Electrical resistivity measurements were performed on single crystals of URu$_2-x$Os$_x$Si$_2$ up to $x$ = 0.28 under hydrostatic pressure up to $P$ = 2 GPa. As the Os concentration, $x$ , is increased, (1) the lattice expands, creating an effective negative chemical pressure $P_{ch}$($x$), (2) the hidden order (HO) phase is enhanced and the system is driven toward a large-moment antiferromagnetic (LMAFM) phase, and (3) less external pressure $P_{c}$ is required to induce the HO to LMAFM phase transition. We compare the $T(x)$, $T(P)$ phase behavior reported here for the URu$_2-x$Os$_x$Si$_2$ system with previous reports of enhanced HO in URu$_2$Si$_2$ upon tuning with $P$, or similarly in URu$_2-x$Fe$_x$Si$_2$ upon tuning with positive $P_{ch}$($x$). It is noted that pressure, Fe substitution, and Os substitution are the only known perturbations that enhance the HO phase and induce the first order transition to the LMAFM phase in URu$_2$Si$_2$. We present a scenario in which the application of pressure or the isoelectronic substitution of Fe and Os ions for Ru results in an increase in the hybridization of the U-5$f$- and transition metal $d$-electron states which leads to electronic instability in the paramagnetic phase and a concurrent stability of HO (and LMAFM) in URu$_2$Si$_2$. Calculations in the tight binding approximation are included to determine the strength of hybridization between the U-5$f$ electrons and each of the isoelectronic transition metal $d$-electron states of Fe, Ru, and Os.
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