Carrier Concentration in Equilibrium Carrier Concentrations in Equilibrium
Carrier Concentration in Equilibrium
Carrier Concentrations in Equilibrium q Developing the Mathematical model for Electrons and Holes Motivation Ø Since current (electron and hole flow) is dependent on the concentration of electrons and holes in the material, we need to develop equations that describe these concentrations. Ø Furthermore, we will find it useful to relate these concentrations to the average energy (Fermi energy) in the material.
Developing the Mathematical Model for Electrons and Holes units of n and p are [ #/cm 3] q The Density of Electrons is: Probability the state is filled Number of states per cm-3 in energy range d. E q The Density of Hole is: Probability the state is empty Number of states per cm-3 in energy range d. E
How do Electrons and Holes Populate the Bands? q Density of States Concept Quantum Mechanics tells us that the number of available states in a cm 3 per unit of energy, the density of states, is given by: Density of States in Conduction Band Density of States in Valence Band
Developing the Mathematical Model for Electrons and Holes Effective density of states in CB This is known as the Fermi-Dirac integral of order 1/2 or, F 1/2 ( c )
Developing the Mathematical Model for Electrons and Holes q We can further define: Effective Density of States in Conduction Band Effective Density of States in Valence Band q This is a general relationship holding for all materials and results in: at 300 K
Developing the Mathematical Model for Electrons and Holes Fermi-Dirac integrals can be numerically determined or read from tables or. . .
Developing the Mathematical Model for Electrons and Holes q Useful approximations to the Fermi-Dirac integral:
Developing the Mathematical Model for Electrons and Holes q Useful approximations to the Fermi-Dirac integral: Nondegenerate Case
Alternative Expressions for n and p q Although in close form, the follow n and p relationships are not in the simplest form possible. q It in needed to simper form for device analysis. q The alternative-form relationship of n and p can be obtained by recalling that Ei. , the Fermi level for an intrinsic semiconductor, lies close to midgap.
Developing the Mathematical Model for Electrons and Holes q Useful approximations to the Fermi-Dirac integral: When n = ni , Ef = Ei [intrinsic energy], then or and or
Developing the Mathematical Model for Electrons and Holes q Other useful relationships: n p product: and Since and known as the Law of mass Action
Where is Ei ? q Since we started with descriptions of intrinsic materials then it makes sense to reference energies from the intrinsic energy, Ei. q Intrinsic Material: When n = ni , Ef = Ei [intrinsic energy], then or and or
Where is Ei ? q Since we started with descriptions of intrinsic materials then it makes sense to reference energies from the intrinsic energy, Ei. q Intrinsic Material: But, Letting Ev=0, this is Eg /2 or “Midgap” -0. 007 e. V for Si @ 300 K (0. 6% of Eg)
Where is Ei ? q Extrinsic Material: Solving for (Ef - Ei) for and
Where is Ei ? q Extrinsic Material: Note: The Fermi-level is pictured here for 2 separate cases: acceptor and donor doped.
Developing the Mathematical Model for Electrons and Holes q Charge Neutrality: Ø If excess charge existed within the semiconductor, random motion of charge would imply net (AC) current flow. Not possible! - charge Mobile Immobile + charge - charge Immobile Mobile + charge Ø Thus, all charges within the semiconductor must cancel.
Developing the Mathematical Model for Electrons and Holes q Charge Neutrality: Total Ionization case Ø NA¯ = Concentration of “ionized” acceptors = ~ NA Ø ND+ = Concentration of “ionized” Donors = ~ ND
Developing the Mathematical Model for Electrons and Holes q Charge Neutrality: Total Ionization case or and if and and
Developing the Mathematical Model for Electrons and Holes q Example An intrinsic Silicon wafer has 1 x 1010 cm-3 holes. When 1 x 1018 cm-3 donors are added, what is the new hole concentration? if and
Developing the Mathematical Model for Electrons and Holes q Example An intrinsic Silicon wafer has 1 x 1010 cm-3 holes. When 1 x 1018 cm-3 acceptors and 8 x 1017 cm-3 donors are added, what is the new hole concentration?
Developing the Mathematical Model for Electrons and Holes q Example An intrinsic Silicon wafer at 470 K has 1 x 1014 cm-3 holes. When 1 x 1014 cm-3 acceptors are added, what is the new electron and hole concentrations?
Developing the Mathematical Model for Electrons and Holes q Example An intrinsic Silicon wafer at 600 K has 4 x 1015 cm-3 holes. When 1 x 1014 cm-3 acceptors are added, what is the new electron and hole concentrations? X Intrinsic material at High Temperature
Developing the Mathematical Model for Electrons and Holes q Temperature behavior of Doped Materials At intrinsic temperature region At extrinsic temperature region Carrier concentration vs. inverse temperature for Si doped with 10 15 donors/cm 3.
Partial Ionization and Parameter Relationships q Partial Ionization Case degeneracy factor g. A = 4 for Si, Ga. As, Ge and g. D = 2 for Si, Ga. As, Ge and For 1014 cm-3 B in Si : NA¯ = 0. 9998·NA For 1017 cm-3 B in Si : NA¯ = 0. 88·NA For 1017 cm-3 P in Si : ND+ = 0. 94·ND most semiconductors
Charge Neutrality q Partial Ionization Case
What are the Degeneracy Factors Ø The degeneracy factors account for the possibility of electrons with different spin, occupying the same energy level (i. e. a true statement of the Pauli Exclusion principle is that no electron with the same quantum numbers (energy and spin) can occupy the same state). Ø g. D is then = 2 in most semiconductors. Ø g. A is 4 due to the above reason combined with the fact that there actually 2 valence bands in most semiconductors. Thus, 2 spins x 2 valance bands makes g. A = 4
Energy Band Diagram q The energy is plotted as a function of the wave number, k, along the main crystallographic directions in the crystal. Si Ge Ga. As The bottom axis describe different directions of the crystal.
Relationships between Parameters Ef p NA ¯ NA n N D+ ND
Developing the Mathematical Model for Electrons and Holes q Electron concentration vs. Temperature for two n-type doped semiconductors: Silicon doped with 1. 15 x 1016 arsenic atoms cm-3 Germanium doped with 7. 5 x 1015 arsenic atoms cm-3
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