All the key relations of Chapter 3 on one page. Each block links to the chapter where the formula is derived and used. Energies in eV, concentrations as written on the slides (cm⁻³) unless stated. Use your browser’s Print → Save as PDF for a paper copy.
\(E_c,E_v\) band edges · \(E_g=E_c-E_v\) · \(E_F\) Fermi level (\(E_{Fi}\) intrinsic, \(E_{Fn},E_{Fp}\) n/p side or quasi-Fermi levels) · \(m_e^*,m_h^*\) effective masses · \(N_d,N_a\) donor/acceptor concentrations · \(n,p\) electron/hole concentrations · \(\mu\) drift mobility · \(D\) diffusion coefficient · \(\tau\) lifetime · \(L\) diffusion length · \(\varepsilon=\varepsilon_0\varepsilon_r\) · \(V\) applied voltage (> 0 forward) · \(V_\text{th}=V_T=k_BT/e\)
\(k_BT=25.85\) meV at 300 K · \(hc=1.2398\) eV µm · \(e=1.602\times10^{-19}\) C · \(\varepsilon_0=8.854\times10^{-12}\) F/m · \(n_i\) (300 K): Si \(1\times10^{10}\) , GaAs \(2.1\times10^6\) , Ge \(2.3\times10^{13}\) cm⁻³
density of states
\(g(E)=\dfrac{8\sqrt2\,\pi m^{*3/2}}{h^3}(E-E_c)^{1/2}\)
Fermi–Dirac
\(f(E)=\dfrac{1}{1+e^{(E-E_F)/k_BT}}\)
Fermi energy of a metal
\(E_{F0}=\dfrac{h^2}{8m_e}\left(\dfrac{3n}{\pi}\right)^{2/3}\)
electrons in CB
\(n=N_ce^{-(E_c-E_F)/k_BT}\)
holes in VB
\(p=N_ve^{-(E_F-E_v)/k_BT}\)
effective DOS
\(N_{c,v}=2\left(\dfrac{2\pi m^*_{e,h}k_BT}{h^2}\right)^{3/2}\)
mass action law
\(np=n_i^2=N_cN_ve^{-E_g/k_BT}\)
intrinsic Fermi level
\(E_{Fi}=E_v+\tfrac12E_g-\tfrac34k_BT\ln(m_e^*/m_h^*)\)
mean electron KE in CB
\(\tfrac32k_BT\)
validity (non-degenerate)
\(E_F\) more than \(3k_BT\) from the band edges
crystal momentum
\(\psi_k=U_k(x)e^{jkx}\) , \(F=d(\hbar k)/dt\)
Doping and transport (Chapter 3
n-type (\(N_d\gg n_i\) )
\(n=N_d\) , \(p=n_i^2/N_d\)
p-type (\(N_a\gg n_i\) )
\(p=N_a\) , \(n=n_i^2/N_a\)
compensation (\(N_d>N_a\) )
\(n=N_d-N_a\) , \(p=n_i^2/(N_d-N_a)\)
general neutrality
\(n=\tfrac{N_d-N_a}{2}+\sqrt{\left(\tfrac{N_d-N_a}{2}\right)^2+n_i^2}\)
Fermi level shift
\(E_{Fn}-E_{Fi}=k_BT\ln(n/n_i)\) , \(E_{Fp}-E_{Fi}=-k_BT\ln(p/n_i)\)
drift velocity
\(v_d=\mu E\)
conductivity
\(\sigma=en\mu_e+ep\mu_h\)
Einstein relation
\(D=\mu k_BT/e\)
applied voltage
\(\Delta E_F=eV\)
pn junction at equilibrium (Chapter 4
Poisson
\(d^2V/dx^2=-\rho_\text{net}/\varepsilon\)
charge balance
\(N_aW_p=N_dW_n\)
built-in voltage
\(V_o=\dfrac{k_BT}{e}\ln\dfrac{N_aN_d}{n_i^2}\)
depletion width
\(W=\left[\dfrac{2\varepsilon(N_a+N_d)(V_o-V)}{eN_aN_d}\right]^{1/2}\)
peak field
\(E_o=-eN_dW_n/\varepsilon=-2(V_o-V)/W\)
law of the junction
\(p_{no}/p_{po}=n_{po}/n_{no}=e^{-eV_o/k_BT}\)
from work functions
\(eV_o=\Phi_p-\Phi_n=E_{Fn}-E_{Fp}\) (before contact)
Debye length
\(L_D=\sqrt{\varepsilon k_BT/e^2N}\)
injection
\(p_n(0)=p_{no}e^{eV/k_BT}\) , \(n_p(0)=n_{po}e^{eV/k_BT}\)
excess profile (long)
\(\Delta p_n(x')=\Delta p_n(0)e^{-x'/L_h}\) , \(L_h=\sqrt{D_h\tau_h}\)
Shockley equation
\(J=J_{so}\left[e^{eV/k_BT}-1\right]\)
saturation current
\(J_{so}=\left(\dfrac{eD_h}{L_hN_d}+\dfrac{eD_e}{L_eN_a}\right)n_i^2\)
short diode
\(L_h\to\ell_n\) , \(L_e\to\ell_p\) ; transit time \(\tau_t=\ell_n^2/2D_h\)
SCL recombination
\(J_\text{rec}=J_{ro}\left[e^{eV/2k_BT}-1\right]\) , \(J_{ro}=\dfrac{en_i}{2}\left(\dfrac{W_p}{\tau_e}+\dfrac{W_n}{\tau_h}\right)\)
general diode equation
\(J=J_o\left[e^{eV/\eta k_BT}-1\right]\) , \(\eta=1\) (diffusion) to 2 (recombination)
reverse current
\(J_\text{rev}=J_{so}+en_iW/\tau_g\)
direct-gap lifetime
\(\tau=1/(BN)\) (weak injection)
Small-signal model (Chapter 6
depletion capacitance
\(C_\text{dep}=\dfrac{\varepsilon A}{W}=\dfrac{A}{(V_o-V)^{1/2}}\left[\dfrac{e\varepsilon N_aN_d}{2(N_a+N_d)}\right]^{1/2}\)
p⁺n junction
\(C_\text{dep}=A\left[\dfrac{e\varepsilon N_d}{2(V_o-V)}\right]^{1/2}\)
Mott–Schottky
\(N_d=-\dfrac{2}{e\varepsilon A^2}\left[\dfrac{d(1/C^2)}{dV}\right]^{-1}\)
dynamic resistance
\(r_d=\eta V_\text{th}/I\)
diffusion capacitance
long (ac): \(\dfrac{\tau_hI}{2V_\text{th}}=\dfrac{\tau_h}{2r_d}\) ; short: \(\dfrac{\tau_tI}{V_\text{th}}\)
photon energy
\(h\nu\approx E_g\) , \(\lambda[\mu\text{m}]=1.24/E_g[\text{eV}]\)
spectrum
\(I(h\nu)\propto(h\nu-E_g)^{1/2}e^{-(h\nu-E_g)/k_BT}\)
peak
\(h\nu_\text{peak}=E_g+\tfrac12k_BT\)
width
\(\Delta(h\nu)\approx1.8k_BT\) (ideal) to \(3k_BT\)
linewidth
\(\Delta\lambda\approx\lambda_o^2\,\dfrac{3k_BT}{hc}\) (107 nm at 1310 nm)
Varshni
\(E_g(T)=E_{go}-\dfrac{AT^2}{B+T}\) (GaAs: ≈ 0.24 nm/K)
degenerate junction
\(\Delta(h\nu)\approx\Delta E_{Fn}+1.5k_BT\)
Vegard / bowing
\(a=xa_1+(1-x)a_2\) , \(E_g=xE_1+(1-x)E_2-bx(1-x)\)
heterojunction leakage
\(\propto e^{-\Delta E_c/k_BT}\)
infinite well
\(\Delta E_n=\dfrac{h^2n^2}{8m^*d^2}\)
in-plane motion
\(E=E_c+\Delta E_n+\dfrac{\hbar^2(k_y^2+k_z^2)}{2m^*}\)
2D DOS (per sub-band)
\(g_{2D}=\dfrac{m^*}{\pi\hbar^2}\) per unit area
finite well, even
\(\dfrac{k}{m_w}\tan\dfrac{kd}{2}=\dfrac{\kappa}{m_b}\)
finite well, odd
\(-\dfrac{k}{m_w}\cot\dfrac{kd}{2}=\dfrac{\kappa}{m_b}\)
emission
\(h\nu=E_{g1}+E_1+E_1'\)
Efficiency and photometry (Chapter 11
escape cone
\(\sin\theta_c=n_a/n_s\)
extraction (one face)
\(\eta\approx\tfrac12(1-\cos\theta_c)\dfrac{4n_sn_a}{(n_s+n_a)^2}\)
internal QE
\(\eta_\text{IQE}=\dfrac{\tau_r^{-1}}{\tau_r^{-1}+\tau_{nr}^{-1}}\)
output power
\(P_o=\eta_\text{EE}\eta_\text{IQE}h\nu\,(I/e)\)
external QE
\(\eta_\text{EQE}=\dfrac{P_o/h\nu}{I/e}\)
power conversion
\(\eta_\text{PCE}=\dfrac{P_o}{IV}\approx\eta_\text{EQE}\dfrac{E_g}{eV}\)
luminous flux
\(\Phi_v=683\ \text{lm/W}\times P_oV(\lambda)\)
luminous efficacy
\(\eta_\text{LE}=\Phi_v/(IV)\)
resistor drive
\(I_F=(V-V_F)/R\)
net lifetime
\(1/\tau=1/\tau_r+1/\tau_{nr}\)
modulation response
\(P_o(\omega)/P_o(0)=1/\sqrt{1+(\omega\tau)^2}\)
electrical bandwidth
\(f_c=1/(2\pi\tau)\)
optical bandwidth
\(f_\text{op}=\sqrt3/(2\pi\tau)\)
rise / fall time
\(\tau_R=\tau_F=2.2\tau\)
fiber coupling (Lambertian)
\(\eta_c\approx\text{NA}^2\times\min[1,(a/r_s)^2]\)