials caused by direct interband optical transitions. The dc electric
current emerges due to the optical alignment of electron and hole momenta by
linearly polarized electromagnetic wave followed by scattering of carriers at
the edge. The current is formed in a narrow strip near the edge with the width
defined by the mean free path at zero magnetic field, and by the diameter of
cyclotron orbit at large magnetic fields. At zero magnetic field the edge
photocurrent behaves as $\sin 2\alpha$, where $\alpha$ is the angle between
the electric field of the wave and the edge. The current contains
counterflowing electron and hole contributions with different magnitudes due
to electron-hole asymmetry. The excitation spectrum of the edge current
calculated for short-range scatterers shows that the current magnitude is
larger in gapless materials and reaches 1 nA per W/cm2. Under application of a
static magnetic field normal to the sample plane, the edge current is also
excited by the electric field parallel or perpendicular to the edge, and the
net current is nonzero even in structures possessing electron-hole symmetry.
In a narrow strip made of 2D material, the photocurrent is generated in the
whole cross section of the strip and its value integrated over the strip is
nonzero in the strip with asymmetric shape or asymmetric static potential. The
maximum value of photocurrent is already reached in strips with the width of
several mean free paths, which allows one to use a ratchet-like multi-strip
structure for the significant increase of the total photocurrent. Our
estimations show that the total current can reach 1 $\mu$A per W/cm2 in a
3$\times$3 mm2 sample. Such ratchet-like structures can be used as fast
detectors of terahertz and infrared radiation and its polarization.

###### Acknowledgements.

We acknowledge financial support from the Russian Science Foundation (project
17-12-01265). M.V.D. also acknowledges the support from the Basis Foundation
for the Advancement of Theoretical Physics and Mathematics and the Russian
Federation President Grant No. MK- 2943.2019.2.

##  VI Appendix

Equation for the electric current (15) can be also obtained without the
explicit calculation of the distribution function (14). To directly calculate
$J_{y}$, we multiply Eq. (10) by $ev_{y}$, sum up the result over $\bm{p}$ and
integrate by $x$, and take into account that $\sum_{\bm{p}}v_{y}g_{\bm{p}}=0$
since the optical transitions do not induces a bulk electric current. This
yields

$J_{y}^{e}=-e\nu\sum\limits_{\bm{p}}\tau_{1}v_{x}v_{y}[f(\infty,\bm{p})-f(0,\bm{p})]\,,$
(29)

where $\tau_{1}$ is the relaxation time of the first angular harmonics of the
distribution function (also called momentum relaxation time) defined by
$\langle\bm{v}{\rm St}f\rangle=-\langle\bm{v}f\rangle/\tau_{1}$. The
distribution function far from the edge can be readily found from the kinetic
equation (10) with the first term being neglected, which gives $\langle
v_{x}v_{y}f(\infty,\bm{p})\rangle=\tau_{2}\langle
v_{x}v_{y}g_{\bm{p}}\rangle$, where $\tau_{2}$ is the relaxation time of the
second angular harmonics of the distribution function, $\langle v_{x}v_{y}{\rm
St}f\rangle=-\langle v_{x}v_{y}f\rangle/\tau_{2}$. For specular electron
reflection from the edge ($\zeta=1$), the term
$\sum\limits_{\bm{p}}\tau_{1}v_{x}v_{y}f(0,\bm{p})$ vanishes, Eq. (29)
coincides with Eq. (15) and specifies that $\tau^{2}$ is $\tau_{1}\tau_{2}$.

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