1 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 E g (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). E g 1.1 ev for Si, 0.67 ev for Ge and 1.43 ev for GaAs
3 Conduction in Terms of Band Metals An energy band is a range of allowed electron energies. The energy band in a metal is only partially filled with electrons. Metals have overlapping valence and conduction bands
4 Drude Model of Electrical Conduction in Metals Conduction of electrons in metals A Classical Approach: In the absence of an applied electric field (ξ) the electrons move in random directions colliding with random impurities and/or lattice imperfections in the crystal arising from thermal motion of ions about their equilibrium positions. The frequency of electron-lattice imperfection collisions can be described by a mean free path λ -- the average distance an electron travels between collisions. When an electric field is applied the electron drift (on average) in the direction opposite to that of the field with drift velocity v The drift velocity is much less than the effective instantaneous speed (v) of the random motion
5 v cm. 1 In copper while where The drift speed can be calculated in terms of the applied electric field ξ and of v and λ When an electric field is applied to an electron in the metal it experiences a force qξ resulting in acceleration (a) qξ Then the electron collides with a lattice imperfection and changes its direction randomly. The mean time between collisions is q ξ τ q ξ λ v a τ The drift velocity is m m v If n is the number of conduction electrons per unit volume and J is the current density Combining with the definition of resistivity gives q1.6x -19 C σ s n q m e a λ v m e e v cm. s mev kbt e μ J nqν q m λ v q e m e σξ τ 3 τ λ v
6 For an electron to become free to conduct, it must be promoted into an empty available energy state For metals, these empty states are adjacent to the filled states Generally, energy supplied by an electric field is enough to stimulate electrons into an empty state Electron Energy Freedom Empty States States Filled with Electrons Distance Energy Band
7 Band Diagram: Metal T > 0 Fermi filling function Conduction band (Partially Filled) E C E F E 0 Energy band to be filled At T 0, all levels in conduction band below the Fermi energy E F are filled with electrons, while all levels above E F are empty. Electrons are free to move into empty states of conduction band with only a small electric field E, leading to high electrical conductivity! At T > 0, electrons have a probability to be thermally excited from below the Fermi energy to above it.
8 Resistivity (ρ) in Metals Resistivity typically increases linearly with temperature: ρ t ρ o + αt Phonons scatter electrons. Where ρ o and α are constants for an specific material Impurities tend to increase resistivity: Impurities scatter electrons in metals Plastic Deformation tends to raise resistivity dislocations scatter electrons σ 1 ρ nq The electrical conductivity is controlled by controlling the number of charge carriers in the material (n) and the mobility or ease of movement of the charge carriers (μ) μ
9 Temperature Dependence, Metals There are three contributions to ρ: ρ t due to phonons (thermal) ρ i due to impurities ρ d due to deformation ρ ρ t + ρ i + ρ d The number of electrons in the conduction band does not vary with temperature. All the observed temperature dependence of σ in metals arise from changes in μ
10 Scattering by Impurities and Phonons Thermal: Phonon scattering Proportional to temperature ρ Impurity or Composition scattering Independent of temperature Proportional to impurity concentration Solid Solution Two Phase t ρo + ρi ρ at Aci ( 1 ci ) ρ α V ρ V t α + Deformation must be experimentally determined ρ d β β
11 Insulator The valence band and conduction band are separated by a large (> 4eV) energy gap, which is a forbidden range of energies. Electrons must be promoted across the energy gap to conduct, but the energy gap is large. Energy gap º E g Conduction Band Empty Electron Energy Forbidden Valence Band Filled with Electrons Energy Gap Distance
12 Band Diagram: Insulator T > 0 Conduction band (Empty) E gap E C E F Valence band (Filled) E V At T 0, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity. Fermi energy E F is at midpoint of large energy gap (- ev) between conduction and valence bands. At T > 0, electrons are usually NOT thermally excited from valence to conduction band, leading to zero conductivity.
13 Conduction in Ionic Materials (Insulators) Conduction by electrons (Electronic Conduction): In a ceramic, all the outer (valence) electrons are involved in ionic or covalent bonds and thus they are restricted to an ambit of one or two atoms. Eg k T If E g is the energy gap, the fraction of electrons in the conduction band is: B e A good insulator will have a band gap >>5eV and k B T~0.05eV at room temperature As a result of thermal excitation, the fraction of electrons in the conduction band is ~e -00 or -80. There are other ways of changing the electrical conductivity in the ceramic which have a far greater effect than temperature. Doping with an element whose valence is different from the atom it replaces. The doping levels in an insulator are generally greater than the ones used in semiconductors. Turning it around, material purity is important in making a good insulator. If the valence of an ion can be variable (like iron), hoping of conduction can occur, also known as polaron conduction. Transition elements. Transition elements: Empty or partially filled d or f orbitals can overlap providing a conduction network throughout the solid.
15 Conduction by Ions: ionic conduction It often occurs by movement of entire ions, since the energy gap is too large for electrons to enter the conduction band. The mobility of the ions (charge carriers) is given by: μ Z q. D k.b. T Where q is the electronic charge ; D is the diffusion coefficient ; k B is Boltzmann s constant, T is the absolute temperature and Z is the valence of the ion. The mobility of the ions is many orders of magnitude lower than the mobility of the electrons, hence the conductivity is very small: Example: σ n. Z. q.μ Suppose that the electrical conductivity of MgO is determined primarily by the diffusion of Mg + ions. Estimate the mobility of Mg + ions and calculate the electrical conductivity of MgO at 1800 o C. Data: Diffusion Constant of Mg in MgO 0.049cm /s ; lattice parameter of MgO a0.396x -7 cm ; Activation Energy for the Diffusion of Mg + in MgO 79,000cal/mol ; k B 1.987cal/Kk-mol; For MgO Z/ion; q1.6x -19 C; k B 1.38x -3 J/K-mol
16 First, we need to calculate the diffusion coefficient D D D o Q exp kt D cm s 79000cal / mol exp cal / mol Kx( ) K D1.119x - cm /s Next, we need to find the mobility μ Z. q. D k. T B ( 19 carriers / ion)( 1. 6 C)( ( )( ) 9 ) 1. 1 C. cm J. s C ~ Amp. sec ; J ~ Amp. sec.volt μ1.1x -9 cm /V.s
17 MgO has the NaCl structure (with 4 Mg + and 4O - per cell) Thus, the Mg + ions per cubic cm is: 4Mg ions / cell ( cm) + n 6. 4 ions / 7 3 cm 3 σ σ nzqμ. 94 ( C. cm 3 cm. V. s )( )( )( ) C ~ Amp.sec ; V ~ Amp.Ω σ.94 x -5 (Ω.cm) -1
18 Example: The soda silicate glass of composition 0%Na O-80%SiO and a density of approximately.4g.cm -3 has a conductivity of 8.5x -6 (Ω-m) -1 at 150 o C. If the conduction occurs by the diffusion of Na + ions, what is their drift mobility? Data: Atomic masses of Na, O and Si are 3, 16 and 8.1 respectively Solution: We can calculate the drift mobility (μ) of the Na + ions from the conductivity expression σ n q μ Where n i is the concentration of Na + ions in the structure. 0%Na O-80%SiO can be written as (Na O) 0. -(SiO ) 0.8. Its mass can be calculated as: i M M At At i 0. ( ( 3) ( )) ( 1( 81. 1) + ( 16)) g. mol 1 The number of (Na O) 0. -(SiO ) 0.8 units per unit volume can be found from the density
19 n n ρ N M At A. 39 (. 4g. cm ( Na O) ) ( 6. 03x g. mol 3 0. ( SiO) mol units cm 1 3 ) The concentration of Na + ions (n i ) can be obtained from the concentration of (Na O) 0. -(SiO ) 0.8 units n i cm. ( + ) +. ( + ) And μ i μ i σ q n μ 1. 6 i i 6 1 ( 8. 5 Ω m 19 ( C) ( m V 1 s ) 6 m 3 ) This is a very small mobility compared to semiconductors and metals
21 Electrical Breakdown At a certain voltage gradient (field) an insulator will break down. There is a catastrophic flow of electrons and the insulator is fragmented. Breakdown is microstructure controlled rather than bonding controlled. The presence of heterogeneities in an insulator reduces its breakdown field strength from its theoretical maximum of ~ 9 Vm -1 to practical values of 7 V.m -1
22 Energy Bands in Semiconductors Energy Levels and Energy Gap in a Pure Semiconductor. The energy gap is < ev. Energy gap º E g Electron Energy Conduction Band (Nearly) Empty Free electrons Forbidden Valence Band (Nearly) Filled with Electrons Bonding electrons Energy Gap Semiconductors have resistivities in between those of metals and insulators. Elemental semiconductors (Si, Ge) are perfectly covalent; by symmetry electrons shared between two atoms are to be found with equal probability in each atom. Compound semiconductors (GaAs, CdSe) always have some degree of ionicity. In III-V compounds, eg. Ga +3 As +5, the five-valent As atoms retains slightly more charge than is necessary to compensate for the positive As +5 charge of the ion core, while the charge of Ga +3 is not entirely compensated. Sharing of electrons occurs still less fairly between the ions Cd + and Se +6 in the II-VI compund CdSe.
23 Semiconductor Materials Semiconductor Bandgap Energy E G (ev) Carbon (Diamond) 5.47 Silicon 1.1 Germanium 0.66 Tin 0.08 Gallium Arsenide 1.4 Indium Phosphide 1.35 Silicon Carbide 3.00 Cadmium Selenide 1.70 Boron Nitride 7.50 Aluminum Nitride 6.0 Gallium Nitride 3.40 IIB Indium Nitride 1.90 Portion of the Periodic Table Including the Most Important Semiconductor Elements 30 Zn Zin c Cd Cadmium Hg Mercury 5 IIIA IVA VA VIA.811 B Bo ro n Al A luminum Ga G a llium In Ind ium Ti Tha llium C Carbon Si Silic o n Ge Germanium 50 Sn Tin Pb Le a d N Nitrogen P Phosphorus As Arsenic Sb Antimony Bi Bism uth O Oxygen S Su lf u r Se Selenium Te Te llurium () 84 Po Polonium
24 Band Diagram: Semiconductor with No Doping T > 0 Conduction band (Partially Filled) E C Valence band (Partially Empty) E F E V At T 0, lower valence band is filled with electrons and upper conduction band is empty, leading to zero conductivity. Fermi energy E F is at midpoint of small energy gap (<1 ev) between conduction and valence bands. At T > 0, electrons thermally excited from valence to conduction band, leading to measurable conductivity.
25 Semi-conductors (intrinsic - ideal) Perfectly crystalline (no perturbations in the periodic lattice). Perfectly pure no foreign atoms and no surface effects At higher temperatures, e.g., room temperature 300 K), some electrons are thermally excited from the valence band into the conduction band where they are free to move. Holes are left behind in the valence band. These holes behave like mobile positive charges. CB electrons and VB holes can move around (carriers). At edges of band the kinetic energy of the carriers is nearly zero. The electron energy increases upwards. The hole energy increases downwards.
26 Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si positive ion core valence electron Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si Si free electron free hole
27 Semiconductors in Group IV Carbon Silicon Germanium Tin Each has 4 valence Electrons. Covalent bond
28 Generation of Free Electrons and Holes In an intrinsic semiconductor, the number of free electrons equals the number of holes. Thermal : The concentration of free electrons and holes increases with increasing temperature. Thermal : At a fixed temperature, an intrinsic semiconductor with a large energy gap has smaller free electron and hole concentrations than a semiconductor with a small energy gap. Optical: Light can also generate free electrons and holes in a semiconductor. Optical: The energy of the photons (hν) must equal or exceed the energy gap of the semiconductor (E g ). If hν > E g, a photon can be absorbed, creating a free electron and a free hole. This absorption process underlies the operation of photoconductive light detectors, photodiodes, photovoltaic (solar) cells, and solid state camera chips.
29 UV Violet Blue Green Yellow Orange Red Near IR nm ev nm ev nm.9-.5 ev nm ev nm ev nm ev nm ev, nm ev Red Violet Orange Blue Yellow Green
30 E g hν E ω E gg Photoconductivity Conductivity is dependent on the intensity of the incident electromagnetic radiation E hν hc/λ, c λ(m)ν(sec -1 ) Band Gaps: Si ev (Infra red) Ge 0.66 ev (Infra red) GaAs 1.4 ev (Visible red) ZnSe.70 ev (Visible yellow) SiC.86 ev (Visible blue) GaN 3.40eV (Blue) AlN 6.0eV (Blue-UV) BN 7.50eV (UV) Total conductivity σ σ e + σ h nqμ e + pqμ h For intrinsic semiconductors: n p & σ nq(μ e + μ h )
31 Question: How many electrons and holes are there in an intrinsic semiconductor in thermal equilibrium? Define: n o equilibrium (free) electron concentration in conduction band [cm -3 ] p o equilibrium hole concentration in valence band [cm -3 ] Certainly in intrinsic semiconductor: n o p o n i n i intrinsic carrier concentration [cm -3 ] As T then n i As E g then n i n p What is the detailed form of these dependencies? We will use analogies to chemical reactions. The electron-hole formation can be + though of as a chemical reaction.. bond e + h Similar to the chemical reaction The Law-of-Mass-Action relates concentration of reactants and reaction products. For water Where E is the energy released or consumed during the reaction. K O H O [ H ][ OH [ H O] + O n + H + (OH) ] exp i E kt This is a thermally activated process, where the rate of the reaction is limited by the need to overcome an energy barrier (activation energy).
32 By analogy, for electron-hole formation: Where [bonds] is the concentration of unbroken bonds and E g is the activation energy In general, relative few bonds are broken to form an electron-hole and therefore the number of bonds are approximately constant. Two important results: 1).. E g exp kt )... n i n p O O K [ n E o ][ po ] g exp [ bonds] kt n i n [bonds] [bonds] o p o >> n o,p o cons tant E g exp kt The equilibrium np product in a semiconductor at a certain temperature is a constant specific to the semiconductor.
33 Effect of Temperature on Intrinsic Semiconductivity The concentration of electrons with sufficient thermal energy to enter the conduction band (and thus creating the same concentration of holes in the valence band) n i is given by Δ T k E n B i exp For intrinsic semiconductor, the energy is half way across the gap, so that T k E n B g i exp Since the electrical conductivity σ is proportional to the concentration of electrical charge carriers, then T k E B g O exp σ σ
34 Example Calculate the number of Si atoms per cubic meter. The density of silicon is.33g.cm -3 and its atomic mass is 8.03g.mol -1. Then, calculate the electrical resistivity of intrinsic silicon at 300K. For Si at 300K n i 1.5x 16 carriers.m -3, q1.60x -19 C, μ e 0.135m (V.s) -1 and μ h 0.048m.(V.s) -1 Solution n N ρ A Si 8 Si Si atoms. m ASi 3 σ ρ n i q resistivity ( μ + μ ) e h Ω m 3 ( Ω m) 1
35 Example The electrical resistivity of pure silicon is.3x 3 Ω-m at room temperature (7 o C ~ 300K). Calculate its electrical conductivity at 00 o C (473K). Assume that the E g of Si is 1.1eV ; k B 8.6x -5 ev/k T k E C B g.exp σ ) (.exp B g k E C σ ) (.exp B g k E C σ Ω + ). (. ) (. ) (. ln ). (. ) ( ) ( ln ) ( ) ( exp m ev k E k E k E k E k E B g B g B g B g B g σ σ σ σ σ σ σ σ
36 Example: For germanium at 5 o C estimate (a) the number of charge carriers, (b) the fraction of total electrons in the valence band that are excited into the conduction band and E g (c) the constant A in the expression when EE g / Data: Ge has a diamond cubic structure with 8 atoms per cell and valence of 4 ; a nm ; E g for Ge 0.67eV ; μ e 3900cm /V.s ; μ h 1900cm /V.s ; ρ 43Ω-cm ; k B 8.63x -5 ev/k (a) Number of carriers T k B T 5 o C 5 ( )( ev / K)( ) eV 0 03 n σ. q( μ + μ ) 1. 6 ( ). e h n A exp electrons 3 cm There are.5x 13 electrons/cm 3 and.5x 13 holes/cm 3 helping to conduct a charge in germanium at room temperature. k B T
37 b) the fraction of total electrons in the valence band that are excited into the conduction band The total number of electrons in the valence band of germanium is : Valence electrons ( 8atoms / cell )( 4valence electrons / atoms) 7 3 ( x cm) Total valence electrons electrons / cm Fraction excited number excited electrons / cm 3 Total valence electrons / cm (c) the constant A A e n Eg k T B e 19 carriers / cm 3
38 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 k p is momentum 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.