The authors of the above paper call into question recent evidence on the properties of self-interstitials, I, in Ge [Cowern *et al*., Phys. Rev. Lett. **110**, 155501 (2013)]. We show that this judgment stems from invalid model assumptions during analysis of data on B marker-layer diffusion during proton irradiation, and that a corrected analysis fully supports the reported evidence. As previously stated, I-mediated self-diffusion in Ge exhibits two distinct regimes of temperature, *T*: high-*T*, dominated by amorphous-like mono-interstitial clusters—i*-*morphs—with self-diffusion entropy ≈30 *k,* and low-*T*, where transport is dominated by simple self-interstitials. In a transitional range centered on 475 °C both mechanisms contribute. The experimental I migration energy of 1.84 ± 0.26 eV reported by the Münster group based on measurements of self-diffusion during irradiation at 550 °C < T < 680 °C further establishes our proposed i-morph mechanism.

A recent *Applied Physics Review* has discussed self-diffusion and B diffusion during irradiation at high temperature.^{1} Unfortunately, erroneous assumptions in B diffusion data analysis led the authors mistakenly to critique recent work by us that identified two forms of self-interstitial in Ge.^{2} Here, we show that their work when correctly interpreted *confirms* our conclusions. In the following discussion, *C*_{X} and *D*_{X} represent the concentration and diffusivity of species X, *E*_{f}^{X} and *E*_{m}^{X} (*S*_{f}^{X} and *S*_{m}^{X}) its formation and migration energies (entropies), respectively, and *D*_{SD}^{X }= *D*_{X}*C*_{X}^{eq}/*C*_{0}, where C_{0} is the lattice density, is the contribution of X to equilibrium self-diffusion with activation energy *E*_{SD}^{X} and activation entropy *S*_{SD}^{X}. The species described are the vacancy, V, self-interstitial, I, and B-interstitial pair, BI, and we consider two distinct forms of I; *I* and I. The first is the compact I, well known from the literature, the second is the *i-morph—*an extended self-interstitial with properties of a small amorphous pocket.^{2} This entity, in some ways reminiscent of the high-temperature “liquid drop” proposed by Seeger,^{3,4} however, needs to be understood from a fundamentally different perspective; the key feature is an *amorphous*-like property with corresponding energetic and entropic behavior.^{2}

Our evidence has been disputed by Bracht and coworkers.^{1,5} Their objection appears to be based on (a) a misunderstanding of our analysis of long-range BI migration in Ref. 2, (b) an erroneous analysis of BI mediated B diffusion in Ref. 1 and a precursor paper.^{6} To clarify the issues, we first briefly review the disputed analysis of Ref. 2. B diffuses in Ge, as in Si,^{7} via a fast migrating BI pair formed by the reaction B_{s} + I ↔ BI.^{8} A parallel reaction, B_{s} ↔ BI + V, also occurs but has no significant influence under the conditions of Refs. 1 and 2. BI in Ge, as in Si, has a large migration length, *λ—*a quantity closely connected to the difference in Gibbs free energy between BI and I.^{2} This leads to exponential diffusion tails after anneal times short enough that only a fraction of C_{Bs} experiences a reaction with I to form BI. This behavior occurs under both equilibrium and irradiation conditions, with *λ* independent of I supersaturation.^{7,8}

To analyse this behavior, the diffusion of I, BI, and V can be modeled by numerical solution of the coupled equation system as in Refs. 1 and 6 and elsewhere. Under certain conditions, the full system can be reduced to one equation which has an analytical solution involving just *g*, *λ*, position, and time*—*the *g-λ* solution.^{7,9} This is a mathematical approximation to the full equation system that describes the detailed properties of dopant diffusion and is applicable under equilibrium and non-equilibrium conditions. The necessary and sufficient conditions for *accuracy* of this approximation are that *C*_{BI} ≪ *C*_{Bs}, the Fermi level at the diffusion temperature, *T*, is slowly varying in the local region of interest, and there are no significant gradients in *C*_{I} or *C*_{V}. The latter condition prevails if *D*_{B}*C*_{B} ≪ *D*_{I}*C*_{I}^{eq}, and no significant gradients are generated by external processes. It is *not* necessary to assume point-defect equilibrium. When the preceding inequality is relaxed towards *D*_{B}*C*_{B} ∼ *D*_{I}*C*_{I}^{eq}, the rate of the reaction I + B_{s} → BI is slightly modified by “chemical pump” effects but the *g-λ* solution still accurately describes *λ*.^{10}

In the experiments of Ref. 2, all these conditions were satisfied, so the *g-λ* approach could be used to extract accurate *λ* values from our experimental secondary-ion mass spectrometry (SIMS) profiles, thus avoiding the costly use of a general diffusion solver as kernel in least-squares minimization. Following this analysis, we deduced a *T*-dependent free-energy difference between BI and I, indicating that the latter has two distinct forms. The first, dominating I-mediated transport at low *T*, is the simple I. The second, dominating *D*_{SD}^{I} at high *T*, is an extended I; the i-morph.^{2} A transitional region, where both defect forms contribute, exists over a ∼100 °C range centered on 475 °C. Reference 1 cites this analysis, claiming in error that the *g*/*λ* approach is inapplicable under non-equilibrium conditions.

We now consider the analysis of B diffusion in Refs. 1 and 6. Fig. 17 of Ref. 1 presents B profiles in Ge measured by SIMS after proton irradiation of a B-doped Ge superlattice at 550 and 630 °C. To show clearly the detailed B profile shape evolution during diffusion, we have selected and plotted the data for a single marker layer in Fig. 1. The profile shows characteristic exponential-like tails (showing up as almost straight lines on the logarithmic plot of Fig. 1) on each side of the B-doped marker layers. The curves turn up at the edges of the plot owing to overlap of diffusion from neighboring markers. The data at 550 °C show significantly more diffusion than at 630 °C, because at lower *T* both *λ* ( = ${DBI/(k1+C0)}1/2$ in Bracht's notation) and the forward reaction rate *g* ( = $k1\u2212CI$) are larger. The larger *λ* reflects the increased number of BI diffusion jumps per migration event as the thermal energy available for dissociation, BI → B + I, is reduced. The larger *g* reflects the increased number of lattice sites each beam-generated I visits before recombining with V. The static peak represents those B_{s} which have not yet undergone reaction (1)—a statistical effect due to the finite arrival rate of I at B_{s}.

The approach taken in Ref. 1, following Ref. 6, fails to recognize and model these key effects. This seems to be caused by unrealistic assumptions (a) on B clustering during annealing of initially substitutional B, (b) that C_{BI} ≫ C_{Bs}. In relation to point (a), Ref. 6 assumed *a priori* that, at all considered anneal times, *t*, each B marker had a large clustered component, adjusted for each *T*/*t* combination to keep *C*_{Bs} ≤ 5 × 10^{18 }cm^{−2}. This ignores the transient dynamics of B clustering in the MBE-grown doping structure as BI migrates and traps on other B atoms. In the simulations of Ref. 6 (Fig. 17), clustering, unrealistically, actually *decreases* with time. Point (b) is a result of assuming S_{f}^{BI }≈ 30 *k—*a problematic choice as the entropy of *D*_{B}, S_{f}^{BI }+ S_{m}^{BI }≈ 20 *k*^{11} and negative S_{m}^{BI} is highly unlikely. It is unclear why such a large S_{f}^{BI} has been used, unless it is to prevent *D*_{B} from varying as (*p*/*n*_{i})^{2} as the model assumes BI is in a singly positive charge state.^{6} The result of these several choices is that all the simulated profiles have Gaussian shapes at low B concentration (blue curves in Fig. 1). This is a poor fit to the data, which show a clear exponential-like trend, thus directly demonstrating that *C*_{BI} ≪ *C*_{Bs}, refuting assumption (b) above and rendering equation (20) and Fig. 6 of Ref. 6 invalid. This key point is further underscored by the fact that proton irradiation experiments with almost identical Frenkel-pair production rates^{12} to those used in Ref. 1 explicitly show $g\u221d\varphi $, where $\varphi $ is the beam flux^{8} (a test not reported in Refs. 1 and 6). The failure of the assumption *C*_{BI} ≫ *C*_{Bs} is most graphically evident in the lower panel in Fig. 1, where we present data from an earlier study^{8} using very similar processing conditions. The data show essentially the same exponential tails as in Ref. 1, although in this case clustering is entirely absent, all B is available to diffuse, and the static peak represents those B atoms which have escaped interaction with I during the short annealing time. The imposition of *C*_{BI} ≫ *C*_{Bs}, however, identifies essentially all unclustered B as continuously diffusing BI, leading to a Gaussian diffusion profile.

Fully coupled models as in Refs. 1 and 6 easily reproduce observed exponential tails if model parameters are correct. A first step towards this goal is to eliminate the unrealistic saturation of *C*_{BI}. This can be done by reducing *S*_{f}^{BI} from 30 *k* to below 20 k. This then allows extraction of other key parameters, inaccessible with the assumption *C*_{BI} ≫ *C*_{Bs}, such as the charge states of BI and I (from data on the Fermi-level dependence of B diffusion), and *E*_{f}^{I}, *S*_{f}^{I} (from exponential tails, since *E*_{f}^{I}, *S*_{f}^{I} determine the parameter k_{1}^{+} in $\lambda ={DBI/(k1+C0)}1/2$ (Refs. 2 and 6)).

In Refs. 1 and 6, the peak *C*_{B} is ∼10× higher than in earlier experiments.^{2} In this situation, clustering, chemical-pump, and Fermi-level effects may all influence diffusion, so the data in Ref. 1 are a more complex resource for parameter determination than those in Ref. 2. Nevertheless, to illustrate the robustness of *λ* extraction with the *g-λ* approximation, we apply it, outside its strictly applicable range, to the “short-time” (1 h) B data of Refs. 1 and 6. The results, using the same *λ* values as in Ref. 2, are shown in Fig. 1 (red curves).^{13} The fits are essentially perfect—far better than those obtained in Refs. 1 and 6. Moreover, unlike the fits in Ref. 1 they respect the data from Ref. 8, which explicitly showed C_{BI} ≪ C_{Bs}. It should now be clear that our analysis in Ref. 2, where conditions were optimal for *λ* extraction, is extremely robust. Moreover, the B data discussed in Ref. 1 support the analysis in Ref. 2, not refuting it as claimed.^{1} The B *model parameters* used in Ref. 1, however, are far from correct. We now restate the established position:^{2} our data (further supported by high-*T* data in Refs. 1 and 6) show that I-mediated self-diffusion in Ge involves a simple *I* at low *T* (significantly below 475 °C), an i-morph, **I**, with *S*_{SD}^{I}^{ }≈ 30 *k*, E_{SD}^{I}^{ }≈ 6.1 eV at high *T* (significantly above 475 °C), and a transitional region around 475 °C where both are significant. Finally, it should be noted that the open triangle^{14,15} in Fig. 2 shows that B also diffuses via BI under equilibrium conditions. An alternative model based on vacancy exchange^{1} would imply jump lengths of only 0.25 nm.

Having dealt with B diffusion analysis in some detail in this comment, we would like to emphasize that Section V of Ref. 1 also references an elegant analysis of experiments by the Munster group and coworkers on the diffusivity of I in irradiation experiments on Ge isotope superlattices.^{6} That analysis revealed *E*_{m}^{I }= (1.84 ± 0.26) eV. This value far exceeds estimates of 0.5–0.6 eV obtained from perturbed angular correlation measurements at low *T*,^{16,17} 0.6 eV obtained for simple *I* configurations from density functional theory using accurate LDA + U functionals,^{18} and <1 eV indicated by jump rates exceeding ∼1 s^{−1} at RT for *I* directly observed in aberration-corrected TEM.^{19} Thus in retrospect one can see that the 1.84 eV value rules out the simple *I* assumed in Ref. 1 and strongly favors the i-morph mechanism we proposed in Ref. 2. This has vast implications for defect physics which remain to be explored. Finally, taken together with our observed E_{SD}^{I}^{ }= 6.1 eV at high *T*,^{2} *E*_{m}^{I}^{ }= 1.84 eV implies *E*_{f}^{I}^{ }≈ 4.3 eV, in the range of recent atomistic calculations in course of publication.^{20} In conclusion, discussion prompted by conflicting analyses of experiments in Refs. 1, 2, and 6 has significantly progressed understanding of the complex behavior of self-interstitials in Ge.

## References

The doping-independence of λ at 550 °C confirms that, at least at this temperature, the reaction B_{s} + I ↔ BI is charge balanced without the involvement of charge carriers.