Sub-meV linewidth in GaN nanowire ensembles:
absence of surface excitons due to the field-ionization of donors
We observe unusually narrow donor-bound exciton transitions (400 µeV) in the photoluminescence spectra of GaN nanowire ensembles grown on Si(111) substrates at very high (°C) temperatures. The spectra of these samples reveal a prominent transition of excitons bound to neutral Si impurities which is not observed for samples grown under standard conditions. Motivated by these experimental results, we investigate theoretically the impact of surface-induced internal electric fields on the binding energy of donors by a combined Monte Carlo and envelope function approach. We obtain the ranges of doping and diameter for which the potential is well described using the Poisson equation, where one assumes a spatially homogeneous distribution of dopants. Our calculations also show that surface donors in nanowires with a diameter smaller than 100 nm are ionized when the surface electric field is larger than about 10 kV/cm, corresponding to a doping level higher than cm. This result explains the experimental observation: since the (D,X) complex is not stable in GaN, surface-donor-bound excitons do not contribute to the photoluminescence spectra of GaN nanowires above a certain doping level, and the linewidth reflects the actual structural perfection of the nanowire ensemble.
In contrast to epitaxial layers, single-crystal GaN can be grown on Si substrates as well as on amorphous substrates in the form of nanowires with diameters ranging typically between 30 and 100 nm.Calleja et al. (2000); Consonni et al. (2011); Sobanska et al. (2014) The high crystal quality of GaN nanowires facilitates the investigation of fundamental aspects of these nanostructures by purely optical means, such as the role of the surface on their spontaneous emission.Schlager et al. (2008); Corfdir et al. (2009a); Park et al. (2009); Brandt et al. (2010); Pfüller et al. (2010a); Gorgis et al. (2012); Korona et al. (2012) These studies have shown that the nanowire surface may affect both the radiative and nonradiative recombination processes of excitons.
Concerning the latter process, surface recombination has been reported to be the dominant recombination process in GaN nanowires with a diameter of 30 nm or thinner Gorgis et al. (2012) despite the fact that the dangling bond states for the nonpolar surfaces of GaN are situated far from midgap.Segev and Walle (2006); Lymperakis et al. (2013) Regarding the former process, both the wavefunction and energy of electrons and excitons bound to point defects are altered in the vicinity of a surface.Levine (1965); Satpathy (1983); Corfdir and Lefebvre (2012) This surface-induced change of the properties of point defects leads to a distribution of the emission energy of donor-bound excitons in nanowiresCorfdir et al. (2009a); Brandt et al. (2010) and to a modification of the lifetimes of the different radiative transitions involved as compared to the bulk case.Corfdir et al. (2009a) The pinning of the Fermi level at the sidewalls of nanowiresSegev and Walle (2006); Lymperakis et al. (2013) adds further complexity to nanowire-based systems. As shown in Ref. Calarco et al., 2005, this pinning induces radial electric fields within the nanowires and is responsible for their electrical depletion.
The consequences of these surface potentials are manifold. For example, they modify the radiative decay rate of excitonsPfüller et al. (2010a); Lefebvre et al. (2012) and enhance the coupling between free and bound excitons with profound consequences for the exciton decay dynamics.Hauswald et al. (2013) In (In,Ga)N/GaN nanowire heterostructures, the interplay between the polarization and surface potentials may lead to a radial separation of electron and holes, resulting in a dramatic decrease of the internal quantum efficiency in the blue spectral range.Marquardt et al. (2013)
To understand these phenomena, a detailed knowledge of the distribution and the magnitude of surface-induced electric fields as a function of the doping level (N) and the nanowire diameter () is indispensable. Furthermore, we need to understand the transition between the single impurity limit for nanowires with a low background doping and small diameters,Pfüller et al. (2010b); Gorgis et al. (2012) and the bulk-like case of a homogeneous dopant distribution reached for intentionally dopedCalarco et al. (2005) or very large nanowires.Schlager et al. (2008)
In this paper, we present experimental results demonstrating that very high growth temperatures can induce the incorporation of Si into GaN nanowires grown on Si substrates, but can simultaneously result in a sub-meV linewidth of the donor-bound exciton transition for the ensemble. To understand these results, we investigate the role of the surface on the properties of donors in GaN nanowires theoretically. Using a combination of Monte-Carlo and envelope function calculations, we examine the validity of assuming a parabolic potential across the section of nanowires. We compute the binding energy of donors in the presence of surface-induced electric fields and discuss the doping- and diameter ranges for which neutral surface donors are stable and for which they are not. We find that neutral surface donors capable of binding excitons do not exist already for moderate doping levels (for example, cm for a nanowire diameter of 100 nm), which explains the record linewidths observed experimentally.
The paper is organized as follows. Section II is devoted to the experimental methods used in this work. In Section III, the photoluminescence spectra of GaN nanowires grown on Si at a temperature of 870C are presented. In Section IV, we present Monte-Carlo simulations which provide the range of values for and for which it is valid to approximate the radial potential across a nanowire by a parabola. In Section V, we calculate the binding energy of donors located at the nanowire surface, in the presence of surface-induced electric fields. In Section VI, we compare the results of our calculations with the experiments in Section III. Finally, we present our conclusions in Section VII.
Ii Experimental details
The nanowire ensembles studied here formed spontaneously during plasma-assisted molecular beam epitaxy of GaN on Si(111) substrates.Bertness et al. (2008); Fernández-Garrido et al. (2009, 2012); Consonni (2013) The as-received substrates were etched in diluted (5%) HF for 2 min. Prior to growth, the substrates were annealed in the growth chamber for 30 min at 880 °C. After this process, the surface reconstruction characteristic of a clean Si(111) surface appeared upon cooling to temperatures below 860 °C. Subsequently, the substrate temperature was set to the desired value for growth, and the substrate was exposed to the N plasma for 10 min before opening the Ga shutter to induce the spontaneous formation of GaN nanowires. A series of samples was prepared using growth temperatures between 785 and 870 °C. For all samples the active N flux provided by the radio frequency N plasma source was kept constant at scm. In contrast, the impinging Ga flux provided by a solid-source effusion cell was increased from to s cm to compensate for the exponential increase in Ga desorption with increasing temperature.Heying et al. (2000) Due to the high Ga desorption rate, the actual growth conditions were invariably N-rich as required for the spontaneous formation of GaN nanowires by molecular beam epitaxy.Fernández-Garrido et al. (2013) The nanowires thus obtained have similar average lengths (2–3 µm), mean diameters (70–-130 nm), densities ( cm) and coalescence degrees.Brandt et al. (2014)
Photoluminescence spectroscopy was performed by exciting these GaN nanowire ensembles with the 325 nm line of a continuous-wave HeCd laser. The laser beam was attenuated with neutral density filters to a power of about 15 nW and was focused to a 1 µm-spot using a near-UV objective with a numerical aperture of 0.45. We estimate that about 40 nanowires are excited simultaneously. The PL signal was collected using the same objective and dispersed by a monochromator with 80 cm focal length and a grating with 2400 lines/mm. For the current experiment, the spectral resolution was set to 0.25 Å(i.e. about 250 µeV). The dispersed signal was detected by a UV-sensitized liquid nitrogen-cooled charge coupled device.
Iii Donor-bound exciton transitions from a GaN nanowire ensemble with sub-mev linewidth\@dblfloat
Figure 1(a) shows the low-temperature (10 K) photoluminescence spectrum of two ensembles of GaN nanowires formed at a growth temperature of 808 and 870 °C. The spectrum of the nanowire sample grown at 808 °C is dominated by a line centered at 3.4711 eV originating from the recombination of A excitons bound to neutral O donors on N sites [(O,X)]. The shoulders on the high-energy side of the (O,X) line are due to the emission from B excitons bound to neutral O donors [(O,X)] at 3.4743 eV and from free A excitons (X) at 3.4775 eV. The energy position of the free A exciton confirms that the net strain in our nanowire ensemble is virtually zero.Calleja et al. (2000); Robins et al. (2007); Corfdir et al. (2009a); Park et al. (2009); Brandt et al. (2010) Despite this fact, the (O,X) line exhibits a full-width at half-maximum of 1.2 meV. This broadening arises from both the microstrain introduced by nanowire coalescenceJenichen et al. (2011); Kaganer et al. (2012) and the modification of the energy and the wavefunction of neutral donors and donor-bound excitons by the surface, as discussed in Refs. Corfdir et al., 2009a; Brandt et al., 2010.
Figure 1(b) displays a cross-sectional scanning electron micrograph of the interface between the GaN nanowires and the Si substrate for a sample grown at a temperature of 870 °C. A common phenomenon during the high-temperature growth of GaN on Si is the melt-back etching of the Si substrate.Zhu et al. (2013) This etching process arises from the formation of a Ga-Si eutectic alloy and results in the creation of large pits in the substrate. For our sample series, we observed these pits for growth temperatures higher than 850 °C.
The photoluminescence spectrum of the sample shown in Fig. 1(b) is compared to that of the sample grown below this critical temperature in Fig. 1(a). For all samples grown above 850 °C, an additional line is observed at 3.4725 eV. This line corresponds in energy to the recombination of A excitons bound to neutral Si donors on Ga sites [(Si,X)], suggesting that the melt-back etching of the Si substrate is accompanied by the incorporation of Si in the nanowires. The incorporation of Si also manifests itself by an increase in the emission intensity of excitons bound to I basal-plane stacking faults as shown in Fig. 1(b)]. In fact, the formation energy for stacking faults in GaN is reduced with increasing Si concentration as demonstrated theoretically by Chisholm and Bristowe (2000, 2001).
Finally, we observe a significant reduction in the linewidth of the donor-bound exciton lines for all samples grown above 850 °C. For the nanowire ensemble grown at 870C, full-width at half-maxima of 400 and 500 eV are measured for the (O,X) and (Si,X) lines, respectively. All samples grown at or above 850C exhibit this correlation between the occurence of Si meltback-etching, a strong (Si,X) transition, and sub-meV donor-bound exciton linewidths. This finding suggests that the higher donor concentration caused by the additional incorporation of Si is responsible for the narrow linewidth of our high-temperature observed for high growth temperature. To understand this apparently paradoxical result, we investigate theoretically in Sections IV and V the properties of donors in GaN nanowires as a function of the nanowire diameter and dopant concentration.
Iv Surface-induced parabolic potential
In this section, we employ Monte-Carlo simulations to evaluate the potential in nanowires with a diameter between 20 and 200 nm and a donor concentration between and cm. The donors are distributed randomly in the nanowires and we assume that the nanowires are fully depleted. Figure LABEL:fig:FigurePotential(a) shows one realization for the distribution of donors in a nanowire with nm, µm and cm. The simulated nanowire contains 15 donors.
Figures LABEL:fig:FigurePotential(b)–LABEL:fig:FigurePotential(e) show the potential across various cross-sections of the nanowire. The shape of the potential as well as the magnitude of the electric field across the section of the nanowire fluctuates along the nanowire length. For such a low doping density, each individual impurity strongly affects the local potential felt by the electron. In transport experiments such as those reported in Refs. Calarco et al., 2005; Sanford et al., 2010, the electronic properties of a single nanowire are averaged along the nanowire length. Accordingly, Figs. 2(a)–2(c) display the surface potential averaged along the nanowire axis [the -axis in Fig. LABEL:fig:FigurePotential(a)] for a nanowire with nm and equal to 10, 10 and 10 cm. Figures 2(d)–2(e) show the corresponding electric field along the -axis [dashed line in Figs. 2(a)–2(c)] compared to that expected from the Poisson equation for a homogeneous distribution of charges with a concentration equal to in an infinitely long cylindrical nanowire with diameter . For = 10 cm, we obtain a good agreement between the computed and the prediction of the Poisson equation. For donor concentrations equal to and larger than this value, the donor distribution can thus be considered as homogeneous. In contrast, for = 10 cm, there is no longer any correlation between and the straight line predicted by the Poisson equation. For low donor concentrations well below = 10 cm, the shape of is sensitive to the detailed distribution of donors and the surface-induced potential can in no way be reproduced by a parabola.
For obtaining a quantitative criterion for the validity of the assumption of a parabolic potential for each set of values , we fit the evolution of the electric field across the section of the nanowire with the following linear regression:
where is the electric field at the surface of the nanowire and is the position along the -axis. We impose the condition that the electric field on the axis of the nanowire () is equal to zero for symmetry reasons. Note that Eq. 1 applies only when the nanowire is fully depleted. For nm, donors situated in the inner core of GaN nanowires remain most probably unionizedCalarco et al. (2005) and can be derived using the expressions given in Ref. Sanford et al., 2010 or in Ref. Dobrokhotov et al., 2006. We note, however, that there is no significant deviation between the results obtained by Eq. 1 and the expressions in Refs. Sanford et al., 2010; Dobrokhotov et al., 2006 when nm.
For a depleted nanowire and when surface potentials can be described by the Poisson equation, the electric field at the surface of the nanowire is:
where is GaN dielectric constant. To grade the quality of the fit obtained using Eq. 1, we utilize the R-square metric.Jin et al. (2001) The R-square is given by , where and are the residual sum of squares and the total sum of squares, respectively. When the R-square value averaged over ten realizations of the nanowire is larger than 0.9, we consider that the surface potential in a nanowire with a given set (,) can be described by a parabola.
Our findings are summarized in Fig. 3, which shows as a function of predicted by the Poisson equation for donor densities between and cm. Below the dashed line, the R-square values are below 0.9 and the actual surface potential in the nanowire does not agree with the result of the Poisson equation for a homogeneous distribution of donors. When is decreased from 200 to 20 nm, has to be increased from about to a value in excess of cm for surface potentials in the nanowire to be correctly described by the Poisson equation. For a nanowire length of 1.5 m, this condition corresponds to having more than 200 donors per nanowire.
The experimental determination of is a challenging task, and consequently only a few values have been reported.Sanford et al. (2010); Pfüller et al. (2010b) Pfüller et al. (2010b) obtained a value of cm from a statistical analysis of the exciton emission of single GaN nanowires with a diameter below 30 nm. This optical method is not affected by the considerations above. On the other hand, Sanford et al. (2010) deduced values between and cm for nanowires with ranging from 140 to 700 nm from an analysis of photoconductivity experiments based on the validity of the Poisson equation.Sanford et al. (2010) For these low donor concentrations, it is clear from Fig. 3 that Eq. 1 (or similar expressions such as those in Refs. Sanford et al., 2010; Dobrokhotov et al., 2006) fails to reproduce the surface potential in nanowires with 200 nm. It is, however, difficult to anticipate whether this breakdown of the continuum model leads to an over- or an underestimate of in such experiments.
V Binding energy of donors in the presence of internal electric fields
Photoluminescence spectroscopy is the most sensitive method for the detection of impurities in semiconductors. For the case of GaN nanowires, the high sensitivity of photoluminescence spectroscopy has been exemplified in Ref. Pfüller et al., 2010b, which reports the detection of the (D,X) transition even in single nanowires containing, on average, only a single donor. Photoluminescence spectroscopy is, however, completely insensitive to the presence of ionized donors. Because of the specific ratio of the effective masses of electrons and holes, the exciton-ionized donor-complex is not stable in GaN,Suffczynski et al. (1967) i. e., the presence of ionized donors does not lead to a radiative transition observable in the photoluminescence spectra of GaN. Indirect evidence for the coexistence of neutral and ionized donors even at low temperatures has been obtained in the experiments reported in Refs. Pfüller et al., 2010a; Lefebvre et al., 2012.
It is intuitively clear that the electric field associated with the ionized donors affects the energy of electrons bound to neutral donors. For flat-band conditions, the presence of the surfaceLevine (1965); Satpathy (1983) together with the dielectric mismatchDiarra et al. (2007); Pierre et al. (2010) leads to a strong site-dependence of the binding energy of electrons to donors in nanowires.Corfdir and Lefebvre (2012) In particular, the binding energy of donors at the surface of a thick GaN nanostructures is approximately 0.6 times that of donors in bulk.Corfdir and Lefebvre (2012) Since the magnitude of surface-induced electric fields is much stronger at the surface than in the core of nanowires (cf. Fig. 2), the properties of neutral donors at the surface of GaN nanowires should be affected.
Hence, we next calculate the binding energy of electrons to donors in nanowires in the presence of a surface-induced parabolic potential. We reduce the three-dimensional problem of a donor in a nanowire to a one-dimensional one by considering the simplified problem of a donor in a slab of GaN bounded by air. As shown in Ref. Corfdir and Lefebvre, 2012, the properties of donors in a nanowire with a diameter are well reproduced using a nanoslab geometry with a nanoslab thickness . This fact remains true even when is as small as 10 nm.
Regarding the symmetry of the potential for an electron in a nanoslab and in the presence of a donor, we choose to set up the problem using cylindrical coordinates. The Hamiltonian for an electron with coordinates in a nanoslab and in the presence of a donor is given by:
where is the electron effective mass. In Eq. 3,
describes the interaction of the electron with the donor and the image charges of the donor while
accounts for the electron self-energy. The donor and its images are on the -axis with coordinates and , respectively, and corresponds to the on-axis coordinate of the image of the electron. The coordinates and are obtained following Ref. Kumagai and Takagahara, 1989. The potential term is the potential across the nanowire section and we assume it to be parabolic. We choose the following trial wavefunction for the electron:
with being a normalization factor, describing the extent of the electron wavefunction in the plane of the nanoslab and representing the electron envelope function along the confinement axis. The Schrödinger equation is solved numerically using the effective potential formalism (for more details, see Refs. Corfdir et al., 2009b; Corfdir and Lefebvre, 2012) allowing us to deduce the electron wavefunction and binding energy for various values of and . The electron density giving rise to such a surface electric field in nanowires can be obtained using Eq. 2.
Figure 4(a) shows the ground-state wavefunction of an electron bound to a surface donor for a nanoslab of width nm. The electron wavefunction is formed from one lobe of a 2-like wavefunction, in agreement with the early works of Levine (1965) and Satpathy (1983). The binding energy of an electron to such a surface donor is 17.7 meV, significantly larger than the 7 meV predicted by those reports which neglected the dielectric mismatch at the nanoslab surface.Diarra et al. (2007); Pierre et al. (2010); Corfdir and Lefebvre (2012)
Figures 4(b)–4(d) show the evolution of the electron wavefunction for a nanoslab with nm and for various magnitudes of . For kV/cm, the electron remains bound to the donor atom and the 2 shape of its wavefunction is mostly maintained. For kV/cm, the electron wavefunction starts to spread towards the center of the nanoslab and its spatial extent increases significantly [cf. Fig. 4(c)]. For even larger values of , the built-in electric fields dominate over the Coulomb interaction between the electron and the donor: the electron wavefunction is centered in the nanoslab, its spreading reduces and its shape corresponds to a Bessel function of the first kind.
To quantify the change in spatial extent of the electron wavefunction, we utilize the standard deviation of the electron position as a function of as shown in Fig. 5. When is maximum, the parabolic potential that attracts the electron towards the center of the nanoslab counterbalances exactly the attraction exerted by the donor. For nm and , nm for a surface donor. The maximum value of of 15.2 nm is obtained for kV. For even larger values of , the electron is delocalized towards the center of the nanoslab. The surface donor is therefore ionized. We note that in the absence of a surface donor, increasing leads to a monotonous decrease of .
Figure 6(a) shows the ionization field , defined as the value of that maximizes for a given , as a function of . The ionization of surface donors occurs for values of for which is parabolic [cf. Fig. 6(b)]. Second, the evolution of with is not monotonic: as shown in Fig. 6(a), the maximum is 28 kV/cm when nm. Note that this ionization field is three times smaller than that expected for donors in the bulk.Pedrós et al. (2007) When is decreased below 32 nm, decreases due to an enhanced interaction between the electron and the image charges located at the interface of the nanoslab on the opposite side of the surface donor. In contrast, when the is larger than 32 nm, the interaction of the electron with all-but-the-nearest of its images becomes marginal. Therefore, for a given , the larger the value of , the larger the potential difference between the center and the surface of the nanowire, and the smaller will be the ionization field .
As discussed in Section III, the linewidth of the donor-bound exciton transition measured at 10 K from a GaN nanowire ensemble formed at temperatures below 850 °C is comparatively large, namely, about 1–2 meV [cf. Fig. 1(a) and Refs. Calleja et al., 2000; Robins et al., 2007; Corfdir et al., 2009a; Park et al., 2009; Brandt et al., 2010]. We have shown in Section V that the nanowire surface leads to a distortion of the donor wavefunction and, in particular, to a change in electron binding energy. It is safe to assume that the energy of donor-bound excitons (formed of two electrons and a hole orbiting around a charged donorSuffczynski and Wolniewicz (1989)) as well as their recombination energy are affected in an analogous way. Therefore, even in the absence of microstrain, the donor-bound exciton transitions is intrinsically broadened due to the random distribution of donors in nanowires.Corfdir et al. (2009a); Brandt et al. (2010)
The result of our present calculations provides further insight into the optical properties of GaN nanowires. First of all, surface donors in nanowires with of the order of 80–100 nm are ionized when is larger than about 10 kV/cm [cf. Fig. 6(a)]. This field corresponds to donor concentration of cm [Fig. 6(b)], a rather typical value for GaN in general, and close to the values reported by Pfüller et al. (2010b) and Sanford et al. (2010) for GaN nanowires in particular. Since the binding energy of donor-bound excitons is a fraction of that of the corresponding donors,Haynes (1960) surface donor-bound excitons are not stable in nanowires for which is larger than . Second, the large spatial extent of the wavefunction of an electron or of an exciton bound to a surface donor can also contribute to the instability of these complexes in nanowires. As shown in Fig. 3, for a nanowire with nm, increases from 0.9 to 9 kV/cm when increases from to cm. This increase in is accompanied by an increase of from 3.7 to 13.4 nm (cf. Figure 5). The volume probed by the electron wavefunction therefore increases by a factor of nearly 50 as compared to bulk-like donors. Consequently, for GaN nanowires of moderate doping density, the wavefunction of a surface donor-bound electron may certainly probe other impurities, e. g., neutral donors located at the core of the nanowire, and bind to them if energetically favorable. Note that this comment also applies to donor-bound excitons, whose wavefunction in bulk material is typically two times larger than that of the electron bound to a neutral donor.Suffczynski and Wolniewicz (1989)
Our calculations demonstrate that surface donor-bound excitons do not form in nanowires with doping levels exceeding a certain, diameter-dependent threshold. As a consequence, the ”intrinsic” broadening of the donor-bound exciton transition in GaN nanowires should abruptly disappear at this threshold for a given nanowire diameter. For a spontaneously formed GaN nanowire ensemble with its comparatively broad diameter distribution, the transition may be a gradual one, but we would certainly expect a reduction of the linewidth of the donor-bound exciton transition upon an increase of the doping level from low (10 cm) to moderate (10 cm). An abrupt reduction of the linewidth of the donor-bound exciton transition is precisely what we observe experimentally for our GaN nanowire ensembles as soon as Si incorporation sets in at high growth temperatures. Our calculations thus explain the seemingly conflicting results of an increased donor concentration and a reduced linewidth. Assuming that the minimum linewidth of 400 µeV observed for the (O,X) line of the high-temperature GaN nanowire is free from surface contributions, these results allow us to get a glimpse onto the actual structural perfection of GaN nanowire ensembles. Taking into account the instrumental broadening of 250 µeV, the linewidth observed corresponds to a residual inhomogeneous broadening of about 300 µeV, which is probably due to microstrain generated by the coalescence of adjacent nanowires.Jenichen et al. (2011); Kaganer et al. (2012); Brandt et al. (2014) In any case, the crystal quality of these spontaneously formed GaN nanowires on Si is clearly on par with that of free-standing GaN.Paskov et al. (2007)
Acknowledgements.The authors would like to thank Timur Flissikowski for a critical reading of the manuscript. They are indebted to Lutz Geelhaar, Holger T. Grahn and Henning Riechert for continuous encouragement and support. Financial support of this work by the Deutsche Forschungsgemeinschaft within SFB 951 is gratefully acknowledged.
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