Predictive and Efficient Modeling of Hot Carrier Degradation with Drift-Diffusion Based Carrier Transport Models

Chapter 3 A Review of Existing HCD Models

As already discussed, HCD is a major reliability concern in power MOSFETs, but deeply scaled devices are also susceptible to hot-carrier effects [131]. Modeling this phenomenon is difficult as the physical effects are not completely clear and involve many mechanisms occurring simultaneously. Moreover, modeling of HCD requires solving the carrier transport problem from the full-band Boltzmann transport equation which is computationally challenging and often leads to convergence issues in large area and high voltage devices. The models also have to cope with the ever changing semiconductor industry. For example, in the eighties, the focus was on shrinking the device dimensions, while the operating voltages were not significantly reduced. This development has led to devices with high electric fields in which the carriers were energetic enough to break the Si-H bonds in a single interaction [132, 133]. Thus, the supply voltages were scaled to reduce the electric fields in the devices [134]. Due to the low source-drain voltage in extremely scaled devices, HCD was expected to be drastically suppressed as described in [135, 136, 137]. Such a trend, however, was not observed [138] and even ultra-scaled MOSFETs were found to exhibit HCD [139, 137, 140]. This peculiarity was attributed to energy exchange mechanisms, such as the multiple-carrier process and electron-electron scattering [86, 10].

Particularly for ultra scaled devices, HCD becomes complicated due to the dominance of the multiple-carrier process of Si-H bond breakage [136, 135, 139]. However, the single- and multiple-carrier mechanisms are limiting cases and commonly a superposition of the two is present [141, 25]. The interplay between the two bond breakage mechanisms causes a shift in the worst-case conditions for HCD. Typically, the worst-case HCD was observed at $V_{\mathrm {gs}}$ $\thickapprox$ 0.5 $V_{\mathrm {ds}}$ which corresponded to the maximum substrate current [85, 142]. This is no longer valid even for long-channel devices where the worst-case damage is observed at maximum average carrier energy, i.e. conditions corresponding to the maximum gate current [143]. In scaled devices, on the other hand, the dominance of the multiple-carrier process requires the maximum carrier flux instead of the carrier energy for most severe HCD, which is obtained at $V_{\mathrm {gs}}$ $\thickapprox$ $V_{\mathrm {ds}}$ [144, 145, 146]. It should be noted that high energy carriers exist also in ultra scaled devices due to processes like electron-electron scattering, so the single-carrier process cannot be ignored [11, 137]. Similarly, in long-channel devices the multiple-carrier process is important especially at higher stress times [147]. Apart from these characteristics, the temperature behavior of HCD is another important aspect to be understood. On the one hand, in long-channel devices hot-carrier damage decreases as the temperature increases. On the other hand, scaled devices show the opposite trend, exhibiting an increase in HCD with temperature. This trend in scaled devices is observed by increased electron-electron scattering which populates the high energy tail of the carrier distribution function [148, 137, 149, 88]. Thus, a comprehensive model should capture these features of HCD and must be necessarily based on the carrier distribution function. The carrier distribution function is the key aspect as it provides the information on the interaction of carriers with the bonds. In this respect, several attempts have been made to simplify the HCD modeling problem as discussed in the following.

The first attempt for HCD modeling was the lucky electron model [150] which introduced the field-driven approach [151, 152, 77, 153]. According to this model, a defect is produced by an electron having high enough energy to overcome the potential barrier at the interface and enter the SiO (math image) conduction band without being emitted back into the oxide. This simple model predicted that carriers with energy greater than 3.7 $\,$ eV lead to interface state generation. This was deemed incorrect by measurements where the degradation was found to occur even at lower stress voltages [85]. Although the application of the lucky electron model is limited to long channel devices, it is still a popular model due to its simplicity. Takeda et al. proposed an extension to the lucky electron model whereby the degradation, i.e. transconductance and threshold voltage shifts with time are represented by a power law $t^{n}$ . Using this empirical approach, the device lifetime could be easily extrapolated from accelerated hot-carrier stress measurements to real operating conditions. However, the model is of limited applicability and inaccurate to represent HCD in devices where a saturation in degradation is observed [154]. Some other extensions of the lucky electron model are: the Goo model which can capture the saturation of degradation [154], the Dreesen concept which was proposed for lightly doped drain transistors [155, 153], the Woltjer approach which incorporates a field-driven correction in the lucky electron model [76, 77]. In the Woltjer model, consideration of the electric field within the oxide allowed the description of degradation due to interface states in devices with varied dimensions and oxide thicknesses.

Mistry et al. introduced three different modes of degradation, i.e., generation of interface and neutral oxide traps at low gate voltages, interface state generation at mid voltage range, and oxide electron trap build-up at high voltages [156, 152]. Although this model was unable to predict the device lifetime correctly and was of limited use, it led to the idea of multiple competing mechanisms for constituting the overall degradation [152]. An important model by Moens et al. considers two competing mechanisms for representing degradation in LDMOS transistors and used different time exponents to characterize these mechanisms [4].

A majority of these models were empirical or at the best phenomenological and usually had time exponents to fit the experimentally obtained degradation characteristics. Another shortcoming of these models was the consideration of the electric field as the driving mechanism, which is valid only for large devices. Thus, models describing the physical picture behind HCD which were based on energy driven approaches were required [85, 86, 10].

3.1 The Hess Model

The first physics-based model for HCD was proposed by Hess et al. who suggested the existence of two mechanisms

for Si-H bond breakage, i.e. the single- and multiple-carrier processes sketched in Figure 3.1 [21, 135, 157]. The single-carrier mechanism leads to breakage of a Si-H bond when a solitary hot carrier interacts with the bond, transferring its energy and leading to an excitation of one of the bonding electrons to an antibonding state. Thus, a repulsive force is induced on the hydrogen atom and leads to hydrogen release. The bond breakage rate with the single-carrier process is given by [157]:

$(3.1) \begin{equation} R_{\mathrm {SP}}=\int _{\varepsilon _{\mathrm {th}}}^{\infty }I(\varepsilon )P(\varepsilon )\sigma (\varepsilon )d\varepsilon ,\label {eq:SP-rate} \end{equation}$

where $I(\varepsilon )$ is the carrier flux in the energy range $\left [\varepsilon ;\varepsilon +d\varepsilon \right ]$ , $P(\varepsilon )$ the probability of bond breakage, $\sigma (\varepsilon )$ the reaction cross section, and $\varepsilon _{\mathrm {th}}$ is the threshold energy the hot carrier should possess in order to initiate the bond breakage process.

The Hess formalism was successful in explaining the giant-isotope effect, i.e. different desorption rates of the hydrogen and deuterium from Si surfaces, observed experimentally [158, 159, 160]: In experiments with scanning tunneling microscopes (STM), hydrogen- and deuterium-passivated Si surfaces were bombarded by cold carriers from the tip of an STM. It was observed that much higher carrier current densities were required to break the Si-D bond as compared to Si-H. The bond breakage rates for the latter were found to be two orders of magnitude larger than the former at high voltages as shown in Figure 3.2. This effect was termed giant isotope effect and was explained by the multiple-carrier process.

Within this concept, the Si-H bond was modeled as a truncated harmonic oscillator comprising of an energy ladder, i.e. a system of eigenstates in the potential well. While interacting with colder carriers the bond can gain or loose energy, leading to its excitation or de-excitation, respectively, within the energetic states of the ladder. Such subsequent interactions with the carriers were linked to the excitation of the phonon modes. The bond is ruptured when the hydrogen leaves the highest energy level and overcomes the potential barrier $E_{\mathrm {emi}}$ , thereby ending up in the transport mode, see Figure 3.1. The passivation process is defined as hydrogen jumping in the opposite direction over the barrier $E_{\mathrm {pass}}$ . The emission and passivation processes, characterized by the energies $E_{\mathrm {emi}}$ and $E_{\mathrm {pass}}$ , respectively, are assumed to obey an Arrhenius law with corresponding rates $P_{\mathrm {emi}}$ and $P_{\mathrm {pass}}$ . The excitation and decay rates of the bond ( $P_{\mathrm {u}}$ , $P_{\mathrm {d}}$ , respectively) in the potential well are obtained by considering all combinations of phonon absorption (bond heating) and phonon emission (related to decay resulting from multiple-carrier process) which can be induced by the carrier flux [21, 157]

$(3.2) \{begin}{align} P_{\mathrm {u}}= & \int _{\varepsilon _{\mathrm {th}}}^{\infty }I(\varepsilon )\sigma _{\mathrm {ab}}(\varepsilon )\left [1-f_{\mathrm {ph}}\left (\varepsilon -\hbar \omega \right )\right ]\mathrm {d}\varepsilon ,\label {eq:MP-rate}\\ P_{\mathrm {d}}= & \int _{\varepsilon _{\mathrm {th}}}^{\infty }I(\varepsilon )\sigma _{\mathrm {emi}}(\varepsilon )\left [1-f_{\mathrm {ph}}\left (\varepsilon +\hbar \omega \right )\right ]\mathrm {d}\varepsilon ,\nonumber \{end}{align}$

with $I(\varepsilon )$ being the carrier flux, $\sigma _{\mathrm {\mathrm {ab}/emi}}(\varepsilon )$ the reaction cross section for absorption/emission of phonons, $f_{\mathrm {ph}}$ the phonon occupation numbers, and $\hbar \omega$ the distance between energy levels. The bond breakage rate for the multiple-carrier process can then be estimated as:

$(3.3) \begin{equation} R_{\mathrm {MC}}=\left (\frac {E_{\mathrm {B}}}{\hbar \omega }+1\right )\left [P_{\mathrm {d}}+\exp \left (\frac {-\hbar \omega }{k_{\mathrm {B}}T_{\mathrm {L}}}\right )\right ]\left [\frac {P_{\mathrm {u}}+\omega _{\mathrm {e}}}{P_{\mathrm {d}}+\exp \left (-\hbar \omega /k_{\mathrm {B}}T_{\mathrm {L}}\right )}\right ]^{-E_{\mathrm {B}}/\hbar \omega }, \end{equation}$

where $E_{\mathrm {B}}=E_{\mathrm {a}}-E_{\mathrm {emi}}$ is the energy of the last bonded state in the potential well, $\omega _{\mathrm {e}}$ the reciprocal phonon lifetime, $k_{\mathrm {B}}$ the Boltzmann constant, and $T_{\mathrm {L}}$ the lattice temperature.

The most important contribution of the Hess model was the consideration of the bond breakage process as a contribution of different carriers in the ensemble. Thus, the carrier distribution function was considered a major ingredient for a proper description of HCD. The DF enters the rate equations, Equations 3.1 and 3.2, via the carrier flux $I(\varepsilon )$ (see Section 4.4). The isotope effect was explained from the difference in energetics of the Si-H and Si-D bonds which led to different parameters (like phonon lifetime) of the quantum well for the two kind of bonds. The Hess model also considers the statistical distribution of the activation energy, $E_{\mathrm {a}}$ , which was supported by density functional theory calculations [21, 161] and by experiments reporting a double power law of interface state generation as:

$(3.4) \begin{equation} N_{\mathrm {it}}=\frac {p_{1}}{1+\left (t/\tau _{1}\right )^{\alpha _{1}}}+\frac {p_{2}}{1+\left (t/\tau _{2}\right )^{\alpha _{2}}}. \end{equation}$

Here $\tau _{1}$ , $\tau _{2}$ are the characteristic times, $\alpha _{1}$ , $\alpha _{2}$ are the different time slopes, while $p_{1}$ , $p_{2}$ are the probabilities related to the traps.

However, the microscopic Hess model was not translated to the macroscopic or device level and thus, prediction of the device lifetime and degradation characteristics such as transconductance, linear drain current, etc. was not addressed.