### 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 and 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

with the electron and hole trapping times approximated as
The electron () and hole () occupancy are defined as
with 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.

#### 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, and in and 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 and reverse rate considering hole trapping ().2 This special form implies that the occupancy of the trap equilibrates to that of the valence band states at the same energy . For steady state conditions, reads

where and take the role of and in (4.2), respectively. The above equation shows that the trap occupancy is eventually governed by the relative magnitude of and . 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 . In the region below , the hole occupancy decreases by several orders of magnitude per so that the hole capture times by far exceed the corresponding emission times . Therefore, no effective hole trapping can take place in this region, irrespectively of the temperature. By contrast above , the exponential decay of causes an increase in the hole emission times , which gives rise to hole trapping there. That is, 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 and .
As a consequence, the trap occupancy is governed by the position of the substrate Fermi level in equilibrium3.

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 and , the latter is a function of and cannot be taken out of the integrals in the rate expressions (2.45) and (2.46). As shown in Fig. 4.4, and , 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 , the integral (2.46) delivers the largest contribution close to the trap level . Therefore, it makes sense to approximate the rate integral (2.46) by dividing it through and define the resulting expression as . This new quantity incorporates the weak dependences on while the sensitive factors, namely or and the WKB factor, are separated. Due to the small variations of and , shows small changes with compared to and (see Fig. 4.5). This fact justifies approximations[23160106105], in which as a prefactor of the WKB factor is approximated as a constant. Using the above definitions, the rate equation for holes reads:

For the time evolution of charge trapping, becomes the essential factor (cf. Fig. 4.6). It shows an exponential decay with increasing trap depth, which results in a shift of and 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 as a demarcation energy remains valid.

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 is situated above , the defect will capture a hole if the defect is initially occupied by an electron. Vice versa, positively charged defects with below 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 () or emission () time constant. The actual trapping times are given by the WKB factor, which predicts increasing capture () and emission () 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[16116223]. 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[164165] are found to have a spread below . Hence, distributions of energy levels with a spread larger than seem to be unrealistic and must be verified by detailed atomistic investigations.

The following simulations, unless otherwise stated, are carried out on a pMOSFET () with a strongly doped p-poly gate () on top of a thick layer. The traps in the dielectric are uniformly distributed in space while their corresponding levels span a range from to below . The operation temperature is 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 and the threshold voltage shift . The former reveals a logarithmic time behavior, which is preserved for a large time range and by far exceeds the measurement window ranging from to . 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 is given by the dominating time constants in the exponential term of equation (4.2). In the hole trapping region (cf. Fig. 4.3), holds. Then equation (4.2) simplifies to

Assuming a rectangular tunneling barrier for the WKB factor, one obtains
using the definitions
and
The exponential term on the right-hand side of equation (4.16) is characterized by a sharp drop at
which suggests the following approximation
The sharp drop at represents a border between defects which ‘already have’ and ‘still do not have’ captured holes from the substrate until the time . 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 of Fig. 4.8. When the border arrives at the gate, hole trapping stops, which becomes visible as a saturation in and . According to the charge sheet approximation, charges more distant from the substrate oxide interface make a smaller contribution to due to their smaller weighting factors in equation (3.25). This yields the curvature seen in 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).

#### 4.2.4 Time Range of Trapping

The simulations in Fig. 4.8 reveal that hole trapping sets in around 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 . 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 thick device (cf. Fig. 4.10). By contrast, no sign of saturation has been experimentally observed in the stress phase of NBTI so far.

#### 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 . According to measurement data, should follow a quadratic field dependence up to approximately . However, the required scaling factors do not show the correct tendency. Recall that only defects energetically shifted above are capable of capturing holes. Therefore, the amount of trapped charges per unit time is determined by the difference between and before and during the application of the stress voltage. The relative shift of these both energies is directly related to the surface potential according to equation (3.9). The simulations in Fig. 4.11 demonstrate that follows a nearly linear behavior over a wide range of the electric fields. As a result, one obtains a linear dependence for in the NBTI region (indicated in Fig. 4.11), opposed to the quadratic field-acceleration observed in experiments.

#### 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.

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 . 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 .

A remarkable peculiarity of NBTI is the the asymmetry in the slopes during stress () and relaxation (). 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 during stress emit their holes with approximately the same time constants during recovery.

#### 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 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 (see insert of Fig. 4.16) for higher temperatures. In the band diagram, this is associated with a rise of towards the center of the substrate bandgap and thus away from . The relative position of and 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.

Here, 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
denote the quantum numbers and 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 or since they are located closest to the conduction or valence band edges, respectively. Using (4.22), the rate (4.21) can be rewritten as
Due to the -function, the right-hand side of equation (4.22) can be simplified to

The refined variant of the ETM shows an increase in the total hole concentration (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 or — causes the inverse trend but the shift of relative to . From a statistical point of view, traps and band states at an energy will equilibrate, meaning that their occupancies and will equalize. For higher temperatures, the raise of in the band diagram implies a reduction of and in consequence , 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.

#### 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:

The subscripts and 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 , the saturation sets in before . In technologically more relevant device structures with a gate dielectrics thinner than , the degradation due to hole trapping ends before 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 . More importantly, hole trapping is reduced below five decades for oxide thicknesses below while the eMSM data on a thick device (see Fig. 1.4) show at least seven decades.

#### 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- 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 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 (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 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 curves of Fig. 4.20. The shape of these curves is in stark contrast to the wide range of timescales usually involved in NBTI.