# Electrical Transport BW, Chs 4 & 8; YC, Ch 5; S, Chs 4 & 5

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Electrical Transport BW, Chs 4 & 8; YC, Ch 5; S, Chs 4 & 5
What was Lord Kelvin’s name?

Electrical Transport BW, Chs 4 & 8; YC, Ch 5; S, Chs 4 & 5
What was Lord Kelvin’s name? “Lord Kelvin” was his title, NOT his name!!!

Electrical Transport ≡ The study of the transport of electrons & holes (in semiconductors) under various conditions. A broad & somewhat specialized area. Among possible topics: 1. 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. Or deep in the Valence Bands. “Free” Electrons: In the conduction bands “Free” Holes: In the 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(ε) ≈ exp[-(E - EF)/kBT]
NOTE! The 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 distribution function is: f(ε) ≈ 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! 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. We will only briefly discuss this, to get an overview.

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: m* can, in principle, be obtained from the bandstructures. τ can, in principle, be obtained from a combination of Quantum Mechanical & Statistical Mechanical calculations. The scattering time, τ could be treated as an empirical parameter in this quasi-classical approach.

The text by YC The text by S
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! Our Text by BW Ch. 4, calculations are quasi-classical & use Newton’s 2nd Law. Ch. 8, combines quasi-classical & Boltzmann Transport methods. The text by YC The calculations are quasi-classical & use Newton’s 2nd Law. The text by S Most calculations use the Boltzmann transport approach.

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.

The Quasi-classical Approximation m*(d2r/dt2) + (m*/τ)(dr/dt) = -eE
Newton’s 2nd Law An Electron in an External Electric Field Assume that the magnetic field B = 0. Later, B  0 The Quasi-classical Approximation Let r = e- position & use ∑F = ma 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, mpurities, other e- , etc. Usually treated as an empirical, phenomenological parameter However, can τ be calculated from QM & Statistical Mechanics, as we will briefly discuss.

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 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

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

Summary of “Quasi-Classical” Theory of Transport
Macroscopic Microscopic Current Charge Ohm’s Law Resistance The Drift velocity vd is the net electron velocity (0.1 to 10-7 m/s). The 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) In Semiconductors, the resistivity decreases with increasing temperature. Why? The scattering time τ also decreases with increasing temperature T. But, as the temperature increases, the number of conduction electrons also increases. That is, more carriers are able to conduct at higher temperatures.

“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 As we’ve seen, the electron concentration n is strongly temperature dependent! n = n(T) We’ve said that τ is also strongly temperature dependent! τ = τ(T).  So, the conductivity σ is strongly temperature dependent! σ = σ(T)

ρ  (1/σ) Ji = ∑jσijEjσ, σij= σij(B) (i,j = x,y,z)
We’ll soon see that, if a magnetic field B is present also, σ is a tensor: Ji = ∑jσijEjσ, σij= σij(B) (i,j = x,y,z) NOTE: This means that J is not 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 If there are both electrons & holes, the 2 contributions are simply added (qe= -e, qh = +e): σ = e(nμe + pμh), μe = -(eτe)/me , μh = +(eτh)/mh 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 scattering theory (we’ll briefly describe) shows that τ is not a constant, but depends on the particle velocity v: τ = τ(v). If we use the classical free particle energy ε = (½)m*v2, then τ = τ(ε). Seeger (Ch. 6) shows 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: Ionized Impurity Scattering: r = (3/2) Acoustic Phonon Scattering: r = - (½)

Numerical Calculation of Typical Parameters
Calculate the mean scattering time τ & the mean free path for scattering ℓ = vthτ for electrons in n-type silicon & for holes in p-type silicon. vd = μE, J = σE, μ = (qτ)/m* σ = nqμ, (½)(m*)(vth)2 = (3/2) kBT

Carrier Scattering in Semiconductors

Some Carrier Scattering Mechanisms
Defect Scattering Phonon Scattering Boundary Scattering (From film surfaces, grain boundaries, ...)

Some Possible Results of Carrier Scattering
Intra-valley Inter-valley Inter-band

Perturbation Potential
Defect Scattering Ionized Defects Perturbation Potential Charged Defect (Neutral Defects

Scattering from Ionized Defects
(“Rutherford Scattering”) The thermal average Carrier Velocity in the absence of an external E field depends on temperature as: as The Mean Free Scattering Rate depends on the temperature as: So, (1/)  <v>-3  T-3/2 This gives the temperature dependence of the Mobility as:

Carrier-Phonon Scattering
Lattice vibrations (phonons) modulate the periodic potential, so carriers are scattered by this (slow) time dependent, periodic, potential. A scattering rate calculation gives: 1/tph ~ T-3/2 . So

Scattering from Ionized Defects & Lattice Vibrations Together
1/tph ~ T-3/2

Mobility Measurements in n-Type Ge

Electrical Conductivity
Measurements in n-Type Ge

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