Review-QM

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Review-QMs and Density of States
Last time, we used a quantum mechanical, kinetic
model, and solved the Schrodinger Equation for an
electron in a 1-D box. -?(x) standing wave
Extend to 3-D and use Bloch Function to impose
periodicity to boundary conditions -?(x)
traveling wave
We showed that this satisfies periodicity. -
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Review-QMs and Density of States
Plug into H? E? to find energy.
Since ??k, higher energy corresponds to larger
k. For kFxkFykFz (i.e. at the Fermi Wave
Vector), the Fermi Surface (surface of constant
energy, wave vector, temperature) the is a
shpere. This applies to valence band for Si, Ge,
GaAs and couduction band for GaAs.

• lt ?F, filled states
• gt ?F, empty states

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Fermi Surfaces in Real Materials
If kxF?kFy?kFz, then you get constant energy
ellipsoids. Also, if k0 is not lowest energy
state, (e.g. for pz orbitals where k? is the
lowest energy), you dont get a constant energy
sphere.
e.g. Silicon and Germanium CB energy min occurs
at Si - X point, k? along lt100gt directions Ge
- L point, k? along lt111gt directions
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Review-QMs and Density of States
of states in sphere
of electrons 2x Number of states, because of
spin degeneracy (2 electrons per state)
Now we wish to calculate density of states by
differentiating N with respect to energy
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Review-QMs and Density of States
Number of electrons, n, can be calculated at a
given T by integration.
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Review-QMs and Density of States
Eff. DOS CB
Eff. DOS VB
We can now calculate the n,p product by
multiplying the above equations.
(intrinsic carrier concentration)2
For no doping and no electric fields
Add a field and n?p, but npconstant
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Outline - Moving On

1. Finish up Si Crystal without doping
2. Talk about effective mass, m
3. Talk about doping
4. Charge Conduction in semiconductors

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Intrinsic Semiconductors
It is useful to define the intrinsic
Fermi-Level, what you get for undoped materials.
(EF(undoped) Ei, npni)
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Intrinsic Semiconductors
Eg
Si -13 meV 1.12 eV
Ge -7 meV 0.67 eV
GaAs 35 meV 1.42 eV
The energy offset from the center of the band gap
is in magnitude compared to the magnitude of the
bandgap. REMEMBER There are no states at Ei or
EF. They are simply electrochemical potentials
that give and average electron energy
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Intrinsic Semiconductors
For Si and Ge, me gt mh, so ln term lt 0, Ei lt Eg
For GaAs, me lt mh, so ln term lt 0, Ei lt Eg
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Effective Mass
The text book 3.2.4 derives m from QM treatment
of a wave packet.
(This expression can be derived quantum
mechanically for a wavepacket with group velocity
vg)
We can use this quantum mechanical results in
Newtonian physics. (i.e. Newtons Secone Law)
Thus, electrons in crystals can be treated like
billiard balls in a semi-classical sense, where
crystal forces and QM properties are accounted
for in the effective mass.
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Effective Mass
or
So, we see that the effective mass is inversely
related to the band curvature. Furthermore, the
effective mass depends on which direction in
k-space we are looking In silicon, for example
Where ml is the effective mass along the
longitudinal direction of the ellispoids and mt
is the effective mass along the transverse
direction of the ellipsoids
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Effective Mass
Relative sizes of ml and mt are important
(ultimately leading to anisotropy in the
conductivity)
or
Again, we stress that effective mass is inversly
proportional to band curvature. This means that
for negative curvature, a particle will have
negative mass and accelerate in the direction
opposite to what is expected purely from
classical considerations.
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Effective Mass
One way to measure the effective mass is
cyclotron resonance v. crystallographic
direction. -We measure the absorption of radio
frequency energy v. magnetic field strength.
Put the sample in a microwave resonance cavity at
40 K and adjust the rf frequency until it matches
the cyclotron frequency. At this point we see a
resonant peak in the energy absorption.
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Carrier Statistics in Semiconductors
Doping -Replace Si lattice atoms with another
atom, particularly with an extra or deficient
valency. (e.g. P, As, S, B in Si)
eD is important because it tells you what
fraction of the dopant atoms are going to be
ionized at a given temperature. For P in Si, eD
45 meV, leading to 99.6 ionization at RT. Then,
the total electron concentration (for and n type
dopant) is
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Carrier Statistics in Semiconductors
If we look at the Fermi Level position as a
function of temperature (for some sample), we see
that all donor states are filled at T 0 (n 0,
no free carriers), EF ED. At high temperatures,
such that, ni gtgt ND, then n ni and EF Ei. EF
ranges between these limiting values at
intermediate temperatures. (see fig) Npni2 still
holds, but one must substitute n ni ND p
ni2/(ni ND). At room temperature n ND For
Silicon - ni 1010 cm-3. - ND 1013 ? 1016
cm-3 (lightly doped ? heavily doped)
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Carrier Statistics in Semiconductors
This is 60meV/decade. Three decades x 60
meV/decade 180 meV EC EF. EF is very close
to the conduction band edge.
Similar to EF(T), lets look at n(T)
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Charge Conduction in Semiconductors
1-D
3-D (from Statistical Mechanics)
Brownian Motion
Applying a force to the particle directionalizes
the net movement. The force is necessary since
Brownian motion does not direct net current.
q lt 0 for e-, q gt 0 for h
Constant Field leads to an acceleration of
carriers scaled by me.
If this were strictly true, e-s would accelerate
without bound under a constant field. Obviously,
this isnt the case. Electrons are slowed by
scattering events.
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Scattering Processes in Semiconductors

1. Ionized Impurity Scattering
2. Phonon (lattice) Scattering
3. Neutral Impurity Scattering
4. Carrier-Carrier Scattering
5. Piezoelectric Scattering

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Scattering Processes in Semiconductors
3) - Donors and acceptors under freez-out. -
Low T only - Defect polycrystalline Si 4) -
e- - h scattering is insignificant due to low
carrier concentration of one type or another
- e- - e- or h - h dont change mobility since
collisions between these dont change the total
momentum of those carriers. 5) - GaAs
displacement of atoms ? internal electric field,
but very weak. 2) - collisions between carriers
and thermally agitated lattice atoms. Acoustic
- Mobility decreases as temperature increases
due to increased lattice vibration - Mobility
decreases as effective mass increases.
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Scattering Processes in Semiconductors
1) Coloumb attraction or repulsion between charge
carriers and ND or NA-.
Due to all of these scattering processes, it is
possible to define a mean free time, ?m, and a
mean free path lm.
Average directed velocity
For solar cells (and most devices) high mobility
is desirable since you must apply a smaller
electric field to move carriers at a give
velocity.
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Current Flow in Semiconductors
e-
(cm-3)(cm/s) e-/cm-2 s
A/cm-2 C/cm-2 s
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Current Flow in Semiconductors
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Mobility
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Mobility
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Saturation Velocity
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Mobility and Impurities