Crystal size dependence of dipolar ferromagnetic order between ${\mathrm{Mn}}_{6}$ molecular nanomagnets

Letter

Crystal size dependence of dipolar ferromagnetic order between ${Mn}_{6}$ molecular nanomagnets

E. Burzurí, M. J. Martínez-Pérez, M. Muntó, L. A. Barrios, N. Ventosa, O. Roubeau, J. Veciana, G. Aromí, and F. Luis

Phys. Rev. B 106, L180407 – Published 28 November 2022

Abstract

We study how crystal size influences magnetic ordering in arrays of molecular nanomagnets coupled by dipolar interactions. Compressed fluid techniques have been applied to synthesize crystals of ${Mn}_{6}$ molecules (spin $S = 12$ ) with sizes ranging from $28 μ m$ down to 220 nm. The onset of ferromagnetic order and the spin thermalization rates have been studied by means of ac susceptibility measurements. We find that the ordered phase remains ferromagnetic, as in the bulk, but the critical temperature $T_{c}$ decreases with crystal size. Simple magnetostatic energy calculations, supported by Monte Carlo simulations, account for the observed drop in $T_{c}$ in terms of the minimum attainable energy for finite-sized magnetic domains limited by the crystal boundaries. Frequency-dependent susceptibility measurements give access to the spin dynamics. Although magnetic relaxation remains dominated by individual spin flips, the onset of magnetic order leads to very long spin thermalization time scales. The results show that size influences the magnetism of dipolar systems with as many as $10^{11}$ spins and are relevant for the interpretation of quantum simulations performed on finite lattices.

Received 18 July 2022
Revised 9 November 2022
Accepted 14 November 2022

DOI:https://doi.org/10.1103/PhysRevB.106.L180407

Physics Subject Headings (PhySH)

Magnetic interactions Magnetic order Magnetic phase transitions Magnetic susceptibility Magnetization dynamics Molecular magnetism Spin dynamics Transition temperature

Ferromagnets Molecular magnets Nanoclusters

AC susceptibility measurements Crystal growth Specific heat measurements X-ray diffraction

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

E. Burzurí^1,2,*, M. J. Martínez-Pérez³, M. Muntó^4,5, L. A. Barrios⁶, N. Ventosa ^4,5, O. Roubeau³, J. Veciana^4,5, G. Aromí⁶, and F. Luis ^3,†

¹Departamento de Física de la Materia Condensada and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de Madrid, 28049 Madrid, Spain
²IMDEA Nanociencia, Campus de Cantoblanco, 28049 Madrid, Spain
³Instituto de Nanociencia y Materiales de Aragón (INMA), CSIC and Universidad de Zaragoza, Plaza San Francisco s/n 50009 Zaragoza, Spain
⁴Institut de Ciència de Materials de Barcelona (ICMAB-CSIC), Campus UAB, Cerdanyola del Vallès, Spain
⁵Networking Biomedical Research Networking Center on Bioengineering, Biomaterials and Nanomedicine (CIBER-BBN), Spain
⁶Departament de Química Inorgànica and IN2UB, Universitat de Barcelona, Martí i Franquès 1-11, 08028 Barcelona, Spain

^*enrique.burzuri@uam.es
^†fluis@unizar.es

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Vol. 106, Iss. 18 — 1 November 2022

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Figure 1
(a) Structure of the $[{Mn}_{6} O_{4} {Br}_{4} {({Et}_{2} dbm)}_{6}]$ molecule, highlighting the octahedron formed by the six ${Mn}^{3 +}$ ions (purple spheres) and the pyramid with the four ${Br}^{-}$ ions (brown spheres) at the apices. H atoms are omitted for clarity. Color code: Mn, purple; Br, brown; O, red; and C, black. (b) Crystal lattice of ${Mn}_{6}$ clusters that can be approximated to a body-centered tetragonal unit cell (solid black lines). The actual monoclinic cell (see Supplemental Material [25] for details) is shown as dashed lines. For simplicity, only the central octahedron of each ${Mn}_{6}$ molecule is shown.
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Figure 2
(a) Transmission electron microscopy image (top) and crystal size distribution measured by dynamic light scattering (bottom) of sample D. The distribution is centered at 220 nm with a half width at half maximum of 75 nm. (b) X-ray diffraction spectra measured on a powder sample of unprocessed bulk ${Mn}_{6}$ and on samples C and D having average crystal sizes of $7.5 μ m$ and 220 nm, respectively. (c) Magnetization isotherms measured on unprocessed ${Mn}_{6}$ and on the same C and D samples. The fit with a Brillouin function (solid line) gives $S = 12$ and $g = 1.95$ . (d) Molar specific heat of unprocessed ${Mn}_{6}$ and of sample A measured at zero field as a function of temperature. The solid line shows the Schottky contribution due to the magnetic anisotropy of each molecule. In both cases, the results are compatible with a weak uniaxial anisotropy constant $D / k_{B} = 0.013$ K [21, 22].
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Figure 3
(a) In-phase magnetic susceptibility $χ^{'}$ of sample A (average crystal size $28 μ m$ ) measured as a function of temperature for different frequencies $ω / 2 π$ ranging from 15 mHz (blue) to 79 kHz (red). The maximum $χ^{'}$ becomes independent of $ω$ < $0.1$ Hz and saturates at a value close to $1 / ρ N = 0.16$ emu/g (dotted red line), signaling the onset of ferromagnetic order. (b) Same as in (a) for sample B (average crystal size $12 μ m$ ). (c) Arrhenius plot for the frequency-dependent temperature of maximum $χ^{'}$ . The tendency of this temperature to saturate to a constant value points to a magnetic phase transition and provides a method to estimate the critical temperature $T_{c}$ (vertical solid lines). (d) Reciprocal equilibrium susceptibility of all samples, corrected from demagnetizing effects, as a function of temperature. The solid lines are least squares Curie-Weiss fits.
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Figure 4
(a) Critical temperature of ${Mn}_{6}$ molecular crystals, estimated from ac susceptibility measurements and normalized to $T_{c}$ of sample A (solid rhombic dots), as a function of average crystal size $d$ . The plot also shows dipolar energies of ferromagnetic domains [solid circles, see (b)] and critical temperatures determined from Monte Carlo simulations [open circles, see (c)], in both cases normalized to the bulk values. (b) Dipolar energy $E$ of ferromagnetic domains with different lengths $N_{z}$ as a function of domain width $N_{x} = N_{y}$ . The dotted line marks the minimum dipolar energy $E_{\min}$ attainable for each domain size. The inset shows an scheme of the tetragonal lattice used in these energy calculations and in the Monte Carlo simulations, where $N_{x}$ , $N_{y}$ , and $N_{z}$ denote number of sites. (c) Magnetic heat capacity obtained from Monte Carlo simulations on lattices of varying $N_{z}$ and fixed $N_{x} = N_{y} = 5$ sites. The solid lines are least squares fits to Lorentzian curves that allow finding $T_{c}$ . The inset shows the dipolar energy as a function of temperature from which the specific heat curves are derived.
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Figure 5
(a) Out-of-phase magnetic susceptibility component $χ^{''}$ of sample A (average crystal size $28 μ m$ ) measured as a function of temperature at 10 different frequencies spanning eight decades, from $79.433$ kHz (dark red) to $15.85$ mHz (purple). (b) Arrhenius plots for the frequency-dependent temperatures of maximum $χ^{''}$ for all samples. This plot provides information on the temperature dependence of the typical spin thermalization times. Solid lines are least square Arrhenius fits $1 / ω = τ_{0} exp (U / k_{B} T)$ , with $τ_{0}$ an attempt time and $U$ the activation energy. The inset shows $U$ as a function of average crystal size. (c) Isotherms of $χ^{'}$ (top) and $χ^{''}$ (bottom) of sample B as a function of frequency for temperatures across $T_{c} ≃ 0.114$ K. Notice the sudden broadening of these curves that occurs at and below $T_{c}$ .
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Physical Review B

covering condensed matter and materials physics

Crystal size dependence of dipolar ferromagnetic order between ${Mn}_{6}$ molecular nanomagnets

E. Burzurí, M. J. Martínez-Pérez, M. Muntó, L. A. Barrios, N. Ventosa, O. Roubeau, J. Veciana, G. Aromí, and F. Luis

Phys. Rev. B 106, L180407 – Published 28 November 2022

Abstract

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Physical Review B

covering condensed matter and materials physics

Crystal size dependence of dipolar ferromagnetic order between Mn6 molecular nanomagnets

E. Burzurí, M. J. Martínez-Pérez, M. Muntó, L. A. Barrios, N. Ventosa, O. Roubeau, J. Veciana, G. Aromí, and F. Luis

Phys. Rev. B 106, L180407 – Published 28 November 2022

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Crystal size dependence of dipolar ferromagnetic order between ${Mn}_{6}$ molecular nanomagnets