Estimating the spin diffusion length and the spin Hall angle from spin pumping induced inverse spin Hall voltages

Kuntal Roy

Phys. Rev. B 96, 174432 – Published 22 November 2017

Abstract

There exists considerable confusion in estimating the spin diffusion length of materials with high spin-orbit coupling from spin pumping experiments. For designing functional devices, it is important to determine the spin diffusion length with sufficient accuracy from experimental results. An inaccurate estimation of spin diffusion length also affects the estimation of other parameters (e.g., spin mixing conductance, spin Hall angle) concomitantly. The spin diffusion length for platinum (Pt) has been reported in the literature in a wide range of 0.5–14 nm, and in particular it is a constant value independent of Pt's thickness. Here, the key reasonings behind such a wide range of reported values of spin diffusion length have been identified comprehensively. In particular, it is shown here that a thickness-dependent conductivity and spin diffusion length is necessary to simultaneously match the experimental results of effective spin mixing conductance and inverse spin Hall voltage due to spin pumping. Such a thickness-dependent spin diffusion length is tantamount to the Elliott-Yafet spin relaxation mechanism, which bodes well for transitional metals. This conclusion is not altered even when there is significant interfacial spin memory loss. Furthermore, the variations in the estimated parameters are also studied, which is important for technological applications.

Received 12 June 2017

DOI:https://doi.org/10.1103/PhysRevB.96.174432

Physics Subject Headings (PhySH)

Inverse spin Hall effect Spin pumping Spintronics

Condensed Matter, Materials & Applied PhysicsInterdisciplinary Physics

Authors & Affiliations

Kuntal Roy^*

School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907, USA

^*Present address: Department of Electrical Engineering and Computer Science, Indian Institute of Science Education and Research Bhopal, Bhopal 462066, Madhya Pradesh, India; kuntal@iiserb.ac.in; kuntalroy@gmail.com

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Vol. 96, Iss. 17 — 1 November 2017

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Images

Figure 1
(a) A precessing magnetization in a magnetic layer is pumping pure spin current into the adjacent normal metal (NM). If the normal metal has a high spin orbit coupling, a considerable amount of charge current can be produced due to the inverse spin Hall effect (ISHE). Spin potentials are developed at the surfaces marked by 3 and 4, while charge potentials are developed at the surfaces marked by 1 and 2. (b) In an equivalent spin circuit diagram, the voltage source $V_{SP}$ acts as a spin battery, $G_{MIX}$ is the interfacial spin mixing conductance between the magnetic layer and the SHE layer, $G_{I}$ is the spin conductance representing the spin memory loss, and $G_{SHE}$ is the spin conductance of the SHE layer, altered by the spin accumulation in the SHE layer in the presence of spin memory loss.
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Figure 2
Fitting the thickness $t$ (2–65 nm) dependence of (a) effective spin-mixing conductance $g_{eff}^{↑ ↓}$ [Eq. (2)], and (b) a metric ${\tilde{V}}_{ISHE}^{SP} / e R w g_{eff}^{↑ ↓} f = θ_{SH} λ tanh (t / λ)$ [Eq. (5)]. Experimental data points are taken from Ref. [57], precisely for $f = 9$ GHz. In part (a), $g^{↑ ↓}, g_{I}$ , and $g_{SHE}$ are plotted as well.
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Figure 3
Thickness $t$ (2–65 nm) dependence of conductivity and spin diffusion length $λ$ . Due to the Elliot-Yafet spin relaxation mechanism, $λ \propto σ$ , and $σ$ depends on the thickness $t$ . The saturated value $λ_{max}$ ( $σ_{max}$ ) = 3.67 nm $(4.3 \times 10^{6}$ 1/ $Ω$ m) and the lowest value of $λ$ ( $σ$ ) = 0.62 nm $(0.73 \times 10^{6}$ 1/ $Ω$ m) at $t = 2$ nm.
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Figure 4
Thickness $t$ (2–65 nm) dependence of (a) effective spin-mixing conductance $g_{eff}^{↑ ↓}$ [Eq. (2)], and (b) a metric ${\tilde{V}}_{ISHE}^{SP} / e R w g_{eff}^{↑ ↓} f = θ_{SH} λ tanh (t / λ)$ [Eq. (5)] for two cases: $λ \propto σ$ (Elliott-Yafet spin relaxation mechanism) and $λ = λ_{max} = 3.67$ nm (Dyakonov-Perel spin relaxation mechanism).
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Figure 5
(a) The $λ_{max}$ (0.5–15 nm) dependence of $g^{↑ ↓}$ and $g_{SHE}$ for a constant effective spin mixing conductance $g_{eff}^{↑ ↓}$ and constant $g_{I}$ representing the interfacial spin memory loss. $g^{↑ ↓}$ increases with increasing $λ_{max}$ while $g_{SHE}$ follows the opposite trend. (b) The $λ_{max}$ dependence of $g_{I}$ and $g_{SHE}$ for constant values of $g_{eff}^{↑ ↓}$ and $g^{↑ ↓}$ . The spin flip parameter $δ$ is varied from 0 to 6.07 while the interface resistance is kept constant at $R^{*} = 87.3 f Ω m^{2}$ . Hence, the $λ_{max}$ varies from 0.18 to 98 nm. $g_{I}$ increases with increasing $λ_{max}$ while $g_{SHE}$ follows the opposite trend. The thickness $t$ of the SHE layer for both cases is 65 nm.
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Figure 6
The $λ_{max}$ (0.5–15 nm) dependence of the spin Hall angle $θ_{SH}$ while keeping the metric ${\tilde{V}}_{ISHE}^{SP} / e R w g_{eff}^{↑ ↓} f = θ_{SH} λ tanh (t / λ)$ [Eq. (5)] constant. The thickness $t$ is assumed to be 65 nm. The $θ_{SH}$ decreases with increasing $λ_{max}$ keeping $θ_{SH} λ_{max}$ constant.
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