Structural relaxation of the amorphous phase

The phenomenon of the structural relaxation (SR) has been observed in the past in different amorphous semiconductors like Si and Ge [50]. The main macroscopical effect observed as consequence of this phenomenon is the temperature activated increase in time of the resistivity of the material. Even phase-change materials are affected by the drift of the resistivity towards higher values. If this phenomenon in standard PCM devices is not a problem, in highly scaled devices it can represent a problem in terms of reliability and shift of the programming resistance. The physics involved in the drift behavior is still debated. In the past years, several theoretical models have been proposed [51]:

- reduction of the conductivity due to the trap decay, in the trap-assisted conduc-tion model;

- repositioning of the Fermi level by generation of the donor/acceptor defect pairs;

- widening of the energy gap between the Fermi level and the mobility edge, caused by mechanical stress release.

The phenomenon has been observed in as-dep [52] and in melt-quenched phase-change materials meaning that probably all the phenomena proposed have a contribution on the final mechanism and are correlated to each other. Once recovered the amorphous phase in a phase-change material coming from the liquid phase (i.e. through what in a final device is called a RESET operation), the number of dangling bonds varies in time and the structure tends to reach a local order of minimum free energy. The metastable state can relax to this equilibrium state by thermal excitation over an energy barrier (EB). The average time τSR needed for the defect relaxation, overcoming the barrier, is assumed to depend on temperature T according to the Arrhenius law [53]

τ_SR =τ₀e^{kB TEB} (1.22)

It is typical that such relaxations exhibit logarithmical slow temporal dependencies of the refraction index, sound velocity, etc., resembling that of the low temperature specific heat and related quantities. The latter are phenomenologically described in terms of random double-well potentials (DWP) [54]. All the models proposed in the literature takes to the same power law empirical dependency of the resistivity on time

ρ=ρ0

t t0

^ν

(1.23) If we don’t assume any electronic phenomenon, in standard DWP theory the drift coefficient ν can be correlated with the intrinsic fluctuation in the material of the energy barrier EB according to

ν = u0D σEB

(1.24)

where u0 is the dilatation coefficient of the volume of the phase-change material, sup-posed to vary linearly with respect to the cumulative probability distribution of the relaxation process, andD is the relative change of the Fermi energy level.

After the application of a RESET pulse, the threshold voltage necessary to switch the amorphized volume, increases in time. The evolution in time of the threshold voltage is correlated, like the resistivity, to the structural relaxation of the amorphous material. This phenomenon is called threshold recovery or drift of the threshold voltage.

Considering always as hypothesis the DWP theory, we can calculate the equation that describes the drift in time of VT H

V_{T H} =V_{T H0}

1 +βln t τ0

(1.25)

HereVT H0 is the value of the threshold voltage at the characteristic timeτ0 (considered the delay time for the activation of the drift phenomenon), β is the drift parameter.

Instead, in the trap decay model [10] the threshold voltage as a function of time can be properly fitted with a power law similar to the resistance drift, based on a linear dependence between the threshold voltage and amorphous resistance

VT H =VT H0+ ∆VT H

t t0

ν

(1.26) A natural consequence of eq. 1.26 is that the coefficient of resistance drift and threshold drift necessarily must have the same value for the drift coefficient ν. Both the models have been supported with experimental results, with no real possibility to prefer one with respect to the other [55]. Recently, the trap decay model has been investigated with a more detailed description of the parameter ν [56]. Combining the equation of the conduction of a semiconductor (eq. 1.19) and the empirical equation of the drift of the resistivity in time (eq. 1.23), the drift coefficient has been put in correlation with the activation energy of the conduction, and then with the energy gap of the amorphous material, supposed to evolve in time, with a law that depends on the reading temperature (TR), the baking temperature (TB) and the baking time (tB) of the experiment considered for the study

EC(TB, tB)∝kB TRν(TB, TR)lntB

t0

(1.27)

The main result of this study, was the finding that the parameter ν measured during the monitoring of the evolution of the resistivity in time, depends on the experimental conditions, demonstrating how the variation in time of the energy gap of the system can be accelerated increasing the baking temperature, and how this variation can ap-pear different, dependently on the reading temperature [57] (temperature at which is performed the measure of the resistivity, that for reasons related to the experimental procedure, can be different from the baking temperature). In general, if the baking and the reading temperature are equal, we find that

ν = ∆EC

∆E_B (1.28)

that relates the drift coefficient linearly to the variation in time of the activation energy of the conduction ∆EC. This relation has been deeply demonstrated also in experi-mental results, for different phase-change alloys, showing a general trend in which low

1.7 The amorphous phase 29

Fig. 1.17. Experimental demonstration of the correlation between the drift coefficient ν and the activation energy of the conduction, for different phase-change materials. The trend observed confirms that low drifting materials are characterized by low activation energy of electronic conduction [52].

drifting materials are characterized by low activation energy of electronic conduction (Fig. 1.17 [52]).

In the model proposed, the variation of the activation energy of the thermal re-laxation ∆E_B, has been put equivalent to the activation energy of the crystalliza-tion (∆EB = Wcr). The real correlation between these two parameters has been demonstrated by means of simulations but still no experimental results relates the SR phenomenon to the crystallization dynamics. The resistance dependence on E_C was explained by the disordered nature of the amorphous chalcogenide phase, where the energy barrier for thermally activated hopping is randomly distributed in the amor-phous volume. As a result, the current is localized in percolation paths with minimum resistance, hence minimum EC. For increasing thickness of the amorphous region, EC

increases due to the decreasing probability of finding a favorable percolation path with a low critical barrier. The model gives an explanation also to the reduction ofν when, thanks to the incoming crystallization, the resistivity of the material decreases [57].

In fact, as seen before, lower resistivity means lower amorphous volume and then a reduced EC (Fig. 1.18).

Still some lacks remain in the comprehension of the drift mechanism, recently cor-related with the change of the distribution of the bond angles in the phase-change material amorphous matrix. Atomistic calculation reported that both octahedral and tetrahedral coordinations are present for Ge atoms in Ge2Sb2Te5, but the tetrahedral coordination is the majority. The coexistence of octahedral and tetrahedral coordina-tions of Ge atoms increases the uncertainty of SR and might play an important role in resistance drift of a PCM material [58, 59]. This led to the conclusion that reducing Ge concentration in Ge-Sb-Te ternary alloys leads to a less drift (as we will analyze

Fig. 1.18. Drift coefficientν measured and annealed at 300 K (room temperature), as a function of the resistance of the material (a) and as a function of the activation energy of the conduction (b) [57].

later in our experimental results).