[Paper Review] Intrinsic Damping Phenomena from Quantum to Classical Magnets:An ab-initio Study of Gilbert Damping in Pt/Co Bilayer

Farzad Mahfouzi, Kim, Jinwoong+1 more|arXiv (Cornell University)|Sep 14, 2017

Magnetic Properties and ApplicationsMaterials Science3 citations

TL;DR

This study presents an ab initio quantum mechanical framework using Keldysh Green's functions to calculate Gilbert damping in Pt/Co bilayers, resolving the breakdown of the relaxation time approximation in the ballistic regime. It shows that damping remains finite due to finite spin size S, recovers classical torque correlation expressions in the S→∞ limit, and accurately predicts spin diffusion length (1–6 nm) and spin mixing conductance (~20 nm⁻²) in good agreement with experiments when disorder is included via self-consistent dephasing.

ABSTRACT

A fully quantum mechanical description of the precessional damping of Pt/Co bilayer is presented in the framework of the Keldysh Green function approach using {\it ab initio} electronic structure calculations. In contrast to previous calculations of classical Gilbert damping ($α_{GD}$), we demonstrate that $α_{GD}$ in the quantum case does not diverge in the ballistic regime due to the finite size of the total spin, $S$. In the limit of $S ightarrow\infty$ we show that the formalism recovers the torque correlation expression for $α_{GD}$ which we decompose into spin-pumping and spin-orbital torque correlation contributions. The formalism is generalized to take into account a self consistently determined dephasing mechanism which preserves the conservation laws and allows the investigation of the effect of disorder. The dependence of $α_{GD}$ on Pt thickness and disorder strength is calculated and the spin diffusion length of Pt and spin mixing conductance of the bilayer are determined and compared with experiments.

Motivation & Objective

• To develop a quantum mechanical framework for intrinsic Gilbert damping in nano-scale ferromagnets beyond the relaxation time approximation.
• To resolve the divergence of Gilbert damping in the ballistic regime under conventional treatments by incorporating finite spin size S.
• To accurately model the effects of disorder and dephasing while preserving conservation laws (energy, charge, angular momentum).
• To extract spin diffusion length and spin mixing conductance from ab initio calculations and compare them with experimental values.
• To decompose Gilbert damping into spin-pumping and spin-orbit torque correlation contributions using a conserving formalism.

Proposed method

• Employing the Keldysh Green's function formalism within ab initio electronic structure calculations to describe non-equilibrium dynamics in Pt/Co bilayers.
• Using a self-consistent Born approximation to include disorder effects while preserving conservation laws.
• Deriving the Gilbert damping via the torque correlation function, decomposed into spin-pumping and spin-orbit torque correlation contributions.
• Introducing a finite broadening parameter η to model electron relaxation, with η → 0 corresponding to the ballistic limit.
• Applying the conserving formalism to avoid unphysical divergences in the relaxation time approximation.
• Fitting ab initio damping data to the spin diffusion model to extract spin diffusion length and effective spin mixing conductance.

Experimental results

Research questions

• RQ1 How does the Gilbert damping constant behave in the ballistic regime when the magnetic moment is treated quantum mechanically rather than classically?
• RQ2 What is the role of finite spin size S in preventing divergence of the damping rate in the absence of relaxation?
• RQ3 How does the inclusion of self-consistent dephasing preserve conservation laws and improve agreement with experimental damping in disordered systems?
• RQ4 To what extent do spin-pumping and spin-orbit torque correlation contributions contribute to the total Gilbert damping?
• RQ5 Can ab initio calculations quantitatively reproduce the spin diffusion length and spin mixing conductance observed in experiments?

Key findings

• In the ballistic regime, Gilbert damping remains finite due to the finite spin size S, avoiding the unphysical divergence seen in classical treatments.
• In the limit S→∞, the formalism recovers the conventional torque correlation expression for Gilbert damping.
• The relaxation time approximation fails for broadening η > 1 meV, which corresponds to room temperature conditions.
• The calculated spin diffusion length of Pt ranges from 1 to 6 nm in the diffusive regime (λ_int > 0.2 eV), consistent with experimental values of 0.5–10 nm.
• The effective spin mixing conductance is found to be ~20 nm⁻², approximately half of the experimental value (~35–40 nm⁻²), but stable across disorder strengths.
• In the ballistic regime (λ_int < 0.2 eV), the spin diffusion length is ~0.5 nm, in agreement with prior theoretical and experimental observations.