2.2 Hot-Carrier Degradation

In Chapter 1 it is mentioned that basically two types of defects appear in MOSFETs, the interface defects and the oxide defects. So far, it has been discussed how charge carrier exchange between oxide defects and their surrounding affect the device parameters in the context of NBTI degradation. In this chapter, the most important modeling approaches for the degradation mechanism HCD, which is associated with the creation of interface defects, are summarized. Similar to BTI, also HCD is a detrimental mechanism in MOSFETs, which affects device parameters, such as , and on-resistance (). In this context, hot is associated with the kinetic energy of the carriers accelerated by high channel electric fields. Therefore, HCD is best observed at high electric fields along the channel, which is typically achieved by a high drain voltage at stress conditions () and a gate bias close to the operating conditions.

The process itself is known since the 60’s [77] and over the time several modeling approaches have been made in order to reflect the changing impact on MOSFET parameters because of the scaling trend. In this context, several attempts have been made in order to distinguish different HCD modes [78], e.g., hot-carrier injection (HCI). HCI is associated with channel carriers, which are entering the conduction band of the oxide as they overcome the energetic barrier between the substrate and the oxide. This requires a kinetic energy higher than  eV. Although the electric fields in the device increased due to the scaling of device dimensions, the simultaneous reduction of the operating voltages compensated this increase and led to a lower kinetic energy of the carriers. Therefore, HCI has faded from the spotlight and new modeling attempts have been made.

One of the first successful HCD models was the so-called “lucky-electron" model [79], valid for long channels or high electric fields. This concept introduces a threshold energy level which needs to be surmounted by the carriers in order to trigger impact-ionization. As soon as a carrier travels a sufficiently long distance without collisions, this energy can be reached. However, it has been found that even for low operating voltages around 1.5 V in devices with dimensions in the deca-nanometer regime HCD is a severe degradation mechanism [80, 81]. Thus, models were required, which take into account colder carriers as a physical cause for HCD.

2.2.1 Experimental Characterization of HCD

HCD in experiments is often characterized by either using the charge pumping method (Section 3.2) in order to extract the number of interface states or by measurements of the linear drain current shift (). Since defects located at the oxide/substrate interface cause surface scattering, which lowers the carrier mobility and tilts the transfer characteristics, the impact of HCD is much more pronounced in than in . can be measured using the method for extraction discussed in Subsection 2.1.1. The stress and recovery phases are interrupted periodically in order to record an - characteristics, extract and subtract it from the unstressed value. In contrast to BTI and as already mentioned in Subsection 2.1.2 created interface states barely recover. Therefore, the recovery of HCD is negligible, and interruptions of the stress conditions hardly affect the degradation state of .

The temperature dependence of HCD is different than the one of BTI. For long channel devices, for example, the degradation of is less detrimental at increased temperatures. However, for short-channel devices HCD is accelerated at higher temperatures [82]. This channel length and temperature dependent acceleration of degradation is caused by temperature-dependent contributions of scattering effects, which may populate the high energetical fraction of the carrier ensemble [23].

In contrast to the temperature dependence of HCD, the field dependence is independent of the channel length. In general, the degradation of is highly channel electric field dependent [23, 83]. However, investigations have revealed that the peak of the electric field, the peak of the average carrier kinetic energy and the maximum of the created interface defect density do not correlate with each other [84–86].

2.2.2 Hess Model

The Hess model proposes that HCD is determined by the dissociation of neutral hydrogen-passivated dangling bonds at the substrate/oxide interface by channel carriers (see Figure 2.27) [87–90]. This is based on two ideas: a hot carrier with sufficient kinetic energy scatters with a dangling bond and causes it to break (single particle process, shown in the left panel of Figure 2.28), as well as multiple colder carriers cause one bond to dissociate (multiple particle process, shown in the right panel of Figure 2.28). The bond breakage process at the interface between substrate and oxide results in centers [91, 92], which can capture and emit charge carriers and distort device characteristics. For example, the trapped charges act as Coulomb scattering centers and degrade the carrier mobility. The capture and emission dynamics are described by the standard SRH theory.

For the modeling of bond dissociation, the Si-H bond is typically modeled using a truncated harmonic oscillator [23]. This oscillator is characterized by the system of eigenstates as shown in Figure 2.29. The single particle process corresponds to the excitation from one of the eigenstates to the last bonded state and the transition to the transport state. Most probably, the interaction energy excites the bonding electron of to an antibonding state, which consequently leads to the release of the hydrogen atom. The desorption rate of this process can be written as the acceleration integral [90]

with

	desorption rate for the single particle process
	threshold energy
	carrier impact frequency on the surface per unit
	area within the range of ,
	desorption probability
	energy-dependent reaction cross section.

The multiple particle process corresponds to an excitation from one eigenstate to the next with each scattering process between a colder carrier and the Si-H bond, whereby the occupation number obeys a Bose-Einstein distribution. This multivibrational mode excitation is accompanied by the phonon mode decay with the corresponding rates

with

/	total phonon emission / absorption rate
	Threshold energy
	carrier impact frequency on the surface (per unit
	energy and area),
/	scattering cross-sections for bond-phonon
	emission / absorption
	phonon occupation numbers
	phonon energy.

The bond rupture happens from the last bonded level to the transport state. Finally, the bond-breakage rate corresponding to the multiple particle process can be written as

with

	bond-breakage rate corresponding to the multiple particle process
	energy of the last bonded level in the quantum well, dissociation energy
	phonon energy
	Boltzmann constant
	lattice temperature
/	total phonon emission / absorption rate
	phonon reciprocal life-time.

The Hess model was quite revolutionary because it expressed the idea that HCD is controlled by the distribution function, which enters the acceleration integral in Equation 2.40. It consideres the interface traps on a microscopic level. However, it remains unconnected to the device level. For example, the degradation of parameters like and cannot be modeled with the Hess model only.

2.2.3 Energy Driven Paradigm

Rauch and LaRosa suggested an alternative empirical model, which reflects the importance of the carrier energy on the degradation. The so-called energy driven paradigm consideres that in the case of scaled devices with channel lengths less than 180 nm the driving force of HCD is the carrier rather than the electric field [24, 83, 94, 95] as it was in the “lucky-electron" model. One further issue associated with the approach of Rauch and LaRosa is the increasing impact of the electron-electron scattering on HCD at reduced channel lengths because it populates the high energy tail of the carrier distribution function.

The impact ionization rate as well as the rate of hot-carrier induced interface state generation is controlled by terms as the following.

is the carrier distribution function and is the reaction cross section. While the first is strongly decaying with increasing energy, the second one grows power-law-likely. The product of both results in a maximum. As long as this is sufficiently narrow, it can be approximated by a delta-function, which avoids time-consuming calculations of the carrier distribution function. The integral can be substituted by a stress condition related empirical factor.

Although the findings of Rauch and LaRosa are a substantial simplification of the HCD treatment, the energy driven paradigm suffers from some shortcomings. For example, the product is not necessarily narrow. Moreover, the energy driven paradigm does not consider as a distributed quantity and, therefore, it does not capture the strong localization of HCD.

2.2.4 Bravaix Model

The Braivaix model is based on features of both, the Hess model and the findings of Rauch and LaRosa. Important ideas of the Hess model which enter the Bravaix model are the interplay between single and multiple carrier mechanisms as well as the realization that the damage is defined by the carrier distribution function. The latter enters Equation 2.43 via the particle flux in and . The findings of Rauch and LaRosa enter the Bravaix model via the substitution of the acceleration integral by the stress condition related empirical factor discussed in the previous subsection.

The Bravaix model describes the kinetics of the oscillator (Figure 2.29) as a system of rate equations [22, 90, 96]:

with

	occupancy of the oscillator level
	time
/	total phonon emission / absorption rate
	last bonded level
	rate of hydrogen released to the transport state, thermal barrier
	interface-charge density
	concentration of the mobile hydrogen

The phonon emission and absorbtion rate are calculated differently to Equations 2.41 and 2.42 [90]:

with

/	total phonon emission / absorption rate
	drain current density
	cross-section of excitation of a phonon mode
	phonon life-time
	phonon energy
	Boltzmann constant
	lattice temperature

Based on the energy-driven paradigm the integral can be substituted by the empirical factor . With this and for the case of weak bond-breakage rate the equation system 2.44 to 2.46 can be solved for to yield:

The multiple particle process related interface state generation rate (bond-breakage rate corresponding to the multiple particle process) can be written as

with

	ground state, it is assumed that the bond occurs most likely in the
	ground state
	energy of the last bonded level in the quantum well, dissociation energy
	empirical factor substituting the acceleration factor (energy-driven
	paradigm)
	drain current
	thermal energy barrier of hydrogen released to the transport state
	phonon energy
	Boltzmann constant
	lattice temperature
	phonon reciprocal life-time.

Depending on whether the stretching or bending vibrational mode is considered, the values for , and can be chosen [96]. With the findings of all involved persons, the dissociation rate by multiple particle processes is represented quite well. Consequently, the life-time can be estimated for the different regimes like the hot-carrier regime where the single particle mechanism plays the dominant role, the intermediate case where electron-electron scattering leads to a population of the high energetical tail of the charge carriers and the high electron flux where the multiple particle process dominates the bond dissociation.

Nevertheless, the missing carrier transport treatment which allows one to distinguish between the single particle process, electron-electron scattering, and multiple particle process driven modes, affects the model quality. Since these mechanisms affect each other via the carrier distribution function they have to be considered. Moreover, the scheme for the single particle process rate is based on fitting parameters and not on physical mechanisms.

2.2.5 HCD Model Based on the Exact Solution of the Boltzmann Transport Equation

In order to overcome the disadvantages of the Braivaix model, a model based on the exact solution of the Boltzmann transport equation (BTE) has been proposed [23, 93, 97, 98]. This approach captures the physical picture behind HCD more accurately by covering three aspects of HCD: the carrier transport, a microscopic description of the defect creation kinetics and the degraded device simulation. The aspect of the carrier transport is solved by using either a stochastic or a deterministic solver of the BTE. While the stochastic solver employs the Monte Carlo method, the deterministic solver is based on the expansion of the carrier density function in a series of spherical harmonics. The latter appears especially for ultra-scaled devices more appropriate because electron-electron scattering, for example, can be implemented easily without leading to a long computational time.

Furthermore, for a proper HCD treatment one has to consider both types of carriers, minority and majority. Due to impact ionization (II), secondary majority carriers are generated. If the device is operated near or beyond pinch-off conditions, channel carriers with sufficient kinetic energy can trigger impact ionization, thereby, generate secondary majority carriers which are accelerated towards the source due to the channel electric field. As a consequence, additional interface states to those created by the primary channel carriers can be created and result in an additional peak of the interface state density shifted towards the source (see Figure 2.30). This additionally created interface states significantly change the degradation characteristics [99, 100].

For a proper modeling approach, the distribution functions for both, majority and minority charge carriers, are evaluated at each point at the interface. The distribution functions enter the carrier acceleration integral, which controls single particle as well as multiple particle mechanisms.

with

	carrier acceleration integral
	threshold energy
	carrier distribution function
	density-of-states
	reaction cross section
	carrier velocity

For the single particle process, the superposition of electron and hole acceleration integrals weighted with the corresponding attempt frequencies gives the generation rate (bond-breakage rate corresponding to the single particle process)

with

	ground state
/	attempt frequencies for electrons / holes
/	acceleration integral for electrons / holes
	time

For the multiple particle process, the - bond is treated as a truncated harmonic oscillator, which leads to the bond-breakage rate corresponding to the multiple particle process

with

	ground state
	last bonded level
	rate of hydrogen released to the transport state, thermal barrier
	rate of passivation of dangling bonds
/	total phonon emission / absorption rate
	time.

The model is able to capture the degradation of MOSFET parameters like measured in devices with different channel lengths and at different hot-carrier (HC) stress conditions. As will be shown in Chapter 5, also effects associated with oxide defects can be explained. In particular, with a thorough carrier transport treatment and under consideration of secondary generated majority carriers in the channel also recovery after different stress conditions can be modeled properly. However, one open question remains, namely how oxide traps contribute to HCD. As already shown in Figure 2.30, not only but also increases with the stress time. It has been assumed that so-called turn-around effects shown in Figure 2.31 could be the result of the interplay between different defect types [23].

2.2.6 Turn-Around Effects

Stress time-dependent turn-arounds of degradation as shown in Figure 2.31 have been discussed in literature for and [99, 100, 102]. A turn-around means in this regard that the degradation trend changes after a certain stress time. In Figure 2.31, initially decreases while after 10ks it starts to increase.

In the case of degradation, it has been proposed that the turn-around can be explained by two aspects. The first aspect concerns the additionally created interface states by secondary generated carriers as already mentioned in the previous subsection. The second aspect, which is finally responsible for the turn-around itself, is the interplay between contributions of defects at the interface and the oxide [101]. In case of degradation, it has been demonstrated that during stress decreases due to minority charge carriers trapping in the oxide while after a certain stress time it increases due to trapping of majority charge carriers by interface traps, generated during stress. This turn-around effect is caused by the partial compensation of the charge stored in the oxide traps by interface state trapping.

The explanation for the turn-around in is quite similar to the one for the turn-around of . Charges of opposite signs are trapped in different sections of the characterized transistor [102]. As a result, first increases followed by a decrease for longer stress times. In Section 5.1 it will be shown that a turn-around effect was measured. However, the exact interplay between oxide defects and interface defects remains still an open question.