In a diffusive conductor with spin-orbit coupling, the spin Hall effect (SHE) converts charge current into transverse spin current, and the inverse SHE converts spin current back into a transverse voltage. Abanin, Shytov, Levitov and Halperin [Phys. Rev. B 79, 035304 (2009)] showed that in a narrow strip (ℓ≪w≪ℓs\ell\ll w\ll\ell_s, with ℓ\ell the mean free path, ww the strip width, and ℓs\ell_s the spin diffusion length) without k-linear spin-orbit terms, spin diffusion mediates a long-range nonlocal transresistance Rnl(x)∝e−∣x∣/ℓsR_{\mathrm{nl}}(x)\propto e^{-|x|/\ell_s} that can dominate the exponentially smaller Ohmic contribution, with a predicted nonmonotonic, sign-changing dependence on an in-plane magnetic field. The open question is the completion of this program: (1) a full quantitative theory of the nonlocal transresistance when k-linear Rashba and Dresselhaus spin-orbit terms coexist with extrinsic SHE scattering, including SO-induced spin precession and dephasing and the mapping of the relevant parameter regimes, and (2) its experimental realization — identifying material and device regimes where the spin-mediated signal is resolvable above Ohmic and other parasitic nonlocal backgrounds, and extracting the SHE parameters from field-dependent nonlocal measurements. Later work partially narrowed the theory core: a generalized spin-diffusion theory of the nonlocal resistance with Rashba-type k-linear coupling now exists, and ferromagnet-free nonlocal SHE measurements have been demonstrated in other platforms, but the predicted signature has not been observed in the originally proposed diffusive semiconductor geometry and the extrinsic-SHE/Dresselhaus case remains untreated.

Background

The spin Hall effect (SHE) is the generation of a transverse spin current by a longitudinal charge current in a conductor with spin-orbit coupling; the inverse SHE is the reciprocal process, in which a spin current or spin accumulation produces a transverse charge current or voltage. The extrinsic SHE arises from spin-dependent impurity scattering (skew and side-jump mechanisms), while the intrinsic SHE arises from spin-orbit terms in the band Hamiltonian, such as the Rashba term HSO=αz^⋅(σ×p)H_{\mathrm{SO}}=\alpha\hat{z}\cdot(\boldsymbol{\sigma}\times\mathbf{p}) produced by structural inversion asymmetry and the Dresselhaus term produced by bulk inversion asymmetry of the crystal. In diffusive systems, spin density relaxes over the spin diffusion length ℓs=Dsτs\ell_s=\sqrt{D_s\tau_s}, where DsD_s is the spin diffusion constant and τs\tau_s the spin relaxation time; in the presence of k-linear spin-orbit coupling, spins also precess and dephase during diffusion (Dyakonov–Perel mechanism). A nonlocal resistance measurement drives a current through one part of a sample and measures the voltage between remote probes a distance xx away; classical Ohmic charge spreading in a strip of width ww decays as exp⁡(−π∣x∣/w)\exp(-\pi|x|/w), so for w≪ℓsw\ll\ell_s the exponentially longer-ranged spin-diffusion signal can dominate. Abanin, Shytov, Levitov and Halperin [Phys. Rev. B 79, 035304 (2009); arXiv:0708.0455] solved this problem in closed form for the extrinsic SHE in a narrow diffusive strip (ℓ≪w≪ℓs\ell\ll w\ll\ell_s) whose band Hamiltonian contains no k-linear spin-orbit terms: because ∇×E=0\nabla\times\mathbf{E}=0, the SHE spin-current source is localized at the strip edges, the resulting spin density and net spin current propagate along the strip over ℓs\ell_s, and the inverse SHE at remote probes gives a transresistance Rnl(x)=12(βsσ)2wσℓse−∣x∣/ℓsR_{\mathrm{nl}}(x)=\frac{1}{2}\left(\frac{\beta_s}{\sigma}\right)^{2}\frac{w}{\sigma\ell_s}e^{-|x|/\ell_s}, where βs\beta_s is the spin Hall conductivity and σ\sigma the charge conductivity. For n-doped GaAs parameters (ℓs≈9 μm\ell_s\approx9\,\mu\mathrm{m}, w=0.5 μmw=0.5\,\mu\mathrm{m}) this gives Rnl∼10−7 ΩR_{\mathrm{nl}}\sim10^{-7}\,\Omega at ∣x∣∼ℓs|x|\sim\ell_s, vastly above the Ohmic leakage (∼e−πℓs/w∼10−25\sim e^{-\pi\ell_s/w}\sim10^{-25}). An in-plane magnetic field, with Larmor frequency ωB=gμBB/ℏ\omega_B=g\mu_BB/\hbar (where gg is the electron gg-factor and μB\mu_B the Bohr magneton), produces a nonmonotonic, sign-changing and eventually suppressed dependence Rnl(ωB)R_{\mathrm{nl}}(\omega_B) — a Hanle-type signature, by analogy with the precession-driven dephasing of a spin signal in a transverse field that is standard in optical and electrical spin-injection experiments, and the proposed experimental discriminator of the spin-mediated signal. The paper also derived geometry constraints for other materials (strip widths w≪360 nmw\ll360\,\mathrm{nm} for InGaAs, w≪85 nmw\ll85\,\mathrm{nm} for Al, and impractically short ℓs∼10 nm\ell_s\sim10\,\mathrm{nm} for Pt) and gave only a parametric estimate for the intrinsic-SHE (Rashba) case. Crucially, the closed-form theory explicitly excludes k-linear Dresselhaus and Rashba terms, whose SO-induced precession and dephasing (treated earlier for local geometries by Mishchenko, Shytov and Halperin [Phys. Rev. Lett. 93, 226602 (2004)] and by Tse and Das Sarma [Phys. Rev. B 74, 245309 (2006)]) complicate spin transport; and the paper notes that in high-mobility Rashba quantum wells the mean free path ∼1 μm\sim1\,\mu\mathrm{m} enhances Ohmic nonlocality, which may mask the spin-mediated signal. The parameters entering the predictions were taken from existing independent measurements rather than extracted self-consistently from the nonlocal response itself.

Problem Statement

Complete the theory and experimental program of spin-diffusion-mediated nonlocal charge transport in the spin Hall effect regime, whose common objective is an all-electrical, ferromagnet-free, quantitative probe of the spin Hall effect. Specifically: (1) Theory — develop the full quantitative theory of the nonlocal transresistance Rnl(x,B)R_{\mathrm{nl}}(x,B) in the diffusive strip geometry when k-linear spin-orbit terms (Rashba and Dresselhaus coupling) are present together with extrinsic spin Hall scattering, including the spin precession and dephasing they induce (Dyakonov–Perel-type dynamics), the interplay between the extrinsic spin Hall and intrinsic/magnetoelectric spin-charge conversion channels, and the field-dependent (Hanle-type) response; the theory should generalize the closed-form results of Abanin et al. [Phys. Rev. B 79, 035304 (2009)], which are valid only when k-linear terms are negligible and should reduce to the Abanin et al. transresistance when k-linear terms vanish, and should map the parameter regimes (spin precession length versus spin diffusion length ℓs\ell_s versus strip width ww versus mean free path) for the experimentally relevant materials (at least GaAs, InGaAs, Al and Pt). (2) Experiment — establish the material, geometry, and device conditions under which the spin-mediated nonlocal signal is actually resolvable: quantify the crossover between the spin-mediated regime and regimes dominated by Ohmic charge spreading, quasiballistic transport, and other parasitic nonlocal contributions (including the enhanced Ohmic nonlocality of high-mobility quantum wells with mean free path ∼1 μm\sim1\,\mu\mathrm{m}); determine the fabrication and probe-coupling tolerances given the derived width constraints (e.g. w≪360 nmw\ll360\,\mathrm{nm} for InGaAs, w≪85 nmw\ll85\,\mathrm{nm} for Al, and the ℓs∼10 nm\ell_s\sim10\,\mathrm{nm} limitation of Pt); and specify a measurement protocol to validate the theory — targeting the predicted nonmonotonic, sign-changing, oscillatory dependence of the transresistance on the in-plane magnetic field and an extraction scheme for the spin Hall and spin-transport parameters (βs\beta_s, σ\sigma, ℓs\ell_s, τs\tau_s) that fits them self-consistently to field-dependent nonlocal data rather than to separate literature values. No executed measurement is required.

References

  1. D. A. Abanin, A. V. Shytov, L. S. Levitov, B. I. Halperin, "Nonlocal charge transport mediated by spin diffusion in the spin Hall effect regime," Phys. Rev. B 79, 035304 (2009). DOI: 10.1103/PhysRevB.79.035304; arXiv:0708.0455. https://doi.org/10.1103/PhysRevB.79.035304
  2. C. Huang, Y. D. Chong, M. A. Cazalilla, "Anomalous Nonlocal Resistance and Spin-Charge Conversion Mechanisms in Two-Dimensional Metals," Phys. Rev. Lett. 119, 136804 (2017). DOI: 10.1103/PhysRevLett.119.136804; arXiv:1702.04955. https://doi.org/10.1103/PhysRevLett.119.136804
  3. E. G. Mishchenko, A. V. Shytov, B. I. Halperin, "Spin Current and Polarization in Impure Two-Dimensional Electron Systems with Spin-Orbit Coupling," Phys. Rev. Lett. 93, 226602 (2004). DOI: 10.1103/PhysRevLett.93.226602. https://doi.org/10.1103/PhysRevLett.93.226602
  4. W.-K. Tse, S. Das Sarma, "Intrinsic spin Hall effect in the presence of extrinsic spin-orbit scattering," Phys. Rev. B 74, 245309 (2006). DOI: 10.1103/PhysRevB.74.245309. https://doi.org/10.1103/PhysRevB.74.245309
  5. J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, A. H. MacDonald, "Universal Intrinsic Spin Hall Effect," Phys. Rev. Lett. 92, 126603 (2004). DOI: 10.1103/PhysRevLett.92.126603. https://doi.org/10.1103/PhysRevLett.92.126603
  6. S. Takahashi, S. Maekawa, "Nonlocal spin Hall effect and spin–orbit interaction in nonmagnetic metals," J. Magn. Magn. Mater. 310, 2067 (2007). DOI: 10.1016/j.jmmm.2006.10.1124; arXiv:cond-mat/0609192. https://doi.org/10.1016/j.jmmm.2006.10.1124
  7. G. Mihajlović, J. E. Pearson, M. A. Garcia, S. D. Bader, A. Hoffmann, "Negative Nonlocal Resistance in Mesoscopic Gold Hall Bars: Absence of the Giant Spin Hall Effect," Phys. Rev. Lett. 103, 166601 (2009). DOI: 10.1103/PhysRevLett.103.166601. https://doi.org/10.1103/PhysRevLett.103.166601
  8. E. S. Garlid, Q. O. Hu, M. K. Chan, C. J. Palmstrøm, P. A. Crowell, "Electrical Measurement of the Direct Spin Hall Effect in Fe/InxGa1-xAs Heterostructures," Phys. Rev. Lett. 105, 156602 (2010). DOI: 10.1103/PhysRevLett.105.156602. https://doi.org/10.1103/PhysRevLett.105.156602
  9. K. A. Kolwas, G. Grabecki, S. Trushkin, J. Wróbel, M. Aleszkiewicz, Ł. Cywiński, T. Dietl, G. Springholz, G. Bauer, "Absence of nonlocal resistance in microstructures of PbTe quantum wells," physica status solidi (b) 250, 37 (2012). DOI: 10.1002/pssb.201248431; arXiv:1111.2433. https://doi.org/10.1002/pssb.201248431
  10. Chien-Liang Chen, Ching-Ray Chang, Branislav K. Nikolić, "Quantum coherence and its dephasing in the giant spin Hall effect and nonlocal voltage generated by magnetotransport through multiterminal graphene bars," Phys. Rev. B 85, 155414 (2012). DOI: 10.1103/PhysRevB.85.155414; arXiv:1111.4892. https://doi.org/10.1103/PhysRevB.85.155414
  11. Jayakumar Balakrishnan, Gavin Kok Wai Koon, Manu Jaiswal, A. H. Castro Neto, Barbaros Özyilmaz, "Colossal enhancement of spin–orbit coupling in weakly hydrogenated graphene," Nature Physics 9, 284 (2013). DOI: 10.1038/nphys2576. https://doi.org/10.1038/nphys2576
  12. Markus Ehlert, Cheng Song, Mariusz Ciorga, Thomas Hupfauer, Junichi Shiogai, Martin Utz, Dieter Schuh, Dominique Bougeard, Dieter Weiss, "All-electrical detection of spin Hall effect in semiconductors," physica status solidi (b) 251, 1725 (2014). DOI: 10.1002/pssb.201350206. https://doi.org/10.1002/pssb.201350206
  13. Dinh Van Tuan, J. M. Marmolejo-Tejada, X. Waintal, B. K. Nikolić, S. O. Valenzuela, S. Roche, "Spin Hall Effect and Origins of Nonlocal Resistance in Adatom-Decorated Graphene," Phys. Rev. Lett. 117, 176602 (2016). DOI: 10.1103/PhysRevLett.117.176602; arXiv:1603.03870. https://doi.org/10.1103/PhysRevLett.117.176602
  14. Mi-Jin Jin, Seon Young Moon, Jungmin Park, Vijayakumar Modepalli, Junhyeon Jo, Shin-Ik Kim, Hyun Cheol Koo, Byoung-Chul Min, Hyun-Woo Lee, Seung-Hyub Baek, Jung-Woo Yoo, "Nonlocal Spin Diffusion Driven by Giant Spin Hall Effect at Oxide Heterointerfaces," Nano Lett. 17, 36 (2017). DOI: 10.1021/acs.nanolett.6b03050. https://doi.org/10.1021/acs.nanolett.6b03050
  15. Cai-Gan Chen, D. Tian, He-Xin Zhou, D. Hou, Xiao-Feng Jin, "Generation and Detection of Pure Spin Current in an H-Shaped Structure of a Single Metal," Phys. Rev. Lett. 122, 016804 (2019). DOI: 10.1103/PhysRevLett.122.016804. https://doi.org/10.1103/PhysRevLett.122.016804
  16. Tobias Völkl, Denis Kochan, Thomas Ebnet, Sebastian Ringer, Daniel Schiermeier, Philipp Nagler, Tobias Korn, Christian Schüller, Jaroslav Fabian, Dieter Weiss, Jonathan Eroms, "Absence of a giant spin Hall effect in plasma-hydrogenated graphene," Phys. Rev. B 99, 085401 (2019). DOI: 10.1103/PhysRevB.99.085401; arXiv:1809.10475. https://doi.org/10.1103/PhysRevB.99.085401
  17. Gregory M. Stephen, Owen A. Vail, Jennifer E. DeMell, Aubrey T. Hanbicki, Patrick J. Taylor, Adam L. Friedman, "Nonlocal Measurement as a Probe of the Spin Hall Effect in Topological Insulators," Phys. Rev. Applied 16, 034007 (2021). DOI: 10.1103/PhysRevApplied.16.034007. https://doi.org/10.1103/PhysRevApplied.16.034007