Marder M.P. Condensed Matter Physics

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Quantum Hall Effect

783

hu}

/2 remains <&/<£>o even after impurities and electron-electron interactions

are considered. The large energy gap between the first and second Landau

levels makes this assumption increasingly plausible as the induction strength

B increases.

Assume that the densities of localized states constituting the shaded regions

in Figure 25.8(A) are rigid and do not change as B changes.

As B increases, the total number of states in the Landau level increases in propor-

tion. If the numbers of localized states in the tail of the Landau level do not change,

virtually all of the increase occurs through a rise in the number of extended states

at the center of the level. So long as \i lies within the region of localized states,

changes in the total number of charges N contained in the current loop must con-

tinue to be given by Eq. (25.51). The accuracy with which the Hall conductivity

is quantized depends upon the accuracy with which the two assumptions about

energy levels are obeyed; experimentally, the accuracy is at the level of parts per

billion.

Figure 25.9. If the electric field across a sample is expressed as a time-dependent vector

potential, it can be made to disappear from everything but the boundary conditions, and

the boundary conditions return to their starting values at time intervals T. Dots indicate

electrons traveling around wire, while reservoirs at

/ÂQ

and ß\ maintain voltages 0 and V\.

Second Explanation: Gauge Invariance. A different way to think about the

quantum Hall effect was proposed by Laughlin (1981). Consider a single electron

moving in a thin strip and in a strong uniform magnetic induction B, shown in

Figure 25.9. The strip should be large enough so that its outer edges in the y

direction are out of the range of B. Draw an imaginary perfect rectangle within the

strip but at the top and bottom passing beyond the magnetic induction, and consider

the quantum-mechanical problem restricted to the area within the rectangle. In the

absence of any potential difference along y, specializing to the gauge

A=yxB,

(25.108)

784 Chapter

25.

Magnetism of Ions and Electrons

Schrödinger's equation is

-h

1 (h d exB\

-2nT^

{l¥y

U(r)

is a potential that

(7) = 0 includes effects of the

*■

' ' crystal, and impurities.

(25.109)

Now introduce a slight complication that for the moment will seem artificial. There

is some boundary condition that must be imposed when solving the Schrödinger

equation; whatever it may be, it can be indicated as a linear condition

E[^(JC,L)]=0, (25.110)

where

is some operator that gives the boundary conditions. Now define t/>

(?) to

be eigenfunctions of the equation such that

£[V>

(;c, L)e

}=0. (25.111)

These eigenfunctions can certainly be found for any applied field and any 7, al-

though it is not clear why one should care.

Now put an electric field E(r) onto the sample, and assume it points only in the

y direction. This field is not just the applied field, but could also include the field

generated by Coulomb charges within the strip. Employ the method of Section

16.3,

which derives electron fields from time-dependent vector potentials, as in

Eq. (16.30). The Hamiltonian becomes

-h

1 fh d exB

1 (H

exB

, - V rr,^

2m dx 2m \ i dy

V>(r) = 0. (25.112)

Because

of the explicit time dependence in the vector potential, this equation

does

not follow from the time-dependent Schrödinger

equation.

However, as in

Section

16.3,

the errors introduced in this way are exponentially small.

The electric field can be eliminated formally from the problem by writing

ieV

4>,

(25.113)

where V is the electric potential defined by

£ = -W. (25.114)

Because near y = 0 (the bottom of the rectangle)

and

y = L (the top of

the

rectangle)

there is no magnetic field and the strip is metallic, the voltage is constant in space to

high accuracy. In particular, the electric potential at the top of the strip, V(L)

—

is independent of

The function

therefore obeys precisely the same equation as

ij)

in the absence of a field, but it obeys boundary condition

B[e-

ietVl/n

4>(x,L)} = 0. (25.115)

In other words, ip is of the form indicated in Eq. (25.111) with

7 = -^. (25.116)

Quantum Hall Effect 785

Now consider what happens when

eVit

~h~

This condition occurs after a time

2vr. (25.117)

T=-£-. (25.118)

The boundary condition has returned to its value at t = 0, so all the single-particle

states

are the same as the states

tp.

The occupation of the states might be differ-

ent. That is, as time progresses from 0 to T, each state / must evolve until finally

at time 7 it turns into some other state /'. All the quantum states at t = 7 must

be identical to those at t = 0, but they are not necessarily occupied in the same

way. In fact, if some Landau level is incompletely filled, the occupation of states

should be expected to change. However, if the Fermi level lies within the region of

localized states between Landau levels, the occupation of states at t = T must be

precisely what it was at t = 0. First consider the localized states. They are trapped

in potential wells, with wave functions that are exponentially small at the system

boundaries, so as the boundary condition indicated by Eq. (25.116) changes, they

change exponentially little. Next consider the extended states within a filled Lan-

dau level. All of these states are occupied at t = 0, and when t arrives at T they

must all remain occupied.

In short, one can be quite sure that the system at t = 7

indistinguishable from

the system at t = 0. Because the voltage at top and bottom is imposed without

allowing current to flow along y, the most that could have happened is that an

integer number v of electrons was transported through the sample in the x direction,

as indicated in Figure 25.9. The current that has passed through the system is

therefore

= -= = —r

, (25.119)

T h

which means that the conductance is

= v— = —. (25.120)

25.5.2 Fractional Quantum Hall Effect

The integer quantum Hall effect depends crucially upon the presence of disorder,

which allows the Fermi level to lie in a gap between extended states and thereby

produces plateaus in the transverse conductivity. Tsui et al. (1982) created two-

dimensional electron gases in GaAs-AlGaAs with electron mobilities [Eq. (19.65)]

on the order of

5 •

-1

, 100 times larger than the mobilities in samples

where the integer Hall effect was observed. The importance of impurities was

786

Chapter

25.

Magnetism of Ions and Electrons

100

50 -

« 25

2 10

«2

50 100

Induction ß (kG)

200

Figure 25.10. Measurement of fractional quantum Hall effect in GaAs/GaAlAs het-

erostructure with mobility of 4-10

s~' and electron density of

1.45-10

-2

at a temperature of 150 mK. [Data of Boebinger, Chang, Stornier, and Tsui, published by

Chang (1990),

185.]

correspondingly less, but instead of eliminating plateaus in the Hall voltage, a host

of new plateaus appeared, as illustrated by the data in Figure 25.10.

The physics underlying the fractional quantum Hall effect is in some ways

quite different from the physics underlying the integer effect. Locking the voltage

into plateaus is due to the formation of a quantum

liquid,

in which the electrons

form a highly correlated state to minimize the effects of Coulomb repulsion.

A general explanation of the fractional quantum Hall effect can be found by

returning to the second argument used for the integer effect. Suppose that the elec-

tron ground state is a sum of many determinental wave functions and that for any

given boundary condition of

the

type described in Eq. (25.115) there exist q degen-

erate ground state solutions. Suppose furthermore that at every time interval T the

ground-state wave function travels between these degenerate states in sequence.

Then the ground-state wave function does not return to itself completely until after

a time q7. Repeating the argument that led to Eq. (25.120), the conductivity will

p e

q h '

where p is the integer number of electrons that leaves the sample in time q7.

(25.121)

Quantum Hall Effect 787

It is obvious to ask in the wake of the argument about the type of microscopic

state which supports this degeneracy. In addition, with few exceptions, only odd

denominators p have ever been seen. This fact also calls out for a microscopic

explanation.

Laughlin Wave Function. Laughlin (1983) proposed a generalization of the

free-electron state in which the density of states within filled levels is eB/phc,

and which therefore has conductivity p times less than that of the single Landau

level. A first observation leading to this state is that in a strong magnetic field, the

Coulomb interaction between electrons becomes increasingly unimportant. The

characteristic minimum distance between electrons is

1/'sjn

= ^hc/eB, so the ra-

tio of Coulomb to magnetic energies is

^/n m*ce

1.93-107

yjB/T.

e°

is the static dielectric constant.

e°hio

e°hs/eBhc e°m

(25.122)

In a metal, the Coulomb energy would be much larger than the magnetic energy at

any realistic field strength, but in gallium arsenide, with e° = 12.5 and

= .061m,

magnetic fields of a few Tesla are sufficient for the magnetic energy to dominate.

The significance of reaching the high-field limit is that the true ground state of the

system of interacting electrons can accurately be written using only states in the

lowest Landau level as basis functions. States in higher Landau levels cost energy

, and Coulomb repulsion is too small an energy to force any electron to make

the transition.

In preparation for constructing the ground state of the interacting electrons, it

is useful to write the Schrodinger equation for free electrons in a magnetic field in

the symmetric gauge, with B pointing along

—z:

1 fh d exB\

1 (h d eyB\

' dy 2c ) 2m \ i dx 2c )

Define the magnetic length

ip(r) = 0. (25.123)

/2Hc y x

h = \ , and define variables y = — and i =—. (25.124)

V eB l

The Hamiltonian becomes

\ f) \

/1 n \

i/>

ip£..

(25.125)

\_d_ \

nd_ :

i dy J \ i dx

Let

= x + iy, and define

= e^

2/2

cf)(z, z). (25.126)

Then the eigenvalue equation for

<fi

l^z-^_

±A

£4>.

(25.127)

788

Chapter

25.

Magnetism of Ions and Electrons

In the partial derivatives, z and its complex conjugate z are treated as independent

variables. Therefore it is easy to construct the many degenerate wave functions that

constitute the first Landau level by writing

<Kz, z) = f(z) => ^(z, z)

f{z)e-^l\ (25.128)

where

is an arbitrary power series in z. Because df(z)/dz

0, all functions of

the form (25.128) have eigenvalue £ =

HLO

/2.

Because the ground state of the true many-body Hamiltonian, with Coulomb

interactions included, can be constructed out of functions living in the first Landau

level, the many-electron wave function must be of the form

* = /(zo

. .

. ZN-\)e~ ^i=o I

, (25.129)

where

is an analytic function of all its arguments. Laughlin (1983) proposed that

one examine a many-body wave function of a simpler form, called a Jastrow wave

function, similar to Eq. (15.114) for the theory of liquid helium, and similar to the

wave function that describes the ground state of superconductors. The form is

= Y[

(

)e~ ^i=o

|2/2

. (25.130)

The function

can be determined by considering constraints imposed by rota-

tional symmetry. The Hamiltonian commutes with the angular momentum opera-

tors

.3_

OZl

(25.131)

and all of these, for different /, commute with one another. Therefore the eigen-

states can be taken to be eigenfunctions of these operators, which means that

Z } = qfl{z)^>h{z)=Z

Where

is an

inte

is to remain in- (25.132)

finitely differentiable at the origin.

l[(zi -zv)

e~ So' M

/2_ (25.133)

/</'

The wave function must be odd under interchange of particles, so q must be odd.

This consequence of Fermi statistics thus explains why odd denominators are most

easily observed in the Hall plateaus. The wave function (25.133) is a plausible

candidate for the ground state of electrons in a strong magnetic field because it is

constructed purely from wave functions in the lowest Landau level, and it vanishes

whenever two particles come close together so as to reduce the effects of Coulomb

repulsion. Guessing the form in Eq. (25.130) is enough to

fix

the wave function,

even without inserting it into Schrodinger's equation and varying parameters.

Quantum Hall Effect 789

When q=\, Eq. (25.133) is precisely an antisymmetric product of N states in

the lowest Landau level:

# =

z7"

ZN-l

ZN-

Vertical lines indicate taking the

Z-<;=o

'l /

determinant of

this

matrix.

Subscripts

on z label particle

number,

while superscripts are

exponents.

(25.134)

To see that Eqs. (25.134) and (25.133) are equal, recall that a determinant is un-

changed when one column is subtracted from another. For example, subtract col-

umn

from column 2. All the vertical entries are of the form

-Z? =

(Z2-Z

m-\

m-l-l

1=0

(25.135)

So zi

— Z]

is a factor of the determinant. Similarly, zi

—

zv is a factor for all / ^ V.

The largest power of

to appear in Eq. (25.133) is z^

, and the largest power

to appear in Eq. (25.134) is the same. The prefactors of Eqs. (25.133) and

(25.134) are polynomials of the same degree and sharing all the same factors, so

they most be equal up to an overall constant.

When q equals an odd integer, the wave function describes a correlated state

where electrons fill a fraction \/q of the states in the lowest Landau level. This

claim may be demonstrated by considering how to find the degeneracy of a Landau

level using functions of

the

type z

exp[—

|z|

/2].

Suppose that the N single-particle

states

-1*172

-W/2

1*172

-W72

(25.136)

are occupied. The density \z

exp[—|z|

/2]|

has a maximum at \z\

= m, so the

functions in (25.136) describe a series of concentric rings of charge, filling an area

uniformly out to a radius of \z\

= N

— 1

PZN. The units of length in Eq. (25.124)

are

IB,

so the area A filled by charge is

■■irNli

2TTNHC

■N

Use

Eq. (25.124) for the magnetic length /g,

and

compare result with Eq.

(25.51).

(25.137)

Thus the area occupied by electrons in a wave function of the form (25.133) can

be determined by finding the radius of maximum density when all but one of the

variables is set to some small value near zero, and the remaining one is allowed

to become large. Set all but zo to values near zero, and search for the maximum

density as a function of

zo-

For general values of q the maximum occurs for

= q(N - I) ^ N

■-

q$o'

Replace

N by Nq in Eq.

(25.137).

(25.138)

In particular, when q = 3, the Landau level is filled one-third of the way to the top,

and the tilling fraction v equals 1/3.

790 Chapter 25. Magnetism of Ions and Electrons

Fractional Charge. Laughlin furthermore proposed that the excitations of the

fractional quantum Hall state carry fractional charge. Imagine constructing an ex-

tremely thin solenoid with uniform induction B inside and no induction outside,

and passing it through the two-dimensional electron gas. Imagine beginning with

no flux through the solenoid, and then slowly increasing B until finally the total flux

in the solenoid is

<&Q.

According to the discussion of Section

25.3.3,

an electron

traveling a circuit around an infinitely thin flux tube with flux of

<£>o

will undergo

a phase change of 2n, and no physical measurement can detect the presence of the

tube.

Therefore the electron gas after the tube reaches flux

<3>o

must be in an eigen-

state of the original Hamiltonian. This state cannot be the ground state because

turning on the flux tube at the origin drives charge away from the origin, and so

one has constructed an excited state.

To see that charge is driven away from the origin, and to calculate how much,

note that turning on the flux tube creates an azimuthal electric field,

1 ô$

(25.139)

c 2nr dt

Traveling far from the flux tube, the current produced by this electric field is per-

pendicular to it, because only a

is nonzero, and the total charge Q transported

away from the flux tube is

1<9$

,xy

~c~di

1 v

■ I dt 2irr j± = - / dt a

CRK

-4>o = —

ve.

(25.140)

(25.141)

C/3

0 100 200 300 400 500 600 700

Current J (pA)

Figure

25.11.

Shot noise in GaAs/GaAlAs heterostructure in magnetic field that puts the

system in the v = 1/3 plateau. The magnitude of the shot noise is consistent with a system

where current is carried by particles of charge e/3, and it is inconsistent with one where

the carriers have charge e. [Source: Saminadayar et al. (1997), p. 2528.]

Problems 791

So for filling fraction v = 1/3, this argument constructs an excited state of the

Hamiltonian in which a charge of precisely e/3 is transported through any large

loop surrounding the origin, and therefore the charge carriers are predicted to have

charge e/3.

Experimental Observation of Fractional Charge. Saminadayar et al. (1997)

tested this idea experimentally. The test is based upon Eq. (18.22), which predicts

that current fluctuations in any current-carrying system contain shot noise propor-

tional to the current, proportional to twice the particle charge, and proportional to

the frequency bandwidth. The results clearly support the claim that charge carriers

in systems with filling fraction u = 1/3 have charge e/3, and they are displayed in

Figure

25.11.

Problems

Classical electrons in a magnetic field—mechanics: Consider a popula-

tion of classical electrons in a two-dimensional box of side length L, all of

which are moving in some direction at velocity v. When a magnetic field is

turned on, the electrons in the center of the box begin to rotate, leading to a

diamagnetic response. However, those electrons within reach of the bound-

ary bounce off it, ultimately traveling around in the opposite direction. Find

the net magnetic response of the whole system; according to Problem 3.2, it

should vanish.

Assume that the centers of the electrons' orbits sit on a finely spaced grid.

Discuss the electrons whose orbits hit the boundary separately from those

whose orbits do not. Use drawings and geometrical arguments rather than

complicated analytical formulae.

Classical electrons in a magnetic field—statistical mechanics: Show that

the free energy of a collection of classical electrons in a box must be indepen-

dent of any static applied magnetic field.

(a) Write down the classical partition function Z for N particles. The particles

interact with each other according to an arbitrary potential U that depends

only upon their positions.

792

Chapter

25.

Magnetism of Ions and Electrons

(b) Show how a spatially uniform static magnetic field is to be included in the

partition function.

can be eliminated from the partition function and therefore cannot affect any

thermodynamic variable.

Tightly bound electrons in two dimensions and in a strong magnetic field:

Suppose that one has a Hamiltonian for a two-dimensional system on a square

lattice of spacing a,

= — + U(R). (25.142)

Suppose that in a basis of Wannier functions, w(R, r), $io takes the form given

in Eq. (8.67),

0 ~ / ,, \1\) t\K ~r 0 \. Measure energies relative to the on-site en- (.Zj.l'+j;

j?(5 ergy

;

recall that

is the set

nearest neigh-

bors

of R.

Now turn on a magnetic field. The Hamiltonian becomes

+ eAlc)

J{=i

rt^

(25.144)

Take the vector potential to be A = (0, x, 0)B.

Expand eigenfunctions of

in the form

i>(T) = -^=y" e-

ieyBR

C^%ü{R, r). w(R, 7) is a Wannier (25.145)

yTV ^f function, and Cg is a

coefficient that needs

to be

determined

in the

remainder

of the problem.

(a) For Wannier functions very well localized compared to the scale on which

the vector potential varies, show that to leading order,

£^(r) = A=Y, C^e-

ieByRx/nc

^ow(R, r). (25.146)

(b) Show that

~ieBaR

/hc _|_ p

ieBaR

/hc

. (25.147)

~ieBaRy/hc

_i_

p JeBaR

/Hc

^ ^R+ay ^ ^R-ay

R R F

n = —, m=—,

= ak

, and £ = —. (25.148)

a a t