Electrical Transport
Ref. Ihn Ch. 10
YC, Ch 5; BW, Chs 4 & 8
• 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  2kBT 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, 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.
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.
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*)
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 vd.
• The scattering time t decreases with increasing temperature
T, i.e. more scattering at higher temperatures. This leads to
higher resistivity.
Recall: NEWTON’S 2nd Law
In the quasi-classical approach,
the left side contains 2 forces:
FE = qE = 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.
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) +qE
m*(d2r/dt2) + (m*/τ)(dr/dt) = qE
• Here, -(m*/τ)(dr/dt)
= - (m*/τ)v =
or
“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, 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)
(2)
• Definition of the mobility μ: vd  μE
• (1) & (2)  The mobility is:
μ  (qτ)/m*
(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*
(7)
σ = (nq2 τ)/m*
Summary of “Quasi-Classical” Theory of Transport
Macroscopic
dq
Current: i 
(Amps)
dt
q   idt
V
i
R
R
L
A
Charge
Ohm’s Law
Microscopic
 di
Current Density: J   (A/m 2 )
dA
 
i   J  dA
Current
 E

J    E where   resistivity

  conductivity


J  n e v d where n  carrier density
vd  drift velocity
Resistance

m
ne2
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.
Two-dimensional (2D) case
Current density j=I/W
where W is the width of the sample
[j] = A/m (instead of A/m2)
Conductivity [] = -1 (not -1m-1)
Specific resistivity [] =  (not m)
Resistivity vs Temperature
• The resistivity is temperature dependent mostly because
of the temperature dependence of the scattering time τ.
E
m
1



2
J
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.
“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)
• if a magnetic field B is present also, σ is a tensor:
Ji = ∑jσijEj, 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 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
  ?


e
e
l  ?
 0 .1 5 m

 em

e
2
m
 1 .1 8 m
/ (V s )
 10
q
*
e
12
sec
v t h e le c  1 . 0 8 x 1 0 5 m / s
l e  v t h e le c 
e
l h  v t h h o le
h

h

m
o

h
 0 .5 9 m
 0 .0 4 5 8 m
h

 hm
q

h
2
o
/ (V s )
 1 .5 4 x 1 0
13
sec
v t h h o le  1 . 0 5 2 x 1 0 5 m / s
 (1 . 0 8 x 1 0 5 m / s ) (1 0
12
s)  10
 (1 . 0 5 2 x 1 0 5 m / s ) (1 . 5 4 x 1 0
13
7
m
s e c )  2 .3 4 x 1 0
8
m
Carrier Scattering in Semiconductors
Some Carrier Scattering Mechanisms
Defect Scattering
Phonon Scattering
Boundary Scattering
(From film surfaces, grain boundaries, ...)
Grain
Grain Boundary
Scattering Mechanisms
Defect Scattering
Crystal
Defects
Neutral
Impurity
Carrier-Carrier Scattering
Alloy
Ionized
Lattice Scattering
Intervalley
Intravalley
Acoustic
Deformation
potential
Optical
Piezoelectric
Nonpolar
Acoustic
Polar
Optical
Some Possible
Results of
Carrier Scattering
1. Intra-valley
2. Inter-valley
3. Inter-band
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: ph ~ T-3/2 . So
Scattering from Ionized Defects &
Lattice Vibrations Together
ph ~ T-3/2
Mobility of 3-dimensional GaAs
The two-dimensional electron gas
Properties of 2D gases
• Electron density: ns  1011-1012 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
Mobility of 2D electron gas in remotely-doped
Ga(Al)As heterostructures
Current record: 30·106 cm2/Vs  mean free path 0.3 mm
- limited by background impurity scattering
Theoretical limit: 100·106 cm2/Vs
Conductivity from Boltzmann’s
transport equation
Formal transport theory
https://nanohub.org/resources/10575
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: f0, i.e. Fermi-Dirac for electrons,
Bose-Einstein for phonons
Non-equilibrium, e.g. in a high electric field or temperature gradient:
f
 f 
 v  r f  F   p f   
t
 t  scat
f  fo
homogeneous electron gas
stationary case
Relaxation Time Approximation
e
t

f  fo
 f 
  
  r,p 
 t  scat
t
Relaxation time