Extended Micro and Nanotechnology Formula Sheet

Semiconductor Fundamentals

Photon energy-wavelength relation: $E=\frac{hc}{\lambda }\Rightarrow \lambda \text{ (nm)}=\frac{1240}{E\text{ (eV)}}$

Intrinsic carrier concentration: $n_i=\sqrt{N_c{N}_v}\exp \left(-\frac{E_g}{2kT}\right)$

Mass action law: $np=n_i^2$

Conductivity: $\sigma =q(n{\mu }_e+p{\mu }_h)=\frac{1}{\rho }$

Drift current density: $J=\sigma E=q(n{\mu }_e+p{\mu }_h)E$

Built-in potential (pn junction): $V_{bi}=\frac{kT}{q}\ln \left(\frac{N_d{N}_a}{n_i^2}\right)$

Diode equation: $I=I_0\left(\exp \left(\frac{qV}{nkT}\right)-1\right)$

Thermal voltage at 300K: $\frac{kT}{q}\approx 25.9\text{ mV}$

Doping & Carrier Transport

n-type (saturation region): $n\approx N_d,p=\frac{n_i^2}{N_d}$

p-type (saturation region): $p\approx N_a,n=\frac{n_i^2}{N_a}$

Resistivity from doping: $\rho =\frac{1}{qn\mu }\text{(for n-type)}$

Sheet resistance: $R_s=\frac{\rho }{t}[\Omega /\square ]$

Resistance of a rectangular film: $R=R_s\frac{L}{W}$

Diffusion & Ion Implantation

Fick’s first law: $J=-D\frac{\partial C}{\partial x}$

Fick’s second law: $\frac{\partial C}{\partial t}=D\frac{{\partial }^{2}C}{\partial x^2}$

Diffusion coefficient (Arrhenius): $D=D_0\exp \left(-\frac{E_A}{kT}\right)$

Constant source (erfc) profile: $C(x,t)=C_s\mathrm{erfc}\left(\frac{x}{2\sqrt{Dt}}\right)$

Limited source (Gaussian) profile: $C(x,t)=\frac{Q}{\sqrt{\pi Dt}}\exp \left(-\frac{x^2}{4Dt}\right)$

Implant dose: $Q=\frac{I\cdot t}{q\cdot A}$

Projected range: depth of peak concentration, set by accelerating voltage

Straggle ($\Delta R_p$): standard deviation of the implant profile

Hall Effect & Characterisation

Hall voltage: $V_H=\frac{I_x{B}_z}{nqd}$

Hall coefficient: $R_H=\frac{V_{H}d}{I_x{B}_z}=\frac{1}{nq}$

Mobility from Hall & resistivity: $\mu =\frac{|R_H|}{\rho }$

Bragg’s law (XRD): $n\lambda =2d\sin \theta$

CV profiling — depletion capacitance: $C=\frac{\epsilon A}{W}$

Doping from CV: $N=\frac{-2}{q\epsilon A^2}{\left(\frac{d(1/C^2)}{dV}\right)}^{-1}$

Alloys & Band Gap Engineering

Band gap of alloy (with bowing): $E_g(x)=(1-x)E_{g1}+x{E}_{g2}-bx(1-x)$

Semiconductor Lasers

Fabry-Perot mode condition: $m\lambda =2nL$

Longitudinal mode spacing: $\Delta \lambda =\frac{{\lambda }^2}{2nL}$

Number of modes in gain bandwidth: $N=\frac{\Delta {\lambda }_{gain}}{\Delta \lambda }$

Threshold condition (gain = loss): $\Gamma g_{th}={\alpha }_i+\frac{1}{2L}\ln \left(\frac{1}{R_1{R}_2}\right)$

where $\Gamma$ = confinement factor, ${\alpha }_i$ = internal loss, $R$ = mirror reflectivity

Bragg condition (DFB): ${\lambda }_B=2n_{eff}\Lambda$

Fresnel reflectivity (cleaved facet in air): $R={\left(\frac{n-1}{n+1}\right)}^2$

Etching

Mean free path (pressure dependence): ${\lambda }_{mfp}=\frac{kT}{\sqrt{2}\pi d^{2}P}$

${\lambda }_{mfp}\propto 1/P$

Cleanroom

Particle count classification: Class $N$ = max $N$ particles ($\ge 0.5\mu$m) per ft³

Optics & Lenses

Lensmaker’s Equation (thin lens in air): $\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

Thin Lens Equation: $\frac{1}{f}=\frac{1}{d_o}+\frac{1}{d_i}$

Magnification: $m=-\frac{d_i}{d_o}$

Lenses in contact: $\frac{1}{f_{net}}=\frac{1}{f_1}+\frac{1}{f_2}$

Compound Microscope Magnification (image at infinity): $M=-\frac{L}{f_o}\cdot \frac{N}{f_e}$ where $L$ = barrel length $-f_o-f_e$, $N$ = 25 cm

Numerical Aperture: $NA=n\sin \theta$

Resolution & Diffraction

Rayleigh Criterion: ${\theta }_{min}=1.22\frac{\lambda }{D}{d}_{min}=\frac{0.61\lambda }{NA}$

Lithography Resolution: $R=k_1\frac{\lambda }{NA}$

Single Slit Diffraction (minima): $a\sin \theta =m\lambda (m=\pm 1,\pm 2,\dots )$

de Broglie & Electron Optics

de Broglie Wavelength: $\lambda =\frac{h}{mv}=\frac{h}{\sqrt{2m_{e}eV}}$

Shortcut: $\lambda \text{ (nm)}=\frac{1.226}{\sqrt{V\text{ (V)}}}$

Quantum Tunnelling

Transmission coefficient: $T\approx e^{-2\kappa d},\kappa =\frac{\sqrt{2m(V_0-E)}}{\hslash }$

AFM Cantilever

$k_N=\frac{Ew{t}^3}{4l^3}{f}_0=\frac{1}{2\pi }\sqrt{\frac{k_N}{0.24m}}$

Thermal Oxidation (Deal-Grove)

$x^2+Ax=B(t+\tau )$

Linear regime: $x\approx \frac{B}{A}(t+\tau )$

Parabolic regime: $x^2\approx Bt$

Rate constants: $\frac{B}{A}={\left(\frac{B}{A}\right)}_0{e}^{-E_A^{lin}/kT},B=B_0{e}^{-E_A^{par}/kT}$

Constants

Constant	Value
$h$	$6.626\times 10^{-34}$ J·s
$\hslash$	$1.055\times 10^{-34}$ J·s
$m_e$	$9.109\times 10^{-31}$ kg
$e$	$1.602\times 10^{-19}$ C
$k_B$	$8.617\times 10^{-5}$ eV/K = $1.381\times 10^{-23}$ J/K
$c$	$3.0\times 10^8$ m/s
${\epsilon }_0$	$8.854\times 10^{-12}$ F/m
$kT$ (300K)	$0.02585eV$
$0^{o}C$	$273.15K$
Eye Near Point	$25cm$
Violet Light	$400nm$
Red Light	$750nm$
STP	$760$ Torr
1 Pascal	$7.5$ mTorr
Min Disc Object by Eye	$70\mu m$