Two Stage Model

6.3 Two Stage Model

The concept of NMP has been used in a slightly modified variant termed multiphonon field-assisted tunneling (MPFAT) [116, 115, 116, 117, 125, 55], which was proposed for ionization of deep impurity centers. The underlying theory accounts for the fact that the emission of charge carriers out of bulk traps is accelerated in the presence of an electric field. This effect is eventually related to the shortened tunneling distance through a triangular barrier when considering thermal excitation of the charge carriers. According to theoretical calculations of Ganichev et al. [117], it yields a field enhancement factor $exp(F 2∕F2c)$ , which is suspected to have a strong impact on hole capture processes in NBTI and has therefore been phenomenologically introduced in the two-stage model (TSM) [61].

6.3.1 Physical Description of the Model

The TSM relies on the Harry-Diamond-Laboratories (HDL) [15] model but is extended by a second stage accounting for the permanent component of NBTI (cf. Fig. 6.3). The defect precursor, an oxygen vacancy according to the HDL model, is capable of capturing substrate holes via the aforementioned MPFAT mechanism. The trap level $Et,1$ of the precursor is located below the substrate valence band and subject to a wide distribution due to the amorphousness of $SiO2$ . Upon hole capture, the defect undergoes a transformation to an $E′$ center, which is visible in ESR measurements [43]. In this new configuration, it features a $Si$ dangling bond associated with a defect level $Et,2$ within or close to within the substrate bandgap in accordance to [162]. The level shift from $Et,1$ to $Et,2$ arises from the change to a new ‘stable’ defect configuration, namely the $Si$ dangling bond. In the $E ′$ center configuration, the defect can be repeatedly charged and discharged by electrons tunneling in or out of its dangling bond. The associated switching behavior¹ is in agreement with the experimental observations made in electrical measurements [15, 16]. Only in the neutral state $3$ , in which the $Si$ dangling bond is doubly occupied by an electron, the $E ′$ center can be annealed, thereby becoming an oxygen vacancy again.

Figure 6.3: The transition state diagram for the trapping dynamics of the TSM. The recoverable component of NBTI constitutes the first stage of the TSM. The precursor (state 1) is transformed to a switching trap (state 2) via an irreversible MPFAT process. In this configuration, the defect can quickly respond to small variations of the gate bias by switching between the states $2$ and $3$ . From the neutral charge state $3$ , the defect can undergo structural relaxation over a thermal barrier and arrives at its initial configuration. The second stage gives an explanation for the permanent component, which is attributed to a hydrogen transition from state $2$ to $4$ . This transition fixes the positive charge (red plus sign) in the defect and creates a new interface state (ellipse with one or two blue arrows).

The second stage involves an amphoteric trap, most probably a $Pb$ center, which has been found to interact with the switching trap as observed in irradiation experiments [43]. That is, a hydrogen is detached from an interfacial $Si$ - $H$ bond and leaves behind a $Pb$ center. In a subsequent reaction, it saturates the dangling bond of the $E ′$ center. This stage fixes the positive charge at the oxide defect and creates a new interface state, whose charge state is controlled by the substrate Fermi level. Since the hydrogen transition is assumed to last much longer than the hole capture or emission process, this stage corresponds to the permanent or slowly recoverable component of NBTI.

Mathematically, the dynamics of this complex mechanism are described by the set of the following rate equations:

∂tf1 = - f1r12 + f3r31, (6.5) ∂tf2 = +f1r12 - f2r23 +f3r32 - f2r24 + f4r42, (6.6) ∂tf3 = +f2r23 - f3r32 - f3r31, (6.7) ∂tf4 = +f2r24 - f4r42, (6.8)

The subscript $i$ of $fi$ stands for the state according to the numbering in Fig. 6.3. The transition rates are denoted as $rij$ , with $i$ and $j$ as the initial and the final states, respectively. The rate $r12$ is derived from the SRH equations (2.69), in which the empirical enhancement factor $exp (Fo2x∕Fc2)$ for the MPFAT transition² $T1→2$ has been phenomenologically introduced. Then the transition rates read

( 2 ) r12 = rcap,h(ΔEb,1,Et,1) exp FFo2x + rem,e(ΔEb,1,Et,1) (6.9) c

with

{ 1 p ( xt) 1, Et > Ev rcap,h(ΔEb, Et) =-TSM-N-- exp - x--- exp(- βΔEb ) , (6.10) τp,0 v p,0 { exp (- β(Ev - Et)), Et < Ev 1 n ( xt ) exp (- β(Ef - Ec)), Et > Ec rem,e(ΔEb, Et) = τTSM-Nc-exp - xn,0- exp(- βΔEb ) exp (- β(E - E )), E < E . (6.11) n,0 f t t c

The quantity $TSM τp,0$ is the equivalent of $SRH τp,0$ in the TSM and can be calculated according to equation (2.76). The barriers $ΔEb,1$ and $ΔEb,2$ are defined analogously to the barrier $ΔEb$ in Fig. 2.5. Therefore, they corresponds to the barrier component, which must be overcome in both directions of the transitions $T1↔2$ and $T2↔3$ , respectively (cf. Fig 2.5). For the transitions between the states $2$ and $3$

r23 = rcap,e(ΔEb,2,Et,2)+ rem,h(ΔEb,2,Et,2) , (6.12) r32 = rem,e(ΔEb,2,Et,2) +rcap,h(ΔEb,2,Et,2) , (6.13)

the capture and emission of electrons as well as holes are taken into account.

( ) { rem,h(ΔEb,Et ) =-T1SM--p-exp - xt-- exp(- βΔEb ) exp(- β(Et - Ef)), Et > Ev (6.14) τp,0 Nv xp,0 exp(- β(Ev - Ef)), Et < Ev 1 n ( x ) {exp (- β(E - E )), E > E rcap,e(ΔEb, Et) =-TSM----exp - --t- exp(- βΔEb ) t c t c (6.15) τn,0 Nc xn,0 1, Et < Ec

The annealing of the defect ( $T3→1$ ) is represented by the rate $r31$ , which is modeled by a structural relaxation over a thermal barrier $ΔEb,3$ .

r31 = ν0 exp(- β ΔEb,3) (6.16)

$ν0$ denotes the attempt frequency, which is usually in the order of $1013s-1$ . The hydrogen transition $T2↔4$ is modeled assuming a field-dependent thermal barrier, as shown in Fig. 6.4.

r24 = ν0 exp (- β(ΔEb,4 - γFox)) (6.17) r42 = ν0 exp (- β(ΔEb,4 - E4 + γFox)) (6.18)

The occupancy of interface traps, present in state 4, is calculated using conventional SRH statistics as implemented in standard device simulators [183].

Figure 6.4: The schematic of the hydrogen transition. The solid line depicts the adiabatic potentials for a hydrogen reaction in a configuration coordinate diagram. When a bias is applied to the gate, the oxide field lifts the energy minimum of state 2 ( $E2$ ) but lowers that of state 4 ( $E4$ ). Since the interfacial $Si$ - $H$ bonds are associated with a dipole moment, the shift of the energy minima depends linearly on the oxide field with a proportionality constant of $γ$ . The applied oxide field gives rise to a reduced forward barrier ( $T2→4$ ) and an increased reverse barrier ( $T4→2$ ). Without loss of generality, the value of $E2$ is set to zero.

6.3.2 Model Evaluation

The following simulations are based on the same numerical scheme as has been presented in Section 3.2 and are used for the ETM and the LSM in Chapter 4 and 5. Each representative trap in this scheme is characterized by its individual set of defect levels and barriers. The generated random numbers are homogeneously distributed for $Et,1$ , $Et,2$ , $ΔEb,1$ , and $ΔEb,2$ while they follow a Fermi-derivative (Gaussian-like) distribution [90] for $ΔEb,4$ and $E4$ . The remaining quantities including $TSM TSM -14 2 σp,0 = σn,0 ~ 3 × 10 cm$ , $Fc ~ 2MV ∕cm$ , $7 vth,p = vth,n ~ 10 cm ∕s$ , $xp,0 = 0.5˚A$ , and $xn,0 = 0.5˚A$ are assumed to be single-valued.

In contrast to previous models, oxide charges (state 2) as well as interface traps (state 4) are incorporated into the TSM so that two states must be considered for the calculation of $ΔVth$ . It is important to note that only a part of the overall degradation during stress is observed within the experimental time window. As demonstrated in Fig. 6.5, a large fraction already occurs before the beginning of the OTF measurement ( $t0 = 1ms$ ) and only a part of the $Vth$ degradation can be monitored by this technique (cf. Section 1.3.2). As a consequence, the measured threshold voltage shift must be calculated as

ΔVth(ts) = Vth(ts)- Vth(t0) . (6.19)

Furthermore, the recovery during relaxation is monitored after $t0 = 1ms$ so that only the tails of the real recovery curve can be assessed experimentally.

Figure 6.5: A comparison of the experimental and simulated degradation during stress (left panel) and relaxation (right panel). The data are extracted from eMSM measurements [25, 26], which employs OTF during stress and determines the $Vth$ shift from the recorded drain current during the relaxation phase. Note that OTF only assesses the change of threshold voltage referred to the first measurement point, but disregards the degradation accumulated before. Analogously, the NBTI recovery may start already before $1ms$ but the MSM technique can only monitor the tails of the degradation curves.

The TSM [61] has been compared to a large set of measurement data, including various combinations of stress voltages and temperatures. For illustration, a fit to the eMSM (cf. Section 1.4) data at $150∘C$ is depicted in Fig. 6.6. The findings of this model are evaluated in the following:

Hole trapping during stress sets in before the first OTF measurement point ( $1ms$ ) is determined and no sign of saturation appears until $2s$ . Detrapping during relaxation extends over the whole experimental time window ranging from $1ms$ to $1000s$ .
$ΔVth(t)$ follows a logarithmic time behavior during the whole stress phase.
During the initial phase of recovery, a logarithmic time dependence is clearly recognizable. However, this behavior is obscured for relaxation times larger than $1s$ where the $ΔVth(t)$ curve starts to level off. This is due to the fact that the TSM also accounts for the permanent component of NBTI.
The logarithmic slopes of $ΔVth$ during the stress and the relaxation phase exhibit a ratio of $~ 2.5$ in agreement with [61]. This asymmetry is demonstrated by the temporal change of the trap occupancy in Fig. 6.6 (right). After the level shift from $Et,1$ to $Et,2$ , when the defect transforms from an oxygen vacancy to an $E′$ center, the trap level $Et,2$ lies closer to the substrate valence bandedge than $Et,1$ . This results in higher trapping rates for hole capture ( $r32$ ) and emission ( $r23$ ) so that the subsystem of states $2$ and $3$ is only weakly affected by $r31$ during recovery. Therefore, this reduced system is close to equilibrium, meaning that
$f2r23 = f3r32 (6.20)$
with $f2 +f3 = 1$ . Then the occupation $f3$ is given by
$f3 = --r---(Δ1E---,E--)-= 1-+-exp(β1(E----E-)) (6.21) 1+rcaepm,,hh(ΔEbb,,22,Ett,,22)- t,2 f$
and can be interpreted as the electron occupancy $f t,2$ of the $E′$ center with the defect level $E t,2$ . According to the above equation, the defect occupancy in the whole defect system of the TSM follows the substrate Fermi level, which thus controls the annealing rate $r 31$ and can slow down the recovery. Due to this ‘occupancy effect’, the TSM is able to capture the asymmetry between stress and relaxation.
The simulations fit the experimental stress and relaxation curves for a wide range of gate biases meaning that the TSM correctly reproduces the field dependence seen in measurements.
An equal quality of fits (not shown here) has been achieved for other temperatures with the same set of parameters, meaning that the TSM is able to describe the temperature dependence observed experimentally.

Figure 6.6: Left: An evaluation of simulation data (lines) against measurement data of a thin $SiON$ device (symbols) for 8 different stress voltages at a temperature of $150∘C$ . The field acceleration as well as the asymmetry between stress and relaxation have been nicely reproduced. Right: Averaged trap occupancy during stress and recovery for a single trap. The dotted lines refer to oxide defects in state $2$ or $3$ , while the solid lines include the positively charged defects (state $3$ ) only. The ratio between the slope of the stress ( $A s$ ) and the relaxation ( $A r$ ) curve yields $A ∕A ~ 2.5 s r$ , as observed experimentally in [61].

The TSM is found to satisfy all criteria of Table 6.2 and therefore seems to properly describe NBTI degradation. Besides that, it also agrees well with the observation of a field-dependent recovery, which is demonstrated by the measurements shown in Fig. 6.7. Due to the occupancy effect, the substrate Fermi level $Ef$ controls the portion of neutral $E‘$ centers (state 3) which can return to state $1$ by structural relaxation and contribute to the NBTI recovery. Interestingly, this field dependence is compatible with the finding that the emission times of ‘anomalous defects’ are field-sensitive.

Table 6.2: Checklist for an NBTI model. The TSM fulfills all criteria established in Section 1.4. From this perspective the model can be justifiably regarded as a reasonable explanation for NBTI.

Figure 6.7: The field dependence of recovery [61]. Five devices were stressed for $6000s$ under the same conditions ( $T = 125∘C$ , $Vs = - 2.0V$ ). The following recovery phase (left and right panel) was interrupted for a period of time during which $VG$ was switched from the recovery voltage of $- 0.13V$ to $Vread$ for $2s$ (middle panel). The experimental data are marked by symbols, while the simulations are represented by lines (dotted: $ΔNox + ΔNit$ , solid: $ΔVth$ due to $ΔQox + ΔQit$ , dashed: $ΔVth$ due to $ΔQit$ ). The measurements demonstrate that the recovery is clearly affected by variations of the recovery bias. This effect is reminiscent of the field-dependent emission times seen in TDDS. The agreement of the simulations with the measurements shows that the TSM can explain the field-dependent NBTI recovery.

The distributions of trap levels obtained from the model calibration are depicted in Fig. 6.8. The trap levels $Et,1$ of the precursors (state $1$ ) are uniformly distributed between $- 1.68eV$ and $- 0.25eV$ in qualitative agreement with the values in [39]. The defects located the highest have also the highest substrate hole capture rates $r12$ and therefore have already been transformed $E ′$ centers (states $2$ and $3$ ) after a stress time of $1000s$ . In this new configuration, they feature a trap level $Et,2$ in the range between $- 0.62eV$ and $0.39eV$ in qualitative agreement with the values published in [39]. According to equation (6.21), the occupancy of the $Et,2$ levels is determined by the substrate Fermi energy and thus the number of neutralized defects in the $E ′$ center configuration (state $3$ ) increases with a lower energies. Since only defects in this state transformed to a precursor (state $1$ ) again, the number of traps in state $2$ diminished towards the substrate Fermi energy (cf. Fig. 6.8). The donor levels of the interface states have been assumed to be uniformly distributed and are located within the lower part of the substrate bandgap consistent with [64].

Figure 6.8: A histogram of energetically distributed defects after a stress time of $1000s$ . The numbers in the white filled boxes denote the state of the defects. The area in the plain red color gives the number of precursors (state $1$ ) with the trap level $Et,1$ . The rest of the defects is in the $E′$ center configuration, in which they can be positively charged (purple striped pattern) or neutralized (plain purple color) and have a trap level of $Et,2$ . The occupied (plain) and unoccupied (striped) interface states are depicted as beige areas. The beige striped area gives the density of interface states originating from the $Pb$ centers.

6.3.3 Quantum Mechanical Simulations

So far, classical calculations of the band diagram have been performed to obtain the interface quantities, such as the position of the bandedges ( $EC$ , $EV$ ), the Fermi level ( $EF$ ), and the electric field ( $F$ ) within the dielectric. These quantities enter the expressions of the rates and will significantly alter them due to their exponential dependences. However, the SRH rates (6.9) used in Section 6.3.1 are valid for a three-dimensional electron gas [55] but this assumption breaks down for an inversion layer of MOS structures. In the one-dimensional triangular potential well in the channel, quasi-bound states build up and form subbands, which correspond to the new initial or final energy levels for the charge carriers undergoing an NMP transitions. The quantum mechanical transition rates are obtained following the derivation in Section 2.5.2 but using the DOS for one-dimensionally confined holes. Then the rate equation (2.59) modifies to

E∫v( ep(E) ) ∂tft = (1- ft)cp(E)fn(E)- ftfp(E ) cp(E )Dp,c3D(E)fp(E )dE . (6.22) -∞

It is noted here that the exact shape of $Dp,c3D(E)$ does not enter this derivation and consequently the DOS can be expressed as

D (E ) = ˜D (E) Θ(E - E ) . (6.23) p,c3D p,c3D p,0

Using the above expression, the rate equation (6.22) simplifies to

( ) Ep∫,0 ∂tft = (1- ft)epcp((EE))fn(E)- ftfp(E ) cp(E)D˜p,c3D (E )fp(E )dE (6.24) -∞

and the capture time $τcap,h$ can be identified as

Ep,0 ∫ cp(E )˜Dp,c3D(E)fp(E)dE --1-- = -∞---------------------- p . (6.25) τcap,h Ep∫,0 D˜p,c3D(E)fp(E )dE ◟---∞-------◝◜----------◞ ≡ σTpSMvth,p

Analogously to the derivation in Section 2.5.2, the barrier dependence for the NMP transition is incorporated in the cross section $σTpSM$ (cf. Fig. 6.9).

{ σTSM = σTSM exp (- x ∕x ) exp (- βΔEb ), Et > Ep,0 , (6.26) p p,0 t p,0 exp (- βΔEb ) exp (- β(Ep,0 - Et)), Ep,0 < Ev

where $Ep,0$ corresponds to the first bound state. Inserting the modified cross section in equation (2.65) and (2.66) yields

{ --1---p- 1, Ep,0 > Ev rcap,h(ΔEb, Et) = τTpS,0M Nv exp (- xt∕xp,0) exp(- βΔEb ) exp(- β(Ep,0 - Et)), Ep,0 < Ev (6.27)

and

1 p {exp (- β(E - E )), E > E rem,h(ΔEb,Et ) =-TSM----exp (- xt∕xp,0) exp (- βΔEb ) t f p,0 v . (6.28) τp,0 Nv exp (- β(Ep,0 - Ef)), Ep,0 < Ev

Figure 6.9: The band diagram of a pMOSFET. According to classical considerations the substrate holes initially lie at the valence band, however, they are concentrated around the first bound state $E1$ when quantum confinement is taken into account. The shift of the initial energy level from $Ev$ to $E1$ reduces the corresponding NMP barriers (red lines) and thus enhances the hole capture rates. Note that the energy difference between $Ev$ and $E1$ varies with the oxide field and thus affects the field dependence of $τcap$ and $τem$ .

These rates have been used to incorporate the aforementioned quantum effects into the TSM, which has been evaluated against the same set of experimental data. For a proper comparison with the classical variant of TSM, only the NMP parameters ( $Et,1$ , $Et,2$ , $ΔEb,1$ , and $ΔEb,2$ ) have been optimized while all other parameters have been held fixed. The simulated degradation curves show a good agreement with experimental data (see Fig. 6.10) so that the quantum mechanically refined variant of the TSM still fulfills all criteria listed in Table 6.2. It is noted here that these simulations yield an the uppermost trap levels $Et,1$ , which have been shifted downwards by about the same energy as the separation of $E1$ and $Ev$ (ranging between $169$ and $277meV$ ). This can be explained when considering that, first, $Ev$ is replaced $Ep,0$ in the rate equations (6.27) and (6.28) and, second, the NMP barrier for hole capture is reduced by this energy difference. From this it follows that also the trap levels $Et,1$ must be shifted down by approximately the same energy in order to obtain hole capture rates of an equal magnitude. In summary, it has been assured that also the quantum mechanically refined variant of the TSM can explain the NBTI data and must therefore be considered as a reasonable NBTI model.

Figure 6.10: The temporal evolution of the trapped charges including quantization effects. A pMOSFET with a gate thickness of $1.75nm$ is subjected to two different temperatures ( $50∘C$ left panels, $150∘C$ tight panels) and three different gate voltages for $1s$ during the first phase termed trapping/stress (left hand side). After the gate voltage is removed, the second phase called detrapping/relaxation phase (right-hand side) sets in. Symbols mark measurement data while solid lines belong to simulation data. Note that the temperature and field dependence is well reproduced simultaneously for relaxation phase. The slight tendency of the simulations to underestimate the measurements at high stress temperatures during the stress phase appear in the classical as well as in the quantum mechanical simulations. They may be traced back to the mobility degradation of the drain current during the MSM measurements [26].

6.3.4 Capture and Emission Time Constants

The criteria in Table 4.1 have been successfully satisfied by the TSM. With this respect, the TSM should be regarded as model qualified to describe NBTI. However, these criteria only evaluate the degradation produced by an ensemble of defects but do not consider whether the behavior of a single defect is correctly reproduced. For this reason, the TSM will be investigated using the time constant plots in the following. Since the TDDS measurements cannot capture the permanent component of NBTI, the transition state diagram must be reduced to stage one. This means that the equation (6.8) and the rates $r24$ and $r42$ in equation (6.6) must be omitted. Since the trap level $Et,2$ is assumed to lie closer to the valence band edge than $Et,1$ , the accociated rate $r23$ and $r32$ are much larger than $r12$ . Thus the fast switching between state $2$ and $3$ produces noise, which is undesired for the analysis of $τem,h$ in the time constant plots. Therefore, a compact rate expression for the transition $2 ↔ 3 → 1$ is sought. One can calculate the corresponding emission time $τ21$ as the mean first passage time in continuous time Markov chain theory [131] (discussed in Section 3.2).

-1- ---r23r31---- τ21 = r21 = r23 + r32 + r31 (6.29)

Assuming $r32 ≫ r31$ and $r23 ≫ r31$ , the above expression can be simplified to

r21 = r31-1r-- . (6.30) 1+ r3223

Since in a pMOSFET the trapping dynamics are dominated by the hole capture and emission from the valence band, the electron rates $rcap,e(ΔEb,1,Et,2)$ , $rcap,e(ΔEb,2,Et,2)$ , and $r23 = rem,e(ΔEb,2,Et,2)$ can be neglected.

( ) ( ) { --1---p- -xt- F-2ox 1, Et,1 > Ev 1∕τcap,h = τTpS,0M Nv exp - xp,0 exp (- βΔEb,1) exp F2c exp (- β(Ev - Et,1)), Et,1 < Ev (6.31) 1∕τem,h = ν0 exp (- βΔEb,3)ft,2 (6.32) ft,2 = ---------1--------- (6.33) 1 + exp(β(Et,2 - Ef))

This reduced variant of the TSM is fitted against the TDDS data and will be evaluated according to the list of criteria established in Section 1.3.4.

The simulated $τcap$ in the time constant plot of Fig. 6.11 has a wrong curvature. This can be traced back to the fact that $τcap$ is proportional to $exp(- F 2∕F 2) ox c$ according to equation (6.31).
The NMP barriers allow for a marked temperature activation of $τcap$ .
The TSM gives field-insensitve $τ em$ as demonstrated in Fig. 6.11. However, this is only true for defects with a trap level $E t,2$ above $E f$ . In this case, $f t,2$ approximates to $1$ and $τ em$ is given by $ν-1 exp(β ΔE ) 0 b,3$ .
The TSM also allows for field-sensitive $τem$ if $Et,2$ is located slightly below $Ef$ at the relaxation voltage. Then the field dependence is caused by the term $ft,2$ in equation (6.32).
In both defects, $τem$ is thermally-activated.

Figure 6.11: The TSM optimized against the TDDS data of ‘normal’ (left) and ‘anomalous’ (right) defects. Symbols mark measurement data while solid lines belong to simulations. Depending on the position of the trap level $Et,2$ , the TSM can explain both defect behaviors. However, it does not give the correct curvature of $τcap$ due to the field enhancement factor.

As demonstrated in the previous section, the TSM is indeed an important improvement of the NBTI model. Regarding the time constant plots (cf. Table 6.3), the introduction of the state $3$ gives an explanation for ‘normal’ as well as the ‘anomalous’ defect behavior. However, the TSM predicts a wrong curvature of $τcap$ and thus cannot be reconciled with the TDDS data. As a consequence, it can be concluded that the TSM performs well for stress and relaxation curves but fails to describe the behavior of single defects.

Table 6.3: The same as in Table 6.1 but including the TSM. In contrast to previous models the TSM can give an explanation for the ‘normal’ as well as the ‘anomalous’ defect behavior.

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