ECE 440 Lecture 12 : Diffusion of Carriers Class Outline: Band Bending Diffusion Processes Diffusion and Drift of Carriers
Things you should know when you leave Key Questions How do I calculate kinetic and potential energy from the bands? How does diffusion work in semiconductors? What is the diffusion current? What is the relationship between diffusion and mobility?
Band Bending The energy bands are not constant in fields! Total Electron Energy T.E. K.E. + P.E. P.E. Instead, they move as a function of position. E ref We need energy equal to the band gap to break bonds and ecite carriers to the conduction band. If we only impart enough energy to promote an electron then it simply sits in the conduction band. Etra energy allows carriers to move.
Band Bending We can use elementary physics to determine the potential energy Electron Kinetic Energy Total Electron Energy T.E. K.E. + P.E. Potential Energy P.E. Hole Kinetic Energy The potential energy: qv E ref But previously we said: E c E ref V Electrostatic potential M. J. Gilbert ECE 440 Lecture 1 2 9/21 09
Band Bending But we can still determine more Total Electron Energy T.E. K.E. + P.E. Electron Kinetic Energy Potential Energy P.E. Hole Kinetic Energy Electrostatic potential By definition, we know: E ref V E 0
Band Bending Let s try an eample E c 2 3 Consider the following band diagram: Assume that: E i E f 1 It is silicon maintained at 300 K. E f E i E g /4 at ± L and E f E i E g /4 at 0. Choose the Fermi level as the reference energy. E v 1 2 3 -L 0 L Let s sketch the electrostatic potential inside the semiconductor: V -L 0 L
Band Bending There is still more information that we can determine We can find the electric field E V 1 de q d 1 dev q d c 1 dei q d E What is the resistivity in the > L portion of the semiconductor? -L 0 L We know that : E f E i E g /4 0.28 ev From the bands, we know that in this region it is p- type.
Band Bending Finally, let s determine the kinetic and potential energies at the different points 2 3 For electrons: Total energy increases as we move up in a band. KE E E c. PE E c E ref E c E f. For holes: Total energy increases as we move up in a band. 2 3 KE Ev E. PE E ref E v E f E v. Carrier Electron 1 KE (ev) PE (ev) Electron 2 Electron 3 Hole 1 Hole 2 Hole 3 E c E i E f E v 1 1
Diffusion Processes What happens when we have a concentration discontinuity?? Consider a situation where we spray perfume in the corner of a room If there is no convection or motion of air, then the scent spreads by diffusion. This is due to the random motion of particles. Particles move randomly until they collide with an air molecule which changes it s direction. If the motion is truly random, then a particle sitting in some volume has equal probabilities of moving into or out of the volume at some time interval. T 0 Shouldn t the same thing happen in a semiconductor if we have spatial gradients of carriers? T 1 0 T 2 0 T 3 0
Diffusion Processes Let s shine light on a localized part of a semiconductor Now let s monitor the system Assume thermal motion. Carriers move by interacting with the lattice or impurities. Thermal motion causes particles to jump to an adjacent compartment. After the mean-free time (τ c ), half of particles will leave and half will remain a certain volume. t 0 t τ 1024 c t 2τ c t 6τ c c t 3τ Process continues until uniform concentration. We must have a concentration gradient for diffusion to start.
Diffusion Processes How do we describe this physical process?? We want to calculate the rate at which electrons diffuse in a simple onedimensional eample. Consider an arbitrary electron distribution λ λ λ Divide the distribution into incremental distances of the mean-free path (λ). Evaluate n() in the center of the segments. Electrons on the left of 0 have a 50% chance of moving left or right in a time, τ c. Same is true for electrons to the right of 0. Net # of electrons moving from left to right in one τ c.
Diffusion Processes So we have a flu of particles The rate of electron flow in the + direction (per unit area): λ φn 2 τ c ( n n ) 1 2 Since the mean-free path is a small differential length, we can write the electron difference as: n ( ) ( ) In the limit of small Δ, or small n n + 1 n 2 λ φ n mean-free path between collisions Diffusion coefficient (cm 2 /sec) λ λ
Diffusion Processes But we already epected this Define the carrier flu for electrons and holes: And the corresponding current densities associated with diffusion Carriers move together, currents opposite directions.
Diffusion and Drift of Carriers How do we handle a concentration gradient and an electric field? e- h+ E n() p() The total current must be the sum of the electron and hole currents resulting from the drift and diffusion processes Drift Diffusion Where are the particles and currents flowing? φ p (diff and drift) φ n (diff) ( ) J n J p J + Electrons Holes φ n (drift) J p (diff and drift) J p (diff) J n (drift) Dashed Arrows Particle Flow Solid Arrows Resulting Currents
Diffusion and Drift of Carriers A few etra observations φ p (diff and drift) φ n (diff) φ n (drift) J p (diff and drift) J p (diff) J n (drift) Dashed Arrows Particle Flow Solid Arrows Resulting Currents Diffusion currents are in opposite directions. Drift currents are in the same direction. Currents depend on: Relative electron and hole concentrations. Magnitude and directions of electric field. Carrier gradients. J J n n ( ) q n( ) E( ) ( ) qµ p( ) E( ) p qd p ( ) dn d dp µ n + qd Diffusion currents can be large even n if the carriers are in the minority by several orders of magnitude. Not true for drift currents. ( ) d
Diffusion and Drift of Carriers Can we relate the diffusion coefficient to the mobility? We can by using what we know about drift, diffusion, and band bending In equilibrium, no current flows. Any fluctuation that would begin a diffusion current also sets up an electric field which redistributes the carriers by drift. Solve for the electric field E(): It s equilibrium, so we know n(): n( ) n e i ( E E ) F k b T i Assuming E is non-zero
Diffusion and Drift of Carriers These relations are called the Einstein relations The balance of drift and diffusion currents creates a built-in electric field to accompany any gradient in the bands. Gradients in the bands can occur at equilibrium when: the band gap varies. alloy concentration varies. dopant concentrations vary. E V 1 de q d 1 dev q d c 1 q de d i