Viscoelastic anomaly accompanying anti-crossing behaviour in liquid As₂Se₃

The As_1−xSe_x system gained significant attention as a glass former [1] where the glass forming ability was sensitively dependent on x, as discussed in terms of rigidity theory [1–3]. It is therefore interesting to investigate the dynamics of the melt via measurements of the dynamic structure factor, $S(Q, E)$, where E is energy transfer, over the meso-scale length scales associated with glass formation, and to relate these results to possible implications for the glass formation.

The structure of liquid As₂Se₃ has been investigated both by experiment [4, 5] and calculation [6, 7]. At lower temperatures, just above the melting point, the liquid has semiconducting properties with a network structure where three-fold coordinated As and two-fold coordinated Se are covalently bonded. The system has asymmetrical bonding: atoms in this network interact with up to the second and third nearest neighbours through weak bending and torsional forces, respectively. At higher temperatures (>1300 K) and pressures (P  >  5 MPa), liquid As₂Se₃ undergoes a semiconductor to metal transition. Ab initio molecular dynamics (AIMD) simulations for liquid As₂Se₃ [6, 7] revealed that the number of the nearest neighbour atoms at an As site decreases with increasing T and two-fold coordinated As becomes dominant in a metallic region. X-ray diffraction experiments on liquid As₂Se₃ [4, 5] confirmed the simulation result that the average nearest neighbour coordination number $\bar{r}$ decreases with increasing T. However, there has not been experimental work to investigate the liquid dynamics on shorter (nm and smaller) length scales, where they may be sensitive to changes in bonding and the semiconductor-metal transition.

Here we investigate the dynamics of an As₂Se₃ melt at a high pressure as T is increased. We use inelastic x-ray scattering (IXS) as a direct probe of the dynamic structure factor on nm length and ps time scales. We observe a surprising anomaly that we discuss in terms of an anti-crossing model applicable to a rattling motion of a guest atom in a large cage (e.g. [8]), and then relate our results, especially, the disappearance of the anomaly at higher temperatures, to rigidity theory [9, 10] using the atomic configurations obtained by reverse Monte Carlo (RMC) modeling [5].

The IXS experiments were conducted at the high-resolution IXS beamline (BL35XU) of SPring-8 in Japan [11]. Backscattering at the Si (11 11 11) reflection provided a beam of approximately 10¹⁰ photons s⁻¹ in a 0.8 meV bandwidth onto the sample. The energy of the incident beam and the Bragg angle of the backscattering were 21.747 keV and approximately 89.98°, respectively. We used twelve spherical analyzer crystals at the end of the 10 m horizontal arm. The spectrometer resolution was 1.5–1.8 meV depending on analyzer crystal, as was determined by measurements of polymethyl methacrylate. The Q resolution, ${\Delta}Q$, was set to be 0.45 and 1.0 nm⁻¹ (full width) for $Q \leqslant 11$ and Q  >  11 nm⁻¹, respectively.

The As₂Se₃ sample of 99.999% purity and 0.1 mm thickness was mounted in a single-crystalline sapphire Tamura-type cell [12] which was placed in a high-pressure vessel. The vessel was filled with He gas (99.999% purity) at 6 MPa to stabilize the liquid state. Further information on the set-up of IXS experiments at high pressures is described in [13]. IXS spectra of liquid As₂Se₃ were measured at temperatures from 773 K to 1673 K. The background spectra were measured at 1273 K and 6 MPa using an empty cell. After background subtraction with the absorption correction and integration with respect to E, we deduced the normalized dynamic structure factor $S(Q, E)/S(Q)$ of liquid As₂Se₃ from the observed spectra.

The ultrasonic sound velocity of liquid As₂Se₃ was measured at $T \leqslant 1273$ K, using a quartz cell with a standard pulse echo technique. A PZT transducer operating at about 8.9 MHz was used to produce and detect ultrasonic waves. The time interval required for sound to travel between a delay line end and a reflector was measured using the time function of a LeCroy LT262 oscilloscope. By this method, changes in the sound velocity  >0.2 m s⁻¹ could be easily resolved. The details of the ultrasonic measurements are described in [14].

Figure 1 shows $S(Q, E)/S(Q)$ at 1073 and 1473 K. At 1073 K, $S(Q, E)/S(Q)$ at $Q \leqslant 2.6$ nm⁻¹ exhibits a single peak of the width slightly larger than the resolution. This result suggests that the inelastic excitation corresponding to the longitudinal acoustic mode is merged into the broad quasi-elastic component. At $Q \geqslant 2.9$ nm⁻¹ the inelastic excitation becomes visible as a shoulder on both sides of the quasi-elastic peak. On the other hand, $S(Q, E)/S(Q)$ at 1473 K exhibits a single broad peak at each Q.

Zoom In Zoom Out Reset image size

To obtain the excitation energy of the longitudinal acoustic mode, we carried out the least-square fitting using the model function composed of a Lorentzian and the damped harmonic oscillator (DHO) [15] as,

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle S(Q,E)/S(Q) &= \left[B(E')F_{\rm model}(Q,E')\right] {\otimes} R(E-E'), \nonumber \\ F_{\rm model}(Q,E) &= \frac{A_0}{\pi} \frac{{\Gamma}_{\rm L}}{{\Gamma}_{\rm L}^2+ E^2} + \frac{A_1}{\pi{\beta}} \frac{4{\Gamma_1}{\sqrt{{\Omega}_1^2-{\Gamma_1}^2}}} {(E^2-{\Omega}_1^2)^2+4E^2{\Gamma}_1^2}, \nonumber \\ B(E) &= \beta{E} / \left[1-\exp(-\beta{E}) \right], \nonumber \label{eqDHO} \nonumber \end{align} \tag{ 1 }$$

where $\beta=(k_{\rm B}T){}^{-1}$. A₀ and $\Gamma_{\rm L}$ are the amplitude and the linewidth of the Lorentzian, respectively. A₁, $\Gamma_1$ and ${\Omega}_1$ are the amplitude, the linewidth and the excitation energy for the inelastic excitation. The symbol ${\otimes}$ indicates convolution with the resolution function, $R(E)$. The optimized curves of these components are shown in figures 1(b) and (d). While we recognize that this simple model can not hope to capture the entire dynamics of the system, practically it provides good fits to the observed spectra.

Figure 2 shows the total structure factor obtained by x-ray diffraction, $S(Q)$, the dispersion relation of the excitation energy $\Omega_1$ of the acoustic mode, and its linewidth $\Gamma_1$ at 1073 and 1473 K. $\Omega_1$ disperses with increasing Q, being consistent with the Q dependence of the acoustic mode. The linewidth at 1473 K agrees well with a simple Q² dependence, However, at 1073 K, there is significant deviation at low Q.

Zoom In Zoom Out Reset image size

Figure 3 shows the Q dependence of $\Omega_1$ and $\Gamma_1$ at each T. $\Omega_1$ at the lowest Q disperses as slowly as the ultrasonic sound speed c_s, and c_s gradually slows down with increasing T. As shown in figures 3(a)–(f), $\Omega_1$ abruptly increases between 2.6 and 2.9 nm⁻¹ for $T \leqslant 1373$ K, whereas such a change is fully suppressed at $T \geqslant 1473$ K. Meanwhile, the linewidth shows an increase at the same momentum transfer, though the peak in the linewidth is extended over a wider Q range, as shown in figures 3(g)–(j).

Zoom In Zoom Out Reset image size

The abrupt change observed in $\Omega_1$ is different from Q dependence of the hydrodynamics–viscoelastic transition so far reported, where the acoustic excitation energy gradually increases with Q [16–21]. We interpret this in terms of an anti-crossing effect as is well known from the dynamics of crystals (see, e.g. [22]). This leads to an effective linewidth increase due to mode mixing, with both modes becoming visible simultaneously, and which can appear as broadening. In particular, an increase in $\Gamma_1$ is observed at approximately 2.3 nm⁻¹ as shown in figures 3(g)–(j). Black circles deviate from the broken line at $T \leqslant 1373$ K but they follow the broken line fairly well at 1673 and 1473 K. $\chi^2/N$ for the optimized Q² curve was a factor of four larger at $T \leqslant 1373$ K than at $T \geqslant 1473$ K.

We analysed the experimentally obtained dispersion curve using a linear chain model with a rattler motion [8] that can represent an anti-crossing behaviour in a dispersion curve. This linear chain is constituted from a cage wall and a guest atom of mass M and m, respectively, and the lattice constant between the cage walls is d. The neighbouring cages interact with a force constant K₁ while that between the guest atom and the wall is K₂. To make an anti-crossing at $Q_{\rm c}=2.7$ nm⁻¹, we chose d  =  0.24 nm, and took $K_2/m=(K_1/M)(1-\cos Q_{\rm c}d)$. See the detailed discussion in the supplementary material, available at (stacks.iop.org/JPhysCM/30/28LT02/mmedia). The least-square fits using this model are shown by the red broken curves in figures 3(a)–(f) and a discrete change at 2.7 nm⁻¹ is well reproduced. The fixed d value is close to the covalent bond length in As₂Se₃. When a covalently bonded network is stabilized in liquid As₂Se₃, stretching, bending and torsional forces are expected to work on each atom in the network. When a covalent bond is regarded as a force between cage walls, bending and torsional forces may cause an effect equivalent to a rattler motion.

For covalently bonded chalcogenide glasses such as As_xSe_1−x, the glass forming ability is known to depend on the average nearest neighbour coordination number, $\bar{r}$, of constituent atoms, suggesting that connectivity between atoms plays a crucial role for stabilizing disordered network glasses. Rigidity theory for the network structure assumes that the system is the most stable when a number of mechanical constraints per atom, n_mc, is equivalent to spatial degrees of freedom [1–3]. This condition gives $r_{\rm c}=2.4$ as a critical value of $\bar{r}$. Rigidity theory explains the high glass forming ability of As_0.4Se_0.6 among As_xSe_1−x by $\bar{r}=r_{\rm c}$ at x  =  0.4 when As and Se maintain three-fold and two-fold coordinated configurations, respectively. Further, rigidity theory has predicted that covalent glasses undergo a stiffness transition from flexible to inflexible structures when $\bar{r}$ crosses a critical value r_c from $\bar{r} < r_{\rm c}$.

To investigate relationship of the observed anti-crossing behaviour in liquid As₂Se₃ with a stiffness transition in the glassy system, we estimated n_mc in the liquid. In the original rigidity theory for network glasses, bond stretching and bond bending constraints are obtained on the assumption that covalent bonds are stiff. To apply this theory to melts, T-dependent constraints have been considered using a two-state thermodynamic functions [9], or partial bond angle distributions (partial BADs) obtained by molecular dynamics simulations [10, 23]. Referring to these studies, we investigated partial BADs in atomic configurations obtained by our own RMC modeling for a system of 5000 particles [5]. In this modeling, partial coordination number fractions were constrained by the prediction of the AIMD simulation [6, 7]. Figure 4 shows partial and total BADs at 673 K and 1473 K. We chose the cutoff distance of 0.27 nm for each bond, as used in the AIMD simulation [6, 7]. As shown in the figure, BAD at 1473 K becomes broader than that at 673 K. We counted bond bending constraints per each atom, n_bending, limiting bond angle ranges for each atomic triplet. We chose the angle ranges as 90–110° for As–As–As, 80–130° for Se–As–As, 80–120° for Se–As–Se (a central As) and 70–150° for a central Se from partial BADs at 673 K. We assumed that the bond angle ranges at a central As is narrower than at a central Se because As form a pyramid. On the other hand, bond stretching constraints per atom was taken r/2 (where r is the coordination number with a cutoff distance  =  0.27 nm at each atom) as in the original rigidity theory. We could obtain a number of mechanical constraints per atom using the equation, $0.5 r + n_{\rm bending}$ [9, 10]. Averaging the constraint numbers, we obtained n_mc at each T.

Zoom In Zoom Out Reset image size

Figure 5(a) shows n_mc (black circles) averaged over all atoms as a function of T. Also shown are n_mc's averaged at As and Se sites by triangles and red circles, respectively. Black circles at $T \leqslant 1000$ K are approximately three, equal to a dimensional number, and n_mc decreases to two at 1573 K with increasing T. A melt-quenched glass is regarded as an undercooled liquid with a long (effectivley infinite, for this experiment) relaxation time. A cage structure where atoms thermally vibrate around their equilibrium positions within a picosecond period in liquid As₂Se₃ near the melting point must strongly correlated with a network in the glassy state. Hence it is expected that rigidity of the cage in liquid As₂Se₃ is closely correlated with a stiffness transition in glassy As_xSe_1−x. In figure 5(b), we plot an average coordination number (black circles) obtained from n_mc using the mean field approximation, $n_{\rm mc}=2.5 \bar{r}-3$ [2]. Black circles at $T \leqslant 1000$ K are consistent with the ideal value of 2.4 and it decreases to two at 1673 K. Although $\bar{r}$ obtained by x-ray diffraction experiments [5] is larger than 2.4 at T  <  1200 K partly because of experimental errors, the present analysis implies that defects with higher coordination numbers contribute to it as reported in glassy As₂Se₃ [24].

Zoom In Zoom Out Reset image size

A stiffness transition in glassy As_xSe_1−x has been investigated [25–27], in terms of the intermediate phase consisting of rigid nanodomains and inter-domain space predicted by advanced rigidity theory [3], and the phase has been confirmed in chalcogenide glasses [28, 29]. Here we considered that a rigid nanodomain can originate from a cage in a liquid. Figure 5(e) shows the half width half maximum $\Delta Q_{\rm FSDP}$ of the FSDP in $S(Q)$ of liquid As₂Se₃ [5]. The coherence length $\xi_{\Delta Q}$ is defined by $\pi/\Delta Q_{\rm FSDP}$, which represents the average size of a cage in the liquid state. $\xi_{\Delta Q}$ decreases from 1.8 nm at 673 K to 0.9 nm at 1373 K in liquid As₂Se₃.

Next, we estimate a structural relaxation time $\tau_{\mu}$ in liquid As₂Se₃. 2$\Gamma_1$ approximately follows 2$\Gamma_1 = 2W_{\rm IXS} Q^2$, where W_IXS is a constant up to 5 nm⁻¹ as predicted by hydrodynamics equations [30]. We carried out the least-square fitting for 2$\Gamma_1$ at $Q \leqslant 5$ nm⁻¹, as shown in figures 3(g)–(i), and determined 2W_IXS as in figure 5(f). The visco-elastic theory predicts the Maxwell relation [30], $2W_{\rm IXS}= \Delta_{\mu}^2 \tau_{\mu}/Q^2$, where $\Delta_{\mu}^2$ is the relaxation strength. We approximated $\Delta_{\mu}^2$ as $(c_{\rm p}^2-c_{\rm s}^2)Q^2$, where c_p is the dynamical sound speed obtained by IXS. We take $c_{\rm p}=\Omega_1/Q$ at 2.9 nm⁻¹ and obtain $\tau_{\mu}$ as shown in figure 5(g). $\tau_{\mu}$ of approximately 0.2 ps at $T \leqslant 1373$ K shifts to 0.5 ps at 1473 K, being correlated with the T dependence of $\Omega_1$ at 2.9 nm⁻¹ and a energy gap at 2.7 nm⁻¹ that was obtained by the optimized parameters⁹, as shown in figures 5(c) and (d), respectively.

We compare $\tau_{\mu}$ with the lifetime of the stretching mode $\tau_{\rm st}=\pi/\Gamma_2$ in liquid As₂Se₃ obtained by IXS. In $S(Q, E)/S(Q)$, $\Gamma_2$ was determined by fitting the second inelastic excitation at high Q using the model function with two damped harmonic oscillators¹⁰. As shown in figure 5(g), $\tau_{\rm st}$ is approximately 1 ps at $T \leqslant 1273$ K. This means that a cage structure in the liquid is stable during 1 ps at $T \leqslant 1273$ K, and the cage will be transformed into a rigid nanodomain in the glass. From $\tau_{\mu}$ of 0.2 ps much shorter than $\tau_{\rm st}$ at $T \leqslant 1273$ K, $\tau_{\mu}$ should be the relaxation time between cages. $\tau_{\mu}$ approaches $\tau_{\rm st}$ at $T \geqslant 1473$ K. The coincidence of $\tau_{\mu}$ and $\tau_{\rm st}$ at 1673 K implies that cages are not distinguished from each other because of weakening of the covalent bonds as metallic properties are enhanced. From these results in liquid As₂Se₃, it is expected that rigidity of a cage with a nanometer size and a picosecond life time may change, as T is elevated.

In summary, the IXS experiments revealed that the acoustic excitation energy in liquid As₂Se₃ exhibits an anomaly for $T \leqslant 1373$ K that has a strong similarity to an anti-crossing behaviour in a solid. This behaviour disappears for $T \geqslant 1473$ K. The temperature dependence is well correlated with a decrease in the number of mechanical constraints (n_mc) estimated, from 3 to 2.2, as determined from RMC simulations. In analogy with rigidity theory used in network glasses, we interpret that an atomic time scale rigidity of cages in liquid As₂Se₃ is correlated with n_mc obtained from RMC simulation. We estimate the size and lifetime of a cage in liquid As₂Se₃ at 1373 K as approximately 1 nm and 1 ps, respectively. The present results suggest that seeds of rigid nanodomains in the glassy state may originate in the liquid state.

The authors would like to thank Japan Society for the Promotion of Science (JSPS) for a Grant-in-Aid for Scientific Research (A) (20244061). The synchrotron radiation experiments were performed at the SPring-8 with the approval of the Japan Synchrotron Radiation Research Institute (JASRI) (Proposals 2012A1156, 2012A1155, 2012A1154, 2009B1286, and 2005B0414).