^{1}, Edward S. Piekos

^{1}and Leslie M. Phinney

^{1,a)}

### Abstract

Accurate knowledge of thermophysical properties is needed to predict and optimize the thermal performance of microsystems. Thermal conductivity is experimentally determined by measuring quantities such as voltage or temperature and then inferring a thermal conductivity from a thermal model. Thermal models used for data analysis contain inherent assumptions, and the resultant thermal conductivity value is sensitive to how well the actual experimental conditions match the model assumptions. In this paper, a modified data analysis procedure for the steady state Joule heating technique is presented that accounts for bond pad effects including thermal resistance, electrical resistance, and Joule heating. This new data analysis method is used to determine the thermal conductivity of polycrystalline silicon (polysilicon) microbridges fabricated using the Sandia National Laboratories SUMMiT V™ micromachining process over the temperature range of 77–350 K, with the value at 300 K being 71.7 ± 1.5 W/(m K). It is shown that making measurements on beams of multiple lengths is useful, if not essential, for inferring the correct thermal conductivity from steady state Joule heating measurements.

The authors would like to thank Allen Gorby at Sandia for assistance with various aspects of the experimental setup and Patrick E. Hopkins, Justin R. Serrano, and C. Thomas Harris for their technical review of this document. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under Contract No. DE-AC04-94AL85000.

I. INTRODUCTION

II. THEORY

A. Modified theory

III. SIMULATIONS

IV. EXPERIMENTAL SETUP

V. RESULTS

VI. SUMMARY AND CONCLUSIONS

### Key Topics

- Thermal conductivity
- 66.0
- Electrical resistivity
- 30.0
- Silicon
- 12.0
- Electric measurements
- 9.0
- Data analysis
- 8.0

## Figures

Schematic of the polysilicon microbridge devices tested. A long, slender beam is suspended above a substrate. The bond pads support each end of the beam and provide an anchor to the substrate.

Schematic of the polysilicon microbridge devices tested. A long, slender beam is suspended above a substrate. The bond pads support each end of the beam and provide an anchor to the substrate.

Cross section showing the computational model for the bond pad fabricated using the SUMMiT V™ process. The material layers are polysilicon, silicon dioxide, silicon nitride, and aluminum. The beam plane represents the location where the microbridge attaches to the bond pad.

Cross section showing the computational model for the bond pad fabricated using the SUMMiT V™ process. The material layers are polysilicon, silicon dioxide, silicon nitride, and aluminum. The beam plane represents the location where the microbridge attaches to the bond pad.

(a) Predicted temperatures in a SUMMiT V™ bond pad due to thermal resistance at an ambient temperature of 295 K. In the simulation, the polysilicon thermal conductivity is 70 W/(m K) and the applied power is 1 mW. (b) Temperature and thermal conductivity dependent thermal resistance of the bond pad. Equation (11) is fit to each set of constant thermal conductivity data.

(a) Predicted temperatures in a SUMMiT V™ bond pad due to thermal resistance at an ambient temperature of 295 K. In the simulation, the polysilicon thermal conductivity is 70 W/(m K) and the applied power is 1 mW. (b) Temperature and thermal conductivity dependent thermal resistance of the bond pad. Equation (11) is fit to each set of constant thermal conductivity data.

(a) Predicted temperatures in a SUMMiT V™ bond pad due to Joule heating at an ambient temperature of 295 K. In the simulation, the polysilicon thermal conductivity is 70 W/(m K) and the applied current is 1 mA. (b) Temperature and thermal conductivity dependent Joule heating of the bond pad. Equation (12) is fit to each set of constant thermal conductivity data.

(a) Predicted temperatures in a SUMMiT V™ bond pad due to Joule heating at an ambient temperature of 295 K. In the simulation, the polysilicon thermal conductivity is 70 W/(m K) and the applied current is 1 mA. (b) Temperature and thermal conductivity dependent Joule heating of the bond pad. Equation (12) is fit to each set of constant thermal conductivity data.

Theoretical temperature profiles calculated according to basic steady state theory (Eq. (3) ) and modified theory that accounts for thermal resistance and Joule heating in the bond pad (Eq. (8) ). Experimental data was obtained at 303 K and 298 K for the 200 *μ*m and 400 *μ*m long beams, respectively, using Raman thermometry. ^{ 36 } The applied power was 10.8 mW for the 200 *μ*m beam and 4.0 mW for the 400 *μ*m beam. The value of *k* is taken to be 71 W/(m K) in Eqs. (3) and (8) .

Theoretical temperature profiles calculated according to basic steady state theory (Eq. (3) ) and modified theory that accounts for thermal resistance and Joule heating in the bond pad (Eq. (8) ). Experimental data was obtained at 303 K and 298 K for the 200 *μ*m and 400 *μ*m long beams, respectively, using Raman thermometry. ^{ 36 } The applied power was 10.8 mW for the 200 *μ*m beam and 4.0 mW for the 400 *μ*m beam. The value of *k* is taken to be 71 W/(m K) in Eqs. (3) and (8) .

Schematic of the steady state thermal conductivity measurement system. The polysilicon microbridges are packaged and housed inside a radiation shielded cryostat that is evacuated to a pressure below 1 mTorr and cooled with a pressurized 10 L LN_{2} dewar. A heating element in the cryostat is used to maintain a constant temperature as set using the temperature controller. A computer is used to control a data acquisition (DAQ) unit that is connected to the current source and digital multimeter. ^{ 11 }

Schematic of the steady state thermal conductivity measurement system. The polysilicon microbridges are packaged and housed inside a radiation shielded cryostat that is evacuated to a pressure below 1 mTorr and cooled with a pressurized 10 L LN_{2} dewar. A heating element in the cryostat is used to maintain a constant temperature as set using the temperature controller. A computer is used to control a data acquisition (DAQ) unit that is connected to the current source and digital multimeter. ^{ 11 }

Thermal conductivity from 77–350 K of polysilicon microbridges calculated from (a) Eq. (6) , (b) Eq. (6) and accounting for additional resistance, and (c) Eq. (10) and accounting for additional resistance and bond pad heating effects. The plotted values are averaged over the two chips.

Thermal conductivity variation with beam length from experimental (*T* = 303 K) data showing the effect of data analysis procedure. The uncorrected data show clear length dependence and underpredict thermal conductivity. With electrical resistance, thermal resistance and Joule heating corrections, the length dependence is minimized resulting in a thermal conductivity of 71 W/(m K).

Thermal conductivity variation with beam length from experimental (*T* = 303 K) data showing the effect of data analysis procedure. The uncorrected data show clear length dependence and underpredict thermal conductivity. With electrical resistance, thermal resistance and Joule heating corrections, the length dependence is minimized resulting in a thermal conductivity of 71 W/(m K).

Measured temperature dependent thermal conductivity of polysilicon (filled circles) compared to reported values in the literature for single crystalline silicon (filled shapes) and polysilicon (unfilled shapes).

Measured temperature dependent thermal conductivity of polysilicon (filled circles) compared to reported values in the literature for single crystalline silicon (filled shapes) and polysilicon (unfilled shapes).

Reported values of doped polysilicon thermal conductivity at 300 K as a function of layer thickness.

Reported values of doped polysilicon thermal conductivity at 300 K as a function of layer thickness.

## Tables

Dimensions of geometric features used in the computational model.

Dimensions of geometric features used in the computational model.

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