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This work presents an algorithm for simulating more accurate temperature distribution in two-phase liquid cooling for three-dimensional integrated circuits than the state of-the-art methods by utilizing local multi-linear interpolation techniques on heat transfer coefficients between the microchannel and silicon substrate, and considering the interdependence between the thermal conductivity of silicon and temperature values. The experimental results show that the maximum and average errors are only 9.7% and 6.7% compared with the measurements, respectively.

Due to the high power density and the ill of heat dissipation capability, the operating temperature of three-dimensional integrated circuits (3-D ICs) is higher than that of two-dimensional (2-D) ICs. Recently, to effectively remove the heat in 3-D ICs, the advanced liquid cooling system has been widely discussed [^{2}) without burning down. Reference [

Different from the above single-phase liquid cooling systems, the two-phase liquid cooling removes the heat via heat absorption during the evaporation process [

This work provides an accurate temperature profile estimate for the two-phase liquid cooling system of 3-D IC. The inputs are the design structure and material files, and the power distribution in device layers. Different from incorporating the fitting curves for correlating HTCs at the wall surface in STEAM [

By using a local multi-linear interpolation technique with the measurement database to approximate HTCs during the thermal simulation procedure for the two-phase liquid cooling system, the accuracy can be reasonably improved.

Considering the interdependence between the operating temperature and thermal conductivity of silicon does improve the simulation accuracy for two-phase liquid cooling systems.

The remainder of this paper is organized as follows. Section 2 reviews the thermal model for at wophase liquid-cooled 3-D IC. With the thermal model, the proposed thermal simulation algorithm is detailed in Section 3. The experimental results are presented in Section 4, and finally, conclusions are drawn in Section 5.

The structure of a liquid-cooled 3-D IC is shown in

interconnects and dielectric. Each device layer is a thin layer on the top surface of stacked silicon substrate or silicon bulk. Functional blocks of each tier are modeled as power generating sources attached to the thin layer that is close to the top surface of the stacked silicon substrateor the silicon bulk. Microchannels are etched on the silicon substrate as heat sink, and a cover plate layer is placed on the back-side of silicon substrate. The material of cover plate is either Si or Pyrex glass. Through silicon vias are placed in the silicon substrate and cover plate layer. All the boundary surfaces of a 3-D IC are assumed to be adiabatic as boundary conditions.

According to the heat transfer equation, the temperature profile T(r) in solids can be governed by

and

Here,

Using the finite difference method and the duality between thermal and electrical quantities, (1) can be transformed to a SPICE compatible equivalent thermal circuit containing thermal resistances and power sources. For a specific control volume (thermal grid) as shown in

Here,

The temperature Ti, j,k at the central node of each solid grid Vi, j,k need to satisfy the following modified nodal analysis (MNA) system.

Here,

Given a thermal grid in liquids as shown in

Here,

In (4), the second term on left-hand side represents the heat entering from the microchannel sidewalls into each two-phase thermal grid, and other terms represent heat removal denoted by the latent heat difference of vaporization between the two interfaces

Based on a similar approach as STEAM [

Here,

The flow chart of our proposed algorithm for solving the vector

1) In the beginning, the thermal conductivities and power densities are stamped into the matrix. Since HTCs in two-phase liquid grids are unknown, based on the inlet condition of refrigerant, these unknown parameters are initially guessed.

2) After the MNA system is established, Super LU solver [

3) With the temperature values adjacent to the two-phase liquid grids and HTCs, each heat flux density

4) With the temperature values at solid grids of silicon, the temperature dependent thermal conductance

5) By using the homogeneous equation for pressure drop [

6) The results obtained by 3)?5) are sent back to 2), and the whole procedure is repeated until

1) Pre-build a regular table of HTCs by the interpolation method on irregular measurement data points: Directly using the measurement data points of HTCs (3899 data points) [

To simply illustrate the pre-building procedure, we assume that the measurement data

where

The interpolation form (6) indicates that the more the measured point is close to the desired point, the more its influence is. For building the regular HTC table, the formula is presented as follows.

Here,

2) Calculate HTCs by the multi-linear interpolation method on regular points at built HTC table: Utilizing the built regular HTC table, HTCs can be efficiently calculated by the multi-linear interpolation technique during the simulation procedure. There are many ways for solving interpolation method on regular grids, such as bilinear interpolation, bicubic interpolation, spline interpolation, etc. Here, the multi-linear interpolation on six variables is adopted.

Again, for simplicity, the same example shown in

Here,

Therefore, in each iteration, the HTC at an arbitrary point e can be calculated by using the above multi-linear interpolation method with six parameters as

where each

For air-cooling or single-phase liquid cooling 3-D ICs, the gradient of temperature along z-axis (normal to the device surface) could be low. However, the refrigerant temperature is almost unchanged (in the situation, the liquid is not dried out), so the gradient of temperature along z-axis might be high in some cases. As a result, the temperature-dependent thermal conductivity should be considered. In this work, the formula of thermal conductivity versus temperature presented in [

Here,

By applying the backward Euler method into (1) and (2), we have

where

The proposed method is implemented in C++ language and tested on Intel Processor i7-3770 3.40 GHz CPU with 20GB RAM. Three pseudo chips with their measured data obtained by [

The structure of pseudo chips is shown in

chosen refrigerant is R236-fa. Their power maps are shown in Figures 8(a)-(c). ^{2}, the inlet temperature is 304.1 K, and the mass flux is 933 kg/m^{2}s. For the point hotspot power

map case shown in ^{2} except that the grid 34 is 202 W/cm^{2}. Its inlet temperature is 304.1 K, and mass flux is 713 kg/m^{2}s. The power density of row hotspot power map as shown in ^{2} from row#1 to row#4, and 116W/cm^{2} in row#5 (grids 51 - 57). Its inlet temperature is 303.5 K, and mass flux is 895 kg/m^{2}s.

The temperature comparison of col#4 between the proposed method and the measured data is shown in Figures 8(d)-(f) The blue dotted lines are the simulation results with only applying the proposed multi-linear interpolation to calculate HTCs, and the red solid lines are obtained by simultaneously utilizing the HTC interpolation technique and considering the temperature-dependent effect of silicon thermal conductivity. The results show good tendency on predicting the temperature distribution of device layers by using the multi-linear interpolation techniques on calculating HTCs during the simulation procedure. Moreover,

To further demonstrate the proposed techniques, we also implement the STEAM model with three HTC correlation functions [

Finally, to demonstrate the robustness of the proposed method for 3-D ICs, a realistic 4-tier 3-D multiprocessor system-on-chip (3-D MPSoC) [^{2}s. The error is compared with a commercial tool, ANSYS CFD. As shown in

The case is constant workload and simulation time is 0.2 second with power density 96.4 W/cm^{2}. Comparing with the ANSYS results, the maximum and average error of “HTC interpol.” are about 12.6% and 7.8%, and the maximum and average error of “HTC interpol. & TDTCS” are about 10.7% and 7.3%. It shows that considering the temperature-dependent effect on thermal conductivity is also important through these results and the similar trend with ANSYS result as well.

An effective and accurate thermal simulation method for two-phase liquid cooling 2-D/3-D ICs has been developed and implemented. The experimental results have demonstrated its accuracy improvement and effectiveness for the real designs.

Test case | HTC correlation in STEAM | Proposed | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

[ | [ | [ | HTC interpol. | HTC interpol. & TDTCS | |||||||||||

Error (%) | Time (s) | Error (%) | Time (s) | Error (%) | Time (s) | Error (%) | Time (s) | Error (%) | Time (s) | ||||||

avg | max | avg | max | avg | max | avg | max | avg | max | ||||||

Uniform | 10.9 | 15.6 | 29.4 | 23.3 | 27.8 | 29.1 | 10.1 | 12.9 | 29.3 | 7.6 | 10.3 | 33.2 | 6.7 | 9.7 | 35.4 |

Point hotspot | 10.5 | 13.6 | 29.6 | 15.2 | 18.3 | 27.5 | 6.7 | 9.8 | 28.3 | 6.3 | 9.5 | 36.5 | 5.3 | 9.3 | 38.9 |

Row hotspot | 11.3 | 15.2 | 28.7 | 13.7 | 16.2 | 30.6 | 7.5 | 10.1 | 29.6 | 6.7 | 9.3 | 35.9 | 6.3 | 8.2 | 37.6 |

3-D MPSoC | 10.9 | 14.1 | 96.5 | 11.7 | 13.2 | 94.7 | 8.6 | 11.5 | 95.6 | 5.9 | 8.3 | 120.9 | 4.8 | 7.7 | 125.6 |

†3-D MPSoC is compared with ANSYS CFD.

The authors would like to thank Professor Suresh V. Garimella and Professor Stefan S. Bertsch for providing the measurement data. This work was supported in part by Ministry of Science and Technology of Taiwan under Grants MOST 105-2221-E-009-168, MediaTek Inc., and Industrial Technology Research Institute.

Chiou, H.-W. and Lee, Y.-M. (2016) Thermal Simulation for Two-Phase Liquid Cooling 3D-ICs. Journal of Computer and Communications, 4, 33- 45. http://dx.doi.org/10.4236/jcc.2016.415003