6.3 Two Stage Model

The concept of NMP has been used in a slightly modified variant termed multiphonon field-assisted tunneling (MPFAT) [11611511611712555], 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 [1516]. 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)
                 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 [2526], 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:


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
          ∫ 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)
                 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.

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.