Download SOLID-STATE PHYSICS 3, Winter 2008 O. Entin-Wohlman Conductivity and conductance

SOLID-STATE PHYSICS 3, Winter 2008
O. Entin-Wohlman
1.
ELECTRONIC TRANSPORT PROPERTIES
2.
WEAK LOCALIZATION
Conductivity and conductance
One of the characteristic properties of a conductor is its electrical conductivity, usually
denoted by σ. In ordinary conductors, the conductivity is described very well by the Drude
formula,
σ=
ne2 τ
,
m
(2.1)
in which m is the electronic (band-structure) mass and e is the electronic charge. The
quantities that dominate the magnitude of the conductivity are n, the electronic density
(number of electrons per unit volume) and τ , the mean free time in-between collisions (of
the conducting electrons) with defects that are always present in the system. These defects
include static impurities, structural defects, localized magnetic moments, etc., which scatter
the electron elastically, and phonons, other electrons or other excitations, which scatter the
electron inelastically. The latter depend on the temperature T , and their scattering efficiency
increases as T is raised. However, at very low temperatures the dominant scattering is elastic,
and then τ does not depend on the temperature. Naturally, τ decreases as the temperature
or the amount of disorder in the system are increased.
Attempting to re-write the Drude formula for the conductivity, Eq. (2.1), in terms of the
diffusion coefficient,
D=
vF2 τ
,
d
(2.2)
where vF is the velocity of an electron having the Fermi energy, and the d is the system
dimensionality, yields the Einstein relation,
σ = e2
h nd i
D.
mvF2
(2.3)
The quantity in the square brackets is basically (up to factors of order unity) the density of
states per unit volume per unit energy at the Fermi level. We denote this quantity by N (0).
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For the free electron gas (i.e., in the absence of any interactions),


2
kF



, d=3 ,


 ~vF

k
F
N (0) '
, d=2,
~vF





 1 , d=1 ,

(2.4)
~vF
where kF is the wave vector of the electron at the Fermi level. In other words, the Drude
formula can be written in the form
σ = e2 DN (0) .
(2.5)
♣Exercise. Find the density of states of free fermion gas having a linear dispersion relation,
Ek ∝ k.
We now introduce the mean free path, `, the basic length step in the diffusive motion of the
conducting electrons,
` = vF τ .
(2.6)
Then, combining together Eqs. (2.2), (2.3), and (2.4), we see that we can re-write the Drude
conductivity in the form


 kF (kF `) ,

h e2 i 
×
σ'
kF ` ,

h



`,


d=3 ,


d=2,



d=1 .
(2.7)
First we see that the Drude conductivity has different units in each dimension. This is
natural since the conductance, G, of a d− dimensional system of length-scale L is related to
its conductivity by
G = σLd−2 ,
(2.8)
and the conductance has dimensions of 1/Ω in any dimension. Secondly we see that the unit
of the conductance (in any dimensionality) is [e2 /h] ' [10kΩ]−1 . Interestingly enough, the
unit of the conductance is given in terms of the quantum Planck constant.
Diffusion of classical and quantum-mechanical particles
In the (semi-classical) Boltzmann approach, consecutive scattering events of a certain particle are assumed to be independent of each other, i.e., collisions are un-correlated. This
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implies that multiple scattering of a particle at a particular scattering center is not taken
into account. Consequently, if there is a finite probability for such multiple scattering to
occur, the basic assumption of the independence of scattering events breaks down and the
validity of the Boltzmann equation results, notably the Drude conductivity, becomes, at
least, questionable. We therefore wish to estimate the probability for multiple scattering
events.
To investigate this point we consider the diffusive motion of a particle in a d−dimensional
disordered system (namely, a system containing various scattering centers, in particular
elastic ones). Let the particle be located at r = 0 at time t = 0. Because of its diffusive
motion, the particle will be located at some later time t within a smooth volume, whose size
is determined by the probability distribution P (r, t) for the particle to be located at point
r at time t. That probability distribution obeys the diffusion equation
∂P
− D∇2 P = 0 ,
∂t
D = vF2 τ /d is the diffusion coefficient .
(2.9)
The solution of the diffusion equation (in an infinite volume) is given by
P (r, t) =
1
2
e−r /4Dt .
d/2
(4πDt)
(2.10)
♣Exercise. Find the average distance that the particle reaches after a time t.
At a time t much longer than the relaxation time τ , t >> τ , we may ignore the exponential
in Eq. (2.10). Then P (r, t) gives the volume covered by the diffusing particle,


1



 (Dt)1/2 , d = 1 



1
1
P (r, t) '
.
'
,
d=2
Dt


Vdiff



 1 3/2 , d = 3 

(2.11)
(Dt)
One may view the diffusion volume also as the probability to return to the origin after a time
t. Generally, the probability to return to the origin will serve us to estimate the probability
for multiple scattering: assuming that the particle was scattered by an impurity at the origin
(r = 0), we need to know its probability to come back to that impurity and to be again
scattered by it. The diffusion equation solution provided us with the classical expression
for the probability to return to the origin. Now we wish to perform a quantum mechanical
calculation for that probability.
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A quantum-mechanical particle can diffuse from a certain point 1 to another point 2 via
many ‘trajectories’ or ‘tubes’, each of them having a typical ‘width’ given by the Fermi
wavelength λF ,
λF =
1
.
mvF
(2.12)
That width is the amount of uncertainty in the exact location of the particle. We note that
a diffusion (i.e., classical) picture for the electron is valid as long as the mean-free-path,
` = vF τ , is much longer than λF = 1/kF , namely,
kF ` 1 , or EF τ 1 .
(2.13)
When the inequalities (2.13) do not hold, we need to calculate the motion of the electron
from the Schrödinger equation. However, here we will adopt a heuristic simple treatment.
Let us consider the probability of a quantum particle to go from a one point in space
(denoted “1”) to another (denoted “2”), by a diffusion process. The electron can take many
paths between 1 and 2. In a classical calculation, we sum all over all the probabilities of
the various paths; in quantum mechanics, we sum over the amplitudes of the various paths,
and only then compute the total probability, by taking the absolute value squared. For
example, suppose that there are n optional diffusion paths between 1 and 2, and suppose
the (complex) amplitude of each one of them is An . Then according to quantum mechanics,
the total probability, P, to diffuse from 1 to 2 is given by
P = |A1 + . . . + An |2
X
X
= |A1 |2 + . . . + |An |2 + 2ReA∗1 A2 + . . . =
|Ai |2 + 2
ReA∗i Aj .
i
(2.14)
i6=j
The first term is just the sum over the probabilities of each path separately; this is the
classical result for the total probability. The second term is specific to quantum mechanics
and describes interference among the paths. That term arises since in quantum mechanics a
particle has a ‘wavy’ character. Usually, such interference terms do not contribute, since each
of the amplitudes Ai has a different phase, and the various phases are distributed randomly.
Therefore, the second sum in Eq. (2.14) disappears, and the classical result coincides with
the quantum-mechanical one. In any event, the second term in Eq. (2.14) is the one which
is neglected in the Boltzmann equation approach. Under many circumstances, this neglect
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is justified, since each trajectory (path) carries a different phase, and on the average the
interference is destructive, and the quantum mechanical correction is unimportant.
We note that the mere existence of the quantum mechanical additional term in the probability results from the assumption of coherent motion of the particle. In other words,
quantum interference is relevant as long as the wave function of the particle maintains its
phase. However, the quantum-mechanical phase is easily destroyed by many processes, in
particular inelastic scattering events, that change the energy of the particle. Therefore,
there is a characteristic time, usually denoted by τφ , during which the particle maintains
its quantum-mechanical phase. This time is called the ‘dephasing time’. Once the phase is
destroyed, quantum interference disappears. One expects the dephasing time to be infinite
at zero temperature, and to decrease as the temperature is increased. We will therefore
assume that we are considering very low temperatures, such that the time over which the
particle retains its phase coherence, τφ , is much longer than the relaxation time, τφ >> τ .
We mentioned that usually the quantum mechanical interference addition to the probability
vanishes. There is, however, one particular exception in which the quantum contribution–
the interference term in the probability–cannot be ignored. This happens when point 1 and
point 2 coincide (within the uncertainty distance λF ). Then the closed path can be traversed
in two opposite directions, clockwise and anticlockwise. In that case, the probability is just
the return probability (the probability to return to where the particle came from). The
clockwise path and the anticlockwise path have the same phase and therefore interfere
in-between themselves constructively. If we denote the clockwise probability by A1 and
the anticlockwise one by A2 then we have A1 = A2 ≡ A. According to Eq. (2.14) the
classical result will be 2|A|2 , while the quantum-mechanical one is 4|A|2 . It follows that the
quantum mechanical probability to return to the origin is twice that obtained classically.
Consequently, quantum-mechanical diffusion is slower than the classical one. In other words,
quantum-mechanical interference effects reduce the Drude (Boltzmann) conductivity.
This remarkable result follows from the interference of the clockwise and anti-clockwise
trajectories, that begin and end roughly (up to order of λF ) at the same point. This pair of
trajectories are related to one another via time-reversal symmetry.
Estimate of the weak-localization correction to the conductivity
Let us estimate the quantum-mechanical correction to the diffusion coefficient and to the
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Drude conductivity. We already know from the discussion above that this change is negative,
and is due to interference. Two paths interfere effectively when they are within a distance of
order λF from one another. We hence look at a d-dimensional ‘tube’ of diameter λF . During
an infinitesimal time dt the diffusing particle will move a distance vF dt and so it will cover
a volume given by dV ' λd−1
F vF dt. On the other hand, the maximal attainable volume is
given by Eq. (2.11) above. It follows that the probability for a particle to stay in the closed
tube is given by the ratio of these two volumes. The relative change in the conductivity,
caused by interference, is hence
δσ
' −vF λd−1
F
σD
Z
τ
τφ
dt
.
(Dt)d/2
(2.15)
We have put the lower limit of the integration to be τ , as one cannot discuss diffusion
processes on time scales less than the mean free time in-between elastic collisions; the upper
limit of the integration is τφ because for times longer than it phase-coherence is lost, and
quantum interference is not relevant any more.
Working out the integral in Eq. (2.15), using λF = kF−1 and introducing the mean-free-path,
` = vF τ , we find
δσ
' −(kF `)1−d
σD
Z
1
τφ /τ






dx
'−

xd/2




h
1/2 i
1
,
1 − ττφ
(kF `)2
τ
1
ln τφ ,
(kF `)
h 1/2
i
τφ
−
1
τ
,


d=3 ,



d=2,




d=1 .
(2.16)
As we have discussed above, the phase coherence time, τφ , diverges at zero temperature.
Let us start with a two-dimensional system. Such a system can be realized very easily
nowadays in a two-dimensional electron gas, formed at the interface between say, GaAs and
AlGaAs, or simply in thin films. At two dimensions, (d = 2), Eq. (2.7) and (2.16) give
e2 τφ
∆σ ' − ln
.
(2.17)
h
τ
d=2
Since τφ , the time over which the phase coherence is lost, diverges as the temperature tends
to zero, Eq. (2.17) implies that the quantum corrections to the Drude conductivity diverge
at very low temperatures. In other words, the conductivity of very thin films at very low
temperatures diverges logarithmically with T , as has been indeed found in experiments.
Since the phase coherence time, τφ , is very long at low temperatures, we may ask what is
the distance that an electron can traverse during that time. As the motion of the electron
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is diffusive, the distance it covers during a time τφ is given by
p
Dτφ . In other words, we
can define a length, `φ , over which the electron loses its quantum-mechanical phase,
`φ =
p
Dτφ .
Since by definition the mean free path, `, is given by ` =
(2.18)
√
Dτ , Eq. (2.17) can be re-written
in the form
∆σ '−
d=2
e2 `φ
ln
.
h
`
(2.19)
Hence, the phase coherence length `φ can be implied from simple measurements of the
conductivity of thin films. Obviously this statement holds as long as the length of the actual
system, say L, is such that L `φ . At sufficiently low temperatures, `φ can be longer than
the size of our two-dimensional system. When this is the case, we need to replace `φ in Eq.
(2.19) by L. We then find the interesting result: the conductivity of a thin film depends on
its length.
What happens in a bulk, three-dimensional system? Putting d = 3 in Eq. (2.16), we find
1
e2 1
−
.
(2.20)
'−
∆σ h ` `φ
d=3
In other words, the weak-localization corrections to the Drude conductivity are small, their
magnitude being of about e2 /h`, and depend only weakly on the temperature, the size of
the system, etc. On the other hand, when we try to put d = 1 in Eq. (2.16), we find a huge
result,
∆σ '
d=1
e2 ` − `φ .
h
(2.21)
Namely, the Drude theory is simply not valid for one-dimensional wires at low temperatures.
Again, in expressions (2.20) and (2.21) we need to replace `φ by the size of the system, L,
when L < `φ .
We see that (a) the weak-localization correction is larger in two dimensions as compared to
three, and does not have any small parameter at all in one dimension, and (b) in both one
and two dimensions it diverges as the temperature tends to zero. Making the reasonable
assumption that
1
' Tp ,
τφ
7
(2.22)
we find that at two dimensions the conductivity diverges logarithmically with the temperature T as the latter is decreased.
The fact that the correction to the conductivity diverges (at dimension less than three)
means that our Drude picture is not adequate at two and less dimensions. In fact, at zero
temperature, a two-dimensional system is insulating.
Size-dependence of weak localization corrections
Our discussion above pertains to an unbounded system. The diffusing electron maintains
p
its phase-coherence during a time τφ , and therefore traverses a distance `φ ' Dτφ before
it loses its phase. The previous discussion is thus valid as long as the size of the system,
denoted L, is such that L `φ . Let us now see in detail what happens when this condition
breaks down at a certain temperature, and the corrections to the conductivity depend on
the size of the system.
We return to Eq. (2.15) and put it into the form
Z `φ
Z
vF λd−1
x
1 2−d `φ
δσ
F
'−
dx d ' −
kF
dxx1−d .
σD
D
x
k
`
F
`
`
(2.23)
But note that the upper limit of the integration here should be replaced by L once the
phase-coherence length `φ becomes longer than the system size.
Performing the integration at two dimensions (the relevant case, for obvious reasons), we
obtain

`φ
δσ
1  ln ` ,
'−
σD
kF `  ln L ,
`

for L > `φ , 
for L < `φ . 
(2.24)
It follows that, at low enough temperatures, the conductivity of a two-dimensional sample depends on its size. For future purposes, we write down the result at one and three
dimensions as well,


δσ
'−

σD
1
2 kF `
`φ
`
1−
−1 ,
`
`φ

, d=3 ,
d=1 .
(2.25)
Again, for L < `φ , we need to replace `φ in the expressions above by the size of the system,
L.
Note that at zero temperature `φ is always longer than the size of the system. Using the
explicit form for the Drude conductivity,
σD =
N e2 τ
' e2 `kFd−1 ,
m
8
(2.26)
we can summarize the size dependence of the conductivity, due to weak localization corrections, in the form
−δσ ' e2











− , d=3 ,


L
ln ` , d = 2 ,



L−` , d=1 .
1
L
1
`
(2.27)
The effect of a magnetic field on weak localization corrections
As we have seen, weak-localization corrections originate from quantum coherence, i.e., from
the ‘wavy’ character of the electron that allows it to interfere with itself. The interference is
constructive because the state with k and that of −k are degenerate, due to time-reversal
symmetry. Magnetic fields break this symmetry, and degrade this constructive interference.
Let us ignore for the time-being the coupling of the magnetic field to the spin of the electron,
namely, neglect the Zeeman interaction. Then the magnetic field just modifies the electron
velocity. Introducing the vector potential A, such that the magnetic field, B, is given by
B = ∇ × A, the Hamiltonian of the electron is
e 2
1 p − A + V (r) ,
H=
2m
c
(2.28)
where V (r) is the potential energy. The Schrödinger equation is thus
HΨ(r) = EΨ(r) .
We can perform a (formal) gauge transformation on the wave function,
ie Z r
e
Ψ(r) = exp
d` · A Ψ(r)
.
c
(2.29)
(2.30)
e is then the same as in the absence
The Schrödinger equation satisfied by the wave function Ψ
of the vector potential,
p2
2m
e
e
+ V (r) Ψ(r)
= E Ψ(r)
.
(2.31)
We may interpret the results (2.30) and (2.31) as if in the presence of a (constant) magnetic
field, all that happens is that the wave function (at site r) accumulates a phase factor, given
by the line integral
ie
c
Z
r
d` · A ,
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(2.32)
which starts at an arbitrary point and ends at r. This phase factor is related to the partial
magnetic flux accumulated along the path starting at the arbitrary point and ending at r.
This observation is usually not so helpful for a practical solution, except when the electron
is confined to move along one-dimensional trajectories. Inspecting Eq. (2.32), we see that
the phase factor is the flux of the magnetic field (times 2π), divided by
Φ0 =
2πc
,
e
(2.33)
which is the unit of the flux quantum. The above result is a manifestation of the AharonovBohm effect, and the phase is sometimes referred to as the Aharonov-Bohm phase.
♣Exercise. Find the wave functions and the energy spectrum of an electron confined to move
on a one-dimensional ring, of a radius R, penetrated by a magnetic field B, directed at angle
θ with respect to the normal to the plane of the ring. In particular, draw the eigen energies
as function of the magnetic field.
Let us now return to our discussion of the quantum probability to return to the origin, vs.
the classical one, Eq. (2.14), and in particular, to the two time-reversed paths that start
and end at the same point. In the presence of a magnetic field, each of these trajectories will
acquire a phase factor. In fact, since each of these paths starts and ends at the same point,
that phase factor will be simply the magnetic flux, contained within the path [in units of
the flux quantum, Eq. (2.33)]
Φ = SB ,
(2.34)
where S is the area enclosed by the path. However, since one of these paths runs clockwise
while the other runs anticlockwise, the phase factor on each of them will be the same, but
with the reversed sign. The clockwise amplitude becomes
A → A1 = Aei2πΦ/Φ0 ,
(2.35)
A → A2 = Ae−i2πΦ/Φ0 ,
(2.36)
while the anticlockwise is now
The total probability to return to the origin coming from this pair is hence
4πΦ 2
2
2
2 2πΦ
= 2A 1 + cos
.
|A1 + A2 | = 4A cos
Φ0
Φ0
10
(2.37)
It follows that, since the magnetic field causes some destructive interference, the quantum
probability to return to the origin is reduced, and as a result the quantum diffusion is less
slow. In other words, the magnetic field reduces the weak localization correction. This is
called ‘anti-weak-localization’. Moreover, we see that the effect of the magnetic field enters
via a periodic function. When measured as function of the magnetic field, the conductivity
(or the conductance) will oscillate as the magnetic field is increased, and the oscillation
period is determined by the area encompassed by the time-reversed paths.
We can estimate the anti-weak-localization correction due to a magnetic field as follows.
The area encompassed by the diffusing electron during a time t is about Dt. Therefore the
. It follows
change in the probability entering the integrand in Eq. (2.15) is 2 1 − cos 4πBDt
Φ0
that the change in the weak localization relative correction to the conductivity is [see Eq.
(2.15)]
∆σ(B)
' vF λd−1
F
σD
Z
τφ
τ
dt 4πBDt 1 − cos
.
(Dt)d/2
Φ0
(2.38)
Let us work out the outcome of this expression for two dimensions. We denote
x≡
4πBDτφ
.
Φ0
(2.39)
Then,
∆σ(B)
1
'
σD
kF `
Z
1
τ /τφ
dy
1
(1 − cos xy) '
y
kF `
Z
0
x
dy
(1 − cos y) .
y
(2.40)
(Note that τ /τφ is a very small number at low enough temperatures, for which our considerations are confined.) We see that for x 1, the cosine term may be expanded, while for
x 1 its contribution to the integral averages to zero. Hence we find that for weak magnetic
fields ∆σ(B) ∝ B 2 , while for relatively strong ones it shows logarithmic dependence on the
field.
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