Conductivity Semiconductors & Metals Chemistry 754 Solid State Chemistry Lecture #20 May 14, 2003 References Conductivity There are many references that describe electronic conductivity in metals and semiconductors. I used primarily the following texts to develop this lecture. The Electronic Structure and Chemistry of Solids P.A. Cox, Oxford University Press, Oxford (1987). Solid State Physics H. Ibach and H. Luth,, Springer-Verlag Verlag,, Berlin (1991). Physical Properties of Semiconductors C.M. Wolfe, N. Holonyak,, Jr., G.E. Stillman,, Prentice Hall, Englewood Cliffs, NJ (1989).
Resistivities of Real Materials Compound Resistivity (Ω-cm) Compound Resistivity (Ω-cm) Ca 3.9 10-6 Si ~ 0.1 Ti 42 10-6 Ge ~ 0.05 Mn 185 10-6 ReO 3 36 10-6 Zn 5.9 10-6 Fe 3 O 4 52 10-6 Cu 1.7 10-6 TiO 2 9 10 4 Ag 1.6 10-6 ZrO 2 1 10 9 Pb 21 10-6 Al 2 O 3 1 10 19 Most semiconductors in their pure form are not good conductors, they need to be doped to become conducting. Not all so called ionic materials like oxides are insulators. Microscopic Conductivity We can relate the conductivity, σ,, of a material to microscopic parameters that describe the motion of the electrons (or other charge carrying particles such as holes or ions). where σ = ne(e (eτ/m*) µ = eτ/m* e σ = neµ n = the carrier concentration (cm -3 ) e = the charge of an electron = 1.602 10-19 C τ = the relaxation time (s) {the time between collisions} m* = the effective mass of the electron (kg) µ = the electron mobility (cm 2 /V-s)
Metals, Semiconductors & Insulators Energy E F Energy E F Energy Conduction Band E F Valence Band DOS Metal DOS Semimetal DOS Semiconductor /Insulator In a metal the Fermi level cuts through a band to produce a partially filled band. In a semiconductor/insulator there is an energy gap between the filled bands and the empty bands. The distinction between a semiconductor and an insulator is artificial, but as the gap becomes large the material usually becomes a poor conductor of electricity. A semimetal results when the band gap goes to zero. Resistivity and Carrier Concentration The resistivities of real materials span nearly 25 orders of magnitude. This is due to differences in carrier concentration (n) and mobility (µ).( Let s first consider carrier concentration. The carrier concentration only includes electrons which can easily be excited from occupied states into empty states. The remaining electrons are localized. In the absence of external excitations (light, voltage, etc.) the excitation must be thermal, this is on the order of kt (~ 0.03 ev at RT) Only electrons whose energies are within a few kt of E F can contribute to the electrical conductivity. Generally this means that E F should cut a band to achieve appreciable carrier concentration. Alternatively impurities/defects are introduced to partially populate a band.
Fermi-Dirac Function The Fermi-Dirac function gives the fraction of allowed states, f(e), at an energy level E, that are populated at a given temperature. f(e) = 1/[1 + exp{(e-e F )/kt kt}] where the Fermi Energy, E F, is defined as the energy where f(e) = 1/2. That is to say one half of the available states are occupied. T is the temperature (in K) and k is the Boltzman constant (k = 8.62 10-5 ev/k) As an example consider f(e) for T = 300 K and a state 0.1 ev above E F : f(e) = 1/[1 + exp{(0.1 ev)/((300k)(8.62 10-5 ev/k)}] f(e) = 0.02 = 2% Consider a band gap of 1 ev. f(1 ev) ) = 1.6 10-17 See that for even a moderate band gap (Silicon has a band gap of 1.1 ev) ) the intrinsic concentration of electrons that can be thermally excited to move about the crystal is tiny. Thus pure Silicon (if you could make it) would be quite insulating. Fermi Dirac Function Metals and Semiconductors f(e) as determined experimentally for Ru metal (note the energy scale) f(e) for a semiconductor
Carrier Mobility Recall the expression for carrier mobility: µ = eτ/m* e where, e = electronic charge m* = the effective mass τ = the relaxation time between scattering events What factors determine the effective mass? m * depends upon the band width, which in turn depends upon orbital overlap. What entities scatter the carriers and reduce the mobility? A defect or impurity (τ( increases as purity increases) Lattice vibrations, phonons (τ( decreases as temp. increases) What is the meaning of k? In our development of the electronic band structure from a linear combination of atomic orbitals the variable k was used to determine the phase of the orbitals. What exactly is k? Wavevector It tells us the how the phases of the orbitals change when translational symmetry is applied. Quantum Number Identifies a particular electronic wavefunction (that can hold 2 electrons with opposite spin). Crystal Momentum In free electron theory k is proportional to the momentum of the electron in the k th wavefunction.
Crystal Momentum To better understand the meaning of k, consider an electron at the outer edge of the Brillouin zone, where k = π/a. The phase of the electronic wavefunction changes sign every unit cell (similar to a p-orbital changing phase at its nodal plane) a λ λ = 2a a = λ/2 k = π/a a = π/k Combining these two relationships gives: λ/2 = π/k k = 2π/λ 2 λ = 2π/k The wavelength of the wavefunction is inversely proportional to k. Crystal Momentum Now consider the DeBroglie relationship (wave-particle duality of matter) λ = h/p p = h/λ p = hk/2 /2π where., p is the momentum of the wavepacket, h is Planck s constant, 6.626 10-34 J-s The momentum of an electron is directly proportional to k. k is a measure of the crystal momentum of an electron in the ψ K wavefunction.
From the ideas on the previous 2 slides one can derive the following relationships to describe the properties of a conduction electron: Velocity v = hk/2 /2πm m = (2π/h)(( /h)(de/dk) Energy E = (h/2π)(k 2 /2m * ) Effective Mass m * = (2π/h)( 2 (1/{d 2 E/dk 2 }) de/dk dk The first derivative of the E vs. k curve. d 2 E/dk 2 The second derivative of the E vs. k curve. Quantity Wide Band Narrow Band de/dk dk Large Small µ High Low Velocity Fast Slow m* Light Heavy Wide (disperse) bands are better for conductivity. Bandstructure & DOS for Cu E F cuts the very wide (disperse) s band, giving rise to a large carrier concentration, along with high mobility. This combination gives rise to high conductivity.
Temperature Dependence-Metals Recall that σ = ne 2 τ/m* In Metals The carrier concentration, n, changes very slowly with temperature. τ is inversely proportional to temperature (τ τ α 1/T), due to scattering by lattice vibrations (phonons). Therefore, a plot of σ vs. 1/T (or ρ vs. T) is essentially linear. Conductivity goes down as temperature increases. Scattering by Impurities and Phonons Phonon scattering Proportional to temperature Impurity scattering Independent of temperature Proportional to impurity concentration
Bandstructure for Ge CB minimum VB maximum p-bands s-band No mixing at Γ. E F falls in the (0.67 ev) ) band gap. Carrier concentration and conductivity are small. Ge is an indirect gap semiconductor, because the uppermost VB energy and the lowest CB energy occur at different locations in k-space. Direct & Indirect Gap Semiconductors Ge Si GaAs Figure taken from Fundamentals of Semiconductor Theory and Device Physics, by S. Wang Direct Gap Semiconductor: : Maximum of the valence band and minimum of the conduction band fall at the same place in k-space. α α (hν-e g ) 1/2 Indirect Gap Semiconductor: : Maximum of the valence band and minimum of the conduction band fall different points in k-space. A lattice vibration (phonon) is involved in electronic excitations, this decreases the absorption efficiency. α α (hν-e g ) 2
Doping Semiconductors The Fermi-Dirac function shows that a pure semiconductor with a band gap of more than a few tenths of an ev would have a very small concentration of carriers. Therefore, impurities are added to introduce carriers. n-doping Replacing a lattice atom with an impurity (donor) atom that contains 1 additional valence electron (i.e. P in Si). This e - can easily be donated to the conduction band. E F Conduction Band e - p-doping Replacing a lattice atom with an impurity (acceptor) atom that contains 1 less valence electron (i.e. Al in Si). This atom can easily accept an e - from the VB creating a hole. E F Conduction Band e - Valence Band Valence Band Common Semiconductor Structures Diamond Fd-3m (Z=8) C, Si, Ge, Sn Sphalerite F-43m (Z=4) GaAs, ZnS, InSb Chalcopyrite I-42d (Z=4) CuFeS 2, ZnSiAs 2
Properties of Semiconductors Compound Si Structure Diamond Bandgap (ev) 1.11 (I) e - mobility (cm 2 /V-s) 1,350 h + mobility (cm 2 /V-s) 480 Ge Diamond 0.67 (I) 3,900 1,900 AlP Sphalerite 2.43 (I) 80 --- GaAs Sphalerite 1.43 (D) 8,500 400 InSb Sphalerite 0.18 (D) 100,000 1,700 AlAs Sphalerite 2.16 (I) 1,000 180 GaN Wurtzite 3.4 (D) 300 --- Temperature Dependence-Semiconductors Recall that σ = ne 2 τ/m* In Semiconductors The carrier concentration increases as temperature goes up, due to excitations across the band gap, E g. n is proportional to exp{-e g /2kT}. τ is inversely proportional to temperature The exponential dependence of n dominates, therefore, a plot of ln σ vs. 1/T is essentially linear. Conductivity increases as temperature increases.
p-n Junctions In the middle of the junction E F falls midway between the VB & CB as it would in an intrinsic semiconductor. When a p-type and an n-type semiconductor are brought into contact electrons flow from the n-doped semiconductor into the p-doped semiconductor until the Fermi levels equalize (like two reservoirs of water coming into equilibrium). This causes the conduction and valence bands to bend as shown above. Applications of p-n Junctions Rectifier: Reverse Bias LED MOSFET Transistor Photovoltaic Cell