Lecture 3 Semiconductor Physics (II) Carrier Transport Thermal Motion Carrier Drift Carrier Diffusion Outline Reading Assignment: Howe and Sodini; Chapter 2, Sect. 2.4-2.6 6.012 Spring 2009 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 τ c collison time [s] In between collisions, carriers acquire high velocity: v th thermal velocity [cms 1 ]. but get nowhere! 6.012 Spring 2009 Lecture 3 2
Characteristic length of thermal motion: λ mean free path [cm] λ = v th τ c Put numbers for Si at room temperature: τ c 10 13 s v th 10 7 cms 1 λ 0.01 µm For reference, state-of-the-art production MOSFET: L g 0.1 µm Carriers undergo many collisions as they travel through devices 6.012 Spring 2009 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: v(t) = a t = ± qe m n,p t 6.012 Spring 2009 Lecture 3 4
But there is (on the average) a collision every τ c and the velocity is randomized: net velocity in direction of field τ c The average net velocity in direction of the field: v = v d = ± qe 2m n,p τ c = ± qτ c 2m n,p E time This is called drift velocity [cm s -1 ] Define: µ n, p = qτ c 2m n,p mobility[cm 2 V 1 s 1 ] Then, for electrons: and for holes: v dn = µ n E v dp = µ p E 6.012 Spring 2009 Lecture 3 5
Mobility - is a measure of ease of carrier drift If τ c, longer time between collisions µ If m, lighter particle µ At room temperature, mobility in Si depends on doping: 1400 1200 mobility (cm 2 /Vs) 1000 800 600 400 holes electrons 200 0 10 13 10 14 10 15 10 16 10 17 10 18 10 19 10 20 N d + N a total dopant concentration (cm 3 ) 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, µ n > µ p 6.012 Spring 2009 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: J n drift = qnvdn = qnµ n E drift J p = qpvdp = qpµ p E Check signs: E E v dn - + v dp J n drift J p drift x x 6.012 Spring 2009 Lecture 3 7
Total Drift Current Density : drift drift drift J = J n + J p = q( nµ 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: σ = 1 = q( nµ n + pµ p ) ρ 6.012 Spring 2009 Lecture 3 8
Resistivity is commonly used to specify the doping level In n-type semiconductor: In p-type semiconductor: ρ n ρ p 1 qn d µ n 1 qn a µ p 1E+4 1E+3 Resistivity (ohm.cm) 1E+2 1E+1 1E+0 1E-1 1E-2 n-si p-si 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 2009 Lecture 3 9
Numerical Example: Si with N d = 3 x 10 16 cm -3 at room temperature µ n 1000 cm 2 / V s ρ n 0.21Ω cm n 3X10 16 cm 3 Apply E = 1 kv/cm v dn 10 6 cm / s << v th J n drift qnvdn = qnµ n E = σ E = E ρ J n drift 4.8 10 3 A / cm 2 Time to drift through L = 0.1 µm fast! t d = L v dn = 10 ps 6.012 Spring 2009 Lecture 3 10
3. Carrier Diffusion Diffusion = particle movement (flux) in response to concentration gradient n Elements of diffusion: x 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 2009 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: For Holes: dn F n = D n dx dp F p = D p dx D n 2 s -1 ] D p hole diffusion coefficient [cm 2 s -1 ] D measures the ease of carrier diffusion in response to a concentration gradient: D F diff D limited by vibration of lattice atoms and ionized dopants. 6.012 Spring 2009 Lecture 3 12
Diffusion Current Diffusion current density =charge carrier flux J diff dn n = qdn dx J diff dp p = qdp dx Check signs: n F n p Fp J ndiff J pdiff x x 6.012 Spring 2009 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 µ = kt q In semiconductors: D n µ n = kt q = D p µ p At room temperature: kt/q thermal voltage kt q 25mV For example: for N d = 3 x 10 16 cm -3 µ n 1000 cm 2 / V s D n 25cm 2 / s µ p 400 cm 2 / V s D p 10 cm 2 / s 6.012 Spring 2009 Lecture 3 14
Total Current Density In general, total current can flow by drift and diffusion separately. Total current density: J n = J drift n J p = J drift p + J diff n + J diff p dn = qnµ n E + qd n dx dp = qpµ p E qd p dx J total = J n + J p 6.012 Spring 2009 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 drift drift dφ J E J dx Diffusion current: produced by concentration gradient diffusion dn dp J, dx dx Diffusion and drift currents are sizeable in modern devices Carriers move fast in response to fields and gradients 6.012 Spring 2009 Lecture 3 16
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