Kinetics of island growth in the framework of “planar diffusion zones” and “3D nucleation and growth” models for electrodeposition

Abstract

In the electrochemical deposition of thin films, the measurement of the current-time curve does not allow for a direct determination of the nucleus growth law, electrode surface coverage, and mean film thickness. In this work, we present a theoretical approach suitable to gain insight into these quantities from the knowledge of nucleation density, solution parameters, and current-time behavior. The model applies to both isotropic and anisotropic growth rates of nuclei and a study on the effect of nucleus shape and aspect ratio on the kinetics is presented. Computer simulations and experimental results from literature are also discussed in the framework of the present approach.

Appendix

Let Q be a point at height h from the electrode surface and O its projection on the electrode surface. The area of the capture zone of point Q for nuclei born at time t^′ < t is equal to πr(t, t^′)² where $r\left(t,{t}^{\prime}\right)\equiv \overline{ON}$ and the point N, on the electrode surface, is such that a nucleus born at N will reach point Q at time t (Fig. 10). Since the points on the nucleus surface satisfy the equation, $\frac{x^2+{y}^2}{A^2\left(t-{t}^{\prime}\right)}+\frac{z^2}{B^2\left(t-{t}^{\prime}\right)}=1$, we get

$$ \frac{r{\left(t,{t}^{\prime}\right)}^2}{A^2\left(t-{t}^{\prime}\right)}+\frac{h^2}{B^2\left(t-{t}^{\prime}\right)}=1, $$

(21)

namely

$$ r{\left(t,{t}^{\prime}\right)}^2={A}^2\left(t-{t}^{\prime}\right)-\frac{h^2}{\alpha^2}, $$

(22)

where $\alpha =\frac{B}{A}$. Since r(t, t^′)² > 0 and ${t}^{\prime }<{t}_{\mathrm{mx}}^{\prime }=t-\frac{h^2}{A^2{\alpha}^2}=t-\frac{h^2}{B^2}$, ξ_c(h, t) is eventually computed by summing over the population of nuclei,

$$ {\xi}_{\mathrm{c}}\left(h,t\right)=H\left({t}_{\mathrm{mx}}^{\hbox{'}}\right)\pi {\int}_0^{t_{\mathrm{mx}}^{\hbox{'}}}r{\left(t,{t}^{\prime}\right)}^2\overset{\cdot }{N}\left({t}^{\prime}\right)\mathrm{d}{t}^{\prime}\kern3em =H\left({t}_{\mathrm{mx}}^{\hbox{'}}\right)\pi {\int}_0^{t-\frac{h^2}{B^2}}\left({A}^2\left(t-{t}^{\prime}\right)-\frac{h^2}{\alpha^2}\right)\overset{\cdot }{N}\left({t}^{\prime}\right)\mathrm{d}{t}^{\prime }. $$

(23)

Fig. 10

Illustration of the “capture zone” method for ellipsoidal nucleus. $\overline{ON}$ is the radius of the capture zone for point Q. Nuclei located at distance $d>\overline{ON}$ are unable to transform Q up to time t. In the framework of the two-rate approach of [7], the capture zone (red disk) is located at height h and is related to the radial growth rate of the nuclei, p_r

For site-saturation nucleation Eq. 23 gives Eq. 14:

$$ {\xi}_{\mathrm{c}}\left(h,t\right)=\pi {N}_0{A}^2t\left(1-\frac{h^2}{B^2t}\right)H\left(1-\frac{h^2}{B^2t}\right). $$

(24)

In what follows, we illustrate the two-rate model of [7]. In this method, growth rates p_r and p_h are defined for “on-plane” and “normal“directions, respectively. These quantities provide the time dependence of the coordinates, r and h, of a point lying on the nucleus surface at time t:

$$ r\left(t,w\right)={\int}_w^t{p}_{\mathrm{r}}\left(\xi \right)\mathrm{d}\xi $$

(25)

$$ h\left({t}^{\prime },w\right)={\int}_{t^{\prime}}^w{p}_{\mathrm{h}}\left(\xi \right)\mathrm{d}\xi . $$

(26)

In these equations, t^′ is the birth time of the nucleus, and w > t^′ is the time of arrival of the nucleus at height h, i.e., the semi-axis of the ellipsoid at time w equals h: B²(w − t^′) = h². Equations 25 and 26 provide h = h(t^′, w) that can be inverted to get w = w(t^′, h). Next, by considering the 2D phase transition on the layer at height h, we obtain (Fig. 10): ${\xi}_{\mathrm{c}}\left(h,t\right)=H\left({t}_{\mathrm{mx}}^{\prime}\right)\pi {\int}_0^{t_{\mathrm{mx}}^{\prime }}r{\left(t,w\left({t}^{\prime },h\right)\right)}^2\dot{N}\left({t}^{\prime}\right)\mathrm{d}{t}^{\prime }$ with $w\left({t}_{\mathrm{mx}}^{\prime },h\right)=t$, which becomes, using Eq. 25,

$$ {\xi}_{\mathrm{c}}\left(h,t\right)=H\left({t}_{\mathrm{mx}}^{\prime}\right)\pi {\int}_0^{t_{\mathrm{mx}}^{\prime }}\dot{N}\left({t}^{\prime}\right){\left[{\int}_{w\left({t}^{\prime },h\right)}^t{p}_{\mathrm{r}}\left(\xi \right) d\xi \right]}^2\mathrm{d}{t}^{\prime }, $$

(27)

that is the expression derived in [6]. Next, we apply Eq. 27 to the ellipsoidal and cone-shaped nuclei in diffusional growth. To this purpose, the p_r and p_h functions have to be determined.

• Ellipsoidal nuclei. In this case, p_r and p_h (Eqs. 25 and 26) should satisfy Eq. 21. It is possible to show that ${p}_{\mathrm{r}}(t)=\frac{A}{2\sqrt{t-w}}$ and ${p}_{\mathrm{h}}(t)=\frac{B}{2\sqrt{t-{t}^{\prime }}}$ are the appropriate functions. Use of these rates in Eq. 27 provides

$$ {\xi}_{\mathrm{c}}\left(h,t\right)=H\left(t-\frac{h^2}{B^2}\right)\pi {\int}_0^{t-\frac{h^2}{B^2}}\dot{N}\left({t}^{\prime}\right){\left[{\int}_{w\left({t}^{\prime },h\right)}^t\frac{A}{2\sqrt{\xi -w}}\mathrm{d}\xi \right]}^2\mathrm{d}{t}^{\prime }. $$

(28)

Moreover, Eq. 26 gives $h\left({t}^{\prime },w\right)=B\sqrt{w-{t}^{\prime }}$ which implies $w\left({t}^{\prime },h\right)=\frac{h^2}{B^2}+{t}^{\prime }$. Performing the integral in Eq. 28, we eventually obtain

$$ {\xi}_{\mathrm{c}}\left(h,t\right)=H\left(t-\frac{h^2}{B^2}\right)\pi {\int}_0^{t-\frac{h^2}{B^2}}\dot{N}\left({t}^{\prime}\right){\left[A\sqrt{t-\frac{h^2}{B^2}-{t}^{\prime }}\right]}^2\mathrm{d}{t}^{\prime }. $$

(29)

For site-saturation nucleation, Eq. 29 gives

$$ {\xi}_{\mathrm{c}}\left(h,t\right)=H\left(t-\frac{h^2}{B^2}\right)\pi {N}_0{\int}_0^{t-\frac{h^2}{B^2}}{\left[A\sqrt{t-\frac{h^2}{B^2}-{t}^{\prime }}\right]}^2\delta \left({t}^{\prime}\right)\mathrm{d}{t}^{\prime }, $$

(30)

that leads to Eq. 24.

Cone-shaped nuclei. In the case of right circular cones ${p}_{\mathrm{r}}=\frac{\mu_1}{\sqrt{t}}$, ${p}_{\mathrm{h}}=\frac{\mu_2}{\sqrt{t}}$ (with constant μ_i) and $w\left({t}^{\prime },h\right)={\left(\frac{h}{2{\mu}_2}+\sqrt{t^{\prime }}\right)}^2$. For site saturation nucleation, Eq. 27 becomes (${t}_{\mathrm{mx}}^{\prime }=\sqrt{t}-\frac{h}{2{\mu}_2}$)

$$ {\xi}_{\mathrm{c}}\left(h,t\right)=H\left({t}_{\mathrm{mx}}^{\prime}\right)\pi {N}_0{\int}_0^{t_{\mathrm{mx}}^{\prime }}\delta \left({t}^{\prime}\right){\left[{\int}_{{\left(\frac{h}{2{\mu}_2}+\sqrt{t^{\prime }}\right)}^2}^t\frac{\mu_1}{\sqrt{\xi }}\ \mathrm{d}\xi \right]}^2\mathrm{d}{t}^{\prime }=4\pi {N}_0{\mu}_1^2\kern0.5em {\left[\sqrt{t}-\frac{h}{2{\mu}_2}\right]}^2 $$

(31)

The volume of the deposit is equal to

$$ V(t)={\int}_0^{h_{\mathrm{mx}}}\left(1-{e}^{-{\xi}_{\mathrm{c}}\left(h,t\right)}\right)\ \mathrm{d}h=2{\mu}_2\sqrt{t}{\int}_0^1\left(1-{e}^{-4\pi {N}_0{\mu}_1^2t{\xi}^2}\right)\ \mathrm{d}\xi . $$

(32)

The current density is obtained from the time derivative of Eq. 32, $J(t)= zF\frac{\rho }{M}\frac{\mathrm{d}V(t)}{\mathrm{d}t}$, as

$$ J(t)=\frac{zF\rho}{M}\frac{\mu_2}{\sqrt{t}}\left(1-{e}^{-4\pi {N}_0{\mu}_1^2t}\right) $$

(33)

that is Eq. 19.