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The potential of chemical bonding to design crystallization and vitrification kinetics

Controlling a state of material between its crystalline and glassy phase has fostered many real-world applications. Nevertheless, design rules for crystallization and vitrification kinetics still lack predictive power. Here, we identify stoichiometry trends for these processes in phase change materials, i.e. along the GeTe-GeSe, GeTe-SnTe, and GeTe-Sb2Te3 pseudo-binary lines employing a pump-probe laser setup and calorimetry. We discover a clear stoichiometry dependence of crystallization speed along a line connecting regions characterized by two fundamental bonding types, metallic and covalent bonding. Increasing covalency slows down crystallization by six orders of magnitude and promotes vitrification. The stoichiometry dependence is correlated with material properties, such as the optical properties of the crystalline phase and a bond indicator, the number of electrons shared between adjacent atoms. A quantum-chemical map explains these trends and provides a blueprint to design crystallization kinetics.


II.
Further information on the chemical bond descriptors

III.
Pros and cons to study crystallization from the as-deposited vs. the melt-quenched state

IV.
Relationship between the size of the Peierls distortion and the optical properties

V.
Optical properties of the amorphous state and its potential relationship to the crystallization speed

VI.
Detailed information on the determination of the onset temperature To

VII. PTE diagrams and measurement of the minimum time for crystallization
The supplementary information summarizes crucial information about the methods employed to derive the data presented and conclusions drawn. At first, all data displayed in figures 1 -4 are summarized in one table. Subsequently, further information is provided regarding the chemical bonding descriptors ES and ET and how they were obtained for the solids investigated here.
Furthermore, pros and cons to study crystallization from the as-deposited vs. the melt-quenched state are discussed, followed by a figure which relates the minimum time for crystallization with the reflectance of the amorphous phase. Next, the determination of the onset temperature for the glass transition is depicted in a figure. Finally, the determination of PTE diagrams and the minimum crystallization time are explained, and all PTE diagrams measured are shown. The crystallization time was measured as explained in detail in section VII. The reflectance of the amorphous and the crystalline phase was measured in the POT using the probe laser. In case crystallization could not be observed in the POT, due to a lack of a discernible reflectance change, measurements of crystallization time were not feasible. These samples are marked with an asterisk in the column for the minimum crystallization time. Measurements of the glass transition temperature were carried out utilizing the FDSC as explained in detail in section VI, values for the melting temperature were taken from published phase diagrams 1

II. Further information on the chemical bond descriptors
The Effective Coordination Number (ECoN) describes a distance-weighted average over all neighbors 2 and is well-suited to characterize trends in the atomic arrangement for systems which undergo local distortions. The value of ECoN for a single atom within a crystal can be calculated according to: where dj is the distance to the j-th neighbor, and dr is an effective distance defining the first coordination shell and the sum extending over all neighbors, .
The ECoN averaged ES can be calculated in an almost identical fashion, where each ES value is weighted by the ECoN contribution corresponding to the respective bonding partner and its distance: Supplementary Fig. 1 Electrons shared. Sketch describing how to determine the number of electrons shared with neighboring atoms.

III. Pros and cons to study crystallization from the as-deposited vs. the melt-quenched state
The minimum time for crystallization differs between melt-quenched and as-deposited glassy states as has been shown by numerous studies [3][4][5] . In the last decade a number of reasons have been identified for this difference. Apparently, subcritical nuclei can be frozen in upon melt-quenching 4 . These nuclei presumably facilitate nucleation and hence speed up the crystallization process. Similar subcritical nuclei and thus a faster crystallization process can be realized upon deposition by other methods like pulsed laser deposition due to the increased kinetic energy of the atoms compared to sputter deposition. At present, the study of the formation of such sub-critical nuclei is beyond the scope of abinitio molecular dynamics simulations, which employ quenching times which are about 4 orders of magnitude shorter than the experimentally accessible quenching times. Hence, it seems that these computations provide a better model of the as-deposited state of sputtered samples. This is one reason why we focus on the as-deposited state here, since we plan to link the findings presented here to quantum-chemical calculations of the as-deposited amorphous state in the near future. GeSe hence has a significantly larger size of the Peierls distortion than GeTe. This has a pronounced impact on the resulting dielectric function. Both increasing ET (above 0) and increasing ES (above 1) leads to a reduction of the maximum in the absorption, i.e. 2(). This can be explained by a growing dissimilarity of the wave function of the valence and conduction band states. The reduction in the maximum of 2() reduces the optical reflectance measured.
The same conclusion is also reached by considering the changes of the band gap with increasing ES.
Increasing the Peierls distortion (and hence ES) increases the band gap and thus shifts the maximum of 2() to higher energies. Since the optical sum rule has to be fulfilled, the height of the maximum will decrease with increasing amplitude of the distortion, reducing the optical reflectance.

V. Optical properties of the amorphous state and its potential relationship to the crystallization speed
As can be seen in Supplementary Fig. 3, there are significant differences in the reflectance of the In this context, the finding reported in Fig. 3b also raises questions. There it was shown that amorphous  Supplementary Fig. 4. Therefore, this onset temperature To is related to the glass transition Tg at that heating rate of 60,000 K/min and should show a similar dependence on stoichiometry. For each material, the onset temperature To is averaged over ten measurements.
In conventional DSC (Diamond DSC, PerkinElmer) the same heating and measuring procedure was applied to the same compounds as in FDSC measurements, employing a heating rate of 40 K/min. The apparent glass transition temperature Tg of the as-deposited material was measured by an onset construction. Please note that this apparent glass transition temperature is not the standard glass transition temperature as it is measured from the as-deposited amorphous phase and not from a standard glass. In the conventional DSC measurements, the glass transition was either visible and the signal became endothermic prior to crystallization, or crystallization was observed before the enthalpy relaxation reached its minimum so that an onset temperature similar to the To obtained from FDSC measurements described above was not obtained.
The reduced onset temperature Trg shows the same trend as the reduced glass transition temperature Trg as Supplementary Fig. 5 demonstrates. Hence, the stoichiometry dependence of the reduced glass transition temperature can be derived from the stoichiometry dependence of the reduced onset temperature.
Supplementary Fig. 4 Excess specific heat. Temperature dependence of the excess specific heat capacity at constant pressure measured at a heating rate of ϑ = 60,000 K/min measured by FDSC. The measurement of a crystalline reference is subtracted in each trace. The temperature is calibrated to the onset temperature of melting for an Indium flake at the same heating rate used for measurements. The maximum endothermal signal of the glass transition is indicated by an arrow. For Se-rich samples the glass transition is well discernible.

VII. PTE diagrams and measurement of the minimum time for crystallization
The minimum time for crystallization  has been determined with the phase change optical tester (POT), a pump-probe laser setup that allows for a continuous measurement of sample reflectance, while the sample is exposed to pump laser pulses. When a rectangular laser pulse hits the sample, the Plotting the contrast in a 2D-map using the applied pulse length and pulse power for the x-and y-axis yields a power-time-effect diagram (PTE-diagram). Since an increased pulse power is correlated with an increased temperature in the sample, the y-axis reflects the temperature within the material. Thus, the pulse length is correlated with the evolution of the material with time. The effective central power density of the laser is given by: Where is the diameter (1/e 2 ) of the laser beam of 2.3 µm. In Supplementary Fig. 6, an exemplary PTE-diagram is shown. Here, an as-deposited amorphous sample has been used, pump pulses of varying length and power have been applied and the reflectance before and after each pulse has been analyzed to obtain the PTE-diagram. For each measurement point, the sample has been moved to illuminate a new region of the sample. PTE-diagrams of each material investigated are shown in Supplementary Fig. 8 and 9. From the PTE diagrams it is evident, that crystallization (as indicated by a positive contrast) does not set in immediately, but only after a certain time has passed. Remarkably, the onset of crystallization is rather constant with respect to pulse power above a certain threshold. As shown in Supplementary Fig. 6, the onset of crystallization involves some stochasticity, i.e. pulse to pulse variation. Consequently, defining a minimum time  it takes the sample to form crystallites should also consider the stochasticity of crystallization. To obtain , the onset of crystallization is investigated in detail. First, based on the PTE-diagrams, suitable pumpparameters (pulse length and pulse power) have been selected to determine  from repeated measurements of the change in sample reflectance upon local heating. Using these parameters, the pulse length interval has been divided into 40 equidistant (on a logarithmic scale) segments and the contrast has been measured 40 times for each segment. For each measurement, the sample has been moved to a new position, so a total of 1,600 positions has been probed. From these experiments two reflectance levels can be identified, the lower level, where no crystallites have formed (Rmin), whereas the upper level corresponds to measurements, where the crystallites completely cover the illuminated area (Rmax). With these two levels, Rmin and Rmax, a threshold value Rthr is calculated as follows: Contrast values above Rthr are counted as crystallization events, whereas contrast values below Rthr are not. Consequently, the probability of crystallization Pcryst can be deduced: where Ntot and Ncryst are the total number of measurements at a given pulse length t and the number of contrast values above Rthr, respectively. The result of plotting the probability of crystallization versus the applied pulse length t for GeSn0.5Te0.5 at a laser power of 70 mW is presented in Supplementary   Fig. 7.
From the probability of nucleation, the minimum time for crystallization is deduced by fitting a Gompertz function to the data and calculating the pulse length where crystallization is observed with a probability of 50%.
Supplementary Fig. 7 Probability of crystallization plotted versus the applied pulse length at a laser power of 70 mW for GeSn0.5Te0.5. The red solid line is a fit to the data (Gompertz function), the black dashed line indicates the pulse length where a probability of crystallization of 50% is observed, termed minimum time for crystallization .