Strain-modulated anomalous circular photogalvanic effect in p-type GaAs

The influence of spatial strain distribution on the anomalous circular photogalvanic effect (ACPGE) is investigated in the p-type GaAs material. By tuning the position of exerted stress, it is experimentally observed that the uniform strain related ACPGE behaves like the sine function, which resembles the non-strain situation. Whereas the gradient strain related ACPGE shows the unimodal function line shape. To explain the observations, a new theoretical model is constructed based on spin splitting of energy bands. It is demonstrated that the ACPGE could purely derive from the spin splitting effect. Besides, the combination effect of spin splitting and inverse spin Hall effect on the ACPGE is also investigated. This work reveals the importance of bands spin splitting on ACPGE, which has not been considered before.


Introduction
Spin-orbit coupling (SOC), describing the interaction between spin and momentum of a quantum particle, influences the property of physical systems distinctly.In condensed matters, SOC is crucial for (quantum) spin Hall effect, topological state, spin-orbit qubits and so on [1].In III-V group semiconductors, such as GaAs, the energy bands are spin splitted by SOC when structure and bulk inversion asymmetries are considered.In consequence, a circularly polarized light can excite spin-polarized photocurrent in the system, which is named as circular photogalvanic effect (CPGE).The CPGE was originally observed in (001)-grown InAs/AlGaSb, GaAs/AlGaAs quantum wells (QWs) and (113)A-grown GaAs/AlGaAs QWs [2,3].In recent years, the CPGE were observed in layered transition-metal dichalcogenides, topological insulators, organic-inorganic hybrid perovskites and oxide two-dimensional electron gases [4][5][6][7][8].On the other hand, the CPGE is suppressed under normal incidence when the system symmetry is C 2v and above.However, the helicity dependent photocurrents are observed when the excitation light spot moving along the perpendicular bisector of detecting electrodes, this phenomenon is called anomalous CPGE (ACPGE).ACPGE was first reported in AlGaN/GaN heterostructures, then was observed in InGaAs/AlGaAs QWs, topological insulators and Weyl semimetals [9][10][11][12].Recently, we discussed the variation of ACPGE in p-doped GaAs bulk material with the doping concentration [13].Although the ACPGE have been observed in distinctly different material systems, their theoretical origins fall into the same frame: the Gaussian profile of the excitation light spot generates an inhomogeneous spin density distribution, then the SOC provides the spin transverse force for the ACPGE formation when the spin density diffuses.As can be seen, the existing theoretical model disregards the detailed energy band properties.But it has been demonstrated the bands spin splitting is crucial for the CPGE.Therefore, it is necessary to make clear whether the spin splitting influences the ACPGE.The strain modulated investigations could help for this purpose.In this work, the strain modulated ACPGE in p-type GaAs is investigated both experimentally and theoretically.By measuring the photocurrent distributions with the variation of exerted stress position, the ACPGE line-shapes under spatial uniform and gradient strains are obtained.It is found that the uniform strain related ACPGE behaves like a sine function, which resemble the non-strain situation.Whereas the linear gradient strain related ACPGE shows the unimodal function line-shape.In theory, we construct a new model for ACPGE considering the spin splitting of the energy bands generated by SOC.It is observed that the circularly polarized light could excite a special carrier distribution in k space, which leads to the current distribution in real space and hence the ACPGE.The model explains the experimental observations well, and it indicates that band splitting is a non-negligible factor for the ACPGE generation.We also discuss the implications and potential applications of the findings.

Experimental method
The sample here is p-type GaAs bulk material cut along [110] crystallographic orientation with a length of 25 mm and a width of 4 mm.The 2 µm thick p-type GaAs layer with beryllium doping (concentration: 3.1 × 10 17 cm −3 ) is grown by molecular beam epitaxy on a semi-insulating GaAs (001) substrate with a 300 nm GaAs buffer.Two pair of indium electrodes are made and annealed at 420 • C to achieve good ohmic contact, as shown in figure 1(a).The x-, y-, and z-axes are set along [100], [010], and [001] crystallographic orientations, respectively.The x ′ -and y ′ -axes are set along the long and short axes of the sample, respectively.The origin of coordinates is set on the middle of the electrodes.The external stress is provided by a home-made strain apparatus based on a micrometre.When the sharp edge at the front of the micrometre is tuned towards the sample, the sample experiences a tension strain along the x ′ -axis and a compression strain along the z-axis.The location of the applied stress, named as strain centre in this article, can be changed as well.The detailed description of this strain related technique can be found in [14].
As for the photocurrent measurement, a tunable femtosecond Ti: sapphire laser is utilized for the excitation.The laser beam with a diameter of 0.8 mm is vertically irradiated on the sample after being modulated by a chopper, polarizer, and photo-elastic modulator (PEM).The function of PEM is to periodically yield circularly polarized light at 50 kHz.The ACPGE currents excited by the circularly polarized light are collected by the standard lock-in amplification technology with PEM and chopper as the reference signals.An electric displacement platform is used to realize the light spot scanning along the perpendicular bisector of detecting electrodes.To investigate the influences of strain on ACPGE, the strain centres are selected to locate at x ′ = −1, 0, and 1 mm, respectively.Then the ACPGE are recorded under these strain conditions.All the experiments are conducted at 300 K.
Before the analysis of ACPGE, we discuss the photocurrent spectra without considering the excitation polarization at first.It is clear the photocurrent reaches maximum at 908 nm regardless of whether the stress is applied.The detailed analysis of this spectra has been finished in our previous work [13].As a result, the 915 nm wavelength is selected for the ACPGE to avoid the error caused by the photocurrent sharp change at 908 nm.We also observed the photocurrents do not vary with the position of the strain centre.Therefore, the light absorption difference induced by the different strain centres can be ignored.

Experimental results
The strain-modulated and strain-free ACPGE currents are measured as a function of the spot location when the strain centre is at x ′ = 1, 0, and −1 mm, respectively.As shown in figure 2(a), the ACPGE currents can be effectively modulated by the variation of the strain position.To demonstrate the impact of pure external The pure strain-induced ACPGE currents and the corresponding strain distribution.The blue, green, and red lines correspond to the data with the strain centre located at x ′ = −1, 0, and 1 mm, respectively.strain, the strain-induced ACPGE currents are plotted in figure 2(b), obtained by subtracting the strain-free ACPGE currents from the strain-modulated currents.Figure 2(b) also shows the strain strength distribution with different positions of the applied stress.When the stress was applied at the strain centre x ′ 0 , maximum strain is obtained here, and the strain decreases gradually towards both ends along the x ′ -direction, which can be described as ε(x ′ ) = 3hJ 0 F(x ′ )/(2a 2 ) [15], where 2a = 20 mm is the distance between the fixed plates at both ends of the sample, h = 500 µm is the sample thickness.J 0 is the deformation of strain centre in the z-direction, which is 0.1 mm here.F(x ′ ) describes the strain distribution along the x ′ -axis.Therefore, the strain conditions in different crystallographic orientations can be approximatively described by [14].
The ACPGE currents induced by strain can be further decomposed into two parts: the one related to the uniform strain, and the other to the gradient strain.Figure 3(a) shows the uniform strain induced ACPGE currents.It is the average of the ACPGE currents with the strain centre at x ′ = −1 and 1 mm, respectively.As shown in figure 3(a), the strain ε xy is a constant 6.75 × 10 −4 from x ′ = −1-1 mm.The corresponding ACPGE currents behave like a sine function.On the other hand, the gradient strain induced ACPGE is the minus of the ACPGE with the strain centre at x ′ = 1 and −1 mm, respectively.As shown in figure 3(b), the ACPGE currents at this condition demonstrate the unimodal function distribution.It is noted that the line-shape of the ACPGE under uniform strain is similar to that under no-strain modulation.Next, the ACPGE characteristics are interpreted theoretically.

Theoretical models and analyses
To explain the observed experimental results, an ACPGE model with the Dresselhaus and Rashba SOC is developed.Considering the bulk inversion asymmetry (BIA) of GaAs, the Hamiltonian of the SOC is a BIA-related Dresselhaus term with a material constant λ [16].The strain adds two types of conduction band spin splitting, namely the Rashba SOC term of the Hamiltonian with material constant C 3 /2 and the Dresselhaus SOC term of the Hamiltonian with material constant D [16], and they both vanish with zero strain.The electric field also contributes to the Rashba term of the Hamiltonian with material constant α [17].The above-mentioned Hamiltonians are, , , . ( The total Hamiltonian is the sum of the terms in equation ( 1).As a result, the two-fold spin degenerate conduction bands are splitted into two branches.When light is incident normally and there is no in-plane electric field, for point k at the two branches, the difference in the optical matrix element of the transition between the valence band and the spin-splitting conduction bands at the k point between right-handed circularly polarid light (σ+) and left-handed circularly polarized light (σ−) is given by the following expression: where ± corresponds to the two branches of conduction bands, p = s| p x |X , in which |s and |X are orbital wave functions at Γ point of the conduction and valence bands, respectively; γ 3 is the effective mass; T, K, and S related to the wave function of the valence band are functions of the wave vectors (both S and T are proportional to k 2 ), while C is related to that of the conduction band.The parameters and calculations are provided in the Supplementary Information and in [18].In this paper, as far as the BIA, the strain along the [110] strip and E z are concerned in the SOC Hamiltonian, the whole system shows a permutation symmetry of x and y, which means that C, K, S, and T are invariant with an exchange of k x and k y .Then, the influences of strain on the ACPGE currents are discussed.Owing to the symmetry of the p-type GaAs sample, the deformation parallel to the [110] crystallographic orientation led to nonvanishing strain components [14].Note that the surface electric field E z shows no contribution to ∆M.In the following calculations, we assume that the Dresselhaus SOC is much larger than the Rashba SOC.Under this assumption, C could be regarded approximately as an even function of k x and k y .
Note that the two branches of conduction bands show opposite ∆M at the same k points.However, for photon with specified energy, the wave vectors corresponding to the excited carriers in the two branches of the conduction band are different, which means that the net contribution of the two branches of conduction bands gives rise to a nonvanishing ∆M.
In this case, ∆M was proportional to k 2 x − k 2 y .The distribution of ∆M in k-space is shown in figure 4. Clearly, the optical transition probabilities induced by σ+ are greater than those of σ− in the [100] crystallographic orientation (red areas in figure 4(a)), while in the [010] crystallographic orientation ∆M is opposite (green areas in figure 4(a)).Considering the carrier at k has a velocity of ⃗ v = ∑ i ν i î,with i = x, y, z and ν i = hk i /m 0 , his reduced Planck constant, î is unit vector.The distribution of ∆M results in the photocurrent difference ∆j k between the σ+ and σ− in k-space, whose directions are denoted by the arrows in figure 4(a).
The intensity distribution of circularly polarized light is f (r), where r is the real-space coordinate, which can also be written as (x, y, z).The difference in photogenerated carriers between σ+ and σ− is , where a is a dimensional constant, and τ P is the spin relaxation time.The optical matrix element between σ+ and σ−, as shown in equation ( 2) results in directed charge current ∆j in real space, given by The ẑ component of the ∆j in equation ( 3) is ignored in the following discussion because we collect and measure the currents in the (001) crystallographic plane.
Thus, in the absence of strain, the directed charge current ∆j induced by bulk inversion asymmetry in real space is given by the following expression: with where f x and f y are the derivatives of the light intensity distribution in the x-and y-directions, respectively, which vary with the crystallographic orientation.Uniform-strain-induced ∆j in real space is expressed as follows: with The distribution of ∆j given by equations ( 4) and ( 6) in real space are shown in figure 4(b), which shows a cross-petal shape with the radial components vanishing in <110> crystallographic orientation.Note that in the absence of strain and the presence of uniform strain, ∆j determined by A or B is proportional to the odd term of k z , which means that the ∆j of k z and −k z would cancel each other.However, for the p-GaAs sample, the electric field in the z-direction disrupts this balance of ∆j of k z and −k z , and introduces a net current that has a similar form to equations ( 4) and ( 6), but substitutes E z and E z • k 2 z for the k z and k z 3 terms in equations ( 4) and ( 6), respectively.The ACPGE currents are proportional to ∆j.Thus, ACPGE currents increase with increasing E z , but when E z is sufficiently large, the rapid decrease in relaxation time leads to a decrease in the total ACPGE current [13].
We have developed an ACPGE model for the bulk inversion asymmetric GaAs.The difference in optical matrix elements ∆M between σ+ and σ− is calculated firstly.Then we calculate the difference in the charge currents ∆j between σ+ and σ− in the real space, whose direction and magnitude vary with the crystallographic orientation.In the following, by equivalenting the directed charge currents to the electric dipoles, we would explain the dependence of ACPGE currents on the spot position and strain.
The ∆j in equations ( 4) and ( 6  crystallographic orientation are negligible.Note that the electrical potential around a directed current is similar to that of a dipole.Thus, ∆j can be simplified to electric dipoles, as shown in figure 4(c).For a dipole ⃗ P, the electrical potential at l is given by V (l) = ⃗ P • ⃗ l/ ( 4π ε 0 l 3 ) , with ⃗ P ∝ ∆j, and ε 0 is the dielectric coefficient of GaAs.The mechanism of the ACPGE is shown in figure 4(c).Electric dipoles 1 and 2 (electric dipoles 3 and 4) respectively have current components in the −y ′ (+y ′ ) direction.When the circularly polarized light spot irradiates on the left side of the original point, electric dipoles 1 and 2 are closer to the electrodes than electric dipoles 3 and 4. Thus, the potential in the electrodes generated by electric dipoles 1 and 2 is greater than that generated by electric dipoles 3 and 4, which induces a net current from electrode a to b, known as the ACPGE current.When the circularly polarized light spot irradiates the right side of the original point, electric dipoles 3 and 4 dominate the electric potential measured across the electrodes, which generates a net current from electrode b to a.When the light spot is located at the origin, the ACPGE currents vanish because there is no component of current on the y ′ -axis.Thus, the functions of the ACPGE currents and the spot positions are sine-like with zero at the origin.When the electric dipole is located at (x ′ n , y ′ n ), l an and l bn are the distances between the electric dipole and electrodes a and b, respectively.It follows that the ACPGE currents collected by the electrodes can be described as In the absence of strain, ∆j is a λ-related expression only with respect to bulk inversion asymmetry.The ∆j given by equation ( 4) is shown in figure 4(b), which has different senses of current along the [100] and [010] crystallographic orientations, flowing toward and away from the centre of the spot, respectively.This means that the ACPGE currents would vanish when the electrode is in the [100] or [010] crystallographic orientation, whereas the ACPGE currents are detectable in the [110] and [1 10] crystallographic orientations.The ACPGE currents induced by the ∆j are shown in figure 4(c).The ACPGE currents of p-type GaAs cut along the [100] and [110] crystallographic orientation, shown in figure 5, agree with the characteristics described in equation ( 4).
The distribution of ∆j as described by equation ( 6) is shown in figure 4(b).The uniform-strain-induced ACPGE currents contributed by ∆j, as given in equation ( 6), are sine-like functions of the light spot positions, which have a line shape likes the ACPGE currents measured in the absence of strain and applied electric fields, as shown in figure 3(a).When a uniform strain of 10 −4 order is applied, the variation in the SOC coefficient was approximately 55%, which is obtained by dividing the uniform-strain-induced ACPGE currents by the ACPGE currents without strain.The strain-enhanced SOC coefficients of previous studies show an approximately 20% enhancement in splitting upon the application of approximately 2 × 10 −4 strain in two-dimensional GaAs [19] and up to 55% with a shear strain of 2.7 × 10 −4 in GaN [20].The variation in SOC splitting measured in our experiment is similar to that reported in previous studies.
Our study shows that a gradient strain parallel to the [110] crystal direction leads to additional ACPGE currents perpendicular to the gradient direction, as shown in figure 3(b).The origin of the gradient-strain-induced ACPGE currents is discussed here.First, we discuss the influence of the slant of energy bands induced by strain [21].The shifts in the energy bands caused by the strain at the strain centre   b) illustrate the gradient-strain-induced ACPGE currents when the strain centres are located at x ′ = −1 and 1 mm, respectively.The triangles and yellow dots indicate the positions of the strain centres and electrodes, respectively.The blue and red arrows are the electric dipoles equivalent to j, whose thickness indicates the magnitude of the electric dipole.The strain gradient breaks the symmetry of j, making the magnitudes of the electric dipoles 3, 4 different from 1, 2, which leads to additional currents perpendicular to the gradient.The gradient-strain-induced ACPGE currents are denoted by black arrows, the directions of which depend on the position of the strain centres.
are ∆E c = −5.24meV, ∆E HH = −2.59meV, and ∆E LH = 4.28 meV, therefore the effective electric fields induced by the gradient strain are less than 9.52 mV cm −1 .When voltages of 0 V and ±5 V (corresponding to electric fields of 0 V cm −1 and ±0.33 V cm −1 ) were applied across the strip electrodes, the dependence of the ACPGE currents on the light spot positions, shown in figure 6, reveals that the voltage cannot effectively modulate the ACPGE current.Thus, the gradient-strain-induced ACPGE currents could not be attributed to the slant of the energy bands.
No studies have clarified the influence of gradient strain the SOC so far.Therefore, we propose a phenomenological model to describe the mechanism of the gradient-strain-induced ACPGE currents.Gradient strain results in nonuniform changes in ∆j and destroys the symmetry in the gradient direction, as shown in figure 7.In figure 7(a), in the presence of the gradient strain, electric dipoles 1 and 2 and electric dipoles 3 and 4, which are equivalent to ∆j, have different amplitudes, so additional ACPGE currents perpendicular to the gradient are generated.Critically, the sign of the gradient-strain-induced ACPGE currents varies with the strain centre.The gradient strain-induced change in the electric dipoles shown in figure 7(b) is opposite to that shown in figure 7(a).As a result, when the centre of the light spot is located at (x ′ 0 , 0), the gradient-strain-induced ACPGE currents read sgn (−x ′ 0 ) ( ⃗ P 0 • ⃗ l a0 /l 3 a0 − ⃗ P 0 • ⃗ l b0 /l 3 b0 )/4π ε 0 , where ⃗ l a0 and ⃗ l b0 are the distances between (x ′ 0 , 0) and electrodes a and b, respectively, and ⃗ P 0 is parallel to the [1 10] crystallographic orientation.
The strain modulates the equivalent electric dipole.Therefore, the strain-free and strain-modulated ACPGE currents can be written as: The first term of equation ( 9) is a sine-like function of the light-spot positions, and the second term is a unimodal function of the spot positions that reaches its maximum at the origin of coordinates.In the absence of strain, ⃗ P 0 is zero and ⃗ P relates to λ.I ACPGE is a sine-like function of spot positions.For a uniform strain parallel to the [110] crystallographic orientation, ⃗ P 0 is zero and − → P is related to the Dresselhaus SOC  Further, let us consider the spin splitting and inverse spin Hall effect (ISHE) together.Generally, the ISHE-related photocurrent difference between σ+ and σ− in k-space is given by ∆j k ∝ γ∆M⃗ v × ⃗ S, here γ is the spin-orbit coupling coefficient, and ⃗ S is the spin of electrons at k.The ISHE-related current in real space ∆j ISHE is also given by equation (3).
When spin splitting is dominated by the surface electric field (E z ), the in-plane spin (determined by E z ) and the z-direction movements (determined by k z ) of electrons result in an in-plane ISHE current ∆j ISHE shown in figure 9(a), which is similar to the ∆j induced by BIA spin splitting shown in figure 4(b).The surface electric field plays an important role in p-GaAs 13 .So, the observed ACPGE currents can also attribute to the ISHE-related ∆j ISHE .
When there is no electric field, ISHE (γ∆M⃗ v × ⃗ S) can give rise to a cross-petal-shaped ∆j ISHE with components along the tangential directions shown in figure 9(b).In contrast to the radial cross-petal-shaped ∆j ISHE , this tangential cross-petal-shaped ∆j ISHE can be observed by the ACPGE along [100] crystallographic orientation.As shown in figure 9(c), when the light spot is on the left side of the sample, component 2 of the ∆j ISHE dominates over others, which causes the ACPGE current to flow from electrode a to electrode b.However, when the spot is on the right side of the sample, the ∆j ISHE component 4 is dominant.Therefore, when the light spot is on different sides of the sample, the signs of the ACPGE current are opposite.This can also explain the ACPGE currents observed along the [100] crystallographic orientation in figure 5.
In the aspect of physics, as mentioned in the introduction part, the ACPGE have been observed in many condensed matter systems, the ISHE is deemed as the universal physics origin.Both extrinsic and intrinsic mechanisms could contribute to the ISHE.The extrinsic mechanism is related to the spin deflection scattering, such as skew-scattering and side-jump, whereas the intrinsic mechanism is related to the energy band properties [22].In this work, we utilize the external strain to tuning the spin splitting of energy bands, then the strain modulated ACPGE are observed in GaAs bulk material.We find out a new physics mechanism related to spin splitting, which could contribute to the ACPGE.This is different from the well established spin transverse force explanation.The spin degeneracy lifted by SOC is more obvious in semiconductor low-dimensional structures, therefore we expect that the charge-spin conversion driven by spin splitting could be found in various QWs, superlattices and nanowires structures.On the other hand, the strain modulated carrier transportation properties of novel condensed matter systems such as topological insulators, Majorana fermions and twisted bilayer materials might be also worthy to investigate from the physics perspective.In the aspect of applications, SOC and symmetry breaking are the core for spin-orbitronics, such as spin transistor, spin-orbit qubits and spin-orbit torque devices.In this work, we demonstrate the strain can effectively manipulate the spin related currents.Therefore, to further explore spin-orbitronics devices, the unavoidably intrinsic and unintentional strain in the material systems should be noticed, especially for the low-dimensional structures.For instance, the intrinsic shear-strain reaching 1 × 10 −3 has been observed in InGaAs/InAlAs superlattices, according to our previous studies [23].On the other hand, enlightened by the quantum emitters based on quantum dots, which can be modulated by the lead magnesium niobate-lead titanate piezoelectric material [24,25], novel spin-orbitronics devices with active strain modulation might be promising as well.

Conclusions
The influences of spatial strain distribution on the ACPGE are investigated in the p-type GaAs.By varying the position of the strain centre, the uniform-strain-induced ACPGE and gradient-strain-induced ACPGE are distinguished.The line-shapes are sine-like function and unimodal function, respectively.It is also observed that there exists an approximately 55% variation of SOC coefficient with a uniform strain of 10 −4 order applied along the [110] direction in p-type GaAs.Further, a theoretical model based on the SOC induced bands spin splitting is constructed.The model explains the experimental observations well.In addition, the ISHE mechanism including spin splitting is discussed.When the spin splitting is dominated by surface electric field or BIA, the ACPGE currents show different distribution characteristics.This work reveals the importance of bands spin splitting to the ACPGE.The possible implications and potential applications of these findings are discussed as well.

Figure 1 .
Figure 1.(a) Schematic diagram of the ACPGE experimental apparatus.(b) The ordinary photocurrent spectra are measured in the absence and presence of strain (εxy = 7.5 × 10 −4 ), the light spot is at the origin point and a voltage of 1 V is applied to the dotted electrodes.

Figure 2 .
Figure 2. (a) The black dots indicate the dependence of the ACPGE currents measured in the absence of strain as a function of the light spot positions.The coloured cubes indicate the ACPGE currents measured in the presence of strain, which are referred to as the strain-modulated ACPGE currents.The positions of the strain centres are indicated by the vertical dash-dot lines.(b)The pure strain-induced ACPGE currents and the corresponding strain distribution.The blue, green, and red lines correspond to the data with the strain centre located at x ′ = −1, 0, and 1 mm, respectively.

Figure 3 .
Figure 3. (a) The distribution of the uniform strain (black line) and uniform-strain-induced ACPGE currents (red line).(b) The distribution of the gradient strain (black line) and gradient-strain-induced ACPGE currents (red line).

Figure 4 .
Figure 4. (a) The ∆M between the σ+ and σ− in k-space.(b) The directed charge currents ∆j in real space.The grey dotted circle indicates the light spot; the orange lines denote the magnitudes of ∆j; and the blue arrows indicate the directions of the ∆j.(c) Illustration of the mechanism of ACPGE currents.The blue arrows are the electric dipoles equivalent to ∆j.The ACPGE currents are collected by the two dotted electrodes (the yellow dots).
) are shown in figure 4, whose components flow toward the centre of the light spot in the [010] and [0 10] crystallographic orientation, and flow away from the centre of the light spot in the [100] and [ 100] crystallographic orientation.It should be noted that the components of ∆j in <110>

Figure 5 .
Figure 5.The light spot position dependencies of ACPGE current are measured in samples cut along [100] and [110] crystallographic orientations.

Figure 6 .
Figure 6.The ACPGE currents are measured at different light spot positions when 0 V and ±5 V voltages are applied to the strip electrodes.

Figure 7 .
Figure 7. (a) and (b) illustrate the gradient-strain-induced ACPGE currents when the strain centres are located at x ′ = −1 and 1 mm, respectively.The triangles and yellow dots indicate the positions of the strain centres and electrodes, respectively.The blue and red arrows are the electric dipoles equivalent to j, whose thickness indicates the magnitude of the electric dipole.The strain gradient breaks the symmetry of j, making the magnitudes of the electric dipoles 3, 4 different from 1, 2, which leads to additional currents perpendicular to the gradient.The gradient-strain-induced ACPGE currents are denoted by black arrows, the directions of which depend on the position of the strain centres.

Figure 9 .
Figure 9.With conduction band spin splitting and ISHE, the distribution of ∆j in real-space, (a) in the presence of surface electric field in the z-direction, (b) in the absence of surface electric field in the z-direction.(c) Illustration of the mechanism of ACPGE currents when the sample cut along [100] crystallographic orientation.The blue arrows are the electric dipoles equivalent to ∆j.The ACPGE currents are collected by the two dotted electrodes.