**10. Average growth rate for the columns with a height of 1000 μm**

Figure 14 shows the dependence of the average growth rate (in the right ordinate) on the voltages employed to grow a micrometer Ni column, 1000 ± 10 μm in height. The average growth rate was calculated by dividing the height of the columns (i.e. 1000 ± 10 μm) by the growth duration. The time taken by the step motor should be deducted from the total duration of the process. From Fig. 14, the growth rate at 3.2 V is 0.114 μm s-1 and it increases from 0.114 ± 0.004 to 1.76 ± 0.06 μm s-1 with increasing the voltages from 3.2 to 4.6 V. The standard deviation increases with an increase of voltages. Voltages less than 3.2 V or higher than 4.6 V were ignored because of impractical tiny rate in the former and unsatisfactory appearance for the deposits obtained in the latter. In Fig. 14, the average current responsible for LECD was also measured and plotted (in the left ordinate) against the voltages. It increases from 0.225 ± 0.021 to 1.881 ± 0.046 mA with increasing the voltage from 3.2 to 4.6 V. The current is almost nine fold for the MAGE conducted at 4.6 V in comparison to that at 3.2 V. With respect to error bars in Fig. 14, the standard deviation increases with voltages. The growth rate estimated from the data of average current is consistent with that evaluated from column height divided by the electroplating duration.

Fig. 14. A plot of the average current and average growth rate for the columns against the electrical bias employed in the MAGE

#### **11. Concentration of nickel ions in the location proceeding LECD**

As shown in Fig. 14, the stabilized local potential was in the range from -550 to -635 mV for the intermittent MAGE conducted at voltages ranging from 3.2 to 4.6V (vs. SCE). The

Mass Transfer Within the Location Where Micro Electroplating Takes Place 225

unchanged except the concentration of nickel ions. The potential was measured using the same reference microelectrode depicted in Fig. 12. The relationship between the potential measured and the concentration of nickel ions prepared was plotted in Fig. 13. The curve representing the dependence of potential on the concentration of nickel ions obeyed equation (11-2) is in good agreement with that based on experimental measurements.

Mass transport of nickel ions in the electroplating process is theoretically governed by the Nernst-Planck equation [13]. According to the Nernst-Planck equation, the flux (in mol s-l m-2)

> *i i i i ii i Z F J D C DC C RT* =− ∇ − ∇ +

potential gradient, *Zi* and *Ci* are the charge (dimensionless) and concentration (in mol m-3) of species *i*, respectively, and *v* stands for the velocity (in m s-1) of the solution flow under stirring. In equation (12-1) the flux is expressed in detail with three terms on the right-hand side to describe the contribution of diffusion, migration, and convection, respectively. The diffusion coefficient (*Di*) of nickel ions in the solution is 8.157×10-10 m2 s-1 [14]. The concentration of nickel ions in the bulk solution of watts bath is 1.445 ×103 mol m-3. The convection term in Equation (12-1) could be ignored in the LECD process without stirring. In an attempt to calculate the flux of nickel ions transported in the LECD process, the geometric transport in a specific electric field should be considered. As soon as the voltages are applied, nickel ions nearby the electrodes migrate to the cathode surface to discharge and are consumed. Reduction of nickel ions into metallic nickel within the local region leads to depletion of the nickel ions. This depletion causes a concentration gradient as compared this location to the around surroundings. The gradient offers a driving force for diffusion of nickel ions from the bulk solution to the depletion zone. Neglecting the initial stage in LECD, the micrometer columns are steadily deposited in an egg-head on their top [15]. We presume the nickel ions migrate into a boundary of semi-sphere rather than egg-head for simplifying the calculation. The semi-spherical boundary responsible for migration is illustrated in Fig. 16a. The nickel ions discharge and are consumed in the electric field surrounded with a cone. The boundary of the cone responsible for further supply of nickel ions by diffusion is illustrated in Fig. 16b. In Fig. 16, r is the average radius (in m) of the microcolumn, g marks the gap (also in m) between the microanode and the top of the column deposited. The area for the semi-sphere (*Amig*, in m2) and for the cone (*Adiff,* in m2)

> <sup>2</sup> *A r mig* = 2π

<sup>6</sup> 125 10 / 2 2 ( )( ) <sup>2</sup> *dif <sup>r</sup> A rg*

The figure 125 x 10-6 in equation (12-3) is the diameter of the tip-end disk of the microanode. The extent of r can be evaluated through examining the columns with SEM, and g is

<sup>−</sup> × +

= + (12-3)

π

replaced by 10-5 m in this work because of its initial setting.

φ

ν

(12-1)

(12-2)

φis the

for the specific ions transported and to be deposited (assigned as *Ji*) is represented as

Where *Di* is the diffusion coefficient (in m2 s-1), ∇*Ci* is the concentration gradient, ∇

can be estimated in the following:

**12. Supply of nickel ions from bulk solution to the LECD location** 

concentration of nickel ions at the local site taking place LECD could be estimated by the Nernst equation as shown in the following.

$$E = E^0 + \frac{RT}{\nu\_e F} \ln \left\{ c\_{ox} \;/ \left. c\_{Red} \right\} \right. \tag{11.1}$$

Where *E0* is the standard potential of the electrode, *R* is gas constant, *T* is absolute temperature, *νe* is the valence number of the metal and *F* is Faraday's constant. C*ox* and C*Red*  are the concentrations of oxidation species and reduction species.

By substituting *E0* with -0.25V (i.e. the standard EMF for 1.0 M nickel ions [12]), *T* with 328 K (55°C), *R* with 8.3144 joules/degree-mole (gas constant), *F* with 96487 Coulomb/mole and *ν<sup>e</sup>* = 2, we gained the equation against SHE as follows.

$$E = -0.491 + \frac{0.065}{2} \log\left\{ \text{Ni}^{2+} \right\} \quad \text{VS.} \quad \text{SHE} \tag{11.2}$$

In place of *E* in equation (11-2) with the data of local potentials measured under various voltages, the steady-state concentration of nickel ions remained at the location after LECD can be calculated. The concentration of nickel ions is plotted against the voltages employed in intermittent MAGE, as shown in Fig. 15. It reveals nickel ions remaining in the LECD vicinity decrease suddenly when increasing the voltages from 3.2 to 4.6 V.

Fig. 15. The concentration of nickel ions calculated from stabilized local potentials at the location where the LECD takes place

A blank test was carried out to verify the correspondence between the electric potential and the concentration of nickel ions present in the solution. The bath conditions were set

concentration of nickel ions at the local site taking place LECD could be estimated by the

Where *E0* is the standard potential of the electrode, *R* is gas constant, *T* is absolute temperature, *νe* is the valence number of the metal and *F* is Faraday's constant. C*ox* and C*Red* 

By substituting *E0* with -0.25V (i.e. the standard EMF for 1.0 M nickel ions [12]), *T* with 328 K (55°C), *R* with 8.3144 joules/degree-mole (gas constant), *F* with 96487 Coulomb/mole and *ν<sup>e</sup>*

{ } 0.065 <sup>2</sup> 0.491 log VS. SHE <sup>2</sup>

In place of *E* in equation (11-2) with the data of local potentials measured under various voltages, the steady-state concentration of nickel ions remained at the location after LECD can be calculated. The concentration of nickel ions is plotted against the voltages employed in intermittent MAGE, as shown in Fig. 15. It reveals nickel ions remaining in the LECD

Fig. 15. The concentration of nickel ions calculated from stabilized local potentials at the

A blank test was carried out to verify the correspondence between the electric potential and the concentration of nickel ions present in the solution. The bath conditions were set

location where the LECD takes place

*e RT EE c c* ν*F*

are the concentrations of oxidation species and reduction species.

vicinity decrease suddenly when increasing the voltages from 3.2 to 4.6 V.

= 2, we gained the equation against SHE as follows.

{ } <sup>0</sup> ln / *ox Red*

= + (11-1)

*E Ni* <sup>+</sup> =− + (11-2)

Nernst equation as shown in the following.

unchanged except the concentration of nickel ions. The potential was measured using the same reference microelectrode depicted in Fig. 12. The relationship between the potential measured and the concentration of nickel ions prepared was plotted in Fig. 13. The curve representing the dependence of potential on the concentration of nickel ions obeyed equation (11-2) is in good agreement with that based on experimental measurements.

### **12. Supply of nickel ions from bulk solution to the LECD location**

Mass transport of nickel ions in the electroplating process is theoretically governed by the Nernst-Planck equation [13]. According to the Nernst-Planck equation, the flux (in mol s-l m-2) for the specific ions transported and to be deposited (assigned as *Ji*) is represented as

$$J\_i = -D\_i \nabla \mathbf{C}\_i - \frac{Z\_i F}{RT} D\_i \mathbf{C}\_i \nabla \phi + \mathbf{C}\_i \nu \tag{12-1}$$

Where *Di* is the diffusion coefficient (in m2 s-1), ∇*Ci* is the concentration gradient, ∇φ is the potential gradient, *Zi* and *Ci* are the charge (dimensionless) and concentration (in mol m-3) of species *i*, respectively, and *v* stands for the velocity (in m s-1) of the solution flow under stirring. In equation (12-1) the flux is expressed in detail with three terms on the right-hand side to describe the contribution of diffusion, migration, and convection, respectively. The diffusion coefficient (*Di*) of nickel ions in the solution is 8.157×10-10 m2 s-1 [14]. The concentration of nickel ions in the bulk solution of watts bath is 1.445 ×103 mol m-3. The convection term in Equation (12-1) could be ignored in the LECD process without stirring.

In an attempt to calculate the flux of nickel ions transported in the LECD process, the geometric transport in a specific electric field should be considered. As soon as the voltages are applied, nickel ions nearby the electrodes migrate to the cathode surface to discharge and are consumed. Reduction of nickel ions into metallic nickel within the local region leads to depletion of the nickel ions. This depletion causes a concentration gradient as compared this location to the around surroundings. The gradient offers a driving force for diffusion of nickel ions from the bulk solution to the depletion zone. Neglecting the initial stage in LECD, the micrometer columns are steadily deposited in an egg-head on their top [15]. We presume the nickel ions migrate into a boundary of semi-sphere rather than egg-head for simplifying the calculation. The semi-spherical boundary responsible for migration is illustrated in Fig. 16a. The nickel ions discharge and are consumed in the electric field surrounded with a cone. The boundary of the cone responsible for further supply of nickel ions by diffusion is illustrated in Fig. 16b. In Fig. 16, r is the average radius (in m) of the microcolumn, g marks the gap (also in m) between the microanode and the top of the column deposited. The area for the semi-sphere (*Amig*, in m2) and for the cone (*Adiff,* in m2) can be estimated in the following:

$$A\_{\rm mig} = \text{2}\,\pi r^2 \tag{12.2}$$

$$A\_{\rm dif} = 2\pi (r+g)(\frac{125 \times 10^{-6} \,/\, 2+r}{2})\tag{12-3}$$

The figure 125 x 10-6 in equation (12-3) is the diameter of the tip-end disk of the microanode. The extent of r can be evaluated through examining the columns with SEM, and g is replaced by 10-5 m in this work because of its initial setting.

Mass Transfer Within the Location Where Micro Electroplating Takes Place 227

Fig. 17. Transport rate as a function of voltages adopted in the intermittent MAGE to display

Figure 18 displays the plot of average current measured at steady state in the LECD and the current efficiency against the voltages employed. It is seen in Fig. 18 the average current increases from 0.2 to 1.8 mA, whereas the current efficiency decreases from 53 to 23 % with increasing the biases in the range from 3.2 to 4.6 V. The standard deviation of the average current also increases with the electrical biases. Observation suggested the increase in average current is proportional to the augmenting in the growing rate of columns. The decrease in the current efficiency responds to the phenomenon that bubbles evolve much more generously when increasing the biases. This enlargement in bubbles evolution implies reducing hydrogen ions contributes much more than nickel ions. The higher standard deviation in the average current at higher corresponding biases reflects the greater variation

The weight of a single micrometer column is so slight and beyond the detection limit of a usual balance. Three columns fabricated at the same conditions were gathered to overcome this difficulty, thus an average weight for a single column could be estimated (*Westimated by weighing*). The current efficiency (*η*) for the LECD conducted under specific

y eighing

= (13-1)

 *estimated b w calculated from current*

In which the numerator (*Westimated by weighing*) is the average weight obtained by the aforementioned for a single micrometer column and the denominator (*Westimated by weighing*) was

the balance between the supply and consumption rate of the nickel ions within the

in the diameter of the columns and in the roughness of their surface morphology.

*W W* η

**13. Consumption of nickel ions in the LECD location** 

conditions can be estimated by equation (13-1)

localization taking place LECD

Fig. 16. Schematic models to illustrate the region involving mass transportation of nickel ions caused by (a) migration and (b) diffusion

The transport rate of nickel ions to the location exerted LECD can be arrived at from a product between the flux and the area of the plane perpendicular to the transport direction.

$$\text{Transport rate} (\text{mol }/\text{s}) = \text{J} (\text{mol }/\text{s} \cdot \text{m}^2) \times A (\text{m}^2) \tag{12.4}$$

An instance is given to explain calculating the fluxes for the transport of nickel ions by migration and diffusion, respectively, in the process performed at 3.2 V.

$$J\_{\rm mig} = -\frac{2 \times 96487}{8.314 \times 328} \times 8.157 \times 10^{-10} \times \mathcal{C}\_{\rm local} \,/\, 10^{-3} \times \frac{\text{voltage}}{10 \times 10^{-6}}\tag{12-5}$$

$$J\_{\rm dif} = -8.157 \times 10^{-10} \times \frac{(1.445 - C\_{\rm local}) / 10^{-3}}{r \times 10^{-6}} \tag{12-6}$$

Replacing the voltage in equation (12-5) with 3.2 V, *Clocal* in equation (12-6) with the value read from Fig. 15 (i.e., 0.01530 M), and r with the radius (that is, 36.1μm) of columns measured in the SEM micrograph, we obtain the fluxes of nickel ions contributed by migration and diffusion separately as follows.

$$J\_{3.2Vmig} = -\frac{2 \times 96487}{8.314 \times 328 K} \times 8.157 \times 10^{-6} \times 0.01530 mol \text{ / } L \times 3.2 V \text{ / } 10 \,\mu m \tag{12-7}$$

$$f\_{3.2Vdf} = -8.157 \times 10^{-6} cm^2 \text{ / s} \times \frac{(1.445 - 0.01530)mol \text{ / L}}{36.1 \mu m} \tag{12.8}$$

From equation (12-4), we can estimate the total transport using equation (12-9)

$$\text{Transport rate}\_{\text{total}} = \text{J}\_{\text{dif}} \cdot \text{A}\_{\text{dif}} + \text{J}\_{\text{mig}} \cdot \text{A}\_{\text{mig}} = 2.77 \times 10^9 \,\text{mol} \,/\text{ s} \tag{12.9}$$

The same treatment can be used to calculate the corresponding transport rate for the LECD performed under other voltages. The average supplying rate calculated from equation (12-9) is plotted with the voltages, as shown in Fig. 17.

Fig. 16. Schematic models to illustrate the region involving mass transportation of nickel

The transport rate of nickel ions to the location exerted LECD can be arrived at from a product between the flux and the area of the plane perpendicular to the transport direction.

An instance is given to explain calculating the fluxes for the transport of nickel ions by

2 96487 8.157 10 /10 8.314 328 10 10 *mig local voltage J C* − −

10

<sup>−</sup> =− × × <sup>×</sup>

*<sup>C</sup> <sup>J</sup>*

<sup>×</sup> =− × × × × <sup>×</sup> <sup>×</sup>

(1.445 ) /10 8.157 10

6

2 96487 8.157 10 0.01530 / 3.2 /10 8.314 328 *Vmig J mol <sup>L</sup> <sup>V</sup> <sup>m</sup>*

(1.445 0.01530) / 8.157 10 / 36.1 *Vdif mol L J cm <sup>s</sup>*

The same treatment can be used to calculate the corresponding transport rate for the LECD performed under other voltages. The average supplying rate calculated from equation (12-9)

6 2

From equation (12-4), we can estimate the total transport using equation (12-9)

Replacing the voltage in equation (12-5) with 3.2 V, *Clocal* in equation (12-6) with the value read from Fig. 15 (i.e., 0.01530 M), and r with the radius (that is, 36.1μm) of columns measured in the SEM micrograph, we obtain the fluxes of nickel ions contributed by

migration and diffusion, respectively, in the process performed at 3.2 V.

*K*

*dif*

migration and diffusion separately as follows.

3.2

is plotted with the voltages, as shown in Fig. 17.

3.2

2 2 *Transport rate mol s J mol s m A m* ( /) ( / ) ( ) = ⋅× (12-4)

10 3

6

−

10 *local*

<sup>×</sup> <sup>−</sup> = − × ×× <sup>×</sup> <sup>×</sup> (12-7)


μ*m* <sup>−</sup> <sup>−</sup> =− × × (12-8)

*r* <sup>−</sup> <sup>−</sup> 6

(12-5)

(12-6)

μ

−

3

ions caused by (a) migration and (b) diffusion

Fig. 17. Transport rate as a function of voltages adopted in the intermittent MAGE to display the balance between the supply and consumption rate of the nickel ions within the localization taking place LECD
