Odd-parity electronic order near the semiconductor limit

Abstract: Identifying materials platforms in which dilute carriers experience strong Coulomb interactions is a central challenge in the search for interaction-driven quantum phases. In such systems, weak carrier screening can promote a variety of collective instabilities beyond the conventional Fermi liquid paradigm, including superconductivity, Wigner crystallization, and odd-parity electronic order. Experimental realizations of such dilute, strongly interacting electronic systems remain rare in crystalline materials. Here we report a spontaneous odd-parity phase transition in the phosphide semiconductor family $\textit{Ln}$Cd$_3$P$_3$ ($\textit{Ln}$ = La, Ce, Pr, Nd). Using optical second harmonic generation, we observe the onset of bulk inversion and rotational symmetry breaking accompanied by the emergence of an in-plane polar axis. Second harmonic microscopy reveals three domain variants related by 120$^\circ$ rotations, while ultrafast transient reflectivity measurements uncover a pronounced electronic reconstruction across the transition. Remarkably, the ordered phase appears only in lightly self-hole-doped compounds and is absent in insulating SmCd$_3$P$_3$, indicating an essential role for itinerant carriers despite their extremely low concentration. Guided by density functional theory, we develop a four-band model of the valence states and show that modest interactions can stabilize odd-parity electronic order. The resulting phase combines a spontaneous Fermi surface distortion with a momentum-dependent bilayer polarization that breaks inversion symmetry. Our results establish a route to interaction-driven parity breaking in dilute-carrier semiconductors and identify honeycomb bilayer systems as a promising platform for odd-parity electronic phases.