3B5 General

Photon energy equation?

De Broglie relations?

Derivation of de Broglie relation?

When does constructive interference occur?

When path difference is nλ —n is integer

Energy, momentum and wavelength of photons vs electrons?

Alternative expression for kinetic energy of electron?

What is the purpose of the Schrodinger equation?

can determine the correct shape of the wavefunction in the presence of external forces or potentials as a function of time

How does an equilibrium band diagram differ?

Fermi level is constant (straight line)

Equilibrium contact potential equation?

Total width of junction equation?

Why does depletion region in p-n junction essentially act as capacitor?

depletion region contains NO mobile carriers

therefore insulating

therefore acts as capacitor

What is forward bias of p-n junction?

+ve voltage to p-side

What happens when you apply +ve voltage to p-side of p-n junction?

Energy of n-side electrons increases

Energy of p-side electrons decreases

E_F lower on p-side

depletion width, barrier, peak field reduced

capacitance increased (because of depletion width)

What happens when you apply -ve voltage to p-side of p-n junction?

Energy of p-side electrons increases

Energy of n-side electrons decreases

E_F higher on p-side

depletion width, barrier, peak field increased (easier to go p-side to n-side than n-side to p-side now)

capacitance decreased (because of depletion width increase)

Visual representation of why depletion width is wider on lower-doped side of p-n junction?

Wider on the lower-doped side because fewer charges, so depletion region extends further into that side to balance the same total charge from other side

Condition for Schottky Barrier for n-type semiconductor in contact with metal?

And all other conditions that follow?

n-type & metal

Φ_m > Φ_SC → Schottky Barrier

Φ_m < Φ_SC → Ohmic Contact

p-type & metal

Φ_m < Φ_SC → Schottky Barrier

Φ_m > Φ_SC → Ohmic Contact

What is Φ measured between?

Φ = E_vac - E_F

What is affinity, X, measured between?

X = E_vac - E_C

What’s a heterojunction?

interface between two layers or regions of different semiconductors

Heterojunction band diagram?

pn junction band diagram?

Heterojunction E_F assumptions?

p+ : E_F - E_V ≈ 0

∴ Φ ≈ E_vac - E_V

n : E_C - E_F ≈ 0

∴ Φ ≈ X

What is the aim of n-type doping?

increase number of electrons in CB (conduction band)

Which group do we add as donor dopants to n-dope a group IV element (e.g. silicon)?

And examples of common dopants?

group V (5 electrons—1 extra electron)

• phosphorous

• arsenic

• antimony

Which group do we add as acceptor dopants to p-dope a group IV element (e.g. silicon)?

And examples of common dopants?

group III (3 valence electrons—1 extra hole)

• boron

• aluminium

• gallium

Equation relating dopant densities and number of electrons?

n = N_D - N_A

E_V in p-type semiconductor?

E_V = 0 eV

(defined for convenience)

p-type vs n-type Schottky barrier?

Equation for energy of electron in a state?

What energy does the equation E=eV represent?

Energy gained by electron when accelerated across a potential difference

Experimental evidence for Einstein postulate E = hv?

Photoelectric effect

Max energy of electrons emitted from metal surface in vacuum linearly increases with photon frequency, suggesting that E = hv

[increasing intensity linearly increases number of photons per second, hence number of electrons emitted per second]

Experimental evidence for de Broglie postulate, λ = h/p?

Electron diffraction

Beam of electrons incident on regular crystal structure metal sample, diffraction pattern with peaks formed when constructive interference due to this angle:

nλ = 2 d sinθ

with λ = h/p

Probability postulate for Schrodinger’s equation?

Probability of finding the particle is |Ψ|²

Total integrated probability over all space is 1

∴ ∫₀^L |Ψ|² dx = 1

Equation for energy of electron in one electron system?

Assumptions of free electron model?

• Electrons in the solid behave like a gas of free particles (free to move)

• Removal of valence electron to be free carrier leaves behind a positively charged ionic core—charge density of these ionic cores is spread uniformly through the solid (with potential V₀)

• No interaction between free electrons

E-k equation?

And when can you use this?

Only for 1D

(also you can use with m*)

What do the spherical shells in k-space represent?

Wavevectors with the same magnitude and ∴ same energy

Same wavenumber (k) —2 electrons at each point

Find number of states between k and (k + dk) with |k|?

Number of states in volume × volume of shell

How to derive g(E) dE?

Why is free electron model good for metals but not other crystals?

In a non-metal crystal, when will the electron propagate as a travelling wave and when will it not?

• λ doesn’t meet Bragg reflection criteria:

  -travelling wave

  -solution from free electron theory will apply to a first approximation

• λ does meet Bragg reflection criteria

  -wave becomes standing wave ‘locked on’ to periodicity of the lattice

  -electron can’t propagate as travelling wave through crystal

  -solution from free electron theory will not apply

For which wavenumbers does Bragg reflection occur?

What is Bloch’s theorem? How does this change the E-k diagram (Brillouin zones)

For a periodic potential, V(x) = V(x + Na)

Solution to the TISE is a plane wave multiplied by a function with the same periodicity as the lattice: Ψ = 𝐴 exp(j𝑘x) u(x)

Impossible to distinguish between electron with wavenumber k and one with wavenumber k+n2π/a

∴ can move any point on E–k diagram by ±n2π/a without changing the physical significance

Can condense all information in E–k diagram into 1st Brillouin zone

Equation for effective mass of electron?

Where is the intrinsic Fermi level?

Intrinsic material:

halfway

Derive intrinsic Fermi level, derive law of mass action, derive Nc and Nv

Equations for how doping changes the Fermi level in p-type and n-type?

Will you get up and read 2023 Q1 crib right now?

Yes

What is N and how do you find it?

N = states occupied

What is n and how do you find it?

n = number density of free electrons

How to find probability that a state is occupied when we have a limited number of states due to the Pauli exclusion principle?

Fermi-Dirac distribution

When can Boltzmann approximation to the Fermi-Dirac be used?

E >> E_F

OR

E_C - E_F >> k_BT

Probability that a state is occupied is very low → finite number of states is no longer the factor limiting occupation

For which semiconductors is direct recombination most likely?

Semiconductors with direct band gaps

For indirect gap: electron must lose energy and dissipate momentum (the creation of a phonon)

What does each part of the continuity equation represent?

Haynes-Shockley experiment?

Pulse of holes (minority carriers) created in n-type bar (e.g. laser)

Electric field applied

Pulse drifts in field and spreads out by diffusion

Drift mobility of holes = from run time

Diffusion coefficient = from pulse broadening

Hole lifetime = from total decay of carriers

Haynes-Shockley gives minority carrier mobility

Hall effect gives majority carrier mobility

Drift and diffusion for majority/ minority carriers?

DRIFT = only for majority

DIFFUSION = minority carriers have great effect

Reason why E_F might be above midway point in intrinsic semiconductor?

N_V > N_C

In n-type and p-type doping, where are the extra electrons and holes desired?

n-type: increase electrons in CB

p-type: increase holes in VB

What is electron affinity?

energy required by electron at bottom of CB to escape from semiconductor into vacuum

In n-type Schottky barrier diode, which barrier is fixed, and what is its value?

Which barrier varies, how does it vary, and what is its value?

Fixed barrier: e(ϕ_m - X_sc)

electron flow from M to SC

Variable barrier: e(ϕ_m - ϕ_sc)

electron flow from SC to M

varies with applied bias

Draw Schottky barrier band diagram?

And way to remember?

And metal and semiconductor band diagrams separately?

SC pulled down ↓ (yoink)

N-type Ohmic contact band diagram?

And way to remember?

SC pulled up ↑(yank)

N-type Schottky barrier diode with forward and reverse bias?

Reverse: barrier for electrons from SC to M is higher → e(V₀ + V_rev)

NMOS vs PMOS?

NMOS: electrons carry current between S & D

PMOS: holes carry current between S & D

How does MOSFET work?

• Voltage applied to gate

• Induces Electric field in gate insulator

• ^ extends into S-D channel

• Changes in this E field cause modulation in resistivity of semiconductor underneath gate insulator

• ∴ V on gate can control current between S & D

What is MOSFET depletion mode?

transistor normally ON

current between S & D

until V (of appropriate magnitude) turns it off

[like JFET—typically in depletion mode, but MOSFET can do both. get you a girl who can do both]

What is MOSFET enhancement mode?

transistor normally OFF

no current between S & D

until sufficiently large threshold voltage applied to gate

Is enhancement or depletion MOSFET better for energy saving?

normally OFF i.e. enhancement

Band diagram of n-type

Lorentz force equation?

And hand rule?

Hall effect explanation in p-type semiconductor?

And derive Hall coefficient?

Hall effect explanation in n-type semiconductor?

And derive Hall coefficient?

What is the Laplacian, ∇², equal to?

[in maths DB]

Important to remember when differentiating TISE solution?

j comes down to front!!

Quick equation to find (dk what to call it but example 9.A eqn)

Equation for electrostatic potential difference between 2 side of pn junction (i.e. contact potential)?

Prove the law of mass action?

Prove intrinsic Fermi level position?

Equation to find Fermi level with respect to intrinsic Fermi level using n_i (for n-type and p-type)?

Equation for effective density of states in CB?

Equation for effective density of states in VB?

Units of mobility?

m² V^-1 s^-1

How to determine direction of Hall voltage?

Find direction of F_y

holes/ electrons build up at surface that F_y points to

E_y in +ve to -ve direction (towards e, away from h)

V_H in opposite direction to E_y (voltage always decreases in applied E field direction)

Groups III, IV & V of periodic table?

Semiconductor Arrhenius plot regions and explanations?

• 0K: no carrier in any band, no ionised dopants —behaves like insulator

• Small temp rise: donor impurities begin to ionise (some electrons in CB)

• Flat region: all donors ionised, saturation region. Extrinsic conduction dominates

• High temps: sufficient thermal energy for electrons VB → CB. Intrinsic conduction dominates

Bandgap and Arrhenius plot relationship?

Intrinsic region (left hand side): slope α E_g

Is de Broglie for photon or electron?

TRICK QUESTION! It’s both ;)

Band diagram and Fermi level of metal?

3 Haynes-Shockley equations?

What is ρ in the Poisson equation?

And what is it equal to?

charge density due to ionised acceptors on p-side: -e N_A

due to ionised donors on n-side: e N_D

What does N_D>>n_i imply?

heavy n doping

Have you reviewed 2024 Q1?

yes, ma’am

Band diagram when there’s an internal electric field but no current flow (as drift is balanced by diffusion)?

And why would there be an internal electric field?

N_D varies with position

How to do drift and diffusion current balance (both holes and electrons)?

How to draw E-field variation for Schottky/ Ohmic for both n- and p-type?

summary: triangle on bending side

also gradient on non-bending side

up triangle if yank (up)

down triangle if yoink (down)

Band diagram and depletion width of forward-biased p-n junction?

Band diagram and depletion width of reverse-biased p-n junction?

Frequency of LED light at T = 0 K vs at T > 0 K ?

T=0K : LED would emit light with frequency corresponding to the band gap: hf=Eg

No thermal energy is available to excite electrons to CB

• Electrons recombine exactly across the band gap

• All photons have the same frequency

T>0K : thermal fluctuations broaden the spectral width of the emitted light

• Thermal energy is available: electrons and holes gain random K.E

• Don’t always recombine exactly at CB boundary

• Emitted light isn't a single frequency anymore — it’s broadened into a small range

Frequency uncertainty is Δf= E / h

Equation for group velocity?

VB at 0K?

Fully occupied

Electrons want to be in lowest possible energy states because no thermal energy to excite them

No electrons have enough energy to jump across bandgap