Quasi-Classical Approach What was Lord Kelvin’s name? Electrical Transport Quasi-Classical Approach What was Lord Kelvin’s name?

Electrical Transport Quasi-Classical Approach What was Lord Kelvin’s name? “Lord Kelvin” was his title, NOT his name!!!

Velocity density recombination an electric field Electrical Transport ≡ The study of the transport of electrons under various conditions. A broad & somewhat specialized area. Among possible topics: Current (drift & diffusion) 2. Conductivity 3. Mobility 4. Hall Effect 5. Thermal Conductivity 6. Saturated Drift Velocity 7. Derivation of “Ohm’s Law” 8. Flux equation 9. Einstein relation 10. Total current density 11. Carrier recombination 12. Carrier diffusion 13. Band diagrams in an electric field

Definitions & Terminology Bound Electrons (Holes): Electrons which are immobile or trapped at defect or impurity sites. “Free” Electrons: In Conduction band “Free” Holes: In Valence Bands “Free” charge carriers: Free electrons or holes.

Note: It is shown in many Solid State Physics texts that: Only free charge carriers contribute to the current! Bound charge carriers do NOT contribute to the current! As discussed earlier, only charge carriers within  kBT of the Fermi energy EF contribute to the current.

The Fermi-Dirac Distribution f(E) ≈ exp[-(E - EF)/kBT] NOTE! Energy levels within ~  2kBT of EF (in the “tail”, where it differs from a step function) are the ONLY ones which enter conduction (transport) processes! Within that tail, instead of a Fermi-Dirac Distribution, the function is: f(E) ≈ exp[-(E - EF)/kBT] (A Maxwell-Boltzmann distribution)

BUT, note that, in transport phenomena, they are NOT at equilibrium! Only charge carriers within 2 kBT of EF contribute to the current:  Because of this, as briefly discussed last time, the Fermi-Dirac distribution can be replaced by the Maxwell-Boltzmann distribution to describe the charge carriers at equilibrium. BUT, note that, in transport phenomena, they are NOT at equilibrium!  The electron transport problem isn’t as simple as it looks!

The electron transport problem isn’t as simple as it looks! Because they are not at equilibrium, to be rigorous, for a correct theory, we need to find the non-equilibrium charge carrier distribution function to be able to calculate observable properties. In general, this is difficult. Rigorously, this must be approached by using the classical (or the quantum mechanical generalization of) Boltzmann Transport Equation.

A “Quasi-Classical” Treatment of Transport This approach treats electronic motion in an electric field E using a Classical, Newton’s 2nd Law method, but it modifies Newton’s 2nd Law in 2 ways: 1. The electron mass mo is replaced by the effective mass m* (obtained from the Quantum Mechanical bandstructures). 2. An additional, (internal “frictional” or “scattering” or “collisional”) force is added, & characterized by a “scattering time” τ

In this theory, all Quantum Effects are “buried” in m* & τ. Note that: 1) m* can, in principle, be obtained from the bandstructures. 2) τ can, in principle, be obtained from a combination of Quantum Mechanical & Statistical Mechanical calculations. 3) The scattering time, τ could be treated as an empirical parameter in this quasi-classical approach.

Justification of this quasi-classical approach is found with a combination of: The Boltzmann Transport Equation (in the relaxation time approximation). We’ll briefly discuss this. Ehrenfest’s Theorem from Quantum Mechanics. This says that the Quantum Mechanical expectation values of observables obey their classical equations of motion!

Notation & Definitions (notation varies from text to text) v (or vd)  Drift Velocity This is the velocity of a charge carrier in an E field E  External Electric Field J (or j)  Current Density Recall from classical E&M that, for electrons alone (no holes): j = nevd (1) n = electron density A goal is to find the Quantum & Statistical Mechanics average of Eq. (1) under various conditions (E & B fields, etc.).

EV(k)  EV(0) - (ħ2k2)/(2m*) In this quasi-classical approach, the electronic bandstructures are almost always treated in the parabolic (spherical) band approximation. This is not necessary, of course! So, for example, for an electron at the bottom of the conduction bands: EC(k)  EC(0) + (ħ2k2)/(2m*) Similarly, for a hole at the top of the valence bands: EV(k)  EV(0) - (ħ2k2)/(2m*)

Recall: NEWTON’S 2nd Law In the quasi-classical approach, the left side contains 2 forces: FE = -eE = electric force due to the E field FS = frictional or scattering force due to electrons scattering with impurities & imperfections. Characterized by a scattering time τ. Figure 4-5. Caption: The bobsled accelerates because the team exerts a force.

Newton’s 2nd Law: An Electron in an External Electric Field The Quasi-classical Approximation Let r = e- position & use ∑F = ma (N’s 2nd Law!): m*a = m*(d2r/dt2) = - (m*/τ)(dr/dt) -eE or m*(d2r/dt2) + (m*/τ)(dr/dt) = -eE Here, -(m*/τ)(dr/dt) = - (m*/τ)v = “frictional” or “scattering” force. Here, τ = Scattering Time. τ includes the effects of e- scattering from phonons, impurities, other e- , etc. Usually treated as an empirical, phenomenological parameter However, in principle, τ can be calculated from QM & Statistical Mechanics,

Fs = - (m*/τ)(dr/dt) = - (m*v)/τ m*(d2r/dt2) = m*(dv/dt) = Fs – Fe With this approach:  The entire transport problem is classical! The scattering force: Fs = - (m*/τ)(dr/dt) = - (m*v)/τ Note that Fs decreases (gets more negative) as v increases. The electrical force: Fe = qE Note that Fe causes v to increase. Newton’s 2nd Law: ∑F = ma m*(d2r/dt2) = m*(dv/dt) = Fs – Fe

m*(d2r/dt2) = m*(dv/dt) = Fs – Fe Newton’s 2nd Law: ∑F = ma m*(d2r/dt2) = m*(dv/dt) = Fs – Fe Define the “Steady State” condition, when a = dv/dt = 0  At steady state, Newton’s 2nd Law becomes Fs = Fe (1) At steady state, v  vd (the drift velocity) Almost always, we’ll talk about Steady State Transport (1)  qE = (m*vd)/τ

σ = (nq2 τ)/m* (7) qE = (m*vd)/τ or vd = (qEτ)/m* (1) So, at steady state, qE = (m*vd)/τ or vd = (qEτ)/m* (1) Using the definition of the mobility μ: vd  μE (2) (1) & (2)  The mobility is: μ  (qτ/m) (3) Use definition of current density J, along with (2): J  nqvd = nqμE (4) Using the definition of the conductivity σ gives: (3) & (6) J  σE (This is Ohm’s “Law” ) (5) (4) & (5)  σ = nqμ (6)  The conductivity in terms of τ & m* σ = (nq2 τ)/m* (7)

Summary : “Quasi-Classical” Theory of Transport Macroscopic Microscopic Current Charge Resistance Ohm’s “Law” Drift velocity vd is the net electron velocity (0.1 to 10-7 m/s). Scattering time τ is the time between electron-lattice collisions.

Electronic Motion The charge carriers travel at (relatively) high velocities for a time t & then “collide” with the crystal lattice. This results in a net motion opposite to the E field with drift velocity vd. The scattering time t decreases with increasing temperature T, i.e. more scattering at higher temperatures. This leads to higher resistivity.

Resistivity vs Temperature The resistivity is temperature dependent mostly because of the temperature dependence of the scattering time τ. In Metals, the resistivity increases with increasing temperature. Why? Because the scattering time τ decreases with increasing temperature T, so as the temperature increases ρ increases (for the same number of conduction electrons n)

“Quasi-Classical” Steady State Transport Summary (Ohm’s “Law”) Current Density: J  σE (Ohm’s “Law”) Conductivity: σ = (nq2τ)/m* Mobility: μ = (qτ)/m*; σ = nqμ The electron concentration n is temperature dependent! n = n(T). τ is also temperature dependent! τ = τ(T). So the conductivity σ is strongly temperature dependent! σ = σ(T)

J = σE, σ = (nq2τ)/m* J = nqvd, vd = μE μ = (qτ)/m*, σ = nqμ We’ll soon see that, if a magnetic field B is present also, σ is a 2nd rank tensor: Ji = ∑jσijEjσ, σij= σij(B) (i,j = x,y,z) NOTE: This means that J isn’t necessarily parallel to E! In the simplest case, σ is a scalar: J = σE, σ = (nq2τ)/m* J = nqvd, vd = μE μ = (qτ)/m*, σ = nqμ Note that the resistivity is simply the inverse of the conductivity: ρ  (1/σ)

More Details τ(ε)  τo[ε/(kBT)]r The scattering time τ  the average time a charged particle spends between scatterings from impurities, phonons, etc. Detailed Quantum Mechanical theory shows that τ is not a constant, but depends on the particle velocity v: τ = τ(v). If we use classical free particle energy ε = (½)m*v2, then τ = τ(ε). It can be shown that that τ has the approximate form: τ(ε)  τo[ε/(kBT)]r where τo= classical mean time between collisions & the exponent r depends on the scattering mechanism: