Magnetic and phononic dynamics in the two-ladder quantum magnet (C5H9NH3)2CuBr4

Abstract

In quantum magnetic materials it is common to observe both static and dynamic lattice effects on the magnetic excitation spectrum. Less common is to find that the magnetic correlations have a significant impact on the phonon spectrum. Can such an interplay occur in a structurally soft system with comparable elastic and magnetic energy scales? Here we study the metal-organic material (C5H9NH3)2CuBr4 (Cu-CPA), in which an explanation of the low-lying excitations depends crucially on a full understanding of both the spin and lattice subsystems. We report high-resolution neutron spectroscopy enabled by large, deuterated single-crystals that reveal how both sectors are affected by the recently discovered structural phase transition. By measuring over several Brillouin zones, we disentangle the vibrational contribution to the spectrum in order to obtain an accurate estimate of the quasi-one-dimensional magnetic signal. The low-energy magnetic excitations are dominated by two gaps, $Δ$ b = 0.41 meV and $Δ$ a = 0.55 meV, which contribute with equal intensity ratios, confirming that Cu-CPA realizes a two-ladder spin Hamiltonian, and we deduce the magnetic interaction parameters of both ladders. The phonon spectrum contains a highly localized mode at an anomalously low-energy around 2 meV. This characteristic frequency drops by approximately 5 percent as magnetic correlations become established with decreasing temperature, and we connect this behavior with the location and structure of the cyclopentylammonium rings.

Figures (13)

• Figure 1: Excitation spectrum and lattice structure of Cu-CPA. (a) Full excitation spectrum measured at $T = 0.05$ K at the IN5 instrument, shown with no background subtraction. The measurements were performed with $E_i = 2.70$ meV and the data were integrated perpendicular to the ladder direction over the ranges $Q_H \in [-0.85, 0.85]$ r.l.u. and $Q_L \in [-2, 4]$ r.l.u. The dispersive modes with high intensity at half-integer values of $Q_K$ are the magnetic ("triplon") excitations of the $S = 1/2$ antiferromagnetic Heisenberg spin ladders. The dispersive modes appearing at integer values of $Q_K$ are acoustic phonons, which have low intensity at this measurement temperature. Intensity contributions arising from a low-lying local (optical) phonon and from the instrument background are also marked. (b) Low-temperature structure of Cu-CPA, showing the two inequivalent spin ladders (denoted as "Ladder $a$" and "Ladder $b$"). For clarity, only half of the crystallographic c-direction is shown. Following the nomenclature introduced in the structural study of this material philippe_metal-organic_2024, the lighter and darker red and blue colors denote inequivalent Cu$^{2+}$ ions on each ladder rung. Here we have shown the cyclopentylammonium (CPA) molecules in gray, projected behind the copper-tetrabromine units, in order to enhance the visibility of the ladders.
• Figure 2: Low-energy phonon dispersion of Cu-CPA, measured with $E_i = 5.11$ meV at $T = 30$ K. (a) Constant-energy slice integrated over the range (1.5, 1.7) meV to identify two rings in ($Q_K$,$Q_L$), around the point $\mathbf{Q} = (0~2~0)$, which are approximately circular in absolute units (Å$^{-1}$). (b) Intensity, $I(Q_K,\omega)$, obtained by integration over $Q_H$ in the range [$-0.85$, 0.85] r.l.u. and $Q_L \in [-2, 4]$ r.l.u. to show the phonon contribution as a function of $Q_K$. Red circles represent phonon 2, which has stronger intensity along $Q_K$ and is shown in panel (c); blue diamonds represent phonon 1, which has stronger intensity along $Q_L$ and is shown in panel (d). The black, dashed line correspond to the fit of the acoustic phonon, and the red, respectively blue shaded area correspond to a $1\sigma$ confidence interval. (c,d) Characteristic constant-energy cuta prepared by integration over small energy and wave-vector intervals around $Q_L = 0$ and $Q_K = 2$, which yield the intensity profiles as functions of $Q_K$ (c) and $Q_L$ (d). (e) Deduction of two phonon dispersions from multiple constant-energy cuts. Dashed lines represent the best fits to these phonons and shaded areas the $\sigma$ fitting uncertainties. Gray symbols were excluded from the velocity-fit.
• Figure 3: Low-energy triplon dispersion of Cu-CPA, measured with $E_i = 2.27$ meV at $T = 0.05$ K. (a) Intensity, $I(Q_K,\omega)$, obtained by integration over $Q_H \in [-0.85, 0.85]$ r.l.u. and over the full range of $Q_L$ to reveal the two dispersive triplon branches. Bragg peaks and phonon contributions have been masked. (b) Constant- Q cut with $Q_K = 1.5$ r.l.u. and integration width $dQ_K = \pm 0.02$ r.l.u., integrating over $Q_H \in [-0.85, 0.85]$ r.l.u. and over the range $Q_L \in [-2, 4]$ r.l.u. The two peaks were fitted with Lorentzians, whose center positions (dashed lines) were taken to estimate the two gaps and whose integrated intensities are shown in the legend. The fitting range (blue circles) was reduced to avoid the elastic line, the extension of its tail to $\hbar\omega \simeq 0.25$ meV, and a dispersionless feature around 1.2 meV (data points not included in the fit are shown as gray circles). The hatched area represents a constant background contribution.
• Figure 4: Full magnetic excitation spectrum of Cu-CPA. (a) Double triplon dispersion determined by subtraction of the background and of the complete phonon contribution. The measurements were performed with $E_i = 2.70$ meV. We integrated the data over the ranges $Q_H \in [-0.85, 0.85]$ r.l.u. and $Q_L \in [-2, 4]$ r.l.u. Red circles (Ladder $b$) and blue diamonds (Ladder $a$) show the center positions of Lorentzian (Gaussian) fits to selected constant- Q (constant-energy) scans. Solid lines show best fits obtained by parameterizing the ladder dispersion relation as a function of $\alpha$barnes1994, as detailed in App. \ref{['app:sec:parametrized_SMA']}. (b) Dynamical structure factor, $S(Q,\omega)$, calculated by density-matrix renormalization-group (DMRG) methods Schmidiger2013 for a two-leg ladder with leg-to-rung interaction ratio $\alpha = 1$ and with the energy scale selected to fit the gap $\Delta_b$ measured in experiment. (c) Bandwidth-to-gap ratio, $W/\Delta$, of the ladder triplon, computed for a range of $\alpha$ values by ED and DMRG and compared with the results deduced for Ladders $a$ and $b$. We draw attention to the remarkably wide regime of linear dependence (up to $\alpha = 1.6$) where $W/\Delta \approx 3 \alpha$.
• Figure 5: Elastomagnetic effects. (a) Scattered intensity obtained by integration over the momentum ranges $Q_H \in [-0.85, 0.85]$ r.l.u., $Q_K \in [1.88, 1.92]$ r.l.u., and $Q_L \in [-2, 4]$ r.l.u., shown as a function of energy transfer at our highest and lowest temperatures to characterize the evolution of the two distinct low-lying vibrational modes associated with the CPA molecules. The lower mode shifts to lower energies upon cooling, whereas the upper mode remains at a constant energy but appears to lose intensity. (b) Temperature-dependence of the center positions of the two vibrational modes (triangles and squares) required to fit the scattered intensity, obtained from Lorentzian profiles. The dashed lines serve as guides to the eye. In most cases, the error bars are smaller than the symbol sizes. The downward shift in the lower mode (squares) below 20 K suggests its association with the onset of magnetic correlations.