Nearly isotropic piezoresistive response due to charge detour conduction in nanoparticle thin films

Piezoresistive responses of nanoparticle thin-film strain sensors on flexible polyimide substrates were studied. Disordered interparticle tunneling introduces microscopic detour of charge conduction so as to reduce gauge factors. The disorder also results in large resistance change when current flows in the direction perpendicular to a unidirectional strain, reducing response anisotropy. For practical usages, stability and endurance of these strain sensors are confirmed with 7 × 104 bending cycles. Cracks form in devices under prolonged cyclic bending and slightly reduce gauge factor.


I. The longitudinal and transverse gauge factors for a trangular NP network.
It is worthy to briefly analyze the anisotropic repsonse of NP monolayer under a unidirectional strain by accounting the irregularity of the NP network. The direction of interparticle tunneling and of the strain would always have an angle θ (Figure S1a).
Therefore the parallel and perpendicular projections of interparticle position vector to the strain direction respectively become ! a  = a cosθ 1+ε in which a = 2r + s is the center-to-center distance of the two neighboring NPs (Figure S1b). Assuming that r does not change under the stress, one has Δs = " a  + " a ⊥ − a  cos 2 θ − sin 2 θν ( ) εa .
Again because of the randomness of NP network, the tunneling direction and the bias direction in general may have an angle φ as shown in Figure S1c. φ would vary for different tunneling events, so it is convinient to define a distribution function ( ) φ P for describing the macroscopic current flow. For NPs in a triangular lattice, there are 6 nearest neighbors and each neighbor occupies an angle of about π/3 (Figure S1d). An estimation for φ distribution function is uniform centered by the bias direction with a cutoff angle φ c  π 6 as shown in Figure S1e. Under this assumption, we calculate the device resistance change as Therefore when the current direction and strain direction are parallel, namely θ = φ , the gauge factor reads When the strain direction is perpendicular to the current direction, one has θ = π 2 − φ , and Here g 0 = β 2r + s ( ) is the gauge factor under an isotropic strain. The assumption ν = 0.3 gives g  = 0.89g 0 and g ⊥ = −0.19g 0 ; a negative resistance change in the perpendicular direction and a large ratio g  g ⊥ ≈ 5 is expected. By choosing ν = 0, we could get positive resisitance change in both directions, g  = 0.91g 0 and g ⊥ = 0.087g 0 , and a large anisotropy ratio g  g ⊥ 10 is also expected.   The optical image of "12 μm and 4 μm-devices" on a same chip. We note that the conductivity of the Au electrodes is much higher than that of the AuNP film. Although the NP film covers the whole chip, the small region bounded by the 4 electrodes contributes mostly the device resitance. (e) The optical image of a 10 μm-device. The AuNP film is patterned using optical lithography and standard lift-off technique.
Significant cracks form in the electrodes after bending test.

III. Film thickness measured by atomic force microscopy
Thickness of AuNP films was checked by using atomic force microscopy (AFM, Caliber, Veeco Instruments). Figure S3 shows a typical AFM image of a MPA-modified AuNP film. This image was scanned near the edge of a man-made scrape of the film. Its cross-sections show that film thickness is about 35 nm, i.e., about 3 layers of 12-nm nanoparticles.

IV. Determination of interpartical spacing
Due to insulating PI substrate, it is difficult to get a high resolution image using electron microscopy. Nevertheless, the interparticle spacing can be determined by samples fabricated on silicon substrates using the same deposition methods. In Figure   S4(a), we present the transmission electron microscopy (TEM) image of the deposited MUA-AuNPs on carbon coated TEM grid. The interparticle spacing, determined by using the high-resolution images yield a histrogram as illustrated in Figure S4(b), with an average value of 1.9 nm. We note that the resistivity of AuNP films deposited on PI substrate is similar to that of films deposted on silicon substrate. Therefore the interparticle spacing in the two cases should be similar.

VIII. The anisotrpic response
Video named "xy bending test.mov" shows the anisotropic response of the devices under a unidirectional strain. To demonstrate this, we fabricated two strain sensors, whose current flow directions are parallel to the short side (blue device) or long side (red device) of one chip. A multimeter (Keithley 2400, USA) was used to register the device current at a constant voltage bias (0.1 V). The blue device has a larger response when the chip is bended along the short side. On the contrary, the red device repsonses lager when the bending direction is along the long side.

IX. Theretical analysis on the detour picture
To gain more understanding on this detour picture, here we introduce a simple theoretical approach encompassing disorder in interparticle tunneling to derive the longitudinal and transverse responses to the unidirectional strain. First, we assume the charge conduction can be fully described by a continuous electric potential, φ in two-dimensional (2D) square domain: x, y = 0 -L, with L the sample size. Second, we assume that the charge conduction follows the Ohm's law with a conductivity matrix, σ. In general, the local charge current density follows with i(j)= x and y. In absence of magnetic field, the off-diagonal elements of the conductivity matrix vanish. The current conservation requires that Here we used the contraction convention by dropping the summation symbol i ∑ and used the notation that ∂ ∂x i → ∂ i and σ ii → σ i .
When there is strain, ε applied in direction-"1" ("1"=x or y), the conductivity changes as follows, δσ 1 = g 0 εσ 1 = ασ 1 and δσ 2 = 0 . Here for convenience we assume Poisson's ratio ν = 0 . The change in the conductivity results in the change in potential, δφ by the current conservation.
In the homogenous case that σ x = σ y and ∂ x σ x = ∂ y σ y = 0 , the potential follows the Laplace equation ∂ i 2 φ = 0 . The gauge factors are g  = δ J  ε J = g 0 and g ⊥ = δ J ⊥ ε J = 0 , as can be intuitively expected. For the disorder case we employed numerical calculations by assuming different strength of disorder and the results are described in the maintext.

X. Real time response and fatigue test
In video named "cyclic bending test.mov" we present the testing results of our MHA-AuNP strain sensors using a homemade automatic cyclic testing platform. The bending sequences were fully programed for giving static and time-varying strain to the device, and a multimeter was applied as the readout of real time responses. In the first part the device undergoes a continuous cyclic bending using the full range, its response repeats nicely with this fast bending sequences. In the second part, we The fatigue test introduced some defects and cracks on the Au/Ni electrodes and AuNP films as illustrated in Figure S11. On the electrodes, long and straight cracks could be as wide as 10s nm. On the contrary, some isolated defects and small cracks, typically 10 nm in width would be found on the AuNP films.

XI. Cyclic bending test of MPA device under environment control
A MPA strain sensor and the bending platform (as Fig. S6) were placed in a polymethylmethacrylate (PMMA) box to provide a stable environment (humidity and temperature) for cyclic fatigue test. The relative humidity in the box was 53.5 ± 0.3 % and temperature was 23.9 ± 0.5 °C while performing the test. In contrast to the results of MHA devices (Fig. 5), R 0 (resistance in absence of strain) of the MPA sensor slowly increase with the bending cycle number N. Meanwhile, the gauge factor of the MPA sensor slowly decrease. We supposed that such behavior was related to the formation of cracks in the MPA-AuMP film. As can be seen in Figure S12(c) and S12(d), many cracks formed after N = 20000. These cracks may increase the path of current in perpendicular direction, i.e., increase the detour path, and thus increase the resistance and meanwhile decrease the gauge factor.
Such cracks effect was not obvious for MHA devices (such as results in Fig. 5).
We believe it is because the longer MHA molecules between AuNP network have stronger binding energy than the shorter MPA molecules. Thus, MHA-AuNP films have much lower probability of crack formation, as can be seen in Fig. S11(b).