Semiconductors

A pure semiconductor is a material with poor electrical conductivity. However the conductivity can be increased by order of magnitude by the addition of a small concentration of dopant atoms or by applying an electric field. Semiconductors form the basis of computation, communication, and display technologies. They are an important component of consumer electronics and transporation technologies. There is a strong increase in the use of semiconductors in solar cells and light-emitting diodes. The most commonly produced lasers are semiconductor lasers.

Outline

Role of semiconductors in technology
Band structure of semiconductors

Conduction band E_c, valence band E_v, band gap E_g
Direct and indirect band gaps W
Absorption and emission of photons and phonons
Examples of semiconductor band structures

Electrons and holes

Effective mass W
Holes W
Crystal momentum W
Ohm's law

Boltzmann approximation

Intrinsic semiconductors W
Effective density of states N_c, N_v
The density of electrons in the conduction band n = N_cexp((μ - E_c)/k_BT)
The density of holes in the valence band p = N_vexp((E_v - μ)/k_BT)
Law of mass action: np = N_cN_vexp(-E_g/k_BT)
The intrinsic carrier density n_i =(N_cN_v)^1/2exp(-E_g/2k_BT)
Chemical potential of intrinsic semiconductors
Electronic component of the internal energy of intrinsic semiconductors
Electronic component of the specific heat of intrinsic semiconductors
Electronic component of the Helmholtz free energy of intrinsic semiconductors
Electronic component of the entropy of intrinsic semiconductors
Table summarizing the thermodynamic properties of semiconductors in the Boltzmann approximation

Extrinsic semiconductors

Doping W
Extrinsic carrier densities
Chemical potential in extrinsic semiconductors W

Semiconductor devices

pn junctions
diodes

Light emitting diodes
Solar cells
Laser diodes

Transistors

Transport in semiconductors

Ohm's law: &sigma = (neμ_n + peμ_p)
Mobility W
Seebeck effect W
Peltier effect W
Thermopower W
Bloch oscillations W

Reading
Kittel chapter 8: Semiconductor Crystals or R. Gross und A. Marx: Halbleiter

For the exam you should

be able to look at a bandstructure diagram for a semiconductor and be able to identify the conduction band, the valence band, the energy gap, whether the semiconductor is direct or in direct, and to be able to determine the effective masses of electrons and holes.
be able to define donors, acceptors, intrinsic semiconductors, extrinsic semiconductors, holes, and Bloch oscillations.
know the formulas n = N_cexp((μ-E_c)/k_BT) and p = N_vexp((E_v-μ)/k_BT). From these formulas you should be able to calculate the density of electrons in the conduction band, the density of holes in the valence band, the intrinsic carrier concentration n_i, and the chemical potential for an intrinsic semiconductor or extrinsic semiconductor.
be able to describe how a light emitting diode, a solar cell, and a laser diode works.
know how thermal transport is coupled to electrical transport in semiconductors. Be able to describe the Seebeck effect and the Peltier effect.

Resources

Semiconductors in Wikipedia: Si, Ge, GaAs, GaN, InP
Semiconductor database on NSM
Program "Fermi" that calculates the carrier concentration as a function of temperature. From the Solid State Simulations project at Cornell.
"Poission" program from the Soild State Simulations project that models pn junctions.
Principles of semiconductor devices, B. Van Zeghbroeck, On-line book.
Nobel prizes

Semiconductor Research Leading to the Point Contact Transistor, John Bardeen, Nobel Prize 1956.
Transistor Technology Evokes New Physics, William B. Shockley, Nobel Prize 1956.
Surface Properties of Semiconductors, Walter H. Brattain, Nobel Prize 1956.
Quasi-Electric Fields and Band Offsets: Teaching Electrons New Tricks, Herbert Kroemer, Nobel Prize in Physics 2000.
Double Heterostructure Concept and its Applications in Physics, Electronics and Technology, Zhores I. Alferov, Nobel Prize in Physics 2000.
Turning Potential into Reality: The Invention of the Integrated Circuit, Jack S. Kilby, Nobel Prize in Physics 2000.

Student presentations

Properties	Si	Ge	GaAs
Bandgap *E_g*	1.12 eV	0.66 eV	1.424 eV
Effective density of states in conduction band (300 K) *N_c*	2.78 × 10²⁵ m^-3	1.04 × 10²⁵ m^-3	4.45 × 10²³ m^-3
Effective density of states in valence band (300 K) *N_v*	9.84 × 10²⁴ m^-3	6.0 × 10²⁴ m^-3	7.72 × 10²⁴ m^-3
Effective mass electrons *m^/m₀**	m_l^* = 0.98 m_t^* = 0.19	m_l^* = 1.64 m_t^* = 0.082	m^* = 0.067
Effective mass holes *m^/m₀**	m_lh^* = 0.16 m_hh^* = 0.49	m_lh^* = 0.044 m_hh^* = 0.28	m_lh^* = 0.082 m_hh^* = 0.45
Crystal structure	diamond	diamond	zincblende
Density	2.328 g/cm³	5.3267 g/cm³	5.32 g/cm³
Atoms/m³	5.0 × 10²⁸	4.42 × 10²⁸	4.42 × 10²⁸

Doped semiconductors

For a doped semiconductor, the density of electrons in the conduction band is,

the density of holes in the valence band is,

the density of ionized donors is,

and the density of ionized acceptors is,

The factor of 4 is valid in the formula for the acceptors if the semiconductor has a light hole and a heavy hole band as Si and Ge do.

The four quantities n, p, N_d, and N_a can only be determined if the chemical potential, μ, is known. Typically, μ must first be determined from the charge neutrality condition,

n + N_a^- = p + N_d⁺.

The chemical potential can be found by solving the charge neutrality condition numerically. One way to do this is to program the formulas for n, p, N_d⁺, and N_a^- in a spreadsheet. Then choose a temperature and calculate n, p, N_d⁺, N_a^- for every value of the chemical potential between E_v and E_c. For one of these μ values, the charge neutrality condition will be satisfied.

When n + N_a^- and , p + N_d⁺ are ploted as a function of μ, the chemical potential is where the two lines cross. A plot like the one below can be generated by pressing the button below the plot.

The Fermi energy can be calculated as a function of temperature by determining where n + N_a^- = p + N_d⁺ for every temperature. A plot like the one below can be generated by pressing the button below the plot.

Normally it is not necessary to determine μ numerically and the following approximation is sufficient.

n-type N_d > N_a:

n = N_d - N_a
p = n_i²/n
μ = E_c - k_BTln(N_c/(N_d - N_a))

p-type N_a > N_d:

p = N_a - N_d
n = n_i²/p
μ = E_v + k_BTln(N_v/(N_a - N_d))







513.001 Molecular and Solid State Physics