Unit 25: Solids
Energy Bands in Solids (Qualitative Ideas)
When atoms are brought together to form a solid, their discrete atomic energy levels interact and split into a large number of closely spaced levels. With the enormous number of atoms (~10²³) in a macroscopic crystal, these levels merge into continuous ranges called energy bands. Between bands there may be ranges of energy that no electron can occupy; these are known as forbidden gaps.
Formation of Energy Bands
Consider a single isolated atom: its electrons occupy discrete energy levels (e.g., 1s, 2s, 2p). When two atoms approach, each level splits into two due to the Pauli exclusion principle. As more atoms join, the splitting continues, producing N distinct levels for each original atomic level, where N is the number of atoms. For N ≈ 10²³, the spacing between adjacent levels becomes vanishingly small, forming a quasi‑continuous band.
Valence Band
The valence band is the highest energy band that is completely filled with electrons at absolute zero temperature (0 K). Electrons in this band are bound to their respective atoms and do not contribute to electrical conduction.
Conduction Band
Above the valence band lies the conduction band. At 0 K this band is empty (or partially empty) in insulators and semiconductors, but electrons can be thermally excited into it, where they become free to move throughout the crystal and conduct electricity.
Forbidden Energy Gap
The forbidden energy gap (or band gap, Eg) is the energy difference between the top of the valence band (Ev) and the bottom of the conduction band (Ec):
E_g = E_c - E_v
The magnitude of Eg determines the electrical properties of the material.
Band Width and Interatomic Spacing
The width of an energy band depends on the overlap of atomic wavefunctions, which is governed by the distance between neighboring atoms (interatomic spacing, a). Decreasing a increases orbital overlap, broadening the bands; increasing a reduces overlap, narrowing the bands. This relationship explains why applying pressure (which reduces a) can change a material from an insulator to a metal.
Difference Between Metals, Insulators and Semiconductors
Materials are classified according to the size of their forbidden gap and the relative positions of the valence and conduction bands.
Metals
In metals the valence band and conduction band either overlap or the conduction band is partially filled even at 0 K. Consequently there is no forbidden gap (Eg ≈ 0). Electrons can move freely, giving metals their high electrical conductivity. Typical examples: copper (Cu), silver (Ag), aluminium (Al).
Insulators
Insulators possess a large forbidden gap, typically greater than 3 eV. At room temperature thermal energy (kT ≈ 0.025 eV) is insufficient to excite electrons across the gap, so the conduction band remains empty and the material conducts poorly. Examples: glass, rubber, diamond (Eg ≈ 5.5 eV).
Semiconductors
Semiconductors have a small forbidden gap, on the order of 1 eV. At 0 K they behave like insulators, but at room temperature a significant fraction of electrons gain enough thermal energy to jump the gap, creating electron‑hole pairs that enable moderate conductivity. Common elemental semiconductors: silicon (Si, Eg = 1.1 eV) and germanium (Ge, Eg = 0.7 eV).
An important characteristic of semiconductors is that their conductivity increases with temperature (opposite to metals), because more electrons are thermally excited across the gap.
Comparison Table
| Property | Metals | Insulators | Semiconductors |
|---|---|---|---|
| Band Gap (Eg) | ≈ 0 eV (overlap) | > 3 eV | ≈ 0.5–3 eV (typically ~1 eV) |
| Valence/Conduction Band Relation | Overlap or partially filled conduction band | Valence band full, conduction band empty | Valence band full, conduction band empty at 0 K |
| Electrical Conductivity (σ) | High (10⁶–10⁸ S/m) | Very low (10⁻¹⁰–10⁻⁸ S/m) | Moderate (10⁻⁶–10⁴ S/m), rises with T |
| Examples | Cu, Ag, Al | Glass, rubber, diamond | Si, Ge, GaAs |
Intrinsic and Extrinsic Semiconductors
Semiconductors can be pure (intrinsic) or modified by intentional impurity addition (extrinsic, or doping).
Intrinsic Semiconductors
An intrinsic semiconductor is a chemically pure crystal with no intentional impurities. At any temperature above 0 K, electron‑hole pairs are generated thermally:
n_e = n_h = n_i
where n_e is the free‑electron concentration, n_h the hole concentration, and n_i the intrinsic carrier concentration. The conductivity σ of an intrinsic semiconductor is given by:
σ = e (n_e μ_e + n_h μ_h) = e n_i (μ_e + μ_h)
Here e is the elementary charge, and μ_e, μ_h are the electron and hole mobilities, respectively. Since n_i depends strongly on temperature:
n_i = \sqrt{N_c N_v}\; \exp\!\left(-\frac{E_g}{2kT}\right)
where N_c and N_v are the effective density of states in the conduction and valence bands, k is Boltzmann’s constant, and T is absolute temperature. Consequently, the conductivity of intrinsic semiconductors rises exponentially with temperature.
Extrinsic Semiconductors (Doping)
By introducing impurity atoms (dopants) into the crystal lattice, the carrier concentration can be increased by many orders of magnitude. Doping creates either extra electrons (n‑type) or extra holes (p‑type).
N‑type Semiconductors
Donor impurities belong to Group V of the periodic table (e.g., phosphorus (P), arsenic (As)). Each donor atom has five valence electrons; four form covalent bonds with the host silicon atoms, leaving one loosely bound electron that can be easily excited into the conduction band. The donated electron increases n_e while the hole concentration remains low.
The Fermi level (EF) shifts upward, closer to the conduction band:
E_F \approx E_c - kT \ln\!\left(\frac{N_D}{n_i}\right)
where N_D is the donor concentration.
P‑type Semiconductors
Acceptor impurities are from Group III (e.g., boron (B), gallium (Ga)). Each acceptor has three valence electrons; after forming three covalent bonds, it creates a vacant bond that can accept an electron from the valence band, leaving behind a mobile hole. This raises the hole concentration n_h.
The Fermi level moves downward, nearer to the valence band:
E_F \approx E_v + kT \ln\!\left(\frac{N_A}{n_i}\right)
where N_A is the acceptor concentration.
Effect of Doping on Conductivity
Because dopants supply carriers directly, the conductivity of an extrinsic semiconductor is largely temperature‑independent (over a moderate range) and is given by:
σ ≈ e (N_D μ_e) for n‑type
σ ≈ e (N_A μ_h) for p‑type
Thus, even a small dopant concentration (~10¹⁵ cm⁻³) can increase σ by factors of 10³–10⁶ compared with the intrinsic material.
Comparison of Intrinsic vs. Extrinsic Properties
| Property | Intrinsic | Extrinsic (N‑type) | Extrinsic (P‑type) |
|---|---|---|---|
| Dominant Carrier | Electrons = Holes (n_i) | Electrons (≈ N_D) | Holes (≈ N_A) |
| Fermi Level Position | Near mid‑gap | Close to conduction band | Close to valence band |
| Temperature Dependence of σ | Strong (∝ exp(−E_g/2kT)) | Weak (mainly mobility variation) | Weak (mainly mobility variation) |
| Typical Dopant Concentration | 0 (pure) | 10¹⁴–10¹⁸ cm⁻³ | 10¹⁴–10¹⁸ cm⁻³ |
| Example | Pure Si | Si doped with P (N_D = 10¹⁶ cm⁻³) | Si doped with B (N_A = 10¹⁶ cm⁻³) |
Illustrative Example: Silicon at 300 K
For silicon (Eg = 1.12 eV, N_c ≈ 2.8×10¹⁹ cm⁻³, N_v ≈ 1.04×10¹⁹ cm⁻³):
n_i = \sqrt{N_c N_v}\; \exp\!\left(-\frac{E_g}{2kT}\right)
≈ 1.5×10¹⁰ cm⁻³
Thus, an intrinsic Si wafer has about 1.5×10¹⁰ free electrons and the same number of holes per cubic centimeter at room temperature.
If doped with phosphorus at N_D = 1×10¹⁶ cm⁻³, the electron concentration becomes ≈ N_D (since N_D ≫ n_i), and the conductivity is:
σ ≈ e N_D μ_e
≈ (1.6×10⁻¹⁹ C)(1×10¹⁶ cm⁻³)(1350 cm²/V·s)
≈ 3.5 S/cm
which is roughly five orders of magnitude larger than the intrinsic conductivity of Si (≈ 4.4×10⁻⁶ S/cm at 300 K).
Summary
This chapter has shown how the collective interaction of atoms in a solid leads to the formation of energy bands, how the size of the forbidden gap distinguishes metals, insulators, and semiconductors, and how intentional doping modifies carrier concentrations and the Fermi level to produce n‑type and p‑type extrinsic semiconductors. Understanding these concepts is essential for grasping the operation of modern electronic devices such as diodes, transistors, and solar cells.