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Lecture 3 Electron and Hole Transport in Semiconductors Review Lecture 3 Electron and Hole Transport in Semiconductors In this lecture you will learn: • How electrons and holes move in semiconductors • Thermal motion of electrons and holes • Electric current via drift • Electric current via diffusion • Semiconductor resistors ECE 315 – Spring 2005 – Farhan Rana – Cornell University Review: Electrons and Holes in Semiconductors holes + electrons + + As Donor A Silicon crystal lattice + impurity + As atoms + ● There are two types of mobile charges in semiconductors: electrons and holes In an intrinsic (or undoped) semiconductor electron density equals hole density ● Semiconductors can be doped in two ways: N-doping: to increase the electron density P-doping: to increase the hole density ECE 315 – Spring 2005 – Farhan Rana – Cornell University 1 Thermal Motion of Electrons and Holes In thermal equilibrium carriers (i.e. electrons or holes) are not standing still but are moving around in the crystal lattice because of their thermal energy The root-mean-square velocity of electrons can be found by equating their kinetic energy to the thermal energy: 1 1 m v 2 KT 2 n th 2 KT vth mn In pure Silicon at room temperature: 7 vth ~ 10 cm s Brownian Motion ECE 315 – Spring 2005 – Farhan Rana – Cornell University Thermal Motion of Electrons and Holes In thermal equilibrium carriers (i.e. electrons or holes) are not standing still but are moving around in the crystal lattice and undergoing collisions with: • vibrating Silicon atoms • with other electrons and holes • with dopant atoms (donors or acceptors) and other impurity atoms Mean time between collisions = c In between two successive collisions electrons (or holes) move with an average velocity which is called the thermal velocity = vth 12 In pure Silicon, c 0.110 s 0.1 ps 7 vth 10 cm s Brownian Motion Mean distance traveled between collisions is called the mean free path vth c In pure Silicon, 107 0.11012 10-6 cm 0.01m ECE 315 – Spring 2005 – Farhan Rana – Cornell University 2 Drift: Motion of Electrons Under an Applied Electric Field Silicon slab E L V E L +-V • Force on an electron because of the electric field = Fn = -qE • The electron moves in the direction opposite to the applied field with a constant drift velocity equal to vdn • The electron drift velocity vdn is proportional to the electric field strength vdn E vdn n E cm2 • The constant n is called the electron mobility. It has units: cm2 V - s • In pure Silicon, 1500 n V - s ECE 315 – Spring 2005 – Farhan Rana – Cornell University Drift: Motion of Holes Under an Applied Electric Field Silicon slab E L V E L +-V • Force on a hole because of the electric field = Fp = qE • The hole moves in the direction of the applied field with a constant drift velocity equal to vdp • The hole drift velocity vdp is proportional to the electric field strength vdp E vdp p E cm2 • The constant p is called the hole mobility. It has units: cm2 V - s • In pure Silicon, 500 p V - s ECE 315 – Spring 2005 – Farhan Rana – Cornell University 3 Derivation of Expressions for Mobility Electrons: Force on an electron because of the electric field Fn qE F q E Acceleration of the electron a n mn mn Since the mean time between collisions is c , the acceleration lasts only for a time period of c before a collision completely destroys electron’s velocity q c So in time c electron’s velocity reaches a value a c E mn q c This is the average drift velocity of the electron, i.e. vdn E mn q c Comparing with v dn n E we get, n mn Holes: q c Similarly for holes one gets, p mp Special note: Masses of electrons and holes (mn and mp) in Solids are not the same as the mass of electrons in free space which equals 9.11031 kg ECE 315 – Spring 2005 – Farhan Rana – Cornell University Mobility Vs Doping More doping (n-type of p-type) means more frequent collisions with charged donor and acceptor impurity atoms and this lowers the carrier mobility /V-s) 2 Mobility (cm Mobility Dopant concentration (1/cm3) Note: Doping in the above figure can either be n-type or p-type ECE 315 – Spring 2005 – Farhan Rana – Cornell University 4 Drift Current Density of Electrons Consider electrons moving under an applied electric field: E vdn n E Flux Density: Flux density is the number of particles crossing a unit area surface per second It has units cm-2-s-1 Unit area surface Density: n Velocity: vdn Area Time Flux density: nvdn Volume = 1 x (vdn x 1) ECE 315 – Spring 2005 – Farhan Rana – Cornell University Drift Current Density of Electrons Electrons Drift Current Density: Check directions Electron flux density from drift n v dn E vdn Electron drift current density drift is, Jn drift Jn drift vdn n E Jn q electron flux density drift q n vdn q n n E Jn q n n E drift Coulombs Amps Jn has units: cm2 - s cm2 ECE 315 – Spring 2005 – Farhan Rana – Cornell University 5 Drift Current Density of Holes Holes Drift Current Density: Check directions The hole drift current density is J drift , p E drift v Jqp hole flux density dp drift qpvdp qp p E Jp vdp p E drift Jp q p p E drift Coulombs Amps Jp has units: cm2 - s cm2 ECE 315 – Spring 2005 – Farhan Rana – Cornell University Conductivity and Resistivity Total Drift Current Density: drift drift drift The total drift current density J is the sum of J n and Jp drift drift drift J Jn Jp q n n p p E E The quantity is the conductivity of the semiconductor: q n n p p Conductivity describes how much current flows when an electric field is applied. Another related quantity is the resistivity which is the inverse of the conductivity, 1 Units of conductivity are: Ohm-1-cm-1 or -1-cm-1 or S-cm-1 Units of resistivity are: Ohm-cm or -cm or S-1-cm ECE 315 – Spring 2005 – Farhan Rana – Cornell University 6 Example: A Semiconductor Resistor Silicon slab For a resistor we know that, V I 1 R L - V We also know that, + Area A I Jdrift A E A I V A A V 2 L L L L 1 1 2 R where q n n p p A A Lessons: • Knowing electron and hole densities and mobilities, one can calculate the electrical resistance of semiconductors • n-doping or p-doping can be used to change the conductivity of semiconductors by orders of magnitudes ECE 315 – Spring 2005 – Farhan Rana – Cornell University Diffusion Diffusion of ink in a glass beaker Why does diffusion happen? ECE 315 – Spring 2005 – Farhan Rana – Cornell University 7 Diffusion and Diffusivity There is another mechanism by which current flows in semiconductors ……. • Suppose the electron density inside a semiconductor is not uniform in space, as shown below n(x) electron flux in +x direction electron flux in -x direction d nx slope dx x • Since the electrons move about randomly in all directions (Brownian motion), as time goes on more electrons will move from regions of higher electron density to regions of lower electron density than the electrons that move from regions of lower electron density to regions of higher electron density d nx • Net electron flux density in +x direction dx d nx D n dx 2 -1 • The constant Dn is called the diffusivity of electrons (units: cm -s ) ECE 315 – Spring 2005 – Farhan Rana – Cornell University Diffusion Current Density Electrons Diffusion Current Density: d nx Check directions Electron flux density from diffusion Dn dx nx diff Electron diffusion current density J n is, diff x Jn q electron flux density d nx Elec. flux q Dn dx diff Jn Holes Diffusion Current Density: d px Check directions Hole flux density from diffusion Dp dx px diff Hole diffusion current density J p is, diff x Jp q hole flux d px Hole flux q Dp dx diff Jp diff diff Coulombs Amps Jn and J p has units cm2 - s cm2 ECE 315 – Spring 2005 – Farhan Rana – Cornell University 8 Einstein Relations Einstein worked on other things besides the theory of relativity…….. • We introduced two material constants related to carrier transport: 1) Mobility 2) Diffusivity •Both are connected with the transport of carriers (electrons or holes) •It turns out that their values are related by the Einstein relationships Einstein Relation for Electrons: D K T Example: n 2 In pure Silicon, n 1500 cm V - s n q 500 cm2 V - s Einstein Relation for Holes: p 2 Dp K T This implies, Dn 37.5 cm s D 12.5 cm2 s p q p • K is the Boltzman constant and its value is: 1.38x10-23 Joules K • KT has a value equal to 0.0258 Volts at room temperature (at 300 oK) q ECE 315 – Spring 2005 – Farhan Rana – Cornell University Total Electron and Hole Current Densities Total electron and hole current densities is the sum of drift and diffusive components Electrons: drift diff Jn x Jn x Jn x d nx q nx E x q D n n dx Holes: drift diff Jp x Jp x Jp x d px q px E x q D p p dx Electric currents are driven by electric fields and also by carrier density gradients ECE 315 – Spring 2005 – Farhan Rana – Cornell University 9 Thermal Equilibrium - I There cannot be any net electron current or net hole current in thermal equilibrium ……… what does this imply ?? Consider electrons first: drift diff Jn x Jn x Jn x 0 d n x q n x E x q D o 0 1 o n n dx d logn x q 1 can also be written as: o Ex dx KT dx Since the electric field is minus the gradient of the potential: Ex dx d logn x q dx We have: o dx KT dx qx KT The solution of the above differential equation is: no x constant e But what is that “constant” in the above equation ??? ECE 315 – Spring 2005 – Farhan Rana – Cornell University Thermal Equilibrium - II qx KT We have: no x constant e Note: one can only measure potential differences and not the absolute values of potentials Convention: The potential of pure intrinsic Silicon is used as the reference value and assumed to be equal to zero.
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