Band Theory of Solids Garcia Chapter 24

1. Band Theory of Solids (Garcia Chapter 24)

3. �Realistic� Potential in Solids ni are integers Example: 2D Lattice

4. �Realistic� Potential in Solids For one dimensional case where atoms (ions) are separated by distance d, we can write condition of periodicity as

5. �Realistic� Potential in Solids Multi-electron atomic potentials are complex Even for hydrogen atom with a �simple� Coulomb potential solutions are quite complex So we use a model one-dimensional periodic potential to get insight into the problem

6. Bloch�s Theorem Bloch�s Theorem states that for a particle moving in the periodic potential, the wavefunctions ?(x) are of the form uk(x) is a periodic function with the periodicity of the potential The exact form depends on the potential associated with atoms (ions) that form the solid

7. Bloch�s Theorem Bloch�s Theorem imposes very special conditions on any solution of the Schr�dinger equation, independent of the form of the periodic potential The wave vector k has a two-fold role: It is still a wave vector in the plane wave part of the solution It is also an index to uk(x) because it contains all the quantum numbers, which enumerate the wavefunction

8. Bloch�s Theorem What is probability density of finding particle at coordinate x?

9. Bloch�s Theorem The probability of finding an electron at any atom in the solid is the same!!! Each electron in a crystalline solid �belongs� to each and every atom forming the solid

10. Covalent Bonding Revisited When atoms are covalently bonded electrons supplied by atoms are shared by these atoms since pull of each atom is the same or nearly so H2, F2, CO, Example: the ground state of the hydrogen atoms forming a molecule If the atoms are far apart there is very little overlap between their wavefunctions If atoms are brought together the wavefunctions overlap and form the compound wavefunction, ?1(r)+?2(r), increasing the probability for electrons to exist between the atoms

11. Covalent Bonding Revisited

12. Schr�dinger Equation Revisited If a wavefunctions ?1(x) and ?2(x) are solutions for the Schr�dinger equation for energy E, then functions -?1(x), -?2(x), and ?1(x)�?2(x) are also solutions of this equations the probability density of -?1(x) is the same as for ?1(x)

13. Consider an atom with only one electron in s-state outside of a closed shell Both of the wavefunctions below are valid and the choice of each is equivalent If the atoms are far apart, as before, the wavefunctions are the same as for the isolated atoms Band Theory of Solids

14. Band Theory of Solids The sum of them is shown in the figure These two possible combinations represent two possible states of two atoms system with different energies Once the atoms are brought together the wavefunctions begin to overlap There are two possibilities Overlapping wavefunctions are the same (e.g., ?s+ (r)) Overlapping wavefunctions are different

15. Tight-Binding Band Theoryof Solids Garcia Chapter 24.4 and 24.5

16. Electron in Two Separated Potential Wells

17. Potential Wells Moved Closer

18. Tight-Binding Approximationfirst two states in infinite and finite potential wells

19. Symmetric and Anti-symmetric Combinations of Ground State Eigenfunctions

20. Six States for Six Atom Solid

21. Splitting of 1s State of Six Atoms

22. Atoms and Band Structure Consider multi-electron atoms: The outer electrons (large n and l) are �closer� to each other than the inner electrons Thus, the overlap of the wave-functions of the outer electrons is stronger than overlap of those of inner electrons Therefore, the bands formed from outer electrons are wider than the bands formed from inner electrons Bands with higher energies are therefore wider!

23. Splitting of Atomic Levels in Sodium

25. Splitting of Atomic Levels in Carbon

26. Occupation in Carbon at Large Atomic Separation

27. Actual Occupation of Energy bands in Diamond

28. Insulators, Semiconductors, Metals The last completely filled (at least at T = 0 K) band is called the Valence Band The next band with higher energy is the Conduction Band The Conduction Band can be empty or partially filed The energy difference between the bottom of the CB and the top of the VB is called the Band Gap (or Forbidden Gap)

29. Can be found using computer In 1D computer simulation of light in a periodic structure, we found the frequencies and wave functions Allowed modes fall into quasi-continuous bands separated by forbidden bands just as would be expected from the tight binding model Computer simulation can give exact solution in simple cases

30. Insulators, Semiconductors, Metals Consider a solid with the empty Conduction Band If apply electric field to this solid, the electrons in the valence band (VB) cannot participate in transport (no current)

31. Insulators, Semiconductors, Metals The electrons in the VB do not participate in the current, since Classically, electrons in the electric field accelerate, so they acquire [kinetic] energy In QM this means they must acquire slightly higher energy and jump to another quantum state Such states must be available, i.e. empty allowed states But no such state are available in the VB!

32. Insulators, Semiconductors, Metals Consider a solid with the half filled Conduction Band (T = 0K) If an electric field is applied to this solid, electrons in the CB do participate in transport, since there are plenty of empty allowed states with energies just above the Fermi energy This solid would behave as a conductor (metal)

33. Band Overlap Many materials are conductors (metals) due to the �band overlap� phenomenon Often the higher energy bands become so wide that they overlap with the lower bands additional electron energy levels are then available

34. Band Overlap Example: Magnesium (Mg; Z =12): 1s22s22p63s2 Might expect to be insulator; however, it is a metal 3s-band overlaps the 3p-band, so now the conduction band contains 8N energy levels, while only have 2N electrons Other examples: Zn, Be, Ca, Bi

35. Band Hybridization In some cases the opposite occurs Due to the overlap, electrons from different shells form hybrid bands, which can be separated in energy Depending on the magnitude of the gap, solids can be insulators (Diamond); semiconductors (Si, Ge, Sn; metals (Pb)

36. Insulators, Semiconductors, Metals There is a qualitative difference between metals and insulators (semiconductors) the highest energy band �containing� electrons is only partially filled for Metals (sometimes due to the overlap) Thus they are good conductors even at very low temperatures The resisitvity arises from the electron scattering from lattice vibrations and lattice defects Vibrations increases with temperature ? higher resistivity The concentration of carriers does not change appreciably with temperature

37. Insulators, Semiconductors, Metals The difference between Insulators and Semiconductors is �quantitative� The difference in the magnitude of the band gap Semiconductors are �Insulators� with a relatively small band gap At high enough temperatures a fraction of electrons can be found in the conduction band and therefore participate in transport

38. Insulators vs Semiconductors There is no difference between Insulators and Semiconductors at very low temperatures In neither material are there any electrons in the conduction band � and so conductivity vanishes in the low temperature limit

39. Insulators vs Semiconductors Differences arises at high temperatures A small fraction of the electrons is thermally excited into the conduction band. These electrons carry current just as in metals The smaller the gap the more electrons in the conduction band at a given temperature Resistivity decreases with temperature due to higher concentration of electrons in the conduction band

40. Holes Consider an insulator (or semiconductor) with a few electrons excited from the valence band into the conduction band Apply an electric field Now electrons in the valence band have some energy sates into which they can move The movement is complicated since it involves ~ 1023 electrons

41. Concept of Holes Consider a semiconductor with a small number of electrons excited from the valence band into the conduction band If an electric field is applied, the conduction band electrons will participate in the electrical current the valence band electrons can �move into� the empty states, and thus can also contribute to the current

42. Holes from the Band Structure Point of View If we describe such changes via �movement� of the �empty� states � the picture can be significantly simplified This �empty space� is a Hole �Deficiency� of negative charge � holes are positively charged Holes often have a larger effective mass (heavier) than electrons since they represent collective behavior of many electrons

43. Holes We can �replace� electrons at the top of eth band which have �negative� mass (and travel in opposite to the �normal� direction) by positively charged particles with a positive mass, and consider all phenomena using such particles Such particles are called Holes Holes are positively charged and move in the same direction as electrons �they replace�

44. Hole Conduction To understand hole motion, one requires another view of the holes, which represent them as electrons with negative effective mass To imagine the movement of the hole think of a row of chairs occupied by people with one chair empty To move all people rise all together and move in one direction, so the empty spot moves in the same direction

45. Concept of Holes If we describe such changes via �movement� of the �empty� states � the picture will be significantly simplified This �empty space� is called a Hole �Deficiency� of negative charge can be treated as a positive charge Holes act as charge carriers in the sense that electrons from nearby sites can �move� into the hole Holes are usually heavier than electrons since they depict collective behavior of many electrons

46. Conduction