ECE 875: Electronic Devices

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ECE 875Electronic Devices

• Prof. Virginia Ayres
• Electrical Computer Engineering
• Michigan State University
• ayresv_at_msu.edu

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Lecture 04, 15 Jan 14
Chp. 01 Crystals Reciprocal space (k-space)
1st Brillouin zone (Wigner-Seitz) Energy
levels E-k Approximating by a parabola Same
Constant energy surfaces
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P. 10-plus for a given set of direct primitive
cell basis vectors a, b, and c, the set of
reciprocal k-space lattice vectors a, b, c
are defined (3D)
P. 11 the general reciprocal lattice vector is
defined G ha kb lc
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Therefore R ma nb pc ? (mnp) plane in
direct space G k ha kb lc ? (hkl)
plane in reciprocal space
That is why when you show that G . R 2 p x
integer (Pr. 05a) it is also a relationship with
a set of planes in the direct lattice. Helpful
(p. 11)
This is 2p dij relationship is used as an
alternative (better) definition to find the
reciprocal basis vectors ... Easy to use it to
evaluate the reciprocal basis vectors ... in 1D
or 2D. Harder in 3D, so the answer (preceding
side) is given in textbooks for you
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For 1.5(a)
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Used to show that
When e-s described as waves y(r,k)
are equal
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Lecture 04, 15 Jan 14
Chp. 01 Crystals Reciprocal space (k-space)
1st Brillouin zone (Wigner-Seitz) Energy
levels E-k Approximating by a parabola Same
Constant energy surfaces
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Motivation
Electronics Transport e-s moving in an
environment Correct e- wave function in a
crystal environment Bloch function Sze y(r,k)
exp(jk.r)Ub(r,k) y(r R,k) Correct E-k
energy levels versus direction of the
environment minimum Egap Correct
concentrations of carriers n and p Correct
current and current density J moving
carriers I-V measurement J Vext direction versus
internal E-k Egap direction Fixed e-s and
holes C-V measurement
(KE PE) y(r,k) E y(r,k)
x Probability f0 that energy level is occupied
q n, p velocity Area
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E-k energy band diagrams very useful. How to
derive one
Step 01
Step 02 minimize the energy E(k)
ECE 802 Nanoelectronics
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• After someone
• specifies y(r,k)
• specifies V(r) for a particular crystal
• Gets a general form solution for E as a function
  of k from Conservation of Energy
• Adjusts y(r,k) so that the energy E(k) is the
  minimum energy possible
• Solves for the specific crystal system E(k)
• Get E-k diagram

E
k
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Looking at k
E
k
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1st Brillouin zone for fcc primitive cell based
crystalsWigner-Seitz cell
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2D example of how to find a Wigner Seitz cell
k-space SAED diffraction pattern
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2D example of how to find a Wigner Seitz cell
Pick center
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2D example of how to find a Wigner Seitz cell
Nearest neighbors
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2D example of how to find a Wigner Seitz cell
Perpendicular bisectors (represents a plane)
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2D example of how to find a Wigner Seitz cell
Next nearest neighbors
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2D example of how to find a Wigner Seitz cell
Perpendicular bisectors
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2D example of how to find a Wigner Seitz cell
Wigner Sietz cell is the shaded area (in 2D) Can
do this in direct space or reciprocal space
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This Wigner-Sietz cell in reciprocal space is the
1st Brillouin zone for all fcc primitive
cell-based crystals
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Looking at E Egap
E
k
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Full expression for E as a function of k can be
complicated for Si, etc.
1D polyacetylene
This simple 1D example still has a complicated
full expression for E(k)
Plot E(k) shows metallic behavior in certain
direction in k-space
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Therefore Use a parabola to approximate E(k) in
the region of lowest EC or highest EV
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Example
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Conduction band minimum
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How many conduction band minima?
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Answer6 conduction band minima
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Could re-write kx, ky and kz in terms of
longitudinal and transverse
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The parabola approximation and the equivalent
constant energy surface ellipsoid (cigar shaped
minima) description are the same
Parabola
Ellipsoid
Wikipedia ellipsoid. Set b c
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b
a
c b
Google Image Result for http--www_mathworks_com-he
lp-releases-R2013b-matlab-ref-ellipsoid1_gif
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(No Transcript)
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Note these are not the real numbers for Si!