ELEC311( 물리전자, Physical Electronics) Course Outlines: by Professor Jung-Hee Lee Lecture notes are prepared with PPT and available before the class (http://abeek.knu.ac.kr). The topics in the notes are from Chapter 1, 2, 4, and 5 in the main text introduced below, but the course covers the materials not only in the text, but also in various references. The remaining topics in Chapter 3, 6, 7, and 8 will be discussed in Electronic Devices in next semester. Main Text: Modern Semiconductor Devices for Integrated Circuits by Chenming Calvin Hu ( 2010, UC Berkeley) References: 1) Solid State Electronic Devices by Ben G. Streetman and Sanjay Kumar Banerjee (2006, U of Texas ar Austin) 2) Semiconductor Device Fundamentals by Robert F. Pierret(1996, Purdue University) 3) An Introduction to Semiconductor Devices by Donald Neamen (2006, U of New Mexico) 4) Principles of Semiconductor Devices by Sima Dimitrijev (2006, Griffith University) Grading: Three Exams (30 % each), Homework (10 %)
Chapter 1 Electrons and Holes in OBJECTIVES Semiconductors 1. Provides the basic concepts and terminology for understanding semiconductors. 2. Understand conduction and valence energy band, and how bandgap is formed 3. Understand carriers (electrons and holes), and doping in semiconductor 4. Use the density of states and Fermi-Dirac statistics to calculate the carrier concentration
Transistor inventors John Bardeen, William Shockley, and Walter Brattain (left to right) at Bell Telephone Laboratories. (Courtesy of Corbis/Bettmann.)
Crystal Lattice Unit cell : a small portion of any given crystal that can be used to reproduce the crystal Primitive cell : the smallest unit cell possible periodic atomic arrangement in the crystal, or symmetric array of points in space A two-dimensional lattice showing translation of a unit cell by r = 3a +2b. Simple cubic lattice and its unit cell (a is the lattice constant) It is possible to analyze the crystal as a whole by investigating a representative volume (e.g. unit cell).
Simple 3-D Unit Cell Simple cubic: only 1/8 of each corner atom is inside the cell contains 1 atom in total Body centered cubic (bcc): an atom at the center of the cube in addition to the atoms at each corner contains 2 atoms Face centered cubic (fcc): contains an atom at each face of the cube in addition to the atoms at each corner ½ of each face atom lies inside the fcc contains 4 atoms
Packing of Hard Spheres in an FCC Lattice Homework #1: 1. What is the maximum fraction of the FCC lattice volume that can be filled with atoms by approximating the atoms as hard spheres? 2. Do the same calculation for the simple cubic and body-centered cubic.
Crystallographic Planes and Directions Miller indices: A set of integers with no common integral divisors that are inversely proportional to the intercepts of the crystal planes along the crystal axes. These indices are enclosed in parenthesis (hkl). Miller Indexing procedure: 1. Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions. 1, 2, 3 2. Take the reciprocals 1, 1/2, 1/3 3. Clear fractions using an appropriate multiplier 6, 3, 2 4. Reduce to lowest terms (already there) and enclose the whole-number set in parenthesis (632) Special facts: The plane that is parallel to a coordinate axis is taken to be infinity. Thus, intercepts at,, 1, for example, result in (001) plane. For a negative axis, a minus sign is placed over the corresponding index number so that an intercept at 1, -1, 2 is designated a (221) plane. A group of equivalent planes is referenced through the use of { }. Example. A (214) crystal plane
Crystallographic planes [Miller indices for the three most important planes in cubic crystals] Because there is no crystallographic difference between the (100), (010), (001) planes, they are uniquely labeled as {100}. {110} plane intersects two axes at a and is parallel to the 3 rd axis. {111} plane intersects all the axes at a. Equivalence of the cube faces ({100} planes) by rotation of the unit cell within the cubic lattice.
Crystallographic Directions The direction perpendicular to (hkl) plane is labeled as [hkl]. A set of equivalent directions is labeled as <hkl>; e.g. <100> represents [100], [010], [001], [100] and so on. Miller Convention Summary Convention (hkl) {hkl} [hkl] <hkl> Interpretation Crystal plane Equivalent planes Crystal direction Equivalent directions [Important directions in cubic crystals]
(a) A system for describing the crystal planes. Each cube represents the unit cell (b) Silicon wafers are usually cut along the (100) plane. This sample has a (011) flat to identify wafer orientation during device fabrication. (c) Scanning tunneling microscope view of the individual atoms of silicon (111) plane.
Semiconductor Lattice (The diamond lattice) Diamond lattice unit cell (Si, Ge, C): Two interpenetrating FCC lattices (the 2 nd FCC lattice displaced ¼ of a body diagonal along a body diagonal direction relative to the 1 st FCC lattice) 8 Si atoms in unit cell (volume=a 3, a=5.43å ) ~5 10 22 atoms/cm 3 Zincblende lattice unit cell (GaAs, InP ): Identical to diamond lattice unit cell, but 2 FCCs are different atoms. i.e. Ga locates on one of the two interpenetrating FCC sub-lattice and As populates the other FCC sub-lattice. Atoms in the diamond and zincblende lattices have 4 nearest neighbors.
Diamond lattice structure: (a) a unit cell of the diamond lattice constructed by placing atoms ¼, ¼, ¼ from each atom in an fcc; (b) top view (along any <100> direction) of an extended diamond lattice.the colored circles indicate one fcc sublattice and the black circles indicate the interpenetrating fcc. 3 4 a Homework #2: What is the maximum fraction of the diamond lattice volume that can be filled with atoms by approximating the atoms as hard spheres? Find the number density (atoms/cm 3 ) and density (g/cm 3 ) of the Si lattice.
Semiconductors, Insulators, and conductors - Every solid has its own characteristic energy band structure. - The band structure is responsible for electrical characteristics. Elemental solids with even atomic numbers (and therefore even numbers of electrons) 9 ev E c E c 1.1 ev E c such as Zn and Pb known as semimetal E v E v Elemental solids with odd atomic numbers (SiO 2 ) (Si) (Conductor) (and therefore odd numbers of electrons) such as Au, Al, and Ag The valence band of Si is completely filled with electrons at 0 K and the conduction band is empty good insulator at 0 K. What will happen if temperature increases? Semiconductor Difference between semiconductor and Insulator - E g,insulator >> E g,semiconductor - Semiconductor can be N or P-type with low resistivity through impurity doping. Insulator Conductor 10 18 10 16 10 14 10 12 10 10 10 8 10 6 10 4 10 2 1 10-2 10-4 10-6 10-8 [Conductivity] Insulator < Semiconductor < Metal Resistivity [Wm]
Periodic Table
Semiconductor Materials Semiconductor - elemental semiconductor : group Ⅳ - compound semiconductor : group Ⅲ &Ⅴ, group II & VI etc. - Binary = two elements - Ternary = three elements - Quaternary = four elements (InGaAsP)
Bond Model of Electrons and Holes Bonding Forces in Solids 1) Ionic Bonding in NaCl Na is surrounded by 6 nearest neighbor Cl atoms. Na (Z=11): [Ne]3s 1 Cl (Z=17): [Ne]3s 2 3p 5 Each Na atom gives up its outer 3s electron to a Cl atom Crystal is made up of ions with the electronic structures of the inert atoms, Ne and Ar (Ar (Z18): [Ne]3s 2 3p 6 ) Ionic bonding attraction These Coloumbic forces pull the lattice together until a balance is reached with repulsive forces. Features of ionic solid Tightly bonded electrons good insulators The energy levels in outer orbits are either totally filled or totally empty Very stable
2) Metallic Bonding Sea of electrons N a + Coulombic forces between N a+ and electron sea In a metal atom the outer electronic shell is only partially filled. (N a + has only one electron in the outer shell.) These electrons are loosely bound and are given up easily in ion formation. Significant number of free electrons excellent thermal/electrical conductor The solid is made up of ions with closed shells immersed in a sea of free electrons.
3) Covalent Bonding in Si Electrons are essentially attached to their own nuclei but they are being shared by two nuclei at the same time. Covalent bonding in Ge, Si, or C diamond lattice The bonding forces arise from a quantum mechanical interaction between the shared electrons. Covalent bonding is stable; Either insulators or semiconductors The silicon crystal structure in a twodimensional representation at 0 K.( no free electron to conduct electric current at 0 K). Sharing the outermost electrons lower excitation energy absorption infrared (IR) range Sensitive to the temperature change (an idealized lattice at 0 K)
At elevated temperature, a covalent electron breaks loose, becomes mobile and can conduct electric current (conduction electron). It also creates a void or a hole represented by the open circle. The hole also move about as indicated by the arrow and thus conduct electric current. Doping of a semiconductor is illustrated with the bond model. (a) As (V) is a donor. (b) B (III) is an acceptor.
4) van der Waals Solid Shows quite weak boding force ex. Pairing of inert gas ions, such as He-He molecules Solid Bond Type Bond Energy [kj/mol] Bond Length [nm] NaCl Ionic 748 0.282 Al Metallic 326 0.152 C-C Covalent 370 0.154 FeO Covalent 509 0.216 Ar van der Waals 1 0.382
Energy Band Model 2p 2s Ψ 1 1s 3s Isolated atoms 3p wave function.... Ψ 2 No interaction between electron wave function odd or antisymmetry combination Quantum numbers: n = 1, 2, 3, 4, l = 0, 1, 2, 3,.n-1 s, p, d, f, g, m = -l, -(l-1),.0, 1,.l s = +1/2, -1/2 Atomic configuration of Si atom: 14electrons 1s 2, 2s 2, 2p 6, 3s 2, 3p 2 Ψ 1 Ψ 2 antibonding energy level bonding energy level even or symmetry combination + antibonding orbital Energy level splitting due to exclusion principle ( No two electrons in a given interacting system may have the same quantum state) Diatoms: two atoms close to each other 1 2 (LCAO ) LCAO: linear combinations of the individual atomic orbitals + + ㅡ bonding orbital The potential energy has been lowered because an electron here would be attracted by two nuclei, rather than just one.
There must be at most one electron per level after there is a splitting of discrete energy levels of the isolated atoms into new levels belonging to the pair rather than no individual atoms E 3 E 2 3p 3p 3s 3s Energy level splitting due to Pauli exclusion principle E 1 If, instead of 2 atoms, one brings together N atoms, there will be N distinct LCAO and N closely-spaced energy levels in a band. In solids, where N is very large, so that the split energy levels form essentially continuous band of energies. 3p + 3s 2s+2p 1s E g E 2 E 1 E 3 conduction band valence band core band N-Atoms
Energy band for Si If N atoms are brought into close proximity, there is no significant change in the core, but the energy state of the valence electrons changes Ec: lowest conduction band energy Ev: highest valence band energy E g = Ec - Ev : Band gap energy Ec Ev 3p sp 3 hybridization 3s 2p core completely filled 2s 1s Lattice constant of Si atom at equilibrium
E-k Diagram for a Free Electron The wave function of plane wave Plot the E-k dependence jkx ( x) c1e c2e 2 d 2 2 k 0, k : 2 dx The wave function of free electron (V = 0) satisfies the Schrödinger wave equation, 2 d m 2 2 E V 2 ( ) 0 dx when 2m 2m 2 2 E k from E k 2 Relate the E-k dependence to the classical kinetic energy, E kin = mv 2 /2 Because free electron has no potential energy E = E kin jkx satisfies the wave equation wave number (or wave vector or propagation constant). The wave function of free electron is exactly same as the wave function of the plane jkx jkx wave; ( x) c e c e 1 2 E k diagram for a free electron E p mv k 2m 2m 2 2 2 2 2 p = k The E-k dependence of a free electron is identical to the classical dependence of kinetic energy on velocity.
Energy Gap and Energy Bands in Semiconductors and Insulators Energy band for free electron: Parabolic E-k dependence. 2 E k 2m 2 Wave function of electrons in the crystal can be modified by the periodic crystal potential, U(k x, x), as k jk x ( x) U ( k, x) e Energy band for semiconductor and insulator: still similar to the free electron energy band, but two slightly modified parabolic bands, conduction band and valence band, with energy gap, E g. x x modulates the wave function according to the periodicity of the lattice Free electron approximation Tight binding approximation E k diagram. E x diagram.
At T 0 K, there is no broken covalent bonds and the valence band is full and the conduction band is empty. At elevated temperature, The semiconductors becomes insulator, because there are no electrons in conduction band and the electrons in the valence band are immobile they are tied in the covalent bonds. some of covalent bonds are broken because a sufficient thermal energy is delivered to a valence electrons, these electrons jumps up into the conduction band, leaving empty states behind in the valence band, called holes. The semiconductors becomes conductive. Insulator: large E g Semiconductor: small E g The electrons at higher energy levels in E C will have kinetic energies according to the upper E- k branch Kinetic energy of the holes as current carriers
Direct and Indirect Semiconductor Allowed values of energy can be plotted vs. the propagation constant, k. Since the periodicity of most lattices is different in various direction, the E-k diagram must be plotted for the various crystal directions (complex). -Direct bandgap: a minimum in the conduction band and a maximum in the valence band for the same k value -Indirect bandgap: a minimum in the conduction band and a maximum in the valence band at a different k value GaAs : the minimum conduction band energy and maximum valence band energy occur at the same k-value. direct band gap semiconductor semiconductor lasers and other optical devices Si, Ge, GaP, AlAs : indirect band gap semiconductor A transition must necessarily include and interaction with the crystal so that crystal momentum is conserved.