Diffusion-drift-reaction model for positron trapping and annihilation at interfaces with space charge in bulk materials

ABSTRACT

The exact solution of a diffusion-drift-reaction model for the trapping and annihilation of positrons at interfaces with adjacent space-charges in bulk materials is presented. Closed-form expressions are obtained for the mean positron lifetime and for the intensity of the positron lifetime component associated with trapping at interfaces. The exact solutions can be conveniently applied for the analysis of experimental data. Compared to available thin-film models, the present approach takes into account that drift usually occurs only in a part of the diffusion zone. The model clearly reveals that for typical drift-zone parameters, positron drift has a substantial impact on the positron annihilation characteristics, that is, disregarding existing drift could lead to misjudgement of experimental positron annihilation data.

KEYWORDS:

• Positron annihilation
• positron diffusion
• positron drift

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Here, the case is considered where $e^{+}$ trapping occurs from one side of the interface. For trapping from both sides, the number density reads: $1/\left(2x_2\right)$.

2 The same is true for the series of fast decay rates ${\lambda }_{0,j}$, which follow from the sequence poles of Equation (26). The corresponding transcendental equation ${\gamma }^{\star }{\hat{x}}_2\tan \left({\gamma }^{\star }{\hat{x}}_2\right)=\alpha {\hat{x}}_2/D$ is identical to that given in [8].

3 Of course, the model is applicable to any other set of parameters as well (see below).

4 $v=3\times {10}^4$ ms${}^{-1}$, ${\tau }_f=220$ ps, ${\tau }_t=270$ ps, $D=3\times {10}^{-4}$ m${}^2$s${}^{-1}$, $\alpha =3\times {10}^3$ ms${}^{-1}$, $x_0=0$, $x_1=30$ nm.

5 Note: For $p\to -{\tau }_f^{-1}$: $\delta =\text{β}$ and, therefore, $(\exp (-\text{β}{x}_1\left)\sqrt{1-{\tanh }^2\left(\delta x_1\right)}\right)$ in FN reduces to $(1-\tanh (\text{β}{x}_1\left)\right)$.

6 That the pole is removable was explicitly shown for the model on diffusion- and reaction-limited trapping at voids [9]; see the Appendix for [9] under https://arxiv.org/abs/1904.07493.

7 W corresponds to $x_1$ in the present model.

8 It should be noted that the absolute value of the drift velocities as well as the depletion width between n- and p-type doping vary only by a few %. Thus, for a better comparison, the same absolute values, corresponding in good approximation to both situations, have been chosen.