Pauli Exclusion Principle Electrons in a single atom occupy discrete levels of energy. No two energy levels or states in an atom can have the same energy. Each energy level can contain at most two electrons -- one with clockwise spin and one with counterclockwise spin. If two or more atoms are brought together, their outer (i.e., valence) energy levels must shift slightly so they will be different from one another. If many (e.g., N) atoms are brought together to form a solid, the Pauli Exclusion Principle still requires that only two electrons in the entire solid have the same energy. There will be N distinct, but only slightly different valence energy levels, forming a valence band.
When a solid is formed, the different split energy levels of electrons come together to form continuous bands of energies (electron energy band).
The extent of splitting depends on the interatomic separation and begins with the outer most electron shells as they are the first to be perturbed as the atoms coalesce.
Band Theory: Two Approaches There are two approaches to finding the electron energies associated with atoms in a periodic lattice. Approach #1: Bound Electron Approach (Solve single atom energies!) Isolated atoms brought close together to form a solid. Approach #2: Unbound or Free Electron Approach (E = p 2 /2m) Free electrons modified by a periodic potential (i.e. lattice ions). Both approaches result in grouped energy levels with allowed and forbidden energy regions. Energy bands overlap for metals. Energy bands do not overlap (or have a gap ) for semiconductors.
Band Theory: Bound Electron Approach For the total number N of atoms in a solid (10 23 cm 3 ), N energy levels split apart within a width ΔE. Leads to a continuous band of energies for each initial atomic energy level (e.g. 1s energy band for 1s energy level). Two atoms Six atoms Solid of N atoms
The electrical properties of a solid material are a consequence of its electron band structure, that is, the arrangement of the outermost electron bands and the way in which they are filled with electrons. Four different types of band structure are possible at 0 o K: Outermost band is only partially filled with electrons (eg, Na, Cu) (Cu has a single 4s valence electron, however for a solid comprised of N atoms, the 4s band is capable of accommodating 2N electrons. Thus, only half of the 4s band is filled.) Overlap of an empty band and a filled band (eg. Mg). (An isolated Mg atom has 2 electrons in its 3s level. When a solid is formed, the 3s and 3p energy levels overlap.) A completely filled valence band is separated from the conduction band by a relatively wide energy band gap (insulators). A completely filled valence band is separated from the conduction band by a relative narrow energy band gap (semiconductors)
Electron Energy E f Freedom Empty States States Filled with Electrons Distance Energy Band The Fermi Level corresponds to the Highest Occupied Molecular Orbital (HOMO). For a Li crystal with N atoms there are 3N electrons. The 1s band is filled and the 2s band is half-filled. The energy difference between adjacent states is infinitesimally small. The electrons near the Fermi level can move from filled states to empty states with no activation energy (metals). Electron Energy Conduction Band Empty Forbidden Valence Band Filled with Electrons Distance Energy Gap If valence band is filled, no empty space are available above the filled states. Electrons can be promoted to the conduction band Lowest Unoccupied Molecular Orbital LUMO with an activation energy > energy gap (band gap). Difference between semiconductors and insulators is due to the size of the bandgap.
Temperature Effect When the temperature of the metal increases, some electrons gain energy and are excited into the empty energy levels in a valence band. This condition creates an equal number of empty energy levels, or holes, vacated by the excited electrons. Only a small increase in energy is required to cause excitation of electrons. Both the excited electrons (free electrons) and the newly created holes can then carry an electrical charge.
Fermi Energy is the energy corresponding to the highest filled state at 0 o K. Only electrons above the Fermi energy can be affected by an electric field (free electrons). Band Diagram: Fermi-Dirac Filling Function The Fermi-Dirac function gives the fraction of allowed states, f FD (E), at an energy level E, that are populated at a given temperature. f E = ( ) ( ) where the Fermi Energy, E F, is defined as the energy where f FD (E F ) = 1/2. That is to say one half of the available states are occupied. T is the temperature (in K) and k B is the Boltzman constant (k B = 8.62 x10-5 ev/k) 1 FD E E F kt e + 1
Probability of electrons (fermions) to be found at various energy levels. f E = 1 ( ) ( ) FD E E F kt e + 1 For E E F = 0.05 ev f(e) = 0.12 For E E F = 7.5 ev f(e) = 10 129 Exponential dependence has HUGE effect! Temperature dependence of Fermi-Dirac function shown as follows: Step function behavior smears out at higher temperatures.
Fermi-Dirac Function Metals and Semiconductors f(e) as determined experimentally for Ru metal (note the energy scale) f(e) for a semiconductor
Velocity of the electrons In metals, the Fermi energy gives us information about the velocities of the electrons which participate in ordinary electrical conduction. The amount of energy which can be given to an electron in such conduction processes is on the order of micro-electron volts (see copper wire example), so only those electrons very close to the Fermi energy can participate. The Fermi velocity of these conduction electrons can be calculated from the Fermi energy. Speed of light: 3x10 8 m.s -1
Fermi Energies, Fermi Temperatures, and Fermi Velocities Element Fermi Energy ev Fermi Temperature x10^4 K Fermi Velocity 10^6 m/s Li 4.74 5.51 1.29 Na 3.24 3.77 1.07 K 2.12 2.46 0.86 Rb 1.85 2.15 0.81 Cs 1.59 1.84 0.75 Cu 7.00 8.16 1.57 Ag 5.49 6.38 1.39 Au 5.53 6.42 1.40 Be 14.3 16.6 2.25 Mg 7.08 8.23 1.58 from N. W. Ashcroft and N. D. Mermin, The Fermi temperature is the temperature associated with the Fermi energy by solving E F = k B T F for, where m is the particle mass and k B is Boltzmann's constant.
Electrical Conductivity (σ) 1 ( Ω ) 1 σ = m ρ Conductivity is the ease of conduction. Ranges over 27 orders of magnitude! Metals 10 7 (Ω.cm) -1 Semiconductors 10-6 -10 4 (Ω.cm) -1 Insulators 10-10 -10-20 (Ω.cm) -1 (a) Charge carriers, such as electrons, are deflected by atoms or defects and take an irregular path through a conductor. (b) Valence electrons in the metallic bond move easily. (c) Covalent bonds must be broken in semiconductors and insulators for an electron to be able to move. (d) Entire ions must diffuse to carry charge in many ionically bonded materials. Electronic conduction: Flow of electrons, e - and electron holes, h + Ionic conduction Flow of charged ions, Ag +
Microscopic Conductivity We can relate the conductivity of a material to microscopic parameters that describe the motion of the electrons (or other charge carrying particles such as holes or ions). From the equations RA V V = IR and ρ = = L L If J=Current density (I/A) Ampere/m 2 ; and ξ=electric field intensity (V/L) then 1 ( ) 1 σ = Ω m ρ We can also determine that the current density is Where n is the number of charge carriers (carriers/cm 3 ); q is the charge on each carrier (1.6x10-19 C); and ν is the average drift velocity (cm/s) at which the charge carriers move. Therefore J J = I A = σξ ρ nqν
J = nqν = σξ then σ = nq ν ξ ν ξ = μ = σ = nqμ mobility cm V.s 2 The charge q is a constant. Electrons are the charge carriers in metals. Electrons and holes are both carriers of electricity in semiconductors. Electrons that hop from one defect to another or the movement of ions are both the carriers of electricity in ceramics.
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
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
2 v 10 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) 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 The drift velocity is v = a τ = = 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 q=1.6x10-19 C σ = s a n q m = 2 e qξ m e λ v 8 1 v 10 cm. s mev kbt e gives e J = nqν = μ = q m λ v 1 2 = 2 = σξ q τ e m e 3 2 τ = λ v
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
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.
Conduction in Materials --Classical approach 1. Drude Model 2. Temperature dependent conductivity 3. Matthissen s Rule 4. Hall effect 5. Skin effect (HF resistance of a conductor) 1. Heat Capacity and thermal conductivity
Classic Model (Drude Model) electron act as a particle Collision: the scattering of an electron by (and only by) an ion core Between collision: electrons do not interact with each other or with ions An electron suffers a collision with probability per unit time τ -1, (τ -1 scattering rate). Electrons achieve thermal equilibrium with their surrounding only through collisions u (a) Vibrating Cu + ions Fig. 2.2 (a): A conduction electron in the electron gas moves about randomly in a metal (with a mean speed u) being frequently and randomly scattered by by thermal vibrations of the atoms. In the absence of an applied field there is no net drift in any direction. (b): In the presence of an applied field, Ex, thereisanet drift along the x-direction. This net drift along the force of the field is superimposed on the random motion of the electron. After many scattering events the electron has been displaced by a net distance, Δx, from its initial position toward the positive terminal Ex Δx (b) V From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca
Relaxation time approximation and Mobility E x Δx A v dx J x Velocity gained along x v x1 -u x1 Last collision Present time Fig. 2.1: Drift of electrons in a conductor in the presence of an applied electric field. Electrons drift with an average velocity vdx in the x-direction.(e x is the electric field.) v x1 -u x1 t 1 Electron 1 Free time t time From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca t 2 Electron 2 t time v x1 -u x1 v dx = μ E eτ m e x J = en μ E σ = ne μ μ = x = σ E (Ohm s law) t 3 Electron 3 t time Fig. 2.3: Velocity gained in the x-direction at time t from the electric field (Ex) for three electrons. There will be N electrons to consider in the metal. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca
Temperature dependence of Conductivity for Metal S = π a 2 l=uτ u a Conduction by free electrons and scattered by lattice vibration Electron Fig. 2.4: Scattering of an electron from the thermal vibrations of the atoms. The electron travels a mean distance l = u τ between collisions. Since the scattering cross sectional area is S, in the volume Sl there must be at least one scatterer, Ns(Suτ) = 1.
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
Matthiessen s Rule Strained region by impurity exerts a scattering force F = - d(pe) /dx τ I τ Τ Fig. 2.5: Two different types of scattering processes involving scattering from impurities alone and thermal vibrations alone. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca
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
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca 2000 1000 Monel-400 Iron Inconel-825 NiCr Heating Wire Tungsten Resistivity (nω m) 100 Tin Platinum Nickel ρ T Copper Silver 10 100 1000 10000 Temperature (K) Fig. 2.6: The resistivity of various metals as a function of temperature above 0 C. Tin melts at 505 K whereas nickel and iron go through a magnetic to non-magnetic (Curie) transformations at about 627 K and 1043 K respectively. The theoretical behavior (ρ ~ T) is shown for reference. [Data selectively extracted from various sources including sections in Metals Handbook, 10th Edition, Volumes 2 and 3 (ASM, Metals Park, Ohio, 1991)]
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
From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca 100 10 ρ T Resistivity (nω m) 1 0.1 0.01 0.001 0.0001 ρ = ρ R ρ T 5 ρ (nω m) 3.5 2 1.5 ρ T 5 1 0.5 ρ = ρ R 0 0 20 40 60 80 100 T (K) 0.00001 1 10 100 1000 10000 Temperature (K) 3 2.5 ρ T Fig.2.7: The resistivity of copper from lowest to highest temperatures (near melting temperature, 1358 K) on a log-log plot. Above about 100 K, ρ T, whereas at low temperatures, ρ T 5 and at the lowest temperatures ρ approaches the residual resistivity ρ R. The inset shows the ρ vs. T behavior below 100 K on a linear plot ( ρ R is too small on this scale).
Electron conduction in nonmetals Insulators Semiconductors Alumina Many ceramics Diamond Inorganic Glasses Mica Polypropylene PET PVDF Soda silica glass Borosilicate Pure SnO 2 Amorphous Intrinsic Si SiO 2 Intrinsic GaAs As 2 Se 3 Degenerately Doped Si Te Conductors Metals Graphite NiCr Superconductors Alloys Ag 10-18 10-15 10-12 10-9 10-6 10-3 10 0 10 3 10 6 10 9 Conductivity (Ωm) -1 10 12 Figure 2.24: Range of conductivites exhibited by various materials From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca
Control of Electrical Conductivity By controlling the number of charge carriers in the material (n) By controlling the mobility or ease of movement of the charge carriers (μ) σ = nqμ
Example: Calculation of Drift velocity of electrons in copper (valence =1) Assuming that all of the valence electrons contribute to current flow, (a) calculate the mobility of an electron in copper and (b) calculate the average drift velocity for electrons in a 100cm copper wire when 10V are applied. Data: conductivity of copper = 5.98 x 10 5 (Ω.cm) -1 ; q = 1.6x10-19 C ; lattice parameter of copper = 0.36151x10-7 cm ; copper has a FCC structure The equation that we need to apply is: σ = nqμ n is the number of carriers. q is the charge. μ is the mobility Number of carriers (n): The valence of copper is 1. Therefore, the number of valence electrons equals the number of copper atoms in the material. n ( atoms / cell )( electrons / atoms) cell _ volume 4 1 ( 0. 36151 10 22 = = = 8. 466 10 7 3 cm) electrons / cm 3
Part (a) μ σ nq 5 1 1 5. 98 10 Ω. cm 22 3 ( 8. 466 10 electrons / cm )( 1. 6 10 = = 19 C) = 2 cm 44. 2 Ω. C = cm 44. 2 V. s 2 Part (b) The electric field intensity is: Note : 1 amp = 1Coulomb / sec ond V 10 V ξ = = = 0. 1V. cm L 100cm 1 The equation we need to use is ν = μξ 2 cm V v = ( 44. 2 )( 0. 1 ) = 4. 42cm / s V. s cm What does it mean? The drift velocity is the average velocity that a particle, such as an electron, attains, due to an electric field. The electron moves at the Fermi speed and it has only a tiny drift velocity superimposed by the applied electric field.
In metals, the Fermi energy gives us information about the velocity of the electrons which participate in ordinary electrical conduction. The amount of energy which can be given to an electron in such conduction processes is on the order of micro-electron volts, so only those electrons very close to the Fermi energy (E F ) can participate. v F = 2E m F
Element Fermi Energy ev Fermi Temperature x 10^4 K Fermi Velocity x 10^6 m/s Li 4.74 5.51 1.29 Na 3.24 3.77 1.07 K 2.12 2.46 0.86 Rb 1.85 2.15 0.81 Cs 1.59 1.84 0.75 Cu 7.00 8.16 1.57 Ag 5.49 6.38 1.39 Au 5.53 6.42 1.40 Be 14.3 16.6 2.25 Mg 7.08 8.23 1.58 Ca 4.69 5.44 1.28 Sr 3.93 4.57 1.18 Ba 3.64 4.23 1.13 Nb 5.32 6.18 1.37 Fe 11.1 13.0 1.98 Mn 10.9 12.7 1.96 Zn 9.47 11.0 1.83 Cd 7.47 8.68 1.62 Hg 7.13 8.29 1.58 Al 11.7 13.6 2.03 Ga 10.4 12.1 1.92 In 8.63 10.0 1.74 Tl 8.15 9.46 1.69 Sn 10.2 11.8 1.90 Pb 9.47 11.0 1.83 Bi 9.90 11.5 1.87
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
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 (2-10 ev) between conduction and valence bands. At T > 0, electrons are usually NOT thermally excited from valence to conduction band, leading to zero conductivity.
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 2 A good insulator will have a band gap >>5eV and 2k B T~0.025eV at room temperature As a result of thermal excitation, the fraction of electrons in the conduction band is ~e -200 or 10-80. 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.
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 2+ ions. Estimate the mobility of Mg 2+ ions and calculate the electrical conductivity of MgO at 1800 o C. Data: Diffusion Constant of Mg in MgO = 0.0249cm 2 /s ; lattice parameter of MgO a=0.396x10-7 cm ; Activation Energy for the Diffusion of Mg 2+ in MgO = 79,000cal/mol ; k B =1.987cal/K=k-mol; For MgO Z=2/ion; q=1.6x10-19 C; k B =1.38x10-23 J/K-mol
First, we need to calculate the diffusion coefficient D D = D o Q exp kt D = cm 0. 0239 s 2 79000cal / mol exp 1. 987cal / mol Kx( 1800 + 273) K D=1.119x10-10 cm 2 /s Next, we need to find the mobility μ = Z. q. D k. T B = ( 19 carriers / ion)( 1. 6 10 C)( 1. 1 10 23 ( 1. 38 10 )( 1800 + 273) 10 2 9 ) = 1. 12 10 C. cm J. s 2 C ~ Amp. sec ; J ~ Amp. sec.volt μ=1.12x10-9 cm 2 /V.s
MgO has the NaCl structure (with 4 Mg 2+ and 4O 2- per cell) Thus, the Mg 2+ ions per cubic cm is: 4Mg ions / cell ( 0. 396 10 cm) 2+ 22 n = = 6. 4 10 ions / 7 3 cm 3 σ = σ = nzqμ = 22. 94 10 ( 6. 4 10 6 22 2 C. cm 3 cm. V. s )( 2)( 1. 6 10 19 )( 1. 12 10 9 ) C ~ Amp.sec ; V ~ Amp.Ω σ = 2.294 x 10-5 (Ω.cm) -1
Example: The soda silicate glass of composition 20%Na 2 O-80%SiO 2 and a density of approximately 2.4g.cm -3 has a conductivity of 8.25x10-6 (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 23, 16 and 28.1 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. 20%Na 2 O-80%SiO 2 can be written as (Na 2 O) 0.2 -(SiO 2 ) 0.8. Its mass can be calculated as: i M M At At i = 0. 2 ( 2( 23) + 116 ( )) + 0. 8 ( 1( 281. 1) + 2( 16)) = 60. 48g. mol 1 The number of (Na 2 O) 0.2 -(SiO 2 ) 0.8 units per unit volume can be found from the density
n n = = ρ N M At A = 2. 39 10 ( 2. 4g. cm 22 ( Na 2 O) ) ( 6. 023x10 1 60. 48g. mol 3 0. 2 ( SiO2 ) 0. 8 23 mol units cm 1 3 ) The concentration of Na + ions (n i ) can be obtained from the concentration of (Na 2 O) 0.2 -(SiO 2 ) 0.8 units n i = 2 39 10 = 3 18 10 0 2 2 1 0 8 1 2.. cm. ( + ) +. ( + ) 0. 2 2 22 21 3 And μ i μ = i σ q n = μ = 1. 62 10 i i 6 1 ( 8. 25 10 Ω m 19 ( 1. 60 10 C) ( 3. 186 10 14 2 m V 1 s 1 1 21 ) 10 6 m 3 ) This is a very small mobility compared to semiconductors and metals
Conduction in ionic crystal and glasses Mobile charges contribute to conduction E Vacancy aids the diffusion of positive ion Anion vacancy acts as a donor (a) Interstitial cation diffuses Na + ig. 2.27: Possible contributions to the conductivity of ceramic and lass insulators (a) Possible mobile charges in a ceramic (b) A Na+ on in the glass structure diffuses and therefore drifts in the irection of the field. (E is the electric field.) From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca E 0 σ = σ 0 exp kt E (b) Conductivity 1/(Ωm) 1 10-1 1 10-3 1 10-5 1 10-7 1 10-9 1 10-11 1 10-13 24%Na2O-76%SiO2 Pyrex SiO 2 PVC As 3.0 Te 3.0 Si 1.2 Ge 1.0 glass 12%Na 2 O-88%SiO 2 PVAc 1 10-15 1.2 1.6 2 2.4 2.8 3.2 3.6 4 10 3 /T (1/K) Fig. 2.28: Conductivity vs reciprocal temperature for various low conductivity solids. (PVC = Polyvinyl chloride; PVAc = Polyvinyl acetate.) Data selectively combined from numerous sources. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca Hopping process
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 ~10 9 Vm -1 to practical values of 10 7 V.m -1
Energy Bands in Semiconductors Energy Levels and Energy Gap in a Pure Semiconductor. The energy gap is < 2 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 +2 and Se +6 in the II-VI compund CdSe.
Semiconductor Materials Semiconductor Bandgap Energy E G (ev) Carbon (Diamond) 5.47 Silicon 1.12 Germanium 0.66 Tin 0.082 Gallium Arsenide 1.42 Indium Phosphide 1.35 Silicon Carbide 3.00 Cadmium Selenide 1.70 Boron Nitride 7.50 Aluminum Nitride 6.20 Gallium Nitride 3.40 IIB 65.37 Indium Nitride 1.90 Portion of the Periodic Table Including the Most Important Semiconductor Elements 30 Zn Zin c 112.40 48 Cd Cadmium 200.59 80 Hg Mercury 5 IIIA IVA VA VIA 10.811 B Bo ro n 26.9815 13 Al A luminum 69.72 31 Ga G a llium 114.82 49 In Ind ium 204.37 81 Ti Tha llium 12.01115 6 C Carbon 28.086 14 Si Silic o n 72.59 32 Ge Germanium 50 Sn Tin 118.69 207.19 82 Pb Le a d 14.0067 7 N Nitrogen 30.9738 15 P Phosphorus 74.922 33 As Arsenic 121.75 51 Sb Antimony 208.980 83 Bi Bism uth 8 15.9994 O Oxygen 32.064 16 S Su lf u r 78.96 34 Se Selenium 127.60 52 Te Te llurium (210) 84 Po Polonium
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.
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 (T @ 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.
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
Semiconductors in Group IV Carbon, Silicon, Germanium, Tin Each has 4 valence Electrons. Covalent bond
Compound Semiconductors (Group III and Group V) The III-V semiconductors are prominent for applications in optoelectronics with particular importance for areas such as wireless communications because they have a potential for higher speed operation than silicon semiconductors. The compound semiconductors have a crystal lattice constructed from atomic elements in different groups of the periodic chart. Each Group III atom is bound to four Group V atoms, and each Group V atom is bound to four Group III atoms, giving the general arrangement shown in Figure. The bonds are produced by sharing of electrons such that each atom has a filled (8 electron) valence band. The bonding is largely covalent, though the shift of valence charge from the Group V atoms to the Group III atoms induces a component of ionic bonding to the crystal (in contrast to the elemental semiconductors which have purely covalent bonds).
Representative III-V compound semiconductors are GaP, GaAs, GaSb, InP, InAs, and InSb. GaAs is probably the most familiar example of III-V compound semiconductors, used for both high speed electronics and for optoelectronic devices. Optoelectronics has taken advantage of ternary and quaternary III-V semiconductors to establish optical wavelengths and to achieve a variety of novel device structures. The ternary semiconductors have the general form (A x ;A 1-x )B (with two group III atoms used to filled the group III atom positions in the lattice) or A(B x ;B 1-x ) (using two group V atoms in the Group V atomic positions in the lattice). The quaternary semiconductors use two Group III atomic elements and two Group V atomic elements, yielding the general form (A x ;A 1-x )(B y ;B 1-y ). In such constructions, 0<x,y<1. Such ternary and quaternary versions are important since the mixing factors (x and y) allow the band gap to be adjusted to lie between the band gaps of the simple compound crystals with only one type of Group III and one type of Group V atomic element. The adjustment of wavelength allows the material to be tailored for particular optical wavelengths, since the wavelength of light is related to energy (in this case the gap energy E g ) by λ= hc/e g, where h is Plank's constant and c is the speed of light.
Hall effect and Hall devices q = +e q = -e v v B B F = qv B B F = qv B (a) (b) Fig. 2.16 A moving charge experiences a Lorentz force in a magnetic field. (a) A positive charge moving in the x direction experiences a force downwards. (b) A negative charge moving in the -x direction also experiences a force downwards. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca
Hall effect in Semiconductor + J y = 0 v hx v ex eey B z ee y E y J x E x + J x y x z + ev hx B z ev ex B z + + + B z A V Fig. 2.26: Hall effect for ambipolar conduction as in a semiconductor where there are both electrons and holes. The magnetic field Bz is out from the plane of the paper. Both electrons and holes are deflected toward the bottom surface of the conductor and consequently the Hall voltage depends on the relative mobilities and concentrations of electrons and holes.(e is the electric field.) From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca
Thermal Conduction Metal HEAT HOT δt dq dt COLD A HEAT HOT COLD δx Fig. 2.19: Heat flow in a metal rod heated at one end. Consider the rate of heat flow, dq/dt, across a thin section δ x of the rod. The rate of heat flow is proportional to the temperature gradient δ T/δ x and the cross sectional area A. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca Electron Gas Vibrating Cu + ions Fig. 2.18: Thermal conduction in a metal involves transferring energy from the hot region to the cold region by conduction electrons. More energetic electrons (shown with longer velocity vectors) from the hotter regions arrive at cooler regions and collide there with lattice vibrations and transfer their energy. Lengths of arrowed lines on atoms represent the magnitudes of atomic vibrations. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca
The failure of Drude model It can explain the electric conduction, not thermal conductivity and heat capacity Electronic heat capacity (Drude) 3 C el = 2 R Experimental result C V = γ T + αt 3 Need more sophisticated model (Sommerfeld or Quantum mechanics)
Thermal conductivity, κ (W K -1 m -1 ) 450 400 300 200 100 0 κ σ = T C WFL Be W Mg Mo Ni Brass (Cu-30Zn) Bronze (95Cu-5Sn) Steel (1080) Pd-40Ag Hg 0 10 20 30 40 50 60 70 Electrical conductivity, σ, 10 6 Ω -1 m -1 Al Ag-3Cu Ag-20Cu Au Ag Cu Fig. 2.20: Thermal conductivity, κ vs. electrical conductivity σ for various metals (elements and alloys) at 20 C. The solid line represents the WFL law with C WFL 2.44 108 W Ω K-2. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca Thermal conductivity, κ (W K -1 m -1 ) 50000 10000 1000 100 10 Aluminum Copper Brass (70Cu-30Zn) Al-14%Mg 1 10 100 1000 Temperature (K) Fig. 2.21: Thermal conductivity vs. temperature for two pure metals (Cu and Al) and two alloys (brass and Al-14%Mg). Data extracted from Thermophysical Properties of Matter, Vol. 1: Thermal Conductivity, Metallic Elements and Alloys, Y.S. Touloukian et. al (Plenum, New York, 1970). From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca
Non-metal Phonon Diamond Good thermal conductor Polymer- -Bad thermal conductor Equilibrium Hot Cold Energetic atomic vibrations Fig. 2.22: Conduction of heat in insulators involves the generation and propogation of atomic vibrations through the bonds that couple the atoms. (An intuitive figure.)
Thermal conductivity and resistance Q Hot L (a) ΔT A Cold Q Q = ΔT/θ ΔT Fig. 2.23: Conduction of heat through a component in (a) can be modeled as a thermal resistance θ shown in (b) where Q = ΔT/θ. θ (b) Q From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca Pure metal Nb Fe Zn W Al Cu Ag κ (W m -1 K - 52 80 113 178 250 390 420 1 ) Metal alloys Stainless Steel 55Cu-45Ni 70Ni-30Cu 1080 Bronze Brass (63Cu- Dural (95Al- Steel (95Cu- 37Zn) 4Cu-1Mg) 5Sn) κ (W m -1 K - 12-16 19.5 25 50 80 125 147 1 ) Ceramics Glass- Silica-fused S 3 N 4 Alumina Saphire Beryllia Diamond and glasses borosilicate (SiO 2 ) (Al 2 O 3 ) (Al 2 O 3 ) (BeO) κ (W m -1 K - 0.75 1.5 20 30 37 260 ~1000 1 ) Polymers Polypropylene PVC Polycarbonate Nylon Teflon Polyethylene Polyethylene 6,6 low density high density κ (W m -1 K - 0.12 0.17 0.22 0.24 0.25 0.3 0.5 1 )