**2. Theory (methodology)**

#### **2.1 Reciprocal space structures of 1D, 2D, and 3D crystals**

Electron diffraction is one of the most powerful tools for investigating crystalline structures, particularly nanomaterials and material surfaces owing to their larger atomic scattering cross sections *σ* than those obtained with X-ray diffraction,

**Figure 1.** *Concept of our approach toward realizing atomically ordered 3D structures.*

*Creation and Evaluation of Atomically Ordered Side- and Facet-Surface Structures of Three… DOI: http://dx.doi.org/10.5772/intechopen.92860*

for instance, *σ* � 4 Mb for a 15 keV electron and 0.002 Mb for a Cu K (�8 keV) X-ray to a Si atom [10, 11]. According to the kinematic diffraction theory, a diffraction pattern can be understood as a reciprocal structure pattern reflecting a crystalline structure (**Figure 2**) in a reciprocal space map (RSM). By comparing an experimental reciprocal structure pattern in an RSM with patterns of candidate crystalline structures, we can analyze the characteristics of crystalline materials, such as dimensionality, atomic structure, orientation, size, and strain [4, 12–16].

For instance, an ideal one-dimensional (1D) material with lattice constant *a* in the *x* direction in real space (**Figure 2(a)**) shows a characteristic reciprocal structure pattern consisting of reciprocal lattice planes with a reciprocal lattice unit length of *<sup>a</sup>*<sup>∗</sup> <sup>¼</sup> <sup>2</sup>*π=<sup>a</sup>* in the *<sup>x</sup>* direction (**Figure 2(b)**). This can be explained by the Bragg interference condition *a* cos *θout* � *a* cos *θin* ¼ *nλ*, that is, *k* ! *out* � *k* ! *in* <sup>þ</sup> *na*<sup>∗</sup> <sup>∙</sup> *<sup>e</sup>* ! *<sup>x</sup>* ¼ 0. Here, *λ* is wavelength, and *k* ! *in* and *k* ! *out* are incident ! ! 

and outgoing wave-number vectors, respectively, satisfying *k*<sup>0</sup> ¼ *k in* <sup>¼</sup> *<sup>k</sup> out* ¼ 2*π=λ*. Since the intersection of reciprocal lattice planes with an Ewald sphere of radius *k*<sup>0</sup> produces a diffraction pattern (**Figure 2(c)**), diffraction rings are observed as the diffraction patterns of 1D materials.

A 2D material with lattice constants *a* and *b*, that is, unit vectors *a* ! and *b* ! (**Figure 2(d)**), shows reciprocal lattice rods with reciprocal lattice unit vectors of *a* ! <sup>∗</sup> and *b* ! <sup>∗</sup> , corresponding to the intersection of orthogonal reciprocal lattice planes (**Figure 2(e)**). The intersection of reciprocal lattice rods with an Ewald sphere produces a pattern of diffraction spots arrayed on arcs. A reciprocal lattice rod at position *ha*!<sup>∗</sup> þ *kb* !<sup>∗</sup> is called an ð Þ *h k* rod, which produces an ð Þ *h k* 2D diffraction

#### **Figure 2.**

*Schematics of (a)–(c) 1D, (d) and (e) 2D, and (f) and (g) 3D crystals: real space structures (a), (d), and (f) and corresponding reciprocal space structures (b), (c), (e), and (g). 1D, 2D, and 3D crystals show reciprocal lattice planes, reciprocal lattice rods, and reciprocal lattice points, respectively.*

prepared by a conventional lithographic method can be used to form 3D

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grated circuits have been individually developed from surface science.

structures and their physical properties.

**2.1 Reciprocal space structures of 1D, 2D, and 3D crystals**

*Concept of our approach toward realizing atomically ordered 3D structures.*

Electron diffraction is one of the most powerful tools for investigating crystalline structures, particularly nanomaterials and material surfaces owing to their larger atomic scattering cross sections *σ* than those obtained with X-ray diffraction,

**2. Theory (methodology)**

**Figure 1.**

**92**

nanostructures. Although considerable attention has been devoted to controlling the size, shape, and positioning in research on 3D patterning, little attention has been focused on the atomic ordering of arbitrarily oriented surfaces on 3D patterned substrates. The realization of perfect surfaces on 3D structures is required to produce high-quality samples. Since material growth starts on a surface, the surface condition clearly determines the structural and physical properties of the grown material. Although techniques for studying 2D surfaces, namely, surface science, have been intensively developed and established, little attention has been devoted to controlling the atomic ordering and structures of side-surfaces on 3D architectures. Fabrication techniques for 3D nanoscale structures that are promising for 3D inte-

The subject of study should be changed from 2D planar surfaces to 3D assembly surfaces to enable atomically ordered nanofabrication on vertical side-surfaces and/or tilted facet-surfaces in 3D space (**Figure 1**). For this purpose, a simple and accurate structure evaluation technique is required. Currently, scanning electron microscopy (SEM) is widely used for the observation of 3D nanostructures. However, SEM cannot be used to evaluate structures involving atomic ordering. On the other hand, transmission electron microscopy (TEM) is a powerful technique for examining the atomic structure of 3D nanomaterials, but it is destructive and not convenient. An alternative technique is a diffraction method. Reflection high-energy electron diffraction (RHEED) enables the examination of surface properties such as atomic ordering, surface roughness (flatness), and surface homogeneity [4]. Low-energy electron diffraction (LEED) is also a conventional and nondestructive surface observation technique [5]. To apply these techniques to 3D structured samples with various oriented surfaces, instead of 2D planar samples, an appropriate alignment of the incident electron beam considering the configuration in 3D space is indispensable [6–9]. In this chapter, we first show how to obtain and evaluate RHEED and LEED patterns from 3D structured samples by explaining the principle of diffraction. Then, we demonstrate creating and evaluating atomically ordered side- and facetsurface structures of 3D silicon nano-architectures. Finally, we discuss novel structures that have been constructed on 3D patterned Si to form 3D interconnected

spot. A 3D material with unit vectors *a* !, *b* ! , and *c* ! (**Figure 2(f)**) shows reciprocal lattice points with unit vectors of *a* !∗ , *b* !∗ , and *c* !<sup>∗</sup> (**Figure 2(g)**). The intersection of reciprocal lattice points with an Ewald sphere produces a pattern of diffraction

spots arrayed on a lattice. A reciprocal lattice point at position *ha*!<sup>∗</sup> , *kb* !∗ , and *l c*!<sup>∗</sup> produces an ð Þ *hkl* 3D diffraction spot.

From diffraction patterns (e.g., diffraction rings or spots) projected on a detection screen, the original reciprocal structure pattern (e.g., reciprocal lattice planes, rods, or points) in a 3D RSM can be regenerated by changing the incident angle of the electron beam to the crystalline material (i.e., the direction of *k* ! *in*) or the beam energy (i.e., *k*0), as illustrated in **Figure 3** for a 3D material with azimuth angle rotation [12–15]. For example, **Figure 4(a)** and **(b)** show a series of transmission electron diffraction patterns of a nanocrystal on a Si 001 ð Þ substrate as a function of the azimuth angle around the substrate surface normal direction [14]. Indeed, the conversion of diffraction spots on a lattice (indicated by orange arrows) in different diffraction patterns generates reciprocal lattice points in a 3D RSM, as indicated by orange arrows in **Figure 4(c)**–**(e)**. In this 3D RSM, we can recognize the existence of a certain 3D crystalline structure, in this case the structure is *α*-FeSi2, among candidate crystal structures [17, 18]; the crystalline orientations are *α*-FeSi2ð Þ 110 ∥ Si 001 ð Þ and *α*-FeSi2½ � 001 ∥ Si 110 h i from the reciprocal structure pattern.

**Figure 4.**

**Figure 5.**

**95**

*a*∗

*Ep is 15.0* keV *(k*<sup>0</sup> <sup>¼</sup> *62.7* <sup>Å</sup>�<sup>1</sup>

*(a) and (b) RHEED patterns of* α*-FeSi2*ð Þ 110 ½ � 001 ∥ *Si*ð Þ 001 h i 110 *at azimuth angles* ϕ *= 0*° *and 45*°*, respectively.*

*Creation and Evaluation of Atomically Ordered Side- and Facet-Surface Structures of Three…*

*DOI: http://dx.doi.org/10.5772/intechopen.92860*

*of RHEED patterns. In (a) and (b), diffraction spots indicated by orange and red arrows are assigned to reciprocal lattice points of 3D* α*-FeSi2 nanocrystals and reciprocal lattice rods of a 2D Si*ð Þ 001 *substrate surface, respectively. Reciprocal lattice rods [e.g.,* ð Þ 00 *and* ð Þ 01 *] lie on Laue zones (e.g., L0 and L1). DB denotes the direct beam position.*

*Schematics of 2D diffraction spots and reciprocal lattice rods of (a) substrate surface and (b) side-surface of 3D*

*fabricated material. In both cases, diffraction spots are elongated in the surface normal direction.*

<sup>0</sup> <sup>¼</sup> <sup>2</sup>π*=a*<sup>0</sup> <sup>≈</sup>2π*=*3*:*<sup>84</sup> <sup>Å</sup>�<sup>1</sup> h i *is the reciprocal lattice unit length of Si*ð Þ <sup>001</sup> *<sup>1</sup>*�*1.*

*). (c) Top view and (d) and (e) side views of the 3D RSM regenerated from a series*

In **Figure 4(a)** and **(b)**, there are also diffraction spots on arcs indicated by red arrows. These spots are often observed under a glancing condition of an electron beam nearly parallel to a substrate surface in RHEED with a primary energy *Ep* of typically 10–15 keV. The conversion of the spots generates reciprocal lattice rods perpendicular to the substrate surface in a 3D RSM, as indicated by red arrows in **Figure 4(c)**–**(e)**. The existence of spots on arcs or reciprocal lattice rods implies that the surface perpendicular to the rods is atomically well-ordered; in this case, it implies the formation of a well-defined clean Si 001 ð Þ substrate surface.

The width of a reciprocal lattice rod is finite (full width at half maximum ≃ 2*π=D*) and reflects the crystalline domain size *D*. Thus, the intersection of the rod with an Ewald sphere is elongated along the rod direction, that is, the surface normal direction. Indeed, in **Figures 4(a)** and **(b)** and **5(a)**, there are the surface spots (red arrows) elongated along the Si 001 ½ � substrate surface normal direction.

#### **2.2 Diffraction patterns from side-surfaces of 3D structured sample**

One of the advantages of RHEED is its ability to observe atomically ordered surfaces in any direction, as described later in Sections 3.2 and 4. RHEED has been used to investigate crystalline structures on planar substrate surfaces. Recently, for

#### **Figure 3.**

*Schematic of relationship between diffraction spots and reciprocal lattice points intersecting with a partial Ewald sphere in 3D RSM, for 3D crystal.*

*Creation and Evaluation of Atomically Ordered Side- and Facet-Surface Structures of Three… DOI: http://dx.doi.org/10.5772/intechopen.92860*

#### **Figure 4.**

spot. A 3D material with unit vectors *a*

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produces an ð Þ *hkl* 3D diffraction spot.

lattice points with unit vectors of *a*

!, *b* ! , and *c*

reciprocal lattice points with an Ewald sphere produces a pattern of diffraction

energy (i.e., *k*0), as illustrated in **Figure 3** for a 3D material with azimuth angle rotation [12–15]. For example, **Figure 4(a)** and **(b)** show a series of transmission electron diffraction patterns of a nanocrystal on a Si 001 ð Þ substrate as a function of the azimuth angle around the substrate surface normal direction [14]. Indeed, the conversion of diffraction spots on a lattice (indicated by orange arrows) in different diffraction patterns generates reciprocal lattice points in a 3D RSM, as indicated by orange arrows in **Figure 4(c)**–**(e)**. In this 3D RSM, we can recognize the existence of a certain 3D crystalline structure, in this case the structure is *α*-FeSi2, among candidate crystal structures [17, 18]; the crystalline orientations are *α*-FeSi2ð Þ 110 ∥

, and *c*

From diffraction patterns (e.g., diffraction rings or spots) projected on a detection screen, the original reciprocal structure pattern (e.g., reciprocal lattice planes, rods, or points) in a 3D RSM can be regenerated by changing the incident angle of

In **Figure 4(a)** and **(b)**, there are also diffraction spots on arcs indicated by red arrows. These spots are often observed under a glancing condition of an electron beam nearly parallel to a substrate surface in RHEED with a primary energy *Ep* of typically 10–15 keV. The conversion of the spots generates reciprocal lattice rods perpendicular to the substrate surface in a 3D RSM, as indicated by red arrows in **Figure 4(c)**–**(e)**. The existence of spots on arcs or reciprocal lattice rods implies that the surface perpendicular to the rods is atomically well-ordered; in this case, it

!∗ , *b* !∗

spots arrayed on a lattice. A reciprocal lattice point at position *ha*

the electron beam to the crystalline material (i.e., the direction of *k*

Si 001 ð Þ and *α*-FeSi2½ � 001 ∥ Si 110 h i from the reciprocal structure pattern.

implies the formation of a well-defined clean Si 001 ð Þ substrate surface.

**2.2 Diffraction patterns from side-surfaces of 3D structured sample**

**Figure 3.**

**94**

*Ewald sphere in 3D RSM, for 3D crystal.*

The width of a reciprocal lattice rod is finite (full width at half maximum ≃ 2*π=D*) and reflects the crystalline domain size *D*. Thus, the intersection of the rod with an Ewald sphere is elongated along the rod direction, that is, the surface normal direction. Indeed, in **Figures 4(a)** and **(b)** and **5(a)**, there are the surface spots (red arrows) elongated along the Si 001 ½ � substrate surface normal direction.

One of the advantages of RHEED is its ability to observe atomically ordered surfaces in any direction, as described later in Sections 3.2 and 4. RHEED has been used to investigate crystalline structures on planar substrate surfaces. Recently, for

*Schematic of relationship between diffraction spots and reciprocal lattice points intersecting with a partial*

! (**Figure 2(f)**) shows reciprocal

!<sup>∗</sup> (**Figure 2(g)**). The intersection of

!∗ , *kb* !∗

!

, and *l c* !∗

*in*) or the beam

*(a) and (b) RHEED patterns of* α*-FeSi2*ð Þ 110 ½ � 001 ∥ *Si*ð Þ 001 h i 110 *at azimuth angles* ϕ *= 0*° *and 45*°*, respectively. Ep is 15.0* keV *(k*<sup>0</sup> <sup>¼</sup> *62.7* <sup>Å</sup>�<sup>1</sup> *). (c) Top view and (d) and (e) side views of the 3D RSM regenerated from a series of RHEED patterns. In (a) and (b), diffraction spots indicated by orange and red arrows are assigned to reciprocal lattice points of 3D* α*-FeSi2 nanocrystals and reciprocal lattice rods of a 2D Si*ð Þ 001 *substrate surface, respectively. Reciprocal lattice rods [e.g.,* ð Þ 00 *and* ð Þ 01 *] lie on Laue zones (e.g., L0 and L1). DB denotes the direct beam position. a*∗ <sup>0</sup> <sup>¼</sup> <sup>2</sup>π*=a*<sup>0</sup> <sup>≈</sup>2π*=*3*:*<sup>84</sup> <sup>Å</sup>�<sup>1</sup> h i *is the reciprocal lattice unit length of Si*ð Þ <sup>001</sup> *<sup>1</sup>*�*1.*

#### **Figure 5.**

*Schematics of 2D diffraction spots and reciprocal lattice rods of (a) substrate surface and (b) side-surface of 3D fabricated material. In both cases, diffraction spots are elongated in the surface normal direction.*

3D fabricated materials, the authors have demonstrated that not only substrate surfaces but also surfaces inclined from or perpendicular to a substrate plane exhibit atomically ordered structures where surface spots elongated along directions inclined from or perpendicular to the substrate normal direction appear in RHEED patterns (**Figure 5(b)**) [6, 7, 9]. We emphasize that a simple surface property of 3D fabricated materials, that is, a surface direction, can be confirmed by the elongation of surface spots.

Low-energy electron diffraction (LEED) with a typical *Ep* of 50–100 eV focuses the interference caused by the backward scattering of an electron incident to an atom, while RHEED focuses the interference caused by forward scattering. Both LEED and RHEED are sensitive to surface structures. A LEED pattern at the normal incidence corresponds to a top view of reciprocal lattice rods intersecting with an Ewald sphere from the surface normal direction. The diffraction spots move to the 00 ð Þ spot arising from the 00 ð Þ rod with increasing *Ep* [8, 18]. For an inclined surface, the arrangement of diffraction spots in alignment changes that in an arc, the center of which is the 00 ð Þ spot. The diffraction spots also move to the 00 ð Þspot with increasing *Ep*. Thus, we can confirm the surface direction of 3D fabricated materials by the arrangement and *Ep*dependent motion of diffraction spots.
