Distinguishing the inverse spin Hall effect photocurrent of electrons and holes by comparing to the classical Hall effect

Yang Zhang; Yang Zhang; Yu Liu; Yu Liu; Xiao lin Zeng; Xiao lin Zeng; Jing Wu; Jing Wu; Jin ling Yu; Yong hai Chen; Yong hai Chen

doi:10.1364/OE.387692

1. Introduction

The generation, manipulation, and detection of spin currents is one of the key aspects of the field of spintronics [1–3]. Among the several possibilities to create and control spin currents, spin Hall effect (SHE) and inverse spin Hall effect (ISHE) have attracted more and more attention [4–7]. In order to further study on the application of ISHE on devices such as selecting electrons or holes as carriers when developing an electron-spin device, it is necessary to understand how electrons and holes work in ISHE and distinguish the contributions between electrons and holes in ISHE. While it is difficult to compare the ISHE between electrons and holes by theoretical calculation because of not only the extrinsic spin Hall effect affected by different doping and defects but also the intrinsic spin Hall effect affected by the stress and built-in electric field [8–13]. Although both minority and majority carrier transport parameters in silicon photovoltaics can be obtained by using optical Hall effect measurement [14], the ISHE currents of electrons and holes in the same sample have never been distinguished before. Most electrical and optical methods can only measure the ISHE of electrons or holes alone, or the total ISHE contribution of electrons and holes in materials [15,16].

In recent years, two experimental studies have been conducted on the ISHE of semiconductors by circularly polarized light, in which the contributions of electrons and holes are different in these experiments [17]. In the photo-induced anomalous Hall effect (PAHE), the spin-polarized electrons and holes can be driven along opposite directions by an electric field to produce a spin flow, which in turn produces a vertical charge current due to ISHE [17–19]. In the anomalous circular photogalvanic effect (ACPGE), due to the Gaussian distribution of the light spot, the spin-polarized electrons and holes have a density gradient in the radial direction of the Gaussian spot and move in the same direction, and the diffusion effects cause radial spin flow. Finally, a vortex current is generated due to ISHE [20–25]. This difference makes the distinguishment of electrons’ and holes’ actions in ISHE possible by the cooperative measurement of PAHE and ACPGE. However, there are different models for PAHE and ACPGE. Besides, both PAHE and ACPGE are related to the optical transition from the conduction band to the valence band, and strongly depend upon the lifetime, drift and diffusion of photo-generated carriers, which are irrelevant to ISHE [18–20]. Therefore, how to rule out these parameters irrelevant to ISHE in PAHE and ACPGE is a problem that needs to be addressed before using PAHE and ACPGE to clarify the role of electrons and holes in ISHE.

In this paper, we propose an optical method to solve this problem by introducing a magnetic field perpendicular to the surface of the investigated sample. Under the magnetic field, the ordinary photon-generated carriers are deflected by the Lorentz force, thus simulating the process of generating current due to the inverse spin Hall effect. Then by comparing the charge current related to the ordinary photon-generated carriers deflected by the Lorentz force and the spin-polarized photon-generated carriers deflected by the inverse spin Hall effect, we can eliminate the irrelevant process of ISHE and successfully separate the contribution of electrons and holes to ISHE through the cooperative measurement of PAHE and ACPGE. What’s more, we can also obtain the spin-orbit coupling strength of holes is larger than the spin-orbit coupling strength of electrons through some approximations and further derivation. This conclusion is consistent with previous theoretical and experimental studies.

2. Sample structure and experimental method

The experimental setup is described as follows. A Ti-sapphire laser is used as the excitation light source. The incident light passes through the chopper, polarizer, photo-elastic modulator (PEM) and lens and the phase retardation of PEM is set to be λ/4 which can generate a modulated circularly polarized light with a fixed modulating frequency at 50 kHz. An optical chopper with a rotation frequency of 223 Hz is used. The Gaussian profile light beam irradiates vertically on the sample with a diameter of about 2 mm and the power is 2 mW. The strip electrodes are used to apply an electric field. And the circularly polarized light photocurrent and ordinary photocurrent are collected through the two circle electrodes by two lock-in amplifiers with the synchronization frequencies set to be 50 kHz and 223 Hz, respectively. Likewise, a vertical magnetic field is applied perpendicularly to the surface of the sample. We used an electromagnet (East Changing Technologies, EM4) to provide a variable magnetic field from −0.5 T to 0.5 T. The magnitude of the magnetic field is directly obtained by its own Gauss meter inside the electromagnet. The pole diameter of the electromagnet is 100 mm and our sample is 12 mm. So the magnetic field passes through the sample uniformly. The experimental setup diagram is shown in Fig. 1(a). As shown in Fig. 1(b), an un-doped In_0.15Ga_0.85As/Al_0.3Ga_0.7As MQW is grown by molecular beam epitaxy. A 200 nm buffer layer is initially deposited on (001) SI-GaAs substrate, followed by ten periods of 10 nm In_0.15Ga_0.85As/10 nm Al_0.3Ga_0.7As quantum well. Then, a 50 nm Al_0.3Ga_0.7As layer and a 10 nm GaAs cap layer are deposited. The sample is cleaved into a narrow strip along the GaAs [110] direction with a width of 4 mm and a length of 12 mm. The geometry is shown in Fig. 2.

Fig. 1. (a) Diagram of the experimental setup; (b) The structural of InGaAs/AlGaAs un-doped multiple quantum well.

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Fig. 2. (a) The current generated by the diffusion of the spin flow because of the Gaussian distribution of the light spot (ACPGE). (b) The deflection of the ordinary diffused photon-generated carriers because of the Gaussian distribution of the light spot under the magnetic field. (c) The schematic diagram of electric current induced by spin drift generated by an external electric field and by the illumination of a circularly polarized light (PAHE). (d) The schematic diagram of the classic Hall effect of the ordinary photon-generated carriers. In these four figures, the green (yellow) dots with black arrows denote spin-polarized electrons (holes), the purple arrows denote the spin currents, the gray arrows denote the electric fields and blue arrows denote the magnetic field, and the red arrows denote the electric currents.

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3. Results and discussion

3.1 The ACPGE experimental configuration

Figure 2(a) shows the experimental configurations of ACPGE. In the ACPGE experimental configuration, a circularly polarized light with a Gaussian distribution arouses an inhomogeneous spin polarization. Spin diffusion occurs when there is a spin density gradient. The electron spin density gradient generates a diffused spin polarization current in the sample plane, which is proportional to the gradient of spin polarization density and the helicity of the incident light. Due to the inverse spin Hall effect, a detectable vortex current is generated around the light spot [as shown in Fig. 2(a)] [22,24]. In the case of optical injection, the spin diffusion and drift caused by carriers’ motion is equivalent to the diffusion and drift of spin-polarized photo-generated carriers [26–30]. Theoretically, we can use the following equation to express the ACPGE current

(1)$${J_{ACPGE}} = {C_\textrm{1}}({D_e}{\gamma _{es}}{\eta _{es}}{\tau _e} + {D_{hh}}{\gamma _{hhs}}{\eta _{hhs}}{\tau _{hh}})F({\rm x},r)e{\nabla _r}{G_0}(r)$$

Here ${J_{ACPGE}}$ is expressed as the ACPGE current we have measured. ${D_e}$ and ${D_{hh}}$ are the diffusion coefficient of the electrons and heavy holes, respectively. ${\gamma _{es}}$ and ${\gamma _{hhs}}$ are the inverse spin Hall coefficient of electrons and heavy holes, respectively. ${\eta _{es}}$ and ${\eta _{hhs}}$ are the circular polarization of electrons and heavy holes, respectively. e is the elementary charge and ${G_0}(r)$ is the generation rate of the photon-generated carriers corresponding to the transition from the first sub-band of heavy hole to the first conduction sub-band. ${\tau _e}$ and ${\tau _{hh}}$ are the lifetime of the photon-generated electrons and heavy holes, respectively. $F({\rm x},r)$ is a function that depends on the position (${\rm x}$) and size of the light spot ($r$). ${C_\textrm{1}}$ is a constant in the ACPGE experimental configuration. For doped semiconductor materials, the contributions of minority carriers are dominated with illumination. Recently, some researches have obtained both minority and majority carrier transport parameters for the first time and the results are consistent with literatures [14,31]. In our experiments, we chose to investigate an un-doped GaAs quantum well. The discussions of the contribution of minority carriers or majority carriers are not concerned because the density of photo-generated electrons is equal to holes in intrinsic semiconductors. By adding a chopper in the optical path and using a lock-in amplifier, only the photo-generated carriers and related Hall current can be detected.

Coincidentally, due to the influence of the Lorentz force of the magnetic field, which is perpendicular to the surface of the sample, the ordinary photo-generated carriers in the ACPGE experimental configuration will also show a transverse deflection [as shown in Fig. 2(b)]. Similar to the vortex current generated by the Gaussian profile of the circularly polarized light, due to the presence of the magnetic field, the vortex current excited by the ordinary light can also be expressed by Eq. (2),

(2)$$J_{_{Hall}}^{ACPGE} = {C_\textrm{1}}({D_e}{\mu _e}{\tau _e} + {D_{hh}}{\mu _{hh}}{\tau _{hh}})eBF({\rm x},r){\nabla _r}{G_0}(r)$$

Here $J_{_{Hall}}^{ACPGE}$ is the measured vortex current due to the Lorentz force, ${\mu _e}$ and ${\mu _{hh}}$ are the Hall mobility of electrons and heavy holes, respectively. ${\tau _e}$ and ${\tau _{hh}}$ are the lifetime of the photon-generated electrons and heavy holes, respectively. $B$ is the magnetic induction in Hall effect. Although the deflection direction of electrons and heavy holes are opposite, the Hall current of them is in the same direction.

By using two lock-in amplifiers with the synchronization frequencies set to be 223 Hz and 50 kHz, respectively, we can simultaneously measure the normal photocurrent and spin-polarized photocurrent induced by the ordinary light and circularly polarized light. The red line in Fig. 3 is the ACPGE current amplitude varying with light spot position and the black line is the ordinary photocurrent amplitude that is normalized by the magnetic field varying with light spot position. In Fig. 3, the ACPGE currents reverse the sign from the left to right side, just like a sine curve. In order to exclude the effect of the light spot position, we extract the amplitude of the two sine curves through data fitting. By comparing the amplitudes of the two curves, we obtain that ${J_{ACPGE}}/(J_{_{Hall}}^{ACPGE}/B) = 0.0015T$. It suggests that there is not a directed current but a current swirling around the center of the light spot. Also, it can be seen that, as the light spot is moving from the left side of the two electrodes to the right side, the normalized photocurrent reverses its direction. This observation is consistent with ACPGE current [18,21,22], which further confirms our model. The exciting light in Fig. 3 is 975 nm, which corresponds to the transition of the first sub-band of the heavy hole to the first conduction sub-band (1hh-1e). The InGaAs / AlGaAs MQW determines the transition energy from valence to conduction band. The transition of the first sub-band of the light and heavy hole to the first conduction sub-band (1L/hh-1e) corresponding excitation wavelength is 925 nm and 975 nm respectively. Therefore, by considering the Einstein relation $\frac{D}{\mu } = \frac{{{K_B}T}}{e}$, and by transforming Eqs. (1) and (2), we can get the following Eq. (3),

(3)$${J_{ACPGE}}/(J_{_{Hall}}^{ACPGE}/B) = \frac{{({\mu _e}{\gamma _{es}}{\eta _{es}}{\tau _e} + {\mu _{hh}}{\gamma _{hhs}}{\eta _{hhs}}{\tau _{hh}})}}{{({\mu _e}^2{\tau _e} + {\mu _{hh}}^2{\tau _{hh}})}}$$

Fig. 3. The slope of the ordinary photocurrent versus the magnetic field as a function of the position of the light spot (black line), and the ACPGE photocurrent varies with the position of the light spot (red line). Inset: schematic of the ACPGE experimental configuration, the laser beam was scanned along the x axis (red arrow). Yellow circles are the measuring electrode.

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3.2 The PAHE experimental configuration

Figure 2(c) shows the experimental configurations of PAHE. Unlike the usual AHE which occurs in magnetic materials, our sample is a non-magnetic semiconductor quantum well. The photo-induced anomalous Hall effect just borrows the name of anomalous Hall effect, but it is essentially an inverse spin Hall effect [20,21,24,29,30]. Then, under the PAHE experimental configuration, the spin-polarized photo-generated carriers that are excited by circularly polarized light drift under the external electric field, resulting in a spin current proportional to the external electric field. Due to the inverse spin Hall effect, there is a detectable lateral charge current perpendicular to the spin current [as shown in Fig. 2(c)]. By theoretical deduction, we can express PAHE current by Eq. (4).

(4)$${J_{PAHE}} = {C_\textrm{2}}({\mu _{hh}}{\gamma _{hhs}}{\eta _{hhs}}{\tau _{hh}} - {\mu _e}{\gamma _{es}}{\eta _{es}}{\tau _e})eE{G_0}$$

Here E is the transverse electric field. ${C_\textrm{2}}$ is the constant in the PAHE experimental configuration. In the coexistence of the external electric field and a perpendicular magnetic field, a charge current excited by an ordinary light can be detected in the same measurement direction of the Lorentz force [as shown in Fig. 2(d)]. This classical Hall current is also proportional to the external electric field and can also be expressed by Eq. (5).

(5)$$J_{_{Hall}}^{PAHE} = {C_\textrm{2}}({\mu _{hh}}^\textrm{2}{\tau _{hh}} - {\mu _e}^\textrm{2}{\tau _e})eEB{G_0}$$

The black and red lines shown in Fig. 4 are the classical Hall current that is normalized by the magnetic field and the PAHE current varying with the transverse electric field, respectively. The exciting light is 975 nm and the light spot irradiated in the center of sample and the ACPGE current is zero. The classical Hall current and PAHE current change linearly with the transverse electric field in Fig. 4 and it is consistent with the models we have given in Eqs. (4) and (5). We can get the slopes of two linear curves through data fitting. And by comparing the slopes of the two curves, we obtain that ${J_{PAHE}}\textrm{/(}J_{_{Hall}}^{PAHE}\textrm{/}B\textrm{)} = 0.00015T$ Therefore, by comparing the normal photocurrent due to the classical Hall effect with the spin-polarized photocurrent due to the PAHE, we can obtain Eq. (6)

(6)$${J_{PAHE}}\textrm{/(}J_{_{Hall}}^{PAHE}\textrm{/}B\textrm{)} = \frac{{({\mu _{hh}}{\gamma _{hhs}}{\eta _{hhs}}{\tau _{hh}} - {\mu _e}{\gamma _{es}}{\eta _{es}}{\tau _e})}}{{({\mu _{hh}}^\textrm{2}{\tau _{hh}} - {\mu _e}^\textrm{2}{\tau _e})}}$$

Fig. 4. The slope of the ordinary photocurrent versus the magnetic field as a function of the applied transverse electric field (black line), and the dependence of the PAHE photocurrent on the applied transverse electric field (red line). Inset: schematic of the PAHE experimental configuration, the laser beam irradiated in the center of sample (red circle). Yellow circles are the measuring electrode and Yellow strips are electrodes for applying the transverse electric field.

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3.3 The cooperation of ACPGE and PAHE currents normalized by the classical Hall current

Because drift and diffusion are responsible for PAHE and ACPGE respectively, i.e., the vertical current and the vortex current are measured in PAHE and ACPGE, respectively. Therefore, it is not advisable to directly compare ACPGE current with PAHE current. To obtain quantitative information concerning the parameters that characterize spin transport and spin-dependent scattering in the wells, we can compare the ACPGE current and PAHE current normalized by the classic Hall current. From Eqs. (3) and (6), we can derive Eqs. (7) and (8)

(7)$$\frac{{{J_{ACPGE}}/(J_{_{Hall}}^{ACPGE}/B)}}{{{J_{PAHE}}\textrm{/(}J_{_{Hall}}^{PAHE}\textrm{/}B\textrm{)}}} = \frac{{({\mu _e}{\gamma _{es}}{\eta _{es}}{\tau _e} + {\mu _{hh}}{\gamma _{hhs}}{\eta _{hhs}}{\tau _{hh}})({\mu _{hh}}^\textrm{2}{\tau _{hh}} - {\mu _e}^\textrm{2}{\tau _e})}}{{({\mu _{hh}}{\gamma _{hhs}}{\eta _{hhs}}{\tau _{hh}} - {\mu _e}{\gamma _{es}}{\eta _{es}}{\tau _e})({\mu _e}^2{\tau _e} + {\mu _{hh}}^2{\tau _{hh}})}}$$

(8)$$\frac{{{J_e}^{ishe}}}{{{J_{hh}}^{ishe}}} = \frac{{{\mu _e}{\tau _e}{\gamma _{es}}{\eta _e}_s}}{{{\mu _{hh}}{\tau _{hh}}{\gamma _{hhs}}{\eta _{hhs}}}} = \frac{{\alpha (\beta + 1) + (\beta - 1)}}{{\alpha (\beta + 1) - (\beta - 1)}}$$

Here we have defined $\frac{{{J_{ACPGE}}/(J_{_{Hall}}^{ACPGE}/B)}}{{{J_{PAHE}}\textrm{/(}J_{_{Hall}}^{PAHE}\textrm{/}B\textrm{)}}} = \alpha$ and ${(\frac{{{\mu _e}}}{{{\mu _{hh}}}})^2}\frac{{{\tau _e}}}{{{\tau _{hh}}}} = \beta$. The ${J_e}^{ishe}$ and ${J_{hh}}^{ishe}$ are the photocurrent of electrons and heavy holes induced by ISHE, respectively. Thus, according to Eq. (8), we can calculate the ratio of the photocurrent of electrons and holes induced by ISHE for different $\alpha$ and $\beta$, as shown in Fig. 5. The different colors in Fig. 5 represent the ratio of $\frac{{{J_e}^{ishe}}}{{{J_{hh}}^{ishe}}}$ and it can be seen that the photocurrent of electrons and heavy holes due to ISHE will be approximately equal when $\frac{{{J_{ACPGE}}/(J_{_{Hall}}^{ACPGE}/B)}}{{{J_{PAHE}}\textrm{/(}J_{_{Hall}}^{PAHE}\textrm{/}B\textrm{)}}} = \alpha > > 1$. What’s more, if $\frac{{{J_{ACPGE}}/(J_{_{Hall}}^{ACPGE}/B)}}{{{J_{PAHE}}\textrm{/(}J_{_{Hall}}^{PAHE}\textrm{/}B\textrm{)}}} = \alpha < < 1$, in the PAHE experimental configuration, the photocurrent of electrons and heavy holes due to ISHE flows in the same direction, while in the ACPGE experimental configuration, the photocurrent of electrons and heavy holes flow in the opposite direction.

Fig. 5. The graphics model for the relationship between the photocurrent of electrons and heavy holes induced by inverse spin Hall effect for different $\alpha$ and $\beta$ that we defined.

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In our experiment, $\frac{{{J_{ACPGE}}/(J_{_{Hall}}^{ACPGE}/B)}}{{{J_{PAHE}}\textrm{/(}J_{_{Hall}}^{PAHE}\textrm{/}B\textrm{)}}} = \alpha \cong 10$ and we can infer that the photocurrent of electrons and heavy holes induced by ISHE is in the same direction and in the opposite direction in ACPGE and PAHE experimental configuration, respectively, when excited by a laser of 975 nm, which corresponds to the transition 1hh-1e.

Then, according to reference [30–34], it is suggested that the lifetimes of photo-generated electrons and holes are approximately equal which is ${\tau _e}\textrm{ = }{\tau _{hh}}$. We know that the mobility of photo-generated electrons and holes highly depend on dopant concentration [14]. In an un-doped GaAs Quantum well, the mobility of photo-generated electrons is an order of magnitude larger than that of photo-generated holes [35]. So Eq. (8) can be simplified as

(9)$$\frac{{{J_e}^{ishe}}}{{{J_{hh}}^{ishe}}} = \frac{{{\mu _e}{\gamma _{es}}{\eta _e}_s}}{{{\mu _{hh}}{\gamma _{hhs}}{\eta _{hhs}}}} = \frac{{\textrm{11}}}{\textrm{9}}$$

Here ${\eta _e}_s$ and ${\eta _{hhs}}$ in Eq. (9) is the circularly polarization of photo-generated electrons and holes, respectively. For doped semiconductor materials, the total density of electrons and holes are not equal even though the photo-generated electrons and holes are equal [14]. Also our sample is an intrinsic semiconductor and the total density of photo-generated electrons and holes should be equal. We know that the circularly polarization is proportional to the spin relaxation time of the carriers. According to references [35,36], the spin relaxation time of the holes at the top of the valence band can be comparable to the electrons for un-doped quantum wells, which means ${\eta _e}_s \approx {\eta _{hhs}}$. Then we can easily conclude that the spin-orbit coupling coefficient of holes is one order of magnitude larger than the spin-orbit coupling coefficient of electrons. This result is consistent with some theoretical calculations and experimental results [36–41].

4. Conclusions

In conclusion, we propose a new optical method of distinguishing the photocurrent of electrons and holes induced by the inverse spin Hall effect. This method takes advantage of the different moving directions of photon-carriers under diffusion and electric field drift. By comparing the inverse spin Hall photocurrent obtained from the two experimental configurations, we find that the inverse spin Hall effect in the un-doped multi-quantum well is not only affected by electrons but also by holes and the contribution of holes is also important. It is a very interesting and important conclusion. In previous studies, it was thought that the electrons play a major role in the spin Hall effect due to the short spin relaxation time of holes in the wells. Now we find that the contribution of photo-generated holes also plays an important role.

Besides, we also study the deflection direction of electrons and holes due to the inverse spin Hall effect in the two different experiments. The photocurrent of the electrons and heavy holes due to the inverse spin Hall effect is in the same direction in ACPGE experimental configuration and in the opposite direction in PAHE experimental configuration, respectively. if the density of spin photo-generated electrons and holes have been measured, the ratio of the spin-orbit coupling coefficient of electrons and holes could be directly obtained through the method we introduced. As an important factor, it can be used as a reference for selecting electrons or holes as carriers when developing an electron-spin device.

Funding

National Key Research and Development Program of China (2018YFE0204001, 2018YFA0209103, 2016YFB0400101, 2016YFB0402303); National Natural Science Foundation of China (61627822, 61704121, 61991430).

Disclosures

The authors declare no conflicts of interest.

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Distinguishing the inverse spin Hall effect photocurrent of electrons and holes by comparing to the classical Hall effect

Abstract

1. Introduction

2. Sample structure and experimental method

3. Results and discussion

3.1 The ACPGE experimental configuration

3.2 The PAHE experimental configuration

3.3 The cooperation of ACPGE and PAHE currents normalized by the classical Hall current

4. Conclusions

Funding

Disclosures

References

Cited By

Figures (5)

Equations (9)

Optics Express