In the last section of this article, we focus on the so-called H-bar geometry, which allows for measurements of the inverse spin Hall effect without the need to generate a spin polarized charge current. This setup was first described and studied theoretically by Hankiewicz et al. . Corresponding measurements were reported by Brüne et al.  for HgTe microstructures and by Mihajlović et al.  for Au Hall bars. Kolwas et al.  carried out measurements in PbTe quantum wells and a similar setup was also used by Balakrishnan et al.  to experimentally determine the strength of spin–orbit coupling in graphene. However, reports for semiconductors are still rare.
We modified the original proposal  and employed for our experimental devices a double H-bar geometry. Figure 6a shows the corresponding design, where a charge current jy is driven in the middle branch B. In the presence of spin–orbit fields, the scattering of charge carriers is spin-dependent, thus generating a spin current js perpendicular to jy (DSHE). While spins accumulate at the channel boundaries, the spin current js can flow along the bridging channel, which connects all branches of the “H”. Scattering of charge carriers gives then rise to a charge current oriented perpendicular (y-direction) to the spin current js (ISHE). In consequence a net charge accumulation and hence a nonlocal resistance can be detected in the adjacent branches A and C. The concept of the double H-bar geometry allows for a validity check, because the charge accumulation, which is induced by the ISHE, is similar for both branches A and C, and depends on the polarity and magnitude of the applied charge current jy .
Figure 6. (a) Sketch of a double H-bar geometry used for detection of ISHE. A charge current jy is driven in the middle branch B. Induced by DSHE (1), a perpendicular spin current js flows along the connecting part of the “H”. Due to ISHE (2) a nonlocal charge imbalance can finally be detected in the adjacent branches A and C. (b) SEM images of one of our p-GaAs H-bar samples and of the central region of this device which consists of four “H”-branches (width 350 nm).
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The SEM image (Fig. 6b) shows one of our p+-doped (2 × 1018 cm−3) GaAs nanostructures fabricated from a single wafer, which was grown by MBE on a (001) GaAs substrate. Four separate “H”-branches with a width of 350 nm are electrically connected by a bridging channel, which is w = 900 nm wide. The distance between the centers of two adjacent branches is L = 700 nm, whereas the height of the conduction channel measures t = 300 nm.
All ISHE measurements were carried out at T = 50 K. We passed a constant dc current IB in the middle branch B and monitored simultaneously the local signal UB and the nonlocal signals UA and UC (see Fig. 6a). Both nonlocal signals UA and UC have the same magnitude and sign and scale linearly with the applied current IB in the range of IB = ±100 nA–10 µA (not shown here), furthermore UA and UC invert fully for opposite polarity of IB. This bias dependence of the nonlocal signals is consistent with the ISHE in a H-bar geometry. The spin Hall angle αSH can be calculated from the nonlocal spin Hall resistance [42, 45], which is given by
Here, Rsq = ρ/t = 60 × 103 Ω is the sheet resistance of our samples. We assume that the spin diffusion length is of order λsf = 1 µm. For , we calculate αSH ≥ 3.1 × 10−1 as a lower limit for the spin Hall angle (note that αSH increases for larger values of λsf). The deduced value for αSH is then at least 2 orders of magnitude larger than values predicted by theory for p-GaAs  and from previous experiments in n-GaAs (αSH = 10−3–10−4) [10, 17-19]. Hence, it is very doubtful that the measured nonlocal signal is fully induced by the ISHE, despite the fact that the spin–orbit coupling is expected to be stronger in p-GaAs compared to n-GaAs .
In order to identify different contributions to the measured nonlocal signal, we performed measurements in an external magnetic field. The spin polarization generated by means of DSHE (see Fig. 6a) points along the ±z-direction. Therefore, an in-plane magnetic field Bx or By along the ±x- or ±y-direction should induce precession of the spins, what would result in an oscillatory behavior of the nonlocal signal . However, no such oscillations were observed in our experiments (not shown here), suggesting that the ISHE is not the origin of the signal. More information on its possible origins was obtained from measurements in out-of-plane magnetic field Bz applied along the ±z-direction. No spin precession, i.e., no change in the nonlocal signal, should be observed if the signal stems from ISHE. However, if other mechanisms than the ISHE contribute to the nonlocal signal, a distinct magnetic field dependence of the nonlocal signals can be expected. In these experiments, the magnetic field Bz was swept in the range of ±10 T and all measurements were again carried out at T = 50 K and for different values of the applied current (±100 nA–10 µA). Typical data of local (UB) and nonlocal signals (UA, UC) for a current IB = ±5 µA are displayed in Fig. 7a and b.
Figure 7. (Inset) Measurement configuration: A current IB is driven in the middle branch B while an out-of-plane magnetic field Bz is swept in the range of ±10 T. We monitor both local signal UB and nonlocal signals UA and UC. (a) Magnetic field dependence of the local signal UB (monitored in the middle branch B) for a current IB = ±5 µA at T = 50 K. (b) Nonlocal signals UA and UC for the same measurements.
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Both local and nonlocal signals show distinct magnetic field dependencies and depend on the polarity of IB, but only the local signal UB fully reverses with the latter. A strong and approximately quadratic dependence of UB on Bz is most likely due to positive magnetoresistance (PMR), which can be observed in non-magnetic materials. The magnitude of the PMR yields ≈12% at Bz = ±10 T. For further analysis, the nonlocal signals UA and UC are split up in symmetric and antisymmetric parts with respect to Bz.
First, we consider the antisymmetric parts of the nonlocal signals (Fig. 8a), which are on the one hand independent of the polarity of IB and on the other hand of opposite sign for both branches A and C, respectively. These features exclude the ISHE as origin, as it is illustrated in Fig. 6a. Furthermore, the magnitude (determined at Bz = 10 T) scales quadratically with the applied current IB (not shown here), suggesting a thermically induced effect, which depends on the dissipated power. Since the dimensions of our samples are very small (cross-section ∼0.1 µm2), it is likely that the middle branch B is heated when a current is passed through (e.g., Pelectrical = 22 µW at IB = 5 µA). In consequence a heat current flows along the bridging channel, as illustrated in the inset of Fig. 8a. The charge carriers are then deflected in the perpendicular magnetic field Bz due to the Lorentz force, what gives rise to a measurable charge carrier accumulation in the branches A and C . This phenomenon, which is known as Ettingshausen–Nernst effect , is independent of the polarity of IB and was also reported for other SHE experiments .
Figure 8. (a) Antisymmetric part (with respect to Bz) of the nonlocal signals UA and UC for IB = ±5 µA (see Fig. 7b). (Inset) Ettingshausen–Nernst effect: A heat current flows from the middle branch B into the adjacent branches and induces a nonlocal charge imbalance, which is of opposite sign for A and C. (b) Symmetric part (with respect to Bz) of the nonlocal signals UA and UC for IB = ±5 µA. (Inset) van der Pauw theorem: In the diffusive regime the current density spreads into the adjacent branches A and C and induces a nonlocal charge imbalance of identical sign for both branches.
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We now analyze the symmetric parts of nonlocal signals (Fig. 8b), which have both the same polarity and invert for opposite polarity of IB. Since the symmetric part of both nonlocal signals UA and UC scales quadratically with Bz, we assume a correlation with the local signal UB, which shows a similar magnet field dependence (Fig. 7a). This can be explained by diffusive transport, where the current density spreads from the middle branch B into the bridging wire, as illustrated in the inset of Fig. 8b. The resulting nonlocal resistance can be calculated by means of the van der Pauw theorem 
For L = 700 nm, we derive from Eq. (7) a nonlocal resistance , which is comparable to experimental data (RNL = 1.3 kΩ, determined at Bz = 0 T). Hence, the measured bias dependence of the nonlocal signals (without magnetic field), for which we calculated a huge value of the spin Hall angle, can mainly be explained by diffusive charge carrier transport. This effect could be reduced by fabricating samples with w ≪ λsf since this diminishes the contribution of diffusive transport to the nonlocal resistance by several orders of magnitude  (e.g., for w = 200 nm).
Here, we reported on ISHE experiments in p-GaAs nanostructures with a double H-shaped geometry, where we passed a current IB through the middle branch of the double “H”. We could observe clear nonlocal signals in the adjacent branches which scaled with the current IB, however, the calculated spin Hall angle was larger than expected. Measurements in an external out-of-plane magnetic field indicated that thermal effects (Ettingshausen–Nernst effect ) and diffusive transport (van der Pauw theorem ) dominate the measured nonlocal signals. The first effect is relevant due to the small dimensions of the samples, i.e., the width of the current-carrying branch, but can be easily identified, as it does not depend on the polarity of IB. The diffusive transport strongly depends on the width w of the bridging channel and can easily be confused with the ISHE, since it also scales with IB. Hence, the unavoidable presence of these both effects requires careful analysis of the measured nonlocal signals, in order to determine the ISHE-related contribution.