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3. Two-dimensional systems
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Image from IBM-Almaden
• Introduction
•Great technological importance: microelectronics industry based
on electrons at interfaces semiconductor-semiconductor, insulator
or metal.
•Type I: natural layered structures, e.g., graphite (with C
nanostructures)•Type II: artificial structures, heterojunctions
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• Contents
•Surfaces and interfaces•Junctions (metal-metal,
metal-semiconductor, semiconductor-semiconductor,
metal-oxide-semiconductor (MOS))•Quantum wells and
supperlattices.•Quantum Hall effect. •Applications
•Semiconductor transistors (bipolar, field-effect,
modulation-doped devices)•Opto-electronic devices (solar cells,
photodetectors, light-emitting diodes, semiconductor lasers)
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•Summary
• Surfaces and interfaces
•Material interfaces:
• Boundary between material and vacuum : - Break of lattice
symmetry
- Bloch wave picture break down, but it is recovered far from
interface
• Boundary between one type of material and another :- Much more
complicated.- Many possibilities
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Work function ! The amount of energy required to take an
electron from the Fermi level within a material and remove it to
infinity.
Fermi level, EF The energy of the highest occupied single
particle state in the material at T=0 , must be negative.
Ws,Work required to remove an electron through the surface
layer:
!= difference between energy at infinity and energy of bound
electron = -EF +Ws.
•Important concepts:
• Surfaces and interfaces
Vacuum potential level, relative to the band bottom
Vacuum level,
Band width, W
!vac
EF
vacuum
W
!
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• Surfaces and interfaces• Why is there a surface layer effect?
Charge spills out into “empty space”, resulting in a dipole layer
pointed toward the metal. • Surface layer effect can depend on
particular crystal face and polarity of bonding. • Single atomic
layers of junk can strongly affect ! by altering the surface charge
layer. • In practice, work function is measured empirically
-photoemission, thermionic emission. (~ 5eV for Au)
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+
-
dipolar
barrier
electronic charge
positive
background
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• Surfaces and interfaces
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• Surfaces and interfaces•Surface states
Image from IBM-Almaden
Occupied free surface states of Cu (111), confined by a ring of
iron atoms.
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Surface states
“Spillage” of electrons across interface implies bound
states
tied to the interface.
General idea: interruption of background charge from ion
cores leads can lead to bound states at the interface.
• Tamm states - general consequence of breaking
periodic potential.
Label surface normal z. Surface states are bound in z, but may
be
free in x and y.
Disorder (impurity, unsatisfied chemical bond, vacancy) can lead
to
surface states that are localized in all 3 directions.
Surface states may be empty or full, depending on Fermi
level.
Surface states Image from IBM-Almaden
Occupied free surface states of Cu (111), confined by a ring
of
iron atoms.
“Spillage” of electrons across interface " bound states tied to
the interface. • Tamm states -general consequence of breaking
periodic potential.
Surface states are bound in z, but may be free in x and y.
Disorder (impurity, unsatisfied chemical bond, vacancy) can lead
to surface states that are localized in all 3 directions.
Surface states may be empty or full, depending on Fermi
level.
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Tamm states
Tamm model:
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Tamm states
Remember the Kronig-Penney model?
V(x)
x
-V0
b
a
Tamm model:
V(x)
x
-V0
b
a
V1
Tamm states
Solving the K-P model in a crystal with periodic boundary
conditions led to bands of allowed states, each with real values
of k.
Those allowed single-particle states are Bloch waves, and are
“free”
or “delocalized” in the sense that the wavefunctions extend
throughout the sample.
Tamm model: to meet b.c., k has to become complex, allowing
wavefunction to exponentially decay away from surface.
Results:
• For large crystal, complex part of k is small for states in
the
middle of a band; decay length is long compared to crystal
size.
These states are relatively unaffected.
• A new state appears, one for each band, in the gap. State
has
large complex component of k, and is spatially localized at
sample edge.
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Tamm states
Remember the Kronig-Penney model?
V(x)
x
-V0
b
a
Tamm model:
V(x)
x
-V0
b
a
V1
Tamm states
Solving the K-P model in a crystal with periodic boundary
conditions led to bands of allowed states, each with real values
of k.
Those allowed single-particle states are Bloch waves, and are
“free”
or “delocalized” in the sense that the wavefunctions extend
throughout the sample.
Tamm model: to meet b.c., k has to become complex, allowing
wavefunction to exponentially decay away from surface.
Results:
• For large crystal, complex part of k is small for states in
the
middle of a band; decay length is long compared to crystal
size.
These states are relatively unaffected.
• A new state appears, one for each band, in the gap. State
has
large complex component of k, and is spatially localized at
sample edge.
Results: • For large crystal, complex part of k is small for
states in the middle of a band; decay length is long compared to
crystal size. These states are relatively unaffected. • A new state
appears, one for each band, in the gap. State has large complex
component of k, and is spatially localized at sample edge.
Kronig-Penney model
•K-P model in a crystal with periodic boundary " bands of
allowed states, eachw ith real values of k.
•Allowed single-particle states are Bloch waves, and are
“delocalized” ; the wavefunctions extend throughout the sample.
Tamm model: to meet b.c., k has to become complex, "
wavefunction to exponentially decay away from surface.
• Surfaces and interfaces
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•Proposed in previous lecture for infinite crystal
• JunctionsMetal-metal junction
Join two metals with different work functions:
Conventional way of drawing: shift bands to allow EF to be
uniform across sample
•Electrons flow from system of higher chemical potential to that
of lower chemical potential.
•Total electrochemical potential (including voltage) must end up
being uniform across junction.
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EF
!
EF
!
Bands effectively bend because of double charged layer
interface.
Double charge layer because departing electrons leave behind ion
cores. Thickness of charge layer called “depletion width”; atomic
scale in metals.
++++
----
-
• Junctions Metal-metal
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• Junctions
Before contact:
After contact:
Depletion width much larger than in metal case.
• Schottky barrier makes it difficult to inject electrons from
metal into semiconductor" nonlinear IV behavior – diode (used for
devices). • Very small barriers can result in almost Ohmic
contact.
Metal-semiconductor
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EV
EF
EC++++++++++++++
vacuum
+++++++++++++++++
Schottky
barrier
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• Junctions Metal-semiconductor
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• Junctions Metal-semiconductor
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ΦS > ΦM ΦS < ΦMOhmic contact
Blocking contact
Ohmic contact
Depletion layer
Energy barrier
Electron affinity
Facile charge flow between metal and semic.(Chemical potential !
in conduction band)
Barrier heightEB,n = ΦM − χS
Electron accumulation layer
Shottky barrier
= (ΦM − ΦS) + (EC − EF )
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• Junctions Metal-semiconductor
Shottky barrier
depletion region
accumulation region
strong dependence for wide-gap semiconductor (ZnS)but very weak
for small-gap semiconductor (Si, GaAs)
Shottky barrier involving n-type and p-type semiconductors
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The Fermi level is “pinned” at a particular energy level in a
band of interface states which lie in the gap
Barrier height strong for wide-gap semiconductor (ZnS)
But barrier height not so dependent on metal workfunction,
• Junctions Metal-semiconductorMetal induced gap states
(MIGS)
The filling of some interface states created dipoles " vacuum
levels of metal and semic. displaced δM
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• Junctions Metal-semiconductorEffect of external potential V on
Shottky barrier (n-type semic.)
Now electrochemical potential is constantη = µ− eV
Forward bias (semic. +)Reverse bias (semic. -)
Reduce barrier Increase barrier
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• Junctions Metal-semiconductor
"A Schottky barrier is a rectifying contact (non-ohmic):•
Forward V "allow I large (e- form semic. to metal)
j = jo[eeV/kBT − 1]
• Reverse V " only small current density due to thermionic
emission over barrier)
j0 = AT 2e−EB,n/kBT
A = 4πem!ek2B/h
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Richardson-Dushman eq.
•Doping heavily the surface " n+ (or p+) layer " allow electron
tunnelling through barrier"similar I for both V polarities "
quasi-ohmic contacts
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• Junctions Metal-semiconductor
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direct gap(c-GaAs)
indirect gap
(c-Si, Ge
Semiconductor-semiconductor
•Two possibilities : 1) Heterojunctions (semiconductors with
different band-gaps) 2) Homojunctions (the same semiconductors with
different dopings)
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Semiconductor-semiconductor junctions
Another commonly used structure: GaAs/AlGaAs interface
Molecular electronics?
Contacts in molecular electronic devices remain poorly
understood.
Charge transfer at the interface with metals undoubtedly plays
an
important role.
Remember, while the charge layer length scales are often
relatively
small in metal-metal joints, for example, even a two-atom
thick
space charge layer can be considerable in a small molecule!
Commonly used structure:GaAs/AlGaAs interface
For example: the p-n junction
•If different " " band-bending
(n-type)(intrinsic)
Electrons confined in GasAs condunction band potential well (2D
electron gas)
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• Junctions
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Metal-semiconductor junctions
++ + + +++++++++
Schottky barrier!
• Schottky barrier makes it difficult to inject electrons
from
metal into semiconductor. Result: nonlinear IV behavior –
diode.
• Very small barriers can result in (almost) Ohmic contact.
Very much an art form.
• Schottky barriers used in a number of devices.
Semiconductor-semiconductor junctions
Many possibilities exist.
One of the most commonly used: the p-n junction
p-type n-type
Depletion
region
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• Junctions Semiconductor-semiconductor
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Homojunctions: p-n junction
• Host: c-Si; n: e- carriers in cond. band ; p: holes in valence
band• Contact " diffusion of e- into p; holes into n " ! equal
Mirror images; V for q>0, band for e- ,
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• Junctions Semiconductor-semiconductor
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• In V bias high & high p-n doping "tunneling of e- and h
through "c barrier • Depletion layer d decrease " enhance junction
current
Large reverse V " conduction band in n lower the band in p " for
narrow d, tunnelling of e- form p valence into conduction band :
Zener breakdown
• Junctions Semiconductor-semiconductor
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Leo EsakiThe Nobel Prize in Physics 1973
• In very heavily doped p-n junction semiconductor deg. "cond.
band min. of n layer below valence band max of p-layer (! in one or
other bands out of d)
The tunel diode (first quantum electron device)
•d very small "tunneling of e- from cond. band in n layer to
valence band of p layer " for small V large I
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• Junctions
Metal-oxide-semiconductor (MOS)
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