Carrier Modeling - Microelectronic Circuits - Lecture Slides | ECE 3040, Study notes of Electrical and Electronics Engineering

Download Carrier Modeling - Microelectronic Circuits - Lecture Slides | ECE 3040 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity! Pierret, Chapter 2 “Carrier Modeling” Jeff Davis FALL 2007 ECE3040 References • Prof. Alan Doolittle’s Notes – users.ece.gatech.edu/~alan/index_files/ECE3040index.htm • Prof. Farrokh Ayazi’s Notes – users.ece.gatech.edu/~ayazi/ece3040/ • Figures for Require Textbooks – (Pierret and Jaeger) Quantum Numbers N = principle quantum number (N = 1,2,3,..) Size of orbital N>=(L+1) L = angular quantum number (L = 0,1,2,3,4,5,6,7 OR s,p,d,f,g,h, i, k) M = magnetic quantum number ( orientation of the orbitals) |M|<=L S = Angular momentum of electron spin (+1/2 or -1/2) Designates shell Designates subshell M can be negative! Pauli Exclusion Principle NO TWO ELECTRONS CAN HAVE THE SAME 4 QUANTUM NUMBERS!!! (If overlapping wave functions) 1. Hydrogen (H) 1s 2. Helium (He) 1s2 3. Lithium (Li) 1s22s 4. Beryllium (Be) 1s22s2 5. Boron (B) 1s22s22p 6. Carbon (C) 1s22s22p2 Carbon has six electrons.. its electrons could have the following set of quantum numbers electron 1: (N=1,L=0(s),M=0,S=+1/2) electron 2: (N=1,L=0(s),M=0,S=-1/2) electron 3: (N=2,L=0(s),M=0,S=+1/2) electron 4: (N=2,L=0(s),M=0,S=-1/2) electron 5: (N=2,L=1(p),M=0,S=+1/2) electron 6: (N=2, L=1(p), M=0,S=-1/2) Examples sp? Hybrid Orbitals Zz z x a y y s Bx z Zz y | y Py P: x — x Hybridization |” * SE z z - x x y ” Different orientation of orbitals gives a tetrahedral arrangement La — Bb en rH a ‘Crbitals ina free C atom {| \ ——<— spo ‘sp? Silicon n=1: Complete Shell 2 “s electrons” n=2: Complete Shell 2 “2s electrons” 6 “2p electrons” n=3: 2 “3s electrons” Only 2 of 6 “3p electrons” 4 empty states 4 Valence Shell Electrons What about silicon? T=0K EC or conduction band EV or valence band “Band Gap” where ‘no’ states exist + For (Ethermal=kT)>0 Carrier Movement Under Bias Direction of Current Flow Direction of Current Flow Ec Ev Electron free to move in conduction band “Hole” movement in valence band + Ec Ev Electron free to move in conduction band “Hole” movement in valence band Carrier Movement Under Bias Direction of Current Flow Direction of Current Flow For (Ethermal=kT)>0 + Ec Ev Carrier Movement Under Bias Electron free to move in conduction band “Hole” movement in valence band Direction of Current Flow Direction of Current Flow For (Ethermal=kT)>0 Energy Dependence of Bandgap Energy - Example GaN Ramirez-Flores, G., H. Navarro-Contreras, A. Lastrae-Martinez, R.C. Powell, J.E. Greene, Temperature-dependent optical band gap of the metastable zinc-blende structure beta -GaN, Phys. Rev. B 50(12) (1994), 8433-8438. Definition of Intrinsic Carrier Concentration, ni Intrinsic Carrier Concentration •Intrinsic carrier concentration is the number of electron (=holes) per cubic centimeter populating the conduction band (or valence band) is called the intrinsic carrier concentration, ni •ni = f(T) that increases with increasing T (more thermal energy) ni~2e6 cm-3 for GaAs with Eg=1.42 eV, ni~1e10 cm-3 for Si with Eg=1.1 eV, ni~2e13 cm-3 for Ge with Eg=0.66 eV, ni~1e-14 cm-3 for GaN with Eg=3.4 eV At Room Temperature (T=300 K) Ec Ev Carrier Movement in Free Space masselectronmechelectronicq timetvelocityvforceF dt dv mqEF o o !! !!! ="= ,arg ,,, Newton’s second law of motion! F=ma Carrier Movement Within the Crystal Electron sees a periodic potential due to the atomic cores masseffectiveelectronm echelectronicq timetvelocityvforceF dt dv mqEF n n ! ! !!! ="= * * ,arg ,,, masseffectiveholem echelectronicq timetvelocityvforceF dt dv mqEF p p ! ! !!! == * * ,arg ,,, Carrier Movement Within the Crystal Effective Mass used to estimate “mobility” of carrier in the lattice Silicon Germanium GaAs mn*= 0.33mo mn*= 0.22mo electron holes mn*= 0.063mo mh*= 0.5mo mh*= 0.31mo mh*= 0.5mo A.K.A. Conductivity Effective Mass Example: P, As, Sb in Si Extrinsic, (or doped material): Concept of a Donor “adding extra” electrons Concept of a Donor “adding extra” electrons: Band diagram equivalent view Example: B, Al, In in Si Extrinsic, (or doped material): Concept of an acceptor “adding extra” holes + Hole Movement Another valence electron can fill the empty state located next to the Acceptor leaving behind a positively charged “hole”. + Hole Movement The positively charged “hole” can move throughout the crystal (really it is the valance electrons jumping from atom to atom that creates the hole motion). + Hole Movement The positively charged “hole” can move throughout the crystal (really it is the valance electrons jumping from atom to atom that creates the hole motion). Summary of Important terms and symbols Bandgap Energy: Energy required to remove a valence electron and allow it to freely conduct. Intrinsic Semiconductor: A “native semiconductor” with no dopants. Electrons in the conduction band equal holes in the valence band. The concentration of electrons (=holes) is the intrinsic concentration, ni. Extrinsic Semiconductor: A doped semiconductor. Many electrical properties controlled by the dopants, not the intrinsic semiconductor. Donor: An impurity added to a semiconductor that adds an additional electron not found in the native semiconductor. Acceptor: An impurity added to a semiconductor that adds an additional hole not found in the native semiconductor. Dopant: Either an acceptor or donor. N-type material: When electron concentrations (n=number of electrons/cm3) exceed the hole concentration (normally through doping with donors). P-type material: When hole concentrations (p=number of holes/cm3) exceed the electron concentration (normally through doping with acceptors). Majority carrier: The carrier that exists in higher population (ie n if n>p, p if p>n) Minority carrier: The carrier that exists in lower population (ie n if n<p, p if p<n) Other important terms (among others): Insulator, semiconductor, metal, amorphous, polycrystalline, crystalline (or single crystal), lattice, unit cell, primitive unit cell, zincblende, lattice constant, elemental semiconductor, compound semiconductor, binary, ternary, quaternary, atomic density, Miller indices, various notations, etc... How do we calculate the electron or hole density in equilibrium? Parking Lot Analogy If we have a lot with 100 spaces and the probability of a single space being occupied is 25%, on average how many parking spaces should be occupied. Change percentage Change # of parking spaces If we have a lot with 100 spaces and the probability of a single space being occupied is 50%, on average how many parking spaces should be occupied. If we have a lot with 200 spaces and the probability of a single space being occupied is 25%, on average how many parking spaces should be occupied. How do electrons and holes populate the bands? Quantum Mechanics tells us that the number of available states in a cubic cm per unit of energy, the density of states, is given by: eV cm StatesofNumber unit EE EEmm Eg EE EEmm Eg v vpp v c cnn c ! " # $ % & ' ( ) = * ) = 3 32 ** 32 ** , )(2 )( , )(2 )( ! ! + + Density of States Concept h = planck’s constant= 6.63e-34 [J-sec] Remember Plancks Constant! h = reduced planck’s constant (pronounced “h-bar”)= h/2π Ephoton = hν = hω Photon frequency Photon angular frequency Density of States Concept Thus, the number of states per cubic centimeter between energy E’ and E’+dE is otherwise and and 0 ,EE’ if )dE(E’g ,EE’ if )dE(E’g vv cc ! " At higher temperatures, higher energy states can be occupied, leaving more lower energy states unoccupied (1-f(E)). Probability of Occupation (Fermi Function) Concept Expected Electron Concentration Thus, the density of electrons (or holes) occupying the states in energy between E and E+dE is: otherwise and and 0 ,EE if dEf(E)]-(E)[1g ,EE if dEf(E)(E)g vv cc ! " Electrons/cm3 in the conduction band between Energy E and E+dE Holes/cm3 in the valence band between Energy E and E+dE Expected Electron Concentration Decreasing (Ec-Ef) increases electron concentration Decreasing (Ef-Ev) increases electron concentration Developing the mathematical model for electrons and holes The density of electrons is: != bandconductionofTop bandconductionofBottom E E c dEEfEgn )()( ! "= bandvalenceofTop bandvalenceofBottom E E v dEEfEgp )](1)[( The density of holes is: Probability the state is filled Probability the state is empty Number of states per cm-3 in energy range dE Number of states per cm-3 in energy range dE Note: units of n and p are #/cm3 Developing the mathematical model for electrons and holes ! " + " = bandconductionofTop c f E E kTEE cnn dE e EEmm n /)(32 ** 1 2 !# ! " # + = 0 )( 2/1 32 2/3** 1 )(2 $ $ % $$ d e kTmm n c nn ! !" == # = # = bandconductionofTop c cf c c ELet EEwhen kT EE and kT EE Letting 0,$ $$ This is known as the Fermi-dirac integral of order 1/2 or, F1/2(ηc) Developing the mathematical model for electrons and holes bandvalencetheinstatesofdensityeffectivethe kTm N and bandconductiontheinstatesofdensityeffectivethe kTm N p v n c 2/3 2 * 2/3 2 * 2 )( 2 2 )( 2 ! ! " # $ $ % & = ! " # $ % & = ! ! ' ' We can further define: This is a general relationship holding for all materials and results in: Katcm m m xN Katcm m m xN o v v o n c 3001051.2 3001051.2 3 2 3 * 19 3 2 3 * 19 ! ! "" # $ %% & ' = "" # $ %% & ' = Developing the mathematical model for electrons and holes Useful approximations to the Fermi-dirac integral: Non-degenerate Case kTEE v kTEE c fv cf eNp and eNn /)( /)( ! ! = = Developing the mathematical model for electrons and holes kTEE iv kTEE vi kTEE ic kTEE ci viiv icci enNoreNn and enNoreNn /)(/)( /)(/)( !! !! == == When n=ni, Ef=Ei (the intrinsic energy), then kTEE i kTEE i fi if enp and enn /)( /)( ! ! = = Developing the mathematical model for electrons and holes Other useful Relationships: n - p product kTE vci kTE vc kTEE vci kTEE vi kTEE ci G Gvc ivci eNNn eNNeNNn eNnandeNn 2/ //)(2 /)(/)( ! !!! !! = == == Developing the mathematical model for electrons and holes Charge Neutrality: Total Ionization Case N-A = Concentration of “ionized” acceptors ~ = NA N+D = Concentration of “ionized” donors ~ = ND ( ) ( ) 0=!+! nNNp DA Developing the mathematical model for electrons and holes Charge Neutrality: Total Ionization Case ( ) 022 =!!! iAD nNNnn ( ) ( ) 0=!+! nNNp DA ( ) 0 2 =!+"" # $ %% & ' ! nNN n n DA i 2 2 2 2 2222 i DADA i ADAD n NNNN porn NNNN n +! " # $ % & ' + ' =+! " # $ % & ' + ' = 2 inpn and = Developing the mathematical model for electrons and holes If ND>>NA and ND>>ni D i D N n pandNn 2 !! If NA>>ND and NA>>ni A i A N n nandNp 2 !! Example: An intrinsic Silicon wafer at 470K has 1e14 cm-3 holes. When 1e14 cm-3 acceptors are added, what is the new electron and hole concentrations? ND=0 p = 1x1014 2 + 1x1014 2 ! "# $ %& 2 + 1x1014( ) 2 p = 1.62x1014 cm'3 ( N A ' N D n = 1x1014( ) 2 1.62x1014 = 6.2x1013cm'3 Developing the mathematical model for electrons and holes Example: An intrinsic Silicon wafer at 600K has 4e15 cm-3 holes. When 1e14 cm-3 acceptors are added, what is the new electron and hole concentrations? ND=0 p = 1x1014 2 + 1x1014 2 ! "# $ %& 2 + 4x1015( ) 2 p = 4x1015 cm'3 = n i ( N A ' N D n = 4x1015( ) 2 4x1015 = 4x1015cm'3 = n i ) Intrinsic Material at HighTemperature Developing the mathematical model for electrons and holes E 1 — c -eorrOr” } Er - Donor-doped t . 3kT £ E,-}---------------—------------------------- 1 E, ~ ‘aa e Acceptor-do ped 3kT E, 1 1 sf! i {rt --= sol ce 103 10!4 1015 1916 19!7 { 1918 1919 1020 -3 Na or Ny (cm ”) Figure 2.21 Fermi level positioning in Si at 300 K as a function of the doping concentration. The solid E, lines were established using Eq. (2.38a) for donor-doped material and Eq. (2.38b) for accep- tor-doped material (kT = 0.0259 eV, and n; = 10!°/cm?).