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Chapter 3 Energy Bands and Charge Carriers in Semiconductors • What holds atoms together in a solid? • Why are some materials good conductors, whereas others are not? • How does current flow in a solid? 3.1 Bonding Forces and Energy Bands in Solids Bonding Forces in Solids Ionic bonding Alkali halides such as NaCl, LiF, KBr, KCl are ionic solids formed by ionic bonding. stable and hard crystals; high vaporization temperatures; good insulators. complete transfer of valence electrons Covalent bonding Semiconductors such Si, Ge, ZnS and insulators such as diamond are formed by covalent bonding, where each atom share its valence electrons with its neighboring atoms. hard; high melting points; insulators. sharing of valence electrons among neighboring atoms Metallic bonding Valence electrons are contributed to the crystal as a whole. The bonding force is the attractive force between the positive ions and the electron gas. Less strong as ionic and covalent bonding; good conductors. Sharing of valence electrons within the whole crystal Molecular bonding e.g. organic solids, ice, inert gas crystals. The bonding force is van de Waals forces. Weak bonds, low melting and boiling points. NOTE: • Only valence electrons participate in bond formation! • When solids are formed from isolated atoms or molecules, the total energy of the system is reduced! Energy Bands —What will happen when two isolated atoms (e.g., H) are brought together? Wave functions Energy levels • The formation of new bonding and antibonding orbitals. • Energy degeneracy is brokenthe splitting of energy level 1s and 2s • The lowering of energy of the bonding state gives rise to the cohesion of the system. • These results can be obtained by solving the Schrödinger equation with the LCAO approximation. LCAO liner combination of atomic orbitals. • What is wrong with the shape of the wave function in Fig. 3-2? —What will happen when many (N) Si atoms are brought together to form a solid? • Energy bands are formed • Conduction band • Valence band • Forbidden band (band gap Eg) Electronic configuration of Si 1s22s22p63s23p2 Metals, Semiconductors, and Insulators —Every solid has its own characteristic energy band structure. —In order for a material to be conductive, both free electrons and empty states must be available. • Metals have free electrons and partially filled valence bands, therefore they are highly conductive (a). • Semimetals have their highest band filled. This filled band, however, overlaps with the next higher band, therefore they are conductive but with slightly higher resistivity than normal metals (b). Examples: arsenic, bismuth, and antimony. • Insulators have filled valence bands and empty conduction bands, separated by a large band gap Eg(typically >4eV), they have high resistivity (c ). • Semiconductors have similar band structure as insulators but with a much smaller band gap. Some electrons can jump to the empty conduction band by thermal or optical excitation (d). Eg=1.1 eV for Si, 0.67 eV for Ge and 1.43 eV for GaAs Direct and Indirect Semiconductors • The real band structure in 3D is calculated with various numerical methods, plotted as E vs k. k is called wave vector p is momentum see example -31 • p k • For electron transition, both E and p (k) must be conserved. • A semiconductor is direct if the maximum of the conduction band and the minimum of the valence band has the same k value • A semiconductor is indirect if the …do not have the same k value • Direct semiconductors are suitable for making light-emitting devices, whereas the indirect semiconductors are not. See Appendix III for more data on semiconductor materials 3.2 Charge Carriers in Semiconductors Electrons and Holes Ec the bottom of the conduction band Ev the top of the valence band EHP an electron-hole pair • At 0K, a semiconductor is an insulator with no free charge carriers • At T > 0K, some electrons in the valence band are excited to the conduction band • The electrons in the conduction band are free to move about via many available states • An empty state in the valence band is referred as a hole The concept of hole The total current in a volume with N electrons N J (q) v i 0 i The total current with the jth electron missing N A valence band (E vs k ) diagram with all states filled J (q) v i (q)v j qv j i The net result: a positive charge moving with velocity vj • A hole is an imaginary positive charge moving in the valence band • The energy of a hole increases downward in a normal band diagram • The total current flow in a semiconductor is the sum of electron current and hole current Energy Band Diagram under Electrical Field What is the direction of the electrical field? Left or Right • (E, k) band diagram vs (E, x) simplified band diagram • Total energy is the sum of potential energy and kinetic energy • Band edges Ec and Ev correspond to electron potential energy • Energies higher in the band correspond to additional kinetic energy of the electron • Electron and hole lose kinetic energy to heat by scattering (section 3.4.3) Effective Mass —The effective mass of an electron in a band with a given (E, k) relationship is defined as m 2 * (3-3) d 2 E /dk 2 2 1 2 1 p2 k2 For free electrons, E mv 2 2 m 2m m* = m • The effective mass is inversely proportional to the curvature of the band • The electrons near the top of the valence band have negative effective mass • In general m* is different in each direction and is a tensor; appropriate averages are needed for various calculation purposes (e.g. density of state effective mass vs conductivity effective mass, section 3.4.1) • The introduction of m* will simplify calculations • electron effective mass is denoted by me* ; hole effective mass is denoted by me* Realistic Band Structures in Semiconductors • GaAs is a direct semiconductor • For holes we have light hole band, heavy hole band and split-off band • Si is an indirect semiconductor • Si has six equivalent conduction band minima at X along six equivalent <100> directions • The constant energy surface for silicon in one of the six conduction bands is a ellipsoid ml is the longitudinal effective mass mt is the traverse effective mass Intrinsic Semiconductor —a perfect semiconductor crystal with no impurities or lattice defects n conduction band electron concentration (electrons per cm3) p valence band hole concentration n=p=ni (3-6) ri recombination rate of EHP; gi generation rate n0 , p0 concentrations at equilibrium; r constant ri=rn0p0 = rni2 =gi EHP generation in an intrinsic semiconductor (3-7) Extrinsic Semiconductor a doped semiconductor crystal whose equilibrium carrier concentrations n0 and p0 are different from the intrinsic carrier concentration ni The consequences of doping • new donor or acceptor levels are created in the band gap • conductivities can be vastly increased (n0 or p0 >> ni ) • semiconductor becomes either n-type or p-type (either n0 >> p0 or p0 >> n0 ) For Si and Ge • Group V elements such as As, P, Sb are donor impurities • Group III elements such as B, Al, Ga and In are acceptor impurities The donor binding energy for GaAs—an example From Bohr model, the ground state energy of an “extra” electron of the donor is m*q 4 E , 2K 2 2 where K 4 r0 (3-8) Compare with the room temperature (300K) thermal energy E=kT≈26meV All donor electrons are freed to the conduction band (ionized) Compare with the intrinsic carrier concentration in GaAs (ni=1.1 x 106 /cm3) We will have an increase in conduction electron concentration by 1010 if we dope GaAs with 1016 S atoms/cm3 “Band Gap Engineering” ( 3.1.5 & 3.2.5 )