Presentation on theme: "ECSE-6230 Semiconductor Devices and Models I Lecture 4"— Presentation transcript:
1 ECSE-6230 Semiconductor Devices and Models I Lecture 4 Prof. Shayla SawyerBldg. CII, Rooms 8225Rensselaer Polytechnic InstituteTroy, NYTel. (518)Fax. (518)March 22, 2017March 22, 20171
2 Outline Carrier Concentration at Thermal Equilibrium IntroductionFermi Dirac StatisticsDonors and AcceptorsDetermination of Fermi LevelDopant CompensationMarch 22, 2017March 22, 20172
3 Carrier Concentration Introduction One of most important properties of a semiconductor is that it can be doped with different types and concentrations of impuritiesIntrinsic material-no impurities or lattice defectsExtrinsic-doping, purposely adding impuritiesN-type mostly electronsP-type mostly holesMarch 22, 2017March 22, 2017333
4 Carrier Concentration Introduction To calculate semiconductor electrical properties, you must know the number of charge carriers per cm3 of the materialMust investigate distribution of carriers over the available energy statesStatistics are needed to do soFermi-Dirac statisticsDistribution of electrons over a range of allowed energy levels at thermal equilibriumMarch 22, 2017March 22, 2017444
5 Fermi-Dirac Distribution Probability that an available energystate at E will be occupied by anelectron at absolute temperature TMathematically,EF (Fermi Energy) is the energy at which f(E) = 1/2The transition region in (E - EF)from f(E) =1 to f(E) = 0 is within3 k T.When T 0,E is discontinous at E = EF.March 22, 20175
6 Fermi-Dirac Distribution To apply the Fermi-Dirac distribution, we must recall that f(E) is the probability of occupancy of an available state at E.Where can we find available states?March 22, 20176
7 Carrier Concentration At Thermal Equilbrium Number of electrons (occupied conduction band levels) given by:Density of states g(E) can be approximated by the density near the bottom of the conduction bandMC is the number of equiv. minimawhereMarch 22, 20177
8 Carrier Concentration At Thermal Equlibrium The integral can be evaluated asFor the valence band, consider light and heavy holes for the density of states effective mass for holes (mdh)and use similar equationWhere NC is the effective density of states in the conduction band given by:March 22, 20178
9 Carrier Concentration At Thermal Equlibrium: Intrinsic For intrinsic material lies at some intrinsic level Ei near the middle of the band gap, electron and hole concentrations areLaw of mass action: product of maj. and min. carriers is fixedMarch 22, 20179
10 email@example.com www.rpi.edu/~sawyes/courses.html Donors and AcceptorsDoping by substituting Si atoms with Column III or V of the Periodic Table.Very dilute doping level, typical 1014 to 1018 cm-3, results in discrete energy levels.Donor level is neutral if filled with e-,positively charged if empty.e.g., P, As, and Sb in Si.Acceptor level is neutral if empty,negatively charged if filled with e-.e.g., B and Al in Si.
11 firstname.lastname@example.org www.rpi.edu/~sawyes/courses.html Donors and Acceptors“Hydrogen-like” Model to describe dopant atom ionization.Hydrogen AtomGround state (n=1) ionization energy of hydrogen is eV.To estimate ionization energy of donors, replace m0 with m* and0 and S (e.g., 11.70 for Si).ED = (0 /S )2 ( m*/ m0 ) EH ~ eV for Ge,0.025 eV for Si,0.007 eV for GaAsEA ~ eV for Ge,0.05 eV for Si,0.05 eV for GaAskT~0.026eVComparable to thermal energies so ionizationis complete at room temperature
14 Determination of Fermi Level Intrinsic Semiconductor - EF ~ Eg / 2Extrinsic Semiconductor - EF adjusted to preservespace charge neutralitySpace Charge Neutralityn0 + NA- = ND+ + p0Total Neg. Charges = Total Positive Chargeselectrons and ionized acceptors=holes and ionized donors100% ionization assumed.Ionized Concentration of DonorsWhen impurities are introduced:where gD is the ground state degeneracy of donor impuritygD = 2 (i) electrons with either spin(ii) no electrons at all
15 Determination of Fermi Level Ionized Acceptorswhere gA is the ground state degeneracy of acceptor impuritygA = 4 for Ge, Si, and GaAs because(i) Acceptor levels can receive electrons with either spin and(ii) Valence band double degeneracy.Space Charge NeutralityN-type Semiconductor is assumed.n=ND++p ~ ND+ therefore
16 email@example.com www.rpi.edu/~sawyes/courses.html Charge NeutralitySince the material must balance electrostatically, the Fermi level must adjust such that charge neutrality remains.The Fermi level therefore can be calculated for a set given ND, ED, NC, and TMarch 22, 20171616
17 Graphical Determination of Fermi Level Graphical solution of the space charge neutrality equation.No need to assume 100% ionization.Need to know the donor (or acceptor) level.Degenerate DopingImpurity levels are broadened intoimpurity bands, thus reducing theionization energy of the dopants.Ex. Phosphorus in Silicon.EC - ED(ND) = x10-8 (ND)1/3[eV] for ND > 1018 cm-3
18 firstname.lastname@example.org www.rpi.edu/~sawyes/courses.html Dopant CompensationWhen both n- and p-type (donor and acceptor) impurities are present,the space charge neutrality condition n0 + NA- = ND+ + p0holds, even when the impurities are deep levels.In an n-type semiconductor where ND>>>NAFermi level can be obtained from
19 email@example.com www.rpi.edu/~sawyes/courses.html Dopant CompensationIn an p-type semiconductor where NA>>>NDFermi level can be obtained from
20 firstname.lastname@example.org www.rpi.edu/~sawyes/courses.html Example ProblemA hypothetical semiconductor has an intrinsic carrier concentration of 1.0 x 1010 cm-3 at 300 K, it has a conduction and valence band effective density of states NC and NV both equal to 1019 cm-3.What is the band gap Eg?If the semiconductor is doped with Nd = 1x1016 donors/cm3 , what are the equilibrium electron and hole concentrations at 300K?
21 email@example.com www.rpi.edu/~sawyes/courses.html Example ProblemA hypothetical semiconductor has an intrinsic carrier concentration of 1.0 x 1010 cm-3 at 300 K, it has a conduction and valence band effective density of states NC and NV both equal to 1019 cm-3.c) If the same piece of semiconductor, already having Nd = 1x1016 donors/cm3, is also doped with Na= 2x1016 acceptors/cm3 , what are the new equiliblrium electron and hole concentrations at 300 K?Consistent with your answer to part (c), what is the Fermi level position with respect to the intrinsic Fermi level, EF – Ei?