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. 11
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constant if jk À k 0 j t
0
°k À k Þ ¼
if jk À k 0 j > t
0

and t ( kF , the Fermi radius. How does  vary with E F now?

8.3 The probability of an electron being scattered from k to k 0 with the emis-
sion of a phonon q of energy 0!q is proportional to fk °1 À fk 0 ) °1 þ fq Þ,
while the probability of the reverse process occurring, in which an
electron scatters from k 0 to k with the absorption of a phonon is propor-
tional to fk 0 °1 À fk Þfq . Are these expressions equal in equilibrium?

8.4 A monovalent simple cubic metal has the lattice potential

2V°cos gx þ cos gy þ cos gzÞ;
314 Semiclassical theory of conductivity in metals
and a magnetoresistance which is found to saturate for all directions of
the applied magnetic ¬eld B. The e¬ect of a small strain on this metal,
however, is to cause the magnetoresistance no longer to saturate for
certain directions of B. Estimate V, explaining your reasoning carefully.

8.5 Express the exact thermal conductivity i 0 (de¬ned as the solution of
Eq. (8.5.4) when j ¼ 0) in terms of p, S, and the approximate value i
given by Eq. (8.5.4) when E ¼ 0:

8.6 Is it meaningful to discuss ˜˜the limiting value of S=T for pure silver as
T ! 0,™™ where S is the thermopower? If not, why not?

8.7 Calculate the thermopowers of the metals de¬ned in Problems 8.1 and
8.2.

8.8 An alternative approach to the kinetic theory mentioned in Section 8.2
argues that each electron receives an extra velocity of eE=m by the time
it has a collision, and thus has an average drift velocity over a long time
of eE=2m. How can this apparent contradiction be resolved?

8.9 The room-temperature resistivity of Cu is about 2.0 m cm. Use a
simple free-electron gas model to ¬nd a relaxation time  that gives
you this resistivity at the right density of conduction electrons. Next,
calculate the resistance of a thin Cu ¬lm by applying an electric ¬eld in
the plane of the ¬lm. Use the Fuchs“Sondheimer boundary conditions to
obtain a solution for the Boltzmann equation from which you can cal-
culate the total current. What is the apparent resistivity of the ¬lm at
thickness d ¼ 0:1, d ¼ 0:5, d ¼ , and d ¼ 5 (with  the mean free
path) for specularity coe¬cients S ¼ 0, S ¼ 0:2, and S ¼ 0:8?
Chapter 9
Mesoscopic physics




9.1 Conductance quantization in quantum point contacts
In Chapter 8 we discussed the Boltzmann equation and the approach to
describing transport properties, such as electrical conductivity, that it pro-
vides. In general, this approach works very well for most common metals and
semiconductors, but there are cases where it fundamentally fails. This hap-
pens, for example, when the wave nature of the electron manifests itself and
has to be included in the description of the scattering. In this case, interfer-
ence may occur, which can a¬ect the electrical conduction. We recall that the
Boltzmann equation describes the electron states only through a dispersion
relation of the Bloch states of an underlying perfect crystal lattice, a prob-
ability function, and a scattering function that gives the probability per unit
time of scattering from one state to another. All these quantities are real,
and do not contain any phase information about the electron states.
Consequently, no wave-like phenomena can be described. The question
then arises as to when the phase information is important. This really boils
down to a question of length scales. We have earlier talked about the mean
free path of an electron, which is roughly the distance it travels between
scattering events. A simple example is given by scattering o¬ static impurities
that have no internal degrees of freedom. In this case the electron scattering is
elastic, since an electron must have the same energy before and after a scat-
tering event. Furthermore, in the presence of impurity scattering the phase of
an electron wavefunction after a scattering event is uniquely determined by
the phase before the scattering event. The wavefunction will in general su¬er
a phase shift as a consequence of the scattering, but this phase shift is not
random and can be calculated for any wavefunction given the impurity
potential. In view of this, we will now be more careful and speci¬cally talk
about the elastic mean free path ˜e as (roughly) the distance an

315
316 Mesoscopic physics
electron travels between elastic scattering events. Note that the elastic mean
free path is only weakly temperature dependent (through the temperature
dependence of the Fermi distribution of the electrons).
Inelastic scattering, on the other hand, randomizes the phase of the elec-
tron wavefunction. A good example is provided by electron“phonon scatter-
ing. Consider such an event within the framework of perturbation theory in
the electron“phonon interaction. In the scattering process, an initial electron
Bloch state and phonon couple through some interaction. For a while, there
will then be some complicated intermediate state made up of a multitude of
electron Bloch states and phonon modes. Eventually the system settles into a
direct product of another electron Bloch state and phonons consistent with
energy and crystal momentum conservation. The electron in the intermediate
state can have any energy for a time consistent with the Uncertainty
Principle, but the time that the electron spends in intermediate states is not
well speci¬ed. When the ¬nal electron Bloch state emerges, its phase is then
unrelated to the initial phase of the electron wavefunction. Thus inelastic
scattering inherently makes the phase before and after the scattering event
incoherent. In the presence of scattering that breaks the phase coherence, it is
useful to introduce a phase breaking length ˜ . We can think of this as the
distance an electron will travel while maintaining phase coherence.
Since inelastic scattering is typically much more strongly temperature
dependent than elastic scattering (again, think of electron“phonon scatter-
ing), one can change the phase breaking length by varying the temperature. If
we make the phase breaking length comparable to, or even smaller than, the
system size, we enter an area where new phenomena, due to manifestations
of the wave nature of the electron states, can occur. This is the area of
mesoscopic physics. ˜˜Meso™™ means something like ˜˜in the middle™™ or ˜˜inter-
mediate,™™ and mesoscopic systems are larger than microscopic systems,
which are of the order of maybe a Bohr radius and where we only have a
few particles. Macroscopic systems contain perhaps 1023 particles and are
very much larger than ˜e and ˜ .
For a speci¬c example, which will serve as a useful and illustrative model
for making the transition from macro to meso, we consider a conductor of
length L, width W, and thickness d. We start by taking L, W, and d all much
greater than both ˜e and ˜ . This system is depicted in Fig. 9.1.1. In a
standard experiment, to which we will return several times, we inject a cur-
rent into the system by connecting it to a potential di¬erence across two
terminals, e.g., 1 and 6. We then measure the resistance by noting the voltage
drop along some section of the system at a ¬xed net current through the
system. Let us assume that we inject a current into terminal 1, and draw
317
9.1 Conductance quantization in quantum point contacts




Figure 9.1.1. A system of length L, width W, and thickness d, with attached
terminals numbered as shown.


the current out at terminal 6. We can then measure the voltage across two
other terminals, such as 4 and 5. It is a property of a macroscopic system that
the resistance we measure across probes 4 and 5 is the same regardless of
whether we inject the current into probes 1, 2 or 3. In fact, the measured
resistance is the same between any two probes separated by the same distance
along the ¬‚ow of the current. This is precisely due to the lack of phase
coherence of the electron states. When we measure the voltage between,
say, probes 4 and 5, we are measuring the lowest cost in energy to remove
an electron from probe 5 and inject it at probe 4. This energy is due to the
di¬erence in electrochemical potential between the probes. When we remove
an electron from probe 5, its phase is completely random because of the small
˜ , and we have no way of ¬guring out whence (i.e., from which probe) the
electron came. Similarly, an electron injected at probe 4 rapidly loses its
phase and so is indistinguishable from other electrons at the electrochemical
potential at that probe.
Now we take our model system into the mesoscopic regime. We do this by
shrinking d and W until both are less than the phase breaking length ˜ . As
these lengths shrink, the energy states for motion along these directions will
become discrete, with the separations between allowed energy eigenvalues
growing as W À2 and d À2 , respectively. For example, if we consider periodic
boundary conditions, the electron energies can be written as

02 k2 22 02 n2 22 02 l 2
x
E°kx ; n; lÞ ¼ þ þ2 ;
W 2 me
2me d me
318 Mesoscopic physics
where n and l are integers. We can then separate the energy values according
to which sub-band n and l they belong. For the moment we assume that d is
small enough that only the lowest sub-band l ¼ 0 is occupied, and that only a
small number of sub-bands n 6¼ 0 are occupied. In real mesoscopic systems, d
can be of the order of one nanometer, with W ranging from perhaps a few
nanometers to a few hundred nanometers. We also for now restrict our
system to having only two external terminals, a source (S) and a drain (D),
at which current is injected and drawn out, respectively, as in Fig. 9.1.2.
These kinds of system are rather easy to fabricate at the interface (known
as a heterojunction) between two di¬erent semiconductors such as GaAs and
GaAlAs grown by molecular beam epitaxy. Electrons (which are supplied by
donors implanted some distance away from the heterojunction) are con¬ned
to move in the plane of the heterojunction. The source and drain can be made
by doping heavily with donors in some small regions. The conducting chan-
nel connecting the source and the drain can be controlled by evaporating
small metallic gates in the region between them, as shown schematically in
Fig. 9.1.3. By applying a negative potential, or gate voltage VG , to the gates,




Figure 9.1.2. Schematic of a simple two-terminal device. An electron injected in one
terminal has a probability R of being re¬‚ected and a probability T of being trans-
mitted through the device to the other terminal.




G
S D
G
Figure 9.1.3. Schematic top view of a two-terminal device with gate electrodes.
319
9.1 Conductance quantization in quantum point contacts
the channel between source and drain through which the conduction elec-
trons must pass can be made as narrow as we please. Heterojunctions can be
grown cleanly enough to make the elastic mean free path of the order of or
larger than the dimensions of the device. By cooling down to liquid-helium
temperatures, the same can be achieved for the phase breaking length.
By measuring the voltage VSD between source and drain as a function of
the gate voltage VG at ¬xed current ISD , one can plot the conductance g in
units of e2 =h as a function of VG . In the extreme limit where L ( ˜e , the so-
called ballistic limit in which electrons traverse the entire device without
scattering elastically (or inelastically), one ¬nds a remarkable result. The
conductance shows a clear staircase-like behavior with steps at g ¼ Me2 =h,
with M an even integer. As the device is made longer and longer, the steps
become noisier and noisier until they can no longer be discerned, even though
each noisy curve is reproducible if the gate voltage is ramped up and down.
This kind of behavior is sketched in Fig. 9.1.4.
Our ¬rst aim here is to understand the electrical conductance of this sys-
tem. To this end, we ¬rst have to describe the electron states, and then how
we drive a current through the system. Landauer pioneered mesoscopic phy-
sics with his insight that conduction should fundamentally be regarded as a
scattering process in which we describe the electron states locally in our
mesoscopic system and consider separately the means by which current is
driven by an applied electrochemical potential di¬erence. So what does this
mean? Let us start with the electron states. We formally separate the meso-
scopic part of the system, which we term the device and is the block of
nominal length L, width W, and height d, from the terminals that are used
to connect the device to the sources of electrochemical potential. Inside the
device, the electron states maintain their phase coherence, since L ( ˜ . The




Figure 9.1.4. Conductance g (in units of e2 =hÞ vs. gate voltage VG for a two-terminal
ballistic device. At low temperatures (bold curve) the electrons pass through the
device without any scattering, resulting in quantized conductance. At higher tem-
peratures, the electrons scatter inelastically from phonons in the device, smearing out
the quantization of the conductance (light curve).
320 Mesoscopic physics
two terminals of the system, S and D, are connected to very large reservoirs
kept at thermodynamic equilibrium at electrochemical potentials  þ Á
and , respectively. These potentials are well de¬ned deep in the reservoirs
(far away from the actual device). The reservoirs inject electrons with energies
up to the respective electrochemical potentials into the device through the
terminals, and electrons ¬‚owing through the device exit it into the terminals
and thereby enter the reservoirs. Dissipative processes within these reservoirs
quickly thermalize electrons and randomize their phases, so that electrons
entering the device from the reservoirs have random phases. Consequently,
interference terms between di¬erent electron states average to zero and can be
ignored. The natural way to describe the electrons that enter and leave the
device is by using a basis of scattering states. In the scattering theory that we
learn in basic quantum mechanics, we have well de¬ned (asymptotic) incom-
ing and outgoing electron states, which are connected by a scattering matrix,
or S-matrix. The matrix elements s of the S-matrix give the probability
amplitude that an incoming electron in state is scattered into the outgoing
state . Here, we have incoming states into the device from the reservoirs
through the terminals, and these states can be scattered into outgoing states
from the device into the terminals and the reservoirs by some potential in the
device. The assertion that di¬erent states do not interfere with one another
simpli¬es the discussion quite substantially, since we then do not have to
work with complex transmission amplitudes (the elements of the S-matrix
itself), but only the real probabilities of scattering from an incoming state to
an outgoing one. The incoming and outgoing states are con¬ned by some
potential V°yÞ (we can neglect the dependence on z since we are only con-
sidering the lowest sub-band of motion along z). For now, we are only
considering a con¬ning potential without any additional more complicated
scattering inside the device itself. In the presence of the con¬ning potential,
the incoming and outgoing parts of the electron states can then locally be
written as

¼ eÆikx hn °yÞ;
n;k °x; yÞ °9:1:1Þ

where n is the sub-band index, and hn °yÞ is the transverse part of the wave-
function of the sub-band, or channel, n. This is a description of the electron
states in the terminals “ we do not attempt to describe the states in the device
itself. The corresponding energy eigenvalue is

02 k2
E n;k ¼ þ En:
2m
321
9.1 Conductance quantization in quantum point contacts
Note that this is perfectly general, and given a good model for V° yÞ, we
can determine E n . These states are solutions of the Schrodinger equation in
¨
the terminals, but are not appropriate scattering states. Those can be con-
structed by making linear combinations of the states given in Eq. (9.1.1). A
state °s; mÞ incoming from the source into the device in channel m with unit
amplitude is

eiksm x hs;m °yÞ

where we now also attach an index s to the sub-band wavefunction (for ease
of notation we do not include the wavevector index). This will allow for
generalization later, when the incoming sub-bands from di¬erent terminals
are not necessarily the same. The probability that this state is scattered into
an outgoing state °d; nÞ in the drain is then Tds;nm , and the probability that the
state is re¬‚ected into an outgoing state eÀiksm x hs;m °yÞ in the source is Rss;mm .
The scattering is elastic, so all states connected by Tds;nm and Rss;mm have the
same energy eigenvalues. Note that in the absence of an external magnetic
¬eld there must exist an outgoing state in the same terminal with the same
energy but opposite wavenumber.
We adopt the standard convention of positive incoming velocities, and at
the same time change our convention about the sign of the charge on the
electron. From now on, we shall make explicit the negative nature of this
charge by writing it as Àe, with e a positive quantity. The incoming current
is;m carried by the state °s; mÞ into the device is then

e @E s;m °kÞ

is;m ¼ Àevs;m °kÞ ¼ À :
0 @k k¼ksm

This incoming state is scattered into states that are outgoing in the terminals
and carry the current out of the device. Current conservation must be strictly
obeyed, so the outgoing currents in all terminals add up to the incoming
current. The transmitted outgoing current in state °d; nÞ is then evsm Tds;nm ,
and the outgoing re¬‚ected current in state °s; m 0 Þ is evsm Rss;m 0 m .
Suppose now that we have applied an electrochemical potential di¬erence
Á between the source and the drain, so that source and drain are at elec-
trochemical potentials  þ Á and , respectively. With fs;m °E k Þ denoting
the occupation numbers of the incoming states, the incoming current in the
source is
X
Is ¼ Àe fs;m °E k Þvs;m :
km
322 Mesoscopic physics
Similarly, the incoming current from the drain is
X
Id ¼ Àe fd;n °E k Þvd;n :
kn


We can ¬nd the total current between source and drain by examining
the current ¬‚owing near the source. Here, the total current consists of the
di¬erence between the net incoming current from the source (incoming
from source minus current re¬‚ected into the source) and the part of the
current from the drain that is transmitted to the source. The total
current is then obtained by summing over all channels in source and
drain:
 
X X X X
I ¼ Àe fs;m °E k Þvs;m 1 À Rss;m 0 m þ e fd;n °E k Þvd;n Tsd;m 0 n :
m0 m0
km kn


Current conservation dictates that incoming current minus re¬‚ected current
equals transmitted current,
X X
1À Rss;m 0 m ¼ Tds;m 0 m ;
m0 m0


so we can write the total current as
X X X X
I ¼ Àe fs;m °E k Þvs;m Tds;m 0 m þ e fd;n °E k Þvd;n Tsd;m 0 n : °9:1:2Þ
m0 m0
km kn


We ¬rst change the sums over wavevectors to integrals. When we do that, we
have to insert the density of states in k-space, which in one dimension is
just a constant, 1=2. We then change the integration variable to energy and
insert a factor of dk=dE due to this change. Since the velocity is propor-
tional to dE=dk, we see that the density-of-states factor precisely cancels
with the velocity in the integrals. This happens only because we can in meso-
scopic physics consider separate, e¬ectively one-dimensional, conducting
channels. Because of this cancellation, all that is left in Eq. (9.1.2) are
occupation numbers and transmission and re¬‚ection probabilities. All
the speci¬cs of the device, such as length and width, have disappeared.
That makes the Landauer approach particularly powerful, simple, and
beautiful.
323
9.1 Conductance quantization in quantum point contacts
The total current in the device is then
° °
X X X X
e e
I ¼À fs;m °EÞ Tds;m 0 m þ fd;n °EÞ Tsd;m 0 n : °9:1:3Þ
dE dE
h h
0 0
m m n m

We now make the assumption that the driving force (in this case the electro-
chemical potential di¬erence) is su¬ciently small that in calculating the
current we need only consider the leading term, which is proportional to Á.
We can then ignore the energy dependence of the transmission probabilities
and evaluate them at the electrochemical potential . It is then convenient
to de¬ne total transmission probabilities Tsm and Tdn for each channel by
summing out the scattered channels. Thus
X
Tsm  Tds;m 0 m
m0
°9:1:4Þ
X
Tdn  Tsd;m 0 n ;
m0

where it is understood that the sums are evaluated at the electrochemical
potential . Using Eq. (9.1.4) we then obtain for the total current
°
eX
I ¼À dE½ fs;m °EÞTsm À fd;m °EÞTdm Š:
hm

In the absence of an external magnetic ¬eld, the system is invariant under
time reversal. This imposes constraints on the S-matrix, with the consequence
that Tsm ¼ Tdm  Tm . This simpli¬es the expression for the total current,
which now becomes
°
eX
I ¼À dE½ fs;m °EÞ À fd;m °Eފ
T
hm m
° þÁ
eX eX
¼À dE ¼ À T Á:
Tm
hm m
hm 


This is a remarkable and simple result: the total current is just the driving
force times the transmission probability at the electrochemical potential times
a universal constant. As we have discussed earlier, the voltage measured
between source and drain is just the electrochemical potential di¬erence,
divided by the electron charge. The measured resistance is then
h
P
Rsd ¼ ÀÁ=°eIÞ ¼ ;
2e2 m Tm
324 Mesoscopic physics
where the additional factor of 2 comes from the two degenerate spin direc-
tions. For the case of a ballistic channel, Tm  1, so we obtain

h
Rsd ¼ ; °9:1:5Þ
e2 M

where M is the total number of channels (including spin degeneracy) con-
nected by source and drain to the device. Equation (9.1.5) predicts a quan-
tized conductance, just as is observed in measurements on quantum point
contacts. The quantum of conductance is e2 =h, and the quantum number is
M, the number of current-carrying channels in the system. For the conduc-
tance to be quantized, the channel has to be smaller than ˜e so that electrons
traverse the device ballistically and Tm ¼ 1. If the channel becomes longer
than ˜e , or some scatterer is introduced into the channel, the transmission
probabilities will in general be less than unity, and the conductance is no
longer quantized. The steps start to degrade and become noisy, but the I“V
curves are retraced if the gate voltage is swept up and down. If, on the other
hand, the channel is made larger than ˜ , there will be random inelastic
scattering in the channel. The conductance will similarly not be quantized,
but in this case there will be thermal noise on the I“V curve which will not be
repeatable if the gate voltage is cycled.
One may wonder how it is that a ballistic device has a ¬nite conductance.
First of all, even though there is no scattering in the device itself, there must
be inelastic scattering in the reservoirs in order for the electrons to therma-
lize. Each channel has a ¬nite conductance due to this contact resistance, and
a mesoscopic device with a ¬nite number of channels cannot have in¬nite
conductance. As the number of channels grows, so does the total conduc-
tance and we recover the perfect conductor (with in¬nite conductance) in the
limit of an in¬nite number of channels.


¨
9.2 Multi-terminal devices: the Landauer“Buttiker formalism
We hinted in the previous section at the fact that resistance in a mesoscopic
device in general depends on which terminals are used as source and drain,
and which terminals are used as voltage probes. In order to demonstrate this,
we need to generalize the Landauer formula to multi-terminal devices. This
generalization was developed by Markus Buttiker, and is a fundamental
¨
cornerstone of mesoscopic physics. A simple introductory example is given
by a four-probe measurement, as depicted in Fig. 9.2.1. In such a measure-
ment, the current ¬‚ows between source and drain, and the electrochemical
325
9.2 Multi-terminal devices: the Landauer“Bu¨ttiker formalism




Figure 9.2.1. Schematic of a four-terminal device with current source (S) and drain
(D), and two extra terminals 1 and 2.


potential di¬erence between terminals 1 and 2 is measured. An ideal volt-
meter has in¬nite internal resistance, and so the proper boundary condition
to be imposed on terminals 1 and 2 is that the net current in (or out) of such
terminals should be zero. On the other hand, there must be some well de¬ned
electrochemical potential associated with each of the probes 1 and 2 in order
for a measurement of the electrochemical potential di¬erence to make sense.
Furthermore, electrons injected from one terminal into the system have in
general ¬nite probabilities of ending up at any of the other terminals. For
example, electrons injected from the source may end up going into terminal 2.
Since terminal 2 is also connected to a reservoir, just like the source and the
drain, it too injects electrons into the system. The net current in or out of this
terminal depends on the balance between incoming and outgoing currents.
We can adjust this balance by changing the electrochemical potential of the
reservoir to which this terminal is attached. In the end, we must then ensure
that the electrochemical potentials at terminals 1 and 2 are self-consistently
adjusted to yield zero net currents at these terminals.
Let us now formally state this for a general multi-terminal system with
terminals 1; 2; 3; . . . ; N. The terminals are in contact with reservoirs at well
de¬ned electrochemical potentials i , with i ¼ 1; 2; . . . ; N. Carriers are
injected from the reservoir in all states with energies up to i into terminal
i. Electrons injected from the terminals have random phases and do not
interfere with one another. An electron injected in state °i; mÞ (channel m
in terminal i) has a probability Tji;nm of being scattered to outgoing state ° j; nÞ
(channel n in terminal j), and a probability Rii;m 0 m of being re¬‚ected into an
outgoing state °i; m 0 Þ. In looking solely for the linear response, we can ignore
the energy dependence of the scattering and re¬‚ection probabilities and eval-
uate them at a common energy 0 . We take this to be the lowest of the
326 Mesoscopic physics
electrochemical potentials i , i ¼ 1; 2; . . . ; N. Then all states in the device
with energies less than 0 are occupied and do not contribute to a net current
in the device (why?). Terminal i then injects a current

X
Mi
e
Ii;injected ¼ À °i À 0 Þ;
h
m¼1


where Mi is the total number of occupied channels at terminal i. The net
current at terminal i is the di¬erence between this injected current and the
sum of the re¬‚ected current at the terminal and the current scattered into this
terminal from other terminals. This net current is
 
X
e
Ii ¼ À °Mi À Rii Þi À Tij j : °9:2:1Þ
h j°6¼iÞ

P
Here we have again used the de¬nition Tij ¼ mm 0 Tij;m 0 m for the total trans-
P
mission probability and have de¬ned Rii ¼ mm 0 Rii;m 0 m for the re¬‚ection
probability, even though this risks confusion with the resistance Rij . Since
the reference potential 0 is common to all reservoirs, it cancels out in
Eq. (9.2.1). Current conservation dictates that the injected current in terminal
i equal the re¬‚ected current and the total transmitted current to other term-
inals. In other words,
X
Mi ¼ Rii þ Tij : °9:2:2Þ
j°6¼iÞ


If we insert this into Eq. (9.2.1) we can write the net current Ii as
eX
Ii ¼ À T ° À j Þ: °9:2:3Þ
h j°6¼iÞ ij i

Equations (9.2.2) and (9.2.3) express current conservation at a terminal and a
relation between total current and driving forces. These are the mesoscopic
versions of Kirchho¬ ™s Laws.
Time reversal symmetry imposes constraints on the scattering matrix that
connects incoming and outgoing states. In the absence of magnetic ¬elds, we
can reverse the directions of all incoming and outgoing states, and because of
time reversal symmetry we must then have Tij ¼ Tji . In the presence of a
magnetic ¬eld, we can reverse the direction of all velocities if we also reverse
the sign of the ¬‚ux È penetrating the system, so in the more general case of an
327
9.2 Multi-terminal devices: the Landauer“Bu¨ttiker formalism
applied magnetic ¬eld, we have Tij °ÈÞ ¼ Tji °ÀÈÞ. This symmetry leads to a
reciprocity theorem that relates resistances when the current and voltage
probes are interchanged.
Armed with Eqs. (9.2.1) and (9.2.3) we can now go ahead and calculate the
resistance Rij;kl due to a chemical potential di¬erence k À l between two
terminals k and l when a current I ¬‚ows from source (terminal i) to drain
(terminal j). This amounts to solving the system of linear equations
Eq. (9.2.1) for the electrochemical potentials k and l under the conditions
that Im ¼ 0 for m 6¼ i; j, Ii ¼ ÀIj  I, and i ¼ 0 þ Á, j ¼ 0 . Once we
have those, the resistance Rij;kl is obtained by just using Ohm™s law. The voltage
V between the terminals k and l is the electrochemical potential di¬erence
between the two terminals, divided by Àe, and so V ¼ ÀÁ=e, and
Rij;kl ¼ V=I. The currents are proportional to the applied electrochemical
potential di¬erence, so it will cancel from the expression for the resistance. In
the end, the resistance will be h=e2 (which is a unit of resistance) times some
combination of the transmission probabilities (which are dimensionless). We
¬rst work this out explicitly for the three-terminal case, and then sketch an
approach for a general N-terminal case. So let us assume that we have a three-
terminal device, with terminal 1 the source and terminal 3 the drain, and we are
measuring a voltage between terminals 1 and 2. Thus, I1 ¼ ÀI3 ¼ I, and I2 ¼ 0.
We will calculate the resistance R13;12 due to the voltage between probes 2 and 1
with a current ¬‚owing from probe 1 to probe 3. Equations (9.2.1) then become

2e
I ¼À ½°M1 À R11 Þ1 À T12 2 À T13 3 Š
h
2e
°9:2:4Þ
0 ¼ À ½°M2 À R22 Þ2 À T21 1 À T23 3 Š
h
2e
ÀI ¼ À ½°M3 À R33 Þ3 À T31 1 À T32 2 Š;
h

where the factor of 2 comes from summing over spin channels. Using
Eq. (9.2.2), we can solve for 3 from the second line in Eq. (9.2.4) to obtain

T21 þ T23 T
3 ¼ 2 À 21 1 :
T23 T23

We insert this into the third line of Eq. (9.2.4), and after collecting some
factors we obtain

2e
ÀI ¼ À D°2 À 1 Þ;
hT23
328 Mesoscopic physics
where we have de¬ned D ¼ T31 T21 þ T31 T23 þ T32 T21 . Then

1 À 2 hT
R13;12 ¼ À ¼ 2 23 :
2e D
eI

Note that we could also have inserted the expression for 3 into the ¬rst line
of Eq. (9.2.4) and obtained the same result. That is a consequence of having
fewer independent chemical potentials than we have equations to solve.
The fact that the i are overdetermined poses computational di¬culties in
the general case. Normally we would solve a set of linear equations like
Eq. (9.2.3) by inverting a matrix, in this case the matrix

e
^
Tij  ½°Mi À Rii Þij À Tij Š
h

(where it is understood that Tii ¼ 0). However, only the N À 1 electrochemi-
cal potential di¬erences can be independent, and not the N electrochemical
potentials at all reservoirs, and so the system of linear equations is singular,
and the matrix cannot be inverted. The approach we take to get around this
di¬culty is ¬rst to note the obvious fact that setting all electrochemical
potentials equal will yield a solution with all currents equal to zero. What
is less obvious is that there may also be other sets of electrochemical poten-
tials that yield zero currents. These are said to form the nullspace of the
^
matrix Tij . Formally, the nullspace of an N ‚ N singular matrix A consists
of all N-dimensional vectors x such that A Á x ¼ 0. Now these are most
certainly not the solutions that we are interested in “ in fact they are the
problem rather than the solution! Technically, we need to separate out the
^
˜˜nullspace™™ from the domain of the linear mapping de¬ned by the matrix Tij ;
that is, we need to weed out all the sets of electrochemical di¬erences that
yield all zero currents from those that yield the ¬nite currents Ii that we have
speci¬ed. A general and very powerful way of doing this (which can also be
e¬ciently implemented on computers) is provided by a technique known as
^ ^
the singular value decomposition of the matrix Tij . This involves writing Tij as
a product of three matrices, one of which is diagonal and has only positive or
zero elements (these are the singular values). The nontrivial solutions for the
i in terms of the currents can then, after some lengthy manipulations, be
expressed in terms of another diagonal matrix that also has only positive or
zero elements. Having thus formally obtained the electrochemical potentials
as functions of the currents, one can calculate the resistance. While we have
omitted some of the details in this description, we can nevertheless note that
the general procedure has been as follows:
329
9.3 Noise in two-terminal systems
The current at the source is I and the current at the drain is ÀI.
1.
2. The currents at all other terminals are set to zero.
3. We write down the linear equation (9.2.1) using these boundary conditions.
4. We solve this equation for the nontrivial electrochemical potentials by using
singular value decomposition.
5. With electrochemical potentials obtained as functions of the applied current, we
calculate the resistance between voltage probes.


The main point here is that the ¬nal expression for the resistance will involve
transmission probabilities from all terminals i; j; k; and l in a combination that
depends on which probes are source and drain, and which are voltage probes.
This has the implication that the resistance will now depend on how the
measurement is conducted and not just on a material parameter (resistivity)
and geometry. As a consequence of the phase coherence in the mesoscopic
system, an electron carries with it phase information from its traversal of the
system, and the probability that an electron will reach one terminal depends
on where it was injected.
The multi-terminal Landauer“Buttiker formula has become the standard
¨
approach for analyzing mesoscopic transport in areas ranging from quantum
point contacts to the quantum Hall e¬ect and spin-dependent tunneling
transport. The basic physics underlying weak localization can also be under-
stood from the point of view of the Landauer“Buttiker formalism, although
¨
in this case it does not easily lead to quantitative predictions. It is very
intuitive, simple, and powerful. It also satis¬es symmetries that lead to
some speci¬c predictions referred to as reciprocities, which have been veri¬ed
experimentally. We will discuss some other consequences when we introduce
a magnetic ¬eld in Chapter 10.


9.3 Noise in two-terminal systems
Any signal we ever measure has to be detected against a background of noise.
Usually in practical applications, such as telecommunications, noise is a
nuisance and we try to suppress it as much as possible. If the signal-to-
noise ratio is low “ it does not matter how large the actual signal amplitude
is “ the signal may be drowned in noise. But noise also contains information
about the physical processes occurring in a system. Di¬erent processes have
di¬erent kinds of noise, and by carefully analyzing the noise we can gather
useful information. Here we look at some of the noise sources in a meso-
scopic system and their characteristics. For simplicity, we here consider only
two-terminal systems.
330 Mesoscopic physics
Noise is most conveniently analyzed in terms of its spectral density S°!Þ,
which is the Fourier transform of the current“current correlation function,
°1
S°!Þ ¼ 2 dt ei!t hÁI°t þ t0 ; TÞÁI°t0 ; TÞi:
À1


Here ÁI°t; TÞ is the time-dependent ¬‚uctuation in the current for a given
applied voltage V at a given temperature T. In a two-terminal electric system
there are two common noise sources. The ¬rst one is the thermal noise, or
Johnson noise, of a device of conductance G ¼ 1=R at temperature T. It is
due to thermal ¬‚uctuations, and can be derived using the ¬‚uctuation-dissipa-
tion theorem. For low frequencies (low, that is, compared with any charac-
teristic frequency of the system, and such that 0! ( kTÞ, the Johnson noise
has no frequency dependence, and is said to be white. The spectral density is
given by

S ¼ 4kTG:

The other common noise source is shot noise, which occurs when the current
is made up of individual particles. When the transits of the particles through
the device are uncorrelated in time, these processes are described as Poisson
processes, and their characteristic noise is known as Poisson noise. The
Poisson noise is also white for low frequencies, with a spectral density that
is proportional to the current:

SPoisson ¼ 2eI:

Shot noise is a large contributor to noise in transistors, but in mesoscopic
systems correlations can suppress it quite dramatically, giving it a spectral
density much below that of Poisson noise. In the Landauer“Buttiker form-
¨
alism, the current is due to transmission of electrons occupying scattering
states. The Pauli principle forbids multiple occupancy of these states, which
in two-terminal systems necessarily correlates the arrival of electrons to the
source from a single scattering state.
Let us now look at this more quantitatively. At ¬nite temperatures, the
Landauer two-terminal formula is
°1
e
I¼ dE½ fs °EÞ À fd °EފTn °EÞ;
h 0
331
9.3 Noise in two-terminal systems
where fs °EÞ and fd °EÞ are the Fermi distribution functions of states injected
from the source and drain, respectively, and are given by

1
fs °EÞ ¼
1 þ e°EÀÀeVÞ=kT
1
fd °EÞ ¼ :
1 þ e°EÀÞ=kT

It is not a di¬cult exercise to evaluate the spectral density by inserting a
current operator and using the relation between incoming and outgoing
currents given by the Landauer formalism. With the linear response assump-
tion (so that the transmission probabilities are taken to be independent of
energy and evaluated at the common chemical potential) the result is

e2 X
S¼2 ½2Tn kT þ Tn °1 À Tn ÞeV coth°eV=2kTފ:
2
°9:3:1Þ
hn

This expression contains several interesting results. In the limit eV=kT ! 0 it
reduces to the Johnson noise, and it is reassuring that we recover the central
result of the ¬‚uctuation-dissipation theorem from the Landauer“Buttiker ¨
formalism. The second term is the shot noise. This one has some peculiar
characteristics in a mesoscopic system. In the limit of zero temperature, the
shot noise part of Eq. (9.3.1) becomes

e2 X
Sshot °T ! 0Þ ¼ 2eV Tn °1 À Tn Þ:
hn

States for which Tn ¼ 1 or Tn ¼ 0 do not contribute to the shot noise. Shot
noise represents ¬‚uctuations due to the uncorrelated arrivals of electrons. If
Tn ¼ 1, that transmission channel is ˜˜wide open,™™ fully transmitting a steady
stream of electrons without any ¬‚uctuations, which are suppressed by the
Pauli Exclusion Principle. Similarly, if Tn ¼ 0 there is no shot noise simply
because there are no electrons at all arriving in that channel. The maximum
noise that a single channel can contribute apparently occurs for Tn ¼ 0:5,
when the channel is half-way between closed and open, and there is max-
imum room for ¬‚uctuations.
This characteristic of the noise can be veri¬ed experimentally. Consider a
two-terminal quantum point contact and its conductance as a function of
gate voltage. Suppose the initial gate voltage is such that the conductance is
quantized. There is then an integer number of channels for which Tn ¼ 1
332 Mesoscopic physics
while for all others Tn ¼ 0, and so the shot noise is zero. As the gate voltage is
changed, the conductance moves towards a transition region where the con-
ductance changes value. This happens when a new channel is opened or the
highest-lying channel (in energy) is being pinched o¬. As this happens, the
corresponding transmission probability goes from zero to unity (or vice
versa), and the shot noise increases and goes through a maximum. As the
conductance levels o¬ on a new plateau, the shot noise vanishes. The e¬ect of
small, nonzero temperatures is just to round o¬ the shot noise curve.
The shot noise in quantum point contacts has been measured, and the
predictions described above have been veri¬ed, again demonstrating the sim-
plicity and power of the Landauer“Buttiker formalism.
¨


9.4 Weak localization
In general, the conductance of a metallic system increases monotonically as
the temperature is reduced, but there are cases in which the conductance
exhibits a maximum and then decreases as the temperature is reduced further.
One such example is the Kondo e¬ect, which will be the subject of Chapter
11. This phenomenon is the consequence of interactions between a local spin
and the spins of conduction electrons. Systems that exhibit the Kondo e¬ect
are invariant under time reversal, and the conductance maximum is caused
by the onset of strong interactions between conduction electrons and the
local spin. There are, however, other systems that exhibit a maximum in
the conductance but for which, in contrast to Kondo systems, the conduc-
tance maximum is closely related to issues of time reversal symmetry. One
such example occurs when the conduction electrons in manganese are elasti-
cally scattered by impurities. As the temperature is decreased below a few
kelvin the conductance decreases. Furthermore, at temperatures well below
this conductance maximum the system exhibits negative magnetoresistance,
which is to say that if an external magnetic ¬eld is applied the conductance
increases. This indicates that the cause of the conductance maximum some-
how depends on time reversal symmetry, and is destroyed if that symmetry is
broken. Further evidence of this is given by the fact that if some small
amount of gold is added, the magnetoresistance is initially positive as the
external ¬eld is applied. Gold is a heavy element and has strong spin“orbit
scattering, which destroys time reversal invariance by a subtle e¬ect called the
Aharonov“Casher e¬ect.
This phenomenon of decreasing conductance, or increasing resistance, at
low temperatures in the presence of time reversal symmetry is another man-
ifestation of the long-range phase coherence of the electron wavefunctions. In
333
9.4 Weak localization
this case, the coherence leads to interference e¬ects in which an electron
interferes destructively with itself. In order for this to be possible at all, the
phase breaking length must be long enough that the electrons di¬use through
elastic scattering while maintaining their phase coherence for some reason-
able distance. That is why the temperature has to be low in order for the
e¬ect, which is called weak localization, to be observable.
In principle, all the physics of weak localization is contained within the
Landauer“Buttiker formalism. In this case, the transmission probabilities Tn
¨
must show some strong behavior for certain channels for which the inter-
ference e¬ects must somehow reduce Tn . Note that while the Landauer“
Buttiker formalism assumes that di¬erent electrons have no phase relations
¨
and so do not interfere with each other, it certainly leaves open the possibility
that each electron state can interfere with itself on its path from one terminal
to another. However, in the case of weak localization the Landauer“Buttiker ¨
formalism does not easily lend itself to practical calculations. In fact, in order
to deal correctly with the problem, one has to use rather sophisticated many-
body perturbation techniques. Instead, we will here give some more intuitive
arguments for what lies behind weak localization.
We consider an electron as it traverses a mesoscopic system from source to
drain. In this case, there is a rather high density of impurities, so the electron
scatters frequently. As a consequence, the electron performs a random walk
through the system. In many respects this is similar to the case for ˜˜normal™™
electron transport in which there is no phase coherence. The motion of a
random walker can, at times long compared with a typical time between
collisions, be described as a classical di¬usion problem. According to the
Einstein relation, the di¬usion constant D0 is proportional to the mobility,
and hence to the conductance. In three dimensions, the probability that a
particle has moved a net distance r in a time t is given by

exp °Àr2 =4D0 tÞ
P3 °r; tÞ ¼ :
°4D0 tÞ3=2

From this equation, we can ¬nd the probability amplitude that the particle
returns to its original position. In general, there may be many di¬erent paths
that take the electron back to the origin during some in¬nitesimal time
interval dt at some time t. Let us for simplicity consider two such paths
with probability amplitudes 1 °r ¼ 0; tÞ and 2 °r ¼ 0; tÞ. To ¬nd the
probability we have to take the squared modulus of the sum of the prob-
ability amplitudes, j 1 °r ¼ 0; tÞj2 þ j 2 °r ¼ 0; tÞj2 þ *°r ¼ 0; tÞ 2 °r ¼ 0; tÞ þ
1
1 °r ¼ 0; tÞ 2
* °r ¼ 0; tÞ. In ˜˜normal™™ macroscopic systems, the electron
334 Mesoscopic physics
su¬ers inelastic collisions which randomize the phase along each path.
Consequently, there is no phase relation between probability amplitudes
1 °r ¼ 0; tÞ and 2 °r ¼ 0; tÞ, and the interference terms vanish as we add up
contributions from all possible paths. In mesoscopic systems, the phase is
preserved for all paths shorter than ˜ , since then the electron returns to the
origin within the phase breaking time  ¼ ˜ =vF . Most of those paths, how-
ever, also have random relative phases and the interference terms vanish.
However, there is now one class of paths for which the interference terms
do not vanish. These are paths that are related by time reversal. In the
absence of a magnetic ¬eld such paths have probability amplitudes that are
precisely complex conjugates of each other, 1 °r ¼ 0; tÞ ¼ * °r ¼ 0; tÞ, and
2
these paths interfere constructively. This means that the particle has an
enhanced probability of returning to the origin, compared with incoherent
classical di¬usion. As a consequence, the probability that the particle has
moved some net distance in a time t is reduced. This reduces the di¬usion
constant, and, according to the Einstein relation, reduces the conductivity.
We can make the argument more quantitative by considering an electron
wavepacket at the Fermi surface. In order for the wavepacket to be able to
interfere it must have a spatial extent Áx of the order of its wavelength,
Áx % F . In a time t, the wavepacket thus sweeps out a volume
 % dÀ1 vF t, where d is the spatial dimensionality of the system. This
F
wavepacket can interfere with itself provided it returns to the origin at
some time t. The probability for this to happen is

 dÀ1 vF t
F
P°r ¼ 0; tÞ % ¼ :
°4D0 tÞd=2 °4D0 tÞd=2

Each such event decreases the e¬ective di¬usion. We now need to sum over
all such events. These can occur only at times less than the phase breaking
time  , since for longer times the phases of the two time-reversed paths have
been randomized and will no longer interfere. We must also insert some
minimum time 0 below which there are on average no collisions, denying
the electron any chance of returning. This lower limit is of the order of the
elastic scattering time e ¼ ˜e =vF . In three dimensions we put d ¼ 3 to ¬nd
° 
22 vF ½°e ÞÀ1=2 À ° ÞÀ1=2 Š
F
dtP3 °r ¼ 0; tÞ % : °9:4:1Þ
°4D0 Þ3=2
e


This probability that an electron can return to the origin with some memory
of its original phase will be of roughly the same magnitude as the fractional
335
9.4 Weak localization
reduction in conductance Á= caused by the interference of the time-
reversed paths. If we substitute the Drude formula D0 ¼ vF ˜e =d we ¬nd
the fractional increase in the resistivity to be

Á= % ½1 À °e = Þ1=2 Š=°kF ˜e Þ2 : °9:4:2Þ

By including the temperature dependence of the elastic relaxation time
(which yields the temperature dependence of the elastic mean free path)
and the phase breaking time we also get an estimate of the overall tempera-
ture dependence of the increase in resistivity.
In two dimensions, the result is more striking, since the denominator in the
expression for P2 °r ¼ 0; tÞ decays only as tÀ1 rather than as tÀ3=2 . An integra-
tion analogous to that in Eq. (9.4.1) gives the fractional increase in resistivity
to be


1
Á= % ; °9:4:3Þ
ln
kF ˜e e

which can be quite signi¬cant if the ratio of phase breaking time to elastic
relaxation time is large. In two dimensions, the probability of a random
walker returning to the origin is much larger than in three dimensions,
which leads to a dramatic enhancement in the resistivity.
Note that weak localization depends sensitively on the phase relation
between time-reversed paths. If this relation is altered, the weak localization
is in general suppressed. One way to achieve this is to apply an external
magnetic ¬eld. This adds a so-called Aharonov“Bohm phase to the path of
an electron. For closed paths, this phase is equal to the 2 times the number
of ¬‚ux quanta enclosed by the path, and the sign is given by the sense of
circulation of the path. If the Aharonov“Bohm phase added to one path is
Á, then the time reversed path gets an added phase ÀÁ, and the inter-
ference term between these two paths now has an overall phase factor. Since
di¬erent pairs of time-reversed paths will have di¬erent, and, on average
random, phase factors, the net e¬ect of the magnetic ¬eld is to wipe out
rapidly the weak localization as the contributions to the increase in resistance
from di¬erent pairs of paths are added together. Hence, the resistance of a
system in the weak localization regime is observed to decrease (˜˜negative
magnetoresistance™™) as an external magnetic ¬eld is applied.
There are also other phenomena associated with phase coherence. One
such is the existence of the so-called universal conductance ¬‚uctuations. If
the conductance of a mesoscopic system is measured as a function of some
336 Mesoscopic physics
external control parameter, for example a magnetic ¬eld or impurity con¬g-
uration, there are seen to be ¬‚uctuations in the conductance. The speci¬c
¬‚uctuations are sample-dependent and reversible as the external control
parameter is swept up and down, but the magnitude is universal and is of
the order of the conductance quantum e2 =h. The root cause of the ¬‚uctuations is
interference between di¬erent paths between two points in the sample. As the
control parameter is varied, the precise interference patterns change but do
not disappear, in contrast to weak localization. The universal conductance
¬‚uctuations are, however, suppressed in magnitude by processes that destroy
time-reversal invariance. In fact, one can use universal conductance ¬‚uctua-
tions to detect the motion of single impurities in a sample, as this will lead to
observable changes in the conductance ¬‚uctuation pattern.


9.5 Coulomb blockade
We close this chapter by brie¬‚y discussing a phenomenon known as Coulomb
blockade. Although it does not per se depend on wavefunction coherence, it is
intimately related to small (nano-scale) devices, and so one can make the case
that it is a mesoscopic e¬ect. It is a quantum phenomenon in that, while it
does not depend directly on 0, it does rely for its existence on the quantized
nature of electric charge. It re¬‚ects the fact that the capacitance of mesoscale
devices can be so small that the addition of a single electron may cause an
appreciable rise in voltage.
We consider a small metallic dot, like a tiny pancake, connected to two
leads. We here explicitly take the connections to the leads to be weak. This
means that there are energy barriers that separate the leads from the dot, and
through which the electrons have to tunnel in order to get on and o¬ the dot.
This allows us, to a reasonable approximation, to consider the electron states
on the dots as separate from the electron states in the leads. As a conse-
quence, it is permissible to ask how many electrons are on the dot at any
given time. If the leads had been strongly coupled to the dot, then electron
eigenstates could simultaneously live both on the dot and in the leads, and we
would not have the restriction that only an integer number of electrons could
reside on the dot. Also, the resistances at the junctions with the dot have to be
large enough that essentially only one electron at a time can tunnel on or o¬.
The condition for this is that the junction resistances be large compared with
the resistance quantum RQ ¼ h=e2 . The tunneling rates are then low enough
that only one electron at a time tunnels. Note also that it is important that
we consider the dot to be metallic, so that there is a fairly large number of
electrons on the dot, and the available states form a continuum. Technically,
337
9.5 Coulomb blockade
we can then consider the operator for the number of electrons to be a classi-
cal variable, just as we did in the liquid-helium problem when we replaced the
y
number operator a 0 a0 for the condensate with the simple number N0 .
Experimentally, the dots can also be made out of semiconductors, in which
case one may have a very small number of electrons “ of the order of ten or so
“ on the dots. In that case, the discrete spectrum of eigenstates on the dots
has to be considered more carefully, and there may be interesting and com-
plicated correlation e¬ects between the electrons.
We now imagine that we connect the leads to some source of potential
di¬erence V, and we monitor the current that ¬‚ows from one lead, through
the dot, and into the other lead. What we ¬nd is that for most values of V the
current is totally negligible, while for some discrete set of voltages the con-
ductance through the dot is rather high, resulting in a conductance vs. bias
voltage curve that looks rather like a series of evenly spaced delta-functions.
It turns out that it is rather easy to come up with a qualitative picture that is
even quantitatively rather accurate. We model the dot and the junctions
according to Fig. 9.5.1, in which C1 and C2 , and V1 and V2 are the capaci-
tances and voltages across each junction, respectively. The total voltage
applied by the voltage supply is V ¼ V1 þ V2 , and the charge on each junc-
tion is Q1 ¼ n1 e ¼ C1 V1 and Q2 ¼ n2 e ¼ C2 V2 , with n1 and n2 the number of
electrons that have tunneled onto the island through junction 1, and the
number of electrons that have tunneled o¬ the island through junction 2,
respectively. Because the tunneling rates across each junction may di¬er, Q1
and Q2 are not necessarily equal. The di¬erence is the net charge Q on the
island,
Q ¼ Q2 À Q1 ¼ Àne;




Figure 9.5.1. Equivalent electrostatic circuit of an island connected to a voltage
source through two tunneling junctions.
338 Mesoscopic physics
with n ¼ n1 À n2 an integer. The island itself has a capacitance Ctot ¼
C1 þ C2 , obtained by grounding the external voltage sources and connecting
a probe voltage source directly to the island.
The electrostatic energy of the system (junctions, island, and voltage source)
consists of electrostatic energy E s stored in the junctions, minus the work W
done by the voltage source in moving charges across the junction. As an
electron tunnels o¬ the island through junction 2, there is a change in voltage
across junction 1. Charge then ¬‚ows to junction 1 from the voltage source,
and the voltage source does work. Similarly, if an electron tunnels onto the
island through junction 1, the voltage across junction 2 changes and charge
¬‚ows from junction 2 to the voltage source. Coulomb blockade occurs if
there is some minimum voltage V that has to be supplied in order to have
an electron tunnel onto or o¬ the island. If V is less than this threshold, no
current can ¬‚ow through the island.
In order to look at this quantitatively, we ¬rst note that we can write
C1 V1 þ C2 °V À V2 Þ C2 V þ ne
V1 ¼ ¼ °9:5:1Þ
Ctot Ctot
C2 V2 þ C1 °V À V1 Þ C1 V À ne
V2 ¼ ¼ : °9:5:2Þ
Ctot Ctot

These two equations give us the voltage across one junction as an electron
tunnels through the other junction. The electrostatic energy stored in the
junctions is

C1 V1 þ C2 V2 C1 C2 °V1 þ V2 Þ2 þ °C1 V1 À C2 V2 Þ2 C1 C2 V 2 þ Q2
2 2
Es ¼ ¼ ¼ :
2 2Ctot 2Ctot

If an electron of charge Àe tunnels out of junction 2, then the charge Q on the
island increases by þe. According to Eq. (9.5.1), the voltage V1 then changes
by Àe=Ctot . To compensate, a charge ÀeC1 =Ctot ¬‚ows from the voltage
source. If we now consider n2 electrons tunneling o¬ the island through
junction 2, the work done by the voltage source is then Àn2 eVC1 =Ctot .
Next, we apply the same reasoning to electrons tunneling onto the island
through junction 1. The result is that for n1 electrons tunneling onto the
island, the voltage source does an amount of work equal to Àn1 eVC2 =Ctot .
For a system with a charge Q ¼ Àne ¼ Àn2 e þ n1 e on the island, the total
energy is then
1 eV
E°n1 ; n2 Þ ¼ E s À Ws ¼ ½C1 C2 V 2 þ Q2 Š þ ½C n þ C2 n1 Š:
Ctot 1 2
2Ctot
339
Problems
We can now look at the cost in energy for having an electron tunnel onto or
o¬ an initially neutral island (Q ¼ 0). If we change n2 by þ1 or À1, the energy
changes by
 
ee
EÆ ¼ Æ VC1 :
2
Ctot 2

Similarly, if we change n1 by þ1 or À1, the change in energy is
 
ee
EÆ ¼ Ç VC2 :
1
Ctot 2

Because the ¬rst term in each of these expressions is inherently positive, the
energy change is also positive when V is small, and the process of electron
transfer will not occur. It will not be until we reach a threshold voltage of
V ¼ e2 =2C1 and V ¼ e2 =2C2 , respectively, that a reduction in energy will
accompany the transfer. In other words, until the threshold voltage has
been reached, the charge on the island cannot change, which means that
no current ¬‚ows through the system. This prevention of conduction by the
requirement that the charging energy be negative is called Coulomb block-
ade. We note that for symmetric barriers, for which C1 ¼ C2 , the threshold
voltage for an initially neutral island is V ¼ e=Ctot .
Quantum dots can in principle be made into single-electron transistors, and
logical circuits can be constructed with single-electron transistors as building
blocks. However, for this to be practically useful, one has to ensure that the
charging energy e2 =2C ) kT, where T is the temperature. For a dot of size
20 ‚ 20 nm2 , which can be fabricated by electron beam lithography, the
capacitance is of the order of 10À17 F, and then the phenomenon will be
observable only at temperatures less than about 10 K. However, dots or
atomic clusters of size 1 nm or less would have capacitances of the order of
or less than 10À19 F, in which case the charging energy is of the order
of electron volts. This opens up the possibility of very compact integrated
circuits and computers.


Problems
9.1 It was stated at the end of the paragraph following Eq. (9.1.1) that ˜˜in
the absence of an external magnetic ¬eld there must exist an outgoing
state in the same terminal with the same energy but opposite wave-
number.™™ Why is this?
340 Mesoscopic physics




Figure P9.1. Schematic of the ballistic quantum point contact for Problem 9.2.


9.2 Calculate the conductance of a ballistic quantum point contact in a
semiclassical two-dimensional electron gas. Assume that a barrier parti-
tions the electron gas with Fermi energies s and d on each side of the
barrier, respectively, with a corresponding di¬erence n in densities, as in
Fig. P9.1. A constriction of width w lets electrons cross from one side to
the other. First calculate the net ¬‚ux through the constriction. This is due
to the excess density n at the source incident with speed vF on the
constriction, and averaged over angle  of incidence. This will give you
the current I through the constriction as a function of n. The chemical
potential di¬erence is  ¼ eV, with V the source-to-drain voltage. In
the expression for the conductance, n= can be taken to be the density
of states of the two-dimensional electron gas.

9.3 A model of a smooth quantum point contact is the saddle-point poten-
tial,

V°x; yÞ ¼ V0 À 1 m!2 x2 þ 1 m!2 y2 ;
x y
2 2

where the curvature of the potential is expressed in terms of the frequen-
cies !x and !y . This potential is separable, and one can solve for the
transmission probabilities. With the reduced variable
 
E À °n þ 1Þ0!y À V0
n ¼ 2 ;
2
0!x

where n denotes the transverse channels, the transmission probabilities
are
1
Tnm ¼ nm :
1 þ eÀn
Plot Tnn as a function of °E À V0 Þ=°0!x Þ for di¬erent values of !y =!x
ranging from <1 to >1 for the three lowest channels n ¼ 0; 1; 2. Set up
the expression for the total conductance, given these transmission prob-
abilities. Under what conditions (at zero temperature) would you say
341
Problems
that the conductance is quantized? [Hint: Calculate the maximum and
minimum slope of the conductance vs. Fermi energy. How wide and ¬‚at
are the plateau regions of the conductance? What would you require of
!x and !y in order to say that the conductance is well quantized?] (This
problem was posed by M. Buttiker.)
¨

9.4 Derive the set of equations for a four-terminal system that correspond to
those given for a three-terminal system in Eqs. (9.2.4).

9.5 A tunneling device with transmission probabilities Tn ( 1 of resistance
100  is to be connected in series with a 25  resistor. The system, con-
sisting of tunneling device and resistor, will operate at room temperature.
Assume that the noise spectrum is white for any bandwidth under con-
sideration. Under what conditions will the noise power of the system be
dominated by Johnson and shot noise, respectively? [Hint: start with
Eq. (9.3.1) and obtain an expression for the noise in the limit of Tn ( 1
for the tunneling device. You must also add the Johnson noise from the
resistor.]

9.6 Fill in the missing steps that lead from Eq. (9.4.1) to Eq. (9.4.2).
Chapter 10
The quantum Hall e¬ect




10.1 Quantized resistance and dissipationless transport
The Hall e¬ect has long been a standard tool used to characterize conductors
and semiconductors. When a current is ¬‚owing in a system along one direc-
tion, which we here take to be the y-axis, and a magnetic ¬eld H is applied in
a direction perpendicular to the current, e.g., along the z-axis, there will be an
induced electrostatic ¬eld along the x-axis. The magnitude of the ¬eld E is
such that it precisely cancels the Lorentz force on the charges that make up
the current. For free electrons, an elementary calculation of the type indi-
cated in Section 1.8 yields the Hall resistivity H ¼ ÀH=0 ec, and apparently
provides a measure of the charge density of the electrons. For Bloch elec-
trons, as we saw in Section 8.3, the picture is more complicated, but H is still
predicted to be a smoothly varying function of H and of the carrier density.
In some circumstances, however, the semiclassical treatment of transport
turns out to be inadequate, as some remarkable new e¬ects appear.
In a two-dimensional system subjected to strong magnetic ¬elds at low
temperatures, the response is dramatically di¬erent in two respects. First,
the Hall resistivity stops varying continuously, and becomes intermittently
stuck at quantized values H ¼ Àh=je2 for a ¬nite range of control parameter,
e.g., external magnetic ¬eld or electron density. In the integer quantum Hall
e¬ect, j is an integer, j ¼ 1; 2; . . . ; and in the fractional quantum Hall e¬ect,
j is a rational number j ¼ q=p, with p and q relative primes and p odd.
(In addition, there exists a fractional quantum Hall state at  ¼ 5=2 and
possibly other related states. The physics of these is, however, very di¬erent
from that of the ˜˜standard™™ odd-denominator fractional quantum Hall states
and will not be discussed here.) Second, at the plateaus in H at which it
attains these quantized values, the current ¬‚ows without dissipation. In other
words, the longitudinal part of the resistivity tensor is zero. The resistivity

342
343
10.1 Quantized resistance and dissipationless transport
and conductivity are both tensor quantities, and it happens that the lon-
gitudinal conductivity also vanishes at these plateaus. This may sound a little
strange, but is a simple consequence of the dissipationless transport in two
dimensions in crossed electric and magnetic ¬elds.
As we shall see, these two observations, a quantized Hall resistance and
dissipationless transport, can be understood if the system is incompressible
(that is, it has an energy gap separating the ground state from the lowest
excited state) and if there is disorder, which produces a range of localized
states. Our ¬rst task will be to ask what causes the incompressibility and
energy gap. In the integer quantum Hall e¬ect, the energy gap (which is
responsible for the incompressibility) is a single-particle kinetic energy gap
due to the motion of single particles in an external ¬eld. It is not necessary to
introduce electron“electron interactions in order to explain the integer quan-
tum Hall e¬ect. In the fractional quantum Hall e¬ect, on the other hand, the
energy gap and the ensuing incompressibility are entirely due to electron“
electron interactions. This, and the absence of any small parameter in the
problem that would permit a perturbation expansion, makes it a very di¬cult
system to study theoretically.
The presence of disorder is necessary in order to explain the plateaus in the
quantized Hall resistivity. Disorder gives us a range of energies within which
states are localized, and as the Fermi energy sweeps through these states the
Hall resistivity exhibits a plateau. In the integer quantum Hall e¬ect, the
disorder dominates over the electron“electron interactions. In the fractional
e¬ect the strengths are reversed. The fractional e¬ect occurs only in samples
that are very clean, and which consequently have a very high electron mobi-
lity. There is, of course, no sharp division between the integer and the frac-
tional quantum Hall e¬ect, and there is no magical amount of disorder at
which the fractional quantum Hall e¬ect is destroyed. Which plateaus, and
thus which fractional or integer quantum Hall states, will be observed
depends on how much disorder there is in the system and what the tempera-
ture is. If we start by imagining a very clean system in the limit of zero
temperature, the Hall resistivity vs. control parameter will exhibit a series
of plateaus corresponding to all fractional and integer Hall states, but the
extent of each plateau becomes very small. As we start to add impurities to
the system, the fractional quantum Hall states with the smallest energy
gaps are destroyed, since the perturbations introduced by the disorder
become larger than the smallest energy gaps. The corresponding plateaus
disappear and neighboring plateaus grow in size. At su¬cient disorder, all
fractional quantum Hall plateaus have vanished and we are left with only
the plateaus of the integer quantum Hall states. Similarly, increasing the
344 The quantum Hall effect
temperature will destroy the quantum Hall e¬ect, as it is, strictly speaking, a
zero-temperature phenomenon. By this we mean that the quantization of the
Hall conductance and vanishing of the longitudinal resistance are exact only
in the limit of low temperatures. As the temperature is raised, the weakest
fractional quantum Hall states will start to disappear. As the temperature is
increased further, the integer quantum Hall states will eventually su¬er the
same fate.



10.2 Two-dimensional electron gas and the integer quantum Hall effect
We start by considering a two-dimensional gas of N noninteracting electrons
in an external magnetic ¬eld and with no disorder. Let the area in the xy-
^
plane be A and the magnetic ¬eld be B ¼ B z. The Hamiltonian of this system
is simply

!2
1X N
e
H0 ¼ pj þ A°rj Þ ;
2m* j¼1 c


where A°rj Þ is the vector potential at the position rj of electron j, the charge
on the electron is now taken to be Àe, and m* is the band mass of the
electron, e.g., m* % 0:07me in GaAs. The ¬rst thing we have to do is to ¬x
a gauge for the vector potential, and there are two common choices for this
depending on which symmetry we want to emphasize. The ¬rst choice is the
so-called Landau gauge, A ¼ Bx^ . This gauge is translationally invariant
y
along the y-axis and so the single-particle eigenstates can be taken to be
eigenstates of py . This choice of gauge is convenient for rectangular geo-
metries with the current ¬‚owing along the y-axis. The other choice is the
^
^
symmetric gauge A ¼ 1 B°x^ À yxÞ ¼ 1 Br/. As the last equality shows, this
y
2 2
gauge is rotationally invariant about the z-axis, and the single-particle eigen-
states can be taken to be eigenstates of the z-component of angular momen-
tum. This choice of gauge is convenient for circular geometries (quantum
dots) and is the gauge in which the Laughlin wavefunction for fractional
quantum Hall states is most easily represented.
For now, we use only the Landau gauge, and with this choice the
Hamiltonian H0 becomes
!
1X @2 @2 0e @ e2 2 2
H0 ¼ À 02 2 À 02 2 þ 2 þ B xj : °10:2:1Þ
Bxj
@yj c2
@xj @yj
2m* j ic
345
10.2 Two-dimensional electron gas and the integer quantum Hall effect
In the absence of potentials that break the translational invariance along the
y-axis, we can write the single-particle states as

kn °x; yÞ ¼ kn °xÞeiky : °10:2:2Þ

We apply periodic boundary conditions along a length Ly on the y-axis. The
admissible values of k are then given by k ¼ 2ik =Ly , with ik ¼ 0; Æ1; Æ2; . . .
By applying the Hamiltonian (10.2.1) to the wavefunction (10.2.2) we obtain
the single-particle Schrodinger equation
¨
!
02 d 2 1
À þ m*!c °x À xk Þ kn °xÞ ¼ E kn kn °xÞ;
2 2
°10:2:3Þ
2m* dx2 2

where !c ¼ eB=°m*cÞ is the cyclotron frequency,¬¬¬¬¬¬¬¬¬¬¬¬¬ k ¼ À°0c=eBÞk, which
p and x
we write as À˜B k, with ˜B the magnetic length, 0c=eB. This is the character-
2

istic length scale for the problem, and is about 10 nm for magnetic ¬elds of 5
to 10 T. Equation (10.2.3) is, for each allowed value of k, the equation for a
harmonic oscillator centered at the position x ¼ xk , and so the energy eigen-
values are

E nk ¼ °n þ 1Þ0!c n ¼ 0; 1; 2; . . . °10:2:4Þ
2


Surprisingly, the energy eigenvalues do not depend on the momentum 0k
along the y-axis, but only on the index n, the so-called Landau level index,
and all states with the same quantum number n form a Landau level. This
means that there is a huge degeneracy in energy. The center points of the
states are xki ¼ À˜2 ki , and the centers of two neighboring states along the x-
B
axis are separated by a distance Áx ¼ 2˜2 =Ly . If the system has a width Lx
B
we can ¬t Lx =Áx states in one Landau level across this width. Each Landau
level thus contains Lx =Áx ¼ Lx Ly =°2˜2 Þ states, which is the degeneracy of
B
each Landau level. Another way to think of this is that each state occupies an
area 2˜2 , and the degeneracy is just the total area A ¼ Lx Ly divided by the
B
area per state.
The degeneracy, or the area per state in units of 2˜2 , leads us to de¬ne a
B
very useful quantity, the ¬lling factor , which is a conveniently scaled mea-
sure of the density of the system. The ¬lling factor is de¬ned as  ¼ 2˜2 ,B
with  now the number of electrons per unit area. Thus, when  ¼ 1, all the
states in the lowest Landau level n ¼ 0 that lie within the area A are ¬lled.
Another way to look at the ¬lling factor, which is especially useful when we
deal with the fractional quantum Hall e¬ect, is that it is a measure of the
346 The quantum Hall effect
number of electrons per ¬‚ux quantum. For this system the ¬‚ux quantum is
È0 ¼ hc=e. It is double the ¬‚ux quantum 0 ¼ hc=2e introduced in Section
7.9 because we are now dealing with single electrons rather than electron
pairs. The total ¬‚ux piercing the system is È ¼ BA ¼ È0 BAe=°hcÞ ¼
È0 A=°2˜2 Þ, so the number of ¬‚ux quanta NÈ0 is A=°2˜2 Þ. Thus the number
B B
of electrons per ¬‚ux quantum is A=NÈ0 ¼ 2˜B  ¼ .
2

With the single-particle energy spectrum given by Eq. (10.2.4), the density
of states for the system of noninteracting particles consists of a series of
-functions of weight A=°2˜2 Þ at the energies °n þ 1Þ0!c , as depicted in
B 2
Fig. 10.2.1. If we plot the ground state energy E 0 °Þ of the N independent
electrons as a function of ¬lling factor  we obtain a piecewise linear plot with
slope °n þ 1Þ0!c and with discontinuities of magnitude 0!c in the slope at
2
integer ¬lling factors, as shown in Fig. 10.2.2. As we add more electrons to a
system, we occupy states in the lowest Landau level n 0 that still has vacant
states available. These states are all degenerate and each extra electron adds




Figure 10.2.1. The density of states of a noninteracting two-dimensional electron gas
in a magnetic ¬eld.




Figure 10.2.2. Ground-state energy E vs. ¬lling factor  for a noninteracting two-
dimensional electron gas in a magnetic ¬eld. As the nth Landau level is being ¬lled,
the energy increases by °n þ 1Þ0!c per particle. When the nth Landau level is precisely
2
¬lled, adding a new electron will require °n þ 3Þ0!c , causing the slope of the curve to
2
change discontinuously.
347
10.2 Two-dimensional electron gas and the integer quantum Hall effect
an energy °n 0 þ 1Þ0!c . When the last available state in this Landau level has
2
been ¬lled, the next electron will need an energy °n 0 þ 1 þ 1Þ0!c , and so the
2
slope increases discontinuously by 0!c . Since the ground-state energy has
angles at integral , this implies that the zero-temperature chemical potential,
 
@E 0
¼ ;
@N B


has discontinuities at integer ¬lling factors, as shown in Fig. 10.2.3. Finally,
we use the fact that the isothermal compressibility  is related to the chemical
potential  through

d
À1 ¼ 2 ;
d

with the derivative taken at constant (here T ¼ 0) temperature. At the integer
¬lling factors, the slope of  vs.  approaches in¬nity, and so the compres-
sibility vanishes there. The compressibility measures the energy cost of
˜˜squeezing™™ the system in¬nitesimally. The compression is created by excit-
ing particles from just below the Fermi energy to just above the Fermi energy
in order to make a long-wavelength density perturbation. For a compressible
system, this costs only an in¬nitesimal energy. However, when the system is
said to be incompressible, compressing the system in¬nitesimally requires a
¬nite energy. This is what happens at integer ¬lling factors: one set of Landau
levels is completely ¬lled, and particles can only be excited by crossing the
energy gap 0!c to the next Landau level.
Let us now turn to the response of the system to a transverse electric ¬eld.
In the absence of any external potential (including disorder), we can easily
calculate the current carried by each single-particle state. The operator that




Figure 10.2.3. The chemical potential for the two-dimensional electron gas in a
magnetic ¬eld has discontinuous jumps whenever a Landau level has been ¬lled.
348 The quantum Hall effect
describes the current is
 
e e
J¼À pþ A
m* c

for an electron of charge Àe. This operator can be thought of as being propor-
tional to a derivative of the Hamiltonian with respect to the vector potential.
This is a very useful observation, and we turn it into a formal device by
introducing a ¬ctitious vector potential a ¼ À°qÈ0 =Ly Þ^ ¼ À½qhc=°eLy ފ^ .
y y
Here q is a dimensionless parameter, È0 is the ¬‚ux quantum hc=e, and we
have applied this ¬ctitious vector potential along the y-axis in order to relate
it most easily to Jy . We note that r ‚ a ¼ 0, so a does not correspond to any
physical magnetic ¬eld through the system. However, if we imagine making
the system a loop in the yz-plane by tying together the ends along
the y-direction, qhc=e could be due to a real magnetic ¬eld piercing

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