Article pubs.acs.org/JPCB
Dynamical Heterogeneity in the Supercooled Liquid State of the Phase Change Material GeTe Gabriele C. Sosso,*,†,⊥ Jader Colombo,‡ Jörg Behler,¶ Emanuela Del Gado,‡,∥ and Marco Bernasconi§ †
Department of Chemistry and Applied Biosciences, ETH Zurich, Vladimir-Prelog-Weg 1-5 CH-8093 Zurich, Switzerland Faculty of Informatics, Università della Svizzera Italiana, Via G. Buffi 13, CH-6900 Lugano, Switzerland ‡ Department of Civil, Environmental and Geomatic Engineering, ETH Zurich, CH-8903 Zurich, Switzerland ¶ Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780 Bochum, Germany, and § Dipartimento di Scienza dei Materiali, Università di Milano-Bicocca, Via R. Cozzi 53, I-20125 Milano, Italy ⊥
S Supporting Information *
ABSTRACT: A contending technology for nonvolatile memories of the next generation is based on a remarkable property of chalcogenide alloys known as phase change materials, namely their ability to undergo a fast and reversible transition between the amorphous and crystalline phases upon heating. The fast crystallization has been ascribed to the persistence of a high atomic mobility in the supercooled liquid phase, down to temperatures close to the glass transition. In this work we unravel the atomistic, structural origin of this feature in the supercooled liquid state of GeTe, a prototypical phase change compound, by means of molecular dynamic simulations. To this end, we employed an interatomic potential based on a neural network framework, which allows simulating thousands of atoms for tens of ns by keeping an accuracy close to that of the underlying first-principles framework. Our findings demonstrate that the high atomic mobility is related to the presence of clusters of slow and fast moving atoms. The latter contain a large fraction of chains of homopolar Ge−Ge bonds, which at low temperatures have a tendency to move by discontinuous cage-jump rearrangements. This structural fingerprint of dynamical heterogeneity provides an explanation of the breakdown of the Stokes−Einstein relation in GeTe, which is the ultimate origin of the fast crystallization of phase change materials exploited in the devices.
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INTRODUCTION Phase change materials are of great technological interest for applications in rewritable optical media (Digital Versatile Disc, Blu-ray discs) and in novel nonvolatile electronic memory devices named phase change memories (PCM).1−4 Both applications rest on a fast and reversible transformation between the crystalline and amorphous phases which represent the two states of the memory that can be discriminated because of the large contrast in the optical reflectivity and electrical conductivity. The phase change is induced by heating, either due to laser irradiation in the optical discs or to Joule effect in PCMs. The materials of choice for both PCM and Blu-ray discs lie on the pseudobinary GeTe-Sb2Te3 tie-line with the Ge2Sb2Te5 (GST) composition used for PCM and Ge8Sb2Te11 used for the optical discs. However, the GeTe binary compound with different doping,5 and other tellurides6,7 alloys are also under scrutiny for their higher crystallization temperature of interest for applications at high temperatures, e.g., in automotive electronics. The GeTe compound has been the subject of intensive investigations also because it shares many properties with the most studied Ge2Sb2Te5 alloy. One of the key properties that make phase change materials attractive for © 2014 American Chemical Society
applications in non volatile memories is the high crystallization speed which allows for a full crystallization in PCM devices on the time scale of 10−100 ns. The atomistic origin of this peculiar property is, however, still a matter of debate. In phase change materials, the crystallization temperature and the glass transition temperature Tg turn out to be very similar, according to conventional differential scanning calorimetry (DSC).8 However, due to the fast heating rate in the set process of PCM, the amorphous phase is brought far above Tg, turning into the supercooled liquid before the onset of crystallization could take place.9 The fragility of the supercooled liquid has then been proposed as the key feature that boosts the crystallization speed at the conditions of operation of the devices.10 A supercooled liquid is classified as “fragile” if the viscosity η remains very low down to temperatures close to Tg, where a steep, super-Arrhenius behavior is observed up to the high value of η expected at Tg.11 On the contrary the viscosity of an ideal strong liquid follows an Arrhenius behavior from the Received: July 23, 2014 Revised: October 27, 2014 Published: October 30, 2014 13621
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melting down to Tg. The crystal nucleation rate and the speed of crystal growth can thus be very high because the atomic mobility is large (the viscosity is low) down to temperatures close to Tg where the thermodynamical driving force for the crystallization is also large. A further enhancement of the atomic mobility at low temperatures is also due to the breakdown of the Stokes−Einstein relation (SER) between η and the self-diffusion coefficient D which is another typical feature of fragile liquids. The inverse proportionality between D and η embodied in the SER is based on macroscopic hydrodynamics that treats the liquid as a continuum. The breakdown of SER in fragile liquids is then ascribed to the failure of the continuum approximation due the emergence of dynamical heterogeneities, which consist of spatially localized regions in which atoms move substantially faster than the average and regions of slow moving atoms.12,13 Experimental evidence of the fragility and of the breakdown of the SER in phase change materials came from ultrafast DSC measurements on Ge2Sb2Te5 films.10 A non-Arrhenius behavior of the crystallization speed in actual PCM devices has also been recently reported.14,15 On the other hand, theoretical insight into the fragility of supercooled liquid GeTe has been gained by large scale molecular dynamics (MD) simulations.16,17 Independent calculations of the viscosity and the self-diffusion coefficient revealed the breakdown of SER 16 and the persistence of a high atomic mobility at high supercooling. Besides, the analysis of the crystallization kinetic suggested17 that crystal nucleation in supercooled liquid GeTe is not rate limiting in the temperature range 500−675 K of interest for device operation, and that the velocity of growth of supercritical nuclei u can be well described by the classical expression3,18 u = 4D/λ[1 − exp(−(Δμ)/(kBT))] where λ is a typical jump distance and Δμ is the difference in free energy between the liquid and the crystal. These results demonstrate that the fast crystallization does indeed originate from a high atomic mobility down to temperatures close to Tg. In this paper, we report on MD simulations that allowed us to identify the occurrence of dynamical heterogeneity in supercooled liquid GeTe leading to the breakdown of the SER and ultimately to the high mobility at low temperatures responsible for the high crystallization speed. To this end, we performed large scale simulations (4096 atoms for several tens of nanoseconds) by using an interatomic potential that we generated previously19 by fitting the energies of a large database of atomic configurations computed within density functional theory (DFT). The fitting was made by means of a neural network (NN) method20 which provided an interatomic potential with an accuracy close to that of the underlying DFT framework whose reliability in describing structural and dynamical properties of GeTe and other phase change materials has been validated in several previous works.21−23 Dynamical heterogeneities (DH) in supercooled liquids have been experimentally observed in a number of materials, in particular colloidal glasses,24 and they have been studied via atomistic simulations25 in several model systems, ranging from hard spheres and Lennard-Jones glasses to liquid silicon modeled by the Stillinger-Weber potential.26−28 In the case of supercooled liquid GeTe, an isoconfigurational analysis29 allowed us to associate the clustering of fast moving atoms with peculiar structural features consisting of the presence of chains of homopolar Ge−Ge bonds.
Article
RESULTS AND DISCUSSION
Dynamical Heterogeneity. We analyzed the dynamics of supercooled liquid GeTe at three temperatures, T = 1000, 600, and 500 K, in between the melting point (998 K30) and the crystallization temperature of 450 K31 that we identify with Tg. As the breakdown of the SER arises at ∼700 K,16 we aim at comparing the dynamical properties of the liquid in the hydrodynamic regime (1000 K) with those of the supercooled liquid in the temperature range where the extent of the deviations from the SER is significant. In particular, our previous results suggest that the viscosity of the supercooled liquid differs from the value obtained from the SER by about 1 order of magnitude at 600 K and at least 3 orders of magnitude at 500 K.16 Molecular dynamics trajectories of a 4096-atom cells were generated by using the NN potential. Details on the numerical simulations can be found in the Supporting Information. Since GeTe spontaneously crystallizes on the time scale of about 1 ns in the temperature range 500−600 K, the analysis has been restricted to about 200 ps after equilibration at the target temperature. In the search for footprints of DH, we analyzed the relaxation dynamics of our system by computing the incoherent intermediate scattering function32 Fs(q , t ) = ⟨Φs(q , t )⟩ with
Φs(q , t ) ≐
1 N
N
∑ exp{iq ⃗ ·[→⎯rj(0) −→⎯rj(t )]} j
(1)
where the sum runs over all the j atoms having position rj⃗ (t) at time t, ⟨...⟩ denotes a time average, and q⃗ is a vector in reciprocal space. In an isotropic system, Fs depends only on the magnitude q of the vector q⃗. This function could be obtained from scattering data, unfortunately not available for GeTe, and provides information on the atomic motion resolved in space and time. For a fixed value of q, Fs(q, t) decays from unity to zero over the time t in which the atoms move over a distance of the order of 2π/q.33 The incoherent scattering function Fs(q0, t) is reported in Figure 1a for q0 = 2.1 Å−1, corresponding to the main peak of the structure factor (see Figure S1). At 1000 K, Fs(q0, t) follows an exponential relaxation, as we expect for a liquid in the hydrodynamic regime, in which all the atoms experience the same diffusive dynamics with a single characteristic time scale. At lower temperatures, Fs(q0, t) displays a two-step decay with an intermediate plateau, which is a typical feature of supercooled liquids that becomes more evident at the lowest temperature. The intermediate times over the plateau encompass a period during which atoms are trapped in the cages formed by their neighbors. The relaxation toward the plateau is known as the β-relaxation regime. At longer times, Fs(q, t) decays to zero following the so-called α-relaxation regime which involves structural, possibly cooperative, rearrangements on a longer time scale. In this regime Fs(q, t) displays a stretched exponential decay, as observed in several other systems.13 To unravel the emergence of DH in supercooled GeTe we have computed the dynamical susceptibility χ4(q, t),34,35 which is defined as the variance of the intermediate scattering function as follows: χ4 (q , t ) = N[⟨|Φs(q , t )|2 ⟩ − ⟨Φs(q , t )⟩2 ] 13622
(2)
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giving rise to dynamical heterogeneities, we have performed an iso-configurational analysis (ISOCA) of the atomic motion. This technique can be used to obtain a spatially resolved map of dynamical heterogeneities.29,36 The analysis consists of performing a number (50, in our case) of MD simulations starting from the same atomic configuration but randomly choosing the initial velocities for each run from a Maxwell− Boltzmann distribution at a given temperature. This ensemble average eliminates the fluctuations due to the initial atomic momenta, inevitably present once the analysis is restricted to a single trajectory. This analysis allows us to link the dynamical heterogeneities to particular structural features present in the liquid. We quantify the tendency of each atom i to move by computing the so-called dynamical propensity (DP) over the characteristic times t* that correspond to the maxima of χ4(q, t) for q = 1.5 Å−1: ( ri(0) − ri(⃗ t *))2 ⃗ MSDαi
DP( i t *) =
ISO
(3)
where αi denotes the atomic species of atom i (either Ge or Te), MSDαi is the mean squared displacement of that species for the same time t*, and ⟨...⟩ISO stands for the average over the NISO trajectories of the isoconfigurational ensemble. The normalization factor MSDαi is necessary because the two species have different mobilities: in fact the ratio between the diffusion coefficient of Ge and Te atoms DGe/DTe increases upon cooling from 1.3 at 1000 K to 3.5 at 500 K (see Figure 2 of ref 16). Atoms whose tendency to move is close to the average for their species have DP ≈ 1, whereas deviations from unity correspond to atoms that tend to be more or less mobile than the average. The choice q = 1.5 Å−1 corresponds to a length scale between the first and second coordination shell at which χ4* is close to its maximum, while t* is 2, 25, and 55 ps at 1000, 600, and 500 K respectively. We remark that different choices of q could in principle reveal different dynamical processes ongoing on different length and time scales. However, we have verified that our results are consistent over the wide q range in which χ*4 develops a significant peak (see Figure 1c). We have also verified that our results are insensitive to the choice of the initial configuration (i.e., different time origin) used as starting point for the isoconfigurational analysis. The analysis of the spatial distribution of DP gives a measure of the dynamical heterogeneities in the liquid. To this end, we introduce a DP density
Figure 1. (a) Incoherent intermediate scattering function Fs(q, t) and (b) dynamical susceptibility χ4(q, t) of liquid GeTe at different temperatures, calculated at q0 = 2.1 Å−1, which corresponds to the main peak of the static structure factor (see Figure S1, Supporting Information). (c) Dependence of the maximum of the dynamical susceptibility χ*4 (lines) and the corresponding time t* (points) as a function of q.
This indicator quantifies fluctuations from the mean degree of correlation in single atom displacements over the time and length scales given by t and q, as shown in Figure 1b. The function χ4(q0, t) of liquid GeTe starts from zero at short times, reaches a maximum between the β and the α-relaxation regimes, and finally approaches the asymptotic value χ4(q, t → ∞) = 1. The height of the peak grows upon cooling and it is expected to be proportional to the number of atoms involved in cooperative motions in the spatial regions of slow and fast moving atoms.32 The height of the peak χ4* and the corresponding time t* as a function of q are reported in Figure 1c. At 1000 K, χ*4 remains close to one along the whole q interval considered, confirming that DH are negligible in the ordinary liquid. At low temperatures, instead, χ4* develops a high peak over a relatively wide q range at a characteristic time t* which is a decreasing function of q, as spatial correlations over larger length scales develop over longer time scales. The functions χ4(q, t) and Fs(q, t) shown in Figure 1 are averaged over the two atomic species. We have verified that the same behavior is observed once these functions are resolved for each species. Note, however, that the relaxation time is longer for Te atoms than for Ge atoms due to the higher diffusion coefficient of Ge with respect to Te, a difference that is barely noticeable at high temperature but that becomes more evident close to Tg.16 Isoconfigurational Analysis. To search for a possible structural origin of the clustering of fast and slow moving atoms
ρ(DP)i =
1 Nj
Nj
∑ j , rij < rd
DPj (4)
where the sum runs over the neighbor atoms j of atom i up to a certain cutoff distance rd, which we set equal to the length scale we are probing by choosing q = 1.5 Å−1. In Figure 2a we show the projection onto the xy-plane of the initial configurations of the ISOCA at 1000 and 500 K, where each atom has been colored according to its DP density. While at 1000 K slow (blue) and fast (red) atoms are randomly distributed, at 500 K slow and fast moving atoms tend to cluster in spatially separated domains. Atoms with a DP at least one standard deviation away from the mean of the DP distribution are identified as most immobile (MI) or most mobile (MM, see Figure S2). This choice ensures that MI and MM domains contain a representative fraction of atoms (about 10%) at each 13623
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Figure 2. (a) Color map of the distribution of dynamical propensity (ρ(DP), see text) at 1000 and 500 K. (b) Number of clusters, average cluster size, and maximum cluster size (number of particles) of most immobile (MI) and most mobile (MM) atoms at different temperatures.
Table 1. Ge−Ge Homopolar Bonds as a Function of Temperaturea T [K]
tot.
MI
MM
NMI
1000 600 500
1254 1066 1056
18 15 7
60 96 104
415 491 490
600 CR
987
11
106
NMM
MM [%]
Gechains
Clength
NC
8.1 4.8 2.4
29.8 36.5 39.5
59 82 88
3.5 4.1 4.2
17 20 21
1.9
33.2
101
3.6
28
MI [%]
366 404 414 During Crystallization 644 487
a
We report the total number of Ge−Ge bonds in the whole system (tot.), the number of Ge−Ge bonds in most immobile (MI) and most mobile (MM) clusters, the total number of atoms in the MI and MM clusters (NMI and NMM), and the fraction of Ge−Ge bonds, formed by atoms inside the MI and MM clusters, with respect to the total number of bonds formed by atoms inside the MI and MM clusters (MI [%] and MM [%]). For the MM regions only, we also report the number of atoms in Ge−Ge chains (Gechains), the average chain length (Clength) and the number of Ge−Ge chains (NC). A chain contains at least a Ge trimer; dimers are not considered as chains. The data at 600 CR refer to ISOCA performed at 600 K at a longer time when a fraction of about 10% atoms form crystalline nuclei as discussed in our previous work.17
MM regions have the characteristic property of containing a fraction of homopolar Ge−Ge bonds higher than the average and much higher than in the MI regions. We assume that a Ge−Ge bond is formed when its length is below 3.0 Å (see refs 17 and 19 for partial pair correlation functions at different temperatures). Variations of the bonding cutoff in the range 2.8−3.2 Å provide very similar results. The fraction of homopolar Ge−Ge bonds in the MM regions increases upon cooling in spite of the fact that the total number of Ge−Ge bonds in the whole of the system decreases (see Table 1). We have found that most of Ge−Ge bonds cluster in chains (depicted in purple in Figure 3b), which are particularly
temperature (see Table 1). We underline that our results do not change significantly with respect to the criterion by which we choose MI and MM atoms, consistently to what has been reported in previous works.37 The MI (or MM) atoms are defined to belong to the same cluster once they are separated by a distance shorter than 3.6 Å. The number of MI and MM clusters at different temperatures and their average and maximum size (number of atoms) are given in Figure 2b. Large clusters up to ∼200 (MI) and ∼100 (MM) atoms are found at 500 K as depicted in Figure 3a. We have not found particular features that characterize the structure of the supercooled liquid in the MI regions. On the contrary, the 13624
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Figure 3. (a) Initial configuration of the ISOCA at 500 K showing most immobile (blue) and most mobile (red) clusters. The Ge−Ge homopolar chains in most mobile regions (purple) are highlighted in panel b. (c) Distribution of the length of Ge−Ge chains in the MM regions reported as stacked histograms for different temperatures. (d) Probability distribution of the number of nearest neighbors Ge atoms within the chains, averaged over all the chains at all temperatures.
bound to the Ge−Ge chains belong to the MM regions themselves since they are more mobile than the average of Te atoms. However, since the dynamical propensity (eq 3) is normalized by the MSD of each atomic species, Te atoms in MM regions are much less mobile than Ge atoms as the average diffusion coefficient of Te is a factor 3.5 lower than the diffusion coefficient of Ge at 500 K.16 The fraction of Ge and Te atoms in MI and MM domains is also only slightly dependent on temperature (see Table S1). From the data in Table 1, we can conclude that atoms around Ge−Ge chains keep moving faster than the average and their contribution to the regions of MM atoms becomes more and more important upon cooling. In other words, by decreasing temperatures atoms slow down, but those around Ge−Ge chains slow down at a lower pace and progressively enrich the clusters of fast moving atoms. The dynamical heterogeneities in GeTe thus originate from structural heterogeneities in the form of chains of Ge−Ge homopolar bonds. A similar correlation between structural heterogeneities and fragility has been discussed recently for GeAsSe compounds.40 The fragility of these alloys is minimal along the pseudobinary tie-line GeSe2−As 2Se3. Deviations in stoichiometry from the pseudobinary line are proposed to lead to an increasing fraction of homopolar bonds responsible for the increased fragility of the supercooled liquid. These features suggest that in chalcogenide liquids very specific structural heterogeneities, such as the Ge−Ge homopolar bond chains detected here, could originate the DH underlying the breakdown of the SER relation and hence the fragile behavior of this class of systems.
numerous in the MM regions at low temperatures. We remark that the chains of Ge−Ge bonds are not isolated from the rest of the liquid network. In fact, the coordination number of Ge atoms involved in Ge−Ge mobile chains is about four (see Figure S3a), and Te atoms bonded to the chains belong to the MM domains as well. The length of the chains of Ge−Ge bonds in the MM regions slightly increases by decreasing temperature as shown in Figure 3c and in Table 1, in spite of the fact that both the number of chains and their length decreases by decreasing temperature in the model as a whole (Figure S4). Most of the chains are linear, self-loops and branched chains are rare (see Figure S3b); in fact, most Ge atoms forming homopolar bonds have only two nearest neighbor Ge atoms (inside the chains) or one nearest neighbor Ge atom (at chains end) as shown by the distribution reported in Figure 3d. Note that Ge−Ge chains are by no means an artifact of the NN potential. In fact, similarly long chains are found in fully DFT simulations of the supercooled liquid, albeit with a small 216-atom cell, as shown in Figure S4 of Supporting Information. We remark that the data in Table 1 refer to the properties of the MI and MM regions only (but for the total number of homopolar bonds). By decreasing temperature, the clustering of fast and slow moving atoms in spatially separated regions increases, but the average mobility still changes smoothly with temperature as it follows an Arrhenius behavior down to 500 K.16 The enrichment of the MM regions with Ge−Ge chains is smooth as well. Sharper changes in the mobility is expected at lower temperatures, closer to Tg where, however, the relaxation times of the liquid might become longer than the incubation time for crystallization and anyway longer that our affordable simulation time. The Te atoms 13625
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Figure 4. (a) Time evolution of the cage-jump indicator J(t) for an atom that undergoes a cage jump (jump, red curve) and an atom that keeps on rattling within its cage (caged, blue curve). (b) Example of string motion of mobile atoms. The arrows depict atomic displacements within t*. Data refer to simulations at 500 K over the characteristic time t* = 55 ps.
clear indication of whether an atom moved significantly or not from SA to SB. The function J(t) develops a peak when the atom escapes its cage because the distance dAB between the average position in the two subtrajectories gets its maximum value. We define a cage-jump as the event for which dAB ≥ 3.0 Å, i.e. the average bond length. Results are rather insensitive to changes of this threshold in a reasonable range. A plot of the evolution of J(t) for representative atoms is given in Figure 4a. We remark that the indicator J(t) is specifically sensitive to jump-like motion, whereas diffusive motions with a very short residence time in the cages do not give rise to specific features in J(t). We can therefore identify the fraction of atoms in the MM regions that undergo a cage-jump motion by simply counting the atoms jumping in the time t* as revealed by J(t). Results are reported in Table 2, where we compare the fraction of jumping atoms
The isoconfigurational analysis at 600 K has been repeated at longer times when the system starts to crystallize. The map of the dynamical propensity clearly shows regions of immobile atoms corresponding to the crystalline nuclei, but also the part of the sample still in a supercooled liquid state keeps displaying dynamical heterogeneities with regions of MM atoms around Ge−Ge homopolar chains (see Figure S5 and Table 1). We note that the Ge−Ge chains are dynamical objects. As we shall see in the next section, they do break and reform on a time scale roughly equal to t*. These findings suggest that the presence of MM domains and specifically of the Ge−Ge homopolar chains allows the supercooled liquid to remain significantly mobile in spatially confined regions around the crystalline nuclei. Such enhanced mobility boosts the kinetic prefactor that applies both to the nucleation rate and the crystal growth velocity.17 Diffusion Mechanism of Most Mobile Atoms at Low Temperatures. The incoherent scattering function in Figure 1 suggests the emergence of a β-relaxation regime at low temperature which is customarily ascribed to rattling motion of atoms in the cage formed by the atoms in the first coordination shell. Longer time α-relaxation processes are related to jumps of atoms outside the cage on the time scale t* corresponding to the maximum of the susceptibility χ4 (cf. Figure 1 and Figure 2).38,39 To detect the occurrence of cage jumps we compute the function J(t) introduced in ref 41, which is roughly constant in time when the atom rattles within the boundaries of its cage, but it develops a peak when the atom escapes its cage. The function is built by dividing the trajectory in two subtrajectories and it is defined by
Table 2. Fraction of Atoms That Undergo a Cage-Jump within t*
ζ(t ) =
2 (t * − t )t t*
all [%]
MI [%]
MM [%]
Ge−Ge chains in MM [%]
500 600
14.4 21.3
12.6 19.7
24.2 30.1
37.1 44.5
with respect to different subregions of the systems, identified by means of the dynamical propensity obtained by the ISOCA analysis. The results have been averaged over all the trajectories of the isoconfigurational ensemble. It turns out that atoms belonging to MM regions display a much more pronounced tendency to experience a cage-jump within t* than all the other atoms, which means that intercage jumps significantly influences the atomic motion in the MM regions. Strikingly, this finding is even more evident for atoms within the Ge−Ge mobile chains. These results are non trivial, as they have been obtained by comparing single atom dynamics events with subsets of atoms defined by the dynamical propensity obtained by the ISOCA. Further insight into the dynamical properties of the Ge−Ge bonds involved in the homopolar chains within the MM domains can be obtained by looking at both the continuous Cc(t) and the intermittent Ci(t) bond time correlation functions42−44 at different temperatures, reported in Figure S6 of the Supporting Information. The temporal decay of Cc(t) allows an estimate of the mean bond lifetime τcb, while the decay of Ci(t) gives a measure of the time scale of local structural relaxation τib. Both quantities are described in detail in the Supporting Information and reported in Table S2. It turns out
J(t ) = ζ(t ) ⟨dA 2(tB)⟩tB∈ SB ⟨dB2(tA)⟩tA ∈ SA with
T [K]
(5)
where t* is the length of the trajectory over which the analysis is performed, which corresponds to the maximum of χ4. SA = {0;t} and SB = {t;t*} are the two subtrajectories before and after time t and dj(tk) is the distance between the atom position at time tk in the subset Sk and the average position along the trajectory in the subset Sj. The average over time tk is taken over the subset Sk, and the prefactor ζ(t) removes the effect of the scarce sampling as J(t) approaches the boundaries of the interval [0,t*]. Thus, J(t) keeps track of the spatial separation between the average atomic position within the two subtrajectories SA and SB as a function of time, providing a 13626
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that τcb does not depend dramatically on temperature, τcb being equal to 0.26 and 1.04 ps at 1000 and 500 K respectively. Larger variations with temperature are instead found for τib whose values are remarkably close t* (cf. Figure 1b) . This finding suggests that the time scale of structural relaxation is indeed related to the change of the local atomic environments due to bonds breaking in a cage-jump fashion. Ge−Ge bonds within mobile homopolar chains rattle on the short time scale of τcb, and eventually break via cage-jumps on the much longer time scale given by τib ∼ t*. Thus, the fact that a consistent fraction of MM atoms undergoes a cage-jump within the characteristic time t* clearly indicates that there is a strong connection between structure and dynamics in the supercooled liquid. In particular, the finding that the Ge−Ge homopolar chains tend to move via cage-jumps over the time scale typical of structural relaxation supports the idea that Ge−Ge homopolar chains have a specific role in the heterogeneous dynamics of this material. Another peculiar dynamical feature of a number of supercooled liquids is string-like motion, which has been reported in a variety of model systems.45−47 Apart from occasional events involving Ge atoms in homopolar chains (Figure 4b), we could seldom observe string-like motion on the time scale given by the characteristic time t*. Nevertheless, even our large simulations can still suffer from finite size effect. In particular, we can not rule out that string-like motion or other cooperative effects might contribute to the dynamics of the system, but are not found in our simulation cell due to insufficient statistics. However, dynamical heterogeneities at the temperatures investigated here emerge even in the lack of string-like motions.
cooperative motion via string-like atomic displacements have also been observed. The emerging picture is that dynamical heterogeneity originates from the chemical specificity of GeTe, which leads to structural heterogeneities in the form of the Ge− Ge chains that preserve and enhance the atomic mobility even close to the glass transition temperature. In summary, we provide an atomistic, structural interpretation of the breakdown of the SER in the supercooled liquid phase of GeTe, which is in turn responsible for the large atomic mobility of this system at low temperatures that boosts its crystallization kinetics. This picture, with the due differences for the specific structural features, may be relevant to other phase change alloys.
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ASSOCIATED CONTENT
S Supporting Information *
Results of dynamical heterogeneity in the supercooled liquid state of the phase change material GeTe. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*(G.C.S.) E-mail: [email protected]. Present Address ∥
Institute for Soft Matter Synthesis and Metrology and Department of Physics, Georgetown University, Washington, DC, 20057
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thankfully acknowledge the computational resources provided by Cineca (Casalecchio di Reno, Italy) through the Program EU-FP7 Prace and the LISA Initiative and by CSCS (Switzerland) under Project No. s477. M.B. acknowledges funding from the European Union Seventh Framework Programme FP7/2007-2013 under Grant Agreement No. 310339. J.C. and E.D.G. are supported by the Swiss National Science Foundation (Grant No. PP00P2_126483/1).
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CONCLUSIONS In this work, we have provided evidence, via large scale molecular dynamics simulations, of dynamical heterogeneities in supercooled liquid GeTe, underlying the breakdown of the SER between viscosity and diffusivity. This behavior is responsible for the persistence of high atomic mobility in the supercooled liquid phase down to very low temperatures. Such a feature is indeed the ultimate source of the high crystallization speed exploited in non volatile memory devices based on phase change materials. Evidences of the presence of dynamical heterogeneities come from the analysis of the intermediate scattering function and its susceptibility. Isoconfigurational analysis provided a vivid picture of the spatially separated domains of fast and slow moving atoms growing in size upon supercooling. We have shown that the dynamical heterogeneities are associated with a structural fingerprint, as the fast moving atoms cluster in regions with a significant concentration of chains of homopolar Ge−Ge bonds. The evidence that wrong (homopolar) bonds are the source of dynamical heterogeneities in GeTe is in line with recent findings that show how the fragility of GeAsSe alloys increases by moving away from the pseudobinary GeSe2−As2Se3 tie-line.40 Wrong bonds are in fact expected to increase by moving off the pseudobinary line leading to an increase of the structural/ dynamical heterogeneities responsible for a higher fragility. In supercooled liquid GeTe, at low temperatures the atomic motion does not follow a continuous diffusive path, as a substantial fraction of atoms rattles in the cage formed by their neighbors and undergoes discontinuous intercage jumps. It turns out that cage jumps are more frequent in the case of atoms involved in Ge−Ge mobile chains. Rare examples of
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REFERENCES
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Dynamical Heterogeneity in the Supercooled Liquid State of the Phase Change Material GeTe Gabriele C. Sosso,*,†,⊥ Jader Colombo,‡ Jörg Behler,¶ Emanuela Del Gado,‡,∥ and Marco Bernasconi§ †
Department of Chemistry and Applied Biosciences, ETH Zurich, Vladimir-Prelog-Weg 1-5 CH-8093 Zurich, Switzerland Faculty of Informatics, Università della Svizzera Italiana, Via G. Buffi 13, CH-6900 Lugano, Switzerland ‡ Department of Civil, Environmental and Geomatic Engineering, ETH Zurich, CH-8903 Zurich, Switzerland ¶ Lehrstuhl für Theoretische Chemie, Ruhr-Universität Bochum, Universitätsstrasse 150, D-44780 Bochum, Germany, and § Dipartimento di Scienza dei Materiali, Università di Milano-Bicocca, Via R. Cozzi 53, I-20125 Milano, Italy ⊥
S Supporting Information *
ABSTRACT: A contending technology for nonvolatile memories of the next generation is based on a remarkable property of chalcogenide alloys known as phase change materials, namely their ability to undergo a fast and reversible transition between the amorphous and crystalline phases upon heating. The fast crystallization has been ascribed to the persistence of a high atomic mobility in the supercooled liquid phase, down to temperatures close to the glass transition. In this work we unravel the atomistic, structural origin of this feature in the supercooled liquid state of GeTe, a prototypical phase change compound, by means of molecular dynamic simulations. To this end, we employed an interatomic potential based on a neural network framework, which allows simulating thousands of atoms for tens of ns by keeping an accuracy close to that of the underlying first-principles framework. Our findings demonstrate that the high atomic mobility is related to the presence of clusters of slow and fast moving atoms. The latter contain a large fraction of chains of homopolar Ge−Ge bonds, which at low temperatures have a tendency to move by discontinuous cage-jump rearrangements. This structural fingerprint of dynamical heterogeneity provides an explanation of the breakdown of the Stokes−Einstein relation in GeTe, which is the ultimate origin of the fast crystallization of phase change materials exploited in the devices.
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INTRODUCTION Phase change materials are of great technological interest for applications in rewritable optical media (Digital Versatile Disc, Blu-ray discs) and in novel nonvolatile electronic memory devices named phase change memories (PCM).1−4 Both applications rest on a fast and reversible transformation between the crystalline and amorphous phases which represent the two states of the memory that can be discriminated because of the large contrast in the optical reflectivity and electrical conductivity. The phase change is induced by heating, either due to laser irradiation in the optical discs or to Joule effect in PCMs. The materials of choice for both PCM and Blu-ray discs lie on the pseudobinary GeTe-Sb2Te3 tie-line with the Ge2Sb2Te5 (GST) composition used for PCM and Ge8Sb2Te11 used for the optical discs. However, the GeTe binary compound with different doping,5 and other tellurides6,7 alloys are also under scrutiny for their higher crystallization temperature of interest for applications at high temperatures, e.g., in automotive electronics. The GeTe compound has been the subject of intensive investigations also because it shares many properties with the most studied Ge2Sb2Te5 alloy. One of the key properties that make phase change materials attractive for © 2014 American Chemical Society
applications in non volatile memories is the high crystallization speed which allows for a full crystallization in PCM devices on the time scale of 10−100 ns. The atomistic origin of this peculiar property is, however, still a matter of debate. In phase change materials, the crystallization temperature and the glass transition temperature Tg turn out to be very similar, according to conventional differential scanning calorimetry (DSC).8 However, due to the fast heating rate in the set process of PCM, the amorphous phase is brought far above Tg, turning into the supercooled liquid before the onset of crystallization could take place.9 The fragility of the supercooled liquid has then been proposed as the key feature that boosts the crystallization speed at the conditions of operation of the devices.10 A supercooled liquid is classified as “fragile” if the viscosity η remains very low down to temperatures close to Tg, where a steep, super-Arrhenius behavior is observed up to the high value of η expected at Tg.11 On the contrary the viscosity of an ideal strong liquid follows an Arrhenius behavior from the Received: July 23, 2014 Revised: October 27, 2014 Published: October 30, 2014 13621
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melting down to Tg. The crystal nucleation rate and the speed of crystal growth can thus be very high because the atomic mobility is large (the viscosity is low) down to temperatures close to Tg where the thermodynamical driving force for the crystallization is also large. A further enhancement of the atomic mobility at low temperatures is also due to the breakdown of the Stokes−Einstein relation (SER) between η and the self-diffusion coefficient D which is another typical feature of fragile liquids. The inverse proportionality between D and η embodied in the SER is based on macroscopic hydrodynamics that treats the liquid as a continuum. The breakdown of SER in fragile liquids is then ascribed to the failure of the continuum approximation due the emergence of dynamical heterogeneities, which consist of spatially localized regions in which atoms move substantially faster than the average and regions of slow moving atoms.12,13 Experimental evidence of the fragility and of the breakdown of the SER in phase change materials came from ultrafast DSC measurements on Ge2Sb2Te5 films.10 A non-Arrhenius behavior of the crystallization speed in actual PCM devices has also been recently reported.14,15 On the other hand, theoretical insight into the fragility of supercooled liquid GeTe has been gained by large scale molecular dynamics (MD) simulations.16,17 Independent calculations of the viscosity and the self-diffusion coefficient revealed the breakdown of SER 16 and the persistence of a high atomic mobility at high supercooling. Besides, the analysis of the crystallization kinetic suggested17 that crystal nucleation in supercooled liquid GeTe is not rate limiting in the temperature range 500−675 K of interest for device operation, and that the velocity of growth of supercritical nuclei u can be well described by the classical expression3,18 u = 4D/λ[1 − exp(−(Δμ)/(kBT))] where λ is a typical jump distance and Δμ is the difference in free energy between the liquid and the crystal. These results demonstrate that the fast crystallization does indeed originate from a high atomic mobility down to temperatures close to Tg. In this paper, we report on MD simulations that allowed us to identify the occurrence of dynamical heterogeneity in supercooled liquid GeTe leading to the breakdown of the SER and ultimately to the high mobility at low temperatures responsible for the high crystallization speed. To this end, we performed large scale simulations (4096 atoms for several tens of nanoseconds) by using an interatomic potential that we generated previously19 by fitting the energies of a large database of atomic configurations computed within density functional theory (DFT). The fitting was made by means of a neural network (NN) method20 which provided an interatomic potential with an accuracy close to that of the underlying DFT framework whose reliability in describing structural and dynamical properties of GeTe and other phase change materials has been validated in several previous works.21−23 Dynamical heterogeneities (DH) in supercooled liquids have been experimentally observed in a number of materials, in particular colloidal glasses,24 and they have been studied via atomistic simulations25 in several model systems, ranging from hard spheres and Lennard-Jones glasses to liquid silicon modeled by the Stillinger-Weber potential.26−28 In the case of supercooled liquid GeTe, an isoconfigurational analysis29 allowed us to associate the clustering of fast moving atoms with peculiar structural features consisting of the presence of chains of homopolar Ge−Ge bonds.
Article
RESULTS AND DISCUSSION
Dynamical Heterogeneity. We analyzed the dynamics of supercooled liquid GeTe at three temperatures, T = 1000, 600, and 500 K, in between the melting point (998 K30) and the crystallization temperature of 450 K31 that we identify with Tg. As the breakdown of the SER arises at ∼700 K,16 we aim at comparing the dynamical properties of the liquid in the hydrodynamic regime (1000 K) with those of the supercooled liquid in the temperature range where the extent of the deviations from the SER is significant. In particular, our previous results suggest that the viscosity of the supercooled liquid differs from the value obtained from the SER by about 1 order of magnitude at 600 K and at least 3 orders of magnitude at 500 K.16 Molecular dynamics trajectories of a 4096-atom cells were generated by using the NN potential. Details on the numerical simulations can be found in the Supporting Information. Since GeTe spontaneously crystallizes on the time scale of about 1 ns in the temperature range 500−600 K, the analysis has been restricted to about 200 ps after equilibration at the target temperature. In the search for footprints of DH, we analyzed the relaxation dynamics of our system by computing the incoherent intermediate scattering function32 Fs(q , t ) = ⟨Φs(q , t )⟩ with
Φs(q , t ) ≐
1 N
N
∑ exp{iq ⃗ ·[→⎯rj(0) −→⎯rj(t )]} j
(1)
where the sum runs over all the j atoms having position rj⃗ (t) at time t, ⟨...⟩ denotes a time average, and q⃗ is a vector in reciprocal space. In an isotropic system, Fs depends only on the magnitude q of the vector q⃗. This function could be obtained from scattering data, unfortunately not available for GeTe, and provides information on the atomic motion resolved in space and time. For a fixed value of q, Fs(q, t) decays from unity to zero over the time t in which the atoms move over a distance of the order of 2π/q.33 The incoherent scattering function Fs(q0, t) is reported in Figure 1a for q0 = 2.1 Å−1, corresponding to the main peak of the structure factor (see Figure S1). At 1000 K, Fs(q0, t) follows an exponential relaxation, as we expect for a liquid in the hydrodynamic regime, in which all the atoms experience the same diffusive dynamics with a single characteristic time scale. At lower temperatures, Fs(q0, t) displays a two-step decay with an intermediate plateau, which is a typical feature of supercooled liquids that becomes more evident at the lowest temperature. The intermediate times over the plateau encompass a period during which atoms are trapped in the cages formed by their neighbors. The relaxation toward the plateau is known as the β-relaxation regime. At longer times, Fs(q, t) decays to zero following the so-called α-relaxation regime which involves structural, possibly cooperative, rearrangements on a longer time scale. In this regime Fs(q, t) displays a stretched exponential decay, as observed in several other systems.13 To unravel the emergence of DH in supercooled GeTe we have computed the dynamical susceptibility χ4(q, t),34,35 which is defined as the variance of the intermediate scattering function as follows: χ4 (q , t ) = N[⟨|Φs(q , t )|2 ⟩ − ⟨Φs(q , t )⟩2 ] 13622
(2)
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giving rise to dynamical heterogeneities, we have performed an iso-configurational analysis (ISOCA) of the atomic motion. This technique can be used to obtain a spatially resolved map of dynamical heterogeneities.29,36 The analysis consists of performing a number (50, in our case) of MD simulations starting from the same atomic configuration but randomly choosing the initial velocities for each run from a Maxwell− Boltzmann distribution at a given temperature. This ensemble average eliminates the fluctuations due to the initial atomic momenta, inevitably present once the analysis is restricted to a single trajectory. This analysis allows us to link the dynamical heterogeneities to particular structural features present in the liquid. We quantify the tendency of each atom i to move by computing the so-called dynamical propensity (DP) over the characteristic times t* that correspond to the maxima of χ4(q, t) for q = 1.5 Å−1: ( ri(0) − ri(⃗ t *))2 ⃗ MSDαi
DP( i t *) =
ISO
(3)
where αi denotes the atomic species of atom i (either Ge or Te), MSDαi is the mean squared displacement of that species for the same time t*, and ⟨...⟩ISO stands for the average over the NISO trajectories of the isoconfigurational ensemble. The normalization factor MSDαi is necessary because the two species have different mobilities: in fact the ratio between the diffusion coefficient of Ge and Te atoms DGe/DTe increases upon cooling from 1.3 at 1000 K to 3.5 at 500 K (see Figure 2 of ref 16). Atoms whose tendency to move is close to the average for their species have DP ≈ 1, whereas deviations from unity correspond to atoms that tend to be more or less mobile than the average. The choice q = 1.5 Å−1 corresponds to a length scale between the first and second coordination shell at which χ4* is close to its maximum, while t* is 2, 25, and 55 ps at 1000, 600, and 500 K respectively. We remark that different choices of q could in principle reveal different dynamical processes ongoing on different length and time scales. However, we have verified that our results are consistent over the wide q range in which χ*4 develops a significant peak (see Figure 1c). We have also verified that our results are insensitive to the choice of the initial configuration (i.e., different time origin) used as starting point for the isoconfigurational analysis. The analysis of the spatial distribution of DP gives a measure of the dynamical heterogeneities in the liquid. To this end, we introduce a DP density
Figure 1. (a) Incoherent intermediate scattering function Fs(q, t) and (b) dynamical susceptibility χ4(q, t) of liquid GeTe at different temperatures, calculated at q0 = 2.1 Å−1, which corresponds to the main peak of the static structure factor (see Figure S1, Supporting Information). (c) Dependence of the maximum of the dynamical susceptibility χ*4 (lines) and the corresponding time t* (points) as a function of q.
This indicator quantifies fluctuations from the mean degree of correlation in single atom displacements over the time and length scales given by t and q, as shown in Figure 1b. The function χ4(q0, t) of liquid GeTe starts from zero at short times, reaches a maximum between the β and the α-relaxation regimes, and finally approaches the asymptotic value χ4(q, t → ∞) = 1. The height of the peak grows upon cooling and it is expected to be proportional to the number of atoms involved in cooperative motions in the spatial regions of slow and fast moving atoms.32 The height of the peak χ4* and the corresponding time t* as a function of q are reported in Figure 1c. At 1000 K, χ*4 remains close to one along the whole q interval considered, confirming that DH are negligible in the ordinary liquid. At low temperatures, instead, χ4* develops a high peak over a relatively wide q range at a characteristic time t* which is a decreasing function of q, as spatial correlations over larger length scales develop over longer time scales. The functions χ4(q, t) and Fs(q, t) shown in Figure 1 are averaged over the two atomic species. We have verified that the same behavior is observed once these functions are resolved for each species. Note, however, that the relaxation time is longer for Te atoms than for Ge atoms due to the higher diffusion coefficient of Ge with respect to Te, a difference that is barely noticeable at high temperature but that becomes more evident close to Tg.16 Isoconfigurational Analysis. To search for a possible structural origin of the clustering of fast and slow moving atoms
ρ(DP)i =
1 Nj
Nj
∑ j , rij < rd
DPj (4)
where the sum runs over the neighbor atoms j of atom i up to a certain cutoff distance rd, which we set equal to the length scale we are probing by choosing q = 1.5 Å−1. In Figure 2a we show the projection onto the xy-plane of the initial configurations of the ISOCA at 1000 and 500 K, where each atom has been colored according to its DP density. While at 1000 K slow (blue) and fast (red) atoms are randomly distributed, at 500 K slow and fast moving atoms tend to cluster in spatially separated domains. Atoms with a DP at least one standard deviation away from the mean of the DP distribution are identified as most immobile (MI) or most mobile (MM, see Figure S2). This choice ensures that MI and MM domains contain a representative fraction of atoms (about 10%) at each 13623
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Figure 2. (a) Color map of the distribution of dynamical propensity (ρ(DP), see text) at 1000 and 500 K. (b) Number of clusters, average cluster size, and maximum cluster size (number of particles) of most immobile (MI) and most mobile (MM) atoms at different temperatures.
Table 1. Ge−Ge Homopolar Bonds as a Function of Temperaturea T [K]
tot.
MI
MM
NMI
1000 600 500
1254 1066 1056
18 15 7
60 96 104
415 491 490
600 CR
987
11
106
NMM
MM [%]
Gechains
Clength
NC
8.1 4.8 2.4
29.8 36.5 39.5
59 82 88
3.5 4.1 4.2
17 20 21
1.9
33.2
101
3.6
28
MI [%]
366 404 414 During Crystallization 644 487
a
We report the total number of Ge−Ge bonds in the whole system (tot.), the number of Ge−Ge bonds in most immobile (MI) and most mobile (MM) clusters, the total number of atoms in the MI and MM clusters (NMI and NMM), and the fraction of Ge−Ge bonds, formed by atoms inside the MI and MM clusters, with respect to the total number of bonds formed by atoms inside the MI and MM clusters (MI [%] and MM [%]). For the MM regions only, we also report the number of atoms in Ge−Ge chains (Gechains), the average chain length (Clength) and the number of Ge−Ge chains (NC). A chain contains at least a Ge trimer; dimers are not considered as chains. The data at 600 CR refer to ISOCA performed at 600 K at a longer time when a fraction of about 10% atoms form crystalline nuclei as discussed in our previous work.17
MM regions have the characteristic property of containing a fraction of homopolar Ge−Ge bonds higher than the average and much higher than in the MI regions. We assume that a Ge−Ge bond is formed when its length is below 3.0 Å (see refs 17 and 19 for partial pair correlation functions at different temperatures). Variations of the bonding cutoff in the range 2.8−3.2 Å provide very similar results. The fraction of homopolar Ge−Ge bonds in the MM regions increases upon cooling in spite of the fact that the total number of Ge−Ge bonds in the whole of the system decreases (see Table 1). We have found that most of Ge−Ge bonds cluster in chains (depicted in purple in Figure 3b), which are particularly
temperature (see Table 1). We underline that our results do not change significantly with respect to the criterion by which we choose MI and MM atoms, consistently to what has been reported in previous works.37 The MI (or MM) atoms are defined to belong to the same cluster once they are separated by a distance shorter than 3.6 Å. The number of MI and MM clusters at different temperatures and their average and maximum size (number of atoms) are given in Figure 2b. Large clusters up to ∼200 (MI) and ∼100 (MM) atoms are found at 500 K as depicted in Figure 3a. We have not found particular features that characterize the structure of the supercooled liquid in the MI regions. On the contrary, the 13624
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Figure 3. (a) Initial configuration of the ISOCA at 500 K showing most immobile (blue) and most mobile (red) clusters. The Ge−Ge homopolar chains in most mobile regions (purple) are highlighted in panel b. (c) Distribution of the length of Ge−Ge chains in the MM regions reported as stacked histograms for different temperatures. (d) Probability distribution of the number of nearest neighbors Ge atoms within the chains, averaged over all the chains at all temperatures.
bound to the Ge−Ge chains belong to the MM regions themselves since they are more mobile than the average of Te atoms. However, since the dynamical propensity (eq 3) is normalized by the MSD of each atomic species, Te atoms in MM regions are much less mobile than Ge atoms as the average diffusion coefficient of Te is a factor 3.5 lower than the diffusion coefficient of Ge at 500 K.16 The fraction of Ge and Te atoms in MI and MM domains is also only slightly dependent on temperature (see Table S1). From the data in Table 1, we can conclude that atoms around Ge−Ge chains keep moving faster than the average and their contribution to the regions of MM atoms becomes more and more important upon cooling. In other words, by decreasing temperatures atoms slow down, but those around Ge−Ge chains slow down at a lower pace and progressively enrich the clusters of fast moving atoms. The dynamical heterogeneities in GeTe thus originate from structural heterogeneities in the form of chains of Ge−Ge homopolar bonds. A similar correlation between structural heterogeneities and fragility has been discussed recently for GeAsSe compounds.40 The fragility of these alloys is minimal along the pseudobinary tie-line GeSe2−As 2Se3. Deviations in stoichiometry from the pseudobinary line are proposed to lead to an increasing fraction of homopolar bonds responsible for the increased fragility of the supercooled liquid. These features suggest that in chalcogenide liquids very specific structural heterogeneities, such as the Ge−Ge homopolar bond chains detected here, could originate the DH underlying the breakdown of the SER relation and hence the fragile behavior of this class of systems.
numerous in the MM regions at low temperatures. We remark that the chains of Ge−Ge bonds are not isolated from the rest of the liquid network. In fact, the coordination number of Ge atoms involved in Ge−Ge mobile chains is about four (see Figure S3a), and Te atoms bonded to the chains belong to the MM domains as well. The length of the chains of Ge−Ge bonds in the MM regions slightly increases by decreasing temperature as shown in Figure 3c and in Table 1, in spite of the fact that both the number of chains and their length decreases by decreasing temperature in the model as a whole (Figure S4). Most of the chains are linear, self-loops and branched chains are rare (see Figure S3b); in fact, most Ge atoms forming homopolar bonds have only two nearest neighbor Ge atoms (inside the chains) or one nearest neighbor Ge atom (at chains end) as shown by the distribution reported in Figure 3d. Note that Ge−Ge chains are by no means an artifact of the NN potential. In fact, similarly long chains are found in fully DFT simulations of the supercooled liquid, albeit with a small 216-atom cell, as shown in Figure S4 of Supporting Information. We remark that the data in Table 1 refer to the properties of the MI and MM regions only (but for the total number of homopolar bonds). By decreasing temperature, the clustering of fast and slow moving atoms in spatially separated regions increases, but the average mobility still changes smoothly with temperature as it follows an Arrhenius behavior down to 500 K.16 The enrichment of the MM regions with Ge−Ge chains is smooth as well. Sharper changes in the mobility is expected at lower temperatures, closer to Tg where, however, the relaxation times of the liquid might become longer than the incubation time for crystallization and anyway longer that our affordable simulation time. The Te atoms 13625
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Figure 4. (a) Time evolution of the cage-jump indicator J(t) for an atom that undergoes a cage jump (jump, red curve) and an atom that keeps on rattling within its cage (caged, blue curve). (b) Example of string motion of mobile atoms. The arrows depict atomic displacements within t*. Data refer to simulations at 500 K over the characteristic time t* = 55 ps.
clear indication of whether an atom moved significantly or not from SA to SB. The function J(t) develops a peak when the atom escapes its cage because the distance dAB between the average position in the two subtrajectories gets its maximum value. We define a cage-jump as the event for which dAB ≥ 3.0 Å, i.e. the average bond length. Results are rather insensitive to changes of this threshold in a reasonable range. A plot of the evolution of J(t) for representative atoms is given in Figure 4a. We remark that the indicator J(t) is specifically sensitive to jump-like motion, whereas diffusive motions with a very short residence time in the cages do not give rise to specific features in J(t). We can therefore identify the fraction of atoms in the MM regions that undergo a cage-jump motion by simply counting the atoms jumping in the time t* as revealed by J(t). Results are reported in Table 2, where we compare the fraction of jumping atoms
The isoconfigurational analysis at 600 K has been repeated at longer times when the system starts to crystallize. The map of the dynamical propensity clearly shows regions of immobile atoms corresponding to the crystalline nuclei, but also the part of the sample still in a supercooled liquid state keeps displaying dynamical heterogeneities with regions of MM atoms around Ge−Ge homopolar chains (see Figure S5 and Table 1). We note that the Ge−Ge chains are dynamical objects. As we shall see in the next section, they do break and reform on a time scale roughly equal to t*. These findings suggest that the presence of MM domains and specifically of the Ge−Ge homopolar chains allows the supercooled liquid to remain significantly mobile in spatially confined regions around the crystalline nuclei. Such enhanced mobility boosts the kinetic prefactor that applies both to the nucleation rate and the crystal growth velocity.17 Diffusion Mechanism of Most Mobile Atoms at Low Temperatures. The incoherent scattering function in Figure 1 suggests the emergence of a β-relaxation regime at low temperature which is customarily ascribed to rattling motion of atoms in the cage formed by the atoms in the first coordination shell. Longer time α-relaxation processes are related to jumps of atoms outside the cage on the time scale t* corresponding to the maximum of the susceptibility χ4 (cf. Figure 1 and Figure 2).38,39 To detect the occurrence of cage jumps we compute the function J(t) introduced in ref 41, which is roughly constant in time when the atom rattles within the boundaries of its cage, but it develops a peak when the atom escapes its cage. The function is built by dividing the trajectory in two subtrajectories and it is defined by
Table 2. Fraction of Atoms That Undergo a Cage-Jump within t*
ζ(t ) =
2 (t * − t )t t*
all [%]
MI [%]
MM [%]
Ge−Ge chains in MM [%]
500 600
14.4 21.3
12.6 19.7
24.2 30.1
37.1 44.5
with respect to different subregions of the systems, identified by means of the dynamical propensity obtained by the ISOCA analysis. The results have been averaged over all the trajectories of the isoconfigurational ensemble. It turns out that atoms belonging to MM regions display a much more pronounced tendency to experience a cage-jump within t* than all the other atoms, which means that intercage jumps significantly influences the atomic motion in the MM regions. Strikingly, this finding is even more evident for atoms within the Ge−Ge mobile chains. These results are non trivial, as they have been obtained by comparing single atom dynamics events with subsets of atoms defined by the dynamical propensity obtained by the ISOCA. Further insight into the dynamical properties of the Ge−Ge bonds involved in the homopolar chains within the MM domains can be obtained by looking at both the continuous Cc(t) and the intermittent Ci(t) bond time correlation functions42−44 at different temperatures, reported in Figure S6 of the Supporting Information. The temporal decay of Cc(t) allows an estimate of the mean bond lifetime τcb, while the decay of Ci(t) gives a measure of the time scale of local structural relaxation τib. Both quantities are described in detail in the Supporting Information and reported in Table S2. It turns out
J(t ) = ζ(t ) ⟨dA 2(tB)⟩tB∈ SB ⟨dB2(tA)⟩tA ∈ SA with
T [K]
(5)
where t* is the length of the trajectory over which the analysis is performed, which corresponds to the maximum of χ4. SA = {0;t} and SB = {t;t*} are the two subtrajectories before and after time t and dj(tk) is the distance between the atom position at time tk in the subset Sk and the average position along the trajectory in the subset Sj. The average over time tk is taken over the subset Sk, and the prefactor ζ(t) removes the effect of the scarce sampling as J(t) approaches the boundaries of the interval [0,t*]. Thus, J(t) keeps track of the spatial separation between the average atomic position within the two subtrajectories SA and SB as a function of time, providing a 13626
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that τcb does not depend dramatically on temperature, τcb being equal to 0.26 and 1.04 ps at 1000 and 500 K respectively. Larger variations with temperature are instead found for τib whose values are remarkably close t* (cf. Figure 1b) . This finding suggests that the time scale of structural relaxation is indeed related to the change of the local atomic environments due to bonds breaking in a cage-jump fashion. Ge−Ge bonds within mobile homopolar chains rattle on the short time scale of τcb, and eventually break via cage-jumps on the much longer time scale given by τib ∼ t*. Thus, the fact that a consistent fraction of MM atoms undergoes a cage-jump within the characteristic time t* clearly indicates that there is a strong connection between structure and dynamics in the supercooled liquid. In particular, the finding that the Ge−Ge homopolar chains tend to move via cage-jumps over the time scale typical of structural relaxation supports the idea that Ge−Ge homopolar chains have a specific role in the heterogeneous dynamics of this material. Another peculiar dynamical feature of a number of supercooled liquids is string-like motion, which has been reported in a variety of model systems.45−47 Apart from occasional events involving Ge atoms in homopolar chains (Figure 4b), we could seldom observe string-like motion on the time scale given by the characteristic time t*. Nevertheless, even our large simulations can still suffer from finite size effect. In particular, we can not rule out that string-like motion or other cooperative effects might contribute to the dynamics of the system, but are not found in our simulation cell due to insufficient statistics. However, dynamical heterogeneities at the temperatures investigated here emerge even in the lack of string-like motions.
cooperative motion via string-like atomic displacements have also been observed. The emerging picture is that dynamical heterogeneity originates from the chemical specificity of GeTe, which leads to structural heterogeneities in the form of the Ge− Ge chains that preserve and enhance the atomic mobility even close to the glass transition temperature. In summary, we provide an atomistic, structural interpretation of the breakdown of the SER in the supercooled liquid phase of GeTe, which is in turn responsible for the large atomic mobility of this system at low temperatures that boosts its crystallization kinetics. This picture, with the due differences for the specific structural features, may be relevant to other phase change alloys.
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ASSOCIATED CONTENT
S Supporting Information *
Results of dynamical heterogeneity in the supercooled liquid state of the phase change material GeTe. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*(G.C.S.) E-mail: [email protected]. Present Address ∥
Institute for Soft Matter Synthesis and Metrology and Department of Physics, Georgetown University, Washington, DC, 20057
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thankfully acknowledge the computational resources provided by Cineca (Casalecchio di Reno, Italy) through the Program EU-FP7 Prace and the LISA Initiative and by CSCS (Switzerland) under Project No. s477. M.B. acknowledges funding from the European Union Seventh Framework Programme FP7/2007-2013 under Grant Agreement No. 310339. J.C. and E.D.G. are supported by the Swiss National Science Foundation (Grant No. PP00P2_126483/1).
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CONCLUSIONS In this work, we have provided evidence, via large scale molecular dynamics simulations, of dynamical heterogeneities in supercooled liquid GeTe, underlying the breakdown of the SER between viscosity and diffusivity. This behavior is responsible for the persistence of high atomic mobility in the supercooled liquid phase down to very low temperatures. Such a feature is indeed the ultimate source of the high crystallization speed exploited in non volatile memory devices based on phase change materials. Evidences of the presence of dynamical heterogeneities come from the analysis of the intermediate scattering function and its susceptibility. Isoconfigurational analysis provided a vivid picture of the spatially separated domains of fast and slow moving atoms growing in size upon supercooling. We have shown that the dynamical heterogeneities are associated with a structural fingerprint, as the fast moving atoms cluster in regions with a significant concentration of chains of homopolar Ge−Ge bonds. The evidence that wrong (homopolar) bonds are the source of dynamical heterogeneities in GeTe is in line with recent findings that show how the fragility of GeAsSe alloys increases by moving away from the pseudobinary GeSe2−As2Se3 tie-line.40 Wrong bonds are in fact expected to increase by moving off the pseudobinary line leading to an increase of the structural/ dynamical heterogeneities responsible for a higher fragility. In supercooled liquid GeTe, at low temperatures the atomic motion does not follow a continuous diffusive path, as a substantial fraction of atoms rattles in the cage formed by their neighbors and undergoes discontinuous intercage jumps. It turns out that cage jumps are more frequent in the case of atoms involved in Ge−Ge mobile chains. Rare examples of
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