Transmission-Line Model for Materials with Spin-Momentum Locking

Open Access

Transmission-Line Model for Materials with Spin-Momentum Locking

Shehrin Sayed, Seokmin Hong, and Supriyo Datta

Phys. Rev. Applied 10, 054044 – Published 20 November 2018

Abstract

We provide a transmission-line representation for channels exhibiting spin-momentum locking (SML) that can be used for both time-dependent and steady-state transport analysis on a wide variety of materials with spin-orbit coupling such as topological insulators, Kondo insulators, transition metals, semimetals, oxide interfaces, and narrow band-gap semiconductors. This model is based on a time-dependent four-component diffusion equation obtained from the Boltzmann transport equation assuming linear response and elastic scattering in the channel. We classify all electronic states in the channel into four groups ( $U^{+}$ , $D^{+}$ , $U^{-}$ , and $D^{-}$ ) depending on the spin index [up ( $U$ ), down ( $D$ )] and the sign of the $x$ component of the group velocity ( $+, -$ ) and assign an average electrochemical potential to each of the four groups to obtain the four-component diffusion equation. For normal metal channels, the model decouples into the well-known transmission-line model for charge and a time-dependent version of the Valet-Fert equation for spin. We first show that, in the steady-state limit, our model leads to simple expressions for charge-spin interconversion in SML channels in good agreement with existing experimental data on diverse materials. We then use the full time-dependent model to study spin-charge separation in the presence of SML, a subject that has been controversial in the past. Our model shows that the charge and spin signals travel with two distinct velocities, resulting in well-known spin-charge separation, which is expected to persist even in the presence of SML. However, our model predicts that the lower velocity signal is purely spin, while the higher velocity signal is largely charge with an additional spin component proportional to the degree of SML, which has not been noted before. Finally, we note that our model can be used within standard circuit simulators such as SPICE to obtain numerical results for complex geometries.

Received 15 July 2017
Revised 15 April 2018

DOI:https://doi.org/10.1103/PhysRevApplied.10.054044

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Electrical generation of spin carriers Electrical spin injection Rashba coupling Spin Hall effect Spin polarization Spin-orbit coupling Spintronics Topological Hall effect

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Shehrin Sayed^1,*, Seokmin Hong², and Supriyo Datta^1,†

¹School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA
²Center for Spintronics, Korea Institute of Science and Technology, Seoul 02792, Republic of Korea

^*ssayed@purdue.edu
^†datta@purdue.edu

Article Text

Click to Expand

References

Click to Expand

Issue

Vol. 10, Iss. 5 — November 2018

Subject Areas

Reuse & Permissions

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Structure and corresponding axes of the channel with spin-momentum locking, for which a transmission-line model is derived. The model has two components: (b) charge [which corrseponds to Eq. (2)] and (c) spin [which corrseponds to Eq. (3)], with a coupling between them described by the degree of SML $p_{0}$ [see Eq. (1)]. The coupling between the charge and spin is modeled by dependent voltage and current sources with $V_{m} = η_{c} V_{s}$ , $V_{n} = η_{s} V_{c} + r_{m} I_{em}$ , and $I_{n} = γ_{s} I_{c} - g_{m} V_{em}$ .
Reuse & Permissions
Figure 2
(a) Structure of the spin-momentum-locked channel in the presence of an external contact. Both (b) charge and (c) spin transmission-line models are modified as compared to Fig. 1, which now correspond to Eqs. (6) and (7), respectively. The dependent sources are $V_{m}^{'} = η_{c} V_{s} + (G_{0} R_{B} / 2) (v_{s} / α + p_{f} v_{c})$ , $V_{n}^{'} = η_{s} V_{c} + r_{m} I_{em} + (G_{0} R_{B} / 2) (α v_{c} + p_{f} v_{s})$ , and $I_{n} = γ_{s} I_{c} - g_{m} V_{em}$ . Note that the model reduces to that shown in Fig. 1 in the limit $G_{0} \to 0$ .
Reuse & Permissions
Figure 3
(a) Setup to observe the charge-current- $I_{c}$ -induced spin voltage $v_{s}$ in the SML channel. The charge terminal of contact 3 is kept open and spin terminals of contacts 1 and 2 are grounded. (b) $v_{s} / I_{c}$ vs $p_{0}$ for $i_{tot}^{s} = 0$ from SPICE simulation and comparison with Eq. (19). (c) Setup to observe the spin-current- $i_{tot}^{s}$ -induced charge voltage difference $Δ V_{c}$ across the SML channel. The charge terminal of contact 3 is kept open and spin terminals of contacts 1 and 3 are grounded. (d) $Δ V_{c} / i_{tot}^{s}$ vs $p_{0}$ for $I_{c} = 0$ from SPICE simulation and comparison with Eq. (20). The SPICE setup is shown in Fig. 6 of Appendix pp2. The parameters are $α = 2 / π$ , $G_{0} \approx 0.05 G_{B}$ , $M + N = 100$ and the scattering rate per unit mode $= 0.04$ per lattice points. We assume reflection with spin flip to be the dominant scattering process in the channel.
Reuse & Permissions
Figure 4
(a) Setup similar to that in Fig. 3 but short-circuit charge current between contacts 1 and 2 is being observed. (b) IREE length ( $λ_{IREE}$ ) vs $p_{0} λ$ from SPICE simulation and comparison to Eq. (23) and experiments on different interfaces with Rashba SOC: $Ag / Bi$ [40], $Cu / Bi$ [41], ${La Al O}_{3} / {Sr Ti O}_{3}$ ( $LAO / STO$ ) [20], and a TI channel ${Bi}_{2} {Se}_{3}$ [42]. The backscattering length $λ$ is estimated from measured sheet resistance or resistivity. The degree of spin-momentum locking $p_{0}$ is estimated from the Rashba coupling coefficient and the Fermi velocity using Eq. (24). The details of estimations are given in Appendix pp2. The SPICE simulation parameters are $α = 2 / π$ , $G_{0} \approx 0.05 G_{B}$ , and $M + N = 100$ . We assume the reflection with spin flip to be the dominant scattering process in the channel.
Reuse & Permissions
Figure 5
(a) Two Fermi circles of a Rashba channel with spin-momentum locking having opposite spin polarizations at a given energy $E_{F}$ . The left (right) half of each circle represents negative (positive) group velocity. The large circle corresponds to the large number of modes $M$ in the channel and has a net spin polarization along $+ \hat{z}$ ( $- \hat{z}$ ) on right (left) half. The small circle corresponds to a smaller number of modes $N$ and has net spin polarization along $- \hat{z}$ ( $+ \hat{z}$ ) on the right (left) half. We classify all electronic states into four groups based on the spin index [up ( $U$ ), down ( $D$ )] and the sign of the $x$ component of the group velocity ( $+$ , $-$ ). (b) We assign four average electrochemical potentials for these four groups: $μ (U^{+})$ , $μ (U^{-})$ , $μ (D^{+})$ , and $μ (D^{-})$ . External contact is modeled as up ( $v_{u}$ ) and down ( $v_{d}$ ) spin voltages applied to up and down states of the channel through up ( $g_{u}$ ) and down ( $g_{d}$ ) spin conductances per mode per unit length, respectively. Four different currents ( $i_{U}^{+}$ , $i_{D}^{+}$ , $i_{U}^{-}$ , and $i_{D}^{-}$ ) enter the four different groups in the channel. The contact can be either normal metal $g_{u} = g_{d}$ or ferromagnet $g_{u} \neq g_{d}$ .
Reuse & Permissions
Figure 6
SPICE setup for dc simulation. The setup connects the transmission-line circuit models in a distributed manner. Block 1 corresponds to the model in Fig. 1 and Block 2 corresponds to the model in Fig. 2. All the external contacts have $p_{f} = 0$ . All the terminals are two-component terminals: charge ( $c$ ) and spin ( $s$ ). All indicated spin terminals are grounded except the spin terminal of contact 3.
Reuse & Permissions

Physical Review Applied