Bloch theorem, Bands Ekn, gaps, metals vs. insulators
From last time
A semiconductor is defined by the density of carriers : High enough for
interesting conductivity; Low enough to be controlled by
temperature and other factors
Exponential variation with temperature implies energy gap
Qualitative understanding of semiconductors and semimetals from band picture
Bands in real semiconductors - Si, Ge, GaAs, ...
All are FCC with 8 valence electrons per primitive cell
Actual bands - nearly-free-electron-like
What is the same, different about Si, Ge, GaAs, ...
Optical properties - comparison of Si, Ge, GaAs, ...
Why is your computer chip made of Si,
but the laser in your CD player is made of GaAs
(in the future GaN?)
Direct “vertical” transition vs “indirect” weak transition
Mobile carriers in a crystal
Key point: the motion of electrons in crystals is affected
by the periodic lattice
The key ideas are already included in the energy Ekn
and the wavefunction psikn, where n is
the band index and k is the "crystal momentum".
If we apply small additional electric field E and/or
magnetic field B, the
electrons can respond by changes in crystal momentum k, energy E, and/or band index n.
Changes in E and k are related: k can change only
if E changes or if k changes along a constant energy surface in 2 nad 3 dimensions
Equation of motion and effective mass
Motion with force F applied
F = hbar (dk/dt) - same as for free electrons!
BUT this is strange - acceleration is not the same
Acceleration = dv/dt which shows electron acts like it has
effective mass m*, where (hbar2/2m*) = d2E/dk2
Not a violation of Newton's laws - Why?
Electrons and holes
What about empty states in the valence band?
Everything is upside down!
Analysis on slides and in Kittel (p 191-205) shown that
one can decribe the properties as "electrons" and "holes"
with "3effective masses" M*
1. khole = - kmissing electron
2. Ehole = - Emissing electron
3. vhole = + vmissing electron
4. m*hole = - m*missing electron > 0
5. qhole = -qmissing electron = + |e| (positive!)
Both electrons and holes can conduct electricity
Current is the sum of electron current and hole current
Law of mass action and equilibrium concentration of electrons and holes
(Kittel p 205-208)
Derivation of densities of carriers
Law of mass action
product np = constant
(depends upon the crystal and the temperature)
Determines directly n and p for an intrinsic crystal where n=p
Extrinsic cases - dping - more later
Motion in a magnetic field Hall effect
Cyclotron resonance measures m*
Hall effect measures signs densities and mobilites (if one
combines conductivty information and knows that only simple bands
contribute)