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Introduction: Where Can Atoms Sit in a Crystal?

In the previous section, we learned that a space group defines the complete set of symmetry operations for a crystal. But knowing the symmetry operations is only half the story. The equally important question is: where in the unit cell can atoms actually sit?

Not all positions within a unit cell are equivalent. Some positions lie on symmetry elements (rotation axes, mirror planes, inversion centers), while others do not. Atoms placed at different positions experience different local symmetry environments, which directly affects their physical properties — bonding geometry, vibrational modes, electronic states, and spectroscopic signatures.

Wyckoff positions provide a systematic classification of all distinct sites within a unit cell for a given space group. Named after the American crystallographer Ralph Wyckoff, this classification tells you exactly how many equivalent atoms are generated when you place one atom at a given site, and what symmetry that site possesses.

The Big Picture: A Wyckoff position answers three questions at once: (1) How many equivalent atoms does this site generate in the unit cell? (2) What is the local symmetry at this site? (3) How many free coordinates do I need to specify? For our Mn:CdSe project, Wyckoff positions tell us exactly where Cd, Se, and Mn atoms sit and what happens to the crystal symmetry when we substitute Mn for Cd.

General vs Special Positions

Every point inside a unit cell falls into one of two categories based on its relationship to the symmetry elements of the space group.

General Position

A general position is a point $(x, y, z)$ that does not lie on any symmetry element — no rotation axis passes through it, no mirror plane contains it, and no inversion center coincides with it. The only symmetry operation that maps this point onto itself is the identity $E$ .

Key properties of a general position:

Maximum multiplicity: every symmetry operation generates a distinct new position, so the multiplicity equals the order of the space group (point-group order × centering multiplicity).
Three free coordinates: the position requires specifying all three fractional coordinates $(x, y, z)$ .
Site symmetry = 1: only the identity operation leaves the site invariant.

For $F\bar{4}3m$ , the general position has multiplicity 96. Placing an atom at a general position like $(0.13, 0.27, 0.41)$ creates 96 symmetry-equivalent copies in the conventional unit cell.

Special Position

A special position lies on one or more symmetry elements. Some symmetry operations map the point onto itself rather than generating a new distinct point. This reduces the multiplicity.

Key properties of a special position:

Reduced multiplicity: fewer than the maximum number of equivalent points are generated, because some operations leave the site unchanged.
Constrained coordinates: some or all coordinates are fixed by symmetry (e.g., $(0, 0, 0)$ or $(x, x, x)$ ).
Site symmetry > 1: the set of symmetry operations that leave the site invariant forms a non-trivial group (a subgroup of the point group).

Analogy

Think of a kaleidoscope. If you place an object at a random position, you see the maximum number of reflections. But if you place it exactly on one of the mirrors, some reflections coincide, and you see fewer distinct images. Special positions in a crystal work the same way — placing an atom on a symmetry element reduces the number of symmetry-generated copies.

Wyckoff Letters and Multiplicity

Each distinct type of site in a space group is labeled with a Wyckoff letter and a multiplicity. The naming convention is systematic:

Letters start from 'a' for the position with the highest site symmetry (lowest multiplicity).
Letters proceed alphabetically (b, c, d, ...) as the site symmetry decreases and multiplicity increases.
The last letter in the list is always the general position with the highest multiplicity.

A Wyckoff position is written as multiplicity + letter, for example:

4a — multiplicity 4, Wyckoff letter a (highest site symmetry)
4c — multiplicity 4, Wyckoff letter c
96i — multiplicity 96, Wyckoff letter i (general position)

Multiplicity and Centering

The multiplicity listed in the International Tables is for the conventional cell. For a face-centered (F) lattice, the conventional cell contains 4 lattice points. So the multiplicities for F-centered space groups are always multiples of 4.

When using a primitive cell in VASP, divide the conventional multiplicity by the centering factor:

\text{primitive multiplicity} = \frac{\text{conventional multiplicity}}{\text{centering factor}}

Centering	Factor	Example
P (primitive)	1	Multiplicity unchanged
I (body-centered)	2	Conventional 8 → primitive 4
F (face-centered)	4	Conventional 4 → primitive 1
R (rhombohedral)	3	Conventional 6 → primitive 2

Practical Consequence

In the primitive FCC cell of CdSe, a Wyckoff position with conventional multiplicity 4 corresponds to just 1 atom. This is why the POSCAR for primitive CdSe zinc blende contains only 2 atoms (1 Cd + 1 Se), even though the conventional cell has 8 atoms (4 Cd + 4 Se).

Site Symmetry

The site symmetry of a Wyckoff position is the set of symmetry operations from the space group that leave that particular point invariant. These operations form a subgroup of the crystal's point group.

Formally, for a point $\mathbf{r}_0$ in the unit cell, the site-symmetry group is:

G_{\mathbf{r}_0} = \{ \{R \mid \mathbf{t}\} \in G \;|\; R\mathbf{r}_0 + \mathbf{t} = \mathbf{r}_0 + \mathbf{T} \}

where $\mathbf{T}$ is some lattice vector. In other words, the site symmetry includes every operation that maps the point back to itself or to a lattice-equivalent copy of itself.

Relationship Between Multiplicity and Site Symmetry

There is a fundamental relationship between multiplicity, the order of the point group, and the order of the site-symmetry group:

\text{multiplicity} = \frac{|G_{\text{point}}| \times Z}{|G_{\mathbf{r}_0}|}

where $|G_{\text{point}}|$ is the order of the point group (24 for $T_d$ ), $Z$ is the number of lattice points per conventional cell (4 for F), and $|G_{\mathbf{r}_0}|$ is the order of the site-symmetry group.

For the 4a position in $F\bar{4}3m$ :

\text{multiplicity} = \frac{24 \times 4}{24} = 4

The site symmetry has order 24 (the full $T_d$ point group), giving the minimum multiplicity of 4.

Why Site Symmetry Matters

Crystal field splitting: The site symmetry determines how atomic orbitals split in the crystal environment. A Mn atom at a $T_d$ site has its d-orbitals split into $e$ and $t_2$ sets.
Selection rules: Spectroscopic transitions depend on the symmetry of the initial and final states. Site symmetry determines which transitions are allowed.
Tensor properties: The site symmetry constrains the form of physical tensors (e.g., electric field gradients at nuclear sites, strain tensors).
Magnetic ordering: The possible magnetic structures are constrained by the site symmetry of the magnetic atoms.

Zinc Blende Wyckoff Positions

Space group $F\bar{4}3m$ (No. 216) has the following Wyckoff positions, listed in order of decreasing site symmetry:

Wyckoff	Mult.	Site Sym.	Coordinates	Typical Atom
4a	4	-43m (T_d)	(0, 0, 0)	Cd in CdSe
4b	4	-43m (T_d)	(1/2, 1/2, 1/2)	— (empty in zinc blende)
4c	4	-43m (T_d)	(1/4, 1/4, 1/4)	Se in CdSe
4d	4	-43m (T_d)	(3/4, 3/4, 3/4)	— (empty in zinc blende)
16e	16	.3m (C_3v)	(x, x, x)	Interstitial sites
24f	24	2.mm (C_2v)	(x, 0, 0)	—
24g	24	2.mm (C_2v)	(x, 1/4, 1/4)	—
48h	48	..m (C_s)	(x, x, z)	—
96i	96	1 (C_1)	(x, y, z)	General position

The 4a Position: Cd Atoms

Cadmium atoms in zinc blende CdSe occupy the 4a Wyckoff position at $(0, 0, 0)$ . The face-centered translations generate the four equivalent Cd positions in the conventional cell:

(0, 0, 0), \quad (\tfrac{1}{2}, \tfrac{1}{2}, 0), \quad (\tfrac{1}{2}, 0, \tfrac{1}{2}), \quad (0, \tfrac{1}{2}, \tfrac{1}{2})

The site symmetry is $\bar{4}3m$ (= $T_d$ ), the full point group of the space group. This is the maximum possible site symmetry. All three coordinates are fixed — there are zero free parameters.

The 4c Position: Se Atoms

Selenium atoms occupy the 4c Wyckoff position at $(\frac{1}{4}, \frac{1}{4}, \frac{1}{4})$ . The face-centered translations generate:

(\tfrac{1}{4}, \tfrac{1}{4}, \tfrac{1}{4}), \quad (\tfrac{3}{4}, \tfrac{3}{4}, \tfrac{1}{4}), \quad (\tfrac{3}{4}, \tfrac{1}{4}, \tfrac{3}{4}), \quad (\tfrac{1}{4}, \tfrac{3}{4}, \tfrac{3}{4})

The site symmetry is also $\bar{4}3m$ (= $T_d$ ), identical to the Cd site. Each Se atom is tetrahedrally coordinated by 4 Cd atoms at a distance of:

d_{\text{Cd-Se}} = \frac{\sqrt{3}}{4} \cdot a = \frac{\sqrt{3}}{4} \times 6.052\,\text{\AA} \approx 2.620\,\text{\AA}

Empty Positions: 4b and 4d

The 4b position at $(\frac{1}{2}, \frac{1}{2}, \frac{1}{2})$ and the 4d position at $(\frac{3}{4}, \frac{3}{4}, \frac{3}{4})$ are unoccupied in the zinc blende structure. If all four positions (4a, 4b, 4c, 4d) were filled with the same atom, you would get the FCC structure. If 4a and 4c are filled with the same atom, you get diamond (as in Si or Ge).

Structure Relationships

The zinc blende structure is closely related to other important crystal structures:

Diamond: fill 4a + 4c with same element → $Fd\bar{3}m$ (gains inversion symmetry)
NaCl (rock salt): fill 4a + 4b with different elements → $Fm\bar{3}m$ (octahedral coordination)
Fluorite (CaF₂): fill 4a with cation and 8c with anion → $Fm\bar{3}m$

Multiplicity and the Unit Cell

Understanding how multiplicity translates to the number of atoms in different cell choices is critical for setting up VASP calculations correctly.

Conventional Cell

The conventional (cubic) unit cell of zinc blende CdSe contains:

4 Cd atoms from Wyckoff position 4a
4 Se atoms from Wyckoff position 4c
Total: 8 atoms per conventional cell

The formula units per conventional cell is $Z = 4$ , meaning there are 4 CdSe formula units.

Primitive Cell

The primitive FCC cell (rhombohedral) has 1/4 the volume of the conventional cell. Dividing all multiplicities by 4:

1 Cd atom (4a ÷ 4 = 1)
1 Se atom (4c ÷ 4 = 1)
Total: 2 atoms per primitive cell

This matches our POSCAR from the previous section, which specified exactly 2 atoms with the primitive FCC lattice vectors.

Counting Atoms: General Formula

For any crystal structure, the total number of atoms in the unit cell is:

N_{\text{total}} = \sum_{i} m_i \cdot o_i

where $m_i$ is the multiplicity of the $i$ -th Wyckoff position and $o_i$ is its occupancy (usually 1 for fully occupied sites).

Quick Check

When building a POSCAR, always verify that the number of atoms matches what you expect from the Wyckoff multiplicities. A mismatch usually indicates an error in the coordinates or the cell choice.

Mn Doping Implications

The central application of this textbook is Mn-doped CdSe. When a manganese atom replaces a cadmium atom at the 4a Wyckoff position, profound changes occur in the crystal symmetry.

Substitution at the 4a Site

In pure CdSe, all 4a sites are equivalent by symmetry. When we place Mn at one of the four Cd positions in the conventional cell, we break the translational symmetry that relates the four FCC sublattice points. The Mn atom is chemically distinct from Cd, so the face-centered translation operations no longer map identical atoms onto each other.

Symmetry Reduction

Consider a 2×2×2 supercell of the conventional cell, containing 64 atoms (32 Cd + 32 Se). Replacing one Cd with Mn gives Cd₃₁MnSe₃₂. The symmetry changes are dramatic:

Property	Pure CdSe	Mn:CdSe (single substitution)
Space group	F-43m (#216)	Depends on supercell and Mn position
Point group	T_d (24 operations)	Reduced (often C_1 or C_3v)
Symmetry operations	24	Typically 1–6
Irreducible k-points	~10 (6×6×6 grid)	~100+ (same grid)
Computational cost	Baseline	10–20x more expensive

The exact residual symmetry depends on which Cd site in the supercell is substituted and the shape of the supercell. If Mn is placed at the origin of the supercell, some symmetry operations that pass through the origin may survive.

Effect on Electronic Structure

The $T_d$ site symmetry of the Cd position splits the Mn 3d orbitals into:

3d \xrightarrow{T_d} e \oplus t_2

$e$ states (2-fold degenerate): $d_{z^2}$ and $d_{x^2-y^2}$ orbitals, pointing between the ligands
$t_2$ states (3-fold degenerate): $d_{xy}$ , $d_{xz}$ , and $d_{yz}$ orbitals, pointing toward the ligands

In the tetrahedral crystal field (unlike octahedral), the $e$ orbitals lie below the $t_2$ orbitals. For Mn²⁺ with 5 d-electrons in a high-spin configuration, all five orbitals are singly occupied:

\text{Mn}^{2+}:\quad e^2 \, t_2^3 \quad (S = 5/2, \text{ high-spin})

Why High-Spin?

The tetrahedral crystal field splitting $\Delta_{\text{tet}}$ is typically much smaller than the octahedral splitting ( $\Delta_{\text{tet}} \approx \frac{4}{9}\Delta_{\text{oct}}$ ). For Mn²⁺ in CdSe, the splitting is so small that the exchange energy (Hund's rule) dominates, and all spins are parallel. The resulting $S = 5/2$ state gives rise to the strong paramagnetism that makes Mn:CdSe a diluted magnetic semiconductor.

Occupancy and Disorder

In real crystals, Wyckoff positions are not always fully occupied by a single atomic species. Two important generalizations are partial occupancy and mixed occupancy.

Partial Occupancy

A site has partial occupancy when it is not fully occupied. For example, occupancy = 0.5 means the site is occupied only 50% of the time (on average across the crystal). This represents a type of configurational disorder.

In the context of Mn:CdSe with low doping concentration:

The 4a site might have occupancy 0.95 Cd + 0.05 Mn (for 5% doping)
This is a statistical description — in any given unit cell, the site is fully occupied by either Cd or Mn, but the average over the entire crystal gives fractional occupancies

Mixed Occupancy (Site Mixing)

When two or more atomic species share the same Wyckoff position with fractional occupancies that sum to 1, this is called mixed occupancy or chemical disorder. X-ray diffraction experiments measure this as an average scattering factor:

\bar{f} = \sum_j o_j \cdot f_j

where $o_j$ is the occupancy and $f_j$ is the atomic scattering factor of species $j$ .

Implications for DFT

DFT calculations (including VASP) require explicit atomic positions— every site must be occupied by exactly one atom. Partial occupancy cannot be directly represented. Instead, we use several approaches:

Supercell approach: Build a large supercell and explicitly place dopants at specific sites. This is the most straightforward method and what we will use for Mn:CdSe.
Virtual Crystal Approximation (VCA): Replace the disordered site with a virtual atom having averaged properties. Less accurate but much cheaper.
Special Quasirandom Structures (SQS): Generate supercells where the dopant arrangement mimics the short-range correlations of a random alloy.
Coherent Potential Approximation (CPA): A mean-field approach that handles disorder without supercells. Not available in standard VASP but implemented in some KKR-CPA codes.

Supercell Size Matters

When modeling dilute doping (e.g., 3% Mn in CdSe), you need a supercell large enough that the dopant does not interact with its periodic images. For 3% doping, you need at least a 2×2×2 conventional supercell (64 atoms) to place one Mn, giving 3.125% doping. For more dilute concentrations, 3×3×3 or larger cells may be needed, which dramatically increases computational cost.

VASP Connection

Understanding Wyckoff positions is essential for correctly building crystal structures for VASP calculations. Here we show how to construct a POSCAR for Mn-doped CdSe.

Pure CdSe POSCAR (Primitive Cell)

📝text

1CdSe zinc blende (primitive cell)
26.052
3  0.0  0.5  0.5
4  0.5  0.0  0.5
5  0.5  0.5  0.0
6Cd Se
71  1
8Direct
9  0.000  0.000  0.000   ! Cd at 4a (origin)
10  0.250  0.250  0.250   ! Se at 4c

Mn-Doped CdSe POSCAR (2x2x2 Conventional Supercell)

To model Mn doping, we build a 2×2×2 supercell of the conventional cubic cell and replace one Cd atom with Mn. This gives a doping concentration of 1/32 = 3.125%:

📝text

1Cd31MnSe32 - Mn:CdSe zinc blende 2x2x2 supercell
212.104
3  1.0  0.0  0.0
4  0.0  1.0  0.0
5  0.0  0.0  1.0
6Cd Mn Se
731 1 32
8Direct
9! --- Cd atoms (31 total, one replaced by Mn) ---
10! Original 4a positions + FCC translations, expanded in 2x2x2
11  0.000  0.000  0.250   ! Cd
12  0.000  0.250  0.000   ! Cd
13  0.000  0.250  0.250   ! Cd
14  0.000  0.000  0.500   ! Cd
15  0.000  0.250  0.500   ! Cd
16  0.000  0.500  0.250   ! Cd
17  0.000  0.500  0.500   ! Cd
18  0.250  0.000  0.000   ! Cd
19  0.250  0.000  0.250   ! Cd
20  0.250  0.250  0.000   ! Cd
21  0.250  0.250  0.250   ! Cd
22  0.250  0.000  0.500   ! Cd
23  0.250  0.250  0.500   ! Cd
24  0.250  0.500  0.000   ! Cd
25  0.250  0.500  0.250   ! Cd
26  0.250  0.500  0.500   ! Cd
27  0.500  0.000  0.000   ! Cd
28  0.500  0.000  0.250   ! Cd
29  0.500  0.250  0.000   ! Cd
30  0.500  0.250  0.250   ! Cd
31  0.500  0.000  0.500   ! Cd
32  0.500  0.250  0.500   ! Cd
33  0.500  0.500  0.000   ! Cd
34  0.500  0.500  0.250   ! Cd
35  0.500  0.500  0.500   ! Cd
36  0.000  0.000  0.750   ! Cd
37  0.000  0.250  0.750   ! Cd
38  0.000  0.500  0.750   ! Cd
39  0.250  0.000  0.750   ! Cd
40  0.250  0.250  0.750   ! Cd
41  0.250  0.500  0.750   ! Cd
42! --- Mn atom (replaces Cd at origin) ---
43  0.000  0.000  0.000   ! Mn (substituting Cd at 4a)
44! --- Se atoms (32 total) ---
45  0.125  0.125  0.125   ! Se
46  0.125  0.125  0.375   ! Se
47  0.125  0.375  0.125   ! Se
48  0.125  0.375  0.375   ! Se
49  0.125  0.125  0.625   ! Se
50  0.125  0.375  0.625   ! Se
51  0.125  0.125  0.875   ! Se
52  0.125  0.375  0.875   ! Se
53  0.375  0.125  0.125   ! Se
54  0.375  0.125  0.375   ! Se
55  0.375  0.375  0.125   ! Se
56  0.375  0.375  0.375   ! Se
57  0.375  0.125  0.625   ! Se
58  0.375  0.375  0.625   ! Se
59  0.375  0.125  0.875   ! Se
60  0.375  0.375  0.875   ! Se
61  0.625  0.125  0.125   ! Se
62  0.625  0.125  0.375   ! Se
63  0.625  0.375  0.125   ! Se
64  0.625  0.375  0.375   ! Se
65  0.625  0.125  0.625   ! Se
66  0.625  0.375  0.625   ! Se
67  0.625  0.125  0.875   ! Se
68  0.625  0.375  0.875   ! Se
69  0.875  0.125  0.125   ! Se
70  0.875  0.125  0.375   ! Se
71  0.875  0.375  0.125   ! Se
72  0.875  0.375  0.375   ! Se
73  0.875  0.125  0.625   ! Se
74  0.875  0.375  0.625   ! Se
75  0.875  0.125  0.875   ! Se
76  0.875  0.375  0.875   ! Se

INCAR Settings for Doped Structures

When running VASP on doped structures, the INCAR requires special attention to symmetry and spin:

📝text

1# INCAR for Mn:CdSe
2# Symmetry: be cautious with doped structures
3SYMPREC = 1E-6       # Tight tolerance for broken symmetry
4ISYM = 0             # Turn off symmetry entirely (safest for defects)
5
6# Spin polarization: MANDATORY for Mn (d5, S=5/2)
7ISPIN = 2            # Spin-polarized calculation
8MAGMOM = 31*0.0 5.0 32*0.0   # 31 Cd (0), 1 Mn (5.0 muB), 32 Se (0)
9
10# Electronic convergence
11ENCUT = 400          # Plane-wave cutoff (eV)
12EDIFF = 1E-6         # Electronic convergence criterion
13
14# DFT+U for Mn d-electrons (if using)
15LDAU = .TRUE.
16LDAUTYPE = 2
17LDAUL = -1 2 -1      # Cd: no U, Mn: d-electrons, Se: no U
18LDAUU = 0.0 4.0 0.0  # U value for Mn d-electrons
19LDAUJ = 0.0 1.0 0.0  # J value for Mn d-electrons

Generating Structures with Python

You can automate supercell construction and doping using the pymatgen library:

🐍python

1from pymatgen.core import Structure, Lattice
2from pymatgen.transformations.standard_transformations import (
3    SupercellTransformation,
4    SubstitutionTransformation,
5)
6
7# Build primitive CdSe zinc blende
8a = 6.052  # lattice constant in angstroms
9lattice = Lattice.cubic(a)
10
11# Define zinc blende structure (conventional cell)
12cdse = Structure(
13    lattice,
14    ["Cd", "Cd", "Cd", "Cd", "Se", "Se", "Se", "Se"],
15    [
16        [0.0, 0.0, 0.0], [0.5, 0.5, 0.0],
17        [0.5, 0.0, 0.5], [0.0, 0.5, 0.5],  # Cd at 4a
18        [0.25, 0.25, 0.25], [0.75, 0.75, 0.25],
19        [0.75, 0.25, 0.75], [0.25, 0.75, 0.75],  # Se at 4c
20    ],
21)
22
23print(f"Conventional cell: {cdse.num_sites} atoms")
24
25# Create 2x2x2 supercell
26supercell_transform = SupercellTransformation([[2, 0, 0], [0, 2, 0], [0, 0, 2]])
27supercell = supercell_transform.apply_transformation(cdse)
28print(f"Supercell: {supercell.num_sites} atoms")
29
30# Substitute one Cd with Mn
31# This replaces the Cd atom closest to the origin
32supercell.replace(0, "Mn")
33
34# Verify composition
35print(f"Composition: {supercell.composition}")
36# Output: Cd31 Mn1 Se32
37
38# Write POSCAR
39supercell.to(filename="POSCAR_MnCdSe", fmt="poscar")
40print("POSCAR written successfully")

Checking Your Structure

After generating a doped POSCAR, always verify: (1) the total atom count matches expectations, (2) the Mn-Se bond lengths are reasonable (~2.5-2.7 angstroms), and (3) the MAGMOM tag in INCAR has the correct number of entries (must equal total atoms in POSCAR, in the same species order).

Summary

Wyckoff positions provide a complete classification of all symmetry-distinct atomic sites within a space group, telling us the multiplicity, site symmetry, and coordinate constraints for each possible site.

Key Takeaways

General positions have maximum multiplicity and no special site symmetry. Special positions lie on symmetry elements, reducing multiplicity and constraining coordinates.
Wyckoff letters (a, b, c, ...) label distinct site types in order of decreasing site symmetry. The notation is "multiplicity + letter" (e.g., 4a).
Site symmetry determines the local environment of an atom — crystal field splitting, selection rules, and allowed tensor components.
In zinc blende CdSe ( $F\bar{4}3m$ ), Cd occupies the 4a position at $(0,0,0)$ and Se occupies the 4c position at $(\frac{1}{4},\frac{1}{4},\frac{1}{4})$ , both with $T_d$ site symmetry.
Mn doping at the 4a site breaks the space-group symmetry of CdSe, reducing symmetry operations and increasing computational cost in DFT.
In VASP, Mn:CdSe requires a supercell approach with ISPIN = 2 and appropriate MAGMOM settings. Use ISYM = 0 or very tight SYMPREC for doped structures.

Concept	Key Detail
Wyckoff position	Symmetry-distinct site labeled by multiplicity + letter
General position	Maximum multiplicity, site symmetry = 1
Special position	Reduced multiplicity, site on symmetry element
Site symmetry	Subgroup of point group leaving site invariant
4a in F-43m	(0,0,0), site symmetry T_d, Cd in CdSe
4c in F-43m	(1/4,1/4,1/4), site symmetry T_d, Se in CdSe
Mn at 4a	Breaks F-43m symmetry, splits 3d into e + t_2
MAGMOM	Must specify initial magnetic moments for Mn (5.0 muB)

Coming Next: In Building Supercells, we will learn how to systematically construct supercells from primitive cells, choose appropriate supercell sizes for doping studies, and handle the computational trade-offs between accuracy and cost.