Common Crystal Structures

A Catalog of Crystal Structures

Throughout this chapter, we have built a rigorous toolkit: lattice vectors, Bravais lattices, symmetry operations, point groups, space groups, Wyckoff positions, and supercells. Now we bring everything together by examining the most important crystal structures encountered in condensed matter physics and materials science.

Each structure below is not merely an arrangement of atoms — it embodies a specific set of symmetries that determine the material's electronic, optical, and mechanical properties. Understanding these prototype structures is essential because they appear again and again across thousands of known compounds.

Intuition: Think of crystal structures as architectural blueprints. Just as a "Colonial" or "Tudor" style tells you the floor plan, window shapes, and roof lines, names like "zinc blende" or "perovskite" immediately tell a physicist the lattice type, coordination numbers, symmetry group, and approximate properties.

The Diamond Structure

The diamond structure is the arrangement of carbon atoms in diamond, and it is also adopted by the elemental semiconductors silicon (Si) and germanium (Ge). It consists of two interpenetrating FCC lattices, offset by one-quarter of the body diagonal.

Construction

Start with an FCC lattice and place an identical atom at each lattice point. Then place a second identical atom at a displacement of $\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right)$ from every lattice point. The basis is:

Since the FCC conventional cell has 4 lattice points, the diamond conventional cell contains $4 \times 2 = 8$ atoms.

Key Properties

• Space group: $Fd\bar{3}m$ (#227) — the highest-symmetry tetrahedral structure.
• Coordination number: 4 — each atom is tetrahedrally bonded to four nearest neighbors, reflecting $sp^3$ hybridization.
• Bond angle: $109.47°$ — the ideal tetrahedral angle.
• Inversion symmetry: Yes — the diamond structure has inversion symmetry, which has consequences for optical selection rules.

POSCAR for Diamond Si

1Si diamond structure
21.0
3   5.431  0.000  0.000
4   0.000  5.431  0.000
5   0.000  0.000  5.431
6Si
78
8Direct
9  0.000  0.000  0.000
10  0.500  0.500  0.000
11  0.500  0.000  0.500
12  0.000  0.500  0.500
13  0.250  0.250  0.250
14  0.750  0.750  0.250
15  0.750  0.250  0.750
16  0.250  0.750  0.750

The Zinc Blende Structure

The zinc blende (or sphalerite) structure is the binary analog of diamond. It is identical to the diamond structure except that the two FCC sublattices are occupied by different atomic species. This is the structure adopted by CdSe in its cubic polymorph — the focus of our Mn:CdSe project.

Construction

The basis consists of two different atoms:

For CdSe: Cd occupies the (0, 0, 0) sublattice and Se occupies the $\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right)$ sublattice (or vice versa).

Symmetry Differences from Diamond

Because the two sublattices hold different atoms, the zinc blende structure has lower symmetry than diamond:

• Space group: $F\bar{4}3m$ (#216) instead of $Fd\bar{3}m$ (#227).
• No inversion symmetry: This is critical! The lack of inversion symmetry enables piezoelectricity and second-harmonic generation.
• Point group: $T_d$ (24 operations) instead of $O_h$ (48 operations).

Common Zinc Blende Semiconductors

The Wurtzite Structure

The wurtzite structure is the hexagonal alternative to zinc blende. Many II-VI and III-V semiconductors can crystallize in either structure, and CdSe is a particularly important example because both polymorphs are commonly synthesized.

Construction

Wurtzite is based on the hexagonal close-packed (HCP) lattice with a two-atom basis. The lattice vectors are:

The four atoms in the conventional cell are at:

where $u \approx 3/8 = 0.375$ is the internal parameter. For an ideal wurtzite structure, $c/a = \sqrt{8/3} \approx 1.633$ and $u = 3/8$.

Key Properties

• Space group: $P6_3mc$ (#186).
• Coordination: Tetrahedral (same as zinc blende), but the second-nearest-neighbor arrangement differs.
• Stacking sequence: ABAB... along the c-axis (HCP stacking), compared to ABCABC... for zinc blende (FCC stacking).
• Polar axis: The c-axis is a polar axis, giving wurtzite structures spontaneous polarization — important for piezoelectric devices.

The Rock Salt Structure

The rock salt (NaCl) structure is one of the most common ionic crystal structures. Unlike zinc blende, it features octahedral coordination — each atom is surrounded by six nearest neighbors of the opposite type.

Construction

The rock salt structure consists of two interpenetrating FCC lattices, but now the second lattice is offset by $\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ instead of the $\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right)$ offset of zinc blende:

Key Properties

• Space group: $Fm\bar{3}m$ (#225) — the full cubic symmetry including inversion.
• Coordination number: 6 — each atom is octahedrally coordinated.
• Inversion symmetry: Yes — present at the midpoint between neighboring atoms.
• Madelung constant: $\alpha_M = 1.7476$ — higher than zinc blende (1.6381), reflecting stronger electrostatic stabilization.

POSCAR for Rock Salt NaCl

1NaCl rock salt structure
21.0
3   5.640  0.000  0.000
4   0.000  5.640  0.000
5   0.000  0.000  5.640
6Na  Cl
74   4
8Direct
9  0.000  0.000  0.000
10  0.500  0.500  0.000
11  0.500  0.000  0.500
12  0.000  0.500  0.500
13  0.500  0.500  0.500
14  0.000  0.000  0.500
15  0.000  0.500  0.000
16  0.500  0.000  0.000

The Perovskite Structure

The perovskite structure, with general formula $ABX_3$, is one of the most versatile and important structures in modern materials science. It underlies high-temperature superconductors, ferroelectrics, multiferroics, and the revolutionary perovskite solar cells.

Construction

In the ideal cubic perovskite:

• A cation (large): sits at the corners $(0, 0, 0)$
• B cation (small): sits at the body center $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$
• X anion: sits at the face centers $(\frac{1}{2}, \frac{1}{2}, 0)$, $(\frac{1}{2}, 0, \frac{1}{2})$, $(0, \frac{1}{2}, \frac{1}{2})$

The B cation is octahedrally coordinated by 6 X anions, while the A cation is 12-fold coordinated (cubo-octahedral). The unit cell contains exactly $1A + 1B + 3X = 5$ atoms.

The Goldschmidt Tolerance Factor

The stability and symmetry of the perovskite structure are predicted by the tolerance factor:

where $r_A$, $r_B$, and $r_X$ are the ionic radii.

Important Perovskite Materials

Structure-Property Relationships

The crystal structure is not just geometry — it directly determines the material's electronic, optical, and physical properties. Understanding these connections is one of the central goals of materials science.

Coordination and Band Gap

The coordination number influences the electronic bandwidth (how much the atomic orbitals overlap) and therefore the band gap:

• Tetrahedral (CN = 4): Narrower bands, often larger band gaps. Typical of semiconductors (diamond, zinc blende, wurtzite).
• Octahedral (CN = 6): Broader bands due to more overlap. Ranges from wide-gap insulators (MgO) to narrow-gap semiconductors (PbS).
• Cubo-octahedral (CN = 12): Very broad bands, typical of metals (FCC Cu, Al).

Symmetry and Optical Properties

Bonding Character

The ratio of ionic to covalent bonding varies across structures and determines many properties:

As ionicity increases: band gaps tend to increase, structures favor higher coordination numbers, and the material becomes more rigid but more brittle.

CdSe Polymorphs: Zinc Blende vs. Wurtzite

CdSe is remarkable because it readily crystallizes in both the zinc blende (cubic) and wurtzite (hexagonal) phases. Understanding the differences between these polymorphs is directly relevant to our Mn:CdSe simulation project.

The energy difference between the two polymorphs is remarkably small — only about 1-2 meV per atom. This means that synthesis conditions (temperature, solvent, ligands) can easily tip the balance. In bulk, wurtzite is the thermodynamically stable phase at room temperature, but zinc blende is readily synthesized and is metastable for long periods.

DFT Energy Comparison

A standard DFT exercise is to compute the total energy per formula unit for both polymorphs and confirm the small energy difference:

This comparison requires careful convergence of both calculations with the same ENCUT, k-point density, and exchange-correlation functional.

Mn:CdSe — Connecting to Our Project

With the structural knowledge from this entire chapter, we can now precisely describe our target system: Mn-doped zinc blende CdSe.

Why Zinc Blende CdSe?

1. Technological importance: CdSe quantum dots are among the most widely studied semiconductor nanocrystals, used in displays (QLED TVs), solar cells, biological imaging, and LEDs.
2. Cubic symmetry: The zinc blende structure's high symmetry simplifies both the crystallography and the DFT calculations.
3. Well-characterized: Extensive experimental data exists for comparison with DFT results.

Why Mn as a Dopant?

1. Dilute magnetic semiconductor (DMS): Mn introduces localized magnetic moments ($S = 5/2$, $d^5$ configuration) into the nonmagnetic CdSe host. This creates a material with coupled electronic and magnetic properties.
2. Substitutional doping: Mn replaces Cd on the cation sublattice. Since $\text{Mn}^{2+}$ and $\text{Cd}^{2+}$ have similar ionic radii (0.80 vs. 0.97 Angstrom), the substitution causes minimal lattice distortion.
3. Half-filled d-shell: The $d^5$ configuration is particularly stable (Hund's rule maximum spin), giving Mn:CdSe a large, well-defined magnetic moment of $5 \mu_B$ per Mn atom.
4. sp-d exchange: The interaction between the band electrons (s, p character) and the localized Mn d-electrons produces giant magneto-optical effects, making Mn:CdSe quantum dots promising for spintronic applications.

Structural Setup Summary

VASP Connection: Setting Up Structures

Let us write POSCAR files for two key structures that you will frequently encounter in the literature and in your own calculations.

Diamond Silicon POSCAR (Primitive Cell)

Using the primitive FCC vectors with a 2-atom basis:

1Si diamond (primitive cell)
21.0
3   0.000  2.716  2.716
4   2.716  0.000  2.716
5   2.716  2.716  0.000
6Si
72
8Direct
9  0.000  0.000  0.000
10  0.250  0.250  0.250

Zinc Blende CdSe POSCAR (Conventional Cell)

1CdSe zinc blende (conventional cell)
21.0
3   6.077  0.000  0.000
4   0.000  6.077  0.000
5   0.000  0.000  6.077
6Cd  Se
74   4
8Direct
9  0.000  0.000  0.000
10  0.500  0.500  0.000
11  0.500  0.000  0.500
12  0.000  0.500  0.500
13  0.250  0.250  0.250
14  0.750  0.750  0.250
15  0.750  0.250  0.750
16  0.250  0.750  0.750

Mn:CdSe POSCAR (2x2x2 Supercell with Substitution)

To create the doped structure, we take the 2x2x2 supercell from Section 9 and replace one Cd atom with Mn. The POSCAR header changes to:

1Mn:CdSe 2x2x2 zinc blende - 1 Mn substituting Cd at (0,0,0)
21.0
3  12.154000   0.000000   0.000000
4   0.000000  12.154000   0.000000
5   0.000000   0.000000  12.154000
6Cd  Mn  Se
731   1  32
8Direct
9  0.000  0.000  0.500   ! Cd  1  (formerly Cd 2)
10  0.000  0.250  0.250   ! Cd  2
11  0.000  0.250  0.750   ! Cd  3
12  ...                    ! (remaining 28 Cd atoms)
13  0.000  0.000  0.000   ! Mn  1  (substituted for Cd at origin)
14  0.125  0.125  0.125   ! Se  1
15  0.125  0.125  0.625   ! Se  2
16  ...                    ! (remaining 30 Se atoms)

The critical change is grouping atoms by species (Cd, then Mn, then Se) and updating the atom counts. VASP requires atoms of the same species to be listed contiguously.

Chapter 1 Summary

This chapter has taken you on a complete journey through the architecture of crystals in real space — from the most basic definition of a crystal to the practical construction of supercells for DFT simulations. Let us review the key concepts from each section.

Chapter Roadmap

The Big Picture

Every concept in this chapter connects to the others:

1. Lattice vectors define the unit cell geometry and determine the Bravais lattice type.
2. Symmetry operations of the lattice determine the point group and space group.
3. Wyckoff positions tell us where atoms sit, consistent with the space group symmetry.
4. Crystal structures (zinc blende, wurtzite, etc.) are specific combinations of Bravais lattice + basis + space group.
5. Supercells allow us to break symmetry for defect simulations while maintaining periodic boundary conditions.

What Comes Next

In Chapter 2: Mathematics of Symmetry — Group Theory Essentials, we formalize symmetry using the language of group theory. You will learn:

• How symmetry operations form mathematical groups with precise axioms
• Representations and character tables that classify how physical quantities transform under symmetry
• Selection rules that predict which transitions are allowed or forbidden
• How VASP uses symmetry internally to accelerate calculations

Then in Chapter 3, we move from real space to reciprocal space, introducing Brillouin zones, k-points, and the framework for understanding band structures.

Coming Next: Chapter 2 begins with What Is a Group? — where we discover that the symmetry operations you learned in this chapter form elegant mathematical structures with deep physical consequences.