The Kissinger analysis of the crystallization dynamics 30

We already presented the amorphous phase as the phase, in which the minimum of the free energy is found recovering the structural long range order of the crystalline phase.

The temperature is the fundamental parameter to define the stability of this phase and the characteristic time (recrystallization time) necessary to fully recrystallize the ma-terial. The direct study of the crystallization dynamics on full sheet wafers can give a first important sight on the material behavior in temperature. As already presented in the CNT, the nucleation rate and the growth speed depend on the temperature. Both mechanisms are involved when we increase the temperature of the phase-change mate-rial, and they contribute to reduce the resistivity of the material sheet. The convolution of the two phenomena, gives rise to a total crystallization dynamics as function of the temperature (presented in section 1.5.3). We can define then a specific temperature at which the complete recrystallization of the material occurs, under given thermal profile and conditions, and that represents for us the crystallization temperature of the material (T_C)

TC =T|vg⊗Inucl=f ast (1.29)

The crystallization temperature of phase change materials tends to vary considerably as a function of material composition. This is not necessarily the temperature at which crystallization is most likely, but instead is the lowest temperature at which the crystallization process becomes “fast” [13]. While the crystallization temperature by itself does not reveal how “slowly” the programmed amorphous state would be lost for slightly lower or much lower temperatures, it sets a definitive and easily measured upper bound on the final data retention vs. temperature curve for a new phase-change material.

Thermodynamics can be used only to calculate the driving force for transformation but it cannot say how fast a transformation will proceed. The study of how fast process

1.8 The liquid phase 31

Fig. 1.19. Transformation from initial to final state through an activated state of higher free energy.

occurs belongs to kinetics. In Fig. 1.19 is shown the free energy curve of a single atom, which transforms from initially metastable state into a state of lower free energy. If G_I and G_F are the free energies of initial and final states, the driving force for the transformation will be ∆G = GF −GI. However, before the free energy of the atom can decrease fromGI toGF the atom must pass trough a so-called activated state with a free energy ∆G_A above G_I. The energies shown in Fig. 1.19 are average energies associated with large number of atoms. As a result of random thermal motion of the atoms the energy of any particular atom will vary with time and occasionally it may be sufficient for the atom to reach the activated state. This process is known as thermal activation, and the energy barrier ∆GA is called activation energy of the process [60].

To better qualify the crystallization dynamics of a phase-change material we can extract from the full sheet materials analysis, the activation energy of the crystalliza-tion (Wcr) through the Kissinger analysis [61, 62]. Applying to the material a ramp of temperature of known increasing rate (vT), and knowing that the speed of the crys-tallization depends on the crystalline fraction itself (α^′(t) =β(T)(1−α(t))), when the crystallization process reaches its maximum reaction rate (α^′′(t) = 0) at the crystal-lization temperature TC as defined above, we have the relation

ln vT

TC2

∝ − Wcr

kBTC

(1.30) where we have supposed that the speed of the reaction β is temperature activated with the energy barrier Wcr, being the reaction under test, the crystallization process.

1.8 The liquid phase

The physics of liquid chalcogenides has not been further investigated in the last years, even if the liquid phase has a key role in the life of a PCM device. Every transition from the crystalline phase to the amorphous phase involves the reaching of the melting temperature Tm in part of the volume of the phase-change material. It has been found from XRD measurements on Ge₂Sb₂Te₅ samples [63] that liquid and amorphous phase

Fig. 1.20. Temperature dependence of electric resistivity of liquid Sb2Te3(left) and GST (right) [66].

are similar in terms of structure, being the liquid phase considerably more structurally disordered. But this disorder observed in experimental results, can be explained by accounting for the diffraction intensity coming from the distribution of valence electrons among the ionic cores. What is supposed, is that in reality the liquid structure, reflects a remanence of bonds and that the crystalline and the amorphous states are the result of a sort of “freezing of this bonds”. Concerning the local structure a remarkable temperature-dependent behavior in the liquid phase of GeTe is found. At temperatures just above the melting point the structure is described to be driven by a reentrant Peierls distortion, exhibiting short and long bonds, similar to the trigonal crystalline ground state of GeTe. This distortion slowly disappears with higher temperatures accompanying a semiconductor to metal transition [64].

One consequence of the transition to the liquid phase is the change of the shear viscosity of the material [23, 65]. Viscosity is a measure of the response of the liquid to a suddenly applied shear stress and is related to the corresponding relaxation time by the Maxwell formula

η=Melτstress (1.31)

whereMelis the high-frequency elastic shear modulus andτstressis the average response time of the system to the applied stress. The cooling of the melted volume provides an exponential increase of the viscosity, and the atomic rearrangement to provide a long range order typical of the crystalline phase becomes more and more difficult decreasing the temperature. It means that a critical cooling rate exists (Rc), that must be exceeded to permit the crystallization

Rc = γkBTm2

Vmη (1.32)

where Vm is the molar volume and γ is a constant. This empirical relation embodies the fact that the measured Rc decreases as the viscosity of the melt increases and its specific form is roughly in accord with experiment. The notion of a critical cooling rate implies that it should be possible in principle to reach the recrystallization, from any liquid, but the speed would change dependently on the phase-change material used.

Only in recent years some resistivity measurements have been performed on liquid Ge2Sb2Te5 and Sb2Te3 (used as phase-change material precursor for GST deposition) [66]. In these studies was found a negative temperature dependence in the electric resistivity of both materials, arguing that they are likely to be semiconductors in their