Lecture 3 Semiconductor Physics (II) Carrier Transport Outline • Thermal Motion •Carrier Drift •Carrier Diffusion Reading Assignment: Howe and Sodini; Chapter 2, Sect. 2.4-2.6 6.012 Spring 2007 Lecture 3 1 1. Thermal Motion In thermal equilibrium, carriers are not sitting still: • Undergo collisions with vibrating Si atoms (Brownian motion) • Electrostatically interact with each other and with ionized (charged) dopants Characteristic time constant of thermal motion: ⇒ mean free time between collisions τc ≡ collison time [s] In between collisions, carriers acquire high velocity: −1 vth ≡ thermal velocity [cms ] …. but get nowhere! 6.012 Spring 2007 Lecture 3 2 Characteristic length of thermal motion: λ≡mean free path [cm] λ = vthτc Put numbers for Si at room temperature: −13 τc ≈ 10 s 7 −1 vth ≈ 10 cms ⇒ λ ≈ 0.01 µm For reference, state-of-the-art production MOSFET: Lg ≈ 0.1 µm ⇒ Carriers undergo many collisions as they travel through devices 6.012 Spring 2007 Lecture 3 3 2. Carrier Drift Apply electric field to semiconductor: E ≡ electric field [V cm-1] ⇒ net force on carrier F = ±qE E Between collisions, carriers accelerate in the direction of the electrostatic field: qE v(t) = a •t =± t mn,p 6.012 Spring 2007 Lecture 3 4 But there is (on the average) a collision every τc and the velocity is randomized: net velocity τc in direction of field time The average net velocity in direction of the field: qE qτc v = vd =± τ c =± E 2mn,p 2mn,p This is called drift velocity [cm s-1] Define: qτ c 2 −1 −1 µn,p = ≡ mobility[cm V s ] 2mn,p Then, for electrons: vdn = −µnE and for holes: vdp = µpE 6.012 Spring 2007 Lecture 3 5 Mobility - is a measure of ease of carrier drift • If τc ↑, longer time between collisions ⇒ µ ↑ • If m ↓, “lighter” particle ⇒ µ ↑ At room temperature, mobility in Si depends on doping: 1400 1200 electrons 1000 Vs) / 2 800 600 mobility (cm holes 400 200 0 1013 1014 1015 1016 1017 1018 1019 1020 + −3 Nd Na total dopant concentration (cm ) • For low doping level, µ is limited by collisions with lattice. As Temp ->INCREASES; µ-> DECREASES • For medium doping and high doping level, µ limited by collisions with ionized impurities • Holes “ heavier” than electrons – For same doping level, µn > µp 6.012 Spring 2007 Lecture 3 6 Drift Current Net velocity of charged particles ⇒ electric current: Drift current density ∝ carrier drift velocity ∝ carrier concentration ∝ carrier charge Drift current densities: drift Jn =−qnvdn = qnµ nE drift J p = qpvdp = qpµ p E Check signs: E E vdn vdp - + drift drift Jn Jp x x 6.012 Spring 2007 Lecture 3 7 Total Drift Current Density : drift drift drift J = Jn + Jp = qn( µ n + pµ p )E Has the form of Ohm’s Law E J = σ E = ρ Where: σ ≡ conductivity [Ω-1 • cm-1] ρ≡resistivity [Ω • cm] Then: 1 σ = = qnµ + pµ ρ ( n p ) 6.012 Spring 2007 Lecture 3 8 Resistivity is commonly used to specify the doping level • In n-type semiconductor: 1 ρn ≈ qNdµn • In p-type semiconductor: 1 ρp ≈ qNaµ p 1E+4 1E+3 1E+2 1E+1 p-Si 1E+0 n-Si 1E-1 Resistivity (ohm.cm) 1E-2 1E-3 1E-4 1E+12 1E+13 1E+14 1E+15 1E+16 1E+17 1E+18 1E+19 1E+20 1E+21 Doping (cm-3) 6.012 Spring 2007 Lecture 3 9 Numerical Example: 16 -3 Si with Nd = 3 x 10 cm at room temperature 2 µn ≈ 1000 cm / V • s ρn ≈ 0.21Ω•cm n ≈ 3X1016 cm−3 Apply E = 1 kV/cm 6 vdn ≈10 cm / s << vth E J drift ≈ qnv = qnµ E = σ E = n dn n ρ drift 3 2 J n ≈ 4.8×10 A/ cm Time to drift through L = 0.1 µm L td = = 10 ps vdn fast! 6.012 Spring 2007 Lecture 3 10 3. Carrier Diffusion Diffusion = particle movement (flux) in response to concentration gradient n x Elements of diffusion: • A medium (Si Crystal) • A gradient of particles (electrons and holes) inside the medium • Collisions between particles and medium send particles off in random directions – Overall result is to erase gradient 6.012 Spring 2007 Lecture 3 11 Fick’s first law- Key diffusion relationship Diffusion flux ∝ - concentration gradient Flux ≡ number of particles crossing a unit area per unit time [cm-2 • s-1] For Electrons: dn F =−D n n dx For Holes: dp F =−D p p dx 2 -1 Dn ≡ electron diffusion coefficient [cm s ] 2 -1 Dp ≡ hole diffusion coefficient [cm s ] D measures the ease of carrier diffusion in response to a concentration gradient: D ↑⇒Fdiff ↑ D limited by vibration of lattice atoms and ionized dopants. 6.012 Spring 2007 Lecture 3 12 Diffusion Current Diffusion current density =charge × carrier flux diff dn Jn = qDn dx dp J diff =−qD p p dx Check signs: n p Fn Fp J Jndiff pdiff x x 6.012 Spring 2007 Lecture 3 13 Einstein relation At the core of drift and diffusion is same physics: collisions among particles and medium atoms ⇒ there should be a relationship between D and µ Einstein relation [will not derive in 6.012] D kT = µ q In semiconductors: D kT D n = = p µn q µ p kT/q ≡ thermal voltage At room temperature: kT ≈ 25mV q 16 -3 For example: for Nd = 3 x 10 cm 2 2 µn ≈ 1000 cm / V • s ⇒ Dn ≈ 25cm / s 2 2 µ p ≈ 400 cm / V • s ⇒ Dp ≈ 10 cm / s 6.012 Spring 2007 Lecture 3 14 Total Current Density In general, total current can flow by drift and diffusion separately. Total current density: dn J = Jdrift + Jdiff = qnµ E + qD n n n n n dx dp J = Jdrift + Jdiff = qpµ E − qD p p p p p dx Jtotal = Jn + Jp 6.012 Spring 2007 Lecture 3 15 What did we learn today? Summary of Key Concepts • Electrons and holes in semiconductors are mobile and charged – ⇒ Carriers of electrical current! • Drift current: produced by electric field dφ Jdrift ∝ EJdrift ∝ dx • Diffusion current: produced by concentration gradient dn dp Jdiffusion ∝ , dx dx • Diffusion and drift currents are sizeable in modern devices • Carriers move fast in response to fields and gradients 6.012 Spring 2007 Lecture 3 16.