Noesis
2.1.5Band Theory & Carrier Physics

Direct vs indirect band gap materials

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🇮🇳 Yeh note Hinglish mein padho →

WHAT is a band gap in EE–kk space?

Figure — Direct vs indirect band gap materials

WHY does momentum matter? (Derive the selection rule)

We derive the constraint on an interband transition from two conservation laws.

Step 1 — Energy conservation. Electron drops from conduction energy Ec(kc)E_c(k_c) to valence energy Ev(kv)E_v(k_v): Ec(kc)−Ev(kv)=ℏωphoton  (±Ephonon)E_c(k_c) - E_v(k_v) = \hbar\omega_{\text{photon}} \;(\pm E_{\text{phonon}})

Why this step? The emitted photon (and any phonon) must carry away exactly the energy the electron loses.

Step 2 — Crystal-momentum conservation. ℏkc=ℏkv+qphoton  (±qphonon)\hbar k_c = \hbar k_v + q_{\text{photon}} \;(\pm q_{\text{phonon}})

Why this step? Total crystal momentum is conserved in the transition, just like linear momentum.

Step 3 — Estimate the photon momentum. A photon of near-gap energy has wavelength λ∼1 μm\lambda \sim 1\,\mu\text{m}: qphoton=2πλ∼2π10−6 m≈6×106 m−1q_{\text{photon}} = \frac{2\pi}{\lambda} \sim \frac{2\pi}{10^{-6}\,\text{m}} \approx 6\times 10^{6}\,\text{m}^{-1}

Compare with the width of the Brillouin zone: kBZ∼πa∼π5×10−10 m≈6×109 m−1k_{\text{BZ}} \sim \frac{\pi}{a} \sim \frac{\pi}{5\times 10^{-10}\,\text{m}} \approx 6\times 10^{9}\,\text{m}^{-1}

Why this step? We need to see how big qphotonq_{\text{photon}} is compared to the momentum change ∣kc−kv∣|k_c - k_v| across the zone.

Step 4 — The ratio. qphotonkBZ≈6×1066×109≈10−3≈0\frac{q_{\text{photon}}}{k_{\text{BZ}}} \approx \frac{6\times10^{6}}{6\times10^{9}} \approx 10^{-3} \approx 0

Why this step? Because a photon's momentum is ~1000× too small to bridge the zone, we conclude:

WHY the absorption edges differ

Because indirect absorption is weak, silicon needs hundreds of microns of material to absorb sunlight, while a direct-gap GaAs cell absorbs the same light in ~1 μm.

Worked Examples

Common Mistakes (Steel-manned)

Flashcards

What defines a direct band gap in EE–kk space?
The conduction band minimum and valence band maximum occur at the same value of kk.
Why can a photon alone cause a direct transition?
A near-gap photon's momentum is ~10−310^{-3} of the Brillouin-zone width, so a vertical (kc≈kvk_c\approx k_v) transition automatically conserves crystal momentum.
What extra particle must an indirect transition involve, and why?
A phonon, to supply the crystal-momentum difference ℏΔk\hbar\Delta k that the photon cannot provide.
Give two direct-gap and two indirect-gap semiconductors.
Direct: GaAs, GaN (also InP, CdTe). Indirect: Si, Ge (also GaP).
Why is silicon a poor light emitter?
Indirect gap → radiative recombination needs a phonon (three-body process), so it is slow and mostly non-radiative (heat).
Absorption edge for a direct gap?
α∝(hν−Eg)1/2\alpha \propto (h\nu - E_g)^{1/2}.
Absorption edge for an indirect gap?
α∝(hν−Eg±Eph)2\alpha \propto (h\nu - E_g \pm E_{ph})^2, split into phonon-absorption/emission branches.
Are band-gap magnitude and type related?
No — they are independent. Wide gaps can be direct (GaN) or indirect (AlAs).
Why does GaAs need only ~1 μm to absorb light but Si needs hundreds of μm?
Direct absorption is strong (large α\alpha); indirect (phonon-assisted) absorption is weak, so Si needs a much longer optical path.
Recall Feynman: explain to a 12-year-old

Imagine an electron is a kid who has to jump down from a top bunk (high energy) to a bottom bunk (low energy). But there's a rule: the kid must land in the same spot left-to-right, not just lower down. If the two beds are lined up (direct), the kid just drops and sings a note (emits light) — easy! If the bottom bed is shifted sideways (indirect), the kid can't just fall; someone has to shove them sideways at the same time. That shove is a "phonon" — a shaking of the whole bed frame. Needing a drop AND a shove at the exact same instant almost never happens, so indirect kids fall silently as heat instead of singing light. That's why we build LED "singers" out of direct materials like GaAs, not silicon.

Connections

  • Band Theory Basics — origin of E(k)E(k) and Brillouin zones
  • Phonons and Lattice Vibrations — the momentum source for indirect transitions
  • Recombination Mechanisms — radiative vs. Shockley–Read–Hall
  • LEDs and Laser Diodes — why they use direct-gap materials
  • Optical Absorption in Semiconductors — Tauc plots and absorption edges
  • Silicon Solar Cells — consequences of the indirect gap

Concept Map

defines

VBM and CBM same k

VBM and CBM different k

governs

governs

tiny momentum ~1000x too small

yields

photon alone works

needs phonon for momentum

supplies momentum

fast radiative

slow inefficient

examples

examples

Band structure E of k

Band gap Eg between VBM and CBM

Direct gap

Indirect gap

Energy conservation

Interband transition

Crystal momentum conservation

Photon

Selection rule kc approx kv

Phonon

Efficient light emission

GaAs InP GaN

Si Ge GaP

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, har electron crystal ke andar do cheezein rakhta hai: uski energy (EE) aur uska crystal momentum (ℏk\hbar k). Jab electron conduction band se valence band mein girta hai (recombination), toh dono conserve hone chahiye. Ab twist yeh hai — photon (light ka packet) ke paas energy toh bahut hoti hai, par momentum almost zero. Toh agar conduction band ka lowest point aur valence band ka highest point same kk par hain, electron seedha neeche gir kar light emit kar deta hai. Isko bolte hain direct band gap — jaise GaAs, GaN. Yeh materials LED aur laser banane ke liye perfect hain.

Lekin agar CBM aur VBM alag kk par hain (jaise Silicon aur Germanium mein), tab electron ko sirf neeche nahi girna, balki side mein bhi shift karna padta hai kk-space mein. Yeh sideways shift photon nahi de sakta — iske liye chahiye ek phonon (lattice ki vibration). Ab ek saath teen cheezein (electron + photon + phonon) milna bahut rare event hai, isliye indirect gap mein recombination slow hota hai aur zyaadatar heat ke roop mein energy waste hoti hai. Isiliye Silicon se LED nahi banti — woh light ki jagah garmi deta hai.

Ek important baat yaad rakhna: gap ka size (EgE_g) aur gap ka type (direct/indirect) do alag cheezein hain. GaN ka gap bhi bada hai (3.43.4 eV) aur woh direct bhi hai; AlAs bada hai par indirect. Type sirf isse decide hota hai ki kk-space mein extrema kahan baithe hain. Aur yahi reason hai ki GaAs solar cell sirf 1 micron mein light absorb kar leta hai, jabki Silicon ko hundreds of microns chahiye — kyunki indirect absorption kamzor hoti hai. Exam mein Tauc plot yaad rakhna: α2\alpha^2 vs hνh\nu straight line = direct, α\sqrt{\alpha} vs hνh\nu straight = indirect.

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Connections

Recombination and generation mechanismsHardware · 2.1.11Photodiodes and solar cellsHardware · 2.3.5Energy bands - valence band and conduction bandHardware · 2.1.1Band gap and its meaning for conductivityHardware · 2.1.2