Overview¶

1. General¶

Heracles is a two-terminal (te, be) compact model for a metal/HfO₂-ZrO₂(HZO)/metal ferroelectric capacitor (FeCap). It combines:

Mixed-phase HZO: a fraction p_ferro of the film area is orthorhombic (ferroelectric, switches), the remainder 1-p_ferro is monoclinic (linear dielectric, non-switching).
A field- and phase-dependent electrode depletion/screening layer (imperfect metal screening → internal depolarization field, source of imprint/wake-up asymmetry).
A physical interfacial low-κ "dead layer" (t_int, eps_int) in series.
Polarization switching kinetics via a two-state Eyring/Arrhenius (transition-state) rate model — not Landau-Khalatnikov.
Two independent leakage/conduction paths: trap-assisted tunneling (TAT) through the FE layer and through the interface layer, plus a direct Fowler-Nordheim (FN) channel through the interface layer.

2. Equivalent Circuit / Node Topology¶

Three series stages between te and be:

Depletion/screening layer (te→n_depletion): voltage-defined branch enforcing charge balance between bound polarization charge and free screening charge.
Ferroelectric core (n_depletion→n_interface): switching capacitance + polarization current, in parallel (at the te–n_interface level) with the non-switching monoclinic dielectric and the FE-layer TAT leakage.
Interface dead layer (n_interface→be): linear capacitance + TAT + FN leakage.

3. State Variable and Switching Kinetics¶

p ≡ V(n1) ∈ [0,1] is the occupation probability of the "down" polarization state (bistable two-well model, not Landau-Khalatnikov/Preisach).

Driving force / barrier shift (from E_fe = V(fecap)/t_fe):

\[w_e = (E_{fe} - E_{off})\, d_e\]

Transition rates (Eyring/transition-state theory, attempt frequency kT/h):

\[k_+ = \frac{k_BT}{h}\exp\!\left(-\frac{(w_b - w_e)q}{k_BT}\right), \qquad k_- = \frac{k_BT}{h}\exp\!\left(-\frac{(w_b + w_e)q}{k_BT}\right)\]

Relaxation ODE (imposed via I(polarization) <+ ddt(p) + k_-\,p - k_+\,(1-p) = 0):

\[\dot p = k_+ (1-p) - k_-\, p, \qquad \tau = \frac{1}{k_++k_-},\quad p_{eq}=\frac{k_+}{k_++k_-}\]

This is a single Debye-type relaxation, not a distribution of nucleation sites (unlike NLS/Preisach-type HZO models, e.g. Fengler-type switching-time-distribution approaches). Device-to-device/domain-to-domain dispersion must be introduced externally (e.g. via the Monte-Carlo parameters or paralleled sub-capacitor instances), not captured intrinsically.

Polarization charge:

\[P = 2(p_s+p_{s,mc})\,p \quad [\mathrm{C/m^2}],\ \ P\in[0,\,2p_s]\]

\[i_p = \frac{d}{dt}\big[2(p_s+p_{s,mc})\,p\big]\]

4. Depletion / Internal Screening Layer¶

Effective screening (depletion) width, separately for the two polarization states (asymmetric fixed interface charge → imprint):

\[t_{depl,u} = \max\!\left(10^{-12},\ \left|\frac{\varepsilon_0\varepsilon_{fe}E_{fe} + Q_{fix,u}}{q\,(n_{depl}+n_{depl,mc})}\right|\right)\]

\[t_{depl,d} = \max\!\left(10^{-12},\ \frac{\varepsilon_0\varepsilon_{fe}E_{fe} + Q_{fix,d}}{q\,(n_{depl}+n_{depl,mc})}\right)\]

\[C_{depl,u}=\frac{\varepsilon_0\varepsilon_{depl}}{t_{depl,u}},\quad C_{depl,d}=\frac{\varepsilon_0\varepsilon_{depl}}{t_{depl,d}},\quad C_{depl,tot} = p\,C_{depl,d} + (1-p)\,C_{depl,u}\]

Charge-balance (voltage-defined) branch, sets the depletion-node voltage directly from polarization and FE-cap charge:

\[V_{depletion} = \frac{2(p_s+p_{s,mc})\,p - (p_s - p_{s,mc}) + C_{fe}\,V_{fecap}}{C_{depl,tot}}\]

5. Capacitances (linear parts)¶

\[C_{fe} = \frac{\varepsilon_0\varepsilon_{fe}}{t_{fe}+t_{fe,mc}}, \qquad C_{de} = \frac{\varepsilon_0\varepsilon_{de}}{t_{fe}+t_{fe,mc}}, \qquad C_{int} = \frac{\varepsilon_0\varepsilon_{int}}{t_{int}+t_{int,mc}}\]

C_fe and C_de share the same physical thickness t_fe (mixed-phase film of one thickness, split electrically by area fraction p_ferro/1-p_ferro), so there is no independent monoclinic-phase thickness parameter.

6. Leakage / Conduction Currents¶

6.1 Trap-Assisted Tunneling (`itat` macro, applied separately to FE and interface layers)¶

Multi-phonon/WKB-type trap conduction model: attenuation length from WKB tunneling through a triangular/parabolic barrier,

\[\lambda_{xtm} = \left[2\sqrt{\phi_b\cdot 2 m_{eff}m_0 q/\hbar^2}\cdot 4\pi^2\,\right]^{-1}\]

followed by a field- and temperature-dependent trap-centroid position x_tm (closed-form solution of a quadratic in bias v), an occupation/barrier-lowering term w_tr_c, and a capture time constant

\[\tau_c = \mathrm{clamp}\!\left(10^{-15},\,10^{6},\ \frac{1}{D_{trap}\,n_{tr}\,\exp\!\left(-x_{tm}/\lambda_{xtm} - w_{tr,c}\,k_BT/q\right)}\right)\]

\[i_{tat} = \tanh_{lim}(v,0.1)\cdot \frac{q\,n_{tr}\,x_c}{2\,\tau_c}\]

6.2 Fowler-Nordheim tunneling (interface layer only)¶

Standard FN equation (Sze & Ng):

\[A_{FN} = \frac{q^2}{8\pi\hbar\,\phi_{b,int}}, \qquad B_{FN} = \frac{8\pi}{3}\frac{\sqrt{2\,m_{eff,int}m_0\,(q\,\phi_{b,int})^3}}{\hbar^2 q^2}\]

\[i_{leak,int} = \tanh_{lim}(v_{int},0.1)\cdot A_{FN}\,E_{int}^2\,\exp\!\left(-\frac{B_{FN}}{|E_{int}|}\right)\]

7. Branch Current Assembly (all scaled by instance area)¶

\[I_{interface} = \big[C_{int}\dot V_{int} + i_{tat,int} + i_{leak,int}\big]\cdot(\mathrm{area}+\mathrm{area}_{mc})\]

\[I_{fecap} = \big[C_{fe}\,p_{ferro}\,\dot V_{fecap} + \dot P\big]\cdot(\mathrm{area}+\mathrm{area}_{mc})\]

\[I_{dielectric} = C_{de}(1-p_{ferro})\,\dot V_{dielectric}\cdot(\mathrm{area}+\mathrm{area}_{mc})\]

\[I_{leakage} = i_{tat,fe}\cdot(\mathrm{area}+\mathrm{area}_{mc})\]

\[V_{depletion} = \text{(charge-balance expression, §4)}\]