Elastic Tunneling

4.2 Elastic Tunneling

Tewksbury[23] assumed elastic tunneling of electrons and holes into and out of oxide defects as the mechanism responsible for charge trapping. The use of his model has been suggested by Huard et al. [58] in order to explain the recoverable part of the NBTI degradation. In this section, Tewksbury’s model will be reviewed, extended for an application of present-day devices with small gate thicknesses, and referred to as the elastic tunneling model (ETM) in the following.

The approach applied in this thesis relies on rate equations for tunneling into one trap. The required rate expressions $re(Et)$ and $rh(Et)$ are given by the equations (2.45) and (2.46), which already incorporate all elastic tunneling transitions between one trap and the numerous band states. Due to their analytical complexity, these rate equations are solved numerically using the numerical iteration scheme presented in Section 3.2. The used simple first-order partial differential equation reads

{ ∂f = fn(Et) re(Et) (1- ft) - fp(Et) re(Et) ft, Et ≥ Ec , (4.6) tt fn(Et) rh(Et) (1- ft) - fp(Et) rh(Et) ft, Et ≤ Ev

with the electron and hole trapping times approximated as

τcap,e(Et,xt) = fn(Et)1re(Et) , (4.7) ----1----- τem,e(Et,xt) = fp(Et) re(Et) , (4.8) τem,h(Et,xt) = fn(Et)1rh(Et) , (4.9) ----1----- τcap,h(Et,xt) = fp(Et) rh(Et) . (4.10)

The electron ( $fn$ ) and hole ( $fp$ ) occupancy are defined as

f (E) = f (E) , n FD (4.11) fp(E) = 1- fFD(E)

with $fFD$ being the Fermi-Dirac distribution. The above rate equation features the same structure as that of equation (4.1). Therefore, the same mathematical implications as for the phenomenological model hold true for the ETM. Furthermore, equation (4.6) covers all four basic transitions illustrated Fig. 4.2 so that it can be viewed as a comprehensive description of elastic tunneling in MOSFETs.

Figure 4.2: The basic tunneling transitions in a MOSFET for traps located above the conduction or below the valence band edge in the dielectric. Defects (gray filled ellipses) become positively charged in the left figure while they are neutralized in the right figure. The red arrows indicate a hole tunneling process, which always starts out from an empty state (marked by a red circle). It is noted here that this ‘empty state’ is located in the valence band and thus should be correctly referred to as a ‘hole’. However, the conduction and valence band states only differ in their effective masses — apart from their energetical position. These masses only affect the transport properties, such as the carrier mobilities, but marginally change the tunneling rates (2.45) and (2.46). In this respect, the terms ‘empty states’ and ‘holes’ are interchangeable and thus used synonymously in the remainder of this thesis.

4.2.1 The Behavior of A Single Trap

In this chapter, the investigations are focused on NBTI in pMOSFETs since these devices attract a large industrial interest at the moment. Therefore, only hole tunneling from the substrate valence band will be addressed in the following. Naturally, the basic statements will also remain valid for electron tunneling in the case of PBTI in nMOSFETs and for the different dielectric materials used in modern device technologies. For the trapping dynamics, $fFD$ and $2 ζWKB,v(Ex,xt)$ in $1∕τcap,h(Et)$ and $1∕τem,h(Et)$ of equation (4.6) are the most sensitive factors. These quantities determine the basic properties of the ETM and will be discussed in detail in this section.

The trapping dynamics are described by the rate equation (4.6), which has the same forward $re$ and reverse $rh$ rate considering hole trapping ( $Et ≤ Ev$ ).² This special form implies that the occupancy of the trap $ft$ equilibrates to that of the valence band states at the same energy $Et$ . For steady state conditions, $ft$ reads

ft(t) = 1+τcap,h(Et,xt1)∕τem,h(Et,xt) , (4.12)

where $τcap,h$ and $τem,h$ take the role of $τcap$ and $τem$ in (4.2), respectively. The above equation shows that the trap occupancy is eventually governed by the relative magnitude of $τcap,h$ and $τem,h$ . Both time constants are most strongly affected by the exponential decay of the Fermi-Dirac distribution. This dependence is pointed out in Fig. 4.3, where the hole capture and emission times are plotted with respect to the trap energy $Et$ . In the region below $Ef$ , the hole occupancy $fp$ decreases by several orders of magnitude per $1eV$ so that the hole capture times $τcap,h$ by far exceed the corresponding emission times $τem,h$ . Therefore, no effective hole trapping can take place in this region, irrespectively of the temperature. By contrast above $Ef$ , the exponential decay of $fn$ causes an increase in the hole emission times $τem,h$ , which gives rise to hole trapping there. That is, $Ef$ can be regarded as a demarcation energy between both regions. This fact is also reflected in the equilibrium solution (4.12) when using the definitions of $τcap,h$ and $τem,h$ .

ft(t) =-------1-------- 1 + fp(Et)∕fn(Et) (4.13) -----1----- = 1 + eβ(Et-Ef)

As a consequence, the trap occupancy is governed by the position of the substrate Fermi level in equilibrium³.

Figure 4.3: The hole capture ( $τcap,h$ ) and emission times ( $τem,h$ ) for two different temperatures as a function of the trap level $Et$ . The simulated device is subject to heavy stress conditions, at which tunneling ‘through’ the dielectric actually play a crucial role and should be taken into account. Nevertheless, $VG$ is set to a high gate voltage for illustration purposes only. The Fermi-Dirac distribution predicts a decrease in $fn$ above $Ef$ , which leads to exponentially rising $τem,h$ . Analogously, $fp$ decays and $τcap,h$ rises below the $Ef$ . As a result, $τem,h$ is larger than $τcap,h$ in the region above the $Ef$ so that hole trapping can occur there. By contrast, hole trapping is inhibited below $Ef$ since $τem,h ≪ τcap,h$ . For higher temperatures, the exponential decay becomes weaker, which is reflected in smaller slopes of the $τcap,h$ and $τem,h$ .

As mentioned before, the most sensitive factors in the tunneling rates of equation (4.6) are the Fermi-Dirac distribution and the WKB factor. In contrast to $fn$ and $fp$ , the latter is a function of $Ex$ and cannot be taken out of the integrals in the rate expressions (2.45) and (2.46). As shown in Fig. 4.4, $DOSp,1D+2D(Ex)$ and $|Mv0,tb(Ex,xt)|2$ , the matrix element without the WKB factor, are subject to small variations so that they weakly affect the tunneling rates. Since the WKB factor falls off quickly with increasing $Ex$ , the integral (2.46) delivers the largest contribution close to the trap level $Et$ . Therefore, it makes sense to approximate the rate integral (2.46) by dividing it through $ζ2WKB,v(Et,xt)$ and define the resulting expression as $τEpT,0M (Et)$ . This new quantity incorporates the weak dependences on $Ex$ while the sensitive factors, namely $fn$ or $fp$ and the WKB factor, are separated. Due to the small variations of $DOSp,1D+2D (Ex)$ and $|Mv0,tb(Ex,xt)|2$ , $τEpT,0M (Et)$ shows small changes with $Et$ compared to $fFD(Et)$ and $ζW2KB,v (Et,x)$ (see Fig. 4.5). This fact justifies approximations[23, 160, 106, 105], in which $τETM p,0$ as a prefactor of the WKB factor is approximated as a constant. Using the above definitions, the rate equation for holes reads:

∂f = f ---1-----ζ2 (E ,x) (1- f ) - f ----1----ζ2 (E ,x) f (4.14) tt n τEp,T0M(Et) WKB,v t t t p τEpT,0M (Et) WKB,v t t t ◟----------◝◜----------◞ ◟----------◝◜----------◞ =1∕τem,h(Et,xt) =1∕τcap,h(Et,xt)

For the time evolution of charge trapping, $2 ζWKB,v(Et,xt)$ becomes the essential factor (cf. Fig. 4.6). It shows an exponential decay with increasing trap depth, which results in a shift of $τcap$ and $τem$ towards larger times (cf. Fig. 4.7). Note that the energy of the tunneling charge carrier also impacts the WKB factor and the capture and the emission times. Since the energy dependence of the WKB factor enters both the capture and the emission times, their relative magnitude remains unaffected and the above argumentation of $Ef$ as a demarcation energy remains valid.

Figure 4.4: The functional form of $DOSp,1D+2D(Ex)$ , $|Mv0,tb(Ex,xt)|2$ , the $ζ2WKB,v(Ex)$ , and their product for flat band conditions. For better visibility, all functions are normalized to their maximal values. While $DOSp,1D+2D (Ex)$ and $|Mv0,tb(Ex, xt)|2$ remain within one order of magnitude, the WKB factor drops significantly. Since only the product of the three quantities enters the integrand of the rate equation (2.45) and (2.46), the largest contribution to the integral comes from the region slightly below the trap level $Et$ .

Figure 4.5: The quantity $τEpT,0M$ as a function of the trap level for different gate biases. Note that $τETM p,0$ remains within one or two orders of magnitude. The peaks close to $Ev$ can be traced back to a small kinetic energy of the holes within a classical picture. Quantum mechanically, the amplitude of the channel wavefunction at the interface is reduced for smaller charge carrier energies. This results in a decreased overlap of the wavefunction in the matrix element, a smaller tunneling probability, and ultimately in larger $τETM p,0$ close to the band edges.

Figure 4.6: The WKB factor (solid line) versus its approximated variant through a rectangular barrier (dashed line) as a function of the trap level. A comparison between both coefficients reveals that the difference between them can be neglected for small tunneling distances. Especially, the rectangular approximation yields reasonable estimates of the accurate WKB factor for trap levels close to the valence band edge. Nevertheless, all simulation results in this chapter are computed using the accurate WKB factor. Note that the temperature does not enter the calculation of the matrix element and in consequence the WKB factor. This is why to first order elastic tunneling is expected to be a temperature independent process. However, there are small effects due to the temperature dependent shape of the channel wavefunction. They are not considered in the derivation of Section 2.5.2 but will be later shown to marginally affect the temperature dependence of hole trapping (see Section 4.2.7).

Figure 4.7: The spatial dependence of the hole capture ( $τcap,h$ ) and emission ( $τem,h$ ) times constants on the trap depths. The more distant the traps are located from the interface, the larger become $τcap,h(Et)$ and $τem,h(Et)$ . This can be ascribed to the properties of the WKB factor, which decreases exponentially with the spatial depth of traps. Hence, the tunneling rates are reduced but the tunneling times are increased for deeper traps. Note that the crossing between $τcap,h$ and $τem,h$ always coincides with $Ef$ , irrespectively of the tunneling distance. As a consequence, the energetical border to the trapping region does not vary with the trap depth.

In conclusion, the following findings have been made: The equilibrium charge state of an oxide defect is directly determined by the Fermi level in the substrate. When $Et$ is situated above $Ef$ , the defect will capture a hole if the defect is initially occupied by an electron. Vice versa, positively charged defects with $Et$ below $Ef$ will emit their holes. However, this relationship does not make any statement about the time point when the tunneling transitions occur. This is solely given by the respective capture ( $τcap,h$ ) or emission ( $τem,h$ ) time constant. The actual trapping times are given by the WKB factor, which predicts increasing capture ( $τcap,h$ ) and emission ( $τem,h$ ) times for larger tunneling distances.

4.2.2 Spatially and Energetically Distributed Traps

In the previous sections, only the behavior of single traps has been addressed but NBTI is actually caused by charging or discharging of a multitude of defects. This means that the degradation has to be understood as a superposition of several trapping events. It must be considered that the individual defects differ in their properties, such as their spatial depth and their energetical position within the oxide bandgap. The defects in this model are assumed to be bulk traps, which are scattered across the whole dielectric. The distribution in the trap levels can be attributed to variations in bond length and angles, which impact the energy levels of the defect orbitals according to quantum mechanical considerations[161, 162, 23]. Therefore, large variances up to a few electron Volts have been assumed in the ETM. However, the defect levels of the hydrogen interstitial[163] and the oxygen vacancy[164, 165] are found to have a spread below $0.5eV$ . Hence, distributions of energy levels with a spread larger than $1eV$ seem to be unrealistic and must be verified by detailed atomistic investigations.

The following simulations, unless otherwise stated, are carried out on a pMOSFET ( $17 -3 Nd = 5× 10 cm$ ) with a strongly doped p-poly gate ( $20 -3 Na = 2 × 10 cm$ ) on top of a $3nm$ thick $SiO2$ layer. The traps in the dielectric are uniformly distributed in space while their corresponding levels span a range from $0.3$ to $4.3eV$ below $Ev$ . The operation temperature is $∘ 125 C$ so that this value lies in the center of the temperature range relevant for NBTI. For simplicity, only charge injection from the valence band has been taken into account. Nevertheless, this does not affect the general findings of the model discussion.

4.2.3 Time Behavior during Stress

In the following, the ETM will be tested whether it is consistent with the experimental findings presented in Section 1.4. For this model evaluation, the temporal behavior during the stress phase is the most essential criterion. It is depicted in Fig. 4.8 as the evolution of trapped charges $ΔQt$ and the threshold voltage shift $ΔVth$ . The former reveals a logarithmic time behavior, which is preserved for a large time range and by far exceeds the measurement window ranging from $1μs$ to $1ks$ . Recall that the behavior of one defect is described by the rate equation (4.6), which has the form of an ordinary first-order differential equation (4.1). It has been pointed out in Section 4.1 that the change in the occupancy $Δft(t)$ is given by the dominating time constants in the exponential term of equation (4.2). In the hole trapping region (cf. Fig. 4.3), $τcap,h ≪ τem,h$ holds. Then equation (4.2) simplifies to

( ) Δf (t,x ,E ) = f (0)- f (t → ∞ )×exp ------t----- . (4.15) t t t ◟t-----t◝◜------◞ τcap,h(xt,Et) =Δft,max

Assuming a rectangular tunneling barrier for the WKB factor, one obtains

( ) Δft (t,xt,Et ) = Δft,max × exp - ----t----e-xt∕xp,0 (4.16) ˜τEpT,0M (Et )

using the definitions

ETM ETM ˜τp,0 (Et) = τp,0 /fp(Et) (4.17)

and

-------ℏ-------- xp,0 = 2√2mp(Ev,sub-Ev,ox) . (4.18)

The exponential term on the right-hand side of equation (4.16) is characterized by a sharp drop at

xB(t) = xp,0 ln(t∕˜τEp,T0M) , (4.19)

which suggests the following approximation

{ Δft,max, x < xB(t) Δft(t,x) = 0, x > xB(t) . (4.20)

The sharp drop at $xB(t)$ represents a border between defects which ‘already have’ and ‘still do not have’ captured holes from the substrate until the time $t$ . This border moves away from the substrate towards deep into the dielectric as time progresses (see Fig. 4.9). Its shift follows a time logarithmic law according to equation (4.19) and results in straight lines in $ΔQt (t)$ of Fig. 4.8. When the border arrives at the gate, hole trapping stops, which becomes visible as a saturation in $ΔQt (t)$ and $ΔVth(t)$ . According to the charge sheet approximation, charges more distant from the substrate oxide interface make a smaller contribution to $ΔVth$ due to their smaller weighting factors $(1- xt∕tox)$ in equation (3.25). This yields the curvature seen in $ΔVth(t)$ plots of Fig. 4.8. However, the resulting curves still roughly follow a logarithmic time behavior. When disregarding the oxide field and temperature dependence for the time being, this fact might be misinterpreted as an agreement with the experimentally observed logarithmic time behavior (1.8).

Figure 4.8: Left: A simulation of the trapped charges $ΔQt$ as a function of stress time for different gate biases ( $T = 125∘C$ ). Charge trapping occurs over up to approximately 15 decades and shows a logarithmic time dependence. The slopes in the plot increase linearly with the oxide field, which cannot be reconciled with the quadratic field dependence obtained from experiments. Right: The time evolution of the threshold voltage shift $ΔVth$ for different gate biases. The resulting curves roughly approximate a logarithmic time dependence in a small experimental window from $1μs$ to $1ks$ .

Figure 4.9: The hole filling of traps during the stress phase ( $VG = - 1V$ ). Single traps are represented by the small circles located in the lower part of the oxide bandgap, where the occupied and the empty states are depicted as purple and white filled circles, respectively. The hole occupation of traps in the band diagram is recorded for a series of stress times and demonstrates the filling of traps with time. One can recognize a tunneling hole front, which starts from the substrate (right) and penetrates deep into the dielectric (towards the left). For demonstration purposes, the simulations were performed with the upper edge of the trap band shifted slightly above the substrate valence band. Note that traps located above the Fermi level do not have captured holes. This is due to the fact that these traps are energetically located within the substrate bandgap and thus have no corresponding energy level which can serve as a hole source in a tunneling process. Furthermore, hole trapping from the poly-gate has been neglected in these simulations but this aspect will be addressed later in Section 4.2.8.

4.2.4 Time Range of Trapping

The simulations in Fig. 4.8 reveal that hole trapping sets in around $1ps$ and lasts over approximately 15 decades. In this context one should consider that the electrical characterization methods used in NBTI have a limited time resolution of about $1μs$ . Therefore, they can only assess the accumulated degradation within a small measurement window while a large part of the real degradation is invisible in the experimental data. However, note that despite the wide time range, hole trapping is limited to a few seconds in a $2nm$ thick device (cf. Fig. 4.10). By contrast, no sign of saturation has been experimentally observed in the stress phase of NBTI so far.

Figure 4.10: The time evolution of hole trapping for different thicknesses of the dielectric ( $VG = - 1V$ , $T = 125∘C$ ). For comparison with Section 4.2.8 uppermost trap level has been shifted $- 0.3eV$ below the substrate valence band edge in this simulation. One can observe an early saturation for devices with an oxide thickness equal or smaller than $2nm$ , not seen in experiments. The inset shows that the time point of saturation is shifted out of the measurement window for devices with an oxide thickness notably exceeding a value of $2nm$ .

4.2.5 Oxide Field Dependence

The simulated curves in Fig. 4.8 have the same shape irrespective of the applied gate bias and can be made overlap by multiplying them with appropriate scaling factors $s(Fox,T)$ . According to measurement data, $s(Fox,T )$ should follow a quadratic field dependence up to approximately $8MV ∕cm$ . However, the required scaling factors do not show the correct tendency. Recall that only defects energetically shifted above $Ef$ are capable of capturing holes. Therefore, the amount of trapped charges per unit time is determined by the difference between $Ev$ and $Ef$ before and during the application of the stress voltage. The relative shift of these both energies is directly related to the surface potential $φs$ according to equation (3.9). The simulations in Fig. 4.11 demonstrate that $φs$ follows a nearly linear behavior over a wide range of the electric fields. As a result, one obtains a linear dependence for $s(F,T )$ in the NBTI region (indicated in Fig. 4.11), opposed to the quadratic field-acceleration observed in experiments.

Figure 4.11: The surface potential $φs$ versus the applied gate bias and the oxide field for the simulated pMOSFET ( $T = 125∘C$ ). This quantity is proportional to the scaling factor $s(F,T)$ in the simulated curves of Fig. 4.8 and thus determines the field-acceleration of the ETM. In the region relevant for NBTI, $φs$ shows a nearly linear dependence on the oxide field.

4.2.6 Time Behavior during Relaxation

In the simulated relaxation data in Fig. 4.12, the degradation returns back to its pre-stress value, meaning that all defects charged during the stress phase also take part in the recovery phase via hole emission. In these simulations it has been presumed that the trapped holes can only tunneling back to the substrate. However in devices with a small oxide thickness, the trapped charges can in principle be emitted to the poly-gate contact. This case will be addressed in detail in Section 4.2.8.

Figure 4.12: The same as Fig. 4.8 but for the relaxation phase. Note the accelerated recovery in the $ΔQt$ curve for heavier stress conditions, which shifts the end of the relaxation phase below $1s$ .

The simulations in Fig. 4.12 exhibit a logarithmic time behavior as observed in experiments (see Section 1.4). As for the stress phase, hole tunneling is determined by the WKB factor so that the tunneling times increase exponentially with larger $xt$ . This gives rise to a tunneling electron⁴ front, which starts out from the substrate and continues deep into the dielectric as illustrated in Fig. 4.13. The annihilation of trapped holes becomes visible as straight lines in Fig. 4.12, consistent with the experimental findings for the recovery phase. Against intuition, devices stressed at a higher gate bias recover faster. This can be traced back to the fact that spatially deeper traps have a reduced tunneling barrier when they are shifted down in the band energy diagram during relaxation (see Fig. 4.14). The above behavior has not been observed in measurements. This deviation from experimental data could, in principle, also originate from an additional permanent or slowly recovering component which is not accounted for in the present simulations. As compared to the stress phase, the time span for the relaxation phase is shortened for higher gate biases so that the recovery already ends before $1s$ .

Figure 4.13: The same as Fig. 4.9 but for the relaxation phase. The tunneling electron front annihilates the trapped holes within the oxide, starting from the substrate.

Figure 4.14: Hole trapping (left) and detrapping (right) during stress and recovery, respectively. The tunneling barrier during relaxation (indicated by the grey area) is appreciably reduced compared to stress, which is associated with shorter tunneling times and results in an accelerated recovery.

A remarkable peculiarity of NBTI is the the asymmetry in the slopes during stress ( $srelax(T,Fox,s)$ ) and relaxation ( $sstress(T,Fox,s)$ ). It is related to the fact that the recovery phase exceeds the duration of the stress phase by a couple of decades. However, the ETM (see Fig. 4.15) predicts that the recovery proceeds at equal pace or even faster than the degradation during stress does. This is due to the fact that traps charged with $τcap,h$ during stress emit their holes with approximately the same time constants $τem,h$ during recovery.

Figure 4.15: Trapping (solid) and detrapping (dashed) curves ( $V = - 1V G$ , $T = 125∘C$ ) for different stress times. Each pair of curves is normalized to the last stress value. Note that both curves approximately cover the same number of decades, which is in disagreement with the experimentally obtained asymmetry in the slopes during the stress and the relaxation phase.

4.2.7 Investigation of the Temperature Dependence using a Quantum Refinement

Another important issue concern the temperature dependence of the ETM. Recall that the experimental data exhibit an increased degradation at higher temperatures, which was thought to go hand in hand with an enhanced ‘total’ hole concentration $ptot$ ⁵. However, the simulations in Fig. 4.16 reveal an inverse tendency for the ETM. The discrepancy might be attributed to a shift of the hole centroid into the substrate and an associated reduction in the ‘interfacial’ hole concentration $pif$ (see insert of Fig. 4.16) for higher temperatures. In the band diagram, this is associated with a rise of $Ef$ towards the center of the substrate bandgap and thus away from $Ev$ . The relative position of $Ef$ and $Ev$ at the interface is the quantity which enters the calculation of the tunneling rates (2.46) and yields a reduction in the amount of hole trapping for higher temperatures. In a quantum mechanical treatment, the substrate holes are described by channel wavefunctions, which are spread over the whole channel region and penetrate deep into the dielectric. Tunneling and thus charge trapping are eventually induced by the overlap of the hole and trap wavefunctions. The ETM must be refined in order to account for the quantum mechanical nature of the confined charge carriers in the channel. Again equation (2.34) is taken as a starting point in the following derivation.

∑ 2π 2 r = ℏ--|Mc ∕v,tb(Ex,xt)| δ(Et - Eb) (4.21) b

Here, $b$ stands for the initial states. The sum over these states is split into components parallel and perpendicular to the semiconductor-oxide interface, where the charge carriers are confined in the latter direction. As derived in the Appendix A.4, the number of states in an one-dimensional confined electron gas is

∑ ∞∫ ∫E ∑ Ep∫,0 Ep∫,0 ∑ = Ayz ( dE Dn,2D dEx jn δ(Ex - En,jn)+ dE Dp,2D dEx jp δ(Ex - Ep,jp)) , (4.22) b En,0 En,0 -∞ E

$jn,jp = 0,1,2,...$ denote the quantum numbers and $En ∕p,jn∕p$ the respective eigenstates of the confined states in the conduction or the valence band, respectively. Note that the integrals run from the first confined eigenstate $En,jn$ or $Ep,jp$ since they are located closest to the conduction or valence band edges, respectively. Using (4.22), the rate (4.21) can be rewritten as

∞ E 2π ∫ ∫ ∑ 2 r =Ayz-ℏ ( dE Dn,2D dEx δ(Ex - En,jn) |Mc,tb(Ex,xt)| δ(Et - E) En,0 En,0 jn E∫p,0 E∫p,0 (4.23) + dE D dE ∑ δ(E - E ) |M (E ,x )|2 δ(E - E)) . p,2D x j x p,jp v,tb x t t - ∞ E p

Due to the $δ$ -function, the right-hand side of equation (4.22) can be simplified to

2π ∫Et ∑ r =Ayz--(Dn,2D dEx δ(Ex - En,jn) |Mc,tb(Ex,xt)|2 ℏ En,0 jn Ep,0 (4.24) ∫ ∑ 2 + Dp,2D dEx δ(Ex - Ep,jp) |Mv,tb(Ex,xt)|) . Et jp

Figure 4.16: The time evolution of trapped holes during stress (left) and relaxation (right) for different temperatures. The applied gate bias has been set to $- 2V$ . Surprisingly, the classical simulations predict a decrease in the amount of trapped holes for higher temperatures. This seems to be related to the reduction in the interfacial hole concentration $pit$ shown in the insert of the left plot.

The refined variant of the ETM shows an increase in the total hole concentration $ptot$ (see inset of Fig. 4.17), which would suggest an increased degradation for higher temperatures. Contrary to this ad hoc hypothesis, the simulations in Fig. 4.17 yield a reduced degradation, which is still in contrast to the experimental observations (see Section 1.4). Actually, not the change in the hole concentration — may it be $pit$ or $ptot$ — causes the inverse trend but the shift of $Ef$ relative to $Ev$ . From a statistical point of view, traps and band states at an energy $Et$ will equilibrate, meaning that their occupancies $fp$ and $ft$ will equalize. For higher temperatures, the raise of $Ef$ in the band diagram implies a reduction of $fp$ and in consequence $ft$ , which is related to the amount of trapped charges. It is important to note here that the density of states only affects the rates but not the equilibrium trap occupancies. In conclusion, the inverse temperature dependence cannot be ascribed to the deficiency of the classical variant of the ETM but it is inherent to elastic tunneling itself. Furthermore, the curves in Fig. 4.17 are shifted approximately two decades towards earlier times compared to the classical variant, which worsens the problem of the early saturation during stress.

Figure 4.17: The same as in Fig. 4.16 but for the quantum mechanical variant of the ETM. The improved model delivers the same tendency as its classical variant even though the total hole concentration $ptot$ (see the insert of the left plot) would suggest an increased hole trapping for higher temperatures. In comparison to the classical variant, a shift of the whole set of curves towards earlier times is observed.

4.2.8 Charge Injection from the Gate

So far, the existing description of charge trapping has been restricted to charge injection from the substrate. In device structures with thicker gate dielectrics, the large tunneling distances from the gate towards the defects are associated with small rates so that the presence of the gate contact as a source or sink of charge carriers could be neglected. As the oxide thickness of modern semiconductor devices has entered the nanometer range, this assumption has lost its justification. Therefore, the ETM needs to be extended by charge carrier injection from the gate. As illustrated in Fig. 4.18, this can be achieved by introducing additional terms into the rates equation:

∂tft = (fn,s(Et) re,s(Et)+ fn,g(Et) re,g(Et)) (1- ft) - (fp,s(Et) re,s(Et)+ fp,g(Et) re,g(Et)) ft + (f (E ) r (E )+ f (E ) r (E )) (1 - f) - (f (E ) r (E )+ f (E ) r (E )) f (4.25) n,s t h,s t n,g t h,g t t p,s t h,s t p,g t h,g t t

Figure 4.18: A schematic of the rates considered in the conventional and the extended ETM for a trap located below the substrate bandgap. The gray filled circles are hole states in the substrate, the dielectric, or the gate. The ellipse marks a trap within the dielectric. Tunneling rates from and towards the substrate are represented by the arrows with solid lines and already taken into account in the conventional ETM. For thinner gate dielectrics, the charge injection from the gate gains importance since the associated rates increase due to the shorter tunneling distances. These rates, indicated by the arrows with the dashed lines, must be incorporated the extended ETM.

The subscripts $s$ and $g$ refer to quantities from the substrate or the poly-gate, respectively. The simulations presented in Fig. 4.19 underline the importance of the gate contact when thin dielectrics are considered. Recall that tunneling in the conventional ETM can be envisaged as a tunneling hole front that starts at the substrate interface and continues towards the gate. Traps located in the second half of the dielectric are located closer to the gate and thus have shorter tunneling distances to the gate. For these traps, electron injection from the gate outbalances hole trapping from the substrate. Hence, the presence of the gate interface establishes a spatial border to the penetrating hole front. When this border is reached, the tunneling hole front stops, which causes an even earlier saturation during the stress phase. At this point it is important to note that, depending on the spatial and energetical distribution of traps, the band bending in the gate can also trigger electron injection from the gate. A comparison of both models is presented in Fig. 4.19. One may notice that the timescale for trapping is dramatically reduced in the case of the extended ETM. For devices with relatively thick gate dielectric of $4nm$ , the saturation sets in before $1s$ . In technologically more relevant device structures with a gate dielectrics thinner than $2.5nm$ , the degradation due to hole trapping ends before $1μs$ and would even be not assessable with ultra-fast MSM measurements[27]. The fact that only traps with short tunneling times participate during the stress phase is also reflected in a fast removal of trapped charges during the relaxation phase. The complete removal of positive charges during the relaxation phase is achieved within approximately the same timescales than hole trapping during stress. By contrast, the relaxation seen in experiments is often described as a long-lasting process, which exceeds $105s$ . More importantly, hole trapping is reduced below five decades for oxide thicknesses below $3nm$ while the eMSM data on a $1.7nm$ thick device (see Fig. 1.4) show at least seven decades.

Figure 4.19: The same as in Fig. 4.10 but for the conventional (solid line) as well as the extended (dashed line) ETM ( $VG = - 1V$ , $T = 125∘C$ ). It is noted that the uppermost trap level is located $0.3eV$ below the substrate valence band edge in this simulation. In the case of the extended ETM, one can recognize an early saturation, which reflects the stopping of the hole front in the dielectric. As pointed out in the inset, this time point of saturation is moved towards an earlier time, which may lie within typical measurement windows for NBTI (ranging from $1μs$ to $10ks$ ). One should keep in mind that only devices with an oxide thickness thicker than $5.5nm$ show a logarithmic time behavior beyond $100ks$ , which corresponds to the largest investigated stress times.

4.2.9 Width of the Trap Band

Up to this point, only broad distributions of trap levels have been addressed. However, it is speculated that some high- $κ$ dielectrics have a crystalline structure[166], which is characterized by small temperature-induced variations of bond distances and angles. These lattice distortions yield a small spreading of defect levels, seen as a small trap band in Fig. 4.20. In Fig. 4.20, the time evolution of $ΔQt$ is plotted for a narrow distribution of defect levels. During stress, the onset of hole trapping is shifted towards earlier times for increasing gate voltages. This behavior can be traced back to different defects involved in charge trapping at different $VG$ (see Fig. 4.21). As the gate bias is increased, defects located closer to the substrate interface are moved into the region around the Fermi level. Due to their reduced tunneling distances, they feature shorter trapping times and therefore give rise to an earlier onset in the $ΔQt (t)$ curves. Since the defects in the active trapping region are spatially concentrated to a small region, their corresponding tunneling distances are limited to a small range. Thus the distribution of trapping times is sharply peaked, which becomes visible as sudden jumps in the $ΔQt (t)$ curves of Fig. 4.20. The shape of these curves is in stark contrast to the wide range of timescales usually involved in NBTI.

Figure 4.20: The time evolution of the normalized $ΔQt$ ( $T = 125∘C$ ) for stress (left) and relaxation (right) presuming a narrow distribution of defect levels ( $0.1eV$ ) centered around $- 0.75eV$ below the substrate valence band edge. For this assumption, the ETM predicts short jumps or drops in $ΔQt$ for both phases, which are limited to only a very few decades. This cannot be reconciled with NBTI data with a relaxation phase that spans over 10 decades. In the case of $VG = 0.5V$ , the defect levels are not moved into the region around the Fermi level so that no trapping can occur.

Figure 4.21: The band diagram for two different gate voltages. The crossing point between the Fermi level and the band of trap levels (shown as a purple and a blue region for a low and a high gate bias, respectively) is linked to the earliest trapping events and the beginning of charge trapping. When the gate bias is increased, the crossing point is shifted closer to the substrate interface ( $x1 → x2$ ) where traps with smaller tunneling time constants ( $τ2 ≪ τ1$ ) are situated. This leads to an earlier onset of charge trapping for higher gate voltages.

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