Transcript Lecture 22

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)
8. Flux equation
2. Conductivity
9. Einstein relation
3. Mobility
10. Total current density
4. Hall Effect
11. Carrier recombination
5. Thermal Conductivity
12. Carrier diffusion
6. Saturated Drift Velocity 13. Band diagrams in an
electric field
7. Derivation of “Ohm’s Law”
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
• 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)
• 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.
• 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.).
• 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
nd
2
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.
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.
• 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
• 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
dq
Current: i 
(Amps)
dt
q   idt
V
i
R
R
L
A
Microscopic
Current Density: J 
i   J  dA
Charge
J
Ohm’s Law
E


Current
  E where   resistivity
  conductivity
J  n e vd
Resistance
di
(A/m2 )
dA
m
ne 
2
where n  carrier density
vd  drift velocity
where   scattering time
• 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.
Page 15
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.
Page 16
Resistivity vs Temperature
• The resistivity is temperature dependent mostly because
of the temperature dependence of the scattering time τ.
FE
ma
E
m
1
e
e
 



2
J
ne vd
ne (a )
n
ne 
• 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.
Page 17
“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)
• 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
• 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
 ?
l ?
me*  1.18 mo
mh  0.59mo
 e  0.15 m 2 /(V  s )
 h  0.0458 m 2 /(V  s )
e me
 h mh
12
e 
 10
sec
h 
 1.54 x10 13 sec
q
vthelec  1.08 x105 m / s
q
vthhole  1.052 x105 m / s
le  vthelec  e  (1.08 x105 m / s )(10 12 s )  10 7 m
lh  vthhole  h  (1.052 x105 m / s )(1.54 x10 13 sec)  2.34 x10 8 m
Carrier Scattering in Semiconductors
Phys 320 - Baski
Page 22
Some Carrier Scattering Mechanisms
Defect Scattering
Phonon Scattering
Boundary Scattering
(From film surfaces, grain boundaries, ...)
Grain
Grain Boundary
Page 23
Some Possible
Results of
Carrier Scattering
1. Intra-valley
2. Inter-valley
3. Inter-band
Page 24
Defect Scattering
Ionized Defects
Perturbation Potential
Charged Defect
(Neutral Defects
Page 25
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:
Page 26
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/ph ~ T-3/2 . So
Page 27
Scattering from Ionized Defects &
Lattice Vibrations Together
1/ph ~ T-3/2
Page 28
Mobility Measurements in n-Type Ge
Page 29
Electrical Conductivity
Measurements in
n-Type Ge
Page 30