^{1,a)}and A. Shakouri

^{1,b)}

### Abstract

The paper discusses the possibility to apply network identification by deconvolution (NID) method to the analysis of the thermal transient behavior due to a laser delta pulse excitation in a pump-probe transient thermoreflectance experiment. NID is a method based on linear RC network theory using Fourier’s law of heat conduction. This approach allows the extraction of the thermal time constant spectrum of the sample under study after excitation by either a step or pulse function. Furthermore, using some mathematical transformations, the method allows analyzing the detail of the heat flux path through the sample, starting from the excited top free surface, by introducing two characteristic functions: the cumulative structure function and the differential structure function. We start by a review of the theoretical background of the NID method in the case of a step function excitation and then show how this method can be adjusted to be used in the case of a delta pulse function excitation. We show how the NID method can be extended to analyze the thermal transients of many optical experiments in which the excitation function is a laser pulse. The effect of the semi-infinite substrate as well as extraction of the interface and thin film thermal resistances will be discussed.

The authors would like to thank Professor S. Dilhaire and K. Fukutani for their valuable help and enlightening discussions. This work was supported by the Interconnect Focus Center, one of the five research centers funded under the Focus Center Research Program, a DARPA and Semiconductor Research Corporation program (Grant No. 59771-444040).

I. INTRODUCTION

II. THEORY

A. Lumped element modeling

B. Step power excitation

C. Delta pulse power excitation

D. Temperature variation at the top free surface of the structure under study

III. RESULTS AND DISCUSSION

A. Effect of the time range in thermal transient analysis

B. Effect of the starting time of the thermal transient measurement

C. Effect of a semi-infinite substrate on the NID results

D. Effect of the metal/SC layer interface thermal resistance on the NID results

E. Effect of normalization on the NID results

F. What is the time range needed in a PPTTR experiment to extract the thermal properties of the thin SC layer using NID method?

G. Can we use NID method to extract simultaneously both the thermal conductivity of the SC layer and the metal/SC layer interface thermal resistance from a single PPTTR experimental signal?

IV. SUMMARY

### Key Topics

- Thermal analysis
- 15.0
- Interface structure
- 14.0
- Metallic thin films
- 14.0
- Silicon
- 14.0
- Thermal properties
- 14.0

## Figures

Schematic diagram of the RC one-port circuit of one layer (a) and the full structure (b).

Schematic diagram of the RC one-port circuit of one layer (a) and the full structure (b).

Schematic diagram of the sample structures with a finite size silicon substrate [case (a)] and a semi-infinite silicon substrate [case (b)].

Schematic diagram of the sample structures with a finite size silicon substrate [case (a)] and a semi-infinite silicon substrate [case (b)].

(a) Calculated 1 W normalized temperature transient rise over a time range of 660 ns with a time resolution of 10 ps after an application of a step power function to the top free surface of the two structures, with SiGe alloy (solid line) and Si/SiGe SL (dashed line), deposited on a finite size substrate. is assumed to be zero at the interface metal transducer/SC layer. (b) Calculated temperature decay over the same time range with the same time resolution after application of a delta power function of amplitude of to the top free surface of the same two structures.

(a) Calculated 1 W normalized temperature transient rise over a time range of 660 ns with a time resolution of 10 ps after an application of a step power function to the top free surface of the two structures, with SiGe alloy (solid line) and Si/SiGe SL (dashed line), deposited on a finite size substrate. is assumed to be zero at the interface metal transducer/SC layer. (b) Calculated temperature decay over the same time range with the same time resolution after application of a delta power function of amplitude of to the top free surface of the same two structures.

TCS of the two structures with finite thickness substrate for both step and delta functions excitations over a time range of 660 ns with 10 ps time resolution and starting at 10 ps. Log-lin representation (a) and log-log representation (b).

TCS of the two structures with finite thickness substrate for both step and delta functions excitations over a time range of 660 ns with 10 ps time resolution and starting at 10 ps. Log-lin representation (a) and log-log representation (b).

Cumulative structure functions of the two structures with finite thickness substrate for both step and delta functions excitations over a time range of 660 ns with 10 ps time resolution and starting at 10 ps.

Cumulative structure functions of the two structures with finite thickness substrate for both step and delta functions excitations over a time range of 660 ns with 10 ps time resolution and starting at 10 ps.

Differential structure functions of the two structures with finite thickness substrate for both step and delta functions excitations over a time range of 660 ns with 10 ps time resolution and starting at 10 ps.

Differential structure functions of the two structures with finite thickness substrate for both step and delta functions excitations over a time range of 660 ns with 10 ps time resolution and starting at 10 ps.

TCS in the log-log representation of the structure with Si/SiGe SL layer on a finite size substrate for both step (a) and delta (b) functions excitations over different time ranges: 660 ns (solid line), 500 ns (solid-dashed line), 100 ns (dashed line), 50 ns (short-dashed line), and 13 ns (dotted line) with 10 ps time resolution and starting at 10 ps.

TCS in the log-log representation of the structure with Si/SiGe SL layer on a finite size substrate for both step (a) and delta (b) functions excitations over different time ranges: 660 ns (solid line), 500 ns (solid-dashed line), 100 ns (dashed line), 50 ns (short-dashed line), and 13 ns (dotted line) with 10 ps time resolution and starting at 10 ps.

Differential structure functions of the structure with Si/SiGe SL layer on a finite size substrate for both step function excitation (a) and delta function excitation (b) over a time range of 660 ns with 10 ps time resolution and starting at different times of 10 ps (solid line), 100 ps (solid-dashed line), and 500 ps (dashed line).

Differential structure functions of the structure with Si/SiGe SL layer on a finite size substrate for both step function excitation (a) and delta function excitation (b) over a time range of 660 ns with 10 ps time resolution and starting at different times of 10 ps (solid line), 100 ps (solid-dashed line), and 500 ps (dashed line).

Comparison between the analytically calculated input transient temperature rise after application of a step excitation (a), the integrated (b) and the raw (c) analytically calculated input transient temperature decay after application of a delta excitation, with the reconstructed new transient temperature rise [(a) and (b)] and decay (c) signal based on NID results for the studied structure with Si/SiGe SL layer on a finite size substrate over a time range of 660 ns with 10 ps time resolution and starting at different times of 10 ps (open circles), 100 ps (open squares), and 500 ps (open triangles). In each figure, represents the number of the RC one-ports used for the discretization of the corresponding TCS.

Comparison between the analytically calculated input transient temperature rise after application of a step excitation (a), the integrated (b) and the raw (c) analytically calculated input transient temperature decay after application of a delta excitation, with the reconstructed new transient temperature rise [(a) and (b)] and decay (c) signal based on NID results for the studied structure with Si/SiGe SL layer on a finite size substrate over a time range of 660 ns with 10 ps time resolution and starting at different times of 10 ps (open circles), 100 ps (open squares), and 500 ps (open triangles). In each figure, represents the number of the RC one-ports used for the discretization of the corresponding TCS.

Differential structure functions of the structure with Si/SiGe SL layer on a semi-infinite substrate for both step function excitation (a) and delta function excitation (b) over different time ranges: 500 ns (solid line), 100 ns (solid-dashed line), 50 ns (dashed line), and 13 ns (short-dashed line) with 10 ps time resolution and starting at 10 ps.

Differential structure functions of the structure with Si/SiGe SL layer on a semi-infinite substrate for both step function excitation (a) and delta function excitation (b) over different time ranges: 500 ns (solid line), 100 ns (solid-dashed line), 50 ns (dashed line), and 13 ns (short-dashed line) with 10 ps time resolution and starting at 10 ps.

TCS of the structure with Si/SiGe SL layer on both finite and semi-infinite substrates with and . A delta function excitation is applied over two time ranges, 500 ns (a) and 13 ns (b), with 10 ps time resolution and all starting at 10 ps.

TCS of the structure with Si/SiGe SL layer on both finite and semi-infinite substrates with and . A delta function excitation is applied over two time ranges, 500 ns (a) and 13 ns (b), with 10 ps time resolution and all starting at 10 ps.

Cumulative structure functions of the structure with Si/SiGe SL layer on both finite and semi-infinite substrates with and . A delta function excitation is applied over two time ranges, 500 ns (a) and 13 ns (b), with 10 ps time resolution and all starting at 10 ps.

Cumulative structure functions of the structure with Si/SiGe SL layer on both finite and semi-infinite substrates with and . A delta function excitation is applied over two time ranges, 500 ns (a) and 13 ns (b), with 10 ps time resolution and all starting at 10 ps.

Cumulative structure functions of the structure with Si/SiGe SL layer on both finite and semi-infinite substrates with [(a) and (b)] and [(c) and (d)] for raw and normalized delta function excitation signals over two time ranges, 500 ns [(a) and (c)] and 13 ns [(b) and (d)], with 10 ps time resolution and all starting at 10 ps. (e) Comparison between the cumulative structure functions of normalized signals for and .

Cumulative structure functions of the structure with Si/SiGe SL layer on both finite and semi-infinite substrates with [(a) and (b)] and [(c) and (d)] for raw and normalized delta function excitation signals over two time ranges, 500 ns [(a) and (c)] and 13 ns [(b) and (d)], with 10 ps time resolution and all starting at 10 ps. (e) Comparison between the cumulative structure functions of normalized signals for and .

Cumulative structure functions of the structure with Si/SiGe SL layer on a semi-infinite substrate with for a delta function excitation over different time ranges, (solid line), (solid-dashed line), (dashed line), (short-dashed line), and (dotted line) with 1 ps time resolution and all starting at 1 ps. (a) 150 nm thick and 10 W/m/K thermal conductivity SL. (b) 150 nm thick and 15 W/m/K thermal conductivity SL. (c) 100 nm thick and 15 W/m/K thermal conductivity SL.

Cumulative structure functions of the structure with Si/SiGe SL layer on a semi-infinite substrate with for a delta function excitation over different time ranges, (solid line), (solid-dashed line), (dashed line), (short-dashed line), and (dotted line) with 1 ps time resolution and all starting at 1 ps. (a) 150 nm thick and 10 W/m/K thermal conductivity SL. (b) 150 nm thick and 15 W/m/K thermal conductivity SL. (c) 100 nm thick and 15 W/m/K thermal conductivity SL.

(a) Temperature decays at the top free surface of the structure with an 80 nm thick Si/SiGe SL layer deposited on a semi-infinite silicon substrate and covered by a 30 nm thick Al film with (solid line), (solid-dashed line), and (dashed line) after excitation with a delta laser pulse function of energy of . Cumulative structure functions (b) and differential structure functions (c) corresponding to the temperature decays in (a).

(a) Temperature decays at the top free surface of the structure with an 80 nm thick Si/SiGe SL layer deposited on a semi-infinite silicon substrate and covered by a 30 nm thick Al film with (solid line), (solid-dashed line), and (dashed line) after excitation with a delta laser pulse function of energy of . Cumulative structure functions (b) and differential structure functions (c) corresponding to the temperature decays in (a).

## Tables

Geometrical and thermal properties as well as the calculated thermal resistances and capacitances of the different layers in the structures under study [case (a)].

Geometrical and thermal properties as well as the calculated thermal resistances and capacitances of the different layers in the structures under study [case (a)].

Extracted thermal resistances and capacitances based on NID results for the case of a finite size silicon substrate [case (a)].

Extracted thermal resistances and capacitances based on NID results for the case of a finite size silicon substrate [case (a)].

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