Rate Equations

8.1 Rate Equations

Based on the existence of oxide defects and the band-to-trap transition possibilities, depicted in Fig. 8.1, already a single defect system has to consider all transitions originating from various band states. This means that the whole conduction or valence band has to be considered, instead of only $Ec$ or $Ev$ . On the basis of the statistical description of the recombination of electrons and holes under the release of energy in terms of lattice vibrations (Shockley-Read-Hall theory [122]), the determination of effective rates in and out of a specific defect system is possible. The corresponding rate equations are

∫ ∞ ∂tfT = [(1 − fT) fFD(E)cn(E )− fT (1 − fFD(E ))en(E)]Dc (E )dE (8.1 ) ∫Ec Ev = [(1− fT)fFD(E )ep(E )− fT (1− fFD (E))cp(E )]Dv (E )dE, (8.2 ) −∞

with the trap occupancy in the oxide $fT$ and the Fermi-Dirac distribution $fFD$ , which represents the probability of an occupied quantum state in the substrate. Since the Fermi-Dirac distribution is valid in thermal equilibrium and still a very good approximation during BTI, as there is nearly no channel current [129, 130], the distributions write as $fFD = (1 + exp(β(E − Ef )))−1$ and $f = (1 + exp(β(E − E )))−1 T T f$ . The quantities $c n$ , $e n$ , $e p$ , and $c p$ stand for the coefficients of electron capture, electron emission, hole emission, and hole capture. The density of states (DOS) is split into a conduction band part $Dc$ and a valence band part $Dv$ .

Figure 8.2: Left: The rate equations are described on basis of multiple traps in an oxide, charge carriers in an n-substrate, and the corresponding capture and emission coefficients. Right: The transition barriers of a hole capture and an electron emission process are equivalent and consist of the trap energy difference according to the state in the substrate $ΔET$ and an additional barrier $ΔE B$ .

Assuming detailed balance [122], which means that each process is balanced by its reverse process, both rates have to equal within (8.1) and (8.2). This yields

en(E-)-= cp(E)-= eβ(ET−E ). cn(E ) ep(E)

(8.3)

Combining (8.3) with (8.2)¹ and evaluating the integral finally gives the capture time constant $τc,p$ of the holes

∫ Ev ( ) ∂tfT = (1− fT )fFD(E )ep(E)-− fT(1 − fFD(E )) cp(E )Dv (E )dE −∞ cp(E) ( ) ∫ Ev = (1 − fT)e−β(E−Ef)e−β(ET−E ) − fT (1 − fFD(E )) cp(E )Dv(E )dE ( ) −∞ = (1 − fT)eβ(Ef− ET) − fT σpvth,pp, (8.4)

with the cross section $σp$ and thermal velocity $vth,p$ of the holes with density $p$ . The term outside the brackets can be identified as the capture rate, which can be seen when (8.4) is compared to the simple rate equation of a two-state defect

∂tfT = (1 − pT(t))kf − pT (t)kr

(8.5)

with the rate $kf$ to fill the defect at $ET$ and $kr$ for the reverse rate. Furthermore, $pT$ gives the probability that the defect is actually filled. Consequently, the capture and emission rates can be written as

kc,p = 1∕τc,p = σpvth,pp (8.6) ke,p = 1∕τe,p = σpvth,ppeβ(Ef− ET) (8.7)

or as the relation

1 1 ----= eβ(Ef− ET)---. τe,p τc,p

In addition to a tunneling coefficient of $exp(− xT∕x0)$ to account for the oxide trap depth after [121], the cross section is considered to be thermally activated with a bias independent barrier $ΔEB$ [124]. Putting these assumptions together yields

σp = σp,0e−xT∕x0e−βΔEB,

(8.8)

with a constant prefactor $σp,0$ [131, 124]. With the knowledge that whether the defect level lies below or above $Ev$ , different barriers are obtained after Fig. 8.1, equations (8.6) to (8.8) are now used to calculate the capture rates

{ −β(Ev−ET ) 1∕τc,p = σp,0vth,ppe−xT∕x0e− βΔEB e ET < Ev (8.9) 1 ET > Ev { −β(E −E ) 1∕τ = σ v pe−xT∕x0e− βΔEB e v f ET < Ev (8.10) e,p p,0 th,p e−β(ET− Ef) ET > Ev.

As thermal equilibrium is assumed and the density of states is low enough to rule out quantum effects, the Fermi-Dirac-distribution can be replaced by the Maxwell–Boltzmann-distribution [10]

−β(Ef− Ev) p ≈ pMB = Nve

(8.11)

with $Nv$ as effective valence band weight,

( )3∕2 Nv = 2 mdpkBT-- . 2πℏ2

The trapping barrier $ΔET$ can further be written as a superposition of the energy distance during flatband $ΔET,0 = ET,0 − Ev,0$ and the applied field $Fox$ which changes the relative barrier between semiconductor and oxide, cf. Fig. 8.1 (right) and Fig. 8.2 (right)

ΔET (Fox) = ET (Fox)− Ev (Fox) = ΔET,0 − q0FoxxT.

(8.12)

With the help of (8.11) and (8.12) the time constants in (8.9) and (8.10) finally read as

{ βΔET,0 −βq0FoxxT 1∕τc,p ≈ σp,0vth,ppMBe−xT∕x0e−βΔEB e e , ET < Ev (8.13) 1, ET > Ev { 1∕τe,p ≈ σp,0vth,pNve −xT∕x0e−βΔEB 1, ET < Ev (8.14) e−βΔET,0eβq0FoxxT. ET > Ev

At first only the part of (8.13) and (8.14), which depends on the relative position of $ET$ to $Ev$ is discussed. The temperature dependence here is dominated by the thermal barrier $ΔET,0$ . While the barrier $ΔET,0$ determines hole capture when $ET < Ev$ holds, the barrier $− ΔET,0$ contributes to hole emission only when $ET > Ev$ . So the barriers are either relevant for $τc,p$ or $τe,p$ and do not affect both rates. This is due to the relative position of the energetic defect level and its reservoir, as depicted in Fig. 8.1 (left). When looking at the term $exp(±βq0FoxxT )$ , it can be seen that the applied field either lowers or rises the barrier, but again the field dependence is only included in either $τc,p$ or $τe,p$ . Additional bias dependencies arise from the surface hole concentration, especially below $VTH$ , and the tunneling coefficient [130].

In a typical BTI stress/relaxation sequence all defects are in thermal equilibrium prior to stress. Due to stress the Fermi level $Ef$ is shifted below $Ev$ . For defects with $ET < Ev$ the resulting barrier $ΔET,0$ can only be balanced by the $Fox$ term in (8.13). After (8.12) this means that energetically deeper defects also need to be located deeper in the oxide in order to become charged during stress, i.e. only defects with $ET > q0ψs,str − q0Fox,strxT$ , where $ψs,str$ denotes the potential at the interface, are accessible during stress [130]. When the stress is completely removed, $E f$ is shifted back above $Ev$ and the previously charged defects will by moved back below $Ef$ . According to (8.14) they can be emptied over a small barrier if there is any. Thereby accessible oxide defects now feature $ET < q0ψs,rel − q0Fox,relxT$ during relaxation [130]. Thus, the exact defect level is not of particular interest for the capture and emission process. $ET$ only has to lie inside the accessible energy region, i.e. above $Ev$ for stress and below $E v$ for relaxation. This means that the conditional part of (8.13) and (8.14) only exhibits a small temperature and field dependence.

It is important to realize that it is the thermal barrier $ΔEB$ in (8.13) and (8.14) introduced by Kirton and Uren, which gives the required temperature dependence, though this dependence is not fully correct, as will be shown later. To first order, the capture $τc,p$ and emission times $τe,p$ of the defects are determined by $xT$ and $EB$ , making another fact visible: $τ c,p$ and $τ e,p$ are correlated, while measurement results determining these times during BTI revealed uncorrelated behavior [111, 116]. This rules out the possibility of describing oxide defects by an extended SRH theory.

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