Symmetry regimes for circular photocurrents in monolayer MoSe2

In monolayer transition metal dichalcogenides helicity-dependent charge and spin photocurrents can emerge, even without applying any electrical bias, due to circular photogalvanic and photon drag effects. Exploiting such circular photocurrents (CPCs) in devices, however, requires better understanding of their behavior and physical origin. Here, we present symmetry, spectral, and electrical characteristics of CPC from excitonic interband transitions in a MoSe2 monolayer. The dependence on bias and gate voltages reveals two different CPC contributions, dominant at different voltages and with different dependence on illumination wavelength and incidence angles. We theoretically analyze symmetry requirements for effects that can yield CPC and compare these with the observed angular dependence and symmetries that occur for our device geometry. This reveals that the observed CPC effects require a reduced device symmetry, and that effects due to Berry curvature of the electronic states do not give a significant contribution.


Introduction
2][3][4][5] The particular band structure of 1L-TMDCs, where two nonequivalent valleys appear at the K and K' points of the 2D Brillouin zone, gives rise to valley-dependent optical selection rules.Specifically, when a 1L-TMDC is illuminated with circularly polarized light with a photon energy close to its bandgap, optical transitions can only take place in one of the two valleys, either K or K', depending on the helicity of the circular polarization, leading to a light-induced valley population imbalance. 5Additionally, monolayer TMDCs present a large spin-orbit splitting, which sign changes between the K and K' valleys, causing a coupling between the spin and valley degrees of freedom. 5As a consequence, different optical processes can be used to generate spin and valley polarized photoresponse in TMDCs, such as the valley-Hall effect 6,7 .For this effect, under circularly polarized illumination, charge carriers in different valleys flow to opposite transverse edges when an in-plane electric field is applied, producing a light helicity-dependent Hall voltage.
The recent observation of helicity-sensitive circular photogalvanic effect (CPGE) 8,9 , both for multilayer and monolayer TMDCs 10,11 , opens another route for producing spin-valley transport through a 1L-TMDC phototransistor.Differently from the valley-Hall effect, which relies on applying an in-plane voltage gradient to the TMDC in order to obtain spin-valley current, the CPGE allows to generate a directed spin-valley current even without applying any voltage, bringing new opportunities for the implementation of active spintronic and valleytronic devices.However, a comprehensive study of this effect and its microscopic origin in 1L-TMDCs is still missing.
In this work we investigate for the first time the spectral and electrical behavior of the helicity-dependent circular photocurrent (CPC) in a 1L-TMDC.By evaluating the spectral response of the CPC in a h-BN encapsulated 1L-MoSe2 phototransistor, we show that the CPC amplitude is maximized when the illumination wavelength matches the A exciton resonance, clearly demonstrating that excitonic absorption plays a central role in the generation of the CPC.We also explore the effect of a drain-source voltage (Vds) on the CPC, revealing two different regimes for small (below 0.4 V) and large voltages, with the CPC changing sign between one regime and the other.Further, we find that the CPC presents very different symmetry upon change of the light incidence angle for the two regimes: For small Vds, the CPC is preserved when the incidence angle is switched from ϕ to -ϕ, whereas for large Vds, inverting the illumination angle ϕ causes a change of sign for the CPC, pointing to a separate physical origin.Recently, it was described 11 that Berry-curvature (BC) at the band edges of 1L-TMDCs can give a contribution to CPC (BC-induced circular photogalvanic effect, BC-CPGE).However, we find that BC-CPGE is not compatible with the angular dependences observed for any of the two CPC regimes.Further, we show that CPC can also emerge in this system due to the circular photon drag effect (CPDE), mostly overlooked in prior literature for 1L-TMDCs.Finally, we show how by applying a gate voltage to modify the Fermi energy of the 1L-MoSe2 channel, one can tune the relative strength of the two contributions at a fixed drain-source voltage, achieving control over the intensity and direction of the helicity-dependent photoresponse.

Circular photocurrent in 1L-TMDC for interband transitions.
When spatial inversion symmetry is broken in the 2D plane in a system with time-reversal symmetry, illumination with circular light can generate a DC photocurrent  ⃗ that behaves as a second order response to the electric field. ⃗ can be written as a series expansion in the light wave vector  ⃗ as Jl = χljk Ej Ek* + Tljk qμ Ej Ek* + (…).Here χljk and Tljk are the photogalvanic and photon drag susceptibility tensors and l, j, k, μ label Cartesian coordinates x, y and z.As we present in Suppl.Info.S6, the device symmetries strongly constrain the tensor components, and they can still vanish for high-symmetry configurations, even for broken inversion symmetry.We consider three different symmetry scenarios: D3h (pristine 1L-MoSe2), C3v (1L-MoSe2 with broken out-of-plane mirror symmetry), and single-mirror symmetry (1L-TMDCs in presence of strain or device inhomogeneities).Comparing the dependence of CPC on illumination angles with the symmetry-allowed CPGE and CPDE contributions, we find that for the low-bias regime our observations are only compatible with a device symmetry of, at most, a single mirror plane.For the high-bias regime the CPC effects are also compatible with C3v symmetry.
In previous reports, the CPC in 1L-TMDCs has been associated with a BC-induced CPGE. 11,12In 1L-TMDCs, the BC takes opposite signs at the K and K' valleys, giving rise to counterpropagating valley currents. 5,6,13Thus, when circularly polarized illumination is used to produce a valley population imbalance, a CPC contribution can appear.In Suppl.Info.S6 we derive the CPGE photocurrent using the Fermi Golden rule.This shows that resonant interband transitions can produce a BC contribution to the CPGE 14 .However, this contribution should maximize for incidence perpendicular to the 2D plane 12 , while our experiments only show nonzero CPC at oblique incidence (see below).

Device fabrication, electrical characterization and setup
The fabricated 1L-MoSe2 field-effect transistor is depicted in Fig. 1a and the actual device is shown in the Suppl.Info.
(Figure S2).To improve the device quality and stability 15,16 , the 1L-MoSe2 channel is encapsulated between a bilayer and a bulk h-BN flake, acting as the top and bottom layers respectively.The 2L-BN/1L-MoSe2/bulk-BN stack is directly built onto a SiO2/doped Si substrate with an oxide thickness of 300 nm.The electrodes are fabricated on top of the structure by ebeam lithography (EBL) and e-beam evaporation of Ti (5 nm)/Au (55 nm) (see Methods section).To further avoid the presence of adsorbates and contaminants, the sample is kept in vacuum (10 -4 mbar) during the whole experiment.All experiments were carried out at room temperature.
Electrical characterization of the sample (Suppl.Info.S1) identifies n-type character for the 1L-MoSe2, with a threshold gate voltage of about 20 V and an electron mobility of 17 cm 2 /V.s.In this sample geometry, the bilayer h-BN plays the role of a tunnel barrier, preventing Fermi level pinning at the metal-semiconductor interfaces. 17licity-resolved photovoltage measurements: description and phenomenological formula.
Figure 1a depicts the experimental setup for measuring the helicity-dependent photogalvanic response of the MoSe2 phototransistor.We illuminate the sample at an oblique angle ϕ with respect to the normal vector of the crystal surface and simultaneously measure the photoinduced currents, either directly (Suppl.Info.S5) or as the associated voltages (main text).We used two perpendicular sets of electrodes, giving voltage signals V12 and VAB.For illumination, we used a laser with tunable photon energy.For achieving a uniform illumination power density and well-defined light incidence angles, we used a collimated beam of 0.5 cm diameter, much larger than the studied device.The polarization of the illumination beam was tuned by rotating a λ/4 waveplate over an angle θ: during rotation over 360° the original linear polarization gets modulated twice between left and right circular polarization (see top labels Fig. 1b).
Figure 1b shows V12 and VAB as a function of θ for illumination fixed at 785 nm (1.58 eV, on-resonance with the A 0 exciton transition of monolayer MoSe2 2,18,19 ), incidence angle ϕ = 20° and azimuthal angle α = -45° (defined as the angle between the x axis and the incidence plane, see Fig. 1a).The gate voltage was fixed to Vgate = 0 V.Both voltages clearly show a polarization dependence, with 2θ and 4θ-periodic components.The fingerprint of a CPC contribution is its helicity-dependence, appearing as a signal VCPC ∝ sin(2θ).A 4θ-periodic modulation, VLPC, can also appear due to linear photogalvanic and linear photon drag effects 8 .The total photovoltage VPC can be described phenomenologically as 8,11,20,21  PC =  0 +  (2) +  1 (4) +  2 (4), (1)   where C accounts for the CPC and L1 and L2 account for the linear photogalvanic and photon drag effects.The total linear polarization-dependent contribution can be accounted as L = (L1 2 + L2 2 ) 1/2 .An additional polarization-independent term, V0, (typically smaller or, at most, comparable to C) can also appear due to inhomogeneities or thermal drifts between the two electrodes.We obtain values for C, L1 and L2 by fitting equation (1) to data as in Fig. 1b. Figure 1c shows the power dependence of C, L1 and L2.The three amplitudes increase linearly with the illumination power, confirming that they are due to a second order response to the light electric field, and in agreement with earlier literature for 1L-MoS2. 11nally, we remark that the CPC signal C behaves as reported below for multiple electrode configurations.We can thus rule out that our CPC signals emerge due to properties of specific contacts, or effects from confinement of light between the micron-scale metallic electrodes.We elaborate on this in Suppl.Info.S5.

Spectral response of the CPC
Next, we investigate the spectral response of the observed helicity-dependent photovoltage.We first characterize the spectral features of the monolayer MoSe2 phototransistor by photocurrent spectroscopy 2 (see ref. 19 for detailed discussion about our measurement technique).We illuminate the sample using a linearly-polarized continuous-wave tunable infrared laser and register the photovoltage as a function of the illumination wavelength at a constant drain-source bias, Vds = 1 V.
The resulting photocurrent spectrum (grey line in Figure 2b), shows a prominent peak at 1.58 eV (785 nm), corresponding to the A 0 exciton resonance of MoSe2.A second, less prominent, peak can also be observed at 1.74 eV (713 nm), which results from the B +/-trion transition 2,19 .
Figure 2a shows the helicity-dependent photovoltage of the 1L-MoSe2 device and fits to equation ( 1) for different illumination wavelengths.Figure 2b shows the wavelength dependence of C, L1 and L2.The CPC contribution C is maximal when the illumination is on-resonance with the A 0 exciton transition (λ = 785 nm) and progressively decreases when the illumination is shifted away from the resonance.For the linear photovoltage L a nonzero amplitude appears, also for out-ofresonance illumination.
The observed spectral behavior of C shows that interband excitons play a central role in the CPC photoresponse.However, since excitons are charge-neutral quasiparticles, they must dissociate to produce a nonzero photocurrent.The required dissociation can be assisted by the large in-plane electric fields present in the depletion regions near a metal-semiconductor junction, especially when a bias voltage is applied 7 .Alternatively, a photocurrent can appear in absence of in-plane electric fields if trions are present in the MoSe2, since they have a nonzero net charge, and can contribute to the photocurrent even without dissociating.Since in our system the whole device is illuminated, both dissociated excitons and non-dissociated trions are expected to play a role in the CPC.

Effect of a nonzero drain-source voltage.
To investigate the influence of an in-plane electric field on the CPC we apply a drain-source voltage Vds between the   Figure 3c shows the CPC amplitude C as a function of the wavelength at Vds = 0 V and Vds = 1 V. Interestingly, the wavelength at which C is maximized (in absolute value) changes from 790 nm (1.57eV) at Vds = 0 V to 780 nm (1.59 eV) at Vds = 1 V.This suggests that at low drain-source voltages the dominant charge carriers involved in the CPC are A +/-trions (which have a nonzero charge and therefore do not need to dissociate to participate in the photovoltage), while at large drainsource voltages the transport is dominated by dissociated A 0 excitons. 22

Effect of the illumination angle in the CPC
In order to identify the symmetry properties of the two different CPC regimes (for Vds above and below VT) we test their behavior under different illumination angles.Figure 4 shows the measured helicity-dependent photovoltage V12 AC for different illumination incidence angles ϕ, in the low-Vds (4a) and high-Vds (4b) regimes.Remarkably, these two regimes show a very different behavior: For Vds = 0 V, the CPC shows the same sign and a similar amplitude at ϕ = 20° and ϕ = -20°, while, for Vds = 1 V, inverting the angle of incidence causes the CPC to reverse its sign, pointing to two separate physical mechanisms.Importantly, for both situations C vanishes for incidence normal to the 2D plane, ϕ = 0°, which rules out that BCinduced CPC gives significant contributions to our signals (Suppl.Info.S6).
We further check the symmetry of the measured CPC by characterizing its dependence on the azimuthal angle  (see Fig. 1a).Figures 4c,d show the measured helicity-dependent photovoltages at different azimuthal angles, for |Vds|<VT (4c) and |Vds|>VT (4d).The insets show the dependence of C on α.Again, two different behaviors emerge: For small Vds, C is proportional to sin(2α).We remark that, since the CPC sign is preserved upon inversion of ϕ, it must also be preserved upon a π rotation of α (both operations are equivalent in our system), and therefore, only a π-periodic dependence on α can appear.
For large Vds, C shows a modulation proportional to sin(3[α+ α0]), where α0 is an angle offset (15° in our case).This 3αperiodic signal suggests that C is modulated by the 120°-periodic crystal structure of 1L-MoSe2.The presence of an angular offset α0 is expected since the orientation of the crystal is not necessarily aligned with the electrodes.Oppositely from before, only an α-dependence that gives an exact inversion upon π increase of α can emerge, for consistency with signal inversion when ϕ is reversed.
As discussed in the Suppl.Info S6, when the device symmetry is reduced to, at most, a single-mirror symmetry (which can be expected in a realistic device due to interface effects at the electrodes and in-plane strain gradients), a CPDE photocurrent can have a term proportional to sin(2α)sin 2 (ϕ), consistent with the observed behavior at low Vds.For the large Vds regime, the inversion of the CPC upon sign flip of ϕ is consistent with both CPGE and CPDE terms (or a combination of them) allowed for this symmetry.Further, ϕ-odd terms are also allowed for CPGE and CPDE under the more restrictive C3v symmetry.Notably, the dependence as sin(3α) for the CPC measured at large Vds does not appear in the symmetry analysis.
Such dependence, however, can emerge from inhomogeneities of the transport properties between the armchair and zigzag directions of the 1L-MoSe2 crystal lattice, not considered in the theory.

Effect of the gate voltage in the CPC
Finally, we explore how the CPC is affected by the gate voltage.Figure 5a shows a color map of the CPC amplitude C (derived from V12 AC lock-in signal) as a function of Vds (applied between electrodes A and B) and Vgate, at α = -45 0 and ϕ = 20°.
The two drain-source voltage regimes discussed above can be observed here as the blue (C > 0 mV) and red (C < 0 mV) areas of the map.Once again, when the incidence angle ϕ is changed to -20° (see Figure 5b), the sign of C at large Vds switches from negative to positive, while at small Vds the sign is preserved.
For Vgate below 0 V we see a much weaker influence on the CPC than for Vgate > 0 V.In the latter case we observe a shift of the transition voltage VT towards larger drain-source voltages.This can be explained by an increased trion population, due a higher density of charge carriers in the MoSe2 crystal when the Fermi energy is brought above the edge of the conduction band 22 .Further, an increased gate voltage can also modify the electric field screening, changing the exciton and trion momentum lifetimes and therefore changing their contributions to the CPC. 11,12When the gate voltage is further increased we observe an overall reduction of the CPC, regardless of the value of Vds, which we associate to a decrease of the carrier momentum lifetime, due to an enhanced electron-electron scattering.Also, the probability of exciton absorption is expected to decrease at large gate voltages, due to the reduced density of unoccupied states in the conduction band.

Summary and conclusions
In conclusion, the two observed regimes for the CPC can be well-described by CPGE and CPDE for a reduced device symmetry.Although effects of higher order in the light electric field could also be allowed by symmetry, the linearity of C with illumination power confirms that the measured signal is dominated by second-order effects.
Importantly, although a Berry-curvature CPGE could be allowed for a low-symmetry device, it is not observed here, as confirmed by the fading of C for incidence normal to the crystal plane.Further, our results indicate a transition from exciton-to trion-dominated transport between the two regimes, but the influence of the excitonic character on CPC is an open question.

Device Fabrication
We mechanically exfoliate atomically thin layers of MoSe2 and h-BN from their bulk crystals on a SiO2 (300 nm)/doped Si substrate.The monolayer MoSe2 and bilayer h-BN are identified by their optical contrasts with respect to the substrate 23 and their thickness is confirmed by by Atomic Force Microscopy (see SI, Figure S2).Using a polymer-based dry pick-up technique, described in detail in ref. 24 , we pick up the bilayer h-BN flake using a PC (Poly(Bisphenol A)carbonate) layer attached to a PDMS stamp.Then we use the same stamp to pick up the MoSe2 flake directly in contact with the h-BN surface and we transfer the whole stack onto a bulk h-BN crystal, exfoliated on a different SiO2/Si substrate.After the final transfer step, the PC layer is detached from the PDMS, remaining on top of the 2L-BN / MoSe2 / bulk-BN stack, and must be dissolved using chloroform.Next, to further clean the stack, we anneal the sample in Ar/H2 at 350 ℃ for 3 hrs.For the fabrication of electrodes, we pattern them by electron-beam lithography using PMMA as the e-beam resist, followed by ebeam evaporation of Ti(5 nm)/Au(75nm) at 10 -6 mbar and lift-off in Acetone at 40 o C.

Electrical characterization
The DC electrical characterization of the studied device is discussed in detail in section S1 of the Supporting Information.
The highly-doped Si substrate is used as the back-gate electrode in order to tune the density of charge carriers in the MoSe2 channel.To eliminate the effect of environmental adsorbates, all the electrical measurements are performed in vacuum (~10 -4 mbar).We measure the source-drain current as a function of the source-drain and back-gate voltages in four-terminal geometry of electrodes, using the side contacts of the Hall-bars 25 as voltage probes.These measurements allow us to obtain a reliable estimation of conductivity and field effect mobility of charge carriers in the monolayer MoSe2 channel.Further I-V characteristics are measured in 3-terminal geometry to evaluate quality the electrical contacts at the metal-semiconductor interface, as further discussed in section S1.

ASSOCIATED CONTENT
The following associated content can be found in the supporting information to this article:

S1. AFM characterization
We measure the height profile of the BN-encapsulated MoSe2 on a SiO2/Si substrate by AFM.The thickness of both of the MoSe2 the top h-BN flakes are measured as 0.7 nm (Figure S1b) which corresponds to monolayer MoSe2 and bilayer h-BN, in agreement with the reported values in literature 1,2 .The bottom h-BN has a thickness of 7.65 nm (21-22 layers).The AFM images also reveal the presence of bubbles due to trapped molecules in the h-BN/MoSe2 interface.Reportedly, the accumulation of the interface contaminants in these bubbles ensures a perfectly clean interface at the bubble-free regions, and is a signature of the good adhesion between the two layers.

S3. Electrical characterization of the 1L-MoSe2 phototransistors
The DC electrical characterization of the sample is performed in the dark while keeping the sample in vacuum (10 - 4 mbar).In order to obtain the electrical transport properties of the MoSe2 channel, we perform four-terminal measurements in Hall-bar geometry.We apply a source-drain current on the contacts 1 and 2 (See figure 1a in the main text) and measure the voltage drop along the channel using the Hall contacts A and C. We remark that using the contacts that only partially cover the channel is preferable for the characterization of the intrinsic electrical properties of the MoSe2 channel, since this allows to prevent the formation of depletion regions near the metal contacts. 4gure S3a shows a transfer characteristic for the 1L-MoSe2 phototransistor, presenting a clear n-type behavior.We extract the threshold gate voltage (Vth) of 19 V as the gate voltage at which the conductivity starts to increase.We estimate a field-effect mobility of about 17 cm 2 /V.s from the linear fit to the transfer curve, for the range of gate voltage (Vgate > Vth) with linear dependence of conductivity.Figure S3b shows the four-terminal I-V characteristics of the phototransistor.The ohmic response of the channel can be readily observed from the linearity of the obtained I-Vs.The inset in Figure S3b shows the square resistance of the MoSe2 channel, Rsq, obtained as the slope of the linear fit to the I-V divided by the length-to-width ratio of the MoSe2 channel, as a function of the gate voltage.
In our device geometry, encapsulation of the MoSe2 channel with h-BN reduces the influence of the adsorbate molecules on the MoSe2 surface and prevents charge scatterings due to interface impurities and the Si substrate, which largely reduces the hysteresis in the charge transport measurements.Moreover, the bilayer h-BN plays the role of a tunnel barrier for injection of charge carriers, preventing the level pinning at the contacts.Figure S5 shows a colormap of the CPC amplitude C as a function of the drain-source and gate voltages for  = 0 o .The value of C remains near zero regardless of the applied voltages.This allows us to rule out that the dominant contribution to our observed CPC signals is a Berry phase-induced CPGE, since it should become maximal for normal incidence.This measurement also rules out that our signals have a significant contribution from the valley-Hall effect, since such effect would appear as a nonzero contribution to the CPC linear with the drain-source voltage.The absence of the valley-Hall effect in our device can be understood since this effect has been reported for studies on highly n-doped devices, and it increases with the gate voltage.In our device, the 1L-MoSe2 channel only starts to open for Vgate > 20 V. Thus, a much larger doping could be required for observing the valley-Hall effect.

S5. Comparison of the photovoltage and photocurrent measurements and consistency checks
Figure S6 shows the helicity-dependent open-circuit photovoltage and short-circuit photocurrent measured in two sets of electrodes: [A, B] and [1, 2].These results are representative for a wider range of checks that we performed, where we always found a linear relation between the observed values for C, L1 and L2 in the current and voltage signals.This photoresponse can thus be measured equivalently as current or voltage signals on our device.In addition, for C we observed no dependence on the orientation of the linear polarization for the laser beam incident on the /4 plate.Finally, Figs.2b and 3c (main text) show that the spectral dependence of C is preserved for two different sets of electrodes, further ruling out a role for specific contacts or standing-wave effects between electrodes.Our full range of consistency checks allows us to rule out that effects at specific electrodes, and effects from confining light between the micron-scale metallic electrode structure, give a significant contribution to the helicity-dependent signals that we analyze.

I. General description of photogalvanic and photon drag effects
6][7] More specifically, they are characterized by a DC current generated by a time-varying electric field, with amplitude proportional to the square of the applied field.This current is generated by photoelectrons which are excited by optical (vertical in the band structure) transitions and, depending on its microscopic origin, can depend on the polarization (linear PCE and PDE) or the helicity (circular PGE and PDE) of the applied field.Recent studies on derivations of PGE and PDE rely on the nonlinear susceptibility 8 , Floquet theory 9 and the kinetic equation approach.
Let us consider the situation of a 2D material illuminated by a monochromatic light source, with complex electric field defined as the plane wave where the subindices i, j and k stand for the Cartesian coordinates,  is the angular frequency and the wave vector  ⃗ can be expressed in spherical coordinates as  ⃗ = −(sin() cos() , sin() sin() , cos()).
We assume that the incident light forms a polar angle  and an azimuthal angle α with the 2D plane (see Figure 1 in the main text).We can write down the electric field (as well as the vector potential  ⃗ =  ⃗⃗ /ω) as The light-induced current density  ⃗ inside the material can be generically written in series of the Cartesian components (l, j, k) of the electric field  ⃗⃗ .Per component of  ⃗ this gives  l =      −+ ⃗⃗ ⃗ + σ  (2 ′ )      −2+2 ⃗⃗ ⃗ +   (2)     * + ⋯ , where   is a second rank tensor and   (2) , and   (2 ′ ) are third rank tensors.The first and second terms in the right correspond respectively to a linear AC current (at optical frequency) in response to the electric field and an AC current of twice the frequency of the radiation, responsible for second harmonic generation.The relevant term for us is the third term, which corresponds to a DC current  l DC in response to the oscillating field: By doing a Taylor expansion over the wave vector q we can rewrite    as   DC =   (2) (,  ⃗)    * =   ()    * +   ()      * + ⋯ Here   =   (2) (, 0) does not depend on the radiation wave vector  ⃗ and is responsible for the photogalvanic effect,  ⃗ PGE , while   accounts for the photon drag effect  ⃗ PDE , linear with factors ql.

Requirement of inversion symmetry breaking
We now show that the absence of inversion symmetry is necessary for getting nonzero photogalvanic and photon drag effect.First, we note that   DC is antisymmetric (changes its sign) under inversion of the spatial coordinates x, y, z  -x, -y, -z, while the object     * is symmetric under that transformation.In consequence, if the inversion transformation is a symmetry of the studied system,   DC cannot have any dependence on     * and   must be zero.In other words, the photogalvanic effect can only emerge in systems with broken inversion symmetry.

Linear and circular photogalvanic and photon drag effect
Next, we observe that, since the current density must be real, it cannot change under complex conjugation.In consequence, from equation (7) where   is a second rank pseudo-tensor and l and s stand for Cartesian coordinates.Thus   CPGE can be expressed as 5 It is convenient to separate   PDE in a similar fashion into its circular and linear polarization sensitive components,   CPDE and   LPDE .For   CPDE we get: At this point it is worth noting that we have not still made any assumption about the physical origin of    .Thus, equations ( 12) and ( 13) are completely general and must hold regardless of the underlying physical mechanism.

II. Symmetry arguments for the CPC in monolayer TMDCs.
In the following, we use symmetry arguments to determine the nonzero components of the tensors   and   , as defined in equations ( 12) and ( 13).This allows to extract constraints for the dependence of   CPGE and   CPDE on the illumination angles α and ϕ.We remark again that, since equations ( 12) and ( 13) must hold regardless of the physical origin of the CPC, the discussion below is completely general.
In order of decreasing symmetry we analyze three cases: D3h, C3v, and Single mirror-plane symmetry.We find that crystal structures belonging to the high-symmetry class D3h cannot support any CPGE.For the case of C3v symmetry with an oblique incidence angle ϕ, only   can have a nonzero value, which then gives a nonzero CPGE, but always with the property that it flips signs upon reversal of ϕ (in conflict with a BC origin).Systems with only one mirror symmetry can not only have nonzero   but also nonzero   and   , allowing for BC-CPGE.We conclude that our experimental results for low source-drain voltage are only compatible with, at most, one mirror-plane symmetry, since otherwise   and   cancel out and, therefore, the photocurrent cannot be preserved upon inversion of the incidence angle, ϕ.

D3h symmetry
The scenario of a 1L-TMDC system with ideal mirror symmetry with respect to the crystal plane (symmetric environments and external fields) give the system the high D3h symmetry.This has a three-fold rotation symmetry around the z axis, defined by the operator C3, three two-fold axes perpendicular to C3, a mirror plane in the xy plane, defined by  ℎ = ( 1 0 0 0 1 0 0 0 −1 ) and three mirror vertical planes with respect to the xy plane σv.The improper rotation is σhC3.
In the section directly below here on C3v symmetry we derive that the CPGE current is ) .
This result for C3v can be extended to the case for D3h by adding the requirement for the additional mirror symmetry  ℎ .This brings that a pseudo-vector ( ⃗⃗ ×  ⃗⃗ * ) becomes − ℎ ( ⃗⃗ ×  ⃗⃗ * ), and gives the condition   = 0. Consequently, all CPGE current contributions cancels out for the D3h symmetry.) .
A direct extension of this analysis shows that the additional mirror plane σh does not impose further constrains on  ⃗  .Thus, CPDE photocurrents can be present for this symmetry.

C3v symmetry
If we assume a 1L-TMDC crystal symmetry in the plane, but drop the assumption of mirror symmetry with respect to the crystal plane (relevant, for example, for a 1L-TMDC sustained on a substrate), the system has C3v symmetry.This corresponds to a three-fold rotation symmetry around the z axis C3, and three mirror planes perpendicular to the xy plane.The CPGE photocurrent is given by    =    ( ⃗⃗ ×  ⃗⃗ * )  .(42) In a 2D crystal the Berry curvature has only a nonzero component, perpendicular to the xy plane (the Berry curvature behaves as a pseudoscalar).In a N-band system, the BC of the n th band comes from all the other N -1 bands.
For a simple two-band approximation, F stands for the conduction band (CB) and The first term of | 2 | is from Berry curvature Ω   ( ⃗⃗ ) and shows that this contribution to the CPGE is independent of α, and maximal for normal incidence, ϕ = 0.As discussed in the main text and in section S4, we do not find any contribution to CPC that satisfies this angular dependence.It is worth noting that, from the general definition of  l CPGE , equation (12), we find that a CPGE contribution changing as cos(ϕ) sin(2θ) is associated with the matrix elements   and   .The symmetry arguments discussed above confirm that these matrix elements can only be nonzero if the device symmetry is reduced to, at most, a single mirror plane.Therefore, the D3h symmetry of 1L-MoSe2 must be reduced (for example from device asymmetries or strain gradients) in order to allow for a Berry curvature-induced CPGE (BC-CPGE).

Fig 1 -
Fig 1 -(a) Schematic experimental setup.The helicity of the laser excitation is controlled by rotating the quarter-wave plate angle, θ.(b) Helicity-dependent photovoltage of the contacts [1, 2] (blue) and [A, B] (orange) as a function of the quarter-waveplate angle θ for λ = 785 nm, ϕ = 20°, Vds = 0, Vgate = 0 and α = 45°.The black lines are fits to the phenomenological equation (1).(c) Power dependence of C, L1 and L2 (extracted from fits to equation (1)).The solid lines are linear fits to the experimental data.The vertical dashed line indicates the power used during the experiments, 0.8 mW.

Fig 2 -
Fig 2 -Spectral evolution of the circular photocurrent.(a) VAB as a function of the waveplate angle, θ (for ϕ = 20°, Vds,12 = 0, Vgate = 0 and α = 45°) under different illumination wavelengths, from 700 nm to 825 nm.For clarity, the traces have been vertically shifted in steps of 0.5 mV.The solid lines are fits to equation (1).(b) Photocurrent spectrum of the 1L-MoSe2 crystal (grey, solid line) and spectral dependence of the fitting parameters C, L1 and L2 (red, dark blue and pale blue lines, see legend).
electrodes A and B and measure the transverse voltage between the electrodes 1 and 2, while keeping λ = 785 nm, Vgate = 0, ϕ = 20° and α = 45°.For improving the signal-to-noise ratio, we now use a chopper to modulate the laser intensity at 331 Hz and lock-in detection of the AC photovoltage V12 AC .Figure 3a,b show the helicity dependence of V12 AC at different Vds and the associated dependence of C and L on Vds.Unlike the case of the valley-Hall effect (where the anomalous Hall voltage

Fig 3 -
Fig 3 -Helicity-dependent photovoltage,  12 AC for different drain-source voltages, Vds.(a)  12 AC as a function of θ for different drain-source voltages with λ = 785 nm, ϕ = 20°, Vgate = 0 and α = 45°.For clarity, the measurements have been vertically shifted in steps of 5 μV and the polaritazion-independent offset, V0, has been substracted (see equation (1)).(b) C and L parameters as a function of the drain-source voltage.(c) CPC amplitude, C, as a function of the wavelength for Vds = 0 V (orange circles) and Vds = 1 V (green squares).For an easier visualization, the data for Vds = 0 V has been multiplied by 10.

Fig 5 -
Fig 5 -(a) Colormap of the CPC amplitude, C for λ = 785 nm as a function of the drain-source and gate voltages, Vds and Vgate for ϕ = 20°.(b) Same as (a) for an incidence angle ϕ = -20°.
S1. AFM characterization S2.Optical microscopy images of the fabrication process and the final device S3.Electrical characterization of the 1L-MoSe2 phototransistors S4.Color map of the CPC amplitude as a function of Vds and Vgate for illumination at normal incidence S5.Comparison of the photovoltage and photocurrent measurements and consistency checks S6.Theoretical analysis of the photogalvanic and photon drag effects CONTENT S1.AFM characterization S2.Optical microscopy images of the fabrication process and the final device S3.Electrical characterization of the 1L-MoSe2 phototransistors S4.Color map of the CPC amplitude as a function of Vds and Vgate for illumination at normal incidence S5.Comparison of the photovoltage and photocurrent measurements and consistency checks S6.Theoretical analysis of the photogalvanic and photon drag effects

Fig
Fig S1 (a) AFM image of the BN-encapsulated MoSe2 on SiO2/Si substrate.The dashed lines highlight the edge of the flakes.(b) Height profile along the red and blue lines indicated in panel (a), corresponding to the edges of the monolayer MoSe2 and the bilayer h-BN flakes.For clarity, The profiles are offsetted to the same zero level.

Fig
Fig S2 Enhanced-contrast optical images of the device fabrication process (a) Fully encapsulated 1L-MoSe2 crystal before the fabrication of the contacts.(b) Final device.(c-e) Zoom in of the region indicated by a dotted square in panels (a) and (b) at the different stages of the fabrication process.(c) Exfoliated 1L-MoSe2 flake on SiO2 before processing.(d) Same flake shown in (c) after encapsulation with top bilayer h-BN and bottom multilayer h-BN.(e) Final device with the fabricated contacts on top of the BN/MoS2/BN stack.The electrodes used for the measurements of the main text are highlighted in green.The dashed white line indicates the edges of the 1L-MoSe2 flake.

Fig
Fig S3 Electrical characterization of the channel in 4-terminal geometry.(a) Channel conductivity as a function of gate voltage.The red line is a linear fit to the data for Vgate > Vth.(b) I-V characteristics of the channel and the square resistances, estimated for the gate voltages of 40 to 60 V (shown in the legend).

Fig
Fig S5 Colormap of the CPC amplitude, C as a function of the drain-source and gate voltages, Vds and Vgate for normal incidence angle, ϕ = 0 degrees.

Fig
Fig S6 Helicity-dependent open-circuit photovoltage (red circles) and short-circuit photocurrent (blue circles) for electrodes [1, 2] (a) and [A, B] (b) at Vgate = 0 V and Vds = 0, as a function of the waveplate angle (θ).The black solid lines are fittings to the phenomenological equation (1) in the main text.Except for a scale factor, the θ dependence of the photovoltage and photocurrent are very similar, as expected from the linear I-V of the semiconductor channel.

)
⃗⃗ * becomes ( ⃗⃗ ×  ⃗⃗ * ) and  ⃗  becomes  ⃗  .Since  should remain the same under the rotational symmetry, we have  ⃗  = ( ⃗⃗ ×  ⃗⃗ * we get   * =   .Therefore, the real part of   is symmetric under coordinate exchange while its imaginary part is antisymmetric under this operation.This allows us to rewrite the photogalvanic current as follows, The first term in || 2 can be related to Berry curvature (BC) for the electronic Bloch states of the n th band: [  ( ⃗⃗ ) −   ′ ( ⃗⃗ )] for the valance band (VB) with the definition of Berry curvature,