Research article Special Issues

Towards low-voltage organic thin film transistors (OTFTs) with solution-processed high-k dielectric and interface engineering

  • Received: 16 May 2015 Accepted: 03 November 2015 Published: 24 November 2015
  • Although impressive progress has been made in improving the performance of organic thin film transistors (OTFTs), the high operation voltage resulting from the low gate capacitance density of traditional SiO2 remains a severe limitation that hinders OTFTs'development in practical applications. In this regard, developing new materials with high-k characteristics at low cost is of great scientific and technological importance in the area of both academia and industry. Here, we introduce a simple solution-based technique to fabricate high-k metal oxide dielectric system (ATO) at low-temperature, which can be used effectively to realize low-voltage operation of OTFTs. On the other hand, it is well known that the properties of the dielectric/semiconductor and electrode/semiconductor interfaces are crucial in controlling the electrical properties of OTFTs. By optimizing the above two interfaces with octadecylphosphonic acid (ODPA) self-assembled monolayer (SAM) and properly modified low-cost Cu, obviously improved device performance is attained in our low-voltage OTFTs. Further more, organic electronic devices on flexible substrates have attracted much attention due to their low-cost, rollability, large-area processability, and so on. Basing on the above results, outstanding electrical performance is achieved in flexible devices. Our studies demonstrate an effective way to realize low-voltage, high-performance OTFTs at low-cost.

    Citation: Yaorong Su, Weiguang Xie, Jianbin Xu. Towards low-voltage organic thin film transistors (OTFTs) with solution-processed high-k dielectric and interface engineering[J]. AIMS Materials Science, 2015, 2(4): 510-529. doi: 10.3934/matersci.2015.4.510

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  • Although impressive progress has been made in improving the performance of organic thin film transistors (OTFTs), the high operation voltage resulting from the low gate capacitance density of traditional SiO2 remains a severe limitation that hinders OTFTs'development in practical applications. In this regard, developing new materials with high-k characteristics at low cost is of great scientific and technological importance in the area of both academia and industry. Here, we introduce a simple solution-based technique to fabricate high-k metal oxide dielectric system (ATO) at low-temperature, which can be used effectively to realize low-voltage operation of OTFTs. On the other hand, it is well known that the properties of the dielectric/semiconductor and electrode/semiconductor interfaces are crucial in controlling the electrical properties of OTFTs. By optimizing the above two interfaces with octadecylphosphonic acid (ODPA) self-assembled monolayer (SAM) and properly modified low-cost Cu, obviously improved device performance is attained in our low-voltage OTFTs. Further more, organic electronic devices on flexible substrates have attracted much attention due to their low-cost, rollability, large-area processability, and so on. Basing on the above results, outstanding electrical performance is achieved in flexible devices. Our studies demonstrate an effective way to realize low-voltage, high-performance OTFTs at low-cost.


    This paper is concerned with the initial boundary value problem

    $ {utt(x,t)uxx(x,t)+μ1(t)ut(x,t)+μ2(t)ut(x,tτ(t))=0in Ω×]0,+[,u(0,t)=u(L,t)=0on ]0,+[,u(x,0)=u0(x),ut(x,0)=u1(x)on Ω,ut(x,tτ(0))=f0(x,tτ(0))in Ω×]0,τ(0)[,
    $
    (1)

    where $ \Omega = ]0,L[ $, $ 0<\tau(t) $ are a non-constant time delay, $ \mu_1(t), \mu_2(t) $ are non-constant weights and the initial data $ (u_0,u_1,f_0) $ belong to a suitable function space.

    This problem has been first proposed and studied in Nicaise and Pignotti [22] in case of constant coefficients $ \mu_1, \mu_2 $ and constant time delay. Under suitable assumptions, the authors proved the exponential stability of the solution by introducing suitable energies and by using some observability inequalities. Some instability results are also given for the case of the some assumptions is not satisfied.

    With a weight depending on time, $ \mu_1(t), \mu_2(t) $ and constant time delay, this problem was studied in [2], where the existence of solution was made by Faedo-Galerkin method and a decay rate estimate for the energy was given by using the multiplier method.

    W. Liu in [19] studied the weak viscoelastic equation with an internal time varying delay term. By introducing suitable energy and Lyapunov functionals, he establishes a general decay rate estimate for the energy under suitable assumptions.

    F. Tahamtani and A. Peyravi [29] investigated the nonlinear viscoelastic wave equation with source term. Using the Potential well theory they showed that the solutions blow up in finite time under some restrictions on initial data and for arbitrary initial energy.

    Global existence and asymptotic behavior of solutions to the viscoelastic wave equation with a constant delay term was considered by M. Remil and A. Hakem in [28].

    Global existence and asymptotic stability for a coupled viscoelastic wave equation with time-varying delay was studied in [3] by combining the energy method with the Faedo-Galerkin's procedure.

    The stabilization problem by interior damping of the wave equation with boundary or internal time-varying delay was studied in [23] by introducing suitable Lyapunov functionals.

    Energy decay of solutions for the wave equation with a time varying delay term in the weakly nonlinear internal feedbacks was considered in [11].

    For problems with delay in different contexts we cite [9,10,30,32] with references therein. In absence of delay ($ \mu_2(t) = 0 $), the problem (1) is exponentially stable provided that $ \mu_1(t) $ is constant, see, for instance [5,6,16,17,21] and reference therein.

    Time delay is the property of a physical system by which the response to an applied force is delayed in its effect, and the central question is that delays source can destabilize a system that is asymptotically stable in the absence of delays, see [7]. In fact, an arbitrarily small delay may destabilize a system that is uniformly asymptotically stable in the absence of delay unless additional control terms have been used, see for example [8,12,31]

    By energy method in [24] was studied the stabilization of the wave equation with boundary or internal distributed delay. By semigroup approach in [27] was proved the well-posedness and exponential stability for a wave equation with frictional damping and nonlocal time-delayed condition. Transmission problem with distributed delay was studied in [18] where was established the exponential stability of the solution by introducing a suitable Lyapunov functional.

    Here we consider a wave equation with non-constant delay and nonlinear weights, thus, the present paper is a generalization of the previous ones. The remaining part of this paper is organized as follows. In the section 2 we introduce some notations and prove the dissipative property of the full energy of the system. In the section 3, for an approach combining semigroup theory (see [21] and [4]) with the energy estimate method we prove the existence and uniqueness of solution. In section 4 we present the result of exponential stability.

    We will need the following hypotheses:

    (H1) $ \mu_1:\mathbb{R}_+ \rightarrow ]0,+\infty[ $ is a non-increasing function of class $ C^1(\mathbb{R}_+) $ satisfying

    $ |μ1(t)μ1(t)|M1,0<α0μ1(t),t0,
    $
    (2)

    where $ \alpha_0 $ and $ M_1 $ are constants such that $ M_1>0 $.

    (H2) $ \mu_2:\mathbb{R}_+ \rightarrow \mathbb{R} $ is a function of class $ C^1(\mathbb{R}_+) $, which is not necessarily positives or monotones, such that

    $ |μ2(t)|βμ1(t),
    $
    (3)
    $ |μ2(t)|M2μ1(t),
    $
    (4)

    for some $ 0 < \beta < \sqrt{1-d} $ and $ M_2>0 $.

    We now state a lemma needed later.

    Lemma 2.1 (Sobolev-Poincare's inequality). Let $ q $ be a number with $ 2\leq q \leq +\infty $. Then there is a constant $ c_* = c_*(]0,L[,q) $ such that

    $ \left\| \Psi \right\|_q \leq c_* \left\| \Psi_x \right\|_2, \quad \mathit{\mbox{for}}\; \Psi \in H_0^1(]0,L[). $

    Lemma 2.2 ([13][16]). Let $ E:\mathbb{R}_+ \rightarrow \mathbb{R}_+ $ be a non increasing function and assume that there are two constants $ \sigma>-1 $ and $ \omega>0 $ such that

    $ \int_{S}^{+\infty} E^{1+\sigma}(t)\,dt \leq \frac{1}{\omega}E^\sigma(0)E(S),\ 0\leq S < +\infty. $

    Then

    $ E(t)=0 tEσ(0)ω|σ|, if1<σ<0,E(t)E(0)(1+σ1+ωσt)1σ t0, ifσ>0,E(t)E(0)e1ωt t0, ifσ=0.
    $

    As in [23], we assume that

    $ τ(t)W2,+([0,T]),  for T>0
    $
    (5)

    and there exist positive constants $ \tau_0,\tau_1 $ and $ d $ satisfying

    $ 0<τ0τ(t)τ1, t>0
    $
    (6)

    and

    $ τ(t)d<1, t>0.
    $
    (7)

    We introduce the new variable

    $ z(x,ρ,t)=ut(x,tτ(t)ρ), xΩ,ρ]0,1[,t>0.
    $
    (8)

    Then

    $ \tau(t)z_t(x,\rho,t) + (1-\tau'(t)\rho)z_\rho(x,\rho,t) = 0, \ x\in \Omega,\ \rho \in ]0,1[,\ t > 0 $

    and problem (1) takes the form

    $ {utt(x,t)uxx(x,t)+μ1(t)ut(x,t)+μ2(t)z(x,1,t)=0inΩ×]0,+[,τ(t)zt(x,ρ,t)+(1τ(t)ρ)zρ(x,ρ,t)=0inΩ×]0,1[×]0,+[,u(0,t)=u(L,t)=0on]0,+[,u(x,0)=u0(x),ut(x,0)=u1(x)onΩ,z(x,ρ,0)=ut(x,τ(0)ρ)=f0(x,τ(0)ρ)inΩ×]0,1[.
    $
    (9)

    We define the energy of the solution of problem (9) by

    $ E(t)=12ut2L2(Ω)+12ux2L2(Ω)+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdx,
    $
    (10)

    where

    $ ξ(t)=ˉξμ1(t)
    $
    (11)

    is a non-increasing function of class $ C^1(\mathbb{R}_+) $ and $ \bar{\xi} $ be a positive constant such that

    $ β1d<ˉξ<2β1d.
    $
    (12)

    Our first result states that the energy is a non-increasing function.

    Lemma 2.3. Let $ (u,z) $ be a solution to the problem (9). Then, the energy functional defined by (10) satisfies

    $ E(t)μ1(t)(1ˉξ2β21d)ut2L2(Ω)μ1(t)(ˉξ(1τ(t))2β1d2)z(x,1,t)2L2(Ω)0.
    $
    (13)

    Proof. Multiplying the first equation (9) by $ u_t(x,t) $, integrating on $ \Omega $ and using integration by parts, we get

    $ 12ddt(ut2L2(Ω)+ux2L2(Ω))+μ1(t)ut2L2(Ω)+μ2(t)Ωz(x,1,t)utdx.
    $
    (14)

    Now multiplying the second equation (9) by $ \xi(t)z(x,\rho,t) $ and integrate on $ \Omega \times ]0,1[ $, to obtain

    $ \tau(t)\xi(t)\int_{\Omega} \int_0^1\!\! z_t(x,\rho,t)z(x,\rho,t)\,d\rho\,dx = -\frac{\xi(t)}{2}\! \int_{\Omega} \int_0^1\!\!\! (1- \tau'(t)\rho)\frac{\partial}{\partial \rho}(z(x,\rho,t))^2\,d\rho\,dx. $

    Consequently,

    $ ddt(ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdx)=ξ(t)2Ω10(1τ(t)ρ)ρ(z(x,ρ,t))2dρdx+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdx=ξ(t)2Ω(z2(x,0,t)z2(x,1,t))dx+ξ(t)τ(t)2Ω10z2(x,1,t)dρdx+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdx.
    $
    (15)

    From (10), (14) and (15) we obtain

    $ E(t)=ξ(t)2ut2L2(Ω)ξ(t)2z(x,1,t)2L2(Ω)+ξ(t)τ(t)2z(x,1,t)2L2(Ω)+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdxμ1(t)ut2L2(Ω)μ2(t)Ωz(x,1,t)utdx.
    $
    (16)

    Due to Young's inequality, we have

    $ μ2(t)Ωz(x,1,t)utdx|μ2(t)|21dut2L2(Ω)+|μ2(t)|1d2z(x,1,t)2L2(Ω).
    $
    (17)

    Inserting (17) into (16), we obtain

    $ E(t)(μ1(t)ξ(t)2|μ2(t)|21d)ut2L2(Ω)(ξ(t)2ξ(t)τ(t)2|μ2(t)|1d2)z(x,1,t)2L2(Ω)+ξ(t)τ(t)2Ω10z2(x,ρ,t)dρdxμ1(t)(1ˉξ2β21d)ut2L2(Ω)μ1(t)(ˉξ(1τ(t))2β1d2)z(x,1,t)2L2(Ω)0.
    $

    Lemma 2.4. Let $ (u,z) $ be a solution to the problem (9). Then the energy functional defined by (10) satisfies

    $ \|u_t(x,t)\|_{L^2(\Omega)}^{2} < -\frac{1}{\sigma}E'(t), $

    where $ \sigma = a_0\left( 1-\frac{\bar{\xi}}{2}- \frac{\beta}{2\sqrt{1-d}} \right) $.

    Proof. From Lemma 2.3, we have that

    $ E(t)μ1(t)(1ˉξ2+β21d)ut2L2(Ω)+μ1(t)(ˉξ(1τ(t))2+β1d2)z(x,1,t)2L2(Ω)0
    $

    and from (H1), we obtain

    $ 0a0(1ˉξ2+β21d)ut2L2(Ω)μ1(t)(1ˉξ2+β21d)ut2L2(Ω)E(t)
    $

    and the lemma is proved.

    For the semigroup setup we $ U = (u,u_t,z)^T $ and rewrite (9) as

    $ {Ut=A(t)U,U(0)=U0=(u0,u1,f0(,,τ(0)))T,
    $
    (18)

    where the operator $ \mathcal{A}(t) $ is defined by

    $ AU=(v,uxxμ1(t)vμ2(t)z(x,1,t),1τ(t)ρτ(t)zρ(x,ρ,t))T.
    $
    (19)

    We introduce the phase space

    $ \mathcal{H} = H_0^1(\Omega)\times L^2(\Omega)\times L^2(\Omega \times ]0,1[) $

    and the domain of $ \mathcal{A} $ is defined by

    $ D(A(t))={(u,v,z)TH/v=z(,0)  in Ω},
    $
    (20)

    where

    $ H = H^2(\Omega)\cap H_0^1(\Omega)\times H_0^1(\Omega)\times L^2(\Omega; H_0^1(]0,1[)). $

    Notice that the domain of the operator $ \mathcal{A}(t) $ is independent of the time $ t $, i.e.,

    $ D(A(t))=D(A(0)),t>0.
    $
    (21)

    $ \mathcal{H} $ is a Hilbert space provided with the inner product

    $ U,ˉUH=Ωuxˉuxdx+Ωvˉvdx+ξ(t)τ(t)Ω10zˉzdρdx,
    $
    (22)

    for $ U = (u,v,z)^T $ and $ \bar{U} = (\bar{u},\bar{v},\bar{z})^T $.

    Using this time-dependent inner product and the next theorem we will get a result of existence and uniqueness.

    Theorem 3.1. Assume that

    (i) $ Y = D(\mathcal{A}(0)) $ is dense subset of $ \mathcal{H} $,

    (ii) (21) holds,

    (iii) for all $ t \in [0,T] $, $ \mathcal{A}(t) $ generates a strongly continuous semigroup on $ \mathcal{H} $ and the family $ \mathcal{A}(t) = \left\{ \mathcal{A}(t)/ t \in [0,T] \right\} $ is stable with stability constants $ C $ and $ m $ independent of $ t $ (i.e., the semigroup $ (S_t(s))_{s\geq 0} $ generated by $ \mathcal{A}(t) $ satisfies $ \| S_t(s)u \|_{\mathcal{H}} \leq Ce^{ms} \| u \|_{\mathcal{H}} $, for all $ u \in \mathcal{H} $ and $ s\geq 0 $),

    (iv) $ \partial_t \mathcal{A}(t) $ belongs to $ L_{*}^{\infty}([0,T],B(Y, \mathcal{H})) $, which is the space of equivalent classes of essentially bounded, strongly measurable functions from $ [0,T] $ into the set $ B(Y, \mathcal{H}) $ of bounded operators from $ Y $ into $ \mathcal{H} $.

    Then, problem (18) has a solution $ U \in C([0,T],Y) \cap C^1([0,T], \mathcal{H}) $ for any initial datum in $ Y $.

    Our goal is then to check the above assumptions for problem (18).

    First, we prove $ D(\mathcal{A}(0)) $ is dense in $ \mathcal{H} $.

    The proof is the same as the one Lemma $ 2.2 $ of [25], we give it for the sake of completeness.

    Let $ (f,g,h)^T $ be orthogonal to all elements of $ D\mathcal{A}(0) $, namely

    $ 0 = \langle (u,v,z)^T,(f,g,h)^T \rangle_{\mathcal{H}} = \int_{\Omega} u_x f_x\,dx + \int_{\Omega} v g\,dx + \xi(t)\tau(t) \int_{\Omega} \int_0^1 z h\,d\rho\,dx, $

    for all $ (u,v,z)^T \in D(\mathcal{A}(0)) $.

    We first take $ u = v = 0 $ and $ z \in D(\Omega \times ]0,1[) $. As $ (0,0,z)^T \in D(\mathcal{A}(0)) $, we get

    $ \int_{\Omega} \int_0^1 zh\,d\rho\,dx = 0. $

    Since $ D(\Omega \times ]0,1[) $ is dense in $ L^2(\Omega \times ]0,1[) $, we deduce that $ h = 0 $. In the same manner, by taking $ u = z = 0 $ e $ v \in D(\Omega) $ we see that $ g = 0 $.

    The above orthogonality condition is then reduced to

    $ 0 = \int_{\Omega} u_xf_x\,dx, \quad \forall (u,v,z)^T \in D(\mathcal{A}(0)). $

    By restricting ourselves to $ v = 0 $ and $ z = 0 $, we obtain

    $ 0 = \int_{\Omega} u_xf_x\,dx, \quad \forall (u,0,0)^T \in D(\mathcal{A}(0)). $

    Since $ D(\Omega) $ is dense in $ H_0^1(\Omega) $ (equipped with the inner product $ \langle \cdot, \cdot \rangle_{H_0^1(\Omega} $), we deduce that $ f = 0 $.

    We consequently

    $ D(A(0) is dense in H.
    $
    (23)

    Secondly, we notice that

    $ ΦtΦsec2τ0|ts|,t,s[0,T],
    $
    (24)

    where $ \Phi = (u,v,z)^T $ and $ c $ is a positive constant and $ \| \cdot \| $ is the norm associated the inner product (22). For all $ t,s \in [0,T] $, we have

    $ Φ2tΦ2secτ0|ts|=(1ec2τ0|ts|)(ux2L2(Ω)+v2L2(Ω))+(ξ(t)τ(t)ξ(s)τ(s)ecτ0|ts|)Ω10z2(x,ρ,t)dρdx.
    $

    It is clear that $ 1 - e^{\frac{c}{\tau_0}|t-s|} \leq 0 $. Now we will prove $ \xi(t)\tau(t) - \xi(s)\tau(s)e^{\frac{c}{\tau_0}|t-s|} \leq 0 $ for some $ c>0 $. To do this, we have

    $ \tau(t) = \tau(s) + \tau'(r)(t-s), $

    where $ r \in ]s,t[ $.

    Hence $ \xi $ is a non increasing function and $ \xi>0 $, we get

    $ \xi(t)\tau(t) \leq \xi(s)\tau(s) + \xi(s)\tau'(r)(t-s), $

    which implies

    $ \frac{\xi(t)\tau(t)}{\xi(s)\tau(s)} \leq 1 + \frac{|\tau'(r)|}{\tau(s)}|t-s|. $

    Using (5) and $ \tau' $ is bounded, we deduce that

    $ \frac{\xi(t)\tau(t)}{\xi(s)\tau(s)} \leq 1 + \frac{c}{\tau_0}|t-s| \leq e^{\frac{c}{\tau_0}|t-s|}, $

    which proves (24) and therefore $ (iii) $ follows.

    Now we calculate $ \langle \mathcal{A}(t)U,U \rangle_t $ for a fixed $ t $. Take $ U = (u,v,z)^T \in D(\mathcal{A}(t)) $. Then

    $ A(t)U,Ut=Ωvxuxdx+Ω(uxxμ1(t)vμ2(t)z(,1))vdxξ(t)Ω10(1τ(t)ρ)zρ(x,ρ)z(x,ρ)dρdx.
    $

    Integrating by parts, we obtain

    $ A(t)U,Ut=μ1(t)v2L2(Ω)μ2(t)Ωz(,1)vdxΩ10(1τ(t)ρ)ρz2(x,ρ)dρdx.
    $

    Since

    $ \left(1- \tau'(t)\rho \right)\frac{\partial}{\partial \rho} z^2(x,\rho) = \frac{\partial}{\partial \rho} \left( \left(1- \tau'(t)\rho \right) z^2(x,\rho) \right) + \tau'(t)z^2(x,\rho), $

    we have

    $ 10(1τ(t)ρ)ρz2(x,ρ)dρ=(1τ(t))z2(x,1)z2(x,0)+τ(t)10z2(x,ρ)dρ.
    $

    So we get

    $ A(t)U,Ut=μ1(t)v2L2(Ω)μ2(t)Ωz(x,1)vdx+ξ(t)2z(x,0)2L2(Ω)ξ(t)(1τ(t))2z(x,1)2L2(Ω)ξ(t)τ(t)2Ω10z2(x,ρ)dρdx.
    $

    Therefore, from (16) and (17), we deduce

    $ A(t)U,Utμ1(t)(1ˉξ2β21d)v2L2(Ω)μ1(t)(ˉξ(1τ(t))2β1d2)z(x,1,t)2L2(Ω)+ξ(t)|τ(t)|2τ(t)τ(t)Ω10z2(x,ρ)dρdx.
    $

    Then, we have

    $ A(t)U,Utμ1(t)(1ˉξ2β21d)v2L2(Ω)μ1(t)(ˉξ(1τ(t))2β1d2)z(x,1,t)2L2(Ω)+κ(t)U,Ut,
    $

    where

    $ \kappa(t) = \frac{\sqrt{1+ \tau'(t)^2}}{2\tau(t)}. $

    From the (13), we obtain

    $ A(t)U,Utκ(t)U,Ut0,
    $
    (25)

    which means that the operator $ \tilde{\mathcal{A}} = \mathcal{A}(t) - \kappa(t)I $ is dissipative.

    Moreover, $ \kappa'(t) = \frac{\tau'(t)\tau''(t)}{2\tau(t)\sqrt{1+\tau'(t)^2}} - \frac{\tau'(t)\sqrt{1+\tau'(t)^2}}{2\tau(t)^2} $ is bounded on $ [0,T] $ for all $ T>0 $ (by (5) and (12)) and we have

    $ \frac{d}{dt}\mathcal{A}(t)U = \left(0,0,\frac{\tau''(t)\tau(t)\rho-\tau'(t)(\tau'(t)\rho-1)}{\tau(t)^2}z_{\rho} \right)^T, $

    with $ \frac{\tau''(t)\tau(t)\rho-\tau'(t)(\tau'(t)\rho-1)}{\tau(t)^2} $ bounded on $ [0,T] $ by (5) and (12). Thus

    $ ddt˜A(t)L([0,T],B(D(A(0)),H)),
    $
    (26)

    the space of equivalence classes of essentially bounded, strongly measurable functions from $ [0,T] $ into $ B(D(\mathcal{A}(0)), \mathcal{H}) $.

    Now, we will show that $ \lambda I - \mathcal{A}(t) $ is surjective for fixed $ t>0 $ and $ \lambda > 0 $. For this purpose, let $ F = (f_1,f_2,f_3)^T \in \mathcal{H} $, we seek $ U = (u,v,z)^T \in D(\mathcal{A}(t)) $ solution of

    $ \left( \lambda I - \mathcal{A}(t) \right)U = F, $

    that is verifying following system of equations

    $ {λuv=f1,λvuxx+μ1(t)vμ2(t)z(,1)=f2,λz+1τ(t)ρτ(t)zρ=f3.
    $
    (27)

    Suppose that we have found $ u $ with the appropriated regularity. Then

    $ v=λuf1.
    $
    (28)

    It is clear that $ v \in H_0^1(\Omega) $. Furthermore, by (27) we can find $ z $. From (20), we have

    $ z(x,0)=v(x),  for xΩ.
    $
    (29)

    Following the same approach as in [22], we obtain, by using equation for $ z $ in (27),

    $ z(x,\rho) = v(x)e^{-\vartheta(\rho,t)} + \tau(t)e^{-\vartheta(\rho,t)} \int_{0}^{\rho} f_3(x,s)e^{\vartheta(s,t)}\,ds, $

    if $ \tau'(t) = 0 $, where $ \vartheta(\ell,t) = \lambda \ell \tau(t) $, and

    $ z(x,\rho) = v(x)e^{\zeta(\rho,t)} + e^{\zeta(\rho,t)} \int_{0}^{\rho} \frac{\tau(t)f_3(x,s)}{1-s\tau'(s)} e^{-\zeta(s,t)}\,ds, $

    otherwise, where $ \zeta(\ell,t) = \lambda \frac{\tau(t)}{\tau'(t)}\ln(1- \ell \tau'(t)) $.

    From (28), we obtain

    $ z(x,ρ)=λu(x)eϑ(ρ,t)f1(x,ρ)eϑ(ρ,t)+τ(t)eϑ(ρ,t)ρ0f3(x,s)eϑ(s,t)ds,
    $
    (30)

    if $ \tau'(t) = 0 $, and

    $ z(x,ρ)=λu(x)eζ(ρ,t)f1(x,ρ)eζ(ρ,t)+eζ(ρ,t)ρ0τ(t)f3(x,s)1sτ(s)eζ(s,t)ds,
    $
    (31)

    otherwise.

    In particular, if $ \tau'(t) = 0 $ and from (30), we have

    $ z(x,1)=λu(x)eϑ(1,t)f1(x,1)eϑ(1,t)+τ(t)eϑ(1,t)10f3(x,s)eϑ(s,t)ds,
    $
    (32)

    and if $ \tau'(t) \neq 0 $ and from (31), we have

    $ z(x,1)=λu(x)eζ(1,t)f1(x,1)eζ(1,t)+eζ(1,t)10τ(t)f3(x,s)1sτ(s)eζ(s,t)ds.
    $
    (33)

    By using (27) and (28), the function $ u $ satisfies

    $ λ2uuxx+μ1(t)v+μ2(t)z(,1)=f2+λf1.
    $
    (34)

    Solving the equation (34) is equivalent to finding $ u \in H^2(\Omega) \cap H_0^1(\Omega) $ such that

    $ Ω(λ2uη+uxηx+μ1(t)vη+μ2(t)z(,1)η)dx=Ω(f2+λf1)ηdx,
    $
    (35)

    for all $ \eta \in H_0^1(\Omega) $.

    Consequently, the equation (35) is equivalent to the problem

    $ Υ(u,η)=L(η),
    $
    (36)

    where the bilinear form

    $ \Upsilon: H_0^1(\Omega) \times H_0^1(\Omega) \rightarrow \mathbb{R} $

    and the linear form

    $ L: H_0^1(\Omega) \rightarrow \mathbb{R} $

    are defined by

    $ \Upsilon(u, \eta) = \int_{\Omega} \left( \lambda^2 u\eta + u_x \eta_x \right)\,dx + \int_{\Omega} \lambda u \left( \mu_1(t) + \mu_2(t)N_1 \right)\eta\,dx $

    and

    $ L(\eta) = \int_{\Omega} \left( \mu_1(t)f_1 \eta + \mu_2(t)N_2 \right)\eta\,dx + \int_{\Omega} (f_2 + \lambda f_1)\eta\,dx, $

    where

    $ N_1 = \left\{ eϑ(1,t),ifτ(t)=0,eζ(1,t),ifτ(t)0
    \right. $

    and

    $ N_2 = \left\{ f1(x,1)eϑ(1,t)+τ(t)eϑ(1,t)10f3(x,s)eϑ(s,t)ds,ifτ(t)=0,f1(x,1)ezeta(1,t)+ezeta(1,t)10τ(t)f3(x,s)1sτ(t)eζ(s,t)ds,ifτ(t)0.
    \right. $

    It is easy to verify that $ \Upsilon $ is continuous and coercive, and $ L $ is continuous. So applying the Lax-Milgram theorem, we deduce that for all $ \eta \in H_0^1(\Omega) $ the problem (36) admits a unique solution

    $ u \in H_0^1(\Omega). $

    Applying the classical elliptic regularity, it follows from (35) that

    $ u \in H^2(\Omega). $

    Therefore, the operator $ \lambda I - \mathcal{A}(t) $ is surjective for any $ \lambda > 0 $ and $ t>0 $. Again as $ \kappa(t)>0 $, this prove that

    $ λI˜A(t)=(λ+κ(t))IA(t) is surjective,
    $
    (37)

    for any $ \lambda >0 $ and $ t>0 $.

    Then, (24), (25) and (37) imply that the family $ \tilde{\mathcal{A}} = \left\{ \tilde{\mathcal{A}}(t)/ t \in [0,T] \right\} $ is a stable family of generators in $ \mathcal{H} $ with stability constants independent of $ t $, by Proposition $ 1.1 $ from [14]. Therefore, the assumptions $ (i)-(iv) $ of Theorem 3.1 are verified by (21), (24), (25), (26), (37) and (23), and thus, the problem

    $ {˜Ut=˜A(t)˜U,˜U(0)=U0=(u0,u1,f0(,,τ(0)))T
    $
    (38)

    has a unique solution $ \tilde{U} \in C\left( [0,+\infty[, D(\mathcal{A}(0)) \right) \cap C^1\left( [0,+\infty[, \mathcal{H} \right) $ for $ U_0 \in D(\mathcal{A}(0)) $. The requested solution of (18) is then given by

    $ U(t) = e^{\int_0^t \kappa(s)\,ds}\tilde{U}(t) $

    because

    $ Ut(t)=κ(t)et0κ(s)ds˜U(t)+et0κ(s)ds˜Ut(t)=et0κ(s)ds(κ(t)+˜A(t))˜U(t)=A(t)et0κ(s)ds˜U(t)=A(t)U(t),
    $

    which concludes the proof.

    The existence and uniqueness are obtained by the following result.

    Theorem 3.2 (Global solution). For any initial datum $ U_0 \in \mathcal{H} $ there exists a unique solution $ U $ satisfying

    $ U \in C([0,+\infty[, \mathcal{H}) $

    for problem (18).

    Moreover, if $ U_0 \in D(\mathcal{A}(0)) $, then

    $ U \in C([0,+\infty[, D(\mathcal{A}(0))) \cap C^1([0,+\infty[, \mathcal{H}). $

    Proof. A general theory for equations of type (18) has been developed using semigroup theory [14], [15] and [26]. The simplest way to prove existence and uniqueness results in to show that the triplet $ \left\{ (\mathcal{A}, \mathcal{H}, Y) \right\} $, with $ \mathcal{A} = \left\{ \mathcal{A}(t)/ t \in [0,T] \right\} $, for some fixed $ T>0 $ and $ Y = \mathcal{A}(0) $, forms a CD-systems (or constant domain system, see [14] and [15]). More precisely, the following theorem gives the existence and uniqueness results and is proved in Theorem $ 1.9 $ of [14] (see also Theorem $ 2.13 $ of [15] or [1]).

    In this section we shall investigate the asymptotic behavior of problem (1). The stability result will be obtained using Lemma 2.2.

    Theorem 4.1 (Stability Result). Let $ (u_0,u_1,f_0(\cdot,-,\tau(0))) \in H_0^1(\Omega) \times L^2(\Omega) \times L^2(\Omega \times ]0,1[) $. Assume that the hypotheses (H1), (H2) and (5)-(7) hold. Then problem (1) admits a unique solution

    $ uC([0,+[,H10(Ω))C1([0,+[,L2(Ω)),
    $
    $ zC([0,+[,L2(Ω)×]0,1[).
    $

    Proof. From now on, we denote by $ c $ various positive constants which may be different at different occurrences.

    Given $ 0 \leq S < T < \infty $ we start by multiplying the first equation of (9) by $ uE^q $ and then integrating over $ (S,T) \times \Omega $, we obtain

    $ TSEqΩu(uttuxx+μ1(t)ut+μ2(t)z(x,1,t))dxdt=0.
    $

    Notice that

    $ u_{tt}u = \left( u_t u \right)_t - u_t^2, $

    using integration by parts and the boundary conditions we know that

    $ 0=[Eq(t)Ωuutdx]TSTSqEq1(t)E(t)ΩuutdxdtTSEq(t)ut2L2(Ω)dt+TSEq(t)ux2L2(Ω)dt+TSEq(t)Ωμ1(t)uutdxdt+TSEq(t)Ωμ2(t)uz(x,1,t)dxdt.
    $
    (39)

    Similarly, we multiply the second equation of (9) by $ E^q \xi(t) e^{-2\rho\tau(t)}z(x,\rho,t) $ and then integrate over $ \Omega \times (0,1) \times (S,T) $ to see that

    $ 0=TSΩ10Eq(t)ξ(t)e2ρτ(t)z(τ(t)zt+(1ρτ(t))zρ)dρdxdt
    $
    $ =12Ω10TSEq(t)ξ(t)e2ρτ(t)tz2dtdρdx+12TSEq(t)ξ(t)Ω10e2ρτ(t)(1ρτ(t))ρz2dρdxdt.
    $

    Using integration by parts and the boundary conditions we know that

    $ 0=[ξ(t)τ(t)2Eq(t)Ω10e2ρτ(t)z2dρdx]TS12TSqEq1(t)E(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt12TSqEq(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt+12TSEq(t)ξ(t)Ω[e2ρτ(t)(1τ(t))z2(x,1,t)z2(x,0,t)]dxdt+TSEq(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt.
    $
    (40)

    Since $ \mu_1 $ is a non-increasing function of class $ C_1(\mathbb{R}) $, its derivatives is non-positive, which implies that $ \xi'(t) \leq 0 $. This result this

    $ TSqEq(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt0.
    $
    (41)

    Moreover, as

    $ 12TSEq(t)ξ(t)Ωe2ρτ(t)(1τ(t))z2(x,1,t)dxdt0,
    $
    (42)

    then, from (40), (41) and (42), we have that

    $ TSEq(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt[ξ(t)τ(t)2Eq(t)Ω10e2ρτ(t)z2dρdx]TS+12TSqEq1(t)E(t)ξ(t)τ(t)Ω10e2ρτ(t)z2dρdxdt12TSEq(t)ξ(t)Ωz2(x,0,t)dxdt.
    $
    (43)

    Using the definition of $ E $, (39) and (43), we get

    $ γ0TSEq+1dt[Eq(t)Ωuutdx]TS[ξ(t)τ(t)2Eq(t)Ω10e2ρτ(t)z2dρdx]TS+qTSEq1(t)E(t)Ωuutdxdt+qTSξ(t)τ(t)2Eq1(t)E(t)Ω10e2ρτ(t)z2dρdxdt
    $
    $ +2TSEq(t)ut2L2(Ω)dtTSEq(t)Ωμ1(t)uutdxdtTSEq(t)Ωμ2(t)uz(x,1,t)dxdt+12TSξ(t)Eq(t)e2ρτ(t)Ωz2(x,0,t)dxdt,
    $
    (44)

    where $ \gamma_0 = 2\min \{ 1, e^{-2\tau_1} \} $.

    Using the Young and Sobolev-Poincaré inequalities and Lemma 2.3, we find that

    $ [Eq(t)Ωuutdx]TSEq(S)Ωu(x,S)ut(x,S)dxEq(T)Ωu(x,T)ut(x,T)dxcEq+1(S).
    $

    Now, we known that

    $ [ξ(t)τ(t)2Eq(t)Ω10e2ρτ(t)z2dρdx]TSξ(S)τ(S)2Eq(S)Ω10e2ρτ(S)z2(x,ρ,S)dρdxcEq(S)ξ(S)τ(S)Ω10z2(x,ρ,S)dρdxcEq+1(S).
    $

    By (13), we have

    $ TSEq1(t)E(t)ΩuutdxdtcTS(E(t))Eq(t)dtcEq+1(S).
    $

    Similarly,

    $ TSEq1(t)E(t)ξ(t)τ(t)2Ω10e2ρτ(t)z2dρdxdtcEq+1(S).
    $

    From Lemma 2.4, we deduce that

    $ TSEq(t)ut2L2(Ω)dtcTSEq(t)E(t)dtcEq+1(S).
    $

    Now, we get that

    $ |TSEq(t)Ωμ1(t)uutdxdt|μ1(0)|TSEq(t)Ωuutdxdt|c(ε1)TSEq(t)Ωu2tdxdt+ε1TSEq(t)Ωu2xdxdtc(ε1)TSEq(t)(E(t))dt+ε1TSEq(t)E(t)dtc(ε1)Eq+1(S)+ε1TSEq+1(t)dt
    $
    (45)

    and from (H2) we obtain that

    $ |TSEq(t)Ωμ2(t)uz(x,1,t)dxdt|βμ1(0)|TSEq(t)Ωφz(x,1,t)dxdt|c(ε2)Eq+1(S)+ε2TSEq+1(t)dt.
    $
    (46)

    Finally,

    $ 12TSEq(t)ξ(t)Ωz2(x,0,t)dxdtˉξμ1(0)2TSEq(t)ut2L2(Ω)dtcTSEq(t)(E(t))dtcEq+1(S).
    $

    Choosing $ \varepsilon_1 $ and $ \varepsilon_2 $ small enough, we deduce from (45) and (46) that

    $ \int_S^T E^{q+1}\,dt \leq \frac{1}{\gamma}E^{q+1}(S). $

    Since $ E(S) \leq E(0) $ for $ S \geq 0 $, we have that

    $ \int_S^T E^{q+1}\,dt \leq \frac{1}{\gamma}E(0)E^{q}(S). $

    We choose $ q = 0 $, we conclude from Lemma 2.2 that

    $ E(t) \leq E(0)e^{1-\gamma t}. $

    This ends the proof of Theorem 4.1.

    [1] Halik M, Klauk H, Zschieschang U, et al. (2004) Low-voltage organic transistors with an amorphous molecular gate dielectric. Nature 431: 963-966.
    [2] Moon H, Zeis R, Borkent E, et al. (2004) Synthesis, crystal structure, and transistor performance of tetracene derivatives. J Am Chem Soc 126: 15322-15223. doi: 10.1021/ja045208p
    [3] Jung B., Lee K, Sun J, et al. (2010) Air-operable, high-mobility organic transistors with semifluorinated side chains and unsubstituted naphthalenetetracarboxylic diimide cores: high mobility and environmental and bias stress stability from the perfluorooctylpropyl side chain. Adv Funct Mater 20: 2930-2944. doi: 10.1002/adfm.201000655
    [4] Dickey K, Anthony J, Loo Y, (2006) Improving organic thin-Film transistor performance through solvent-vapor annealing of solution-processable triethylsilylethynyl anthradithiophene. Adv Mater 18: 1721-1726. doi: 10.1002/adma.200600188
    [5] Coropceanu V, Cornil J, Demetrio A, et al. (2007) Charge transport in organic semiconductors. Chem Rev 107: 926-952. doi: 10.1021/cr050140x
    [6] Li W, Auciello O, Premnath R, et al. (2010) Giant dielectric constant dominated by Maxwell-Wagner relaxation in Al2O3/TiO2 nanolaminates synthesized by atomic layer deposition. Appl Phys Lett 96: 162907. doi: 10.1063/1.3413961
    [7] Lee J, Kim J, Im S, et al. (2003) Pentacene thin-film transistors with Al2O3+x gate dielectric films deposited on indium-tin-oxide glass. Appl Phys Lett 83: 2689. doi: 10.1063/1.1613997
    [8] Maunoury C, Dabertrand K, Martinez E, et al. (2007) Chemical interface analysis of as grown HfO2 ultrathin films on SiO2. J Appl Phys 101: 034112. doi: 10.1063/1.2435061
    [9] Di C, Yu G, Liu Y, et al. (2006) High-performance low-cost organic field-effect transistors with chemically modified bottom electrodes. J Am Chem Soc 128: 16418-16419. doi: 10.1021/ja066092v
    [10] Di C, Yu G, Liu Y, et al. (2008) High-performance organic field-effect transistors with low-cost copper electrodes. Adv Mater 20: 1286-1290. doi: 10.1002/adma.200701812
    [11] Di C, Liu Y, Yu G, et al. (2009) Interface engineering: an effective approach toward high-performance organic field-effect transistors. Acc Chem Res 42: 1573-1583. doi: 10.1021/ar9000873
    [12] Ma H, Yip H, Huang F, et al. (2010) Interface engineering for organic electronics. Adv Funct Mater 20: 1371-1388. doi: 10.1002/adfm.200902236
    [13] Chua L, Zaumseil J, Chang J, et al. (2005) General observation of n-type field-effect behaviour in organic semiconductors. Nature 434: 194-199. doi: 10.1038/nature03376
    [14] Frank M, Sayan S, Dörmann S, et al. (2004) Hafnium oxide gate dielectrics grown from an alkoxide precursor: structure and defects. J Mater Sci Eng B 109: 6-10. doi: 10.1016/j.mseb.2003.10.020
    [15] Lu Y, Lee W, Lee H, et al. (2009) Low-voltage organic transistors with titanium oxide/polystyrene bilayer dielectrics. Appl Phys Lett 94: 113303. doi: 10.1063/1.3097010
    [16] Fleischli F, Suarez S, Schaer M, et al. (2010) Organic thin-film transistors: the passivation of the dielectric-pentacene interface by dipolar self-assembled monolayers. Langmuir 26: 15044-15049. doi: 10.1021/la102060u
    [17] Wu W, Liu Y, Wang Y, et al. (2008) High-performance, low-operating-voltage organic field-effect transistors with low pinch-off voltages. Adv Funct Mater 18: 810-815. doi: 10.1002/adfm.200701125
    [18] Tang M, Okamot T, Bao Z, (2006) High-performance organic semiconductors: asymmetric linear acenes containing sulphur. J Am Chem Soc 128: 16002-16003.
    [19] Acton O, Osaka I, Ting G, et al. (2009) Phosphonic acid self-assembled monolayer and amorphous hafnium oxide hybrid dielectric for high performance polymer thin film transistors on plastic substrates. Appl Phys Lett 95: 113305. doi: 10.1063/1.3231445
    [20] Gao W, Dickinson L, Grozinger C, et al. (2009) Self-assembled monolayers of alkylphosphonic acids on metal oxides. Langmuir 12: 6429-6435.
    [21] Ma H, Acton O, Ting G, et al. (2008) Low-voltage organic thin-film transistors with π-σ-phosphonic acid molecular dielectric monolayers. Appl Phys Lett 92: 113303. doi: 10.1063/1.2857502
    [22] McElwee J, Helmy R, Fadeev A, et al. (2005) Thermal stability of organic monolayers chemically grafted to minerals. J Colloid Interface Sci 285: 551-556. doi: 10.1016/j.jcis.2004.12.006
    [23] Acton O, Ting G, Ma H, et al. (2008) π‐σ‐Phosphonic acid organic monolayer/sol-gel hafnium oxide hybrid dielectrics for low‐voltage organic transistors. Adv Mater 20: 3697-3701. doi: 10.1002/adma.200800810
    [24] Lu X, Minari T, Kumatani A. et al. (2011) Effect of air exposure on metal/organic interface in organic field-effect transistors. Appl Phys Lett 98: 243301. doi: 10.1063/1.3599056
    [25] Di C, Yu G, Liu Y, et al. (2008) Efficient modification of Cu electrode with nanometer-sized copper tetracyanoquinodimethane for high performance organic field-effect transistors. Phys Chem Chem Phys 10: 2302-2307. doi: 10.1039/b718935d
    [26] Gu W, Jin W, Wei B, at al. (2010) High-performance organic field-effect transistors based on copper/copper sulphide bilayer source-drain electrodes. Appl Phys Lett 97: 243303. doi: 10.1063/1.3526737
    [27] Su Y, Wang C, Xie W, et al. (2011) Low-voltage organic field-effect transistors (OFETs) with solution-processed metal-oxide as gate dielectric. ACS Appl Mater Interfaces 3: 4662-4667. doi: 10.1021/am201078v
    [28] Su Y, Xie W, Li Y, et al. (2013) A low-temperature, solution-processed high-k dielectric for low-voltage, high-performance organic field-effect transistors (OFETs). J Phys D Appl Phys 46: 095105. doi: 10.1088/0022-3727/46/9/095105
    [29] Su Y, Ouyang M, Liu P, et al. (2013) Insights into the interfacial properties of low-voltage CuPc field-effect transistor. ACS Appl Mater Interfaces 5: 4960-4965. doi: 10.1021/am4006447
    [30] Su Y, Wang M, Xie F, et al. (2013) In situ modification of low-cost Cu electrodes for high-performance low-voltage pentacene thin film transistors (TFTs). Org Electron 14: 775-781. doi: 10.1016/j.orgel.2012.12.025
    [31] Su Y, Xie W, Xu J, (2014) Facile modification of Cu source-drain (S/D) electrodes for high-performance, low-voltage n-channel organic thin film transistors (OTFTs) based on C60. Org Electron 15: 3259-3267.
    [32] Su Y, Jiang J, Ke N, et al. (2013) Low-voltage flexible pentacene thin film transistors with solution-processed dielectric and low-cost source-drain (S/D) electrodes. J Mater Chem C 1: 2585-2592. doi: 10.1039/c3tc00577a
    [33] Forrest S, (1997) Ultrathin organic films grown by organic molecular beam deposition and related techniques. Chem Rev 97: 1793-1896. doi: 10.1021/cr941014o
    [34] Yang S, Shin K, Par C, (2015) The effect of gate-dielectric surface energy on pentacene morphology and organic field-effect transistor characteristics. Adv Funct Mater 15: 1806-1814.
    [35] Gao J, Xu J, Zhu M, et al. (2007) Thickness dependence of mobility in CuPc thin film on amorphous SiO2 substrate. J Phys D Appl Phys 40: 5666-5669. doi: 10.1088/0022-3727/40/18/022
    [36] Qi Q, Yu A, Wang L, et al. (2010) Behavior of pentacene initial nucleation on various dielectrics and its effect on carrier transport in organic field-effect transistor. J Nanosci Nanotechnol 10: 7103-7107. doi: 10.1166/jnn.2010.2802
    [37] Chung Y, Verploegen E, Vailionis A, et al. (2011) Controlling electric dipoles in nanodielectrics and its applications for enabling air-stable n-channel organic transistors. Nano Lett 11: 1161-1165. doi: 10.1021/nl104087u
    [38] Nakamura M, Goto N, Ohashi N, et al. (2005) Potential mapping of pentacene thin-film transistors using purely electric atomic-force-microscope potentiometry. Appl Phys Lett 86: 122112. doi: 10.1063/1.1891306
    [39] Wang S, Minari T, Miyadera T, et al. (2007) Bias stress instability in pentacene thin film transistors: contact resistance change and channel threshold voltage shift. Appl Phys Lett 91: 203508. doi: 10.1063/1.2813640
    [40] Kim Y, Jeon D, (2010) Effect of deposition temperature on the morphology and contact resistance of Au on pentacene. J Appl Phys 108: 016101. doi: 10.1063/1.3445268
    [41] Diao L, Frisbie C, Schroepfer D, et al. (2007) Electrical characterization of metal/pentacene contacts. J Appl Phys 101: 014510. doi: 10.1063/1.2424396
    [42] Zaumseil J, Baldwin K, Rogers J, (2003) Contact resistance in organic transistors that use source and drain electrodes formed by soft contact lamination. J Appl Phys 93: 6117-6124. doi: 10.1063/1.1568157
    [43] Watkins N, Yan L, Gao Y, (2002) Electronic structure symmetry of interfaces between pentacene and metals. Appl Phys Lett 80: 4384. doi: 10.1063/1.1485129
    [44] Kang S, Yi Y, Kim C, et al. (2006) Energy level diagrams of C60/pentacene/Au and pentacene/C60/Au. Synth Met 156: 32-37. doi: 10.1016/j.synthmet.2005.10.001
    [45] Zhang D, Liu Y, Liu Y, et al. (2004) The electrical properties and the interfaces of Cu2O/ZnO/ITO p-i-n heterojunction. Physica B: Cond Mat 351: 178-183. doi: 10.1016/j.physb.2004.06.003
    [46] Tseng C, Cheng Y, Lee M, et al. (2007) Study of anode work function modified by self-assembled monolayers on pentacene/fullerene organic solar cells. Appl Phys Lett 91: 233510. doi: 10.1063/1.2823579
    [47] Zhou Y, Fuentes-Hernandez C, Shim J, et al. (2012) A universal method to produce low-work function rlectrodes for organic electronics. Science 336: 327-332. doi: 10.1126/science.1218829
    [48] Yu Y, Zhao Y, Ryu S, et al. (2009) Tuning the graphene work function by electric field effect. Nano Lett 9: 3430-3434.
    [49] Rentenberger S, Vollmer A, Zojer E, (2006) UV∕ozone treated Au for air-stable, low hole injection barrier electrodes in organic electronics. J Appl Phys 100: 053701. doi: 10.1063/1.2336345
    [50] Scheinert S, Grobosch M, Paasch G, et al. (2012) Contact characterization by photoemission and device performance in P3HT based organic transistors. J Appl Phys 111: 064502. doi: 10.1063/1.3693541
    [51] Shibata K, Ishikawa K, Takezoe H, et al. (2008) Contact characterization by photoemission and device performance in P3HT based organic transistors. Appl Phys Lett 92: 023305. doi: 10.1063/1.2834374
    [52] Kumatani A, Li Y, Darmawan P, et al. (2013) On practical charge injection at the metal/organic semiconductor interface. Sci Rep 3: 1026.
    [53] Heimel G, Romaner L, Bredas J, et al. (2006) Interface energetics and level alignment at covalent metal-molecule Junctions: π-conjugated thiols on gold. Phys Rev Lett 96: 196806. doi: 10.1103/PhysRevLett.96.196806
    [54] Heimel G, Romaner L, Zojer E, et al. (2008) The interface energetics of self-assembled monolayers on metals. Acc Chem Res 41: 721-729. doi: 10.1021/ar700284q
    [55] Li H, Paramonov P, Bredas J, (2010) Theoretical study of the surface modification of indium tin oxide with trifluorophenyl phosphonic acid molecules: impact of coverage density and binding geometry. J Mater Chem 20: 2630-2637. doi: 10.1039/b921768a
    [56] Yang R, Park J, Colesniuc C, et al. (2009) Analyte chemisorption and sensing on n- and p-channel copper phthalocyanine thin-film transistors. J Chem Phys 130: 164703. doi: 10.1063/1.3078036
    [57] Ma H, Acton O, Hutchins D, et al. (2012) Multifunctional phosphonic acid self-assembled monolayers on metal oxides as dielectrics, interface modification layers and semiconductors for low-voltage high-performance organic field-effect transistors. Phys Chem Chem Phys 14: 14110-14126.
    [58] Acton O, Ting G, Ma H, et al. (2008) Low-voltage high-performance C60 thin film transistors via low-surface-energy phosphonic acid monolayer/hafnium oxide hybrid dielectric. Appl Phys Lett 93: 083302. doi: 10.1063/1.2975175
    [59] Colleaux F, Ball J, Wobkenberg P, et al. (2011) Bias-stress effects in organic field-effect transistors based on self-assembled monolayer nanodielectrics. Phys Chem Chem Phys 13: 14387-14393. doi: 10.1039/c1cp20769e
    [60] Nanditha D, Dissanayake M, Hatton R, (2007) Operation of a reversed pentacene-fullerene discrete heterojunction photovoltaic device. Appl Phys Lett 90: 113505. doi: 10.1063/1.2713345
    [61] Chu C, Shrotriya V, Li G, et al. (2006) Tuning acceptor energy level for efficient charge collection in copper-phthalocyanine-based organic solar cells. Appl Phys Lett 88: 153504. doi: 10.1063/1.2194207
    [62] Chu C, Sung C, Kekuda D, et al. (2009) Flexible fullerene field-effect transistors fabricated through solution processing. Adv Mater 21: 4845-4849. doi: 10.1002/adma.200901215
    [63] Ishii H, Hayashi N, Ito E, et. (2004) Kelvin probe study of band bending at organic semiconductor/metal interfaces: examination of Fermi level alignment. Phys Stat Sol A 201: 1075-1094. doi: 10.1002/pssa.200404346
    [64] Kuribara K, Wang H, Uchiyama N, et al. (2012) Organic transistors with high thermal stability for medical applications. Nat Commun 3: 723. doi: 10.1038/ncomms1721
    [65] Kraus M, Richler S, Opitz A, et al. (2010) High-mobility copper-phthalocyanine field-effect transistors with tetratetracontane passivation layer and organic metal contacts. J Appl Phys 107: 094503. doi: 10.1063/1.3354086
    [66] Ramanathan S, Park C, McIntyre P, (2002) Electrical properties of ultrathin zirconia films grown by UV ozone oxidation. J Appl Phys 91: 4521-4527. doi: 10.1063/1.1459103
    [67] Gonon P, El Kamel F, (2007) Dielectric response of Cu/amorphous BaTiO3/Cu capacitors. J Appl Phys 101: 073901. doi: 10.1063/1.2716871
    [68] Klauk H, Zschieschang U, Pflaum J, et al. (2007) Ultralow-power organic complementary circuits. Nature 445: 745-748. doi: 10.1038/nature05533
    [69] Kim S, Yoon W, Jang M, et al. (2012) Damage-free hybrid encapsulation of organic field-effect transistors to reduce environmental instability. J Mater Chem 22: 7731-7738. doi: 10.1039/c2jm13329f
    [70] Yang H, Kim S, Yang L, et al. (2007) Pentacene nanostructures on surface-hydrophobicity-controlled polymer/SiO2 bilayer gate-dielectrics. Adv Mater 19: 2868-2872. doi: 10.1002/adma.200700560
    [71] Yang H, Yang L, Ling M, et al. (2008) Aging susceptibility of terrace-like pentacene films. J Phys Chem C 112: 16161-16165. doi: 10.1021/jp8055224
    [72] Nabok D, Puschnig P, Ambrosch-Draxl C, et al. (2007) Crystal and electronic structures of pentacene thin films from grazing-incidence x-ray diffraction and first-principles calculations. Phys Rev B 76: 235322. doi: 10.1103/PhysRevB.76.235322
    [73] Yang C, Yoon J, Kim S, et al. Bending-stress-driven phase transitions in pentacene thin films for flexible organic field-effect transistors. Appl Phys Lett 92: 243305.
    [74] Kang S, Noh Y, Baeg K, et al. (2008) Effect of rubbed polyimide layer on the field-effect mobility in pentacene thin-film transistors. Appl Phys Lett 92: 052107. doi: 10.1063/1.2830694
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