3.4 Interface shift kinetics - Anomalous (Non-Fickian) diffusion kinetics

It is known from Fick's phenomenological laws that, during annealing of a diffusion couple, the displacement of a plane with constant composition (or an abrupt interface) is proportional to $t^{1/2}$ (Fickian or normal shift kinetics with $t$ equal to time) (2.2). However, as was shown above the validity of Fick's laws is limited on the nanoscale, especially with increasing diffusion asymmetry (2.5 and 3.2). Therefore, we revisited the problem of the interface shift kinetics on the nanoscale.

In order that we could systematically study the interface shift kinetics, we divided the problem into two parts. First we studied the completely miscible ($V = 0$ , where $V$ is a solid solution parameter proportional to the mixing energy of the system) and phase separating ($V > 0$ ) systems. (2.4) In these cases only the shift of one interface is to be studied. Second, we investigated ordering systems, where the kinetics of two interfaces is to be followed since a growing new $AB$ ordered phase connects to both the pure $A$ and $B$ matrixes ($A/AB/B$ ). (2.4)

3.4.1 Completely miscible ($V=0$ ) and phase separating ($V>0$ ) systems

3.4.1.1 Computer simulations and theory

To investigate the interface shift, we recorded sequentially the position of the interface of a diffusion couple [7] (2.6.1). Its logarithm versus the logarithm of the time ( $\log p \propto \log t$ ) was plotted. Fitting a straight line to the data, its slope gave the power of the function describing the shift of the interface (called kinetic exponent and denoted by $k_c$ ). Obviously for parabolic (normal Fickian) interface shift $k_c=0.5$ . Thus if the kinetics is non-Fickian, $k_c \neq 0.5$ or the data do not fit on a straight line on the $\log p \propto \log t$ plot. Both the parameters $m'$ and $V$ (or $V/kT$ ) were changed during the calculations.

Figure 3.4 shows the initial values of $k_c$ versus $V/kT$ for different $m'$ values. It can be seen that $k_c$ is almost constant and, as expected, is very close to $0.5$ for small $m'$ . At the same time, the deviation from the square-root kinetics increases with increasing $m'$ for a fixed value of $V/kT$ .

Figure 3.4: Kinetic exponent versus $V/kT$ for different $m'$ values. For small $m'$ values $k_c$ is almost constant and is very close to $0.5$ . The deviation (anomaly) from the Fickian (normal) kinetics increases with increasing $m'$ for a fixed value of $V/kT$ . We can observe both super- and sub-diffusion regimes.

The deviation from the parabolic law is a real ``nano-effect'' because, after dissolving a certain number of layers (long time or macroscopic limit), the interface shift returns to the parabolic behaviour independently of the input parameters.(see e.g. Fig. 3.4)

Figure 3.5: Change of $k_c$ during dissolution ($m'=7$ , $V/kT=0.09$ ). The more the number of layers dissolved, the closer the value of $k_c$ is to $0.5$ .

We have shown [10], that this transition can be understood from the analysis of the atomic currents in the different parts of the sample. In principle three currents can be distinguished: i) $J_\alpha$ in matrix $A$ , where the diffusion is very slow, ii) $J_I$ across the interface region and iii) $J_\beta$ in matrix $B$ , where the diffusion is fast. (see Figs. 3.6 and 3.7) However, $J_\alpha$ can be neglected, because, practically, there is no diffusion in matrix $A$ . Moreover, at the beginning of the kinetics, when the composition gradient is very large, the flux in the $B$ -rich phase ($\beta$ phase) is larger than across the interface ( $J_I < J_\beta$ ). In this stage $J_I$ controls the diffusion. During the process $J_\beta$ becomes smaller and smaller because the tail of the composition profile in the $\beta$ phase grows more and more resulting in the decrease of the gradient of the composition. Although, $J_I$ also decreases with increasing time/number of layers dissolved, $J_\beta$ decreases much faster. As a result, in a certain moment $J_\beta$ becomes smaller than the $J_I$ , and from this point $J_\beta$ is the rate limiting term. Thus the transition time or thickness must be deduced from the condition $J_I = J_\beta$ which has to be fulfilled at the interface.

Figure 3.6: Scheme of the composition profile and the atomic fluxes in the linear ($t_1$ , solid line) and parabolic ($t_2»t_1$ , dashed line) kinetic regimes. In the linear regime $J_\beta»J_I$ , whereas in the parabolic one $J_\beta«J_I$ . The length of the arrows illustrates the intensity of the fluxes. Note that for large $\vert m\vert$ $J_\alpha$ is practically zero as indicated.

Figure 3.7: Change of the characteristic fluxes during intermixing.

It is worth mentioning that, from the analysis of the currents, an atomistic explanation of the phenomenological interface transfer coefficient ($K$ ) can also be made, which has been missing in the reaction diffusion literature. Following the phenomenological definition of $K$ , $J_I = K(c_e-c)$ ($c$ and $c_e$ denote the current and the equilibrium composition at the interface, respectively) and, comparing this to the $J_I$ in the discrete model [10]

$$K \cong \nu z_v \exp (-Q_K/kT)$$ (3.1)

with $Q_K = E_0 + z_l V + mkT/2$ ($E_0$ is the saddle point energy, $z_l$ the lateral coordination number). $K$ is practically proportional to the jump frequency from the $A$ -rich phase to the $B$ -rich one. In fact the magnitude of the finite value of $J_I \cong K$ gives the permeability of the interface and it is determined by the $m$ and $V/kT$ parameters.

Although it is almost exclusively accepted in the literature that linear growth kinetics are the result of interface reaction control, our results suggest that the linear (non-Fickian) growth of a reaction layer on the nanoscale cannot be automatically interpreted by interface reaction. In the light of the above observations it seems desirable to reformulate our results in the form of the linear-parabolic (or Deal and Groves [17]) law routinely used in the interpretation of experimental data for processes showing a transition between interface reaction and diffusion control (see e.g. Ref. [18]). This was done in Ref. [10] and shown that the linear-parabolic transition or crossover thickness can be estimated from the following expression:

$$X'* \cong c_\beta D_\beta / 2K,$$ (3.2)

where $c_\beta$ and $D_\beta$ are the composition and the diffusion coefficients in the $\beta$ phase.

3.4.1.2 Experiments

The main idea in these experiments is to prepare a thin deposit onto the surface of a substrate, where the deposit is thin enough to be able to detect the signal coming from the substrate, i.e. the deposit is ``transparent'' for the experimental technique used. In this case the thickness of the deposit can be calculated from the ratio of the deposit and substrate signal intensities ( $I_{dep}/I_{sub}$ ). During annealing, if the interface remains abrupt, from the change of $I_{dep}/I_{sub}$ ($I_{dep}$ decreases, whereas $I_{sub}$ increases in time) it is possible to determine how the thickness of the deposit decreases in time. The deposit thickness can easily be converted to interface position; i.e. the $\log p \propto log t$ function can be plotted, for which the slope is just equal to $k_c$ .

Figure 3.8: Scheme of the measurement of interface shift.

Since according to the results of computer simulations different $k_c$ values are expected depending on the strength of the diffusion asymmetry ($m'$ ) and the phase separation tendency ($V$ ), we have investigated different systems: an ideal (Ni/Cu) and a phase-separating (Ni/Au) system. Moreover we also wanted to check if the anomalous interface shift kinetics is independent of the sample structure and diffusion mechanisms, thus we have also performed measurements in the amorphous Si/Ge system, which is also ideal. Then, to be able to compare the results for the same system but with different structure, we have repeated the measurements but in crystalline Si/Ge.

3.4.1.2.1 Ni/Cu(111) system ( $m' \cong 5$ , $V \cong 0$ - ideal).

To investigate the interface shift kinetics in the Ni/Cu system we deposited $3-14$ monolayers Ni onto a Cu(111) single crystal. The samples were heat treated in the temperature range of $600-730$  K and the dissolution process was followed by in-situ AES measurements. From the analysis of the change of the Ni($848$  eV) and Cu($920$  eV) signals, we have determined the interface shift kinetics and we obtained that the interface shift is proportional to the time ( $k_c \cong 1$ ) and not to its square-root as predicted by Fick's theory.

Moreover, we have shown that if the interface shift kinetics is proportional to the time, the speed of the interface shift is constant. The speed can be determined from the interface shift kinetics easily. From the interface shift speed an intrinsic diffusion coefficient of Ni in a Ni$_{78}$Co$_{12}$ alloy can be deduced; which is just equal to the $K$ interface transfer coefficient [6]:

$$D \equiv K =2.9 \exp \left(-\frac{(297 \pm 62) \text{kJ/mol}}{RT} \right)\text{m}^2/\text{s}.$$ (3.3)

3.4.1.2.2 Ni/Au(111) system ( $m' \cong 6$ , $V \cong 0.019$  eV - phase separating).

In order to check the validity of the computer simulation results obtained also for phase separating systems, we repeated the above experiments but with the Ni/Au system (solubility $\approx 3 \%$ at about $T = 680$  K, $V = 0.019$  eV). Here $3$  nm thick Ni was deposited onto the surface of a Au(111) single crystal and the dissolution process was investigated by in-situ XPS measurements in the temperature range of $643-733$  K. We have determined kc from the change of the ratio of the integrated Au-4f and Ni-2p core line intensities. The values of $k_c$ at different temperatures are $k_c \cong 0.6 - 0.7$ . These values show that the kinetics is anomalous in this case.

3.4.1.2.3 Amorphous Si/Ge system ( $m' \cong 3$ , $V = 0$ - ideal)

As was presented above, several theoretical and experimental studies of diffusion kinetics on the nanoscale have shown that the time evolution differs from the classical Fickian law. However, all work so far was based on crystalline samples or models. To reveal if there are kinetic anomalies in amorphous systems, the dissolution kinetics of a thin amorphous Si layer into amorphous Ge was carried out. The interface shift was monitored by AES and XPS techniques. Figure 3.9 shows how the interface shifts in time. As can be seen, two domains can be identified corresponding to two different lines having different slopes. This means that the kinetic exponents are different for the two domains. Initially $k_c$ was found to be $0.7 \pm 0.1$ , whereas in the later domain it was $0.5$ . Therefore not only the anomalous part of the diffusion process could be observed but also the transition back to the classical Fickian behaviour was seen.

Figure 3.9: Interface shift (initial minus apparent thickness of the Si film) in dependence of time on a log-log scale. The non-Fickian first part as well as the transition are clearly visible. (The dashed straight line is fitted to the first anomalous part of the data, whereas the solid one to the last Fickian part.)

3.4.1.2.4 Crystalline Si/Ge(111) system ( $m' \cong 4-5$ , $V = 0$ - ideal).

To be able to compare the results in case of the same system but with different structures, we have repeated the previously discussed measurements, but in crystalline Si/Ge in the temperature range of $740-755$  K. $2-4$  nm Si was deposited on the top of a Ge(111) substrate. We have found a $k_c$ value of $0.85 \pm 0.1$ . The diffusion lengths were around $2$  nm in two experiments, which is more than the largest non-Fickian $\to$ Fickian transition length observed for the amorphous system, but we have not found any change of $k_c$ indicating a larger transition length than in the case of the amorphous system. This is consistent with the larger $m'$ of this system.

3.4.2 Ordering systems ($V<0$ ) - Solid state reaction

3.4.2.1 Computer simulations

Computer simulations have shown that, during phase growth, the growth kinetics (also the shift kinetics of the interface bordering the growing ordered phase) may be anomalous (non-Fickian) due to the diffusion asymmetry and not because of the interface reaction control usually mentioned in solid state reactions [19].

3.4.2.2 Experiments

3.4.2.2.1 Crystalline-Co/amorphous-Si system ($m'$ is uncertain, as in the literature there is a contradiction in the value of $m'$ , $V < 0$ - ordering).

CoSi growth has been measured by XRD in the Co/Si system. [19] To measure the kinetics of the growth of an intermetallic layer during solid state reaction (SSR), we have prepared crystalline-Co/amorphous-Si multilayers. In this system, at lower temperatures (in our experiment: $523$  K and $543$  K) the crystalline CoSi is the only growing phase. At slightly higher temperatures (in our experiment: $573$  K and $593$  K) crystalline Co$_2$Si also starts to grow, thus we could measure its growth kinetics simultaneously with the shrinkage of the Co and CoSi layers. The growth and shrinkage of the layers were measured by XRD. The areas of the corresponding peaks have been plotted (which are proportional to the thickness of the corresponding layers in our multilayer structure) as the function of time. (see Fig. 3.10) We have found in all the cases that the kinetics is anomalous. [19] It is (very probably) due to the diffusion asymmetry and not because of the interface reaction control usually mentioned in solid state reactions. (for details see Ref. [19])

Figure 3.10: Change in peak intensities for the growth of Co$_2$Si at $573$  K ( $\blacksquare$ ) and at $593$  K. ($\square$ )

Since, however, these measurements do not provide direct composition profiles and any chemical information, which are important to study the early stages of any Solid State Reaction, we have performed complementary experiments: combined GIXRF (grazing incidence x-ray fluorescence) and EXAFS (extended x-ray absorption fine structure) experiments in a waveguide structure [14] as well as using the $4$ wire resistance method and TEM [19].