Two Dimensional Electron Gas System (2DEG)

8/12/2019 Two Dimensional Electron Gas System (2DEG)

1/16

Chapter7-1

Chapter 7. Two Dimensional Electron Gas System (2DEG)

Si MOSFET (metal-oxide-semiconductor field effect transistor)GaAs HEMT (high electron mobility transistor)

S. M. Sze, Physics of Semiconductor Devices (John Wiley, New York, 1981)

7.1 Two-dimensional electron gas (2-DEG)


2/16

Chapter7-2

momentum lost

by scattering

( )statesteadyfieldscattering

=

dt

pd

dt

pd

momentum relaxation time

vd

E

=em

m

mvd

m

Mobility:

momentum received

from external field

e

E

(cm2/Vs)

phonon scattering

107

106

105

104

10 100 T

impurity scattering

modulation-doped GaAs 2-DEG

doped bulk GaAs


3/16

Chapter7-3

sheet density:

Si MOSFET (metal-oxide-

semiconductor field

effect transistor)

E 1MV cm

positive

gate voltage

e

VV

dn

thg

ox

oxs

=

GaAs HEMT (high electron mobility

transistor)

Schottky

barrier

modulation

doping

hetero junction+

modulation doping

high mobility and

high sheet density

50nm

1011~1012 cm-2

5V09.11


4/16

Chapter7-4

( )( ) ( ) ( )yxEyxyxU

m

AeiEs ,,,

2

2

=

+

++

Effective mass equation:

( )( ) ( ) ( )rErrU

m

AeiEc

=

+

++

2

2

Ec = constant,

A = 0,U

r( )= 0 r( )= ei

k

r

implicitly means the real wavefunction

Es

potential energy(due to space charge etc.)

( ) ( )sEE

mEN =

2

(position-dependent)conduction band energy

vector potential

(magnetic field)

(plane wave)

u

k

r( )ei

k

r

Effective mass equation in 2-DEG system:

( ) ( ) ( )

( )222

2 yxnc

ykxki

n

kkm

EE

ezr yx

+++=

= +

(single band: n = 1)

Two-dimensional density of states:

: a factor of 2 (for spin) included

2D-DOS is constant for all energies exceeding Es.

( ) ( )svs EEm

ggEN = 22

gs: spin degeneracy

gv: valley degeneracy

2.9 1010 1/cm2meV for GaAs (m = 0.07 m0)

Z

EF

2

n =11 (occupied)

n = 2(empty)


5/16

Chapter7-5

(unit step function)

f E( )Ef E( )

Degenerate and non-degenerate 2-DEG:

f E( )= 1e

EEf( )kBT +1(Fermi-Dirac distribution)

(Boltzmann distribution)

degenerate (low temperature) limit

sheet density

de Broglie wavelength :dB = hP

( )

non-degenerate (high temperature) limit

eEs Ef( )kBT >>1

f E( )eEf kBTeE kBT

eEs Ef( )kBT


6/16

Chapter7-6

At low temperatures the current is carried mainly by electrons

having an energy close to the Fermi energy so that the Fermi

wavelength is the relevant length. Other electrons with less kinetic

energy have longer wavelengths but they do not contribute to the

conductance.

1

m=

1

cm

Tmkk BT

222 =

=

collision (momentum

scattering)

(thermal de Broglie wavelength)

coherence length of 2-DEG

An electron in a perfect crystal moves as if it were in vacuum

but with a different mass.

Mean free path (Lm = vfm);

(GaAs: = 3 107cm/s for ns = 5 1011cm-2, m = 100psLm = vfm 30m)

(GaAs: T= 4K T 250nm)

Any deviation from perfect crystallinity

(impurities, lattice vibrations, other electrons)

effectiveness [0, 1]momentum

relaxation time collision

time

m

k

f

f

v

=

IfL


7/16

Chapter7-7

Phase relaxation length (L):1

=

1

c

effectivenessphase

relaxation time

(A) Standard argument

Rigid (static) scatterers do not contribute to phase relaxation;

only fluctuating (dynamic) scatterers contribute to phase

relaxation.

(B) Quantum mechanical argument

The interference is destroyed when a measurement tells us

which path the probability amplitude took.

electron

moving mirror

rigid mirror

x p h

(A) If the position uncertainty x of a moving mirror exceeds an

electron de Broglie wavelength, x > , the phase uncertainty

of the reflected electron wave exceeds 2 and thus theinterference disappears. ( fluctuating scatterer model)

(B) If the momentum uncertainty p of a moving mirror becomes

smaller than (an electron momentum), , the

electron recoil imposed on the mirror allows us to tell the

electron took the horizontal path and thus the interference

disappears. ( which-path measurement model)

k kp


8/16

Chapter7-8

In fact, the argument (A) is included in this QM argument (B).

kp

measurement error of

electron momentum

back action noise of

electron position

L0

1

Visibility

phonon emission

no phonon emission

LB

tells us an electron tookthe lower arm

L0

1

Visibilityphonon emission in

upper arm

phonon emission in lower arm

phonon emission

phonon emission

no degradation

L0

1

Visibilityphonon emission

no degradation

Long wavelength phonons are less effective to dephase.

phonon emission


9/16

Chapter7-9

2

DEG2:constantln~

2

+

f

f

E

E

[B. L. Altschuler et al., J. Phys. C 15, 7367, 1982]

The energy uncertainty of an electron due to random

emission and absorption of phonons with energy :

( )c

22 =

t

0random-walk

The phase uncertainty after :3

1

2~1~~

c

Electron-electron scattering is the dominant source of dephasing

at low temperatures.

=EEf (electron excess energy)

An electron with a small excess energy has very few states

to scatter down into since most states below it are already

full.

Since the average excess energy of electrons is ~ kBTat

low bias, ~ kBTand 1/T2.[A. Yacoby et al., Phys. Rev. Lett. 66, 1938, 1991]


10/16

Chapter7-10

If Lm < L < L , diffusive transport

(phase relaxation length)

High-mobility semiconductors m m( )

L = vfballistic

(phase relaxation length)

Low-mobility semiconductors >> m


11/16

Chapter7-11

slope ~ T2

1/

(sec-

1)


12/16

Chapter7-12

J = e

vdns

=e ns

=e m m

mvd

m= e

E +

vd

B[ ] d

p

dt

scattering

=d

p

dt

field

: current density

: conductivity

: mobility

7.3 Magnetoresistance

(A) Drude model (low field effect)

=

y

x

y

x

yyyx

xyxx

E

E

J

J

xx =yy =1

xy = yx = B

ens

: longitudinal resistance

: transverse (Hall) resistance

V2

V3

I

V1

WI

Ly

x

Vx =V1 V2Vy =V2 V3

Ex =xxJx , Ey =yxJx Jy = 0( )

xx =VxW

ILyx =

Vy

I

I=JxW

Vx =ExL

Vy =EyW

mem

B

B mem

vx

vy

=

Ex

Ey

carrier density and mobility:

ns = e dyx

dB[ ]1

= I edVy dB

= 1e nsxx

= I ensVx W L


13/16

Chapter7-13

transverseresistance

longitudinal

resistance


14/16

Chapter7-14

cyclotron frequency:

Landau levels:

density of states:

If we change the magnetic field (keeping the electron density

constant) or change the electron density by means of a gate

voltage (keeping the magnetic field constant), the position of the

Fermi energy is changed relative to the Landau level peaks.

wrong!

(B) Shubnikov-de Haas (SdH) oscillations (high-field effect )

Energy E

E2

Es

density of statesN(E)

E1

( ) 0:2

== Bm

EN

c

c =eB

m

++=2

1nEE csn

( )h

eBmEN cn

22

==

Intuitively it might appear that the longitudinal resistance is a

minimum whenever the Fermi level coincides with a peak of the

density of states.

(GaAs:B = 2T N(En) = 9.6 1010cm-2)


15/16

Chapter7-15

The electric field only moves a few

electrons from -kfto +kf. The current is

carried by a small fraction of the total

electrons which move withthe Fermi velocity.

(degenerate/low temperature)

single particle point of view:

All the conduction electrons drift along

and contribute to the current.

(non-degenerate/high temperature)

The correct answer is just the opposite. The resistance is a

minimum when the Fermi energy lies between two Landau levels

so that the density of states at the Fermi energy is a minimum.

The resistance is almost zero even for the sample length of ~ 1mm.(enormous suppression of momentum relaxation)

adiabatic transport

7.4 Drift velocity or Fermi velocity?

J = ensvd

collective point of view:

J = e ns

vdvf[ ]

vf

ns vd vf

The solution is edge state of quantum Hall effect.


16/16

Chapter7-16

( )

( )

mmf

df

df

df

md

md

d

eELeEvm

kkFF

mkkF

m

kkF

eEk

m

eEv

m

k

222

~

2~

2~

2

2

2

==

+

===

+

+

Quasi-Fermi level separation:

: quasi-Fermi level for electrons moving

to +x direction

: quasi-Fermi level for electrons moving

to -x direction

the energy that anelectron gains from the

electric field in a mean

free path

Two Dimensional Electron Gas System (2DEG)

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