Research ArticleCHEMICAL PHYSICS

How do phonons relax molecular spins?

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Science Advances  27 Sep 2019:
Vol. 5, no. 9, eaax7163
DOI: 10.1126/sciadv.aax7163
  • Fig. 1 VO(acac)2 structure and spin phonon coupling distributions.

    (A) The geometrical structure of the two VO(acac)2 molecular units inside the crystal’s unit cell. Vanadium atoms are represented in pink, oxygen in red, carbon in green, and hydrogen in white. (B) The spin-phonon coupling distribution relative to the Zeeman energy as function of the phonons’ frequency. (C) The spin-phonon coupling distribution relative to the dipolar spin-spin energy as function of the phonons’ frequency. (D) The spin-phonon coupling distribution relative to the hyperfine energy as function of the phonons’ frequency.

  • Fig. 2 Spin relaxation time as function of the B field for one electronic spin coupled to one nuclear spin.

    The relaxation time, τ, in milliseconds and T = 20 K as a function of the external field in Tesla is reported for the simulations of one electronic spin coupled to one nuclear spin and relaxing due to the phonon modulation of the Zeeman and hyperfine energies (black line and dots). The contribution coming from the sole hyperfine energy modulation is represented by a red line and dots, while the sole Zeeman contribution is reported by the green line and dots. The experimental relaxation time as extracted from AC magnetometry (19) is also reported (blue dots and line).

  • Fig. 3 Spin relaxation time as function of the temperature T.

    The relaxation time, τ, in milliseconds at B = 5 T as a function of the temperature is reported for the simulations of one electronic spin coupled to one nuclear spin and relaxing due to the modulation of the Zeeman and hyperfine energies by harmonic phonons (black line and dots), phonons with 1 cm−1 of line width (red line and dots), and phonons with a linearly T-dependent phonon lifetime (green line and dots). A coefficient γ = 0.1 cm−1/K is chosen. The experimental relaxation time as extracted from AC magnetometry (19) is also reported (blue dots and line).

  • Fig. 4 Spin relaxation time as function of external field for two coupled electronic spins.

    The relaxation time, τ, in milliseconds as a function of the external field in Tesla is reported for the simulations of two electronic spins relaxing due to the phonon modulation of the Zeeman and dipolar energies (black line and dots). Green line and dots represent the relaxation time of two isolated spins, where only the Zeeman energy is modulated by phonons. The contribution coming from the sole dipolar energy modulation is represented by a red line and dots. The experimental relaxation time as extracted from AC magnetometry (19) is also reported (blue dots and line).

  • Fig. 5 Phonon density of states.

    The total phonon density of states (DOSs) as a function of the frequency is reported in black. The total phonon density of states was also decomposed in a pure translational contribution (red line), a rotational contribution (blue line), and an intramolecular contribution (green line), all relative to a single molecule inside the unit cell. The inset shows the details of the density of states in the low-energy part of the spectrum. The Brillouin zone was integrated with a uniform mesh of 643 points. The reported density was smeared with a Gaussian function with breadth of 1.0 cm−1. a.u., arbitrary units.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/9/eaax7163/DC1

    Note S1. Derivation of the nonsecular Redfield equations

    Note S2. Lattice parameters and spin Hamiltonian

    Note S3. K points and phonon’s line width convergence

    Note S4. Number of spins convergence

    Note S5. Effect of numerical noise

    Table S1. Lattice parameters for the VO(acac)2 molecular crystal.

    Table S2. Spin Hamiltonian parameters for VO(acac)2.

    Fig. S1. Spin relaxation time as function of external fields for different values of k points.

    Fig. S2. Spin relaxation time as function of the Gaussian breadth for different external fields.

    Fig. S3. Spin relaxation time as function of the number of coupled spins.

    Fig. S4. Effect of numerical noise on spin relaxation time.

    Spin-phonon coupling coefficients and ab initio calculation' input files

    Reference (32)

  • Supplementary Materials

    The PDF file includes:

    • Note S1. Derivation of the nonsecular Redfield equations
    • Note S2. Lattice parameters and spin Hamiltonian
    • Note S3. K points and phonon’s line width convergence
    • Note S4. Number of spins convergence
    • Note S5. Effect of numerical noise
    • Table S1. Lattice parameters for the VO(acac)2 molecular crystal.
    • Table S2. Spin Hamiltonian parameters for VO(acac)2.
    • Fig. S1. Spin relaxation time as function of external fields for different values of k points.
    • Fig. S2. Spin relaxation time as function of the Gaussian breadth for different external fields.
    • Fig. S3. Spin relaxation time as function of the number of coupled spins.
    • Fig. S4. Effect of numerical noise on spin relaxation time.
    • Reference (32)

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    Other Supplementary Material for this manuscript includes the following:

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