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SCIENCE CHINA Information Sciences, Volume 64 , Issue 2 : 122402(2021) https://doi.org/10.1007/s11432-020-3005-8

Design space for stabilized negative capacitance in HfO$_2$ ferroelectric-dielectric stacks based on phase field simulation

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  • ReceivedMay 8, 2020
  • AcceptedJul 24, 2020
  • PublishedJan 20, 2021

Abstract


Acknowledgment

This work was supported in part by National Natural Science Foundation of China (Grant Nos. 61804003, 61674008) and in part by China Postdoctoral Science Foundation (Grant Nos. 2019T120017, 2018M630034).


References

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  • Figure 1

    (Color online) (a) MFM and (b) MFIM structures, where top electrode is biased by $V_{\rm~App}\sin(2\pi~f\cdot~t)$, and bottom electrode is grounded. (c) Measured and simulated $P$-$E$ characteristics of MFM capacitor with 10 nm HZO film, showing the model calibration. Notice that measured slow switching of FE can be obtained by considering distribution of FE parameters.

  • Figure 2

    (Color online) $P$-$E$ characteristics in the MFIM structure with (a) $T_F$ = 50 nm, (b) $T_F$ = 40 nm, (c) $T_F$ = 30 nm,protect łinebreak (d) $T_F$ = 15 nm, (e) $T_F$ = 5 nm, and (f) $T_F$ = 3 nm. Black square is the $P_F$-$E_F$, red circle is the $P_D$-$E_D$, blue triangle is the overall $P$-$E$ of FE-DE bilayer, green dash is the $P_F$-$E_F$ based on the LGD theory. All the simulation parameters are from Table 1 except for $T_F$.

  • Figure 3

    (Color online) (a) Critical ferroelectric thickness $T_{{\rm~cri},{\rm~A}}$ and $T_{{\rm~cri},{\rm~C}}$ for phase transition of MFIM with various gradient coefficient at equilibrium state ($V_{\rm~App}=0$). (b) Domain patterns at $V_{\rm~App}=0$ at the specified $T_F$ in (a) within region A, B and C, respectively.

  • Figure 4

    (Color online) (a) Domain width versus ferroelectric thickness with different gradient coefficient at equilibrium state ($V_{\rm~App}=0$) for the region B in Figure 3; (b) domain patterns of MFIM for different $T_F$ and different $k$ at $V_{\rm~App}=0$.

  • Figure 5

    (Color online) (a) Voltages ($V_{\rm~DE}$and $V_{\rm~App}$, (b) electric fields, and (c) polarization charges as a function of time in MFIM with $T_F=30$ nm operated at frequency of 1 MHz as shown in Figure 2(c). (d) Polarization domain patterns from the fully negatively polarized state (downward $P_z$, blue) to fully positively polarized state (upward $P_z$, red) corresponding to the stage I, II, III, and IV, respectively.

  • Figure 6

    (Color online) (a) $V_{\rm~DE}$-$V_{\rm~App}$ and (b) $A_{v}$-$V_{\rm~App}$ characteristics with different $T_F$.

  • Figure 7

    (Color online) (a) Voltage window and (b) voltage amplification factor for hysteresis-free operation versus $T_F$ with different $k$.

  • Figure 8

    (Color online) $P$-$E$ characteristics in the MFIM structure with different $T_D$ and $\varepsilon_{D}$, accounting for SiO$_2$, Al$_2$O$_3$, and HfO$_2$, respectively. (a)–(c) and (d)–(f) are the $T_D=1$ nm and 5 nm cases. All the simulation parameters are taken from Table 1 except for $T_D$ and $\varepsilon_{D}$.

  • Figure 9

    (Color online) (a), (b) $P$-$E$ and (c), (d) $V_{\rm~DE}$-$V_{\rm~App}$ characteristics in the MFIM structure with different values of $\varepsilon_{D}/T_D$ respectively. (a), (c) are the $\varepsilon_{D}/T_D$$\propto{3.9}$cases, and (b), (d) are the $\varepsilon_{D}/T_D$$\propto{10}$cases.

  • Figure 10

    (Color online) (a)–(c) $P$-$E$ characteristics in the MFIM structure with different conductivity ($\Gamma=10$, 0.1 and 0.01 S/m)protect łinebreak operated at $f$ = 1M Hz. (d)–(f) $P$-$E$ characteristics with $\Gamma=10$ S/m operated at $f$ = 1k, 10M and 100M Hz. All the simulation parameters are taken from Table 1 except for $\Gamma$ and $f$.

  • Figure 11

    (Color online) (a) Voltage window and (b) voltage amplification for hysteresis-free operation versus frequency with different $\Gamma$.

  • Table 1  

    Table 1Default parameters used in the simulation

    Symbol Quantity Value
    $\alpha$ Landau constant of ferroelectric $-2.9\times10^{8}$ V$\cdot~$m/C
    $\beta$ Landau constant of ferroelectric $-4.0\times10^{8}$ V$\cdot$m$^{5}$/C$^{3}$
    $\gamma$ Landau constant of ferroelectric $6.0\times10^{10}$ V$\cdot~$m$^{9}$/C$^{5}$
    ıtshapek Gradient coefficient of ferroelectric $1\times10^{-10}$ V$\cdot~$m$^{3}$/C
    ıtshape$\Gamma$ Conductivity of polarization switching 10 S/m
    $\varepsilon_{\rm~Fb}$ Dielectric constant of ferroelectric 25
    $\varepsilon_{D}$ Dielectric constant of dielectric 25
    ıtshapeT$_F$ Thickness of ferroelectric 10 nm
    ıtshapeT$_D$ Thickness of dielectric 5 nm
    ıtshapeL Lateral size of simulation region 50 nm
    ıtshapeV$_{\rm~App}$ Amplitude of external applied voltage 10 V
    ıtshapef Sweep frequency of applied voltage $10^{6}$ Hz