Every electron in a crystal wears TWO name-tags at once: how much energy it has, and how fast/which way it is weaving through the atoms (its crystal momentum). The whole "direct vs indirect" story is just this: when an electron drops to a lower energy, whatever catches its leftover energy must ALSO match its momentum tag — and light is great at carrying energy but almost useless at carrying momentum.
This page builds every symbol the parent note throws at you — starting from what a "crystal" even is, ending with the E k
Definition Crystal & lattice constanta crystal is atoms stacked in a pattern that repeats forever, like tiles on a floor. The distance from one repeating unit to the next is the ==lattice constant a a ≈ 5.4 × 1 0 − 10 m
Look at the figure: the dots are atoms, and a a size of the momentum world an electron lives in.
Why do we care that it repeats ? Because repetition is what forces electrons into bands (Section 3) and gives momentum a special "crystal" flavour (Section 4). A single atom has sharp energy levels; a repeating grid smears them into ranges. Repetition is the source of everything.
See Band Theory Basics for how the repeating grid produces bands in the first place.
E Energy E E electron-volts (eV) : one eV is the energy an electron gains falling through a 1-volt battery, about 1.6 × 1 0 − 19
Picture energy as height . A high electron sits on an upper shelf; a low one sits on a lower shelf. When an electron falls from a high shelf to a low shelf, it loses energy — and that lost energy has to go somewhere (that "somewhere" is the whole point of the topic).
Momentum is normally "mass times velocity" — how hard something is to stop. Inside a repeating crystal, an electron behaves like a wave, and a wave is described by how tightly its ripples are packed . That packing is the wavevector k
k ℏ k k m − 1 k k = 0 ℏ k crystal momentum , where ℏ
The figure shows two waves: a stretched one (small k k
Now combine the two name-tags. For each value of k E k E bands .
Definition Valence band, conduction band, band gapE g
Valence band = the lower band, normally full of electrons (they are "at home").Conduction band = the upper band, normally empty (an electron here is "free to roam").Between them is a forbidden zone with no allowed energies: the ==band gap E g
VBM = V alence B and M aximum = the highest point of the lower band.CBM = C onduction B and M inimum = the lowest point of the upper band.
The gap E g shortest vertical energy jump from filled to empty — the minimum ticket price to promote an electron.
Everything above lives on one graph. Horizontal axis = k (the which-way tag). Vertical axis = E (the height tag). The bands are curves on this plane.
Read the figure like a map. LEFT panel: the CBM (bottom of upper curve) sits directly above the VBM (top of lower curve) — same k straight down — pure vertical arrow. That is a direct gap .
RIGHT panel: the CBM has slid sideways to a different k diagonal — it needs both an energy change (vertical) AND a momentum change (horizontal). That is an indirect gap .
The single question "is the fall vertical or diagonal?" is the ENTIRE topic. Everything else explains why vertical is easy and diagonal is hard.
h ν q photon is a particle of light. Its energy is ==h ν ν h λ q = 2 π / λ
Why photons are the star of this topic: a near-gap photon carries a whole eV of energy but a microscopic q (its wavelength ∼ 1 μ m q E k long vertical arrow with almost zero sideways shift . That is why only vertical (direct) transitions are easy.
h ℏ h 6.6 × 1 0 − 34 J⋅s E = h ν ℏ h /2 π = ℏ k ν k
No picture needed — think of h ℏ ×
Definition Phonon and its wavevectorq p h phonon is a quantum of lattice vibration — a ripple of the atoms jiggling in the crystal. Because it is a wave of the whole grid , it can carry a large momentum ℏ q p h (its q p h k little energy E p h
A phonon is the exact opposite of a photon:
Photon: big energy, tiny momentum → good vertical arrow, useless sideways.Phonon: tiny energy, big momentum → good sideways shove, negligible vertical.
An indirect (diagonal) transition needs BOTH the vertical drop AND the sideways shift, so it must recruit a photon and a phonon at once. Needing two partners to show up together is rare → slow, weak.
Deeper on this in Phonons and Lattice Vibrations .
Definition Brillouin zone,Γ X a k first Brillouin zone , and its half-width is k B Z ≈ π / a centre (k = 0 Γ ; a special edge point is the X
Why we need k B Z q k B Z q / k B Z ∼ 1 0 − 3
Definition Absorption coefficientα α m − 1 α α α α
More in Optical Absorption in Semiconductors and its payoff in Silicon Solar Cells .
Momentum world has size k_BZ
Wavevector k which-way tag
Different k means indirect
Alpha large thin material
Alpha small thick material
This map is the parent topic in miniature: the two name-tags (E k α Recombination Mechanisms and LEDs and Laser Diodes for where this leads.
Cover the right side and check you can answer each before reading the parent note.
What does the lattice constant a The repeat distance between identical units in a crystal (a length, ~0.5 nm for Si).
What is E The electron's energy — its "height" tag — measured in electron-volts (eV).
What does the wavevector k How tightly the electron's wave-ripples are packed (units m − 1
What is crystal momentum? ℏ k
What are VBM and CBM? Valence Band Maximum (top of the full lower band) and Conduction Band Minimum (bottom of the empty upper band).
What is the band gap E g The forbidden energy width between VBM and CBM — the minimum jump to free an electron.
On an E k Direct = VBM and CBM at the SAME k k
Energy and momentum of a near-gap photon? Large energy h ν ℏ q q ∼ 1 0 − 3 k B Z
Energy and momentum of a phonon? Tiny energy E p h ℏ q p h
What is k B Z The half-width π / a
What does the absorption coefficient α How strongly a material absorbs light per metre; large for direct gaps, small for indirect.
What is the Γ The centre of the Brillouin zone, k = 0
