Semiconductor crystallography

A crystal structure is defined as a group of atoms, known as the basis, which is attached to every point in a regular periodic array known as a lattice [25]. The regular atomic ordering in a crystal is useful in explaining the macroscopic electronic, magnetic and thermal properties that it exhibits [10]. The arrangement of the basis atoms in a crystal will look identical at points r and r’ given that the crystal translation vector, T, is satisfied:

T = u1a1+ u2a2+ u3a3 (Equation 2.1)

Where: a1, a2 and a3 are noncoplanar vectors and u1, u2 and u3 are arbitrary integers

as seen in figure 2.1 [26]. The primitive cell is defined as being the smallest volume cell and contains 1 lattice point and consequently a single basis as shown in figure 2.2.

Figure 2.1: Diamond fcc lattice structure, where: 𝐚 = 𝐚𝐱= 𝐚𝐲= 𝐚𝐳 & 𝛃 = 𝛂 = 𝛄 = 𝟗𝟎°. The red spheres represent atoms and the silver rod are covalent bonds. The diamond basis contains 2 atoms separated by a quarter diagonal. The tetragonal bonds in a diamond structure are all 109.47°. (Adapted from Kittel) [26]

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Figure 2.2: Diamond fcc lattice and primitive cell (in blue). The primitive cell vectors are: A =𝐚

𝟐[𝐢 + 𝐣], B = 𝐚

𝟐[𝐢 + 𝐤] and C = 𝐚

𝟐[𝐣 + 𝐤]. (Adapted from Kittel)[26].

There are 7 crystal systems which are sub-divided into 14 lattice types known as Bravais lattice. A space group for a particular Bravais lattice is defined by the number of symmetry operations. In a cubic lattice unit cell, the unit cell angles are all 90° and the unit cell length is identical in all directions. Within the cubic crystal system, there are 3 Bravais lattices: primitive cubic (sc or cP), body centred cubic (bcc or cI), face- centred cubic (fcc or cF). The diamond structure is a particular fcc crystal, where two fcc structures are combined into a single lattice, as seen in figure 2.1. Table 2.1 compares the simple fcc structure with the diamond fcc structure. In a diamond basis

there are two identical atoms at position [000] and [1

4 1 4 1

4], hence there are 8 atoms per

unit cell, instead of 4 as is typical in standard fcc lattices. The point group for diamond structures is Oh (Schoenflies notation) [27] and the space group is Fd3̅m (Hermann-

Mauguin notation), where the F indicates face centered cubic, d indicates translational symmetry along the quarter face diagonal i.e.: a glide plane and 3̅ pertains to rotational symmetry along 4 diagonal 3-fold axes and m denotes 2 rotoinversion axes i.e.: a mirror plane [28]. Group IV elements such as carbon, silicon, germanium, α-Sn and group IV alloys such as SiGe have this type of covalent bonded structure.

FCC Diamond FCC

Conventional cell volume a3 a3

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Atoms per unit cell 4 8

Primitive cell volume 1

4(a

3₎ 1

4(a

3₎

Lattice points per unit volume 4 a3 4 a3 Number of nearest neighbours 12 4

Nearest-neighbour distance _a√2

2 a√3 4 Number of second neighbours 6 12

Second neighbour distance a _a√2

2

Table 2.1: Comparison between fcc lattice and diamond fcc lattice [26]

The zinc blende structure is related to the diamond structure where the basis is composed of two different atoms, as is in ZnS where the Zn atoms occupy atomic sites starting at [000] and S atoms occupy sites at a quarter diagonal from the Zn. Many III- V binary compounds exist most stably as a zinc blende fcc lattice except for III-V nitrides which are most stable as a hexagonal, Wurtzite, lattice structures e.g.: boron nitride, gallium nitride and indium nitride. Due to the space group similarities between diamond and zinc blende, zinc blende semiconductors can therefore be “grown” on diamond fcc substrates such as silicon and germanium, where the grown crystal can be expected to maintain its structure and orientation in a process known as epitaxy.

Figure 2.3: InSb unit cells showing the typical Zinc Blende structure in the diamond fcc lattice. 𝐚𝐱= 𝐚𝐲=

𝐚𝐳 is the lattice constant. In figure 2.3(a) the indium atoms (purple) occupy one fcc ‘structure’ at [000] and

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the antimony atoms (blue) occupy the other fcc structure at [𝟏

𝟒 𝟏 𝟒 𝟏

𝟒]. Figure 2.3(b) is of an equivalent ‘sub- lattice’ orientation of InSb where the antimony atom occupies the [000] site and the indium atom occupies the [𝟏

𝟒 𝟏 𝟒 𝟏

𝟒] sites, which is achieved by 90° rotation or translation from (a) to (b). (Adapted from Grundmann) [29].

Figure 2.3 (a) & (b) show unit cells of the III-V compound semiconductor InSb which has a zinc blende structure and a fcc lattice. The point group for zinc blende structures is Td and the space group is F4̅3m, there are 4 three-fold rotoinversion axes as is the

case with all cubic space groups and the m indicates a mirror plane. There are 48 symmetry operations with diamond structure however there are only 24 with zinc blende structures [27]. Unlike diamond structure there is no quarter diagonal glide plane for translational symmetry in zinc blende structures, therefore the [110] and [1̅10] directions are not equivalent. Figure 2.3. (a) & (b) shows two alternative unit cells of InSb where both lattices have the same zinc blende fcc lattice space group, however the indium and antimony atoms occupy opposite locations. Figure 2.3 (a) can be converted to figure 2.3 (b) by applying any of the 24 symmetry operations present in the diamond structure but not present in zinc blende [28]. As will be discussed later on in this investigation, the simultaneous presence of the two sub lattices on a diamond cubic substrate surface leads to the generation of regions in the epitaxial crystal known as inversion domains. Inversion domains are regions with the alternate sub lattice, i.e.: regions where the polar direction changes. Due to the difference in valence electrons between the two elements in a zinc blende fcc lattice, charge neutrality is not maintained at the boundary between sub lattices leading a charged defect [29, 30]. This will be covered later in the defects section.