Crystal structure and magnetic properties of spin-$1/2$ frustrated two-leg ladder compounds (C$_4$H$_{14}$N$_2$)Cu$_2X_6$ ($X$= Cl and Br)

We have successfully synthesized single crystals, solved the crystal structure, and studied the magnetic properties of a new family of copper halides (C$_4$H$_{14}$N$_2$)Cu$_2X_6$ ($X$= Cl, Br). These compounds crystallize in an orthorhombic crystal structure with space group $Pnma$. The crystal structure features Cu$^{2+}$ dimers arranged parallel to each other that makes a zig-zag two-leg ladder-like structure. Further, there exists a diagonal interaction between two adjacent dimers which generates inter-dimer frustration. Both the compounds manifest a singlet ground state with a large gap in the excitation spectrum. Magnetic susceptibility is analyzed in terms of both interacting spin-$1/2$ dimer and two-leg ladder models followed by exact diagonalization calculations. Our theoretical calculations in conjunction with the experimental magnetic susceptibility establish that the spin-lattice can be described well by a frustrated two-leg ladder model with strong rung coupling ($J_0/k_{\rm B} \simeq 116$ K and 300 K), weak leg coupling ($J^{\prime\prime}/k_{\rm B} \simeq 18.6$ K and 105 K), and equally weak diagonal coupling ($J^{\prime }/k_{\rm B} \simeq 23.2$ K and 90 K) for Cl and Br compounds, respectively. These exchange couplings set the critical fields very high, making them experimentally inaccessible. The correlation function decays exponentially as expected for a gapped spin system. The structural aspects of both the compounds are correlated with their magnetic properties. The calculation of entanglement witness divulges strong entanglement in both the compounds which persists upto high temperatures, even beyond 370~K for the Br compound.