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
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.
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 3/22
Functionality of 3ω-method
Alternating current
substr
ate
heate
r
I
..cosω..�� ��I ω
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 4/22
Functionality of 3ω-method
Alternating current
substr
ate
heate
r
I
..cosω..�� ��I ω
Joule heatingP = R · I2
cos2=..cos 2ω..�� ��P 2ω
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 5/22
Functionality of 3ω-method
Alternating current
substr
ate
heate
r
I
..cosω..�� ��I ω
Joule heatingP = R · I2
cos2=..cos 2ω..�� ��P 2ω
Temperature amplitude ∆TAnalytic solutionResistance oscillationFE-Simulation
�� ��R =�� ��R0 +�� ��∆R 2ω
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 6/22
Functionality of 3ω-method
Alternating current
substr
ate
heate
r
I
..cosω..�� ��I ω
Joule heatingP = R · I2
cos2=..cos 2ω..�� ��P 2ω
Temperature amplitude ∆TAnalytic solutionResistance oscillationFE-Simulation
�� ��R =�� ��R0 +�� ��∆R 2ω
Modulated voltageU = I · R
substr
ate
heate
r
V
cosω · cos2ω =cosω · cos3ω�� ��ω 3ω
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 7/22
Functionality of 3ω-method
Alternating current
substr
ate
heate
r
I
..cosω..�� ��I ω
Joule heatingP = R · I2
cos2=..cos 2ω..�� ��P 2ω
Temperature amplitude ∆TAnalytic solutionResistance oscillationFE-Simulation
�� ��R =�� ��R0 +�� ��∆R 2ω
Modulated voltageU = I · R
substr
ate
heate
r
V
cosω · cos2ω =cosω · cos3ω�� ��U ω 3ω
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 8/22
Functionality of 3ω-method
Alternating current
substr
ate
heate
r
I
..cosω..�� ��I ω
Joule heatingP = R · I2
cos2=..cos 2ω..�� ��P 2ω
Temperature amplitude ∆TAnalytic solutionResistance oscillationFE-Simulation
�� ��R =�� ��R0 +�� ��∆R 2ω
Modulated voltageU = I · R
substr
ate
heate
r
V
cosω · cos2ω =cosω · cos3ω�� ��U ω 3ω
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 9/22
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
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 10/22
Strategy for nanostructured materials
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 11/22
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
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 12/22
Geometry import to generate mesh
Example for 2d:
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 13/22
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
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 14/22
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
dlay = 100 nm shape2
dlay = 100 nm shape3
dlay = 100 nm shape4
How to enhance the sensitivity for change in κlay?
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 15/22
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
dlay = 100 nm shape1
dlay = 100 nm shape2
dlay = 100 nm shape3
dlay = 100 nm shape4
How to enhance the sensitivity for change in κlay?
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 16/22
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
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 17/22
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 platform
dlay = 400 nmYSZ substrate
PCMO layer
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 18/22
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 platform
dlay = 400 nm
dlay = 400 nm, sink 2000 nm
YSZ substrate
PCMO layer
Cu heat sink
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 19/22
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.
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 20/22
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!
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 21/22
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 22/22
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))
Introduction Functionality of 3ω-method Geometry configurations Numerical analysis Outlook
Manuel Feuchter, Marc Kamlah – Comsol Conference 2011 October 26, 2011 23/22