In this section, the proposed reliability prediction model is explained. The leakage current of a planar DRAM storage capacitor is simulated using the possible leakage current mechanisms of metal/insulator/metal (MIM) structures (Figure 1a): (i) thermionic emission, (ii) Fowler–Nordheim tunneling, (iii) Poole–Frenkel emission (PFE), (iv) trap-assisted tunneling (TAT), (v) trap-to-trap tunneling, and (vi) direct tunneling. Figure 1b shows the structure of the simulated storage capacitor, which has a TiN/ZrO_{2}/TiN stack.

The leakage currents of ZrO

_{2}-based storage capacitors are affected mainly by PFE and TAT owing to the high defect density of ZrO

_{2} [

13,

14]. In addition to the PFE and TAT, the proposed model solved the electron and hole continuity equations coupled with the Poisson equation:

where

$\epsilon $ is the electrical permittivity,

q is the elementary electronic charge,

N_{D} is the concentration of ionized donors,

N_{A} is the concentration of ionized acceptors,

${\rho}_{\mathrm{trap}}$ is the charge density contributed by traps,

R_{net,n} and

R_{net,p} are the electron and hole net recombination rates,

G_{net,n} and

G_{net,p} are the electron and hole net generation rates,

${\overrightarrow{J}}_{n}$ is the electron current density,

${\overrightarrow{J}}_{p}$ is the hole current density, and

n and

p are the electron and hole densities, respectively. The trap-assisted charge transport was calculated using the Shockley–Read–Hall (SRH) recombination rate:

where

N_{TRAP} is the trap density,

E_{TRAP} is the energy of the trap,

c_{n} and

c_{p} are the electron and hole capture rates, and

g_{n} and

g_{p} are the electron and hole degeneracy factors, respectively. All of the used tunneling models, such as the elastic/inelastic TAT and trap-to-trap tunneling, are nonlocal models. Only the PFE model was used as a local model and considered to increase the emission rate of electrons injected through tunneling. The electron capture rate for the phonon-assisted (inelastic) transition from the conduction band is [

15]

where

V_{TRAP} is the interaction volume of the trap,

S is the Huang–Rhys factor,

$\hslash \mathsf{\omega}$ is the energy of the phonon involved in the transition,

$\alpha $ is a dimensionless parameter,

l is the number of phonons emitted in the transition,

f_{B} is the Bose–Einstein occupation of the phonon state,

$z=2S\sqrt{{f}_{\mathrm{B}}\left({f}_{\mathrm{B}}+1\right)}$,

$\mathsf{\chi}=\sqrt{{l}^{2}+{z}^{2}}$,

$\u2206E$ is the dissipated energy,

E_{F,n} is the Fermi energy,

T_{n} is the electron temperature,

m_{t} is the relative tunneling mass, and

g_{c} is the prefactor for the Richardson constant at the interface or contact. The electron capture rate for the elastic transition from the conduction band is [

16]

where

$f\left(x\right)=1/\left(1+\mathrm{exp}\left(-x\right)\right).$ The electron capture rate for the trap-to-trap tunneling is [

17,

18]

where transitions occur between a localized state

i with an energy of

${E}_{\mathrm{TRAP}}^{i}$ and neighboring localized states

j with energies of

${E}_{\mathrm{TRAP}}^{j}$,

${W}_{\mathrm{OPT}}$ is the trap optical ionization energy,

${W}_{\mathrm{T}}$ is the trap thermal ionization energy,

${Q}_{0}=\sqrt{2\left({W}_{\mathrm{OPT}}-{W}_{\mathrm{T}}\right)}$,

r_{i,j} is the spatial distance between traps

i and

j involved in the transition,

f_{j} is the localized trap

j occupation probability, and

C_{f} is a multiplication factor. The electron capture rate for the PFE model is [

17]

where

${v}_{\mathrm{th}}^{n}$ is the electron thermal velocity,

${\sigma}_{0}^{n}$ is the electron capture cross section, and

${\epsilon}_{\mathrm{PFE}}$ is an adjustable parameter. The emission rates were computed following the principle of detailed balance.

Figure 1c shows that our simulation results matched well with experimental data under various temperature conditions. The main parameters were:

T_{INS} = 8 nm, conduction band offset (CBO) = 1.90 eV,

E_{TRAP} = 1.1 eV, and

N_{TRAP} = 1 × 10

^{19} cm

^{−3}. It is worth noting that the experimental data measured at a low electric field were ignored in our simulation as they were attributed to deep traps [

19]. Only the shallow trap level, which provided the dominant leakage path formed by oxygen vacancies, was considered [

20].