Neutron scattering evidence for isolated spin-$\frac{1}{2}$ ladders in ${\left({\text{C}}_5{\text{D}}_{12}\text{N}\right)}_2{\text{CuBr}}_4$

Inelastic neutron scattering was used to determine the spin Hamiltonian for the singlet ground-state system of fully deuterated BPCB, ${\left({\text{C}}_5{\text{D}}_{12}\text{N}\right)}_2{\text{CuBr}}_4$. A two-leg spin-1/2 ladder model, with $J_{\perp }=\left(1.09\pm 0.01\right)\text{meV}$ and $J_{\parallel }=\left(0.296\pm 0.005\right)\text{meV}$, accurately describes the data. The experimental limit on the effective interladder exchange constant is $\left|J_{\text{int}}^{\text{eff}}\right|\lesssim 0.006\text{meV}$, and the limit on total diagonal, intraladder exchange is $\left|J_F+J_{F^{\prime }}\right|\le 0.1\text{meV}$. Including the effects of copper to bromide covalent spin transfer on the magnetic form factor, the experimental ratios of intraladder bond energies are consistent with the predictions of continuous unitary transformation.

Figure 1

(Color online) Various projections of the previously determined structure of BPCB (Ref. 6). (a) Exchange interactions entering the Hamiltonian in Eq. (1). Black lines indicate the legs and rungs of the ladder. [(b)–(d)] BPCB crystal structure, except hydrogen, projected along each of the crystallographic axes.

Figure 2

(Color online) False color image of raw ${\lambda }_i=5\text{Å}$ neutron scattering data measured with ${\mathbf{c}}^{\ast }\parallel {\mathbf{k}}_i$. Intensity is integrated in the $l$ direction. (a) $T=1.4$. (b) 25 K, used as background. (c) Intensity at $T=1.4\text{K}$ after background subtraction. For presentation purposes only, the data in (c) were smoothed as described in the text (see Sec. 3B).

Figure 3

(Color online) Trajectories in $\left(h,0,l\right)$ plane for the three sample orientations measured at $\hslash \omega =0.8$ and 1.5 meV, with ${\lambda }_i=5\text{Å}$.

Figure 4

(Color online) Bandwidth over band-gap $\left(W/\Delta \right)$ ratio for spin excitations versus normalized intraladder exchange interactions, computed using Eqs. (4, 5). The experimental result of $W/\Delta =0.73$ for BPCB constrains intraladder exchange interactions to lie on the dashed line.

Figure 5

$H_{c2}$ as a function of $J_F/J_{\perp }$, for $W/\Delta =0.73$. The dashed lines correspond to $H_{c2}=13.8$ and 14.6 T, as reported in different Refs. 22, 35. Several other measurements of the upper critical field lie in this range (Refs. 36, 37, 38, 39). Detail shown in the inset indicates that $\left|J_F/J_{\perp }\right|\le 0.05$, so $\left|J_F/k_B\right|\le 0.4\text{K}$.

Figure 6

(Color online) Global fit to a spin-ladder model. (a) color map of smoothed (see text), background-subtracted data, with $\varphi =\angle {\mathbf{k}}_i$, ${\mathbf{c}}^{\ast }=-10°$.The color scale is the same one used in Fig. 2c; (b) color map of the corresponding fit; [(c)–(h)] constant energy cuts, with an energy window of 0.1 meV. The solid lines represent the global fit to the data when including interladder coupling effects.

Figure 7

(Color online) False color images of neutron scattering intensity for three sample orientations (rows): [column (a)] smoothed (see text), background-subtracted data; [columns (b)–(e)] simulations with the phenomenological dispersion relation, Eq. (3), but only one bond energy (see top) modulating the intensity for each column; [column (f)] fit including all bond energies. For each sample orientation, select sections of data with unique values of $h$ and $l$ are shown. The color scale is the same one used in Fig. 2c.

Figure 8

(Color online) Solid black lines show bond-energy ratios for $J_F$ and $J_{\perp }$, as a function of $\beta =J_F/J_{\perp }$, for several values of $\alpha =J_{\parallel }/J_{\perp }$, from 0.05 to 0.5, every 0.05, as estimated by the continuous unitary transformation. The experimental value and corresponding errors are shown as horizontal lines.

Figure 9

Ratio of leg to rung bond energies versus $\alpha =J_{\parallel }/J_{\perp }$, as calculated using the continuous unitary transformation. Experimental values are shown as horizontal/vertical solid lines, with error bars indicated by dashed lines.

Figure 10

Reduced ${\chi }^2$ for the global fit to BCPB inelastic neutron scattering data as a function of $\gamma =J_{\text{int}}^{\text{eff}}/J_{\perp }$. The dashed line is a quadratic fit for points around the minimum, which indicates that $\gamma =0.002\pm 0.006$.

Figure 11

The difference between the best-fit one-dimensional dispersion relation and Gaussian fits to cuts through raw data resolved versus wave-vector transfer along ${\mathbf{c}}^{\ast }$. No systematic $l$ dependence is observed indicating that to within error a one-dimensional model is appropriate for BPCB $\left(\gamma =0.002\pm 0.006\right)$.