Direct vs indirect band gap materials: Question bank

For the deeper machinery see Band Theory Basics, Phonons and Lattice Vibrations, Recombination Mechanisms, Optical Absorption in Semiconductors, LEDs and Laser Diodes and Silicon Solar Cells. Hinglish version: yahaan padho.

True or false — justify

TF1. A direct-gap material always has a smaller band gap than an indirect one.

False. Gap magnitude and gap type are independent — GaN is direct and wide (

3.4

eV), while AlAs is indirect and also wide. Type is set by where in

k

-space the band extrema sit.

TF2. In a direct transition no phonon is ever involved.

False in the strict sense. A direct transition does not require a phonon for momentum, but phonons still exist and can participate in fine details (e.g. thermal broadening). The point is that a photon alone already conserves crystal momentum.

TF3. Because photons carry momentum

p = h / λ

, a photon can bridge any

k

-space gap.

False. The number that matters is the ratio

q_{photon} / k_{BZ} \sim 1 0^{- 3}

; on the scale of a zone-wide jump the photon's momentum is effectively zero, so only near-vertical transitions are photon-driven.

TF4. Silicon cannot absorb light at all because it is indirect.

False. Indirect absorption is weak and phonon-assisted, not forbidden — that is exactly why Si solar cells work, they just need hundreds of microns of path length.

TF5. Indirect emission is suppressed much more strongly than indirect absorption.

True. Both need a phonon, but for absorption there is plenty of incoming light to make the improbable event happen often; for emission the electron must wait for a phonon while faster non-radiative paths steal it first.

TF6. The absorption edge shape (

α \propto (h ν - E_{g})^{1/2}

(h ν - E_{g})^{2}

) can tell you the gap type without knowing the crystal structure.

True. A

α^{2}

-vs-

h ν

line implies direct; a

α^{1/2}

-vs-

h ν

(i.e.

α \propto (h ν - E_{g})^{2}

) line implies indirect — the power law is a fingerprint of the joint density of states.

TF7. A phonon carries a lot of momentum but very little energy compared to a near-gap photon.

True. Phonons reach wavevectors near the zone edge (

\sim k_{BZ}

) yet have energies of only tens of meV — the perfect complement to a photon, which is the reverse.

TF8. At

k = 0

(the

Γ

point) every semiconductor is automatically direct.

False. Direct means the VBM and CBM coincide in

k

. The VBM is usually near

Γ

, but if the CBM sits elsewhere (Si's is near

X

), the material is indirect even though one extremum is at

Γ

Spot the error

SE1. "A photon supplies the momentum, so the electron in silicon just needs the right-colour light to recombine radiatively."

The photon supplies energy, not the ~zone-wide momentum change silicon needs; a phonon must supply

ℏΔ k

, making it a rare three-body event.

SE2. "GaAs is used in LEDs because its band gap is exactly the right size for visible light."

The gap size sets the colour, but the reason GaAs works at all is that it is direct, giving fast radiative recombination — an indirect material of the same gap would barely glow.

SE3. "Since indirect recombination produces heat, indirect materials get hot faster and are bad for solar cells."

The opposite functional point: indirect gaps make Si a poor emitter but a fine absorber (long carrier lifetime lets carriers be collected), which is why Si dominates photovoltaics.

SE4. "The absorption edge of an indirect material is sharp because you need an exact phonon energy."

It is actually soft/gradual and split into two branches (

\pm E_{p h}

) — phonon-absorption turns on below

E_{g}

and phonon-emission above, smearing the edge over

\sim 2 E_{p h}

SE5. "Crystal momentum

ℏ k

is just ordinary momentum, so it obeys the same conservation as a billiard ball with no exceptions."

Crystal momentum is conserved only up to a reciprocal-lattice vector

G

(Umklapp) — but for the interband transitions here that subtlety doesn't rescue the photon, which is still ~1000× too weak.

SE6. "Making the film thicker fixes any weak-emitter problem in silicon."

Thickness helps absorption (more path length), but it does nothing for the emission bottleneck, which is the low radiative probability per event, not a path-length issue.

SE7. "Since a direct gap needs no phonon, direct-gap emission is temperature-independent."

Radiative rate still varies with temperature (carrier distributions, non-radiative competitors, gap shrinkage), but it does not depend on phonon availability the way indirect emission does — that is the real contrast.

Why questions

WY1. Why is crystal momentum conservation, not just energy conservation, the deciding factor between direct and indirect behaviour?

Energy can always be matched by choosing the photon frequency; it is the momentum mismatch (

k_{c} \neq = k_{v}

) that a photon cannot fix, forcing the phonon and making the process rare.

WY2. Why does the indirect absorption edge follow a

(h ν - E_{g})^{2}

law while the direct one follows

(h ν - E_{g})^{1/2}

The indirect process integrates over all phonon-assisted paths, adding an extra energy integration that raises the density-of-states power from

1/2

2

WY3. Why can a phonon do a job the far more energetic photon cannot?

Because momentum and energy are separate currencies: the phonon is momentum-rich but energy-poor, exactly matching the electron's need to change

k

while barely changing

E

WY4. Why does GaAs absorb sunlight in ~1 μm while Si needs hundreds of μm?

Direct (vertical) transitions have a large absorption coefficient

α

; indirect ones are phonon-limited and weak, so light penetrates far deeper before being absorbed.

WY5. Why is "the gap is forbidden" a misleading way to describe indirect emission?

It is improbable, not forbidden — the transition happens with low probability whenever a suitable phonon is present, so it occurs, just slowly and mostly non-radiatively.

WY6. Why do we compare

q_{photon}

k_{BZ}

rather than to some absolute momentum scale?

Because the relevant question is whether the photon can move an electron across the zone to a different extremum;

k_{BZ}

is precisely the size of that possible

k

-jump.

WY7. Why does making silicon into nanostructures (porous Si, quantum dots) let it emit light despite being indirect?

Confinement smears out the wavefunctions in

k

-space, relaxing the strict

k

-conservation rule so that near-vertical radiative transitions gain probability (see Recombination Mechanisms).

Edge cases

EC1. What happens to the absorption edge exactly at

h ν = E_{g}

for a direct gap?

α \to 0

(h ν - E_{g})^{1/2} \to 0

— absorption switches on smoothly from zero right at the gap, with no phonon offset.

EC2. What happens below

E_{g}

in an indirect material at finite temperature?

A phonon-absorption branch lets weak absorption begin at

h ν = E_{g} - E_{p h}

, because an existing lattice vibration donates the missing momentum-and-energy.

EC3. What is the limiting case as temperature

T \to 0

for the phonon-absorption branch of an indirect gap?

It vanishes — with no thermally populated phonons to absorb, only the phonon-emission branch (starting at

E_{g} + E_{p h}

) survives.

EC4. What if VBM and CBM sit at nearly the same

k

but not exactly (a "quasi-direct" gap)?

The transition is weakly direct-like: small residual

Δ k

means the photon almost conserves momentum, giving intermediate absorption strength between the two ideal cases.

EC5. What happens to the direct/indirect distinction under strong strain or alloying (e.g. Ge under tension, SiGe)?

Strain shifts the band extrema in

k

-space and can convert an indirect gap toward direct (tensile Ge), because type depends on extremum positions that strain moves.

EC6. What is the degenerate case where the "vertical transition" argument alone determines everything?

When VBM and CBM coincide at the same

k

(pure direct): momentum is conserved automatically, so only energy conservation remains as a constraint and the photon does all the work.

Recall One-line self-test

Which single conserved quantity separates fast radiative (direct) from slow non-radiative (indirect) recombination? ::: Crystal momentum $ℏ k$ — a photon conserves energy easily but supplies almost no momentum, so a matched $k$ (direct) allows a photon-only jump while a mismatched $k$ (indirect) demands a phonon.