EEE 352 Master Packet 1
EEE 352 Master Packet
Cheat Sheet (2 pages) — Identification Table — 80 Final-Exam Problems + Full Solutions +
Rubrics
2-Page Allowed Cheat Sheet (Compressed, Exam-Ready)
Goal: ultra-dense, minimal narrative, maximum formulas + triggers. Print-friendly.
Page 1 — Crystals → QM → Bands
Constants / quick conversions Uncertainty:
• k = 8.617 × 10−5 eV/K, kT ≈ 0.0259 eV at 300 K ℏ ℏ
∆x ∆p ≥ , ∆E ∆t ≥
• q = 1.602 × 10−19 C 2 2
• h = 6.626 × 10−34 J · s, ℏ = 2π
h
= 1.055 × 10−34 J · s
• 1240
hc ≈ 1240 eV · nm ⇒ λ(nm) = E(eV) Schrödinger (1D, time-independent):
ℏ2 d2 ψ
− + V (x)ψ = Eψ
A) Crystal structure essentials 2m dx2
Probability: |ψ|2 , normalization: |ψ|2 dx = 1.
R
Miller indices (planes): intercepts in units of a → re-
ciprocals → clear fractions → reduce. Infinite well (width L):
Plane: (hkl), direction: [uvw]. r
Cubic equivalence: (100) ≡ (010) ≡ (001) (permuta- n 2 π 2 ℏ2 2 nπx
En = , ψn = sin
tions/signs). 2mL2 L L
Plane spacing (cubic):
Scaling: E ∝ 1/L2 .
a Finite barrier (tunneling):
dhkl = √
h2 + k 2 + l2 r
2m(V0 − E)
Atomic density: α= , T ∼ e−2αL
ℏ2
Natoms per cell
N=
a3 C) Bands (why bandgaps exist)
Surface atom density:
Key assumption: periodic potential V (x + a) = V (x).
#atoms on plane Bloch theorem:
ρs =
plane area
ψk (x) = eikx uk (x), uk (x + a) = uk (x)
Orientation ⇒ anisotropy (etch/growth, dangling bonds,
surface energy).
BZ edge (1D): k = ±π/a.
Bragg (XRD):
From E(k):
nλ = 2d sin θ
−1
d2 E
1 dE ∗ 2
B) QM core tools v= , m =ℏ
ℏ dk dk 2
Photon: Direct vs indirect bandgap:
hc
E = hν = • Direct: CB min and VB max at same k ⇒ strong
λ
de Broglie: absorption/emission near Eg .
h • Indirect: phonon required ⇒ weak near-edge absorp-
λdB = tion/emission (Si).
p
Cheat Sheet + Identification Table + 80 Problems (Solutions + Rubrics)
,EEE 352 Master Packet 2
Page 2 — Statistics, Doping, Equilibrium, Opto & LEDs
D) Fermi-Dirac, DOS, carriers F) Conductivity / resistivity
Fermi-Dirac:
1 1 1
f (E) = , f (EF ) = σ = q(nµn + pµp ), ρ=
1 + e(E−EF )/kT 2 σ
Boltzmann approx (nondegenerate): |E − EF | ≳ 3kT :
f (E) ≈ e−(E−EF )/kT
Effective DOS: G) Equilibrium vs nonequilibrium
3/2 3/2
2πm∗n kT 2πm∗p kT
Nc = 2 , Nv = 2 Equilibrium rules:
h2 h2
Nondegenerate carriers: • One Fermi level everywhere: EF = constant
• Drift + diffusion cancel ⇒ net current = 0
n = Nc e−(Ec −EF )/kT , p = Nv e−(EF −Ev )/kT
Illumination/injection: quasi-Fermi levels EF n , EF p
Intrinsic: p
−Eg /(2kT ) (split).
ni = Nc Nv e
Mass action (equilibrium):
np = n2i
H) Optical + LEDs (ABC)
Intrinsic level:
∗
mp
Ec + Ev 3
Ei = + kT ln Absorption near edge:
2 4 m∗n
• Direct: strong at Eg
• Indirect: weak near Eg (phonon required)
E) Doping, ionization, neutrality
Recombination:
Fully ionized (typical at 300K in Si unless stated):
n2i R = AN + BN 2 + CN 3
n-type: n ≈ ND − NA , p=
n
n2i
p-type: p ≈ NA − ND , n= IQE:
p
Fermi shift (nondegenerate): BN 2
IQE =
AN + BN 2 + CN 3
n p
EF − Ei = kT ln = −kT ln
ni ni
Charge neutrality (general): Quantum well levels (infinite well approx):
n + NA− = p + ND
+
n2 π 2 ℏ2
Partial ionization (freeze-out): En =
2m∗ L2
+ ND
ND = , gD ≈ 2
1 + gD e(EF −ED )/kT Emission:
NA
NA− = , gA ≈ 4
1 + gA e(EA −EF )/kT 1240
Regimes: freeze-out / extrinsic / intrinsic. Eemit ≈ Eg + E1e + E1h , λ(nm) =
Eemit (eV)
Cheat Sheet + Identification Table + 80 Problems (Solutions + Rubrics)
, EEE 352 Master Packet 3
Pure Identification Table (Problem Trigger → What To Do Immediately)
If the prompt says / shows... Topic Do this immediately Key equations / rules
Intercepts → plane indices Miller indices Convert intercepts to a units → recip- (hkl) procedure
rocals → clear fractions
Need (010) but only Cubic symmetry Pick equivalent plane (100) (100) ≡ (010) ≡ (001)
(100)/(110)/(111) available √
Plane spacing request Crystallography Use cubic spacing formula dhkl = a/ h2 + k2 + l2
Surface atom density compare Surface density Count atoms on 2D surface cell / area ρs = #/area
planes
XRD peak at θ with λ Bragg/XRD Apply Bragg’s law to get d nλ = 2d sin θ
Why bandgap exists? Band theory Say periodic potential + Schrödinger Bloch theorem
Allowed/forbidden energies in Kronig–Penney Use | cos(ka)| ≤ 1 idea KP + zone boundaries
lattice
E–k parabola, heavier/lighter? Effective mass Flatter ⇒ m∗ larger m∗ = ℏ2 /(d2 E/dk2 )
Velocity direction from E–k Group velocity Use slope sign v = (1/ℏ)dE/dk
Weak absorption near Eg Indirect gap Conclude phonon-assisted direct vs indirect
Intrinsic sample / where is EF ? Intrinsic stats EF = Ei ; if m∗ similar ⇒ midgap Ei formula
√
Two materials same ni Bandgap inference If Nc , Nv similar ⇒ same Eg ni = Nc Nv e−Eg /2kT
Compute n, p from EF Carrier stats Use nondegenerate forms n = Nc e−(Ec −EF )/kT , p =
Nv e−(EF −Ev )/kT
Given doping at 300K Extrinsic doping Assume full ionization unless warned n ≈ ND − NA , p = n2i /n
+
Low T / dopant level given Freeze-out Use ND (EF ) + neutrality + neutrality + ionization
n(EF ), p(EF )
n(T ) flat Extrinsic regime n ≈ ND (donor-limited) extrinsic formulas
n(T ) rises fast Intrinsic regime n ≈ ni (T ) ni (T ) exponential
Band bending diagram, equilib- Equilibrium rule EF flat; drift cancels diffusion one EF , net J = 0
rium
Illuminated solar cell Nonequilibrium quasi-Fermi levels split E F n , EF p
Resistivity depends on what? Transport identify n, p, µ σ = q(nµn + pµp )
2 2 2
Quantum well energy levels Confinement infinite well if stated; E ∝ 1/L2 En = n2mπ∗ Lℏ2
Emission wavelength Opto device compute Eemit then λ λ = 1240/E
BN 2
LED efficiency / droop Recombination Use ABC model IQE = AN +BN 2 +CN 3
Cheat Sheet + Identification Table + 80 Problems (Solutions + Rubrics)
EEE 352 Master Packet
Cheat Sheet (2 pages) — Identification Table — 80 Final-Exam Problems + Full Solutions +
Rubrics
2-Page Allowed Cheat Sheet (Compressed, Exam-Ready)
Goal: ultra-dense, minimal narrative, maximum formulas + triggers. Print-friendly.
Page 1 — Crystals → QM → Bands
Constants / quick conversions Uncertainty:
• k = 8.617 × 10−5 eV/K, kT ≈ 0.0259 eV at 300 K ℏ ℏ
∆x ∆p ≥ , ∆E ∆t ≥
• q = 1.602 × 10−19 C 2 2
• h = 6.626 × 10−34 J · s, ℏ = 2π
h
= 1.055 × 10−34 J · s
• 1240
hc ≈ 1240 eV · nm ⇒ λ(nm) = E(eV) Schrödinger (1D, time-independent):
ℏ2 d2 ψ
− + V (x)ψ = Eψ
A) Crystal structure essentials 2m dx2
Probability: |ψ|2 , normalization: |ψ|2 dx = 1.
R
Miller indices (planes): intercepts in units of a → re-
ciprocals → clear fractions → reduce. Infinite well (width L):
Plane: (hkl), direction: [uvw]. r
Cubic equivalence: (100) ≡ (010) ≡ (001) (permuta- n 2 π 2 ℏ2 2 nπx
En = , ψn = sin
tions/signs). 2mL2 L L
Plane spacing (cubic):
Scaling: E ∝ 1/L2 .
a Finite barrier (tunneling):
dhkl = √
h2 + k 2 + l2 r
2m(V0 − E)
Atomic density: α= , T ∼ e−2αL
ℏ2
Natoms per cell
N=
a3 C) Bands (why bandgaps exist)
Surface atom density:
Key assumption: periodic potential V (x + a) = V (x).
#atoms on plane Bloch theorem:
ρs =
plane area
ψk (x) = eikx uk (x), uk (x + a) = uk (x)
Orientation ⇒ anisotropy (etch/growth, dangling bonds,
surface energy).
BZ edge (1D): k = ±π/a.
Bragg (XRD):
From E(k):
nλ = 2d sin θ
−1
d2 E
1 dE ∗ 2
B) QM core tools v= , m =ℏ
ℏ dk dk 2
Photon: Direct vs indirect bandgap:
hc
E = hν = • Direct: CB min and VB max at same k ⇒ strong
λ
de Broglie: absorption/emission near Eg .
h • Indirect: phonon required ⇒ weak near-edge absorp-
λdB = tion/emission (Si).
p
Cheat Sheet + Identification Table + 80 Problems (Solutions + Rubrics)
,EEE 352 Master Packet 2
Page 2 — Statistics, Doping, Equilibrium, Opto & LEDs
D) Fermi-Dirac, DOS, carriers F) Conductivity / resistivity
Fermi-Dirac:
1 1 1
f (E) = , f (EF ) = σ = q(nµn + pµp ), ρ=
1 + e(E−EF )/kT 2 σ
Boltzmann approx (nondegenerate): |E − EF | ≳ 3kT :
f (E) ≈ e−(E−EF )/kT
Effective DOS: G) Equilibrium vs nonequilibrium
3/2 3/2
2πm∗n kT 2πm∗p kT
Nc = 2 , Nv = 2 Equilibrium rules:
h2 h2
Nondegenerate carriers: • One Fermi level everywhere: EF = constant
• Drift + diffusion cancel ⇒ net current = 0
n = Nc e−(Ec −EF )/kT , p = Nv e−(EF −Ev )/kT
Illumination/injection: quasi-Fermi levels EF n , EF p
Intrinsic: p
−Eg /(2kT ) (split).
ni = Nc Nv e
Mass action (equilibrium):
np = n2i
H) Optical + LEDs (ABC)
Intrinsic level:
∗
mp
Ec + Ev 3
Ei = + kT ln Absorption near edge:
2 4 m∗n
• Direct: strong at Eg
• Indirect: weak near Eg (phonon required)
E) Doping, ionization, neutrality
Recombination:
Fully ionized (typical at 300K in Si unless stated):
n2i R = AN + BN 2 + CN 3
n-type: n ≈ ND − NA , p=
n
n2i
p-type: p ≈ NA − ND , n= IQE:
p
Fermi shift (nondegenerate): BN 2
IQE =
AN + BN 2 + CN 3
n p
EF − Ei = kT ln = −kT ln
ni ni
Charge neutrality (general): Quantum well levels (infinite well approx):
n + NA− = p + ND
+
n2 π 2 ℏ2
Partial ionization (freeze-out): En =
2m∗ L2
+ ND
ND = , gD ≈ 2
1 + gD e(EF −ED )/kT Emission:
NA
NA− = , gA ≈ 4
1 + gA e(EA −EF )/kT 1240
Regimes: freeze-out / extrinsic / intrinsic. Eemit ≈ Eg + E1e + E1h , λ(nm) =
Eemit (eV)
Cheat Sheet + Identification Table + 80 Problems (Solutions + Rubrics)
, EEE 352 Master Packet 3
Pure Identification Table (Problem Trigger → What To Do Immediately)
If the prompt says / shows... Topic Do this immediately Key equations / rules
Intercepts → plane indices Miller indices Convert intercepts to a units → recip- (hkl) procedure
rocals → clear fractions
Need (010) but only Cubic symmetry Pick equivalent plane (100) (100) ≡ (010) ≡ (001)
(100)/(110)/(111) available √
Plane spacing request Crystallography Use cubic spacing formula dhkl = a/ h2 + k2 + l2
Surface atom density compare Surface density Count atoms on 2D surface cell / area ρs = #/area
planes
XRD peak at θ with λ Bragg/XRD Apply Bragg’s law to get d nλ = 2d sin θ
Why bandgap exists? Band theory Say periodic potential + Schrödinger Bloch theorem
Allowed/forbidden energies in Kronig–Penney Use | cos(ka)| ≤ 1 idea KP + zone boundaries
lattice
E–k parabola, heavier/lighter? Effective mass Flatter ⇒ m∗ larger m∗ = ℏ2 /(d2 E/dk2 )
Velocity direction from E–k Group velocity Use slope sign v = (1/ℏ)dE/dk
Weak absorption near Eg Indirect gap Conclude phonon-assisted direct vs indirect
Intrinsic sample / where is EF ? Intrinsic stats EF = Ei ; if m∗ similar ⇒ midgap Ei formula
√
Two materials same ni Bandgap inference If Nc , Nv similar ⇒ same Eg ni = Nc Nv e−Eg /2kT
Compute n, p from EF Carrier stats Use nondegenerate forms n = Nc e−(Ec −EF )/kT , p =
Nv e−(EF −Ev )/kT
Given doping at 300K Extrinsic doping Assume full ionization unless warned n ≈ ND − NA , p = n2i /n
+
Low T / dopant level given Freeze-out Use ND (EF ) + neutrality + neutrality + ionization
n(EF ), p(EF )
n(T ) flat Extrinsic regime n ≈ ND (donor-limited) extrinsic formulas
n(T ) rises fast Intrinsic regime n ≈ ni (T ) ni (T ) exponential
Band bending diagram, equilib- Equilibrium rule EF flat; drift cancels diffusion one EF , net J = 0
rium
Illuminated solar cell Nonequilibrium quasi-Fermi levels split E F n , EF p
Resistivity depends on what? Transport identify n, p, µ σ = q(nµn + pµp )
2 2 2
Quantum well energy levels Confinement infinite well if stated; E ∝ 1/L2 En = n2mπ∗ Lℏ2
Emission wavelength Opto device compute Eemit then λ λ = 1240/E
BN 2
LED efficiency / droop Recombination Use ABC model IQE = AN +BN 2 +CN 3
Cheat Sheet + Identification Table + 80 Problems (Solutions + Rubrics)
