Course overview Me: Dr Luke Wilson Office: E17 open door policy email: luke.wilson@sheffield.ac.uk The course: Physics and applications of semiconductors 10 lectures aim is to allow time for at least one revision lecture Each lecture will start with a quick, anonymous test for feedback
Syllabus 1. Brief review of semiconductor properties. The p-n junction Energy band diagrams Depletion and junction capacitance Current-voltage characteristics Breakdown 3. Transistors Bipolar devices Field effect transistors New semiconductor materials 4. Light emitting diodes (LEDs) Properties and operating characteristics Rate equations and recombination processes Example materials systems 5. Semiconductor lasers Underlying principles of operation Rate equations and gain Determination of operating characteristics Comparison with other laser types Homostructure/heterostructure comparison Practical design issues 6. Photodetectors Photoconductive/photovoltaic operation Photodiode spectral and temporal response Avalanche photodiodes
Review of semiconductor properties At T0 EMPTY HALF FULL FULL FULL E G - Bandgap CONDUCTION BAND VALENCE BAND Conductor / metal Semiconductor / insulator Diamond E G ~6eV Silicon E G ~1eV Number of conduction band electrons: n~exp(-e G /kt) kt~1/40ev @ T300K Semiconductors may have resistivity varying over several orders of magnitude
At T 0 E Electron (+ve mass, -ve charge) k E G Hole (+ve mass, +ve charge) Effective mass from dynamics of wave packet, moves at group velocity v g d ω dk 1 h de dk dk de 1 h v g Electric field applied so wave packet moves distance dx in time dt, change in energy de efdx http://upload.wikimedia.org/wikipedia/commons/c/c1/wave_packet_%8no_dispersion%9.gif
dk dt dk de 1. ef de dt hv g dx dt dx dt v so h g dk dt ef Acceleration dv g 1 d de 1 d de dk 1 d E dt h dt dk h dk dk dt h dk dk dt Sub in for dk/dt: dv g 1 dt h d E ef dk Now, like Fma Effective mass: m* d h E dk
Types of semiconductor DIRECT BAND GAP INDIRECT BAND GAP k0 c.b. c.b. k0 PHONON E G DIR PHOTON E G INDIR E G DIR PHOTON E G INDIR v.b. v.b. E G DIR < E G INDIR e-h recombination involves only a PHOTON FAST PROCESS E G DIR > E G INDIR e-h recombination involves a PHOTON and a PHONON SLOW PROCESS
Carrier statistics Electrons fermions, therefore use Fermi-Dirac distribution for state occupation P(E,T) P ( E, T ) e 1 E E kt F + 1 Fermi energy (E F ) where half of the available levels are occupied For E-E F >> kt (ie energies well above the Fermi energy), the Boltzmann approximation can be used many available states, so exclusion principle not important B ( E T ) P, E E kt e Can then use this to find number of electrons by multiplying by the density of states F
Doping Method for increasing free electron or hole density Thermal excitation produces very few electrons/holes by excitation across the bandgap (intrinsic carriers) n~e -E G/kT e -0 ~1x10-9 (Si @ T300K) With doping, carriers are produced by adding known amounts of certain impurities: c.b. v.b. c.b. v.b. Fermi level (E F ) E F E D E A nn D e -E D/kT N D Doping density n-type ELECTRONS pn A e -E A/kT N A Doping density p-type HOLES n-type n (majority) >> p (minority) p-type p (majority) >> n (minority) Carriers from doping are known as extrinsic carriers
Doping A very important (and quite simple!) equation: np n i In equilibrium, we can assume that the number of dopants equals the number of majority carriers i.e. n n N D p p N A So, if we want the number of minority carriers: n p n N i A p n n N i D All in terms of known properties of a particular semiconductor
Causes of current - drift and diffusion We ll consider reasons for a net flow of electrons or holes 1. An electric potential gradient dv/dx DRIFT. A particle number density gradient dn/dx DIFFUSION Drift If an electric field E accelerates carriers of charge e and effective mass m*, their acceleration is given by Force, F m * a, F ee m * a so acceleration a Distance moved in direction of E is given by ½aτ, where τ is the time between collisions Now, average velocity (called drift velocity) given by avg. distance / avg. time between collisions v d Define Then e m * 1 x τ xe τ eτd μ m * v μe d, v d eτd.e m * ee m * τ d is mean time between collisions Mobility (notice for high µ need high τ d, small m*)
Diffusion If the concentration of a particular carrier type is not uniform then thermal motion will try to produce a net movement to correct this. This motion is called diffusion t 0 t 0 When diffusion occurs for charged particles the resultant motion acts like a current 1D diffusion equation: dn J ed dx D is the diffusion coefficient D kt e μ
Real semiconductors Band gaps at T300K and corresponding emission wavelengths E G hν, νc/λ, E G hc/λ GROUP IV Si 1.1eV (1100nm) Ge 0.66eV (1900nm) III-V BINARIES AlAs.eV (560nm) AlP 3.0eV (410nm) AlSb 1.58eV (780nm) GaAs 1.4eV (870nm) GaP.6eV (550nm) GaSb 0.75eV (1650nm) InP 1.35eV (90nm) InAs 0.36eV (3400nm) InSb 0.17eV (7300nm) GaN 3.5eV (350nm) Visible light 400nm<λ<700nm very little choice using binary compounds
However, can use an alloy of two binary semiconductors: e.g. (GaAs) + (GaP) GaAs 1-x P x where (0 x 1) -Provides a continuously tunable band gap Both the direct and indirect band gaps vary with composition GaAs 1-x P x is a ternary (three) compound Can also have quaternary compounds with four elements e.g. Ga x Al y In 1-x-y P (0 x,y 1)
Recombination of electrons and holes in optoelectronic devices Electrons and holes (density n and p per unit volume) are generated at a rate G gen per unit volume either optically or electrically (generally assume np) Recombine at rate R rec Therefore rate equation: dn dt G gen R rec (0 in equilibrium) R rec can be written as a sum of a number of terms: R rec R sp +R nr +R L +R st (1) R sp spontaneous recombination rate (photon emission) Electrons and holes recombine across band gap. Energy is conserved by emitting a photon: c.b. Recombination rate depends on density of both electrons and holes hνe G Rsp Bnp Bn if np B is a constant v.b.
() R nr non-radiative recombination rate electrons and holes recombine but do not emit a photon main processes: (a) Deep states impurities or defects produce states within the band gap. Electrons may reach valence band by a cascade process and emission of phonons c.b. c.b. (b) Auger electron and hole recombine but transfer energy to a third particle Single particle process Rate An (A is proportional to Impurity density) v.b. v.b. Three particle process Rate Cn 3 R nr An+Cn 3
(3) R L leakage rate - Carriers may escape from the important region of the device e.g. in a quantum well electrons may be thermally excited into the barrier R L Dn (4) R st stimulated emission rate - Photon production only in lasers Can define a radiative efficiency η r as the fraction of created electrons and holes which produce a photon. Assuming R st 0 this is just the ratio of R sp to total recombination rate: Rsp i.e. ηr R + R + R sp nr L For an efficient light producing device a large η r is required, hence R sp >>R nr, R L It is also useful to define a carrier lifetime τ R rec n τ dn dt G gen n τ
Therefore 1 τ R n rec 1 n ( R + R + R ) sp nr L And substituting in 1 n 3 ( Bn + An + Cn + Dn) Which gives 1 τ Bn + A + D + Cn 3 Hence, τ is not constant but varies with n
Lecture 1 - Conclusions Review of basic semiconductor properties Valence and conduction bands Band gap Effective mass Direct and indirect band gaps Doping Carrier drift/diffusion Real Semiconductors Alloy semiconductors Electron and hole recombination mechanisms