1. Nanoelectronics Chapter 5 Electrons Subjected to a Periodic Potential – Band Theory of Solids 1Q.Li@Physics.WHU@2015.3

2. 2 STM image of a nickel surface

3. 5.1 Crystalline Materials Solid materials may be classified as crystalline, polycrystalline or amorphous. Crystalline solid consists of a periodic array of atoms called the lattice. Polycrystalline solid has a well-defined structure in each of many small regions, each region differs from its neighboring regions. Amorphous solid does not exhibit any sort of regularity, such as glass and plastic. Q.Li@Physics.WHU@2015.33

4. 5.1 Crystalline Materials Most interested electronic materials are crystalline solids, e.g., semiconductors, such as Si, GaAs, and conductors, such as Al, Cu and Au Here, we consider the effect of periodic lattice on electronic properties. Q.Li@Physics.WHU@2015.34

5. Crystal Types The fundamental property of a crystal is regularity in its atomic structure; the atoms in a crystal arranged in a regular (periodic / repeated) array. A lattice: is a set of points that form a periodic structure. we need to understand its basis. Here, we consider: simple cubic (sc), body- centered cubic (bcc) and face-centered cubic (fcc). Q.Li@Physics.WHU@2015.35

6. Crystal Types Q.Li@Physics.WHU@2015.36 The 7 lattice systems (From least to most symmetric) 14 Bravais lattices

7. Simple cubic (sc) is the simplest lattice Q.Li@Physics.WHU@2015.37 A sc lattice consists of points equally spaced at the corners of a 3D cube. Fundamental translation vectors identical Atomic arrangement at r ’ and r u 1, u 2 and u 3 are integers.

8. Simple cubic (sc) is the simplest lattice The set of three vectors: form a parallelepiped. The parallelepiped with the smallest volume is called the primitive cell. The primitive cell has only one lattice point, and a crystal can be constructed from repetitions of the primitive cell. However, usually other unit cells is more convenient to work with. Q.Li@Physics.WHU@2015.38

9. Simple cubic (sc) is the simplest lattice Not necessary that a single atom is located at each lattice point. a group of atoms, called a basis, is placed at each lattice point. Q.Li@Physics.WHU@2015.39

10. Body-Centered Cubic (bcc) Q.Li@Physics.WHU@2015.310 Primitive translation vectors Examples: Na and W lattices

11. Face-Centered Cubic (fcc) Q.Li@Physics.WHU@2015.311 Primitive translation vectors

12. Face-Centered Cubic (fcc) Q.Li@Physics.WHU@2015.312 How many atoms in a conventional cell? Examples: metals such as Cu, Au, Ag, Ni and semiconductors such as Si, GaAs and Ge The volume of conventional cell is a 3. How about the primitive cell? The basis consists of two atoms The two atoms are identical  diamond structure: Si, Ge and C The two atoms are not identical  zinc blende structure GaAs and AlAs

13. Diamond crystal structure Q.Li@Physics.WHU@2015.313 The diamond crystal structure belongs to the face-centered cubic lattice, with a repeated 2-atom pattern. 2 fcc merge along ¼ diagonal direction.

14. 5.2 Electrons in a Periodic Potential Consider a one-dimensional example Q.Li@Physics.WHU@2015.314 Ionized atom / ion Coulomb potential of an electron:

15. 5.2 Electrons in a Periodic Potential Obviously, the potential is periodic In three-dimensional system: Q.Li@Physics.WHU@2015.315 Potential energy is periodic r is the position vector of the electron, T is crystal translation vector

16. 5.2 Electrons in a Periodic Potential Bloch’s Theorem: Q.Li@Physics.WHU@2015.316 If the potential is periodic, The solution of Schrodinger’s eq.    Electrons can propagate through a perfect periodic medium without scattering

17. 5.3 Kronig-Penney Model of Band Structure Q.Li@Physics.WHU@2015.317 If we set V 0 as infinitely large, it will become delta function.

18. 5.3 Kronig-Penney Model of Band Structure Q.Li@Physics.WHU@2015.318 where

19. 5.3 Kronig-Penney Model of Band Structure Q.Li@Physics.WHU@2015.319 where

20. In one dimension: The Bloch wavevector k to be determined: a is the period of the lattice: Q.Li@Physics.WHU@2015.320 5.3 Kronig-Penney Model of Band Structure

21. Q.Li@Physics.WHU@2015.321 5.3 Kronig-Penney Model of Band Structure

22. If 0 < E < V 0 If E > V 0 Q.Li@Physics.WHU@2015.322 5.3 Kronig-Penney Model of Band Structure where

23. Q.Li@Physics.WHU@2015.323 5.3 Kronig-Penney Model of Band Structure γ(E)

24. Q.Li@Physics.WHU@2015.324 5.3 Kronig-Penney Model of Band Structure

25. Dispersion diagram Because:  Q.Li@Physics.WHU@2015.325 5.3 Kronig-Penney Model of Band Structure Note: first Brillouin Zone second Brillouin Zone

26. Free electron continuous band  Electron energy band with gap Q.Li@Physics.WHU@2015.326 5.3 Kronig-Penney Model of Band Structure

27. 5.3.1 Effective Mass We view an electron quantum mechanically as a wavepacket, with electron’s velocity being its group velocity for free-electron model consider the importance of electrons near bandedges: Q.Li@Physics.WHU@2015.327

28. 5.3.1 Effective Mass Effective mass can be defined as or Q.Li@Physics.WHU@2015.328 agree with classical mechanics

29. Band Structures of Ge, Si and GaAs Q.Li@Physics.WHU@2015.329

30. Silicon Band Structure Q.Li@Physics.WHU@2015.330

31. Germanium Band Structure Q.Li@Physics.WHU@2015.331

32. First Brillouin Zone Q.Li@Physics.WHU@2015.332

33. 5.3.1 Effective Mass To simplify the Schrodinger’s equation, we can use m * to replace m Effective mass is different in different direction Q.Li@Physics.WHU@2015.333

34. 5.3.1 Effective Mass For SiO 2 Effective mass is modeled as Depends on interface Q.Li@Physics.WHU@2015.334

35. 5.4 Band Theory of Solid One electrons in partially filled energy band can contribute to conduction Q.Li@Physics.WHU@2015.335

36. 5.4 Band Theory of Solid Q.Li@Physics.WHU@2015.336

37. 5.4.1 Doping in Semiconductors Q.Li@Physics.WHU@2015.337 donor

38. 5.4.1 Doping in Semiconductors Q.Li@Physics.WHU@2015.338

39. 5.4.1 Doping in Semiconductors Q.Li@Physics.WHU@2015.339 Acceptor, B, typically E a – E v is about 45 meV

40. 5.4.2 Interacting systems Model Q.Li@Physics.WHU@2015.340

41. 5.4.2 Interacting systems Model Q.Li@Physics.WHU@2015.341

42. 5.4.2 Interacting systems Model Q.Li@Physics.WHU@2015.342

43. Direct Bandgap Semiconductors Q.Li@Physics.WHU@2015.343

44. Indirect Bandgap Semiconductors Q.Li@Physics.WHU@2015.344