Experimental Quantification of Spin-Phonon Coupling in Molecular Qubits using Inelastic Neutron Scattering

Electronic spin superposition states enable nanoscale sensing through their sensitivity to the local environment, yet their sensitivity to vibrational motion also limits their coherence times. In molecular spin systems, chemical tunability and atomic-scale resolution are accompanied by a dense, thermally accessible phonon spectrum that introduces efficient spin relaxation pathways. Despite extensive theoretical work, there is little experimental consensus on which vibrational energies dominate spin relaxation or how molecular structure controls spin-phonon coupling (SPC). We present a fully experimental method to quantify SPC coefficients by combining temperature-dependent vibrational spectra from inelastic neutron scattering with spin relaxation rates measured by electron paramagnetic resonance. We apply this framework to two model S = 1/2 systems, copper(II) phthalocyanine (CuPc) and copper(II) octaethylporphyrin (CuOEP). Two distinct relaxation regimes emerge: below 40 K, weakly coupled lattice modes below $50~\mathrm{cm}^{-1}$ dominate, whereas above 40 K, optical phonons above $185\mathrm{cm}^{-1}$ become thermally populated and drive relaxation with SPC coefficients nearly three orders of magnitude larger. Structural distortions in CuOEP that break planar symmetry soften the crystal lattice and enhance anharmonic scattering, but also raise the energy of stretching modes at the molecular core where the spins reside. This redistributes vibrational energy toward the molecular periphery and out of plane, ultimately reducing SPC relative to CuPc and enabling room-temperature spin coherence in CuOEP. Although our method does not provide mode-specific SPC coefficients, it quantifies contributions from distinct spectral regions and establishes a broadly applicable, fully experimental link between crystal structure, lattice dynamics, and spin relaxation.

💡 Research Summary

This paper introduces a fully experimental framework for quantifying spin‑phonon coupling (SPC) in molecular qubits by combining temperature‑dependent inelastic neutron scattering (INS) with spin‑lattice relaxation rates measured by pulse electron paramagnetic resonance (EPR). The authors apply the method to two prototypical S = ½ copper(II) complexes: copper(II) phthalocyanine (CuPc) and copper(II) octaethylporphyrin (CuOEP).

First, the crystal structures of both compounds are characterized by X‑ray and neutron diffraction, revealing that CuOEP’s β‑ethyl substituents induce a saddled, non‑planar macrocycle and a softer lattice (thermal expansion roughly twice that of CuPc).

INS measurements on powder samples (≈500 mg) are performed on the VISION spectrometer at ORNL, covering the full vibrational spectrum from 5 cm⁻¹ to 3100 cm⁻¹ in under 30 minutes per temperature point. The spectra are recorded from 5 K to 300 K, allowing the authors to track anharmonic shifts and linewidth broadening of individual phonon peaks. Density‑functional theory (DFT) calculations are used for mode assignment; key observations include a red‑shift of the overall phonon density in CuOEP relative to CuPc and a systematic hardening of the symmetric Cu–N stretching modes in CuOEP.

Spin‑lattice relaxation times (T₁) are measured by pulse EPR on highly diluted samples (1 % Cu in diamagnetic Zn analogues) to suppress spin‑spin interactions. Below 7 K the relaxation follows a linear 1/T₁ ∝ T dependence, characteristic of the direct (one‑phonon) process. Between 7 K and 40 K the temperature dependence follows a Raman (two‑phonon) law with a power‑law exponent near –2, indicating that thermally populated phonons dominate the transition. Above ~40 K a second change in slope appears, signalling the onset of a different Raman regime involving higher‑energy phonons.

To extract the characteristic phonon energies that drive the Raman processes, the authors fit the 1/T₁ data with a “local‑mode” model that assumes the two phonons involved have the same energy (valid because the Zeeman splitting (~0.14 cm⁻¹ at 0.3 T) is negligible compared with phonon energies). The fits require two local modes in addition to the direct term. For CuPc the low‑temperature Raman mode is 42 ± 6 cm⁻¹ and the high‑temperature mode is 265 ± 14 cm⁻¹; for CuOEP they are 37 ± 5 cm⁻¹ and 236 ± 13 cm⁻¹, respectively.

Comparing these energies with the INS spectra shows that the low‑energy regime (< 50 cm⁻¹) corresponds to weakly coupled lattice vibrations that dominate relaxation below 40 K, while the high‑energy regime (> 185 cm⁻¹) corresponds to optical phonons that become thermally populated above 40 K and drive relaxation with SPC coefficients roughly three orders of magnitude larger.

Structural analysis explains why CuOEP exhibits longer coherence at room temperature. The β‑ethyl groups break planar symmetry, soften the crystal lattice (enhancing anharmonic scattering) but simultaneously raise the energy of core Cu–N stretching modes. Consequently, vibrational energy is redistributed toward peripheral and out‑of‑plane motions, reducing the overlap between spin‑relevant electronic orbitals and vibrational displacements. This reduction in SPC allows CuOEP to retain measurable T₁ up to 300 K, whereas CuPc’s T₁ becomes too short to detect above ~180 K.

The authors emphasize that, although the method does not yield mode‑specific coupling constants, it provides quantitative SPC coefficients for distinct spectral regions and establishes a broadly applicable experimental link between crystal structure, lattice dynamics, and spin relaxation. This framework bridges the gap between theory‑driven predictions and experimental observation, offering concrete design guidelines for future molecular qubits: (i) suppress low‑energy lattice modes to limit direct‑process relaxation, (ii) shift core‑stretching modes to higher energies to weaken high‑temperature Raman relaxation, and (iii) exploit structural distortions that increase lattice softness without enhancing coupling to the spin‑bearing metal center.