CHEM 548: Materials Chemistry

Lecture 17: Band Structures of Graphene, Primitive Cubic, and FCC Crystals

Mon 11/02/2026 · Meeting 19

Reading: Woodward, Karen, Evans, and Vogt, Solid State Materials Chemistry
Ch. 6 §6.4.2, §§6.5–6.6

Learning Objectives: Graphene

By the end of this lecture, you will be able to:

Construct reciprocal lattice vectors for non-orthogonal hexagonal systems and identify the hexagonal first Brillouin zone
Locate special k-points (Gamma, M, K, K’) in hexagonal Brillouin zones and convert between reciprocal lattice and Cartesian coordinates
Distinguish sigma and pi band contributions in graphene and apply orbital orthogonality at high-symmetry points
Predict bonding character of π orbitals at Gamma, M, and K points using phase relationship analysis
Explain semi-metallic behavior, Dirac points, and linear dispersion as unique features of graphene’s electronic structure

Welcome to our lecture on graphene - arguably the most famous 2D material. This lecture introduces hexagonal lattices, where unit cell axes are no longer orthogonal.

Hexagonal reciprocal lattice: real space vectors A and B have equal length with 120 degree angle; A* perpendicular to B, B* perpendicular to A — the reciprocal lattice vectors make a 60 degree angle
First Brillouin zone: a hexagon — the Wigner-Seitz cell, points closer to the origin than any other lattice point
Special k-points: Gamma at the center where k equals zero, M at the hexagon edge center, K and K-prime at the hexagon corners — where we analyze the band structure
Graphene structure: honeycomb of fused benzene rings; two carbons per unit cell means eight total bands from 2s, 2px, 2py, 2pz
Pi bands: like benzene, the key features come from pi overlap of 2pz orbitals perpendicular to the sheet — orthogonal to sigma orbitals throughout the Brillouin zone
Crystal orbitals: at gamma, pi bonding everywhere (lowest) and pi antibonding everywhere (highest); at K the three neighbors of each carbon have zero pz coefficient — completely nonbonding, so the bands touch precisely at K
Semi-metallic behavior: eight electrons per unit cell fill exactly four bands; at K and K-prime the bands just touch with zero gap — Dirac points, and any doping converts graphene to a metallic conductor

To describe graphene's bands we first need the reciprocal lattice of a hexagonal system.

Reciprocal Space: 2D Hexagonal Lattice

Real-space 2D hexagonal lattice with vectors a and b of equal length at 120 degrees and the translation T equals u a plus v b; a click adds the reciprocal-vector criteria with a-star perpendicular to b and b-star perpendicular to a.

The Reciprocal Lattice and First Brillouin Zone

Real-space and reciprocal-space hexagonal lattices side by side; a click adds the Cartesian vector formulas and shades the hexagonal first Brillouin zone around a reciprocal lattice point.

And if we were to draw the entire lattice of reciprocal space points, it would look like this.

It still has that hexagonal symmetry, but the lattice vectors now point in somewhat different directions

[advance: add the vector formulas and the first Brillouin zone]

First Brillouin zone: the locus of points closest to the origin
Construction: draw lines from the origin to all nearest lattice points, then bisect each line — this yields a hexagon for a 2D hexagonal lattice
Cartesian coordinates: B points along Y with magnitude A; vector A has the same length but isn't along X — a 30-60 right triangle, components (√3/2)A in X and -A/2 in Y
A dot A* has to be 2 pi, and B dot B* has to be 2 pi — that gives us these lattice vectors in Cartesian coordinates
I'll leave it for you to confirm that both dot products give 2 pi

Inside that hexagonal zone a few special points carry names we will use all lecture.

Special Points in a 2D Hexagonal Lattice

The hexagonal first Brillouin zone with the special points Gamma, M, K, and K-prime labeled, and tables giving their coordinates in reciprocal lattice vectors and Cartesian wave vectors.

Graphene

Graphene: Band Structure

Carbon's relevant atomic orbitals are 2S, 2PX, 2PY, and 2PZ.

Eight bands total with two carbons per unit cell; five lowest shown — the three lowest bands at M or K are sigma bands
At Γ, 2S is orthogonal to 2P — the lowest crystal orbital is purely 2S bonding character, with two degenerate bands (2PX and 2PY) above it
Recurring pattern: at Γ, valence S orbitals form bonding crystal orbitals, while valence P orbitals form orthogonal antibonding crystal orbitals
These sigma bands aren't our focus — like benzene, graphene's key electronic features arise from π overlap
π and π* bands come from 2PZ overlap, perpendicular to the plane; 2PZ is orthogonal to the other three everywhere in the first Brillouin zone — we only need to consider π overlap
Antibonding sigma bands exist at higher energy (not shown) but aren't important for properties

Let's now examine the π and π* bands' crystal orbitals at key Brillouin zone points.

Graphene π Crystal Orbitals at Γ, M, and K

The pi and pi-star crystal orbitals of graphene drawn at the Gamma point beside the band structure; clicks advance the same pictures to the M point and then the K point where the bands touch.

At Γ, there are two bands from the two PZ orbitals in the unit cell (one per carbon).

Basis set: the π bonding MO (lower band) and π* antibonding MO (upper band)
π bonding at Γ: looking down at the lobe emerging from the plane, each carbon bonds with its three nearest neighbors — the lowest energy crystal orbital
π* at Γ: each carbon antibonding with its three nearest neighbors — the most antibonding crystal orbital in the entire first Brillouin zone, considerably higher energy

Now let's go to M.

[advance: move the crystal-orbital pictures to the M point]

At M, the reciprocal space lattice vector is (1/2)A* + 0B* — phase changes moving in the A direction, stays constant in B
Bonding MO: each carbon forms two π bonding and one π antibonding interaction — a weakly bonding crystal orbital
Antibonding band: same phase pattern — two antibonding and one bonding per carbon, weakly π antibonding
At M there's still considerable energy separation between the two bands

What about at K?

[advance: move the crystal-orbital pictures to the K point]

At K, k = (1/3)A* + (1/3)B* — phase changes once every three unit cells in both directions; moving three unit cells in any direction returns to the same phase
Harder to visualize — only the real part of the wave function is shown (there's also an imaginary part at K)
Starting from any carbon, the pz coefficient on its three neighbors is zero — non-bonding, for both π and π*, so they touch precisely at K
Electron filling: four valence electrons per carbon, eight per unit cell — fills four bands, the three sigma bands and the π band, leaving π* empty

Seen in three dimensions, that touching at K has a name and a consequence.

Dirac Points and Semi-Metallic Graphene

Three-dimensional plot of graphene's pi and pi-star bands over the Brillouin zone, touching at six Dirac points on the zone boundary at K and K-prime.

Here's a 3D plot visualizing graphene's π and π* band structure.

Γ at the center (KX = KY = 0): large energy separation between the low-energy π band and high-energy π* band
At K the bands touch — degenerate; the same occurs at K'
Around the zone periphery, six points where upper and lower bands just touch — Dirac points
Perfect semi-metal: the Fermi level doesn't cut through a band like metals, yet there's no gap like semiconductors
Doping converts graphene to a metallic conductor — p-type lowers the Fermi level to cut through bands below this plane, n-type raises it into the conduction band; any deviation from perfect electron count makes graphene metallic
Unique feature: moving away from Dirac points, π* energy increases linearly with k while π decreases linearly — implications in Chapter 10

Let's check our understanding of the graphene band structure with a few questions.

Check Your Understanding: π and π* Orbitals

Question 3 asking which atomic orbitals primarily form the pi and pi-star bands of graphene near the Fermi level, then its answer: the 2pz orbitals.

Check Your Understanding: Graphene's Unit Cell

Question 4 asking how many carbon atoms are in the conventional unit cell of graphene, then its answer: two.

Check Your Understanding: Where the Bands Touch

Question 5 asking at which high-symmetry point the pi and pi-star bands of graphene become degenerate, then its answer: the K and K-prime points.

Check Your Understanding: The π Band at Γ

Question 6 asking for the crystal orbital character of the lowest energy pi band at the Gamma point, then its answer: fully bonding.

Learning Objectives: 3D Band Structures

By the end of this lecture, you will be able to:

Construct 3D reciprocal lattice vectors using cross products and identify special k-points (Gamma, X, M, R) in primitive cubic Brillouin zones
Predict band dispersion for s and p orbitals by analyzing nearest-neighbor bonding patterns at high-symmetry points
Explain how p-orbital directionality causes band splitting and determines which bands remain degenerate at specific k-points
Apply sigma and pi overlap analysis to rationalize subtle energy variations between k-points
Predict Peierls distortions in half-filled p-band systems and calculate Fermi level positions from valence electron count

Welcome to 3D band structures. We start with the simplest case: primitive cubic alpha-polonium.

Reciprocal lattice vectors use cross products: A-star equals B cross C times 2 pi over unit cell volume; for orthogonal lattices the reciprocal vectors are also orthogonal, same directions as real space — primitive cubic stays primitive cubic
Special k-points: Gamma at origin, X at half A-star, M at half A-star plus half B-star, R at half of all three vectors
Four bands: one atom per unit cell with 6s and 6p orbitals — lower band from 6s, upper three from 6p; DFT sets zero energy at the Fermi level by convention
6s band: at Gamma six bonding neighbors, lowest; at X two antibonding plus four bonding; at M four antibonding plus two bonding; at R six antibonding, highest
P orbital directionality breaks the degeneracy except at Gamma and R — 6px drops at X when sigma becomes bonding; 6pz remains high at X and M, drops at R with sigma bonding
Fermi level: six valence electrons fill 6s completely plus two-thirds of the p bands; bismuth's half-filled p bands trigger a Peierls distortion — three short bonds and three long bonds, lowering occupied states
Spin-orbit coupling in heavy elements removes p-orbital degeneracy at Gamma and R

First, the reciprocal lattice vectors in three dimensions.

Reciprocal Space Vectors in 3D

A 3D primitive cubic real-space lattice beside the reciprocal-space definitions: a-star equals two pi over V times b cross c, and cyclic formulas for b-star and c-star.

To define the first Brillouin zone in three dimensions, we need three reciprocal space lattice vectors.

Three real space lattice vectors (A, B, C) with fractional coordinates (X, Y, Z)
Definition: A* = (B × C) × 2π/V_cell, with similar relationships for B* and C*, where V_cell is the unit cell volume
If real space vectors are orthogonal, the reciprocal vectors are also orthogonal and point in the same directions as their real space counterparts
This follows because the cross product of two vectors yields a vector perpendicular to both, with length equal to the parallelogram area they define

We won't derive these reciprocal space lattice vectors in detail, but it's useful to understand their origin

For the primitive cubic lattice those vectors give a simple cubic Brillouin zone with four named points.

First Brillouin Zone: Primitive Cubic

The cubic first Brillouin zone of a primitive cubic lattice with special points Gamma, X, M, and R labeled, and tables of their coordinates and Cartesian wave vectors.

Band Structure of a Primitive Cubic Metal (α-Po): the 6s Band

Calculated band structure of alpha-polonium along Gamma, X, M, R, Gamma with four bands; clicks add 6s crystal-orbital pictures at Gamma, X, M, and R below the plot.

The simplest 3D material is arguably a primitive (simple) cubic metal.

Alpha-polonium: the only element with this structure; shown here is its calculated band structure from DFT, which we'll use for the rest of this chapter
Zero energy at the Fermi level by convention — different from the extended Hückel calculations used previously
Four bands: one polonium atom per unit cell, valence 6s and three 6p orbitals — you can recognize that the lower band comes from 6s and the upper three from 6p

[advance: show the 6s crystal orbital at Γ]

At gamma: all atoms must have the same phase from one unit cell to the next; the base set is just a single atomic orbital, the 6s orbital on polonium — six nearest neighbor bonding interactions, the lowest point in the 6s band

[advance: add the 6s crystal orbital at X]

At x: phases change along the x-axis only — two neighbors antibonding, the other four still bonding; higher in energy than gamma

[advance: add the 6s crystal orbital at M]

At m: phases change on translation in x or y — four neighbors in the xy plane antibonding, two in z still bonding; more antibonding than bonding, higher still

[advance: add the 6s crystal orbital at R]

At r: phase changes on translation in any of the three directions — all six nearest neighbor interactions antibonding, the highest energy crystal orbital in the 6s band
This could have been pretty easily extrapolated from what we learned in one and two dimensions, just looking at these hydrogen atom sheets

Now for the p orbitals, starting with 6px.

The $6p_x$ Band

Alpha-polonium band structure with the 6px crystal orbital drawn at Gamma; clicks add the 6px pictures at X, M, and R showing sigma bonding turn on at X.

So let's consider the six p orbitals.

Start with 6px at gamma: just like our fluorine atom chain, the sigma interaction between p orbitals is going to be anti-bonding — relatively high energy
At the gamma point all three p orbitals are degenerate — the same antibonding sigma interaction

[advance: add the 6px crystal orbital at X]

At x: translating in the x direction changes the phase — the orbitals have become sigma bonding, so the energy should drop

[advance: add the 6px crystal orbital at M]

At m: phases change in x and y — the sigma interactions are still bonding; x-bonding, y antibonding, so higher energy

[advance: add the 6px crystal orbital at R]

At r: bonding x, antibonding y, antibonding z
Which band is 6px? Going from Γ to X changes σ antibonding to σ bonding — a substantial energy drop; only band 1 drops from Γ to X, so band 1 is the 6px band
Why lowest at X, rising at M and R: the weaker π interactions — bonding with the four neighbors at Γ and X; at M bonding along z but antibonding along y, net non-bonding; at R antibonding in both perpendicular directions — this accounts for the subtle rise of the 6px band

The 6pz orbital tells a different story.

The $6p_z$ Band

Alpha-polonium band structure labeled 6pz; clicks add the 6pz crystal-orbital pictures at Gamma, X, M, and R showing sigma antibonding persist until R.

Now let's work through the same k points for the six p z orbital.

[advance: show the 6pz crystal orbital at Γ]

The other p bands are not all on top of the 6px band — and we can see why from the 6pz overlap
At gamma: sigma anti-bonding character and pi bonding character

[advance: add the 6pz crystal orbital at X]

At x: orbitals displaced by one unit cell in x change phase, but because it's a 6pz orbital we still have sigma anti-bonding interactions everywhere — 6pz does not go down in energy from gamma to x

[advance: add the 6pz crystal orbital at M]

At m: we now change phases in both the x and y directions

[advance: add the 6pz crystal orbital at R]

At r: a change in phase on translation in any direction — back to sigma bonding, pi anti-bonding interactions
At r and at gamma all three orbitals are degenerate; at the other points they're not — simply the directionality of the p orbitals

So which of these three bands is the p z orbital?

[advance: hold the full picture while we pick out the band]

Hopefully you can see it's band number three: sigma star at gamma, x, and m — relatively high energy, and only band three does that
It goes up from gamma to x to m as the pi interactions go from bonding to non-bonding to anti-bonding, then drops sharply at the r point

Putting all three orbitals together completes the polonium band structure.

Assembling Polonium's Band Structure

Alpha-polonium band structure beside a grid of crystal-orbital pictures for the pz, px, and s bands at Gamma, X, and R.

Putting this together for polonium's band structure: we didn't examine py orbital overlap, but by deduction, this band is the py band.

Recurring pattern: at Γ, s orbitals typically have bonding overlap while p orbitals have antibonding overlap, with large energy separation between them
Fermi level placement: polonium is group 16 with six valence electrons — 6s filled, four electrons left for 6p, so the p bands are two-thirds filled
Bismuth, one element left (lower atomic number): three valence p electrons, making the p orbitals half-filled
A half-filled band is susceptible to distortion — in one dimension a Peierls distortion, and bismuth metal indeed exhibits something analogous to a Peierls distortion

Here is what that distortion looks like in three dimensions.

Peierls Distortion in 3D

Cubic alpha-polonium with 6p bands two-thirds filled beside rhombohedral bismuth with 6p bands half filled, three short bonds at 3.07 angstroms and three long bonds at 3.53 angstroms.

The Band Structure of Bismuth

Published band structure of bismuth from a pseudopotential calculation, showing split bands and a gap opening at the Fermi energy along several symmetry lines.

Check Your Understanding: The 6s Band

Question 2 asking for the energy ordering of the alpha-polonium 6s band at the high-symmetry k-points, then its answer: Gamma below X below M below R.

Check Your Understanding: Identifying the $6p_x$ Band

Question 3 asking how the 6px band can be uniquely identified along the Gamma to X path, then its answer: the 6px band drops significantly in energy while 6py and 6pz do not.

3. In the band structure of alpha-Polonium, the 6px, 6py, and 6pz orbitals are degenerate at Γ. How can the 6px band be uniquely identified along the Γ → X path, where X = (1/2, 0, 0)? A. The 6px band rises significantly in energy, while the 6py and 6pz bands drop. B. All three p-bands remain degenerate along the Γ → X path. C. The 6px band drops significantly in energy, while the 6py and 6pz bands do not. D. The 6px band hybridizes with the 6s band at X, while the others do not. Take a moment and think about it.

[advance: reveal the answer]

C. The 6px band drops significantly in energy, while the 6py and 6pz bands do not. That's right! The k-vector at X introduces a phase change along the x-direction, which turns the 6px σ-anti-bonding interaction (at Γ) into a σ-bonding one (at X).

Before we go on to face-centered cubic crystals, your homework.

Learning Objectives: FCC Band Structures

Band Structures of Face-Centered Cubic Crystals

Understand the first Brillouin zone structure for face-centered cubic lattices
Analyze the electronic band structure of FCC metals using aluminum as a representative example
Examine band structures of diamond structure materials (carbon, silicon, germanium)
Distinguish between valence bands, conduction bands, and band gaps in semiconductors
Explain the progression from insulator to semiconductor to metal down Group 14 elements
Understand how orbital overlap influences bonding/anti-bonding states and metallic vs semiconducting behavior

Electronic Band Structures | Materials Science & Chemistry

First Brillouin Zone: Face-Centered Cubic

The truncated-octahedron first Brillouin zone of the FCC lattice with points Gamma, X, W, K, and L, beside the FCC cell with its rhombohedral primitive cell outlined.

The first Brillouin zone for a face-centered cubic (FCC) Bravais lattice looks much more complicated than simple cubic.

This complexity arises because the Brillouin zone corresponds to the primitive unit cell of FCC
The cubic-centered unit cell has four atoms; the dark lines show the rhombus primitive unit cell — equal-length edges, 60° angles between them — which explains why the zone appears so complex
The shape is a truncated octahedron
Γ is always at (0,0,0) at the center
L point: the FCC equivalent of the R point in primitive cubic — (1/2) of each of the three reciprocal space lattice vectors, so phase changes occur from one unit cell to the next in any real space lattice direction

Aluminum is our first face-centered cubic example.

Aluminum

For aluminum with cubic close-packed (FCC) metallic structure, the primitive cell contains one aluminum atom, giving four bands: a 3s band and three 3p bands.

At Γ: large separation between the s and p bands
3s at Γ: all interactions in phase and bonding — each aluminum has 12 nearest neighbors, all bonding; the lowest energy crystal orbital in the entire structure
p orbitals at Γ: drawn parallel to the rhombohedral unit cell's long axis; because p orbitals have nodal planes and change phase from one side to the other, at Γ there are antibonding interactions between p orbitals — all three degenerate, the highest energy crystal orbital in the first Brillouin zone
At other special points and along lines between them, sp mixing produces crystal orbitals between these two extremes
Very wide bands spanning ~24-25 eV; three valence electrons fill the bands only 37.5% (3/8)
The Fermi level cuts through these wide bands — a recipe for delocalized electronic conductivity

From an FCC metal we turn to an FCC insulator: diamond.

Diamond

Diamond: Band Structure

With two atoms per primitive unit cell, we have doubled the number of bands—eight total.

Lower four bands: bonding MOs between the two carbons in the unit cell; like aluminum, at Γ the s and p orbitals don't mix — a completely bonding crystal orbital with carbon 2s character, and higher-energy carbon 2p states that are largely antibonding
"Antibonding in the bonding MO band": the two-atom motif within the unit cell has bonding interactions, but this same basis has antibonding interactions with neighboring unit cells' basis sets — at Γ_p it's a mixture
Upper four bands: antibonding combinations of the two-atom basis — one antibonding s band and three antibonding p bands
Filling: eight valence electrons per unit cell — enough to fill the bottom four bands, with a 5.5 eV gap to the next four empty bands
Nomenclature: the filled, mostly bonding states are the valence band; the empty, mostly antibonding states are the conduction band — the 5.5 eV gap makes diamond an insulator
Indirect band gap: valence band maximum at Γ, conduction band minimum between Γ and X

Diamond is not alone; the elements below carbon adopt the same structure.

Diamond-like Group 14 Elements

The diamond structure beside a table of Group 14 elements showing bond distance growing from 1.54 to 2.81 angstroms and band gap shrinking from 5.5 to 0.1 eV, with a p-block periodic table.

Moving down group 14: elements immediately below carbon—silicon and germanium—also crystallize with the diamond structure.

Bonds lengthen moving down the periodic table (as expected), and the band gap decreases substantially
Si and Ge: ~1 eV band gaps, ideal for electronic devices — silicon is one of the most important elements for modern computer technology
Tin is polymorphic: gray tin has the diamond structure with longer bonds and only 0.1 eV band gap — practically a semimetal; white tin is tetragonal, properties closer to a metal than a semiconductor
Lead: metallic with a close-packed structure, not a semiconductor
Insulator (top of group 14) to semiconductors (Si, Ge, Sn) to metal (Pb) isn't unique to group 14 — it occurs across much of the p-block. How can we understand this from a bonding perspective?

The silicon and germanium band structures hold the answer.

Silicon and Germanium

Silicon and germanium band structures are qualitatively similar to diamond—both are indirect band gap semiconductors with valence band maximum at Γ and conduction band minimum elsewhere.

Key observation: the S* band energy at Γ — the crystal orbital from the antibonding combination of 2s orbitals in the basis at Γ
In carbon, S* is much higher than the antibonding p bands; in silicon, S* is still higher but not by much; in germanium, S* drops below the antibonding p bands, quite close to the valence band top; in tin, the gap almost disappears
What's happening: as bond distance increases and orbital overlap decreases, bonding states become slightly less bonding while antibonding states become substantially less antibonding — shrinking the gap between antibonding and bonding bands
Lead: S* drops below bonding p states (if calculated with diamond structure) — antibonding states fill before bonding states complete filling, totally destabilizing the tetrahedral coordination of the diamond structure; no more semiconducting behavior
Bottom line: decreasing orbital overlap drives the insulator→semiconductor→metal crossover — the low-coordination covalent bonding arrangement becomes less stable

Two final questions check the FCC ideas.

Check Your Understanding: Why Aluminum Is a Metal

Question 4 asking why aluminum is considered a metal based on its band structure, then its answer: its Fermi level cuts through wide, partially filled bands.

Check Your Understanding: Down Group 14

Question 10 asking for the primary physical reason for the insulator to semiconductor to metal trend down Group 14, then its answer: bond distances increase, decreasing orbital overlap and making anti-bonding states less anti-bonding.

10. What is the primary physical reason for the trend from insulator (C) to semiconductor (Si, Ge) to metal (Pb) down Group 14? A. Bond distances increase, decreasing orbital overlap and making anti-bonding states less anti-bonding. B. The number of valence electrons increases down the group. C. Relativistic effects become dominant for carbon and silicon. D. The elements become more electronegative down the group. Take a moment and think about it.

[advance: reveal the answer]

A. Bond distances increase, decreasing orbital overlap and making anti-bonding states less anti-bonding. That's right! This decrease in overlap causes the anti-bonding S* band to drop in energy, shrinking and eventually closing the band gap.

That completes today's lecture on the band structures of graphene, primitive cubic, and face-centered cubic crystals. Your homework is now the Lecture 17 practice questions on the course website.

Practice questions

Work the Lecture 17 practice questions before the next class. They cover the hexagonal reciprocal lattice and its first Brillouin zone, graphene’s σ and π bands and its Dirac points, the primitive cubic band structure of alpha-polonium and the Peierls distortion of bismuth, and the face-centered cubic band structures of aluminum, diamond, silicon, and germanium.

Open the Lecture 17 practice questions

Every question carries a worked explanation, so you can check your reasoning as you go.

The homework for this lecture now lives in the online practice bank.

Work through it before the next class
It covers the reciprocal lattice of a hexagonal system and its special k-points
It covers the pi and pi-star crystal orbitals of graphene at Gamma, M, and K and the semi-metallic behavior at the Dirac points
It covers the 6s and 6p bands of primitive cubic alpha-polonium and the Peierls distortion that gives bismuth its rhombohedral structure
It covers the face-centered cubic band structures of aluminum, diamond, silicon, and germanium along with the insulator to semiconductor to metal trend down Group 14

Every question has a worked explanation attached, so you can check your reasoning as you go.