Photon drag effect

Similarly to the PGE and SPGE, the photon drag effect (PDE) is also manifested as the generation of an electric current under irradiation with a light pulse. However, the mechanism of the PDE differs significantly from the PGE. Specifically, the PDE originates from the momentum transfer from the incident photons to free conduction electrons [10,50]. In semiconductors, the PDE is associated with the momentum-selective interband transitions and with different carrier mobility at the corresponding levels [51–53].

The PDE is the second-order nonlinear optical phenomenon that can be phenomenologically described by the following constitutive equation (see, for example, [54]):

𝑗_𝑖^{(𝑃𝐷𝐸)}∝ 𝜒_{𝑖𝑘𝑙𝑚}^{(𝑃𝐷𝐸)}𝑘𝑘𝐸𝑙𝐸𝑚∗, (1.3)

where χ^(PDE) is the tensor of the fourth rank; k is the wave vector; E is the vector of the electric field of the light wave, k, l, m are the indices corresponding to the selected coordinate system.

In the isotropic medium the PDE photocurrent can be estimated in terms of light pressure according to the formula [7]:

𝑗^{(𝑃𝐷𝐸)}∝ 𝑒𝑡𝑝ℏ𝑘 𝑀^∗

𝐴𝑎𝑏𝑠𝐼

ℏ𝜔 , (1.4)

where tp is the electron relaxation time, M* is the effective mass of the electron, and A^abs is the relative absorption. It is worth noting that a similar expression can be obtained from the equations of conservation of energy and momentum [7].

The electron drag by photons has first described in Ref. [55]. However, the developed model is applicable only if the photon energy is much higher than the temperature of the crystal lattice. The transitions between two subbands have been also considered.

Ref. [10] reports PDE in p-type germanium excited by a pulsed CO² laser with a peak power of 2 kW. An interesting feature of the photocurrent is a change of the current sign when the temperature changes from room temperature to the temperature of liquid nitrogen. The obtained experimental regularities were well described by the developed theory. The PDE in germanium was also investigated in Ref. [50], which also showed that the CO2 laser radiation can transmit a sufficient pulse to produce a photocurrent in a four-centimeter rectangular rod. Studies of PDE in germanium continued in Refs. [54,56–58].

29 PDE has also been studied in other materials. For example, the theory of the PDE photocurrent at optical interband transitions in tellurium has been developed in Ref. [59]. The theory predicts the temperature dependence of the PDE photocurrent. The experimental study of PDE in tellurium [60] involved complex measurements were performed to calculate and interpret all components of the χ^(PDE) tensor in a tellurium single crystal under irradiation with the CO2 laser. Three mechanisms of the appearance of the photocurrent were identified in addition to PDE including optical rectification and structural imperfections of the crystal. The experimental results obtained were consistent with the theory of PDE.

Experimental studies of PDE were also carried out in bismuth [61,62], two-dimensional electron gas [51,63–66], gallium arsenide [9,46,67,68], ZnTe single crystals [69], silver [70] and gold nanostructures [70–74], graphene [75]. As a result of experimental and theoretical studies, the properties of the PDE photocurrent in 2D materials have been studied. In particular, it was shown that the PDE photocurrent linearly depends on the intensity of the exciting radiation (see, for example, [66]). It has been also found that the dependence of the photocurrent on the angle of incidence is an odd function (for example, [70]). Experiments had shown that the transverse photocurrent has sin(2Φ) dependence on the polarization plane azimuth Φ of the excitation beam (see, for example, [54,60]). One can see that other mechanisms (e.g. SPGE and PGE) show similar polarization dependences of photocurrent.

In addition to the momentum of photons, their angular momentum can also be transferred to charge carriers. This manifests itself in the dependence of the photocurrent on degree of circular polarization of the excitation beam and often referred to as the circular photon drag effect (CPDE). One of the first works devoted to the study of the CPDE was [76], where it was pointed out that it is possible to generate a CPDE photocurrent in a tellurium crystal at a CO² laser irradiation. It is worth noting that in the classical approximation, the CPDE can be considered as the Hall effect in the field of a light electromagnetic wave [49]. Phenomenologically, the CPDE photocurrent can be described by the following constitutive equation [77]:

𝑗_𝑖^{(𝑃𝐷𝐸)}∝ 𝑖𝜒_𝑖𝑘𝑙^{(𝑃𝐷𝐸)}[𝐸𝐸^∗]_𝑙𝑘_𝑘, (1.5)

where χ^(CPDE) is the pseudotensor of the third rank, which is odd under time inversion. Numerous experimental and theoretical studies have shown that the CPDE photocurrent is directly proportional to the degree of circular polarization.

For example, in Ref. [78], the observation of a circular photocurrent in graphene was reported. A photocurrent was generated in a graphene monolayer upon oblique irradiation with a CO² laser. The polarization was controlled by a rotating a quarter-waveplate, and the longitudinal and transverse photocurrents were measured. Figure 1.7 shows the obtained dependences of the photocurrent on the angle of incidence of radiation. It can be seen that the photocurrent dependences are odd functions of the incident angle, i.e. at normal incidence, the photocurrent is equal to zero. The dependences of the CPGE on angle φ are presented in Figure 1.8.

30

Figure 1.7. Typical dependences of the a) transverse and b) longitudinal PDE photocurrents on the angle of incidence of the exciting radiation. The dependences were obtained in Ref [78] on graphene layers excited by the radiation of a CO2 laser.

Figure 1.8. Typical dependences of the longitudinal and transverse PDE photocurrent on angle φ. The dependences were obtained in Ref. [78] on graphene layers excited by the radiation of a CO2 laser.

In Ref. [78] the experimental dependence of the transverse photocurrent on rotation angle φ of the quarter-wave plate can be described as

𝐽𝑦=𝐽𝑦,𝑐𝑖𝑟𝑐sin 2𝜑+𝐽𝑦,𝑙𝑖𝑛sin 4𝜑+𝐽𝑦,𝑐𝑜𝑛𝑠, (1.6) where J^y,circ and J^y,lin are responsible for the circular and linear contributions, respectively, J^y,const is the constant component of the photocurrent, which was

31 observed in some measurements. The longitudinal photocurrent is described by the following equation:

𝐽_𝑥=𝐽_{𝑥,𝑐𝑖𝑟𝑐}cos 4𝜑+𝐽_{𝑥,𝑐𝑜𝑛𝑠}. (1.6) Based on the results obtained, the authors conclude that the photocurrent in graphene is due to the circular Hall effect (CPDE). In addition, it was found that in graphene on a SiC substrate the photocurrent exhibits resonance at frequencies that coincide with longitudinal optical phonons in SiC. It should be noted that PDE can also result in the shown in Figure 1.8 dependences, which are typical for the photocurrent having both linear and circular contributions.

The Ref. [79] is devoted to the study of CPDE in quantum wells grown in zinc blende in the (110) direction. The generation of the photocurrent was caused by transitions between subbands induced by infrared and terahertz radiation. It is reported that the circular photocurrent at normal incidence was due to CPGE, but at an oblique incidence, it changed sign, which indicates the generation of a circular photocurrent by the mechanisms of CPGE and CPDE. In addition to the experiments, in Ref. [79] a microscopic theory was developed based on the sensitivity of the spin orientation to the wave vector and the subsequent asymmetric spin relaxation.

In addition, the possibility of generating a CPDE photocurrent in bulk tellurium [11], two-dimensional metal photonic crystals [80], and nanoporous gold films [81]

has been demonstrated. The angular and polarization dependences of the PDE photocurrent obey the same regularities as the SPGE and PGE photocurrents.