MTLE-6120: Advanced Electronic Properties of Materials Classical - PowerPoint PPT Presentation

1 MTLE-6120: Advanced Electronic Properties of Materials Classical Drude theory of conduction Contents: ◮ Drude model derivation of free-electron conductivity ◮ Scattering time estimates and Matthiessen’s rule ◮ Mobility and Hall coefficients ◮ Frequency-dependent conductivity of free-electron metals Reading: ◮ Kasap: 2.1 - 2.3, 2.5

2 Ohm’s law ◮ Local Ohm’s law: current density driven by electric field � j = σ � E ◮ Current in a sample of cross section A is I = jA ◮ Voltage drop across a sample of length L is V = EL ◮ Ohm’s law defines resistance R ≡ V I = EL jA = σ − 1 L A ◮ Units: Resistance in Ω , resistivity ρ = σ − 1 in Ωm , conductivity σ in (Ωm) − 1

3 Typical values at 293 K d ρ σ [ (Ωm) − 1 ] ρ d T [K − 1 ] Substance ρ [ Ωm ] 1 . 59 × 10 − 8 6 . 30 × 10 7 Silver 0.0038 1 . 68 × 10 − 8 5 . 96 × 10 7 Copper 0.0039 5 . 6 × 10 − 8 1 . 79 × 10 7 Tungsten 0.0045 2 . 2 × 10 − 7 4 . 55 × 10 6 Lead 0.0039 4 . 2 × 10 − 7 2 . 38 × 10 6 Titanium 0.0038 6 . 9 × 10 − 7 1 . 45 × 10 6 Stainless steel 0.0009 9 . 8 × 10 − 7 1 . 02 × 10 6 Mercury 0.0009 5 − 8 × 10 − 4 1 − 2 × 10 3 Carbon (amorph) -0.0005 4 . 6 × 10 − 1 Germanium 2.17 -0.048 6 . 4 × 10 2 1 . 56 × 10 − 3 Silicon -0.075 1 . 0 × 10 12 1 . 0 × 10 − 12 Diamond 7 . 5 × 10 17 1 . 3 × 10 − 18 Quartz 10 23 − 10 25 10 − 25 − 10 − 23 Teflon Note 1 /T = 0 . 0034 K − 1 at 293 K ⇒ approximately ρ ∝ T for the best conducting metals.

4 Temperature dependence 4 +2% Ni Resistivity [10 -8 m] 3 + 1% Ni 2 Cold worked Pure copper 1 0 0 100 200 300 T emperature [K] ◮ Linear at higher temperatures ◮ Residual resistivity (constant at low T) due to defects and impurities

5 Drude model setup ◮ Fixed nuclei (positive ion cores) + gas of moving electrons ◮ Electrons move freely with random velocities ◮ Electrons periodically scatter which randomizes velocity again ◮ Average time between collisions: mean free time τ ◮ Average distance travelled between collisions: mean free path λ ◮ In zero field, drift velocity (averaged over all electrons) � v d ≡ � � v � = 0 + + + + + but electrons are not stationary: e - e - e - e - � v 2 � = u 2 + + + + + e - e - e - ◮ Current density carried by electrons: e - + + + + + � j = n ( − e ) � v d = 0 e - e - e - e - where n is number density of electrons + + + + +

6 Apply electric field ◮ Electron starts at past time t = − t 0 with random velocity � v 0 ◮ Force on electron is � F = ( − e ) � E ◮ Solve equation of motion till present time t = 0 : m d � v d t = ( − e ) � E v 0 − e � Et 0 � v = � m ◮ Need to average over all electrons ◮ Probability that electron started at − t 0 and did not scatter till t = 0 is P ( t 0 ) ∝ e − t 0 /τ = e − t 0 /τ /τ (normalized) ◮ Probability distribution of initial velocities satisfies � d � v 0 P ( � v 0 ) = 1 (normalized) � d � v 0 P ( � v 0 ) � v 0 = 0 (random)

7 Drift velocity in electric field ◮ Drift velocity is the average velocity of all electrons � v d ≡ � � v � � � � � ∞ v 0 − e � Et 0 ≡ d � v 0 P ( � v 0 ) d t 0 P ( t 0 ) � m 0 � � ∞ � � ∞ d t 0 P ( t 0 ) e � Et 0 = d � v 0 P ( � v 0 ) � v 0 d t 0 P ( t 0 ) − d � v 0 P ( � v 0 ) m 0 0 � ∞ e � e − t 0 /τ Et 0 = 0 · 1 − 1 · d t 0 τ m 0 � ∞ = − e � E t 0 d t 0 e − t 0 /τ mτ · 0 �� ∞ � = − e � E n ! x n d xe − ax = mτ · τ 2 a n +1 0 = − e � Eτ m

8 Drude conductivity ◮ Current density carried by electrons: � � − e � = ne 2 τ Eτ � � j = n ( − e ) � v d = n ( − e ) E m m ◮ Which is exactly the local version of Ohm’s law with conductivity σ = ne 2 τ m ◮ For a given metal, n is determined by number density of atoms and number of ‘free’ electrons per atom ◮ e and m are fundamental constants ◮ Predictions of the model come down to τ (discussed next) ◮ Later: quantum mechanics changes τ , but above classical derivation remains essentially correct!

9 Classical model for scattering ◮ Electrons scatter against ions (nuclei + fixed core electrons) ◮ Scattering cross-section σ ion : projected area within which electron would be scattered ◮ WLOG assume electron travelling along z ◮ Probability of scattering between z and d z is − d P ( z ) = P ( z ) σ ion d z n ion � �� � d V eff + + + + + where n ion is number density of ions and d V eff is the volume from which ions can scatter electrons + + + + + ◮ This yields P ( z ) ∝ e − σ ion n ion z ◮ ⇒ Mean free path + + + + + 1 λ = e - n ion σ ion + + + + +

10 Classical estimate of scattering time ◮ From Drude model, τ = σm/ ( ne 2 ) ◮ Experimentally, σ ∝ T − 1 ⇒ τ ∝ T − 1 ◮ From classical model, τ = λ/u , where u is average electron speed ◮ λ = 1 / ( n ion σ ion ) should be T -independent � 2 mu 2 = 3 1 ◮ Kinetic theory: 2 k B T ⇒ u = 3 k B T/m ◮ Therefore classical scattering time τ = λ 1 ∝ T − 1 / 2 u = � n ion σ ion 3 k B T/m gets the temperature dependence wrong

11 Comparisons for copper ◮ Experimentally: σ = 6 × 10 7 (Ωm) − 1 (at 293 K) 4 A) 3 = 8 . 5 × 10 28 m − 3 n = n ion = (3 . 61 ˚ ne 2 = 6 × 10 7 (Ωm) − 1 · 9 × 10 − 31 kg τ = σm 8 . 5 × 10 28 m − 3 (1 . 6 × 10 − 19 C) 2 = 2 . 5 × 10 − 14 s ◮ Classical model: A) 2 ∼ 3 × 10 − 20 m 2 σ ion ∼ π (1 ˚ 1 1 8 . 5 × 10 28 m − 3 · 3 × 10 − 20 m 2 ∼ 4 × 10 − 10 m λ = ∼ n ion σ ion � � 3 · 1 . 38 × 10 − 23 J / K · 293 K 3 k B T = 1 . 2 × 10 5 m / s u = = 9 × 10 − 31 kg m τ = λ u ∼ 3 × 10 − 15 s ◮ Need σ ion to be 10x smaller to match experiment

12 What changes in quantum mechanics? 1. Electron velocity in metals is (almost) independent of temperature ◮ Pauli exclusion principle forces electrons to adopt different velocities ◮ ‘Relevant’ electrons have Fermi velocity v F (= 1 . 6 × 10 6 m/s for copper) 2. Electrons don’t scatter against ions of the perfect crystal ◮ Electrons are waves which ‘know’ where all the ions of the crystal are ◮ They only scatter when ions deviate from ideal positions! ◮ Crude model σ ion = πx 2 for RMS displacement x 2 kx 2 = 1 ◮ Thermal displacements 1 2 k B T √ ◮ Spring constant k ∼ Y a ∼ (120 GPa)(3.6˚ A/ 2 ) ∼ 30 N/m ◮ σ ion = πk B T ∼ 4 × 10 − 22 m 2 (at room T ) k ◮ 1 k n ion πk B Tv F ∼ 1 . 7 × 10 − 14 s (at room T ) τ = n ion σ ion v F = ◮ Correct 1 /T dependence and magnitude at room T (expt: 2 . 5 × 10 − 14 s)!

13 Matthiessen’s rule ◮ Perfect metal: τ T ∝ T − 1 due to scattering against thermal vibrations (so far) ◮ Impurity and defect scattering contribute τ I ∝ T 0 ◮ Scattering rates (not times) are additive, so net τ given by τ − 1 = τ − 1 + τ − 1 + · · · T I ◮ Resistivity ρ ∝ τ − 1 ∼ ρ 0 + AT with residual resistivity ρ 0 due to τ I 4 +2% Ni Resistivity [10 -8 m] 3 + 1% Ni 2 Cold worked Pure copper 1 0 0 100 200 300 T emperature [K] Is the experimental data strictly ρ 0 + AT ?

14 Mobility ◮ Drude conductivity in general σ = nq 2 τ = n | q | µ m where n is the number density of charge carriers q with mobility µ = | q | τ m effectively measuring the conductivity per unit (mobile) charge ◮ In metals, q = − e since charge carried by electrons (so far) ◮ In semiconductors, additionally q = + e for holes and σ = e ( n e µ e + n h µ h ) ◮ Semiconductors have typically higher µ , substantially lower n and σ

15 Hall effect ◮ Apply magnetic field perpendicular to current: voltage appears in third direction ◮ Hall coefficient defined by E y V H /W = V H d R H = = j x B z I/ ( Wd ) B z IB z ◮ Simple explanation in Drude model ◮ Average driving force on carriers now F = q ( � � v d × � E + � B ) = q ( E x ˆ x − ( v d ) x B z ˆ y ) ◮ Steady-state current only in ˆ x ◮ ⇒ E y = ( v d ) x B z develops to cancel F y

16 Hall coefficient in metals ◮ Note E y = ( v d ) x B z , while j x = nq ( v d ) x ◮ Eliminate ( v d ) x to get E y = 1 R H ≡ j x B z nq ◮ In particular, q = − e for electronic conduction ⇒ R H = − 1 / ( ne ) ◮ Compare to experimental values: Experiment R H [m 3 /C] Drude R H [m 3 /C] Metal − 5 . 5 × 10 − 11 − 7 . 3 × 10 − 11 Cu − 9 . 0 × 10 − 11 − 10 . 7 × 10 − 11 Ag − 2 . 5 × 10 − 10 − 2 . 4 × 10 − 10 Na +6 . 0 × 10 − 11 − 5 . 8 × 10 − 11 Cd +2 . 5 × 10 − 11 − 2 . 5 × 10 − 11 Fe ◮ Good agreement for ‘free-electron’ metals ◮ Wrong sign for some (transition metals)!