Two-Dimensional Electron Gas (2DEG) in a Magnetic Field

In classical physics, an electron orbits around the magnetic field at

a well-defined radius

r

=

p /e B

with angular frequency ω = eB/m .

In quantum physics the energy E

=

ħω

is quantized into discrete

levels

En

(Landau levels). And a classical orbit becomes a probability distri-

bution

(the absolute square of the electron wave function).

Classical picture

(Quantum picture on Slide 6)

Landau Levels in Two Dimensions

The continuous 2D density of states contracts into discrete levels.

As the B-field increases, the level spacing increases, and each level sweeps up a larger part of the continuum.

ContinuumThe spacing between Landau levels is the same as for a harmonic oscillator, including the zero point energy:

En =

(n+½) ·

ħωc

The corresponding angular frequency is the cyclotron frequency ωc

, which contains the effective mass m*

:

ωc =

e B/m*

m*m*me

The magnetic moment

is related to the angular momentum of a rotating electron. There are two types of angular momentum, spin (left) and

orbital (center , right).

Introduce Magnetic Interactions into the Schrödinger Equation

The electric potential

generates the electric field E : E = /r

The magnetic potential A produces the magnetic field B :

B = /r ×A

A enters the Schrödinger equation in the same way as .

Make the substitutions

Energy E

(+i ħ

/t +

e )

Momentum p

( i ħ

/r + e A)

in the classical equation for the kinetic energy E: E = p2 /2m

That leads to the Schrödinger equation in a magnetic field:

(iħ

/t + e)

= 1/2m

(

i ħ

/r + eA)2

Time-independent potentials produce energy levels En

: i ħ

/t

= En

(In a solid one has m = m*

= effective mass, e= V0

=

inner potential.)

Landau’s Solution of the Schrödinger Equation in a Magnetic Field

A constant field Bz

is described by the vector potential A = (0,

Bz

x , 0) .B = /r A

Bz

= /x Ay /y Ax

Bx

= By= 0In two dimensions x,y

the Schrödinger equation takes the form:

En

= 1/2m*

[-iħ/r + eA]2

= 1/2m*

[-ħ2 2/x2

+ (-iħ

/y +

eBz

x)2

]

The trial wave function (x,y) = exp(i ky) (x)converts the y-derivative into a multiplication with i k .

After dividing by

exp(i ky) one obtains a one-dimensional Schrödinger equation for (x)

:En

= 1/2m*

[-ħ2 2/x2

+ (ħk + eBz

x)2

]

This becomes the Schrödinger equation of a harmonic oscillator

, if one

rewrites 1/2m*

(ħk + eBz

x)2

as ½ f (x-x0

)2 with the “force constant”

f .Then one can use the familiar energy levels En of the harmonic oscillator to obtain the Landau levels and their wave functions n

(

Lect.11

, p.

4

).

The same B-field can be created by other vector potentials, such asA =

½ (-Bz

y , Bz

x , 0) . This ambiguity is called gauge symmetry. It plays a fundamental role in our understanding of particle physics.

Wave Functions of Electrons in Landau Levels

The radial wave functions are those of a harmonic oscillator, except that the count starts at n=1 instead of n=0.

n

=

1 n =

2 n =

3

n

=

1

n

=

2

n

=

3

Classical probability

Magnetic Flux Quantization

The magnetic flux is quantized

in units of h/e

.

Thus, a magnetic field really can be viewed as composed of individual field lines, as shown in the previous slide. Each line carries one flux quantum h/e. The B-field is the flux density (flux quanta per area).

Flux quantization can be observed directly in superconductors, where the flux quantum is h/2e

because of electron pairs with charge 2e

:

Regular array of flux quanta crossing the surface of a superconductor (white dots). This STM image is taken with a very small applied voltage, less than the energy gap of the superconductor. Superconducting regions are dark, because electrons cannot tunnel inside a gap. The magnetic field of a flux quantum destroys superconductivity and allows tunneling, creating bright spots for the flux quanta.

Integer vs. Fractional Quantum Hall Effect

Integer

Quantum Hall Effect

:n electrons

circle around

one

flux quantum

(more electrons than flux quanta).

Fractional

Quantum Hall Effect n

=

1/m :One

electron circles around

m

flux quanta (more flux quanta than electrons).

Each flux quantum gets a fraction of the electron.

n

=

2

n

=

1/3

The normal Hall

effect gives the line. xy

Ey

/jx

Bz

The quantum Hall

effect gives steps. xy

=

h/e2

·

1/n

for n =

1,2,3,…

Hall Effect vs. Quantum Hall Effect

n=1

n=2

Ohmic

resistivity

xy

[h/e2]xx

xy

Video: Landau Level Filling vs. Quantum Hall Effect

Each of the plateaus has a very precise value of the Hall resistivity, which is determined purely by the fundamental constants h and e. For n=1 one obtains the value xy

=

h/e2

= 25.

8128…

k

. It can be measured so precisely that the quantum Hall effect has become the resistance standard.

The Quantum Hall Resistance Standard

n=1

n=2

xy

[h/e2]xx

Resistivity = =

Resistance = = =V Voltage E · lI Current j · A

Resistivity vs. Resistance in 2D and 3D

Since resistivity does not contain a length in 2D, the quantum Hall effect becomes independent of the shape of the sample.

in 2D: Resistivity =

Resistance = in 3D: Resistivity = m

Resistance = samein

2D

E

Electric Fieldj Current Density

A

length

l

Vanishing Ohmic

Resistivity

The Ohmic

resistivity

xx

nearly vanishes at the plateaus. (It looks like a superconductor, but the resistance is not exactly zero.) That helps making accurate measurements.

Ohmic

resistivity

xy

[h/e2]xx

Edge States Carry the Current

The Fractional Quantum Hall Effect

When the electron density is reduced or the B-field increased beyond the n

=

1 plateau

, additional plateaus appear at fractional values of n

, such as n

=

2/3, 3/5

.

Quantized Conductance

Attach nanotubes

to a STM tip and dip them into a liquid metal electrode.

Conductance Quantum: G0

=

2 e2/h

1 /13 k( factor 2 for spin ,

)

Each

wave

function =

band =

“channel”

contributes G0 to the conductance.

Quantum conductance: G =

G0

•TG0

=

2 e2/h per channel, T

1=

transmission at the contacts

Energy to switch one bit: E =

kBT • ln2

Time to switch one bit: t =

h / E

Energy to transport a bit: E =

kBT • f/c • d

at the rate f over a distance d

Limits of Electronics from Information Theory

Birnbaum

and Williams, Physics Today, Jan. 2000, p. 38. Landauer, Feynman Lectures on Computation .