Physics Feud 2

Direct semiconductor

Conduction band minimum and valence band maximum are at the same location in k-space

indirect semiconductor

Conduction band minimum and valence band maximum are at different locations in k-space

Photoexcitation of electrons

• A photon with an energy greater than E_g can excite can electron from the VB to the CB.

• When a photon breaks a Si-Si bond, a free electron and a hole in the Si-Si bond are created.

Direct Photon Absorption

• absorption of photon which excites electrons from valence band to

  conduction band and generates electron-hole pairs

• photon energy is same as band gap

• No change in momentum (k_photon = Δk ≈ 0)

Indirect Absorption

• absorption of photon assisted by absorption/emission of phonon that excites electrons from valence band to conduction band and generates electron-hole pairs – satisfying momentum and energy conservation laws.

• has to have phonon for conservation of momentum

Thermal generation of electron hole pairs

• atoms are constantly vibration above 0 K

• Some of the thermal vibrations of atoms may break bonds, thus create electron-hole pairs

Hole

• corresponds to empty valence band wavefunction

• collective motion of all other electrons

• effective positive charge

• move due thermal energy activation

Recombination

• Hole in valence band meets electron in conduction band and electron fills hole site

• excess energy may be released as photon or lost as lattice vibrations

What happens to the system energy as a hole moves downward?

The energy increases

General formula for force on a crystal

How can electron motion be interpreted as hole motion?

An unbalanced electron moving left is equivalent to a hole moving right

Expression for effective electron mass

The effect of lattice-lattice interactions are wrapped into into ∇kε(k), so now apart of the effective mass

What does a thinner band parabola represent?

A smaller effective electron mass

Energy expression using semiconductors

What statistics do we use for semiconductors

Boltzmann, electrons act like classical particles in semiconductors

Why can we assume electrons in a semiconductor behave classically?

non zero energy in the DoS (conduction band and valence band) is very far from fermi level, so the fermi dirac distribution can be approximated by boltzmann stats

Electron concentration in conduction band formula (intrinsic)

Effective density of states at conduction band edge (intrinsic)

Hole concentration in valence band formula (intrinsic)

Effective density of states at valence band edge (intrinsic)

Mass action Law

When can we use the mass action law?

• At thermal equilibrium

• dark (no illumination)

• applies to both intrinsic and extrinsic semiconductors

Fermi Energy for intrinsic semiconductors

Fermi Energy for intrinsic semiconductors if if Nc = Nv, or m_e* = m_h*

In the middle of the band gap

Where is Fermi Energy for intrinsic semiconductors if m_e* < m_h*

Above the middle of the band gap

Where is Fermi Energy for intrinsic semiconductors if m_e* > m_h*

Below the middle of the band gap

Fermi level for n-type semiconductor

closer to the conduction band

Fermi level for p-type semiconductor

closer to the valence band

Elements that are electron acceptors (p-type dopants)

Group 13 elements (boron, aluminum, gallium, indium…)

Elements that are electron donors (n-type dopants)

Group 15 elements (phosphorous, arsenic, antimony…)

N-type doping

• donor energy levels are just below E_c around ionized donor sites

• Thermal excitation at room temperature can ionize donors and donates a free electron in conduction band

• Donor atom concentration is often much higher than n_i so thus n ≈

  N_d

• hole concentration is really small

Assumptions we can make with n-type doping

• At RT, donors are fully ionized: Nd ≈ Nd⁺

• Heavy doping: Nd ≫ ni

General expression for concentration of Occupied Donors:

g is typically 2 for donors

General Expression for Concentration of Ionized Donors at T:

g is typically 2 for donors

P-type doping

• Acceptor energy levels just above E_v around ionized acceptor sites

• Thermal excitation at room temp can excite electron into acceptor level, and create a free hole in valence band

• acceptor atom concentration often much higher than p_i,so p ≈ N_a

• electron concentration in CB is small

General expression for concentration of Occupied Acceptors:

usually g = 4 for acceptors

General Expression for Concentration of Ionized Donors at T:

usually g = 4 for acceptors

Compensation doping

doping semiconductor with both donors and acceptors

Majority carrier for n-type compensation doping

minority carrier for n-type compensation doping

Majority carrier for p-type compensation doping

minority carrier for p-type compensation doping

General expressions for compensation doping

Degenerate semiconductors

• semiconductor is excessively doped, where n or p is comparable or greater than Nc or N_v

• Becomes more metal-like than semiconductor like

• larger number of donors/acceptors form band overlapping with CB/VB

• No longer valid: (i) Mass action law; (ii) Boltzmann approximation

• Need to use: (i) Fermi-Dirac statistics; (ii) Sommerfeld model.

Conductivity of semiconductor

Current density of a semiconductor

Electron and hole drift velocities

Electron and hole drift mobilities

n-type conductivity

p-type conductivity

Temperature Dependence of conductivity plot

behavior below saturation temperature (n-type)

electron concentration is controlled by the ionization of the donors (via thermal excitation of electron from E_d to E_c).

n ≈ N_d⁺

behavior between saturation temperature and intrinsic temperature (n-type)

electron concentration is equal to donor concentration due to full donor ionization

n ≈ N_d⁺ ≈ N_d

behavior above intrinsic temperature (n-type)

thermally generated electrons from VB to CB exceed the number of electrons from ionized donors, semiconductor behaves as if its intrinsic

n_i >>N_d

Temperature dependence of electron concentration (n-tyoe)

electron concentration for high temp n-type semiconductor

The doped semiconductor behave like intrinsic semiconductor.

Fermi level for high temp n-type semiconductor

The carrier concentration is fully controlled by the doping concentration.

electron concentration for room temp n-type semiconductor

The carrier concentration is fully controlled by the doping concentration.

electron concentration for low temp n-type semiconductor

The doped semiconductor behave as if E_d is the “valence band”

Fermi level for high temp n-type semiconductor

2 sources of temp dependence drift mobility

1. Scattering of electrons by lattice vibrations at high T

2. scattering of electrons by ionized impurities at low T (columbic potenital from ion scatters electrons)

Lattice Scattering Limited mobility

• Effective at high temps

• higher temps increase scattering cross section (S) which reduces drift mobility

Ionized impurity scattering limited mobility

• Effective at lower temperature

• higher temps reduce scattering cross sections which increases drift mobility

Effective mobility

Hall coefficient for holes only

What must the net force applied to each electron or hole must be equal to under steady state in the hall effect

0

Formula for retarding force in hall effect

what should the current along the y direction be equal to in the hall effect at steady state

0

Ambipolar hall coefficient

Effect low level photo injection has on charge carriers in n-type

• does not significantly affect majority n_n

• but it drastically affects the minority p_n

What determines the recombination time of excess carriers during photo injection

Minority carrier lifetime

Excess Minority Carrier Concentration in n-type semiconductor =

what does a short Minority carrier lifetime mean?

fasts switching pn junction

what does a long Minority carrier lifetime mean?

persistent luminescence

Stages in transient photoconductivity

1. Thermal Eq: carriers and lattices are interacted with one another ONLY through thermal excitation for long enough time, homogenous fermi level.

2. Transient: excess carriers are generated (for example, under illumination), and evolve with time.

3. Steady State: the external excitation is applied for long enough time until the excess carriers no longer change with time. It is still a non-equilibrium state, and the Fermi energy is not constant throughout the system.

Equation for steady state photoconductivity

η is quantum efficiency (# of electrons generated per incident photon)

Metal Metal Junctions

• electrons from higher lvl tunnel to metal with lower fermi lvl

• tunneling occurs until the fermi lvls across both metals are equal

• at equilibrium, there is a contact potential across the interface

Band bending under E field for electrons

• drift opposite direction of electric field

• As V(x) decreases, electron PE increases

Band bending under E field for holes

• drift same direction of electric field

• As V(x) increases, hole PE decreases

Flux density

Fick’s first law

electrons and holes diffuse from high-concentration regions to low-concentration regions.

Diffusion current density

Electron Diffusion and electric current

in opposite directions

Hole Diffusion and electric current

in same direction

Einstein Relations

Total current density

diffusion and drift directions for photo excitation at the left side of semiconductor

Electron diffusion (flux): to the right

Hole diffusion (flux): to the right

Electron diffusion current: to the left

Hole diffusion current: to the left

diffusion and drift directions for photo excitation at the left side of semiconductor with E field left to right

Electron drift (flux): to the left

Hole drift (flux): to the right

Electron drift current: to the right

Hole drift current: to the right

Ideal PN Junction

band bends and causes contact potential with e field pointing towards p side

Transport of electrons and holes in p-n junction

Electron Diffusion: to the left

Electron drift: to the right

Hole diffusion: to the right

Hole drift: to the left

No net current

Continuity Eq for holes in n-type

Continuity Eq for electrons in p-type

Photoconductor with surface photogeneration

Assumptions:

(1) Infinitely long semiconductor

(2) Steady states: ∂/∂t = 0

(3) Negligible electric field: Ex ~ 0

(4) Surface photogeneration: Gph ≠ 0 only when x = 0

Diffusion length

The mean distance diffused by the excess carriers before recombination.

Bias graph for pn diode

Space charge region

• region near a p-n junction depleted of mobile carriers, leaving behind fixed charged ions.

• contains net charge from immobile ions

• creates the electric field

Built in Potential/Voltage from SCR

Forward bias on PN junction

• E_Fp shifts down away from E_Fn

• p and n side bands shift relative (closer) to each other (less bending)

• built in potential and depletion width decreases

• net current from p to n

hole and electron drift and diffusion for forward bias of PN diode

• Electron Diffusion: Larger and to the left (less of a barrier)

• Electron Drift: Smaller and to the right (because potential is smaller)

• Hole diffusion: Larger and to the right

• Hole Drift: Smaller and to the left

• Net current: to the right (n side)

Reverse bias on PN junction

• E_Fp shifts up away from E_Fn

• p and n side bands shift relative (away) to each other (more bending)

• built in potential and depletion width increases

• net current from n to p