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ECE 875: Electronic Devices

ECE 875: Electronic Devices. Prof. Virginia Ayres Electrical & Computer Engineering Michigan State University ayresv@msu.edu. Course Content: Core: Part I: Semiconductor Physics Chapter 01: Physics and Properties of Semiconductors – a Review Part II: Device Building Blocks

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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. Course Content: Core: Part I: Semiconductor Physics Chapter 01: Physics and Properties of Semiconductors – a Review Part II: Device Building Blocks Chapter 02: p-n Junctions Chapter 03: Metal-Semiconductor Contacts Chapter 04: Metal-Insulator-Semiconductor Capacitors Part III: Transistors Chapter 06: MOSFETs VM Ayres, ECE875, S14

  3. Course Content: Beyond core: VM Ayres, ECE875, S14

  4. Lecture 02, 10 Jan 14 VM Ayres, ECE875, S14

  5. Crystal Structures: Motivation: Electronics: Transport: e-’s moving in an environment Correct e- wave function in a crystal environment: Block function: Y(R) = expik.ay(R) = Y(R + a) 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 = EY x Probability f0 that energy level is occupied q n, p velocity Area VM Ayres, ECE875, S14

  6. Unit cells: A Unit cell is a convenient but not minimal volume that contains an atomic arrangement that shows the important symmetries of the crystal Why are Unit cells like these not good enough? Compare: Sze Pr. 01(a) for fcc versus Pr. 03 VM Ayres, ECE875, S14 Non-cubic

  7. fcc lattice, to match Pr. 03 VM Ayres, ECE875, S14

  8. 14 atoms needed VM Ayres, ECE875, S14

  9. Crystal Structures: Motivation:Electronics: Transport: e-’s moving in an environment Correct e- wave function in a crystal environment: Block function: Y(R) = expik.ay(R) = Y(R + a) Periodicity of the environment: Need specify where the atoms are Unit cell a3 for cubic systems sc, fcc, bcc, etc. OR Primitive cell for sc, fcc, bcc, etc. OR Atomic basis Think about: need to specify: Most atoms Fewer atoms Least atoms VM Ayres, ECE875, S14

  10. A primitive Unit cell is theminimal volume that contains an atomic arrangement that shows the important symmetries of the crystal Example: What makes a face-centered cubic arrangement of atoms unique? Hint: Unique means unique arrangement of atoms within an a3 cube. VM Ayres, ECE875, S14

  11. Answer: Atoms on the faces VM Ayres, ECE875, S14

  12. Answer: Atoms on the faces Also need two corner atoms that give maximum dimension of volume of a3 cubic Unit cell VM Ayres, ECE875, S14

  13. Answer: Atoms on the faces Also need two corner atoms that give maximum dimension of volume of a3 cubic Unit cell This arrangement of 8 atoms does represent the fcc primitive cell But: specifying the arrangement of 8 atoms is a complicated description. There is a simpler way. VM Ayres, ECE875, S14

  14. Switch to a simpler example: • How many atoms do you need to describe this simple cubic structure? • Want to specify: • atomic arrangement • minimal volume: a3 for this structure Start: 8 atoms,1 on each corner. Do you need all of them? VM Ayres, ECE875, S14

  15. Simpler Example: Answer: 4 atoms and 3 vectors between them give the minimal volume = l x w x h. 4 red atoms Specify 3 vectors: a = l = ax + 0y + 0z b = w = 0x + ay + 0z c = h = 0x + 0y + az Minimal Vol = a . b x c Specify the atomic arrangement as: one atom at every vertex of the minimal volume. h w l VM Ayres, ECE875, S14

  16. Return to fcc primitive cell example: 8 atoms: Simpler description: 4 atoms and 3 vectors between them give the volume of a non-orthogonal solid (parallelepiped) p.11: Volume = a . b x c Specify the atomic arrangement as: one atom at every vertex. rotate b c a VM Ayres, ECE875, S14

  17. Better picture of the fcc parallelepiped: tilt rotate Ashcroft & Mermin VM Ayres, ECE875, S14

  18. This is what Sze does in Chp.01, Pr. 03: VM Ayres, ECE875, S14

  19. Picture and coordinate system for Pr. 03: For a face centered cubic, the volume of a conventional unit cell is a3. Find the volume of an fcc primitive cell with three basis vectors: (000) -> (a/2,0,a/2): a = a/2 x + 0 y + a/2 z (000) -> (a/2,a/2,0): b = a/2 x + a/2 y + 0 z (000) -> (0,a/2,a/2): c = 0 x + a/2 y + a/2 z z c a y b (000) x VM Ayres, ECE875, S14

  20. VM Ayres, ECE875, S14

  21. = Volume of fcc primitive cell VM Ayres, ECE875, S14

  22. Sze, Chp.01, Pr. 03: For a face centered cubic, the volume of a conventional unit cell is a3. Find the volume of an fcc primitive cell with three basis vectors: (000) -> (a/2,0,a/2): a = a/2 x + 0 y + a/2 z (000) -> (a/2,a/2,0): b = a/2 x + a/2 y + 0 z (000) -> (0,a/2,a/2): c = 0 x + a/2 y + a/2 z a, b and c are the primitive vectors of the fcc Bravais lattice. P. 10: “Three primitive basis vectors a, b, and c of a primitive cell describe a crystalline solid such that the crystal structure remains invariant under translation through any vector that is the sum of integral multiples of these basis vectors. In other words, the direct lattice sites can be defined by the set R = ma + nb + pc.” Translational invariance is great fordescribing an e- wave function acknowledging the symmetries of its crystal environment: Block function: Y(R) = expik.ay(R) = Y(R + a) VM Ayres, ECE875, S14

  23. Formal definition of a Primitive cell, Ashcroft and Mermin: “A volume of space that when translated through all the vectors of a Bravais lattice just fills all the space without either overlapping itself of leaving voids is called a primitive cell or a primitive Unit cell of the lattice.” VM Ayres, ECE875, S14

  24. Steps for fcc were: 1. four atoms a = a/2 x + 0 y + a/2 z b = a/2 x + a/2 y + 0 z c = 0 x + a/2 y + a/2 z 2. three vectors between them Anywhere: R = ma + nb + pc 3. minimal Vol = a3/4 (parallelepiped) atom at each vertex of the minimal volume VM Ayres, ECE875, S14

  25. Typo, p. 08: No! Figure 1 shows conventional Unit cells! VM Ayres, ECE875, S14

  26. Conventional Unit Cells Vol. = a3 Primitive Unit Cells: Smaller Volumes Vol = a3/4 VM Ayres, ECE875, S14

  27. Primitive cell for fcc is also the primitive cell for diamond and zincblende: Conventional cubic Unit cell Primitive cell for: fcc, diamond and zinc-blende VM Ayres, ECE875, S14

  28. P. 08: the diamond (and zinc-blende) lattices can be considered as two inter-penetrating fcc lattices. The two interpenetrating fcc lattices are displaced (¼, ¼, ¼) x aNote: also have pairs of atoms displaced (¼, ¼, ¼) x a: a = lattice constant VM Ayres, ECE875, S14

  29. P. 08: the diamond (and zinc-blende) lattices can be considered as two inter-penetrating fcc lattices. The two interpenetrating fcc lattices are displaced (¼, ¼, ¼) x aNote: also have pairs of atomsdisplaced (¼, ¼, ¼) x a: a = lattice constant VM Ayres, ECE875, S14

  30. Example: What are the three primitive basis vectors for the diamond primitive cell? (000) -> (a/2,0,a/2): a = a/2 x + 0 y + a/2 z (000) -> (a/2,a/2,0): b = a/2 x + a/2 y + 0 z (000) -> (0,a/2,a/2): c = 0 x + a/2 y + a/2 z How to make it diamond: two-atom basis VM Ayres, ECE875, S14

  31. z y x Picture and coordinate system for example problem: (000) VM Ayres, ECE875, S14

  32. Answer: Three basis vectors for the diamond primitive cell: (000) -> (a/2,0,a/2): a = a/2 x + 0 y + a/2 z (000) -> (a/2,a/2,0): b = a/2 x + a/2 y + 0 z (000) -> (0,a/2,a/2): c = 0 x + a/2 y + a/2 z Same basis vectors as fcc Same primitive cell volume a3/4 Make it diamond by specifying the atomic arrangement as: a two-atom basis at every vertex of the primitive cell. Pair a 2nd atom at (¼ , ¼, ¼) x a with every fcc atom in the primitive cell: (000) VM Ayres, ECE875, S14

  33. Rock salt VM Ayres, ECE875, S14

  34. Rock salt can be also considered as two inter-penetrating fcc lattices.Discussion: Lec 03 13 Jan 14 VM Ayres, ECE875, S14

  35. Direct space (lattice) Direct space (lattice) Conventional cubic Unit cell Primitive cell for: fcc, diamond, zinc-blende, and rock salt Rock salt VM Ayres, ECE875, S14

  36. Direct space (lattice) Direct space (lattice) Reciprocal space (lattice) Conventional cubic Unit cell Primitive cell for: fcc, diamond, zinc-blende, and rock salt Reciprocal space = first Brillouin zone for: fcc, diamond, zinc-blende, and rock salt VM Ayres, ECE875, S14

  37. HW01: Direct lattice Reciprocal lattice Reciprocal lattice Reciprocal lattice Needed fordescribing an e- wave function in terms of the symmetries of its crystal environment: Block function: Y(R) = expik.ay(R) = Y(R + a) VM Ayres, ECE875, S14

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