Anharmonic lattice dynamics and superionic transition in AgCrSe₂

The structural phase transition in ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$ was tracked by $T$-dependent X-ray diffraction (XRD) measurements on powders, performed at the Advanced Photon Source (APS) sector 11-ID-C (35) with incident energy 105.708 keV (Fig. 1B). By comparing the XRD patterns at 300 and 600 K, we can see that most Bragg intensities decrease as temperature increases, as a result of a larger Debye–Waller factor at high $T$, combined with the disordering of Ag ions. The suppression of the (003), (006), and (015) Bragg intensities above $T_{od}$ reveals the disordering of Ag ions across $\alpha$ and $\beta$ sites (Fig. 1B, Inset). From our power-law fits, a fully reversible and near second-order transition (SI Appendix, Fig. S1) occurs around a nominal temperature 500 K, both on heating and on cooling (SI Appendix, Figs. S2 and S3). The offset from the reported $T_{od}\hspace{0.17em}=$ 475 K likely originates from the temperature gradient between sample and sensor.

To understand the change in lattice dynamics across the superionic transition, we used INS to measure the dynamical structure factor, S($\left|\mathbf{Q}\right|,E$), which directly probes microscopic excitations (phonons, magnons). INS measurements were performed on powder samples with the cold neutron chopper spectrometer (CNCS) (36), using $E_i\hspace{0.17em}=$ 4 and 12 meV, with elastic line energy resolution of 112 and 673 $\mathrm{\mu }$eV full width at half maximum (FWHM), respectively. Fig. 2 and SI Appendix, Fig. S4 present S($\left|\mathbf{Q}\right|,E$) obtained with $E_i\hspace{0.17em}=$ 12 meV at nominal temperatures 10, 40, 300, 460, 530, 600, and 700 K. At 10 and 40 K (SI Appendix, Fig. S4 A and B), we clearly observe spin-wave dispersions up to 2 meV emerging from magnetic Bragg peaks at $\left|\mathbf{Q}\right|\hspace{0.17em}=$ 0.2 ${\AA }^{-1}$ and 2.0 ${\AA }^{-1}$, as discussed by Damay et al. (17). Clear spin waves are no longer seen in the paramagnetic phase, but a magnetic quasielastic signal persists up to $\left|\mathbf{Q}\right|\hspace{0.17em}=$ 0.6 ${\AA }^{-1}$ even at room temperature, revealing damped fluctuations of spin correlations. Meanwhile, one can clearly see weak vertical streaks extending from the Bragg peaks at 0 meV (white) to 3.5 meV near $\left|\mathbf{Q}\right|$ = 1.8, 3.3 ${\AA }^{-1}$, as well as a horizontal band of intensity around 3.5 meV (Fig. 2A and SI Appendix, Fig. S4 A and B). The vertical streaks and band become more intense as temperature rises to 300 K, which confirms that the signals originate from phonons. From the DFT-calculated phonon dispersion (SI Appendix, Fig. S6B), the vertical streaks match the dispersive long-wavelength acoustic phonons, while the flat horizontal band around 3.5 meV originates from very flat TA dispersions away from the zone center. For $T$ below but close to $T_{od}$, the flat signal dampens significantly (Fig. 2B). At $T$ $>$ $T_{od}$, the flat mode completely broadens, revealing extreme phonon damping, and eventually merges with the quasielastic signal (Fig. 2 C and D). On the other hand, the dispersive portions of acoustic phonon dispersions remain clear, especially near the (018) Bragg peak at $\left|\mathbf{Q}\right|$ = 3.3 ${\AA }^{-1}$ even at $T\hspace{0.17em}=$ 600 K, far into the superionic regime (Fig. 2D). Consequently, two distinct behaviors are concurrent for different components of the acoustic phonons: the persistence of the dispersive acoustic phonons and the damping of the nondispersive acoustic phonons.

The damping of the flat phonon branch at 3.5 meV was further investigated by integrating S($\left|\mathbf{Q}\right|,E$) over $\left|\mathbf{Q}\right|$ (Fig. 2E). A peak can be seen at 3.5 meV at low temperatures, and it softens (shifts to lower energy) on warming to $T_{od}$ and smears out above the transition. This behavior follows the expected evolution for a damped harmonic oscillator (DHO) with increasing damping, as shown in Fig. 2F. A well-defined phonon mode (e.g., at 300 K) can be considered underdamped, where the DHO peak remains sharp, shown as the purple curve. As $T$ increases, the DHO width increases, and the energy slightly softens, shown in blue. At $T$ $>$ $T_{od}$, the mode is overdamped, so that its spectral width becomes larger than its frequency, and the oscillation is no longer well defined. In this case, the spectral weight of the DHO is considerably shifted and leads to a central peak near zero frequency, shown in red. Consequently, at $E$ $<$ 2 meV, the S(E) behavior is no longer given by the Gaussian function of the instrument resolution (response to the elastic scattering from Bragg and incoherent static signal), but instead comprises a Lorentzian signal (QENS signal, discussed in detail in Fig. 5) in addition to the Gaussian component. The Lorentzian component is seen as a wide tail to the elastic line, whose width and intensity increase with temperature. The energy width of this Lorentzian is inversely proportional to the residence time of Ag ions on a site between hops, and thus the Lorentzian broadening indicates faster hopping of Ag ions in the superionic phase. Thus, we observe not only the damping of phonon-like oscillations, but also the transfer of spectral weight from vibrational dynamics to a diffusive dynamic.

It was proposed in ref. 19 that the damping of the 3.5-meV peak corresponds to a complete disappearance of the TA (shear) modes, akin to a liquid state, and that the LA phonons would thus dominate thermal transport. However, since the 3.5-meV modes are quite flat in energy across all Q at $T$ $<$ $T_{od}$, they have very small group velocities ($v_g\hspace{0.17em}=$ $d\omega /d\mathbf{q}$, where $E=\hslash \omega$), and hence an already small contribution to ${\kappa }_{lat}$ in the ordered phase [${\kappa }_{lat}=\frac{1}{3}\sum C_V{v}_g^2\tau$, where $C_V$ is the heat capacity, and $\tau$ is the phonon lifetime for each phonon mode (37)]. The main contribution of TA phonons to ${\kappa }_{lat}$ actually comes from the dispersive part of the acoustic branches, and these still persist even at $T$ $>$ $T_{od}$. Thus, it is crucial to identify the role of various modes and understand how they contribute to ${\kappa }_{lat}$.

Next, we consider the evolution of the phonon density of states over the whole frequency range to contrast the strong damping behavior of the 3.5-meV modes with the higher-frequency vibrations of the stiff framework structure. The DOS was measured with INS using the wide angular-range chopper spectrometer (ARCS) (38), using $E_i\hspace{0.17em}=$ 80 and 20 meV. The 80-meV $E_i$ gives the whole spectrum and the 20-meV data provides better resolution at the 3.5-meV peak. We merged the two datasets at $E\hspace{0.17em}=$ 9 meV (Materials and Methods). As shown in Fig. 3A, at 300 K, a sharp peak is seen at 3.5 meV, similar to that observed in $S\left(E\right)$ data from the CNCS. This peak drastically broadens at 450 K and smears out completely once ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$ enters the superionic regime (between 450 and 520 K). However, an overall milder broadening can be seen on warming for all features above 5 meV, which is a consequence of anharmonicity and increasing atomic vibration amplitudes. The mean energy, evaluated as the first moment of the phonon DOS, shows little change: less than 0.5 meV from 300 to 650 K, indicating that the overall crystalline framework remains stiff through the transition.

In Fig. 3B, we compare the experimental DOS to harmonic phonon simulations and AIMD simulations carried out at 300 and 600 K. All simulations are corrected to consider the neutron weighting factors and are convolved with the estimated energy resolution function of ARCS. The harmonic DOS exhibits sharp peaks as in low-$T$ INS data. The DOS from AIMD at 600 K qualitatively matches the overall broadening of features seen in INS data. Most importantly, the simulations closely reproduce the extreme damping of the 3.5-meV peak as temperature increases. Further, we calculated the atomic decomposition of the DOS and projected along the $c$ axis (Fig. 3C) or onto the $a$–$b$ plane (Fig. 3D). This decomposition clearly reveals that the 3.5-meV peak arises overwhelmingly from Ag in-plane vibrations, and Ag atoms have much larger vibration amplitude than Cr and Se atoms (SI Appendix, Fig. S11). As Ag atoms are weakly bound to Cr and Se atoms, they tend to vibrate in-plane where they experience few constraints. Therefore, we may describe these local vibrations as “2D rattler modes,” which give rise to the large peak at 3.5 meV in the DOS. This behavior reflects the intercalated layer nature of ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$, as discussed in previous studies (19, 23). The existence of such a mode in the normal phase could lead to low ${\kappa }_{lat}$ (39) below $T_{od}$. Further, the weak bonding between ${\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$ layers and the intercalants creates easy hopping pathways for Ag between $\alpha$ and $\beta$ sites at $T$ $>$ $T_{od}$ and enables fast long-range diffusion on the half-filled Ag sublattice. In the superionic regime, the in-plane Ag vibrations around either $\alpha$ or $\beta$ sites are strongly disrupted and damped, leading to the smearing of the 3.5-meV peak in the vibrational spectrum.

After describing how the flat portion of the TA branch breaks down in the superionic phase, we now establish the persistence of the dispersive TA phonons. The definite determination of phonon polarizations and dispersions requires momentum-resolved measurements, which we obtained from IXS measurements on small single crystals (SI Appendix, Fig. S5) using the high-energy resolution inelastic X-ray spectrometer (HERIX) spectrometer ($E_i\hspace{0.17em}=$ 23.7 keV) at the APS (40, 41). IXS spectra were collected in multiple Brillouin zones at three temperatures, 80, 300, and 560 K, across $T_{od}$. The experimental phonon dispersions are shown as markers in Fig. 4 A–C, overlaid on the S($\mathbf{Q}$, E) from DFT (gray scale). The agreement between IXS data and DFT is quite good at low $T$, especially at 80 K. At 300 and 560 K, the phonon energies are shifted, reflecting some anharmonic renormalization compared to the harmonic DFT results. This is seen especially for $\mathbf{Q}$ along $\left[1,\hspace{0.17em}0,\hspace{0.17em}L\right]$. We evaluate the renormalization using mode Grüneisen parameters (SI Appendix, Fig. S6). The long-wavelength TA phonons have large negative values, possibly explaining their stiffening as the lattice expands with temperature. Additional intrinsically anharmonic renormalization can also be partially captured by AIMD (SI Appendix, Fig. S7), using the temperature-dependent effective potential (TDEP) method.

In Fig. 4A, we show the dispersions for the transverse modes along the q = [$-$2H, H, 0] (0 $\le$ H $\le$ 1) direction, for both in-plane ($a$) and out-of-plane ($c$) polarizations. The TA phonons stiffen a little from 80 to 300 K. Still, all phonon peaks in the IXS spectra remain well defined. However, in the superionic phase ($T$ $>$ 560 K), TA modes along the flat portion become extremely damped. For instance, at Q $=$ 0, 1.5, 0 (zone boundary, H $=$ 0.5), the IXS spectrum is captured with an overdamped DHO profile, with no obvious phonon peak, as shown in red in Fig. 4F. We use a rectangle with red gradient to denote the intensities of this DHO (energy range lower-bound cutoff is one-sixth of the maximum DHO intensity in all plots). Contrary to the overdamped behavior of the flat portion, the dispersive portion of TA at small q remains well defined even at 560 K. The same behavior is seen for other measured directions [1, 0, L] and [$-$2+H, 1, 0] (Fig. 4 B and C). The constant-Q spectra at small q from IXS (Fig. 4 D and E) and corresponding AIMD simulations (Fig. 4 G and H) further illustrate the selective breakdown of the TA phonon by contrasting small and large q (Fig. 4 F and I). Compared to the case of ${\mathrm{C}\mathrm{u}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$ (22), in which nondispersive phonons arise from a low-lying flat optical branch ($\sim 8$ meV), the flat low-energy dispersion around 3.5 meV in ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$ is contributed by the top of the TA branch. Systematic trends for TA phonons at 300 and 600 K, obtained from DHO fits from zone center to zone boundary, are shown in Fig. 4 J and K. At 300 K the TA phonon energies increase with $H$ near the zone center and become constant near the zone boundary, mapping out the TA dispersion along [1 $-$ 2H, 1+H, 0]. At 560 K, similar behaviors are seen at $H$ $<$ 0.2 (dispersive portion), but modes at $H$ $>$ 0.2 are severely damped, representing the breakdown of nondispersive TA phonons in the superionic phase. The persistence of the long-wavelength dispersive TA phonons is thus clearly established from IXS.

We further corroborate through simulations that the dispersive portions of the TA branches persist when their phonon eigenvectors are not dominated by Ag atoms. We note that the harmonic approximation predicts nondegenerate TA branches along the [1 $-$ 2H, 1+H, 0] direction, while IXS yields nearly degenerate TA branches. This may reflect limitations of the harmonic approximation in systems with large anharmonicity, such as ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$. The Ag contribution to the TA modes at varying $\mathbf{q}$ was evaluated from the modulus of the computed eigenvectors. The Ag contributions for the nondispersive part of the TA branch (at short wavelengths) are 78%, 64%, and 83% at Q $=$ 0, 1.5, 0, Q $=$ 1, 0, 5.5, and Q $=$ −1.5, 1, 0, respectively. This large contribution from a single atom type gives these modes an optical character, similar to the picture of decoupled Einstein modes, reflecting the weak force constants between Ag ions and other atoms. On the other hand, the Ag contribution to the dispersive TA modes (long wavelengths) is 39% at Q $=$ 0.9, 1.05, 0, 45% at Q $=$ 1.05, 0, 4.5, and 31% at Q $=$ −1.9, 1, 0. Meanwhile, we note that the diffusion of the Ag atoms is via site hopping. Ag atoms still vibrate at both $\alpha$ and $\beta$ sites between the intersite hops (as shown below using AIMD). Yet the random distribution of Ag among half-filled $\alpha$/$\beta$ sites yields strong disorder, which is expected to effectively scatter short-wavelength modes.

In summary, our IXS measurements on single crystals show that only the nondispersive short-wavelength TA modes break down in the superionic phase, corresponding to an optic-like character with strong Ag participation, while the dispersive long-wavelength TA modes are less sensitive and remain well defined, with limited damping at high temperature. A mixed lattice dynamic is therefore observed in ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$ where the framework of ${\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_6$ octahedra remains crystalline and the dispersive TA persists, while the order of the Ag sublattice “melts” in the superionic phase. We emphasize the importance of momentum-resolved techniques to track phonon modes with different polarizations and their temperature dependence. We also note that due to their low phonon group velocities, the flat portions of the TA dispersions contribute little to ${\kappa }_{lat}$. The change in the experimentally reported thermal conductivity across $T_{od}$ is indeed quite small (19, 26).

We now focus on the superionic behavior by investigating the diffusion of Ag ions via the QENS signal around the elastic line in $S\left(\mathbf{Q},E\right)$. QENS is a powerful technique to probe stochastic diffusion of atoms. A strong QENS signal is seen in ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$ owing to its large ionic mobility at $T$ $>$ $T_{od}$ (30), as shown in Fig. 5A. The total signal consists of three portions: the QENS contribution in the form of a Lorentzian centered at zero energy transfer, a purely elastic delta function convolved with the instrumental (Gaussian) resolution, and a linear background. At 300 K, the QENS signal is very weak, and we mainly see the elastic line with the background signal. The QENS intensity increases with $T$ and saturates around 530 K, whereas the width of the Lorentzian tail continues broadening to 0.83 $\hspace{0.17em}\pm \hspace{0.17em}$ 0.30 meV at 750 K, indicating a faster diffusion process upon heating (SI Appendix, Fig. S12).

To rationalize the QENS measurements, we use AIMD simulations to track the trajectories of Ag ions and correlations in their motions. In Fig. 5 B and C, we plot the in-plane probability distribution of Ag ions at 300 and 600 K, starting from the same configuration where all of the Ag ions occupy $\alpha$ sites. While most Ag ions are distributed around their equilibrium positions at 300 K, simulations at 600 K clearly show the delocalization of Ag ions across both $\alpha$ and $\beta$ sites, as observed in ref. 42. Some Ag ions are observed to hop to different $\alpha$ sites when the previously occupied Ag ions at those sites hop away, enabling long-range diffusion. The diffusion distances are limited by the simulation supercell size, but a large enough simulation box should reproduce the long-range diffusion reported experimentally (30). All of the dynamical processes seen in experiments have been reproduced, although a longer trajectory (currently 13 ps long) is required for Ag ions to equally populate the $\alpha$ and $\beta$ sites. The weak bonding at room temperature and diffusive behavior of Ag ions above $T_{od}$ can be seen from the mean-square displacement (MSD). As shown in SI Appendix, Fig. S11, at 300 K, the MSD of Ag atoms is larger than those of Cr and Se but stays constant during long simulation times. The MSD increases dramatically when the temperature reaches 600 K. While the Ag ions exhibit hopping dynamics, Cr and Se atoms retain oscillatory dynamics around their respective equilibrium positions, with vibration amplitudes characteristic of regular crystals (SI Appendix, Fig. S14), showing the rigid nature of the framework sublattice, which is an important reason for the persistence of dispersive TA modes. The diffusion constant estimated from the linear portion of the MSD at 600 K is 1.5 $\times \hspace{0.17em}{\mathrm{10}}^{-5}$ ${\mathrm{c}\mathrm{m}}^2\cdot {\mathrm{s}}^{-1}$, in qualitative agreement with the experimentally measured value (30), further confirming the superionic behavior.

The diffusion pathway and the associated energy barrier were investigated from the free-energy landscape computed from AIMD trajectories of Ag ions (Materials and Methods), as shown in Fig. 5 D and E. The shape of the energy landscape in ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$ is unusual (43, 44). A very low and flat energy barrier about 0.11 eV in ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$ is found across an extended region between $\alpha$ and $\beta$ sites, indicating a “frustrated environment” with no single optimal diffusion pathway, similar to recent findings in Li superionics (45). As we show in Fig. 5C and SI Appendix, Fig. S13, a quasi-2D probability distribution of Ag ions can be seen and represents the diffusion channel. The weak bonding of Ag ions enables large atomic displacements and low vibration frequencies, which results in a low free-energy barrier, matching the observations in Li SICs in ref. 46. In the “concerted migration” scheme, reported by ref. 3, multiple Li ions hop simultaneously along the diffusion channel and produce a similar shallow and flat energy landscape, but this scheme prefers a high Li concentration environment with the high-energy site occupied. We note that the hopping mechanism is different in ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$, where the collective hopping might originate from the strong repulsion between nearby Ag ions discussed below.

To get more insight into such ion–ion interactions, we investigated the atomic correlations which describe microscopic interactions with characteristic length. In Fig. 5F and SI Appendix, Fig. S18, we plot the atomic PDF obtained from both XRD experiments (11-ID-C at APS with ${\mathrm{E}}_i\hspace{0.17em}=$ 105.708 keV) and AIMD simulations. Data at 300 and 600 K are represented in blue and yellow, respectively. The blue dashed lines in Fig. 5F indicate the Ag–Ag bond lengths in the theoretical relaxed structure at 0 K. The orange dashed lines are the Ag–Ag bond for the same structure but with Ag atoms occupying both $\alpha$ and $\beta$ sites: An extra peak is clearly expected at $d=2.3$ Å. We first compare the reduced PDF ($G\left(r\right)\hspace{0.17em}=\hspace{0.17em}4\pi r{\rho }_0\left(g\left(r\right)-1\right)$) from the XRD in SI Appendix, Fig. S18A and from AIMD simulations in SI Appendix, Fig. S18E. The experimental PDF shows several important features: Upon heating, all of the peaks broaden considerably, but the temperature dependence of the first peak at around 2.5 Å is much less pronounced compared to the other peaks, which is the superposition of the nearest-neighbor Cr–Se and Ag–Se pairs. Our PDF from AIMD reproduces the experimental features quite well overall. We therefore proceed to analyze the behavior of individual atom pairs by decomposing the computed PDF. As seen in SI Appendix, Fig. S18B–D, the distributions of bonds involving at least one Ag atom change dramatically with temperature, but the PDF of Cr–Cr, Cr–Se, and Se–Se bonds varies little upon heating, consistent with the rigid nature of the ${\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_6$ octahedra framework (SI Appendix, Fig. S18F–H). Second, the normal PDF peak shape is expected to be a Gaussian within the harmonic approximation, when the atoms vibrate around their equilibrium position in a harmonic potential. Deviations from this Gaussian profile can be explained by large-amplitude anharmonic vibrations in ${\mathrm{A}\mathrm{g}\mathrm{C}\mathrm{r}\mathrm{S}\mathrm{e}}_2$. Most importantly, in Fig. 5F, the intensity and the lineshape of the Ag–Ag pair correlations smear out dramatically, and the predicted nearest $\alpha$–$\beta$ correlation length at 2.3 Å never appears. The circumvention of this bond length reveals a strong repulsion between nearest-neighbor Ag ions, which, because of the half-filled nature of the Ag sublattice, indicates necessarily correlated hopping dynamics of the Ag ions. This repulsion will tend to assist the hopping of other Ag atoms at $\alpha$ sites once one Ag atom hops to the $\beta$ sites and a low free-energy barrier is achieved.