Figure 1

(Color online) Schematic representation of the ${\text{Cu}}^{2+}$ ion arrangement and the SE/SSE spin-exchange paths in ${\text{BiCu}}_2{\text{PO}}_6$. The two crystallographically nonequivalent copper atoms are indicated by blue and cyan circles, and the oxygen atoms by red circles.Reuse & Permissions

Figure 2

(Color online) Seven ordered spin states (i.e., FM, ${\text{AF}}_1$, ${\text{AF}}_2$, ${\text{AF}}_3$, ${\text{AF}}_4$, ${\text{AF}}_5$, and ${\text{AF}}_6$) of ${\text{BiCu}}_2{\text{PO}}_6$ employed to extract the six spin-exchange parameters $J_1-J_6$ and their total spin-exchange energies in terms of $J_1-J_6$. Only the ${\text{Cu}}^{2+}$ sites are shown for simplicity, and the up-spin and down-spin ${\text{Cu}}^{2+}$ sites are represented by filled and empty circles, respectively. For each ordered state, the three numbers in the parenthesis, from left to right, refer to the relative energies obtained from the $\text{GGA}+U$ calculations with $U=4$, 6, and 8 eV, respectively.Reuse & Permissions

Figure 3

(Color online) Analysis of the magnetic susceptibility and spin gap of ${\text{BiCu}}_2{\text{PO}}_6$ using the $J_1-J_2-J_4$ frustrated spin ladder mode: (a) experimental (circles) and fitted (red line) magnetic susceptibility. The typical gapped behavior (black solid line) is shown after removing the paramagnetic contribution (black dotted line). (b) Evolution of the normalized spin gap $\Delta /J_1$ as a function of the frustration ratio $J_2/J_1$ (with $J_4/J_1$ fixed at 0.25) and the rung to leg ratio $J_4/J_1$ (with $J_2/J_1$ fixed at 0.50).Reuse & Permissions

Figure 4

(Color online) (a) Map of $S\left(Q,\omega \right)$ measured for ${\text{BiCu}}_2{\text{PO}}_6$ at 11 K on the three-axis cold spectrometer 4F1 (color online). (b) Experimental $Q$ scan measured at 4 meV; the energy of the lowest magnetic excitation in ${\text{BiCu}}_2{\text{PO}}_6$ at $T=1.7\text{K}$(from the thermal triple axis spectrometer 2T in counts per 2 minutes of acquisition time). (c) Magnetic structure factor $S\left(Q,{\omega }_0\right)$ at the energy ${\omega }_0$ of the lowest magnetic excitation using the $J_1-J_2-J_4$ and $J_1-J_2-J_3$ frustrated spin ladder models. The curves 1 and 2 are obtained from the $J_1-J_2-J_4$ model and the curve 3 from the $J_1-J_2-J_3$ model (curve 1: $J_4/J_1=1$ and $J_2/J_1=0.75$; curve 2: $J_4/J_1=0.25$ and $J_2/J_1=0.75$; and curve 3: $J_3/J_1=1$ and $J_2/J_1=0.75$). The intensity is in arbitrary units. (d) Mapping of $S\left(Q,{\omega }_0\right)$ as a function of the frustration ratio $\alpha =J_2/J_1$. The color scale represents the amplitude of $S\left(Q,{\omega }_0\right)$.Reuse & Permissions