Advertisement

Disordered Semiconductors on Mechanically Flexible Substrates for Large-Area Electronics

  • Peyman Servati
  • Arokia Nathan
Part of the Springer Handbooks book series (SPRINGERHAND)

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 ΔVT 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 Tg (\(<{\mathrm{250}}\,{\mathrm{{}^{\circ}\mathrm{C}}}\)). 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 (75C) 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 > 1015 Ω cm) and breakdown field (\(> {\mathrm{5}}\,{\mathrm{MV/cm}}\)) have been deposited using conventional plasma-enhanced chemical vapor deposition (PECVD ) at 120C [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 ΔT upon cooling from the process temperature [44.2], thermal stress σth = YfΔCΔT develops in the film [44.23]. Here, ΔC is the difference between CTEs of the substrate and the film, and Yf 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 150C 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 > 108), 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 μband. 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 U0 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 = 2ZI [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.

For a small degree of disorder, Anderson’s criteria may not hold for states with large overlap energies (i. e., the extended states). Mott and Davis [44.26] stipulated that, for a given value of U0, 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 U0 ≈ 5B. 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 EC 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 103–104, signifying the different transport mechanisms on the two sides of the edge.
Fig. 44.1

Sketch of (a) DOS in valence and conduction bands, and (b) hole and electron mobilities (After [44.29])

In the extended states, the carriers have a finite mobility, and thus take part in conduction. This prevails at room temperature for a-Si:H and takes the form \(\sigma_{0}=0.06q^{2}/\hbar r(E)\) [44.26], where 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 EF < EC, the conductivity depends on the Fermi energy and reads
$$\sigma=\sigma_{0}\exp\left(\frac{E_{\mathrm{F}}}{k_{\mathrm{B}}T}\right),$$
(44.1)
where kB is Boltzmann’s constant and T the temperature, predicting Arrhenius behavior for the conductivity. Here, EF is defined as negative for the traps with respect to the conduction band edge EC. This change in conductivity is due to a change in the concentration of carriers, excited to the extended states nband with a band mobility μband such that
$$\sigma=q\mu_{\mathrm{band}}n_{\mathrm{band}}\;,$$
(44.2)
where
$$n_{\mathrm{band}}=N_{\mathrm{b}}\exp\left(\frac{E_{\mathrm{F}}}{k_{\mathrm{B}}T}\right),$$
(44.3)
with \(N_{\mathrm{b}}=\sigma_{0}/q\mu_{\mathrm{band}}\). From the given value for σ0, 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 1015–1016 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 (CVf) 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 EC, behave as acceptor-like states, while the states in the lower part of the gap, closer to EV, 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.

In a-Si:H, the number of donor-like states closer to the valence band is much higher than the number of acceptor-like states. As a result, following the neutrality condition, the position of the Fermi energy EF in an intrinsic a-Si:H sample in the dark lies closer to EC 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
$$g(E)=\frac{N_{\mathrm{t}}}{k_{\mathrm{B}}T_{\mathrm{t}}}\exp\left(\frac{E}{k_{\mathrm{B}}T_{\mathrm{t}}}\right),$$
(44.4)
where Nt is the total acceptor-like states in the conduction band tail, and Tt the associated slope of the exponential state distribution.

44.2.3 Effective Carrier Mobility

Due to the high density of tail and deep states, only a small number of carriers are thermally excited to the extended states and contribute to conduction as described in (44.2 ). The total carrier density n is the sum of the excited and trapped carriers such that
$$n=n_{\mathrm{band}}+n_{\mathrm{t}}\;.$$
(44.5)
To obtain the trapped carrier density nt as a function of the Fermi energy EF, we integrate the product of the DOS and the probability of occupation of a state f ( E )  over the mobility gap
$$n_{\mathrm{t}}=\int_{E_{\mathrm{V}}}^{0}g(E)f(E)\mathrm{d}E\;.$$
(44.6)
In equilibrium, the probability of occupation can be described by the Fermi–Dirac function,
$$f(E)=\frac{1}{1+\exp\left[(E-E_{\mathrm{F}})/k_{\mathrm{B}}T\right]}\;.$$
(44.7)
Equation (44.6 ) can be numerically solved and approximated along the lines given by Shaw and Hack [44.40] as
$$n_{\mathrm{t}}(E_{\mathrm{F}})=N_{\mathrm{t}}\exp\frac{E_{\mathrm{F}}}{k_{\mathrm{B}}T_{\mathrm{t}}}u\left(\frac{T_{\mathrm{t}}}{T},E_{\mathrm{F}}\right).$$
(44.8)
The underlying assumption is that EF moves no closer than a few kBT 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 Tt ∕ T. For T ≪ Tt, \(u(T_{\mathrm{t}}/T,E_{\mathrm{F}})\) 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 Tt ∕ 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 μD of the carriers as the band mobility reduced by the fraction of time that the carrier is trapped,
$$\mu_{\mathrm{D}}=\mu_{\mathrm{band}}\frac{\tau_{\mathrm{band}}}{\tau_{\mathrm{band}}+\tau_{\mathrm{trap}}}\;,$$
(44.9)
where τband and τtrap are the times that carriers spend in the extended and localized states, respectively.
Carriers accumulated by the field effect in TFTs also demonstrate similar transport properties. The field-effect mobility μFE in TFTs is conventionally defined as
$$\mu_{\mathrm{FE}}=\frac{g}{qn}\;,$$
(44.10)
where g is the conductance of the film, and qn 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 nband and trapped carriers nt. The μFE is conventionally retrieved from measurement of the transistor current (IDS,lin) in the linear regime (VDS = 0.1 V), by using the following
$$\mu_{\mathrm{FE}}=\frac{L}{WC_{i}V_{\mathrm{DS}}}\frac{\partial I_{\mathrm{DS},\mathrm{lin}}}{\partial V_{\mathrm{GS}}}\;,$$
(44.11)
where W and L are the channel width and length, respectively, C i the gate capacitance, and VGS and VDS 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 EF. Consequently, when nband and nt are averaged over the volume of the semiconducting film to obtain the μFE described by (44.10), the mobility becomes a function of device parameters (e. g., layer thicknesses and bias conditions).

The bias dependence is also evident in temperature-dependence measurements of mobility and conductivity in a-Si:H and organic TFTs. Figure 44.2 illustrates the temperature dependence of μFE 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 Ea turns out to be bias-dependent (inset of Fig. 44.2 ). More importantly, an anomaly arises in which the μFE and Ea 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 μFE for the same bias.
Fig. 44.2

Temperature dependence of μFE for a-Si:H, PQT-12 [44.44], pentacene and PTV [44.43]. Inset shows the bias dependence of the activation energy Ea of μFE in these materials

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

We reconsider the densities of trapped and free carriers according to (44.3) and (44.8). Since both densities vary exponentially with the Fermi energy, the density of band carriers in terms of trapped carriers can be written as
$$n_{\mathrm{band}}=\theta n_{\mathrm{t}}^{T_{\mathrm{t}}/T}\;,$$
(44.12)
where \(\theta=N_{\mathrm{b}}/(N_{\mathrm{t}}u)^{T_{\mathrm{t}}/T}\). Equation (44.12 ) has been found to hold empirically for a wide range of disordered semiconductor systems and operating conditions, including organic semiconductors [44.46, 44.47].
According to (44.2) and (44.12 ), the conductivity σ ( n )  as a function of carrier concentration can be written as
$$\sigma(n)=\frac{q}{\mu_{\mathrm{eff}}N_{0}}\left(\frac{n}{N_{0}}\right)^{T_{\mathrm{t}}/T}\;,$$
(44.13)
where μeff is the effective mobility defined at a reference concentration as
$$\mu_{\mathrm{eff}}\equiv\frac{\sigma(n=N_{0})}{qN_{0}}=\mu_{\mathrm{band}}\frac{N_{\mathrm{b}}}{N_{0}}\left(\frac{N_{0}}{N_{\mathrm{t}}u}\right)^{T_{\mathrm{t}}/T}\;.$$
(44.14)
Consequently, the conductivity at any carrier concentration is given by μeff and Tt ∕ T. Here, μeff represents the effective carrier mobility at concentration N0 and Tt ∕ T describes the change in conductivity with carrier concentration.
The significance of this representation becomes clear when we use it to obtain the current–voltage characteristics of a TFT (Fig. 44.3). Using the gradual channel approximation, we can write
$$I_{\mathrm{DS},\mathrm{lin}}=\frac{W}{L}V_{\mathrm{DS}}\int_{0}^{\delta}\sigma(y)\mathrm{d}y\;,$$
(44.15)
where 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]
$$I_{\mathrm{DS},\mathrm{lin}}=\mu_{\mathrm{eff}}\zeta\frac{W}{L}(C_{i}V_{\mathrm{GT}})^{\alpha-1}V_{\mathrm{DS}}\;,$$
(44.16)
where \(V_{\mathrm{GT}}=V_{\mathrm{GS}}-V_{\mathrm{thr}}\), Vthr the threshold voltage, \(\alpha=2T_{\mathrm{t}}/T\) the saturation current–voltage characteristics power parameter, and
$$\zeta=\frac{(2\epsilon k_{\mathrm{B}}T_{\mathrm{t}}N_{0})^{1-\alpha/2}}{\alpha-1}\;.$$
(44.17)
Here, ζ is just a function of Tt ∕ 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
$$f(\phi)\equiv\frac{\mu_{\mathrm{FE}}}{\mu_{\mathrm{eff}}}=\zeta(\alpha-1)(C_{i}V_{\mathrm{GT}})^{\alpha-2}\;.$$
(44.18)
Representation of μeff according to (44.14) is valid only when there is an exponential relationship between the carrier concentration n and the Fermi energy EF 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 Vthr 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 N0 must be selected such that the Fermi energy EF0 associated with N0 resides well above the deep states. This requires the charge accumulated in the channel Qchannel to be higher than the charge C i Vthr needed to turn on the device. If the carrier concentration at the semiconductor interface is N0, Qchannel = C i V0, where \(V_{0}=(2\epsilon k_{\mathrm{B}}T_{\mathrm{t}}N_{0})^{1/2}/C_{i}\). Thus, for Qchannel > C i Vthr, we conclude V0 > Vthr or \(N_{0}> C_{i}^{2}V_{\mathrm{thr}}^{2}/(2\epsilon k_{\mathrm{B}}T_{\mathrm{t}})\), indicating the lower limit for N0. Using typical values, Vthr = 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 \(\zeta=\left({\mathrm{4.1\times 10^{-16}}}\alpha\right)^{(1-\alpha/2)}/(\alpha-1)\).
Fig. 44.3

Simplified schematic of a TFT

Fig. 44.4

Density of trapped carriers as a function of the Fermi energy, showing the reference concentration N0 and transport (conduction) band

We now revisit the temperature dependence of μeff by recasting (44.14) in the following manner
$$\mu_{\mathrm{eff}}=\mu_{\mathrm{eff}0}\exp\left(\frac{-E_{\mathrm{a}0}}{k_{\mathrm{B}}T}\right),$$
(44.19)
where
$$\begin{aligned}\displaystyle&\displaystyle\mu_{\mathrm{eff}0}\equiv\mu_{\mathrm{band}}\frac{N_{\mathrm{b}}}{N_{0}}\quad\text{and}\\ \displaystyle&\displaystyle E_{\mathrm{a}0}\equiv k_{\mathrm{B}}T_{\mathrm{t}}\ln\left(\frac{N_{\mathrm{t}}u}{N_{0}}\right)=-E_{\mathrm{F}0}\;.\end{aligned}$$
(44.20)
Equation (44.19 ) predicts Arrhenius behavior for the effective mobility with an activation energy of \(E_{\mathrm{a}0}=-E_{\mathrm{F}0}\) (Fig. 44.4). The activation energy of μeff is not bias-dependent and corresponds to the energy needed for carriers to thermalize from EF0 to the mobility edge.
We now examine the relation between the two activation energies Ea and Ea0. To do so, we look at the temperature dependence of μFE and μeff, viz.,
$$\begin{aligned}\displaystyle\frac{\partial\mu_{\mathrm{FE}}}{\mu_{\mathrm{FE}}\partial T}&\displaystyle=\frac{\partial\mu_{\mathrm{eff}}}{\mu_{\mathrm{eff}}\partial T}+\frac{\partial f(\phi)}{f(\phi)\partial T}\\ \displaystyle&\displaystyle=-\frac{E_{\mathrm{a}0}}{k_{\mathrm{B}}T^{2}}+\frac{2T_{\mathrm{t}}}{T^{2}}\ln\left|\frac{V_{\mathrm{GT}}}{V_{0}}\right|\;,\end{aligned}$$
(44.21)
which yields
$$E_{\mathrm{a}}=E_{\mathrm{a}0}-2k_{\mathrm{B}}T_{\mathrm{t}}\ln\left|\frac{V_{\mathrm{GT}}}{V_{0}}\right|\;.$$
(44.22)
Equation (44.22) describes the bias dependence observed for Ea in the inset of Fig. 44.2.

Amorphous Organic Semiconductors

The concept of effective mobility and transport band may be generalized to accommodate different amorphous organic semiconductors despite differences in the underlying transport mechanism. This generalization follows from the relation \(n_{\mathrm{band}}=\theta n^{T_{\mathrm{t}}/T}\) (44.12 ) for organic semiconductors for a wide range of temperatures and carrier concentrations [44.46]. For this relation to hold, the distribution of trapped and band carriers must be exponential. Although evidence of a Gaussian trap distribution has been reported for organic materials [44.49], the Gaussian distribution is effectively seen as an equivalent exponential distribution due to the small variation in the Fermi energy because of the large tail state distribution. In addition, 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 EF0, the hopping band is located at
$$E_{\mathrm{0hopping}}=k_{\mathrm{B}}T_{\mathrm{t}}\ln\left[\frac{\gamma^{3}}{N_{0}}\left(\frac{2T}{3T_{\mathrm{t}}}\right)^{3}\right],$$
(44.23)
where 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 kBTtln(BC ∕ 3π3) higher than that predicted by Monroe [44.30]. Here, BC ≈ 2.8 is the critical number for percolation in three-dimensional amorphous systems. Table 44.1 summarizes the values for μeff at room temperature Tt, and Ea0 at \(N_{0}={\mathrm{10^{17}}}\,{\mathrm{cm^{-3}}}\), determined from the results presented for different disordered materials reported in literature.
Table 44.1

Extracted transport parameters (μeff, Tt, and Ea0) for a selection of disordered semiconductors at \(N_{0}={\mathrm{10^{17}}}\,{\mathrm{cm^{-3}}}\)

Semiconductor

μeff at 300 K (cm2 ∕  ( V s ) )

Tt (K)

Ea0 (eV)

Reference

a-Si:H

0.98

350

0.23

Our measurements

PQT-12

\({\mathrm{3.9\times 10^{-2}}}\)

326

0.24

[44.44]

Pentacene

\({\mathrm{9.0\times 10^{-4}}}\)

385

0.31

[44.43]

PTV

\({\mathrm{1.1\times 10^{-5}}}\)

380

0.42

[44.43]

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 sp3 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].

In multi-component oxides, trap-limited conduction (TLC ) and percolation are observed to play a role at low and high gate voltages, respectively [44.57, 44.58]. The percolation arises from the presence of potential barriers above the conduction band, which lead to spatially distributed perturbations in the minima of conduction band having an average potential barrier of φB0 and variance of σB. As a result, based on the effective mobility model, the activation energy of TLC conduction for these materials can be written as
$$E_{0\text{TLC}}=-E_{\text{F0}}-\mathrm{q}\phi_{\mathrm{B}0}+\frac{\left(\mathrm{q}\sigma_{\mathrm{B}}\right)^{2}}{2}\;,$$
(44.24)
At high voltage, as the Fermi energy gets close to the mobility edge, the mobility approaches saturation and percolation dominates the characteristics. A new power law dependence for current voltage characteristics sets in. In contrast, at low temperatures variable range hopping plays a critical role in determining the transport in these materials [44.59].

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

Strain gauges are thin-film transducer elements embedded on a substrate Fig. 44.5. The strain in the substrate leads to a change in the geometry, and therefore, the resistance of the gauge, which is detected and measured by external circuitry. The sensitivity of the resistance R to strain ε is referred to as the gauge factor k, which can be written as
$$k\equiv\frac{\Updelta R}{\epsilon R}=\frac{\Updelta L}{\epsilon L}-\frac{\Updelta W}{\epsilon W}-\frac{\Updelta t}{\epsilon t}+\frac{\Updelta\rho}{\epsilon\rho}\;,$$
(44.25)
where 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.
Fig. 44.5

Schematic of a thin-film strain gauge

Usually, the gauge length is oriented in a direction where strain is largest to achieve the highest gauge factor. This is referred to as the longitudinal orientation, where the current flows parallel to the strain. Similarly, the transverse orientation is defined when the gauge length is oriented in a direction perpendicular to the maximum strain (Fig. 44.6 ). The modulation of resistivity in the different orientations with the strain and correspondingly stress can be summarized by the following expression
$$\begin{aligned}\displaystyle&\displaystyle\frac{\Updelta\rho_{i}}{\rho}=\gamma_{ij}\epsilon_{j}=\pi_{ij}\tau_{j}\\ \displaystyle&\displaystyle\text{with }i=1,2,3\text{ and }j=1,2,\ldots,6\;,\end{aligned}$$
(44.26)
where 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].)
Fig. 44.6

Longitudinal and transverse wires under uniaxial stress and strain components in the 1-direction

In metal gauges, the sensitivity of 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 Δρ ∕ 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
$$k_{\mathrm{l}}=2-\frac{\Updelta(\gamma\mu)_{\mathrm{l}}}{\epsilon\gamma\mu}\;,$$
(44.27)
where 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].
These results can be validated using beam-deflection (cantilever) experiments performed on strain gauges integrated on glass and silicon substrates. Figure 44.7 illustrates the schematic of a beam-deflection system. One end of the sample is clamped (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 L0 reads
$$\epsilon(x)=\pm\frac{3t_{\mathrm{s}}\Delta}{2L_{0}^{2}}\frac{1-x}{L_{0}}\;,$$
(44.28)
with the positive and negative signs denoting tensile and compressive configurations, respectively.
Fig. 44.7

Schematic of a beam-deflection experiment

Figure 44.8 illustrates the result of the beam-deflection experiment on a 150 nm-thick molybdenum strain gauge with 30 turns, line width of 20 μm, and length of 1100 μm. The strain gauges are biased with a constant current of 100 μA and the voltage drop across its terminals is measured. The gauges are subjected to a sequence of loading/unloading steps in tensile configuration to eliminate systematic errors associated with slowly varying transients. By averaging the value of the voltage modulation, we retrieve a gauge factor of approximately kl ≈ 2.06 for the longitudinal strain gauges.
Fig. 44.8

Normalized change in output voltage of a longitudinal Mo strain gauge

The gauge factor for semiconductors is much higher due to the higher elastoresistance coefficients. Dössel [44.70] has related the gauge factor to the elastoresistance coefficients as follows
$$\begin{aligned}\displaystyle k_{\mathrm{l}}&\displaystyle=\gamma_{1}-(\nu_{\mathrm{s}}+\nu_{\mathrm{h}})\gamma_{2}\quad\text{and}\\ \displaystyle k_{\mathrm{t}}&\displaystyle=-\nu_{\mathrm{s}}\gamma_{1}+(1-\nu_{\mathrm{h}})\gamma_{2}\;,\end{aligned}$$
where
$$\nu_{\mathrm{h}}=\nu_{\mathrm{f}}\frac{1-\nu_{\mathrm{s}}}{1-\nu_{\mathrm{f}}}\;,$$
(44.29)
and \(\gamma_{1}=2-\Delta(\gamma\mu)_{\mathrm{l}}/\gamma\mu\epsilon\) and \(\gamma_{2}=\Delta(\gamma\mu)_{\mathrm{t}}/\gamma\mu\epsilon\) are the elastoresistance coefficients, and νs and νf denote the Poisson’s ratios for the substrate and the film, respectively.
The same beam-deflection experiment can also be performed on n+ 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.
Fig. 44.9

Change in voltage of longitudinal and transverse n+ μc-Si:H and longitudinal Mo gauges under tensile and compressive strains

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 × 1019–1020. 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

Figure 44.10 illustrates measurement results for the transient drain current of a longitudinal TFT subject to a sequence of tensile loading/unloading steps. We see immediate changes in current superimposed on the intrinsic transient response of the TFT. The measured change in current ΔID decreases with decreasing displacement. Correspondingly, we define the sensitivity
$$S_{\epsilon}(I_{\mathrm{D}})=\frac{1}{I_{\mathrm{D}}}\left.\frac{\partial\Updelta I_{\mathrm{D}}}{\partial\epsilon}\right|_{\epsilon=\epsilon_{0}},$$
(44.30)
where ID is the unstrained value of current and ε0 a reference strain value.
Fig. 44.10

Change in TFT current in deflection experiments

Figure 44.11 depicts measurement results for the change in drain current of the longitudinal, shear, and transverse TFTs (Fig. 44.12 ) under tensile and compressive strains. The TFTs have \(W/L={\mathrm{400}}\,{\mathrm{\upmu{}m}}/{\mathrm{400}}\,{\mathrm{\upmu{}m}}\) and are biased in the linear regime with constant VGS = 20 V and VDS = 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.
Fig. 44.11

Normalized change in TFT current as a function of tensile and compressive strain

Fig. 44.12

Longitudinal, transverse, and shear TFTs

The measured value for S ε  ( ID,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.

In addition to TFT orientation, we observe that the gate bias alters the magnitude of strain-induced change in current ΔID. To examine the impact of bias, deflection experiments were performed for different values of the gate bias VGS in the range 4–20 V in 1 V steps. Figure 44.13 illustrates the measured S ε  ( ID )  as a function of VGS for TFTs of different orientations. Solid symbols denote measurement data for the linear regime (VDS = 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.
Fig. 44.13

Bias dependence of the TFT current sensitivity

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

From the results of Fig. 44.13, we identify two distinct components underlying the strain-induced modulation of current: the high-bias (VGS > 7 V) S ε H  ( ID )  and low-bias (VGS < 7 V) S ε L  ( ID )  components such that
$$S_{\epsilon}(I_{\mathrm{D}})=S^{\mathrm{H}}_{\epsilon}(I_{\mathrm{D}})+S^{\mathrm{L}}_{\epsilon}(I_{\mathrm{D}})\;.$$
(44.31)
At high biases, S ε  ( ID )  of longitudinal, transverse, and shear TFTs gradually approach constant values. As seen from Table 44.2 , the extracted values for S ε H  ( ID )  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.
Table 44.2

Values for S ε H  ( ID ) , S ε  ( μeff ) , and S ε  ( Vthr )  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

Superimposed on the high-bias component is the low-bias component S ε L  ( ID )  which manifests itself as a bias-dependent positive shift in S ε  ( ID ) . 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 Vthr 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  ( IDS,lin )  as
$$\begin{aligned}\displaystyle S_{\epsilon}(I_{\mathrm{DS},\mathrm{lin}})&\displaystyle=S_{\mathrm{G}}+S_{\epsilon}(\mu_{\mathrm{eff}})-S_{\epsilon}(V_{\mathrm{thr}})\frac{\alpha V_{\mathrm{thr}}}{V_{\mathrm{GT}}}\\ \displaystyle&\displaystyle\quad+S_{\epsilon}(\alpha)\left(\alpha\ln\left|\frac{V_{\mathrm{GT}}}{V_{0}}\right|-\frac{\alpha}{2}-\frac{1}{\alpha-1}\right),\end{aligned}$$
(44.32)
where \(V_{0}=(2\epsilon k_{\mathrm{B}}T_{\mathrm{t}}N_{0})^{1/2}/C_{i}\), and
$$\begin{aligned}\displaystyle S_{\mathrm{G}}=S_{\epsilon}(W)-S_{\epsilon}(L)+(\alpha-1)S_{\epsilon}(C_{i})\;.\end{aligned}$$
Here, SG 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  ( IDS,lin )  on S ε  ( μeff ) , S ε  ( Vthr ) , and S ε  ( α ) , respectively, which represent strain-induced modulation of the different TFT parameters (μeff, Vthr, and α).
As seen in (44.32 ), the modulation in mobility S ε  ( μeff )  is directly reflected in the change of current. In contrast, the strain-induced change in threshold voltage ΔVthr ∕ Vthrε is scaled, and by a factor of αVthr ∕ VGT. This is particularly visible at low biases and its effect decreases with increasing gate bias VGT. Thus, at high biases, the impact of threshold-voltage modulation is minimal, which yields
$$S^{\mathrm{H}}_{\epsilon}(I_{\mathrm{D}})=S_{\mathrm{G}}+S_{\epsilon}(\mu_{\mathrm{eff}})\;.$$
(44.33)
In contrast, the low-bias component can be written as
$$S^{\mathrm{L}}_{\epsilon}(I_{\mathrm{D}})=-S_{\epsilon}(V_{\mathrm{thr}})\frac{\alpha V_{\mathrm{thr}}}{V_{\mathrm{GT}}}\;.$$
(44.34)
It is important to note that the last term on the right-hand side of (44.32) represents the impact of strain-induced modulation of 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  ( ID )  seen in Fig. 44.13. This implies that modulation in Vthr and μeff are the dominant contributors to the observed changes in current.
The activation energy for the temperature dependence of S ε L  ( ID )  and S ε H  ( ID )  is defined as
$$E_{i}=\frac{\partial S^{i}_{\epsilon}(I_{\mathrm{D}})}{\partial(1/(k_{\mathrm{B}}T))}\quad\text{with }i=\text{H or L}\;.$$
(44.35)
The values for EH and EL 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  ( ID )  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  ( ID )  identifies the role of deep states in the gap, which is in agreement with our previous finding that S ε  ( ID )  ∝ S ε  ( Vthr )  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  ( μeff )  and S ε  ( Vthr )  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.

References

  1. 44.1
    S.R. Forrest: Nature 428, 911 (2004)CrossRefGoogle Scholar
  2. 44.2
    S. Wagner, H. Gleskova, J. C. Sturm, Z. Suo: In: Technology and Applications of Amorphous Silicon, ed. by R.A. Street (Springer, Berlin 2000) p. 222Google Scholar
  3. 44.3
    S.E. Shaheen, R. Radspinner, N. Peyghambarian, G.E. Jabbour: Appl. Phys. Lett. 79, 2996 (2001)CrossRefGoogle Scholar
  4. 44.4
    R.A. Street, M. Mulato, R. Lau, J. Ho, J. Graham, Z. Popovic, J. Hor: Appl. Phys. Lett. 78, 4193 (2001)CrossRefGoogle Scholar
  5. 44.5
    P. Servati, Y. Vygranenko, A. Nathan, S. Morrison, A. Madan: J. Appl. Phys. 96, 7575 (2004)CrossRefGoogle Scholar
  6. 44.6
    S. Soltanian, A. Servati, R. Rahmanian, F. Ko, P. Servati: J. Mater. Res. 30, 121 (2015)CrossRefGoogle Scholar
  7. 44.7
    L. Collins: IEE Rev. 49(3), 42 (2003)CrossRefGoogle Scholar
  8. 44.8
    A.B. Chwang, M.A. Rothman, S.Y. Mao, R.H. Hewitt, M.S. Weaver, J.A. Silvernail, K. Rajan, M. Hack, J.J. Brown, X. Chu, L. Moro, T. Krajewsky, N. Rutherford: Appl. Phys. Lett. 83(3), 413 (2003)CrossRefGoogle Scholar
  9. 44.9
    A. Nathan, A. Kumar, K. Sakariya, P. Servati, S. Sambandan, D. Striakhilev: IEEE J. Solid-State Circuits 39(9), 1477 (2004)CrossRefGoogle Scholar
  10. 44.10
    J.K. Jeong: J. Mater. Res. 28(16), 2071 (2013)CrossRefGoogle Scholar
  11. 44.11
    H.Y. Jung, Y. Kang, A.Y. Hwang, C.K. Lee, S. Han, D.-H. Kim, J.-U. Bae, W.-S. Shin, J.K. Jeong: Sci. Rep. 4, 3765 (2014)CrossRefGoogle Scholar
  12. 44.12
    A. Nathan, S. Lee, S. Jeon, J. Robertson: J. Disp. Technol. 10(11), 917 (2014)CrossRefGoogle Scholar
  13. 44.13
    M.J. Powell, C. van Berkel, A.R. Franklin, S.C. Deane, W.I. Milne: Phys. Rev. B 45(8), 4160 (1992)CrossRefGoogle Scholar
  14. 44.14
    D. Knipp, R.A. Street, A. Volkel, J. Ho: J. Appl. Phys. 93, 347 (2003)CrossRefGoogle Scholar
  15. 44.15
    S. Alexander, P. Servati, G.R. Chaji, S. Ashtiani, R. Huang, D. Striakhilev, K. Sakariya, A. Kumar, A. Nathan, C. Church, J. Wzorek, P. Arsenault: J. Soc. Info. Disp. 13(7), 587 (2005)CrossRefGoogle Scholar
  16. 44.16
    G.L. Graff, R.E. Williford, P.E. Burrows: J. Appl. Phys. 96(4), 1840 (2004)CrossRefGoogle Scholar
  17. 44.17
    J.A. Rogers, Z. Bao, A. Dodabalapur, A. Makhija: IEEE Electron Device Lett. 21(3), 100 (2000)CrossRefGoogle Scholar
  18. 44.18
    C.-S. Yang, L.L. Smith, C.B. Arthur, G.N. Parsons: J. Vac. Sci. Technol. B 18(2), 683 (2000)CrossRefGoogle Scholar
  19. 44.19
    H. Gleskova, S. Wagner, V. Gašparík, P. Kováč: Appl. Surf. Sci. 175/176, 12 (2001)CrossRefGoogle Scholar
  20. 44.20
    D. Stryahilev, A. Sazonov, A. Nathan: J. Vac. Sci. Technol. A 20(3), 1087 (2002)CrossRefGoogle Scholar
  21. 44.21
    M. Meitine, A. Sazonov: Mater. Res. Soc. Symp. Proc. 769, H6.6.1 (2003)Google Scholar
  22. 44.22
    W.A. MacDonald: J. Mater. Chem. 14, 4 (2004)CrossRefGoogle Scholar
  23. 44.23
    K.L. Chopra: Thin Film Phenomena (McGraw–Hill, Toronto 1969)Google Scholar
  24. 44.24
    E.Y. Ma, S. Wagner: Appl. Phys. Lett. 74, 2661 (1999)CrossRefGoogle Scholar
  25. 44.25
    A. Madan, P.G. Le Comber, W.E. Spear: J. Non-Cryst. Solids 20, 239 (1976)CrossRefGoogle Scholar
  26. 44.26
    N.F. Mott, E.A. Davis: Electronic Processes in Non-Crystalline Materials (Oxford Univ. Press, Oxford 1971)Google Scholar
  27. 44.27
    C. Popescu, T. Stoica: In: Thin Film Resistive Sensors, ed. by P. Ciureanu, S. Middelhoek (IOPP, New York 1992) p. 37Google Scholar
  28. 44.28
    P.W. Anderson: Phys. Rev. 109(5), 1492 (1958)CrossRefGoogle Scholar
  29. 44.29
    M.H. Cohen, H. Fritzsche, S.R. Ovshinsky: Phys. Rev. Lett. 22(20), 1066 (1969)CrossRefGoogle Scholar
  30. 44.30
    D. Monroe: Phys. Rev. Lett. 54(2), 146 (1985)CrossRefGoogle Scholar
  31. 44.31
    R.A. Street: Hydrogenated Amorphous Silicon (Cambridge Univ. Press, New York 1992)Google Scholar
  32. 44.32
    P. Servati, A. Nathan, G.A.J. Amaratunga: Phys. Rev. B 74, 245210 (2006)CrossRefGoogle Scholar
  33. 44.33
    W.E. Spear, P.G. Le Comber: J. Non-Cryst. Solids 8–10, 727 (1972)CrossRefGoogle Scholar
  34. 44.34
    M.J. Powell: Philos. Mag. B 43(1), 93 (1981)CrossRefGoogle Scholar
  35. 44.35
    C.-Y. Huang, S. Guha, S.J. Hudgens: Phys. Rev. B 27(12), 7460 (1983)CrossRefGoogle Scholar
  36. 44.36
    J.D. Cohen, D.V. Lang, J.P. Harbison: Phys. Rev. Lett. 45(3), 197 (1980)CrossRefGoogle Scholar
  37. 44.37
    M. Hirose, T. Suzuki, G.H. Döhler: Appl. Phys. Lett. 34(3), 234 (1979)CrossRefGoogle Scholar
  38. 44.38
    P. Viktorovitch, G. Moddel: J. Appl. Phys. 51(9), 4847 (1980)CrossRefGoogle Scholar
  39. 44.39
    M. Shur, M. Hack: J. Appl. Phys. 55(10), 3831 (1984)CrossRefGoogle Scholar
  40. 44.40
    J.G. Shaw, M. Hack: J. Appl. Phys. 64(9), 4562 (1988)CrossRefGoogle Scholar
  41. 44.41
    C. Tanase, E.J. Meijer, P.W.M. Blom, D.M. de Leeuw: Phys. Rev. Lett. 91(21), 216 (2003)CrossRefGoogle Scholar
  42. 44.42
    T. Tiedje, A. Rose: Solid State Commun. 37, 49 (1980)CrossRefGoogle Scholar
  43. 44.43
    M.J.C.M. Vissenberg, M. Matters: Phys. Rev. B 57(20), 12964 (1998)CrossRefGoogle Scholar
  44. 44.44
    A. Salleo, T.W. Chen, A.R. Völkel, Y. Wu, P. Liu, B.S. Ong, R.A. Street: Phys. Rev. B 70, 115 (2004)Google Scholar
  45. 44.45
    B.S. Ong, Y. Wu, P. Liu, S. Gardner: J. Am. Chem. Soc. 126, 3378 (2004)CrossRefGoogle Scholar
  46. 44.46
    A.J. Campbell, M.S. Weaver, D.G. Lidzey, D.D.C. Bradley: J. Appl. Phys. 84(12), 6737 (1998)CrossRefGoogle Scholar
  47. 44.47
    D. Natali, M. Sampietro: J. Appl. Phys. 92(9), 5310 (2002)CrossRefGoogle Scholar
  48. 44.48
    P. Servati, D. Striakhilev, A. Nathan: IEEE Trans. Electron Devices 50(11), 2227 (2003)CrossRefGoogle Scholar
  49. 44.49
    C. Tanase, E.J. Meijer, P.W.M. Blom, D.M. de Leeuw: Org. Electron. 4, 33 (2003)CrossRefGoogle Scholar
  50. 44.50
    F.R. Shapiro, D. Adler: J. Non-Cryst. Solids 74(2/3), 189 (1985)CrossRefGoogle Scholar
  51. 44.51
    S.D. Baranovskii, T. Faber, F. Hensel, P. Thomas: J. Phys.: Condens. Matter 9, 2699 (1997)Google Scholar
  52. 44.52
    S. Lee, A. Nathan, Y. Ye, Y. Guo, J. Robertson: Sci. Rep. 5, 13467 (2015)CrossRefGoogle Scholar
  53. 44.53
    S. Lee, A. Nathan: Appl. Phys. Lett. 101, 11 (2012)Google Scholar
  54. 44.54
    K. Ghaffarzadeh, A. Nathan, J. Robertson, S. Kim, S. Jeon, C. Kim, U.-I. Chung, J.-H. Lee: Appl. Phys. Lett. 97, 143510 (2010)CrossRefGoogle Scholar
  55. 44.55
    S. Jeon, S.-E. Ahn, I. Song, C.J. Kim, U.-I. Chung, E. Lee, I. Yoo, A. Nathan, S. Lee, J. Robertson, K. Kim: Nat. Mater. 11, 301 (2012)CrossRefGoogle Scholar
  56. 44.56
    S. Lee, S. Jeon, R. Chaji, A. Nathan: Proc. IEEE 103, 644 (2015)CrossRefGoogle Scholar
  57. 44.57
    S. Lee, K. Ghaffarzadeh, A. Nathan, J. Robertson, S. Jeon, C. Kim, I.-H. Song, U.-I. Chung: Appl. Phys. Lett. 98, 20 (2011)Google Scholar
  58. 44.58
    S. Lee, S. Jeon, A. Nathan: J. Disp. Technol. 9, 883 (2013)CrossRefGoogle Scholar
  59. 44.59
    S. Lee, A. Nathan, J. Robertson, K. Ghaffarzadeh, M. Pepper, S. Jeon, C. Kim, I.-H. Song, U.-I. Chung, K. Kim: Temperature dependent electron transport in amorphous oxide semiconductor thin film transistors, Technical Digest, IEEE Int. Electron Device Meet. (IEDM) 2011 (2011), 14.6.1Google Scholar
  60. 44.60
    A. Nathan, H. Baltes: Microtransducer CAD (Springer, Wien 1999)CrossRefGoogle Scholar
  61. 44.61
    C. Herring, E. Vogt: Phys. Rev. 101(3), 944 (1956)CrossRefGoogle Scholar
  62. 44.62
    V.A. Gridchin, V.M. Lubimsky, M.P. Sarina: Sens. Actuators A 49(1/2), 67 (1995)CrossRefGoogle Scholar
  63. 44.63
    A. Dévényi, A. Belu, G. Korony: J. Non-Cryst. Solids 4, 380 (1970)CrossRefGoogle Scholar
  64. 44.64
    W.E. Spear, M. Heintze: Philos. Mag. B 54(5), 343 (1986)CrossRefGoogle Scholar
  65. 44.65
    B. Welber, M.H. Brodsky: Phys. Rev. B 16(8), 3660 (1977)CrossRefGoogle Scholar
  66. 44.66
    B.A. Weinstein: Phys. Rev. B 23(2), 787 (1981)CrossRefGoogle Scholar
  67. 44.67
    R. Zallen, W. Paul: Phys. Rev. 155(3), 703 (1967)CrossRefGoogle Scholar
  68. 44.68
    D. Lazarus: Phys. Rev. B 24(4), 2282 (1981)CrossRefGoogle Scholar
  69. 44.69
    G. Arlt: J. Appl. Phys. 49(7), 4273 (1978)CrossRefGoogle Scholar
  70. 44.70
    O. Dössel: Sens. Actuators 6(3), 169 (1984)CrossRefGoogle Scholar
  71. 44.71
    K. Rajanna, S. Mohan: Phys. Status Solidi (a) 105(2), K181 (1988)CrossRefGoogle Scholar
  72. 44.72
    W. Germer: Sens. Actuators 7(2), 135 (1985)CrossRefGoogle Scholar
  73. 44.73
    H. Gleskova, S. Wagner, W. Soboyejo, Z. Suo: J. Appl. Phys. 92(10), 6224 (2002)CrossRefGoogle Scholar
  74. 44.74
    P. Servati, A. Nathan: Appl. Phys. Lett. 86(7), 033504 (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Electrical and Computer EngineeringUniversity of British ColumbiaVancouverCanada
  2. 2.Dept. of Electrical EngineeringUniversity of CambridgeCambridgeUK

Personalised recommendations