Electrical Transport. Ref. Ihn Ch. 10 YC, Ch 5; BW, Chs 4 & 8
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1 Electrical Transport Ref. Ihn Ch. 10 YC, Ch 5; BW, Chs 4 & 8
2 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
3 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! Only charge carriers within 2k B T of the Fermi energy E F contribute to the current.
4 The Fermi-Dirac Distribution NOTE! The energy levels within ~ 2k B T of E F (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 - E F )/k B T] (A Maxwell-Boltzmann distribution)
5 Only charge carriers within 2 k B T of E F contribute to the current: Because of this, 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.
6 A Quasi-Classical Treatment of Transport This approach treats electronic motion in an electric field E using a Classical, Newton s 2 nd Law method, but it modifies Newton s 2 nd Law in 2 ways: 1. The electron mass m o 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.
7 Notation & Definitions (notation varies from text to text) v (or v d ) 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 = nev d (1) n = electron density A goal is to find the Quantum & Statistical Mechanics average of Eq. (1) under various conditions (E & B fields, etc.).
8 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: E C (k) E C (0) + (ħ 2 k 2 )/(2m*) Similarly, for a hole at the top of the valence bands: E V (k) E V (0) - (ħ 2 k 2 )/(2m*)
9 Electronic Motion Electrons 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 v d. The scattering time t decreases with increasing temperature T, i.e. more scattering at higher temperatures. This leads to higher resistivity.
10 Recall: NEWTON S 2 nd Law In the quasi-classical approach, the left side contains 2 forces: F E = qe = electric force due to the E field F S = frictional or scattering force due to electrons scattering with impurities & imperfections. Characterized by a scattering time τ. Assume that the magnetic field B = 0. Later, B 0
11 The Quasi-classical Approximation Let r = e - position & use F = ma m*a = m*(d 2 r/dt 2 ) = - (m*/τ)(dr/dt) +qe m*(d 2 r/dt 2 ) + (m*/τ)(dr/dt) = qe or Here, -(m*/τ)(dr/dt) = - (m*/τ)v = frictional or scattering force. τ = Scattering Time. τ includes the effects of e - scattering from phonons, impurities, other e -, etc. Usually treated as an empirical, phenomenological parameter However, τ can be calculated from QM & Statistical Mechanics, as we will briefly discuss.
12 With this approach: The entire transport problem is classical! The scattering force: F s = - (m*/τ)(dr/dt) = - (m*v)/τ Note that F s decreases (gets more negative) as v increases. The electrical force: F e = qe Note that F e causes v to increase. Newton s 2 nd Law: F = ma m*(d 2 r/dt 2 ) = m*(dv/dt) = F s + F e Define the Steady State condition, when a = dv/dt = 0 At steady state, Newton s 2 nd Law becomes F s = -F e (1) At steady state, v v d (the drift velocity) Almost always, we ll talk about Steady State Transport (1) qe = (m*v d )/τ
13 So, at steady state, qe = (m*v d )/τ or v d = (qeτ)/m* (1) Definition of the mobility μ: v d μe (2) (1) & (2) The mobility is: μ (qτ)/m* (3) Using the definition of current density J, along with (2): J nqv d = 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 = (nq 2 τ)/m* (7)
14 Summary of Quasi-Classical Theory of Transport Macroscopic Current: q i R i dq dt id t V R L A (Amps) Microscopic di 2 Current Density: J (A/m ) da i J da E J E where resistivity conductivity J nevd where n carrier density drift velocity ne m 2 v d where scattering time The Drift velocity v d is the net electron velocity (0.1 to 10-7 m/s). TheScattering time τ is the time between electron-lattice collisions. Charge Ohm s Law Resistance Current
15 Two-dimensional (2D) case Current density j=i/w where W is the width of the sample [j] = A/m (instead of A/m 2 ) Conductivity [] = -1 (not -1 m -1 ) Specific resistivity [] = (not m)
16 Resistivity vs Temperature The resistivity is temperature dependent mostly because of the temperature dependence of the scattering time τ. E m J 2 ne 1 n 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) InSemiconductors, 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.
17 Quasi-Classical Steady State Transport Summary (Ohm s Law ) Current density: J σe (Ohm s Law ) Conductivity: σ = (nq 2 τ)/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)
18 if a magnetic field B is present also, σ is a tensor: J i = j σ ij E j, σ ij = σ ij (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 = (nq 2 τ)/m* J = nqv d, v d = μe μ = (qτ)/m*, σ = nqμ If there are both electrons & holes, the 2 contributions are simply added (q e = -e, q h = +e): σ = e(nμ e + pμ h ), μ e = -(eτ e )/m e, μ h = +(eτ h )/m h Note that the resistivity is simply the inverse of the conductivity: ρ (1/σ)
19 More Details The scattering time τ the average time a charged particle spends between scatterings from impurities, phonons, etc. Detailed Quantum Mechanical scattering theory 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*v 2, then τ = τ(ε). Seeger (Ch. 6) shows that τ has the approximate form: τ(ε) τ o [ε/(k B T)] r where τ o = classical mean time between collisions & the exponent r depends on the scattering mechanism: Ionized Impurity Scattering: r = (3/2) Acoustic Phonon Scattering: r = - (½)
20 Numerical Calculation of Typical Parameters Calculate the mean scattering time τ & the mean free path for scattering l = v th τ for electrons in n-type silicon & for holes in p-type silicon. v d = μe, J = σe, μ = (qτ)/m* σ = nqμ, (½)(m*)(v th ) 2 = ( 3 / 2 ) k B T e e th elec *? l? m e 1.18 m o m h 0.59m o m / ( V s ) h m / ( V s ) m m q q e e 12 h h sec h 1.54x10 sec v x m s v x m s / th / hole l e v th e (1.0 8 x1 0 m / s )(1 0 s ) 1 0 m elec 5 l h v th h (1.052x10 m hole / )( se c ) s x x m
21 Carrier Scattering in Semiconductors
22 Some Carrier Scattering Mechanisms Defect Scattering Phonon Scattering Boundary Scattering (From film surfaces, grain boundaries,...) Grain Grain Boundary
23 Scattering Mechanisms Defect Scattering Carrier-Carrier Scattering Lattice Scattering Crystal Defects Impurity Alloy Intravalley Intervalley Neutral Ionized Acoustic Optical Acoustic Optical Deformation potential Piezoelectric Nonpolar Polar
24 Some Possible Results of Carrier Scattering 1. Intra-valley 2. Inter-valley 3. Inter-band
25 Ionized Defects Defect Scattering Perturbation Potential Charged Defect Neutral Defects
26 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:
27 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: ph ~ T -3/2. So
28 Scattering from Ionized Defects & Lattice Vibrations Together ph ~ T -3/2
29 Mobility of 3-dimensional GaAs
30 The two-dimensional electron gas
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32 Properties of 2D gases Electron density: n s cm -2 Dispersion relation: Wave function: Density of states: Fermi energy as a function of electron density: Fermi wavevector: Fermi wavelength: Fermi velocity: Ref. Ihn Ch. 9
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35 Mobility of 2D electron gas in remotely-doped Ga(Al)As heterostructures Current record: cm 2 /Vs mean free path 0.3 mm - limited by background impurity scattering Theoretical limit: cm 2 /Vs
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40 Conductivity from Boltzmann s transport equation Formal transport theory
41 Boltzmann Transport Equation for Particle Transport Distribution Function of Particles: f = f(r,p,t) --probability of particle occupation of momentum p at location r and time t Equilibrium Distribution: f 0, i.e. Fermi-Dirac for electrons, Bose-Einstein for phonons Non-equilibrium, e.g. in a high electric field or temperature gradient: f t v r f F p f f t scat f f o homogeneous electron gas stationary case Relaxation Time Approximation f f f t r,p scat o e t Relaxation time t
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