Semiconductor Physics II
Low-dimensional Systems
SS 2012
Gregor Koblmüller
Mo, 15:00-15:45 h
Lecture slides, other info:
www.wsi.tum.de
„teaching“
Login: student
Password: fkii
Address: WSI
Room S315
Tel: 289-12779
Gregor.Koblmueller
@wsi.tum.de
Orientation
1.
Review of Semiconductor Physics I – Band structures, DOS
 from 3D to 2D, 1D, and 0D
2.
Growth and Fabrication of Semiconductor Nanostructures
3.
Electrical Transport in low-dimensional structures
4.
Coulomb Blockade, Single Electron Transistor
5.
Optical properties of Quantum Dots (QDs)
6.
Optical control of Excitons and Spins in QDs
7.
Nanophotonic Devices (LEDs, QD-Lasers,…)
8.
Novel Low-dimensional Systems (Nanowires, 2D layers –
Graphene, MoS2,…)
Reminder: Modulation doped 2DEG
Example: Si-doped AlGaAs/GaAs
Unstable state
Thermal equilibrium
e- move toward undoped side
until electrostatic field bends
the band edges so that Fermi
level becomes constant
across both materials
 2DEG
• Charge separation leads to space charge layers
• Schrödinger and Poisson equation important
to describe quantization of e- in potential well
QW-like 2DEG formed by:
(1) Band offsets
(2) electrostatic potential
Polarization-induced 2DEG
Example: polar nitrides (AlGaN/GaN)
nitrides exhibit strong internal
polarization-induced E-fields
(spontaneous + piezoelectric)
 several MV/cm !!!
Polarization-induced 2DEG
Example: polar nitrides (AlGaN/GaN)
surface states participate in
screeing of polarization field
and contribute to band bending
and 2DEG formation
surface states occupied with (native) Oxygen atoms act as donors  2DEG
Polarization-induced 2DEG
Example: polar nitrides (AlGaN/GaN)
2DEG is an explicit function of the surface barrier, AlGaN
thickness, and the bound positive charge at the interface
Polarization-induced 2DEG
Example: polar nitrides (AlGaN/GaN)
huge
2DEG is an explicit function of the surface barrier (i.e., AlGaN
composition, AlGaN thickness, and the bound positive charge
at the interface
Polarization- vs. Modulation-doped 2DEG
Comparison: nitrides (AlGaN/GaN) and arsenides (AlGaAs/GaAs)
• No doping required for 2DEG in AlGaN/GaN
• much higher sheet charge and higher conduction band discontinuity for
AlGaN/GaN heterostructure
Electron Scattering in AlGaAs/GaAs 2DEG
….and the effect on transport regime ?
at low T: conductance determined
by electrons at Fermi energy
(kT << EF –E0)  Fermi gas
Intercontact distance L < coherence length
 mesoscopic, ballistic transport
Electron Scattering in AlGaAs/GaAs 2DEG
….and the effect on transport regime ?
at low T: conductance determined
by electrons at Fermi energy
(kT << EF –E0)  Fermi gas
Intercontact distance L < coherence length
 mesoscopic, ballistic transport
Regimes of Electrical Transport
Classical regime:
Quantum regime:
Inelastic scattering
processes negligible
(no loss of
coherence)
Transport depends
on geometry and
contacts !
 Resistance determined by contact geometry in ballistic transport
Quantumballistic Transport in ideal 1D system
model: stronger confinement through potential barriers in z-and y-direction,
propagation only in x-direction!
Source
Drain
under bias:
µD = EF
µS = EF+eVSD
µS
Current through 1D system:
µD
EF
I  n.e.vx  D1D ( E ).eVSD .e.vx
1D density of states
Quantumballistic Transport in ideal 1D system
Reminder: Change in density of states (DOS) from 2D 1D
…..via device size reduction
DOS
1  dE (k x ) 

D1D ( E )  
  dk x 
1
Quantumballistic Transport in ideal 1D system

 I n,m  n.e.vx  D
D n,m1D ( E ) 
v n ,m x
n,m
1D

( E ).eVSD .e.v n,m x …current through each subband
1  dE (k x ) 


  dk x 
1
1  dE (k x ) 
 …group velocity
 
  dk x 
for small eVSD:
µS
Ek x , n , m   n , m 
f
dE
lim  f ( E , µS )  f ( E , µD )  eV ( EF  E )
f ( E , µS )  f ( E , µD )  eVSD
2
 k
2m
2
x
*
Linear regime: I  V
T  0

 I n ,m  n.e.vx  D
n,m
1D

( E ).eVSD .e.v
n,m
1 2e 2
 eV .e. 
.V


h
1
x
µD
…each subband contributes the same current
independent of subband index n,m and energy !!!
Quantum of Conductance
Gn ,m
2e 2
1

 40µS ,
 25k …Landauer formula
h
Gn,m
Each channel (n.m)/subband gives same value Gn,m to the total current
2
2
e
 G
h
T (E
n,m
F
, n, m)
T…transmission coefficient (Tn,m =1…ideal)
…quantum ballistic transport with finite conductance
Sweeping
through each
s
subband
..
..
1
2
3 4
5 6
7
8
9 10 11
Consecutive filling of subbands
Systems for Quantized Conductance
The simplest device: Quantum point contact (QPC)
First good observation in High-mobility AlGaAs/GaAs 2DEG
 fabricated with top-metal gate with 250 nm wide opening
…point contacts defined by electrostatic
depletion of 2DEG underneath gate
 control of width of point contact
 conduction only through QPC
Systems for Quantized Conductance
The simplest device: Quantum point contact (QPC)
First good observation in High-mobility AlGaAs/GaAs 2DEG
 fabricated with top-metal gate with 250 nm wide opening
Pinch-off
…point contacts defined by electrostatic
depletion of 2DEG underneath gate
 control of width of point contact
 conduction only through QPC
Quantized Conductance in QPC
Operating principle
Quantumballistic Transport
but
Quantumballistic Transport
Deviations at (a) higher T
(b) 1D- wire length very large
L. Worschech, et. al., APL 75, 587 (1999).
A. Kristensen, et. al., JAP 83, 607 (1998).
Landauer quantization for length <10 mm
 mean free path of electrons >10 mm
Deviations from Quantumballistic Transport
but
Example Systems for Quantized Conductance
A. Top Gates on 2DEG systems
Lithographically defined island
Meirav et al. App. Phys. Lett., 54, 268, (1988)
Example Systems for Quantized Conductance
A. Top Gates on 2DEG systems – Deviations from universal conductance fluctuations
Example Systems for Quantized Conductance
B. Cleaved Edge Overgrowth (CEO)
Example Systems for Quantized Conductance
C. Nanowires
e.g. TU Delft –
L.P. Kouwenhoven
InAs NWs
Example Systems for Quantized Conductance
D. Carbon Nanotubes
Example Systems for Quantized Conductance
E. Edges States in QHE (Quantum Hall Effect) – presence of high B-field !
Summary
• Quantum ballistic transport in ideal 1D system:
Gn ,m
- low-T, low-bias, high-purity
systems (QPC, NW, C nanotube,
metallic wires, QHE)
2e 2
1

 40µS ,
 25k …Landauer formula
h
Gn,m
Filling of each subband with electons gives same value Gn,m to the total current
Sweeping
through each
s
subband
..
..