Modelling a Resonant Near Field

Scanning Microwave Microscope

(RNSMM) Probe

Nadia Smith, Kevin Lees, Andrew Gregory, Robert Clarke

NPL, National Physical Laboratory, UK

18th September 2014

• Develop Micrometre-scale Near-Field

Scanning Microwave Microscope

(NSMM) metrology.

• Allow traceable measurements of

complex permittivity on micron scales.

• A dielectric resonator (RNSMM) design

may be employed which improves

sensitivity.

• Scan surface complex permittivity of

composites, electro-ceramics,

semiconductors, thin films.

EMINDA Electromagnetic Characterisation of Materials for

Industrial Applications up to Microwave Frequencies

• Develop traceable electromagnetic

and functional materials metrology.

• Enable uptake of these materials

within EU industry, especially

important for electronics and ICT-

related companies.

• NPL research of key advanced

measurement techniques to give

broader infrastructure for EM materials

metrology in Europe.

http://projects.npl.co.uk/eminda/

Modelling a RNSMM

A RNSMM consists of two parts:

• Microwave resonator Dimensions ~ cm

• Probe/tip that interacts with sampleDimensions ~ 100 𝜇𝑚

Simplified model of probe and sample:

• Understand the probe/sample interaction of a RNSMM

• Account for non-flatness and non-uniformity of specimens in RNSMM

measurements

2D simplification

Sphere: 0.1 mm diameter

Cylinder: 0.08 mm diameter

0.32 mm long shaft

Disc: 4 mm diameter

2 mm thickness

Gaps of

interest:

0 - 10 𝝁m

5 mm

Assumption: Probe cylinder is

sufficiently long that resonator and

other parts of the device can be

neglected so that only the probe and

the sample need to be considered.

Governing equations

𝐄 = −∇𝑉

∇ ⋅ 𝐉 = −𝑗𝜔𝜌

𝐉 = 𝜎 + 𝑗𝜔𝜀0𝜀𝑟 𝐄 + 𝐉𝐞

A high frequency alternating current is applied to the probe, which

generates an electric field around the probe.

This field is affected by the sample, and the effect can be sensed by

looking at the current running through the probe.

J: current density

E: electric field

𝜌: charge density

𝜔: angular frequency

𝜀0: permittivity of vacuum

𝜀𝑟: relative permittivity of the material

𝜎: electrical conductivity of the material

𝐉𝐞: externally generated current.

These equations are completed with the following boundary conditions:

• Terminal: V=1 V, on the top boundary of the probe.

• Ground plane: V=0 V on the bottom of the sample.

• Electric insulation: 𝐧 ⋅ 𝐉 = 0 elsewhere.

The result of interest is capacitance of the system, which can be extracted

directly as a lumped parameter from the model.

Use of COMSOL Multiphysics

Required the AC/DC module, specifically the electric currents capability in the

frequency domain.

The use of a terminal node enabled the computation of the lumped parameters

of the system: capacitance and resistance.

Model run at the operating frequencies of the NPL RNSMM: 240 MHz, 1.5 GHz,

and 3.9 GHz.

As the probe is small compared with the microwave wavelength, the

capacitance is effectively frequency independent, so for all the frequencies run

the capacitance results are found to be the same

The mesh in the gap between the sphere and the sample, and in the first few

microns of the sample, has to be very fine.

Sample configurations

to be investigated

Geometric imperfections: Valley / Hill Layered or inhomogeneous materials

Two varying parameters:

• h: height of the dip into (or bump off) the

surface. Range: 1, 3, 6, and 10 µm

• Θ: angle into (or off) the sample.

Range: 30°60°90°

Two varying parameters:

• r and t: set to 1, 2, 5 and 10 times the probe

radius, keeping the overall radius and

thickness of the sample fixed.

• The value of t has also been set to 1 μm to

set up a thin film model.

Analytical solution when the probe is treated as a sphere (no cylinder)

in contact with a sample which is uniform and perfectly flat

Validation against analytical model

-50 -40 -30 -20 -10 0-5

-4

-3

-2

-1

0x 10

4

z, m

Ez,

V/m

Comsol

Analytical

C. Gao and X.-D. Xiang,

Quantative microwave near-field

microscopy ofdielectric properties,

Review of Scientific Instrument, 69

(11), 3846-3851 (1998).

Results and discussion

0 2 4 6 8 10 50005

6

7

8

9

10

11

12

13

Distance between probe/sample (m)

Capacitance (

fF)

UF1 (=1)

UF2 (=3.8)

UF3 (=10.59)

UF4 (=23.9)

UF5 (=300)

Uniform material with a flat surface

0 2 4 6 8 10 5000

6

8

10

12

14

16

18

Distance between probe/sample (m)

Capacitance (

fF)

All M2

r=0.05mm

r=0.1mm

r=0.25mm

r=0.5mm

All M1

Concentric material with a flat surface

M1: 𝜀 = 3.8

M2: copper

5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

Real capacitance C` of probe (fF)

Imagin

ary

capacitance C

`` o

f pro

be (

fF)

Mapping * onto C*

`

``

1 2 3 4 5 6 7 8 9 10

0

1

2

3

45 6 7 8 9 10

5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Real Capacitance (C') of probe (fF)

Imagin

ary

Capacitance (

C'')

of pro

be (

fF)

Mapping * onto C*

`

tan

1 2 3 4 5 10 20 30 50 100 200 3000

0.2

0.4

0.6

0.8

1

tan

Mappings from the

complex permittivity

plane to the complex

capacitance plane for

lossy dielectrics

(imaginary permittivity

is non-zero)

Enables to read off

the material

complex

permittivity

corresponding to a

given complex

capacitance.

Summary

Very useful tool to provide insight into the effects of likely

imperfections in the real measurement technique.

Modelling studies of this kind can help answer questions such as:

• how accurately does the separation between the probe and

the sample need to be known?

• how flat does the surface need to be?

• if the sample is composed of two materials, how accurately

can the permittivity of either of them be measured?

Simple capacitive models of probe/sample interaction cannot

capture many of the effects described above, and this fact in turn

demonstrates the importance of this form of modelling for

RNSMM metrology to be made fully traceable.

Acknowledgments

This project was funded by the ‘EMINDA’ EURAMET

joint research project which received its funding from

the European Community’s Seventh Framework

Programme, ERA-NET Plus.