Characterization of supercooled liquid Ge 2 Sb 2 Te 5 and its crystallization by ultrafast-heating calorimetry

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1 Characterization of supercooled liquid Ge 2 Sb 2 Te 5 and its crystallization by ultrafast-heating calorimetry J. ORAVA, 1,2 A. L. GREER, 1* B. GHOLIPOUR, 3 D. W. HEWAK, 3 AND C. E. SMITH 4 1 Department of Materials Science and Metallurgy, University of Cambridge, Pembroke St., Cambridge CB2 3QZ, UK; * alg13@cam.ac.uk 2 Centre for Material Science, University of Pardubice, Studentska 95, Pardubice, Czech Republic 3 Optoelectronics Research Centre, University of Southampton, Southampton SO17 1BJ, UK 4 Mettler-Toledo Ltd., Beaumont Leys, Leicester LE4 1AW, UK 1. Introduction The work is concerned with the kinetics in supercooled liquid Ge 2 Sb 2 Te 5 (GST), its characterization over a broad temperature range, and comparison with other glass-forming liquids. We give more information here on the methods used and in particular on checks of their validity. 2. Kissinger method applied to crystallization of glasses We apply the Kissinger method to obtain the temperature dependence of crystal growth velocity. In this section we examine carefully the validity of this approach Influence of nucleation and growth The Kissinger method 1 has been widely used to analyse the crystallization kinetics of glasses. The gradient of a plot of the kind shown in Fig. 2 in the main text gives the activation energy for crystallization. Kelton 2 has pointed out that this activation energy is essentially that for crystal growth. This is the assumption we make in the present work, but this may seem at odds with the well accepted view that the crystallization of GST is nucleation-driven 3. The distinction between nucleation-driven and growth-driven crystallization has implications (in optical recording) for the scaling of complete erasure time with mark size 3, but can be misleading in considering other studies of chalcogenide crystallization. In amorphous thin films (as used in the present work, and in most of the crystallization studies in the literature), there is no crystalline background surrounding a glassy mark, and so the crystallization must always involve both nucleation and growth. Isothermal experiments conducted near to T g often clearly show nucleation and growth, and allow their distinct rates to be determined, but the observed kinetics can be very different from those on continuous heating. In their study of laser-induced crystallization of amorphous GST, Weidenhof et al. 4 showed that after an incubation time is accounted for, the isothermal crystallization proceeds with an Avrami coefficient of 2.5 to 3.0. This suggests growth in 2 to 3 dimensions at 1 NATURE MATERIALS 1

2 constant rate from a fixed number of nuclei 5. Nucleation clearly precedes growth, and this is even more likely to be true when GST crystallizes on continuous heating, because of the different ways in which the nucleation and growth rates vary with temperature. An example can be found in the calculations for GST by Russo et al. based on classical kinetics of nucleation and growth 6. They show that nucleation becomes progressively slower than growth for temperatures greater than about 180 C ( 450 K). Our ultra-fast DSC data are from 450 to 650 K. Because of factors such as these, it is generally considered that on continuous heating to crystallize a glass, nucleation has finished before the main transformation of the volume by growth. This is fundamentally why Kissinger analyses of crystallization kinetics are primarily influenced by the temperature dependence of the growth rate, as analysed for example by Kelton 2. It is still the case that the rate of crystallization is affected by the number (N per unit volume) of nucleation centres. N is most likely to saturate on heating, at a value determined by a number density of heterogeneities, and thereby would be insensitive to the heating rate. In principle, though, N could vary with heating rate. The kinetic coefficient, on which our JMA analysis is based, is proportional to, where N is the (fixed) number of nuclei per unit volume and is the temperature-dependent growth rate (see equation (S2) later). Thus the values of U inferred from K are relatively insensitive to N. The crystallization rate revealed by the Kissinger plot in Fig. 2 ranges over some 6 orders of magnitude, when conventional as well as ultra-fast differential scanning calorimetry (DSC) results are included. If the rate were governed by N, rather than by U, then N would have to change over some 18 orders of magnitude! We conclude that, even though N is not guaranteed to be constant over the range of our study, its variation would introduce only a small error when the activation energy for crystal growth is equated with that for overall crystallization derived on continuous heating. Interestingly, if nucleation is still occurring during the main transformation, there may be only a small effect on the temperature dependence of the crystallization rate. When nucleation overlaps with growth, in contrast with the expression above,. The effective activation energy determined by the Kissinger method is then simply a weighted average of those for nucleation and growth:. As shown in the calculations of Russo et al., 6 at the lower temperatures (~450 K) where some influence of nucleation is most likely, the temperature dependence of the nucleation rate is rather similar to that of growth, so the overall temperature dependence of crystallization rate (given by the weighted average) is not strongly affected Kinetic law for crystallization The rate of crystallization in glasses or glass-forming liquids at constant temperature is taken to follow Johnson-Mehl-Avrami (JMA) kinetics, which can be expressed as 2 NATURE MATERIALS

3 SUPPLEMENTARY INFORMATION, (S1) where X is the transformed fraction at time t, K is a temperature-dependent kinetic coefficient and n is the Avrami exponent. The use of this equation and the origins of K and n have been reviewed, for example, by Christian 5. JMA kinetics (equation (S1)) applies to transformations involving nucleation and growth, and is valid when the centres of growth are randomly dispersed and when the growth rate is constant (i.e. dependent on temperature, not on time or crystal radius 7 ). The latter condition is satisfied when, as for GST itself, the crystallization does not involve any change in composition. Following the discussion in the previous section, we assume that growth occurs on a fixed number of nucleation sites, giving (for 3D growth) an Avrami exponent of in equation (S1) 5. In that case, the rate of crystallization at any instant is dependent on X and on T, but not on thermal history, the condition that must be satisfied for the JMA kinetics to be valid under non-isothermal conditions 7. Assuming spherical growth of crystallites, the kinetic coefficient takes the form:, (S2) where N is the number of nuclei per unit volume and is the temperature-dependent growth rate. In the Kissinger method 1 the temperature at which the reaction rate is maximum (the peak temperature in a DSC trace) is measured as a function of heating rate Φ. As illustrated in Fig. 2, the variation of with Φ can be used to derive Q. Henderson 7 showed that with JMA kinetics, the peaks of crystallization exotherms on heating in DSC correspond to a transformed fraction that is always close to 63%. The constancy of this value validates the use of Kissinger method to determine the the temperature dependence of crystallization rate. Importantly, Henderson also showed that the method does not rely on K having an Arrhenius temperature dependence Validity of the method at high heating rate The Kissinger method works by comparing crystallization rates at different temperatures, characterizing each heating rate by the conditions at the peak of the DSC exotherm. In addition to Henderson s analysis of the method 7, it has also been tested by numerical calculations 2,8, in which linear heating is approximated as a series of short isothermal steps, and JMA kinetics is applied during each step. Such tests 2,8 have taken to be Arrhenius, and examined the range of heating rate found in conventional DSC. It was found that an activation energy used as input for the calculations is accurately recovered from application of the Kissinger method to the shifts in the calculated DSC peak temperatures. NATURE MATERIALS 3

4 Even so, it is worthwhile to test the method further for the case of very high heating rates. In the present work, the fastest heating gives peaks in the temperature range where the crystallization rate as a function of temperature approaches a broad maximum (i.e. a very low effective activation energy). The crystallization exotherms become very broad in this limit, and it reasonable to suspect that there would be problems in characterizing a given heating rate only by the conditions at the peak temperature. We perform numerical calculations analogous to those in refs 2 and 8. The linear heating is approximated as a series of short isothermal steps (in the present work, separated by a temperature jump of 0.05 K), and JMA kinetics (equation (S1), with ) is applied during each step. We take the temperature dependence of the kinetic coefficient K to be markedly non-arrhenius (matching that given by the Cohen and Grest expression for in Section 3, equation (S6)), and we adjust the absolute values of K to obtain simulated DSC peak temperatures similar to those found in experiment. Typical simulated DSC traces are shown in Fig. S1. Figure S1 Simulated DSC peaks for crystallization of amorphous Ge 2 Sb 2 Te 5 (GST) on ultra-fast heating. These differential scanning calorimetry traces are labelled with the heating rate in. From such traces it is possible to test whether the maximum in the DSC traces corresponds to a given fraction transformed. Table S1 shows that at lower Φ, the peak temperatures are always very close to the temperatures at which a volume fraction of 4 4 NATURE MATERIALS

5 SUPPLEMENTARY INFORMATION 0.63 has transformed. This is in agreement with Henderson 7 and helps to understand why the Kissinger method is valid for conventional DSC. At higher Φ, however, Table S1 shows that the values deviate systematically from, suggesting a breakdown in the method. A direct test of the method is to simulate DSC exotherms and to see whether the shift of their peak temperatures with Φ, analysed according to the Kissinger method, reproduces the temperature dependence of U put into the numerical calculations. If the method works perfectly, then the plot of would give a curve whose gradient at any temperature would match the gradient of, i.e. the activation energy derived from the Kissinger plot would match that of the input. By adjusting the absolute magnitude of U, the two curves can be compared (Fig. 2). When the two curves are superposed at low temperatures ( ), it is evident that the gradients do match, thus validating the Kissinger method at the heating rates used in conventional DSC. But at higher heating rates (higher ), there is a clear deviation, the Kissinger plot giving an effective activation energy that is erroneously high. Fortunately, the true temperature dependence of U (and ) can still be obtained from the Kissinger plot, by fitting it using simulations of the kind shown in Fig. S1, as described in the main text. Table S1 Simulations of DSC heating traces for crystallization of GST. At different heating rates Φ, these give the peak temperatures and the temperatures at which the fraction crystallized is The two temperatures diverge noticeably at higher heating rates , , , , , Thermal lag in DSC In any DSC, there is a thermal lag (the sample temperature trails behind that of the sensor). If not corrected for, this gives erroneously high temperature values on DSC traces. The lag is expected to be proportional to heating rate (and to the heat capacity of the sample), and leads to a systematic error in the determination of activation energy by the Kissinger method: in conventional DSC, measured values of activation energy are as much as 20% too low 8. Since this error is significant even at comparatively low heating rates, clearly it is a 5 NATURE MATERIALS 5

6 great concern for ultra-fast DSC. With this background, care is taken in the present work to eliminate such effects. The apparent variation in the melting temperature of a 1 µg indium sample was measured over a wide range of heating rates. The thermal lag influences the onset temperature T on of the melting endotherm, which increases linearly with heating rate. We find that the lag is as great as at. Lags of this magnitude, even if uncorrected, are clearly insignificant compared to the peak shifts of analysed in the Kissinger plot (Fig. 2). The most interesting aspect is that the lag is so small, presumably because of the small sample mass used in ultra-fast DSC (and lack of a sample pan). For the GST samples, the mass is less than for the indium, but the thermal contact is likely to be worse. A particular problem is that the thermal contact with our samples is variable. For that reason, several experimental runs were undertaken at any given heating rate, and as explained in the main text, the data with the smallest lag are used in the subsequent analysis. The spread of the data (range of peak temperatures) can be seen in Fig. 2: it is not wide enough to disrupt the characterization of the essential temperature dependence of the crystallization. It is also of interest to estimate whether high heating rates can lead to significant temperature differences within a DSC sample (i.e. the temperature of the top surface lagging behind that of the bottom surface). Such differences could contribute to thermal lag and to distortion of DSC exotherms. Their significance can be assessed using the dimensionless Biot number:, (S3) where h is the heat-transfer coefficient between the heater surface (DSC plate) and the sample, L is the sample thickness, and κ is the thermal conductivity of the sample material. If, the temperature differences within the sample are negligible compared to the step in temperature between the heater and the sample. Estimation of h is based on experience with metal casting for which h at the metal-mould interface varies widely, depending for example on the extent of the air gap. Typical values 9 of likely represent an upper bound for our case. For our thin films L = 270 nm. For amorphous GST, κ can be as low as (ref. 10). Taking these values, the upper-bound estimate of Bi is to These values, clearly less than 0.1, indicate that the predominant contribution to the lag is poor thermal contact between the DSC and the sample, not poor heat conduction in the sample itself. It is important to check that the range of h suggested above is consistent with the lag in overall sample temperature. To heat a plate of thickness L at a heating rate Φ requires a heat difference between the DSC plate and the sample, ΔT: 6 6 NATURE MATERIALS

7 SUPPLEMENTARY INFORMATION, (S4) where is the heat capacity per unit volume. From the specific heat capacity of GST 11, we estimate that. Taking this value, and the values of L and h quoted above, we find for the maximum that ΔT is in the range 1 to 10 K. This range is comparable with the spread in measured peak temperatures (Fig. 2) and the lags detected in melting the test samples of indium, suggesting that the assumed range of h is reasonable. 3. Cohen and Grest description of the viscosity of GST Our application of the Kissinger method enables us to fit the temperature dependence of the kinetic coefficient for crystal growth (Fig. 2). To implement this fitting we need a functional form for the temperature dependence. The DSC data span the temperature range 410 K to 650 K (0.59 < < 0.93). We need a functional form that not only permits fitting within this range, but also has a firm physical basis making it reasonable to extrapolate outside this range. In particular, we wish to extrapolate to and to. The extrapolation to permits the scaling of the data to match the viscosity value obtained by Akola and Jones 12 for that temperature. Cohen and Grest 13 have applied free-volume theory to analyse the glass transition. Their extended version of the theory considers solid-like and liquid-like cells in the liquid and applies percolation theory. They derive a generalized equation for the viscosity η that shows excellent agreement with experiment, over 12 orders of magnitude in η, for a wide variety of systems:. (S5) In high-temperature liquids, at and above, it is reasonable to assume that. But on lowering the temperature down to, the crystal growth may become faster than would be predicted from this simple inverse scaling with viscosity; indeed the present work provides evidence for decoupling of this kind. Ediger et al. 14 suggested that the decoupling can be represented by, where. Thus we can derive:. (S6) This is expressed in simpler form as equation (1) in the main text, where. Our fit to the data in Fig. 2 gives:,,, with the quality of 7 NATURE MATERIALS 7

8 fit given by. (The value of the fitting parameter A is of no direct significance since absolute values are not determined.) The characteristic temperature is such that the free volume at high temperature is proportional to. Cohen and Grest 13 found that is 10% to 17% higher than the measured glass-transition temperature in the systems they studied. For our fitting of GST, is 11.5% higher than ( ), lying within the expected range. 4. Transposition of crystal-growth data on to the Angell plot Having obtained the temperature dependence of from the Kissinger plot in Fig. 2, we then use this kinetic coefficient to compare supercooled liquid GST with other glassforming liquids over a wide temperature range. This is done by transposing the data onto an Angell plot (Fig. 3). As the abscissa of this plot is an inverse temperature scale normalized with respect to the glass-transition temperature, the transposition requires knowledge of, which for GST is unfortunately not well characterized. Kalb et al. 15 used atomic-force microscopy of annealed, partially crystallized GST to measure the crystal growth velocity in the temperature range 388 K to 418 K. They found an effective activation energy of. As this activation energy is much higher than that for the isoconfigurational viscosity in a slightly lower temperature range ( at 333 K to 373 K) 16, Kalb et al. conclude that their crystal growth measurements are all in the supercooled liquid regime above. While this is likely, the measurements must be very close to, and may not fully reflect the strong temperature dependence expected for what must be a liquid of high fragility. Certainly other measurements of crystal growth rates in apparently the same temperature range give a very different value of activation energy 17. We conclude that must be less than, but close to, 388 K. This is supported by Morales- Sanchez et al. 18 who from a variety of measurements including calorimetry, concluded that for GST. Later work has suggested higher values. Kalb et al. 19 used DSC to study the glass transition and crystallization in pre-annealed samples of GST. The pre-annealing leads to an endothermic peak in DSC scans just before crystallization. From this peak, Kalb et al. suggested that is about 455 K (i.e. 10 K below the crystallization temperature on continuous heating at ). There is a well established procedure (equal-area construction) to extract the from DSC traces of pre-annealed glasses. The DSC trace is idealized as a simple step at between two horizontal lines (constant specific heat) representing the glass and the supercooled liquid. The temperature of the step is selected to give equal areas under the ideal and actual DSC traces. Unfortunately, it is not possible to perform the equal-area construction on the DSC trace for GST in ref. 19 as the endothermic peak overlaps with the onset of crystallization. We can note that is likely to be no higher that the onset of the endothermic peak, 413 K for GST 19. If the endothermic peak is large (as 8 8 NATURE MATERIALS

9 SUPPLEMENTARY INFORMATION would be true for a well-annealed glass), and if the difference in specific heat between the glass and supercooled liquid (i.e. the height of the step) is small, then the true may be at a temperature considerably lower than the onset of the endothermic peak. We conclude that the data in ref. 19 do not permit a close estimate of, and that they would still be consistent with the lower values given in the earlier works 15,18. We take the value, within the range of values in these works. 5. Estimating crystal growth rate from viscosity The Kissinger plot in Fig. 2 gives the temperature dependence of, but not absolute values. As noted briefly in the main text, it would be desirable to use measured values of crystal growth rate to calibrate our calorimetric data, thereby estimating absolute values. There are many studies of crystallization of GST near to 4,6,17,20 25, but unfortunately they show much scatter. Values obtained for the activation energy of crystallization range from 1.6 to 3.2 ev/atom. And values of crystal growth velocity 6,15,23, when extrapolated to permit comparison, vary by as much as five orders of magnitude at a given temperature. Reasons for these disparities include such factors as electron-beam effects in transmission electron microscopy studies 22. In any case, we cannot easily use crystal-growth data from near to obtain the values of over a wide temperature range from our DSC data. Instead we exploit the well accepted 14 close coupling of diffusion, crystal growth and viscous flow in equilibrium liquids (i.e. at the melting temperature and above). We estimate the effective diffusion coefficient D from the viscosity η using the Stokes-Einstein relation:, (S7) where a is an effective atomic diameter or jump distance, and k is Boltzmann s constant. This relation appears to work rather well for high-temperature liquids, but on cooling towards the real diffusivity exceeds the value that would be predicted from the viscosity, by as much as three orders of magnitude for a fragile liquid such as o-terphenyl 26. For growth governed by diffusion processes at the crystal-liquid interface (not by long-range transport of solute or heat), (which can be regarded as the limiting velocity) is given by. (S8) Using the average interatomic spacing from the metastable rocksalt structure of GST determined by Nonaka et al. 27, we estimate. We use equations (S7) and (S8) to obtain the value of at from the viscosity value of 1.1 to obtained by Akola and Jones 12 in their molecular-dynamics simulation of GST at that temperature. 9 NATURE MATERIALS 9

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