Question
How does band theory explain why some solids conduct electricity (metals), some do not (insulators), and some conduct partially (semiconductors)? What determines the band gap?
(JEE Main, CBSE 12 — band theory appears in solid state chemistry and is cross-linked with semiconductor physics questions)
Solution — Step by Step
In an isolated atom, electrons occupy discrete energy levels. When atoms come together in a solid (where ), each energy level splits into closely spaced levels, forming a continuous energy band.
The two most relevant bands:
- Valence band (VB): the highest energy band that is fully or partially occupied by electrons at 0 K
- Conduction band (CB): the next higher energy band, which is empty or partially filled
The energy gap between the top of VB and bottom of CB is the band gap ().
In metals like Cu, Ag, Al, the valence band and conduction band overlap (or the valence band is only partially filled). Electrons can move freely into available empty states with no energy barrier.
(overlapping bands)
This is why metals conduct electricity at all temperatures — free electrons are always available. Conductivity decreases with temperature because lattice vibrations scatter electrons more.
In materials like diamond ( eV) and rubber, the band gap is so large (> 3 eV) that thermal energy at room temperature ( eV) cannot promote electrons from VB to CB.
The VB is completely filled, the CB is completely empty, and no conduction occurs. Even at high temperatures, the number of thermally excited electrons is negligible.
Silicon ( eV) and germanium ( eV) have small band gaps. At 0 K, they behave like insulators. But at room temperature, some electrons gain enough thermal energy to jump from VB to CB.
Each electron that jumps to CB leaves behind a hole in VB. Both electrons (in CB) and holes (in VB) can carry current. This is why semiconductor conductivity increases with temperature — opposite to metals.
Doping introduces additional energy levels:
- n-type (donor doping, e.g., P in Si): extra electrons near CB, donor level just below CB
- p-type (acceptor doping, e.g., B in Si): extra holes near VB, acceptor level just above VB
flowchart TD
A["Solid Material"] --> B{"Band gap size?"}
B -->|"Eg = 0 or bands overlap"| C["Conductor (Metal)"]
B -->|"Eg > 3 eV"| D["Insulator"]
B -->|"Eg < 3 eV"| E["Semiconductor"]
E --> F{"Doped?"}
F -->|"No"| G["Intrinsic semiconductor"]
F -->|"Group 15 element added"| H["n-type (excess electrons)"]
F -->|"Group 13 element added"| I["p-type (excess holes)"]
C --> J["σ decreases with temperature"]
E --> K["σ increases with temperature"]Why This Works
The band theory naturally emerges from quantum mechanics. When atoms are brought close together, their wavefunctions overlap, and the Pauli exclusion principle forces each energy level to split into distinct levels. The key insight is that electrical conduction requires partially filled bands — fully filled bands cannot carry net current because for every electron moving one way, there is another moving the opposite way.
Metals have partially filled bands (free electron movement), insulators have full VB with a huge gap to empty CB, and semiconductors sit in between with a gap small enough for thermal excitation.
Common Mistake
Students often say “semiconductors have free electrons at room temperature, so they are like metals.” Wrong. In semiconductors, the number of free carriers is tiny ( for Si) compared to metals (). That is a trillion-fold difference. Semiconductors are useful precisely because their conductivity can be controlled by doping and temperature — metals cannot be tuned this way.
Band gap values to remember: Diamond = 5.5 eV (insulator), Si = 1.1 eV, Ge = 0.67 eV (semiconductors). The 3 eV boundary is approximate — the real distinction is whether thermal excitation at working temperature produces significant carriers.