Granular and columnar nickel–carbon composites may exhibit large strain sensitivity, which makes them an interesting sensor material. Based on experimental results and morphological characterization of the material, we develop a model of the electron transport in the film and use it to explain its piezoresistive effect. First we describe a model for the electron transport from particle to particle. The model is then applied in Monte Carlo simulations of the resistance and strain properties of the disordered films that give a first explanation of film properties. The simulations give insights into the origin of the transverse sensitivity and show the influence of various parameters such as particle separation and geometric disorder. An important influence towards larger strain sensitivity is local strain enhancement due to different elastic moduli of metal particles and carbon matrix.

In our associated paper, part 1

Nickel–carbon films offer high stability and large gauge factors
(

The morphology was analyzed by means of transmission electron
microscopy (TEM). The films consist of columnar nickel particles
encapsulated in several atomic layers of graphite-like
carbon. A schematic diagram of the columnar structure is shown in
Fig.

To understand the strain sensing properties of the granular film,
we first consider the charge-transport mechanism. Due to the
heterogeneous conductivity, i.e., highly conductive metal
particles and relatively poorly conductive barriers of carbon,
electrons will tunnel between metal particles. The nature of the
piezoresistive effect in granular metals is the following: strain
affects electron tunneling distances, leading to changes in
conductivity and thus results in strain sensitivity

This effect occurs in several composite materials with local
regions of different conductivity, e.g., in diamond-like carbon
(

Although all of the materials mentioned above have been investigated
in terms of their strain sensitivity in the longitudinal direction,
only in some cases has this been extended to the transverse
sensitivity
(

Simplified representation of a columnar nickel–carbon film on a substrate.

We set out to find a model for the thin film that allows us to explain
the enhanced longitudinal and transverse gauge factors and some
parameters they depend on. First, mechanisms of electron transport
are investigated. Then, we propose a model for the geometric
structure of the material. For this, we represent the columnar
geometry (Fig.

Literature on granular metals describes their electrical
conductivity for several domains depending on temperature and
electrical coupling (field strength)

For a simplified approach at a given temperature, the conductive
mechanism can be described by the Arrhenius-form tunneling conductivity

For a simple model approach, we consider a granular metal–matrix system with a matrix material that has an isotropic resistivity. The metallic intra-particle resistance is lower than the matrix resistance by several orders of magnitude, as evidenced by the much higher resistivity of metal–carbon samples compared to purely metallic thin films. Thus, the conductance across the sample can be viewed as electron transport from one metal particle to the next through an interjacent area of carbon.

Percolated films with a large metal content will behave like metallic
films. They have gauge factors of

The effective conductance between two neighboring particles is the
sum of the conductance through the matrix material and the electron
tunneling conductance between the particles. Both conductivities
depend on the particle separation distance

Resistor model from one particle to the next.

Hexagonal cluster networks in different orientations

In terms of an electrical resistance network, the tunneling
resistance

The particle resistance

The resistance calculation accounts for the strain influence on
the particle separation. Our model does not cover the gauge factor
of a significantly percolated film with its metallic conduction
– as the resistance will be constant in this case and

Finally, the gauge factor can be derived from the resistance law

TEM image and model representation of the disordered film.

Analytical results for a single particle-to-particle junction.

To consider not only the electrical properties described
previously but also the heterogeneous mechanical properties of the
metal–carbon composite material and the geometric disorder of
particles in the film, a global model built from a network of
particles will be derived. We assume that a large number of
columnar metal particles with some given diameter is arranged in
a certain pattern (see Fig.

Using the TEM images we can find a mean separating distance

The direction of the particle-to-next-neighbor paths does not follow any particular order and is assumed to be equally distributed for all directions for any sufficiently large area of the thin film.

A regular hexagonal grid includes preferred directions for
electron transport, which might exist locally in the actual thin
film, but are not found on a large scale. To account for this, we
introduce the angle

The circular column model is an approximation of the real material. If the model geometry is chosen with particle separation distances equal to the actual material, larger void areas appear compared to the slightly irregularly-formed columns of the real sample. This does not affect the simulation results as far as gauge factors are concerned, but introduces an offset of the metal content: for the same separation distances, the model film will have a lower metal content than the real material.

When a strain is applied to the network, a softer matrix material
will result in locally enhanced strain

We now have a model that contains a representation of the film's geometrical structure, its mechanical response to strain, and the particle-to-particle electrical conduction. It allows us to find results for the longitudinal and transverse gauge factor for columnar metal-in-insulator films for different particle diameters and separation distances.

The numerical model is set up according to the following steps:

Randomly generate normal (or log-normal) distribution of column diameters

Generate coordinates

For each column, find the six nearest columns.

Add virtual nodes that represent the electrodes at

Resistance evaluation. Perform the following calculations for (i) an unstrained system and (ii) all strained systems of interest, e.g.
longitudinal strain

Calculate the inter-particle resistance

Build a system of linear equations for this resistor network.

Calculate the total resistivity using the Gauss–Jordan algorithm.

Repeat for different strains.

Calculate gauge factors

By simply modifying the coordinate-generating algorithm in step 1, different models for the particle distribution (two- or three-dimensional) can be implemented. All other steps can be carried out unchanged.

The numerical calculation is implemented in Python 2.7 using the

In order to evaluate the resistance and gauge factor of the randomly
distributed metal particles, the steps laid out above are repeated

The gauge factor values found in these random experiments have
a rather large variance. The idea of the Monte Carlo simulation is
that this uncertainty can be reduced by many repetitions and
subsequent averaging of the value. Thus, we can simply take the
mean value of each of the gauge factors

Several model parameters for the simulations are chosen based on physical reasoning; remaining parameters that are unknown or uncertain are found by parameter fitting.

The mean diameter

For the electrical parameters, the particle resistance is found by
assuming that on average, electrons travel through the diameter of
the column and traverse a small part of the column vertically
before the next inter-column transport occurs. With a path length
of

For the tunneling mechanism, the exponential coefficient

Parameters

The strain applied to the thin film is a global value. Locally, on
the scale of the metal columns and separation walls, we expect the
strain to be inhomogeneous because of their different elastic
moduli. In literature, there is a common value for bulk nickel:

For the calculation, the film is subjected to uniaxial strain. This
is equivalent to straining the film by bending a sample onto
a constant radius as described in part 1

The globally applied strain used for calculations is typically

The strained system for our calculations is then derived by
changing the individual separation distances

After comparing results with an increasing number of particles (

The number of repetitions

Simulations are carried out as described earlier. As expected, the
total resistance of particle networks is increased when the system
is subjected to a longitudinal strain

First, we demonstrate the resistance law and its characteristics by plotting gauge factors vs. mean particle separation for different distributions; then the effect of elastic moduli of the materials and the influence of particle diameters will be shown in simulation results of gauge factors vs. metal content.

Strain changes separation distances between particles and thus
affects resistances between particles. From the resistance law

In Fig.

Looking at the gauge factor dependence on the mean particle
separation

With small

Only in the medium region around 2.3

Large

Longitudinal and transverse gauge factors as a function of mean particle
separation, shown for different standard deviations (SD) of

In addition to the analytical function, the simulation allows us to
analyze the transverse gauge factor

The

In the

The

Directions of local current flow in a hexagonal particle arrangement. For
a global current in the

Longitudinal and transverse gauge factors for different elastic moduli of
matrix (

Mechanical strain enhancement for incompressible metal particles, i.e.,

The changes in the

The results show that even for a hexagonal grid without any
disorder in diameter (

With a growing disorder due to varying diameters, the transverse sensitivity is enhanced, because detours within the possible conduction paths are becoming increasingly favorable. Since straight paths are more and more likely to contain some increased separating distances, detours along shorter separating distances become more viable.

It becomes apparent that a transverse sensitivity ratio of

Gauge factors

The influence of elastic moduli of matrix (

The underlying effect is a mechanical and geometrical one: for

The column diameter of about 15

If we assume a constant column diameter for all values of metal
content

Results by

In Fig.

Compared to the constant-diameter example, this result better
reflects our experiments (

Still, the gauge factors obtained experimentally for low metal
content (

An experimentally found characteristic of our nickel–carbon films
is a gauge factor maximum vs. metal content with a relatively wide
range of elevated gauge factors. At the same time, a substantial
transverse sensitivity ratio in the range of 0.5 is seen. These
general properties are reproduced by simulations of the strain
sensitivity for a relative SD of tunneling
distances of about 10

The base value of the transverse sensitivity ratio for a film in
a two-dimensional, perfectly hexagonal particle arrangement is

The model highlights the possible gauge factor enhancement caused
by mechanical properties of the matrix material: with a relatively
low elastic modulus of matrix vs. metal particles, significant
local strain enhancement will occur. This amplified change in
separation distance will result in a larger gauge factor

As has been found experimentally before, the model shows that the
particle separation distances should be carefully tuned by choice
of metal content to achieve the optimum of possible gauge
factors. The simulated

Simulation code and a corresponding “Readme” file with instructions are availably in the Supplement to this article.

All plotted data are available in the Supplement to this article.

The supplement related to this article is available online at:

UW performed TEM analysis and developed the initial version of the simulation. GS contributed to the model and simulation in many discussions and helped preparing the manuscript. SS implemented the simulation and prepared the manuscript.

The authors declare that they have no conflict of interest.

The authors thank Michael Huth (Goethe University Frankfurt) for helpful discussions. This work was funded by the Federal Ministry of Education and Research of Germany under the “IngenieurNachwuchs” program (project “Nanocermet”, funding reference number: 03FH006IX4). Edited by: Ryutaro Maeda Reviewed by: two anonymous referees