Two-dimensional electron gas (2DEG) in Ch. 17

• a gas of electrons free to move in two dimensions, but tightly confined in the third dimension

• In MOSFETs the transistor is in inversion mode beneath the gate oxide ~ 2DEG

• High-electron-mobility transistors (HEMTs) ~ heterojunction between two semiconductors to

confine electrons to a triangular quantum well

The hole consentration at the surface

is far larger than the electron concen-

tration in the interior

EC

EV

EF

gate oxide

EC

EF

EV

N n N n

+ -

In a heterojunction, the Fermi level should be independent

of position by transfering electrons from the N-side to the

n-side of the interface. A depletion layer of positively ionized

donors is left behind on the N-side.

• HEMTs exhibit very high mobility by utilizing an intentionally undoped channel to avoid the

ionized impurity scattering.

• The mean free path is proportional to the mobility ranging up to ~ mm.

• Elastic scattering with missing or displaced atoms (dislocations), foreign atoms (impurities) or

walls ~ random walk motion of electrons with

• Inelastic scattering ~ an electron traveling through the crystal can induce the lattice distortion

(electron-phonon scattering) or the Coulomb interaction between two traveling electrons can

lead to the exchange of energy (electron-electron interaction) ~

n : electron density in 2DEG

, , initial final initial final initial finalE E k k k k

*

*

* *

* *

*

* e-e scattering

e-ph scattering

, initial final initial finalE E k k

Diffusive

Ballistic

transport ~ ,

transport ~ ,

i e

i e

L

L

(1) Weak Localization

• Drunken sailor's random walk ~ At each corner he makes a random turn because he does

not remember where he came from. There is a finite probability that the sailor returns (by

accident) back to the original bar.

• Electrons in a metal film with scattering centers ~ We consider a path which is a closed

loop. When an electron wave propagates from 0 via 1, 2, 3, .. back to 0, the electron can, at the

same time, also propagate from 0 via 10, 9, 8, .. back to 0 on the reversed path ("time-reversal

symmetry"). At point 0 the two complementary waves interfere. The combined return

probability of the two complementary waves is |2A|²=4|A|². For two uncorrelated electron waves

this return probability has only the value 2|A|². Therefore we have an additional "coherent back

scattering" of 2|A|². ~ Electrons tend to remain at their initial site, resulting a reduction of the

conductance (and an increase of the resistance) of the system ~ "Weak localisation (WL)"

Experiments on WL

• Application of magnetic field breaks time-reversal symmetry. If we apply a magnetic field

perpendicular to the film then our electron wave picks up a magnetic phase shift. For a closed

loop this phase shift is equal to where F is the enclosed magnetic. Since our two

complementary waves (along the closed loop) surround the flux in opposite direction their

magnetic phase shifts have different signs and the two amplitudes have no longer the same

phase. Thus they no longer interfere constructively ~ the magnetoresistance would decrease

with the magnetic field (negative magneto-resistance).

/eF

The resistance of a pure disordered Mg film is plotted versus 1/B, the inverse magnetic field, in a semi-log plot (upper curve). The second curve shows the corresponding results after the Mg is covered with a fraction of a monolayer of Au. The Au nuclei rotate the electron spins and cause a destructive interference, equivalent to a positive magneto-resistance ~ Anti-weak-localization

Weak localization effect With the help of coherent backscattering, additional resistance occurs through the interaction between the two time-reversed paths.

• 45 parallel NW’s • Crossover from WAL to WL • Lf = 260 nm • spin relaxation length of Lso ~ 200 nm

InAs NW’s A. E. Hansen et al., PRB (05)

0

2 2The characteristic B field for the negative magnetoresistance, ~ c

hB

e

F

(2) Universal conductance fluctuations

Lee & Stone, PRL (1985)

“UCF’s are caused by the quantum inter-ference of multiply scattered electronic wavefunctions in a weakly disordered conductor.”

N N

Lf

L

rms(GN) = 0.4 – 0.7 e2/h

Y.-J. Doh et al., J. Korean Phys. Soc. (09)

rms(GN) ~ e2/h

Fluctuations as a function of perpendicular magnetic field of the conductance of a 310 nm long and 25 nm wide Au wire at 10 mK. The trace appears random, but is completely reproducible from one measurement to the next.

(3) Aharonov-Bohm oscillations

• A quantum mechanical phenomenon in which an electrically charged particle is affected by

an electromagnetic field (E, B), despite being confined to a region in which both the

magnetic field B and electric field E are zero, caused by the coupling of the electromagnetic

potential with the complex phase of a charged particle's wavefunction

Schematic of double-slit experiment in which Aharonov–Bohm effect can be observed: electrons pass through two slits, interfering at an observation screen, with the interference pattern shifted when a magnetic field B is turned on in the cylindrical solenoid.

• Electromagnetic theory implies that a particle with electric charge q travelling along some path P in a region with zero magnetic field B, but non-zero A (by ), acquires a phase shift φ, given in SI units by

0 B A

path

qd A r

• Therefore particles, with the same start and end points, but travelling along two different routes will acquire a phase difference Δφ determined by the magnetic flux ΦB through the area between the paths (via Stokes' theorem and ), and given by:

B A

0

2B Bq

F F

F

Quantum Hall effect

• The Hall resistance, which is the Hall voltage divided by the channel current, in 2DEG is

quantized to the ratio of fundamental constant at low temperature and high magnetic field.

Hall effect

2

2 or where integer (1, 2, 3,...) or rational fraction (1/3, 2/5, 3/7, 2/3, 1/5, ...)=H

H H

x

V h eR

I e h

In the Drude model, an electron is accelerated by the electric field, , for an average relaxation time, ,

or mean free time before scattering ~

The average drift velocity of electron is

x

x x

E

F eE mx

2

2

2

The current density is where is a carrier density

Conductivity, , and resistivity, , are defined by and ~

When the magnetic field i

,

xx

xx x

x

x x

x x

eEv a

m

n

jj E

nej nev E

m

ne m

m neE j

s applied along the -direction, the Lorentz force is added.

( ) x x y z

y y x z

z

F e E v Be

F e E v B

F E v B ˆzzBB

v

• The accumulated charges generate a transversal electric field to make Fy to be zero.

22

0

For the sample with dimensions of length ( ), width ( ), and thickness ( ) in 3-dim,

with = [ ]xx

y y x z y x z

x x x x

x x x x

y y x z

x x x

H

F e E v B E v B

L w t

V E L E L E L mcm

I j wt j wt nene Ewt

m

V E w v B w

I j wt nev

LR

wt

Rw

22

~ Carrier density is obtained from the Hall resistance

For 2DEG system, the resistance and Hall resistance are given by

with = =

x x x xxS

x x SS x

xx

H

S

z zB Bn

nett

V E L E L m R

LI j w n en e Ew

wm

LR

w

R et

= [ ]

where is the number of carriers in unit area

zH

S

y y x z

x x S Sx

n

V E w v B w

I j w n e

R

v

B

n ewR

2

2 2 22 2

For the 2DEG system with the lateral dimensions of , the kinetic energy of an electron is given by

2 2

with the boundary conditions of ( ) ( ) and ( ) ( )

exp( ) exp( (

K x y

x x

L

kE k k

m m

x x L y y L

ik x ik x

2 2 22

2 2 2 2 2

2 2)) exp( ) 1 2 or and 0,1,2,...

Total number of states neglecting spin with is

( ) 2 1( )

4 422

Density of state

2)

(S

x x x y

K

n nL ik L k L n k k n

L L

E E

k k N E k mEN E L

mEn E

hL

L

2

is given by

~

Under the high magnetic field, the kinetic energy

( ) 2( ) constant density of states in 2D

of an electron is quantized to Landau

EG system with 0

level due to the

cyclotron mo

Sdn E mD E B

dE h

tion like as a quantum harmonic oscillator with

~ cyclotron frequency

The Landau levels are discrete in magnetic field wi

1 where

th the

seperation f

2

o

LL C C

LL C

eBE n

m

E

xk

yk

C

2 2 ~ linearly proportional

The number of states in a Landau level is

2 2( )

When the Landau level separation is much bigger than the thermal fluctuation energy,

o

,

t C CLL

C B

m m eBD E

h h

eBN

h

T

Bm

k

and the Fermi level is residing between two Landau levels, the Landau levels from bottom to the -th level

are filled and the levels above are empty, which indicates the total number of carriers are Sn

2

2

The Hall resistance is given by

The Hall conductance is also quantized to be

S

H

B B h

n

eB

h

hR

e

e

h

e e eB

Landau level quantization

• Quantized energy levels of the cyclotron orbits of charged particle in magnetic field

• The Landau levels are degenerate with the number of electrons per level directly proportional

to the strength of the applied magnetic field

2

Hamiltonian of a two-dimensional system of non-interacting particles with charge and spin is

01 ˆ ˆˆ ˆ where is the electromagnetic vector potential giving 0

2

Choice of vect

q S

H p qA Am

B

B A =

or potential for a given magnetic field is free without influencing the physical properties

0

ˆexcept the overall phase of the wave function. We choose the Landau gauge such that

0

1ˆ

2ˆ

A Bx

mH

2 2 22 2

22

0

2ˆ ˆ1 1 ˆˆ ˆ ˆ ˆ2 2 2 2

ˆwhere ~ same as the 1-dim quantum harmonic oscillato

ˆˆ 1ˆ

2

r with a coordinate shift of

Thus the energy of the

2

yxC

C

x xy y

y

C

C

p pp qA p qBx k qBx

m m m m

kqB

k

xm m

pm x

m m

2DEG system applied with the magnetic field along the z-axis are identical to those

of the quantum harmonic oscillator1

, : 02

n CE n n