# ELEG240 Review Equations ELEG240 Review Equations

Microscopic Ohm’s Law:

, or

Electrostatics:

Integral form of Gauss’s LawDifferential form

(Maxwell 1st)

note: ε = εrε0

Relation between electric field and voltage

(one-dimensional)

Definition of capacitance:

Q = CV(total charge on one plate equals capacitance x voltage)

Magnetostatics:

Magnetic field from current (Biot-Savert Law):

integration along entire current line

No “magnetic” charges (only magnetic loops):

(Maxwell 2nd)

Integral form of Ampere’s LawDifferential form (Maxwell 4th, static)

Electrodynamics (Maxell’s 3rd and 4th equations)

comes from , inductor formula

comes from adding term based on

Derivation of light (electromagnetic) wave equation:

Postulate Ex, By, only nonzero:

(forward and backward propagating)

(phasor notation)

=3x108 meters/second

Power flow of electromagnetic wave:

P = EB/μ0(power/area)

Light bending (Snell’s Law):

Lasers:

The electric field of a light wave passing through two partial mirrors is

,

where R is the power reflectivity of either mirror. By setting the denominator equal to zero, via a negative imaginary component of the refractive index, the ratio goes to infinity, and we have lasing, which results in two conditions necessary,

Magnetic vector potential (antenna generation):

(Un)Phased array antenna:

Power maximum when

Phased array antenna:

E1(t) → E1(t-T)

E2(t) → E2(t+T)

Power maximum when

Diffration gratings:

Photons:

E = hf

p = h/λ

Electrons as waves (quantum mechanics):

λe = h/p = h/(mv)

Wavefunction:

Ψ = ψ(x,y,z)w(t)( probability of electron in dxdydz)

w(t) is always e-iωt

Schrodinger’s equation for time-independent wavefunction:

If U=0, in one dimension,

Free electron theory of metals:

Leads to construction in k-space of discrete states spaced by π/L, or in other words, the k-space volume per state is (π/L)3

2 electrons per k-state, leads to Fermi energy, or highest energy electron in metal:

At zero T, states filled to , empty above; at non-zero T,

Work function energy is the difference between the Fermi energy and the energy to get just outside the solid, or the energy it takes to lift an electron out of the solid.

Molecular and crystalline compound formation:

Covalently bonded compounds are formed by adding different atoms so that group numbers add to 8 or multiples of 8. When compounds are formed, atom with higher group number gets more of the shared electronic charge, resulting in higher ionicity. Group IV materials such as carbon, germanium, and silicon can form an elemental (they are the only type of atom) covalently bonded solid since they add to 8. Hence only group IV elements can be non-metallic.

Band gaps:

For crystalline materials, the periodic (in space) potential energy due to the regular lattice of atoms leads to there being no solution for Schrodinger’s equation for certain ranges of energy. When the appropriate wavefunctions are plugged into Schrodinger’s equation with a periodic potential, in one dimension, the following equation results:

,

where in the E vs. k relationship now there are gaps in E. Near k=0,

Thus the three main effects of electron (and hole) dynamics in a crystal:

1. Band gaps.
2. Electron moves under field with an effective mass.

And

Indirect and direct band gaps:

3.Band gaps may be direct (minimum of conduction band at same k as maximum of valence band) or indirect (minimum of conduction band at different k as maximum of valence band). This is important for optoelectronics, since the k of a photon is nearly zero compared to the k of an electron, and thus indirect materials cannot emit light, and inefficiently absorb light.

p and n-type semiconductors:

If a dopant atom is placed in a crystal of different group number, it either has too many or too few valence electrons for the bond. For example, a group V atom in silicon such as As has 5 valence electrons and so one too many, and one can get free and conduct. A group III atom in silicon would act as an acceptor of electrons from the valence band, and thus would leave a hole behind, that can act like a positively charged carrier. Electrons: n type; holes: p-type.

The (symmetric) pn diode: electron current=hole current, total is

, , , W=light power

is essentially temperature independent. Note that

= 0.026 eV

= 0.026 volts

This equation is modified when forward bias injected carriers recombine in the junction:

(when recombination occurs in junction, such as in LED)

Recombination of electrons and holes can produce heat or light (photons). The photon energy is roughly the band gap.

Asymmetric pn diodes:

p-n+:

, where

p+n-:

Bipolar transistors:

n+p-n or p+n-p (emitter-base-collector) double junction (two types). The asymmetric current flow leads to more emitter current than base current, hence current gain.

Heterojunction bipolar transistors:

Achieves asymmetric current flow across base-emitter junction via difference in band gap, allowing doping to be high in base for high speed:

(same doping in emitter and base)

MOSFETs:

A variable conducting channel (channel runs from drain to source) depending upon the gate voltage. With a resistive load on the channel, leads to voltage gain.

[charge/(gate area)]

Note

, go back to chapter 1 to get channel resistance=.