3.225 1 Electronic Materials Silicon Age: • Communications • Computation • Automation • Defense • ……… Factors: • Reproducibility/Reliability • Miniaturization • Functionality • Cost • ………… © H.L. Tuller-2001 Pervasive technology 3.225 2 What Features Distinguish Different Conductors? • Magnitude: agnitude! • metal; semiconductor; insulator • Carrier type: • electrons vs ions; • negative vs positive • Mechanism: • wave-like • activated hopping • Field Dependence: • Linear vs non-linear © H.L. Tuller-2001 varies by over 25 orders of m 1 3.225 3 How Do We Arrive at Properties That We Want? • Crystal Structure: • diamond vs graphite • Composition • silicon vs germanium • Doping • n-Si:P vs p-Si:B • Microstructure • single vs polycrystalline • Processing/Annealing Conditions •Ga 1+x As vs Ga 1-x As © H.L. Tuller-2001 3.225 4 • Interconnect • Resistor • Insulator • Non-ohmic device – diode, transistor • Thermistor • Piezoresistor • Chemoresistor • Photoconductor • Magnetoresistor What is the Application? © H.L. Tuller-2001 2 3.225 5 Origin of Conduction Range of Resistivity Why? © E.A. Fitzgerald-1999 3.225 6 Response of Material to Applied Potential I V e- V I Linear, Ohmic Rectification, Non-linear, Non-Ohmic V=IR V=f(I) Metals show Ohmic behavior microscopic origin? © E.A. Fitzgerald-1999 R 3 3.225 7 Microscopic Origin: Can we Predict Conductivity of Metals? • Drude model: Sea of electrons – all electrons are bound to ion atom cores except valence electrons – ignore cores – electron gas © E.A. Fitzgerald-1999 Schematic model of a crystal of sodium metal. From: Kittel, Introduction to Solid State Physics, 3rd Ed., Wiley (1967) p. 198. C. 3.225 8 Does this Microscopic Picture of Metals Give us Ohm’s Law? F=-eE E F=ma m(dv/dt)=-eE v =-(eE/m)t v,J,σ,I t t E No, Ohm’s law can not be only from electric force on electron! Constant E gives ever-increasing v © E.A. Fitzgerald-1999 4 3.225 9 Equation of Motion - Impact of Collisions Assume: • probability of collision in time dt = dt/τ • time varying field F(t) v(t+dt) = (1- dt/τ) {v(t) +dv} = (1- dt/τ) {v(t) + (F(t)dt)/m} ≈ v(t) + (F(t)dt)/m - v(t) dt/τ (for small dt) ⇒ dv(t)/dt + v(t)/τ = F(t)/m Note: erm proportional to velocity corresponds to frictional damping term © H.L. Tuller-2001 T 3.225 10 Hydrodynamic Representation of e- Motion dp t dt pt Ft F t () () () () =− + + + τ 1 Response (ma) p=momentum=mv Drag Driving Force Restoring Force dp t dt pt eE () () ≈− − τ Add a drag term, i.e. the electrons have many collisions during drift 1/τ represents a ‘viscosity’ in mechanical terms © E.A. Fitzgerald-1999 2 5 3.225 11 In steady state, dp t dt () = 0 pt p e t () ( ) = ∞ − 1 τ p E ∞ =− τ p t -eEτ τ If the environment has a lot of collisions, mv avg =-eEτ v avg =-eEτ/m µ τ = e m © E.A. Fitzgerald-1999 E µ−= Define v Mean-free Time Between Collisions, Electron Mobility − e 3.225 12 v d E j = I/A A dx What is the Current Density ? n (#/vol) © H.L. Tuller-2001 • # electrons crossing plane in time dt = n(dxA) = n(v d dtA) • # charges crossing plane per unit time and area = j •Ohm’s Law: Dimensional analysis: (A/cm 2 )/(V/cm)=A/(V-cm)= (ohm-cm) -1 = Siemens/cm-(S/cm) ( ) () ( EmnevnedtAedtAvnj d d τ 2 =−=−= ( EjmneEj ==⇒= τσσ 2 ) ) 6 3.225 13 Energy Dissipation - Joule Heating Frictional damping term leads to energy losses: • Power absorbed by particle from force F: P = W/t = (F•d)/t = F•v • Electron gas: P/vol= n(-eE)•(-eτE/m) = ne 2 τE 2 /m = σ E 2 = jE = (I/A)(V/l) = IV/vol • Total power absorbed: 2 /R = I 2 R How much current does a 100 W bulb draw? I = 100W/115V = 0.87A © H.L. Tuller-2001 P = IV = V 3.225 14 Predicting Conductivity using Drude n theory from the periodic table (# valence e- and the crystal structure) n theory =A V Zρ m /A, where AV is 6.023x10 23 atoms/mole ρ m is the density Z is the number of electrons per atom A is the atomic weight For metals, n theory ~10 22 cm -3 If we assume that this is correct, we can extract τ © E.A. Fitzgerald-1999 7 3.225 15 • τ~10 -14 sec for metals in Drude model Extracting Typical τ for Metals © E.A. Fitzgerald-1999 3.225 16 Thermal Velocity • So far we have discussed drift velocity v D and scattering time τ related to the applied electric field • Thermal velocity v th is much greater than v D kTmv th 2 3 2 1 2 = m kT v th 3 = Thermal velocity is much greater than drift velocity x x x L=v D τ © E.A. Fitzgerald-1999 8 3.225 17 Resistivity/Conductivity Pessimist vs Optimist L W I V t R = ρ L/Wt = ρ L/A ⇒ρ(οhm-cm) σ = 1/ρ ⇒ σ (οhm-cm) -1 ⇒σ (Siemens/cm) (Test your dimensions: σ=E/j=neµ) Ohms/square ⇒ Note, if L=W, then R= ρ /t independent of magnitude of L and W. Useful for working with films of thickness, t. R R R © H.L. Tuller-2001 R=V/I; 3.225 18 How to Make Resistance Measurements R s R c1 R c2 I V V/I = R c1 + R s + R c2 I. s >> R c1 + R c2 ; no problem II. For R s ≤ R c1 + R c2 ; major problem ⇒ 4 probes © H.L. Tuller-2001 For R 9 3.225 19 How to Make Resistance Measurements R s R c4 R c1 I V 14 v 23 R c2 R c3 4 probe method: Essential feature - use of high impedance voltmeter to measure V 23 ⇒ no current flows through R c2 & R c3 ⇒ therefore no IR contribution to V 23 R s (2-3) = v 23 /I = σ -1 (d 23 /A) = ρ (d 23 /A) (Note: ρ-resistivity is inverse of σ−conductivity) © H.L. Tuller-2001 3.225 20 How to Make Resistance Measurements - Wafers I V d d R R+dR x j = I/2πR 2 ; V = IR = Iρd/A = jρd V 23 = ⌠ 2d (I/2πR 2 ) ρ dR = (- Iρ/ 2πR) 2d = Iρ/4πd ⌡ d d ρ = (2πd/I) V 23 ; ρ = (π/ln2) V/I for d >> x Si © H.L. Tuller-2001 I d 10 . Total power absorbed: 2 /R = I 2 R How much current does a 10 0 W bulb draw? I = 10 0W /11 5V = 0.87A © H.L. Tuller-20 01 P = IV = V 3.225 14 Predicting Conductivity using Drude n theory from. can extract τ © E.A. Fitzgerald -19 99 7 3.225 15 • τ ~10 -14 sec for metals in Drude model Extracting Typical τ for Metals © E.A. Fitzgerald -19 99 3.225 16 Thermal Velocity • So far we. Tuller-20 01 R=V/I; 3.225 18 How to Make Resistance Measurements R s R c1 R c2 I V V/I = R c1 + R s + R c2 I. s >> R c1 + R c2 ; no problem II. For R s ≤ R c1 + R c2 ;