3.1.1.

Power density distribution

Power density distribution²⁵ $E(x,y,z)$ is the set of all power densities at location $z$ for all coordinates $(x,y)$. The power density distribution of the MWE (Fig. 5) is the collection of pixels from 0 to 6 in both $x$ and $y$ axes.

3.1.2.

Power density

Power density²⁵ $E(x_p,y_p,z)$ is the power density at location $z$ for coordinate $(x_p,y_p)$. The power density of the MWE (Fig. 5) is $E(\mathrm{2,4},z)=10$.

3.1.3.

Maximum power density

Maximum power density²⁵ $E_{\max }(z)$ is the maximum of the power density distribution $E(x,y,z)$ at location $z$ for coordinate $(x_{\max },y_{\max },z)$. The maximum power density of the MWE (Fig. 5) is $E_{\max }(z)=12$ for coordinates (3, 4) and (4, 3).

3.1.4.

Power

Power²⁵ $P(z)$ is the sum of the power density $E(x_p,y_p,z)$ at location $z$ evaluated over all coordinates $(x,y)$ of the power density distribution $E(x,y,z)$. $P(z)$ is mathematically defined as

The power of the MWE (Fig. 5) is $P(z)=128$.

3.1.5.

Clip-level power density

Clip-level power density²⁵ $E_{\eta \mathrm{CL}}(z)$ is the fraction $\eta$ of the maximum power density $E_{\max }(z)$ at location $z$. $E_{\eta \mathrm{CL}}(z)$ is mathematically defined as

3.1.6.

Clip-level power

Clip-level power²⁵ $P_{\eta }(z)$ is the sum of the power density $E(x_p,y_p,z)$ at location $z$ evaluated only over locations $(x,y)$ for which $E(x,y,z)>E_{\eta \mathrm{CL}}(z)$. $P_{\eta }(z)$ is mathematically expressed as

The clip-level power of the MWE (Fig. 5) is $P_{\eta }(z)=96$ considering a clip-level of $\eta =0.80$.

3.1.7.

Fractional power

Fractional power²⁵ $f_{\eta }(z)$ is the fraction of the clip-level power $P_{\eta }(z)$ for a given $\eta$ to the total power $P(z)$ in the distribution at location $z$. $f_{\eta }(z)$ is mathematically expressed as

3.1.8.

Beam centroid

Beam centroid²⁶ $(\overline{x}(z),\overline{y}(z))$ is the coordinate of the first-order moment of the power density distribution $E(x,y,z)$ at location $z$. $(\overline{x}(z),\overline{y}(z))$ is mathematically defined as

The first-order moment (Fig. 6) computes the value of each pixel weighted by its corresponding column index (if the moment is calculated about the $x$ axis) and by its corresponding row index (if the moment is calculated about the $y$ axis). For instance, the first-order moment about the $x$ axis is calculated by multiplying the values of each pixel in the first column (index 0) by 0, in the second column (index 1) by 1, in the third column (index 2) by 2, and so on [Fig. 6(a)]. Similarly, the first-order moment about the $y$ axis is calculated by multiplying the values of each pixel in the first row (index 0) by 0, in the second row (index 1) by 1, in the third row (index 2) by 2, and so on [Fig. 6(b)]. The values of each pixel are subsequently added. The beam centroid of the MWE (Fig. 5) is $\overline{x}(z)=3$ and $\overline{y}(z)=3$.

Fig. 6

Visual explanation of the first-order moment. (a) The first-order moment about the $x$ axis is computed by multiplying the values of each pixel in the original power density distribution by its corresponding column index. (b) The first-order moment about the $y$ axis is calculated by multiplying the values of each pixel in the original power density distribution by its corresponding row index.

3.1.9.

Beam positional stability

Beam positional stability²⁶ $({\mathrm{\Delta }}_x(z),{\mathrm{\Delta }}_y(z))$ is four times the standard deviation of the measured beam positional movement at location $z$. $({\mathrm{\Delta }}_x(z),{\mathrm{\Delta }}_y(z))$ is expressed mathematically as

3.1.10.

Beam width

Beam width²⁶ $(d_{\sigma x}(z),d_{\sigma y}(z))$ is four times the square root of the second-order moment of the power density distribution $E(x,y,z)$ about the beam centroid $(\overline{x}(z),\overline{y}(z))$ at location $z$. $(d_{\sigma x}(z),d_{\sigma y}(z))$ is mathematically expressed as

The second-order moment (Fig. 7) computes the value of each pixel weighted by the square of the distance from its corresponding column index to the beam $x$ centroid (if the moment is calculated about the $x$ axis) and by the square of the distance from its corresponding row index to the beam $y$ centroid (if the moment is calculated about the $y$ axis). For example, the second-order moment about the $x$ axis is calculated by multiplying the values of each pixel in the first column (index 0) by ${(0-3)}^2=9$, in the second column (index 1) by ${(1-3)}^2=4$, in the third column (index 2) by ${(2-3)}^2=1$, and so on [Fig. 7(a)]. Likewise, the second-order moment about the $y$ axis is calculated by multiplying the values of each pixel in the first row (index 0) by ${(0-3)}^2=9$, in the second row (index 1) by ${(1-3)}^2=4$, in the third row (index 2) by ${(2-3)}^2=1$, and so forth [Fig. 7(b)]. The values of each pixel are subsequently added. The bean width of the MWE (Fig. 5) is $d_{\sigma x}(z)=4.36$ and $d_{\sigma y}(z)=4.34$.

Fig. 7

Visual explanation of the second-order moment. (a) The second-order moment about the $x$ axis is computed by multiplying the values of each pixel in the original power density distribution by the square of the distance from its corresponding column index to the beam $x$ centroid. (b) The second-order moment about the $y$ axis is calculated by multiplying the values of each pixel in the original power density distribution by the square of the distance from its corresponding row index to the beam $y$ centroid.

3.1.11.

Beam aspect ratio

Beam aspect ratio²⁶ $\epsilon (z)$ quantifies the squareness of a power density distribution $E(x,y,z)$ at location $z$. $\epsilon (z)$ is mathematically defined as

The beam aspect ratio of the MWE (Fig. 5) is $\epsilon (z)=0.99$, which suggests a fairly square shape.

3.1.12.

Clip-level irradiation area

Clip-level irradiation area²⁵ $A_{\eta }^i(z)$ is the irradiation area at location $z$, for which the power density $E(x_p,y_p,z)$ exceeds the clip-level power density $E_{\eta \mathrm{CL}}(z)$. $A_{\eta }^i(z)$ is mathematically expressed as

3.1.13.

Clip-level average power density

Clip-level average power density²⁵ $E_{\eta \mathrm{ave}}(z)$ is the spatially averaged power density of the distribution at location $z$. $E_{\eta \mathrm{ave}}(z)$ is mathematically defined as

The clip-level average power density of the MWE (Fig. 5) is $E_{\eta \mathrm{ave}}(z)=10.67$ considering a clip-level of $\eta =0.80$.

3.1.14.

Flatness factor

Flatness factor²⁵ $F_{\eta }(z)$ is the ratio of the clip-level average power density $E_{\eta \mathrm{ave}}(z)$ to the maximum power density of the distribution $E_{\max }(z)$ at location $z$. $F_{\eta }(z)$ is mathematically defined as

3.1.15.

Beam uniformity

Beam uniformity²⁵ $U_{\eta }(z)$ is the normalized root mean square deviation of the power density distribution $E(x,y,z)$ from its clip-level average power density $E_{\eta \mathrm{ave}}(z)$ at location $z$. $U_{\eta }(z)$ is mathematically expressed as

Fig. 8

Visual explanation of one of the steps to calculate the beam uniformity. The expression ${[E(x,y,z)-E_{\eta \mathrm{ave}}(z)]}^2$ for pixel $(\mathrm{0,6})$ is ${[0-10.67]}^2=144$, approximately. Similarly for pixel (1, 5), the expression is approximately ${[2-10.67]}^2=75$. The double sum yields 6.00.

3.1.16.

Plateau uniformity

Plateau uniformity²⁵ $U_p(z)$ is the quantitative measure of the homogeneity of nearly flat-top profiles. $U_p(z)$ is mathematically expressed as

Fig. 9

Histogram of the power density distribution in Fig. 4. The histogram is divided into 256 bins since the power density distribution in Fig. 4 has a total of 65,536 data points. The inset image shows the detailed view of the histogram peak near $E_{\max }(z)$, the normal 3 mixture fit, and a visual explanation of the FWHM.

3.1.17.

Edge steepness

Edge steepness²⁵ $S_{\epsilon ,\eta }(z)$ is the normalized difference between the clip-level irradiation areas $A_{\epsilon }^i(z)$ and $A_{\eta }^i(z)$. $S_{\epsilon ,\eta }(z)$ is mathematically defined as

4.1.1.

Measurement unit of power

The measurement unit of power can be converted from ADC to $W$ by correlating the calculated power $P(z)$ (28,576,928.9160 ADC) with the output laser power. Since the entire optical system has a loss of $\sim 5\%$, the output laser power should be verified by a calibrated power meter. The power meter used in this study is a PowerMax-Pro HP from Coherent. It features a thin-film detector with a calibration uncertainty of $\pm 2\%$ at 810 nm, a spectral compensation accuracy of $\pm 5\%$, a power linearity of $\pm 2\%$, and a spatial uniformity of $\pm 5\%$.

The output laser power was measured three consecutive times, and the average value was confirmed as 700 W. Therefore, the conversion factor of power is $\frac{700\mathrm{W}}{\mathrm{28,576,928.9160}\mathrm{ADC}}=2.4495\times {10}^{-5}\frac{\mathrm{W}}{\mathrm{ADC}}$.

4.1.2.

Measurement unit of length

The measurement unit of length can be converted from px to mm by correlating the pixel resolution ($256\mathrm{px}\times 256\mathrm{px}$) with the size of the measuring window ($35.1\mathrm{mm}\times 35.1\mathrm{mm}$). The conversion factor of length is $\frac{35.1\mathrm{mm}}{256\mathrm{px}}=0.1371\frac{\mathrm{mm}}{\mathrm{px}}$.

4.1.3.

Measurement unit of power density

The measurement unit of power density can be converted from $\frac{\mathrm{ADC}}{\mathrm{px}}$ to $\frac{\mathrm{W}}{{\mathrm{mm}}^2}$ by combining the conversion factors of power and length. The conversion factor of power density is: $2.4495\times {10}^{-5}\frac{\mathrm{W}}{\mathrm{ADC}}÷(0.1371\frac{\mathrm{mm}}{\mathrm{px}}{)}^2=0.0013\frac{\mathrm{W}}{\mathrm{ADC}}\frac{{\mathrm{px}}^2}{{\mathrm{mm}}^2}$.

4.2.1.

Beam power

The power $P(z)$ has an average value of 700 W with a standard deviation of 0.1796%, and the clip-level power $P_{\eta }(z)$ has an average value of 641.17 W with a standard deviation of 0.2915% considering a clip level of $\eta =0.80$.

4.2.2.

Beam power density

The maximum power density $E_{\max }(z)$ has an average value of $1.8387\frac{\mathrm{W}}{{\mathrm{mm}}^2}$ with a standard deviation of 0.4033%, the clip-level power density $E_{\eta \mathrm{CL}}(z)$ has an average value of $1.4710\frac{\mathrm{W}}{{\mathrm{mm}}^2}$ with a standard deviation of 0.4033%, and the clip-level average power density $E_{\eta \mathrm{ave}}$ has an average value of $1.6079\frac{\mathrm{W}}{{\mathrm{mm}}^2}$ with a standard deviation of 0.1895%.

4.2.3.

Beam position

The coordinate of maximum $(x_{\max },y_{\max },z)$ has an average value of (17.6173 mm, 27.4063 mm) with a standard deviation of (57.0124%, 0.1582%). The high variation of the $x$ coordinate can be explained by looking at the coordinate of each individual measurement (Table 2). It is clear that one half of the measurements are centered at around 8.0889 mm, whereas the other half are centered at around 27.1458 mm. Considering each individual group, the new standard deviation is 0.0000% for the group at 8.0889 px and 0.3571% for the group at 27.1458 mm. The beam centroid $(\overline{x}(z),\overline{y}(z))$ is located at the coordinate (17.6311 mm, 17.4117 mm) with a beam positional stability $({\mathrm{\Delta }}_x(z),{\mathrm{\Delta }}_y(z))$ of (0.2832 mm, 0.0000 mm).

Table 2

Coordinate of maximum at location z for each individual measurement.

4.2.4.

Effective beam size

The beam width $(d_{\sigma x}(z),d_{\sigma y}(z))$ has an average value of (24.2375 mm, 24.3706 mm) with a standard deviation of (0.0555%, 0.0685%) [Fig. 11(a)]. The clip-level irradiation areas $A_{\epsilon }^i(z)$ and $A_{\eta }^i(z)$ have an average value of $463.4029{\mathrm{mm}}^2$ and $397.8185{\mathrm{mm}}^2$, respectively, with a corresponding standard deviation of 0.2351% and 0.3165%.

Fig. 11

Beam width definition and robustness of the LAB process. (a) The beam width $d_{\sigma x}(z)$ has an average value of 24.2375 mm. (b) Example of a die of size $13.00\mathrm{mm}\times 13.00\mathrm{mm}$ and a beam width $d_{\sigma x}(z)$ of the same dimensions. The solder bumps at the periphery receive insufficient laser energy, which can lead to open-circuit issues. (c) The beam width $d_{\eta x}(z)$ has an average value of 19.9557 mm, 17.67% smaller than $d_{\sigma x}(z)$. (d) Example of a die of size $13.00\mathrm{mm}\times 13.00\mathrm{mm}$ and a beam width $d_{\eta x}(z)$ of the same dimensions. The solder bumps at the periphery receive sufficient laser energy, which reduces the risk of open-circuit issues.

One of the strong advantages of the LAB process is the selective heating capability, in which only the target device and its immediate surroundings are irradiated by the laser beam. In practical terms, this is achieved by setting the beam size to match the dimensions of the target device. However, the definition of beam size as described in Sec. 3.1.10²⁶ could pose a threat to the robustness of the LAB process since the edges of the target device could receive an insufficient amount of laser power, as discussed in Sec. 1. To get this point across, consider a die of size $13.00\mathrm{mm}\times 13.00\mathrm{mm}$ and a beam width of the same dimensions, as defined by Eqs. (9) and (10) [Fig. 11(b)]. It is evident that this arrangement has a high probability of yielding open-circuit issues at the peripheral solder bumps since the edges of the target device do not receive sufficient laser energy.

At this point, the reader might realize that occasionally the ISO standards do not apply to their specific application, and that new definitions that suit their needs can be proposed. In this sense, we propose a new beam width definition that avoids this issue by combining the concepts of beam width and clip-level power density—the clip-level beam width $(d_{\eta x}(z),d_{\eta y}(z))$. The clip-level beam width is calculated considering only contiguous pixels whose values are greater than $E_{\eta \mathrm{CL}}(z)$. Using this definition, the beam width has an average value of (19.9557 mm, 19.9288 mm) with a standard deviation of (0.3365%, 0.0552%) [Fig. 11(c)]. Once again, consider a die of size $13.00\mathrm{mm}\times 13.00\mathrm{mm}$ and a beam width of the same dimensions, but this time as defined by the clip-level beam width [Fig. 11(d)]. With this approach, one can be sure that the entire area of the target device receives sufficient laser energy and a robust solder interconnection is secured. Refer to Appendix A for the full definition of $(d_{\eta x}(z),d_{\eta y}(z))$.

4.2.5.

Beam shape

The beam aspect ratio $\epsilon (z)$ has an average value of 1.0055, which is expected since the laser beam has a square shape. The fractional power $f_{\eta }(z)$ has an average value of 0.9160, which indicates that 91.60% of the total power $P(z)$ has a value greater than the clip-level power $P_{\eta }(z)$. The flatness factor $F_{\eta }(z)$ has as average value of 0.8744, only 12.56 percent points smaller than a perfectly flat-top power density distribution. The beam uniformity $U_{\eta }(z)$ has an average value of 0.0219, which represents a variation of only $\pm 2.19\%$ rms from the clip-level average power density $E_{\eta \mathrm{ave}}(z)$. The plateau uniformity $U_p(z)$ has an average value of 0.0284, which is just 2.84 percent points higher than a perfectly flat-top power density distribution. The edge steepness $S_{\epsilon ,\eta }(z)$ has an average value of 0.1415, which indicates a power density distribution with fairly vertical edges.

During the LAB process, it is important to understand the influence that the laser emission onto a target device has on its neighboring components. This influence is expected to become more critical as the number of devices per substrate increases and the clearance between them decreases. Therefore, it is valuable to quantitatively determine a keep-out-zone outside of which the laser beam has negligible impact and devices are safe from undesired laser radiation. Because the edge steepness $S_{\epsilon ,\eta }(z)$ does not provide a straightforward explanation on this matter, we propose a new definition—the clip-level edge width. Here again, the reader might appreciate the possibility of proposing new definitions that better suit their specific application. The clip-level edge width $(d_{\epsilon \eta x}(z),d_{\epsilon \eta y}(z))$ is calculated considering contiguous pixels whose values are in-between $E_{\epsilon \mathrm{CL}}(z)$ and $E_{\eta \mathrm{CL}}(z)$ (Fig. 12). Using this definition, the edge width has an average value of (0.7514 mm, 0.8556 mm) with a standard deviation of (7.5791%, 0.3823%). Refer to Appendix B for the full definition of $(d_{\epsilon \eta x}(z),d_{\epsilon \eta y}(z))$.

Fig. 12

Clip-level edge width definition. The edge width $d_{\epsilon \eta x}(z)$ has an average value of 0.7514 mm with a standard deviation of 7.5791%.