Charge Carriers in Semiconductors • At T = 0K, a semiconductor is an insulator with no free charge carriers. • At T > 0K, some electrons in the valence band are excited to the conduction band, where they are free to move about. • The empty state in the valence band left behind by a promoted electron is referred to as a hole. • Ec – bottom of conduction band. • Ev – top of valence band. • At 0K, valence band of a semiconductor is completely filled with electrons while its conduction band is empty. • Electrons in the valence band cannot conduct electrical current because there are no available states to transfer to. • This causes a semiconductor to behave like an insulator at 0 deg K. • At T > 0K, some electrons in the valence band of the semiconductor gain enough energy to overcome the band gap and jump up to the conduction band, becoming free to conduct electricity. • The empty state left behind by the excited electron in the valence band is known as a 'hole.' • Holes in the valence band, like electrons in the conduction band, are also free to move about. • In a semiconductor, both electrons in the conduction band and holes in the valence band can be utilized to conduct electricity, which is why both of them are also known as 'charge carriers'. 1 Intrinsic Semiconductors • Pure semiconductor materials, no impurity atoms. • Charge carriers are solely the electrons and holes created in pairs by thermal energy. • Electrons in the valence band gain thermal energy and are promoted into the conduction band as temperature increases, leaving behind corresponding holes in the valence band: “electron-hole” pair. • The number of electrons in the conduction band equals the number of holes in the valence band. • The band gap determines how temperature variation affects conductivity. • Intrinsic conduction: process that results from the band structure of a pure element or compound. 2 Intrinsic Carrier Density versus Temperature in GaAs, Si and Ge Intrinsic Semiconductors…. • One-to-one correspondence between the electrons in the conduction band and the holes in the valence band in an intrinsic semiconductor. • In intrinsic semiconductors, density of intrinsic electrons (ni) = density of intrinsic holes (pi). •Conductivity σ of an intrinsic semiconductor is: σ = Neqe(µe+µh) and Ne = N0 exp(-Eg/2kT), where Ne = number of conduction electrons, qe = electron charge, µe = mobility of the conduction electrons, µh = mobility of the valence holes. N0 = material constant. Thus, σ = σ0 exp(-Eg/2kT) where σ0 = N0qe(µe+µh). 3 Intrinsic Semiconductors…. • σ = σ0 exp(-Eg/2kT) OR lnσ = lnσ0 -Eg/2kT • Plot of ln σ versus 1/T is a straight line whose slope is Eg/2k. •Band gap of an intrinsic semiconductor may be determined experimentally by observing how its conductivity varies with temperature. •The mobility of both electrons and holes decreases linearly with an increase in temperature • Number of mobile charge carriers increases exponentially with an increase in temperature. • With increasing temperature, the exponential increase in the number of carriers is a more dominating factor than the linear decrease in carrier mobility, so the conductivity of an intrinsic semiconductor always increases as temperature increases. •Ohm’s law: I = V/R where R= electrical resistance. •R depends on resistivity (ρ), an intrinsic property of the material. R = ρl/A and σ = 1/ρ. • σ = electrical conductivity. •Electric field E = V/l. • Ohm's law: j = σ E where j = I/A = current density. • Conductivity:107 (Ω-m) typical of metals, 10-20 (Ωm) for good electrical insulators and in the range 10-6 to 104 (Ω-m) for semiconductors. • Mobility (µ): a quantity relating dependence of drift velocity of charge carriers to the applied electric field. σ = n |e| µe + p |e| µh where p is the hole concentration and µh the hole mobility. Electrons move much faster than holes: µe > µh 4 Extrinsic Semiconductors ¾Semiconductors doped with foreign atoms to alter their intrinsic electron and hole concentrations. ¾Doped semiconductors: Semiconductors, which contain impurities (foreign atoms) incorporated into the crystal structure of the semiconductor. ¾Sources of impurities: Either (a) Added on purpose to provide free carriers in the semiconductor OR (b) Incorporated unintentionally, due to lack of control during the growth of the semiconductor. ¾Free carriers are generated when impurity atoms give off electrons or holes. Donors Donor Impurity Atom – donates an electron to the conduction band without creating a hole in the valence band; resulting material is an ntype semiconductor. The donor energy level is filled prior to ionization. Ionization causes the donor to be emptied, yielding an electron in the conduction band and a positively charged donor ion. 5 Acceptors Acceptor Impurity Atom – generates a hole in the valence band without generating an electron in the conduction band; resulting material is an p-type semiconductor. The acceptor energy is empty prior to ionization. Ionization of the acceptor corresponds to the empty acceptor level being filled by an electron from the filled valence band. This is equivalent to a hole given off by the acceptor atom to the valence band. 6 Dopant Atoms and Band Gap Energy Levels • Addition of impurity atoms to an intrinsic semiconductor to form an extrinsic semiconductor basically creates new energy levels within the band gap of the semiconductor. •Donor atoms: the extra electrons loosely bound to the donor atoms are not restricted to the energy levels allowed for the host atoms. They occupy energy levels forbidden to the electrons of the host atoms. This energy level is the “donor level” Ed, located in the upper half of the forbidden band. • Acceptor atoms: Similarly,holes from the acceptor atoms occupy “acceptor level” Ea, located in the lower half of the forbidden band. E E Ec Ed Ec Ei Ei Ev Ea Ev Ionization of a shallow donor Ionization of a shallow donor 7 Ionization • Is a process by which free charge carriers (electrons and holes) are produced in an extrinsic semiconductor. Ionization Energy – energy needed to move a donor electron from the donor level (Ed) to the conduction band. Or energy needed to move a valence band electron from the valence band to the acceptor level (Ea). • An extrinsic semiconductor will only have free carriers if the impurity atoms are ionized. • Shallow impurities: those whose ionization energies are ≤ kT. Deep impurities require larger energies to ionize. • If ionization energy > 5kT very minimal ionization will occur. 8 Complete Ionization -When all donor and acceptor atoms have become ionized. -Occurs at about room temperature for extrinsic semiconductors. -At absolute zero, the opposite of complete ionization occurs, i.e., freeze-out. -Freeze-out: when all donor and acceptor atoms are neutrally charged, i.e., no donor electrons are elevated into the conduction band and no electrons from the valence band are elevated into the acceptor level. -Between absolute zero and room temperature – various levels of partial ionization of donor/acceptor atoms exist. Ionization Energies 9 Temperature Dependence •Extrinsic semiconductors exhibit conductivities that behave differently with temperature variation. •Conductivity of an extrinsic semiconductor depends on the type of impurity atoms that it has been doped with. Conductivity of Extrinsic Semiconductors - Conduction in extrinsic semiconductors occurs at lower temperatures than for intrinsic semiconductors. - Conductivities can be vastly increased (no or po >> ni ) - Semiconductor becomes either n-type or p-type. - N-type semiconductor: extrinsic with more shallow donor impurity atoms. - P-type semiconductor: extrinsic with more shallow acceptor impurity atoms. -Compensated semiconductor: extrinsic with equal amounts of shallow donors and acceptors. No net free charge carriers. 10 Carrier Distribution Functions 11 • Distribution or probability density functions describe the probability with which one can expect particles to occupy the available energy levels in a given system. • Our interest is in probability density function of electrons, called the Fermi function. • Probability density functions derived in statistical thermodynamics. • Other distribution functions: Bose-Einstein distribution function, Maxwell Boltzmann distribution. • Fermi-Dirac distribution function provides the probability of occupancy of energy levels by Fermions. •As Fermions are added to an energy band, they will fill the available states in an energy band. •Electrons are Fermions. F-D function provides the probability that an energy level at energy, E, in thermal equilibrium with a large system, is occupied by an electron. • States with the lowest energy are filled first, followed by the next higher ones. 12 •At 0 K, the energy levels are all filled up to a maximum energy, which we call the Fermi level. • No states above the Fermi level are filled. •At higher temperature, transition between completely filled states and completely empty states is gradual rather than abrupt. • The system: characterized by its temperature T and Fermi energy EF. •At 0K, electrons are distributed in the lowest possible energy levels. • At higher temperature, electrons gain energy and transfer to higher energy levels. •Change in occupancy of the energy levels as temperature increases is quantified by the FermiDirac Distribution Function f(E). •It gives probability that an energy level is occupied by an electron at absolute temperature T. The Fermi-Dirac Distribution: f (E) = 1 1 + (exp{[ E − E f ] / kT }) 13 Fermi-Dirac Distribution Function ƒ(E) - describes the statistical behavior of electrons among available energy states ƒ(E): - gives the probability that an allowed energy state is occupied by an electron - ratio of filled to total quantum states at any energy E where: N(E) = number of particles per unit volume per unit energy; g(E) = number of quantum states per unit volume per unit energy The F-D function at three different temperatures 14 •The function = 1 for energies, which are more than a few times kT below EF. • The function = 1/2 for E = EF. • The function decreases exponentially for energies which are a few times kT larger than EF. • At T = 0 K, the function is a step function, the transition is more gradual at finite temperatures and more so at higher temperatures. Fermi-Dirac Distribution Function at T = 0 K 15 Fermi-Dirac Distribution Function at T > 0K Density of states • Finding # of available states at each energy helps calculate the density of carriers in a semiconductor. • Number of electrons at each energy - then obtained by multiplying the number of states with the probability that a state is occupied by an electron. • Number of energy levels – very large, depends on semiconductor size. • Just calculate the number of states per unit energy and per unit volume. 16 Calculation of the Density of States • Density of states in a semiconductor equals the density per unit volume and energy of the number of solutions to Schrödinger's equation. • Assume semiconductor can be modeled as an infinite quantum well in which electrons with effective mass, m*, are free to move. The energy in the well is set to zero. •The semiconductor is assumed a cube with side L. Calculation of the Density of States • Solve Schrödinger's wave equation. •Solution where V(x) = 0 are sine and cosine functions: 17 Schrödinger equation: Total energy (classical) E = K.E. + PE Define wavefunction, Ψ, multiply through to convert expression into a wave equation: Introduce operator Apply to a plane wave to provide the square of the momentum, p: k = wavenumber = 2π /λ. Replace p2 by this operator. Yields the timeindependent Schrödinger equation: Where V(x) = 0, solutions are sine and cosine functions: 18 A and B - constants to be determined. Ψ= 0 at the infinite barriers of the well. Ψ = 0 at x = 0 so that only sine functions can be valid solutions. Ψ = 0 at x = L Yields possible values for the wavenumber, kx. Similar results for y and z direction. Each possible solution ≡ a cube in k-space with size nπ/L. 19 •Calculate volume of 1/8 a sphere with radius k then divide it by the volume corresponding to a single solution. •Gives total number of solutions with a different value for kx, ky and kz and with a magnitude of the wavevector less than k. Results: Factor of two - two possible spins of each solution. Use chain rule to obtain the density per unit energy: The kinetic energy E of a particle with mass m* is related to the wavenumber, k, by: 20 The density of states per unit volume and per unit energy, g(E), is: The density of states is zero at the bottom of the well as well as for negative energies Density of States for electrons in Semiconductor •Same analysis for electrons in a semiconductor. • Use effective mass of electrons to account for effect of periodic potential on the electron. • Minimum energy = energy at the bottom of the conduction band, Ec. • Below EC, density of states is zero. • Density of states for electrons in the conduction band is given by: 21 The density of states for holes in the valence band is given by: Carrier Densities: Electrons •To find density of electrons in semiconductor: •Find density of available states for electrons. •Find probability that each of these states is occupied. • Density of occupied states per unit volume and energy, n(E), is given by the product of the density of states in the conduction band, gc(E) and the Fermi-Dirac probability function, f(E). 22 Carrier Densities: Holes • Holes = empty states in the valence band. •Probability of having a hole = probability that a particular state is not filled. •Hole density per unit energy, p(E), equals: • gv(E) is the density of states in the valence band. Carrier Densities •To obtain density of carriers, integrate the density of carriers per unit energy over all possible energies within a band. •Approximate solution: use simple particle-in-a box model, where one assumes that the particle is free to move within the material. 23 The carrier density (electrons) in a semiconductor: Where gc(E) is the density of states in the conduction band and f(E) is the Fermi function. 24 Carrier Density Integral Density of states, gc(E), Density per unit energy, n(E), Probability of occupancy, f(E). Carrier density, no, equals the crosshatched area. 25 26 Carrier density at zero Kelvin At T = 0 K, f(E) = 1 for all E < EF f(E) = 0 for all E > EF. and integration yields: This expression can be used to approximate the carrier density in heavily degenerate semiconductors provided kT << (EF - Ec) > 0 Non-degenerate semiconductors •Non-degenerate semiconductor one with EF at least 3kT away from either band edge. •This definition allows the fFD(E) to be replaced by fMB(E) - a simple exponential function. •The carrier density integral can then be solved analytically yielding: 27 Non-degenerate semiconductors where Nc is the effective density of states in the conduction band. The Fermi energy, EF, is obtained from: •M-B distribution applies to non-interacting particles, which can be distinguished from each other. •Provides the probability of occupancy for noninteracting particles at low densities, e.g. atoms in an ideal gas. •The Maxwell-Boltzmann distribution function is given by: 28 Probability of occupancy vs energy of the F-D, B-E and M-B distribution. Assumes EF = 0 All almost equal for large E (a few kT beyond EF). F-D = 100% for E ~ a few kT below EF. Similarly for holes, one can approximate the hole density integral as: where Nv is the effective density of states in the valence band. The Fermi energy, EF, is obtained from: 29 Degenerate Semiconductors A useful approximate expression applicable to degenerate semiconductors was obtained by Joyce and Dixon and is given by: for electrons and by: for holes. 30