This simulation visualises how electrons in a semiconductor occupy a filled valence band and an empty conduction band separated by a forbidden bandgap E_g. It computes the intrinsic carrier density from n_i = √(N_c·N_v)·exp(−E_g/2kT), tracks the temperature-dependent gap with a Varshni-style approximation E_g(T) = E_g0 − 4×10⁻⁴T²/(T+600), and locates the Fermi level relative to mid-gap.
Material buttons switch between silicon, germanium, GaAs and diamond, each with their own E_g0, N_c and N_v. The doping buttons (intrinsic, n-type, p-type) plus the donor concentration and temperature sliders shift the Fermi level and recompute electron and hole densities. Three views — band diagram, E-k dispersion and carrier-versus-temperature — illustrate the physics underpinning every diode, transistor and solar cell.
What is a semiconductor band structure?
It is the allowed range of electron energies in a crystal, split into a filled valence band and an empty conduction band with a forbidden bandgap between them. The width of that gap, E_g, governs whether a material conducts. The simulation draws these bands plus the Fermi level for four common materials.
What does the bandgap E_g represent?
E_g is the minimum energy an electron needs to jump from the valence band into the conduction band, leaving behind a hole. Silicon is about 1.12 eV, germanium 0.66 eV, GaAs 1.42 eV and diamond 5.47 eV. A larger gap means far fewer thermally excited carriers at a given temperature.
How does doping change the picture?
Adding donor atoms (n-type) supplies extra electrons and pushes the Fermi level up towards the conduction band; acceptor atoms (p-type) create holes and pull it down towards the valence band. The simulation shifts the dashed Fermi line accordingly and recomputes the majority and minority carrier densities.
The temperature slider (100–700 K) changes the thermal energy kT, which strongly raises the intrinsic carrier density and slightly narrows the gap via the Varshni term. The dopant slider sets N_d on a log scale from 10¹² to 10²⁰ cm⁻³, controlling how heavily the material is doped and therefore how far the Fermi level moves.
At equilibrium the mass-action law n·p = n_i² always holds. For an n-type sample the simulation solves n = N_d/2 + √(N_d²/4 + n_i²), then sets p = n_i²/n. The intrinsic density itself comes from n_i = √(N_c·N_v)·exp(−E_g/2kT), where k is Boltzmann's constant, 8.617×10⁻⁵ eV/K.
In a direct-gap material like GaAs the conduction-band minimum sits at the same crystal momentum k as the valence-band maximum, so electrons and holes recombine easily by emitting light. In indirect materials like silicon and germanium the minima are offset in k, shown in the E-k view as conduction valleys at ±k, requiring a phonon to conserve momentum.
Near a band edge the energy varies roughly as E(k) ≈ E_edge ± ℏ²k²/2m*, a parabola whose curvature is set by the effective mass m*. The simulation draws a conduction parabola plus heavy- and light-hole valence bands, so a sharper curve corresponds to a lighter, more mobile carrier.
It is a clear teaching model rather than a full band-structure calculation. The parabolic-band, Boltzmann-statistics and Varshni approximations reproduce the correct trends and order-of-magnitude carrier densities, but it omits degenerate Fermi-Dirac statistics, incomplete dopant ionisation and the detailed multi-valley shapes that a code such as DFT would produce.
The intrinsic density carries an exp(−E_g/2kT) factor, so heating the crystal exponentially increases the number of thermally excited electron-hole pairs. The carrier-versus-temperature view plots this on a log scale and shows how a heavily doped sample stays near N_d at low temperature before the intrinsic curve eventually overtakes it.
With equal electron and hole populations and comparable density-of-states, the chemical potential lands close to the middle of the gap. The simulation measures E_F from mid-gap and computes its shift as kT·ln(n/n_i) for n-type or −kT·ln(p/n_i) for p-type, so adding carriers moves it towards the nearer band.
Every diode, transistor, LED, laser and solar cell relies on controlling band edges and Fermi levels through doping. Choosing silicon for logic, GaAs for fast or light-emitting devices and wide-gap diamond for power and high-temperature electronics all follows directly from the bandgaps and band structures this simulation illustrates.