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 :

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 °EÞ

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

h

2e

°9:2:4Þ

0 ¼ À ½°M2 À R22 Þ2 À T21 1 À T23 3

h

2e

ÀI ¼ À ½°M3 À R33 Þ3 À T31 1 À T32 2 ;

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

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 °EÞTn °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=2kTÞ:

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

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 =°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 þ ½C n þ C2 n1 :

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 Þ^ .

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