Electronic structure of Solids

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Bohr quantized the atom…
An atom has a set of energy levels
Some (but not all) occupied by
electrons
Not really dealing with isolated
atoms, but 3D solids
As atoms approach each other, each
affects the other
Energy levels are altered, splitting into
bands
Each atom in the system produces
another energy level in the band
structure
Electronic structure of Solids
Outer levels begin to interact
Overlapping levels
Electron energy
gas
Broadening of energy
levels as atoms approach
Degenerate: All electrons
in an orbital have the same
(lowest) energy
Solid
Gas
E=0
Electron energy
N=5
N=4
N=3
N=2
3.67Å
Internuclear distance
continuum
E=0
Electron energy
Overlapping bands
N=4
N=3
N=2
Energy bands for solid sodium at internuclear distance of 3.67Å
Solid
Gas
E=0
Electron energy
N=5
N=4
N=3
N=2
3.67Å
Internuclear distance
Immediate implication for X-Ray microanalysis…
Electron transitions from split levels (bands) will result in
photon emission energies that do not reflect the discrete
degenerate level…
Conduction band:
First empty band above
the highest filled band
Electron
energy
Valence band:
Outermost band
containing electrons
Conduction
band
Empty band
Bandgap
Partially full
Bandgap
Full
Bandgap
Full
nucleus
Outermost
band
containing
elelectrons
Valence band
Transitions from the
valence band involved
in characteristic X-ray
emission will be energy
shifted depending on
bond lengths, etc.
Conduction
band
Empty band
Electron
energy
Bandgap
Partially full
N=2 (L)
Resulting X-Rays will
not be monochromatic
These will be Kα X-rays
for ultra-light elements
N=1 (K)
nucleus
Valence band
Classification of solids:
Conductors
Insulators
Semiconductors
Conductors:
Outermost band not completely filled
Essentially no band gap
overlap
lots of available energy states if field is applied
Metals and Alkali metals
Insulators:
Conduction band
Empty
Valence band full or nearly full
Eg
Wide band gap with empty
conduction band
Essentially no available energy
states to which electron energies
can be increased
Wide bandgap
Valence band
Full
Dielectric breakdown at high potential
Semiconductors:
Zinc blende
(FCC ZnS)
Similar to insulators but narrow band gap
At electrical temperatures some electrons can be
promoted to the conduction band
Most are cubic
Diamond FCC (single element)
Conduction band
Almost Empty
Eg
Valence band
Almost Full
Conduction band
Empty
bandgap
Valence band
Full
T > 0K
T = 0K
Some common band gaps:
Element
gap (ev)
Ge
0.6
Si
1.1
GaAs
1.4
SiO2
9.0
Mark McClure, UNCPembroke
S
Zn
Semiconductors are either
intrinsic or extrinsic
Intrinsic Semiconductors:
Pure state
Example: Covalently bonded, tetravalent Si lattice
Promotion of an electron to the conduction band leaves “hole” in the
valence band = electron-hole pair
Apply an electric field and the electron will migrate to +
The hole will migrate to – (that is, the electron next to the hole will be
attracted to the +, leaving a hole toward -)
+
-
Ec
Eg
Ev
Net propagation of hole
Extrinsic Semiconductors:
Doped with impurity atoms
p-type
n-type
n-type
Dope Si with something like pentavalent antimony (5 valence electrons)
Narrows the band gap relative to Si
easy to promote Sb electron
Majority carriers are electrons in conduction band
Minority carriers are holes in valence band
Lattice doped with donor atoms
localized energy levels just below conduction band
Ed
Ec
Ev
Si
Si lattice with n-type dopant
Sb
p-type
Dope Si with something like trivalent indium (3 valence electrons)
Incomplete bonding with Si
Nearby electron from Si can fill hole
Majority carriers are holes in the valence band
Minority carriers are electrons in the conduction band
Lattice doped with acceptor atoms
localized energy levels just above valence band
Ec
Ea
Ev
Si
Si lattice with p-type dopant
In
Fermi Level:
That energy level for which there is a 50% probability of
being occupied by an electron
Conduction band
Intrinsic
Ec
Eg
Ef
Ev
Valence band
Conduction band
n-type
Eg
Ev
Valence band
Recombination
Electron-hole pairs not long lasting
Electron encountering hole can “fall” into it
Free time = microsecond or less
Ec
Ef
The p-n junction
Single crystal of semiconductor
Make one end p-type (dope with acceptors)
Make the other end n-type (dope with doners)
The junction of the two leads to rectification
Current only passed in one direction (diode)
In the region of the junction
Recombination = depletion of region with few charge carriers
Results in “built-in” electric field
p
-
Depletion width
W
Direction of built-in field
Space-charge layers
+
+
+
+
+
+
n
Energy band diagram for p-n junction at equilibrium
Ecp
eV0
Ecn
Efp
Efn
Evp
Evn
Apply eV0 to get diffusion
p
-
Depletion width
W
Direction of built-in field
Space-charge layers
+
+
+
+
+
+
n
Energy band diagram for p-n junction – applied forward bias
Ecp
eV
Ecn
Efp
Efn
Evp
Evn
Depletion width reduced
Built-in field reduced
Apply small V to get diffusion
Barrier height reduced
Diffusion current increased
+
p
-
-
Depletion width
W
Direction of built-in field
Space-charge layers
+
+
+
+
+
+
If Vforward = V0
No barrier
n
Pass large
current in
one direction
Energy band diagram for p-n junction – applied reverse bias
Ecp
eV
Evp
Ecn
Depletion width increased
Built-in field increased
Barrier height increased - Diffusion current decreased
p
-
Evn
+
Depletion width
W
Direction of built-in field
Space-charge layers
+
+
+
+
+
+
Becomes
Capacitor
n
No current
passed
So:
Can use reversed bias p-n junction as voltage regulator
Zener diode
Voltage too high? Overcome gap energy and pass current
Can use forward bias p-n junction for rectification
AC → DC transformer
Analog-to-digital conversion
LED
Recombination – “tune” bandgap to achieve photon emission
at the required wavelength
GaAs (IR) GaInN (blue) GaAsP (red) YAG:Ce (white)
Ternary and quaternary compounds allow precise bandgap engineering
PIN diode (p and n sections separated by high resistance material)
light detection
X-ray detection
electron detection
-Each of these serve to excite electron-hole pairs
-Bias properly and get amplification rather than simple propagation
Bipolar transistor = pair of merged diodes - NPN or PNP
base
N
P
collector
base
N
P
emitter
collector
N
P
emitter
Three voltages (NPN)
Collector = + relative to base (collects
electrons)
Emitter = - relative to base (emits electrons)
Small adjustments of the current on the base
results in large changes in collector current.
= current amplifier
Amplify weak signals
Use small currents to
switch large ones
Simple optical encoding:
Generate sine wave by LED passing ruled slide
Phototransistor sees varying light intensity
current output varies with base current
Diode rectifies
AC→DC
Square waves
Digital output to counter
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