Energy Band Theory: Valence Band, Conduction Band, Bandgap

Every semiconductor device — diode, BJT, MOSFET, GaN HEMT — is a machine for controlling electrons using energy barriers set by crystal structure. Energy band theory is the reason silicon can be turned into a switch or amplifier while a similar-looking piece of glass or copper cannot. Without a quantitative picture of the valence band, conduction band, and bandgap, doping concentrations, leakage currents, and temperature coefficients in a datasheet are just numbers with no physical anchor.

Why Bands Exist

In an isolated atom, electrons occupy discrete energy levels. When atoms are packed into a crystal lattice (as in silicon's diamond structure, covalently bonded), the outer electron orbitals of neighboring atoms overlap and interact. The Pauli exclusion principle forbids identical electron states, so the discrete levels of ~10²³ atoms/cm³ split and smear into continuous ranges of allowed energy — bands — separated by ranges of forbidden energy — gaps.

Two bands matter for electronics:

• Valence band (VB): the highest band that is normally filled with electrons at 0 K — these are the electrons involved in covalent bonding.
• Conduction band (CB): the next band up, empty at 0 K. Electrons here are free to move through the lattice and carry current under an applied field.

The energy gap between the top of the VB (E_v) and bottom of the CB (E_c) is the bandgap, E_g = E_c − E_v, measured in electron-volts (eV).

Classifying Materials by Band Structure

Representative bandgaps at 300 K:

Ge:    0.66 eV
Si:    1.12 eV
GaAs:  1.42 eV
GaN:   3.4  eV
SiC:   3.26 eV
SiO2:  ~9   eV  (insulator)

Wide-bandgap materials (GaN, SiC) tolerate higher electric fields and temperatures before breakdown — this is precisely why they dominate high-voltage, high-frequency power devices, a direct consequence of E_g being larger than silicon's.

Carrier Generation: Crossing the Gap

At absolute zero, a semiconductor's valence band is completely full and conduction band completely empty — no free carriers, no current, behaves like an insulator. At any T > 0, thermal energy statistically excites some electrons across E_g from VB to CB. Each electron promoted leaves behind a hole — a missing electron in the VB that behaves as a mobile positive charge carrier. This is intrinsic carrier generation, and it is the basis of the intrinsic carrier concentration n_i.

The probability that a state at energy E is occupied follows the Fermi-Dirac distribution:

f(E) = 1 / (1 + exp[(E − E_F)/(k·T)])

where E_F is the Fermi level and k·T ≈ 0.0259 eV at 300 K. Because E_g ≫ k·T for silicon (1.12 eV vs. 0.026 eV), only the exponential tail of this distribution reaches the conduction band — which is why n_i is small (~1.5×10¹⁰ cm⁻³ for Si at 300 K) compared to the ~5×10²² cm⁻³ atomic density, and why intrinsic conductivity is weak and strongly temperature-dependent.

Approximately, n_i scales as:

n_i ∝ T^(3/2) · exp(−E_g / (2·k·T))

This exponential is the reason reverse-leakage current in diodes and BJTs roughly doubles every 8–10 °C rise for silicon — a direct, quantifiable consequence of band structure, not an empirical curiosity.

Worked Example: Comparing Thermal Excitation Across the Gap for Si and GaAs

Estimate the relative intrinsic carrier concentration of Si (E_g = 1.12 eV) vs. GaAs (E_g = 1.42 eV) at T = 300 K, ignoring the T^(3/2) prefactor difference (assume it's similar for both — a simplification).

k·T = 0.0259 eV
ratio = exp[−(E_g,Si)/(2kT)] / exp[−(E_g,GaAs)/(2kT)]
      = exp[ (E_g,GaAs − E_g,Si) / (2kT) ]
      = exp[ (1.42 − 1.12) / (2 × 0.0259) ]
      = exp[ 0.30 / 0.0518 ]
      = exp(5.79)
      ≈ 327

So silicon's intrinsic carrier concentration should be roughly 300× higher than GaAs's at the same temperature, from the bandgap difference alone.

Check: published n_i values at 300 K are ≈1.5×10¹⁰ cm⁻³ (Si) and ≈2×10⁶ cm⁻³ (GaAs), a ratio of ≈7500. Our simplified exponential-only estimate (327×) is in the right direction (Si ≫ GaAs) but understates the ratio because the T^(3/2) prefactor and effective-mass-dependent density-of-states terms (which we dropped) also differ significantly between materials. This is expected — the exponential term dominates the trend, but a full n_i calculation requires the material-specific effective masses too. The qualitative result (larger E_g → dramatically fewer intrinsic carriers) still holds and is the useful design takeaway.

Doping: Moving the Fermi Level Without Changing the Bands

Adding donor atoms (Group V, e.g., phosphorus, in silicon) contributes extra electrons that sit in shallow donor levels just below E_c, easily ionized at room temperature into the conduction band — this is n-type material, and it pushes E_F up, closer to E_c. Acceptor atoms (Group III, e.g., boron) create shallow levels just above E_v that accept electrons from the valence band, generating holes — p-type material, with E_F pulled down toward E_v.

Crucially, doping does not change E_g or the band structure itself — it only changes the population of carriers and the position of E_F within the existing gap. This distinction matters when reading device physics: mobility, effective mass, and E_g are material properties; carrier concentration and conductivity are the tunable, doping-dependent properties built on top of that fixed band structure.

Design and Diagnostic Implications

• Reverse leakage and I_S scale with E_g: wider-bandgap devices (SiC, GaN) have dramatically lower intrinsic leakage and can operate at higher junction temperatures than silicon for the same leakage budget.
• Breakdown voltage correlates with E_g: larger bandgap generally supports higher critical electric field before impact ionization causes avalanche breakdown — the physical reason SiC/GaN power devices rate to much higher voltages per unit thickness than Si.
• Temperature sensitivity of V_F, V_BE, I_S all trace back to the exponential T-dependence of n_i through E_g — this is why analog designers track ~−2 mV/°C for silicon junction voltage as a rule of thumb.
• Optoelectronic wavelength selection (LEDs, photodiodes) is a direct bandgap engineering problem: photon energy hν must be ≥ E_g for absorption, or emitted photon energy ≈ E_g for direct-bandgap LED emission, so E_g sets the color/wavelength.
• Direct vs. indirect bandgap (not covered quantitatively here) determines whether a material can efficiently emit light — Si is indirect (poor LED material), GaAs and GaN are direct (efficient LEDs/lasers).

Key Takeaways

• Energy bands arise from the overlap of atomic orbitals in a crystal lattice; the valence band is filled, the conduction band is empty at 0 K, separated by the bandgap E_g = E_c − E_v.
• Conductors have E_g ≈ 0, semiconductors ~0.6–2.5 eV, insulators >~5 eV — this single parameter is the first-order classifier of electrical behavior.
• Thermal energy excites electrons across E_g, generating electron-hole pairs; intrinsic carrier concentration n_i depends exponentially on −E_g/(2kT), making leakage and conductivity strongly temperature- and bandgap-dependent.
• Doping shifts the Fermi level and carrier populations but does not alter E_g or the underlying band structure — those are fixed material properties.
• Wider-bandgap materials (SiC, GaN) support higher breakdown fields, lower leakage, and higher operating temperatures than silicon, directly motivating their use in power and RF devices.