Extending spin-lattice relaxation theory to three-phonon processes

Abstract

Spin-lattice relaxation theory has been developed over almost a century, but some cardinal assumptions on the nature of the interactions involved have never been fully verified. This includes the weak coupling approximation, which makes it possible to describe spin dynamics perturbatively and leads to the canonical description of spin relaxation in terms of one- and two-phonon processes. Here, we extend the first-principles theory of spin relaxation to three-phonon processes and apply it to the vdW crystal of a spin-1/2 Chromium nitride complex. Results show that three-phonon contributions to spin relaxation only become relevant at temperatures inaccessible to experiments for this molecule, thus providing unprecedented evidence for the validity of the weak spin-phonon coupling assumption in spin relaxation theory. At the same time, we numerically show that a relatively small increase in spin-phonon coupling would lead to a crossover between three- and two-phonon processes' efficiency at room temperature, illustrating the possibility for three-phonon effects in molecular materials as well as paving the way to a systematic exploration of strong coupling in spin systems.

Figures (8)

• Figure 1: A spin transition from the state $| a\rangle$ to $| b \rangle$ (red solid line), due to a resonant one-phonon emission (blue wavy line) process as dictated by the generator $R^{(2)}_{ba}$.
• Figure 2: A spin transition from the state $| a\rangle$ to $| b \rangle$ (red solid line), due to a two-phonon process as dictated by the superoperator $R^{(4)}_{ba}$, where one phonon is absorbed and one is emitted simultaneously (blue wavy lines), and the entire process is mediated through intermediate virtual transitions (red dashed lines) to an excited spin state $| c \rangle$.
• Figure 3: A spin transition from the state $| a\rangle$ to $| b \rangle$ (red solid line), due to a three-phonon process as dictated by the superoperator $R^{(6)}_{ba}$, where one phonon is absorbed and two are emitted simultaneously (blue wavy lines), and the entire process is mediated through intermediate virtual transitions (red dashed lines) to the excited spin states $| c \rangle$ and $| d \rangle$.
• Figure 4: Molecular structure of CrN(pyrdtc)$_{\mathrm 2}$. Color code: blue for Cr, magenta for N, dark gray for C, yellow for S, light gray for H
• Figure 5: $T_1$ computed with three-phonon (green line) as well as two-phonon (purple line) processes is reported for the temperature range $T =$ 5-400 K. The two-phonon contributions show a good agreement with the experimental data (red points).