Tirtho Biswas Cal Poly Pomona 10 th February Review From one to many electron system Noninteracting electrons first approximation Solve Schroidinger equation With subject to Boundary conditions ID: 546057
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Slide1
Quantum Mechanics
Tirtho
Biswas
Cal Poly Pomona
10
th
FebruarySlide2
Review
From one to many electron system
Non-interacting
electrons (first approximation)
Solve
Schroidinger
equation
With subject to
Boundary conditions
Obtain
Energy
eigenstates
Include
degeneracy
(density of states)
Obtain
ground state
configuration according to
Pauli’s exclusion principle
Excited states
Thermodynamics (later)Slide3
Free Electron
Loosely bound
How does the spectrum of a free particle in a box look like?
Almost continuous band of states
How do you think the spectrum will change if we add a potential to the system?
No change
The spectrum will still be almost continuous, but the spacing will decrease
The
spectrum will still be almost
continuous
, but the spacing will
decrease
The spectrum will separate into different “bands” separated by “gaps”.Slide4
Kronig
-Penney Model
How to model an electron free to move inside a lattice?
Periodic potential wells
c
ontrolled by three important parameters: Height of the potential barrier Width of the potential barrier Inter-atomic distanceIs there a clever way of solving this problem? SymmetryBloch’s theorem: If V(x+a) = V(x) thenSlide5
Dirac-
Kronig
-Penney Model
Simplify life to get a basic qualitative picture
What strategy to adopt in solving SE?
Solve it separately in different regions and then matchWhat is the wave function in Region II? Slide6
Matching Boundary conditions
Wavefunction
is
coninuous
The derivatives are discontinuous if there is a delta function:
Condition from
wavefunction
continuity Slide7
Lets calculate the
derivatives
What about region II?
Slide8
Discontinuity of derivatives gives is
Eventually one finds
depends on the property of the
materialSlide9
Energy Gap
Depending upon the value of
, there are values of k for which the |RHS|>1 => no solutions
There are ranges in energy which are forbidden!
Larger the , the bigger the band gaps
With increasing energy the band gaps start to shrink
Slide10
Energy Bands
No object is really infinite…we can connect the two ends to form a wire, for instance.
= a
can then only take certain discrete values
LHS = cos N states in a given band, one solution of z, for every value of . Let’s not forget the spin => 2N stateshttps://phet.colorado.edu/en/simulation/band-structure Slide11
If each atom has q valence electrons,
Nq
electrons around
q = 1 is a conductor…little energy to excite
q
=2 is an insulator…have to cross the band gap
Doping (a few extra holes or electrons) allows to control the flow of current…semiconductors
Applications of semiconductorsIntegrated circuits (electronics)Photo cells DiodesLight emitting diodes (LED)Solar cell…