Models for the Bias Temperature Instability

6.2 Models for the Bias Temperature Instability

As stated in the beginning of this chapter, BTI is, as its name suggests, a gate voltage, device temperature and time dependent threshold voltage shift ΔVth, which indicates a charge buildup in the oxide of the investigated MOS structures. As the magnitude of ΔVth over stress time ts can be expressed by a power law [130, 131], this spawned a debate over the correct power law exponent. At this stage of research recovery after bias temperature stress has only been superficially investigated and the reaction-diffusion (RD) models [130, 132] could explain the published measurement data. All flavours of the RD models assume a charge buildup at the semiconductor-oxide-interface through hydrogen-based reaction and diffusion. It was soon noticed that the measurement techniques and all time transients differ from publication to publication and that time delays and transients need to be well defined. Additionally, measurement and ΔVth extraction techniques for BTI have been investigated [123]. Utilizing well defined experiments the quantitative relations between stress time, recovery time, device temperature and gate bias or oxide field can be identified. As such it was, for example, found that ΔVth depends quadratically on the magnitude of the oxide field. However, the authors of [115] had to introduce a permanent component of BTI, due to the limited recovery time in the experiments. This permanent component is still under debate and is necessary to reproduce recovery traces, due to the stark asymmetry of stress and recovery over time as can also be seen in Figure 6.1. By asymmetry we mean that the recovery of 1mV of ΔVth after BTI stress takes longer than the buildup of 1mV of ΔVth during stress, provided the device temperature is kept constant. Thus when applying a model to experimentally recorded ΔVth data, the data outside the measurement window is accounted for by a permanent non-recoverable component although a plateau of recovery has to date never been, reproducibly, observed [133]. With the published amount of data, it was finally conclusively found that any reaction diffusion model, particularily on the microscopic scale, cannot possibly explain all the observed BTI characteristics [112]. Finally, with the emergence of time dependent defect spectroscopy (TDDS), it was possible to investigate the charge capture and emission of single defects in small-area MOSFETs and to identify a multiphonon process as the charge exchange process between the substrate and the defects in the oxide.

In the course of this thesis three different models for BTI have been implemented into the drift diffusion simulator MinimosNT. First a simple two well model, second the Two-Stage-Model [134] and last a four state non-radiative multiphonon (NMP) model [4]. In contrast to the four-state NMP model alone, which can reproduce the recoverable component, the Two-Stage-Model predicts both the recoverable and the permanent component of BTI. Thus, the four-state NMP model is often combined with a two well model, since the two well model is used to reproduce the permanent component of BTI. The permanent degradation is usually modelled as a buildup of interface states, whereas the recoverable component is modelled as formation and annealing of oxide traps. Thus the two well model describes the time evolution of Nit, whilst the four-state NMP model describes the time evolution of N_ot. The Two-Stage model, in contrast to the other two models, includes a description of N_ot (recoverable) and Nit (permanent) over time. Since the four state NMP model combined with the two well model can explain more BTI characteristics than the other two models alone, this is the model which will be explained in detail.

6.2.1 Phenomenological Models

In order to model the weak temperature and the quadratic oxide field dependence of the charge capture process causing the threshold voltage shifts many phenomenological models have been developed. In addition to that many authors, e.g. [135], tried to explain the charge capture process by extending the successful SRH [136] trapping model with a tunneling coefficient to account for the fact that a charge trapping defect is not necessarily located directly at the oxide interface. However, it was shown that these models cannot sufficiently explain the physics involved in BTI [113], especially the capture and emission time constants found using TDDS. Nevertheless, a double-well model serves as a good example how interface defects caused by BTI can be modelled.

Figure 6.7:

A phenomenological model for BTI. In the precursor state 1 the defect is neutral and can, by being thermally activated, undergo a transition into the active trapping state 2, where it can capture and emit charge carriers. The energy barrier which must be overcome by the defect can, in this model, be lowered by an oxide field. In this model the parameters γ and ν must be found experimentally.

Starting from the SRH model introduced in Section 2.2.2, one can explain the formation of new defects causing a shift in threshold voltage by introducing a model for the trap concentration Nit. To capture the oxide field dependence and the temperature dependence of the formation and annealing of defects an Arrhenius-type law is often employed. In a two-well model for Nit, a defect can be in two states. It can either be active, where trapping and detrapping of charge carriers is governed by SRH statistics (charge trapping), or inactive, where transitions are modelled by defect reaction rates. The barrier height E_a determining the defect reactions rates is assumed to be oxide field dependent and temperature activated. For the model in Figure 6.7 the defect reaction rates are

k₁₂	= ν₀ exp,	(6.2)
k₂₁	= ν₀ exp,	(6.3)

where ν₀ ≈ 10¹³s^-1 is the attempt frequency.

Another often employed approach is to explain the defect buildup by initially electrical inactive traps and hole trapping instead. To this end the trap is assumed to reside in the oxide, a distance xT away from the semiconductor-oxide interface and the trapping kinetics are described by the SRH model and elastic tunneling [137]. The modified SRH rates per trap read

k₁₂ = k₁₂^SRHλ(x T,E_ox) and k₂₁ = k₂₁^SRHλ(x T,E_ox),

(6.4)

where the function λ denotes the elastic tunneling coefficient, usually a modified WKB approximation [138, 139, 140], depending on the depth of the trap and on the oxide field energy barrier. In [113] it has been shown that these simple models are insufficient to describe all the observed characteristics of BTI, especially those found by time dependent defect spectroscopy on nanoscale devices.

6.2.2 Non-radiative Multiphonon Transitions

The first researchers to extend the SRH theory [136] formulated a charge trapping theory which was loosely based on non-radiative multiphonon transitions (NMP) [141]. These models considered charge carrier tunneling and a thermally activated, oxide field dependent process as demonstrated in the previous section [142]. In this model [142] the capture and emission time constants, which are obtained from the reaction rates using Markov-Chain theory [143], read

τc	= τ₀ _λ(xT)exp,	(6.5)
τe	= τ₀ _λ(xT) exp,	(6.6)

where E_t is the trap level, E_C and E_V are the respective band edges and θ denotes the Heaviside function [113]. When fitting this model to capture and emission times obtained through TDDS it was realized that one can either obtain a fit for τc or τe but not for both [113]. Since none of the above models can fully capture the findings obtained by TDDS experiments a new model was required [29]. This new model has been fully based on non-radiative multiphonon theory, which shall be explained in the following.

Basic Theory

The non-radiative multiphonon theory is based on the possible scattering between multiple phonons with a single electron, which eventually gets trapped in the process. It is theorized that this particular electron can, depending on the configuration and bonds of the surrounding atoms, not only emit photons to loose this gained energy, but now has sufficent energy to occupy a previously unoccupied state (charge trapping). What the atomic configuration at the defect site is, is still not completely clear [112]. In order to derive the reaction rates necessary to describe the charge trapping process lots of approximations to the full Schrödinger equation (cf. Equation (2.1)) describing the many particle problem are required. First the Born-Approximation is applied to separate the equation into two loosely coupled equations: one for the system of electrons and one for the system of nuclei. Per electron i and nucleus j the system of equations is

ψ_i(r,R)	= V _i(R)ψ_i(r,R),	(6.7)
ϕ_iJ(R)	= E_iJϕ_iJ(R),	(6.8)

where V _i(R) denotes the adiabatic potential, which corresponds to the energy of a certain atomic configuration and ϕ_iI is the vibrational wave function corresponding to the states i and I. Next, to describe the complex scattering process, first order time-dependent perturbation theory and the Franck-Condon approximation [144, 145] are applied. Putting all these approximations together the reaction rates between the electronic state i, the electronic state j and the respective states I and J of the nuclei can be written as [146]

kij = Aijfij

(6.9)

where

A = 2π ℏ-1|⟨ψ |V ′|ψ ⟩|2 ij i j

(6.10)

is the electronic matrix element, V ′ is the adiabatic perturbation operator, ⟨ψ_i| is the electronic wave function corresponding to state i. In Equation (6.9) f_ij is the so-called lineshape function which formally equates to,

( ) ∑ 2 fij = avgI |⟨ϕiI|ϕjJ⟩|δ(EiI - EjJ) , J

(6.11)

where the operator avg denotes the thermal average operator over all initial states of the nuclei I. The lineshape function describes the likelihood of a transition from the nuclei state I to J and the electronic matrix element describes the electronic transition probability. In the NMP theory both need to be non-vanishing in order to have a defect reaction. For the lineshape function to be non-vanishing the energies of the final and initial state need to be very close. In Equation (6.11) this has been approximated by the Dirac-Delta. Thus the Franck-Condon factor |⟨ϕ_iI|ϕ_jJ⟩|² is interpreted as a transition probability, determined by the overlap of the two vibrational wave functions ϕ_iI and ϕ_jJ (cf. Figure 6.8).

Figure 6.8:

Definition of the parameterized adiabatic potentials V _i and V _j. Upon parameterizing the adiabatic potential V _i(R), it is possible to analytically calculate the classic transition barriers ΔV _ij and the intersection points q₁ and q₂, where usually one intersection point dominates the lineshape function (here q₁).

The lineshape function, its formal derivation, numeric evaluation using data from density functional theory (DFT) calculations and subsequent approximations to Equation (6.11) have been extensively discussed previously [112]. For the following discussion the classic approximation of the line shape function is briefly summarized. Parameterizing the adiabatic potentials (cf. Figure 6.8) using the reaction coordinate concept [147] yields

V _i(q)	= c_i(q - q_i)² + d_i,	(6.12)
V _j(q)	= c_j(q - q_j)² + d_j,	(6.13)

where c is the curvature, d is the potential offset, q_i and q_j denote the positions of the adiabatic potentials (cf. Figure 6.8). Additionally, d_i -d_j is usually expressed as a function of ΔE = E -Et to fit into the conventions used when describing the energy bands in semiconductors. Using the parameterized potentials to find the intersection points (IP) q₁ and q₂ (cf. Figure 6.8) of the adiabatic potentials and the classic approximation of the line shape function in the high temperature limit [112] one obtains

√ -- ( ( ciq21-) ( -ciq22) ) -----ci-- ( ----exp----kBTL------- -----exp---kBTL------) fij = 2√ πkBTL- |ciq1 - cj(q1 - qj - qi)| + |ciq2 - cj(q2 - qj - qi)| ,

(6.14)

where q₁ and q₂ are the intersection points between V _i(q) and V _j(q). Depending on the actual parameters of the adiabatic potentials V _i and V _j, multiple further approximations are possible to obtain simpler formulae for the lineshape function. The barrier heights from the parameterized potentials, which have been used in the previous equation are

ΔV _ij	= ²	(6.15)
	= ²,	(6.16)

where the parametrisation with c_i∕c_j = R² and the Huang and Rhys factor Sℏω = c_i(q_i -q_j)², often found in the literature, has been employed.

Depending on the location of the intersection between V _i and V _j two cases of phonon coupling can be distinguished. When the dominant intersection point q₁ (cf. Figure 6.8) is located between the minima of the potentials (q_i < q₁ < q_j) the process is referred to as strong phonon coupling, otherwise it is referred to as weak phonon coupling [113].

Evaluation of Transition Rates in Semiconductors

To obtain useful transition rates for semiconductors, where a continuum of energy levels must be considered, an integration over the electronic energy is necessary. Employing the definition of the density of states (cf. Equation (2.7)) the rates for capturing a hole (or emitting an electron) from/to the valence or conduction band reads

k₁₂ =	ν^p ∫ _-∞^Evg(E)(1 - fp(x,E,t))A₁₂(E,xT)f₁₂(_ΔE)dE
	+ νⁿ ∫ _Ec^∞g(E)(1 - fn(x,E,t))A₁₂(E,xT)f₁₂(_ΔE)dE.	(6.17)

The rate for absorbing an electron (or emmiting a hole) from/to the valence or conduction band reads

k₂₁ =	ν^p ∫ _-∞^Evg(E)fp(x,E,t)A₂₁(E,xT)f₂₁(_ΔE)dE
	+ νⁿ ∫ _Ec^∞g(E)fn(x,E,t)A₂₁(E,xT)f₂₁(_ΔE)dE.	(6.18)

For repetitive evaluation it is often beneficial to approximate the energy barrier (Equation (6.15)) by expanding Equation (6.15) into a Taylor series about ΔE up to the quadratic term,

ΔV ₁₂

≈

(6.19)

Now, in order to evaluate the transition rates numerically, the electronic matrix elements A_ij need to be evaluated. The evaluation of these functions strongly depends on the quantities available from the employed transport model. Thus when using a Schrödinger-Poisson solver together with the transport model one can directly evaluate integrals of the electronic wave functions. For semi-classic transport models, such as SHE or the drift diffusion model, one needs to suitably approximate the electronic matrix elements. Assuming that the wave function of the trapped charge carrier is strongly localized around the defect site, one can approximate the electronic matrix element by an energy dependent tunneling coefficient and a prefactor, usually a WKB-approximation of charge carrier tunneling [113]. All factors, which are not dependent on energy can be pulled out of the integrand and are summarized in the parameter σ₀. Having a solution of the Boltzmann Transport equation, the integrals (cf. Equation (6.17) and Equation (6.18)) can be directly evaluated. However, for moment based transport models further assumptions regarding the density of states and the energy distribution functions are necessary. For drift diffusion models Maxwellian distribution functions and a parabolic band are usually assumed. With these approximations the non-radiative multiphonon transition rates read

k₁₂ =	σ₀ expv_th^ppexp
	+ σ₀ expv_thⁿnexpexp	(6.20)
k₂₁ =	σ₀ expv_th^ppexpexp
	+ σ₀ expv_thⁿnexp	(6.21)

6.2.3 Structural Relaxation

In [121] the strong temperature dependence of BTI was explicitly shown. Also in TDDS measurements an oxide field independent, but temperature dependent regime was found [105]. To account for these findings it was theorized that this is due to a structural relaxation of the nuclei forming the defect site after the non-radiative multiphonon transition took place. Due to structural relaxation the effective energy barriers for a subsequent NMP transition change. Such a mechanism can also explain the strongly temperature dependent recovery, where a high temperature accelerates the recovery.

Figure 6.9:

Schematic representation of the adiabatic potentials used to define transition rates for a defect with three states. During stress an NMP transition from state 1 to state 2′ is very likely (bold arrow). In relaxation, when there is a small oxide field, the reverse reaction (from 2′ to 1) is more likely. However, the likelihood for the defect to undergo a structural relaxation (red potentials) does not change with the oxide field (relaxation/stress). Once the defect has undergone a transition from state 2′ to state 2 (structural relaxation) it cannot, in this model, directly exchange charges with the substrate.

It was theorized and investigated by [148] that a dimer configuration or a hydrogen bridge can be broken by capturing a hole (or emitting an electron) by an NMP transition and that the resulting configuration can structurally relax such that the inverse NMP transition becomes less likely (cf. Figure 6.9). It is assumed that the process of structural relaxation is solely dependent on the temperature. Thus the transition rates are modelled using an Arrhenius law,

k_2′2 = ν₀ exp ( )
E2-′2-
- kBTL

and k_22′ = ν₀ exp ( )
-E22′
- kBTL

(6.22)

where E_2′2 and E_22′ are the constant barriers and ν₀ ≈ 10¹³s^-1. Structural relaxation was shown to be essential to describe DCIV experiments [149, 150] and the possible field independence of charge emission times τe found in TDDS data.

6.2.4 The four State NMP Model

NMP as a charge capture and emisson process as well as structural relaxation have been combined by [151] into a four state model (cf. Figure 6.11 and Figure 6.10), which was shown to successfully describe the capture and emission time constants found in TDDS experiments. The model is constructed such that a defect can in each state either exchange charges with the substrate or undergo structural relaxation. However, it is noteworthy that this model does not attempt to explain the step-heights of single defects. Instead the ΔVth contribution of each defect is attributed to the interaction between the charged defect and the random discrete dopants in the channel underneath the defect (cf. Chapter 4).

Figure 6.10:

Definition of the adiabatic potentials for the oxide trap model from [151]. The sketch shows a finite state diagram for a single defect close to the valence band. State 1 is the stable and electrically neutral precursor state. Via an NMP transition (intersection between blue and red parabolas) the defect goes into state 2′ and the defect becomes charged but remains meta-stable. A defect in state 2′ can, upon the emission of a hole, undergo a transition back to the neutral and stable precursor state 1. The charge carrier exchange processes are modeled using NMP theory.

Figure 6.11:

The oxide trap model from [151]. The figure shows a finite state diagram for a single defect, in which state 1 is the stable and electrically neutral precursor state. The charge carrier exchange processes with the substrate are modeled using NMP theory. Upon hole capture (red arrow from state 1 to state 2′) the defect becomes positively charged but remains meta-stable. A defect in state 2′ can, upon the emission of a hole, undergo a transition back to the neutral and stable precursor state 1. Alternatively, a defect in state 2′ can undergo the slow process of structural relaxation, become stable and stay positively charged (state 2). In state 2 the defect can either go back into state 2′ or can emit a hole, thus becoming electrically neutral and return to state 1′. In state 1′ the defect is neutral and can either undergo a transition into the stable precursor state 1 by structural relaxation or can capture a hole and thus change into the stable positively charged state 2.

In order to determine whether or not a defect is charged, a system of equations describing the state transitions per defect are required. To this end, the framework of first-order continuous-time Markov-Chains [143] can be directly applied [29]. A single defect can only be in one state at a time. More precisely X_i(t) is the random variable for state i at time t, which is exactly 1 if the defect is in the ith state and 0 otherwise. The condition that any defect has to be in any of its N states, where N is a finite integer number, can be expressed by

N∑ Xi(t) = 1. i

(6.23)

Next, the probabilities for state transitions need to be defined. Since a transition from i to j depends on the occupation probabilities X_i(t) and X_j(t), where X_i(t) = 1 and X_j(t) = 0, one can only define conditional probabilities. Formally, for an infinitesimal small time step dt the conditional transition probability from state i to j reads,

P(Xj (t+ dt) = 1|Xi(t) = 1 ) = kijdt + O(dt),wheredlitm→0 O(dt) = 0.

(6.24)

In the above equation k_ij are the transition rates, which have been covered in the preceding sections. Inserting Equation (6.24) into Equation (6.23) and defining that

pi(t) = P(Xi (t + dt) = 1|Xi(t) = 1)

(6.25)

is the probability for the defect to stay in a certain state, one obtains

p (t+ dt) = k dtp (t)- k dtp (t). j ij i ji j

(6.26)

Equation (6.26) can now be written as

∂tpj(t) = kijpi(t)- kjipj(t).

(6.27)

In addition Equation (6.23) can be reexpressed to

N ∑ p (t) = 1. i i

(6.28)

Equation (6.27) and Equation (6.28) yield N + 1 equations for N unknowns p_i(t). Since the system is overdetermined one can ommit a single equation from the equation system. The p_i(t) are occupancy probabilities, which determine the probability of the defect to be in state i. Thus the X_i(t) are at a single instance in time either 1 or 0 with the probability p_i(t). Sometimes one is only interested in the expecation values of the occupancies. Consequently, applying the expectation operator E{} to Equation (6.27) and Equation (6.28) one obtains

∂_tp_j(t) = k_ijp_i(t) - k_jip_j(t)	⇒ ∂_tf_j(t) = k_ijf_i(t) - k_jif_j(t),	(6.29)
E	⇒∑ _i^Nf_i(t) = 1,	(6.30)

where E{X_i(t)} = f_i(t), meaning that f_i(t) can have any value between zero and one. It is usually referred to as occupancy of state i (cf. Chapter 2).

Evaluation of the Four State Model for the Drift Diffusion Model

Putting all of the above together, one is able to write down the system of equations for the four state NMP model from Figure 6.11 and Figure 6.10. The full, time averaged system for the drift diffusion model reads,

k_12′	= σ₀λ(xT)v_th^ppexp + σ₀λ(xT)v_thⁿN_c exp,	(6.31)
k_2′1	= σ₀λ(xT)v_th^pN_v exp + σ₀λ(xT)v_thⁿnexp,	(6.32)
k_2′2	= ν₀ exp,k_22′ = ν₀ exp,	(6.33)
k_11′	= ν₀ exp,k_1′1 = ν₀ exp,	(6.34)
k_1′2	= σ₀λ(xT)v_th^ppexp + σ₀λ(xT)v_thⁿN_c exp,	(6.35)
k_21′	= σ₀λ(xT)v_th^pN_v exp + σ₀λ(xT)v_thⁿnexp,	(6.36)
1	= f₁ + f_2′ + f₂ + f_1′,	(6.37)
∂_tf₁	= k_2′1f_2′ + k_1′1f_1′- (k_12′ + k_11′)f₁,	(6.38)
∂_tf_2′	= k_12′f₁ + k_22′f₂ - (k_2′1 + k_2′2)f_2′,	(6.39)
∂_tf₂	= k_2′2f_2′ + k_1′2f_1′- (k_22′ + k_21′)f₂,	(6.40)
∂_tf_1′	= k_11′f₁ + k_21′f₂ - (k_1′1 + k_1′2)f_1′,	(6.41)

where ν₀ ≈ 10¹³∕s, λ(xT) denotes the tunneling coefficient and superscripts v and c have been added to account for electron and hole dependent parameters, respectively. This system of equations has more than ten parameters per band edge. One parameter set is the position of the defect in the oxide, where only the trap depth xT directly enters the equations, which together with the dopant positions determines the step-height by electrostatic interaction. The parameters mainly determining the capture and emission times are the energy barriers (cf. Figure 6.11), the trap depth xT, the Huang-Rhys factors Sℏω, S′ℏω′ as well as the curvature ratios R and R′. All other quantities in the equation system are either fixed values, such as the band weights N_c and N_v, or directly obtained from the transport model.

Evaluation of Trap Charge and Recombination Terms

The trapping model is also tightly coupled to the transport model via the trapped charge and charge recombination. The trapped charge Q_t is computed via

Q_t = ±∥q∥(X_2′ + X₂)orE{Q_t} = ±∥q∥(f_2′ + f₂),

(6.42)

where the sign of the charge depends on the type of trap, which can either be donor- or acceptor-like. The recombination per defect can be straightforwardly generalized from Section 2.2.2 and read

Γⁿ{fν n,fν p }	= ∑ _defects ∑ _ν ∫ k_12′p₁(1 - p_2′)(1 - fν n)g_n - k_2′1p_2′(1 - p₁)fν ng_nd³k,	(6.43)
Γ^p{fν n,fν p }	= ∑ _defects ∑ _ν ∫ k_2′1p_2′(1 - p₁)(1 - fν p )g_p - k_12′p₁(1 - p_2′)fν p g_pd³k,	(6.44)

for an acceptor-like defect. For a moment based transport model the above relations simplify to

Rⁿ	= ∑ _defectsk_12′p₁(1 - p_2′)n - k_2′1p_2′(1 - p₁)n,	(6.45)
R^p	= ∑ _defectsk_2′1p_2′(1 - p₁)p - k_12′p₁(1 - p_2′)p.	(6.46)

Evaluation of Capture and Emission Times

For comparison with experiment, the capture and emission times τc and τe need to be calculated. Since the four state NMP model has been derived using first-order continuous-time Markov-Chain theory one can use the concept of first passage times from the theory. In the four state model, there are two pathways for charge capture and emission. Charge capture either proceeds via state 1′ or 2′. Thus, following [113], we define the capture and emission times as

τc

τe

(6.47)

where the first passage times τc(1′), τe(1′) and τc(2′), τe(2′) are

τc(1′)	= ,	τc(2′)	= ,	(6.48)
τe(1′)	= ,	τe(2′)	= .	(6.49)

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