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ECE 875: Electronic Devices

ECE 875: Electronic Devices. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu. Lecture 04, 15 Jan 14. Chp. 01 Crystals: Reciprocal space ( k -space) 1 st Brillouin zone (Wigner-Seitz) Energy levels: E- k Approximating by a parabola

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ECE 875: Electronic Devices

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

  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 VM Ayres, ECE875, S14

  3. 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* VM Ayres, ECE875, S14

  4. 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 VM Ayres, ECE875, S14

  5. For 1.5(a): VM Ayres, ECE875, S14

  6. Used to show that: When e-s described as waves y(r,k) are equal VM Ayres, ECE875, S14

  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 VM Ayres, ECE875, S14

  9. E-k energy band diagrams: very useful. How to derive one: Step 01: Step 02: minimize the energy E(k) ECE 802: Nanoelectronics VM Ayres, ECE875, S14

  10. 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 VM Ayres, ECE875, S14

  11. Looking at k: E k VM Ayres, ECE875, S14

  12. 1st Brillouin zone for fcc primitive cell based crystals:Wigner-Seitz cell VM Ayres, ECE875, S14

  13. 2D example of how to find a Wigner Seitz cell: k-space SAED diffraction pattern VM Ayres, ECE875, S14

  14. 2D example of how to find a Wigner Seitz cell: Pick center VM Ayres, ECE875, S14

  15. 2D example of how to find a Wigner Seitz cell: Nearest neighbors VM Ayres, ECE875, S14

  16. 2D example of how to find a Wigner Seitz cell: Perpendicular bisectors (represents a plane) VM Ayres, ECE875, S14

  17. 2D example of how to find a Wigner Seitz cell: Next nearest neighbors VM Ayres, ECE875, S14

  18. 2D example of how to find a Wigner Seitz cell: Perpendicular bisectors VM Ayres, ECE875, S14

  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 VM Ayres, ECE875, S14

  20. This Wigner-Sietz cell in reciprocal space is the 1st Brillouin zone for all fcc primitive cell-based crystals: VM Ayres, ECE875, S14

  21. Looking at E: Egap: E k VM Ayres, ECE875, S14

  22. 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: VM Ayres, ECE875, S14

  23. Therefore: Use a parabola to approximate E(k) in the region of lowest EC or highest EV VM Ayres, ECE875, S14

  24. Example: VM Ayres, ECE875, S14

  25. Conduction band minimum: VM Ayres, ECE875, S14

  26. How many conduction band minima? VM Ayres, ECE875, S14

  27. Answer:6 conduction band minima VM Ayres, ECE875, S14

  28. Could re-write kx, ky and kz in terms of longitudinal and transverse: VM Ayres, ECE875, S14

  29. The parabola approximation and the equivalent constant energy surface ellipsoid (“cigar shaped minima”) description are the same: Parabola: Ellipsoid: Wikipedia: ellipsoid. Set b = c VM Ayres, ECE875, S14

  30. b a c = b Google Image Result for http--www_mathworks_com-help-releases-R2013b-matlab-ref-ellipsoid1_gif VM Ayres, ECE875, S14

  31. VM Ayres, ECE875, S14

  32. Note: these are not the real numbers for Si! VM Ayres, ECE875, S14

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