review of semiconductor physics
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Review of Semiconductor Physics. Energy bands. Bonding types – classroom discussion The bond picture vs. the band picture. Bonding and antibonding Conduction band and valence band. The band gap is the consequence of Bragg diffraction: Two plane waves e ikx form standing waves - PowerPoint PPT PresentationTRANSCRIPT
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Review of Semiconductor Physics
Energy bands
• Bonding types – classroom discussion• The bond picture vs. the band picture
Bonding and antibondingConduction band and valence band
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The band gap is the consequence of Bragg diffraction:Two plane waves eikx form standing waves
- one peaks near the atoms and the other between them
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Gap
You can draw the E-k just in the 1st BZ.
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• The band picture – Bloch’s Theorem
Notice it’s a theorem, not a law. Mathematically derived.
The theorem:
Physical picture
The eigenstates (r) of the one-electron Hamiltonian
where V(r + R) = V(r) for all R in a Bravais lattice can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais lattice:
)()( ,, rr krk
k ni
n ue
where un,k(r + R) = un,k(r) .Equivalently,
)()( ,, rRr kRk
k ni
n e
- Wave function
)(2
ˆ 22
rVm
H
A periodic function u(r) is a solution, but the Bloch function is general.
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1D case
3D case
- Band structure Indirect gap
2/a
a = 5.43 Å = 0.543 nm
8/a
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Direct gap
a = 5.65 Å = 0.565 nm
2/a
8/a
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Limitations of the band theory
Static lattice: Will introduce phononsPerfect lattice: Will introduce defectsOne-electron Shrödinger Eq: We in this class will live with this
Justification: the effect of other electrons can be regarded as a kind of background.
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Semi-classical theory
Free electron Block electron
ħk is the momentum. ħk is the crystal momentum, which is not a momentum, but is treated as momentum in the semiclassical theory.n is the band index.
m
kE
2)(
22k
*
20
2
2
||)(
mE
kkk
En(k) = En(k+K)
1D
dk
dE
m
kv
1
)(1
kk
v k Em
3D
dk
dE
m
kkv n
n 1)(
*0
)(1||
)(*
0 kkk
kv k nn Em
1D
3D
rkk r ie)( )()( ,, rr k
rkk n
in ue
un,k(r + R) = un,k(r)
If you want to really understand the semiclassical theory and why you can treat the band edges as potentials, read (how?) James, Physical Review 76, 1611 (1949).
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The Bloch (i.e. semiclassic) electron behaves as a particle following Newton’s laws.
(We are back in the familiar territory.)
• With a mass m*
• Emerging from the other side of the first Brillouin zone upon hitting a boundary
Newton’s 1st law: the Bloch electron moves forever
Oscillation in dc field. So far not observed yet. Why?
Newton’s 2nd law:
F = dp/dt = ħdk/dt
Consider one electron, a full band of e’s, and a partial band of e’s
Can get a current w/o applying a voltage?
What is that constant speed if there’s no applied voltage (field/force)?
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Real crystals are not perfect. Defects scatter electrons.
On average, the electron is scattered once every time period . Upon scattering, the electron forgets its previous velocity, and is “thermalized.”
EE
** m
q
m
Fvd *m
q
EE qnqnvJ d *
2
m
nqqn
Mobility
Note:It’s the defects, not the atoms (ions), that are scattering the electrons!How can we ignore the atoms (ions)?!
The Drude (or crude?) model
For a metal, you know n from its valence and atom density, so you know .
= m / q The mean free path l = vth = vthm / q >> a
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Values of k
k = 2n/L, n = 1, 2, 3, …, N
Run the (extra?) mile:Show the above by using the “periodic boundary” condition.
Holes
A vacancy in a band, i.e. a k-state missing the electron, behaves like a particle with charge +q.
Run the (extra?) mile:Show the above.
Discrete but quasi-continuous
L = Na
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Review of Semiconductor PhysicsCarrier Statistics
• Fermi-Dirac distribution
Nature prefers low energy.Lower energy states (levels) are filled first.Imagine filling a container w/ sands, or rice, or balls, or whatever
- Each particle is “still” T = 0 K- Each has some energy, keeping bouncing around T > 0 K
• Density of States
How many states are there in the energy interval dE at E?D(E)dE
1D case derived in class.
The take-home message: D(E) E1/2
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2D caseRun the extra mileDerive D(E) in 2D.Hint: count number of k’s in 2D.
The answer:2
*2
2)(
mL
ED
Or, for unit area2
*
2
1)(
m
ED
The take-home message: D(E) = constant
3D caseRun the extra mileDerive D(E) in 3D.Hint: count number of k’s in 2D.
For unit area, Em
ED3
2/3*
2
)(
2
2)(
The take-home message: D(E) E1/2
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Things we have ignored so far: degeneracies
Spin degeneracy: 2Valley degeneracy: Mc
Mc = 6 for Si
Em
MED c 3
2/3*
2
)(
2
22)(
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Total number of carriers per volume (carrier density, carrier concentration)
Run the extra mileDerive the electron density n.Hint: Fermi-Dirac distribution approximated by Boltzmann distribution.
Results for n and p are given.
DopingOne way to manipulate carrier density is doping.Doping shifts the Fermi level.
np = ni2
p is the total number of states NOT occupied.
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One small thing to keep in mind:Subtle difference in jargons used by EEs and physicists
We use the EE terminology, of course.
EF = EF(T)
Physicists:
Chemical potential (T)
Fermi level
Fermi energy EF = (0)
Same concept
We already used for mobility.
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Before we talk about devices, what are semiconductors anyway?
Why can we modulate their properties by orders of magnitude?
Classroom discussion
Classroom discussion
Na, K, … are metals
Mg, Ca, … are also metals. Why?
Si and Ge are semiconductors. Pb is a metal.
Jezequel & Pollini, Phys Rev B 41, 1327 (1990)
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Real crystals are not perfect. Defects scatter electrons.On average, the electron is scattered once every time period . Upon scattering, the electron forgets its previous velocity, and is “thermalized.”
EE
** m
q
m
Fvd *m
q
EE qnqnvJ d *
2
m
nqqn
Mobility
We have mentioned defect scattering:
Any deviation from perfect periodicity is a defect. A perfect surface is a defect.
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Phonons
Static lattice approximation
Atoms vibrate
Harmonic approximation: springs
Vibration quantized
Each quantum is a phonon.Similar to the photon: E = ħ, p = ħk
Phonons scatter carriers, too.The higher the temperature, the worse phonon scattering.You can use the temperature dependence of conductivity or mobility to determine the contributions of various scattering mechanisms.
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Phonons
Sound wave in continuous media = vk
Microscopically, the solid is discrete.
Phonon dispersion
Wave vector folding, first Brillouin zone.
Watch animation at http://en.wikipedia.org/wiki/File:Phonon_k_3k.gif
Recall that
Crystal structure = Bravais lattice + basis
If there are more than 1 atom in the basis, optical phonons
http://physics-animations.com/Physics/English/phon_txt.htm
If you are serious and curious about photons, readKittel, Introduction to Solid State Physics
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Phonons in the 3D world -- Si
In 3D, there are transverse and longitudinal waves.
E = h = ħ
15 THz 62 meV
When electron energy is low, the electron only interacts with acoustic phonons,
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Optical phonons and transport
At low fields, thd vv E
Tkvm Bth 2
3
2
1 2*
For Si, vth = 2.3 × 107 cm/s
= 38 meV
At high fields, vd comparable to vth
Electrons get energy from the field, hotter than the lattice – hot electrons
E
vd
vsat
When the energy of hot electrons becomes comparable to that of optical phonons, energy is transferred to the lattice via optical phonons.
Velocity saturation
For Si, vsat ~ 107 cm/s
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AlloysCompounds, alloys, heterostructures
InP, GaAs, …, SiC
InxGa1-xAsyP1-y, …, SixGe1-x
Epitaxy
Band structure of alloys
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Topics• Review of Semiconductor physics
- Crystal structure, band structures, band structure modification by alloys, heterostructurs, and strain
- Carrier statistics- Scattering, defects, phonons, mobility, transport in heterostructures
• Device concepts - MOSFETs, MESFETs, MODFETs, TFTs - Heterojunction bipolar transistors (HBT) - Semiconductor processing- Photodiodes, LEDs, semiconductor lasers - (optional) resonant tunneling devices, quantum interference devices,
single electron transistors, quantum dot computing, ...- Introduction to nanoelectronics
We will discuss heterostructures in the context of devices.
More discussions on semiconductor physics will be embedded in the device context.