Photogalvanic effect

The photogalvanic effect (PGE) manifests itself as a generation of the stationary electric current in a homogeneous medium during its uniform irradiation with light. PGE is due to the asymmetry of the elementary processes of generation, recombination, and scattering of charge carriers, i.e. it is not associated with inhomogeneity in the irradiation pattern or a temperature gradient.

Figure 1.2. Illustration of the asymmetry of elastic scattering on a wedge (adapted from [8]).

A simple example of the asymmetry of elementary electronic processes is the elastic scattering of a particle by a wedge-shaped potential (Figure 1.2) [8]. It can be seen that for particles located to the right of the wedge during elastic scattering, the probability of the transition of a particle from a state with momentum p to a state with momentum p' is equal to the probability of a transition from state –p' to state -p. At the same time, for particles on the right side of the wedge, it cannot be said that the transition from the –p state to -p' is equally probable the transition from p to p'. Thus, for particles whose momentum is directed in opposite directions, scattering will occur in different ways. If variable electric field acts on the particles,

19 the field vector of which is oriented vertically, and its amplitude changes according to E = E0cos(ωt). Then the momentum of the elastically scattered electrons will be directed to the left, i.e. an electric current will appear. A similar situation is observed for particles undergoing photoexcitation and recombination in an asymmetric potential well.

As one can see from the example, the fluxes of electrons with opposite momenta are not compensated, instead, the particles reversed in time are compensated.

Therefore, almost any non-centrosymmetry leads to a current.

Observation of the PGE is possible in the crystal without inversion centre, i.e. in those belonging to the following point groups: C1, C2, Cs, C2v, C4, C4v, C3, C3v, C6v, C3h, D3h, Td, D2, D4, D2d, D3, D6, S4, T, O.

In the general case, the PGE can be described by the following constitutive equation [8]:

𝑗_𝑖=𝜒_𝑖𝑘𝑙^(LPGE)𝐸_𝑘𝐸_𝑙^∗+𝑖𝜒_𝑖𝑘^(CPGE)�𝐸×𝐸^∗�

𝑘, (1.1)

where χ^(LPGE) is the tensor of the third rank, χ^(CPGE) is the pseudotensor of the second rank. It can be seen from equation (1.1) that the first and second terms are maximal for the linearly and circular polarized excitation beams, respectively. The second term exists only at elliptical polarized light. Accordingly, the χ^(LPGE) tensor is responsible for the linear photogalvanic effect (LPGE), and χ^(CPGE) is responsible for the circular photogalvanic effect (CPGE). In the general case, CPGE is associated with the conversion of the angular momentum of photons into the translational motion of electrons [8,13]. The nature of LPGE is more complex and associated with the processes of dissipation.

The basis of the LPGE theory was established in Ref. [14]. The mechanism of LPGE can be most clearly demonstrated by a simplified model shown in Figure 1.3 for laser beam polarized along x-axis. If the electron bandstructure of the material has no plane of symmetry perpendicular to the x-axis, each act of photon absorption will be accompanied with generation of an photoelectron having a momentum determined by the excitation conditions, for example, the frequency of light ω. In this case, the momentum distribution of particles will be proportional to

~cos2(ψ) , where ψ the angle between the electron quasimomentum p and electric fields of the light wave E [9,15,16]. Under normal conditions, at a normal incidence of radiation, the momentum of particles moving to the right will be compensated by the momentum of particles moving to the left. However, due to the asymmetric structure of the sample, the excitation of particles moving to the right will be more likely than the excitation of particles moving to the left (Figure 1.3, inset on the right). In addition, recombination of particles moving to the left is more likely than particles moving to the right (Figure 1.3, insert on the left). Thus, due to the asymmetric structure of the sample, the momentum of particles moving to the right is not compensated, and a photocurrent will appear in the sample in the direction of the x axis.

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Figure 1.3. Illustration of LPGE using an example of a sample containing asymmetric potential barriers in the energy structure.

In the general case, LPGE is associated with asymmetry due to electron scattering processes in the crystal lattice lacking inversion centre. A simple example may be asymmetry due to elastic collisions of electrons with impurities of the lattice [8]. In addition, asymmetry can be caused by the scattering of electrons and holes by phonons, excitons, and other disturbances of the lattice periodicity.

There are two mechanisms of LPGE. In the first mechanism, the photocurrent is generated due to the absorption of light on free charge carriers, followed by their scattering by phonons. This is ballistic mechanism of the LPGE. The second mechanism is caused by the displacement of charge carriers in real space during quantum transitions. This is the shift mechanism of the LPGE. Both mechanisms are considered in more detail in Ref. [17].

For the first time, LPGE was discovered on tellurium crystals in the study of optical rectification [18]. Pulse laser radiation with a wavelength of 10.6 μm and a peak power of 3.5 kW was directed to the crystal. The pulse duration was 200 ns.

For experiments, the irradiated ends of the crystal were treated with a solution of chromium trioxide in hydrochloric acid, then washed in hydrochloric acid, and then deionized in water. The photovoltage was measured in two mutually perpendicular directions, perpendicular to the direction of light propagation. The crystal was mounted so that it could rotate 360° around the axis of the laser beam.

The registered photo emf showed a strong dependence on the polarization of light, and the authors identified it as a manifestation of optical rectification. However, it was later established that the photovoltage in tellurium can be constant for a long time, therefore, the observed phenomenon cannot be a consequence of the transitional process [8]. For the same reasons, it can be said that LPGE was detected in GaP crystals [19], although the authors of the work also explained the appearance of the photocurrent by the effect of optical rectification.

21 Very often the crystal symmetry permits simultaneous observation of the LPGE and CPGE. For example, the both effects were observed in bismuth silicate [20], lithium niobate [21], quantum wells p-SiGe [22], InGaAs/AlGaAs [23], InAs/AlGaSb and GaAs/AlGaAs [24,25], and indium nitride films [26], as well as in a two-dimensional electron gas in the MgZnO/ZnO structure [27]. CPGE is a conversion of the angular momentum of photons into the translational motion of free charge carriers. Two microscopic mechanisms of CPGE have been studied: 1) the spin orientation of the carriers by light is accompanied by their directed motion due to the spin-orbit coupling [13]; 2) a purely orbital mechanism, when the photocurrent arises as a result of the interference of various contributions to absorption by free electrons [28]. One can visualize CPGE by considering mechanical systems that convert rotational motion to translational. There are two types of such systems, the first type contains a screw mechanism, such as in an airplane with a propeller, and the second one contains systems that have contact between a circular and flat surface, for example, a car wheel and a road. The electronic analog of screw systems is described by the diagonal components of the χ^(CPGE) tensor, in this case the motion of the charge carriers will occur parallel to the propagation of the exciting radiation.

Accordingly, the wheel analog is described by non-diagonal components of the χ^(CPGE), in this case the direction of motion of the charge carriers is determined by their ratio. The circular photocurrent is propotional to the degree of circular polarization P which is determined by the angle φ between the plane of incidence and the optical axis of the quarter-wave plate, P = sin(2φ) (see, for example, [13]).

For the first time, CPGE in tellurium was predicted in [29], where the photocurrent was calculated for interband and intraband light absorption. It is interesting that the developed theory permits the inverse effect, i.e. according to [29], an electric current in gyrotropic crystals can result in the emission of circularly polarized light. The experimental discovery of CPGE in tellurium crystals has been reported in [30]. A pulsed CO2 laser with a power of 3 kW and a pulse duration of 100 ns was used as a radiation source. The polarization of radiation was determined by the angle φ between the optical axis of the quarter-wave plate and the initial plane of incidence. The samples were tellurium cylinders 0.8 cm long with a cross section of 3.1·10^–2 cm². To register the photoresponse, ring contacts were applied to the cylinders near the ends. As the theory predicted, at a temperature of 300 K the magnitude of the photovoltage was proportional to the degree of circular polarization of the radiation. Figure 1.4 shows the dependence of the photoresponse on the degree of circular polarization. It can be seen that the photovoltage due to the CPGE varies according to ~sin(2φ) and reaches its maximum values at circular polarization.

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Figure 1.4. The dependences of the CPGE photovoltage on the degree of circular polarization from [30] in a tellurium crystal excited by the radiation of a CO2 laser at 1) T = 300 K, 2) T=150 K. Inset shows scheme of experiment.

The dependence of the photocurrent on the external electric field applied to the Pb⁵Ge³O¹¹ crystal was studied in [31]. Like other ferroelectrics, this crystal has the property of switching spontaneous polarization under the influence of an electric field and, thereby, transforming from a left crystal to a right one. In the ferroelectric phase, Pb⁵Ge³O¹¹ belongs to the C³ group, while above the transition temperature this crystal is a representative of the C^3h group. For experiments, a plate cut from Pb⁵Ge³O¹¹ with translucent electrodes was used. The radiation source was a helium-cadmium laser with a wavelength of 440 nm and a power of 5 mW. An important feature of the experiments is that, before measurements, the sample was heated to a temperature above the Curie point of 177° C, and then gradually cooled to room temperature. Immediately after cooling, under laser irradiation, the photocurrent was zero. This was explained by the fact that the crystal domains were oppositely directed, and on average the arising currents were mutually compensated.

However, after exposure of the sample for 10 minutes under the influence of a constant electric field, a photocurrent appeared. The sign of the photocurrent changed at field strength of 16 and -20 kV/cm. The difference in the field strength, as well as the small dark current in the sample was attributed by the authors to a small birefringence of the sample.

CPGE induced by a magnetic field was discovered in Ref. [32]. The experiments were carried out on p-GaAs(Zn) crystals having a parallelepiped shape of 7 × 4 × 1.5 mm in the temperature range 78–300 K. The radiation source was a 5 kW CO2

laser. The radiation hit the sample normally. In the absence of a magnetic field, a photocurrent was practically absent, but when it was turned on a photo-emf pulse was observed. The photo-emf pulse shape almost completely repeated the shape of exciting laser pulse. The amplitude of the detected pulses also depended on the

23 orientation of the quarter-wave plate as sin(2φ), while the sign of the photo-emf changed when the direction of the external magnetic field reversed. It was shown that the nature of the effect is associated with the processes of dissipation, and although it manifests itself as CPGE, its mechanism is more similar to LPGE and can have ballistic and shift components like LPGE. It is also interesting that this effect was not observed in n-GaAs crystals.

In Ref. [21] the authors paid attention to an important problem in detecting a circular photocurrent associated with spatial oscillations of the photovoltaic current. In most crystals, birefringence occurs when light is propagating through a crystal because the phase difference between the ordinary and extraordinary waves changes periodically leading to oscillations of the photocurrent [8]. An exception is the case when radiation propagates along the optical axis of a uniaxial crystal (crystals of classes D², D³, D⁴, D⁶, C³, C⁴, C⁶) or when circular polarization corresponds to the eigenoscillation mode (crystals of classes T and O). It was usually proposed to use samples in the form of thin films or to measure using a periodic system of electrodes to detect spatially oscillating currents [33]. In photorefractive crystals, the detection of such currents is possible by recording diffraction gratings using waves with orthogonal polarization [34]. Instead, it was proposed to use electrodes located on the surface of the crystal perpendicular to the direction of wave propagation. With sufficiently uniform illumination of the entire space between the electrodes, the photocurrent will be determined only by a thin near-surface region. Such a photocurrent will depend only on the state of polarization of the incident radiation, and there will be no spatial oscillations of the photocurrent.

An experimental verification of this idea was performed in the same work on a lithium niobate crystal LiNbO³ [21]. In the experiments the authors obtained a sinusoidal dependence of the photocurrent on the phase shift between the ordinary and extraordinary waves, which indicated the direct detection of a circular photocurrent in lithium niobate without spatial oscillations.

Noteworthy work is Ref. [35], in which the author, on the basis of a phenomenological analysis, predicts the CPGE in gyrotropic isotropic liquids. In that work, it is considered the photocurrent in isotropic non-centrosymmetric media of the limiting symmetry class ∞∞, to which chiral liquids belong, which arise during the passage of elliptically polarized light through them. The resulting expression for the transverse current density was proportional to sin(2φ).

According to Ref. [8], the presence of such a dependence is one of the main signatures of CPGE. Thus, in that work, chiral liquids are proposed as a new photorefractive material.

It is interesting to note Ref. [36]. A feature of this publication is that experiments where the photocurrent was excited in a sample under the influence of uniaxial deformation were carried out. An InN crystal grown on a sapphire substrate with a GaN buffer layer was used as a sample. In the absence of mechanical stress,

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experiments revealed that a photocurrent can be divided into two parts: the current arising due to the spin splitting of energy levels, which can arise due to bulk or structural asymmetry [37]; and the current arising due to the asymmetry of scattering by impurities, called in the work inhomogeneous. As mechanical stress was applied, the photocurrent increased linearly. The authors show that with increasing mechanical stress, the inhomogeneous photocurrent decreases, while the current associated with structural asymmetry increases. This indicates that the greatest contribution to the photocurrent in InN is made by the current due to structural asymmetry. Thus, the spin-dependent photocurrent associated mainly with structural asymmetry was demonstrated in the work. It was shown that the degree of structural asymmetry can be changed, for example, by application of mechanical stress.

Recently much attention has been paid to the study of topological insulators. A topological insulator is a material that is an insulator in volume and is a conductor on the surface due to the spin-split metal states. A number of materials that can be considered topological insulators were predicted in [38]. Topological insulators can have such interesting effects as magnetoelectric polarizability [39], magnetic monopole behavior [40], and CPGE. The theoretical study of CPGE in topological insulators was the subject of papers [41–43]. In addition to theoretical work, an experimental study of CPGE was carried out. An example of such studies can be found in Ref. [44,45], where a Bi²Se³ crystal was used as a topological insulator.

Thus, the observation of the PGE photocurrent is possible in the presence of asymmetry, bulk or structural. It can be observed in bulk non-centrosymmetric crystals or in presence of an external field. Structural asymmetry can be associated with crystal surface (surface photogalvanic effect). Experimentally, the PGE is observed in the form of a photocurrent depending on the polarization of the exciting radiation. Thus, the momentum and angular momentum of photons can be transferred to charge carriers, which leads to the appearance of an electric current, and this has been observed in many different materials, both with linearly and circularly polarized excitation beams.