Enhancement of the photocurrents injected in gapped graphene by the orthogonally polarized two-color laser field

We theoretically investigate the photocurrents injected in gapped graphene by the orthogonally polarized two-color laser field. Depending on the relative phase, the photocurrents can be coherently controlled by deforming the electron trajectory in the reciprocal space. Under the same field strength, the peak photocurrent in the orthogonally polarized two-color field is about 20 times larger than that for linearly polarized light, and about 3.6 times for elliptically polarized light. The enhancement of the photocurrent can be attributed to an obvious asymmetric distribution of the real population in the reciprocal space, which is sensitive to the waveform of the laser field and related to the quantum interference between the electron trajectories. Our work provides a noncontact method to effectively enhance the injected current in graphene. © 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

To overcome these difficulties, graphene is an excellent material to investigate the photocurrent. Graphene, as a monolayer two-dimensional (2D) material, has a honeycomb crystal structure with two sublattices in the unit cell. Accordingly, the first Brillouin zone (BZ) contains two nonequivalent Dirac points K and K ′ (see in Fig. 1(a)) in the reciprocal space. Due to its unique electron transport properties, graphene has aroused the intense interest for electronic device applications [24][25][26][27][28][29]. It has also become a typical platform to study the interactions between intense laser fields and solids because of the broadband and ultrafast optical response, weak screening and high damage threshold. Note that pristine graphene is a zero-bandgap material.
We can also open its bandgap to a narrow gap by different methods including doping [30] or positioning on an incommensurate substrate [31,32]. Gapped graphene is chemically identical to the pristine graphene but its inversion symmetry is broken.
More recently, intense laser fields have been applied to inject the photocurrent in graphene [23,33,34]. By steering the waveform of linearly or elliptically polarized few-cycle pulses, the current can be controlled effectively. However, the current injected by linear and elliptic few-cycle laser pulses are relatively weak. Considering that the current is sensitive to the waveform of the laser field, an alternative solution to enhance the photocurrent is designing the waveform of the laser field. The fields with several different waveforms have been used in related studies. Single-cycle or few-cycle linearly polarized pulses are used to control the electron momentum distribution or valley polarization by steering the carrier envelope phase (CEP) [35,36]. The bicircular pulses are also used to control the asymmetric population [37,38]. In comparison of the linearly or elliptically polarized fields, orthogonally polarized two-color (OTC) fields show a greater asymmetry even with multi-cycle pulses and different circular polarization by steering the relative phase between the fundamental and second-harmonic fields. Due to these properties, the OTC field has been applied widely in interacting with gas medium [39][40][41][42][43][44][45], including high harmonic generation in gas and optical chirality.
In this paper, we propose a method to enhance the photocurrent in gapped graphene by the OTC field. The peak photocurrent in the OTC filed is obviously enhanced compared with linearly and elliptically polarized lights. We attribute the enhancement of the photocurrent to an obvious asymmetric real population in the reciprocal space. The asymmetric population is sensitive to the waveform of the laser field and related to the quantum interference of electron trajectories. In addition, the peak photocurrent can be controlled by changing the bandgap of gapped graphene. The OTC fields can induce a great asymmetry in the region near every K or K ′ point, respectively. This asymmetry is different from the valley polarization induced by bicircular pulses, which is the asymmetry between K and K ′ point.

Theoretical model
In the laser field, electron dynamics can be expressed by the time-dependent Schrödinger equation, which has the following form: The total Hamiltonian H(t) reads where H 0 is the Hamiltonian of the solid in the absence of laser fields, e is the electron charge and F is the electric field. Atomic units are used throughout the paper unless otherwise stated. In our work, gapped graphene is considered as a 2D two-band model with the tight-binding approximation [46,47]. In this model, the hopping parameter is γ = 3.03 eV between nearest neighboring atoms. The Hamiltonian is described by: Here ∆ is the finite gap between the valence band (VB) and the conduction band (CB), and where a = 0.246 nm is the lattice constant of graphene. From the above Hamiltonian H 0 , the energies of CB and VB can be obtained as follows: where the footnotes of c and v stand for CB and VB, respectively. The energy dispersion of gapped graphene with ∆ = 1 eV is shown in Fig. 1 In gapped graphene, the electron dynamics is driven by the laser field. According to the Bloch acceleration theorem [48], for an electron with an initial momentum k, the time evolution of the crystal momentum k(t) can be expressed as: The corresponding adiabatic wave functions are the Houston functions [49], where m = c, v for CB and VB, and Ψ (m) k are the Bloch-band eigenfunctions in the absence of the laser field. The dynamic phase ϕ (D) m and the geometric phase ϕ (B) m [50] are defined by the following expressions: Here is the intraband connection for band m. By solving the 2D two-band density matrix equations (DMEs) [51][52][53][54], the response of the laser-driven electron in a crystal can be described as follows: where n c and n v are the populations of CB and VB, respectively. Π is the off-diagonal element of the density matrix. α = n v − n c is the population difference between VB and CB. T 2 is the dephasing time stemming from the coherence decay in solids. Based on Ref. [55], the dephasing time is 35 fs in our calculation. The phase difference is . Ω is the Rabi frequency, given by Ω(k(t)) = −iF(t) · d cv (k). d cv is the dipole matrix element between the conduction and valence states [56][57][58]: After the pulsed interaction, the real population on CB is defined as N c (k). According to Ref. [33], the residual photocurrent after laser excitation in the first Brillouin zone (BZ) can be estimated as the following: Here we choose the value of ballistic lifetime τ 1 = 40 fs based on previous researches [59][60][61]. The diffusion constant D equals 1.1 × 10 4 cm 2 s −1 and the diffusion length l is about 0.35 µm.

Results and discussion
We apply an OTC laser pulse incident normally to the gapped graphene. The electric field of this pulse is described as where ω is the fundamental frequency. Considering the value of current and the damage threshold of graphene, we choose the fundamental laser with λ = 4700 nm. e k x and e k y are the unit vectors along k x and k y . The field amplitudes F 0x = F 0y = √ I, where I=0.34 TW/cm 2 , and ϕ is the relative phase between the two components. The time-envelope of the pulses has the following expression: where the full width at half maximum τ = 2.5T 0 . T 0 is the optical cycle of second harmonic pulses. For the elliptically and linearly polarized fields, they have the following expressions: The photocurrents along k x induced by OTC, elliptically and linearly polarized fields are shown as blue, orange and purple dot-dash lines in Fig. 2(a), respectively. For the OTC field, the current is tunable with steering the relative phase. When ϕ = −0.5π, the photocurrent reaches the maximum. The direction of the current is reversal at ϕ = 0. For the elliptically polarized light, we also observe the oscillating current by changing the relative phase. However, the peak current is only about 0.38 nA. For the linearly polarized field, the current shows a weak modulation and the maximum is just 0.07 nA. Compared with the currents in the elliptically polarized field and the linearly polarized field, the peak current in the OTC field is about 3.6 and 20 times larger than that in the two else fields, respectively. We also show the photocurrents along k y in Fig. 2(b). Similarly, the peak current in the OTC field is higher than that in the linearly and the elliptically polarized fields.
To understand the mechanism of the peak current enhancement, we show the real population in the linearly (ϕ = −0.5π), elliptically polarized (ϕ = −0.75π) and OTC field (ϕ = −0.5π, 0) in Figs. 3(a)-(d), respectively. The inset figures are the vector potential of the laser field. The excited electrons are mainly distributed around K and K ′ . In Fig. 3(a), one can see a bright area of the population along a straight line whose center is K or K ′ . In Fig. 3(b), the population exhibits bright circles around K and K ′ . In Fig. 3(c), the population follows a C shape. In the OTC field at ϕ = 0, the population exhibits a figure-eight shape in Fig. 3(d). These results clearly show that the population has a similar shape to the waveform of the laser field. Compared with straight lines, circles and the figure-eight shape, the C shape of the population exhibits the most obvious asymmetry in k x direction. Note that the population near every K and K ′ point is asymmetric. To quantify this asymmetry, we choose the real population distribution around K. Taking account of the dashed region (see in Fig. 3(c)), we define the asymmetry along k x and k y as respectively. n 1 (n 2 ) is the sum of the population where −0.25<k x <0 (0<k x <0.25) and n 3 (n 4 ) is the sum of the population where 0.33<k y <0.5 (0.17<k y <0.33) (see in Fig. 4). As illustrated in Fig. 4(a), the maximum of η x is close to 9% in the linearly or elliptically polarized fields. Whereas, the peak η x can be enhanced to 24% in the OTC field. Moreover, η y shown in Fig. 4(b) also exhibits a similar enhancement as a function of the relative phase. In comparison of the photocurrents and the population asymmetry (see Figs. 2 and 4), one can see that they have similar dependence on phase. In other words, the photocurrents and the population asymmetry have a strong relevance. The higher the population asymmetry, the larger the photocurrent output will be. Next we have a deeper insight into the asymmetric population along k x and k y in the following respectively. Along k x , we choose two points P 1 and P 3 around K in the OTC field at ϕ = −0.5π. As indicated in Fig. 5(a), P 1 is located on the bright fringes with large population. P 3 situates to the left of point K where the population is small. In Fig. 5(b), we define the parameter I(t i ) = | χ(t i )| 2 , which is proportional to the ionization probability. The complex amplitude of ionization rate χ(t i ) is defined as [52,53]:   4. The asymmetry of the real population as a function of CEP or the relative phase. η x and η y are defined as n 1 −n 2 n 1 +n 2 × 100% and n 3 −n 4 n 3 +n 4 × 100%, respectively. n 1 , n 2 , n 3 and n 4 are indicated in inset figures.
One can see that I almost maintains zero for electrons with the initial momentum P 3 , which leads to the little accumulation of the population. In contrast, I raises up obviously at time t 1 and t 3 for electrons with the initial momentum P 1 and reaches about 3 at the end of the OTC field. The three points are indicated as blue dots, P 1 , P 2 and P 3 . The field of the vector potential is shown as the blue curve. (b) The parameter I of P 1 , P 2 and P 3 as a function of time. I is defined as | χ(t i )| 2 , which is proportional to the ionization probability. (c) The trajectories of electrons with the three initial momentum P 1 , P 2 and P 3 in the reciprocal space. The real part of d cv (k) is shown as the background figure with a colorbar. (d) The real part of the Rabi frequency and the phase of ionization rate for the three points. When solids are irradiated by a laser pulse, electrons oscillate on VB before excited into CB. In this pre-acceleration process [52], an electron with a large real part of d cv has a higher probability to be excited. We show the real part of d cv in Fig. 5(c). Note that it reaches the maximum near K. Meanwhile, the electron trajectories (k = k 0 − A(t)) of P 1 and P 3 are plotted as green and yellow curves. One can see that the trajectory of P 1 passes through K, but that of P 3 keeps away from K. In consequence, the elelctron at P 3 is hardly to be excited due to the small real part of d cv . Whereas, the electrons passing through K have a large probability to be excited into CB. These electrons populated on CB form a similar shape of the vector potential (see the blue curve in Fig. 5(a)).
Then we focus on the asymmetric population along k y . Different from that in k x direction, the OTC field has a good symmetry along k y . The enhancement of photocurrents in k y direction is due to another mechanism. As illustrated in Figs. 3(c) and (d), one can see many bright fringes of the population. We choose P 1 and P 2 to discuss the enhanced photocurrents along k y . As indicated in Fig. 5(a), P 2 is located in the interval between the fringes with small population. We show the parameter I of electrons with the initial momentum P 2 in Fig. 5(b). Comparing electrons of P 1 and P 2 , they have obvious differences in the excitation process. At time t 2 , I of P 1 raises up from 0.2 to 1.9, while that of P 2 keeps about 0.4. After time t 4 , I of P 1 oscillates around 3, whereas I of P 2 decreases rapidly. Futhermore, the real part of the Rabi frequency between CB and VB is shown in Fig. 5(d). For P 1 and P 2 , the real part of the Rabi frequency peaks at t 1 , t 2 , t 3 and t 4 . Electrons excited at these time dominate the ionization process and interfere with each other. The interference between electron excition at time t 1 and t 2 belongs to the intracycle interference [62]. Similar to Eq. (21), the complex amplitude of electron excition at time t 1 and t 2 can be described by χ(t 1 ) and χ(t 2 ). The intensity of interference is proportional to (1 − e i∆S ), where ∆S is the difference of total phase between t 1 and t 2 . Note that total phase includes S 1 , S 2 and the phase of d cv . When ∆S is an integral multiple of 2π, electron excition is suppressed [62]. As shown in the upper part of Fig. 5(d), the curves depict the total phase of χ(t i ). Note that ∆S is 8.9π for P 1 and 6.0π for P 2 . This makes a drop of I for P 2 at t 2 shown in Fig. 5(b). Similarly, an obvious decrease exists at time t 4 for P 2 because ∆S is 10.1π between t 3 and t 4 . Due to the different gathered phase for electrons, the asymmetric interference fringes of the population arises. They result in the enhancement of the photocurrent along k y .
We also investigate the population in the OTC field at ϕ = 0 in Fig. 6. Similarly, we select P 4 and P 5 on the each side of K along k x in Fig. 6(a). As shown in Fig. 6(b), I of P 4 is almost as large as that of P 5 . This leads to the symmetric population. As a result, the current injected by the OTC field at ϕ = 0 is much lower than that at ϕ = −0.5π. By steering the relative phase, we can control the current effectively. In addition, the photocurrent is related to the bandgap of gapped graphene. In gapped graphene with different bandgaps, the photocurrent induced by the OTC field at ϕ = −0.5π is shown in Fig. 7. The current keeps larger than 1.2 nA when 0 eV <∆< 1 eV and achieves its maximum near ∆ = 1 eV. Once ∆ is more than 1 eV, the current drops rapidly. This may be related to the low ionization probability. It implies that one can control the photocurrent injected in gapped graphene by steering the bandgap except for the relative phase of the OTC field. Besides, we also simulate the current as a function of the bandgap for different intensities and frequencies. The general tendency of the current is similar as the conclusion of Fig. 7. For a higher intensity, the current is increased and achieves its maximum at a smaller gap. The current decreases by using OTC fields with shorter wavelengths.

Conclusion
In summary, we investigate the photocurrent injected in gapped graphene by the OTC field. Compared with the linearly and ellipitically polarized fields, the photocurrent injected by the OTC field increases by 20 times and 3.6 times respectively. We attribute the enhancement to a stronger asymmetry of the population on CB and the quantum interference between electron trajectories. In addition, the photocurrent is also impacted by the bandgap of gapped graphene except for the relative phase of the field. These results indicate that one can inject the photocurrent in gapped graphene by the OTC field effectively and control the strength and direction of the current by steering the relative phase or the bandgap.

Disclosures. The authors declare no conflicts of interest.
Data availability. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.