Numerical analysis of the 3 -method - COMSOL Multiphysics · PDF fileInstitute for Applied...

Institute for Applied Materials (IAM)

Numerical analysis of the 3ω-methodComsol Conference, Stuttgart

Manuel Feuchter, Marc Kamlah | October 26, 2011

KIT – University of the State of Baden-Wuerttemberg andNational Laboratory of the Helmholtz Association

www.kit.edu

http://www.kit.edu

Outline

1 Introduction

2 Functionality of 3ω-method

3 Geometry configurations

4 Numerical analysis

5 Outlook

Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook

Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 2/22

Introduction

substra

te

heate

r

A

Concept of the 3ω-method :

The 3ω-method is an ac technique to determine the thermal conductivityof an amorphous solid.

The 3ω-measurement is performed on a thin electrical conducting metalstrip (heater) deopsited on the sample surface.

Literature: [Cahill & Pohl 1987], [Cahill 1990], [Lee & Cahill 1997], [Kim& Feldman 1999], [Borca-Tasciuc 2001], [Chen 2004], [Olson 2005]...

Why do we work on the 3ω-method ?

Involved in DFG SPP 1386 - Understanding size and interface dependentanisotropic thermal conduction in correlated multilayer structures.

Goal: Minimize thermal conductivity in nanostructured thermoelectricmaterials.

Reliable measurement technique is required.



Functionality of 3ω-method

Alternating current

substr

ate

heate

r

I

..cosω..�� I ω




Alternating current

substr

ate

heate

r

I


Joule heatingP = R · I2

cos2=..cos 2ω..�� P 2ω




Alternating current

substr

ate

heate

r

I




Temperature amplitude ∆TAnalytic solutionResistance oscillationFE-Simulation

�� R =�� R0 +�� ∆R 2ω




Alternating current

substr

ate

heate

r

I





�� R =�� R0 +�� ∆R 2ω

Modulated voltageU = I · R

substr

ate

heate

r

V

cosω · cos2ω =cosω · cos3ω�� ω 3ω




Alternating current

substr

ate

heate

r

I





�� R =�� R0 +�� ∆R 2ω


substr

ate

heate

r

V

cosω · cos2ω =cosω · cos3ω�� U ω 3ω



Geometry configurations

Classic geometry configuration:

Heater is placed on top of (multi)layer-substrate systems.Analytic solutions are available.

Inverse geometry configuration:

New possibilities to determine the thermal conductivity of nanostructuredmaterials.Always the same heater-substrate platform (A)as sample holder.

Known system properties.Optimum reproducibility.

No analytic solutions are available.substra

te

heate

r

A

B

C

D

E



Strategy for nanostructured materials



Finite Element SimulationsCOMSOL Multiphysics

Heat conduction equation has to be solved in the time domain.

Mathematic interface is used in the general form of the partial differentialequation (PDEg).

Example for 2d:

∂2Th

∂x2+∂2Th

∂y 2− 1

Dh

∂Th

∂t+

Q

κh= 0 ,

∂2Ts

∂x2+∂2Ts

∂y 2− 1

Ds

∂Ts

∂t= 0 ,

κmx

∂2Tm

∂x2+ κmy

∂2Tm

∂y 2− ρmcpm

∂Tm

∂t= 0

Q = [%0 · (1 + α · (T − T0))]/(A)

· I 20 · 1

2· (1 + cos(2ωt))

x

y

ab

κn∂Ts

∂n− hbc · (T − T0) = 0



Geometry import to generate mesh

Example for 2d:



Examples

Placed material on top, ideal system:

Two times substrate, contact just over heater.To study the influence of material parameters onthe temperature amplitude.

∆T

[K]

0

0,05

0,1

0,15

0,2

0

0,05

0,1

0,15

0,2

Frequency fc [Hz]

1e+01 1e+02 1e+03 1e+04 1e+05 1e+06

10 100 1.000 10k 100k 1M

Dm · 1

Dm · 2

Dm · 4

Dm · 8

Dm · 1/2

Dm · 1/4

Dm · 1/8

D = κρ·cp

∆T

[K]

0

0,02

0,04

0,06

0,08

0,1

0,12

0

0,02

0,04

0,06

0,08

0,1

0,12

Frequency fc [Hz]

1e+01 1e+02 1e+03 1e+04 1e+05 1e+06

10 100 1.000 10k 100k 1M

Dm · 1

Dm · 2

Dm · 4

Dm · 8

Dm · 1/2

Dm · 1/4

Dm · 1/8

D = κρ·cp



Examples

Thin layer on top:

Bad conducting substrate for asensitivity in ∆T .

Contour shape analysis.

x

y

b

substr

ate

heate

rlayer

dlaya

dsubs

By S. Wiedigen, project partner

∆T

[K]

0,0

1,0

2,0

3,0

4,0

0,0

1,0

2,0

3,0

4,0

Frequency fc [Hz]

10 100 1.000 10.000 100.000 1.000.000

10 100 1.000 10k 100k 1M

dlay = 100 nm

dlay = 200 nm

dlay = 400 nm

dlay = 800 nm

dlay = 1000 nm

STO layer

∆T

[K]

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

Frequency fc [Hz]

1e+01 1e+02 1e+03 1e+04 1e+05 1e+06

10 100 1.000 10k 100k 1M

dlay = 100 nm shape1




How to enhance the sensitivity for change in κlay?



Examples

Thin layer on top:

Bad conducting substrate for asensitivity in ∆T .

Contour shape analysis.

x

y

b

substr

ate

heate

rlayer

dlaya

dsubs

By S. Wiedigen, project partner

∆T

[K]

0,0

1,0

2,0

3,0

4,0

0,0

1,0

2,0

3,0

4,0

Frequency fc [Hz]

10 100 1.000 10.000 100.000 1.000.000

10 100 1.000 10k 100k 1M

dlay = 100 nm

dlay = 200 nm

dlay = 400 nm

dlay = 800 nm

dlay = 1000 nm

PCMO layer

∆T

[K]

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

Frequency fc [Hz]

1e+01 1e+02 1e+03 1e+04 1e+05 1e+06

10 100 1.000 10k 100k 1M





How to enhance the sensitivity for change in κlay?



Examples

Application of heat sink:

Heat sink: Good thermal conducting toattract heat.

Full covering heat sink.

x

y

b

substrate

heaterlayerdlaya

dsubs

heat sink

dsink

∆T

[K]

0,0

1,0

2,0

3,0

4,0

0,0

1,0

2,0

3,0

4,0

Frequency fc [Hz]

10 100 1.000 10.000 100.000 1.000.000

10 100 1.000 10k 100k 1M

Heater-substrate platformYSZ substrate



Examples




x

y

b

substrate

heaterlayerdlaya

dsubs

heat sink

dsink

∆T

[K]

0,0

1,0

2,0

3,0

4,0

0,0

1,0

2,0

3,0

4,0

Frequency fc [Hz]

10 100 1.000 10.000 100.000 1.000.000

10 100 1.000 10k 100k 1M

Heater-substrate platform

dlay = 400 nmYSZ substrate

PCMO layer



Examples




x

y

b

substrate

heaterlayerdlaya

dsubs

heat sink

dsink

∆T

[K]

0,0

1,0

2,0

3,0

4,0

0,0

1,0

2,0

3,0

4,0

Frequency fc [Hz]

10 100 1.000 10.000 100.000 1.000.000

10 100 1.000 10k 100k 1M

Heater-substrate platform

dlay = 400 nm

dlay = 400 nm, sink 2000 nm

YSZ substrate

PCMO layer

Cu heat sink



Outlook

Heat flow analysis for 3d structures.

Multiphysical coupling with respect to mechanical and dielectric properties.

Bottom electrode configurations with additional heaters to measureSeebeck cross-plane.



Acknowledgement:

DFG for funding.

Project partners from the University of Gottingen,Institute for material physicsGroup of C. Jooss and C. Volkert

Supervisor M. Kamlah

THANK YOU FOR YOUR ATTENTION!



Attachment

Alternating current I = I0 cos(ωt)

Joule heating PJH = P02

(1 + cos(2ωt))

Temperature oscillation ∆T(t) = ∆T cos(2ωt + ϕ)

Resistance oscillation R = R0 + ∆R cos(2ωt + ϕ)

Modulated voltage U = I · R

U = I0R0 cos(ωt) + I0∆R2

(cos(3ωt + ϕ) + cos(ωt + ϕ))



Numerical analysis of the 3 -method - COMSOL Multiphysics · PDF fileInstitute for Applied...

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