Origin of gaplike behaviors in ${\mathrm{URu}}_2{\mathrm{Si}}_2$: Combined study via quasiparticle scattering spectroscopy and resistivity measurements

We address two long-standing questions regarding the hidden order in ${\mathrm{URu}}_2{\mathrm{Si}}_2$: Is it associated with the hybridization process, and what are the distinct roles played by the localized and itinerant electrons? Our quasiparticle scattering spectroscopy reveals a hybridization gap ubiquitous in the entire phase space spanned by P and Fe substitutions in ${\mathrm{URu}}_2{\mathrm{Si}}_2$, including the no-order and antiferromagnetic regions, with minimal change upon crossing the phase boundary. This indicates its opening is not associated with the ordering, and thus localized electrons must be the major player. Towards a consistent understanding of all the other gaplike behaviors observed only below transition temperatures, we analyze the electrical resistivity using a model in which gapped bosonic excitations are the dominant scattering source. With their stiffness set to follow an unusual temperature dependence (decreasing with decreasing temperature), this model fits all of our resistivity data well including the jump at the transition. Remarkably, the extracted gap increases slowly with increasing Fe content, similarly to the gap detected by inelastic neutron scattering at $Q_1=(1.4,0,0)$, suggesting a common origin. Such a model can also naturally explain the Hall effect temperature dependence without invoking Fermi surface gapping.

Figure 1

(a) Temperature vs P-content ($T\text{−}x$) phase diagram of ${\mathrm{URu}}_2{\mathrm{Si}}_{2-\mathrm{x}}{\mathrm{P}}_{\mathrm{x}}$, adapted from Ref. [27] and based on the measurements of magnetic susceptibility ($\chi$) and electrical resistivity ($\rho$). AF-I stands for antiferromagnetic order and NO for no-order. $T_{\mathrm{coh}}$ denotes the coherence temperature. (b) $T\text{−}y$ phase diagram of ${\mathrm{URu}}_{2-\mathrm{y}}{\mathrm{Fe}}_{\mathrm{y}}{\mathrm{Si}}_2$ constructed based on thermal expansion coefficient ($\alpha$) and resistivity [17], and magnetic susceptibility measurements [32]. AF-II stands for antiferromagnetic order and CE for coexisting orders. The inset depicts a unit cell of ${\mathrm{URu}}_2{\mathrm{Si}}_2$. In both panels, vertical arrows along the horizontal axis indicate substituent concentrations studied in this work. (c) Hybridization gap (${\mathrm{\Delta }}_{\mathrm{hyb}}$) at $T=2$ K (from Figs. 2 and 3) vs coherence temperature ($T_{\mathrm{coh},\chi }$). The solid lines are a guide to the eyes.

Figure 2

(a)–(c) Normalized differential conductance taken from junctions on ${\mathrm{URu}}_2{\mathrm{Si}}_{2-\mathrm{x}}{\mathrm{P}}_{\mathrm{x}}$ at $T=2$ K (colored symbols) and best-fit curves (solid gray lines). In (a) and (b), data and fit curves are shifted vertically for clarity. (d) Hybridization gap (${\mathrm{\Delta }}_{\mathrm{hyb}}$) and renormalized $f$ level ($\lambda$) extracted from an analysis using the MDC model. Labels denote different regions in the phase diagram.

Figure 3

(a)–(c) Temperature-dependent conductance spectra taken from junctions on ${\mathrm{URu}}_{2-\mathrm{y}}{\mathrm{Fe}}_{\mathrm{y}}{\mathrm{Si}}_2$ (colored symbols) and best-fit curves (solid gray lines). Data and fit curves are shifted vertically for clarity. (d)–(f) Extracted hybridization gap (${\mathrm{\Delta }}_{\mathrm{hyb}}$) and renormalized $f$ level ($\lambda$).

Figure 4

(a) Normalized resistance vs temperature for ${\mathrm{URu}}_2{\mathrm{Si}}_{2-\mathrm{x}}{\mathrm{P}}_{\mathrm{x}}$. (b) Resistivity in the low-temperature region with the phonon contribution subtracted out (colored symbols). ${\rho }_{\mathrm{U}}$ is the resistivity of ${\mathrm{URu}}_2{\mathrm{Si}}_{2-\mathrm{x}}{\mathrm{P}}_{\mathrm{x}}$ and ${\rho }_{\mathrm{Th}}$ represents the resistivity of ${\mathrm{ThRu}}_2{\mathrm{Si}}_2$. Data and fit curves are shifted vertically for clarity. The inset shows the temperature-dependent bosonic excitation gap (left axis, solid line) and stiffness (right axis, dashed line) that are extracted for the parent compound. (c) Blue squares indicate the extracted gap value at zero temperature. The dashed line is a guide to the eye. Red circles represent the gap ratio, $2{\mathrm{\Delta }}_0/k_B{T}_0$, where $T_0$ is the ordering temperature. (d)–(f) Same as (a)–(c) but for ${\mathrm{URu}}_{2-\mathrm{y}}{\mathrm{Fe}}_{\mathrm{y}}{\mathrm{Si}}_2$. The $E_1$ gap for ${\mathrm{URu}}_{2-\mathrm{y}}{\mathrm{Fe}}_{\mathrm{y}}{\mathrm{Si}}_2$ in (f) is from Ref. [54]. Solid lines are a guide to the eye.