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Dynamics of Charge Carriers

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Quantitative Recombination and Transport Properties in Silicon from Dynamic Luminescence

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Abstract

This chapter introduces the essential concepts of carrier recombination and transport in silicon. The differential equations of interest are derived and solutions are given—particularly focusing on harmonically modulated excess carrier generation.

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Notes

  1. 1.

    In spite of their frequent occurence in literature, both terms are ambiguous. While the term continuity equation is likewise used to express both local and temporal conservation of particles, probability, or electrical charge (cf. Eq. 3.13), the term diffusion equation is also used as a synonym for the heat equation which is equivalent to Fick’s second law of diffusion. It will be outlined in this chapter that the continuity equation describing the dynamics of excess carriers in a semiconductor is a special case of the general inhomogeneous diffusion equation.

  2. 2.

    Note that the focus of the present work lies in the measurement and interpretation of total recombination rate rather than in an interpretation in terms of an isolated recombination mechanism.

  3. 3.

    Note that the notation for carrier densities according to Würfel [5] is adhered throughout this work, as it facilitates indexing.

  4. 4.

    If not otherwise specified, the denotation \(\Delta n\) implies equality of excess electron and hole densities.

  5. 5.

    Throughout this work, boundary conditions are solely represented by interface recombination. In the case of electrical current injection to be discussed in Sect. 7.4, carrier recombination and injection through the interface can be accurately separated in most cases. Thus, the problem can be treated with the same boundary conditions as introduced here.

  6. 6.

    Inverse light penetration depth and the band-to-band absorption coefficient do not coincide if significant competing light absorption processes exist. Note that—although in literature the notation of the band-to-band absorption coefficient may differ from the notation of the total absorption coefficient [17, 18]—\(\alpha _{\gamma }\) denotes the band-to-band absorption coefficient throughout this work.

  7. 7.

    Note that for decay time measurements featuring a weighted depth-dependent sensitivity with respect to excess carrier density (e.g. due to luminescence reabsorption or high-level injection measurement conditions) or depth-resolved decay time measurement techniques, the odd mode may not generally be negligible.

  8. 8.

    The condition \(\alpha _{\gamma }d\gg 1\) is also denoted as shallow excess carrier generation in the following.

  9. 9.

    Treatment analogous to [5].

  10. 10.

    The chosen nomenclature is adapted from [29] for the most part.

  11. 11.

    There are diverging perceptions of a quasi-steady-state condition, as will be discussed in Sect. 5.1. Here, quasi-steady-state denotes negligible derivatives with respect to time.

  12. 12.

    Such techniques are denoted as additive here, with an amplitude subscript \(+\).

  13. 13.

    Amplitude-sensitive shall denote the interpretation of a signal amplitude in terms of carrier lifetime via models, assumptions, and calibrations, rather than to exploit the time dependence of a measured amplitude.

  14. 14.

    multiplicative techniques, with an amplitude subscript \(\times \).

  15. 15.

    Dynamic measurement artifacts caused by trapping are not caused by the mismatch between \(\Delta n_e\) and \(\Delta n_h\). Therefore, excess carrier densities are again denoted as \(\Delta n\) in the following.

  16. 16.

    The equality of lifetime and harmonically modulated time shift only holds in case of injection-independent carrier lifetime. This is fully derived in Sect. 6.2.

  17. 17.

    If such a quasi-steady-state condition is not fulfilled, time shifts become smaller than carrier lifetime. This is actually visible on the left of Fig. 3.7: at a ratio \(\mathcal {T}/\tau =100\) the time shift (without trapping artifact) is already measurably lower than true lifetime (cf. Fig. 3.8). Note that a self-consistent lifetime analysis would yield exact lifetime, even if a quasi-steady-state condition was violated (cf. Sect. 5.1).

  18. 18.

    Here, the quasi-steady-state condition denotes a significant variation of excess carrier generation rate only on a timescale much greater than excess carrier recombination lifetime (cf. Sect. 5.1).

  19. 19.

    The transmitted photon current density corresponds to the impinging photon current density \(j_{\gamma ,i}\) multiplied by the substrate interface transmission \(1-R_{\gamma }\).

  20. 20.

    For \(L = \alpha _{\gamma }^{-1}\), the solution given in Eq. 3.64 is not defined. However, this appears to be an irrelevant restriction for virtually any practical application, as this particular solution is valid for an arbitrarily small \(|L- \alpha _{\gamma }^{-1}|\).

  21. 21.

    The depth profile of excess carrier density in Eq. 3.74 may be additionally weighted due to the sensitivity of the applied measurement technique.

  22. 22.

    This form of a quasi-steady-state condition will recur throughout this work.

  23. 23.

    This relation will prove inaccurate (even if the above stated quasi-steady-state condition is satisfied) for injection-dependent lifetime. This is due to the fact that harmonically modulated lifetime measurements are differential measurements (cf. Sect. 6.2).

  24. 24.

    Note that this very short digest of literature does not claim to be comprehensive.

  25. 25.

    The boundary condition Eq. 3.20 is apparently invariant under the transformation \(\Delta n\rightarrow \Delta n^{\prime }\).

  26. 26.

    Note the retransformation \(\Delta n = e^{-\frac{t}{\tau }}\Delta n^{\prime }\), which is incorporated here.

  27. 27.

    This treatment is not restricted to harmonic time modulation of excess carrier generation rate.

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  1. Fraunhofer Institut für Solare Energiesysteme ISE, Freiburg, Germany

    Johannes Giesecke

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Giesecke, J. (2014). Dynamics of Charge Carriers. In: Quantitative Recombination and Transport Properties in Silicon from Dynamic Luminescence. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-06157-3_3

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