Haynes-Shockley Experiment J.R. Haynes and W. Shockley, Bell Telephone Laboratories, “The Mobility and Life of Injected Holes and Electrons in Germanium”, Physical Review, 81, 835 (1951). n-type semiconductor (n0 >>p0 ) • Shine short pulse of laser light on sample • Create localized pulse of excess minority carriers (holes) Three effects: -pulse spreads out in time (diffusion) -pulse travels with drift velocity vD=µE -area of pulse decreases with time because of recombination Phys 365 2002-1: Non equilibrium effects 130 Case (a): No external E field, no recombination • Diffusion only (no drift) • After light pulse generation rate gp is zero. • rp=0 ∂δp ( x, t ) 1 ∂J p =− ∂t q ∂x Continuity equation: (assume minority carrier lifetime is long compared to observation time) ∂δp Assuming only diffusion current: J p = − qD p ∂x ∂δp ( x , t ) ∂ 2δp( x , t ) = Dp ∂t ∂x 2 Solution: δp( x, t ) = 1 4πD p t e  x2  −   4 D pt  • Hole pulse width increases with t Width of the Gaussian pulse: ∆x = 4 D p t Correction in presence of recombination: δp( x, t ) = 1 4πD p t e  x2  −   4 D p t  e −t /τ p Case (b) Include drift (no recombination) • Center of Gaussian pulse moves with vd= µpE • Observe pulse as it passes a fixed contact point • If hole pulse width is ∆x, this is observed as a pulse of width ∆t ~ ∆x / vd on the oscilloscope screen. • Use the relation ∆x = 4 D p t to determine Dp. Phys 365 2002-1: Non equilibrium effects 131 Summary of Haynes-Shockley Experiment: • Diffusivity Dp which is related to the width of the pulse as a function of time • Drift velocity is given by the delay between the launch of the pulse and its detection at some distance along the sample. • Minority carrier lifetime is given by the decay of the integral of the pulse shape Phys 365 2002-1: Non equilibrium effects 132 Junctions (Mostly p-n junctions) Equilibrium Properties • Concentration gradient of electrons leads to net diffusion current of electrons to the p-type side. • Similarly, net flow of holes toward the n-type side. • Near the junction, we get electron-hole recombination, which would result in "depletion" of holes from p-type side and "depletion" of electrons from n-type side. • n-doped region near the junction becomes positively charged due to depletion of electrons • p-doped region becomes negatively charged due to depletion of holes. Built in field: As the electrons and holes move across the junction, the exposed charges that they leave behind set up a built-in electric field which opposes further charge separation. Gauss’ law: In one dimension: Electric potential: ∇ ⋅E = ρ ε rε0 dEx ρ = dx ε r ε 0 Ex = − Phys 365 2002-1: Non equilibrium effects ∂V ∂x 133 ∂ 2V ρ = − Poisson’s equation: εr ε0 ∂x 2 Consider the solution of Gauss’s law for two finite charged slabs of opposite charge Energy bands: Note: electrostatic potential is repulsive to holes on the left and repulsive to electrons on the right. Phys 365 2002-1: Non equilibrium effects 134 Location of Fermi level for electrostatic and thermal equilibrium • In a system in electrostatic and thermal equilibrium, the Fermi level remains at a constant value independent of x. qV Calculation of the Built-in Potential V0 In neutral p-type region: p = ni e ( Ei ( p )− E F ) / kT and p=Na  NA    E ( p ) − E = kT ln i F Therefore:  ni  In neutral n-type region: Phys 365 2002-1: Non equilibrium effects 135 n = ni e ( E F − Ei ( n )) / kT Therefore: and n=Nd N EF − Ei ( n) = kT ln D  ni     By inspection of above figure: qV0= Ei(p)-Ei(n) Therefore:  N  N qV 0 = E i ( p ) − E i ( n ) = kT  ln  A  + ln  D  ni   ni  Therefore: V0 = kT  N A N D ln q  ni 2         Built-in voltage Example: For a Si p-n junction with NA=1018 cm-3 and ND=1015 cm-3 we have kT  N A N D V0 = ln q  ni 2   (1018 )(10 15  = (0.0259 ) ln   (1.45 × 10 10 ) 2      = 0.755V  Higher doping levels => higher barrier height For example, for NA=ND = 1×1018 cm-3 we get: V0=0.93. Comparing this with the band gap of Si of EG= 1.1 eV, we can see that V0 approaches EG in the limit of high doping Phys 365 2002-1: Non equilibrium effects 136 Analysis of the Depletion Region: Abrupt Junction Abrupt junction: • Uniform carrier concentrations in the p and n regions respectively • Transition between n and p is perfectly abrupt. Energy bands: Phys 365 2002-1: Non equilibrium effects 137