5.2 The Level Shift Model

So far, only qualitative statements about the defect behavior could be made based on the switching levels. In order to make quantitative predictions, the ETM must be generalized in a way to account for the levels shift. Recall that the conventional concept of the ETM is based on the assumption that the energy levels for tunneling ‘into’ and ‘out of’ a defect coincide. This is only the case for unrealistic defects which do not deform after a tunneling event. But as proven in the previous Section 5.1, defects do undergo structural relaxation and therefore feature two switching levels, say E0∕+  for hole capture and E+ ∕0  for hole emission for instance, which can even be separated by some electron Volts (see Fig. 5.9). To be precise, the switching levels E0∕+  and E+∕0  actually originate from the one defect orbital and thus must be correctly interpreted as one trap level, which shifts after each charging or discharging event. For the trapping kinetics, this means that only one of these levels can be present in the band diagram at a time. For instance, when the defect in Fig. 5.10 is in its neutral charge state, it has a trap level E0∕+  for hole capture while its corresponding trap level E+ ∕0  remains inactive for the time being. If a hole capture process takes place, the E0∕+  level vanishes and thus the E+∕0  level appears. Based on the considerations above, the ETM must be regarded as a special case of the level shift model (LSM) but with a negligible defect relaxation. Consequently, the formulation of the ETM must be modified in order to account for the level shift. Thus equation (4.6) is rewritten as

∂tft = + fn(Eq1∕q2) re,q1∕q2(Eq1∕q2,xt)(1- ft) - fp(Eq2∕q1)-re,q2∕q1(Eq2∕q1,xt) ft
        ◟    =1∕τcap,◝e◜(Eq ∕q ,xt)   ◞            ◟    =1∕τem,e◝(E◜q ∕q,xt)    ◞
                      1 2                                2 1                      (5.1)
     +  f◟n(Eq1∕q2) rh,q◝1◜∕q2(Eq1∕q2,xt◞)(1- ft) - fp◟(Eq2∕q1)-rh,q2◝∕◜q1(Eq2∕q1,xt)◞ ft ,
             =1∕τem,h(Eq1∕q2,xt)                     =1∕τcap,h(Eq2∕q1,xt)

where the rates are defined as

r     (E     ,x ) = r    (E    ,x ) =  -----1------ζ2    (E     ,x ) ,      (5.2)
 e,q1∕q2  q1∕q2 t    e,q2∕q1  q1∕q2  t     τEnT,0M(Eq1∕q2)  WKB,c  q1∕q2  t
                                          1       2
rh,q2∕q1(Eq2∕q1,xt) = rh,q2∕q1(Eq2∕q1,xt) = τETM(E----)-ζWKB,v(Eq2∕q1,xt) .      (5.3)
                                      p,0    q2∕q1

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Figure 5.9: A schematic of the level shift model. As opposed to the ETM, two distinct defect levels must be considered — namely, E+∕0  and E0∕+  for electron and hole capture, respectively. When the positively charged trap (blue filled circle in gray ellipse) captures a substrate electron with an energy of E+ ∕0  (blue arrow), the defect level E+∕0  vanishes but reappears at E0∕+  . The now neutral defect (red circle in gray ellipse) is only capable of capturing a hole from the silicon valence band (red arrow). Right after this process the defect level has returned to its initial position E+ ∕0  again.

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Figure 5.10: Left: The calculated time constants according to the ETM (T = 50∘C  ). The defect is placed close to the interface (xt = 2˚A  ) of a pMOSFET in the off-state and features one single trap level Et = E+∕0 = E0 ∕+  indicated by the vertical dashed line. In the present case, the trap level lies 2.5eV  below the substrate valence band edge, where hole emission (circle) proceeds much faster than hole capture (box). Note that for trap levels within the substrate bandgap also tunneling from interface states [23] has been taken into account. Right: The same as in the left figure but for the LSM. Note that the form of the capture and emission times remain the same in both models whereas only the energy E+∕0  at which electron capture time is calculated has been changed (E0∕+ = E+∕0  ). According to the equations (5.2) and (5.3), τcap,h  and τem,h  are evaluated at two different trap levels, namely one at E+ ∕0  for electron capture and another at E0∕+  for hole capture. For this case the relative magnitude of τcap,e  and τcap,h  depends on the energy distance of the respective trap level to Ef  .


q1  and q2  denote the two charge states involved in the tunneling process and the trap levels Eq1∕q2   and Eq2∕q1   corresponds to the switching traps introduced in Section 2.3. The above rate equation is reminiscent of the ETM presented in the previous Section 2.5.2. The peculiarity of the LSM is that the particular terms on the right-hand side of equation (5.1) must be evaluated for different energies, namely E+ ∕0  or E0∕+  , depending on the charge state of the defect before the tunneling transition occurs. For instance, the positively charged defect of Fig. 5.9 has a trap level E+ ∕0  , which must be applied for calculation of the electron capture rate 1∕τcap,e(E+ ∕0,xt)  (see Fig. 5.10). By contrast, the neutralized defect features a trap level E0∕+  used for the hole capture rate 1∕τcap,h(E0 ∕+,xt)  . The calculation of the corresponding time constants is illustrated in Fig. 5.10. It is important to note here that the expressions for τcap,e  , τcap,h  , τem,e  , and τem,h  remain the ‘same’ as in the ETM and only change in the energy they are evaluated for. This is due to the fact that the tunneling mechanism itself is not affected by the structural relaxation. Thus, analogously to the ETM, the tunneling process can be described by the tunneling rates (2.45) and (2.46) of the ETM and reasonably approximated by (5.2) and (5.3).

In the following, a new quantity, referred to as the demarcation energy1, E
  d  will be introduced. It determines equilibrium occupancy of the defects and is defined by the condition

τcap,e(E+∕0,Ef)  = τcap,h(E0∕+,Ef)               (5.4)
with

                       ETM
τcap,e(E+ ∕0,Ef)  = -2---τn,0-(E+∕0)----- ,           (5.5)
                 ζWKB,c(EE+T∕0M,xt) fn(E+∕0)
τcap,h(E0∕+,Ef)  = ζ2---τ(pE,0-(,Ex0)∕+)f(E--) .           (5.6)
                  WKB,v 0∕+  t p  0∕+
Assuming Boltzmann statistics, the energy dependences of the electron and hole occupation can be approximated as follows:
        {exp(- β(E - E )), E > E
fn(E) ≈               f         f               (5.7)
        {1,                E < Ef
         1,                E > Ef
fp(E) ≈                                         (5.8)
         exp(- β(Ef - E)), E < Ef
Suppose that E0∕+ > Ef  and E+ ∕0 < Ef  as it has been the case for the E′γ  center. Then only the exponential terms of fn  and fp  enter the time constants. The WKB factors in the expressions (5.5) and (5.6) can be replaced by their approximative variants for a rectangular barrier.
 2                  √2mt-  ∘ -----------
ζWKB,c∕v(E ) ≈ exp(- 2 ℏ  xt  |Ec∕v,ox - E|) .        (5.9)
Since τEnT,0M (E+∕0) ≈ τEpT,0M (E0 ∕+ ) ~ 10-14s- 1  holds, equation (5.4) can be rewritten as
     √2mt   ∘ -----------    (           )
exp(2  ℏ xt   Ec√,ox - E+∘ ∕0)-exp-β(E+∕0 - Ef() =       )
        = exp(2 -2ℏmtxt  E0∕+ - Ev,ox) exp β (Ef - E0∕+) .       (5.10)
Taking the logarithm of this equation and after rearranging some terms, one obtains
Ed  = Ef = E0∕++E2+-∕0-+ αβ (∘Ec,ox---E+∕0 - ∘E0-∕+---Ev,ox)xt      (5.11)
with
       ---
α  = 2√2mt .                       (5.12)
        ℏ
The factor αxt∕(βq0)  in equation (5.11) takes a value of approximately 0.094  at room temperature (  ∘
23 C  ). Thus the last term is negligible compared to the remainder of equation (5.11) and Ed  can be estimated by
     E+ ∕0 + E0∕+
Ed = -----2----- .                     (5.13)
This quantity predicts the electron occupancy of a defect when equilibrium has been reached. For instance, when a stress voltage is applied to the gate of a pMOSFET, Ed  is raised above Ef  and the initially neutral defect can capture a substrate hole during the stress phase. Conversely, when the MOSFET is switched from stress to relaxation, Ed  falls below Ef  and the positively charged defect will become neutralized in equilibrium. As an example, the Ed  level of the oxygen vacancy varies between - 1.7eV  and - 0.5eV  according to the present DFT results. These values lie too low to be shifted above Ef  for defects located close to the interface (< 1nm  ). Therefore, the level Ed  of the E′
 γ  center reveals that the LSM is incompatible with the concept of hole capture into oxygen vacancies. All other defect candidates, investigated by the DFT simulations in this thesis, feature values of Ed  close to Ef  and cannot be ruled out on the basis of the above argument.

While the equilibrium occupation of the defects is given by the demarcation energy, the trapping dynamics directly follow from the electron and hole capture time constants, whose dependence on the Fermi level and the trap depth will be discussed in the following. The ‘interesting’ instance is when the Fermi level is situated inbetween the levels E+ ∕0  and E0∕+  (cf. Fig. 5.10). Using Boltzmann statistics (5.7) and (5.8) and approximative WKB factor (5.9), the capture time constants can be estimated by

                         (  ∘ -----------  )
τcap,e(E+∕0,xt) = τEnT,M0  exp 2α  Ec,ox - E+∕0xt  exp(β(E+∕0 - Ef))       (5.14)
                         (  ∘ -----------  )    (           )
τcap,h(E0∕+,xt) = τEpT,0M  exp 2α  E0∕+ - Ev,oxxt  exp β(Ef - E0∕+) .      (5.15)
In the equations above, the last term, originating from the Fermi-Dirac distribution, has the largest impact on both time constants. For instance, τcap,e  exponentially depends on the energy difference |E0∕+ - Ef| , that is, a higher E0 ∕+  level gives a larger τcap,e  . Analogous considerations hold true for the energy difference |Ef - E0∕+| and τcap,h  . These dependences are also reflected in the exponential branches of the time constant plot in Fig. 5.10. Note that the simulated defect in this figure has been placed only 2˚A  away from the interface where the time taken for the tunnel step can be almost neglected. However, when the defects are assumed to be situated deeper within the oxide, their time constants are increased due to the reduced tunnel probability. As demonstrated in Fig. 5.11 (left), this effect is more pronounced in the middle of the oxide bandgap while it almost diminishes towards the band edges due to the reduced tunneling barrier there. The ‘uninteresting’ instance is when the Fermi level is situated above E0∕+  as well as E+ ∕0  as shown Fig. 5.11 (right). In this case τem,h  is larger than τcap,h  , implying that hole capture is effectively suppressed. Note that analogous consideration holds true for the electron capture when the Fermi level fall below E+∕0  and E0∕+  .


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Figure 5.11: Left: The calculated time constants according to the LSM for various trap depths (T = 50∘C  , VG = 0V  ). The increase in the time constants is related to a larger tunneling distance for deeper traps. However, this effect, incorporated in the WKB factor, becomes weaker for energies closer to the oxide band edges since the tunneling barrier is dramatically lowered there. Right: The calculated time constants (T = 50∘C  , VG = 0V  ) for the case when the Fermi level is situated above both trap levels E0 ∕+  and E+ ∕0  . Since τcap,h  exceeds τem,h  , hole capture is overcompensated by hole emission and thus effectively suppressed.


5.2.1 Model Evaluation

In this section, the LSM will be employed to investigate the impact of the level shift on the trapping dynamics in NBTI experiments. Based on the NBTI checklist established in Section 1.4, it will be tested whether this model is capable of reproducing the NBTI degradation seen in experiments. The following simulations are carried out on a pMOSFET (          17  - 3
Nd = 5 × 10  cm  ) with a strongly doped p-poly gate (          20  - 3
Na = 3× 10  cm  ). The thickness of the oxide layer has been chosen to be 5nm  for demonstration purposes. Thereby, the traps can be homogeneously spread within the dielectric but are still sufficiently separated (2.0nm  ) from the poly interface in order to be able to neglect trapping from the gate. Furthermore, this wide range of trap depths ensures a large distribution of capture and emission times over 14 decades in time (cf. Fig. 5.11). The energy levels of the traps have been assumed to be uniformly distributed with the E0∕+  and the E+∕0  levels being uncorrelated and thus independently calculated using a random number generator. The operation temperature is set to    ∘
125 C  and thus lies in the middle of the range relevant for NBTI. It is noted that only charge injection from the substrate is accounted for in the following simulations for simplicity.

In the following, the basic properties of the LSM will be discussed on the basis of a simple showcase. Therefore, this model is evaluated for a type of defect whose trap level E0∕+  has a wide distribution below Ev  while the E+∕0  counterpart is sharply peaked slightly above Ec  (see Fig. 5.12). At the beginning of the stress phase nearly all defects are neutral and thus occupied by one electron. In this state, the traps are characterized by the hole capture levels E0∕+  located below the substrate valence band. The corresponding electron capture levels E+ ∕0  lie above the substrate conduction band but are inactive for the time being. During the stress phase, the substrate holes must be thermally excited to the defect level E0∕+  in order that a tunneling process can occur. In equation (5.6) their energy-dependent concentration is linked to the factor fp  , which decays exponentially with decreasing energies assuming Boltzmann statistics. This decay would lead to a temporal filling of traps from energetically higher towards lower traps in the band diagram. Furthermore, the tunneling of electrons and holes has an additional trap depth dependence, which is reflected in the WKB factor of equation (5.6). Analogously to the ETM, this would cause a temporal filling of traps starting from close to the interface and continuing deep into the oxide. As demonstrated in Fig. 5.13, the superposition of both effects results in a tunneling hole front which proceeds from high defect levels close to the interface towards lower ones deep in the oxide. The resulting time evolution of the trap occupancies is visualized in Fig. 5.12.


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Figure 5.12: The time evolution of hole trapping during the stress (first row) and the relaxation (second row) phase (Vs = - 3.0V  , T = 150∘C  ). The substrate Fermi level is indicated by the red line and the small circles represent the trap levels E+∕0  and E0∕+  of the single defects. The neutral defects are assumed to have fully occupied defect orbitals and thus can only trap holes. Therefore, they feature hole capture levels E0∕+  , which are located below the center of the substrate bandgap and are said to be ‘active’. By contrast, the positively charged defects are assumed to be able to trap electrons only. Accordingly, they have only electron capture levels E+∕0  above midgap but no energy levels E0 ∕+  , which reappear once these defects are charged again. It is emphasized that the occupancy of one defect is related to which of its trap levels E0∕+  and E+ ∕0  is active at the moment. Therefore, the above figures also reflects the trap occupancies at a certain time. Since the trap levels are recorded at the beginning, the middle, and the end of the stress and the relaxation phase, these figures show the tunneling hole front, which is illustrated by the active trap levels and thus the occupancy of the defects.

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Figure 5.13: Motion of the tunneling hole front shown for a snippet of the hole occupancy in Fig. 5.12 after a stress time       5
ts = 10 s  . The capture time constants are determined by the exponential decay in the WKB factor and the Fermi-Dirac distribution. The latter has been approximated by the Maxwell-Boltzmann distribution and shows an exponential energy dependence a few kBT  away from the Fermi level. As indicated by the green vertical arrow, this dependence leads to a tunneling hole front moving downwards in the band diagram. By contrast, the WKB factor is most strongly affected by the trap depth, resulting in an tunneling hole from the substrate to deep into the oxide (green horizontal arrow). The resulting motion of the tunneling hole front is depicted by red arrow.


The temporal filling is also reflected in the occupancies of the demarcation energies, shown in Fig. 5.14. As already mentioned before, only defects located above Ef  can participate in hole capture. As a consequence, the temporal filling of traps does not proceed below Ef  , which thus marks a border to the tunneling hole front.


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Figure 5.14: Left: The demarcation levels Ed  at the end of the stress phase for the defects shown in Fig. 5.12 (Vs = - 3.0V  , T = 150∘C  ). The white circles represent neutral defects which have not trapped a hole during the stress phase. The purple filling color indicates that the defect is positively charged due to a finished hole capture event. Since there exist neutral defects with an Ed  level above Ef  , hole trapping has not reached saturation after a stress period lasting 105s  . Middle: The same as in the left figure but at the end of the relaxation phase. The figure indicates that not all trapped holes have been removed after a relaxation time of 1010s  . Right: Resulting active area for charge trapping in the LSM. The full red lines show the substrate Fermi levels during stress (Ef,stress  ) and relaxation (Ef,relax  ). The defects situated below Ef,stress  cannot capture a hole during stress while those located above Ef,relax  remain positively charged during relaxation and thus do not contribute to ΔVth  .


After the stress phase, a large part of the hole capture levels has disappeared and is replaced by their corresponding electron capture levels E0∕+  . The latter are assumed to be concentrated in a small trap band slightly above the substrate conduction band. During the recovery phase, electrons in the substrate conduction band must be thermally excited up to the E0∕+  level where the trap depth-dependent tunneling process can take place. According to Fig. 5.12, the defects are found to be filled according to their trap depth, visible as a horizontally moving tunnel front. The small separation of the E+ ∕0  levels on the energy scale results in a narrow distribution of electron capture times. From this it follows that two particular charging events at the upper and the lower edge of the trap band can be hardly resolved in time. As a consequence, no vertical component in the motion of the tunnel front is observed during the recovery phase of Fig. 5.12. Analogously to the stress phase, the tunneling hole front also appears in the occupancies of the demarcation levels displayed in Fig. 5.14. During the relaxation phase, the Ed  levels are shifted below Ef  where they can be neutralized if equilibrium has been reached. However, Fig. 5.14 reveals that the discharging of traps has not been completed even until an unrealistic long relaxation time of 1010s  . It is important to note here that Ef  during stress and relaxation determines the active area in which hole capture is possible. Defects above this area are already unoccupied before stress and thus cannot capture a further hole, while the ones below will remain neutral due to the high hole emission rate. As a result, only defects within this area can change their charge state and thus contribute to the net amount of captured holes and in further consequence to NBTI.

The LSM has been employed to simulate NBTI degradation in a pMOSFET for a wide range of different stress conditions. The calculated stress/relaxation curves for the aforementioned showcase are presented in Fig. 5.15. In contrast to the ETM, they exhibit a marked temperature dependence in addition to the obvious field acceleration. While the former one mainly stems from the temperature dependent Fermi-Dirac distribution in τcap,h  (cf. equation (5.6)), the latter one cannot be simply interpreted by the lowering of the tunneling barrier at higher Fox  . The field acceleration originates form the larger shift of the trap levels at higher Fox  , as visualized in Fig. 5.16. As such, the field acceleration strongly depends on the distribution of the trap levels in space and energy but is not inherent to the LSM itself. For instance, a defect with xt = 0nm  and Ed < Ef  during stress will not be able to capture a hole at all (see Fig. 5.17).


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Figure 5.15: Left: The trapped charges ΔQox(t)  as a function of stress time for different gate biases. Right: The same as for the left figure but for different temperatures.

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Figure 5.16: The trap levels E0∕+  and E+ ∕0  (both second row) after a stress time of   5
10 s  for Vs = - 1.0∕ - 2.5∕- 4.0V  (from left to right) and        ∘
T = 150 C  . The higher position of the E0∕+  levels goes hand in hand with shorter hole capture times and in consequence a larger amount of trapped charges after a stress time of   5
10 s  . From this it can be concluded that the oxide field dependence of the LSM primarily originates from the upwards shift of the trap levels but is only marginally influenced by the reduced tunneling barrier at higher Fox  .

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Figure 5.17: The position and the trap occupancy of Ed  after a stress time of 105s  for Vs = - 1.0∕- 2.5∕- 4.0V  (from left to right) and T = 150∘C  . It can be recognized that a higher Fox  increases the portion of Ed  levels above Ef  and thus a larger number of traps are available for hole capture.


In order to obtain more realistic results, tunneling from interface states [23] has been incorporated into LSM. The obtained degradation curves for defects with E+ ∕0 = Ev + 1.5 1.0eV  and E0∕+ = Ev - 1.75 1.0eV  are depicted in Fig. 5.18. Based on these results, it will be evaluated whether the LSM can satisfactorily reproduce the basic features seen in NBTI experiments. In the NBTI checklist of Table 5.2, these features are formulated as necessary criteria, where each of them will be judged in the following.

The above list provides strong evidence that the LSM cannot be reconciled with the experimental NBTI data. As a consequence, pure tunneling must be discarded as a possible cause for hole trapping in NBTI.


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Figure 5.18: The same as in Fig. 5.15 but for a different distribution of defect levels and including tunneling from interface states. Left: During stress, charge trapping sets in later (10ms  ) when smaller biases are applied to the gate. The degradation curves follow a nearly logarithmic behavior over a wide time range, where the curves obtained for a small stress voltages can be better approximated by a power-law. Furthermore, none of the curves show a sign of saturation until a stress time of 105s  . It is noted that their slopes appear to be insensitive to the applied gate bias — except from small gate biases again. Regarding the relaxation phase, deviations from the time logarithmic behavior can be recognized below 10ms  . Right: For the stress as well as the relaxation phase, a very weak temperature dependence is obtained.

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Figure 5.19: The field acceleration and temperature activation in the LSM during stress. Since the degradation roughly follows a logarithmic behavior in this phase, scaling factors can be extracted from Fig.5.18. s(T )  and s(F  )
   ox  demonstrate that the LSM does not reproduce the quadratic field and temperature dependence seen in experiments.