two dimensional electron gas system (2deg)
TRANSCRIPT
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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)
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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
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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
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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)
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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
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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
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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
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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
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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]
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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
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Chapter7-11
slope ~ T2
1/
(sec-
1)
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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
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Chapter7-13
transverseresistance
longitudinal
resistance
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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)
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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.
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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