Subsections

• 4.2.1 Interaction of Dopants
• 4.2.2 Continuity Equations

4.2 Five-Stream Dunham Diffusion Model

Dunham presented 1992 a general model [81] for the coupled diffusion of dopants with point defects, which includes the reaction of dopant-defect pairs with defects and other pairs, as well as all posible charge states for both dopants and pairs. It consists of five streams, because the comprehensive modeling of dopant behavior requires five differential equations, each treating a different concentration stream: one for the dopant atoms, two for the interstitial and vacancy point-defects, and two for the dopant-vacancy and dopant-interstitial pairs [24].

4.2.1 Interaction of Dopants

In silicon a dopant diffuses via interactions with point-defects, which can be described by a set of reactions. First, there are the dopant-defect pairing reactions

$$A^+ + I^i \Longleftrightarrow (AI)^{i+1},$$ (4.2)

$$A^+ + V^i \Longleftrightarrow (AV)^{i+1},$$ (4.3)

where $A^+$ represent the ionized dopant atoms, $I$ and $V$ represent the interstitials and vacancies, $(AI)$ and $(AV)$ represent the dopant-defect pairs, and $i$ stands for the charge state of the defect or pair as $-$, 0, $+$. Next, the recombination and generation of Frenkel pairs must be considered

$$\vspace*{-8mm} I^i + V^j \Longleftrightarrow (-i-j) e^-,$$ (4.4)

where $e$ are electrons. A Frenkel pair is a vacancy-interstitial pair formed when an atom is displaced from a lattice site to an interstitial site.

Additionally, the pairs can interact directly with the opposite type defect to produce a reaction which is equivalent to a pair dissociation followed by defect recombination

$$(AI)^i + V^j \Longleftrightarrow A^+ + (1-i-j) e^-,$$ (4.5)

$$(AV)^i + I^j \Longleftrightarrow A^+ + (1-i-j) e^-,$$ (4.6)

Finally, two opposite type pairs can recombine leaving two unpaired dopant atoms

$$(AI)^i + (AV)^j \Longleftrightarrow 2A^+ + (2-i-j) e^-.$$ (4.7)

The last three reactions provide an alternative path for the recombination and generation of vacancies and interstitials with the potential for a significant increase of the effective recombination rate for Frenkel pairs.

4.2.2 Continuity Equations

The five continuity equations for the total concentrations $C_X$ (X stands for A$^+$, AI, AV, I, or V), over all charge states for a single donor species are [82]

$$\frac{\partial C_{A^+}} {\partial t}=-R_{AI}-R_{AV}+R_{AI+AV}+2R_{AI+AV},$$ (4.8)

$$\frac{\partial C_{I}} {\partial t}=-\nabla J_{I} +R_{AI}-R_{I+V}-R_{AV+I},$$ (4.9)

$$\frac{\partial C_{V}} {\partial t}=-\nabla J_{V} +R_{AV}-R_{I+V}-R_{AI+V},$$ (4.10)

$$\frac{\partial C_{AI}} {\partial t}=-\nabla J_{AI} +R_{AI}-R_{AI+V}-R_{AI+AV},$$ (4.11)

$$\frac{\partial C_{AV}} {\partial t}=-\nabla J_{AV} +R_{AV}-R_{AV+I}-R_{AI+AV}.$$ (4.12)

$R_{AI}$ and $R_{AV}$ are the net rates of the dopant-defect pairing reactions (4.2) and (4.3) as defined in [81]:

$$R_{AI}=\Big[\sum_i k_{AI}^i K_I^i \Big(\frac{n_i}{n}\Big)^i\Big]\Big[C_{A^+}C_{I^0}-\frac{C_{(AI)^+}}{K_{A+I}^0}\Big],$$ (4.13)

$$R_{AV}=\Big[\sum_i k_{AV}^i K_V^i \Big(\frac{n_i}{n}\Big)^i\Big]\Big[C_{A^+}C_{V^0}-\frac{C_{(AV)^+}}{K_{A+V}^0}\Big].$$ (4.14)

$k_X$ are the forward reaction rate coefficients and $K_X$ are the equilibrium constants. $n$ and $n_i$ are the local and intrinsic carrier concentrations.

The net rate $R_{I+V}$ of Frenkel pair recombination (4.4) is [81]

$$R_{AV}=\Big[\sum_{i,j} k_{I+V}^{i,j} K_I^i K_V^j \Big(\frac{n_i}{n}\Big)^{i+j}\Big]\big[C_{I^0}C_{V^0}-C_{I^0}^*C_{V^0}^*\big],$$ (4.15)

where $^*$ indicates equilibrium values.

Finally, $R_{AI+V}$, $R_{AV+I}$, and $R_{AI+AV}$ are the net rates of the pair-defect (4.5) (4.6) and pair-pair reactions (4.7) [81]:

$$R_{AI+V}=\Big[\sum_{i,j} k_{AI+V}^{i,j} K_{AI}^i K_V^j \Big(\frac... ...)^{i+j}\Big]\big[C_{(AI)^+}C_{V^0}- K_{A+I}^0 C_{I^0}^* C_{V^0}^*C_{A^+} \big],$$ (4.16)

$$R_{AV+I}=\Big[\sum_{i,j} k_{AV+I}^{i,j} K_{AV}^i K_I^j \Big(\frac... ...)^{i+j}\Big]\big[C_{(AV)^+}C_{I^0}- K_{A+V}^0 C_{I^0}^* C_{V^0}^*C_{A^+} \big],$$ (4.17)

$$R_{AI+AV}=\Big[\sum_{i,j} k_{AI+AV}^{i,j} K_{AI}^i K_{AV}^j \Big(... ...C_{(AI)^+}C_{(AV)^+}- K_{A+I}^0 K_{A+V}^0 C_{I^0}^* C_{V^0}^*(C_{A^+})^2 \big].$$ (4.18)

The continuity equations (4.30)-(4.30) also need the fluxes of mobile dopants, defects, and pairs. The total flux of interstitials is [81]

$$J_I=-\Big[\sum_{i} D_{I^i} K_{I}^i \Big(\frac{n_i}{n}\Big)^{i}\Big] \nabla C_{I^0},$$ (4.19)

where $D_{I^i}$ represents the diffusivity of interstitials of charge state $i$. Similarly, the total vacancy flux is [81]

$$J_V=-\Big[\sum_{i} D_{V^i} K_{V}^i \Big(\frac{n_i}{n}\Big)^{i}\Big] \nabla C_{V^0}.$$ (4.20)

The total pair fluxes are [81]

$$J_{AI}=-\Big[\sum_{i} D_{AI^{i+1}} K_{AI}^i \Big(\frac{n_i}{n}\Bi... ...)^+}+ C_{(AI)^+} \Big(\frac{n_i}{n}\Big) \nabla \Big(\frac{n}{n_i}\Big) \Big],$$ (4.21)

$$J_{AV}=-\Big[\sum_{i} D_{AV^{i+1}} K_{AV}^i \Big(\frac{n_i}{n}\Bi... ...^+}+ C_{(AV)^+} \Big(\frac{n_i}{n}\Big) \nabla \Big(\frac{n}{n_i} \Big) \Big],$$ (4.22)

where $D_{AI^i}$ and $D_{AV^i}$ are the diffusivities of dopant-defect pairs with charge $i$.