Semiconductor physics explains how the electrical properties of a material can be controlled through crystal structure, temperature, electric fields, and carefully introduced impurities. These principles form the physical basis of diodes, bipolar transistors, JFETs, MOSFETs, integrated circuits, solar cells, and many sensors.
The equations in this introductory chapter use idealized models. Unless stated otherwise, the semiconductor is assumed to be in thermal equilibrium and close to room temperature.
At approximately 300 K, the band gap is about 1.12 eV for silicon, 0.66 eV for germanium, and 1.42 eV for gallium arsenide. These values vary with temperature.
Learning Objectives
After completing this chapter, you should be able to:- distinguish conductors, semiconductors, and insulators using energy bands;
- explain the roles of electrons and holes;
- describe intrinsic, n-type, and p-type semiconductor material;
- relate carrier concentration and mobility to conductivity;
- distinguish drift current from diffusion current;
- explain the formation and operation of a pn junction;
- connect semiconductor physics to transistor operation.
1. Crystal Structure and Covalent Bonds
Silicon and germanium are group-IV elements with four valence electrons. In a pure crystal, each atom shares electrons with four neighboring atoms through covalent bonds. At very low temperature, these bonds are largely occupied and few mobile charge carriers are available. Thermal energy can break a covalent bond and release an electron into a mobile state. The missing electron behaves as a positively charged mobile entity called a hole. Electrons and holes are therefore generated in pairs in an intrinsic semiconductor.2. Energy Bands
In an isolated atom, electrons occupy discrete energy levels. When many atoms form a crystal, the allowed states broaden into energy bands.- Valence band: the highest range of energies normally occupied by bound electrons.
- Conduction band: a range of higher-energy states in which electrons can move through the crystal.
- Band gap \(E_g\): the forbidden energy range separating the valence and conduction bands.
| Material Class | Band Description | Electrical Behavior |
|---|---|---|
| Conductor | Partly filled band or overlapping bands | Many mobile carriers |
| Semiconductor | Relatively small band gap | Carrier population can be controlled |
| Insulator | Large band gap | Very few mobile carriers under ordinary conditions |
3. Electrons and Holes
An electron in the conduction band carries charge −q. A hole in the valence band can be treated as a mobile carrier with charge +q. Both contribute to electrical current, although their mobilities are generally different. The carrier concentrations are denoted by:- n: concentration of conduction-band electrons;
- p: concentration of valence-band holes.
4. Intrinsic Semiconductors
An intrinsic semiconductor is ideally pure and has no intentional doping. In thermal equilibrium, electrons and holes are generated in equal numbers:\[n=p=n_i\]
Here, \(n_i\) is the intrinsic carrier concentration. It depends strongly on temperature and on the band gap. For nondegenerate material in thermal equilibrium, the mass-action relation is:
\[np=n_i^2\]
Increasing temperature produces more electron-hole pairs and therefore increases the conductivity of an intrinsic semiconductor.
5. Doping
Doping deliberately introduces a small concentration of impurity atoms into the semiconductor crystal. It changes the balance between electrons and holes without giving the material a net macroscopic charge.n-Type Material
Donor atoms such as phosphorus or arsenic have five valence electrons. Four participate in covalent bonds, while the fifth is relatively weakly bound and can enter the conduction band. Electrons are the majority carriers and holes are the minority carriers.p-Type Material
Acceptor atoms such as boron have three valence electrons. They can accept an electron from a neighboring bond, leaving a mobile hole. Holes are the majority carriers and electrons are the minority carriers.| Material | Typical Dopant | Majority Carrier | Minority Carrier |
|---|---|---|---|
| n-type silicon | Donor, e.g. phosphorus | Electrons | Holes |
| p-type silicon | Acceptor, e.g. boron | Holes | Electrons |
6. The Fermi Level
The Fermi level \(E_F\) describes the electron occupancy of available energy states. The Fermi-Dirac probability is:\[f(E)=\frac{1}{1+\exp\!\left(\frac{E-E_F}{kT}\right)}\]
In intrinsic material, the Fermi level lies near the middle of the band gap. Donor doping moves it closer to the conduction band; acceptor doping moves it closer to the valence band. The Fermi level is especially useful for describing equilibrium, contacts, and junctions.
7. Conductivity and Mobility
In a uniform semiconductor, both carrier types contribute to conductivity:\[\sigma=q\left(n\mu_n+p\mu_p\right)\]
Here, \(\mu_n\) and \(\mu_p\) are the electron and hole mobilities. Resistivity is the reciprocal of conductivity:
\[\rho=\frac{1}{\sigma}\]
Mobility measures how readily a carrier responds to an electric field. It is affected by temperature, lattice vibrations, impurity concentration, and crystal defects.
8. Drift Current
An electric field accelerates carriers between scattering events. The average directed motion is called drift. The drift current densities are:\[J_{n,\mathrm{drift}}=qn\mu_nE\]
\[J_{p,\mathrm{drift}}=qp\mu_pE\]
Although electrons physically move opposite to the electric field, conventional electron current is defined in the field direction.
9. Diffusion Current
Carriers also move from regions of high concentration toward regions of low concentration because of random thermal motion. This produces diffusion current. In one dimension:\[J_{n,\mathrm{diff}}=qD_n\frac{dn}{dx}\]
\[J_{p,\mathrm{diff}}=-qD_p\frac{dp}{dx}\]
The diffusion constants D and mobilities μ are related by the Einstein relation:
\[\frac{D}{\mu}=\frac{kT}{q}=V_T\]
At room temperature, the thermal voltage \(V_T\) is approximately 25.9 mV.
10. Formation of the pn Junction
A pn junction is formed when p-type and n-type regions meet within one crystal. Immediately after formation, electrons diffuse from the n side toward the p side, while holes diffuse from the p side toward the n side. Recombination near the interface leaves behind fixed ionized donors and acceptors. The region depleted of mobile majority carriers is called the depletion region. Its fixed charge creates an internal electric field. The resulting drift current opposes further diffusion. At thermal equilibrium, the net current is zero. The built-in potential of an ideal abrupt junction is:\[V_{\mathrm{bi}}=V_T\ln\!\left(\frac{N_A N_D}{n_i^2}\right)\]
\(N_A\) and \(N_D\) are the acceptor and donor concentrations. The depletion region extends farther into the more lightly doped side.
11. Forward and Reverse Bias
Forward Bias
Forward bias makes the p side more positive than the n side. It lowers the junction barrier and allows majority carriers to be injected across the depletion region. The current rises approximately exponentially:\[I=I_S\left[\exp\!\left(\frac{V_D}{nV_T}\right)-1\right]\]
\(I_S\) is the reverse saturation current and n is an ideality factor. The familiar silicon “0.7 V” value is only a practical approximation; the actual forward voltage depends on current, device construction, and temperature.
Reverse Bias
Reverse bias increases the barrier and widens the depletion region. Only a small minority-carrier current flows until breakdown is reached. Zener breakdown is associated with strong-field tunneling in heavily doped junctions, while avalanche breakdown is caused by impact ionization.12. Junction Capacitance
A reverse-biased pn junction behaves partly like a voltage-dependent capacitor because the depletion region separates fixed charge. Increasing reverse bias generally widens the depletion region and reduces the depletion capacitance. Under forward bias, stored injected charge produces diffusion capacitance. Junction capacitances influence switching speed and high-frequency behavior.13. Generation, Recombination, and Carrier Lifetime
Real crystals contain defects and impurity states that can assist generation and recombination. The average time before an excess carrier recombines is described by its carrier lifetime. Carrier lifetime and diffusion length influence diode recovery, bipolar transistor action, photodetectors, and solar cells.14. From Junctions to Transistors
- BJT: uses two pn junctions and controlled minority-carrier injection through a thin base.
- JFET: uses a reverse-biased pn junction to vary the width of a conductive channel.
- MOSFET: uses an insulated gate and an electric field to create or modulate a surface channel.