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Introduction

This example discusses a method to use our spin-circuit models in a distributed manner to simulate 2D spin diffusion and related phenomenon. We use the method to analyze the concept of spin funneling proposed in Ref. [1]. In Ref. [1], it has been proposed that a composite structure constructed with existing materials with spin-orbit coupling (SOC) can inject spins into a ferromagnet (FM) by an order magnitude higher than the SOC material itself (conventional structure). This indicates an effective enhancement in the charge to spin conversion efficiency often quantified with spin Hall angle (θ_SH). The idea is to collect spins from a large area of the SOC material and funnel into a smaller area of the FM using a normal metal (NM) with large spin diffusion length and low resistivity e.g. Cu, Al, Ag, etc. However, the low resistive NM in direct contact with the SOC will shunt a large amount of current leading to effective degradation in charge to spin conversion. The unwanted current shunting is avoided by utilizing a thin layer of ferromagnetic insulator which enables magnon assisted pure spin conduction (PSC). This proposal has received partial experimental support in Ref. [2].

The enhancement caused by "Spin Funneling" is quantified using the following enhancement factor:

which is ~1 for conventional structures and >1 for our proposed composite structure.

Spin-Circuit Model

We have performed 2D simulations using our spin circuit models: (1) GSHE module (for SOC layer), (2) PSC module (both interface and bulk), (3) NM module, (4) FM|NM module (for interface at FM and NM), and (5) FM module (for bulk). We used these modules to construct distributed circuits using standard circuit rules to represent the structures of interest.

In simulation, the length of SOC layer (L_g) is 700 nm and kept larger than the NM and PSC layers.We have varied the FM length (L_f ) from 5 to 20 nm and NM length (L_n) from 100 to 500 nm. Thicknesses of SOC, NM (t_n), and PSC are 6, 20, and 4 nm respectively in most simulations. The length of each small GSHE block is ~1 nm, except the red shaded block which has the same length as the FM.

We have discretized the NM layer in a ladder like structure to capture the 2D spin diffusion. The length of each black and red shaded longitudinal blocks are 1 nm and (L_f + 1 nm)/2, respectively. The thickness of each longitudinal block is half of the thickness of NM layer (t_n/2). The length of transverse blocks is t_n and the thicknesses of transverse NM blocks is the length of the corresponding GSHE blocks. To take into account the current shunting in NM layer, the charge terminals of the left most and right most NM blocks of the bottom row are connected to the charge terminals of the adjacent GSHE blocks.

Each of the PSC blocks are connected (both charge and spin terminals) between bottom NM row and GSHE via interface blocks on both sides. The length of each PSC block is the thickness of the FMI layer and the thickness of each PSC block is the length of corresponding GSHE block right under it.

To simulate the “spin sink”, we have applied ground boundary condition at the spin terminal of the block representing the region under the FM, instead of attaching FM and FM|NM interface modules. Otherwise, CoFeB, Co, and Py parameters have been used for simulation.

Please see Ref. [1] for further details.

Simulation Results

Funnel layer characteristics

Enhancement ratio as a function of NM spin diffusion length (λ_n) for different NM lengths (L_n) is shown below, which saturates roughly at min(λ_n, L_n). W|YIG|Cu|CoFeB structure is simulated.

Click here to download the simulation files.

Resistivity mismatches

To increase the enhancement ratio, the resistivity of the funnel layer has to be much lower than the SOC layer.

Click here to download the simulation files.

Critical thickness of funnel layer

Enhacement ratio as a function of NM layer thickness (t_n). Three different SOC material have been considered: tungsten (W), tantalum (Ta), and platinum (Pt). We have considered four different materials as NM layer: copper (Cu), silver (Ag), aluminum (Al), and gold (Au). Enhancement (>1) caused by spin funneling is observed for the case where W and Ta is used as spin source but degradation (<1) is observed for the case where Pt is used as spin source.

Click here to download the simulation files.

FM parameters

Spin funneling is a mechanism to enhance spin injection into low spin resistive load like FM which acts like a spin sink. We have analyzed the ideal spin sink case along with realistic FM parameters e.g. CoFeB, Co, and Py. Enhancement ratio is higher for FM with smaller length and the enhancement doubles if we make the FM length half.

Click here to download the analysis on FM length.

Click here to download the analysis on different FM.

Spin transparency of the PSC layer

The enhancement ratio of the composite structure depends on the spin transmission efficiency of the PSC layer determined by the magnon resistivity (ρ_m) and the FMI|NM interface conductance (g_s).

Click here to download simulation files for interface conductance analysis.

Click here to download simulation files for magnon resistivities analysis.

Material Parameters

The material parameters used for simulation are documented in the supplementary information of Ref. [1]. They are given below.

SOC material parameters (see Refs. [3]):

Material	λg [nm]	ρg [μΩ-cm]	θSH
W	2.1	200	0.33
Ta	1.8	190	0.15
Pt	1.2	24	0.07

NM material parameters (see Refs. [4]):

Material	λn [nm]	ρn [μΩ-cm]
Cu	500	2.08
Al	600	3.2
Ag	300	5.5
Au	60	5.2

FM material parameters (see Refs. [5]):

Material	pf	λf  [nm]	ρf [μΩ-cm]
CoFeB	0.65	12	168
Co	0.49	5.5	23.1
Py	0.45	38	21

FM|NM interface parameters (see Refs. [6]):

Material	kF  [nm-1]	Gr (x 1015Ω-1-m-2)
CoFeB	10.4	2.6
Py	10.5	0.39
Co	9.6	0.55

Questions / Comments

Please contact Shehrin Sayed ( ssayed AT berkeley DOT edu or ssayed AT lbl DOT gov) for questions or comments regarding this page.

References

[1] S. Sayed, V. Q. Diep, K. Y. Camsari, and S. Datta, "Spin Funneling for Enhanced Spin Injection into Ferromagnets," Sci. Rep. 6, 28868, 2016. (Link)

[2] V. Ostwal, A. Penumatcha, Y.-M. Hung, A. D. Kent, and J. Appenzeller, "Spin-orbit torque based magnetization switching in Pt/Cu/[Co/Ni]₅ multilayer structures," J. Appl. Phys., 122, 21, 213905, 2017. (Link)

[3] Pai et al. Appl. Phys. Lett. 101, 122404, 2012; Liu et al. Science, 336, 555-558, 2012; Liu et al. Phys. Rev. Lett. 109, 096602, 2012.

[4] Jedema et al. Nature 410, 345-348, 2001; Jedema et al. Phys. Rev. B 67, 085319, 2003; Kimura et al. Phys. Rev. B 72, 014461, 2005; Godfrey et al. Phys. Rev. Lett., 96, 136601, 2006; Fukuma et al. Nat. Materials 10, 527–531, 2011.

[5] Huang et al. Appl. Phys. Lett. 92, 242509, 2008; Kimura et al. Phys. Rev. Lett. 100, 066602, 2008; Taniguchi et al. Appl. Phys. Express, 1, 031302, 2008; Bauer et al. Phys. Rev. B, 67, 094421, 2003; Taniguchi et al. Magnetics, IEEE Transactions on 44, 2636-2639, 2008.

[6] Petrovykh et al. Appl. Phys. Lett. 73, 3459-3461, 1998; Li et al. Phys. Rev. Lett, 116, 117602, 2016; Boone et al. J. Appl. Phys. 113, 153906, 2013; Xia et al. Phys. Rev. B 65, 220401(R), 2002.