1 / 32

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

Download Presentation

ECE 875: Electronic Devices

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related