Semiconductor Physics and Devices Chapter 3.

Transcription

1 Introduction to the Quantum Theory of Solids We applied quantum mechanics and Schrödinger s equation to determine the behavior of electrons in a potential. Important findings Semiconductor Physics and Devices Chapter 3. - Electron can take only discrete values of energy. (Energy Quantization) Seong Jun Kang Department of Advanced Materials Engineering for Information and Electronics Laboratory for Advanced Nano Technologies Introduction to the Quantum Theory of Solids Now, we will generalize the concepts to the electron in a crystal lattice. Introduction to the Quantum Theory of Solids The concept of allowed and forbidden electron energy bands in a crystal material. Conduction and valence energy bands in a semiconductor material. Concept of negatively charged electrons and positively charged holes. Electron energy vs. momentum to understand direct and indirect band gap semiconductor. Effective mass of an electron and a hole. Density of quantum states in the allowed energy bands Fermi-Dirac probability function to understand the statistical distribution of electrons and define the Fermi energy level. e - e - e - e - e - Allowed and Forbidden Energy Bands (a) Shows the radial probability density function for the lowest electron energy state of the single, noninteracting hydrogen atom. (b) Same probability density function of the electrons of the two atoms overlap. (interacting) (c) Interaction of two atoms results in the splitting into two discrete energy levels. Allowed and Forbidden Energy Bands Far interatomic distance: single atomic energy level Pushing atoms together: splitting into a band of discrete energy levels r 0 represents the equilibrium interatomic distance in the the crystal. Suppose 1) one-electron atoms 2) Width of allowed energy band is 1 ev 3) At the equilibrium interatomic distance The splitting of the discrete state into two states is consistent with the Pauli exclusion principle. Energy levels are separated by ev The Pauli exclusion principle is the quantum mechanical principle that no two identical fermions (particles with half-integer spin) may occupy the same quantum state simultaneously. Quasi-continuous energy distribution through the allowed energy band

2 Allowed and Forbidden Energy Bands Consider 1) Regular periodic arrangement of atoms 2) Each atom contains more than one electron (up through n=3 energy level) Bands of allowed energies that the electrons may occupy separated by bands of forbidden energies. Allowed and Forbidden Energy Bands Silicon (14 electrons) - Ten electrons occupy core energy level, close to the nucleus - Four valence electrons n = 3 level for the valence electrons Band gap energy E g : between top of the valence band and the bottom of the conduction band. Formation of allowed and forbidden band The Kronig-Penney Model Electron in a Periodic Field of Crystal: Kronig-Penney Model The Kronig-Penney model is an idealized periodic potential representing one dimensional single crystal. Isolated atom Periodic arrangement of multiple atoms Crystal The Kronig-Penney Model The Kronig-Penney model is an idealized periodic potential representing one dimensional single crystal. Coulomb potential for one atom Coulomb potential for periodic crystal Similar to particle in a box system Net potential function of one-dimensional cryst Potential well Kronig-Penney model Bloch theorem Solving periodic potential Bloch theorem states that all one-electron wave functions, for problems involving periodically varying potential energy functions, must be of the form, Region I: looks like a free electron case, where V(x) = 0. The function u(x) is a periodic function with period (a+b). p: periodicity Wave-function: plane wave x periodic function of lattice [Bloch Function] Total wave function in crystal We define parameter

3 Solving periodic potential Region II: looks like a Tunneling case (V 0 > E) Solving periodic potential Solution of Schrodinger equation Region I: Region II: We define parameter Now, we need to determine A, B, C, and D!!! Boundary conditions Applying boundary conditions u and (du/dx) should be continuous Solving periodic potential From the boundary condition [1] [2] We are interesting for E<V 0, therefor is imaginary [3] [4] The solution of the equation above will result in a band of allowed energies. Consider b 0, and the V 0 infinite Solving periodic potential We are ready to understand k-space Main concept of Kronig-Penney model The result gives the relation between the parameter k, total energy E, and the potential barrier V 0. The equation is not a solution of Schrodinger s equation, but gives the conditions for which Schrodinger s equation will have a solution!!

4 Consider a special case, V 0 =0 (free electron). Consider a special case, V 0 0 (electron in crystal). Parabolic relation between energy and momentum for the free particle. The allowed values of the f( a) must be bounded between +1 and -1. Parameter k is a wave number. The parameter is related to the total energy E of the particle. The parameter is related to the total energy E of the particle. Now, we can generate the energy E of the particle as a function of the wave number k. Discrete energy zones are allowed Electron that moves in a periodic potential (Crystal) can only occupy certain allowed energy zone Energy band & Energy gap exist in crystal!!!! Consider the right side of equation. Free electron From Kronig-Penney Model, For free electron case, V 0 becomes zero P=0 In x-direction only (1D) Periodic!! cos is a periodic function in every n2, where n = 0, +/-1, +/-2,. For free electron case

5 Free electron For free electron case Periodicity = 2 /a Electron in crystal Kronig-Penney Model IF electrons move in a periodic potential.. (1D) Always observe discontinuities of the energies when cos(k x a) has a maximum or a minimum Discontinuities in E-k diagram!!! At every k x value above, there is a discontinuities k-space diagram vs band diagram Electron in crystal: Kronig-Penney Model Discontinuities in E-k diagram!!! At every k x value above, there is a discontinuities Electrical conduction in solids The energy band vs. the bond model of atom T = 0 K T > 0 K For Silicon: Band gap ~ 1.1 ev Electrical conduction in solids Bond model of atom to the E versus k energy band T > 0 K T = 0 K At T=0K, the 4N states in the lower band are filled with the valence electrons. All of the valence Electrons are in the valence band. Upper energy band is completely empty. As the temperature increase above 0K, a few valence band electrons may gain enough thermal Energy to break the covalent bond and jump into the conduction band. Thermal Energy

6 Drift current No external force (electric field) + - The movement of an electron in a lattice will be different from that of an electron in a free space. In addition to an externally applied force, there are internal forces in the crystal due to positively charged ions or protons and negatively charged electrons, which will influence the motion of electrons in the lattice. E-field (external) E-field (external) Vacuum Solild Without external force, net drift current density is zero. (symmetric momentum) Drift Current Asymmetric momentum of electrons in conduction band. e: electron charge v d : average drift velocity v i : velocity of ith charge N: volume density (/cm-3) m*: Effective mass Internal force: Interaction with protons & electrons in the lattice. (quantum mechanics) NOT easy to define Internal force Instead, define a new quantity of mass in the lattice Free electron Electron in the bottom of an allowed energy band (parabola approximation) Apply an electric field to the free electrons Second derivative of E with respect to k is inversely proportional to the mass of the particle The motion of the free electron is in the opposite direction to the applied electric field. Apply an electric field to the Electron in the bottom of an allowed energy band (parabola approximation)

7 Concept of the Hole The movement of a valence electron into the empty state is equivalent to the movement of the positively charged empty state itself. The motion equivalent to a positive charge moving in the valence band. Concept of the Hole Drift current density due to electrons in the valence band This charge carrier is called a hole. Valence Band : If Totally occupied Valence Band : If Empty + Filled Drift current density due to electrons in the valence band Drift Current in valence band Positive charges Holes (the same as electron but + charge) Metals, Insulators, and Semiconductors Metals, Insulators, and Semiconductors Insulators Semiconductors Metals Partially full band in conduction band Typical E g = 3.5 to 6 ev Typical E g = 1 ev Controllable resistivity Conduction and valence band overlap at the equilibrium interatomic distance s of Si and GaAs Actual potential is from a three dimensional crystal. - Different traveling direction means different potential s of GaAs E versus k diagram of GaAs. E vs. k diagram for 1D model was symmetric in k. (no new information in the negative axis.) 1) Plot the [100] direction along the normal +k axis. 2) Plot the [111] direction so the +k points to the left. Indiamondorzincblendelattice,themaximainthe valence band energy and minima in the conduction band energy occur at k=0. Direct band gap semiconductor - Transitions between the two allowed bands can take place with no change in crystal momentum. The E versus k diagrams are a function of the k space direction in a crystal. - Useful to optical properties of materials. - GaAs is suitable in semiconductor lasers and optics.

8 s of Si E versus k diagram of Si. The maximum in the valence band energy occurs at k=0. Direct and indirect band gap Light Emission in Direct vs. Indirect Semiconductor The minimum in the conduction band energy occurs not at k=0, but along the [100] direction. The band gap is defined as the difference between these two maximum and minimum. Indirect band gap semiconductor - A semiconductor whose maximum valence band energy and minimum conduction band energy do not occur at the same k value. E g - When electrons make a transition between the conduction and valence bands, we must consider the law of conservation of momentum. - A transition in an indirect band gap material must necessarily include an interaction with the crystal to that crystal momentum is conserved. Second-order transition involving both Phonon + Photon Mainly thermal energy dissipated NO Light Density of states function To consider the number of electrons contributed to the conduction, Density of state & Fermi-Dirac distribution probability Density of states in semiconductor Conduction band The density of allowed quantum states as a function of energy. Black points are the allowed quantum states in k space. Valence band Relate the number of state to the energy. Fermi-Dirac distribution Fermi-Dirac distribution Continuous density of quantum state g(e) function of energy. The number density N(E) is the number of particles per unit volume per unit energy. The function g(e) is the number of quantum states per unit volume per unit energy. Fermi-Dirac distribution gives the probability that a quantum state at the energy E will be occupied by an electron. (or the ratio of filled to total quantum states at any energy E.) The energy E F is called the Fermi energy. The Fermi-Dirac distribution function for T = 0 K. N 0 electrons T = 0 K The electrons fill the state below E F, while the state above E F are empty. If g(e) and N 0 are known, then the Fermi energy E F can be determined. The probability of state being occupied is unity below E F. The probability of state being occupied is zero above E F.

9 Fermi-Dirac distribution At the temperature above 0 K. Density of states The density of occupied electron states at a given energy and temperature can be found by multiplying the density of state g(e) with the Fermi-Dirac distribution f(e, T). f F (E): Probability of a state being occupied 1-f F (E): Probability of a state being empty, The total number of electron N,