C Diffusion-Limited Stress Phase of the Reaction-Diffusion Theory

Appendix C
Diffusion-Limited Stress Phase of the Reaction-Diffusion Theory

In the standard reaction-diffusion theory the kinetic rate equation describing the interface reaction via a hydrogen species $Xit$ [56, 57, 58] in

∂Nit- 1∕a ∂t = kf(N0 − Nit)− krNitXit ,

(C.1)

with $Nit$ and $N0$ as unpassivated and total amount of interface states. Thus $N0 − Nit$ denotes the concentration of passivated interface defects depassivating with the rate $kf$ . The passivation rate $kr$ of the dangling bonds also depends on the hydrogen species $Xit$ with its kinetic exponent $a$ ( $1$ for $H0$ and $H+$ , and $2$ for $H2$ ) [172]. Assuming the quasi-equilibrium regime of the interface reaction ( $∂N ∕∂t ≈ 0 it$ ) as the dominant regime after [59, 66, 17, 71], the rate equation (C.1) can be rewritten as

( ) kfN0-−-Nit a Xit = kr Nit .

(C.2)

The boundary value problem for $Xit$ is as follows:

∂X (x,t) ∂2X (x, t) ∂X (x,t) ---it---- = DX ---it2----± μEox ---it----. ∂t ∂x ∂x

(C.3)

When neglecting charged hydrogen ( $H+$ ), the drift term inside the drift-diffusion process (C.3) vanishes. The remaining diffusion process will be approximately solved on basis of a triangular hydrogen profile¹ [85, 59], as depicted in Fig. C.1 (right).

After the continuity equation the interface states additionally created, $ΔNit (t)$ , are due to the leaving hydrogen $Xit(t)$ , which is shown in Fig. C.1 (middle). The corresponding diffusion front is given by $√----- DX t$ . Comparing with Fig. C.1 (right) yields the area confined by the diffusion front on the one hand and the number of $aXit$ at the interface at the other hand which equals

√ ----- aXit(t)--DX-t-= ΔNit (t). 2

(C.4)

The ratio between the diffusing hydrogen species and resulting interface states is determined by the kinetic exponent $a$ , i.e. each $H2$ leaves $2$ dangling bonds.

To solve equations (C.2) and (C.4) the following assumptions are made: (i) The amount of passivated interface states $N0$ is much larger than the initial value of $Nit$ , $Nit,0$ , and (ii) $ΔNit (t) ≫ Nit,0$ . A schematic picture of the interface shows all necessary quantities and is relations (Fig. C.1 (left)). Inserting these assumptions into (C.2) and comparing with (C.4) gives

( )a 2ΔNit-(t) kf---N0-- a√ DX t ≈ krΔNit (t) .

(C.5)

The approximated number of $ΔNit$ is then

( k ) aa+1 ( a) a+11 --1-- ΔNit (t) = -fN0 -- (DXt)2(a+1) . kr 2

(C.6)

By using atomic hydrogen (a=1) this term simplifies to

∘ ------ ΔNit(t) = kfN0-(DXt )14 , kr 2

(C.7)

while molecular hydrogen (a=2) yields

2 ( kf ) 3 1 ΔNit(t) = k-N0 (DXt )6 . r

(C.8)

Alternatively (C.4) can be formulated via the flux of the hydrogen profile (the gradient right at the interface) and yields a first-order differential equation in time to solve. The results differing by a constant prefactor from the algebraic expressions in (C.6) are summarized in [71].

Figure C.1: Left: For the diffusion-limited part of the RD theory the interface is considered to be nearly fully passivated at $t0$ , i.e. $N0 ≫ Nit,0$ . The additionally created dangling bonds $ΔNit$ are furthermore assumed to dominate the total number of dangling bonds ( $Nit,0 + ΔNit (t) ≈ ΔNit (t)$ ). Note that in fact the $Si–H$ and the silicon dangling bonds are not arranged in two groups like schematically depicted here, but are randomly distributed. Right: Hydrogen profile inside the oxide during a diffusion process. The area under the hydrogen profile with its progressing diffusion front $√ ---- DXt$ equals the number of additionally generated interface states $ΔNit (t)$ given by relation (C.4).

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Appendix CDiffusion-Limited Stress Phase of the Reaction-Diffusion Theory

Appendix C
Diffusion-Limited Stress Phase of the Reaction-Diffusion Theory