1. ECE 875: Electronic Devices
Prof. Virginia Ayres
Electrical & Computer Engineering
Michigan State University

2. 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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3. P. 11: the general reciprocal lattice vector is defined:
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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4. R = ma + nb + pc  (mnp) plane in direct space
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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5. For 1.5(a):
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6. Used to show that: are equal When e-s described as waves y(r,k)
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7. 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
VM Ayres, ECE875, S14

8. 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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9. 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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10. specifies V(r) for a particular crystal
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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11. Looking at k: E k
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12. 1st Brillouin zone for fcc primitive cell based crystals: Wigner-Seitz cell
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13. 2D example of how to find a Wigner Seitz cell:
k-space
SAED diffraction pattern
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14. 2D example of how to find a Wigner Seitz cell:
Pick center
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15. 2D example of how to find a Wigner Seitz cell:
Nearest neighbors
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16. 2D example of how to find a Wigner Seitz cell:
Perpendicular bisectors (represents a plane)
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17. 2D example of how to find a Wigner Seitz cell:
Next nearest neighbors
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18. 2D example of how to find a Wigner Seitz cell:
Perpendicular bisectors
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19. 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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20. This Wigner-Sietz cell in reciprocal space is the 1st Brillouin zone for all fcc primitive cell-based crystals:
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21. Looking at E: Egap: E k
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22. Plot E(k): shows metallic behavior in certain direction in k-space:
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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23. Therefore: Use a parabola to approximate E(k) in the region of lowest EC or highest EV
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24. Example:
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25. Conduction band minimum:
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26. How many conduction band minima?
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27. Answer:6 conduction band minima
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28. Could re-write kx, ky and kz in terms of longitudinal and transverse:
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29. Wikipedia: ellipsoid. Set b = c
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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30. b a
c = b
Google Image Result for http--www_mathworks_com-help-releases-R2013b-matlab-ref-ellipsoid1_gif
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31. VM Ayres, ECE875, S14

32. Note: these are not the real numbers for Si!
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