Non-linear spin filter for non-magnetic materials at zero magnetic field

TL;DR

This paper introduces a novel all-electrical, non-invasive method for detecting spin accumulation in non-magnetic materials at zero magnetic field, utilizing non-linear spin-charge interactions in systems with strong spin-orbit coupling.

Contribution

It presents a new technique exploiting non-linear interactions for spin detection without magnetic fields, demonstrated with ballistic GaAs holes and a quantum point contact.

Findings

Effective spin-to-charge conversion demonstrated

Operates without magnetic field or large magnetic field

Applicable to various spin-orbit coupled systems

Abstract

The ability to convert spin accumulation to charge currents is essential for applications in spintronics. In semiconductors, spin-to-charge conversion is typically achieved using the inverse spin Hall effect or using a large magnetic field. Here we demonstrate a general method that exploits the non-linear interactions between spin and charge currents to perform all-electrical, rapid and non-invasive detection of spin accumulation without the need for a magnetic field. We demonstrate the operation of this technique with ballistic GaAs holes as a model system with strong spin-orbit coupling, in which a quantum point contact provides the non-linear energy filter. This approach is generally applicable to electron and hole systems with strong spin orbit coupling.

Equations

$${\partial }_B{V}_3(\omega ){|}_{B=0}=-\frac{\sigma g{\mu }_B}{2}\left[\frac{2e}{h}\int dE\left(-{\partial }_{E}f(E)\right){\partial }_{E}T(E)\right]\delta {\mu }_s,$$

$$I_{\mathrm{spin},\mathrm{linear}}\simeq \frac{2\hslash {\Omega }_{\mathrm{qpc}}}{\pi g{\mu }_B}\frac{e^2}{h}{\partial }_B{V}_3(\omega ){|}_{B=0},$$

$$V_3(2\omega )=\frac{1}{2}\left[\frac{e}{h}\int dE\left(-{\partial }_{E}f(E)\right){\partial }_{E}T(E)\right](\delta {\mu }_s{)}^2.$$

Full text

Introduction. Spintronics is a technology that uses the spin degree of freedom to manipulate information Wolf et al. (2001); Žutić et al. (2004). A key challenge in spintronics is the generation and detection of spin accumulation Han et al. (2018). In semiconductors, spin accumulation is typically generated by optical excitations Stern et al. (2006); Chang et al. (2007); Matsuzaka et al. (2009); Ando et al. (2010); Wunderlich et al. (2010) or the intrinsic spin Hall effect Brüne et al. (2010); Balakrishnan et al. (2013); Choi et al. (2015); Bottegoni et al. (2017), whilst spin-to-charge conversion (i.e. spin accumulation translating into a charge current or voltage) is achieved through the inverse spin Hall effect Brüne et al. (2010); Balakrishnan et al. (2013); Choi et al. (2015). However, generating/detecting spin accumulation optically or via the spin Hall effect-inverse spin Hall effect pair is challenging for strongly spin-orbit coupled mesoscopic systems with short spin relaxation time and spin diffusion lengths.

Here we adapt the concept of a spin filter, i.e. a device that separates spin species based on their energies, for detecting spin accumulation in strongly spin-orbit coupled mesoscopic systems. The first spin filter was developed by Stern and Gerlach who used an inhomogenous magnetic field to spatially separate electrons with different spins Gerlach and Stern (1922). Spin filters have also been realized in the solid state using spin-dependent transport in mesoscopic devices Chesi et al. (2011); Nichele et al. (2015). These techniques allow a spin current to be converted into a charge current, which is then detected as a voltage signal that depends on the applied magnetic field. Unfortunately, these linear techniques require a large magnetic field, which is impractical and can change the spin signal being probed. In this work, we demonstrate a non-linear technique that requires no magnetic field, and allows fast detection of spin accumulation.

We use GaAs holes as a model system for strongly spin-orbit coupled systems with short spin relaxation time ($<100$ fs) Hilton and Tang (2002) and spin diffusion length much shorter than the typical device dimensions ($\sim 100-1000$ nm, see also Sec. S2 of the Supplementary Material). Semiconductor holes have recently attracted great interest in semiconductor spintronics due to their exceptionally strong spin-orbit interaction Akhgar et al. (2016); Wang et al. (2016); Srinivasan et al. (2017); Marcellina et al. (2018); Hendrickx et al. (2018); Watzinger et al. (2018); Crippa et al. (2018); Hendrickx et al. (2018, 2020). The spin accumulation in strongly spin-orbit coupled ballistic mesoscopic systems is generated as follows. In mesoscopic systems with strong spin-orbit interaction, charge currents are generally accompanied by spin currents Bardarson et al. (2007); Krich and Halperin (2008); Adagideli et al. (2010); Stano and Jacquod (2011); Ramos et al. (2018). When the spin-orbit length is much shorter than both the device dimensions and mean free path, the spin precesses around randomly oriented spin-orbit fields throughout the sample region, giving rise to spin currents with a non-zero average Adagideli et al. (2010). Consequently, different spin species can have different chemical potentials, which give rise to a net spin accumulation whose amount and distribution depend on the sample geometry as well as the strength and form of the spin-orbit interaction. The spin accumulation adjacent to the energy barrier can then be detected through a voltage signal containing contributions linear and non-linear in spin accumulation.

This paper is laid out as follows. We first demonstrate spin-to-charge conversion in the linear regime using an in-plane magnetic field. We then show spin-to-charge conversion in the non-linear regime and confirm that it works even at zero magnetic field, so that is all-electrical and works much faster than linear spin-to-charge conversion. Our method can be generalized for any strongly spin-orbit coupled material such as GaSb, InAs, transition metal dichalcogenides, as well as topological insulators, since non-linear spin-to-charge conversion only requires a finite spin accumulation, regardless of the spin orientation, and an energy barrier. Furthermore, the rapidness of non-linear spin-to-charge conversion enables detection of spin orientation with radio-frequency techniques down to 1 ns Taskinen et al. (2008).

Experimental concept. We use a three-terminal geometry with a quantum point contact (QPC) as an energy-selective barrier (Fig. 1a). Passing a current $I_{\mathrm{sd}}$ in the drive channel between terminals 1 and 2 results in a voltage difference $V_{\mathrm{sd}}$ and a net non-equilibrium spin accumulation $\delta{\mu}_{s}$: Spins with orientation $\sigma_{+}$ have a higher chemical potential (of $\delta{\mu}_{s}$) than $\sigma_{-}$ (Figs. 1b and c). The kink in the drive channel helps direct the spin accumulation towards the QPC Benter et al. (2013). Spin-to-charge conversion occurs if one spin species has a higher transmission probability $T(E)$ through the QPC than the other. In the linear regime, the difference in the transmission probability originates from the difference in the hole’s kinetic energy due to an in-plane Zeeman interaction (Fig. 1d). However, in the non-linear regime, the energy dependence of the transmission probability $T(E)$ through the barrier causes the $\sigma_{+}$ spins to have a higher transmission probability through the QPC (Fig. 1e) than $\sigma_{-}$ even at zero field. In both the linear and non-linear regimes, the charge current through the QPC (Figs. 1f and g) causes a restoring voltage $V_{3}$ to maintain zero net charge current through the QPC with terminal 3 set as a floating probe. While the drive current $I_{\mathrm{sd}}$ oscillates at a frequency $\omega$, the linear and non-linear signals oscillate at the first and second harmonics of $V_{3}$, i.e. $V_{3}(\omega)$ and $V_{3}(2\omega)$ respectively.

Theoretical analysis. Using the transmission probability $T(E)\equiv T(E,B)$ for a QPC Buettiker (1990) (see also Sec. S1 of the Supplementary Material Note (1)), in the linear regime, the spin signal is proportional to the Zeeman splitting of the one-dimensional subbands. This gives rise to a three-terminal voltage $V_{3}(\omega)\equiv V_{3}(\omega,B)$ asymmetric in $B$. The asymmetry $\partial_{B}V_{3}(\omega)|_{B=0}$ is Stano et al. (2012):

[TABLE]

where $\sigma$ is the sign of the spin accumulation, $g$ is the in-plane $g-$factor, $\mu_{B}$ is the Bohr magneton, and $f(E)$ is the Fermi-Dirac distribution. Eq. (1) allows one to quantify the spin accumulation from the voltage asymmetry. The spin current through the QPC is Stano and Jacquod (2011); Nichele et al. (2015)

[TABLE]

where $\hbar\Omega_{\mathrm{qpc}}$ is the QPC saddle potential curvature Buettiker (1990).

In the non-linear regime, the difference in the transmission probability across the QPC is proportional to $\delta\mu_{s}$. Thus, the non-linear component of the spin signal $V_{3}$ is quadratic in $\delta\mu_{s}$:

[TABLE]

Given that $V_{3}(2\omega)$ is independent of the sign of $\delta\mu_{s}$, it is also symmetric in $B$.

Besides quantifying the spin current and accumulation, Eqs. (1)-(3) allow us to verify the spin origin of $V_{3}(\omega)$ and $V_{3}(2\omega)$ via their dependence on the QPC gate voltage $V_{\mathrm{qpc}}$, $B$, and $I_{\mathrm{sd}}$. Furthermore, since $\partial_{E}T(E)$ [Eqs. (1) and (3)] correlates with the QPC transconductance $\partial{G_{\mathrm{qpc}}}/\partial{V_{\mathrm{qpc}}}$, we expect maximal (no) spin signals when $\partial{G_{\mathrm{qpc}}}/\partial{V_{\mathrm{qpc}}}$ is maximal (minimal), where $G_{\mathrm{qpc}}$ is the QPC conductance.

Methods. An image of the device is shown in Fig. 2a. The device is made from an AlGaAs/GaAs heterostructure grown on a (100) GaAs substrate. For the measurements presented here, the two-dimensional hole density is $p=2\times 10^{11}$ cm*-2*, corresponding to a Fermi wavelength $\lambda_{F}=56$ nm, a spin-orbit length $l_{\mathrm{SO}}=35$ nm, and a mobility $\mu=550,000$ cm2 V*-1* s*-1* (see Sec. S2 of the Supplementary Material 111See the Supplementary Material at [URL] for additional details.). Surface gates define a conducting region in the shape of a ‘K’, with length 4 $\mu$m and width 1 $\mu$m, whilst the QPC is 370 nm wide and 210 nm long. When the ‘K-bar’ is defined, the conducting channel in the region is one-dimensional and the transport is ballistic with a mean free path of 4 $\mu$m (see Sec. S3 of the Supplementary Material Note (1) for details). All measurements were performed in a dilution fridge using standard lock-in techniques with $\omega=7$ Hz.

We send a current $I_{\mathrm{sd}}$ through the drive channel, and measure the resulting two-terminal $V_{\mathrm{sd}}\equiv V_{1}-V_{2}$ and three-terminal voltages $V_{3}(\omega)$ between terminals 2 and 3 (see also Fig. 1). Unless otherwise stated, $I_{\mathrm{sd}}$ is kept at 5 nA. Throughout this work, we concentrate our analysis on the second subband. While the first subband is affected by the “0.7 feature” Thomas et al. (1996); Danneau et al. (2008); Micolich and Zülicke (2011); Iqbal et al. (2013); Bauer et al. (2013), the spin signal is small for higher subbands ($N_{\mathrm{qpc}}>3$): The conductance quantization is progressively worse for these subbands, diminishing the spin-to-charge conversion efficiency. Fig. 2b shows how the QPC conductance is tuned by the QPC gate voltage. The two outer dashed lines mark the second and third conductance risers, where the spin-to-charge conversion should be most pronounced. The middle dashed line locates the second QPC plateau, where the spin-to-charge conversion should be suppressed. Fig. 2c shows the two-terminal resistance $V_{\mathrm{sd}}/I_{\mathrm{sd}}$ across the drive channel as a function of $V_{\mathrm{qpc}}$ and $B$. As expected from the Onsager reciprocity relation for electrical current in two-terminal systems, $V_{\mathrm{sd}}/I_{\mathrm{sd}}$ is approximately symmetric in $B$ (the QPC is a small perturbation to the drive channel, see Fig. S4 of the Supplementary Material Note (1)). Fig. 2d shows line cuts of Fig. 2c at the second and third QPC conductance risers and at the second QPC conductance plateau, confirming that $V_{\mathrm{sd}}$ is approximately symmetric in $B$ regardless of $V_{\mathrm{qpc}}$.

Linear spin-to-charge conversion. We now examine the linear three-terminal voltage $V_{3}(\omega)$. Fig. 2e shows $V_{3}(\omega)/I_{\mathrm{sd}}$ as a function of $V_{\mathrm{qpc}}$ and $B$, demonstrating that $V_{3}(\omega)$ is generally asymmetric in $B$. The line cuts of Fig. 2e shown in Fig. 2f reveal that $V_{3}(\omega)/I_{\mathrm{sd}}$ is asymmetric in $B$ on the second and third QPC conductance risers, but almost symmetric on the middle of the second conductance plateau. This is a crucial observation for the linear spin-to-charge conversion: The asymmetry of $V_{3}$ with $B$ is expected only if the spin accumulation is present and the QPC transmission is spin-(Zeeman energy) sensitive. At the QPC conductance plateau, although the spin current is still flowing through the QPC, it is not converted to a charge voltage. The asymmetry in $V_{3}(\omega)$ as a function of $B$ cannot be due to a Hall voltage as the sample was oriented to within $\pm 0.01^{\circ}$ with respect to the magnetic field Yeoh et al. (2010), so that the out-of-plane magnetic field is always $<0.5$ mT.

We next quantify the spin accumulation, spin current and the spin-to-charge conversion efficiency. Fig. 2g shows the asymmetry \Delta(\omega)\equiv\frac{1}{I_{\mathrm{sd}}}\frac{\partial V_{3}(\omega)}{\partial B}\Bigr{|}_{B=0}\propto I_{\mathrm{spin,linear}} of the three-terminal resistance at $I_{\mathrm{sd}}=2,~{}5,~{}10,~{}32$ nA as a function of $V_{\mathrm{qpc}}$. The asymmetry $\Delta(\omega)$ is obtained by performing a linear fit of $V_{3}(\omega)$ against $B$ between $-1~{}\mathrm{T}\leq B\leq 1$ T in Fig. 2g 222The fitting ranges $|B|\leq 0.5,0.75,1,1.25$ T give the same peak positions (see Sec. S5 of the Supplementary Material Note (2)).. There is a clear correlation between $\Delta(\omega)$ and $\partial_{V_{\mathrm{qpc}}}G_{\mathrm{qpc}}/G_{\mathrm{qpc}}$, which indicates linear spin-to-charge conversion (Eq. 1) for currents up to $I_{\mathrm{sd}}=10$ nA. The spin signal is suppressed at large $I_{\mathrm{sd}}$ (e.g. at $I_{\mathrm{sd}}=32$ nA), possibly due to averaging out of spin accumulations at different energies Nichele et al. (2015).

Using the results in Fig. 2g and experimental parameters $I_{\mathrm{sd}}=5$ nA, $g=0.38\pm 0.01$, $\hbar\Omega_{\mathrm{qpc}}=(0.17\pm 0.01)$ meV (see Sec. S3 of the Supplementary Material), $\Delta(\omega)=40$ $\mathrm{\Omega}$/T, $N_{\mathrm{drive}}=14$ (see Sec. S4 of the Supplementary Material Note (2)) and $N_{\mathrm{qpc}}=1.5$, we find that the spin accumulation is $\delta\mu_{s}=1~{}\mu$eV (Eq. S7) while the spin current is $I_{\mathrm{spin,linear}}=37$ pA (Eq. 2). The spin Hall angle Nichele et al. (2015), which measures the spin-to-charge conversion efficiency, is $\Theta\equiv(I_{\mathrm{spin,linear}}/N_{\mathrm{qpc}})/(I_{\mathrm{sd}}/N_{\mathrm{drive}})=6.8\%$. While our spin Hall angle falls within the range of previously reported values Jungwirth et al. (2012); Balakrishnan et al. (2014); Nichele et al. (2015); Sinova et al. (2015), caution must be exercised in the comparison since $\Theta$ is not only determined by the material but also the device details.

Non-linear spin-to-charge conversion. Now that we have established evidence for spin-to-charge conversion in the linear regime, we show that it also occurs in the non-linear regime. As before, we evaluate the dependence of the non-linear signal $V_{3}(2\omega)$ on $B$, $V_{\mathrm{qpc}}$, and $I_{\mathrm{sd}}$. Fig. 3a shows a color map of the non-linear resistance $V_{3}(2\omega)/I_{\mathrm{sd}}^{2}$ as a function of $B$ and $V_{\mathrm{qpc}}$. The non-linear signal $V_{3}(2\omega)$ is symmetric in $B$, contrasting with the linear signal $V_{3}(\omega)$ (Fig. 2e), and in line with Eq. (3). Next, we examine the dependence of $V_{\mathrm{qpc}}$ at $0\leq I_{\mathrm{sd}}\leq 44.1$ nA at $B=0$ T (Fig. 3b). The peak in the non-linear signal coincides with the QPC transconductance since $\partial_{E}T(E)$ is maximal at $T(E)=1/2$ when $B=0$ T, consistent with Eq. (3).

We next compare the linear and non-linear signals. Fig. 4a shows the amplitude $\Delta^{\prime}(\omega)$ of the linear signal relative to the background, i.e. the value of $\Delta(\omega)$ at the second subband subtracted by the lowest minimum (see Fig. 2g), against $I_{\mathrm{sd}}$. The spin current is linear in $I_{\mathrm{sd}}$ (and hence $\delta\mu_{s}$) at low excitation currents ($I_{\mathrm{sd}}\lesssim 5$ nA, see Fig. 4a). For comparison, Fig. 4b shows how $V_{3}(2\omega)$ varies with $I_{\mathrm{sd}}$. We find that the non-linear voltage is proportional to $I_{\mathrm{sd}}^{2}$ for $I_{\mathrm{sd}}\lesssim 7$ nA. While there is a possibility that Joule heating, which causes thermopower Appleyard et al. (1998); Bakker et al. (2010), could contribute to the second-harmonic response, the fact that both the linear and non-linear signals saturate at similar $I_{\mathrm{sd}}$ ($\approx 5$ nA and $\approx 7$ nA for the linear and non-linear signals, respectively) suggests that they are of a spin origin.

To further verify the spin origin of the signals, we consider their dependence on $B$ at low ($I_{\mathrm{sd}}=5$ nA) and high ($I_{\mathrm{sd}}=32$ nA) excitation currents. At low $I_{\mathrm{sd}}$ (Figs. 4a and b), both the linear (Fig. 4c, see also Sec. S5 of the Supplementary Material Note (2)) and non-linear signals (Fig. 4d) gradually decrease at $B\gtrsim 1.4$ T, suggesting that a strong magnetic field suppresses the spin accumulation. In contrast, for high $I_{\mathrm{sd}}$, where the spin-to-charge conversion is inefficient Stano et al. (2012), both the linear and non-linear signals are almost unaffected by the in-plane magnetic field (see also Sec. S6 of the Supplementary Material Note (2)). The consistency between the linear and non-linear signals confirms the reliability of non-linear spin-to-charge conversion. As non-linear spin-to-charge conversion requires no magnetic field (Figs. 3b and 4b), it allows a much faster detection of spin accumulation than linear spin-to-charge conversion 333As an example, the linear spin-to-charge measurements in Fig. 2 took $\sim 30$ hours, as they needed data at both $+B$ and $-B$. The equivalent non-linear spin-to-charge conversion, which only involved measuring $V_{3}$ as a function of $V_{\mathrm{qpc}}$ at $B=0$, took only $\sim 10$ minutes. .

Conclusions and outlook. Using ballistic mesoscopic GaAs holes as a model system, we demonstrate a new all-electrical non-linear technique for spin-to-charge conversion that does not require a magnetic field. We confirm the spin origin of the non-linear signals by calibrating them against linear spin-to-charge conversion. The non-linear spin detection technique allows much faster measurements than linear detection schemes, limited only by the bandwidth of the measurement circuit. Finally, we note that non-linear spin-to-charge conversion is very general: it only requires a spin accumulation regardless of its orientation and an adjacent energy-selective barrier. Our methods should be applicable in materials with very strong spin-orbit interaction such as GaSb, InAs, transition metal dichalcogenides, and topological materials, while its rapid speed will enable time resolved measurements of spin orientation to a 1 ns resolution using radio-frequency techniques.

Acknowledgment. The authors would like to thank Heiner Linke and I-Ju Chen for many enlightening discussions. This work was supported by the Australian Research Council under the Discovery Projects scheme and CE170100039. The device 27F8Q was fabricated using the facilities of the New South Wales 279 Node of the Australian National Fabrication Facility. F. Nichele acknowledges support from European Research Commission, grant number 804273.