Electronic Band Structures electrons in solids: in a periodic potential due to the periodic arrays of atoms electronic band structure: electron states in the energy versus wavevector space reciprocal lattice of a crystal structure in the k -space periodic array of lattices in real space spatial Fourier transformation Reciprocal Lattices and the First Brillouin Zone reciprocal lattice : expression of crystal lattice in Fourier space

reciprocal lattice FT in time FT in space

 Distribution of electrons Ex) covalent bonding crystal Any periodic function can be expanded as a Fourier series. A crystal is invariant under any translation of the form : crystal axes Physical property of the crystal is invariant under : : electron number density

For a 1-D system, expand n(x) in Fourier series 2  p/a : ensures that n(x) has a period a. reciprocal point in the reciprocal lattice or Fourier space of the crystal. in a compact form n p : complex number

The spatial Fourier transform produces the wavevectors. These wavevectors form discrete points in k -space. : reciprocal lattice vector

definition of primitive vectors of reciprocal lattice reciprocal lattice vector In a 3-D system,  Reciprocal lattice vectors : primitive vectors of the crystal lattice

scattering amplitude the difference in phase factor  Diffraction Conditions The set of reciprocal lattice vectors determines possible x-ray reflections. : scattering vector

If diffraction condition : If

 Inversion of Fourier series Sinceis an integer,

reciprocal lattice axis

defined as a Wigner-Seitz unit cell in the reciprocal lattice The waves whose wavevector starts from origin, and ends at this plane satisfies diffraction condition Brillouin construction exhibits all the wavevectors which can be Bragg reflected by the crystal Wigner-Seitz unit cell  Brillouin zone

1 st Brillouin zone 2 nd B. Zone The first brilloun zone is the smallest volume entirely enclosed by planes that are the perpendicular bisection of the reciprocal lattice vectors drawn from the origin

lattice constant : a primitive basis vectors  Reciprocal lattice to simple cubic lattice primitive vectors : volume : reciprocal primitive vectors 1 st Brillouin zone : planes that perpendicular at the center of 6 reciprocal vectors volume :

primitive basis vectors  Reciprocal lattice to BCC lattice primitive vectors : volume : reciprocal primitive vectors : volume :

general reciprocal lattice vector : First Brillouin zone of bcc 1st Brillouin zone : the shortest

primitive basis vectors of fcc  Reciprocal lattice to FCC lattice primitive vectors : volume : reciprocal primitive vectors volume :

Brillouin zone of fcc 1st Brillouin zone : the shortest volume :

One electron model or nearly free electron model : Each electrons moves in the average field created by the other electrons and ions Hamiltonian operator Bloch’s Theorem Recall

periodic potential function corresponding eigenvalue problem In 1-DIn 3-D

one-electron Schrödinger equation : periodicity of the lattice structure : can be expanded as a Fourier series :

solution to Schrödinger equation (eigenfunctions) for a periodic potential must be a special form : Bloch theorem : a periodic function with the periodicity of lattices

 Central Equation one-electron Schrödinger equation : Wavefunction can be expressed as a Fourier series summed over all values of permitted wavevectors:

The coefficient of each Fourier component must be equal : : central equation When free electron as used in the Sommerfeld theory

At the 1 st Brillouin zone boundaries : due to symmetry (from diffraction condition ) 1 st Brillouin zone where Consider the 1-D case when the Fourier components small compare with the kinetic energy of electrons at the zone boundary : Weak-potential assumption

Nontrivial solutions for the two coefficient energy band

near the zone boundary : due to symmetry nontrivial solutions for the two coefficient energy band

standing wave Time-independent states is represented by standing wave standing wavefunctions in a 1-D weak potential at the edge of the first Brillouin zone;the lower part of the figure illustrates the actual potential of electronsin a 1-D weak potential at the edge of the first Brillouin zone;the lower part of the figure illustrates the actual potential of electrons -The upper part of the figure plots the probability density-The upper part of the figure plots the probability density L : length of crystal  Standing Wave

Electron band structure reduced-zone schemeextended-zone scheme completely free electron without band gap

- Calculated energy band structure of copper Copper outermost configuration : 4s4s 3d3d Interband transitions : The absorption of photons will cause the electrons in the s band to reach a higher level within the same band Band Structures of Metals and Semiconductors  Band structure of Metals Electron in the s band can be easily excited from below the E F to above the E F : concuctor

Calculated energy band structure of Silicon Calculated energy band structure of GaAs - Interband transitions : The excitation or relaxation of electrons between subbands - Indirect gap : The bottom of the conduction band and the top of the valence band do not occur at the same k - Direct gap : The bottom of the conduction band and the top of the valence band occur at the same k  Band Structure of Semiconductor

- Energy versus wavevector relations for the carriers e : electron, h : hole E V : the energy at the top of the valencw band E e : the energy at the bottom of the conduction band

 Effective Mass An electron in a periodic potential is accelerated relative to the lattice in an applied electric or magnetic field as if the mass of the electron were equal to an effective mass.

work done  on the electron by the electric field E in the time interval  t and