1 Lectures on the Basic Physics of Semiconductors and Photonic Crystals References 1. Introduction to Semiconductor Physics, Holger T. Grahn, World Scientific (2001) 2. Photonic Crystals, John D. Joannopoulos et al, Princeton University Press (1995)

Hanjo Lim School of Electrical & Computer Engineering Lecture 1 : Overview on Semiconductors and PhCs

Overview 3 Review on the similarity of SCs and PhCs Semiconductors: Solid with periodic atomic positions Photonic Crystals: Structure with periodic dielectric constants Semiconductor: Electron characteristics governed by the atomic potential. Described by the quantum mechanics (with wave nature). Photonic Crystals: Electomagnetic(EM) wave propagation governed by dielectrics. EM wave, Photons: wave nature  Similar Physics. ex) Energy band ↔ Photonic band

4 Review on semiconductors Solid materials: amorphous(glass) materials, polycrystals, (single) crystals - Structural dependence : existence or nonexistence of translational vector, depends on how to make solids - main difference between liquid and solid; atomic motion * liquid crystals (nematic, smetic, cholestoric)  Classification of solid materials according to the electrical conductivity - (superconductors), conductors(metals), (semimetals), semiconductors, insulators - Difference of material properties depending on the structure * metals, semiconductors, insulators : different behaviors

5 So-called “band structure” of materials - metals, semiconductors, insulators * temperature dependence of electrical conductivity, conductivity dependence on doping Classification of Semiconductors - Wide bandgap SC, Narrow bandgap SC, - Elemental semiconductors : group IV in periodic table - Compound semiconductor : III-V, II-VI, SiGe, etc * binary, ternary, quaternary : related to 8N rule(?) * IV-VI/V-VI semiconductors : - band gap and covalency & ionicity

6  Crystal structure of Si, GaAs and NaCl - covalent bonding : no preferential bonding direction - symmetry : - the so-called 8N rule : - ionic bond: preferencial bonding direction (NaCl) Importance of semiconductors in modern technology (electrical industry) - electronic era or IT era : opened from Ge transitor * Ge transistor, Si DRAMs, LEDs and LDs - merits of Si on Ge IT era: based on micro-or nano-electronic devices - where quantum effects dominate * quantum well, quantum dot, quantum wire

7 Crystal = (Bravais) lattice + basis - lattice = a geometric array of points, with integer numbers 3 primitive vectors - Basis = an atom (molecule) identical in composition and arrangement * lattice points : have a well-defined symmetry * position of lattice point basis ; arbitrary - primitive unit cell : volume defined by 3 vectors, arbitrary - Wignez-Seitz cell : shows the full symmetry of the Bravais lattice Cubic lattices - simple cubic(sc), body-centered cubic(bcc), face-centered (fcc) * =lattice constant Report : Obtain the primitive vectors for the bcc and fcc. Crystal Structure and Reciprocal Latiice

8 Wignez-Seitz cells of cubic lattices (sc, bcc, fcc) - sc : a cube - bcc : a truncated octahedron - fcc : a rhombic dodecahedron, * Confer Fig Packing density of close-packed cubics Hexagonal lattice - hexagonal lattice = two dimensional (2D) triangular lattice + c axis - Wignez-Seitz cell of hcp : a hexagonal column (prism)  Note that semiconductors do not have sc, bcc, fcc or hcp structures. - SCs : Diamond, Zinc-blende, Wurtzite structures - Most metals : bcc or fcc structures

9 Diamond structure : Basics of group IV, III-V, II-VI Semiconductors - C : - Diamond : with tetrahedral symmetry, two overlapped fcc structures with tow carbon atoms at points 0, and Zincblende (sphalerite) structure - Two overlapped fcc structures with different atoms at 0 and - Most III-V (parts of II-VI) Semiconductors : Cubic III-V, II-VI - Concept of sublattices : group III sub-lattice, group V sub-lattice  Graphite and hcp structures - Graphite : Strong bonding in the plane weak van der Waals bondding to the vertical direction * Graphite : layered structure with hexagonal ring plane

10 Symmetry operations in a crystal lattice - Translational symmetry operation with integer def) point group : collection of symmetry operations applied at a point which leave the lattice invariant  around a given point - Rotational symmetry n, defined by 2π/n (n=1~6 not 5) - Reflection symmetry - Inersion symmetry def) space group : structure classified by and point operations - Difference btw the symm. of diamond and that of GaAs * Difference between cubic and hexagonal zincblende ex) CdS bulk or nanocrystals, TiO 2 (rutile, anatase)

11 Electron motions in a solid - Nearly free electrons : weak interactions (elastic scattering) between sea of free and lattice of the ions * elastic scattering btw : momentum conservation, why? - lattice : a perfectly regular array of identical objects - free : represented by plane waves, - interaction btw and lattice ↔ optical (x-) ray and grid * Bragg law (condition) : when 2d sinθ = with integer constructive interference (2D rectangular lattice)

12 : position vector defining a plane made of lattice sites. reflection plane, ; inversely proportional to With general (positions of real lattice points), should be satisfied in general. A set of points in real space  a unique set of points with : defined in -space. → Reciprocal lattice vector,  3D Crystal with (triclinic) With should be satisfied simultaneously for the integral values of Let to be determined. Then eq. (2) will be solution of eq. (1) if eq. (3) holds

13 Note that plane and plane, etc. plane Thus should be the fundamental (primitive) vectors of the reciprocal lattice. Note 1) ;scattering vector, crystal momentum, Fourier- transformed space of, called as reciprocal lattice. Note 2) X-ray diffraction, band structure, lattice vibration, etc.

14 Note 3) Reciprocal lattice of a Bravais lattice is also a Bravais lattice. Report : Prove that forms a Fourier-transformed space of Brillouin zone : a Wigner-Seitz cell in the reciprocal lattice. Elastic scattering of an EM wave by a lattice ; Scattering condition for diffraction; : a vector in the reciprocal lattice Take so that they terminate at one of the RL points, and take (1), (2) planes so that they bisect normally respectively. Then any vector that terminates at the plane (1) or (2) will satisfy the diffraction condition.

15 The plane thus formed is a part of BZ boundary. Note 4) An RLV has a definite length and orientation relative to Any wave incident to the crystal will be diffracted if its wavevector has the magnitude and direction resulting to BZ boundary, and the diffracted wave will have the wave vector with corresponding If are primitive RLVs  1 st Brillouin zone. Report : Calculate the RLVs to sc, bcc, and fcc lattices. Miller indices and high symmetry points in the 1 st BZ - (hkl) and {hkl} plane, [hkl] and direction - see Table 2.4 and Fig. 2.7 for the 1 st BZ and high symm. points. - Cleavage planes of Si (111), GaAs (110) and GaN (?).

16 Electronic crystals (conductor, insulator) ex) one-dimensional electronics crystals => periodic atomic arrangement Schroedinger equation : If => plane wave If is not a constant, ; Bloch function ; modulation, ; propagation with If with the lattice constant Basic Concepts of photonic(electromagnetic) crystals

17 Note) Bragg law of X-ray diffraction If constructive reflection of the incident wave (total reflection) ∴ A wave satisfying this Bragg condition can not propagate through the structure of the solids. If one-dimensional material with an atomic spacing is considered, ∴ Strong reflection of electron wave at (BZ boundary)

18 Optical control - wave guiding (reflector, internal reflection) - light generation (LED, LD) - modulation (modulator), add/drop filters PhCs comprehend all these functions => Photonic integrated ckt. Electronic crystals: periodic atomic arrangement. - multiple reflection (scattering) of electrons near the BZ boundaries. - electronic energy bandgap at the BZ boundaries.  Photonic (electromagnetic) crystals: periodic dielectric arrangement. - multiple reflection of photons by the periodic - photonic frequency bandgap at the BZ boundaries. ex) DBR (distributed Bragg reflector): 1D photonic crystal

19 Strong reflection around “Photonic (Electromagnetic) crystals” - concept of PhCs: based on electromagnetism & solid-state physics - solid-state phys.; quantum mechanics Hamiltonian eq. in periodic potential. - photonic crystals; EM waves (from Maxwell eq.) in periodic dielectric materials single Hamiltonian eq. - Exist. of complete PBG in 3D PhCs : theoretically predicted in 1987.