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Analytic description of nanowires II: morphing of regular cross sections for zincblende and diamondstructures to match arbitrary shapes
^{a}Integrated Materials Design Lab (IMDL), Research School of Physics and Engineering, The Australian National University, ACT 2601, Australia, ^{b}Institute of Semiconductor Electronics (IHT), RWTH Aachen University, 52074 Aachen, Germany, ^{c}Integrated Materials Design Centre (IMDC), University of New South Wales, NSW 2052, Australia, and ^{d}Department of Applied Mathematics, Research School of Physics and Engineering, The Australian National University, ACT 2601, Australia
^{*}Correspondence email: solidstatedirk@gmail.com
Setting out from our recent publication [König & Smith (2021). Acta Cryst. B77, 861], we extend our analytic description of the regular cross sections of zincblende and diamondstructure nanowires (NWires) by introducing morphing to arbitrary convex shapes featuring linear interfaces as encountered in experiment. To this end, we provide addon terms to the existing number series with their respective running indices for zincblende (zb) and diamondstructure NWire cross sections. Such addon terms to all variables yield the required flexibility for morphing, with main variables presented by the number of NWire atoms N_{Wire}(d_{Wire}[i]), bonds between NWire atoms N_{bnd}(d_{Wire}[i]) and interface bonds N_{IF}(d_{Wire}[i]). Other basic geometric variables, such as the specific length of interface facets, as well as widths, heights and total area of the are given as well. The cross sections refer to the six highsymmetry zb NWires with lowindex faceting frequently occurring in the bottomup and topdown approaches of NWire processing. The fundamental insights into NWire structures revealed here offer a universal gauge and thus enable major advancements in data interpretation and the understanding of all zb and diamondstructurebased NWires with arbitrary convex cross sections. We corroborate this statement with an exact description of irregular Si NWire cross sections and irregular InGaAs/GaAs coreshell NWire cross sections, where a radially changing unitcell parameter can be included.
Keywords: nanowire cross section; metrology; morphing; zincblende; diamond.
1. Introduction
In recent publications, we derived (König & Smith, 2019) and improved (König & Smith, 2021) the analytical description of six regular zbNWire cross sections relevant to experiment (Weber & Mikolajick, 2017); see Fig. 1. To this end, we described the number of atoms, the number of bonds between such atoms and the number of interface bonds for an NWire slab with a thickness of the periodic (UC) along its growth axis with its interface length, height, width and NWire area. An analytical structural description of the NWire down to the individual bond and atom is a powerful tool for interpreting or predicting (König et al., 2021) any experimental data as a function of NWire size, shape and orientation of its growth axis and interfaces. Here, we aim to extend this analytic description for zb and diamondstructure NWires to arbitrarily convex cross sections featuring linear interfaces, thereby allowing one to fit the analytics of such cross sections to any irregular convex shape encountered in experiment.
Section 2 provides the necessary background information on the nomenclature on how to interpret the images per NWire type, and a brief assignment of primary and secondary parameters to structuredriven phenomena. Section 3 contains the number series of all six different NWire cross sections, as shown in Fig. 1 for uniaxial morphing (C_{2} symmetry). In Section 4, we introduce triaxial morphing to all four hexagonal cross sections (C_{3} symmetry) with three independent run indices, allowing for a vast range of shapes. Combining the morphing algorithm in both sections, virtually any crystalline zbNWire with convex geometry can be described. In Section 5, we show examples of applying the number series and derived secondary parameters to experimental data from the literature for each, irregular Si and coreshell III–V NWires. Appendices A, B and C derive characteristic lengths and areas for cross sections with [110], and [111] growth vectors, respectively. As this work builds upon our previous publications, we refer the reader to König (2016) and König & Smith (2021) for the background information on chosen cross sections, interface energetics, bond densities and further details regarding the basics of associated analytic number theory.
2. General remarks on analytic number series, structural boundary conditions and nomenclature
Table 1 lists the primary and secondary parameters calculated by number series.
All parameters are calculated over an NWire slab presenting the thickness of the UC a_{uc} in the growth direction as per König & Smith (2021) to achieve periodicity (Table 2). In addition, Table 2 lists the amount of atoms and bonds per column (i.e. per atom or bond visible) as a function of NWire axis orientation for a top view onto the thereby allowing atoms and bonds to be counted. Respective images are provided for all NWire types presented here.

The periodicity in the growth direction and the assumption that the length of the NWire exceeds its diameter allows for a highly accurate description of parameters, though, as per mathematical definition, they are correct only for .
On a par with König & Smith (2021), the indexing of NWire type is given as a superscript with its shape and growth direction; see Table 3.

With , we obtain a gauge for the response to internal stress, e.g. by dopant species. The ability of embedding materials or ligands to exert stress (Schuppler et al., 1994; Boyd & Wilson, 1987) onto NWires or vice versa can be described with . The impact of a highly polar surface termination on the zbNWire electronic structure observed as interfacerelated electronic phenomena (Zahn et al., 1992; Campbell et al., 1996; He et al., 2009; König et al., 2014, 2018, 2019, 2021) is assessed by the ratio . The ratio can be useful for detecting facetspecific interface defects. For Si, interfacespecific dangling bond (DB) defects exist, namely, the P _{b0} centre at {001} interfaces and the P _{b1} centre at {111} interfaces (Helms & Poindexter, 1994; Keunen et al., 2011). These DB defects occur in a ratio which reflects and can be detected by (EPR) (Stesmans et al., 2008). For SiNWires, the ratio is therefore a valuable tool for identifying cross sections of the NWires treated in Sections 3.2, 4.1 and 4.2. We illustrate the results on tetrahedral C, Si and Ge NWires (all diamond structure). NWire atoms without interface bonds are shown in grey. Atoms with interface bonds are colourcoded: species with one/two/three interface bonds are red/blue/green; see Fig. 1 for an example. The analytical number series introduced below also hold for zbNWires due to straightforward symmetry arguments (König & Smith, 2021). Material properties resulting from differences in the base cell – A–B for zbstructures versus A–A for diamond structures – are not considered here. This constraint has no impact on the applicability of the analytics of our work, unless the atomic sequence mentioned above is of primary interest when comparing two solids.
The nominal number series describing the highsymmetry NWire cross sections follow a run index i which defines the nominal size of the Morphing of cross sections is introduced by a second class of run indices j_{1},j_{2} ( j,k_{1},k_{2} or k_{1},k_{2},k_{3}) for C_{2} symmetry uniaxial (C_{3} symmetry triaxial) morphing, defining the shape – or more precisely, its deviation from the respective highsymmetry For quandrangle cross sections treated in Sections 3.1 and 3.2, one index j is sufficient to describe the symmetry deviations elaborated here, as is straightforward to see by turning cross sections by 90°. For the four remaining more complex hexagonal cross sections, we introduce two running indices j_{1}, j_{2} to allow for independent morphing from the top and bottom interfaces. Generally, we have j = j_{1} = j_{2} = 0 for the nominal shape of the The morphing indices then span the range of for quadrangle cross sections, and – with one exception (see Section 3.5) – of for hexagonal cross sections, the positive limit of the latter occurring due to their interface planes intersecting at finite distance for (as opposed to parallel interfaces for quadrangle cross sections). An example of morphing is shown in Fig. 2. These limits to lateral run indices for hexagonal cross sections are also valid in triaxial morphing with lateral run indices j,k_{1},k_{2} or k_{1},k_{2},k_{3} (Sections 4 and 5.1), again with one exception (Sections 4.3 and 5.2).
For the hexagonal cross sections, we originally developed an even and an odd series to account for minor deviations from the highsymmetry cross sections in experiment (König & Smith, 2021). The differences between parameters from odd versus even series are outrun by far with the modifications due to morphing. As a consequence, we introduce morphing here only to the even series of all hexagonal cross sections. While adequate number series modifications can also be derived for the odd series of all hexagonal cross sections, their descriptions of NWire cross sections are covered by morphing the even series onto experimental data. An example is the cross sections of Si NWires with a [110] growth axis and {001} plus {111} interfaces; see Fig. 1(c) herein and the experimental data published by Yi et al. (2011). This was treated in König & Smith (2019) with even and odd calculus. With the morphing introduced in Section 3.3 and, in particular, in Section 4.1, we can simply use the even series and morph it exactly onto the experimental image.
3. Morphing cross sections along one symmetry axis
Terms describing the highsymmetry i.e. j = j_{1} = j_{2} = 0) are identical with the respective Equations in König & Smith (2021). Such terms are printed here in grey to distinguish them from terms due to morphing which are printed in black.
(3.1. Nanowires growing along the [001] direction with square and four {001} interfaces
As mentioned briefly in the Introduction, run indices for this are limited to and .
The square shape of the
results in .Fig. 3 shows morphing examples of the square NWire cross sections with growth along the [001] direction and four {001} interfaces.
3.2. Nanowires growing along the [110] direction with a rectangular and two {001} plus two {110} interfaces
As mentioned briefly in the Introduction, run indices for this are limited to and .
Since we morph the j. However,
along the {110} interfaces, does not change withis a function of j by which the ratio of interface bonds between facets becomes
The centre expression shows both number series in their explicit form, while the expression on the right presents the simplified result of their ratio.
As was the case for , the length of the {001} interface remains unchanged;
The width of the rectangular
follows in a straightforward manner from . Morphing has an impact on the length of the {110} interfaces, resulting inThe height of the rectangular
follows in a straightforward manner from .The total
area is given by of this NWire type is shown in Fig. 43.3. Nanowires growing along the [110] direction with a hexagonal and four {111} plus two {001} interfaces
The remaining four NWire types to be investigated all have a hexagonal Introduction, run indices for these cross sections are limited to and , except for the with an exclusive {110} interface and a [111] growth axis; see Section 3.5.
which has a more complex geometry. As mentioned briefly in theFor the number of atoms forming the NWire
we getThe number of bonds between these atoms is described by
The number of interface bonds over all facets is given by
Since the number of {111} facet bonds being added or removed equals the number of {001} facet bonds being removed or added per change in j_{1} or j_{2}, such contributions cancel each other out; see Equation 17 below. For a graphical verification, we refer the reader to Fig. 5, or to Fig. 4 in König & Smith (2021).
The ratio of interface bonds per facet orientation is given by
The lengths of the {001} and {111} facets depend only on one lateral run index , which affects the respective facet. For the {001} facet, a scaled offset of 1 exists for the two irregular triangular areas in the apexes:
For the {111} facet, a scaled offset of −1/4 exists due to the two irregular triangular areas in the apexes:
Due to morphing along the vertical symmetry axis of the cross sections, and thus stays unchanged. The scaled offset due to the two isosceles triangles at the side apexes of the
is which is added to the nominal increment of :Obviously, the height of the j_{1}, j_{2} in steps of per , resulting in
does change withThe total
area is described byThe prefactor presents the area of one X_{6} ring, seen along the 〈110〉 lattice vector, which is straightforward to derive from four such rings filling the zbUC when cut along the {110} plane, covering an area of ; see Appendix A. The offset areas concern the isosceles triangles at the {001} facets with an area of , and the irregular triangles at the {111} facets with , both of which can be found when considering an X_{6} ring seen along the 〈110〉 lattice vector, using geometrical arguments. The total offset area 4/16+2/32 presents the four scalene triangles at the two lower and upper apexes of the plus the two isosceles triangles occurring at the left and rightmost apexes of the cross section^{1}, see Figs. 5 and 16, Appendix A and Fig. 4 in König & Smith (2021).
3.4. Nanowires growing along the [] direction with hexagonal and four {} plus two {111} interfaces
For the number of atoms forming the NWire
we getThe number of bonds between these atoms is described by
The number of interface bonds over all facets is given by
The assignment of interface bonds to the {111} and {11} facets is shown in Fig. 10(b). With this assignment of interface atoms to {111} and {11} facets, we obtain the ratio of interface bonds per facet orientation as
Equation 26 shows the explicit number series in the top line, while the bottom line is the compacted version for the ratio of interface bonds. The following facet lengths depend only on one lateral run index , which is assigned to the facet of interest:
and
For the facets, the smallest unit is the diagonal of the congruent rectangular areas constituting the ) and a vertical scaled length of (see Equation 30), yielding for the scaled diagonal of the rectangle. Due to morphing along the vertical symmetry axis of the cross sections, and thus stays unchanged:
area. These rectangles have a horizontal scaled length of (see Equation 29The height of the
obviously changes with morphing, followingThe length presents a third of the diagonal connecting two opposite corners in 〈111〉 direction through the zbUC, whereby the (111) vector is orthonormal to the plane; . This length is equivalent to the longer side of the rectangle which presents the unit area of NWires growing along the vector class, accounting for the increment in in Equation 30; see Fig. 2. The total area is described by
The scaled coefficient of describes the rectangular unit area of the and 10. For a detailed geometrical derivation of characteristic lengths and areas, see Appendix B.
as discussed above, following from . Facets cut the outermost rectangles along their diagonal, rendering their triangular area . For an illustration of the morphed hexagonal with {111} top and bottom interfaces plus {11} side interfaces, refer to Figs. 23.5. Nanowires growing along the [111] direction with a hexagonal and six {110} interfaces
The smoother geometry of this
allows us to use lateral run indices of for morphing.For the number of atoms forming the NWire
we getThe number of bonds between these atoms is described by
The number of interface bonds over all facets is given by
The facet lengths depending on the respective , are
for the top and bottom facets, and
for the side facets. The scaled coefficient 1/ refers to the side length of the equilateral triangles which form the unit area unit on a {111} plane defining the
This coefficient follows from a {111} plane cut through the zbUC along its corner points, resulting in an equilateral triangle of scaled side length , containing an area equivalent to 12 small equilateral triangles (6 equilateral + 6 isosceles with same area = 12) with a scaled side length of 1/.The width of the j and thus stays unchanged:
is not a function ofThe height of the j_{1} and j_{2} as it is parallel to the symmetry axis along which axialsymmetric morphing occurs:
depends onThe total
area is described byThe scaled coefficient of /24 describes the area per equilateral triangle as the unit area of the shows the of this NWire type.
and follows directly from our discussion of facet lengths above. Fig. 63.6. Nanowires growing along the [111] direction with a hexagonal and six {} interfaces
This i.e. with . For the number of atoms forming the NWire we get
returns to the nominal limitation of lateral (morphing) run indices,The number of bonds between these atoms is described by
The number of interface bonds over all facets is given by
The facet lengths of top and bottom interfaces depend on the respective :
The facet length of side interfaces is
Since the width of this hexagonal
is not a function of , it remains unchanged:For the height of the
we getThe , with the facet orientation rotated by 60° ({110} ). This rotation swaps the scaled coefficients of facet lengths and width on one side and height on the other side when compared to Section 3.5 (see discussion there).
plane has the same orientation as in Section 3.5The total due to the same orientation of the NWire (same growth vector) and thus the same small equilateral triangles as the unit area:
area has the same scaled coefficient of as in Equation 39The outermost area elements at the facets form isosceles triangles (Fig. 7), which have the same area as their equilateral counterparts mentioned above; see also the related discussion in Section 3.5 and the geometrical derivation explained in Appendix C. The of this NWire type is shown in Fig. 7.
 Figure 7 for 
4. Morphing cross sections along three symmetry axes
Such morphings naturally lend themselves to cross sections with hexagonal symmetry. We therefore do not consider square cross sections with 〈001〉 normal vectors on growth plane and facets, as well as rectangular cross sections with 〈110〉 growth vector and {001} and {110} facets.
Depending on the symmetry of the hexagonal j as used in Section 3, and run indices k_{1}, k_{2} for the other two morphing directions, with a different facet orientation {abc} for both run indices k_{1} and k_{2}. This is the case in Sections 4.1 and 4.2.
we have to introduce different lateral number series per facet orientation, with run indicesAs in Section 3, lateral run indices – j, k_{1}, k_{2}, k_{3} – are positive for reductive morphing (cutting into the nominal cross section) and negative for expansive morphing (extending the nominal cross section), with the nominal presented if all lateral run indices are zero. Under the condition that all lateral run indices are equal, i.e. j = k_{1} = k_{2}, or k_{1} = k_{2} = k_{3}, all cross sections will assume a triangular or quasitriangular shape on maximum expansive morphing, as well as on maximum reductive morphing; see Fig. 8(a).
Other, more irregular, shapes can be described in an arbitrary fashion, under the constraint that all facets or singular points where facets meet do not penetrate the nominal hexagonal via a lateral run index have a minimum length or at least a point where adjacent (directly morphed) facets meet. These minimum lengths or singular points are all located on the respective borders (facets) of the nominal considered. By preventing the penetration of such minimum facet lengths or singular points into the nominal we prevent the lateral number series from overlapping with each other, erasing the facet between the two associated morphing sections in the process. Thereby, we obtain a minimum facet length or a common point between two neighbouring morphed facets for maximum reductive morphing. A minimum facet length refers to cross sections morphed in Sections 4.1, 4.2 and 4.4, and a common point between two neighbouring facets refers to morphing in Section 4.3. While such overlap can be dealt with in number theory and crystallography, we point out that – besides its complexity – such a description of NWire cross sections is not beneficial since the free choice of the nominal run index i per and subsequent morphing within these constraints covers virtually any convex NWire shape encountered in experiment. Apart from Section 4.3, where we introduce slightly different limits on run indices to prevent an overlap, such limitations are as follows. All lateral run indices have a minimum value of , resulting in maximum expansive morphing; see Figs. 9(a), 10(a) and 12(a). Run index doublets are limited to j + k_{1} = j + k_{2} = k_{1} + k_{2} = i − 1 in Sections 4.1 and 4.2, and to k_{1} + k_{2} = k_{2} + k_{3} = k_{3} + k_{1} = i − 1 in Section 4.4. The treated in Section 4.3 has a limit on run index doublets of k_{1} + k_{2} = k_{2} + k_{3} = k_{3} + k_{1} = i, because its high symmetry and atom interconnectivity at the corner points allows for facets not being directly morphed to be reduced to singular points on the boundary of the regular (versus minimum facet length for all other three cases). The basic principle of 3axes morphing and related implications for overlap in size and form of cross sections is depicted in Fig. 8.
As a result, all facets not being directly morphedAll number series reflect the variables we presented in Section 3, with additional series for facet lengths, NWire widths and heights, which depend on lateral (morphing) run indices. These are required in particular for finding the right indices to fit experimental values, such as facet lengths, heights or widths of NWire cross sections; see Section 5. As mentioned before in Section 3, the contribution of the nominal to the respective number series is printed in grey to facilitate the decomposition into contributions per run index. For the same reason, most number series will be shown uncompacted, followed by their shortest form.
4.1. 3Axes morphing of nanowires growing along the [110] direction and four {111} plus two {001} interfaces
For the number of atoms in the NWire
we obtainThe number of bonds between these NWire atoms are described by
The total number of interface bonds of the NWire
amounts toAs was the case with axialsymmetric morphing (see Equation 16), the number of {111} facet bonds being added/removed equals the number of {001} facet bonds being removed/added per change in j,k_{1} or j,k_{2}, whereby becomes independent of j.
The ratio of facet bonds at {111} to {001} interfaces is given by
whereby the top row in Equation 51 show the explicit number series per interface orientation and the lower row presents their ratio. The length of the top {001} facet is given by
whereby the analogy of to in clearly visible; see Equation 18. The length of the bottom {001} facet is given by
being equivalent to Equation 18.
The length of the two upper {111} facets depends only on the respective :
In Equation 54, we add or remove one X_{6} ring per change in , as is the case for j in Equation 19, underlining the high symmetry of the NWire The length of the two lower {111} facets depends on the respective and j, the latter limiting such facets from below:
These facet lengths are shown in Fig. 9(b).
Due to , the width of the k_{1},k_{2} only, which is a direct consequence of the morphing limits discussed at the beginning of Section 4:
depends onFor the same reason, the height of the j:
depends only onThe total area of the
naturally depends on all running indices:The underbrace in line 2 of Equation 58 indicates the offset area which is composed of four scalene and two isosceles triangles at the corner points of the see Appendix A and Fig. 16 for their derivation^{2}. The underbraces in line 3 of Equation 58 denote the contribution to maximum extensive morphing per class of lateral run indices j and k_{1},k_{2}, from which the respective area is subtracted when . Fig. 9 shows crystallographical details of this and a couple of examples of triaxial morphing.
4.2. 3Axes morphing of nanowires growing along the [11] direction with four {31} plus two {111} interfaces
For the number of atoms in the NWire
we obtainThe number of bonds between these NWire atoms are described by
The total number of interface bonds of the NWire
amounts toThe assignment of interface bonds to the {111} and {11} facets is shown in Fig. 10(b). With this assignment of interface atoms to {111} and {11} facets, we obtain
for the respective explicit number series in the top row and for the more compact form describing the ratio of interface bonds only. There are four different facet lengths which have to be used with their respective run indices j and , , as required for the facet of interest:
For an illustration of facet lengths, we refer to Fig. 10.
As discussed in Section 4.1 around Equation 56, the width of the depends on k_{1},k_{2} only:
For the same reason, the height of the j:
depends only onThe total
area is presented byAs for Equation 58, we have assigned the contribution to maximum extensive morphing per class of lateral run indices j and k_{1},k_{2} from which the respective area is subtracted when , before converting Equation 69 to its shortest form. For an illustration of irregular 3axes morphing of this refer to Fig. 10.
4.3. 3Axes morphing of nanowires growing along the [111] direction with a hexagonal and six {110} interfaces
This and 4.2. All three morphing areas are identical and subject to their respective run index, which becomes apparent if we look at their interface orientations, which are identical to each other. As a result, we introduce just one class of run indices k_{1}, k_{2}, k_{3}. We also point out that the morphing areas are identical to those depending on j in Section 3.5. The difference occurs by the morphing of opposite areas (referring to a C_{2} symmetry), while here we morph three equal areas – subject to identical run indices – when rotated by 120° (C_{3} symmetry). As there is no overlap in morphing regions for one under the constraint that the other two k indices are ≤0 [see Fig. 11(c)], and the ultimate corner point of the extensive morphing occurs for [see Fig. 11(a)], we can extend the all the way to . Still, the indices k_{1}, k_{2} and k_{3} are restricted over along , where are cyclic permutations of the run indices, viz. [1,2], [2,3], [3,1]. Thereby, we avoid the morphing of the three triangular areas running into each other. As examples, if i = 9 and k_{1} = 9, we have −i ≤ k_{2} + k_{3} ≤ 0; if i = 9 and k_{2} = 7, we have −i ≤ k_{3} + k_{1} ≤ 2, etc.
has a higher symmetry, as opposed to the two previous cases in Sections 4.1For the number of atoms in the NWire
we obtainThe final form of Equation 70 summarizes the terms which depend on into a sum for brevity; we will use this short form in all subsequent equations where applicable. The number of bonds between these NWire atoms is described by
The total number of interface bonds of the NWire
amounts toThere are two types of interface lengths. One represents the facets normal to the morphing vector given by atomic planes being added or subtracted, and depends only on the respective :
Another interface length exists for facets which are modified in their length by the two adjacent morphing regions:
Due to the C_{3} symmetry of the hexagonal its height can be calculated along all three morphing vectors:
The calculation of the , which should be sufficient to assign run indices to an experimental image.
width does not appear to be useful. It would require a discrimination to depart from the nominal width when , adding an increment of , which is somewhat cumbersome in handling experimental data. We therefore rely onto the height of the as per Equation 75The total
area is presented byThe maximum external morphing per assigned in the top row of Equation 76 is straightforward to see in Fig. 11(a), where the three equilateral triangles cover half of the nominal consisting of six of such triangles. Morphing examples are shown in Figs. 11(b) and 11(c).
4.4. 3Axes morphing of nanowires growing along the [111] direction with a hexagonal and six {11} interfaces
This and 4.2, together with the restriction , where are cyclic permutations of the run indices; see beginning of Section 4. The symmetry considerations given in Section 4.3 also apply to this which has exclusive {} facet orientations.
reverts back to the restrictions on the run indices we had for cross sections in Sections 4.1For the number of atoms in the NWire
we obtainThe number of bonds between these NWire atoms are described by
The total number of interface bonds of the NWire
amounts toBy analogy to Equations 73 and 74, we have two types of facet lengths with their respective lateral run indices, namely, the facets normal to the morphing vector given by atomic planes being added or subtracted
and for facets which are modified in their length by the two adjacent morphing regions:
As was the case in Section 4.3, the C_{3} symmetry of the hexagonal allows for its height to be calculated along all three morphing vectors:
The total
area is presented byThe maximum external morphing per assigned in the top row of Equation 83 is straightforward to see in Fig. 12(a), where the three axialsymmetric trapezoids fold back onto the nominal which consists of six such trapezoids, whereby the two central atoms shown in black in Fig. 12(a) are not covered in the folding process. Morphing examples are shown in Figs. 12(b) and 12(c).
5. Application examples
5.1. Si NWires
Si NWires have been shown to grow as monolithic crystals along the [111] axis with atomically flat interfaces when aluminium (Al) is used as a seed catalyst (Moutanabbir et al., 2011). Such NWire cross sections are shown in Fig. 13. We picked two examples from this reference to show the usage and results derived by the morphing algorithms from experimental input. Table 4 shows all parameters and results of the cross sections shown in Figs. 13(b) and 13(c), respectively.

Due to several run indices present, the fitting onto the exact ).
shape requires an iterative process which is well suited to a computer code. Such a code could be added to existing visual software for gauging NWire cross sections – a task we illustrate here in a stepwise fashion as a principal guide. As unitcell parameter for Si, we use = 0.54309 nm (Böer, 1990There are two ways to start an iteration for obtaining the run indices. The first is to start with one (Equation 74) and its two adjacent (Equation 73), rearranging for the three run indices such as i, k_{1},k_{2}. This approach may be more appealing to the experimentalist, and is illustrated on a coreshell NWire in Section 5.2. The more direct starting point for the iteration is given by using (Equation 75) and the , which is at one end of , rearranging only for the two run indices involved.
With the measured height and interface lengths h and a, as listed for Fig. 13(b) in Table 4, we get
yielding and i = 127. The tolerance range for k_{1}, k_{2} and k_{3} originates from the tolerance in length measurement which is in the range nm; see Fig. 13 and Table 4. Such tolerance ranges translate into ranges for and i which serve to minimize the difference to the respective calculated length ; see Equation 85 below and Equation 86 in Section 5.2. With i known, we can proceed with Equation 84(II) to get a start range for the other two , yielding k_{2}(i = 127, c) = and k_{1}(i = 127, e) = . Next, we rearrange Equation 74 for its run indices, viz.
Equation 85 provides us with a sum of , which we can match under consideration of the range of and . For , we get k = −43 … −41 = k_{1} + k_{2}. This range allows for and k_{2} = −17 … −19. Moving on to = f(X = b), we obtain k = −43 … −41 = k_{2} + k_{3}. This range allows for and , with the valid doublets of [k_{2};k_{3}] = [−17;−24], [−18;−23], [−19;−22], [−18;−24], [−19;−23] and [−19;−24]. Arriving at the last interface length , we get k = −45 … −43 = k_{3} + k_{1}, allowing for and k_{1} = −22 … −23, yielding [k_{3};k_{1}] = [−23;−22], [−22;−23], [−22;−22]. It becomes apparent that with ongoing iteration over an increasing number of interface lengths , the indices get narrowed down towards one integer value. We can narrow down the range for the further by reconsidering Equation 84(I) with i = 127 and to match the experimental value h [Fig. 13(b) and Table 4]. This is best achieved using = −22) = 91.8 nm, leaving just 0.4 nm to match the measured value of h. With k_{3} = −22, we can iterate again, obtaining k_{2} = −19 and k_{1} = −23 ± 1 = −23, whereby we chose the centre of the k_{1} range to arrive at a minimum deviation from the measured length parameters of the NWire An iterative computer code would modify k_{1}, k_{2} and k_{3} around their initial values such that the sum of the absolute deviation values of all the NWire length parameters from their measured counterparts is minimized as the criterion to arrive at the best structural fit of the number series; see Equation 86 in Section 5.2. The indices [i, k_{1}, k_{2}, k_{3}] = [127, −23, −19, −22] can then be used in Equations 77 to 83 to calculate the structural results. These are shown in Table 4, together with the results for the in Fig. 13(c).
The flexibility of the above algorithms in describing the NWire et al., 2021) can be tracked and linked to the surface energies of respective facets. Tracking such changes over time with our crystallographic description and the energy intake by local heating should allow the diffusion process to be described in much detail. Such findings can be key to NWire design on demand.
could also be very useful for NWire shapes changing by postgrowth extrinsic means. As an example, an atomic rearrangement at Si NWires due to high current densities inducing local heating (Bahrami5.2. Coreshell III–V zbNWires with different unitcell parameters a_{uc}
III–V NWires are often found to grow along the [111] axis which requires the least energy and have hexagonal cross sections (Joyce et al., 2011; Treu et al., 2015). Here we will focus on coreshell GaAs–In_{0.2}Ga_{0.8}As zbNWires grown by solidstate molecular beam (MBE) along the [111] axis with {110} interfaces, using visual and crystallographic data from Balaghi et al. (2019). Fig. 14 shows the NWire with crystallographic details and the assignment of variables to respective interface lengths and one UC height of the NWire cross section.
Below, we will show how we can derive structural results using Equations 70 to 76 in Section 4.3 to match all interface lengths to their measured value (Table 5), taking the different unitcell parameters for core and shell of NWire, and , into account. The resulting run indices allow us to obtain the main variables , , and for the core and total NWire cross sections, hereafter denoted as , , and A, respectively, to keep the prsentation of variables as simple as possible. The same simplification goes for all and , hereafter denoted as and h, respectively. From these data, we can derive the main variables of the shell by simple differential/additive calculations. The calculation of run indices for the core and shell of the NWire require additional indexing of the run indices to avoid confusion: run indices using the unitcell parameter of the NWire core will be , and run indices using the unitcell parameter of the NWire shell will be .

We set out with the ) and use the lengths of three adjacent interfaces as a starting point, thereby illustrating the second method briefly mentioned in Section 5.1 of how to find the run indices , , , NWire cross sections.
of the NWire core (Fig. 14The convergence criterion (CC) we use for obtaining the run indices which describe the NWire with minimum deviation is given by the sum of absolute deviations of the calculated , and – see Equations 73 to 75 in Section 4.3 – from their respective measured values a to f plus h_{1} to h_{3} for the NWire core, and A to F plus H_{1} to H_{3} for the NWire shell; see Fig. 14(c):
Please observe that below we will work with only one height per (c) readable; the inclusion of all three heights into a computer code featuring Equation 86 is straightforward. As a starting point for all cross sections, we set = 0 (presenting the nominal regular shape of the NWire cross section) and run an iteration scheme with Equation 86 in compound with Equation 87, using i^{α} as a run index to arrive at an educated guess from where to start the fitting procedure.
to keep the explanation of the fitting procedure concise and Fig. 14Our search criterion features a length of an interface which depends on three run indices, , plus its adjacent interface lengths and . We start with choosing = 1 and = 2 to obtain the following convergencies: → d, → e and → c, with c, d and e being the measured interface lengths listed in Table 5:
When treating the (c) – c, d and e change to C, D and E – and set α = shell in Equation 87. The last line of Equation system 87 adds up Equations (I) to (III), thereby eliminating all , yielding the search condition for a regular hexagonal to make an educated guess at a starting value for . The unitcell parameter we use is the value for GaAs, = 0.56533 nm (Böer, 1990), which comprises the NWire core; see Fig. 14. If i^{α} is located nearly halfway between two integer values, we may run the calculations below with both adjacent i^{α} values to see which one has the lower CC as per Equation 86. Once we got , we can run each single Equation 87 (I) to (III) for obtaining and . We then move on to the next and , namely, and , and to the next three measured interface lengths e, f and a. The third index rotation then uses and with the interface lengths a, b and c. As a result, we obtain the final results for the NWire core, , , and . When iterating for the NWire shell we proceed the same way, using the adequate variables as mentioned above.
of the NWire shell, we have to use the interface lengths presented by capital letters in Fig. 14As in Section 5.1, we use the tolerance in length measurement (Fig. 14 and Table 5) to provide narrow ranges for all k^{α} and h^{α} around their calculated value, aiming for a minimum CC while keeping i^{α} constant. For the NWire core, the result of this fitting procedure with CC → min are the run indices and . Replacing a to e and h_{1} to h_{3} in the above process with A to E and H_{1} to H_{3}, and using = 0.57343 nm (Adachi, 2004) as the unitcell parameter of the shell material featuring In_{0.2}Ga_{0.8}As, we get and for the NWire shell. All these indices are shown in Table 5, together with the measured interface length and height of the respective [Fig. 14(c)].
The run indices of the NWire core . Deriving the parameters of the shell and eventually of the entire NWire system requires some additional calculations. After calculating and with , we apply an increment to , viz. , and use the same for a calculation of with . We then calculate the cross sections of the core with and and the same as obtained with . The transition adds one atomic ML to the accounting for the interface region between the core and shell material as seen from the NWire core using :
can be used directly to calculate all of its parameters discussed in Section 4.3We then iterate again as per the above description for the NWire core, but now use , see the third row in Table 5, obtaining and .
With this result, we again apply the increment and use the same for a calculation of with . This transition adds one atomic ML to the core
as well, using the unitcell parameter of the shell material, accounting for the interface region between the core and shell material, as seen from the NWire shell using :With both interface areas, we can now calculate their average value as the most accurate interface area we can obtain:
The reason we use two descriptions of the interface area is given by the transition of the unitcell parameters when going from the core to the shell material. This approach can be further exploited for an MLwise increment in i^{α}, which adds further precision if a radial distribution of the unitcell parameter is known, such as in Fig. 3(a) in Balaghi et al. (2019), and is further discussed in Section 5.2.1. The indices and of the NWire core without and with one ML as interface region for each, and , have further use for other interim data we use to arrive at our final results. Before we can derive , , and of the NWire shell, and , , and A^{full} of the complete NWire, we need to carry out another iteration scheme using and (Equation 86) for the total NWire with . The resulting , and with their respective running indices are then used in straightforward differential calculations.
with for each increment inThe calculation of the number of atoms within the NWire shell is obtained from
thereby eliminating all atoms from the total NWire area in the core region. The calculation of requires , where we use and obtained in Equation 89. In addition, we have to substract , as such bonds belong to the interface region between the core and shell:
From a practical viewpoint, the calculation of delivers two values. For spectroscopic characterization techniques where no carrier recombination is involved, such as Raman, Fouriertransform infrared (FT–IR) or and shell, whereby we count for in addition to including them for the core:
the coreshell interface bonds are considered as dipoles whereby they get counted only once with the core for the complete NWire, resulting in . For spectroscopic characterization techniques where carrier recombination is involved, such as or carrier lifetime spectroscopy, the interface bonds at the coreshell interface can acquire and trap free carriers from the core, as well as from the shell. Hence, these bonds have an impact on the coreSince the number of interface bonds does not directly depend on versus for epitaxial NWire growth, we can drop their dependence on the unitcell parameter in Equation 93. The area of the NWire shell requires the area of the NWire core and the coreshell interface region to be subtracted from the total NWire area, viz.
The final results of the full NWire a_{uc} – follow from the addition of results from core and shell, namely,
– core and shell with their respectiveThe area of the internal interface (Equation 90) is another final result not included in A^{full}, since NWire interfaces behave in a significantly different manner from the core and shell regions in terms of electronic and optical properties, such as carrier transport, recombination and interface dipoles. The calculation of the dimensionless crystallographic parameters , and , which allow for interNWire comparison, are straightforward. Table 5 shows all the measured lengths of the coreshell NWire depicted in Fig. 14, together with selected interim results for the of the NWire core using listed in column 3, and all final results.
5.2.1. Flexibility of calculations for coreshell NWires
We have briefly mentioned above that – due to i → i+1 adding a defined ML (usually atomic ML) to the NWire – we can introduce a unitcell parameter with a radial dependence to the entire coreshell NWire. If such a dependence is known, e.g. for the NWire shell (Balaghi et al., 2019), the precision of the NWire description can be further increased. We note here that nonradial deviations of , such as local inhomogeneities, cannot be accounted for due to the radial layer dependence of all number series with their main run index i.
Since our analytic treatment of zbNWire cross sections works on the basis of smallest area segments coming along with every atom and bond considered, the arrangement of the core and shell to each other is flexible over a wide range. To illustrate the implications, the NWire core does not have to be located in the centre of the NWire shell, nor does any restriction exist for the core and shell NWire cross sections to be morphed independently from each other. This finding can be verified in our above example (Fig. 14 and Table 5). It becomes apparent from Fig. 14 that the NWire core is not aligned with the NWire shell to share the same symmetry centre. We can go further down this path and adapt the outer NWire shape to a different interface orientation, as would be the case for a coreshell NWire growing along the [111] axis, with internal {110} and external interfaces (Fig. 15). Such an NWire requires a partition into three sections, of which two are treated in accord with the core and shell sections in the above numerical example (Fig. 14). The third section describes the outermost shape of the NWire shell, where the change in interface orientation occurs. When assembling the respective variable , , and A for the entire NWire, we calculate the full shell (index `tot' in above example) of the entire NWire with outer interfaces and then simply substract the core with , yielding the above variables for the shell with different faceting at the inner and outer interfaces. Such coreshell NWire descriptions can be chained to describe multicoreshell cross sections by repeating the calculations shown in this section for every coreshell pair.
6. Conclusions
Building on our previous work (König & Smith, 2021), we introduced extensions into analytical number series for zb and diamondstructure NWires for adapting their cross sections to arbitrary shape (morphing), covering the following NWire cross sections: square, 〈001〉 growth axis and interfaces; rectangular, 〈110〉 growth axis and {110} plus {001} interfaces; hexagonal, 〈110〉 growth axis and {001} plus {111} interfaces; hexagonal, 〈〉 growth axis and {111} plus interfaces; hexagonal, 〈111〉 growth axis and {110} interfaces; hexagonal, 〈111〉 growth axis and interfaces. Our extensions provide the exact crystallographic description of zbNWires with arbitrary cross sections as encountered in experiment, and thus are only limited in their precision by measurement tolerances of the imaging technique used. As previously, the results we obtain by our analytics are the number of NWire atoms , the number of bonds between such atoms , the number of NWire interface bonds and areas A. We demonstrated that our analytic description is applicable with the same precision to coreshell NWires with arbitrary shape and interface orientation of the core and shell, under the constraint that they share the same orientation of their growth axis, and have an interfaces roughness below the tolerance limit of the measured interface lengths. The above results are available per core and shell section of the NWire, and internal (coreshell) interface areas are given as well. If a radial distribution of the unitcell parameter can be provided, such data can be included for all mentioned NWire cross sections, adding further flexibility and precision to their description. The description of coreshell NWires can easily be applied to multiple coreshell (layered) NWires if these comply with epitaxial growth and smooth interfaces.
The analytic description of zb and diamondstructure NWire cross sections with arbitrarily convex shape and multiple radial layers (multiple coreshell structures) can provide major advancements in experimental data interpretation and understanding of III–V, II–VI and groupIVbased NWires. The number series allow for a deconvolution of experimental data into environmentexerted, interfacerelated and NWireinternal phenomena. Our method offers an essential tool to predict NWire cross sections and to tune process conditions for tailoring NWires towards desired shape and interface properties, see König & Smith (2019), König & Smith (2021) and König (2016) for details.
APPENDIX A
Geometric details for NWire cross sections with [110] growth vector
We start the analysis with the zbUC projection into the {110} plane which defines the NWire cross sections; see Fig. 16. We then look at unit lengths to calculate the interface lengths, width and height of the For {001} facets, this unit length is . For the {111} facets, its unit length is straightforward to obtain from the 〈 111〉 diagonal through the {110} plane of the UC, which is . Amounting to half of the length of this diagonal, we get . Each facet has length offsets which are required to arrive at the correct interface length, width and height of the For , two identical offsets exist, each being . The total offset is thus ; see Equations 52 and 53. For the {111} facets, two offset lengths and exist.
For deriving , we use the diagonal through the {110} plane of the UC. When going through the {110} plane along the vector, we see that we have four equidistant atomic MLs. These divide the diagonal of the UC into four equal parts which each fit . We therefore get . For deriving and the width offset , we use and the interML distance in the direction, which is . We first note that we can use the intercept theorem to obtain , hence , and thus . Via the opposite corner, the total width offset is then , adding 1/8 to the −4/8 offset in Equation 56^{3}. With obtained, we can calulate using Pythagoras' theorem, viz.
The total offset for the {111} facet lengths is thus . Taking into account, we get . Noting that , we see the offset implemented in Equations 54 and 55 by the substraction of 1/4 from the run index i. The offset in h becomes apparent when looking at the number of atomic MLs increasing in the direction with i as 3, 7, 11, … for i = 1, 2, 3, etc. The −1 ML offset to thus accounts for the offset of −1/4 as per Equation 57.
Next, we work out the areas which form the total as = = . Each of the four corners between the {001} and {111} facets has a scalene triangle as an offset area. We can use translational, reflectory and rotational geometry operations, as illustrated in Fig. 16, to show that = . Adding up all offset areas, we get
The hexagonal rings in the {110} plane are the basic area unit of this amounting to = = , which is equivalent to 1/4 of the {110} plane of the UC. Breaking down further, we arrive at the area for the open isosceles triangles at the {001} facets which each have an area of . These two areas are required to calculate the area as a function of run indices. There are six offset areas presented by one triangle at each corner of the hexagon which can be calculated by using the offset lengths derived in the previous paragraph. Two isosceles triangles exist at the two corners between {111} facets. Their area is straightforward to derive from Fig. 16With the prefactor of the unit area used in Equation 58, we finally get
APPENDIX B
Geometric details for NWire cross sections with [11] growth vector
In contrast to the rather nontrivial growth vector class, the geometrical details of the plane class {111} top and bottom facets. We start with the diagonal through the {111} plane of the UC, as shown in Fig. 17, amounting to . This length is composed of three bonds which lie in the plane, with each being long, plus three bonds sticking out of the plane at an angle of = 54.73° with a projected length in the plane of each. From these two lengths, we get the bigger length of the unit area as = , or alternatively via = . This length presents the increment unit for the height of the see Equation 68. The unit length of the {111} facet along the vectors is given by (see Equations 63 and 64). With these two lengths, we can calculate the unit area required to obtain the total area of the as = = (see Equation 69), whereby we used = . The length is also used as the unit length for the width of the see Equation 67. The diagonal of this rectangular unit area presents the unit length of the facets as ; see Equations 65 and 66.
are rather simple, mostly owing to theAPPENDIX C
Geometric details for NWire cross sections with [111] growth vector
We start our analysis by deriving the unit area, followed by the derivation of unit lengths for the width, height and facets per
type.The unit area used for calculating the total area of NWire cross sections with [111] growth vector is given by the equilateral triangle shown in red in Fig. 18. Its area is straightforward to obtain by projecting the {111} plane cut through the UC into the scheme as shown by the blue large triangle in Fig. 18. This equilateral triangle has a side length of , which is twice as big as the unit length for the facets; . The {111} plane cut through the UC has a total area of and consists of six equilateral triangles plus six isosceles triangles. All of these triangles have the same area, as is apparent from the thin dotted blue lines within the {111} plane of the UC in Fig. 18 with the use of some trivial mirrorsymmetry operations. We thus get the unit area of the as , see the prefactor in Equations 76 and 83. The side length of the unit area follows straight from the fact that this triangle is equilateral as well, resulting in . Hexagonal symmetry shows that is also the unit length of the {110} facets; , see the prefactor in Equations 73 and 74, and 1/3 of the increment in height for cross sections with facets, cf. the prefactor in Equation 82. Looking at from the perspective of the {110} facets, we get the increment in height for cross sections with {110} facets as h_{Δ} = ; see the prefactor in Equation 75. This Equation makes it clear that the transition for is required for morphing to accommodate height changes with .
Footnotes
^{1}We also note here that the offset area was corrected again with respect to König & Smith (2021). This correction amounts to a mere 0.0644, which translates to ca. 1.95 Å^{2} for semiconductors such as Si or GaAs, and hence should have no practical relevance in experiment.
^{2}We also note here that the offset area was corrected again with respect to König & Smith (2021). This correction amounts to a mere 0.0644 which translates to ca. 1.95 Å^{2} for semiconductors such as Si or GaAs, and hence should have no practical relevance in experiment.
^{3}The offset −4/8 originates from 1/2, 3/2, 5/2, … double hexagonal rings, each having a width of , for i = 1, 2, 3, etc., as can be seen from the prefactor of Equation 56.
Acknowledgements
Open access publishing facilitated by University of New South Wales, as part of the Wiley – University of New South Wales agreement via the Council of Australian University Librarians.
Funding information
Funding for this research was provided by: 2018 TheodorevonKàrmàn Fellowship (award to Dirk König).
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