Geometrical effects on spin injection: 3D spin drift diffusion model

Abstract: We discuss a three-dimensional (3D) spin drift diffusion (SDD) model to inject spin from a ferromagnet (FM) to a normal metal (N) or semiconductor (SC). Using this model we investigate the problem of spin injection into isotropic materials like GaAs and study the effect of FM contact area and SC thickness on spin injection. We find that in order to achieve detectable spin injection a small contact area or thick SC samples are essential for direct contact spin injection devices. We investigate the use of thin metal films (Cu) proposed by S.B. Kumar et al. and show that they are an excellent substitute for tunnelling barriers (TB) in the regime of small contact area. Since most tunnelling barriers are prone to pinhole defects, we study the effect of pinholes in AlO tunnelling barriers and show that the reduction in the spin-injection ratio ($γ$) is solely due to the effective area of the pinholes and there is no correlation between the number of pinholes and the spin injection ratio.

• The paper develops a fully 3D spin drift-diffusion model that accurately captures the geometric dependence of spin injection, highlighting limitations of 1D models.
• It shows that spin injection efficiency is critically influenced by contact area, semiconductor height, and the presence of tunneling barriers or pinhole defects.
• The study demonstrates that thin metallic films can effectively replace tunneling barriers and that pinhole effects are determined by the total effective defect area.

Geometrical Effects on Spin Injection: Three-Dimensional Spin Drift Diffusion Model

Introduction

This work formulates and implements a fully three-dimensional (3D) spin drift-diffusion (SDD) model for spin injection from ferromagnets (FM) into normal metals (N) or semiconductors (SC), addressing substantial limitations inherent in one-dimensional (1D) and quasi-2D treatments. By numerically solving the 3D SDD equations with boundary conditions incorporating device geometry, material parameters, and interface properties, the study delivers comprehensive insights into the dependence of spin injection efficiency on geometrical parameters and the presence of tunneling barriers or metallic inserts. The research resolves essential open questions concerning size scaling, the use of metallic buffers, and defects such as pinholes in tunneling barriers, delivering both quantitative predictions and qualitative understanding strictly unavailable from 1D or 2D models.

Theoretical Framework

The SDD model describes spin-polarized transport by separating spin-up and spin-down electrochemical potentials and assembling continuity equations that couple drift and diffusion channels, parameterized by spin polarization, bulk conductivities, and spin diffusion lengths. This model is extended to three spatial dimensions, where spin-dependent boundary conditions at material interfaces and device boundaries are implemented. The use of the fictitious domain finite element method allows accurate and computationally scalable solution of the SDD equations in arbitrarily complex device geometries.

The 3D formalism explicitly incorporates:

• device boundary conditions enforcing zero spin current leakage,
• separate calculation domains for FM and N/SC regions with iterative boundary matching,
• explicit inclusion of all relevant length scales (spin diffusion length, sample geometry, contact area),
• robust convergence checks with respect to mesh size and domain extent,
• direct comparison and recovery of 1D analytic limits.

Device Geometry and Spin Injection

Analysis of prototypical three-terminal spin valves (NiFe/Cu) demonstrates that the spin injection ratio ($\gamma$) increases dramatically for small bridging distances between FM and N, a dependence absent from 1D SDD analysis. $\gamma$ shows no explicit bridging distance dependence in the 1D model, whereas in 3D, $\gamma$ departs from the 1D limit as $d$ approaches or falls below $\lambda_N$ (spin diffusion length in N). These results establish that the 1D SDD model is strictly invalid in real device geometries with finite, comparable structural dimensions.

Systematic variation of contact area and SC height in NiFe/n-GaAs devices reveals that only for extremely small FM/SC contact areas ($\sim 12.5 \, \text{nm}^2$) and thick SC samples ($\gg \lambda_\text{SC}$) does direct contact spin injection yield significant $\gamma$. For experimentally accessible contact areas ($\sim 10^5\, \text{nm}^2$), the spin signal vanishes due to spreading resistance and conductivity mismatch. Notably, the 3D model quantifies the exponential sensitivity of $\gamma$ to contact size, a key limitation in practical device scaling.

Tunneling Barriers, Thin Metal Films, and Spin Injection Optimization

Tunneling barriers (AlO) are evaluated as solutions to the conductivity mismatch problem, affording large interface resistances and decoupling bulk conductivities. In the 3D model, tunneling barriers robustly maintain high $\gamma$ irrespective of contact area and SC thickness, provided barriers are ideal (defect-free).

Importantly, the study rigorously demonstrates that thin metallic films (e.g., 50 nm Cu) can effectively substitute for tunneling barriers in devices with small contact areas. While metallic buffers limit the effective device spin diffusion length, they are not prone to barrier defect modes, thus providing an experimentally attractive alternative in certain device classes.

Pinholes in Tunneling Barriers

A central contribution of the paper is the quantitative and qualitative analysis of pinhole defects in tunneling barriers. Through detailed 3D simulations, the study establishes that the reduction in the spin injection ratio is determined solely by the total effective area of pinholes, not their number or spatial distribution, within experimentally relevant inter-pinhole distances.

This observation is physically justified by modeling the device as parallel spin channels (pinhole and barrier), each characterized by its own spin injection efficiency. The aggregate $\gamma$ is given by the current-weighted average of these channels. The model demonstrates (and simulations confirm) that even a single pinhole can dominate the total spin current by shunting current away from the tunneling barrier, leading to a proportional reduction in $\gamma$ as pinhole coverage increases. This behavior is not predictable by any 1D or simple resistor-tree model; only the full 3D treatment exposes the critical role of pinhole area fraction.

Implications and Future Prospects

The results have critical implications for both design and interpretation of spin injection experiments. They explicitly rule out the reliability of 1D models for quantitative prediction in any geometry where structural length scales are comparable to spin diffusion lengths or where interface inhomogeneity (defects, finite contacts) is relevant. The findings advocate strongly for nanoscale control of contact dimensions and the use of high-quality tunneling barriers or metallic film substitutes, depending on the regime of operation. For device engineering, inevitable process-induced defects such as pinholes must be quantified by area fraction rather than defect count. The model provides a predictive tool for evaluating the impact of geometric optimization, material selection, and imperfections on spintronic device performance.

Extending the 3D SDD approach to additional materials, heterostructures, and complex non-collinear magnetization configurations is possible and motivated. The computational framework can incorporate anisotropic transport, temperature-dependent diffusivity, and spin relaxation from additional mechanisms, enabling systematic exploration of practical device limits and new operational regimes.

Conclusion

This paper presents the first fully 3D SDD model solution for spin injection, delivering essential advances in understanding and predicting the geometric dependence of spin injection in FM/N or FM/SC devices. The analysis rigorously demonstrates the severe limitations of 1D (or quasi-1D) models, highlights critical scaling behavior with contact area and semiconductor height, and identifies both the effectiveness of thin metallic buffers and the precise role of pinhole defects. The study provides actionable insights for optimization, fault modeling, and architectural innovation in spintronic devices.