Pressure-induced charge density wave phase in $\mathrm{A}{\mathrm{g}}_{2\text{−}\delta }\mathrm{Te}$

Considerable excitement was generated by the observation of large and linear positive magnetoresistance in nonmagnetic silver chalcogenides. Renewed interest in these materials was kindled by the discovery that $\mathrm{A}{\mathrm{g}}_2\mathrm{Te}$ in particular is a topological insulator with gapless linear Dirac-type surface states. High-pressure x-ray-diffraction studies, combined with first-principles electronic structure calculations, have identified three phase transitions as the pressure is increased: an isostructural transition identified with an electronic topological transition followed by two structural phase transitions. These recent studies were carried out on nominally stoichiometric $\mathrm{A}{\mathrm{g}}_2\mathrm{Te}$. For the present work we have prepared single-phase self-doped $\mathrm{A}{\mathrm{g}}_{2\text{−}\delta }\mathrm{Te}$ samples with a well-characterized silver deficit ($\delta =2.0\times {10}^{-4}$) for structural and electrical transport measurements over extended ranges of pressure (0–43 GPa), temperature (2–300 K), and magnetic field (0–9 T). The temperature dependence of the resistivity exhibits anomalous behavior at 2.3 GPa, slightly above the isostructural transition, which we postulate is due to Fermi surface reconstruction associated with a charge density wave (CDW) phase. The anomaly is enhanced by the application of a 9-T magnetic field and shifted to higher temperature, implying that the electronic Zeeman energy is sufficient to alter the gapping of the Fermi surface. A peak in the pressure dependence of the resistivity and a sudden drop in the pressure dependence of the mobility, occurring at 2.3 GPa, provide additional evidence for a CDW phase at pressures slightly above the isostructural transition.

Figure 1

(a) Angle-dispersive XRD patterns of $\mathrm{A}{\mathrm{g}}_{2\text{−}\delta }\mathrm{Te}(\delta =2\times {10}^{-4})$ at selected pressures. (b) Lattice parameters as a function of pressure obtained from Rietveld refinement, showing the jump at the phase transition from $P{2}_1/c$ to Cmca at 2.9 GPa and then to Pnma around 12.9 GPa. The inset shows the change of $a/c$ with pressure, with discontinuities at the isostructural transition around 2.1 GPa and at the $P{2}_1/c\to Cmca$ transition at 2.9 GPa. (c) Unit-cell volume obtained from Rietveld refinement as a function of pressure. The solid curves represent fits of the second-order Birch-Murnaghan equation. The inset shows an enlarged plot of the low-pressure region, in which the unit-cell volume is discontinuous around 2.1 GPa due to the isostructural transition and at 2.9 GPa due to the $P{2}_1/c\to Cmca$ transition, where the two phases coexist.

Figure 2

Rietveld profile refinements of the XRD patterns at 1.7, 3.4, 21.9. and 39.4 GPa, representing the $P{2}_1/c$ phase, Cmca phase, coexisting Cmca and Pnma phases, and Pnma phase, respectively.

Figure 3

(a)–(c) Resistivity versus temperature for different pressures at zero field. (d) The temperature derivative of the resistivity dρ/dT; two turning points are seen at 0.3 GPa, denoted $T_{\mathrm{CDW}1}$ and $T_{\mathrm{CDW}2}$. With increasing pressure $T_{\mathrm{CDW}1}$ can be observed up to 2.8 GPa while $T_{\mathrm{CDW}2}$ is no longer evident. The solid horizontal lines indicate points where $d\rho /dT=0$ at 2.3 and 2.8 GPa.

Figure 4

Temperature dependence of the resistivity in magnetic fields $H=9\mathrm{T}$ and $H=0\mathrm{T}$ at 2.8 GPa. The curves are a guide to the eye.

Figure 5

(a), (b) Hall resistivity ${\rho }_{\mathrm{xy}}$ at 2 K as a function of magnetic field at different pressures and 2 K, showing that the dominant carriers are electrons. The Hall coefficient is nonlinear in the CDW phase between 2.3 and 3.5 GPa.

Figure 6

(a) Pressure dependence of the carrier density $n$ at different temperatures. (b) Pressure dependence of the relative change of resistivity with temperature, $\mathrm{\Delta }\rho /\rho =\left[\rho \right(80\mathrm{K})-\rho (300\mathrm{K}\left)\right]/\rho \left(300\mathrm{K}\right)$, showing an anomalous peak at 2.3 GPa, compared with the carrier density at 80 K. The curves are a guide to the eye.

Figure 7

Pressure dependence of resistivity ρ (blue points) and carrier mobility μ (red points) at different temperatures. The resistivity shows a singularity at 2.3 GPa, while the carrier mobility drops at the same pressure. The curves are a guide to the eye.

Figure 8

Magnetoresistance for a series of pressures at 80 K shown as (a) a linear-linear plot and (b) a log-log plot. It exhibits quadratic behavior at low fields changing to a nonsaturating linear behavior at higher fields and is strongly reduced with increasing pressure, reverting to quadratic behavior over the measured field range above 10 GPa. The inset to (a) shows the MR at 9 T as a function of pressure at the same temperatures; the curve is a guide to the eye.

Figure 9

Phase diagram of $\mathrm{A}{\mathrm{g}}_{2\text{−}\delta }\mathrm{Te}$ summarizing the structural and electronic transport results over (a) the full range from 0.4 to 20.9 GPa and (b) a restricted range highlighting the CDW behavior.