Neutron scattering from a coordination polymer quantum paramagnet

Inelastic neutron scattering measurements are reported for a powder sample of the spin-$1∕2$ quantum paramagnet $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$. Magnetic neutron scattering is identified above an energy gap of $1.9\mathrm{meV}$. Analysis of the sharp spectral maximum at the onset indicates that the material is magnetically quasi-one-dimensional. Consideration of the wave vector dependence of the scattering and polymeric structure further identifies the material as a two-legged spin-$1∕2$ ladder. Detailed comparisons of the data to various models of magnetism in this material based on the single-mode approximation and the continuous unitary transformation are presented. The latter theory provides an excellent account of the data with leg exchange $J_{\Vert }=2.0\mathrm{meV}$ and rung exchange $J_{\perp }=3.3\mathrm{meV}$.

Figure 1

Schematic of a two-leg spin-ladder system, indicating the rung interaction $J_{\perp }$ and the rail interaction $J_{\Vert }$. The indexing of spin sites used in Eq. (1) is also shown.

Figure 2

(Color online) Structure of $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ projected along four different directions to illustrate potential interacting spin models. (a) The molecular ladder structure of $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ (Ref. 27), which the ladder extends along the monoclinic $\mathbf{b}$ axis. Interactions along the $(\mathbf{a}\pm \mathbf{b})∕2$ and $\mathbf{c}$ directions could lead to two-dimensional dimer models in the $\mathbf{a}-\mathbf{b}$ or $\mathbf{b}-\mathbf{c}$ planes (b) and (c) or a three-dimensional dimer model (d). Color coding is as follows: Pink, Cu; green, Br; blue, C; red, H; and gray, N.

Figure 3

Scaled wave vector dependence of inelastic neutron scattering from $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ at $T=60\mathrm{K}$ and for various values of energy transfer. The scaling behavior is consistent with double scattering by phonons and incoherent elastic nuclear scattering as described by Eqs. (7, 8).

Figure 4

The energy dependence of $Q$-integrated neutron scattering from $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ at $T=$ (a) 60 and (b) $1.4\mathrm{K}$ before detailed balance correction. In both frames solid symbols show the temperature-independent background determined as described in Sec. 2B1. Background-subtracted data at $T=$ (c) 60 and (d) $1.4\mathrm{K}$ data. The solid line is the phonon scattering contribution at $T=1.4\mathrm{K}$ as determined by Eq. (5).

Figure 5

(Color online) Normalized powder inelastic neutron scattering intensity for $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ at $T=$ (a) 60 and (b) $1.4\mathrm{K}$ after detailed balance correction as described in Sec. 2B1.

Figure 6

(Color online) Normalized inelastic scattering intensity for $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$. (a) Phonon scattering intensity at $T=1.4\mathrm{K}$. (b) Magnetic scattering intensity at $T=1.4\mathrm{K}$ after subtracting the phonon contribution as described in Sec. 2B1.

Figure 7

Energy dependence of the normalized magnetic scattering intensity for $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ averaged over wave vectors $0.5⩽Q⩽1.0{\mathrm{\AA }}^{-1}$. Open circles are experimental data. The dashed line indicates the one-triplon dynamic spin correlation function from the continuous unitary transformation theory. The dotted line shows the spectrum for a two-dimensional model with $B_0=3.82\mathrm{meV}$ and $B_{hk}=1.02\mathrm{meV}$. The solid line shows the spectrum for a quasi-one-dimensional ladder model with $B_0=3.82\mathrm{meV}$, $B_k=1.84\mathrm{meV}$, $B_{hk}=0.08\mathrm{meV}$, and $B_l=0.14\mathrm{meV}$. All models were convolved with the experimental resolution function and an overall constant was fitted to account for any discrepancies in the background subtraction procedure.

Figure 8

Wave vector dependence of normalized magnetic scattering intensity for $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ at $1.4\mathrm{K}$, integrated over energy transfers $1.85⩽\hslash \omega ⩽2.65\mathrm{meV}$. Open circles are experimental data. The solid and dashed lines indicate the dynamic spin correlation function of the one-triplon CUT and SMA 1D alternating spin chain models, respectively, convolved with the experimental resolution function. Multiple scattering was added to the models as calculated from Eq. (7) using the experimentally determined value of $\mathcal{T}=0.40$. Note that data for $Q<0.5{\mathrm{\AA }}^{-1}$ do not include the full energy range due to the kinematic limitations of the experiment (see Fig. 11).

Figure 9

${\chi }^2=(1∕N_{\text{free}}){\sum }_j{(I_j^{\mathrm{obs}}-I_j^{\mathrm{ex}})}^2∕{\sigma }_j^2$ versus the ratio of rail to rung exchange in the comparison of magnetic neutron scattering data for $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ to the continuous unitary transformation theory of one-triplon excitations in a $\mathbf{b}$-axis spin ladder.

Figure 10

(Color online) (a) One-triplon contribution to the powder-averaged dynamic spin correlation function for a spin ladder as calculated by the continuous unitary transformation. Lattice spacings are those appropriate for $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ and the exchange constants were derived by fitting to neutron scattering data as described in the text. (b) The comparison of triplon dispersion relation for the best fit between CUT (solid line) and SMA (dashed line) ladder models.

Figure 11

(Color online) Comparison of inelastic magnetic neutron scattering $\tilde{I}(Q,\omega )$ from $\mathrm{Cu}\left(\text{quinoxaline}\right){\mathrm{Br}}_2$ at $T=1.4\mathrm{K}$ (c) to (a) the one-triplon scattering derived from the continuous unitary transformation and (b) a quasi-one-dimensional spin-ladder model in the single-mode approximation. The models were complemented with incoherent elastic double scattering, according to Eq. (8), with $\mathcal{T}=0.40$ and an overall additive constant was fitted to account for any discrepancies in background subtraction.