# Disordered Semiconductors on Mechanically Flexible Substrates for Large-Area Electronics

## Abstract

Low-temperature-thin-film semiconductors and dielectrics are critical for large-area flexible electronics, including displays, smart skins and imagers. Despite the presence of structural disorder, these materials show promising electronic transport properties that are vital for devices such as thin-film transistors (TFT s) and sensors. This chapter presents an overview of material and transport properties pertinent to large-area electronics on mechanically flexible substrates. We begin with a summary of process challenges for low-temperature fabrication of a-Si:H TFTs on plastic substrates, followed by a description of transport properties of amorphous semiconducting films, along with their influence on TFT characteristics. The TFTs must maintain electrical integrity under mechanical stress induced by bending of the substrates. Bending-induced changes are not limited to alteration of device dimensions and involve modulation of electronic transport of the active semiconducting layer.

There is significant demand for electronic-grade thin-films for a variety of applications such as organic light-emitting diode (OLED ) displays and lighting modules [44.1, 44.2], solar cells [44.3], digital imagers [44.4, 44.5], and wearable sensors [44.6]. Interest in these materials is driven by the promise of low-cost roll-to-roll manufacturing [44.7] and amenability to novel flexible and stretchable product designs. They enable high-performance electronic devices including thin-film transistors (TFTs), sensors, diodes and p-i-n photodetectors on large-area flexible plastic, paper and fabric substrates. For example, OLEDs have become highly attractive as thin emissive devices due to their fast response, high conversion efficiency, wide viewing angle, thin profile, and compatibility with plastic substrates [44.8]. In high-information-content displays, the OLED must be integrated with a TFT circuit that provides stable drive currents and allows for fast content switching [44.9]. Here, TFTs based on low-temperature a-Si:H (solution-processed/vacuum-deposited) organic semiconductors or metal-oxides [44.10, 44.11, 44.12] are needed for building the switches and circuits.

In addition to advances in material deposition and integration technologies, significant research efforts aim to improve the lifetime of TFTs and OLEDs in view of instabilities induced by bias stress and ambient exposure. The instability of TFTs stems from structural and interfacial disorder in the semiconductor and insulator layers, and interfaces. This leads to a shift in the threshold voltage Δ*V*_{T} under prolonged bias stress in a-Si:H [44.13] and organic [44.14] TFTs as well as photo-bias instability for metal oxide TFTs [44.10, 44.11]. To deal with the shift in device parameters, smart pixel circuits and driving schemes have been employed for compensation [44.15]. The lifetime of the OLED, on the other hand, can be enhanced by efficient thin-film encapsulation layers [44.16], which are imperative for active matrix organic light-emitting diode (AMOLED ) displays on plastic substrates.

This chapter presents an overview of material and transport properties pertinent to mechanically flexible electronics, in which amorphous-silicon is used as the example. We begin with an overview of low-temperature fabrication of high-performance a-Si:H transistors and associated processing challenges. The transport properties of disordered semiconducting films are then presented, along with their influence on TFT characteristics. We also examine the impact of external mechanical stress, induced by bending of the flexible substrate, on the characteristics of thin-film devices, including strain gauges and TFTs. The shifts are not only limited to changes in device dimensions but also to a modulation of electronic transport of the active semiconducting layer.

## 44.1 a-Si:H TFTs on Flexible Substrates

Compared to conventional (rigid) glass substrates, mechanically flexible substrates are attractive because of their reduced fragility and weight [44.17]. Polyethylene terephthalate (PET) [44.18] and polyimide [44.19] are examples of such substrates. Most plastic substrates have a low glass-transition temperature *T*_{g} (\(<{\mathrm{250}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\)). Consequently, the device processing temperatures must be reduced without compromising the electronic properties of the deposited materials. Despite the adverse effects of reducing the processing temperature, high-performance low-temperature (\(<{\mathrm{150}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\)) processes have been developed for fabrication of a-Si:H TFTs on flexible substrates [44.18, 44.19, 44.20], and even lower temperature processes (75^{∘}C) are under investigation [44.21].

One of the key challenges in low-temperature fabrication is to achieve a high-quality gate and/or passivation dielectric films, including a-SiO_{ x } [44.21] or a-SiN_{ x }:H [44.20] gate and/or passivation dielectric with low defect density, leakage, and interface state density. Nitrogen rich (*x* = 1.6 to 1.7) a-SiN_{ x } : H films with high resistivity (*ρ* > 10^{15} Ω cm) and breakdown field (\(> {\mathrm{5}}\,{\mathrm{MV/cm}}\)) have been deposited using conventional plasma-enhanced chemical vapor deposition (PECVD ) at 120^{∘}C [44.20].

The fabrication challenges are not limited to the electrical quality of the deposited materials. Most plastic substrates have a large coefficient of thermal expansion (CTE ) (\(> {\mathrm{50}}\,{\mathrm{ppm/{}^{\circ}\mathrm{C}}}\)), which is much higher than that of the TFT layers (\(\approx{\mathrm{4}}\,{\mathrm{ppm/{}^{\circ}\mathrm{C}}}\)) [44.22]. As the deposited films go through a temperature drop Δ*T* upon cooling from the process temperature [44.2], thermal stress *σ*_{th} = *Y*_{f}Δ*C*Δ*T* develops in the film [44.23]. Here, Δ*C* is the difference between CTEs of the substrate and the film, and *Y*_{f} denotes the biaxial elastic modulus of the deposited film. The thermal stress bends the composite structure, giving rise to handling problems. Thus, the substrate needs to be held flat in a frame or glued to a rigid substrate [44.2]. In addition, since most layers are patterned during processing, the thermal stress exhibits a local pattern-dependent distribution, which leads to problems in alignment of consecutive masks. Lowering the process temperature and choosing a substrate with a smaller CTE, e. g., polyethylene naphthalate (PEN) with a CTE of \({\mathrm{13}}\,{\mathrm{ppm/{}^{\circ}\mathrm{C}}}\) [44.22], will help alleviate the thermal stress, and associated alignment issues. It is important to note that thin rigid (steel, aluminum, or glass) substrates are also mechanically flexible and are being considered for device fabrication [44.24]. Although these substrates can withstand higher temperatures, handling issues still remain and depend on the desired degree of mechanical flexibility.

Research on TFT fabrication on flexible substrates have demonstrated successful handling of material processing issues. For instance, TFTs fabricated on plastic substrate at 150^{∘}C have been reported to have parameter values very similar to the high-temperature counterparts in terms of low reverse and high on current (with switching ratio of > 10^{8}), reasonable field-effect mobility \(> {\mathrm{0.8}}\,{\mathrm{cm^{2}/(V{\,}s)}}\), threshold voltage of ≈ 1–2 V, and low subthreshold slope \(<{\mathrm{0.3}}\,{\mathrm{V/dec}}\) [44.20].

## 44.2 Field-Effect Transport in Amorphous Films

The electronic properties of a semiconducting film are primarily determined by its structural disorder and grain size, which strongly depend on deposition conditions and postprocessing treatment [44.25].

The structure of disordered semiconductors such as a-Si:H lacks the long-range periodicity of crystalline silicon. However, the atoms of the materials bond to their neighbors according to the 8-N rule [44.26] for both amorphous and crystalline silicon, leading to short-range structural order. The extent to which this short-range order is conserved is dependent on processing conditions and leads to different degrees of disorder. This range can be as small as a few nanometers in amorphous semiconductors or as large as a few tens of microns in polycrystalline material. Nanocrystalline (nc) and microcrystalline (μc) materials reside between these extremes depending on grain size and stoichiometry.

The observation of an absorption gap and thermally activated conductivity in amorphous semiconductors stems from the short-range order [44.27] and resembles the band-gap properties of an ideal crystalline semiconductor, which has no states in the gap. The edges of the gap are well defined and represent discontinuities in the density of states (DOS ). In contrast, the lack of long-range order in the amorphous semiconductor leads to the presence of states in the gap, and thus, tailing of the band edges.

### 44.2.1 Localized and Extended States

In the ideal crystalline semiconductor, the band states are all extended with a corresponding band mobility *μ*_{band}. In the case of an amorphous semiconductor, however, the structural disorder of the material leads to the presence of localized states for which the mobility vanishes at *T* = 0. *Anderson* [44.28] demonstrated that, for a sufficiently large degree of disorder, all states in the band are localized. In his work, the disorder in the material is modeled by a random but uniform distribution *U*_{0} for the energy of the wells at lattice sites. He found that all of the states of the energy band become localized when \(U_{0}/B> 5\), where *B* is the band width and is given by *B* = 2*Z**I* [44.26]. Here, *Z* is the coordination number, and *I* the overlap energy for the states in neighboring wells, which decreases exponentially with distance based on the tight-binding approximation.

*Mott*and

*Davis*[44.26] stipulated that, for a given value of

*U*

_{0}, the states can be localized or extended, depending on their overlap energy. Consequently, for some materials there are distinct energies which separate the localized states from the extended ones and can be determined from Anderson’s criteria

*U*

_{0}≈ 5

*B*. These energies are referred to as mobility edges (or mobility shoulders) due to the significant change in the mobility expected at these edges. Figure 44.1 illustrates the DOS

*g*(

*E*) for the mobility gap of a-Si:H indicating the extended and localized states. The mobility shoulder

*E*

_{C}of the conduction band separates the localized states from the extended ones. A similar band structure and mobility shoulder is present for the valence band. This graph is compiled from the work of

*Cohen*et al. [44.29], who first demonstrated that the change in mobility at these edges can be of the order of 10

^{3}–10

^{4}, signifying the different transport mechanisms on the two sides of the edge.

*q*is the elementary charge and

*r*(

*E*) the mean distance between the wells that contribute to states with energy

*E*. In the conduction band,

*r*(

*E*) will be close to the lattice constant, which yields \(\sigma_{0}={\mathrm{350}}\,{\mathrm{\Upomega{}^{-1}{\,}cm^{-1}}}\). For Fermi energies

*E*

_{F}<

*E*

_{C}, the conductivity depends on the Fermi energy and reads

*k*

_{B}is Boltzmann’s constant and

*T*the temperature, predicting Arrhenius behavior for the conductivity. Here,

*E*

_{F}is defined as negative for the traps with respect to the conduction band edge

*E*

_{C}. This change in conductivity is due to a change in the concentration of carriers, excited to the extended states

*n*

_{band}with a band mobility

*μ*

_{band}such that

*σ*

_{0}, the band mobility can be found as \(\mu_{\mathrm{band}}\approx{\mathrm{12}}\,{\mathrm{cm^{2}/(V{\,}s)}}\) for a-Si:H.

In the localized states, the conduction is only possible by hopping of the carriers to the neighboring localized states [44.30]. This method of conduction prevails at low temperatures (*T* < 200 K) and for highly disordered materials.

The picture of the mobility gap can be completed by adding defect states in the middle of the gap, as indicated by the dashed lines in Fig. 44.1a. These states are attributed to the broken or unsaturated bonds such as dangling bonds in the network structure. The density of dangling bonds is reduced to 10^{15}–10^{16} cm^{−3} in a-Si:H by hydrogen atoms, which passivate some of these unsaturated bonds [44.27, 44.31].

### 44.2.2 Density of States (DOS)

The density of states (DOS) *g* ( *E* ) , including the density of band tail and defect states, as schematically represented in Fig. 44.1 a, determines the transport properties of the disordered semiconductor [44.32]. The DOS in a-Si:H has been extensively studied by using different experimental techniques such as field-effect measurements [44.33, 44.34], photoconductivity measurements [44.35], deep-level transient capacitance spectroscopy (DLTS ) [44.36], and capacitance–voltage–frequency (*C*–*V*–*f*) characteristics [44.37, 44.38]. It is found that an exponential distribution of the deep and tail states [44.39] can efficiently describe the distribution of the localized states and hold for a wide range of materials with rapidly changing DOS. In the case of a-Si:H, the localized states in the upper half of the mobility gap, closer to *E*_{C}, behave as acceptor-like states, while the states in the lower part of the gap, closer to *E*_{V}, behave as donor-like states. Acceptor-like states are neutral when they are empty, and negatively charged after capturing an electron, whereas the donor-like states are positively charged when they are empty and neutral after capturing an electron.

*E*

_{F}in an intrinsic a-Si:H sample in the dark lies closer to

*E*

_{C}due to the asymmetrical DOS distribution [44.31]. This results in a much stronger electron conduction in a-Si:H, which signifies the role of the conduction band tail in dispersive electron transport. The density of states for the conduction band tail

*g*(

*E*) is written as

*N*

_{t}is the total acceptor-like states in the conduction band tail, and

*T*

_{t}the associated slope of the exponential state distribution.

### 44.2.3 Effective Carrier Mobility

*n*is the sum of the excited and trapped carriers such that

*n*

_{t}as a function of the Fermi energy

*E*

_{F}, we integrate the product of the DOS and the probability of occupation of a state

*f*(

*E*) over the mobility gap

*Shaw*and

*Hack*[44.40] as

*E*

_{F}moves no closer than a few

*k*

_{B}

*T*to the mobility edge. This is true because of the high density of tail states, which tends to pin the movement of the Fermi energy. Here, \(u(T_{\mathrm{t}}/T,E_{\mathrm{F}})\) represents the changes in the trapped-carrier density with normalized ambient temperature

*T*

_{t}∕

*T*. For

*T*≪

*T*

_{t}, \(u(T_{\mathrm{t}}/T,E_{\mathrm{F}})\) is often approximated by \([\sin(\pi T/T_{\mathrm{t}})/\pi T/T_{\mathrm{t}}]\) [44.41] with a value close to 1. The ratio

*T*

_{t}∕

*T*, referred to as the dispersion parameter, characterizes the dispersive transport of electrons in the conduction band tail, and can be obtained from the time dependence of the electron drift mobility in time-of-flight experiments [44.42]. The presence of a high trapped-carrier density in amorphous semiconductors leads to an effective trapped-carrier mobility that is lower than the band mobility.

*Street*[44.31] has defined the drift mobility

*μ*

_{D}of the carriers as the band mobility reduced by the fraction of time that the carrier is trapped,

*τ*

_{band}and

*τ*

_{trap}are the times that carriers spend in the extended and localized states, respectively.

*μ*

_{FE}in TFTs is conventionally defined as

*g*is the conductance of the film, and

*q*

*n*is the field-induced charge in the semiconducting film. The terms

*g*and

*n*are averaged over the device active area and are related to the number of band carriers

*n*

_{band}and trapped carriers

*n*

_{t}. The

*μ*

_{FE}is conventionally retrieved from measurement of the transistor current (

*I*

_{DS,lin}) in the linear regime (

*V*

_{DS}= 0.1 V), by using the following

*W*and

*L*are the channel width and length, respectively,

*C*

_{ i }the gate capacitance, and

*V*

_{GS}and

*V*

_{DS}are the gate–source and drain–source biases, respectively.

According to (44.3) and (44.8), the densities of the band and trapped carriers increase differently with increasing *E*_{F}. Consequently, when *n*_{band} and *n*_{t} are averaged over the volume of the semiconducting film to obtain the *μ*_{FE} described by (44.10), the mobility becomes a function of device parameters (e. g., layer thicknesses and bias conditions).

*μ*

_{FE}for a-Si:H, pentacene and polythienylene vinylene (PTV) [44.43], and poly[5,5

^{′}-bis(3-alkyl-2-thienyl)-2,2

^{′}-bithiophene)] (PQT-12) [44.44, 44.45] TFTs at different gate biases. Here, the activation energy

*E*

_{a}turns out to be bias-dependent (inset of Fig. 44.2 ). More importantly, an anomaly arises in which the

*μ*

_{FE}and

*E*

_{a}become dependent on the gate capacitance

*C*

_{ i }, which is solely a geometrical capacitance. A higher

*C*

_{ i }implies higher carrier accumulation and consequently higher

*μ*

_{FE}for the same bias.

To remove this anomaly, we need to identify the effective carrier mobility such that \(\mu_{\mathrm{FE}}=\mu_{\mathrm{eff}}\times f(\phi)\), where *μ*_{eff} is the effective physical mobility and *f* ( *φ* ) describes the device attributes such as bias and geometry.

*σ*(

*n*) as a function of carrier concentration can be written as

*μ*

_{eff}is the effective mobility defined at a reference concentration as

*μ*

_{eff}and

*T*

_{t}∕

*T*. Here,

*μ*

_{eff}represents the effective carrier mobility at concentration

*N*

_{0}and

*T*

_{t}∕

*T*describes the change in conductivity with carrier concentration.

*y*denotes the location across the channel and

*δ*the channel depth. Using (44.13) for

*σ*and changing the integral parameter from

*y*to

*n*, we find after mathematical manipulation [44.48]

*V*

_{thr}the threshold voltage, \(\alpha=2T_{\mathrm{t}}/T\) the saturation current–voltage characteristics power parameter, and

*ζ*is just a function of

*T*

_{t}∕

*T*and accounts for the carrier distribution across the film. Revisiting the definition of the field-effect mobility as given by (44.11), but this time employing a more meaningful representation of the current, i. e., (44.16), we find

*μ*

_{eff}according to (44.14) is valid only when there is an exponential relationship between the carrier concentration

*n*and the Fermi energy

*E*

_{F}as given by (44.8). This is true for Fermi energy locations below the transport band edge and above the energies of the deep states (the region shown by the solid slope in Fig. 44.4 ). Since the deep states mostly contribute to the threshold voltage

*V*

_{thr}and are filled before the device turns on, the mobility definition of (44.14) is valid for the above-threshold regime. Consequently, the value of the reference concentration

*N*

_{0}must be selected such that the Fermi energy

*E*

_{F0}associated with

*N*

_{0}resides well above the deep states. This requires the charge accumulated in the channel

*Q*

_{channel}to be higher than the charge

*C*

_{ i }

*V*

_{thr}needed to turn on the device. If the carrier concentration at the semiconductor interface is

*N*

_{0},

*Q*

_{channel}=

*C*

_{ i }

*V*

_{0}, where \(V_{0}=(2\epsilon k_{\mathrm{B}}T_{\mathrm{t}}N_{0})^{1/2}/C_{i}\). Thus, for

*Q*

_{channel}>

*C*

_{ i }

*V*

_{thr}, we conclude

*V*

_{0}>

*V*

_{thr}or \(N_{0}> C_{i}^{2}V_{\mathrm{thr}}^{2}/(2\epsilon k_{\mathrm{B}}T_{\mathrm{t}})\), indicating the lower limit for

*N*

_{0}. Using typical values,

*V*

_{thr}= 2 V and \(C_{i}={\mathrm{20}}\,{\mathrm{nF/cm^{2}}}\), we have \(N_{0}> {\mathrm{6\times 10^{16}}}\,{\mathrm{cm^{-3}}}\). For instance, with \(N_{0}={\mathrm{10^{17}}}\,{\mathrm{cm^{-3}}}\), we have for the bias dependence factor \(\zeta=\left({\mathrm{4.1\times 10^{-16}}}\alpha\right)^{(1-\alpha/2)}/(\alpha-1)\).

*μ*

_{eff}by recasting (44.14) in the following manner

*μ*

_{eff}is not bias-dependent and corresponds to the energy needed for carriers to thermalize from

*E*

_{F0}to the mobility edge.

*E*

_{a}and

*E*

_{a0}. To do so, we look at the temperature dependence of

*μ*

_{FE}and

*μ*

_{eff}, viz.,

*E*

_{a}in the inset of Fig. 44.2.

#### Amorphous Organic Semiconductors

*Shapiro*and

*Adler*[44.50] have demonstrated that a transport band is present in which hopping conduction dominates irrespective of the position of Fermi energy. Similar to the mobility edge, the trapped carriers are thermalized to the hopping band [44.30]. Relative to

*E*

_{F0}, the hopping band is located at

*γ*is the effective overlap parameter for electronic states in the band tail.

*Baranovskii*et al. [44.51] have generalized this concept of transport band beyond the exponential DOS assumption and to a broader range of disordered materials with Gaussian or similar rapidly changing distributions. The generalized band concept can also accommodate the percolation-based hopping transport described by

*Vissenberg*and

*Matters*[44.43] for amorphous organic semiconductors, which predicts a hopping band that is just

*k*

_{B}

*T*

_{t}ln(

*B*

_{C}∕ 3

*π*

^{3}) higher than that predicted by

*Monroe*[44.30]. Here,

*B*

_{C}≈ 2.8 is the critical number for percolation in three-dimensional amorphous systems. Table 44.1 summarizes the values for

*μ*

_{eff}at room temperature

*T*

_{t}, and

*E*

_{a0}at \(N_{0}={\mathrm{10^{17}}}\,{\mathrm{cm^{-3}}}\), determined from the results presented for different disordered materials reported in literature.

Extracted transport parameters (*μ*_{eff}, *T*_{t}, and *E*_{a0}) for a selection of disordered semiconductors at \(N_{0}={\mathrm{10^{17}}}\,{\mathrm{cm^{-3}}}\)

#### Amorphous Oxide Semiconductors

Amorphous oxide semiconductors, such as indium zinc oxide (IZO ) or the ternary indium gallium zinc oxide (IGZO), have high band gap and strong ionic electronic structure that can exhibit high mobility (\(> {\mathrm{5}}\,{\mathrm{cm^{2}/(V{\,}s)}}\)) even when fabricated at low temperatures. The high mobility and reduced sensitivity to disorder is attributed to the unique role of the spherical 5s orbitals in the conduction of these materials. While chemical bonds in covalent semiconductors such as amorphous silicon are made of sp^{3} or p orbitals, strained bonds form deep and high density localized states below the conduction band and above the valence band resulting in formation of tail states [44.52, 44.53]. On the other hand the electronic structure in oxides benefits from its strong ionicity. The conduction band minima are made up of spherically extended s orbitals of metal cations and their overlaps with neighboring metal s orbitals are not significantly altered in the disordered oxide. Therefore the electronic levels are not very affected by structural randomness. Carrier transport benefits from the strong ionicity in which charge transfer occurs from metal to oxygen atom. The conduction band minima is almost entirely localized on the metal s orbital and the valence band consists of oxygen 2p states. However, non-bonding state of the metal cation or oxygen defect can be formed in or at conduction band minima in the non-compensated system, allowing vacancies, for example, to act as shallow donors but not as an effective electron trap [44.52]. But these states are not stable; their level relaxes to form either shallow donors or deep occupied states. The presence of oxygen defects leads to persistent conductivity [44.54, 44.55] behavior, in which the post-stress relaxation of the threshold voltage in an oxide TFT shows an initial rapid recovery followed by a slow-decaying component. This is indicative of a slow- decaying recovery process. The relaxation behavior of the sub-threshold slope is also similar. Here, the slow-decaying component points towards an increased density of shallow states near or above Fermi level. To restore its initial state a positive gate pulse can be applied. This has demonstrated to maintain high frame rates in displays and imagers [44.56].

*φ*

_{B0}and variance of

*σ*

_{B}. As a result, based on the effective mobility model, the activation energy of TLC conduction for these materials can be written as

## 44.3 Electronic Transport Under Mechanical Stress

Mechanical stress deforms the structure of the thin film leading to modulation in carrier mobility and density of states, and consequently, modulation of resistance. The change in resistance of a solid with elastic strain or stress is commonly referred to as the *elastoresistance* or *piezoresistance* effect, respectively [44.60]. The magnitude of the change is a function of the electronic properties of the material, the dimensions of the solid, and the direction of current flow.

Structural order in the material influences the elastoresistance effect. In the case of crystalline silicon, anisotropic scattering of electrons in the n-type material leads to a strong orientation dependence of the elastoresistive behavior [44.61]. In polycrystalline silicon, the crystallite size and orientation, and material texture play a critical role in determining the magnitude of the effect [44.62, 44.63]. In amorphous semiconductors, the random network behaves like an isotropic medium, and the anisotropy found in the crystalline material is less visible. However, the elastoresistance coefficients still depend on the relative orientation of the current and applied strain [44.64].

In sputtered amorphous silicon, *Welber* and *Brodsky* [44.65] have reported a decrease in the absorption gap with hydrostatic pressure with a coefficient of about \({\mathrm{-1}}\,{\mathrm{me{\mskip-2.0mu}V/kbar}}\) (\({\mathrm{-1\times 10^{-11}}}\,{\mathrm{e{\mskip-2.0mu}V/Pa}}\)) observed from the shift in the absorption edge. *Weinstein* [44.66] has also reported similar results drawn from photoluminescence experiments. The change in the optical gap is similar in sign and magnitude to that measured for crystalline silicon [44.67]. *Lazarus* [44.68], however, has reported an exponential increase in the resistivity of a-Si:H with increasing hydrostatic pressure at room temperature. The increase in resistivity is ascribed to either a decrease in the number of carriers or a reduction of the mobility with compressive strain [44.64].

In this section, we investigate the impact of mechanical stress on electrical properties of thin-film devices, insight into which is critical for design of mechanically flexible electronics. We begin with metallic and semiconductor thin films for strain-gauge applications and continue with a-Si:H TFTs.

### 44.3.1 Thin-Film Strain Gauges

*R*to strain

*ε*is referred to as the gauge factor

*k*, which can be written as

*ρ*is the resistivity of the material, and

*W*,

*L*, and

*t*are the width, length, and thickness of the strain gauge, respectively. The last term on the right-hand side of (44.25 ) reflects the strain-induced change in resistivity of the sample, whereas the first three terms refer to geometrical changes only.

*γ*

_{ i j }and

*π*

_{ i j }are the elements of the compact matrix of the elastoresistance and piezoresistance coefficients, respectively, and

*ε*

_{ j }and

*τ*

_{ j }denote the strain and stress components, respectively. (Here, we have used compact notation for these tensors [44.60].)

*ρ*to strain is assumed to be negligible, leading to the well-known longitudinal gauge factor of \(k_{\mathrm{l}}=1-2\nu\), where

*ν*is the Poisson’s ratio. However,

*Arlt*[44.69] has shown that the term Δ

*ρ*∕

*ρ*also includes geometrical attributes. This is due to the change in the volume of the wire and the resulting change in carrier density. According to

*Arlt*[44.69], we have for the longitudinal gauge factor

*γ*denotes the number of free electrons per atom and

*μ*the electron mobility. Since the last term on the right-hand side of (44.27) is relatively small for metallic gauges, the gauge factor mainly stems from geometrical changes; it is close to 2 and temperature independent [44.70].

*x*= 0) and the free end is deflected by a displacement

*Δ*.

*Rajanna*and

*Mohan*[44.71] used this method in measurements of both tensile and compressive configurations by simply placing the sample with the films on the top or bottom, respectively (see Fig. 44.7). The strain at location

*x*along a sample with a length

*L*

_{0}reads

*k*

_{l}≈ 2.06 for the longitudinal strain gauges.

*Dössel*[44.70] has related the gauge factor to the elastoresistance coefficients as follows

*ν*

_{s}and

*ν*

_{f}denote the Poisson’s ratios for the substrate and the film, respectively.

^{+}μc-Si:H strain gauges. Figure 44.9 displays the results of tensile and compressive tests on metal and semiconductor gauges, indicating a higher gauge factor. The values are \(k_{\mathrm{l}}=-17.0\) and \(k_{\mathrm{t}}=-3.41\) for longitudinal and transverse semiconductor gauges, respectively. Assuming \(\nu_{\mathrm{s}}=\nu_{\mathrm{f}}=0.23\), we find that \(\gamma_{1}=-22\) and \(\gamma_{2}=-10.9\) for n

^{+}μc-Si:H films.

The longitudinal gauge factor obtained for the n^{+} μc-Si:H gauges corroborate the results of *Germer* [44.72] for phosphorus-doped microcrystalline thin-film samples with a doping density of \(\approx{\mathrm{10^{20}}}\,{\mathrm{cm^{-3}}}\). For the transverse gauge factor, Germer has observed a small (negative or positive) gauge factor for doping densities in the range 5 × 10^{19}–10^{20}. This shows that the transverse gauge factor is highly sensitive to the doping density and other process conditions, which explains the slight difference between our values and that reported by Germer.

### 44.3.2 Strained Amorphous-Silicon Transistors

*I*

_{D}decreases with decreasing displacement. Correspondingly, we define the sensitivity

*I*

_{D}is the unstrained value of current and

*ε*

_{0}a reference strain value.

*V*

_{GS}= 20 V and

*V*

_{DS}= 0.5 V. As seen in the figure, the results for compressive and tensile strain are similar but opposite in sign. For the longitudinal TFT, the current increases with tensile strain with a sensitivity \(S_{\epsilon}(I_{\mathrm{D},\mathrm{l}})=12.5\). In contrast, for the transverse TFT, this is small and negative \(S_{\epsilon}(I_{\mathrm{D},\mathrm{t}})=-1.1\), clearly signifying an orientation dependence. The value of \(S_{\epsilon}(I_{\mathrm{D},\mathrm{s}})=4.5\) for the shear TFT can be explained from the linear superposition of the effects of longitudinal and transverse strain components.

The measured value for *S*_{ ε } ( *I*_{D,l} ) is close to the value of 15 ± 3 reported by *Spear* and *Heintze* [44.64] for intrinsic a-Si:H at room temperature. *Gleskova* et al. [44.73] have found a higher longitudinal sensitivity \(S_{\epsilon}(I_{\mathrm{D},\mathrm{l}})=26\). For the transverse sensitivity, Spear and Heintze have reported a positive value of \(S_{\epsilon}(I_{\mathrm{D},\mathrm{t}})=7\) for intrinsic a-Si:H samples.

*I*

_{D}. To examine the impact of bias, deflection experiments were performed for different values of the gate bias

*V*

_{GS}in the range 4–20 V in 1 V steps. Figure 44.13 illustrates the measured

*S*

_{ ε }(

*I*

_{D}) as a function of

*V*

_{GS}for TFTs of different orientations. Solid symbols denote measurement data for the linear regime (

*V*

_{DS}= 0.5 V) while the open symbols are those for the saturation regime where the gate and drain terminals are shorted. Interestingly, the modulation in the current shifts toward positive values as the gate bias decreases. This is true for TFTs of all orientations, and independent of whether the devices were integrated on glass or silicon substrates.

However, the sensitivity *S*_{ ε } ( *I*_{D,t} ) for the transverse TFT undergoes a sign change. At high biases, the *S*_{ ε } ( *I*_{D,t} ) is generally small and negative (i. e., Δ*I*_{D} is positive for tensile strain). As the gate bias decreases, the *S*_{ ε } ( *I*_{D,t} ) virtually vanishes at approximately 7 V, and subsequently increases to a sizable positive value (i. e., Δ*I*_{D} is negative for tensile strain) at lower voltages (*V*_{GS} < 7 V).

*V*

_{GS}> 7 V)

*S*

_{ ε }

^{H}(

*I*

_{D}) and low-bias (

*V*

_{GS}< 7 V)

*S*

_{ ε }

^{L}(

*I*

_{D}) components such that

*S*

_{ ε }(

*I*

_{D}) of longitudinal, transverse, and shear TFTs gradually approach constant values. As seen from Table 44.2 , the extracted values for

*S*

_{ ε }

^{H}(

*I*

_{D}) are strongly orientation dependent, suggesting the presence of strain-induced modulation in carrier mobility, whose sensitivity we define as \(S_{\epsilon}(\mu_{\mathrm{eff}})=\Updelta\mu_{\mathrm{eff}}/\mu_{\mathrm{eff}}\epsilon\). The mobility change in the longitudinal orientation is higher than that in the transverse orientation.

Values for *S* _{ ε } ^{H} ( *I*_{D} ) , *S*_{ ε } ( *μ*_{eff} ) , and *S*_{ ε } ( *V*_{thr} ) for different orientations

Parameter | Longitudinal | Transverse | Shear |
---|---|---|---|

\(S^{\mathrm{H}}_{\epsilon}(I_{\mathrm{D}})=\Updelta I_{\mathrm{D}}/I_{\mathrm{D}}\epsilon\) | 12.1 | −1.1 | 4.5 |

\(S_{\epsilon}(\mu_{\mathrm{eff}})=\Updelta\mu_{\mathrm{eff}}/\mu_{\mathrm{eff}}\epsilon\) | 11 | −1.1 | 4.0 |

\(S_{\epsilon}(V_{\mathrm{thr}})=\Updelta V_{\mathrm{thr}}/V_{\mathrm{thr}}\epsilon\) | 5 | 4.5 | 4.7 |

*S*

_{ ε }

^{L}(

*I*

_{D}) which manifests itself as a bias-dependent positive shift in

*S*

_{ ε }(

*I*

_{D}) . This component can be attributed to the modulation in threshold voltage. Correspondingly, we define the threshold-voltage sensitivity as \(S_{\epsilon}(V_{\mathrm{thr}})=\Updelta V_{\mathrm{thr}}/V_{\mathrm{thr}}\epsilon\). The change in

*V*

_{thr}can be attributed to a strain-induced change in the density of deep states, which is orientation independent [44.64]. The modulation in threshold voltage leads to a significant change in current at low biases, and can be quantitatively explained by looking at the current–voltage relation in the linear regime. From (44.16) and using partial differentiation with respect to strain, we can write

*S*

_{ ε }(

*I*

_{DS,lin}) as

*S*

_{G}includes the effect of change in device dimensions. The other terms on the right-hand side of (44.32), in order from the left, describe the dependence of

*S*

_{ ε }(

*I*

_{DS,lin}) on

*S*

_{ ε }(

*μ*

_{eff}) ,

*S*

_{ ε }(

*V*

_{thr}) , and

*S*

_{ ε }(

*α*) , respectively, which represent strain-induced modulation of the different TFT parameters (

*μ*

_{eff},

*V*

_{thr}, and

*α*).

*S*

_{ ε }(

*μ*

_{eff}) is directly reflected in the change of current. In contrast, the strain-induced change in threshold voltage Δ

*V*

_{thr}∕

*V*

_{thr}

*ε*is scaled, and by a factor of

*α*

*V*

_{thr}∕

*V*

_{GT}. This is particularly visible at low biases and its effect decreases with increasing gate bias

*V*

_{GT}. Thus, at high biases, the impact of threshold-voltage modulation is minimal, which yields

*α*. This term contains a bias-dependent scaling factor of \(\alpha\ln|V_{\mathrm{GT}}/V_{0}|\), which does not correlate with the observed bias dependence of

*S*

_{ ε }(

*I*

_{D}) seen in Fig. 44.13. This implies that modulation in

*V*

_{thr}and

*μ*

_{eff}are the dominant contributors to the observed changes in current.

*S*

_{ ε }

^{L}(

*I*

_{D}) and

*S*

_{ ε }

^{H}(

*I*

_{D}) is defined as

*E*

_{H}and

*E*

_{L}are found to be 140 meV and 0.58 eV, respectively. The much lower activation energy (140 meV) at high biases indicates that the

*S*

_{ ε }

^{H}(

*I*

_{D}) stems from the shallow states in the conduction-band tail. Again, this corroborates our previous findings that the sensitivity of the current at high biases is associated with the mobility modulation that is principally determined by the tail states.

*Spear*and

*Heintze*[44.64] have found an activation energy of 0.52 eV for intrinsic and doped a-Si:H layers. This corroborates with our low-voltage sensitivity data. The high activation energy for

*S*

_{ ε }

^{L}(

*I*

_{D}) identifies the role of deep states in the gap, which is in agreement with our previous finding that

*S*

_{ ε }(

*I*

_{D}) ∝

*S*

_{ ε }(

*V*

_{thr}) at low biases. Here, the strain is believed to modify the energy of the deep states [44.64].

From (44.33) and (44.34), the values of *S*_{ ε } ( *μ*_{eff} ) and *S*_{ ε } ( *V*_{thr} ) for different orientations can be determined (Table 44.2). The values can be incorporated in a compact model for the different TFT orientations, which can be used for computer-aided design (CAD) of mechanically flexible TFT circuits [44.74].

## Notes

### Acknowledgements

The authors would like to acknowledge Natural Sciences and Engineering Research Council (NSERC) of Canada for their financial support.

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