the amorphisation of this material, is the existence of an incubation time of the nucle-ation, independent of the amorphous volume, required for the nucleation process to be stable.
1.5 Crystallization kinetics
The crystallization mechanism impacts many aspects of the life of a final PCM device.
It affects in particular the programming speed, and the stability of the amorphous phase. To describe the crystallization process, we refer to the the Classical Nucle-ation Theory (CNT) developed by Gibbs in 1878. The main hypothesis of this theory are the unchanged composition of the material during the crystallization and the dif-fusion-limited nature of the process. The free energy of the system, consisting of a cluster of crystalline phase evolving in the surrounding amorphous phase, is expressed as the sum of the bulk contributions of the nucleus and the amorphous phase. These bulk terms are integrated by interfacial contributions, and the main one is given by the product of the interfacial area and specific surface energy. Applying the theory to cluster formation, these surface terms result initially in an increase of the charac-teristic thermodynamic potential and the existence of a critical cluster size (crystal nucleation). Only clusters with sizes larger than the critical size are capable to grow up in a deterministic way to macroscopic sizes (crystal growth). The change of the characteristic thermodynamic potential resulting from the formation of a cluster of critical size is commonly denoted as work of critical cluster formation. This quantity reflects basically the thermodynamic or energetic aspects of nucleation.
1.5.1 Crystal nucleation
We can calculate the free energy necessary to build a spherical nucleus with radius r, in an amorphous volume. The change of G due to the formation of a nucleus can be written as:
∆G=σacA−∆GvV (1.1)
where ∆Gv is the difference between the free energies of amorphous and crystal per unit volume of the crystal (i.e., the thermodynamic driving force for crystallization), σac is the specific interface free energy of the crystal-amorphous interface, whileA and V are the external surface and the volume of the nucleus respectively. This equation has a minimum for a critical nucleus size r0, and allows the calculation of the so called thermodynamic barrier of nucleation (W) that leads the crystallization process. W can be defined as follow:
W = ∆G|∂∆G
∂r =0 (1.2)
whereris the spatial variable in a spherical coordinate system. According to the CNT, the steady-state homogeneous volume nucleation rate depends on W according to the equation
Is ∝e−kB TW (1.3)
where kB is the Boltzmann constant. Eq. 1.3 determines the number of supercritical clusters formed per unit time in a unit volume of the system in the steady-state con-dition, and shows how the nucleation is a statistical process with a given probability
distribution function dependent on the temperature. Some time period is needed for the reconstruction of a stable nuclei distribution toward the time independent distribution described in eq. 1.3 at a given temperature T. During this period, the nucleation rate varies and approaches a steady-state value. The time required to establish steady-state nucleation in the system is commonly denoted as the time-lag of the nucleation, or incubation time, that we will take into account later in our simulation tool.
The existence of foreign solid particles, phase boundaries, material interfaces, etc., may favor nucleation. This effect is mainly due to the diminished W, as compared to that of a homogeneous nucleation, owing to a decrease of the effective surface energy contributions to the work of critical cluster formation. This is the main distinguishing feature of heterogeneous nucleation. To estimate the energy barrier reduction, we can multiply W by a parameter Φ that takes into account the surface deformation and varies from zero to one (Whet=WΦ) [27].
1.5.2 Crystal growth
Another important mechanism that takes part to the crystallization is the crystal growth and it is driven by the reorganization of the atoms along the crystal-amorphous interface. An atom has to overcome an energy barrier UB to abandon its amorphous local order and to start to take part to the crystalline order of the neighboring atoms, lowering the energy of the entire system. To describe this phenomenon, we have to consider the contribution of the probability for the atom to rearrange itself in the crys-talline matrix, and the probability to come back to the amorphous bond configuration.
Subtracting the two probabilities we obtain the final growth speed, that can be written as [28]:
vg = dr
dt ∝∆d
νce−kBTUB −νae−UBkB T+∆Gac
(1.4) where r is the average radius of the crystalline volume, ∆d is the average increase of the radius of the crystalline volume, νc is the frequency of the atomic vibration in the crystalline phase, νa is the frequency of the atomic vibration in the amorphous phase and ∆Gac is the difference between the Gibbs energies of the two different phases. If we suppose that the atomic vibration at the interface is equal in both phases (γa), we can rewrite eq. 1.4 as follow:
vg ∝∆d γa
1−e−∆GackB T
(1.5)
1.5.3 Overall Crystallization Kinetics
Crystal nucleation followed by subsequent growth results in overall crystallization. This process can be described by determining the volume fraction of the transformed phase, α(t). The formal theory of overall-crystallization kinetics under isothermal conditions was developed in the late 30’s by Kolmogorov, Johnson and Mehl, and Avrami and is well known today as JMAK theory. According to this theory the volume fraction of the new phase (crystalline) is given by
α(t) = 1−e−gR0tInucl(t′) Rt′tvg(t′′)dt′′
3
dt′ (1.6)
1.5 Crystallization kinetics 17
Fig. 1.11. Schematic drawing representing (a) nucleation-driven crystallization and (b) growth-driven crystallization during the laser erasure of a written amorphous mark in a crystalline background [29].
where g is the shape factor, which is equal to 4π/3 for spherical crystals, Inucl is the nucleation rate andvg is the growth rate. If bothInucl andvg are constant throughout the transformation (steady-state nucleation) eq. 1.6 can be rewritten as
α(t) = 1−e−gInuclv
g t34
4 (1.7)
When the phase-change material presents a crystallization dominated by a really fast growth speed, we can suppose that the number of growing crystals (N0) and the growing crystalline germs at the crystal-amorphous interface do not change with time and eq. 1.6 transforms to
α(t) = 1−e−gN0v3gt3 (1.8) Avrami proposed that generally the following relation should be used to describe the final crystallization kinetic
α(t) = 1−e−Ktn (1.9)
where the parameters K and n can be estimated by fitting the experimental data of α(t), and are correlated respectively with the rate of nucleation and growth and with the “reaction order” of the crystallization process. The Avrami coefficientn, as observed in eq. 1.7 and eq. 1.8, can give an idea about the final contribution of the nucleation rate and of the growth rate, to the crystallization process. In particular, the transition to a diffusion controlled growth rate and the reduction of the nucleation rate can be concurrent and take to the lowering of n, discriminating between a nucleation-driven (higher n) and a growth-driven crystallization (lower n), situations schematically de-scribed in Fig. 1.11. Eq. 1.9 can be considered as the cumulative distribution function (CDF) relative to the probability to have a given percentage of crystalline volume α, after a given time t at a specific temperature [27].
Fig. 1.12. Example of hexagonal crystal structure and fcc-type crystal structure, in crystalline Ge2Sb2Te5[9].