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QuantumBloch band structureundergraduategraduate

Why Metals Conduct (Band Theory)

Also known as: Electronic band theory · Bloch's theorem

Bring two atoms together and the Pauli principle splits each shared level into two. Bring 10²³ atoms together and each level splits into 10²³ sub-levels so closely spaced they form a continuous 'band'. Electrons fill these bands from the bottom up. If the topmost occupied band is only partly full, electrons sit right next to empty states and an electric field nudges them freely — a metal. If a band is completely full and the next is far above across a wide gap, no nearby empty states exist, electrons can't move, and you have an insulator. A small gap gives a semiconductor.

ψnk(r)=eik⋅r unk(r),unk(r+R)=unk(r)\psi_{n\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k}\cdot\mathbf{r}}\,u_{n\mathbf{k}}(\mathbf{r}),\qquad u_{n\mathbf{k}}(\mathbf{r}+\mathbf{R}) = u_{n\mathbf{k}}(\mathbf{r})
Live simulation
warming up the physics…

Watch discrete atomic levels broaden into bands as atoms come together; the gap slider sets whether the Fermi level lands in a band (metal) or a gap (insulator).

Equivalent forms

Eg=Econd−EvalE_g = E_{\text{cond}} - E_{\text{val}}
σ=ne2τ/m∗\sigma = n e^2 \tau / m^*
f(E)=1e(E−μ)/kBT+1f(E) = \frac{1}{e^{(E-\mu)/k_BT}+1}
Conductivity — a property spanning 30 orders of magnitude across materials — reduces to one yes/no question: is the Fermi level inside a band or inside a gap?