Homogeneous Semiconductors Dopants Use Density of States and
- Slides: 31
Homogeneous Semiconductors • Dopants • Use Density of States and Distribution Function to: • Find the Number of Holes and Electrons.
Energy Levels in Hydrogen Atom
Energy Levels for Electrons in a Doped Semiconductor
Assumptions for Calculation
Density of States (Appendix D) Energy Distribution Functions (Section 2. 9) Carrier Concentrations (Sections 2. 10 -12)
GOAL: • The density of electrons (no) can be found precisely if we know 1. the number of allowed energy states in a small energy range, d. E: S(E)d. E “the density of states” 2. the probability that a given energy state will be occupied by an electron: f(E) “the distribution function” no = S(E)f(E)d. E band
For quasi-free electrons in the conduction band: 1. We must use the effective mass (averaged over all directions) 2. the potential energy Ep is the edge of the conduction band (EC) S(E) = (1/2 p 2) (2 mdse*/ 2)3/2(E - EC)1/2 For holes in the valence band: 1. We still use the effective mass (averaged over all directions) 2. the potential energy Ep is the edge of the valence band (EV) S(E) = (1/2 p 2) (2 mdsh*/ 2)3/2(EV - E)1/2
Energy Band Diagram Eelectron E(x) conduction band EC S(E) EV valence band Ehole x note: increasing electron energy is ‘up’, but increasing hole energy is ‘down’.
Reminder of our GOAL: • The density of electrons (no) can be found precisely if we know 1. the number of allowed energy states in a small energy range, d. E: S(E)d. E “the density of states” 2. the probability that a given energy state will be occupied by an electron: f(E) “the distribution function” no = S(E)f(E)d. E band
Fermi-Dirac Distribution The probability that an electron occupies an energy level, E, is f(E) = 1/{1+exp[(E-EF)/k. T]} – where T is the temperature (Kelvin) – k is the Boltzmann constant (k=8. 62 x 10 -5 e. V/K) – EF is the Fermi Energy (in e. V)
f(E) = 1/{1+exp[(E-EF)/k. T]} 1 T=0 o. K T 1>0 T 2>T 1 f(E) 0. 5 0 EF E All energy levels are filled with e-’s below the Fermi Energy at 0 o. K
Fermi-Dirac Distribution for holes Remember, a hole is an energy state that is NOT occupied by an electron. Therefore, the probability that a state is occupied by a hole is the probability that a state is NOT occupied by an electron: fp(E) = 1 – f(E) = 1 - 1/{1+exp[(E-EF)/k. T]} ={1+exp[(E-EF)/k. T]}/{1+exp[(E-EF)/k. T]} 1/{1+exp[(E-EF)/k. T]} = {exp[(E-EF)/k. T]}/{1+exp[(E-EF)/k. T]} =1/{exp[(EF - E)/k. T] + 1}
The Boltzmann Approximation If (E-EF)>k. T such that exp[(E-EF)/k. T] >> 1 then, f(E) = {1+exp[(E-EF)/k. T]}-1 {exp[(E-EF)/k. T]}-1 exp[-(E-EF)/k. T] …the Boltzmann approx. similarly, fp(E) is small when exp[(EF - E)/k. T]>>1: fp(E) = {1+exp[(EF - E)/k. T]}-1 {exp[(EF - E)/k. T]}-1 exp[-(EF - E)/k. T] If the Boltz. approx. is valid, we say the semiconductor is non-degenerate.
Putting the pieces together: for electrons, n(E) f(E) 1 T=0 o. K T 1>0 T 2>T 1 S(E) 0. 5 0 E EV EF EC n(E)=S(E)f(E) E
Putting the pieces together: for holes, p(E) fp(E) T=0 o. K 1 T 1>0 T 2>T 1 0. 5 S(E) 0 EV EF EC p(E)=S(E)f(E) hole energy E
Finding no and po the effective density of states in the conduction band
Energy Band Diagram intrinisic semiconductor: no=po=ni E(x) conduction band EC n(E) p(E) EF=Ei EV valence band x where Ei is the intrinsic Fermi level
Energy Band Diagram n-type semiconductor: no>po E(x) conduction band EC n(E) p(E) EF EV valence band x
Energy Band Diagram p-type semiconductor: po>no E(x) conduction band EC n(E) p(E) E EVF valence band x
A very useful relationship …which is independent of the Fermi Energy Recall that ni = no= po for an intrinsic semiconductor, so nopo = ni 2 for all non-degenerate semiconductors. (that is as long as EF is not within a few k. T of the band edge)
The intrinsic carrier density is sensitive to the energy bandgap, temperature, and m*
The intrinsic Fermi Energy (Ei) For an intrinsic semiconductor, no=po and EF=Ei which gives Ei = (EC + EV)/2 + (k. T/2)ln(NV/NC) so the intrinsic Fermi level is approximately in the middle of the bandgap.
Space-charge Neutrality Consider a semiconductor doped with NA ionized acceptors (-q) and ND ionized donors (+q). positive charges = negative charges po + ND = no + NA using ni 2 = nopo ni 2/no + ND = no+ NA ni 2 + no(ND-NA) - no 2 = 0 no = 0. 5(ND-NA) 0. 5[(ND-NA)2 + 4 ni 2]1/2 we use the ‘+’ solution since no should be increased by ni no = ND - NA in the limit that ni<<ND-NA
Similarly, po= NA - ND if NA-ND>>ni no/(ND-NA) po/(NA-ND) carrier conc. vs. temperature po= ni intrinsic material 1 freeze-out insufficient energy to ionize the dopant atoms 300 100 600 T (K)
Degenerate Semiconductors …the doping concentration is so high that EF moves within a few k. T of the band edge (EC or EV) impurity band EC ED 1 + + Eg(ND) Eg 0 for ND > 1018 cm-3 in Si High donor concentrations cause the allowed donor wavefunctions to overlap, creating a band at EDn First only the high states overlap, but eventually even the lowest state overlaps. EV This effectively decreases the bandgap by DEg = Eg 0 – Eg(ND).
Degenerate Semiconductors As the doping conc. increases more, EF rises above EC DEg available impurity band states filled impurity band states EC (intrinsic) EF EC (degenerate) ~ ED apparent band gap narrowing: DEg* (is optically measured) Eg* is the apparent band gap: - an electron must gain energy Eg* = EF-EV EV
Electron Concentration in degenerately doped n-type semiconductors The donors are fully ionized: no = ND The holes still follow the Boltz. approx. since EF-EV>>>k. T po = NV exp[-(EF-EV)/k. T] = NV exp[-(Eg*)/k. T] = NV exp[-(Ego- DEg*)/k. T] = NV exp[-Ego/k. T]exp[DEg*)/k. T] nopo = NDNVexp[-Ego/k. T] exp[DEg*)/k. T] = (ND/NC) NCNVexp[-Ego/k. T] exp[DEg*)/k. T] = (ND/NC)ni 2 exp[DEg*)/k. T]
Summary non-degenerate: nopo= ni 2 degenerate n-type: nopo= ni 2 (ND/NC) exp[DEg*)/k. T] degenerate p-type: nopo= ni 2 (NA/NV) exp[DEg*)/k. T]
- Non homogeneous differential equation
- Specific gravity
- Linear density fcc 111
- Linear atomic density
- Physiological density vs arithmetic density
- Nda full dac
- Why is arithmetic density also called crude density?
- Direct and indirect band gap semiconductors
- Semiconductor
- Quarternaries
- Elemental and compound semiconductors
- Carrier concentration formula
- Introduction to semiconductors
- International roadmap for semiconductors
- International technology roadmap
- Continuity equation in semiconductor
- Continuity equation of semiconductor
- Equation of continuity in semiconductors
- Semiconductor
- Konduktor isolator semikonduktor
- Equipement used for making semiconductors
- Continuity equation derivation in semiconductors
- Fundamentals of semiconductor devices anderson solution
- Carrier concentration equation
- The physics of semiconductors
- σsk
- Semiconductor junction devices
- Quasi fermi level
- International roadmap for semiconductors
- Bank of america semiconductors
- Types of semiconductor in physics
- What is semiconductor