Band gap and its meaning for conductivity

WHAT is a band gap?

WHY do gaps exist at all? When atoms come together, discrete atomic levels split into $N$ closely-spaced levels (a band). Between two atomic levels there is empty energy space → that becomes the gap. More physically, electron waves with wavelength matching the crystal spacing get Bragg-reflected; standing waves of two different energies form at the zone boundary, and the energy difference between them is the gap.

WHY the gap controls conductivity

Current flows only when electrons can gain a tiny bit of energy from an applied field and move to a slightly higher momentum state.

HOW the gap enters the numbers: the probability of an electron being thermally kicked across the gap follows a Boltzmann-like factor. The number of conduction electrons scales as $n \propto e^{-E_g/(2k_BT)}$ so conductivity $\sigma \propto e^{-E_g/(2k_BT)}$ .

Formula Why the factor is

$E_g/2$ (derivation sketch) WHAT: number of electrons crossing the gap. HOW: For an intrinsic semiconductor, occupancy of a state at energy $E$ is $f(E)=\frac{1}{1+e^{(E-E_F)}/k_BT}\approx e^{-(E-E_F)/k_BT}$ (tail, since $E-E_F\gg k_BT$ ). The Fermi level $E_F$ sits mid-gap, so an electron reaching $E_c$ must climb $E_c-E_F=E_g/2$ . Hence $n \propto e^{-(E_c-E_F)/k_BT}=e^{-E_g/(2k_BT)}.$ WHY the 2: the gap is shared — you pay half to promote the electron up, and the "hole" it leaves also lives half a gap below $E_F$ .

The three classes (a spectrum, not a wall)

Figure — Band gap and its meaning for conductivity

Worked example Classifying by

$E_g$

Class	$E_g$	At room $T$	Example
Metal / conductor	$E_g=0$ (bands overlap)	huge $n$ , high $\sigma$	Cu, Al
Semiconductor	$\sim 0.1$ – $3$ eV	few carriers, tunable	Si (1.12), Ge (0.66), GaAs (1.42)
Insulator	$\gtrsim 4$ eV	almost none	diamond ( $\approx 5.5$ ), SiO₂ ( $\approx 9$ )

Why this step? $k_BT\approx 0.026$ eV at 300 K. Compare it to $E_g/2$ : for Si, $e^{-1.12/(2\cdot0.026)}\approx e^{-21.5}\approx 5\times10^{-10}$ — small but nonzero. For diamond $e^{-5.5/0.052}\approx e^{-106}$ — utterly negligible → no conduction.

Common mistake Steel-man the classic errors

1. "Full band = insulator because it has no electrons." Why it feels right: we associate "no current" with "no charges." Fix: a full valence band is packed with electrons; it can't conduct because there are no empty states to move into. Emptiness of states, not of electrons, matters.

2. "Metals conduct because they have zero band gap and semiconductors don't conduct at all." Why it feels right: metals are "good," insulators "bad," semiconductors sound in-between/off. Fix: semiconductors conduct a little, and that little is exponentially tunable by $T$ , light, and doping — which is exactly why they run all electronics.

3. Using $e^{-E_g/k_BT}$ (forgetting the 2). Why it feels right: Boltzmann factors usually use the full energy. Fix: $E_F$ is mid-gap, so the climb is only $E_g/2$ . The whole 2 comes from the geometry of the Fermi level.

4. "Higher temperature always means higher conductivity." Why it feels right: it's true for semiconductors. Fix: in metals higher $T$ increases lattice vibrations → more scattering → lower $\sigma$ . Opposite trend!

Recall Feynman: explain to a 12-year-old

Imagine a two-storey car park. The ground floor is completely full of cars — nobody can budge, so no traffic moves. The top floor is empty. If a car wants to drive around (that's electric current), it must jump up to the empty floor. The band gap is how high that jump is. In metals the two floors overlap, so cars roll freely. In insulators the ceiling is so high no car can ever jump — no traffic ever. In semiconductors the jump is just high enough that on a warm day a few cars manage it, and if you heat the place up, more and more make the leap.

Flashcards

What is the band gap

E_g

in terms of band edges?

E_g = E_c - E_v

, the forbidden energy range between valence-band top and conduction-band bottom.

Why can't a completely full band conduct?

There are no empty adjacent states for electrons to move into, so no net momentum change is possible.

How does intrinsic carrier concentration depend on

E_g

and

T

n \propto e^{-E_g/(2k_BT)}

Why is the exponent

E_g/2

and not

E_g

The Fermi level sits mid-gap, so promoting a carrier to

E_c

costs only

E_c-E_F=E_g/2

Typical band gaps of Si, Ge, GaAs, diamond?

Si 1.12 eV, Ge 0.66 eV, GaAs 1.42 eV, diamond ≈5.5 eV.

How does conductivity of a semiconductor change with temperature, and why?

It increases exponentially — more electrons are thermally excited across the gap.

How does a metal's conductivity change with temperature, and why?

It decreases — more lattice vibration causes more electron scattering.

Value of

k_BT

at room temperature (300 K)?

About 0.026 eV (≈1/40 eV).

What distinguishes a semiconductor from an insulator physically?

Only the gap size; small enough (

\lesssim 3

eV) that thermal/optical excitation gives useful carrier numbers.

Why does a band gap exist microscopically?

Electron waves at the Brillouin-zone boundary Bragg-reflect into two standing waves of different energy; the difference is the gap.

Connections

Formation of energy bands from atomic orbitals
Fermi level and Fermi-Dirac distribution
Intrinsic vs extrinsic semiconductors
Doping and carrier concentration
Direct vs indirect band gap
Conductivity, mobility and drift current
Temperature dependence of resistivity in metals

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, har solid me electron ki energy alag-alag "bands" me hoti hai. Do bands ke beech ek forbidden zone hota hai jise hum ==band gap ( $E_g$ )== bolte hain — is range me koi electron reh hi nahi sakta. Ye gap decide karta hai ki material conductor hai, semiconductor hai, ya insulator. Yaad rakho: current chalne ke liye electron ko thodi si energy le kar aage badhna padta hai, aur iske liye uske paas khaali seat honi chahiye same band me.

Ab yahi twist hai — agar band poori tarah bhari hui hai (jaise packed parking), to electron move hi nahi kar sakta, chahe kitne bhi electron ho. Isliye current ke liye ya to bands overlap karein (metal, $E_g=0$ ), ya electron ko thermal energy se gap paar karwa ke upar wali empty band (conduction band) me bhejna padta hai. Semiconductor me gap chhota hota hai (Si me 1.12 eV) isliye kuchh electron room temperature pe bhi jump kar jaate hain. Insulator (diamond, ~5.5 eV) me gap itna bada hai ki koi jump nahi kar paata.

Formula ka core: carriers $n \propto e^{-E_g/(2k_BT)}$ . Yahan 2 kyun aaya? Kyunki Fermi level gap ke beech me baithta hai, to electron ko sirf aadha gap ( $E_g/2$ ) climb karna padta hai. Isi wajah se semiconductor ki conductivity temperature badhne pe badhti hai — garam karo, zyada electron cross karenge. Lekin metal me ulta hota hai: garam karne pe scattering badhti hai, conductivity girti hai. Exam me ye difference bahut poocha jaata hai, isliye pakka yaad rakho.

WHAT is a band gap?

WHY the gap controls conductivity

The three classes (a spectrum, not a wall)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Band Theory & Carrier Physics

Connections