CHEM 548: Materials Chemistry

Lecture 16: Reciprocal Space, 2D Band Structures, and the CuO₂²⁻ Layer

Wed 10/28/2026 · Meeting 18

Reciprocal lattices, the first Brillouin zone, 2D band structures, and the CuO₂²⁻ layer

Reading: Woodward, Karen, Evans, and Vogt, Solid State Materials Chemistry
Ch. 6 §§6.3–6.5

Learning Objectives: 2D Band Structures

Reciprocal lattice vectorsconstruct them from real-space vectors by the perpendicularity and length requirements

First Brillouin zoneidentify it as the Wigner-Seitz cell of reciprocal space

Special k-pointslocate and interpret Γ, X, and M in a 2D square lattice

Orbital phasespredict crystal-orbital phase relationships at different k-points in two dimensions

Band diagramsread and construct them for 2D materials using cuts through reciprocal space

Real and Reciprocal Lattices in 2D

A five by five array of filled lattice points with x and y axis arrows, and the vectors a and b drawn from one point to each of its two nearest neighbours.

real-space lattice
$\mathbf{T} = u\mathbf{a} + v\mathbf{b}$
$\mathbf{a}$, $\mathbf{b}$ = lattice vectors; $u$, $v$ = integers

A five by five array of open reciprocal lattice points with x and y axis arrows, and the vectors a-star and b-star drawn from one point to each of its two nearest neighbours.

reciprocal-space lattice
$\mathbf{G} = h\mathbf{a}^{*} + k\mathbf{b}^{*}$
$\mathbf{a}^{*}$, $\mathbf{b}^{*}$ = lattice vectors; $h$, $k$ = integers

Defining the Reciprocal Lattice Vectors

A rectangular array of reciprocal-space lattice points with the vectors a-star and b-star drawn from one point to each of its two nearest neighbours.

reciprocal-space lattice
$\mathbf{G} = h\mathbf{a}^{*} + k\mathbf{b}^{*}$
$\mathbf{a}^{*}$, $\mathbf{b}^{*}$ = lattice vectors; $h$, $k$ = integers

Reciprocal space lattice vectors

$\mathbf{a}^{*}$perpendicular to $\mathbf{b}$, $\mathbf{a}^{*} \cdot \mathbf{a} = 2\pi$

$\mathbf{b}^{*}$perpendicular to $\mathbf{a}$, $\mathbf{b}^{*} \cdot \mathbf{b} = 2\pi$

Reciprocal space rectangular lattice

$\mathbf{a}^{*} = \dfrac{2\pi}{a}\,\hat{x}$

$\mathbf{b}^{*} = \dfrac{2\pi}{b}\,\hat{y}$

Reciprocal Space in 1D

real-space lattice
$\mathbf{T} = u\mathbf{a}$

A row of five filled dots on a line with a double-headed bracket over one dot spacing labelled a.

reciprocal-space lattice
$\mathbf{G} = h\mathbf{a}^{*}$

A row of five open dots on a line with a double-headed bracket over one dot spacing labelled a-star equals 2 pi over a.

real-space lattice
$\mathbf{T} = u\mathbf{a}$

reciprocal-space lattice
$\mathbf{G} = h\mathbf{a}^{*}$

The same row of open dots, with arrows beneath labelling the centre dot k equals 0 and its neighbours k equals minus pi over a and k equals plus pi over a.

real-space lattice
$\mathbf{T} = u\mathbf{a}$

reciprocal-space lattice
$\mathbf{G} = h\mathbf{a}^{*}$

The boundaries of the first Brillouin zone are ½ a* toward each of the neighboring reciprocal-space lattice points.

To see what the reciprocal lattice has to do with band structures, we go back to one dimension. We did not bother with it there because it is trivial. There is only one direction, so the only thing that differs between the real and the reciprocal lattice is the length of the unit cell. The reciprocal lattice vector is 2 pi divided by the real space lattice vector, the same inverse relationship as before.

[advance: label the k values]

Remember that k also has units of one over distance. When we talked about the first Brillouin zone, we said that to know everything about the electronic structure we need the values of k from minus pi over a to plus pi over a. That interval is exactly the unit cell of the reciprocal lattice. So the first Brillouin zone is the unit cell of the reciprocal lattice, and that is part of why it is so important.

[advance: mark the Brillouin zone boundaries]

To find the first Brillouin zone, we start at a lattice point and go half of the reciprocal lattice vector in either direction.

The First Brillouin Zone in 2D

A rectangular array of reciprocal-space lattice points with a shaded rectangle centred on one point, labelled first Brillouin zone, and the vectors a-star and b-star drawn from that point.

reciprocal-space lattice
$\mathbf{G} = h\mathbf{a}^{*} + k\mathbf{b}^{*}$
$\mathbf{a}^{*}$, $\mathbf{b}^{*}$ = lattice vectors; $h$, $k$ = integers

First Brillouin zone the area in reciprocal space that contains all k points closer to the lattice point it surrounds than to any other reciprocal space lattice point. In crystallography we call this type of cell a Wigner-Seitz cell.

Per Solid State Materials Chemistry, §6.4.1.

Now we take that idea to two dimensions, with a rectangular reciprocal lattice. If we draw a rectangle that bisects both reciprocal lattice vectors, halfway to the next lattice point in each direction, that rectangle is the first Brillouin zone.

[advance: add the formal definition]

Formally, the first Brillouin zone is the set of k points that are closer to the origin than to any other lattice point. Once we move outside the rectangle, we are closer to another lattice point. A unit cell made by bisecting the lattice vectors is called a Wigner-Seitz cell. When the lattice vectors are not orthogonal, as in a hexagonal or oblique lattice, the first Brillouin zone does not have to be a parallelogram, and in that way it differs from the crystallographic unit cell. To represent the band structure in two dimensions we need the energy at every point in this zone, which is a three-dimensional plot, with k along a star, k along b star, and energy. So most of the time we take cuts through it.

Special Points in a 2D Square Lattice

The shaded first Brillouin zone of a square lattice with the centre point Gamma circled in red, X on the lower edge, M at the lower corner, and the a-star and b-star axes drawn from Gamma.

The shaded first Brillouin zone of a square lattice labelled Gamma at the centre, X on the lower edge, and M at the lower corner, with the a-star and b-star axes drawn from Gamma.

label	coordinates
Γ	$0\,\mathbf{a}^{} + 0\,\mathbf{b}^{}$
X	$(1/2)\,\mathbf{a}^{} + 0\,\mathbf{b}^{}$
M	$(1/2)\,\mathbf{a}^{} + (1/2)\,\mathbf{b}^{}$

$\mathbf{a}^{*} = \dfrac{2\pi}{a}\hat{x}$ $\mathbf{b}^{*} = \dfrac{2\pi}{a}\hat{y}$ (square: $b = a$)

$\mathbf{k} = k_x\,\hat{x} + k_y\,\hat{y}$

$\mathbf{a}^{*} = \dfrac{2\pi}{a}\hat{x}$ $\mathbf{b}^{*} = \dfrac{2\pi}{a}\hat{y}$ (square: $b = a$)

$\mathbf{k} = k_x\,\hat{x} + k_y\,\hat{y}$

label	substitute $\mathbf{a}^{}$, $\mathbf{b}^{}$	$(k_x,\ k_y)$
Γ	$0$	$(0,\ 0)$
X	$\tfrac12\cdot\tfrac{2\pi}{a}\hat{x}$	$(\pi/a,\ 0)$
M	$\tfrac12\cdot\tfrac{2\pi}{a}\hat{x} + \tfrac12\cdot\tfrac{2\pi}{a}\hat{y}$	$(\pi/a,\ \pi/a)$

Per Solid State Materials Chemistry, Figure 6.14.

We make those cuts between special high-symmetry points in the Brillouin zone, because that is usually where the bands have their maxima and minima. A square lattice has only three. Gamma is at the center of the zone, X is at the zone boundary in the a star direction, and M is at the corner. The point at the boundary in the b star direction is exactly equivalent to X, because the square lattice has fourfold symmetry. The table gives their coordinates in terms of the reciprocal lattice vectors.

[advance: drop the ring on gamma and give the a-star and b-star definitions]

We want to write these points as wave vectors. In two dimensions the wave vector k has two components, k x along x and k y along y, and each component is a number. Remember that a star is 2 pi over a along x and b star is 2 pi over b along y. In a square lattice a and b are equal, so both have length 2 pi over a.

[advance: rewrite the three points as wave vectors]

Now we substitute. At gamma, k x and k y are both zero. X is one half of a star, and one half of 2 pi over a is pi over a, so at X, k x is pi over a and k y is zero. M is one half of a star plus one half of b star, so k x and k y are both pi over a. We already know what these values mean for the orbitals. When k is zero the phase of the orbital does not change from one unit cell to the next, and when k is pi over a the phase reverses every time we move to a neighboring unit cell.

Band Structure of a 2D H-Atom Sheet

A square lattice of hydrogen 1s orbitals, five by five, all drawn as open circles joined by bonds.

A five by five lattice of hydrogen 1s orbitals all drawn open, every orbital in phase with its neighbours.

Γ point
$k_x = k_y = 0$
bonding

The same lattice with whole rows alternating between open and shaded orbitals: the phase reverses along x and is unchanged along y.

X point
$k_x = \pi/a$ $k_y = 0$
nonbonding

Γ point
$k_x = k_y = 0$
bonding

X point
$k_x = \pi/a$ $k_y = 0$
nonbonding

The same lattice in a checkerboard of open and shaded orbitals: the phase reverses along both x and y.

M point
$k_x = k_y = \pi/a$
antibonding

A plot of energy in electron volts against the k-point path Gamma to X to M to Gamma: the band rises from about minus 22 eV at Gamma through the Fermi level near X to about 53 eV at M, then falls back to Gamma.

Γ point
$k_x = k_y = 0$
bonding

X point
$k_x = \pi/a$ $k_y = 0$
nonbonding

M point
$k_x = k_y = \pi/a$
antibonding

Per Solid State Materials Chemistry, Figure 6.17.

As a simple example, we take a two-dimensional sheet of hydrogen atoms in a square lattice.

[advance: show the gamma-point crystal orbital]

At gamma, k x and k y are both zero, so all the hydrogen 1s orbitals have the same phase. E to the i k x is just one, so when we move to a neighboring unit cell the phase stays the same.

[advance: add the X-point crystal orbital]

At X, k x is pi over a, so every time we move one unit cell in the x direction we change the phase, positive, negative, positive, negative. But k y is still zero, so moving from one unit cell to the next in the y direction does not change the phase. This crystal orbital is basically nonbonding, because each atom has antibonding interactions with two neighbors and bonding interactions with the other two.

[advance: add the M-point crystal orbital]

Finally, at M the phase alternates in both the x and the y directions. Now each hydrogen has four antibonding interactions with its neighbors, so this is the most antibonding crystal orbital. Energy increases from gamma to X to M.

[advance: draw the band structure]

This is one way to represent the band structure of the hydrogen sheet. The y axis is energy, as in one dimension, and the x axis is labeled with the special k points. Gamma is the all-bonding orbital, so it is at the minimum energy. X is antibonding in one direction and bonding in the other, so it is close to nonbonding, and M is antibonding in both directions, so it is the highest. We go from gamma to X to M, and usually we come back from M to gamma. The choice of path is somewhat arbitrary, and the points in between are a line through the first Brillouin zone. Like the hydrogen atom chain, this is a half-filled band, so the Fermi level cuts through at about X, and this sheet would be a metallic conductor.

Learning Objectives: The CuO₂²⁻ Layer

Multi-atom unit cellsconstruct band structures from molecular orbital diagrams

DOS and partial DOSinterpret them to identify orbital contributions

Overlap and symmetrypredict orbital overlap patterns and symmetry-allowed interactions at different k-points

Band widthrelate it to orbital overlap strength and electron delocalization

Propertiesextract metallic versus semiconducting behavior and localized versus delocalized electrons from a band diagram

La₂CuO₄: Home of the CuO₂²⁻ Layer

The layered structure of lanthanum copper oxide: sheets of blue copper-oxygen octahedra stacked with layers of green lanthanum ions between them.

La₂CuO₄, a layered perovskite

The CuO2 sheet seen down the stacking axis: blue copper atoms on a square net, each bridged to four red oxygen atoms, with the square unit cell outlined.

the CuO₂²⁻ sheet viewed down the stacking axis

The CuO₂²⁻ Sheet

A segment of the CuO2 sheet: blue copper atoms on a square net bridged by red oxygen atoms, with one square unit cell shaded and outlined in black.

d-orbital energies, square planar Cu²⁺

Per Solid State Materials Chemistry, Figure 6.20, §6.5.2.

MO Diagram → DOS Plot

Three panels on a shared energy axis from minus 17 to minus 11 electron volts: the extended Hueckel MO diagram of square planar copper at left, the total density of states in the middle with regions one, two, and three labelled and the Fermi level drawn in, and the copper 3d partial density of states at right, resolved into dx2-y2, dxy, dxz and dyz, and dz2 columns.

The same three panels with the copper 3d partial density of states panel at right ringed in red.

IO 2p nonbonding and Cu-O bonding

IICu 3d-O 2p π antibonding

IIICu 3d-O 2p σ antibonding

Per Solid State Materials Chemistry, Figure 6.21.

Let's jump ahead to the results of the band structure calculation and then work backwards to understand them. On the left is the extended Hückel MO diagram for square planar copper two plus. Four d orbitals, d x z, d y z, d x y and d z squared, are at about minus 13 to minus 14 electron volts, and d x squared minus y squared, the sigma antibonding orbital, is several electron volts higher. Below them are mostly oxygen 2p orbitals, some mixed with copper as bonding orbitals. If we broaden these orbitals into bands we get the density of states in the middle. Region one is oxygen 2p nonbonding and copper oxygen bonding, region two comes from the four d orbitals that do not point at the oxygens, and region three is the half-filled sigma star band from d x squared minus y squared, which has one electron.

[advance: highlight the partial density of states]

On the far right is a partial density of states, which shows where each orbital contributes. D x squared minus y squared contributes at the bottom, bonding with oxygen, and at the top, antibonding, and the large splitting is because it has the strongest interaction with oxygen. D x y lies in the same plane as the oxygens and forms pi and pi star interactions with them, so it has the next largest splitting, with bonding character in the oxygen bands and antibonding character in region two. D x z and d y z are similar. D z squared sits almost entirely in one narrow energy window, because its interaction with oxygen is very weak, so it is nonbonding.

[advance: clear the highlight and go back to all three regions]

The relative areas of the regions follow from counting orbitals. Regions two and three both come from copper d orbitals, but region two has four and region three only one, so region three has one fourth of the area. Region one is mostly oxygen 2p. There are two oxygens per copper, each with three 2p orbitals, so six bands, and region one should have about 50 percent more area than region two with its four. That is as far as the MO diagram takes us. To say how wide the bands are and draw the band structure itself, we have to look at the orbital overlap, so let's do that for two bands.

Basis Set Orbitals: the d_xy Band

The unit cell square with the four-lobed copper 3dxy orbital at its top-left corner and the two oxygen 2p orbitals lying perpendicular to the copper-oxygen bonds.

$\mathrm{Cu}\ 3d_{xy} - \mathrm{O}\ 2p\ \pi^{*}$

The d_xy Band at Γ

$\Gamma\ (k_x = 0,\ k_y = 0)$

The copper lattice drawn as a two by two grid of unit cells with one copper 3d xy orbital, two of its four lobes shaded, circled at the top left corner.

Cu 3d_xy

The oxygen lattice drawn as a two by two grid of unit cells with one vertical and one horizontal oxygen 2p orbital circled at the top left corner.

O 2p

A three by three lattice of copper 3d xy orbitals, every orbital in the same phase, with the central orbital ringed in red.

Cu 3d_xy

The oxygen 2p orbitals filled into the lattice, the vertical ones with the shaded lobe pointing up and the horizontal ones with the shaded lobe pointing left.

O 2p

The three by three lattice of copper 3d xy orbitals, every orbital in the same phase.

Cu 3d_xy

The oxygen 2p lattice with a red curve drawn through the upper left cell, tracing the bonding overlap of the copper lobe that points that way.

O 2p

Cu 3d_xy

The oxygen 2p lattice with a red curve drawn through the lower right cell, tracing the antibonding overlap of the opposite copper lobe.

O 2p

Cu 3d_xy

The oxygen 2p lattice back without the overlap curves, every vertical orbital shaded up and every horizontal orbital shaded left.

O 2p

Mixing is symmetry forbidden Cu 3d_xy and O 2p remain nonbonding.

Per Solid State Materials Chemistry, Figure 6.23.

starting at gamma, where k x and k y are both zero. I have put the copper orbitals and the oxygen orbitals on separate diagrams just for clarity.

[advance: fill in all the orbital phases]

At gamma the phase does not change when we move to a neighboring unit cell, so all the copper d x y orbitals have the same phase, and the same is true for the oxygens. The shaded lobe of every vertical oxygen 2p points up, and on every horizontal one it points left. The net interaction between these two sets is zero, and we can check that if we take the copper d x y orbital in the middle, put it on the oxygen lattice, and look at the net interaction.

[advance: highlight the bonding lobe]

The lobe that points this way has a bonding interaction with the oxygen 2p orbitals.

[advance: highlight the antibonding lobe]

But the lobe that points the opposite way, which is also shaded dark, has an antibonding interaction. The other two lobes meet oxygen 2p orbitals of opposite phase, so they also give equal bonding and antibonding overlap. So the net interaction is equal parts bonding and antibonding.

[advance: show that mixing is symmetry forbidden]

So the two sets of orbitals have different symmetries, and symmetry does not allow them to mix. So they stay nonbonding at gamma, and we can call this nonbonding copper d x y at gamma.

The d_xy Band at M

$\text{M}\ (k_x = \pi/a,\ k_y = \pi/a)$

The copper lattice drawn as a two by two grid of unit cells with one copper 3d xy orbital circled at the top left corner.

Cu 3d_xy

The oxygen lattice grid with one vertical and one horizontal oxygen 2p orbital circled at the top left corner.

O 2p

→

π* crystal orbital

A three by three lattice of copper 3d xy orbitals whose phases alternate with every step of one unit cell in x and in y.

Cu 3d_xy

The oxygen lattice grid still empty apart from the two circled orbitals at the top left corner.

O 2p

→

π* crystal orbital

The three by three lattice of copper 3d xy orbitals with phases alternating in both directions.

Cu 3d_xy

The oxygen 2p orbitals filled into their lattice, their phases alternating with every step of one unit cell in x and in y.

O 2p

→

π* crystal orbital

Cu 3d_xy

The oxygen 2p lattice with its alternating phases.

O 2p

→

The copper and oxygen lattices overlaid: every copper 3d xy lobe meets an oxygen 2p lobe of opposite phase, an antibonding pi interaction at every contact.

π* crystal orbital

Per Solid State Materials Chemistry, Figure 6.23.

Basis Set Orbitals: the d_x²−y² Band

The unit cell square with the four-lobed copper 3dx2-y2 orbital at its top-left corner and the two oxygen 2p orbitals pointing along the copper-oxygen bonds.

$\mathrm{Cu}\ 3d_{x^2-y^2} - \mathrm{O}\ 2p\ \sigma^{*}$

The d_x²−y² Band at Γ

$\Gamma\ (k_x = 0,\ k_y = 0)$

A three by three lattice of copper 3d x squared minus y squared orbitals, every orbital in the same phase, with one orbital circled at the top left corner.

Cu 3d_x²−y²

The oxygen 2p sigma orbitals filled into the lattice, the horizontal ones with the shaded lobe pointing left and the vertical ones with the shaded lobe pointing down.

O 2p

→

σ* crystal orbital

Mixing is symmetry forbidden No O 2p character in the Cu 3d_x²−y² band at Γ.

The same three by three lattice of copper 3d x squared minus y squared orbitals, every orbital in the same phase.

Cu 3d_x²−y²

The oxygen 2s orbitals in the same lattice, drawn as circles, the shaded ones and open ones alternating between the two oxygen sites of each cell.

O 2s

→

The copper and oxygen lattices overlaid: each oxygen 2s circle meets a copper lobe of opposite phase, an antibonding sigma interaction.

σ* crystal orbital

O 2s mixes inThe lower-lying O 2s orbitals, previously neglected, have the correct symmetry to mix in an antibonding way with d_x²−y², so the band is not completely nonbonding at Γ.

At gamma we do exactly the same analysis as for the d x y band. All the copper orbitals have the same phase, and so do the oxygen 2p sigma orbitals. On all the horizontal oxygens the shaded lobe points left, and on all the vertical ones it points down. Just as with d x y, everywhere there is a bonding overlap there is an equal and opposite antibonding overlap, so the copper d x squared minus y squared and the oxygen 2p orbitals do not mix at gamma.

[advance: bring in the oxygen 2s orbitals]

But that does not make d x squared minus y squared completely nonbonding at gamma. The oxygen 2s orbitals, which we have neglected so far, have the right symmetry to mix with it in an antibonding way. That antibonding interaction is fairly weak, because the oxygen 2s is not well matched in energy to the copper 3d, but it is not zero.

The d_x²−y² Band at M

$\text{M}\ (k_x = \pi/a,\ k_y = \pi/a)$

The copper lattice drawn as a two by two grid of unit cells with one copper 3d x squared minus y squared orbital circled at the top left corner.

Cu 3d_x²−y²

The oxygen lattice grid with one horizontal and one vertical oxygen 2p sigma orbital circled at the top left corner.

O 2p

→

σ* crystal orbital

A second copper 3d x squared minus y squared orbital added one unit cell along, drawn in the opposite phase to the first.

Cu 3d_x²−y²

The oxygen lattice grid still holding only the two circled orbitals at the top left corner.

O 2p

→

σ* crystal orbital

The full three by three lattice of copper 3d x squared minus y squared orbitals, their phases alternating with every step of one unit cell in x and in y.

Cu 3d_x²−y²

The oxygen 2p sigma orbitals filled into their lattice with the same alternation of phase in x and in y.

O 2p

→

The two lattices overlaid: every copper lobe meets an oxygen lobe of the opposite phase, so every copper-oxygen contact is antibonding.

σ* crystal orbital

Two Bands at Γ, X, and M

Γ X M Cu 3d_x²−y² – O σ* band

A three by three net of copper dx2-y2 orbitals with oxygen p orbitals between them, every copper lobe pointing at an oxygen lobe of the same shade along both axes.

weakly antibonding

The same net with the orbital phase reversed from column to column along x and unchanged along y.

antibonding in x

The same net in a checkerboard: the orbital phase reverses along x and along y.

antibonding in x,y

Cu 3d_xy – O π* band

A three by three net of copper dxy orbitals all in the same phase, with no oxygen p orbitals drawn between them.

nonbonding

The same net with small oxygen p orbitals between the coppers and the phase reversed from column to column along x.

antibonding in x

The same net with oxygen p orbitals along both axes and the phase reversed along x and along y.

antibonding in x,y

Per Solid State Materials Chemistry, Figure 6.24.

Band Structure of the CuO₂²⁻ Sheet

Energy in electron volts from minus 17 to minus 11 plotted along the path Gamma to X to M to Gamma, with a dashed Fermi level cutting the topmost band, beside the total density of states with its three peaks labelled one, two and three. A brace beside the topmost band labels it Cu dx2-y2, the sigma-star band.

$E_\mathrm{F}$ cuts the Cu $d_{x^2-y^2}$ $\sigma^{*}$ band

Per Solid State Materials Chemistry, Figure 6.22.

Now we can draw the band structure. The d x squared minus y squared band is lowest at gamma, rises at X, rises again at M, and comes back down to gamma.

[advance: label the pi star bands]

The copper d x y band does something similar, rising from gamma to M and coming back down to gamma.

[advance: label the d z squared band]

The almost flat band is copper d z squared, which has minimal interaction with the oxygens. D x z and d y z go from nonbonding at gamma to antibonding at X and M, but less antibonding than d x y.

[advance: label the oxygen 2p bands]

Below are the bonding bands that mirror these antibonding ones. The count matches, six oxygen 2p bands, four d bands, and one half-filled band higher than the other d bands, the copper d x squared minus y squared band.

Key Concepts: Band Structure

Six questions to ask of any band structure diagram.

Question	Answer
What is being plotted?	Energy vs. k, where k is the wavevector that gives the phase of the MO’s on moving from one unit cell to the next (as well as the crystal momentum of the electron).
How many lines are there in a band structure diagram?	As many as there are atomic orbitals in the unit cell.
How is the center of gravity energy level of each band determined?	It usually follows from the MO diagram.
How do we determine whether a band runs uphill or downhill?	By comparing the orbital overlap at k=0 and k=π/a.
How do we distinguish metals from semiconductors and insulators?	The Fermi level cuts a band in a metal, whereas there is a gap between the filled and empty states in a semiconductor.
Why are some bands flat and others steep?	This depends on the degree of orbital overlap between building units.

Wide bandslarge intermolecular overlap → delocalized e⁻

Narrow bandsweak intermolecular overlap → localized e⁻

Before we leave two dimensions, here are some things to keep in mind for band diagrams. The plot is energy against k, and k tells us how the phase changes from one unit cell to the next. There is one band for every atomic orbital in the unit cell, and the center of each band usually follows from the MO diagram of the building unit. The orbital overlap at k equals zero and k equals pi over a tells us whether a band runs uphill or downhill. When the Fermi level cuts a band we expect a metal, and a gap means a semiconductor or an insulator. Wide bands mean a lot of overlap between unit cells and delocalized electrons, as long as the band is not completely full or empty, and narrow bands like the d z squared band mean localized electrons. Next we will use these ideas on graphene and then on three-dimensional band structures.

Practice questions

Work the Lecture 16 practice questions before the next class.

Open the Lecture 16 practice questions

CHEM 548: Materials Chemistry

Lecture 16: Reciprocal Space, 2D Band Structures, and the CuO₂²⁻ Layer

Learning Objectives: 2D Band Structures

Real and Reciprocal Lattices in 2D

Defining the Reciprocal Lattice Vectors

Reciprocal Space in 1D

The First Brillouin Zone in 2D

Special Points in a 2D Square Lattice

Band Structure of a 2D H-Atom Sheet

Learning Objectives: The CuO₂²⁻ Layer

La₂CuO₄: Home of the CuO₂²⁻ Layer

The CuO₂²⁻ Sheet

MO Diagram → DOS Plot

Basis Set Orbitals: the dxy Band

The dxy Band at Γ

The dxy Band at M

Basis Set Orbitals: the dx²−y² Band

The dx²−y² Band at Γ

The dx²−y² Band at M

Two Bands at Γ, X, and M

Band Structure of the CuO₂²⁻ Sheet

Key Concepts: Band Structure

Practice questions

Basis Set Orbitals: the d_xy Band

The d_xy Band at Γ

The d_xy Band at M

Basis Set Orbitals: the d_x²−y² Band

The d_x²−y² Band at Γ

The d_x²−y² Band at M