Extrinsic Semiconductors
Pure silicon and germanium are decent insulators at room temperature because their electrons are tightly bound in covalent bonds — far too few carriers exist to conduct electricity well. Extrinsic semiconductors solve this problem through a deliberate process called doping: adding tiny, controlled amounts of impurity atoms to the crystal lattice. The result is a material whose electrical conductivity can be tuned over many orders of magnitude, which is the foundation of every transistor, diode, and solar cell ever made. The magic of doping lies in the periodic table. Silicon has four valence electrons and forms four covalent bonds with its neighbours. If you substitute even one silicon atom per million with a phosphorus atom (five valence electrons), that extra electron has no bond partner and is only weakly held — it can roam freely through the crystal at room temperature. Conversely, swap silicon for boron (three valence electrons) and you create a vacancy in the bonding structure called a hole, which acts like a positive charge carrier. The first type is called an n-type semiconductor (extra negative carriers), and the second is p-type (extra positive carriers). Extrinsic semiconductors are the reason modern electronics can be so precisely engineered. By varying the dopant concentration from roughly 10¹³ to 10²⁰ atoms per cubic centimetre, engineers can smoothly dial conductivity across seven orders of magnitude. Understanding how doping changes the density of free electrons and holes — and how those two densities are coupled through the law of mass action — is one of the most powerful ideas in all of modern physics.
- Donor Impurities and n-type Silicon
- Acceptor Impurities and p-type Silicon
- The Law of Mass Action
- Charge Neutrality and Fermi Level Shift
Donor Impurities and n-type Silicon
Atoms from Group 15 (phosphorus, arsenic, antimony) have five valence electrons. When one replaces a silicon atom, four electrons form the normal covalent bonds and the fifth sits in a shallow energy level just below the conduction band — typically only 0.045 eV below it in silicon. Thermal energy at room temperature (k_B T ≈ 0.026 eV) is enough to ionise almost all donor atoms, releasing electrons into the conduction band. Each ionised donor leaves behind a fixed positive ion in the lattice. These donated electrons are the majority carriers. If the donor concentration is N_D, then in most practical situations the free electron concentration n ≈ N_D, far exceeding the intrinsic value n_i ≈ 1.5 × 10¹⁰ cm⁻³ for silicon at 300 K.
Acceptor Impurities and p-type Silicon
Atoms from Group 13 (boron, aluminium, gallium) have only three valence electrons. When boron replaces silicon, it can only form three covalent bonds, leaving one bond unsatisfied — an empty state in the valence band called a hole. The energy level sits just above the valence band edge (~0.045 eV in silicon), so thermal energy easily promotes a neighbouring valence electron into this acceptor level, completing the bond and leaving behind a mobile hole. Each ionised acceptor becomes a fixed negative ion. The hole concentration p ≈ N_A, the acceptor concentration, which is the majority carrier in p-type material.
The Law of Mass Action
Even after doping, electrons and holes coexist. Their concentrations are linked by the law of mass action: n × p = n_i², where n_i is the intrinsic carrier concentration (a temperature-dependent constant for a given semiconductor). This equation is a consequence of thermodynamic equilibrium — it is the semiconductor analogue of an equilibrium constant in chemistry. So in n-type silicon with n ≈ N_D = 10¹⁶ cm⁻³, the hole concentration collapses to p = n_i²/N_D ≈ (1.5 × 10¹⁰)²/10¹⁶ ≈ 2.25 × 10⁴ cm⁻³. The majority carriers overwhelmingly dominate, but the minority carriers (holes in n-type, electrons in p-type) still exist and matter enormously in device physics.
Charge Neutrality and Fermi Level Shift
The crystal as a whole remains electrically neutral: the charge of ionised donors or acceptors is balanced by the extra carriers. Adding donors shifts the Fermi level E_F upward toward the conduction band; adding acceptors shifts it downward toward the valence band. When E_F is close to the conduction band, many conduction-band states are occupied, confirming high electron concentration. This Fermi-level engineering is what allows p-n junctions to form built-in electric fields, enabling rectification and transistor action.
The Key Equation
- n — Electron concentration (cm⁻³): Number of free electrons per unit volume in the conduction band.
- p — Hole concentration (cm⁻³): Number of holes per unit volume in the valence band.
- n_i — Intrinsic carrier concentration (cm⁻³): The electron (or hole) concentration in a perfectly pure semiconductor; for silicon at 300 K, n_i ≈ 1.5 × 10¹⁰ cm⁻³.
Worked Example
Problem. A silicon crystal at 300 K is doped with phosphorus at a concentration of N_D = 5 × 10¹⁵ cm⁻³. Assume complete ionisation and n_i = 1.5 × 10¹⁰ cm⁻³. Find (a) the electron concentration n, (b) the hole concentration p, and (c) state whether this is n-type or p-type material.
Answer. n = 5 × 10¹⁵ cm⁻³, p = 4.5 × 10⁴ cm⁻³; n-type semiconductor.
Origins · 1 minWilliam Shockley and how this was discovered
Shockley, along with John Bardeen and Walter Brattain, invented the transistor at Bell Labs in 1947 and shared the 1956 Nobel Prize in Physics. Shockley's theoretical framework for understanding doped semiconductors — including how donor and acceptor impurities create excess carriers and shift the Fermi level — provided the intellectual foundation for the entire semiconductor industry. His work on junction transistors directly exploited the properties of n-type and p-type extrinsic semiconductors. Shockley famously drew his own diagram of the first transistor on a Christmas card to send to friends and colleagues in 1947, before the invention had even been publicly announced.
Related Concepts
- Intrinsic Semiconductors — Extrinsic semiconductors are defined in contrast to intrinsic (pure) semiconductors; understanding the undoped case and the intrinsic carrier concentration n_i is essential before studying doping.
- The p-n Junction — Joining p-type and n-type extrinsic semiconductors creates a p-n junction, the basis of diodes and transistors, making it the most direct application of doping concepts.
- Band Theory and the Fermi Level — The Fermi level shifts upon doping and determines carrier concentrations; band theory provides the full quantum mechanical picture underlying extrinsic semiconductor behaviour.