# Photocurrents in bulk tellurium

M. D. Moldavskaya1, L. E. Golub1, S. N. Danilov1, V. V. Bel’kov2, D. Weiss1
and S. D. Ganichev1 1Terahertz Center, University of Regensburg, 93040
Regensburg, Germany  2Ioffe Institute, 194021 St. Petersburg, Russia

###### Abstract

We report a comprehensive study of polarized infrared/terahertz photocurrents
in bulk tellurium crystals. We observe different photocurrent contributions
and show that, depending on the experimental conditions, they are caused by
the trigonal photogalvanic effect, the transverse linear photon drag effect,
and the magnetic field induced linear and circular photogalvanic effects. All
observed photocurrents have not been reported before and are well explained by
the developed phenomenological and microscopic theory. We show that the
effects can be unambiguously distinguished by studying the polarization,
magnetic field, and radiation frequency dependence of the photocurrent. At
frequencies around 30 THz, the photocurrents are shown to be caused by the
direct optical transitions between subbands in the valence band. At lower
frequencies of 1 to 3 THz, used in our experiment, these transitions become
impossible and the detected photocurrents are caused by the indirect optical
transitions (Drude-like radiation absorption).

##  I Introduction

Tellurium is an elementary semiconductor that has been studied from the very
beginning of the history of semiconductor physics. In the 60’s-70’s such
phenomena as quantum effects in cyclotron resonance [1], Shubnikov-de Haas
effect [2], Nerst-Ettingshausen and Seebeck effects [3], surface quantum
states [4, 5], and natural optical activity [6, 7, 8, 9] were detected in Te
crystals. In addition, several novel photoelectric phenomena, including the
circular photogalvanic effect [10, 11], the circular photon drag effect [12],
and the electric current-induced optical activity [10, 13], were discovered
for the first time in bulk Te, for a recent review see [14]. These effects
arise from spin splitting of the valence band at the boundary of the first
Brillouin zone (”camel back” structure).

In recent years, studies of Te have experienced a renaissance due to the
possibility of fabricating 2D Te crystals (tellurene) which exhibit unique
material properties, for reviews see e.g. Refs. [15, 16, 17, 18, 19, 20], and
theoretical proposals for closing the energy gap in Te and the appearance of
Weyl points near the Fermi level with Fermi arcs at the surface by applying
proper strain, see e.g. Refs. [21, 22, 23, 24, 25, 26, 27].

Experimental access to the properties of tellurene as well as to the specific
properties of Weyl fermions should allow studies of photoelectric effects
excited by infrared/THz radiation. The power of the method has already been
demonstrated for other 2D materials, surface states of topological insulators
and Weyl semimetals, see e.g. Refs. [28, 29, 30, 31, 32, 33, 34, 35, 36, 37].
Recently, a transverse CPGE has been observed in bulk unstrained Te at oblique
incidence and has been attributed to Weyl fermions [38]. However, no features
specific to Weyl fermions have been detected and these results can be
explained alternatively by considering optical transitions in the conventional
Te band structure without involving Weyl bands, which are not expected without
a significant strain [21, 22, 23, 24, 25, 26, 27]. In view of the increasing
interest in photoelectric effects excited in 2D and on the surface of 3D Te
crystals, it becomes important to understand the photocurrents excited in bulk
Te that are not related to 2D states or the topological charges of the Weyl
points.

In the present work, we report the observation of three photoelectric
phenomena in Te crystals, which have not been previously addressed either
experimentally or theoretically. These phenomena are: (i) trigonal linear
photogalvanic effect (LPGE) due to intersubband optical transitions in the
valence band, (ii) transverse linear photon drag effect, and (iii) circular
(radiation helicity driven) magnetophotocurrents due to intersubband optical
transitions in the valence band. While the effect (ii) is detected for Drude
absorption (THz frequencies) only, the effects (i) and (iii) are observed at
both infrared frequencies and Drude absorption. The observed phenomena are
characterized by different dependences on radiation frequency and
polarization. Furthermore, the linear and circular magnetophotocurrents depend
linearly on an external magnetic field $B$ and vanish for $B=0$. The
qualitatively different functional behavior allows us to clearly distinguish
and study all these individual effects. The results are well described by the
developed phenomenological and microscopic theories. It is shown that all
phenomena are excited in the bulk of the material and are caused by the
displacement of electrons in real space due to direct intersubband optical
transitions (trigonal LPGE at IR frequencies), asymmetric scattering of
carriers at Drude absorption (trigonal LPGE at THz frequencies), transfer of
linear photon momentum to free carriers (photon drag effect at THz
frequencies), and magnetic field assisted asymmetric scattering (magnetic
field induced LPGE and CPGE).

The paper is structured as follows. In Sec. II we describe the investigated
samples and the experimental technique. In Sec. III we discuss the
experimental results. In the following Secs. IV we perform a symmetry analysis
of the photocurrent excited by radiation propagating along the $c$-axis and
identify different mechanisms of the observed photocurrents excited by
linearly (Sec. IV.1) and elliptically (Sec. IV.2) polarized radiation. In the
following section we discuss possible optical transitions in the studied
experimental arrangement and some details of the band structure. Next, we
present the developed theory and corresponding model pictures for the
photogalvanic effects excited at the inter subband transitions (Secs. V.1 and
V.2) and the intra subband Drude-like transitions (Secs.V.3 and V.4). After
discussing the mechanisms of photogalvanic currents, we consider the linear
photon drag effect (PDE) and the linear magnetic field induced PDE excited at
intraband (Drude like) transitions (Sec. V.6). In Sec. VI we compare the
experimental and theoretical results. Finally, in Sec. VII we summarize the
results. We have also included four appendices. In Appendix A we cover the
complete phenomenology of PGE and PDE currents at normal incidence. In
Appendix B we present the equations that describe the absorption coefficient
at direct intersubband transitions, and Appendices C and D contain the
microscopic derivation of the PDE and MPDE currents.

Figure 1: Experimental setup. Here $z$ is the $c$-axis of Te crystal, and $x$
is parallel to one of three $C_{2}$ rotation axes.

##  II Samples and experimental setup

The measurements were carried out on an $p$-type tellurium single crystal
grown by the Czochralski method in a hydrogen atmosphere. The inset in Fig. 1
shows the sample and the experimental setup. A plate with thickness 0.8 mm was
cut perpendicular to the to $c$-axis. A pair of Ohmic contacts was fabricated
on opposite sides of the hexagon shaped plate 111Note that due to growth
conditions the hexagon was slightly distorted.. This allowed us to measure the
photocurrents along the $y$-direction. Note that we use the coordinate system
$(x,y)$ where $x$ is parallel to one of three $C_{2}$ rotation axes. The
contacts were made of an alloy of tin, bismuth, and antimony with a low
melting temperature (Sn : Bi : Sb = 50 : 47 : 3) [12]. The magnetotransport
measurements were performed in the van der Pauw geometry on a sample cut from
the same tellurium crystal as the sample used for the photocurrent
measurements. The room temperature carrier density was $p=7\times 10^{16}$
cm-3 and the hole mobility $\mu=700~{}$ cm2/(V$\cdot$s). For the effective
mass $m\approx 0.2~{}m_{0}$ (see Refs. [40, 41]) this results in a momentum
relaxation time $\tau\approx 8\times 10^{-14}$ s. Note that the same
parameters were obtained for similar Te crystals used in Ref. [12].

Figure 2:  Azimuth angle dependence of the normalized photocurrent $J_{y}/P$
measured at zero magnetic field (red squares) and $B_{x}=\pm 1.7$ T (blue and
dark yellow squares). Note that the small polarization independent offset
$J_{0}\ll J_{y}$ is subtracted for clarity. The data are obtained at $f=31.2$
THz ($\lambda=9.6\,\,\mu$m). Solid lines are fits to
$(J_{y}-J_{0})/P\propto\sin 2\alpha$. The double arrows at the top illustrate
the state of the polarization for different values of the azimuth angle
$\alpha$. The inset shows the intensity dependence of the photocurrent. Solid
line is the linear fit.

To study the photocurrent in a wide frequency range we used two pulsed laser
systems: a TEA CO2 laser and an optically pumped molecular terahertz laser
[42]. The lasers operated at single frequencies in the range from $f\approx 1$
to 30 THz (corresponding photon energy range from $\hbar\omega=4.4$ to 132
meV, where $\omega=2\pi f$ is the angular frequency). Radiation with
frequencies of about 30 THz was obtained by a line-tunable TEA CO2 laser [43,
44]. We used four frequencies of the laser radiation from 28.3 THz (wavelength
$\lambda=10.6~{}\mu$m, $\hbar\omega=117$ meV) to 32.2 THz
($\lambda=9.3~{}\mu$m, $\hbar\omega=133.3$ meV) 222The number of studied
frequencies was defined by the laser tunability and availability of the
polarizers operating at specific frequencies.. The laser generated single
pulses with a duration of about 100 ns and a repetition rate of 1 Hz. The
radiation power on the sample surface $P$ was about 50 kW. For the low
frequencies (from 1 to 3.3 THz) we used a line-tunable pulsed molecular laser
with NH3 as active media [46, 47, 48]. The laser operated at $f=1.07$ THz
($\lambda=280~{}\mu$m, $\hbar\omega=4.4$ meV) and 3.3 THz
($\lambda=90.5~{}\mu$m, $\hbar\omega=13.7$ meV). The operation mode of the NH3
laser was similar to that of the TEA CO2 laser. The radiation power on the
sample surface was about 5 kW. The peak power of the radiation was monitored
by infrared and terahertz photon-drag detectors [42, 49], as well as by a
pyroelectric power meters. The beam positions and profiles were checked with
pyroelectric camera or thermally sensitive paper. The radiation was focused
onto spot sizes of about 0.5 to 3 mm diameter, depending on the radiation
frequency.

Photocurrents were measured at room temperature by applying polarized
radiation at normal incidence, see the inset in Fig. 2. In experiments with
linearly polarized radiation the in-plane radiation electric field vector
$\bm{E}$ was rotated counter-clockwise with respect to the $y$-axis. The
orientation of the vector $\bm{E}$ is defined by the azimuth angle $\alpha$
($\bm{E}\parallel y$ corresponds to $\alpha=0$) and was varied by rotation of
a $\lambda/2$-plate. To study photocurrents sensitive to the radiation
helicity we used $\lambda/4$ plates. By rotating the $\lambda/4$ plate we
varied the THz radiation helicity according to $P_{\rm
circ}\propto\sin{2\varphi}$ [50], where $\varphi$ is the angle between the
laser polarization plane and the optical axis of the plate. Note that for
$\varphi=0$ the radiation is linearly polarized along the $y$-direction.

The induced photocurrents were detected as a voltage drop across load
resistors $R_{L}=50$ Ohm. The signals were recorded using digital
oscilloscopes. In experiments on magneto-photocurrents an external in-plane
magnetic field $B$ up to 1.7 T was applied along $x$-direction using an
electromagnet.

##  III Experiment

We begin by describing the experimental results obtained under various
experimental conditions. The phenomenological theory and identification of the
photocurrent mechanisms are given in the next section.

First we present the photocurrent excited at frequencies about $30$ THz.
Figure 2 shows the dependence of the normalized photocurrent $J_{y}/P$ excited
by linearly polarized radiation as a function of the orientation of the
electric field vector. The data obtained at $f=31.2$ THz are shown for zero
magnetic field and for magnetic fields $B_{x}=\pm 1.7$ T. All three traces can
be well fitted by

$J_{y}=J_{1}(B_{x})\sin 2\alpha+J_{0}\,\,\,\,,$ (1)

where coefficients $J_{1}(B_{x})$ and $J_{0}$ are fit parameters 333While in
magnetic fields up to about $\pm~{}1.2$ T the current linearly depends on the
magnetic field at higher fields, it possibly tends to deviate from this
behavior. To justify this tendency experiments at higher magnetic fields are
needed, which was out of scope of the present work.. Note that the
polarization-independent offset $J_{0}$ is more than an order of magnitude
smaller than the parameter $J_{1}(B_{x})$ and will not be discussed below. The
magnetic field dependence of the coefficient $J_{1}(B_{x})$ measured for
$f=30.2$ and 28.3 THz are shown in Fig. 3(a). It demonstrates that the
photocurrent depends linearly on $B$ and has a substantial magnitude at zero
magnetic field. Solid lines in Fig. 3(a) show that the coefficient $J_{1}(B)$
within the error bars are well described by the function

$J_{1}=A^{\rm LPGE}P+D^{\rm MLPGE}PB_{x}\,\,\,\,,$ (2)

where $A^{\rm LPGE}$ and $D^{\rm MLPGE}$ are fitting parameters, which, as we
show below, describe the linear photogalvanic and linear magneto-photogalvanic
effects, respectively. The magnitudes of these coefficients measured for four
laser frequencies in the range between 28.3 and 32.2 THz are shown in the
inset in Fig. 3(a). The slope of the straight line in the inset in Fig. 3(a)
depends on the radiation frequency. Measuring the photocurrent as a function
of the radiation power $P$, we found that it scales linearly with $P$, i.e.
depends quadratically on the radiation electric field $E$, see the inset in
Fig. 2.

Figure 3: (a) Magnetic field dependencies of the amplitudes of linear
($J_{1}$, squares) and circular ($J_{\rm circ}$, circles) photocurrents
normalized on the radiation power $P$. The red and blue symbols correspond to
data obtained at $f=28.3$ THz ($\lambda=10.6\,\,\mu$m) and $f=31.2$ THz
($\lambda=9.6\,\,\mu$m), respectively. Inset: 1 – $A^{\rm LPGE}$ in nA/W, 2 –
$D^{\rm MLPGE}$ in nA/(WT), 3 – $D^{\rm MCPGE}$ in nA/(WT). (b) Frequency
dependencies of photocurrents $J_{1}/P$ and its components due to the photon
drag $J_{\rm PDE}$ and photogalvanic $J_{\rm PGE}$ effects measured at zero
magnetic field in the THz frequency range. Lines correspond to the spectral
dependence of the Drude absorption. (c) Magnetic field dependencies of the
amplitudes of linear (squares) and circular (circles) photocurrents $J/P$
measured at $f=3.3$ THz ($\lambda=90\,\,\mu$m).  Figure 4: Azimuth angle
dependence of the normalized photocurrent $(J-J_{0})/P$ measured at zero
magnetic field (red squares) and $B_{x}=\pm 1.7$ T (blue and olive squares).
The data are obtained applying radiation with $f=3.3$ THz
($\lambda=90\,\,\mu$m). Solid lines are fits after
$(J-J_{0})/P\propto\sin[2\alpha-\psi(B_{x})]$. The double arrows at the top
illustrate the state of the polarization for different values of the azimuth
angle $\alpha$. Inset shows magnetic field dependence of phase $\psi$
(circles). Solid line is fit after Eq. (13).

For radiation at lower frequencies ($f=1.07$ and 3.3 THz) the photocurrent
still varies sinusoidally with the double angle $\alpha$ but the experimental
traces become phase shifted, see Fig. 4. Now the data can be well fitted by

$J_{y}=J_{1}(B_{x})\sin[2\alpha-\psi(B_{x})]+J_{0}\,.$ (3)

The magnetic field dependencies of the amplitude $J_{1}(B_{x})$ and the phase
$\psi(B_{x})$ are shown in Fig. 3(c) and the inset in Fig. 4, respectively.
Figure 3(c) reveals that, alike at high frequencies, the coefficient
$J_{1}(B_{x})$ depends linearly on magnetic field $B_{x}$ and has a
substantial amplitude at zero magnetic field.

Figure 3(b) shows the frequency dependence of the photocurrent magnitude
$J_{1}(0)$ at zero magnetic field. For the low frequencies used in the
experiments, $f=1.07$ and 3.3 THz, which correspond to the photon energies of
4.4 and 13.7 meV, optical transitions in Te can only be due to Drude
absorption. It varies with the radiation frequency as $K_{D}\propto
1/(1+\omega^{2}\tau^{2})$, where $K_{D}$ is the absorption coefficient. The
solid line in Fig. 3(b) shows the frequency dependence of $J_{1}\propto K_{D}$
calculated for a momentum relaxation time $\tau=8\times 10^{-14}$ s determined
from magneto-transport measurements. It shows that Drude absorption describes
the data well at low frequencies. At high frequencies, however, the signal
magnitudes are by more than two orders of magnitude larger than would be
expected from the calculated dependence. This observation shows that at
frequencies of about 30 THz, i.e. photon energies of about 125 meV, the
photocurrent is not related to the indirect optical transitions in the valence
band, but stems from direct optical transitions between subbands of the
valence band. Below we discuss this in more detail. Comparing the slopes of
the magnetic field dependencies of $J_{1}(B_{x})$ measured at high frequencies
[28.3 and 31.2 THz, see Fig. 3(a)] with that measured at low frequencies [3.3
THz, see Fig. 3(c)], we obtain that they have comparable magnitudes. Using the
same arguments as above for the zero magnetic field photocurrent we conclude
that also the magneto-photocurrent at high frequencies $\sim 30$ THz stems
from direct intersubband transitions and not the Drude absorption.

Using circularly polarized radiation we also detected a magnetic field induced
circular photocurrent whose direction reverses when the radiation helicity
reverses. Figure 5 shows the dependencies of the photocurrent on the angle
$\varphi$ measured at $f=31.2$ THz. The curves can be fitted by

$J_{y}={1\over 2}J_{1}(B_{x})\sin 4\varphi+J_{\rm circ}(B_{x})\sin
2\varphi+J_{0}\,\,\,,$ (4)

where $J_{1}(B_{x})$ is the same as used in Eq. (1) and $J_{\rm circ}(B_{x})$
is a fitting parameter that corresponds to the magnitude of the circular
photocurrent, which is proportional to $P_{\rm circ}$ and reverses its sign by
changing polarization from $\sigma^{-}$ to $\sigma^{+}$ 444Note that in the
geometry applying $\lambda$-quarter plate the $\sin 4\varphi/2$ corresponds to
the Stokes parameter describing the degree of linear polarization and the
polarization ellipse orientation, which in experiments applying $\lambda$-half
plate is given by $\sin 2\alpha$, see below and Refs. [53, 50]. . The magnetic
field dependencies of the circular photocurrent, measured at two different
radiation frequencies, are shown in Figs. 3(a) and (c). These plots reveal
that the sign of

$J_{\rm circ}(B_{x})=D^{\rm MCPGE}PB_{x}$ (5)

changes by reversing the magnetic field direction; at $B_{x}=0$ this
contribution vanishes.

At first glance, one can attribute the magnetic field induced circular
photocurrent $J_{y}$ to the CPGE current excited along $c$-axis [10, 11],
which is turned towards the sample plane due to the Lorentz force. To examine
this possibility, we performed additional measurements. Using a tellurium
sample with a length of 25 mm and the cross-section of about 20 mm2 we
detected the longitudinal circular photocurrent only. Application of magnetic
field $B_{x}\leq 2$ T neither change the magnitude of the CPGE current nor
causes other photocurrents, see Fig. 6. This contradicts with the scenario
addressed at the beginning of the paragraph, because that should result in the
suppression of the longitudinal photocurrent due to the Lorentz force. This
proves that the measured current Eq. (5) comes from a so far unknown mechanism
of the circular magneto-photocurrent. The microscopic sense of the fitting
parameter $D^{\rm MCPGE}$ is discussed in Sec. VI.

Figure 5: Helicity dependences of the normalized photocurrent $(J-J_{0})/P$
obtained at zero magnetic field (red circles) and $B_{x}=\pm 1.7$ T (blue and
dark yellow circles). The data are measured at $f=31.2$ THz
($\lambda=9.6\,\,\mu$m). Solid lines are fits after Eq. (4). The ellipses on
top illustrate the polarization states at several angles $\varphi$. Vertical
blue and red arrows in the panels indicate left- and right-handed circularly
polarized radiation. Figure 6: Magnetic field dependence of the amplitude of
the longitudinal circular photocurrent normalized on the radiation power $P$.
The inset shows the experimental setup.

##  IV Phenomenological theory and identification of individual photocurrent
contributions

Tellurium is a chiral semiconductor with two enantiomorphs in nature,
dextrarotatory and levorotatory, which are mirror images of each other. They
can be visualized as two screws with opposite threads. We used levorotatory
tellurium, which was determined by measuring its natural optical activity
[12]. The point symmetry group of tellurium is $D_{3}$. This group has a
three-fold rotation axis $C_{3}$, the so-called $c$-axis. We denote this
direction as $z$. There are also three $C_{2}$ rotation axes in the
perpendicular plane $(xy)$.

A complete phenomenological analysis of the photocurrents excited by a light
propagating uniformly along the $c$-axis ($\bm{q}\parallel+z$) in a
homogeneous Te crystal is presented in Appendix A. In experiments the
polarization-dependent photocurrent is measured in the $y$ direction for
either zero magnetic field or a magnetic field applied along the $x$-axis
parallel to one of the $C_{2}$ axes. For these conditions the phenomenological
theory yields:

$\displaystyle j_{y}=$ $\displaystyle\quantity[-\chi\tilde{P}_{\rm
lin}+\Phi_{l}\tilde{P}_{\rm lin}B_{x}-\Phi_{c}P_{\rm circ}B_{x}]_{\rm
PGE}\absolutevalue{\bm{E}}^{2}+$ $\displaystyle\quantity[\tilde{T}P_{\rm
lin}q_{z}-S_{l}B_{x}P_{\rm lin}q_{z}]_{\rm
PDE}\absolutevalue{\bm{E}}^{2}\,\,\,,$ (6)

where $\bm{j}$ is the photocurrent density, $\bm{E}$ is the complex amplitude
of the radiation electric field

$\bm{E}(t)=\bm{E}\exp(-i\omega t)+\bm{E}^{*}\exp(i\omega t)$ (7)

and

$\displaystyle P_{\rm
lin}\absolutevalue{\bm{E}}^{2}=\absolutevalue{E_{x}}^{2}-\absolutevalue{E_{y}}^{2},\,\,$
(8) $\displaystyle\tilde{P}_{\rm
lin}\absolutevalue{\bm{E}}^{2}=E_{x}E_{y}^{*}+E_{x}^{*}E_{y},$ (9)
$\displaystyle P_{\rm
circ}\absolutevalue{\bm{E}}^{2}=i(E_{x}E_{y}^{*}-E_{x}^{*}E_{y})$ (10)

are the Stokes parameters, which describe the polarization of the radiation
[53]. The terms in the brackets $[...]_{\rm PGE}$ describe the photogalvanic
effects (PGE) caused by the chiral symmetry of bulk Te. The first term in this
bracket proportional to the parameter $\chi$ is the trigonal linear
photogalvanic effect (LPGE), the second and the third terms proportional to
$\Phi_{l}$ and $\Phi_{c}$ describe the linear and circular magneto-
photogalvanic currents (MPGE). The terms in the bracket $[...]_{\rm PDE}$
describe the photon drag effect (PDE) caused by the transfer of the linear
momentum of light to the charge carriers. The first term in this bracket
denoted by parameter $\tilde{T}$ is the trigonal PDE and the second one,
proportional to the parameter $S_{l}$, is the magneto-photon-drag effect
(MPDE).

The PGE current amplitudes obtained in experiments are related with the
theoretical values by $[A^{\rm LPGE},D^{\rm MLPGE},D^{\rm
MCPGE}]=2\pi\mathcal{S}/(cn_{\perp})[\chi,\Phi_{l},\Phi_{c}]$, where
$\mathcal{S}$ is the sample aspect ratio and $n_{\perp}$ is the refraction
index of Te for radiation propagating along the $c$-axis.

###  IV.1 Photocurrents in response to linearly polarized radiation

Equation (IV) shows that a proper choice of experimental setup together with
variation of the polarization state can be used to identify the contributions
of different photocurrent mechanisms. In our experiments linearly polarized
light propagates along $z\parallel\bm{c}$ and the orientation of the in-plane
electric field vector $\bm{E}$ is varied by a counterclockwise rotation of a
$\lambda/2$-plate. A zero azimuth angle, $\alpha=0$, corresponds to
$\bm{E}\parallel y$. Under these conditions the Stokes parameters are given by
[50]

$\displaystyle P_{\rm lin}=-\cos 2\alpha,\,\,\,\,\,\tilde{P}_{\rm lin}=-\sin
2\alpha.$ (11)

Consequently, Eq. (IV), which describes the total current in $y$-direction and
represents the sum of the linear photogalvanic and photon drag effects takes
the form

$\displaystyle j_{y}=$ $\displaystyle(\chi\sin 2\alpha-\tilde{T}q_{z}\cos
2\alpha)\absolutevalue{\bm{E}}^{2}+$ $\displaystyle(-\Phi_{l}\sin
2\alpha+S_{l}q_{z}\cos 2\alpha)B_{x}\absolutevalue{\bm{E}}^{2}\,\,.$ (12)

For convenience, we have grouped the zero magnetic field (first bracket) and
magnetic field induced (second bracket) PGE and PDE contributions.

Figure 2 shows that both zero-field (first bracket) and magneto-photocurrents
(second bracket) excited by radiation with $f\approx 30$ THz vary with the
rotation of the electric fied vector $\bm{E}$ sinusoidally with the double
angle $\alpha$. This fact shows that at $B_{x}=0$ the photocurrents are
governed by the trigonal LPGE ($j_{y}=\chi\sin
2\alpha\absolutevalue{\bm{E}}^{2}$) and the photon drag contribution with a
current proportional to $\cos 2\alpha$ is not detectable, see Fig. 2. In the
presence of a magnetic field $B_{x}$, this current is superimposed the linear
MPGE ($j_{y}=\Phi_{l}B_{x}\sin 2\alpha\absolutevalue{\bm{E}}^{2}$), see Figs.
2 and 3(a) (squares) 555Note that the trigonal photocurrents given by
coefficients $\chi$ depend on the polarization plane orientation as the 2nd
angular harmonics because projections of the current onto fixed axes are
measured. If one detects a direction of the photocurrent as a function of the
light polarization then it has a form of the 3rd angular harmonics..

At low frequencies, however, we found that the dependence of the zero-magnetic
field photocurrent on the angle $\alpha$ is phase shifted by the angle $\psi$,
see Fig. 4(a). This shows that at these frequencies the photocurrent is caused
by the superposition of the LPGE and linear PDE, which are proportional to the
sine and cosine of $2\alpha$, respectively. From the fit obtained for zero
magnetic field, where $J_{y}\propto\sin(2\alpha-46^{0})$, see Fig. 4(a), we
can conclude that $\chi/q_{z}\tilde{T}=\tan 46^{0}\approx 1$, hence
$\chi\approx q_{z}\tilde{T}$.

From measurement of the azimuthal dependence of the photocurrent for different
magnetic field strengths we found that the phase shift $\psi(B_{x})$ depends
on the magnitude and sign of $B_{x}$, see inset in Fig. 4. Considering that,
as shown above, $\chi\approx q_{z}\tilde{T}$, we obtain the azimuthal angle
dependence of the magnetic field induced photocurrent in the form

$\displaystyle
j_{y}\propto\sin(2\alpha-\arctan\frac{1-B_{x}S_{l}/\tilde{T}}{1-B_{x}\Phi_{l}/\chi}\,)\,\,.$
(13)

The dependence of the phase $\psi$ on the in-plane magnetic field
$\psi(B_{x})$, see inset in Fig. 4, allows us to extract the ratio of the
parameters defining the linear MPGE and the trigonal PGE, $\Phi_{l}/\chi=-0.5$
T-1, as well as the parameters defining the magneto-photon-drag effect and the
trigonal photon drag effect, $S_{l}/\tilde{T}=-0.2$ T-1.

###  IV.2 Photocurrents in response to elliptically polarized radiation

Experiments show that the circular photocurrent, whose direction is reversed
by changing from $\sigma^{-}$ to $\sigma^{+}$ circularly polarized radiation,
can only be observed in the presence of an external magnetic field, see Fig. 5
and Figs. 3(a) and (c). This is fully consistent with the phenomenological
theory which yields a magneto-induced circular photogalvanic current (MCPGE)

$j_{y}^{\rm MCPGE}=-\Phi_{c}B_{x}P_{\rm circ}\absolutevalue{\bm{E}}^{2}\,\,.$
(14)

In experiments, the polarization state of the radiation is varied by rotating
the $\lambda/4$-plate by an angle $\varphi$ with respect to the $y$-direction,
and the Stokes parameters are given by [50]

$\displaystyle P_{\rm lin}=-(\cos 4\varphi+1)/2\,\,\,\,\,$ (15)
$\displaystyle\tilde{P}_{\rm lin}=-\sin 4\varphi/2$ (16) $\displaystyle P_{\rm
circ}=\sin 2\varphi\,\,.$ (17)

Consequently, the photocurrent in $y$-direction is given by

$\displaystyle j_{y}=$ $\displaystyle[\chi\sin 4\varphi/2-\tilde{T}q_{z}(\cos
4\varphi+1)/2]\absolutevalue{\bm{E}}^{2}$ $\displaystyle-[\Phi_{l}\sin
4\varphi/2-S_{l}q_{z}(\cos 4\varphi+1)/2]B_{x}\absolutevalue{\bm{E}}^{2}$
$\displaystyle-\Phi_{c}B_{x}\sin 2\varphi\absolutevalue{\bm{E}}^{2}\,\,.$ (18)

Here, the last term describes the circular MPGE. All other terms are due to
the photogalvanic and photon drag effects discussed in the previous sections:
they are characterized by the same values of the parameters
$\chi,\tilde{T},\Phi_{l},S_{l}$ as detected in experiments with linearly
polarized radiation and the polarization dependence is simply modified
according to the modification of the corresponding Stokes parameters. The fit
of the experimental data with this function describes the experimental data
well and is shown in Fig. 5. The magnetic field dependencies of the MCPGE
current, extracted from the experimental helicity dependencies, or, in some
measurements, as the half difference between the photocurrent magnitudes in
response to $\sigma^{+}$\- and $\sigma^{-}$\- circularly polarized radiation,
are shown in Figs. 3(a) and (c).

##  V Microscopic theory

Figure 7: Band structure of the valence band of tellurium. At
$\hbar\omega>2\Delta_{2}$, direct intersubband optical transitions are
allowed.

Figure 7 sketches the valence band structure of bulk Te crystals. For the
microscopic picture of the observed photocurrents we need to identify optical
transitions responsible for their formation. As discussed in Sec. III, the
analysis of the photocurrent frequency dependence, see Fig. 3(b), shows that
the photocurrent excited by infrared radiation with photon energies of the
order of 130 meV is driven by the direct intersubband transitions, see red
downward arrows in Fig. 7. Note, that the photon energies are too low to
excite interband transitions ($\hbar\omega\ll E_{g}=335$ meV). For THz
radiation with photon energies in the order of 10 meV, i.e. much smaller than
the energy separation of the subbands in the valence band $2\Delta_{2}\approx
100-125$ meV [55, 56, 57, 58, 59], the photocurrents are caused by the
indirect Drude-like optical transitions.

We develop a theory of photocurrents in tellurium that are induced by both
intersubband optical transitions in the valence band, Fig. 7, and intrasubband
Drude-like transitions. We derive expressions for the trigonal LPGE current
and the photocurrents in the presence of a magnetic field.

In the basis of $\pm 3/2$ states, the valence-band Hamiltonian for the $H$
point of the Brillouin zone has the following form

$\mathcal{H}=\Delta_{2}\sigma_{x}+\beta
k_{z}\sigma_{z}{+\mathcal{A}_{1}k_{z}^{2}+\mathcal{A}_{2}k_{\perp}^{2}}.$ (19)

Here $\Delta_{2}$ is half the gap between the valence subbands at $k=0$, see
Fig. 7, and $\beta$ is a constant that is the same for both $H$ and
$H^{\prime}$ valleys but of opposite sign in levorotatory and dextrarotatory
tellurium. The eigenstates of the Hamiltonian $\mathcal{H}$ have the camel-
back dispersion

$E_{1,2}=\mathcal{A}_{1}k_{z}^{2}+\mathcal{A}_{2}k_{\perp}^{2}\mp\sqrt{\Delta_{2}^{2}+(\beta
k_{z})^{2}}$ (20)

and the envelopes

$\psi_{1,2}={1\over\sqrt{2}}\quantity(\sqrt{1\pm\eta},\pm\sqrt{1\mp\eta}),\quad\eta={\beta
k_{z}\over\sqrt{\Delta_{2}^{2}+(\beta k_{z})^{2}}}.$ (21)

The direct intersubband optical transition matrix element is at $k_{\perp}\neq
0$ nonzero only [57]

$V_{21}=i{eE\over\hbar\omega}{\Delta_{2}\over
E_{g}^{2}}\absolutevalue{L}^{2}(k_{+}e_{-}-k_{-}e_{+})\,.$ (22)

Here $k_{\pm}=k_{x}\pm ik_{y}$, $E_{g}$ is the energy gap between the
conduction band and valence band and $L$ is the interband momentum matrix
element 666In Eq. (22) the factor $\absolutevalue{L}^{2}/E_{g}$ absent in Eq.
(25) of Ref. [57] is restored.. The matrix element $V_{21}$ is quadratic in
$L$ because the intersubband transitions in the valence band occur via virtual
states in the conduction band. The absorption coefficient for intersubband
transitions $K(\omega)$ calculated according to Fermi’s Golden rule with the
matrix element (22) is given in Appendix B.

###  V.1 Trigonal LPGE current at intersubband transitions

First we derive the trigonal LPGE which is responsible for the photocurrent
$J_{1}(B=0)$ excited by infrared radiation. The Hamiltonian (19) describes the
uniaxial model of tellurium. In order to account for the trigonality we add an
additional term to the valence-band Hamiltonian which has the following form
in the basis of the $\pm 3/2$ states [57]:

$\delta\mathcal{H}=i\gamma^{\prime}(k_{+}^{3}-k_{-}^{3})\sigma_{z}.$ (23)

Here $\gamma^{\prime}$ is a real constant equal at both $H$ and $H^{\prime}$
Brillouin zone points. This correction results in an additional term in the
intersubband matrix element $\delta
V_{21}(\bm{k})=(ie/\hbar\omega)\bm{E}\cdot\bm{\nabla}_{\bm{k}}\delta\mathcal{H}$.
This results in two competitive contributions to the LPGE: the shift current
and the ballistic or injection current. The latter occurs when the
interference of light absorption with disorder or phonon scattering is taken
into account [61]. The shift current is due to holes with a wavevector
$\bm{k}$ undergo a spatial shift $\bm{R}_{21}(\bm{k})$ during the light
absorption process. The accumulation of the shifts results in a contribution
to the steady-state photocurrent. In general, the ballistic and shift
photocurrents have the same order of magnitude and equally dependent on the
system parameters. Here we estimate the LPGE current by calculating the shift
contribution.

The shift value for direct optical transitions is given by [62, 63]

$\bm{R}_{21}(\bm{k})=-\bm{\nabla}_{\bm{k}}{\rm arg}(V_{21}+\delta
V_{21})+\bm{\Omega}_{2}(\bm{k})-\bm{\Omega}_{1}(\bm{k}),$ (24)

where ${\rm arg}$ stands for the complex argument, and the Berry connections
in the subbands are
$\bm{\Omega}_{1,2}=i\expectationvalue{\bm{\nabla}_{\bm{k}}}{\psi_{1,2}}$. The
shift photocurrent density reads

$\displaystyle\bm{j}=2e$ (25)
$\displaystyle\times\sum_{\bm{k}}{2\pi\over\hbar}\absolutevalue{V_{21}}^{2}\bm{R}_{21}\delta\quantity[2\sqrt{\Delta_{2}^{2}+(\beta
k_{z})^{2}}-\hbar\omega](f_{1}-f_{2}).$

Here the factor 2 accounts for the two valleys of tellurium, and $f_{1,2}$ are
occupancies of the initial and final states.

Calculating the shift photocurrent for intersubband transitions (ib) we obtain
in accordance with the phenomenological Eq. (IV)

$\displaystyle j_{x}=\chi^{\rm ib}{P}_{\rm lin}\absolutevalue{\bm{E}}^{2},$
(26) $\displaystyle j_{y}=-\chi^{\rm ib}\tilde{P}_{\rm
lin}\absolutevalue{\bm{E}}^{2},$ (27)

where

$\chi^{\rm
ib}=\gamma^{\prime}K(\omega){12ecE_{g}^{2}\over\pi\absolutevalue{L}^{2}(\hbar\omega)^{2}}$
(28)

with $K(\omega)$ being the absorption coefficient for intersubband
transitions, see Appendix B.

###  V.2 Linear and circular MPGE currents at intersubband transitions

Application of an external magnetic field results in an additional
photocurrent superimposed with the trigonal one. This is caused by the change
in the probability of optical transitions for positive and negative
wavevectors resulting in an imbalance of the population of states involved in
radiation absorption. In a magnetic field, the following correction to the
Hamiltonian appears:

$\delta\mathcal{H}_{B}=\gamma_{B}\sigma_{z}(k_{+}^{2}k_{-}B_{-}-k_{-}^{2}k_{+}B_{+}).$
(29)

Here, $\gamma_{B}$ is a real constant that is the same for both $H$ and
$H^{\prime}$ Brillouin-zone points. The constant $\gamma_{B}$ can be obtained
in the 5th order of the $\bm{k}\cdot\bm{p}$ perturbation theory. Accounting
for this term results in additional, magnetic field-dependent spatial shifts
of the holes under elliptical polarization. Calculation of the shift
$\bm{R}_{21}(\bm{k},\bm{B})$ by Eq. (24) with $\delta
V_{21}=(ie/\hbar\omega)\bm{E}\cdot\bm{\nabla}_{\bm{k}}\delta\mathcal{H}_{B}$
and then the photocurrent by Eq. (25) we obtain (at $\bm{q}\parallel+z$) in
accordance with Eqs. (IV) the shift MCPGE current

$\bm{j}=\Phi_{c}^{\rm ib}\bm{B}\times\hat{\bm{z}}P_{\rm
circ}\absolutevalue{\bm{E}}^{2},$ (30)

where the shift contribution to $\Phi_{c}^{\rm ib}$ is given by

$\Phi_{c}^{\rm ib}=-{\gamma_{B}\over 4\gamma^{\prime}}\chi^{\rm ib}$ (31)

with $\chi^{\rm ib}$ the zero-field trigonal LPGE constant for intersubband
transitions, Eq. (28).

Accounting for the correction (29) also changes the selection rules for the
intersubband transitions under linearly polarized or at unpolarized
excitation. The correction to the matrix element is given by

$\displaystyle\delta V_{21}^{\rm lin}=$
$\displaystyle-2{eE\over\hbar\omega}\sqrt{1-\eta^{2}}\gamma_{B}k_{\perp}^{2}B_{\perp}$
(32)
$\displaystyle\times\quantity[2\sin(\alpha-\varphi_{\bm{B}})+\sin(2\varphi_{\bm{k}}-\alpha-\varphi_{\bm{B}})],$

where $\varphi_{\bm{k}}$, $\varphi_{\bm{B}}$, and $\alpha$ are the azimuthal
angles of $\bm{k}_{\perp}$, $\bm{B}_{\perp}$ and $\bm{E}_{\perp}$,
respectively. As a result, the squared matrix element contains a part
$\absolutevalue{V_{21}+\delta V_{21}^{\rm lin}}^{2}\propto
2{\real}(V_{21}^{*}\delta V_{21}^{\rm lin})$ asymmetrical in $\bm{k}$. The
corresponding asymmetrical part of the intersubband transition probability
$W_{21}^{(as)}$ results in a ballistic photocurrent. Its density is calculated
as

$\bm{j}_{\perp}=2e\sum_{\bm{k}}\bm{v}_{\perp}W_{21}^{(as)}(f_{1}-f_{2})(\tau_{2}-\tau_{1}).$
(33)

Here $\tau_{1,2}$ are the momentum relaxation times in the subbands labeled
“1” and “2”, and $\bm{v}_{\perp}=2\mathcal{A}_{2}\bm{k}_{\perp}/\hbar$ is the
hole velocity equal in both subbands. Calculation yields in accordance with
Eq. (IV) and Appendix A

$j_{x,y}=\quantity[\Phi_{l}^{\rm ib}(\pm{P}_{\rm lin}B_{x,y}+\tilde{P}_{\rm
lin}B_{y,x})+{\Lambda}^{\rm ib}B_{x,y}]\absolutevalue{\bm{E}}^{2},$ (34)

where the MLPGE and the polarization-independent MPGE constants $\Phi_{l}^{\rm
ib}$ and ${\Lambda}^{\rm ib}$ are given by

$\Phi_{l}^{\rm ib}=-2{\Lambda}^{\rm ib}={8k_{\rm B}T(\tau_{1}-\tau_{2})\over
3\hbar}\Phi_{c}^{\rm ib}.$ (35)

Here $T$ is the temperature, $k_{\rm B}$ is the Boltzmann constant, and
$\Phi_{c}^{\rm ib}$ is given by Eq. (31).

###  V.3 Trigonal LPGE current at intraband transitions

Now we turn to the photocurrent generated by linearly polarized THz radiation.
For Drude-like intraband optical transitions inside the ground valence subband
which are relevant for THz frequencies, the photocurrent is derived from the
Boltzmann kinetic equation. In this approach, the low symmetry of tellurium is
taken into account in the collision integral. This means that the photocurrent
is generated due to the action of polarized radiation and asymmetric hole
scattering by disorder or phonons.

In order to obtain the asymmetrical scattering probability, we consider the
following terms of different parity in the interband Hamiltonian:

$\matrixelement{\pm 1/2}{\mathcal{H}}{\pm 3/2}=Lk_{\pm}+Qk_{\mp}^{2}.$ (36)

Here $\ket{\pm 1/2}$ are the conduction-band states, $L$ is the interband
matrix element giving rise to intersubband transitions, Eq. (22), and $Q$
fixes the trigonal symmetry of tellurium. Accordingly, the wavefunction
envelope in the ground valence subband has the form

$\psi_{\bm{k}}=\psi_{0}+{(Lk_{+}+Qk_{-}^{2})\ket{1/2}+(Lk_{-}+Qk_{+}^{2})\ket{-1/2}\over
E_{g}\sqrt{2}},$ (37)

where $\psi_{0}$ is the wavefunction caclulated without mixing with the
conduction band. Then, the matrix element of scattering by the disorder
potential
$U_{\bm{k}^{\prime}\bm{k}}=U_{0}\innerproduct{\psi_{\bm{k}^{\prime}}}{\psi_{\bm{k}}}$
gets an asymmetric part:

$\displaystyle U_{\bm{k}^{\prime}\bm{k}}=U_{0}\biggl{\\{}1+i{{\rm
Im}(LQ^{*})\over E_{g}^{2}}$ (38)
$\displaystyle\times\quantity[k_{\perp}{k_{\perp}^{\prime}}^{2}\cos(\varphi_{\bm{k}}+2\varphi_{\bm{k}^{\prime}})-k_{\perp}^{2}{k_{\perp}^{\prime}}\cos(\varphi_{\bm{k}^{\prime}}+2\varphi_{\bm{k}})]\biggr{\\}}.$

Here, $U_{0}$ is the Fourier image of the scattering potential which is
assumed to be independent of $\bm{k}$ and $\bm{k}^{\prime}$ corresponding to
the scattering by short-range elastic impurities, and the notation ‘$\perp$’
denotes the projection onto the $(xy)$ plane. This form of
$U_{\bm{k}^{\prime}\bm{k}}$ allows us to obtain the asymmetric (skew)
scattering probability
$W^{a}_{\bm{k}^{\prime}\bm{k}}=-W^{a}_{\bm{k}\bm{k}^{\prime}}$. This can be
achieved using the next to Born approximation, which yields [64, 65, 29, 66]:

$\displaystyle W^{a}_{\bm{k}^{\prime}\bm{k}}=$
$\displaystyle{(2\pi)^{2}\over\hbar}\mathcal{N}\delta(\varepsilon_{\bm{k}}-\varepsilon_{\bm{k}^{\prime}})$
(39) $\displaystyle\times\sum_{\bm{p}}{\rm
Im}(U_{\bm{k}^{\prime}\bm{p}}U_{\bm{p}\bm{k}}U_{\bm{k}\bm{k}^{\prime}})\delta(\varepsilon_{\bm{k}}-\varepsilon_{\bm{p}}),$

where $\mathcal{N}$ is the concentration of scatterers and
$\varepsilon_{\bm{k}}\equiv E_{1}(\bm{k})$ is the hole dispersion in the
ground valence subband. In the following we assume an isotropic parabolic
dispersion $\varepsilon_{k}=\hbar^{2}k^{2}/(2m)$. The calculation yields the
asymmetrical scattering probability in the form

$\displaystyle W^{a}_{\bm{k}^{\prime}\bm{k}}={2\pi
U_{0}\over\tau}\delta(\varepsilon_{k}-\varepsilon_{k^{\prime}}){{\rm
Im}(LQ^{*})\over E_{g}^{2}}$ (40)
$\displaystyle\times[k_{\perp}{k^{\prime}_{\perp}}^{2}\cos(\varphi_{\bm{k}}+2\varphi_{\bm{k}^{\prime}})-k^{\prime}_{\perp}k_{\perp}^{2}\cos(\varphi_{\bm{k}^{\prime}}+2\varphi_{\bm{k}})],$

where $\tau$ is the relaxation time introduced by
$1/\tau(\varepsilon_{k})=(2\pi/\hbar)\mathcal{N}U_{0}^{2}g(\varepsilon_{k})$
where $g(\varepsilon)$ is the density of states. Note that the value of ${\rm
Im}(LQ^{*})$ is the same in the $H$ and $H^{\prime}$ valleys.

The trigonal PGE constant $\chi^{D}$ in 3D tellurium which describes the
trigonal LPGE current for Drude-like absorption

$\displaystyle j_{x}=\chi^{D}{P}_{\rm lin}\absolutevalue{\bm{E}}^{2}$ (41)
$\displaystyle j_{y}=-\chi^{D}\tilde{P}_{\rm lin}\absolutevalue{\bm{E}}^{2},$
(42)

is calculated analogously to the 2D case in Refs. [65, 29, 66]:

$\chi^{D}=2e^{3}\sum_{\bm{k}}\xi\tau\quantity[\quantity({\tau\over
1+\omega^{2}\tau^{2}}f_{0}^{\prime})^{\prime}+\tau^{\prime}f_{0}^{\prime}{1-\omega^{2}\tau^{2}\over(1+\omega^{2}\tau^{2})^{2}}].$
(43)

Here, $f_{0}$ is the equilibrium distribution function, prime means
differentiation over energy $\varepsilon_{k}$, and the asymmetry parameter is
introduced according to

$\xi(\varepsilon_{k})=\tau\expectationvalue{\sum_{\bm{k}^{\prime}}W^{a}_{\bm{k}\bm{k}^{\prime}}{v_{x}(\bm{k})\quantity[v_{x}^{2}(\bm{k}^{\prime})-v_{y}^{2}(\bm{k}^{\prime})]}},$
(44)

where angular brackets indicate averaging over directions of $\bm{k}$ at a
fixed energy $\varepsilon_{k}$. The calculation gives

$\xi(\varepsilon_{k})=-{16\pi{\rm Im}(LQ^{*})\over
15\hbar^{3}E_{g}^{2}}U_{0}g(\varepsilon_{k})\varepsilon_{k}^{3}.$ (45)

Integrating the first term in Eq. (43) by parts we get

$\chi^{D}=-e^{3}\sum_{\bm{k}}{\xi\tau^{2}{/\varepsilon_{k}}\over
1+\omega^{2}\tau^{2}}f_{0}^{\prime}\quantity(7+{1-\omega^{2}\tau^{2}\over
1+\omega^{2}\tau^{2}}),$ (46)

where we used the energy dependencies
$g(\varepsilon_{k})\propto\sqrt{\varepsilon_{k}}$, $\tau\propto
1/\sqrt{\varepsilon_{k}}$, and $\xi\propto\varepsilon_{k}^{7/2}$.

Equation (46) is valid for any temperature and any relationship between
frequency and relaxation rate. For Boltzmann statistics and high frequency
${\omega\tau\gg 1}$ we get

$\chi{{}^{D}}=-{{96}pe^{3}k_{\rm B}T{\rm Im}(LQ^{*})\over
5\sqrt{\pi}E_{g}^{2}\hbar^{3}\omega^{2}}S_{\rm nB}(T).$ (47)

Here $p$ is the hole concentration, and we introduced the non-Born
dimensionless parameter $S_{\rm nB}(T)=2\pi U_{0}g(k_{\rm B}T)$. The
temperature dependence is ${\chi^{{D}}\propto T^{{3}/2}}$.

###  V.4 Linear and circular MPGE current at intraband transitions

The experiment shows that also at THz frequencies the application of an
external magnetic field $B_{x}$ results in magnetic field induced linear and
circular photocurrents, see Figs. 4, 5, and 3(b). The MPGE current caused by
the Drude-like intraband optical transitions is calculated analogously to Eqs.
(41)–(43) but taking into account the magnetic field in the interband mixing.
The magnetic field changes the interband matrix elements Eq. (36) according
to:

$\matrixelement{\pm 1/2}{\mathcal{H}}{\pm 3/2}=Lk_{\pm}+MB_{\pm}.$ (48)

We see that the forbidden interband transitions are allowed by the magnetic
field. Using the same approach as in Eq. (37) we obtain the the
$\bm{B}_{\perp}$-dependence of the disorder scattering matrix element
$U_{\bm{k}^{\prime}\bm{k}}$. It contains a part responsible for MPGE, given by

$U_{\bm{k}^{\prime}\bm{k}}=U_{0}\quantity[1+{{\rm Re}(LM^{*})\over
E_{g}^{2}}\bm{B}_{\perp}\cdot(\bm{k}_{\perp}+\bm{k}^{\prime}_{\perp})].$ (49)

This part of the scattering matrix element leads to the gyrotropic terms in
the scattering probability already in the Born approximation:

$W_{\bm{k}^{\prime}\bm{k}}=W_{\bm{k}^{\prime}\bm{k}}^{(0)}\quantity[1+2{{\rm
Re}(LM^{*})\over
E_{g}^{2}}\bm{B}_{\perp}\cdot(\bm{k}_{\perp}+\bm{k}^{\prime}_{\perp})],$ (50)

where $W_{\bm{k}^{\prime}\bm{k}}^{(0)}$ is the zero-field symmetrical part.
Note that the value of ${\rm Re}(LM^{*})$ is the same in the $H$ and
$H^{\prime}$ valleys.

To calculate the MPGE current, one has to iterate the kinetic equation for the
hole distribution function $f_{\bm{k}}$

$\partialderivative{f_{\bm{k}}}{t}+{e\over\hbar}\bm{E}(t)\cdot\partialderivative{f_{\bm{k}}}{\bm{k}}=\sum_{\bm{k}^{\prime}}(W_{\bm{k}\bm{k}^{\prime}}f_{\bm{k}^{\prime}}-W_{\bm{k}^{\prime}\bm{k}}f_{\bm{k}})$
(51)

with $\bm{E}(t)$ given by Eq. (7), in the small parameters $\bm{E}$,
$\bm{E}^{*}$ and $\bm{B}$. At the first step, we account for $\bm{E}$ and
obtain the correction to the distribution function in the form

$f^{(E)}_{\bm{k}}=-e\tau_{\omega}f_{0}^{\prime}\bm{E}\cdot\bm{v}_{\bm{k}}.$
(52)

In the following we use the notation $\tau_{\omega}=\tau/(1-i\omega\tau)$.

Then there are two ways to get the current carrying distribution. One is an
account for $\bm{E}^{*}$ at the next step and get the correction
$f^{(EE)}\propto\absolutevalue{\bm{E}}^{2}$, and account for the $\bm{kB}$
terms in the end to obtain the correction
$f^{(EEB)}\propto\absolutevalue{\bm{E}}^{2}B_{\perp}$. In the 2nd step we get
the correction to the distribution function describing an alignment of
electron momenta:

$f^{(EE)}_{\bm{k}}=e^{2}\absolutevalue{\bm{E}}^{2}\tau v_{\perp}^{2}{\rm
Re}(\tau_{\omega}f_{0}^{\prime})^{\prime}(P_{\rm
lin}\cos{2\varphi_{\bm{k}}}+\tilde{P}_{\rm lin}\sin{2\varphi_{\bm{k}}}).$ (53)

Then we include the magnetic field and get the correction $f^{(EEB)}_{\bm{k}}$
from the equation

${f^{(EEB)}_{\bm{k}}\over\tau}+\sum_{\bm{k}^{\prime}}W_{\bm{k}^{\prime}\bm{k}}\quantity(f^{(EE)}_{\bm{k}}-f^{(EE)}_{\bm{k}^{\prime}})=0,$
(54)

which yields

$f^{(EEB)}_{\bm{k}}=-2{{\rm Re}(LM^{*})\over
E_{g}^{2}}(\bm{B}_{\perp}\cdot\bm{k}_{\perp})f^{(EE)}_{\bm{k}}.$ (55)

It contributes to the MPGE current which is calculated as follows:

$\bm{j}=2e\sum_{\bm{k}}\bm{v}_{\bm{k}}f^{(EEB)}_{\bm{k}}.$ (56)

Substituting $f^{(EEB)}_{\bm{k}}$ we obtain in accordance with the
phenomenological Eq. (IV)

$j_{x,y}=\Phi_{l}^{D}(\pm{P}_{\rm lin}B_{x,y}+\tilde{P}_{\rm
lin}B_{y,x})\absolutevalue{\bm{E}}^{2},$ (57)

where the MLPGE constant $\Phi_{l}^{D}$ is given by

$\Phi_{l}^{D}=-e^{3}{8{\rm Re}(LM^{*})\over 5m\hbar
E_{g}^{2}}\sum_{\bm{k}}\varepsilon_{k}^{2}\tau\quantity({\tau\over
1+\omega^{2}\tau^{2}}f_{0}^{\prime})^{\prime}.$ (58)

Now we calculate an additional contribution to the distribution function that
takes into account the $\bm{kB}$ terms in the 2nd step. In doing so, we find a
time-dependent correction to the distribution function
$f^{(EB)}_{\bm{k}}\propto EB$ that satisfies the equation

${f^{(EB)}_{\bm{k}}\over\tau_{\omega}}+\sum_{\bm{k}^{\prime}}W_{\bm{k}^{\prime}\bm{k}}\quantity(f^{(E)}_{\bm{k}}-f^{(E)}_{\bm{k}^{\prime}})=0.$
(59)

The solution is given by

$\displaystyle f^{(EB)}_{\bm{k}}=2{{\rm Re}(LM^{*})\over E_{g}^{2}}$ (60)
$\displaystyle\times{\tau_{\omega}^{2}\over\tau}ef_{0}^{\prime}\quantity[(\bm{B}_{\perp}\cdot\bm{k}_{\perp})(\bm{E}\cdot\bm{v}_{\bm{k}})-(\bm{B}_{\perp}\cdot\bm{E}){k_{\perp}v_{\perp}\over
2}].$

Then we find the quadratic in $E$ and linear in $B$ correction $f^{(EBE)}$
which satisfies the following equation:

${e\over\hbar}\bm{E}^{*}\cdot\partialderivative{f^{(EB)}_{\bm{k}}}{\bm{k}}+c.c.=-{f^{(EBE)}_{\bm{k}}\over\tau}.$
(61)

Substituting $f^{(EBE)}_{\bm{k}}$ to Eq. (56) (instead of
$f^{(EEB)}_{\bm{k}}$) and integrating by parts we get a contribution to the
MPGE current in the form

$\delta\bm{j}=2{e^{2}\over\hbar}\bm{E}^{*}\cdot\sum_{\bm{k}}f^{(EB)}_{\bm{k}}\partialderivative{(\tau\bm{v}_{\bm{k}})}{\bm{k}}+c.c.$
(62)

Since $f^{(EB)}_{\bm{k}}$ is zero on average, we differentiate here $\tau$
only. This yields

$\delta\bm{j}=2e^{2}\sum_{\bm{k}}\tau^{\prime}\quantity[\bm{v}(\bm{E}^{*}\cdot\bm{v})-{v_{\perp}^{2}\over
2}\bm{E}^{*}]f^{(EB)}_{\bm{k}}+c.c.,$ (63)

where
$\tau^{\prime}=\differential\tau/\differential\varepsilon_{k}{=-\tau/(2\varepsilon_{k})}$.
Substituting $f^{(EB)}_{\bm{k}}$ we finally obtain in accordance with Eq. (IV)

$\delta j_{x,y}=\quantity(\pm\Phi_{c}^{D}{P}_{\rm
circ}B_{y,x}+{\Lambda}^{D}B_{x,y})\absolutevalue{\bm{E}}^{2},$ (64)

where the MCPGE and the polarization-independent MPGE constants $\Phi_{c}^{D}$
and ${\Lambda}^{D}$ for Drude-like absorption are given by

$\Phi_{c}^{D}=e^{3}{16\omega{\rm Re}(LM^{*})\over 5m\hbar
E_{g}^{2}}\sum_{\bm{k}}{\tau^{2}\tau^{\prime}\over(1+\omega^{2}\tau^{2})^{2}}\varepsilon_{k}^{2}f_{0}^{\prime},$
(65) ${\Lambda}^{D}=e^{3}{8{\rm Re}(LM^{*})\over 5m\hbar
E_{g}^{2}}\sum_{\bm{k}}{\tau\tau^{\prime}(1-\omega^{2}\tau^{2})\over(1+\omega^{2}\tau^{2})^{2}}\varepsilon_{k}^{2}f_{0}^{\prime}.$
(66)

For Boltzmann statistics, short-range scattering potential and high frequency
$\omega\tau\gg 1$ we get

$\Phi_{l}^{D}=-{24pe^{3}{\rm Re}(LM^{*})\over
5m\hbar\omega^{2}E_{g}^{2}},\quad\Phi_{c}^{D}=-{4\over
3\sqrt{\pi}\omega\tau_{T}}\Phi_{l}^{D},$ (67)

and ${\Lambda}^{D}=-{\Phi_{l}^{D}/4}$, where $\tau_{T}=\tau(k_{\rm B}T)$.

###  V.5 Chiral PDE in tellurium

We consider linearly-polarized light propagation along $z\parallel\bm{c}$. In
this setup, the following PDE current is allowed by symmetry in Te, see Eq.
(IV) and Appendix A:

$j_{x}=\tilde{T}q_{z}\absolutevalue{\bm{E}}^{2}\tilde{P}_{\rm lin},\qquad
j_{y}=\tilde{T}q_{z}\absolutevalue{\bm{E}}^{2}{P}_{\rm lin}.$ (68)

The constant $\tilde{T}$ is chiral, i.e. it has opposite sign in two
enantiomorphic modifications of tellurium. This PDE current is different from
that in $C_{3v}$ symmetric systems where $q_{z}$ is invariant, and we have
$j_{x}\propto q_{z}{P}_{\rm lin}$, $j_{y}\propto q_{z}\tilde{P}_{\rm lin}$.
This results in different dependencies on the light polarization in Te and in
previously studied $C_{3v}$ systems such as surface states in topological
insulators [65, 67, 31].

We calculate the constant $\tilde{T}$ for intraband absorption in the ground
valence subband of Te assuming an isotropic energy dispersion for holes
$\varepsilon_{k}=\hbar^{2}k^{2}/(2m)$. To account for $D_{3}$ symmetry for
intraband transitions, one has to include asymmetric hole scattering in the
kinetic theory, which can be described as scattering by wedge-shape defects.
We denote the corresponding part of the scattering probability as
$W^{w}_{\bm{k}^{\prime}\bm{k}}$. It is asymmetric with respect to an exchange
of the initial and final wavevectors:
$W^{w}_{\bm{k}^{\prime}\bm{k}}=-W^{w}_{\bm{k}\bm{k}^{\prime}}$ [64]. This
probability is similar to $W^{a}_{\bm{k}\bm{k}^{\prime}}$, Eq. (39) but
describes skew scattering of holes with a non-zero $z$-component of the
velocity.

In order to calculate the PDE current, one has to take into account either the
finite wavevector of light $q_{z}$ or the magnetic field $\tilde{\bm{B}}\perp
z$ of the radiation. In the first approach, the PDE current is the sum of
various contributions obtained by iteration of the Boltzmann kinetic equation
in the small parameters $E$, $q_{z}$, $E^{*}$ and $W^{w}$, which we also
denote as ‘$w$’. The analysis shows that the following three iteration
sequences can contribute to the PDE current, $EqEw$, $EwqE$, and $EqwE$. In
the second approach, there are two contributions, $Ew\tilde{B}$ and
$E\tilde{B}w$.

Qualitatively, the PDE current is formed by skew scattering from wedges of
nonequilibrium holes, which have an anisotropic momentum distribution due to
the action of the spatially-dispersive polarized radiation.

We consider a geometry with $\bm{q}\parallel z$, when the electric and
magnetic fields of the radiation, $\bm{E}_{\perp}$ and
$\tilde{\bm{B}}_{\perp}$ are perpendicular to the $z$ axis. The hole
distribution function $f(\bm{k})$ satisfies the Boltzmann kinetic equation
where the field term contains the forces of the electric field and the Lorentz
force of magnetic field $\tilde{\bm{B}}_{\perp}$ of the radiation:

$\displaystyle\partialderivative{f_{\bm{k}}}{t}+iq_{z}v_{z}f_{\bm{k}}$
$\displaystyle+{e\over\hbar}\bm{E}_{\perp}(t)\cdot\partialderivative{f_{\bm{k}}}{\bm{k}_{\perp}}$
$\displaystyle+{e\over\hbar
c}[\bm{v}\times\tilde{\bm{B}}_{\perp}(t)]\cdot\partialderivative{f_{\bm{k}}}{\bm{k}}={\rm
St}[f].$ (69)

Here $\tilde{\bm{B}}_{\perp}(t)=\tilde{\bm{B}}_{\perp}\exp(-i\omega t)+c.c.$,
the second term comes from the ${\bm{v}}\cdot{\bm{\nabla}}$ term taking into
account that the coordinate dependence of $f_{\bm{k}}$ is
$\propto\exp(iq_{z}z)$, and ${\rm St}[f]$ stands for the elastic collision
integral. It describes the isotropization of the distribution over the
isoenergetic surface $\varepsilon_{k}={\rm const}$ and skew scattering by
wedges:

${\rm
St}[f]=-{f_{\bm{k}}-\expectationvalue{f}\over\tau}+\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}f_{\bm{k}^{\prime}}.$
(70)

In the following, we consider either $q_{z}\neq 0$ or $\tilde{B}\neq 0$ in the
kinetic equation because they give contributions to the PDE currents of the
same order.

The microscopic theory developed in Appendix C yields the PDE current
described by the phenomenological Eq. (68) where the constant $\tilde{T}$ is
given by

$\tilde{T}=2e^{3}\sum_{\bm{k}}\quantity[\tau^{\prime}(\xi_{w}-\tilde{\xi}_{w}){{\rm
Im}(\tau_{\omega}^{3})\over\tau}-{(\xi_{w}\tau\sqrt{\varepsilon_{k}})^{\prime}\over\sqrt{\varepsilon_{k}}}{\rm
Im}(\tau_{\omega}^{2})]f_{0}^{\prime}.$ (71)

The imaginary parts read

${\rm
Im}(\tau_{\omega}^{3})={-}{\omega\tau^{4}[(\omega\tau)^{2}-3]\over[1+(\omega\tau)^{2}]^{3}},\quad{\rm
Im}(\tau_{\omega}^{2})={2\omega\tau^{3}\over[1+(\omega\tau)^{2}]^{2}}.$ (72)

We introduced two dimensionless parameters $\xi_{w}$ and $\tilde{\xi}_{w}$
which are nonzero due to the $D_{3}$ point symmetry of Te and describe the
wedge-like character of hole scattering:

$\xi_{w}=\tau\expectationvalue{\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}v_{z}(\bm{k}^{\prime})v_{y}(\bm{k})[v_{x}^{2}(\bm{k}^{\prime})-v_{y}^{2}(\bm{k}^{\prime})]},$
(73)
$\tilde{\xi}_{w}=\tau\expectationvalue{\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}v_{z}(\bm{k}^{\prime})v_{y}(\bm{k}^{\prime})[v_{x}^{2}(\bm{k})-v_{y}^{2}(\bm{k})]}.$
(74)

In contrast to the parameter $\xi$ given by Eq. (45), the values $\xi_{w}$ and
$\tilde{\xi}_{w}$ describe skew scattering of holes propagating oblique to the
$(xy)$ plane.

At high frequencies $\omega\tau\gg 1$ we have

$\tilde{T}={2e^{3}\over\omega^{3}}\sum_{\bm{k}}\quantity[{-}{\tau^{\prime}\over\tau}(\xi_{w}-\tilde{\xi}_{w})-2{(\xi_{w}\tau\sqrt{\varepsilon_{k}})^{\prime}\over\tau\sqrt{\varepsilon_{k}}}]f_{0}^{\prime}.$
(75)

The wedge scattering efficiencies $\xi_{w}$, $\tilde{\xi}_{w}$ are obtained in
the 4th order of $\bm{k}\cdot\bm{p}$ perturbation theory, therefore they
$\propto{\varepsilon_{k}^{9/2}}$. Using $\tau\propto\varepsilon_{k}^{-1/2}$ we
have for Boltzmann statistics

$\tilde{T}={{24}pe^{3}\over\sqrt{\pi}\omega^{3}(k_{\rm
B}T){{}^{2}}}\quantity({17}\xi_{w}+\tilde{\xi}_{w})_{\varepsilon_{k}=k_{\rm
B}T},$ (76)

where $p$ is the hole concentration. The temperature dependence is
$\tilde{T}\propto(k_{\rm B}T)^{{5/2}}$.

###  V.6 Magnetic field induced photon drag effect at intraband transitions

For the considered linearly polarized light propagating along
$z\parallel\bm{c}$ and an external magnetic field $\bm{B}\parallel x$, the
following MPDE current is allowed, see Eq. (IV) and Appendix A:

$j_{x}=B_{x}q_{z}\absolutevalue{\bm{E}}^{2}S_{l}\tilde{P}_{\rm lin},\quad
j_{y}=-B_{x}q_{z}\absolutevalue{\bm{E}}^{2}S_{l}{P}_{\rm lin}.$ (77)

This photocurrent which can also be rewritten in the form
$j_{+}=S_{l}iq_{z}B_{-}E_{+}^{2}$ is allowed in systems of any symmetry. We
calculate the constant $S_{l}$ for intraband absorption in the ground valence
subband of Te.

We discuss the qualitative picture of the MLPDE assuming, for brevity, a
degenerate statistics. Under the action of the radiation electric field, an ac
electric current in the $(xy)$ plane
$\bm{j}_{\perp}(z,t)=\bm{j}_{\perp}(\omega)\exp[i(q_{z}z-\omega t)]+c.c.$
appears. It oscillates in space and time with the amplitude

$\bm{j}_{\perp}(\omega)=\sigma_{\omega}\bm{E}_{\perp},$ (78)

where $\sigma_{\omega}=pe^{2}\tau_{\omega}/m$ is the ac conductivity. The
magnetic field $B_{x}$ leads to cyclotron motion, which causes the Hall
component of this ac current to flow in the $z$-direction:

$j_{z}(\omega)=-\omega_{c}\tau_{\omega}j_{y}(\omega),$ (79)

where $\omega_{c}=eB_{x}/(mc)$ is the cyclotron frequency. This ac current is
accompanied by oscillations of the carrier density $\delta p(z,t)=\delta
p_{\omega}\exp[i(q_{z}z-\omega t)]+c.c.$ Its amplitude is related to the ac
current by the continuity equation:

${-}ei\omega\delta p_{\omega}+iq_{z}j_{z}(\omega)=0.$ (80)

These density oscillations in the presence of the radiation’ electric field
result in a dc current due to rectification [68, 28]:

$\bm{j}_{\perp}={e^{2}\tau\over m}\overline{\delta
p(z,t)\bm{E}_{\perp}(z,t)}={e^{2}\tau\over m}\delta
p(\omega)\bm{E}_{\perp}^{*}+c.c.,$ (81)

where the bar denotes averaging over time and $z$ coordinate. This approach
gives the MLPDE current Eq. (77) with the constant $S_{l}$ given by

$S_{l}=-{pe^{4}\tau^{3}(1-\omega^{2}\tau^{2})\over m^{3}\omega
c(1+\omega^{2}\tau^{2})^{2}}.$ (82)

There is a competing contribution $\Delta\bm{j}$ of the same order which stems
from the radiation magnetic field. It appears as follows: in the presence of
radiation with amplitude $E_{y}$ and the Lorentz force from $\bm{B}\parallel
x$, an ac electric current appears which oscillates along and opposite to the
$z$ axis with an amplitude $j_{z}(\omega)$, Eq. (79). Then, this current is
rotated due to the action of the ac magnetic field $\tilde{\bm{B}}_{\perp}$.
This gives rise to the emergence of a dc ‘Hall’ component

$\Delta j_{y}={e\tau\over mc}\tilde{B}_{x}^{*}j_{z}(\omega)+c.c.\propto
E_{y}\tilde{B}_{x}^{*}+c.c.$ (83)

Noting that $\tilde{\bm{B}}=c\bm{q}\times\bm{E}/\omega$, we obtain the
contribution to Eq. (77) with

${\Delta S_{l}={pe^{4}\tau^{3}(1-\omega^{2}\tau^{2})\over m^{3}\omega
c(1+\omega^{2}\tau^{2})^{2}}.}$ (84)

This qualitative consideration gives the correct magnitude of the MLPDE
current at a constant relaxation time $\tau$ independent of electron energy.
However, a microscopic theory is needed because, in the qualitative
consideration above, the two contributions cancel each other out.

The microscopic theory developed in Appendix D shows that the resulting
coefficient $S_{l}$ in Eq. (77) is nonzero and depends on the dominating
scattering mechanism. In order to calculate the MPDE current one has to take
into account either the finite wavevector of light $q_{z}$ or the magnetic
field $\tilde{\bm{B}}\perp z$ of the radiation. In the first approach, the
MLPDE current is a sum of several contributions obtained by iterating the
Boltzmann kinetic equation in the small parameters $E$, $q_{z}$, $E^{*}$ and
$B_{x}$. The analysis shows that the following three iteration sequences can
contribute to the MLPDE current, $EqEB$, $EBqE$, and $EqBE$. In the second
approach, there are two contributions, $EB\tilde{B}$ and $E\tilde{B}B$.
Microscopic calculations show that all the contributions give rise to the
MLPDE constant $S_{l}$ in Eq. (77) for high frequencies ${\omega\tau\gg 1}$ in
the following form

$S_{l}=a{4e^{4}\over
3m^{3}c\omega^{3}}\sum_{\bm{k}}\varepsilon_{k}(-f_{0}^{\prime})\tau$ (85)

with different pre-factors $a$. All contributions to the pre-factor $a$ are
given in the Table 1.

Contribution | Pre-factor $a$
---|---
$EqEB$ | 0
$EBqE$ | ${1+2r/3}$
$EqBE$ | $-2r/5$
$EB\tilde{B}$ | ${-}1$
$E\tilde{B}B$ | pol./indep.
Table 1: Various contributions to the MLPDE constant pre-factor $a$ in Eq.
(85). The coefficient $r=\differential\ln\tau/\differential\ln\varepsilon$ is
determined by the dominating elastic scattering mechanism.

With Boltzmann statistics and $\tau\propto\varepsilon_{k}^{r}$ we obtain
summing all contributions Eq. (85)

$S_{l}={{16r}\Gamma\quantity(r+5/2)pe^{4}\tau_{T}\over
45\sqrt{\pi}m^{3}c\omega^{3}}\propto T^{r},$ (86)

where $\tau_{T}=\tau(\varepsilon_{k}=k_{\rm B}T)$. At arbitrary frequency, we
obtained for the MLPDE constant $S_{l}$ the following expression, see Appendix
D

$S_{l}={8re^{4}\over
45m^{3}c\omega}\sum_{\bm{k}}\varepsilon_{k}f_{0}^{\prime}\tau\quantity[5{\rm
Re}(\tau_{\omega}^{2})-3\omega{\rm Im}(\tau_{\omega}^{3})].$ (87)

##  VI discussion

Below we discuss the experimental data in the context of the the theory
developed. We compare experimental and theoretical results and present
microscopic models that illustrate the generation of the different
photocurrents described in the previous section. The photocurrents were
obtained for radiation with strongly different photon energies of about 130
meV in experiments with infrared CO2-laser radiation and on the order of
several meV in experiments with THz radiation.

The magnitude of the photocurrent excited by infrared radiation is large
enough to enable direct intersubband transitions sketched in Fig. 7 but about
three times smaller than the forbidden gap, which excludes interband
absorption.

As discussed in Secs. III and V, the photon energy in the THz range is much
smaller than the energy required for any kind of direct optical transitions
(intersubband or interband) and the photocurrent is due to Drude absorption.

The microscopic mechanisms responsible for the photocurrents for both
intersubband and Drude absorption which summarize the results of Sec. V are
given in Table 2. We begin with the photocurrents and magneto-photocurrents
excited by infrared radiation and then consider the photoresponse to THz
radiation.

| Intersubband | Intraband Drude
---|---|---
LPGE | shift | skew scattering by triangles
LPDE | $-$ | skew scattering by wedges
MCPGE | shift |  $\bm{k}\bm{B}$ terms in scattering
MLPGE | $\tau_{1}\neq\tau_{2}$ |  $\bm{k}\bm{B}$ terms in scattering
MLPDE | $-$ | Lorentz force
|  | and photon momentum
Table 2: Microscopic mechanisms responsible for the observed photocurrents for
infrared and THz ranges at inter- and intrasubband absorption, respectively.
The notation ‘shift’ indicates the sum of the shift contribution and the
ballistic contribution is due to the interference of electron-photon
interaction with scattering. Note that the MLPGE current at intersubband
transitions is caused by ballistic propagation of photoholes in both subbands
limited by scattering with different transport times $\tau_{1}\neq\tau_{2}$.

###  VI.1 Infrared radiation induced photocurrent at zero magnetic field

For all the infrared frequencies used in our experiments we observed that the
photocurrent is excited only when the degree of linear polarization is non-
zero. This is seen in experiments with linearly polarized radiation, see Fig.
2(a), and elliptically polarized radiation, see Fig. 5(a). The latter figure
clearly shows that the response to circularly polarized radiation
($\sigma^{+}$ and $\sigma^{-}$) is zero. The photocurrent detected in the
$y$-direction is proportional to the degree of linear polarization
$j_{y}\propto\tilde{P}_{\rm lin}$, see Eq. (IV) and Figs. 2(a) and 5(a).
Consequently, the azimuthal angle dependence of the current in response to
linearly polarized radiation is given by $j~{}\propto\sin(2\alpha)$ or, in
experiments using a $\lambda/4-$plate, $j~{}\propto\sin(4\varphi)$, see Eqs.
(8), (11) and (16). This functional behavior corresponds to the trigonal LPGE
current obtained in Sec. V, see the terms proportional to the parameter $\chi$
in Eqs. (IV), (IV.1), (IV.2) and (27). We emphasize that all other
photocurrents either depend differently on the degree of linear polarization,
$j_{y}\propto P_{\rm lin}\propto\cos(2\alpha)$ (contributions proportional to
the parameter $\tilde{T}$) or vanish at zero magnetic field (contributions
proportional to $\Phi_{l}$, $\Phi_{c}$ and $S_{l}$). We also note that the
theory shows that the circular photocurrent at normal incidence and zero
magnetic field is symmetry forbidden, see Eq. (IV) and Appendix A.

The microscopic theory of the observed LPGE current is developed in Sec. V.1.
Taking into account the shift mechanism of the LPGE we obtained the trigonal
LPGE photocurrent and derived the parameter $\chi$, see Eqs. (27) and (28).
The generation of the LPGE current caused by the shifting of the hole wave
packets in real space is illustrated in Fig. 8. Panel (a) shows the
crystallographic structure of the Te crystal with the Te atoms at the corners
of the triangles when viewed in the direction of the $c$-axis. Optical
transitions between the subbands are sketched in Fig. 8 by downward vertical
arrows. As can be seen from Eqs. (24) and (25), depending on the orientation
of the radiation’ electric field vector with respect to the $y$-axis, these
transitions lead to shifts of the photoexcited holes by $+R$
($\alpha=45^{\circ}$) or $-R$ ($\alpha=-45^{\circ}$), which, consequently,
causes a $dc$ electric current (blue horizontal arrows). For vertical or
horizontal polarization ($\alpha=0$ or $90^{\circ}$) the shift along the $y$
axis is zero. The frequency dependence of this mechanism is $\chi\propto
K(\omega)/\omega^{2}$. In the frequency range studied the frequency dependence
is weak, which agrees with the experimental data, see curve 1 in the inset of
Fig. 3(a).

Figure 8: Illustration of the shift contribution to the trigonal LPGE current
at intersubband transitions. Figure 9:  Illustration of the MLPGE current
$j_{y}\propto B_{x}\tilde{P}_{\rm lin}$ formation at intersubband transitions.
$\Omega_{ph}$ is the optical phonon frequency.

###  VI.2 Infrared radiation induced linear and circular MPGE currents

The application of an in-plane magnetic field ${\bm{B}\parallel x}$ leads to
new photocurrent contributions, which belong to the class of magneto-
gyrotropic photogalvanic effects [30, 50]. Figures 2, 3(a), and 5 demonstrate
that the magnetic field results in the photocurrents being odd in magnetic
field $\bm{B}$ and excited by linearly as well as circularly polarized
infrared radiation.

Figure 2 (blue and olive squares at $\pm B_{x}$) shows that the magnetic field
induced current in the $y$-direction excited by a linearly polarized infrared
radiation is characterized by the same azimuthal angle dependence as the
photocurrent at zero magnetic field: $j_{y}(B_{x})\propto\tilde{P}_{\rm
lin}\propto\sin(2\alpha)$. Analyzing Eq. (IV) we find that this indicates that
the magnetic field induced current is caused by the linear MPGE. Note that
another possible magnetic field induced photocurrent caused by the photon drag
is not detected in the experiments discussed, because it would lead to
$j_{y}(B)\propto P_{\rm lin}\propto\cos(2\alpha)$, and, consequently, to a
phase shift in the azimuthal angle dependence. As follows from the microscopic
theory presented in Sec. V.2, the linear MPGE is described by Eqs. (34) and
(35). The generation of the linear MPGE current at intersubband transitions is
illustrated in Fig. 9. For any value of the photon energy
$\hbar\omega>2\Delta_{2}$, optical transitions are possible for hole states in
the upper subband with wavevectors $\pm k_{z}$ that satisfy the energy
conservation law

$\hbar\omega=2\sqrt{\Delta_{2}^{2}+(\beta k_{z})^{2}}.$

The in-plane wavevector can be arbitrary, and there are pairs of hole states
with a fixed $k_{x}$ and $\absolutevalue{k_{y}}$ that differ by the sign of
$k_{y}$. These transitions result in the depopulation of initial states $i1$
and $i2$ and the population of the final states $f1$ and $f2$ by photoexcited
holes. As a result, four elementary currents are generated, two in the upper
subband and two in the lower subband. This is indicated by horizontal blue
arrows. As shown in Sec. V.2, the transitions $i1\to f1$ and $i2\to f2$ have
different probabilities, which depend on the magnetic field strength and
direction. In Fig. 9 this difference is illustrated by a thick red downward
arrow for transitions from state $i2$ and a thin dashed arrow for transitions
from state $i1$. As a result the balance between the population of states $i1$
and $i2$ and $f1$ and $f2$ is violated and the corresponding currents have
different strength, which is illustrated by sketching the dominating currents
by solid arrows. Furthermore, the magnitude of the contribution of each state
to the total current is determined by the momentum relaxation time of the
corresponding states. Holes in the final states of optical transitions have
more effective momentum relaxation due to the emission of optical phonons
($\hbar\Omega_{\rm ph}=11$ meV in Te [57]), see bent dotted curves in Fig. 9,
than in the initial states ($\tau_{i}>\tau_{f}$). As a result, the
contributions of photoexcited holes to the current is smaller, and the current
in Fig. 9 is dominated by the contribution $j_{i2}$ caused by the depopulation
of the initial state $i2$. The frequency dependence of this mechanism is the
same as for LPGE, Eq. (28): $\Phi_{l,c}\propto K(\omega)/\omega^{2}$. In the
investigated frequency range this gives rise to a weak frequency dependence,
which agrees with the experimental data, see curves 2 and 3 in the inset of
Fig. 3(a), respectively.

The experimentally observed magnetic field induced circular photocurrent
$J_{y}\propto P_{\rm circ}B_{x}$, see Figs. 5(b,c) and 3(a), can also be
explained by the shift mechanism in the developed theory. It shows that such
photocurrent is caused by the photogalvanic effect, see Eqs. (IV), (30), and
(31). The mechanism of its formation is similar to that of the linear PGE. The
only difference is that the probability for optical transitions at positive
and negative $k_{y}$, see Fig. 8(b), depends on the radiation helicity.
Consequently, the resulting current reverses its sign upon reversal of the
magnetic field or/and switching polarization from $\sigma^{+}$ to $\sigma^{-}$
and vice versa. The latter is described by Eqs. (30) and (31) and detected
experimentally, see Figs. 5(b) and (c).

###  VI.3 Terahertz radiation induced trigonal LPGE and photon drag currents

As discussed above the photocurrents excited by infrared radiation are caused
by intersubband optical transitions. At terahertz frequencies the
corresponding mechanisms are inapplicable, because the photon energies are too
small to fulfill the energy conservation law and only indirect optical
transitions can be responsible for the radiation absorption (Drude-like
mechanism) and the photocurrent generation. In this spectral range the
azimuthal angle dependence of the zero-$B$ photocurrent is observed to be
phase shifted as compared to the one discussed for the LPGE, see Fig. 4(a).
This observation indicates that the THz radiation induced photocurrent is
caused by the superposition of the linear photogalvanic
($j_{y}\propto\tilde{P}_{\rm lin}\propto\sin(2\alpha)$) and linear photon drag
effects ($j_{y}\propto P_{\rm lin}\propto\cos(2\alpha)$), see Eq. (IV). As
shown in Sec. IV.1 both effects contribute almost equally to the total
current.

The microscopic theory developed in Sec. V.3 shows that the linear PGE is
caused by asymmetric scattering of holes driven by the THz electric field. The
corresponding current is described by Eqs. (42) and (43) which, for the case
relevant to our experimental conditions, Boltzmann statistics, and
$\omega\tau\gg 1$, takes the form of Eq. (47). The frequency dependence of the
trigonal PGE follows that of the Drude absorption and is for $\omega\tau>1$
given by $j_{y}\propto 1/\omega^{2}$. This is in full agreement with the
experiment, see fits in Fig. 3(b). The model of the LPGE due to intraband
transition is similar to that presented previously for bulk or surface states
of other materials having trigonal symmetry. It has been discussed in detail
in works aimed at THz radiation induced LPGE in GaN quantum wells [69] and
surface states of BiSbTe-based 3D topological insulators [65]. Therefore, it
is applicable to the description of photocurrent in Te, and will not be
presented here.

The microscopic theory of the chiral linear photon drag effect is developed in
Sec. V.5. It confirms that the PGE and PDE photocurrents in Te are
characterized by $\pi/2$-shifted azimuthal angle dependencies. The chiral PDE
current is described by Eqs. (68). Similar to PGE its frequency dependence is
given by that of Drude absorption, $q_{z}\tilde{T}\propto 1/\omega^{2}$ at
$\omega\tau\gg 1$, see Eq. (76), which is in agreement with the experiment,
see fits in Fig. 3(b). The trigonal PDE current has previously been
demonstrated for 2D surface states of BiSbTe 3D topological insulators [67,
31]. The model developed to describe it is applicable to the chiral PDE in Te,
the only difference with respect to the trigonal PDE in 2D systems is that it
is based on scattering by 2D triangle wedges whereas in Te the wedges are
three dimensional (top view triangle but atoms are shifted along $z$-axis). To
avoid repetition this model will not be presented in this paper.

We estimate the chiral PDE current: $J_{\rm
PDE}/P=(2\pi\mathcal{S}/cn_{\perp})\tilde{T}q_{z}$, where $\mathcal{S}$ is the
sample aspect ratio. From Eq. (76) we obtain $\tilde{T}\approx
10^{2}pe^{3}\xi_{w}/[\omega^{3}(k_{\rm B}T)^{2}]$. The wedge scattering
probability is obtained in the 4th order of the $\bm{k}\cdot\bm{p}$
perturbation theory and beyond the Born approximation, therefore we have an
estimate ${\xi_{w}\sim S_{\rm nB}\absolutevalue{LL^{\prime}}^{2}(k_{\rm
B}T/\hbar E_{g})^{4}}$. Here $L^{\prime}$ is the interband momentum matrix
element fixing the chirality of Te
($\absolutevalue{L^{\prime}}\ll\absolutevalue{L}$), and $S_{\rm nB}$ is the
non-Born dimensionless parameter, cf. Eq. (47). This gives an estimate

${J_{\rm PDE}\over P}\approx 10^{2}{2\pi\mathcal{S}pe^{3}\over
c^{2}\omega^{2}}S_{\rm nB}\absolutevalue{LL^{\prime}}^{2}{(k_{\rm
B}T)^{2}\over(\hbar E_{g})^{4}}.$ (88)

For $\mathcal{S}=0.15$, room temperature, $f=1$ THz, $E_{g}=335$ meV, $L=3.3$
eVÅ [27], and taking $\absolutevalue{L^{\prime}}=0.1\absolutevalue{L}$ and
$S_{\rm nB}=10^{-2}$, we obtain $J_{\rm PDE}/P\sim 1$ nA/W which coincides by
an order of magnitude with the measured values.

###  VI.4 Terahertz radiation induced linear and circular MPGE and linear MPDE
currents

Finally, we discuss the magnetic field induced photocurrent excited by THz
radiation. The analysis of the experimental data in the light of
phenomenological theory shows that the magneto-photocurrent induced by
linearly polarized radiation is given by

$j_{y}=(\Phi_{l}\tilde{P}_{\rm lin}-S_{l}q_{z}P_{\rm
lin})B_{x}\absolutevalue{\bm{E}}^{2},$

see Sec. IV.1, Fig. 3, and Eq. (IV). As a result, the MPGE has about 2.5
larger magnitude ($\Phi_{l}$) as compared to the MPDE one ($S_{l}q_{z}$).

MPGE is shown microscopically to be based on the terms in the scattering
probability, which are linear in both the wavevector $\bm{k}$ and the magnetic
field $\bm{B}$, Eq. (50). These terms caused by the change of the scattering
rates by the magnetic field, result in the photocurrent driven by linear
polarizations, see Eqs. (57), (58), and the photocurrent driven by circularly
polarized radiation, see Eqs. (64), (65), also detected in THz experiments,
see Fig. 3(c). The magneto-photogalvanic current is caused by the interband
mixing of states by the magnetic field, see the interband matrix element, Eq.
(48). The $\bm{k}_{\perp}\cdot\bm{B}_{\perp}$ terms (50) are specific for the
$D_{3}$ symmetry of Te because they result in the MPGE currents $j_{y}\propto
B_{x}\tilde{P}_{l}$, $j_{y}\propto B_{x}P_{\rm circ}$ perpendicular to the
magnetic field, while in structure inversion-asymmetric 2D systems (quantum
wells, graphene on a substrate etc.) [70, 71], similar mechanism gives MPGE
photocurrents parallel to the magnetic field at the same polarizations.

The magneto-photon drag effect driven by linearly polarized radiation (linear
MPDE) is caused by the simultaneous action of the spatially-varying electric
field of radiation and the Lorentz force, see Eqs. (81) and (83). A major
contribution of this photocurrent is manifested in THz experiments as a phase
shift in the azimuthal angle dependence, see Fig. 4(b) and (c) and the inset
in this figure. The linear MPDE current is allowed in systems of any symmetry.
Note that the circular magneto-photocurrent solely due to the magneto-
photogalvanic effect and the circular MPDE are forbidden by symmetry in the
investigated experimental arrangement, see Eq. (IV).

Finally, we note that the frequency dependence of all magnetophotocurrents
induced by THz radiation is given by that of the Drude absorption, see Eqs.
(77), (85), which is in agreement with experimental results (not shown).

##  VII summary

Our studies of photocurrents excited by infrared/THz radiation show that
depending on the experimental conditions, the radiation induces a series of
phenomena caused by photogalvanics and, in THz range, the transfer of linear
photon momentum to free carriers. A rich palette of the microscopic mechanisms
of the observed photocurrents is due to the different photocurrent roots
depending on the kind of the optical transitions responsible for the radiation
absorption, polarization state, the presence of an external magnetic field,
and the transfer of linear photon momentum during absorption of radiation. All
observed photocurrents can be described in terms of the developed
phenomenological and microscopic theories, taking into account a shift of the
electron wavepacket in real space or/and asymmetric scattering in a system
without inversion symmetry. The main mechanisms resulting in the observed
photocurrents are summarized in Table 2. We believe that the results of this
study will be useful in future studies of novel Te based materials such as
tellurene and Weyl fermions with surface Fermi arcs, which are expected in Te
crystals under high pressure.

##  VIII Acknowledgments

We thank V. A. Shalygin, K. v. Klitzing, I. Gronwald and I. V. Sedova for
fruitful discussions. The support from the Deutsche Forschungsgemeinschaft
(DFG, project Ga501/19), and the Volkswagen Stiftung Program (97738) is
gratefully acknowledged.

##  Appendix A Phenomenology of PGE and PDE currents at normal incidence

Tellurium is a chiral semiconductor with the point symmetry group $D_{3}$. We
use the coordinate system $(x,y)$ where $x$ is parallel to one of three
$C_{2}$ rotation axes. We study photocurrents at light propagation along
$z\parallel\bm{c}$ when the unit vector in the light propagation direction
$\hat{\bm{e}}\parallel z$. The PGE current components in the presence of the
magnetic field $\bm{B}$ are given by

$\displaystyle j_{x}=\quantity[\chi P_{\rm lin}+\Phi B_{z}\tilde{P}_{\rm
lin}+\Phi_{l}(B_{x}P_{\rm lin}+B_{y}\tilde{P}_{\rm lin})+\Phi_{c}B_{y}P_{\rm
circ}\hat{e}_{z}+\Lambda B_{x}]\absolutevalue{\bm{E}}^{2},$ (89)
$\displaystyle j_{y}=\quantity[-\chi\tilde{P}_{\rm lin}+\Phi B_{z}{P}_{\rm
lin}+\Phi_{l}(B_{x}\tilde{P}_{\rm lin}-B_{y}{P}_{\rm lin})-\Phi_{c}B_{x}P_{\rm
circ}\hat{e}_{z}+\Lambda B_{y}]\absolutevalue{\bm{E}}^{2},$ (90)
$\displaystyle j_{z}=\quantity[\gamma P_{\rm
circ}\hat{e}_{z}+\Phi^{\prime}(B_{y}P_{\rm lin}+B_{x}\tilde{P}_{\rm
lin})+\tilde{\Lambda}B_{z}]\absolutevalue{\bm{E}}^{2}.$ (91)

Here the zero-field current $j_{z}=\gamma P_{\rm
circ}\absolutevalue{\bm{E}}^{2}$ is the longitudinal CPGE and $j_{+}=\chi
E_{-}^{2}$ is the trigonal LPGE. The polarization-independent components along
the magnetic field $j_{x,y}\propto{\Lambda}B_{x,y}$,
$j_{z}\propto\tilde{\Lambda}B_{z}$ and polarization-dependent currents
$j_{+}=\Phi_{l}B_{-}E_{+}^{2}$,
$\bm{j}_{\perp}=\Phi_{c}(\bm{B}\times\bm{\varkappa})\absolutevalue{\bm{E}}^{2}$
are due to an absence of reflection planes in the $D_{3}$ group
($\bm{\varkappa}=P_{\rm circ}\hat{\bm{e}}$ is the photon angular momentum).
The currents $j_{+}=\Phi iB_{z}E_{-}^{2}$ and
$j_{z}=\Phi^{\prime}i(B_{-}E_{-}^{2}-B_{+}E_{+}^{2})/2$ describe the trigonal
MPGE current which requires both absence of reflections and trigonality.

The PDE current which, by definition, accounts for $q_{z}\neq 0$ reads

$\displaystyle j_{x}=\quantity[\tilde{T}\tilde{P}_{\rm lin}+SB_{z}P_{\rm
lin}+S_{l}(B_{x}\tilde{P}_{\rm lin}+B_{y}P_{\rm lin})+S_{c}B_{x}P_{\rm
circ}\hat{e}_{z}+RB_{y}]q_{z}\absolutevalue{\bm{E}}^{2},$ (92) $\displaystyle
j_{y}=\quantity[\tilde{T}P_{\rm lin}-SB_{z}\tilde{P}_{\rm
lin}-S_{l}(B_{x}P_{\rm lin}+B_{y}\tilde{P}_{\rm lin})+S_{c}B_{y}P_{\rm
circ}\hat{e}_{z}-RB_{x}]q_{z}\absolutevalue{\bm{E}}^{2},$ (93) $\displaystyle
j_{z}=\quantity[T+\tilde{S}B_{z}P_{\rm circ}\hat{e}_{z}+S^{\prime}(B_{x}P_{\rm
lin}-B_{y}\tilde{P}_{\rm lin})]q_{z}\absolutevalue{\bm{E}}^{2}.$ (94)

The polarization-independent PDE currents
$j_{z}=Tq_{z}\absolutevalue{\bm{E}}^{2}$,
$\bm{j}_{\perp}=R(\bm{B}\times\bm{q})\absolutevalue{\bm{E}}^{2}$ and helicity-
dependent magneto-induced currents
$j_{z}=\tilde{S}B_{z}\varkappa_{z}q_{z}\absolutevalue{\bm{E}}^{2}$,
$\bm{j}_{\perp}=S_{c}\bm{B}_{\perp}\varkappa_{z}q_{z}\absolutevalue{\bm{E}}^{2}$,
$j_{+}=S_{l}iq_{z}B_{-}E_{+}^{2}$ are allowed in any symmetry. The currents
$j_{+}=\tilde{T}iq_{z}E_{-}^{2}$, $j_{+}=SB_{z}E_{-}^{2}$ and
$j_{z}=S^{\prime}(B_{+}E_{+}^{2}+B_{-}E_{-}^{2})q_{z}\absolutevalue{\bm{E}}^{2}/2$
are allowed in $D_{3}$ symmetry due to both trigonality and absence of
reflection planes.

##  Appendix B Intersubband absorption coefficient

The absorption coefficient for direct optical transitions $K(\omega)$ is
defined as follows

$\displaystyle{K(\omega){I}\over\hbar\omega}$ (95)
$\displaystyle=2\sum_{\bm{k}}{2\pi\over\hbar}\absolutevalue{V_{21}}^{2}\delta\quantity[2\sqrt{\Delta_{2}^{2}+(\beta
k_{z})^{2}}-\hbar\omega](f_{1}-f_{2}).$

Here the factor 2 accounts for two valleys of tellurium,
${I}=cn_{\perp}\absolutevalue{\bm{E}}^{2}/(2\pi)$ is the light intensity,
$f_{1,2}$ are occupations of the initial and final states, and the optical
matrix element is given by Eq. (22). For Boltzmann statistics we have
$f_{1}-f_{2}=[1-\exp(-\hbar\omega/k_{\rm B}T)]f_{1}$. Since the hole
concentration $p=2\sum_{\bm{k}}f_{1}$, we obtain

$K(\omega)={8\pi^{2}e^{2}\absolutevalue{L}^{4}\Delta_{2}^{2}\sqrt{\absolutevalue{\mathcal{A}_{1}}k_{\rm
B}T}\over\hbar
cE_{g}^{4}\hbar\omega\absolutevalue{\mathcal{A}_{2}\beta}}p{1-\exp(-\hbar\omega/k_{\rm
B}T)\over J_{1}\sqrt{1-(2\Delta_{2}/\hbar\omega)^{2}}}\\\
\times{\exp{-{\absolutevalue{\mathcal{A}_{1}}[(\hbar\omega)^{2}-(2\Delta_{2})^{2}]\over
4\absolutevalue{\beta}k_{\rm B}T}+{\hbar\omega/2-\Delta_{2}\over k_{\rm
B}T}}},$ (96)

where $J_{1}$ was introduced in Ref. [57]:

$J_{1}=\int\limits_{0}^{\infty}\differential x\exp[-x^{2}-{\Delta_{2}\over
k_{\rm B}T}+\sqrt{\quantity({\Delta_{2}\over k_{\rm
B}T})^{2}+{x^{2}\beta^{2}\over\absolutevalue{\mathcal{A}_{1}}k_{\rm B}T}}].$
(97)

##  Appendix C Microscopic calculation of PDE current

We solve iteratively the kinetic Eq. (V.5). First, we find the linear in
$\bm{E}$ correction to the distribution function:

$f^{(E)}=e\tau_{\omega}(-f_{0}^{\prime})(\bm{E}_{\perp}\cdot\bm{v}_{\perp}).$
(98)

Then, depending on the contribution under study, we iterate the kinetic
equation in small parameters $q_{z}$, $\bm{E}_{\perp}^{*}$ or
$\tilde{\bm{B}}_{\perp}$.

Account for $q_{z}$ yields the correction $f^{(Eq)}$ which satisfies the
following kinetic equation:

$-i\omega f^{(Eq)}+iq_{z}v_{z}f^{(E)}=-{f^{(Eq)}\over\tau}.$ (99)

It yields

$f^{(Eq)}=-iq_{z}v_{z}\tau_{\omega}f^{(E)}=ieq_{z}\tau_{\omega}^{2}f_{0}^{\prime}v_{z}(\bm{E}_{\perp}\cdot\bm{v}_{\perp}).$
(100)

If we then account for the second power of $\bm{E}_{\perp}$, we find the time-
independent correction $f^{(EqE)}$:

${e\over\hbar}\bm{E}_{\perp}^{*}\cdot\partialderivative{f^{(Eq)}}{\bm{k}_{\perp}}+c.c.=-{f^{(EqE)}\over\tilde{\tau}}.$
(101)

Here, depending on the anisotropic/isotropic part of the filed term in the
left-hand side of this equation, the relaxation time $\tilde{\tau}$ equals to
either $\tau$ or to the energy relaxation time $\tau_{\varepsilon}$. Splitting
the field term to the isotropic and anisotropic in $\bm{k}$ parts, we obtain

$\displaystyle f^{(EqE)}=$ $\displaystyle 2e^{2}q_{z}v_{z}{\rm
Im}\biggl{\\{}\tau\quantity(\tau_{\omega}^{2}f_{0}^{\prime})^{\prime}\quantity(\absolutevalue{\bm{E}_{\perp}\cdot\bm{v}_{\perp}}^{2}-\absolutevalue{\bm{E}_{\perp}}^{2}{v_{\perp}^{2}\over
2})$
$\displaystyle+\tau_{\varepsilon}\absolutevalue{\bm{E}_{\perp}}^{2}\quantity[{v_{\perp}^{2}\over
2}\quantity(\tau_{\omega}^{2}f_{0}^{\prime})^{\prime}+{\tau_{\omega}^{2}f_{0}^{\prime}\over
m}]\biggr{\\}}.$ (102)

For linearly polarized radiation we can rewrite this expression as follows

$\displaystyle f^{(EqE)}=e^{2}q_{z}v_{z}\absolutevalue{\bm{E}_{\perp}}^{2}$
$\displaystyle\times{\rm
Im}\biggl{\\{}\tau\quantity(\tau_{\omega}^{2}f_{0}^{\prime})^{\prime}\quantity[2v_{x}v_{y}P_{\rm
lin}+(v_{x}^{2}-v_{y}^{2})\tilde{P}_{\rm lin}]$
$\displaystyle+\tau_{\varepsilon}\quantity[{v_{\perp}^{2}\over
2}\quantity(\tau_{\omega}^{2}f_{0}^{\prime})^{\prime}+{\tau_{\omega}^{2}f_{0}^{\prime}\over
m}]\biggr{\\}}.$ (103)

Let us now take into account the last perturbation, the scattering by wedges,
and find the current-carrying part of the distribution function $f^{(EqEw)}$.
The asymmetric scattering enters the kinetic equation as an incoming term:

$\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}f^{(EqE)}_{\bm{k}^{\prime}}={f^{(EqEw)}_{\bm{k}}\over\tau}.$
(104)

Solving this algebraic equation, we calculate the $EqEw$ contribution to the
PDE current which flows in the $(xy)$ plane in this geometry:

$\displaystyle\bm{j}_{\perp}^{(EqEw)}=2e^{3}q_{z}\absolutevalue{\bm{E}_{\perp}}^{2}\sum_{\bm{k}}\bm{v}_{\perp}\tau^{2}\quantity[{\rm
Im}(\tau_{\omega}^{2})f_{0}^{\prime}]^{\prime}\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}$
$\displaystyle\times
v_{z}(\bm{k}^{\prime})\quantity{2v_{x}(\bm{k}^{\prime})v_{y}(\bm{k}^{\prime})P_{\rm
lin}+[v_{x}^{2}(\bm{k}^{\prime})-v_{y}^{2}(\bm{k}^{\prime})]\tilde{P}_{\rm
lin}}.$ (105)

Here we took into account that the isotropic part of
$f^{(EqE)}\propto\tau_{\varepsilon}$ does not contribute to the current
because in $D_{3}$ symmetry

$\expectationvalue{\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}v_{z}(\bm{k}^{\prime})\bm{v}_{\perp}(\bm{k})}=0.$
(106)

The nonzero average allowed in $D_{3}$ point symmetry group is the following
dimensionless value

$\displaystyle\xi_{w}=\tau\expectationvalue{\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}v_{z}(\bm{k}^{\prime})v_{y}(\bm{k})[v_{x}^{2}(\bm{k}^{\prime})-v_{y}^{2}(\bm{k}^{\prime})]}$
$\displaystyle=2\tau\expectationvalue{\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}v_{z}(\bm{k}^{\prime})v_{x}(\bm{k})v_{x}(\bm{k}^{\prime})v_{y}(\bm{k}^{\prime})}.$
(107)

Then we obtain a contribution to the PDE constant $\tilde{T}$, Eq. (68), which
reads

$\tilde{T}_{EqEw}=2e^{3}\sum_{\bm{k}}\tau\xi_{w}\quantity[{\rm
Im}(\tau_{\omega}^{2})f_{0}^{\prime}]^{\prime}.$ (108)

Since the density of states is $\propto\sqrt{\varepsilon_{k}}$, we can rewrite
this expression as

$\tilde{T}_{EqEw}=-2e^{3}\sum_{\bm{k}}{(\xi_{w}\tau\sqrt{\varepsilon_{k}})^{\prime}\over\sqrt{\varepsilon_{k}}}{\rm
Im}(\tau_{\omega}^{2})f_{0}^{\prime}.$ (109)

Now we calculate $EqwE$ contribution to the PDE current. We find the time-
dependent correction $f^{(Eqw)}$:

$\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}f^{(Eq)}_{\bm{k}^{\prime}}={f^{(Eqw)}_{\bm{k}}\over\tau_{\omega}}.$
(110)

The solution of this equation reads

$f^{(Eqw)}_{\bm{k}}=ieq_{z}\tau_{\omega}^{3}f_{0}^{\prime}\tau\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}v_{z}(\bm{k}^{\prime})[\bm{E}_{\perp}\cdot\bm{v}_{\perp}(\bm{k}^{\prime})].$
(111)

The static correction $f^{(EqwE)}$ yielding the contribution to the current
satisfies the equation

${e\over\hbar}\bm{E}_{\perp}^{*}\cdot\partialderivative{f^{(Eqw)}}{\bm{k}_{\perp}}+c.c.=-{f^{(EqwE)}\over\tau}.$
(112)

Solving this equation we get the current

$\bm{j}^{(EqwE)}=2e^{2}\sum_{\bm{k}}f^{(Eqw)}\quantity[{\tau\over
m}\bm{E}^{*}+\tau^{\prime}\bm{v}\quantity(\bm{E}_{\perp}^{*}\cdot\bm{v}_{\perp})]+c.c.$
(113)

Substituting here $f^{(Eqw)}$ we see that the first term here does not
contribute because $f^{(Eqw)}$ is zero on average, but the second term yields
a contribution to the current (68) with

$\tilde{T}_{EqwE}=-2e^{3}\sum_{\bm{k}}\tau^{\prime}\tilde{\xi}_{w}{{\rm
Im}(\tau_{\omega}^{3})\over\tau}f_{0}^{\prime}.$ (114)

Here we introduced another wedge-scattering efficiency constant
$\tilde{\xi}_{w}$ which is linearly-independent of $\xi_{w}$:

$\displaystyle\tilde{\xi}_{w}=\tau\expectationvalue{\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}v_{z}(\bm{k}^{\prime})v_{y}(\bm{k}^{\prime})[v_{x}^{2}(\bm{k})-v_{y}^{2}(\bm{k})]}$
$\displaystyle=2\tau\expectationvalue{\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}v_{z}(\bm{k}^{\prime})v_{x}(\bm{k}^{\prime})v_{x}(\bm{k})v_{y}(\bm{k})}.$
(115)

Let us turn now to the $EwqE$ contribution to the PDE current. For its
calculation, we account for wedge scattering in the kinetic equation at the
second step and find the correction $f^{(Ew)}$:

$\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}f^{(E)}_{\bm{k}^{\prime}}={f^{(Ew)}\over\tau_{\omega}}.$
(116)

Solution of this equation reads

$f^{(Ew)}=-e\tau_{\omega}^{2}f_{0}^{\prime}\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}[\bm{E}\cdot\bm{v}(\bm{k}^{\prime})].$
(117)

Then we should take into account $q_{z}$ and then $\bm{E}$. In the next step
we find $f^{(Ewq)}$ which satisfies

$iq_{z}v_{z}f^{(Ew)}=-{f^{(Ewq)}\over\tau_{\omega}},$ (118)

and we get

$f^{(Ewq)}=iq_{z}v_{z}e\tau_{\omega}^{3}f_{0}^{\prime}\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}[\bm{E}\cdot\bm{v}(\bm{k}^{\prime})].$
(119)

Then we search for the correction $f^{(EwqE)}$:

${e\over\hbar}\bm{E}_{\perp}^{*}\cdot\partialderivative{f^{(Ewq)}}{\bm{k}_{\perp}}+c.c.=-{f^{(EwqE)}\over\tau}.$
(120)

It allows for calculation of the corresponding contribution to the PGE
current:

$\bm{j}_{\perp}^{(EwqE)}=2e\sum_{\bm{k}}\bm{v}_{\perp}f^{(EwqE)}.$ (121)

Substituting $f^{(EwqE)}$ and integrating by parts we obtain

$\bm{j}_{\perp}^{(EwqE)}=2e^{2}\sum_{\bm{k}}\bm{v}\tau^{\prime}(\bm{E}^{*}\cdot\bm{v})f^{(Ewq)}+c.c.$
(122)

Substituting here $f^{(Ewq)}$ we get

$\displaystyle\bm{j}_{\perp}^{(EwqE)}$
$\displaystyle=2iq_{z}e^{3}\sum_{\bm{k}\bm{k}^{\prime}}\tau^{\prime}\tau_{\omega}^{3}f_{0}^{\prime}\bm{v}(\bm{E}^{*}\cdot\bm{v})v_{z}W^{w}_{\bm{k}\bm{k}^{\prime}}[\bm{E}\cdot\bm{v}(\bm{k}^{\prime})]+c.c.$
(123)

This yields a contribution to the PDE constant $\tilde{T}$:

$\tilde{T}_{EwqE}=2e^{3}\sum_{\bm{k}}\tau^{\prime}\xi_{w}{{\rm
Im}(\tau_{\omega}^{3})\over\tau}f_{0}^{\prime}.$ (124)

Now we put $q_{z}=0$ and take into account the radiation magnetic field
$\tilde{\bm{B}}$. If we do this after accounting for $E$ and $w$, then we get
a steady-state correction $f^{(Ew\tilde{B})}$ which satisfies the equation

${e\over\hbar
c}(\bm{v}\times\tilde{\bm{B}}^{*})\cdot\partialderivative{f^{(Ew)}}{\bm{k}}+c.c.=-{f^{(Ew\tilde{B})}\over\tau}.$
(125)

Solving this algebraic equation and calculating the current
$\bm{j}^{(Ew\tilde{B})}=2e\sum_{\bm{k}}\bm{v}_{\perp}f^{(Ew\tilde{B})}$,
integrating by parts we get

$\bm{j}^{(Ew\tilde{B})}={2e^{2}\over
mc}\sum_{\bm{k}}\tau(\bm{v}\times\tilde{\bm{B}}^{*})f^{(Ew)}.$ (126)

Substituting $f^{(Ew)}$ we see that this contribution is zero because it is
proportional to an average (106).

Finally, we have one more contribution to the PDE current coming from the
correction to the distribution function $f^{(E\tilde{B}w)}$ obtained by
account for $E$, $\tilde{\bm{B}}_{\perp}$ and then $w$. It is found from the
equation

$\sum_{\bm{k}^{\prime}}W^{w}_{\bm{k}\bm{k}^{\prime}}f^{(E\tilde{B})}_{\bm{k}^{\prime}}={f^{(E\tilde{B}w)}\over\tau},$
(127)

where the steady-state correction $f^{(E\tilde{B})}$ is given by

$f^{(E\tilde{B})}={e^{2}\over
mc}v_{z}\tau\tau_{\omega}f_{0}^{\prime}(\tilde{\bm{B}}^{*}\times\bm{E})+c.c.$
(128)

We see that $f^{(E\tilde{B})}\propto q_{z}\absolutevalue{\bm{E}}^{2}$, i.e. it
is polarization-independent. Moreover, this contribution to the PDE current is
zero because it is also proportional to the value (106).

We obtained that the PDE current is described by the phenomenological Eq. (68)
where the constant $\tilde{T}$ is given by

$\tilde{T}=\tilde{T}_{EqEw}+\tilde{T}_{EqwE}+\tilde{T}_{EwqE},$ (129)

which yields Eq. (71) from the main text.

##  Appendix D Microscopic calculation of MLPDE current

We consider the geometry with $\bm{q}\parallel z$, when the electric and
magnetic fields of the radiation, $\bm{E}_{\perp}$ and
$\tilde{\bm{B}}_{\perp}$ are perpendicular to the $z$ axis, and the external
magnetic field $\bm{B}\parallel x$. The hole distribution function $f(\bm{k})$
satisfies the Boltzmann kinetic equation where the field term contains the
forces of the radiation electric field and two Lorentz forces – of the static
magnetic field $B_{x}$ and the ac radiation magnetic field
$\tilde{\bm{B}}_{\perp}$:

$\displaystyle\partialderivative{f}{t}+iq_{z}v_{z}f$
$\displaystyle+{e\over\hbar}\bm{E}_{\perp}(t)\cdot\partialderivative{f}{\bm{k}_{\perp}}$
$\displaystyle+{e\over\hbar
c}[\bm{v}\times(\bm{B}+\tilde{\bm{B}}_{\perp}(t))]\cdot\partialderivative{f}{\bm{k}}={\rm
St}[f].$ (130)

Here $\tilde{\bm{B}}_{\perp}(t)=\tilde{\bm{B}}_{\perp}\exp(-i\omega t)+c.c.$,
${\rm St}[f]$ stands for the elastic collision integral describing
isotropization of the distribution over the isoenergetic surface
$\varepsilon_{k}={\rm const}$. In what follows we account either for
$q_{z}\neq 0$ or for $\tilde{B}\neq 0$ in the kinetic equation because they
give contributions to the the PDE currents of the same order.

First, we find the linear in $\bm{E}$ correction to the distribution function
(98). Then, depending on the contribution under study, we iterate the kinetic
equation in small parameters $q_{z}$, $\bm{E}_{\perp}^{*}$ or
$\tilde{\bm{B}}_{\perp}$.

Account for $q_{z}$ yields the correction $f^{(Eq)}$ (100). If we then account
for the second power of $\bm{E}_{\perp}$, we find the time-independent
correction $f^{(EqE)}$ (C). Let us now take into account the last
perturbation, the magnetic field $B_{x}$, and find the current-carrying part
of the distribution function $f^{(EqEB)}$. The magnetic field enters the
kinetic equation via the Lorentz force:

${e\over\hbar
c}(\bm{v}\times\bm{B})\cdot\partialderivative{f^{(EqE)}}{\bm{k}}=-{f^{(EqEB)}\over\tau}.$
(131)

The solution reads

$f^{(EqEB)}=-{m\omega_{c}\tau\over\hbar}\quantity(v_{z}\partial_{k_{y}}-v_{y}\partial_{k_{z}})f^{(EqE)},$
(132)

where $\omega_{c}=eB_{x}/(mc)$ is the cyclotron frequency. Then we calculate
the $EqEB$ contribution to the MLPDE current which flows in the $(xy)$ plane
in this geometry:

$\bm{j}_{\perp}^{(EqEB)}=2e\sum_{\bm{k}}\bm{v}_{\perp}f^{(EqEB)}.$ (133)

Substituting $f^{(EqEB)}$ from Eq. (132) and integrating by parts we obtain
that this contribution is zero:

$\bm{j}_{\perp}^{(EqEB)}=0$ (134)

(the simplest way to see this is to calculate the $x$ component of the
current).

Now we calculate $EqBE$ contribution to the MLPDE current. We find the time-
dependent correction $f^{(EqB)}$:

${e\over\hbar
c}(\bm{v}\times\bm{B})\cdot\partialderivative{f^{(Eq)}}{\bm{k}}=-{f^{(EqB)}\over\tau_{\omega}}.$
(135)

The solution of this equation reads

$f^{(EqB)}=-ieq_{z}\omega_{c}\tau_{\omega}^{3}f_{0}^{\prime}\quantity[(v_{z}^{2}-v_{y}^{2})E_{y}-v_{x}v_{y}E_{x}].$
(136)

The static correction $f^{(EqBE)}$ yielding the contribution to the current
satisfies the equation

${e\over\hbar}\bm{E}_{\perp}^{*}\cdot\partialderivative{f^{(EqB)}}{\bm{k}_{\perp}}+c.c.=-{f^{(EqBE)}\over\tau}.$
(137)

Solving this equation we get for the $y$ component of the current

$j_{y}^{(EqBE)}=2e^{2}\sum_{\bm{k}}f^{(EqB)}\quantity[{\tau\over
m}E_{y}^{*}+\tau^{\prime}v_{y}\quantity(\bm{E}_{\perp}^{*}\cdot\bm{v}_{\perp})]+c.c.$
(138)

Substituting here $f^{(EqB)}$ and averaging over directions of $\bm{k}$ we
obtain

$\displaystyle j_{y}^{(EqBE)}=2ie^{3}q_{z}\omega_{c}$ (139)
$\displaystyle\times\sum_{\bm{k}}\tau_{\omega}^{3}f_{0}^{\prime}\tau^{\prime}{v^{4}\over
15}(2\absolutevalue{E_{y}}^{2}+\absolutevalue{E_{x}}^{2})+c.c.$

Noting that $\absolutevalue{E_{x,y}}^{2}=\absolutevalue{\bm{E}}^{2}(1\pm
P_{\rm lin})/2$, and introducing
$r=\differential\ln\tau/\differential\ln\varepsilon_{k}$, we get a
contribution to the polarization-dependent MLPGE current Eq. (77),
$j_{y}^{(EqBE)}=-B_{x}q_{z}\absolutevalue{\bm{E}}^{2}S_{l}^{(EqBE)}{P}_{\rm
lin}$ where $S_{l}^{(EqBE)}$ is given by

$S_{l}^{(EqBE)}=-{8re^{4}\over 15m^{3}c}\sum_{\bm{k}}{\rm
Im}(\tau_{\omega}^{3})f_{0}^{\prime}\tau\varepsilon_{k}.$ (140)

At $\omega\tau\gg 1$ we have ${\rm Im}(\tau_{\omega}^{3})=-1/\omega^{3}$. This
gives

$S_{l}^{(EqBE)}={8re^{4}\over
15m^{3}c\omega^{3}}\sum_{\bm{k}}f_{0}^{\prime}\tau\varepsilon_{k}=-{8r\Gamma(r+5/2)pe^{4}\tau_{T}\over
15\sqrt{\pi}m^{3}c\omega^{3}}.$ (141)

Let us turn now to the $EBqE$ contribution to the MLPDE current. For its
calculation, we account for the Lorentz force in the kinetic equation at the
second step and find the correction $f^{(EB)}\propto E_{\perp}B_{x}$:

${e\over\hbar
c}(\bm{v}\times\bm{B})\cdot\partialderivative{f^{(E)}}{\bm{k}}=-{f^{(EB)}\over\tau_{\omega}}.$
(142)

Solution of this equation reads

$f^{(EB)}=e\omega_{c}\tau_{\omega}^{2}f_{0}^{\prime}E_{y}v_{z}.$ (143)

Then we should take into account $q_{z}$ and then $\bm{E}$. In the next step
we find $f^{(EBq)}$ which satisfies

$iq_{z}v_{z}f^{(EB)}=-{f^{(EBq)}\over\tau_{*}},$ (144)

where $\tau_{*}$ equals to $\tau_{\omega}$ or $i/\omega$ for the
anisotropic/isotropic part of the left-hand side, and we get

$f^{(EBq)}=e\omega_{c}q_{z}E_{y}f_{0}^{\prime}\quantity[-i\tau_{\omega}^{3}\quantity(v_{z}^{2}-{v^{2}\over
3})+{\tau_{\omega}^{2}\over\omega}{v^{2}\over 3}].$ (145)

Then we search for the correction $f^{(EBqE)}$:

${e\over\hbar}\bm{E}_{\perp}^{*}\cdot\partialderivative{f^{(EBq)}}{\bm{k}_{\perp}}+c.c.=-{f^{(EBqE)}\over\tau}.$
(146)

It allows for calculation of the corresponding contribution to the MLPGE
current:

$\bm{j}_{\perp}^{(EBqE)}=2e\sum_{\bm{k}}\bm{v}_{\perp}f^{(EBqE)}.$ (147)

Integration by parts and averaging over directions of $\bm{k}$ yields

$\bm{j}_{\perp}^{(EBqE)}=2e^{2}\bm{E}^{*}\sum_{\bm{k}}\quantity({\tau\over
m}+\tau^{\prime}{v^{2}\over 3})f^{(EBq)}+c.c.$ (148)

We see that only the isotropic part of $f^{(EBq)}$ which contains a factor
$1/\omega$ contributes to the current. Using
$\tau^{\prime}=r\tau/\varepsilon_{k}$, we obtain

$\bm{j}_{\perp}^{(EBqE)}={4e^{3}\omega_{c}q_{z}\over
3m^{2}\omega}\quantity(1+{2r\over
3})\bm{E}^{*}E_{y}\sum_{\bm{k}}f_{0}^{\prime}\varepsilon_{k}\tau\tau_{\omega}^{2}+c.c.$
(149)

In particular, it means that the $EBqE$ contribution to $S_{l}$ is given by

$S_{l}^{(EBqE)}={4e^{4}\over 3m^{3}c\omega}\quantity(1+{2r\over
3})\sum_{\bm{k}}f_{0}^{\prime}\varepsilon_{k}\tau{\rm Re}(\tau_{\omega}^{2}).$
(150)

At $\omega\tau\gg 1$ when ${\rm Re}(\tau_{\omega}^{2})=-1/\omega^{2}$ we have

$S_{l}^{(EBqE)}={4e^{4}\over 3m^{3}c\omega^{3}}\quantity(1+{2r\over
3})\sum_{\bm{k}}(-f_{0}^{\prime})\varepsilon_{k}\tau.$ (151)

For Boltzmann statistics we get

$S_{l}^{(EBqE)}={4pe^{4}\tau_{T}\over
3\sqrt{\pi}m^{3}c\omega^{3}}\quantity(1+{2r\over 3})\Gamma\quantity(r+{5\over
2}).$ (152)

Now we put $q_{z}=0$ and take into account the radiation magnetic field
$\tilde{\bm{B}}$. If we do this after accounting for $E$ and $B_{x}$, then we
get a steady-state correction $f^{(EB\tilde{B})}$ which satisfies the equation

${e\over\hbar
c}(\bm{v}\times\tilde{\bm{B}}^{*})\cdot\partialderivative{f^{(EB)}}{\bm{k}}+c.c.=-{f^{(EB\tilde{B})}\over\tau}.$
(153)

It solution reads

$f^{(EB\tilde{B})}={e^{2}\omega_{c}\tau\over
mc}\tau_{\omega}^{2}f_{0}^{\prime}E_{y}\quantity(\tilde{\bm{B}}^{*}\times\bm{v})_{z}+c.c.$
(154)

Calculating the current
$\bm{j}^{(EB\tilde{B})}=2e\sum_{\bm{k}}\bm{v}_{\perp}f^{(EB\tilde{B})}$, we
obtain

$j_{y}^{(EB\tilde{B})}={4e^{3}\omega_{c}\over
3m^{2}c}E_{y}\tilde{B_{x}^{*}}\sum_{\bm{k}}\varepsilon_{k}\tau_{\omega}^{2}f_{0}^{\prime}\tau+c.c.$
(155)

Using the relation between the radiation magnetic and electric fields
$\tilde{\bm{B}}=c\bm{q}\times\bm{E}/\omega$ we obtain that this contribution
yields $S_{l}$ in the form

$S_{l}^{(EB\tilde{B})}=-{4e^{4}\over
3m^{3}c\omega}\sum_{\bm{k}}\varepsilon_{k}{\rm
Re}(\tau_{\omega}^{2})f_{0}^{\prime}\tau.$ (156)

At $\omega\tau\gg 1$ we have

$S_{l}^{(EB\tilde{B})}={4e^{4}\over
3m^{3}c\omega^{3}}\sum_{\bm{k}}\varepsilon_{k}f_{0}^{\prime}\tau.$ (157)

For Boltzmann statistics and $\tau(\varepsilon_{k})\propto\varepsilon_{k}^{r}$
we get

$S_{l}^{(EB\tilde{B})}={4pe^{4}\over
3\sqrt{\pi}m^{3}c\omega^{3}}\Gamma\quantity(r+{5\over 2})\propto T^{r}.$ (158)

Finally, we have one more contribution to the MPDE current coming from the
correction to the distribution function $f^{(E\tilde{B}B)}$ obtained by
account for $E$, $\tilde{\bm{B}}_{\perp}$ and then $B_{x}$. It is found from
the equation

${e\over\hbar
c}(\bm{v}\times\bm{B})\cdot\partialderivative{f^{(E\tilde{B})}}{\bm{k}}=-{f^{(E\tilde{B}B)}\over\tau},$
(159)

where the steady-state correction $f^{(E\tilde{B})}$ is given by

$f^{(E\tilde{B})}={e^{2}\over
mc}v_{z}\tau\tau_{\omega}f_{0}^{\prime}(\tilde{\bm{B}}^{*}\times\bm{E})_{z}+c.c.$
(160)

Calculating the current density we get

$j_{y}^{(E\tilde{B}B)}=2e\sum_{\bm{k}}v_{y}f^{(E\tilde{B}B)}=2e\omega_{c}\sum_{\bm{k}}v_{z}\tau
f^{(E\tilde{B})}.$ (161)

Substitution of $f^{(E\tilde{B})}$ results in a polarization-independent
contribution only.

For high frequencies $\omega\tau\gg 1$ we obtained Eq. (85) where the pre-
factor $a$ is a sum of various contributions summarized in Table 1 of the main
text. For arbitrary frequencies, the MLPDE constant
$S_{l}=S_{l}^{(EqBE)}+S_{l}^{(EBqE)}+S_{l}^{(EB\tilde{B})}$ is given by Eq.
(87) of the main text.

## References

  * Button  _et al._ [1969] K. J. Button, G. Landwehr, C. C. Bradley, P. Grosse, and B. Lax, Quantum effects in cyclotron resonance in $p$-type tellurium, Physical Review Letters 23, 14 (1969).
  * Bresler  _et al._ [1970] M. S. Bresler, V. G. Veselago, Y. V. Kosichkin, G. E. Pikus, I. I. Farbshtein, and S. S. Shalyt, Energy scheme of the tellurium valence band, Sov. JETP 30, 799 (1970).
  * Saffert  _et al._ [1974] R. Saffert, J. Schapawalow, G. Landwehr, and E. Gmelin, Nernst-ettingshausen and seebeck effect of pure and electron-irradiated tellurium at low-temperatures, phys. stat. sol. (b) 61, 509 (1974).
  * von Klitzing and Landwehr [1971] K. von Klitzing and G. Landwehr, Surface quantum states in tellurium, Solid State Communications 9, 2201 (1971).
  * Englert  _et al._ [1977] T. Englert, K. von Klitzing, R. Silbermann, and G. Landwehr, Influence of surface on galvanomagnetic properties of tellurium, phys. stat. sol. (b) 81, 119 (1977).
  * Nomura [1960] K. C. Nomura, Optical activity in tellurium, Phys. Rev. Lett. 5, 500 (1960).
  * Ivchenko and Pikus [1974] E. Ivchenko and G. E. Pikus, Natural optical activity of semiconductor (tellurium), Sov. Phys. Solid State 16, 1933 (1974).
  * Fukuda  _et al._ [1975] S. Fukuda, T. Shiosaki, and A. Kawabata, Infrared optical-activity in tellurium, phys. stat. sol. (b) 68, K107 (1975).
  * Stolze  _et al._ [1977] H. Stolze, M. Lutz, and P. Grosse, Optical-activity of tellurium, phys. stat. sol. (b) 82, 457 (1977).
  * Ivchenko and Pikus [1978] E. L. Ivchenko and G. E. Pikus, New photogalvanic effect in gyrotropic crystals, JETP Lett. 27, 604 (1978).
  * Asnin  _et al._ [1978] V. M. Asnin, A. A. Bakun, A. M. Danishevskii, E. L. Ivchenko, G. E. Pikus, and A. A. Rogachev, Observation of a photo-emf that depends on the sign of the circular-polarization of the light, JETP Lett. 28, 74 (1978).
  * Shalygin  _et al._ [2016] V. A. Shalygin, M. D. Moldavskaya, S. N. Danilov, I. I. Farbshtein, and L. E. Golub, Circular photon drag effect in bulk tellurium, Phys. Rev. B 93, 045207 (2016).
  * Vorob’ev  _et al._ [1979] L. E. Vorob’ev, E. L. Ivchenko, G. E. Pikus, I. I. Farbshtein, V. A. Shalygin, and A. I. Shturbin, Optical-activity in tellurium induced by a current, JETP Lett. 29, 441 (1979).
  * Shalygin [2022] V. A. Shalygin, Current-induced optical activity: First observation and comprehensive study, in  _Optics and Its Applications_, edited by D. Blaschke, D. Firsov, A. Papoyan, and H. A. Sarkisyan (Springer International Publishing, Cham, 2022) pp. 1–19.
  * Wang  _et al._ [2018] Y. Wang, G. Qiu, R. Wang, S. Huang, Q. Wang, Y. Liu, Y. Du, W. A. Goddard, III, M. J. Kim, X. Xu, P. D. Ye, and W. Wu, Field-effect transistors made from solution-grown two-dimensional tellurene, Nat. Electron. 1, 228 (2018).
  * Wu  _et al._ [2018] W. Wu, G. Qiu, Y. Wang, R. Wang, and P. Ye, Tellurene: its physical properties, scalable nanomanufacturing, and device applications, Chem. Soc. Rev. 47, 7203 (2018).
  * Shi  _et al._ [2020] Z. Shi, R. Cao, K. Khan, A. K. Tareen, X. Liu, W. Liang, Y. Zhang, C. Ma, Z. Guo, X. Luo, and H. Zhang, Two-dimensional tellurium: Progress, challenges, and prospects, Nano-Micro Letters 12, 99 (2020).
  * Zhang  _et al._ [2021] L. Zhang, T. Gong, Z. Yu, H. Dai, Z. Yang, G. Chen, J. Li, R. Pan, H. Wang, Z. Guo, H. Zhang, and X. Fu, Recent advances in hybridization, doping, and functionalization of 2d xenes, Adv. Funct. Mater. 31, 2005471 (2021).
  * Xu  _et al._ [2020] N. Xu, P. Ma, S. Fu, X. Shang, S. Jiang, S. Wang, D. Li, and H. Zhang, Tellurene-based saturable absorber to demonstrate large-energy dissipative soliton and noise-like pulse generations, Nanophotonics 9, 2783 (2020).
  * Yan  _et al._ [2022] Z. Yan, H. Yang, Z. Yang, C. Ji, G. Zhang, Y. Tu, G. Du, S. Cai, and S. Lin, Emerging two-dimensional tellurene and tellurides for broadband photodetectors, Small 18, 2200016 (2022).
  * Agapito  _et al._ [2013] L. A. Agapito, N. Kioussis, W. A. Goddard, III, and N. P. Ong, Novel family of chiral-based topological insulators: Elemental tellurium under strain, Phys. Rev. Lett. 110, 176401 (2013).
  * Hirayama  _et al._ [2015] M. Hirayama, R. Okugawa, S. Ishibashi, S. Murakami, and T. Miyake, Weyl node and spin texture in trigonal tellurium and selenium, Phys. Rev. Lett. 114, 206401 (2015).
  * Murakami  _et al._ [2017] S. Murakami, M. Hirayama, R. Okugawa, and T. Miyake, Emergence of topological semimetals in gap closing in semiconductors without inversion symmetry, Science Advances 3, e1602680 (2017).
  * Ideue  _et al._ [2019] T. Ideue, M. Hirayama, H. Taiko, T. Takahashi, M. Murase, T. Miyake, S. Murakami, T. Sasagawa, and Y. Iwasa, Pressure-induced topological phase transition in noncentrosymmetric elemental tellurium, Proceedings of the National Academy of Sciences 116, 25530 (2019).
  * Zhang  _et al._ [2020] N. Zhang, G. Zhao, L. Li, P. Wang, L. Xie, B. Cheng, H. Li, Z. Lin, C. Xi, J. Ke, M. Yang, J. He, Z. Sun, Z. Wang, Z. Zhang, and C. Zeng, Magnetotransport signatures of Weyl physics and discrete scale invariance in the elemental semiconductor tellurium, Proceedings of the National Academy of Sciences 117, 11337 (2020).
  * Oliveira  _et al._ [2021] J. F. Oliveira, M. B. Fontes, M. Moutinho, S. E. Rowley, E. Baggio-Saitovitch, M. B. Silva Neto, and C. Enderlein, Pressure-induced anderson-mott transition in elemental tellurium, Communications Materials 2, 1 (2021).
  * Glazov  _et al._ [2022] M. M. Glazov, E. L. Ivchenko, and M. O. Nestoklon, Effect of pressure on the electronic band structure and circular photocurrent in tellurium, JETP 135, 575 (2022).
  * Glazov and Ganichev [2014] M. M. Glazov and S. D. Ganichev, High frequency electric field induced nonlinear effects in graphene, Phys. Rep. 535, 101 (2014).
  * Otteneder  _et al._ [2020] M. Otteneder, S. Hubmann, X. Lu, D. A. Kozlov, L. E. Golub, K. Watanabe, T. Taniguchi, D. K. Efetov, and S. D. Ganichev, Terahertz photogalvanics in twisted bilayer graphene close to the second magic angle, Nano Lett. 20, 7152 (2020).
  * Ivchenko and Ganichev [2017] E. L. Ivchenko and S. D. Ganichev, Spin–photogalvanics, in  _Spin physics in semiconductors_ , edited by M. I. Dyakonov (Springer, 2017).
  * Plank and Ganichev [2018] H. Plank and S. D. Ganichev, A review on terahertz photogalvanic spectroscopy of Bi2Te3\- and Sb2Te3-based three dimensional topological insulators, Solid-State Electron. 147, 44 (2018).
  * Ishizuka  _et al._ [2016] H. Ishizuka, T. Hayata, M. Ueda, and N. Nagaosa, Emergent electromagnetic induction and adiabatic charge pumping in noncentrosymmetric Weyl semimetals, Phys. Rev. Lett. 117, 216601 (2016).
  * Golub  _et al._ [2017] L. E. Golub, E. L. Ivchenko, and B. Z. Spivak, Photocurrent in gyrotropic Weyl semimetals, JETP Lett. 105, 782 (2017).
  * de Juan  _et al._ [2017] F. de Juan, A. G. Grushin, T. Morimoto, and J. E. Moore, Quantized circular photogalvanic effect in Weyl semimetals, Nat. Comm. 8, 15995 (2017).
  * Chan  _et al._ [2017] C.-K. Chan, N. H. Lindner, G. Refael, and P. A. Lee, Photocurrents in Weyl semimetals, Phys. Rev. B 95, 041104 (2017).
  * Ma  _et al._ [2017] Q. Ma, S.-Y. Xu, C.-K. Chan, C.-L. Zhang, G. Chang, Y. Lin, W. Xie, T. Palacios, H. Lin, S. Jia, P. A. Lee, P. Jarillo-Herrero, and N. Gedik, Direct optical detection of Weyl fermion chirality in a topological semimetal, Nat. Phys. 13, 842 (2017).
  * Ji  _et al._ [2019] Z. Ji, G. Liu, Z. Addison, W. Liu, P. Yu, H. Gao, Z. Liu, A. M. Rappe, C. L. Kane, E. J. Mele, and R. Agarwal, Spatially dispersive circular photogalvanic effect in a Weyl semimetal, Nat. Mat. 18, 955 (2019).
  * Ma  _et al._ [2022] J. Ma, B. Cheng, L. Li, Z. Fan, H. Mu, J. Lai, X. Song, D. Yang, J. Cheng, Z. Wang, C. Zeng, and D. Sun, Unveiling Weyl-related optical responses in semiconducting tellurium by mid-infrared circular photogalvanic effect, Nat. Comm. 13, 5425 (2022).
  * Note [1] Note that due to growth conditions the hexagon was slightly distorted.
  * Bauer  _et al._ [1973] G. Bauer, H. Kahilert, K. von Klitzing, and G. Landwehr, Time-dependent non-ohmic conductivity and magnetoresistance in p-tellurium at 2 k, Physica Status Solidi (b) 59, 479 (1973).
  * Ribakovs and Gundjian [1977] G. Ribakovs and A. A. Gundjian, Theory of the photon drag effect in tellurium, Journal of Applied Physics 48, 4609 (1977).
  * Ganichev and Prettl [2006] S. D. Ganichev and W. Prettl,  _Intense Terahertz Excitation of Semiconductors_ (Oxford University Press, Oxford, 2006).
  * Ganichev  _et al._ [2003a] S. D. Ganichev, V. V. Bel’kov, P. Schneider, E. L. Ivchenko, S. A. Tarasenko, W. Wegscheider, D. Weiss, D. Schuh, E. V. Beregulin, and W. Prettl, Resonant inversion of the circular photogalvanic effect in n-doped quantum wells, Phys. Rev. B 68, 035319 (2003a).
  * Ganichev  _et al._ [2003b] S. D. Ganichev, P. Schneider, V. V. Bel’kov, E. L. Ivchenko, S. A. Tarasenko, W. Wegscheider, D. Weiss, D. Schuh, B. N. Murdin, P. J. Phillips, C. R. Pidgeon, D. G. Clarke, M. Merrick, P. Murzyn, E. V. Beregulin, and W. Prettl, Spin-galvanic effect due to optical spin orientation in n-type gaas quantum well structures, Phys. Rev. B 68, 081302 (2003b).
  * Note [2] The number of studied frequencies was defined by the laser tunability and availability of the polarizers operating at specific frequencies.
  * Ganichev  _et al._ [1993] S. D. Ganichev, W. Prettl, and P. G. Huggard, Phonon assisted tunnel ionization of deep impurities in the electric field of far-infrared radiation, Phys. Rev. Lett. 71, 3882 (1993).
  * Ganichev  _et al._ [1995] S. D. Ganichev, I. N. Yassievich, W. Prettl, J. Diener, B. K. Meyer, and K. W. Benz, Tunneling ionization of AutolocalizedDX-centers in terahertz fields, Phys. Rev. Lett. 75, 1590 (1995).
  * Ganichev  _et al._ [1998] S. D. Ganichev, E. Ziemann, T. Gleim, W. Prettl, I. N. Yassievich, V. I. Perel, I. Wilke, and E. E. Haller, Carrier tunneling in high-frequency electric fields, Phys. Rev. Lett. 80, 2409 (1998).
  * Ganichev  _et al._ [1985] S. D. Ganichev, Y. V. Terent’ev, and I. D. Yaroshetskii, Photon-drag photodetectors for the far-ir and submillimeter regions, Pis’ma Zh. Tekh. Fiz. 11, 46 (1985), [Sov. Tech. Phys. Lett. 11, 20 (1985)].
  * Bel’kov  _et al._ [2005] V. V. Bel’kov, S. D. Ganichev, E. L. Ivchenko, S. A. Tarasenko, W. Weber, S. Giglberger, M. Olteanu, H. P. Tranitz, S. N. Danilov, P. Schneider, W. Wegscheider, D. Weiss, and W. Prettl, Magneto-gyrotropic photogalvanic effects in semiconductor quantum wells, J. Phys. Condens. Matter 17, 3405 (2005).
  * Note [3] While in magnetic fields up to about $\pm~{}1.2$ T the current linearly depends on the magnetic field at higher fields, it possibly tends to deviate from this behavior. To justify this tendency experiments at higher magnetic fields are needed, which was out of scope of the present work.
  * Note [4] Note that in the geometry applying $\lambda$-quarter plate the $\sin 4\varphi/2$ corresponds to the Stokes parameter describing the degree of linear polarization and the polarization ellipse orientation, which in experiments applying $\lambda$-half plate is given by $\sin 2\alpha$, see below and Refs. [53, 50].
  * Saleh and Teich [2019] B. E. A. Saleh and M. C. Teich,  _Fundamentals of Photonics_ (John Wiley and Sons Ltd., 2019).
  * Note [5] Note that the trigonal photocurrents given by coefficients $\chi$ depend on the polarization plane orientation as the 2nd angular harmonics because projections of the current onto fixed axes are measured. If one detects a direction of the photocurrent as a function of the light polarization then it has a form of the 3rd angular harmonics.
  * Fischer  _et al._ [1973] D. Fischer, E. Bangert, and P. Grosse, Intervalence band transitions in tellurium. I. Polarization ${E\parallel c}$, physica status solidi (b) 55, 527 (1973).
  * Bangert  _et al._ [1973] E. Bangert, D. Fischer, and P. Grosse, Intervalence band transitions in tellurium. II. Polarization ${E\perp c}$, physica status solidi (b) 59, 419 (1973).
  * Averkiev  _et al._ [1984a] N. S. Averkiev, V. M. Asnin, A. A. Bakun, A. M. Danishevskii, E. L. Ivchenko, G. E. Pikus, and A. A. Rogachev, Circular photogalvanic effect in tellurium. I. Theory, Sov. Phys. Semicond. 18, 397 (1984a).
  * Averkiev  _et al._ [1984b] N. S. Averkiev, V. M. Asnin, A. A. Bakun, A. M. Danishevskii, E. L. Ivchenko, G. E. Pikus, and A. A. Rogachev, Circular photogalvanic effect in tellurium. II. Experiment, Sov. Phys. Semicond. 18, 402 (1984b).
  * Yang  _et al._ [2022] D.-Q. Yang, L.-Q. Zhu, J.-L. Wang, W. Xia, J.-Z. Zhang, K. Jiang, L.-Y. Shang, Y.-W. Li, and Z.-G. Hu, Band structure and lattice vibration of elemental tellurium investigated by temperature-dependent mid-and-far infrared transmission and raman spectroscopy, physica status solidi (b) 259, 2100625 (2022).
  * Note [6] In Eq. (22) the factor $\absolutevalue{L}^{2}/E_{g}$ absent in Eq. (25) of Ref. [57] is restored.
  * Sturman [2020] B. I. Sturman, Ballistic and shift currents in the bulk photovoltaic effect theory, Physics-Uspekhi 63, 407 (2020).
  * Belinicher  _et al._ [1982] V. I. Belinicher, E. L. Ivchenko, and B. I. Sturman, Kinetic theory of the displacement photovoltaic effect in piezoelectric, Zh. Eksp. Teor. Fiz. 83, 649 (1982), [JETP 56, 359 (1982)].
  * Leppenen and Golub [2023] N. V. Leppenen and L. E. Golub, Linear photogalvanic effect in surface states of topological insulators, Phys. Rev. B 107, L161403 (2023).
  * Belinicher and Sturman [1980] V. I. Belinicher and B. I. Sturman, The photogalvanic effect in media lacking a center of symmetry, Sov. Phys. Usp. 23, 199 (1980), [Usp. Fiz. Nauk 130, 415 (1980)].
  * Olbrich  _et al._ [2014] P. Olbrich, L. E. Golub, T. Herrmann, S. N. Danilov, H. Plank, V. V. Bel’kov, G. Mussler, C. Weyrich, C. M. Schneider, J. Kampmeier, D. Grützmacher, L. Plucinski, M. Eschbach, and S. D. Ganichev, Room-temperature high-frequency transport of dirac fermions in epitaxially grown Sb2Te3\- and Bi2Te3-based topological insulators, Phys. Rev. Lett. 113, 096601 (2014).
  * Hild  _et al._ [2023] M. Hild, L. E. Golub, A. Fuhrmann, M. Otteneder, M. Kronseder, M. Matsubara, T. Kobayashi, D. Oshima, A. Honda, T. Kato, J. Wunderlich, C. Back, and S. D. Ganichev, Terahertz spin ratchet effect in magnetic metamaterials, Phys. Rev. B 107, 155419 (2023).
  * Plank  _et al._ [2016] H. Plank, L. E. Golub, S. Bauer, V. V. Bel’kov, T. Herrmann, P. Olbrich, M. Eschbach, L. Plucinski, C. M. Schneider, J. Kampmeier, M. Lanius, G. Mussler, D. Grützmacher, and S. D. Ganichev, Photon drag effect in $({\mathrm{Bi}}_{1-x}{\mathrm{Sb}}_{x}){}_{2}{\mathrm{Te}}_{3}$ three-dimensional topological insulators, Phys. Rev. B 93, 125434 (2016).
  * Perel and Pinskii [1973] V. I. Perel and Y. M. Pinskii, Constant current in conducting media due to a high-frequency electron electromagnetic field, Sov. Phys. - Solid State 15, 688 (1973).
  * Weber  _et al._ [2008] W. Weber, L. E. Golub, S. N. Danilov, J. Karch, C. Reitmaier, B. Wittmann, V. V. Bel’kov, E. L. Ivchenko, Z. D. Kvon, N. Q. Vinh, A. F. G. van der Meer, B. Murdin, and S. D. Ganichev, Quantum ratchet effects induced by terahertz radiation in gan-based two-dimensional structures, Phys. Rev. B 77, 245304 (2008).
  * Drexler  _et al._ [2013] C. Drexler, S. A. Tarasenko, P. Olbrich, J. Karch, M. Hirmer, F. Müller, M. Gmitra, J. Fabian, R. Yakimova, S. Lara-Avila, S. Kubatkin, M. Wang, R. Vajtai, P. M. Ajayan, J. Kono, and S. D. Ganichev, Magnetic quantum ratchet effect in graphene, Nature Nanotechnology 8, 104 (2013).
  * Ganichev  _et al._ [2017] S. D. Ganichev, D. Weiss, and J. Eroms, Terahertz electric field driven electric currents and ratchet effects in graphene, Annalen der Physik 529, 1600406 (2017).