Processing math: 68%
Research article

Analysis of drain induced barrier lowering for junctionless double gate MOSFET using ferroelectric negative capacitance effect

  • Received: 04 October 2022 Revised: 15 November 2022 Accepted: 02 December 2022 Published: 14 December 2022
  • We analyze the drain induced barrier lowering (DIBL) of a negative capacitance (NC) FET using a gate structure such as a metal-ferroelectric-metal-insulator-semiconductor (MFMIS) for a junctionless double gate (JLDG) FET. NC FETs show negative DIBL characteristics according to the ferroelectric thickness. To elucidate the cause of such negative DIBL, the DIBLs are obtained by the second derivative method using the 2D potential distribution and drain current-gate voltage curve. The analytical DIBL model is also presented for easy observation of the DIBL of NC FET. It has been found that the results of this analytical DIBL model are very similar to those of the second derivative method. The results of this analytical DIBL model are also in good agreement with the results of TCAD. As a result, it was found that the negative DIBL phenomenon is caused by the change according to the drain voltage of the charge existing in the ferroelectric material. The negative DIBL phenomenon easily occurred as the ferroelectric thickness increased and the thickness of SiO2 used as an insulator decreases.

    Citation: Hakkee Jung. Analysis of drain induced barrier lowering for junctionless double gate MOSFET using ferroelectric negative capacitance effect[J]. AIMS Electronics and Electrical Engineering, 2023, 7(1): 38-49. doi: 10.3934/electreng.2023003

    Related Papers:

    [1] Salim A. Messaoudi, Mohammad M. Al-Gharabli, Adel M. Al-Mahdi, Mohammed A. Al-Osta . A coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities: Existence, uniqueness, blow-up and a large-time asymptotic behavior. AIMS Mathematics, 2023, 8(4): 7933-7966. doi: 10.3934/math.2023400
    [2] Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Maher Nour, Mostafa Zahri . Stabilization of a viscoelastic wave equation with boundary damping and variable exponents: Theoretical and numerical study. AIMS Mathematics, 2022, 7(8): 15370-15401. doi: 10.3934/math.2022842
    [3] Xiongmei Fan, Sen Ming, Wei Han, Zikun Liang . Lifespan estimate of solution to the semilinear wave equation with damping term and mass term. AIMS Mathematics, 2023, 8(8): 17860-17889. doi: 10.3934/math.2023910
    [4] Mohammad Kafini, Jamilu Hashim Hassan, Mohammad M. Al-Gharabli . Decay result in a problem of a nonlinearly damped wave equation with variable exponent. AIMS Mathematics, 2022, 7(2): 3067-3082. doi: 10.3934/math.2022170
    [5] Adel M. Al-Mahdi . The coupling system of Kirchhoff and Euler-Bernoulli plates with logarithmic source terms: Strong damping versus weak damping of variable-exponent type. AIMS Mathematics, 2023, 8(11): 27439-27459. doi: 10.3934/math.20231404
    [6] Noufe H. Aljahdaly . Study tsunamis through approximate solution of damped geophysical Korteweg-de Vries equation. AIMS Mathematics, 2024, 9(5): 10926-10934. doi: 10.3934/math.2024534
    [7] Mohammad Kafini, Mohammad M. Al-Gharabli, Adel M. Al-Mahdi . Existence and stability results of nonlinear swelling equations with logarithmic source terms. AIMS Mathematics, 2024, 9(5): 12825-12851. doi: 10.3934/math.2024627
    [8] Zhiqiang Li . The finite time blow-up for Caputo-Hadamard fractional diffusion equation involving nonlinear memory. AIMS Mathematics, 2022, 7(7): 12913-12934. doi: 10.3934/math.2022715
    [9] Muhammad Shakeel, Amna Mumtaz, Abdul Manan, Marouan Kouki, Nehad Ali Shah . Soliton solutions of the nonlinear dynamics in the Boussinesq equation with bifurcation analysis and chaos. AIMS Mathematics, 2025, 10(5): 10626-10649. doi: 10.3934/math.2025484
    [10] Khaled Kefi, Nasser S. Albalawi . Three weak solutions for degenerate weighted quasilinear elliptic equations with indefinite weights and variable exponents. AIMS Mathematics, 2025, 10(2): 4492-4503. doi: 10.3934/math.2025207
  • We analyze the drain induced barrier lowering (DIBL) of a negative capacitance (NC) FET using a gate structure such as a metal-ferroelectric-metal-insulator-semiconductor (MFMIS) for a junctionless double gate (JLDG) FET. NC FETs show negative DIBL characteristics according to the ferroelectric thickness. To elucidate the cause of such negative DIBL, the DIBLs are obtained by the second derivative method using the 2D potential distribution and drain current-gate voltage curve. The analytical DIBL model is also presented for easy observation of the DIBL of NC FET. It has been found that the results of this analytical DIBL model are very similar to those of the second derivative method. The results of this analytical DIBL model are also in good agreement with the results of TCAD. As a result, it was found that the negative DIBL phenomenon is caused by the change according to the drain voltage of the charge existing in the ferroelectric material. The negative DIBL phenomenon easily occurred as the ferroelectric thickness increased and the thickness of SiO2 used as an insulator decreases.



    We consider the Cauchy problem for the nonlinear damped wave equation with linear damping

    uttΔu+ut=|u|p+|u|q+w(x),t>0,xRN, (1.1)

    where N1, p,q>1, wL1loc(RN), w0 and w0. Namely, we are concerned with the existence and nonexistence of global weak solutions to (1.1). We mention below some motivations for studying problems of type (1.1).

    Consider the semilinear damped wave equation

    uttΔu+ut=|u|p,t>0,xRN (1.2)

    under the initial conditions

    (u(0,x),ut(0,x))=(u0(x),u1(x)),xRN. (1.3)

    In [12], a Fujita-type result was obtained for the problem (1.2) and (1.3). Namely, it was shown that,

    (ⅰ) if 1<p<1+2N and RNuj(x)dx>0, j=0,1, then the problem (1.2) and (1.3) admits no global solution;

    (ⅱ) if 1+2N<p<N for N3, and 1+2N<p< for N{1,2}, then the problem (1.2) and (1.3) admits a unique global solution for suitable initial values.

    The proof of part (ⅰ) makes use of the fundamental solution of the operator (ttΔ+t)k. In [15], it was shown that the exponent 1+2N belongs to the blow-up case (ⅰ). Notice that 1+2N is also the Fujita critical exponent for the semilinear heat equation utΔu=|u|p (see [3]). In [6], the authors considered the problem

    utt+(1)m|x|αΔmu+ut=f(t,x)|u|p+w(t,x),t>0,xRN (1.4)

    under the initial conditions (1.3), where m is a positive natural number, α0, and f(t,x)0 is a given function satisfying a certain condition. Using the test function method (see e.g. [11]), sufficient conditions for the nonexistence of a global weak solution to the problem (1.4) and (1.3) are obtained. Notice that in [6], the influence of the inhomogeneous term w(t,x) on the critical behavior of the problem (1.4) and (1.3) was not investigated. Recently, in [5], the authors investigated the inhomogeneous problem

    uttΔu+ut=|u|p+w(x),t>0,xRN, (1.5)

    where wL1loc(RN) and w0. It was shown that in the case N3, the critical exponent for the problem (1.5) jumps from 1+2N (the critical exponent for the problem (1.2)) to the bigger exponent 1+2N2. Notice that a similar phenomenon was observed for the inhomogeneous semilnear heat equation utΔu=|u|p+w(x) (see [14]). For other results related to the blow-up of solutions to damped wave equations, see, for example [1,2,4,7,8,9,13] and the references therein.

    In this paper, motivated by the above mentioned works, our aim is to study the effect of the gradient term |u|q on the critical behavior of problem (1.5). We first obtain the Fujita critical exponent for the problem (1.1). Next, we determine its second critical exponent in the sense of Lee and Ni [10].

    Before stating the main results, let us provide the notion of solutions to problem (1.1). Let Q=(0,)×RN. We denote by C2c(Q) the space of C2 real valued functions compactly supported in Q.

    Definition 1.1. We say that u=u(t,x) is a global weak solution to (1.1), if

    (u,u)Lploc(Q)×Lqloc(Q)

    and

    Q(|u|p+|u|q)φdxdt+Qw(x)φdxdt=QuφttdxdtQuΔφdxdtQuφtdxdt, (1.6)

    for all φC2c(Q).

    The first main result is the following Fujita-type theorem.

    Theorem 1.1. Let N1, p,q>1, wL1loc(RN), w0 and w0.

    (i) If N{1,2}, for all p,q>1, problem (1.1) admits no global weak solution.

    (ii) Let N3. If

    1<p<1+2N2or1<q<1+1N1,

    then problem (1.1) admits no global weak solution.

    (iii) Let N3. If

    p>1+2N2andq>1+1N1,

    then problem (1.1) admits global solutions for some w>0.

    We mention below some remarks related to Theorem 1.1. For N3, let

    pc(N,q)={1+2N2ifq>1+1N1,ifq<1+1N1.

    From Theorem 1.1, if 1<p<pc(N,q), then problem (1.1) admits no global weak solution, while if p>pc(N,q), global weak solutions exist for some w>0. This shows that pc(N,q) is the Fujita critical exponent for the problem (1.1). Notice that in the case N3 and q>1+1N1, pc(N,q) is also the critical exponent for uttΔu=|u|p+w(x) (see [5]). This shows that in the range q>1+1N1, the gradient term |u|q has no influence on the critical behavior of uttΔu=|u|p+w(x).

    It is interesting to note that the gradient term |u|q induces an interesting phenomenon of discontinuity of the Fujita critical exponent pc(N,q) jumping from 1+2N2 to as q reaches the value 1+1N1 from above.

    Next, for N3 and σ<N, we introduce the sets

    I+σ={wC(RN):w0,|x|σ=O(w(x)) as |x|}

    and

    Iσ={wC(RN):w>0,w(x)=O(|x|σ) as |x|}.

    The next result provides the second critical exponent for the problem (1.1) in the sense of Lee and Ni [10].

    Theorem 1.2. Let N3, p>1+2N2 and q>1+1N1.

    (i) If

    σ<max{2pp1,qq1}

    and wI+σ, then problem (1.1) admits no global weak solution.

    (ii) If

    σmax{2pp1,qq1},

    then problem (1.1) admits global solutions for some wIσ.

    From Theorem 1.2, if N3, p>1+2N2 and q>1+1N1, then problem (1.1) admits a second critical exponent, namely

    σ=max{2pp1,qq1}.

    The rest of the paper is organized as follows. In Section 2, some preliminary estimates are provided. Section 3 is devoted to the study of the Fujita critical exponent for the problem (1.1). Namely, Theorem 1.1 is proved. In Section 4, we study the second critical exponent for the problem (1.1) in the sense of Lee and Ni. Namely, we prove Theorem 1.2.

    Consider two cut-off functions f,gC([0,)) satisfying

    f0,f0,supp(f)(0,1)

    and

    0g1,g(s)={1if0s1,0ifs2.

    For T>0, we introduce the function

    ξT(t,x)=f(tT)g(|x|2T2ρ)=F(t)G(x),(t,x)Q, (2.1)

    where 2 and ρ>0 are constants to be chosen. It can be easily seen that

    ξT0andξTC2c(Q).

    Throughout this paper, the letter C denotes various positive constants depending only on known quantities.

    Lemma 2.1. Let κ>1 and >2κκ1. Then

    Q|ξT|1κ1|ΔξT|κκ1dxdtCT2ρκκ1+Nρ+1. (2.2)

    Proof. By (2.1), one has

    Q|ξT|1κ1|ΔξT|κκ1dxdt=(0F(t)dt)(RN|G|1κ1|ΔG|κκ1dx). (2.3)

    Using the properties of the cut-off function f, one obtains

    0F(t)dt=0f(tT)dt=T0f(tT)dt=T10f(s)ds,

    which shows that

    0F(t)dt=CT. (2.4)

    Next, using the properties of the cut-off function g, one obtains

    RN|G|1κ1|ΔG|κκ1dx=Tρ<|x|<2Tρ|G|1κ1|ΔG|κκ1dx (2.5)

    On the other hand, for Tρ<|x|<2Tρ, an elementary calculation yields

    ΔG(x)=Δ[g(r2T2ρ)],r=|x|=(d2dr2+N1rddr)g(r2T2ρ)=2T2ρg(r2T2ρ)2θ(x),

    where

    θ(x)=Ng(r2T2ρ)g(r2T2ρ)+2(1)T2ρr2g(r2T2ρ)g(r2T2ρ)2+2T2ρr2g(r2T2ρ)g(r2T2ρ).

    Notice that since Tρ<r<2Tρ, one deduces that

    |θ(x)|C.

    Hence, it holds that

    |ΔG(x)|CT2ρg(|x|2T2ρ)2,Tρ<|x|<2Tρ

    and

    |G(x)|1κ1|ΔG(x)|κκ1CT2ρκκ1g(|x|2T2ρ)2κκ1,Tρ<|x|<2Tρ.

    Therefore, by (2.5) and using the change of variable x=Tρy, one obtains

    RN|G|1κ1|ΔG|κκ1dxCT2ρκκ1Tρ<|x|<2Tρg(|x|2T2ρ)2κκ1dx=CT2ρκκ1+Nρ1<|y|<2g(|y|2)2κκ1dy,

    which yields (notice that >2κκ1)

    RN|G|1κ1|ΔG|κκ1dxCT2ρκκ1+Nρ. (2.6)

    Finally, using (2.3), (2.4) and (2.6), (2.2) follows.

    Lemma 2.2. Let κ>1 and >κκ1. Then

    Q|ξT|1κ1|ξT|κκ1dxdtCTρκκ1+Nρ+1. (2.7)

    Proof. By (2.1) and the properties of the cut-off function g, one has

    Q|ξT|1κ1|ξT|κκ1dxdt=(0F(t)dt)(Tρ<|x|<2Tρ|G|1κ1|G|κκ1dx). (2.8)

    On the other hand, for Tρ<|x|<2Tρ, an elementary calculation shows that

    |G(x)|=2T2ρ|x||g(|x|2T2ρ)|g(|x|2T2ρ)122Tρ|g(|x|2T2ρ)|g(|x|2T2ρ)1CTρg(|x|2T2ρ)1,

    which yields

    |G|1κ1|G|κκ1CTρκκ1g(|x|2T2ρ)κκ1,Tρ<|x|<2Tρ.

    Therefore, using the change of variable x=Tρy and the fact that >κκ1, one obtains

    Tρ<|x|<2Tρ|G|1κ1|G|κκ1dxCTρκκ1Tρ<|x|<2Tρg(|x|2T2ρ)κκ1dx=CTρκκ1+Nρ1<|y|<2g(|y|2)κκ1dy,

    i.e.

    Tρ<|x|<2Tρ|G|1κ1|G|κκ1dxCTρκκ1+Nρ. (2.9)

    Using (2.4), (2.8) and (2.9), (2.7) follows.

    Lemma 2.3. Let κ>1 and >2κκ1. Then

    Q|ξT|1κ1|ξTtt|κκ1dxdtCT2κκ1+1+Nρ. (2.10)

    Proof. By (2.1) and the properties of the cut-off functions f and g, one has

    Q|ξT|1κ1|ξTtt|κκ1dxdt=(T0F(t)1κ1|F(t)|κκ1dt)(0<|x|<2TρG(x)dx). (2.11)

    An elementary calculation shows that

    F(t)1κ1|F(t)|κκ1CT2κκ1f(tT)2κκ1,0<t<T,

    which yields (since >2κκ1)

    T0F(t)1κ1|F(t)|κκ1dtCT2κκ1+1. (2.12)

    On the other hand, using the change of variable x=Tρy, one obtains

    0<|x|<2TρG(x)dx=0<|x|<2Tρg(|x|2T2ρ)dx=TNρ0<|y|<2g(|y|2)dy,

    which yields

    0<|x|<2TρG(x)dxCTNρ. (2.13)

    Therefore, using (2.11), (2.12) and (2.13), (2.10) follows.

    Lemma 2.4. Let κ>1 and >κκ1. Then

    Q|ξT|1κ1|ξTt|κκ1dxdtCTκκ1+1+Nρ. (2.14)

    Proof. By (2.1) and the properties of the cut-off functions f and g, one has

    Q|ξT|1κ1|ξTt|κκ1dxdt=(T0F(t)1κ1|F(t)|κκ1dt)(0<|x|<2TρG(x)dx). (2.15)

    An elementary calculation shows that

    F(t)1κ1|F(t)|κκ1CTκκ1f(tT)κκ1,0<t<T,

    which yields (since >κκ1)

    T0F(t)1κ1|F(t)|κκ1dtCT1κκ1. (2.16)

    Therefore, using (2.13), (2.15) and (2.16), (2.14) follows.

    Proposition 3.1. Suppose that problem (1.1) admits a global weak solution u with (u,u)Lploc(Q)×Lqloc(Q). Then there exists a constant C>0 such that

    Qw(x)φdxdtCmin{A(φ),B(φ)}, (3.1)

    for all φC2c(Q), φ0, where

    A(φ)=Qφ1p1|φtt|pp1dxdt+Qφ1p1|φt|pp1dxdt+Qφ1p1|Δφ|pp1dxdt

    and

    B(φ)=Qφ1p1|φtt|pp1dxdt+Qφ1p1|φt|pp1dxdt+Qφ1q1|φ|qq1dxdt.

    Proof. Let u be a global weak solution to problem (1.1) with

    (u,u)Lploc(Q)×Lqloc(Q).

    Let φC2c(Q), φ0. Using (1.6), one obtains

    Q|u|pφdxdt+Qw(x)φdxdtQ|u||φtt|dxdt+Q|u||Δφ|dxdt+Q|u||φt|dxdt. (3.2)

    On the other hand, by ε-Young inequality, 0<ε<13, one has

    Q|u||φtt|dxdt=Q(|u|φ1p)(φ1p|φtt|)dxdtεQ|u|pφdxdt+CQφ1p1|φtt|pp1dxdt. (3.3)

    Here and below, C denotes a positive generic constant, whose value may change from line to line. Similarly, one has

    Q|u||Δφ|dxdtεQ|u|pφdxdt+CQφ1p1|Δφ|pp1dxdt (3.4)

    and

    Q|u||φt|dxdtεQ|u|pφdxdt+CQφ1p1|φt|pp1dxdt (3.5)

    Therefore, using (3.2), (3.3), (3.4) and (3.5), it holds that

    (13ε)Q|u|pφdxdt+Qw(x)φdxdtCA(φ).

    Since 13ε>0, one deduces that

    Qw(x)φdxdtCA(φ). (3.6)

    Again, by (1.6), one has

    Q(|u|p+|u|q)φdxdt+Qw(x)φdxdt=QuφttdxdtQuΔφdxdtQuφtdxdt=Quφttdxdt+QuφdxdtQuφtdxdt,

    where denotes the inner product in RN. Hence, it holds that

    Q(|u|p+|u|q)φdxdt+Qw(x)φdxdtQ|u||φtt|dxdt+Q|u||φ|dxdt+Q|u||φt|dxdt. (3.7)

    By ε-Young inequality (ε=12), one obtains

    Q|u||φtt|dxdt12Q|u|pφdxdt+CQφ1p1|φtt|pp1dxdt, (3.8)
    Q|u||φt|dxdt12Q|u|pφdxdt+CQφ1p1|φt|pp1dxdt, (3.9)
    Q|u||φ|dxdt12Q|u|qφdxdt+CQφ1q1|φ|qq1dxdt. (3.10)

    Next, using (3.7), (3.8), (3.9) and (3.10), one deduces that

    12Q|u|qφdxdt+Qw(x)φdxdtCB(φ),

    which yields

    Qw(x)φdxdtCB(φ). (3.11)

    Finally, combining (3.6) with (3.11), (3.1) follows.

    Proposition 3.2. Suppose that problem (1.1) admits a global weak solution u with (u,u)Lploc(Q)×Lqloc(Q). Then

    Qw(x)ξTdxdtCmin{A(ξT),B(ξT)}, (3.12)

    for all T>0, where ξT is defined by (2.1).

    Proof. Since for all T>0, ξTC2c(Q), ξT0, taking φ=ξT in (3.1), (3.12) follows.

    Proposition 3.3. Let wL1loc(RN), w0 and w0. Then, for sufficiently large T,

    Qw(x)ξTdxdtCT, (3.13)

    where ξT is defined by (2.1).

    Proof. By (2.1) and the properties of the cut-off functions f and g, one has

    Qw(x)ξTdxdt=(T0f(tT)dt)(0<|x|<2Tρw(x)g(|x|2T2ρ)dx). (3.14)

    On the other hand,

    T0f(tT)dt=T10f(s)ds. (3.15)

    Moreover, for sufficiently large T (since w,g0),

    0<|x|<2Tρw(x)g(|x|2T2ρ)dx0<|x|<1w(x)g(|x|2T2ρ)dx. (3.16)

    Notice that since wL1loc(RN), w0, w0, and using the properties of the cut-off function g, by the domianted convergence theorem, one has

    limT0<|x|<1w(x)g(|x|2T2ρ)dx=0<|x|<1w(x)dx>0.

    Hence, for sufficiently large T,

    0<|x|<1w(x)g(|x|2T2ρ)dxC. (3.17)

    Using (3.14), (3.15), (3.16) and (3.17), (3.13) follows.

    Now, we are ready to prove the Fujita-type result given by Theorem 1.1. The proof is by contradiction and makes use of the nonlinear capacity method, which was developed by Mitidieri and Pohozaev (see e.g. [11]).

    Proof of Theorem 1.1. (ⅰ) Suppose that problem (1.1) admits a global weak solution u with (u,u)Lploc(Q)×Lqloc(Q). By Propositions 3.2 and 3.3, for sufficiently large T, one has

    CTmin{A(ξT),B(ξT)}, (3.18)

    where ξT is defined by (2.1),

    A(ξT)=Qξ1p1T|ξTtt|pp1dxdt+Qξ1p1T|ξTt|pp1dxdt+Qξ1p1T|ΔξT|pp1dxdt (3.19)

    and

    B(ξT)=Qφ1p1|ξTtt|pp1dxdt+Qξ1p1T|ξTt|pp1dxdt+Qξ1q1T|ξT|qq1dxdt. (3.20)

    Taking >max{2pp1,qq1}, using Lemmas 2.1, 2.3, 2.4 with κ=p, and Lemma 2.2 with κ=q, one obtains the following estimates

    Qξ1p1T|ξTtt|pp1dxdtCT2pp1+1+Nρ, (3.21)
    Qξ1p1T|ξTt|pp1dxdtCTpp1+1+Nρ, (3.22)
    Qξ1p1T|ΔξT|pp1dxdtCT2ρpp1+Nρ+1, (3.23)
    Qξ1q1T|ξT|qq1dxdtCTρqq1+Nρ+1. (3.24)

    Hence, by (3.19), (3.21), (3.22) and (3.23), one deduces that

    A(ξT)C(T2pp1+1+Nρ+Tpp1+1+Nρ+T2ρpp1+Nρ+1).

    Observe that

    2pp1+1+Nρ<pp1+1+Nρ. (3.25)

    So, for sufficiently large T, one deduces that

    A(ξT)C(Tpp1+1+Nρ+T2ρpp1+Nρ+1).

    Notice that the above estimate holds for all ρ>0. In particular, when ρ=12, it holds that

    A(ξT)CTpp1+1+N2 (3.26)

    Next, by (3.20), (3.21), (3.22), (3.24) and (3.25), one deduces that

    B(ξT)C(Tpp1+1+Nρ+Tρqq1+Nρ+1).

    Similarly, the above inequality holds for all ρ>0. In particular, when ρ=p(q1)q(p1), it holds that

    B(ξT)CTpp1+1+Np(q1)q(p1). (3.27)

    Therefore, it follows from (3.18), (3.26) and (3.27) that

    0<Cmin{Tpp1+N2,Tpp1+Np(q1)q(p1)},

    which yields

    0<CTpp1+N2:=Tα(N) (3.28)

    and

    0<CTpp1+Np(q1)q(p1):=Tβ(N). (3.29)

    Notice that for N\in\{1, 2\} , one has

    \alpha(N) = \left\{\begin{array}{lll} \frac{-p-1}{2(p-1)} &\mbox{if}& N = 1,\\ \frac{-1}{p-1} &\mbox{if}& N = 2, \end{array} \right.

    which shows that

    \alpha(N) \lt 0,\quad N\in \{1,2\}.

    Hence, for N\in\{1, 2\} , passing to the limit as T\to \infty in (3.28), a contradiction follows ( 0 < C\leq 0 ). This proves part (ⅰ) of Theorem 1.1.

    (ⅱ) Let N\geq 3 . In this case, one has

    \alpha(N) \lt 0 \Longleftrightarrow p \lt 1+\frac{2}{N-2}.

    Hence, if p < 1+\frac{2}{N-2} , passing to the limit as T\to \infty in (3.28), a contradiction follows. Furthermore, one has

    \beta(N) \lt 0 \Longleftrightarrow q \lt 1+\frac{1}{N-1}.

    Hence, if q < 1+\frac{1}{N-1} , passing to the limit as T\to \infty in (3.29), a contradiction follows. Therefore, we proved part (ⅱ) of Theorem 1.1.

    (ⅲ) Let

    \begin{equation} p \gt 1+\frac{2}{N-2}\quad\mbox{and}\quad q \gt 1+\frac{1}{N-1}. \end{equation} (3.30)

    Let

    \begin{equation} u(x) = \varepsilon (1+r^2)^{-\delta},\quad r = |x|,\quad x\in \mathbb{R}^N, \end{equation} (3.31)

    where

    \begin{equation} \max\left\{\frac{1}{p-1},\frac{2-q}{2(q-1)}\right\} \lt \delta \lt \frac{N-2}{2} \end{equation} (3.32)

    and

    \begin{equation} 0 \lt \varepsilon \lt \min\left\{1,\left(\frac{2\delta(N-2\delta-2)}{1+2^q\delta^q}\right)^{\frac{1}{\min\{p,q\}-1}}\right\}. \end{equation} (3.33)

    Note that due to (3.30), the set of \delta satisfying (3.32) is nonempty. Let

    w(x) = -\Delta u-|u|^p-|\nabla u|^q,\quad x\in \mathbb{R}^N.

    Elementary calculations yield

    \begin{eqnarray} w(x)& = &2\delta \varepsilon \left[(1+r^2)^{-\delta-1}-2r^2(\delta+1)(1+r^2)^{-\delta-2}+(N-1)(1+r^2)^{-\delta-1}\right]\\ && -\varepsilon^p(1+r^2)^{-\delta p}-2^q\delta^q\varepsilon^qr^q(1+r^2)^{-(\delta+1)q}. \end{eqnarray} (3.34)

    Hence, one obtains

    \begin{eqnarray*} w(x) &\geq & 2\delta \varepsilon \left[(1+r^2)^{-\delta-1}-2(\delta+1)(1+r^2)^{-\delta-1}+(N-1)(1+r^2)^{-\delta-1}\right]\\ &&-\varepsilon^p(1+r^2)^{-\delta p}-2^q\delta^q\varepsilon^q(1+r^2)^{-(\delta+1)q+\frac{q}{2}}\\ & = & 2\delta \varepsilon (N-2\delta-2)(1+r^2)^{-\delta-1}-\varepsilon^p(1+r^2)^{-\delta p}-2^q\varepsilon^q\delta^q(1+r^2)^{-\left(\delta+\frac{1}{2}\right)q}. \end{eqnarray*}

    Next, using (3.32) and (3.33), one deduces that

    w(x)\geq \varepsilon \left[2\delta (N-2\delta-2)-\varepsilon^{p-1}-2^q\varepsilon^{q-1}\delta^q\right](1+r^2)^{-\delta-1} \gt 0.

    Therefore, for any \delta and \varepsilon satisfying respectively (3.32) and (3.33), the function u defined by (3.31) is a stationary solution (then global solution) to (1.1) for some w > 0 . This proves part (ⅲ) of Theorem 1.1.

    Proposition 4.1. Let N\geq 3 , \sigma < N and w\in I_\sigma^+ . Then, for sufficiently large T ,

    \begin{equation} \int_Q w(x)\xi_T\,dx\,dt \geq C T^{\rho(N-\sigma)+1}. \end{equation} (4.1)

    where \xi_T is defined by (2.1).

    Proof. By (3.14) and (3.15), one has

    \begin{equation} \int_Q w(x)\xi_T\,dx\,dt = CT\int_{0 \lt |x| \lt \sqrt{2}T^\rho} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx. \end{equation} (4.2)

    On the other hand, by definition of I_\sigma^+ , and using that g(s) = 1 , 0\leq s\leq 1 , for sufficiently large T , one obtains (since w, g\geq 0 )

    \begin{eqnarray*} \int_{0 \lt |x| \lt \sqrt{2}T^\rho} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx &\geq & \int_{0 \lt |x| \lt T^\rho} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx\\ & = & \int_{0 \lt |x| \lt T^\rho} w(x)\,dx\\ &\geq & \int_{\frac{T^\rho}{2} \lt |x| \lt T^\rho} w(x)\,dx\\ &\geq & C \int_{\frac{T^\rho}{2} \lt |x| \lt T^\rho} |x|^{-\sigma}\,dx \\ & = & C T^{\rho(N-\sigma)}. \end{eqnarray*}

    Hence, using (4.2), (4.1) follows.

    Now, we are ready to prove the new critical behavior for the problem (1.1) stated by Theorem 1.2.

    Proof of Theorem 1.2. (ⅰ) Suppose that problem (1.1) admits a global weak solution u with (u, \nabla u)\in L^p_{loc}(Q)\times L^q_{loc}(Q) . By Propositions 3.2 and 4.1, for sufficiently large T , one obtains

    \begin{equation} CT^{\rho(N-\sigma)+1}\leq \min\{A(\xi_T),B(\xi_T)\}, \end{equation} (4.3)

    where \xi_T , A(\xi_T) and B(\xi_T) are defined respectively by (2.1), (3.19) and (3.20). Next, using (3.26) and (4.3) with \rho = \frac{1}{2} , one deduces that

    \begin{equation} 0 \lt C\leq T^{\frac{-p}{p-1}+\frac{\sigma}{2}}. \end{equation} (4.4)

    Observe that

    \frac{-p}{p-1}+\frac{\sigma}{2} \lt 0 \Longleftrightarrow \sigma \lt \frac{2p}{p-1}.

    Hence, if \sigma < \frac{2p}{p-1} , passing to the limit as T\to \infty in (4.4), a contradiction follows. Furthermore, using (3.27) and (4.3) with \rho = \frac{p(q-1)}{q(p-1)} , one deduces that

    \begin{equation} 0 \lt C\leq T^{\frac{p}{p-1}\left(\frac{\sigma(q-1)}{q}-1\right)}. \end{equation} (4.5)

    Observe that

    \frac{p}{p-1}\left(\frac{\sigma(q-1)}{q}-1\right) \lt 0\Longleftrightarrow \sigma \lt \frac{q}{q-1}.

    Hence, if \sigma < \frac{q}{q-1} , passing to the limit as T\to \infty in (4.5), a contradiction follows. Therefore, part (ⅰ) of Theorem 1.2 is proved.

    (ⅱ) Let

    \begin{equation} \sigma \geq \max\left\{\frac{2p}{p-1},\frac{q}{q-1}\right\}. \end{equation} (4.6)

    Let u be the function defined by (3.31), where

    \begin{equation} \frac{\sigma-2}{2} \lt \delta \lt \frac{N-2}{2} \end{equation} (4.7)

    and \varepsilon satisfies (3.33). Notice that since \sigma < N , the set of \delta satisfying (4.7) is nonempty. Moreover, due to (4.6) and (4.7), \delta satisfies also (3.32). Hence, from the proof of part (iii) of Theorem 1.2, one deduces that

    w(x) = -\Delta u-|u|^p-|\nabla u|^q \gt 0,\quad x\in \mathbb{R}^N.

    On the other hand, using (3.34) and (4.7), for |x| large enough, one obtains

    w(x)\leq C (1+|x|^2)^{-\delta-1} \leq C |x|^{-2\delta-2}\leq C|x|^{-\delta},

    which proves that w\in I_\sigma^- . This proves part (ⅱ) of Theorem 1.2.

    We investigated the large-time behavior of solutions to the nonlinear damped wave equation (1.1). In the case when N\in\{1, 2\} , we proved that for all p > 1 , problem (1.1) admits no global weak solution (in the sense of Definition 1). Notice that from [5], the same result holds for the problem without gradient term, namely problem (1.5). This shows that in the case N\in\{1, 2\} , the nonlinearity |\nabla u|^q has no influence on the critical behavior of problem (1.5). In the case when N\geq 3 , we proved that, if 1 < p < 1+\frac{2}{N-2} or 1 < q < 1+\frac{1}{N-1} , then problem (1.1) admits no global weak solution, while if p > 1+\frac{2}{N-2} and q > 1+\frac{1}{N-1} , global solutions exist for some w > 0 . This shows that in this case, the Fujita critical exponent for the problem (1.1) is given by

    p_c(N,q) = \left\{\begin{array}{lll} 1+\frac{2}{N-2} &\mbox{if}& q \gt 1+\frac{1}{N-1},\\ \infty &\mbox{if}& q \lt 1+\frac{1}{N-1}. \end{array} \right.

    From this result, one observes two facts. First, in the range q > 1+\frac{1}{N-1} , from [5], the critical exponent p_c(N, q) is also equal to the critical exponent for the problem without gradient term, which means that in this range of q , the nonlinearity |\nabla u|^q has no influence on the critical behavior of problem (1.5). Secondly, one observes that the gradient term induces an interesting phenomenon of discontinuity of the Fujita critical exponent p_c(N, q) jumping from 1+\frac{2}{N-2} to \infty as q reaches the value 1+\frac {1}{N-1} from above. In the same case N\geq 3 , we determined also the second critical exponent for the problem (1.1) in the sense of Lee and Ni [10], when p > 1+\frac{2}{N-2} and q > 1+\frac{1}{N-1} . Namely, we proved that in this case, if \sigma < \max\left\{\frac{2p}{p-1}, \frac{q}{q-1}\right\} and w\in I_\sigma^+ , then there is no global weak solution, while if \max\left\{\frac{2p}{p-1}, \frac{q}{q-1}\right\}\leq \sigma < N , global solutions exist for some w\in I_\sigma^- . This shows that the second critical exponent for the problem (1.1) in the sense of Lee and Ni is given by

    \sigma^* = \max\left\{\frac{2p}{p-1},\frac{q}{q-1}\right\}.

    We end this section with the following open questions:

    (Q1). Find the first and second critical exponents for the system of damped wave equations with mixed nonlinearities

    \begin{eqnarray*} \left\{\begin{array}{lllll} u_{tt}-\Delta u +u_t & = & |v|^{p_1}+|\nabla v|^{q_1} +w_1(x) &\mbox{in}& (0,\infty)\times \mathbb{R}^N,\\ v_{tt}-\Delta v +v_t& = & |u|^{p_2}+|\nabla u|^{q_2} +w_2(x) &\mbox{in}& (0,\infty)\times \mathbb{R}^N, \end{array} \right. \end{eqnarray*}

    where p_i, q_i > 1 , w_i\in L^1_{loc}(\mathbb{R}^N) , w_i\geq 0 and w_i\not\equiv 0 , i = 1, 2 .

    (Q2). Find the Fujita critical exponent for the problem (1.1) with w\equiv 0 .

    (Q3) Find the Fujita critical exponent for the problem

    u_{tt}-\Delta u +u_t = \frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}|u(s,x)|^{p}\,ds +|\nabla u|^{q} +w(x),\quad t \gt 0,\,x\in \mathbb{R}^N,

    where N\geq 1 , p, q > 1 , 0 < \alpha < 1 and w\in L^1_{loc}(\mathbb{R}^N) , w\geq 0 . Notice that in the limit case \alpha \to 1^- , the above equation reduces to (1.1).

    (Q4) Same question as above for the problems

    u_{tt}-\Delta u +u_t = |u|^p+\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}|\nabla u(s,x)|^{q}\,ds+w(x)

    and

    u_{tt}-\Delta u +u_t = \frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}|u(s,x)|^{p}\,ds +\frac{1}{\Gamma(1-\beta)}\int_0^t(t-s)^{-\beta}|\nabla u(s,x)|^{q} +w(x),

    where 0 < \alpha, \beta < 1 .

    The author is supported by Researchers Supporting Project RSP-2020/4, King Saud University, Saudi Arabia, Riyadh.

    The author declares no conflict of interest.



    [1] Chen M, Sun X, Liu H, et al. (2020) A FinFET with one atomic layer channel. Nat Commun 11: 1205. https://doi.org/10.1038/s41467-020-15096-0 doi: 10.1038/s41467-020-15096-0
    [2] Maurya RK, Bhowmick B (2021) Review of FinFET Devices and Perspective on Circuit Design Challenges. Silicon 14: 5783-5791. https://doi.org/10.1007/s12633-021-01366-z doi: 10.1007/s12633-021-01366-z
    [3] Park J, Kim J, Showdhury S, et al. (2020) Electrical Characteristics of Bulk FinFET According to Spacer Length. Electronics, 9: 1283. https://doi.org/10.3390/electronics9081283 doi: 10.3390/electronics9081283
    [4] Vashishtha V, Clark LT (2021) Comparing bulk-Si FinFET and gate-all-around FETs for the 5 nm technology node. Microelectron J 107: 104942. https://doi.org/10.1016/j.mejo.2020.104942 doi: 10.1016/j.mejo.2020.104942
    [5] Kim S, Kim J, Jang D, et al. (2020) Comparison of Temperature Dependent Carrier Transport in FinFET and Gate-All-Around Nanowire FET. Applied Sciences 10: 2979. https://doi.org/10.3390/app10082979 doi: 10.3390/app10082979
    [6] Agarwal A, Pradhan PC, Swain BP (2019) Effects of the physical parameter on gate all around FET. Sadhana 44: 248. https://doi.org/10.1007/s12046-019-1232-8 doi: 10.1007/s12046-019-1232-8
    [7] Ma J, Chen X, Sheng Y, et al. (2022) Top gate engineering of field-effect transistors based on wafer-scale two-dimensional semiconductors. J Mater Sci Technol 106: 243-248. https://doi.org/10.1016/j.jmst.2021.08.021 doi: 10.1016/j.jmst.2021.08.021
    [8] Karbalaei M, Dideban D, Heidari H (2021) A sectorial scheme of gate-all-around field effect transistor with improved electrical characteristics. Ain Shams Eng J 12: 755-760. https://doi.org/10.1016/j.asej.2020.04.015 doi: 10.1016/j.asej.2020.04.015
    [9] Lee K, Park J (2021) Inner Spacer Engineering to Improve Mechanical Stability in Channel-Release Process of Nanosheet FETs. Electronics 10: 1395. https://doi.org/10.3390/electronics10121395 doi: 10.3390/electronics10121395
    [10] Saeidi A, Rosca T, Memisevic E, et al. (2020) Nanowire Tunnel FET with Simultaneously Reduced Subthermionic Subthreshold Swing and Off-Current due to Negative Capacitance and Voltage Pinning Effects. Nano Letters 20: 3255-3262. https://doi.org/10.1021/acs.nanolett.9b05356 doi: 10.1021/acs.nanolett.9b05356
    [11] Zhang M, Guo Y, Zhang J, et al. (2020) Simulation Study of the Double-Gate Tunnel Field-Effect Transistor with Step Channel Thickness. Nanoscale Res Lett 15: 128. https://doi.org/10.1186/s11671-020-03360-7 doi: 10.1186/s11671-020-03360-7
    [12] Cao W, Banerjee K (2020) Is negative capacitance FET a steep-slope logic switch? Nat Commun 11: 196. https://doi.org/10.1038/s41467-019-13797-9 doi: 10.1038/s41467-019-13797-9
    [13] Rahi SB, Tayal S, Upadhyay AK (2021) A review on emerging negative capacitance field effect transistor for low power electronics. Microelectron J 116: 105242. https://doi.org/10.1016/j.mejo.2021.105242 doi: 10.1016/j.mejo.2021.105242
    [14] Lukyanchuk I, Razumnaya A, Sene A, et al. (2022) The ferroelectric field-effect transistor with negative capacitance. NPJ Comput Mater 8: 52. https://doi.org/10.1038/s41524-022-00738-2 doi: 10.1038/s41524-022-00738-2
    [15] Lee MH, Wei YT, Huang JJ, et al. (2015) Ferroelectricity of HfZrO2 in Energy Landscape With Surface Potential Gain for Low-Power Steep-Slope Transistors. J Electron Devi Society 3: 377-381. https://doi.org/10.1109/JEDS.2015.2435492 doi: 10.1109/JEDS.2015.2435492
    [16] Alam MA, Si M, Ye PD (2019) A critical review of recent progress on negative capacitance field-effect transistors. Appl Phys Lett 114: 090401. https://doi.org/10.1063/1.5092684 doi: 10.1063/1.5092684
    [17] Li Y, Kang Y, Gong X (2017) Evaluation of Negative Capacitance Ferroelectric MOSFET for Analog Circuit Applications. IEEE T Electron Dev 64: 4317-4321. https://doi.org/10.1109/TED.2017.2734279 doi: 10.1109/TED.2017.2734279
    [18] Lee H, Yoon Y, Shin C (2017) Current-Voltage Model for Negative Capacitance Field-Effect Transistors. IEEE Electr Device L 38: 669-672. https://doi.org/10.1109/LED.2017.2679102 doi: 10.1109/LED.2017.2679102
    [19] Ortiz-Conde A, Garcia-Sanchez FJ, Muci J, et al. (2013) Revisiting MOSFET threshold voltage extraction methods. Microelectron Reliab 53: 90-104. https://doi.org/10.1016/j.microrel.2012.09.015 doi: 10.1016/j.microrel.2012.09.015
    [20] Chebaki E, Djeffal F, Bentrcia T (2012) Two-dimensional numerical analysis of nanoscale junctionless and conventional Double Gate MOSFETs including the effect of interfacial traps. Physica Status Solidi C 9: 2041-2044. https://doi.org/10.1002/pssc.201200128 doi: 10.1002/pssc.201200128
    [21] Farzan J, Sallese JM (2018) Modeling Nanowire and Double-Gate Junctionless Field-Effect Transistors. Cambridge University Press.
    [22] Shalchian M, Jazaeri F, Sallese JM (2018) Charge-Based Model for Ultrathin Junctionless DG FETs, Including Quantum Confinement. IEEE T Electron Dev 65: 4009-4014. https://doi.org/10.1109/TED.2018.2854905 doi: 10.1109/TED.2018.2854905
    [23] Woo J, Choi J, Choi Y (2013) Analytical Threshold Voltage Model of Junctionless Double-Gate MOSFETs With Localized Charges. IEEE T Electron Dev 60: 2951-2955. https://doi.org/10.1109/TED.2013.2273223 doi: 10.1109/TED.2013.2273223
    [24] Dhiman G, Ghosh PK (2017) Threshold Voltage Modeling for Nanometer Scale Junction Less Double Gate MOSFET. International Journal of Applied Engineering Research 12: 1807-1810.
    [25] Jiang C, Liang R, Wang J, Xu J (2015) A two-dimensional analytical model for short channel junctionless double-gate MOSFETs. AIP Adv 5: 057122. https://doi.org/10.1063/1.4821086 doi: 10.1063/1.4821086
    [26] Hoffmann M, Pesic M, Slesazeck S, et al. (2017) Modeling and design considerations for negative capacitance field-effect transistors. 2017 Joint International EUROSOI Workshop and International Conference on Ultimate Integration on Silicon (EuroSOI-ULIS), 1-4. IEEE. https://doi.org/10.1109/ULIS.2017.7962577
    [27] Rassekh A, Sallese J, Jazaeri F, et al. (2020) Negative Capacitance DG Junctionless FETs: A Charge-based Modeling Investigation of Swing, Overdrive and Short Channel Effect. J Electron Devi Society 8: 939-947. https://doi.org/10.1109/JEDS.2020.3020976 doi: 10.1109/JEDS.2020.3020976
    [28] Jazaeri F, Barbut L, Koukab A, et al. (2013) Analytical model for ultra-thin body junctionless symmetric double gate MOSFETs in subthreshold regime. Solid-State Electron 82: 103-110. https://doi.org/10.1016/j.sse.2013.02.001 doi: 10.1016/j.sse.2013.02.001
    [29] Awadhiya B, Kondekar PN, Yadav S, et al. (2021) Insight into Threshold Voltage and Drain Induced Barrier Lowering in Negative Capacitance Field Effect Transistor. Trans Electr Electro Mater 22: 267-273. https://doi.org/10.1007/s42341-020-00230-y doi: 10.1007/s42341-020-00230-y
    [30] Saha AK, Sharma P, Dabo I, et al. (2017) Ferroelectric transistor model based on self-consistent solution of 2D Poisson's, non-equilibrium Green's function and multi-domain Landau Khalatnikov equations. 2017 IEEE International Electron Devices Meeting (IEDM), 13-15. https://doi.org/10.1109/IEDM.2017.8268385
    [31] Lee J (2021) Unified Model of Shot Noise in the Tunneling Current in Sub-10 nm MOSFETs. Nanomaterials 11: 2759. https://doi.org/10.3390/nano11102759 doi: 10.3390/nano11102759
    [32] Ding Z, Hu G, Gu J, et al. (2011) An analytical model for channel potential and subthreshold swing of the symmetric and asymmetric double-gate MOSFETs. Microelectron J 42: 515-519. https://doi.org/10.1016/j.mejo.2010.11.002 doi: 10.1016/j.mejo.2010.11.002
    [33] Rassekh A, Jazaeri F, Sallese J (2022) Nonhysteresis Condition in Negative Capacitance Junctionless FETs. IEEE T Electron Dev 69: 820-826. https://doi.org/10.1109/TED.2021.3133193 doi: 10.1109/TED.2021.3133193
    [34] Khan AI, Radhakrishna U, Chatterjee K, et al. (2016) Negative Capacitance Behavior in a Leaky Ferroelectric. IEEE T Electron Dev 63: 4416-4422. https://doi.org/10.1109/TED.2016.2612656 doi: 10.1109/TED.2016.2612656
    [35] Rassekh, Jazaeri F, Sallese JM (2022) Design Space of Negative Capacitance in FETs. IEEE T Nanotechnol 21: 236-243. https://doi.org/10.1109/TNANO.2022.3174471 doi: 10.1109/TNANO.2022.3174471
    [36] Jung H (2021) Relationship of drain induced barrier lowering and top/bottom gate oxide thickness in asymmetric junctionless double gate MOSFET. International Journal of Electrical and Computer Engineering 11: 232-239. https://doi.org/10.11591/ijece.v11i1.pp232-239 doi: 10.11591/ijece.v11i1.pp232-239
  • This article has been cited by:

    1. Sen Ming, Jiayi Du, Bo Du, Existence of Blow-Up Solution to the Cauchy Problem of Inhomogeneous Damped Wave Equation, 2025, 17, 2073-8994, 1009, 10.3390/sym17071009
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2626) PDF downloads(245) Cited by(1)

Figures and Tables

Figures(8)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog