The spin Hall effect (SHE), which exploits the interplay of charge and spin, has attracted much attention in the field of spintronics. In presence of the spin–orbit coupling, a transverse spin current, which is widely believed to be beneficial for potential applications, is generated via a charge current driven by a voltage gradient.^[1,2] Therefore, from a practical application point of view, large spin Hall angle (SHA), i.e., the converting efficiency from charge current to spin current, is highly desired.^[3]

The exploration for the larger SHA largely relies on the understanding of the underneath mechanisms. It is widely accepted that the SHE and the anomalous Hall effect (AHE) share the same origin,^[4,5] of which three mechanisms have been proposed to account for the left–right asymmetry. Among them, the skew scattering^[6] and the side jump^[7] are realized upon electrons scattered by local potential created by foreign impurities or defects, known as the extrinsic mechanisms. In contrast, the intrinsic mechanism is solely determined by the electronic band structure.^[8,9] Therefore, the intrinsic mechanism leaves limited choices for artificial modification, while the magnitude of extrinsic mechanism, with various combinations of impurities and hosts, can be effectively tuned.^[10–12]

Furthermore, since the source of the scattering centers include foreign impurities, as well as the surface, and crystal defects,^[13] it is possible that the extrinsic mechanism becomes more pronounced in thin films.^[14] A recent experiment employing a H-pattern in 10-nm-thick Au indicated a large SHA of about 0.1, while a relative small SHA was reported by subjecting 60-nm-thick Au to similar measurement.^[15] Besides, a previous paper also reported that the SHA of pure Au, where no impurity is intentionally doped, was strongly reduced with thickness increasing.^[16] All those results appear to imply a sound logic that the surface scattering plays a crucial role at thin films, because the thinner the film, the more efficient the surface scattering.

The single crystal Au thin film, with minimum defects at the interior, manifests itself as a good platform to study the surface scattering. The physical picture is as follows: for the electrons traveling in a thin film with thickness comparable to the mean free path, the electrons should be scattered by both interior defects, and surfaces. The reduced defects indicates a longer mean free path at the interior, thereby ultimately results in the increasing of the surface scattering contribution. In an extreme case of perfect crystal structure, the electrons could only be scattered by the surface. Therefore, one may expect a comparable or even higher SHA in single crystalline Au thin film, if the surface scattering is indeed significant for the SHE as expected. In this paper, we investigate the SHE in single crystalline Au thin film through H-pattern approach, and derive an SHA of 0.08 at room temperature. This magnitude is consistent with previous experimental results obtained from poly-crystalline thin films.^[16–18] Thus, our result supported that the giant SHE is independent from the concentration of interior defects, but originates from the surface scattering.

We use the H-pattern to measure the SHE. Among the copious efforts waged on the SHE investigation, the H-pattern approach, with the capability of employing the spin current generation and detection within one layer, avoids the complex issues originated from the spin current transporting through the interface among layers.^[19–21] In principle, the H-pattern thus allows an accurate determination of the SHA without involving other mechanisms or complicated modeling.

As shown in Fig. 1(a), by applying current I through the left bar, the resultant V_nl is measured at the right bar, thus R_nl defines as V_nl/I. One may expect a non-local resistance R_sH from SHE contribution based on quasi one-dimensional (1D) spin diffusion process:^[22] [see Fig. 1(a)]

Fig. 1.

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Fig. 1. (a) The SHE contribution existed in H-pattern measurement as explained in the text. L stands for the center to center distance of two adjacent bars. w is the horizontal bridge width. Note that the RsH is positive. (b) Scanning electron microscopy image of H-pattern device, with several vertical bars separated by various distance L and connected by central bridge of width w = 40 nm. The Rnl is obtained from bar distance L of 200 nm unless specifically mentioned.

However, H-pattern measurement is not without controversies and complications. Early attempts with 60-nm-thick Au films performed by Mihajlović et al. suggest that the resultant non-local resistance is dominated by the additional undesired diffusive charge contribution (bypass contribution) R_c and ballistic transport contributions R_b instead of SHE R_sH.^[15] Thus, the total R_nl = R_c + R_b + R_sH. Recently, it has been demonstrated that the R_sH can be observed by enlarging device lateral geometry (w > l_s) and reducing film thickness. Remarkably, the temperature dependency of the reported signal is solidly evidenced to be from R_sH, and an SHA of about 0.1 is derived from 10-nm-thick poly-crystalline Au films. However, the quantitative analyses must rely on the assumption that no R_b residue is remaining, and resort to Eq. (1) which is only valid in the frame of w < l_s.

We notice that the ballistic contribution R_b is a secondary effect of the bypass contribution R_c which decays exponentially with w decreasing,^[5] while the SHE contribution, on the other hand, decreases linearly. In principle, with a proper combination of w and L, the R_sH should be larger than R_c and R_b. In addition, the R_sH is always positive due to the underlying mechanism, while R_c + R_b is supposed to remain negative at narrow w. Those different scaling dependencies on w could be the traces that ultimately lead to the unambiguous determination of R_sH. The previous experiments preclude such investigation for the reliability of narrow w devices is hindered by the poly-crystalline nature of the film.

To this end, we here have fabricated H-pattern based on both poly and single crystalline 10-nm Au film with w ranging from 70 nm to 40 nm. Although the single crystalline nature enhances the negative ballistic contribution R_b due to the longer electron mean free path l_e, it also ensures the possibility of fabrication of extremely small devices. Fortunately, the different behaviors of measured R_nl with various w are clearly observed and allow us to separate the R_sH.

Two types of Au film have been prepared in our experiments. The poly-crystalline Au film devices have been patterned into nanometer scale H-pattern by standard lift-off process through electron-beam lithography (EBL) on SiO₂/Si substrate before the film deposition by E-beam evaporation.

The single-crystalline Au films, on the other hand, were prepared on MgO (001) substrate by molecular beam epitaxy in an ultra-high vacuum system with base pressure of 1 × 10⁻⁷ Pa, ahead of any patterning processes. Before the deposition, the MgO (001) substrate was annealed at 500 °C for 60 min in order to form a clean and plain surface. Additionally, the Au film was in-situ annealed at 200 °C for 30 min after the deposition. The crystal quality of annealed MgO substrate and 10-nm Au with (001) were characterized by reflection high-energy electron diffraction (RHEED) technique. Clean and sharp RHEED patterns were obtained and presented as the inset of Fig. 2(a). Once the film preparation is completed, H-pattern structures were fabricated via negative resist-based EBL and Ar ion etching method. An extraordinary narrow horizontal bridge with w = 40 nm is achieved. Figure 1(b) demonstrates the scanning electron microscopy image of a complete device. The electrical measurements were performed by the DC reverse technique in combination with a Keithley 2182 nanovolt meter and 2400 source meter with a bias current ranging from 100 μA to 300 μA.

Fig. 2.

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Fig. 2. (a) Resistivity ρxx of 10-nm thick poly-crystalline (black) and single-crystalline (red) Au film as a function of temperature. Upper inset: the schematic map of the 4-probe measurement configuration. Lower inset: the RHEED pattern of the annealed MgO substrate (left) and the single crystal Au film (right). (b) Temperature dependency of Rnl measured for devices of w = 70 nm with different crystal quality. Inset: schematic map of the non-local measurement.

The value of resistivity ρ_xx is always an essential factor for assessing the film quality. The resistivity measurement of 10-nm thick Au films with different crystalline nature have been performed through 5 K to room temperature as demonstrated in Fig. 2(a). The result from both the poly-crystal and single-crystal film show common temperature dependency, i.e., the ρ_xx decrease linearly with temperature decreasing, as expected for normal metal films. However, their values are largely different. The ρ_xx in single-crystal film is 3.86 × 10⁻⁸ Ω· m at room temperature, nearly three times smaller than that in poly-crystal film of 1.19 × 10⁻⁷ Ω· m, due to the significantly improved crystal quality. Thereby a much longer l_e (3 times) should also be expected in single-crystal film.

Figure 2(b) shows the nonlocal resistance of R_nl, measured from both poly- and single-crystalline sample with w = 70 nm from 5 K to 290 K. We employ the non-local measurement by applying current I through the left bar hence the V₊ and V₋ for detection, as shown as inset of Fig. 2(b). The R_nl, thereby, is defined as V_nl/I. At room temperature, the R_nl for poly-crystalline sample is positive. With temperature decreasing to around 120 K, the sign of R_nl is reversed to negative. Such behavior has been well explained by the combination of the charge diffusive contribution R_c and electron ballistic contribution R_b that spontaneously entangles with R_sH. Phenomenologically one can describe the R_nl by^[5]

Although the magnitude of R_nl in two samples are about the same scale at 5 K, it does not indicate the ballistic contributions can be equally treated. One may notice that the ρ_xx is utterly different between the two types of Au film, such as R_sq. In Fig. 3(a), we presented the R_nl/R_sq as a function of temperature. It is immediately clear that the ballistic contribution is much more significant in single-crystalline sample at 5 K, consistent with previous understanding. Besides, the negative sign of R_nl/R_sq further supports our speculation that R_c + R_b is negative when w is narrow enough. Apparently, if the R_nl is solely consisting of R_c + R_b, R_nl should remain negative with w further reducing.

Fig. 3.

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	Fig. 3. (a) Rnl/Rsq as a function of temperature. Inset: I–V curve obtained form nonlocal measurement in single-crystal Au film devices with the central bridging wire w = 40 nm at 5 K. The linear dependency ensures the Vnl results from the applied current. Temperature dependency of Rnl (b) and Rnl/Rsq (c) for w = 40 nm and w = 50 nm.

To examine the speculation, the nonlocal measurement has been carried out in single crystal devices with w = 50 nm and w = 40 nm. The temperature dependency of R_nl and R_nl/R_sq are shown in Figs. 3(b) and 3(c). With w reducing to 50 nm, the absolute value of the R_nl/R_sq is slightly smaller than that in w = 70 nm due to the reduced lateral geometry, as can be seen from Eq. (2). But the sign of R_nl/R_sq is positive at room temperature, and again reverses to negative upon temperature decreasing. Even more surprisingly, when w = 40 nm, R_nl remains positive within the entire temperature range. In contrast, the negative R_nl is observed in w = 70 nm devices even at room temperature due to the acute ballistic contribution. Such behavior cannot be convincingly explained with previous understandings.^[15]

The inset of Fig. 3(a) shows the result of bias-dependent measurement stems from w = 40 nm sample at 5 K. The linear dependency of the nonlocal voltage on applied current ensures the effectiveness of the data, and our DC reverse technique helps to rule out Joule heating effect or the thermoelectric effects induced thermal voltage. As described by Eq. (2) and demonstrated by R_nl/R_sq in 70 nm, R_c + R_b is negative once w is narrow enough in respect to l_e. Therefore, if the R_nl solely consists of R_c and R_b, or R_sH is negligibly small, for a given device with set lateral geometry and l_e, one should expect that R_nl remains negative once w is below a critical value which is obviously opposite to our observations. It thus evidences that an additional positive contribution from R_sH must exist, and the R_c + R_b is negligibly small, if at all.

Using the obtained R_nl/R_sq from w = 40 nm, which are 0.085 × 10⁻³ at 5 K and 0.15 × 10⁻³ at room temperature, and Eq. (1), we are able to calculate the SHA. With an l_s of 86 nm for a gold film prepared on MgO with a similar resistivity of 2.7μ ·Ω· cm,^[17] we calculated the SHA of 0.08 at room temperature. We note that the value is larger than some of α in previous reports,^[15,23–25] but consistent with the result reported by^[16–18] based on the electrical injection experiments in poly-crystalline 10-nm-thick Au film, supporting the conclusion that the giant SHA is attributed to the surface scattering. Besides our results are also in agreement with the early determination of 8 percent polarization asymmetry in gold foil through double Mott scattering experiment.^[26]

As for other combinations with L longer than 200 nm, the V_nl is practically zero and does not obviously respond to applied current, indicating the resultant R_nl is beyond the resolution of our measurement. In addition, the temperature dependency of R_nl/R_sq with w = 40 nm is also unexpected. Normally the magnitude of SHE-related nonlocal voltage should monotonically decay with increasing temperature due to the decreasing of l_s. But it is important to note the equation (1) varies with l_s in a non-monotonic manner. Such a behavior may be responsible for this discrepancy. Based on Eq. (1) with w = 40 nm and L=200 nm, the R_sH/R_sq is deduced to be increasing with increasing l_s at short l_s, and reaching the maximum value at around l_s = 160 nm before decreasing at longer l_s.

Although one cannot derive the l_s in our sample from current results, however, given the ρ_xx in single crystal is 3 times smaller than that in poly-crystal at 5 K^[27,28] and the l_s for a poly-crystal gold film at 10 K is 63 nm,^[19,29] it is quite reasonable to expect an l_s larger than 160 nm in single-crystalline sample. If this is the case, the obtained SHA is a lower limit. Another possible reason could be the residual R_c + R_b. Despite the R_sH overruns the R_c + R_b in magnitude, but the temperature dependency may be dominated by the latter. However, with b = 20l_e = 40 nm, l_s=70 nm, α = 0.1, the calculated R_sH is about 1.64×10⁻⁴, orders of magnitude larger than the calculated R_c + R_b = −9.65 × 10⁻⁷. Thus, R_c + R_b is very unlikely dominating the temperature dependency of R_nl. In contrast, with w = 50 nm, R_c + R_b is calculated to be −1.66 × 10⁻⁵, comparable with R_sH. Nevertheless, a significantly smaller device should bring deeper insight to this issue, but we are not able to achieve such device currently. Besides the extremely small R_nl probably makes it impractical to acquire a solid signal.

In conclusion, we reported a series of investigations of R_nl in gold H-pattern fabricated based on both poly-crystal film and single-crystal film. Our analysis shows that the positive R_nl in narrow w devices arises from the SHE effect. With the finite signal obtained, we acquire an SHA of 0.08 in 10-nm-thick single-crystal Au film, indicating that a giant SHE originates from the surface scattering as expected, instead of the interior defects. In addition, our results prove that a clean SHE contribution can be unambiguously detected solely by optimizing the device’s geometry without resorting to external magnetic field, additional layers or complicated modeling, thus extending the boundary on the SHE investigation.