# Heatflow Equation, Solved for T

## A) Non-uniform Heat Conductivity on Chip

Here are results from solving the equation del.(kappa del(T))=-H for T. Kappa is defined between mesh points and is nonuniform in the domain for all of the results presented here. The first three figures below show the kappa distribution (1/4115 where blue, 5/4115 where red) and the locations of two separate heat sources for which simulation results are presented:

 Kappa distribution Heat source 1 location Heat source 2 location

As boundary conditions, we set an external temperature of T=300 K on all four sides of the domain and also define a "package kappa" (kpack) value, which might be uniform along the border or be defined individually for each border. The assumption is that kpack is the thermal conductivity between the border mesh point and the T=300 K outer point.

For both heat sources, we solved the equation for four package conductivity setups.

• Case A: kpack=1 (i.e. approximately 4000 times the substrate conductivity) on all four sides.
• Case B: kpack=1/4000 (i.e. same order of magnitude, slightly higher than the substrate conductivity.)
• Case C: kpack=1/4000; k_north=10*kpack(the other three directions =kpack, northern boundary more conductive.)
• Case D: kpack=1/4000, k_north=kpack/10 (the other three directions =kpack, northern boundary less conductive.)

### Solutions for Heater 1:

 Case A Case B Comments: When the package conductivity is decreased, the temperature on the substrate increases (Case A to Case B). The effect of the nonuniform kappa distribution on the substrate is visible in both cases; to the left and bottom of the "chip", the contour lines are much closer together. See experimental data below. Case C Case D Comments: Here the package conductivity on the "northern" (up) border is different from the other three borders. If it is larger (Case D, above right) the mesh points near this border are pulled down to the outside temperature and the gradient of temperature increases between the heat source and this border. If it is smaller (Case C, above left) the opposite effect occurs. Note that the asymmetry observed in the experimental data below can be explained by either effect (substrate kappa nonuniformity and border kappa nonuniformity). The full explanation is possibly a combination, where the contour lines' changing distances are affected on-chip by the kappa distribution and directionality explained by the package conductivity. We need to work on this more to figure out which is which. Further simulations: a) With uniform substrate conductivity, b) with directionally-nonuniform substrate conductivity. Experimental Results:(Note that the simulation is supposed to emulate the middle 13 rows and 11 columns of this chip.)

### Solutions for Heater 2:

 Case A Case B Comments: When the package conductivity is decreased, the temperature on the substrate increases (Case A to Case B). The effect of the nonuniform kappa distribution on the substrate is visible in both cases; to the left and bottom of the "chip", the contour lines are much closer together. See experimental data below. Case C Case D Comments: Similar effects with the heater 1 case is observed. Looking at the experimental data below, it seems like the border conductivity has a larger effect in this case. Experimental Results:(Note that the simulation is supposed to emulate the middle 13 rows and 11 columns of this chip.)

## B) Uniform Kappa Distribution on Chip

Next, we try the same two heaters with a uniform kappa distribution on the surface of the chip, changing the boundary conditions of kappa as above.

### Solutions for Heater 1:

 Case A Case B Comments: When the package conductivity is decreased, the temperature on the substrate increases (Case A to Case B). Now the contour line shapes are obviously affected by the position of the heater on the chip, for the two cases above where the borders are all equally conductive. However, close to the heater the contours remain symmetrical for this case, whereas for the nonuniform kappa case presented above, distortion showed up near the boundary of the kappa change. There are also no more "break" features in the contour lines as was the case where there was a discontinuity in the heat conductivity. Case C Case D Comments: Here the package conductivity on the "northern" (up) border is different from the other three borders. Reducing the conductivity in this border has the same effect as above of reducing the temperature gradient and spreading the heat more in this direction than the others, as seen in Case C (above left); however, the effect has a smoother shape. Experimental Results:(Note that the simulation is supposed to emulate the middle 13 rows and 11 columns of this chip.)

### Solutions for Heater 2:

 Case A Case B Comments: When the package conductivity is decreased, the temperature on the substrate increases (Case A to Case B). Once again, now the contour shapes are governed by the heater position and chip shape only, everything else being uniform. Case C Case D Comments: Similar effects with the heater 1 case is observed. The border effect is also more symmetric when the substrate conductivity distribution is uniform. Experimental Results:(Note that the simulation is supposed to emulate the middle 13 rows and 11 columns of this chip.)

## C) Directionally Non-Uniform Kappa Distribution on Chip

Since the metal networks on our chip are directional (M1 north-south, M2 east-west, mostly) use a uniform distribution in the east-west direction while increase the conductivity in the north-south direction only in the same quarter as in the initial case above.

### Solutions for Heater 1:

 Case A Case B Comments: When the package conductivity is decreased, the temperature on the substrate increases (Case A to Case B). Close to the heater the contours remain symmetric in the east-west direction and there's less temperature drop towards the high-kappa regions in the north-south direction. This seems to emulate the experimental result better; see results below again. Case C Case D Comments: Here the package conductivity on the "northern" (up) border is different from the other three borders. Reducing the conductivity in this border has the same effect as above of reducing the temperature gradient and spreading the heat more in this direction than the others, as seen in Case C (above left). Experimental Results:(Note that the simulation is supposed to emulate the middle 13 rows and 11 columns of this chip.)

### Solutions for Heater 2:

 Case A Case B Comments: When the package conductivity is decreased, the temperature on the substrate increases (Case A to Case B). Once again, now the contour shapes are governed by the heater position and chip shape only, everything else being uniform. Case C Case D Comments: Similar effects with the heater 1 case is observed. The border effect is also more symmetric when the substrate conductivity distribution is uniform. Experimental Results:(Note that the simulation is supposed to emulate the middle 13 rows and 11 columns of this chip.)

## D) Directionally Non-Uniform Kappa Distribution on Chip and on the Northern Border

Since the metal networks on our chip are directional (M1 north-south, M2 east-west, mostly) use a uniform distribution in the east-west direction while increase the conductivity in the north-south direction only in the same quarter as in the initial case above. But on the northern border the left-hand side is more conductive than the right hand side in the E-W direction (not the package conductivity, the first row conductivity). This is to emulate one wide metal line there which carries power to the heater resistors.

### Solutions for Heater 1:

 Case A Case B Comments: With the northern package kappa the same as in the other directions, having nonuniform E-W kappa on the top row did not affect things much. See below... Case C Case D Comments: However, when the conductivity on the northern border is lower than in the other three directions (Case C), we get a distortion towards the side where the E-W conductivity is set higher on the first row. This seems to match the experimental result. Experimental Results:(Note that the simulation is supposed to emulate the middle 13 rows and 11 columns of this chip.)

### Solutions for Heater 2:

 Case A Case B Comments: Not much difference from the directionally-nonuniform kappa case here, except for a slight distortion in the upper left corner in case A because of the discontinuity there. Case C Case D Comments: The corner distortion we obtain when the northern package conductivity is set lower than the other three and the row 1 E-W conductivity is higher than the rest looks like the experimental result. Experimental Results:(Note that the simulation is supposed to emulate the middle 13 rows and 11 columns of this chip.)

Zeynep Dilli