Carrier Concentration in Equilibrium

 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.  Furthermore, we will find it useful to relate the these concentrations to the average energy ( Fermi energy ) in the material. Carrier Concentrations in Equilibrium  Developing the Mathematical model for Electrons and Holes  Motivation 

 The Density of Electrons is: Probability the state is filled Number of states per cm -3 in energy range dE Probability the state is empty Number of states per cm -3 in energy range dE units of n and p are [ #/cm 3 ]  The Density of Hole is: Developing the Mathematical Model for Electrons and Holes

How do Electrons and Holes Populate the Bands?  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

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 Effective density of states in CB

at 300 K  We can further define:  This is a general relationship holding for all materials and results in: Developing the Mathematical Model for Electrons and Holes Effective Density of States in Valence Band Effective Density of States in Conduction Band

Fermi-Dirac integrals can be numerically determined or read from tables or... Developing the Mathematical Model for Electrons and Holes

 Useful approximations to the Fermi-Dirac integral: Developing the Mathematical Model for Electrons and Holes

Nondegenerate Case  Useful approximations to the Fermi-Dirac integral: Developing the Mathematical Model for Electrons and Holes

Alternative Expressions for n and p  It in needed to simper form for device analysis.  The alternative-form relationship of n and p can be obtained by recalling that E i., the Fermi level for an intrinsic semiconductor, lies close to midgap.  Although in close form, the follow n and p relationships are not in the simplest form possible.

 Useful approximations to the Fermi-Dirac integral: When n = n i, E f = E i [intrinsic energy], then or and or Developing the Mathematical Model for Electrons and Holes and

 Other useful relationships: n  p product: known as the Law of mass Action and Since Developing the Mathematical Model for Electrons and Holes

Where is E i ?  Since we started with descriptions of intrinsic materials then it makes sense to reference energies from the intrinsic energy, E i.  Intrinsic Material: When n = n i, E f = E i [intrinsic energy], then or and or and

Where is E i ?  Since we started with descriptions of intrinsic materials then it makes sense to reference energies from the intrinsic energy, E i.  Intrinsic Material: But, Letting E v =0, this is E g /2 or “Midgap” eV for 300K ( 0.6 % of E g )

Where is E i ?  Extrinsic Material: Solving for (E f - E i ) andfor andfor

Where is E i ?  Extrinsic Material: Note: The Fermi-level is pictured here for 2 separate cases: acceptor and donor doped.

 If excess charge existed within the semiconductor, random motion of charge would imply net (AC) current flow.  Not possible!  Thus, all charges within the semiconductor must cancel.  Charge Neutrality: Mobile + charge  Immobile - charge  Immobile + charge  Mobile - charge  Developing the Mathematical Model for Electrons and Holes

 N A ¯ = Concentration of “ionized” acceptors = ~ N A  N D + = Concentration of “ionized” Donors= ~ N D  Charge Neutrality: Total Ionization case Developing the Mathematical Model for Electrons and Holes

and if and if or  Charge Neutrality: Total Ionization case and Developing the Mathematical Model for Electrons and Holes

 Example and if Developing the Mathematical Model for Electrons and Holes An intrinsic Silicon wafer has 1x10 10 cm -3 holes. When 1x10 18 cm -3 donors are added, what is the new hole concentration?

 Example Developing the Mathematical Model for Electrons and Holes An intrinsic Silicon wafer has 1x10 10 cm -3 holes. When 1x10 18 cm -3 acceptors and 8x10 17 cm -3 donors are added, what is the new hole concentration?

 Example Developing the Mathematical Model for Electrons and Holes An intrinsic Silicon wafer at 470K has 1x10 14 cm -3 holes. When 1x10 14 cm -3 acceptors are added, what is the new electron and hole concentrations?

 Example Developing the Mathematical Model for Electrons and Holes An intrinsic Silicon wafer at 600K has 4x10 15 cm -3 holes. When 1x10 14 cm -3 acceptors are added, what is the new electron and hole concentrations?  Intrinsic material at High Temperature X

 Temperature behavior of Doped Materials Developing the Mathematical Model for Electrons and Holes Carrier concentration vs. inverse temperature for Si doped with donors/cm 3. At extrinsic temperature region At intrinsic temperature region

Partial Ionization and Parameter Relationships  Partial Ionization Case g D = 2 for Si, GaAs, Ge and most semiconductors For cm -3 P in Si : N D + = 0.94 · N D g A = 4 for Si, GaAs, Ge and most semiconductors For cm -3 B in Si : N A ¯ = ·N A For cm -3 B in Si : N A ¯ = 0.88 · N A degeneracy factor

Charge Neutrality  Partial Ionization Case

 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).  g D is then = 2 in most semiconductors.  g A is 4 due to the above reason combined with the fact that there are actually 2 valence bands in most semiconductors. Thus, 2 spins x 2 valance bands makes g A = 4 What are the Degeneracy Factors

Energy Band Diagram The bottom axis describe different directions of the crystal. SiGeGaAs  The energy is plotted as a function of the wave number, k, along the main crystallographic directions in the crystal.

Relationships between Parameters EfEf pn NA¯NA¯ ND+ND+ NANA NDND

 Electron concentration vs. Temperature for two n-type doped semiconductors: Developing the Mathematical Model for Electrons and Holes Silicon doped with 1.15 x arsenic atoms cm -3 Germanium doped with 7.5 x arsenic atoms cm -3