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Solid State Physics
Lecture 11 – Semiconductors
Professor Stephen Sweeney
Advanced Technology Institute and Department of Physics
University of Surrey, Guildford, GU2 7XH, UK
[email protected]
Solid State Physics - Lecture 11
Recap from Lecture 10
• E(k) relationship for free electrons is quadratic and
continuous
• In a real crystal, electron wavefunctions are modulated by the
periodic interaction with ions (BLOCH WAVES)
• At critical wave-vectors the electrons are Bragg-scattered
• Leads to concept of BONDING and ANTI-BONDING states
and a BAND GAP (also predicted by the tight binding model)
• CONDUCTORS, INSULATORS and SEMICONDUCTORS
are defined by the ease at which electrons can gain energy
Solid State Physics - Lecture 11
Semiconductors (recap)
As stated earlier, semiconductors are intermediate band gap materials which may
conduct due to thermal excitation of electrons
By increasing the number of conduction electrons
with temperature, the resistance decreases with
increasing temperature (unlike metals). Silicon is
commonly used in thermistors
Carrier density still much lower than a good
metallic conductor, e.g.
Si (300K)*
Cu (300K)
n ~ 1010cm-3
n ~ 1023cm-3
*doping Si with impurities can increase n up to ~1020cm-3
Solid State Physics - Lecture 11
The concept of a hole
Consider a semiconductor:
What happens when an electron is thermally excited from a lower energy (valence)
band to an upper (conduction) band across the band gap?
Empty upper
energy
(conduction)
band
Energy
An unfilled state
is left behind in
the valence band
Band gap (Eg)
Filled lower
energy (valence)
band
Solid State Physics - Lecture 11
The concept of a hole
Consider a semiconductor:
What happens if we now apply an electric field?
Empty upper
energy
(conduction)
band
BUT empty state
also accelerated
by the field
Energy
Electron is
accelerated by
the field
Band gap (Eg)
Filled lower
energy (valence)
band
Empty state
therefore carries
a current – we
call this a HOLE
E
Electrons and holes are known as current CARRIERS
Solid State Physics - Lecture 11
Current flow in a semiconductor
Both electrons and holes carry current in a
semiconductor
• Electrons in conduction band drift in
opposite direction to E-field (ve)
• Holes in valence band drift in same
direction as field (vh)
• Movement of hole is actually the
hopping of electrons in the valence
band into the vacant bond
(source: hyperphysics.phy-astr.gsu.edu)
• Current due to electrons and holes is in
the same direction, i.e.
I total  I e  I h
E
ve
Ie
vh
Ih
Solid State Physics - Lecture 11
Enhancing current flow - doping
Doping of materials to change their properties, e.g. to increase electrical conduction in
a semiconductor such as silicon – technologically very important!
Doping to add electrons is called
“n-type” (negative) doping
Doping to remove electrons is
called “p-type” (positive) doping
Solid State Physics - Lecture 11
Effect of doping
“Pure” semiconductors are known as intrinsic
If we add dopants, it becomes known as extrinsic
Energy
Conduction
Band
Band gap (Eg)
Valence
band
Intrinsic
(undoped)
Extrinsic
n-doped
(donors)
Solid State Physics - Lecture 11
Extrinsic pdoped
(acceptors)
T = 0K
Effect of doping
At high temperatures, impurity donor electrons are thermally excited into the
conduction band and can carry a current. Similarly, electrons in the valence band are
thermally excited into the acceptor states allowing holes to carry a current in the
valence band.
Conduction
Band
electrons free
to conduct
Energy
++++++++++
Band gap (Eg)
static +ve ions
static –ve ions
-------------holes free to
conduct
Intrinsic
(undoped)
Extrinsic
n-doped
(donors)
Solid State Physics - Lecture 11
Extrinsic pdoped
(acceptors)
Valence
band
T = 300K
Temperature dependence of carrier density
Intrinsic (undoped)
semiconductor
• At low temperature carrier
density ~0
• At higher T, carriers are
excited into conduction
band
Extrinsic (doped)
semiconductor
• At low T carrier density
increases as dopant
carriers are thermally
excited
• At intermediate T, carrier
density ~ constant
(=doping density)
• At high T, intrinsic carrier
excitation dominates
(adapted from Rosenberg)
Solid State Physics - Lecture 11
Effect of doping on Fermi level (EF)
Since EF corresponds to the energy at which the probability of a state being filled with
an electron is 50%, it is strongly influenced by doping
For n-type material, EF moves to higher energy, for p-type material, EF moves to
lower energy
Conduction
Band
electrons free
to conduct
Energy
EF
EF
static holes
Band gap (Eg)
static electrons
EF
holes free to
conduct
Intrinsic
(undoped)
Extrinsic
n-doped
(donors)
Solid State Physics - Lecture 11
Extrinsic pdoped
(acceptors)
Valence
band
T = 300K
p-n junctions
What happens if we bring a piece of p-doped and n-doped semiconductor together?
(just showing band edges in these diagrams)
conduction
band
EF
EF
valence
band
n-doped
p-doped
Solid State Physics - Lecture 11
p-n junctions
What happens if we bring a piece of p-doped and n-doped semiconductor together?
(just showing band edges in these diagrams)
conduction
band
Built-in voltage
EF
valence
band
+++
EF
--p-doped
n-doped
Upon contact the excess electrons in the n-doped region diffuse into the p-doped
region, and the excess holes in the p-doped region diffuse into the n-doped region and
recombine. This reduces the concentration gradient at the interface.
The Fermi-levels align (thermal equilibrium)
Ionised donor and acceptors leave a “space charge” depletion region which generates
a built-in voltage which prevents further diffusion
Solid State Physics - Lecture 11
p-n junctions – effect of bias
Unbiassed
Reverse bias
Forward bias
current
Built-in
voltage
+V
-V
n-doped
p-doped
Built-in voltage is due
to “space charge”
layers which prevent
electrons and holes
from diffusing further
n-doped
p-doped
If a positive voltage
difference is applied
across the pn junction,
the potential barrier is
reduced, hence a
current can flow
Solid State Physics - Lecture 11
+V
n-doped
-V
p-doped
If a negative voltage
difference is applied across
the pn junction, the potential
barrier is increased, hence
current cannot flow
p-n junction diode
This rectifying property is put to good use in a diode, e.g. based
on Silicon or Germanium
+V
n-doped
-V
p-doped
Reverse breakdown occurs
when electrons quantum
mechanically tunnel through
thin potential barrier
Solid State Physics - Lecture 11
Direct vs. Indirect Semiconductors
In a DIRECT band gap semiconductor, the lowest
energy state in the conduction band (CB) lies
directly above the highest state in the valence
band (VB)
This means it is easy for an electron in the CB to
lose energy by filling (recombining with) a hole in
the VB by emitting a PHOTON
In an INDIRECT band gap semiconductor,
the lowest energy state in the CB occurs at
a different k-vector to the highest state in
the VB
It is difficult for an electron in the CB to
recombining with a hole in the VB by
emitting a photon (since photons carry
negligible k) – a PHONON is also required
CB
CB

h
h
VB
VB
E
k
Solid State Physics - Lecture 11
Direct vs. Indirect Semiconductors
Electronics is dominated by Silicon due to its abundance and good electrical properties.
BUT Silicon has an indirect band gap hence it is a poor optical material
Photonics is dominated by direct band gap “III-V” semiconductors (compounds and alloys based
on elements from groups III & V of the periodic table – e.g. GaAs). Almost all LEDs and lasers are
based on these.
Si (indirect gap)
GaAs (direct gap)
Solid State Physics - Lecture 11
Common semiconductors
Direct Eg
Indirect Eg
Solid State Physics - Lecture 11
Low Dimensional Systems
By producing semiconductor
layers with reduced
dimensions we can restrict
motion of electrons and make
more efficient devices
Reducing the number of
dimensions changes the
density of states
In the ultimate limit a quantum
dot is a “zero dimensional”
object with discrete energy
states – like an atom
E
E
g(E)dE
Solid State Physics - Lecture 11
E
g(E)dE
E
g(E)dE
g(E)dE
Quantum Well Lasers and LEDs
The quantum wells are
only 4 or 5 atoms thick!
2.5nm
e.g. used in Blu-ray
lasers and LEDs in new
efficient light bulbs
Solid State Physics - Lecture 11
The Future of Solid State Physics
Energy will be a crucial scientific, technological and political issue for many
generations to come.
Solid-State Physics will play a pivotal role in developing new materials and devices
which can generate electricity (e.g. photovoltaics – solar cells) or to allow us to use
it more efficiently (LEDs for lighting, high efficient lasers for the internet etc.)
You can be a part of this… PhD...?
Solid State Physics - Lecture 11