The Band Theory of Solids

1.  V x 0 L Single electron theory: for any electron everything in the crystal represented by an effective periodic potential The Band Theory of Solids ? What are the limitations of the model of the free electron gas We don’t understand the origin of semiconductors and insulators Where are the fingerprints of discrete energy levels of the atoms in solids ? periodic potential Jellium model generalize

2. + + + + + translational invariance of electron density translational invariance of V + + + Jm and Since is fulfilled with |Jm|=1 Applying periodic boundary conditions Bloch functions 1 dimensional consideration: Schroedinger equation invariant under translation J L JN=1 M: integer Note: for arigorous proof of the Bloch Theorem based on group theoretical methods compare e.g.: M. Tinkham, Group Theory and Quantum Mechanics, McGraw-Hill 1964, p.38

3. with J can be written in the form Any function satisfying Transformational property imposed by the translational symmetry group where celebrated Bloch theorem

4. ? How do Bloch functions look like a Bloch functions straightforward generalized to three dimensions

5. Expansion involving the orthogonal set Apply this to We know already: Fourier expansion of translational invariant function where reciprocal lattice vectors Consequences of this representation

6. Bloch waves which differ by are identical Proof:

7. Periodic potential with Periodic potential Let’s gradually “switch on” the VG

8. Reduced Zone Scheme Consider analogy to phonon dispersion -standing waves at zone boundary due to Bragg reflection -degeneracy lifted for 2 atoms/unit cell with -standing electron waves at zone boundary due to Bragg reflection -degenerated lifted for degeneracy

9. Superposition of and @ Standing electron wave: 1 2 reduced with respect to free electron Potential energy of increased with respect to free electron

10. E k Degeneracy lifted Splitting proportional to leading Fourier coefficient VG of the periodic potential

11. Use periodicity of Qualitative discussion more formal approach in 1 dimension Schroedinger equation: and plane wave expansion each coefficient=0

12. where and Translation by a reciprocal lattice vector G Simplification: Leading x-dependent term of periodic potential

13. Consistent approximation for approx. only for G=0 We are interested in the splitting at and for G=0 and for G=0 and V0=0 for simplicity for Study 2 coupled equations for Ck and Ck-G1 1 2 more accurate approach requires qm perturbation theory

14. Free electron energy @ ? What do we learn from the 2 linear equations substitution in Standing waves : 1 From secular equation of 2 we obtain: where

15. E k Solution: 2|VG1|

16. Free electrons in periodic potential “switching on” Bandstructure can be derived from two extreme limiting cases 1 2 From insulated atoms “switching on” interaction for

17. Tight-Binding Approximation LCAO-method: Linear Combination of Atomic Orbitals • Energy splitting: • coupled pendulums • covalent bond in H2-molecule

18. group velocity of wave packet band structure E effective mass k Motion of Electrons in Bands and the Effective mass Free electron in vacuum: Remains useful description for electrons in a crystal in a crystal: Crystal momentum:

19. 0 k 0 k 0 k Bands in general not isotropic effective mass becomes a tensor Consider the free electron to remember this formula: E v m*

20. Current density for homogeneous velocity Currents in Bands and the Concept of Holes Remember: generalized Current density for k-dependent velocity Only one spin species considered In this formula 1st BZ for a fully occupied band

21. Current density carried by a full band is zero: Full band: for each velocity there is also because: for crystals with inversion symmetry in general:

22. k empty behave like positively charged particles Partially filled band: 0 1st BZ k occupied k empty Current density of holes

23. E Metals, Insulators and Semiconductors Only partly filled electronic band can contribute to electric current conduction band EC Eg EF EV valence band core electrons Semiconductor Insulator Metal

24. e.g. Li+ 0 k ? Why is Na or Li a metal e- metal Half filled band SEA OF MOBILE VALENCE ELECTRONS L=Na 1 valence electron/ atom band is filled with 2N electrons Spin degeneracy # of primitive unit cells