Solid State 041
Transcript of Solid State 041
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#1: Review of Solid State Physics
• Types of Solids
– Ionic, Covalent, and Metallic.
• Classical Theory of Conduction
– Current density j, drift velocity vd , resistivity r.
• Band Theory and Band Diagrams – Energy levels of separated atoms form energy “band” when
brought close together in a crystal.
– Fermi Function for how to “fill” bands.
– Metal, Insulator, and Semiconductor band diagrams.
– Donor and Acceptor dopants (Hall Effect).
• Devices
– pn junction, diode
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Types of Solids: Ionic Solid Properties
• Formed by Coulombic attraction between ions.
– Examples include group I alkali cations paired with group VIIhalide anions, e.g. Na+ Cl-.
• Large cohesive energy (2-4 eV/ atom).
– Leads to high melting and boiling points.
• Low electrical conductivity.
– No “free” electrons to carry current.
• Transparent to visible light.
– Photon energy too low to “free” electrons.
• Soluble in polar liquids like water.
– Dipole of water attracts ions.
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Types of Solids: Ionic Solid Crystal Spacing
• Potential Energy: Utot
= Uattract
(+, –) + Urepulse
( –, –)
Repulsive Potential 1/rm
Attractive CoulombicPotential -1/r
Total Potential
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Simple Cubic Body-Centered Cubic Face-CenteredCubic
FCC
structure:
NaClNa+
Cl-
Types of Solids: Example Crystalline Structures
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Types of Solids: Covalent Solid
• Examples include group IV elements (C, Si) and III-V elements
(GaAs, InSb).
• Formed by strong, localized bonds with stable, closed-shell structures.
• Larger cohesive energies than for ionic solids (4-7 eV/atom).
– Leads to higher melting and boiling points.
• Low electrical conductivity.
– Due to energy band gap that charged carriers must overcome in
order to conduct.
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Types of Solids: Example Crystalline Structures
Diamond Tetrahedral sp3 bonding
(very hard!)
Graphite
Planar sp2 bonding
(good lubricant)
Vertical p-bonds Bond angle = 109.5º
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Types of Solids: Metal
• Formed by Coulombic attraction between (+) lattice ions and
( –) electron “gas.”
• Metallic bonds allows electrons to move freely through lattice.
• Smaller cohesive energy (1-4 eV).
• High electrical conductivity.
• Absorbs visible light (non-transparent, “shiny” due to re-emission).
• Good alloy formation (due to non-directional metallic bonds).
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Current: (Amps)dq
dt i
q t id
i R
V
R L A
r
Macroscopic Microscopic
2Current Density: (A/m )
d
dA J i
d i A J
where resistivityconductivity
E
E J r
r
where carrier densitydrift velocity
d
d nv
n e J v
2where scattering timem
ne r
Classical Theory of Conduction (E&M Review)
• Drift velocity vd is net motion of electrons (0.1 to 10-7 m/s).
• Scattering time is time between electron-lattice collisions.
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Classical Theory of Conduction: Electron Motion
• Electron travels at fast velocities for a time and then “collides” with
the crystal lattice.
• Results in a net motion opposite to the E field with drift velocity vd.
• Scatter time decreases with increasing temperature T, i.e. more
scattering at higher temperatures (leads to higher resistivity).
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2
1
( )
E
d
F ma E me e
J ne v ne a ne n
r
• Metal: Resistance increases with Temperature.
• Why? Temp , n same (same # conduction electrons) r • Semiconductor: Resistance decreases with Temperature.
• Why? Temp , n (“free-up” carriers to conduct) r
Classical Theory of Conduction: Resistivity vs. Temp.
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Band Theory: Two Approaches
• There are two approaches to finding the electron energies associated
with atoms in a periodic lattice.• Approach #1: “Bound” Electron Approach (single atom energies!)
– Isolated atoms brought close together to form a solid.
• Approach #2: “Unbound” or Free Electron Approach (E = p2 /2m)
– Free electrons modified by a periodic potential (i.e. lattice ions).
• Both approaches result in grouped energy levels with allowed and
forbidden energy regions.
– Energy bands overlap for metals.
– Energy bands do not overlap (or have a “gap”) for semiconductors.
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Solid of N atomsTwo atoms Six atoms
Band Theory: “Bound” Electron Approach
• For the total number N of atoms in a solid (1023 cm – 3), N energy
levels split apart within a width E.
– Leads to a band of energies for each initial atomic energy level
(e.g. 1s energy band for 1s energy level).
Electrons must occupy
different energies due toPauli Exclusion principle.
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1
1
F FD E
k
E
T
f E
e
Band Diagram: Fermi-Dirac “Filling” Function
• Probability of electrons (fermions) to be found at various energy levels.
• Temperature dependence of Fermi-Dirac function shown as follows:
• At RT, E – EF = 0.05 eV f(E) = 0.12E – EF = 7.5 eV f(E) = 10 – 129
• Exponential dependence has HUGE effect!
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• At T = 0, all levels in conduction band below the Fermi energy EF arefilled with electrons, while all levels above EF are empty.
• Electrons are free to move into “empty” states of conduction bandwith only a small electric field E, leading to high electricalconductivity!
• At T > 0, electrons have a probability to be thermally “excited” from
below the Fermi energy to above it.
Band Diagram: Metal
EF
EC,V
Conduction band(Partially Filled)
T > 0
Fermi “filling”
function
Energy bandto be “filled”
E = 0
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Band Diagram: Insulator
• At T = 0, lower valence band is filled with electrons and upper
conduction band is empty, leading to zero conductivity.
– Fermi energy EF is at midpoint of large energy gap (2-10 eV) between
conduction and valence bands.
• At T > 0, electrons are usually NOT thermally “excited” from valence
to conduction band, leading to zero conductivity.
EF
EC
EV
Conduction band(Empty)
Valence band(Filled)
Egap
T > 0
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Band Diagram: Semiconductor with No Doping
• At T = 0, lower valence band is filled with electrons and upper
conduction band is empty, leading to zero conductivity.
– Fermi energy EF is at midpoint of small energy gap (<1 eV) between
conduction and valence bands.
• At T > 0, electrons thermally “excited” from valence to conduction
band, leading to measurable conductivity.
EF EC
EV
Conduction band(Partially Filled)
Valence band(Partially Empty)
T > 0
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Band Diagram: Donor Dopant in Semiconductor
• For group IV Si, add a group V element
to “donate” an electron and make n-type
Si (more negative electrons!).
• “Extra” electron is weakly bound, with
donor energy level ED just below
conduction band EC.
– Dopant electrons easily promoted to
conduction band, increasing electrical
conductivity by increasing carrierdensity n.
• Fermi level EF moves up towards EC.
• Increase the conductivity of a semiconductor by adding a small amount
of another material called a dopant (instead of heating it!)
EC
EV
EFED
Egap~ 1 eV
n-type Si
B d Di A D i S i d
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Band Diagram: Acceptor Dopant in Semiconductor
• For Si, add a group III element to“accept” an electron and make p-type
Si (more positive “holes”).
• “Missing” electron results in an extra
“hole”, with an acceptor energy levelEA just above the valence band EV.
– Holes easily formed in valence
band, greatly increasing the
electrical conductivity.
• Fermi level EF moves down towards EV.
EA
EC
EV
EF
p-type Si
B d Di O h V i bl
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Band Diagram: Other Variables
Metal Semiconductor
Work Function j Electron Affinity c Surface State
Bending
J i B d Di
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pn Junction: Band Diagram
• Due to diffusion, electrons
move from n to p-side andholes from p to n-side.
• Causes depletion zone at junction where immobilecharged ion cores remain.
• Results in a built-in electricfield (103 to 105 V/cm),which opposes furtherdiffusion.
• Note: EF levels are alignedacross pn junction underequilibrium.
Depletion Zone
pn regions “touch” & free carriers move
electrons
pn regions in equilibrium
holes
EV
EF
EC
EF
EV
EF
EC
+++
++++
++++
+ – – – –
– – – –
– – – –
p-type
n-type
J ti IV Ch t i ti
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• Current-Voltage Relationship
• Forward Bias: current
exponentially increases.
• Reverse Bias: low leakage
current equal to ~Io.
• Ability of pn junction to passcurrent in only one direction is
known as “rectifying”
behavior.
pn Junction: IV Characteristics
/ [ 1]eV kT
o I I e
Reverse
Bias
ForwardBias
J ti B d Di d Bi
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pn Junction: Band Diagram under Bias
Forward Bias Reverse BiasEquilibrium
e – e –
• Forward Bias: negative voltage on n-side promotes diffusion of
electrons by decreasing built-in junction potential higher current.
• Reverse Bias: positive voltage on n-side inhibits diffusion of electrons
by increasing built-in junction potential lower current.
e –
Majority Carriers
p-type n-type p-type n-type p-type n-type
–V +V
S i d t D t D it i H ll Eff t
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• Why Useful? Determines carrier type (electron vs. hole) and
carrier density n for a semiconductor.
• How? Place semiconductor into external B field, push current along one
axis, and measure induced Hall voltage VH along perpendicular axis.
• Derived from Lorentz equation FE (qE) = FB (qvB).
Carrier density n = (current I ) (magnetic field B)
(carrier charge q) (thickness t )(Hall voltage V H
)
Semiconductor: Dopant Density via Hall Effect
Hole Electron
+ charge – charge
BF qv B