NME 220 Molecular and Nanoscale Principles
Contents
Intro to NME
Driving forces and potentials
| Description | Equations |
|---|---|
| Avogadro’s number (Critical parameter of amount of particles to resist fluctuations) |
$N_A = 6.02\times 10^{23} \ \mathrm{mol^{-1}}$ |
| Fick’s law $\vec{J}_{AB} \ [\mathrm{mol/(m^2 s)}]$ - diffusive flux $D_{AB} \ [\mathrm{m^2/s}]$ - binary diffusion coefficient $\nabla c_A$ - concentration gradient |
$\vec{J}_{AB} = -D_{AB}\nabla c_A$ |
Scaling laws
| Description | Equations |
|---|---|
| Power density of engines | $\dfrac{P}{V} = \dfrac{Fv}{V}$ |
| Scaling law of power density | $\dfrac{P}{V} \propto \dfrac{1}{L}$ |
| Terminal velocity of droplets | $v_t = \dfrac{2gr^2(\rho_{\text{sphere}} - \rho_{\text{air}})}{9 \eta}$ |
| Scaling law of terminal velocity | $v_t \propto L^2$ |
Electric transport properties
| Description | Equations |
|---|---|
| Bohr’s radius (Critical length scale parameter for discrete excited electronic states) |
$a_0 = \dfrac{4\pi\varepsilon_0(\frac{h}{2\pi})^2}{m_e e^2}$ |
| Ohm’s law $\vec{J}$ - electric current density (flux) $\sigma \ [\mathrm{Sm^{-1}}]$ - electric conductivity $\nabla V$ - electric potential gradient |
$\vec{J} = \sigma\nabla V$ |
| Ohm’s law $G_e \ [\mathrm{S = \Omega^{-1}}]$ - electric conductance |
$G_e = \dfrac{1}{R} = \dfrac{I}{V}$ |
| Mean free path | $\lambda = \dfrac{k_BT}{\sqrt{2}\pi Pd^2}$ |
| Drude model Microscopic description - electric conductivity in 3D $\sigma$ - electric conductivity $\lambda$ - mean free path $\overline{c}$ - mean electron gas velocity |
$\sigma = \dfrac{\lambda e^2 n\overline{c}}{6k_B T}$ |
| Electric conductivity in 1D $N_e$ - transmission probability, # of electronic states (modes) at Fermi level $L$ - length of conuctor |
$\sigma = \dfrac{2e^2 N_e}{hL}$ |
| Electric conductance in 3D | $G_e = \dfrac{\lambda e^2 n\overline{c}}{6k_B T} \dfrac{A}{L} \propto \dfrac{1}{L}$ |
| Electric conductance in 1D | $G_e = \dfrac{2e^2}{h}N_e$ |
Thermo transport properties
| Description | Equations |
|---|---|
| Volumetric heat capacity at constant pressure | $C_V = \rho c_P$ |
| Thermal diffusivity | $\alpha = \dfrac{k_c}{C_V} = \dfrac{k_c}{\rho c_P}$ |
| Fourier’s law $\vec{q} \ [\mathrm{Wm^{-2}}]$ - heat flux $k_c \ [\mathrm{Wm^{-1}K^{-1}}]$ - thermal conductivity $\nabla T$ - temperature gradient |
$\vec{q} = -k_c \nabla T$ |
| Fourier’s law $\vec{J}$ - heat flux (?) $\alpha$ - thermal diffusivity $\nabla T$ - temperature gradient |
$\vec{J} = \dfrac{\vec{q}}{C_V} = -\alpha\nabla T$ |
| Microscopic description | $k_c = \frac{1}{2}n\overline{c}\lambda k_B$ |
| Thermal conductance in 3D | $G_{th} = k_c \dfrac{A}{L} \propto \dfrac{1}{L}$ |
| Thermal conductance in 1D $N_{ph}$ - # of phonons |
$G_{th} = \dfrac{\pi^2 k_B^2 T}{3h}N_{ph}$ |
Miniaturization effect on surface energy and strain
| Description | Equations |
|---|---|
| Surface stress $\gamma \ [\mathrm{Jm^{-2}}]$ - surface energy per area of solid $\gamma \ [\mathrm{F/m}]$ - surface tension of liquid $\varepsilon$ - strain $\frac{\partial\gamma}{\partial\varepsilon}$ - stored mechanical energy in solid $(\frac{\partial\gamma}{\partial\varepsilon} = 0)$ for liquid |
$f = \gamma + \dfrac{\partial\gamma}{\partial\varepsilon}$ |
| Surface energy/tension | $\gamma = E_{\text{coh, surface}} - E_{\text{coh, inside}}$ |
| Isotropic pressure $f$ - surface stress $D$ - diameter of particle |
$P = \dfrac{4f}{D}$ |
| Scaling law of elastic strain $\varepsilon$ - strain $K$ - bulk modulus |
$\varepsilon = -\dfrac{P}{3K} = -\dfrac{4}{3}\dfrac{f}{K}\dfrac{1}{D} \propto \dfrac{1}{D}$ |
Atomic Theory of Matter
Blackbody radiation
| Description | Equations |
|---|---|
| Wein’s displacement law $\lambda_{\mathrm{max}}$ - maximum of irradiation spectrum |
$\lambda_{\mathrm{max}}T = 2.898\times 10^{-3} \ \mathrm{m \cdot K}$ |
| Stefan-Boltzmann law $I \ [\mathrm{Wm^{-2}}]$ - radiation power $I_\lambda \ [\mathrm{Wm^{-3}}]$ - spectral irradiation $\varepsilon$ - emissivity $\sigma$ - Stefan-Boltzmann constant |
$I = \int_0^\infty I_\lambda \ d\lambda = \varepsilon\sigma T^4 \newline (\sigma = 5.67\times 10^{-8} \ \mathrm{W m^{-2} K^{-4}})$ |
| Radiation power | $P = IA$ |
| Rayleigh-Jeans' average mode energy of photon | $\langle E \rangle = \frac{1}{2}k_BT$ |
| Rayleigh-Jeans radiation law $k_B$ - Boltzmann’s constant |
$I_\lambda(\lambda) = \dfrac{2\pi ck_B T}{\lambda^4} \newline I_\nu(\nu) = \dfrac{8\pi \nu^2 k_B T}{c^3}$ |
| Planck’s average mode energy of photon | $\langle E \rangle = \dfrac{h\nu}{e^{h\nu / k_BT} - 1}$ |
| Planck’s radiation law $u$ - energy density |
$u_\lambda(\lambda) = \dfrac{8\pi hc}{\lambda^5}\dfrac{1}{e^{hc / \lambda k_BT} - 1} \newline u_\nu(\nu) = \dfrac{8\pi h \nu^3}{c^3}\dfrac{1}{e^{h\nu / k_BT} - 1}$ |
| Planck’s law confirms Stefan-Boltzmann law $I$ - radiation intensity $\sigma$ - Stefan-Boltzmann constant |
$I(\nu) = \sigma T^4 \newline \sigma = \dfrac{2\pi^5 k_B^4}{15c^2 h^3}$ |
| Energy of an EM mode | $E_n = nh\nu$ |
Photoelectric effect
| Description | Equations |
|---|---|
| Energy of a photon | $\Delta E = h\nu$ |
| Work function of a metal | $\Phi = e\phi$ |
| Photoelectric effect | $E_k = h\nu - e\phi$ |
| Condition for moving free electrons excited by photon | $h\nu \ge e\phi$ |
| Mass-energy equivalence | $E=mc^2$ |
Wave-particle duality
| Description | Equations |
|---|---|
| Classical linear momentum | $p = mv = \sqrt{2mE}$ |
| Photon linear momentum | $p = \dfrac{h\nu}{c} = \dfrac{h}{\lambda}$ |
| de Broglie wavelength of particle | $\lambda = \dfrac{h}{mv} = \dfrac{h}{\sqrt{2mE_k}}$ |
| Davisson-Germer experiment (electron diffraction constructive interference) |
$n\lambda = 2d\sin\theta$ |
| Bragg’s law | $\dfrac{1}{\lambda} = \dfrac{n}{2d\sin\theta}$ |
| Bragg-de Broglie relation | $\dfrac{1}{\lambda} = \begin{cases} \frac{n}{2d\sin\theta} \cr \frac{p}{h} = \frac{\sqrt{2mE}}{h}\end{cases}$ |
Atomic Model
Bohr’s model of atoms
| Description | Equations |
|---|---|
| Rydberg formula Emission lines of hydrogen $R_H$ - Rydberg constant $n_1 = 1,2,3, … \newline n_2 > n_1$ |
$\dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n_1^2} - \dfrac{1}{n_2^2} \right) \newline R_H = 1.097 \times 10^7 \ \mathrm{m^{-1}}$ |
| Quantization condition of Bohr’s model $n = 1,2,3,…$ $\lambda$ - wavelength of electron $r$ - radius of stable shell |
$2\pi r = n\lambda$ |
| Energy of electron in each shell $n = 1,2,3,…$ |
$E_n = \dfrac{-m_e e^4}{8 \varepsilon_0^2 h^2}\dfrac{1}{n^2}$ |
| Emission and absorption of H atom Emission - $j>1$ Absorption - $i>j$ |
$\begin{aligned}h\nu &= \Delta E \cr &= -\dfrac{m_e e^4}{8\varepsilon_0^2 h^2} \left( \dfrac{1}{n_j^2} - \dfrac{1}{n_i^2} \right) \cr &= -13.6 \ \mathrm{eV} \left( \dfrac{1}{n_j^2} - \dfrac{1}{n_i^2} \right)\end{aligned}$ |
| Reduced mass | $\mu = \dfrac{m_e m_p}{m_e + m_p}$ |
| Planck’s constant | $\hbar = \dfrac{h}{2\pi}$ |
| Rydberg’s constant | $R_H = R_\infty \dfrac{\mu}{m_e} = \dfrac{\mu e^4}{8\varepsilon_0^2 h^3 c}$ |
| Bohr radius | $a_0 = \dfrac{4\pi\varepsilon_0\hbar^2}{m_e e^2} \approx 0.053 \ \mathrm{nm}$ |
| Reduced Bohr radius | $a_0^* = \dfrac{4\pi\varepsilon_0\hbar^2}{\mu e^2} \approx 0.053 \ \mathrm{nm}$ |
| Ionization energy of electron from ground state | $E_I = \dfrac{-\mu e^4}{8\varepsilon_0^2 h^2} = -13.6 \ \mathrm{eV}$ |
Dispersion relations
| Description | Equations |
|---|---|
| Wave equation of traveling wave | $u(x,t) = A\sin(kx-\omega t + \phi)$ |
| Wave number | $k = \dfrac{2\pi}{\lambda}$ |
| Angular frequency | $\omega = 2\pi\nu = \dfrac{2\pi}{\lambda}c = kc$ |
| Period | $T = \dfrac{1}{\nu} = \dfrac{\lambda}{c}$ |
| Dispersion relation of EM wave | $\omega(k) = ck$ |
| Dispersion relation of particle wave | $\omega(k) = \dfrac{\hbar}{2m}k^2$ |
Single slit experiment
| Description | Equations |
|---|---|
| Variables | $a$ - width of the slit $L$ - distance between slit and screen $p$ - integer destructive interference # $p = \pm 1, \pm 2, \pm 3, …$ |
| Destructive interference | $\phi = p\lambda = a\sin\theta$ |
| Constructive interference | $\phi = (p+\frac{1}{2})\lambda = a\sin\theta$ |
| Location of destructive interference | $y_p = p\dfrac{\lambda L}{a}$ |
| Location of constructive interference | $y_p = (p+\frac{1}{2})\dfrac{\lambda L}{a}$ |
Double slit experiment
| Description | Equations |
|---|---|
| Variables | $\delta$ - path difference of two diffracting waves $d$ - distance between two slits $L$ - distance between slit and screen $m$ - integer constructive interference # $m = 0, \pm 1, \pm 2, \pm 3, …$ |
| Constructive interference | $\delta = m\lambda = d\sin\theta$ |
| Destructive interference | $\delta = (m+\frac{1}{2})\lambda = d\sin\theta$ |
| Location of constructive interference | $y_m = m\dfrac{\lambda L}{d}$ |
| Location of destructive interference | $y_m = (m+\frac{1}{2})\dfrac{\lambda L}{d}$ |
| Intensity of macroscopic particle and quantum particle with observation | $I = I_1 + I_2$ |
| Intensity of wave and quantum particle | $I = I_1 + I_2 + 2\sqrt{I_1I_2}\cos\delta$ |
Heisenberg uncertainty principle
| Description | Equations |
|---|---|
| Heisenberg uncertainty principle position-momentum form |
$\Delta x \Delta p \ge \dfrac{\hbar}{2}$ |
| Heisenberg uncertainty principle energy-time form |
$\Delta E \Delta t \ge \dfrac{\hbar}{2}$ |
| Macroscopicity | $10^\mu \propto \dfrac{\tau_p}{\tau_e}$ |
Wave Function
The Schrodinger equation
| Description | Equations |
|---|---|
| Wave function | $\Psi(x, t) = \Psi_0 e^{i(kx-\omega t)}$ |
| Wave function as probability density | $P = \lvert \Psi(x) \rvert^2$ |
| Linear momentum operator | $\hat{p_x} = -i\hbar\frac{\partial}{\partial x} \implies p_x$ as eigenvalue |
| Time operator | $\hat{t} = -i\hbar\frac{\partial}{\partial t} \implies E$ as eigenvalue |
| Hamiltonian operator | $\hat{H} = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x) \implies E$ as eigenvalue |
| Time-dependent wave equation for free particle | $i\hbar\dfrac{\partial}{\partial t}\Psi(x, t) = -\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2}\Psi(x, t)$ |
| Time-dependent Schrodinger equation (particle with potential constraint) |
$i\hbar\dfrac{\partial}{\partial t}\Psi = \left(-\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2} + V(x)\right) \Psi$ |
| Time-independent Schrodinger equation | $\left(-\dfrac{\hbar^2}{2m}\dfrac{\partial^2}{\partial x^2} + V(x)\right) \psi = E\psi$ |
| Time-independent Schrodinger equation (operator form) | $\hat{H}\psi = E\psi$ |
Particle in infinite 1D box
| Description | Equations |
|---|---|
| Potential function | $V(x) = \begin{cases} 0 & (0, L) \cr \infty & (-\infty, 0) \cup (L, \infty) \end{cases}$ |
| SE inside the box | $\dfrac{d^2\psi}{dx^2}+k^2\psi = 0$ |
| General solution | $\psi_n(x) = \sqrt{\dfrac{2}{L}} \sin \left(\dfrac{n\pi x}{L}\right)$ |
| Quantized wave number | $k_n = \dfrac{n\pi}{L}$ |
| Quantized momentum | $p_n = \hbar k_n = \dfrac{n\pi\hbar}{L}$ |
| Quantized energy level | $E_n = \dfrac{p_n^2}{2m} = \dfrac{(n\pi\hbar)^2}{2mL^2} = n^2 E_1$ |
Quantum structures
| Description | Equations |
|---|---|
| General continuous energy spectrum in non-confined dimensions | $E_i = \dfrac{\hbar^2 k_i^2}{2m}$ |
| General discrete energy spectrum in confined dimension | $E_{n_{i}} = \dfrac{\hbar^2\pi^2}{2mL^2}n_i^2$ |
| Total energy | $E = E_x + E_y + E_z$ |
| General wave function | $\psi_{n_x}(x) = \sqrt{\dfrac{2}{L}}\sin\left(\dfrac{n_x\pi x}{L}\right)$ |
The hydrogen atom
| Description | Equations |
|---|---|
| Potential function | $V(r) = \dfrac{-e^2}{4\pi\varepsilon_0 r}$ |
| Energy level of electrons in H atom | $E_n = \dfrac{-m_e e^4}{8\varepsilon_0^2 h^2}\dfrac{1}{n^2}$ |
| Wave function of electrons in H atom | $\psi_{100} = \dfrac{1}{\sqrt{\pi}}\left(\dfrac{1}{a_0}\right)^{3/2}e^{-r/a_0}$ |
| Principle quantum number | $n = 1, 2, 3, …$ |
| Orbital (angular momentum) quantum number | $l = 0, 1, 2, …, n-1$ |
| Magnetic quantum number | $m_l = -l, …, -1, 0, 1, …, l$ |
| Spin quantum number | $m_s = \frac{1}{2}, -\frac{1}{2}$ |
Nanoscience & Technology
Electron transport through 1D quantum wire
| Description | Equations |
|---|---|
| Variables | $\mu$ - electrochemical potential of electrode $M$ - mode number (single node: $M=1$) S - source; D - drain |
| Electrochemical potential difference between source and drain | $eV_{\text{bias}} = \mu_S - \mu_D$ |
| Current in 1D quantum wire (single mode) | $I = \dfrac{2q}{h}(\mu_S - \mu_D) = \dfrac{2q^2}{h}V_{\text{bias}}$ |
| Quantum conductance (single mode) | $G_Q = \dfrac{2q^2}{h}\approx 7.75\times 10^{-5} \ \mathrm{S}$ |
| Quantum resistance (single mode) | $R = \dfrac{h}{2q^2}\approx 12.9 \ \mathrm{k\Omega}$ |
| Current in 1D quantum wire (multi-mode) | $I = \dfrac{2qM}{h}(\mu_S - \mu_D) = \dfrac{2q^2M}{h}V_{\text{bias}}$ |
| Quantum conductance (multi-mode) | $G_Q = \dfrac{2q^2 M}{h}$ |
| Quantum resistance (multi-mode) | $R = \dfrac{h}{2q^2 M}$ |
Particle in finite 1D box
| Description | Equations |
|---|---|
| Finite potential | $V(x) = \begin{cases} V & (-\infty, 0) \cup (L, \infty) \cr 0 & (0, L) \end{cases}$ |
| Wave number | $k = \sqrt{\dfrac{2mE}{\hbar^2}}$ |
| Modified wave number | $k' = \sqrt{\dfrac{2m}{\hbar^2}(V-E)}$ |
| Wave function on the left | $\psi_1(x) = C_1e^{ik’x}$ |
| Wave function in the middle | $\psi_2(x) = A_1\sin(kx) + A_2\cos(kx)$ |
| Wave function on the right | $\psi_3(x) = C_2e^{-ik’x}$ |
Tunneling effect
| Description | Equations |
|---|---|
| Wave number | $k = \sqrt{\dfrac{2mE}{\hbar^2}}$ |
| Modified wave number | $k' = \sqrt{\dfrac{2m}{\hbar^2}(V-E)}$ |
| Wave function on the left | $\psi_1(x) = \psi_{\text{incid}}(x) + \psi_{\text{refl}}(x) \newline \psi_{\text{incid}}(x) = Ae^{ikx} \newline \psi_{\text{refl}}(x) = Be^{-ikx}$ |
| Wave function in the barrier | $\psi_2(x) = Ce^{ik’x} + De^{-ik’x}$ |
| Wave function on the right | $\psi_3(x) = \psi_{\text{trans}}(x) = Fe^{ikx}$ |
| Transmission probability | $\begin{aligned}T(L, V, E) &= \dfrac{\lvert\psi_{\text{trans}}(x)\rvert^2}{\lvert\psi_{\text{incid}}(x)\rvert^2} \cr &\approx 16 \dfrac{E}{V} \left(1-\dfrac{E}{V}\right) e^{-k’L} \cr &\approx \frac{1}{2}e^{2k’L}\end{aligned}$ |
| Work function with trapezoidal approximation at junction | $\Phi = \frac{1}{2}(\Phi_1 + \Phi_2 - \lvert eV_{\text{bias}}\rvert)$ |
| Tunnel current $D_s(E_F)$ - density of states at Fermi level |
$I_t \propto V_{\text{bias}}D_s(E_F)\exp\left(-\sqrt{\frac{8m}{\hbar^2}(\Phi-E)}L\right)$ |
Energy discretization of nanoparticles
| Description | Equations |
|---|---|
| Spacing between energy levels | $\Delta E = \dfrac{4}{3} \dfrac{E_F}{N_e}$ |
| Energy level and thermal noise | $\Delta E \begin{cases} \ll k_BT & \footnotesize\text{not quant. confined, continuous} \cr > k_BT & \footnotesize\text{quant. confined, discrete} \end{cases}$ |
| Conductor-insulator classification | $\Delta E = \dfrac{4}{3} \dfrac{E_F}{N_e} = k_BT$ |
| Number of electrons that can have discrete energy level | $N_e < \dfrac{4}{3}\dfrac{E_f}{k_B T}$ |
Single electron box
| Description | Equations |
|---|---|
| Variables | t - tunnel; g - gate; c - charging |
| Total capacitance | $C_{\text{dot}} = C = C_t + C_G$ |
| Gate voltage | $V_G = \dfrac{q_1}{C_t}+\dfrac{q_2}{C_G}$ |
| System energy | $E_{\text{sys}} = \dfrac{q_1^2}{2C_t} + \dfrac{q_2^2}{2C_G}$ |
| Charging energy | $E_c = \dfrac{e^2}{2C}$ |
| Quantum kinetic energy $V$ - volume of QD $D_s(E)$ - density of states |
$E_k = \dfrac{1}{VD_s(E_F)}$ |
| Electron addition energy | $E_a = E_c + E_k$ |
| Thermal noise requirement of single electron box (Coulomb blockade) | $E_c \gg k_BT \newline E_c > 10 k_BT$ |
| Quantum noise requirement of single electron box | $R_t \gg \dfrac{h}{e^2} \approx 25.8 \mathrm{k\Omega}$ |
| RC time constant | $\tau = R_tC$ |
| Gibbs free energy | $\mathcal{F} = H-TS$ |
| Gibbs free energy at equilibrium (Coulomb parabola potential) |
$\mathcal{F}(n, V_G) = \dfrac{(C_GV_G - ne)^2}{2C} \propto V_G^2$ |
| Coulomb parabola potential conditions | $\mathcal{F} = \begin{cases} 0 & \text{if } C_GV_G = ne \cr E_c & \text{if } C_GV_G = (n+1)e \end{cases}$ |
| Capacitance of quantum dots $d$ - diameter |
$C_{\text{dot}} = G\varepsilon\varepsilon_0 d$ |
| Geometric factor | $G = \begin{cases}2\pi & \text{sphere} \cr 4 & \text{disc}\end{cases}$ |
Single electron transistor
| Description | Equations |
|---|---|
| Charging energy | $E_c = \dfrac{e^2}{2C_{\text{dot}}}$ |
| Charging energy $Q_0$ - polarization charge $ne$ - uncompensated electrons |
$E_c = \dfrac{(Q_0 - ne)^2}{2C_{\text{dot}}}$ |
| Charging voltage | $V_g = \dfrac{E_c}{e} = \dfrac{e}{2C_{\text{dot}}}$ |
| Capacitance of quantum dot | $C_{\text{dot}} = C_{t_1} + C_{t_2} + C_G$ |
Electronic Structure of Molecules
The hydrogen atom (revisited)
| Description | Equations |
|---|---|
| Potential function | $V(r) = \dfrac{-e^2}{4\pi\varepsilon_0 r}$ |
| Energy level of electrons in H atom | $E_n = \dfrac{-m_e e^4}{8\varepsilon_0^2 h^2}\dfrac{1}{n^2} = -13.6 \ \mathrm{eV}\dfrac{1}{n^2}$ |
| Bohr radius | $a_0 = \dfrac{4\pi\varepsilon_0\hbar^2}{m_e e^2} \approx 0.053 \ \mathrm{nm}$ |
| Wave function | $\psi(\mathbf{r}) = R(r)Y(\theta, \phi)$ |
| Angular component of wave function | $Y(\theta, \phi) = \Theta(\theta)\Phi(\phi)$ |
| Wave function in ground state | $\psi_{100} = \dfrac{1}{\sqrt{\pi}}\left(\dfrac{1}{a_0}\right)^{3/2}e^{-r/a_0}$ |
| Wave function in $n,0,0$ state | $\psi_{n00} = \dfrac{R_n(r)}{\sqrt{4\pi}}$ |
| Probability density function | $\rho_{nlm} = \lvert\psi_{nlm}\rvert^2 = \psi\psi^*$ |
| Radial probability density | $P_{nl}(r) = \lvert R_{nl} \rvert^2 r^2$ |
| Number of radial nodes | $n-l-1$ |
Angular momentum
| Description | Equations |
|---|---|
| Angular momentum operator in $z$ direction | $\hat{L}_z = i\hbar (x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x})$ |
| Orbital angular momentum in $z$ direction | $L_z = m_l\hbar$ |
| Angular momentum magnitude operator | $\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2$ |
| Orbital angular momentum | $L = \hbar\sqrt{l(l+1)}$ |
| Spin angular momentum in $z$ direction | $S_z = m_s\hbar$ |
| Spin angular momentum | $S = \hbar\sqrt{s(s+1)}$ |
| Total angular momentum in $z$ direction | $J_z = m_j\hbar$ |
| Total angular momentum | $J = \hbar\sqrt{j(j+1)}$ |
| spin angular momentum quantum number | $s = \frac{1}{2}, m_s = \pm \frac{1}{2}$ |
| Spin-orbit coupling | $j = l \pm s$ |
| Energy degeneracy considering spin | $2n^2$ |
Bonding in molecules
| Description | Equations |
|---|---|
| Bonding (symmetric linear combination) | $\psi_g = \psi_1 + \psi_2$ |
| Anti-bonding (asymmetric linear combination) | $\psi_u = \psi_1 - \psi_2$ |
Electronic states
Note: the mass here are electron mass.
| Description | Equations |
|---|---|
| Fermi-Dirac distribution | $f(E)=\dfrac{1}{\exp\left(\dfrac{E- E_{f}}{k_B T}\right)+1}$ |
| Fermi energy in 3D at 0 K | $E_{F}=\dfrac{\hbar^{2} k_{F}^{2}}{2 m}=\dfrac{\hbar^{2}}{2 m}\left(3 \pi^{2} n_e\right)^{2/3}$ |
| Fermi wave number in 3D at 0 K | $k_{F}=\frac{1}{\hbar} \sqrt{2 m E_{F}}=\left(3 \pi^{2} n_e\right)^{1 / 3}$ |
| Fermi velocity approximation in 3D | $v_F = \dfrac{\hbar k_F}{m} = \dfrac{\hbar}{m}(3\pi^2 n_e)^{1/3}$ |
| Wavelength in 3D at 0 K | $\lambda_F = \dfrac{2\pi\hbar}{mv_F}$ |
| Density of states (general) | $D_s(E)=\dfrac{dN}{dE}\dfrac{1}{V}$ |
| Density of states in 3D at 0 K | $D_s^{\text{3D}}(E)=\dfrac{8 \sqrt{2} \pi m^{3 / 2}}{h^{3}} \sqrt{E}$ |
| Density of states in 2D at 0 K $(0<E<E_1)$ |
$D_s^{\text{2D}}(E)=\dfrac{4\pi m}{h^2}$ |
| Density of states in 1D at 0 K | $D_s^{\text{1D}}(E)=\dfrac{\sqrt{2m}}{h}\dfrac{1}{\sqrt{E}}$ |
| Density of states in 0D at 0 K | $D_s^{\text{0D}}(E)=2$ |
| Number density of electron (general) | $n_e(E) = \dfrac{N}{V} = \displaystyle\int_{0}^\infty f(E) D(E) \ dE$ |
| Number density of electron in 3D at 0 K | $n_e^{\text{3D}} = \dfrac{16 \sqrt{2} \pi m^{3/2}}{3 h^{3}} E_{F}^{3/2}$ |
| Number density of electron in 2D at 0 K | $n_e^{\text{2D}} = \dfrac{4\pi m}{h^2}E_F$ |
| Number density of electron in 1D at 0 K | $n_e^{\text{1D}} = \dfrac{2\sqrt{2m}}{h}\sqrt{E_F}$ |
| Number density of electron in 0D at 0 K | $n_e^{\text{0D}} = 2E_F$ |
Semiconductor carriers
| Description | Equations |
|---|---|
| Band gap | $E_g = E_c - E_v$ |
| Density of state of electrons (conduction band) | $D_c(E) = \dfrac{8\sqrt{2}\pi m_e^{*3/2}}{h^3}\sqrt{E-E_c}$ |
| Density of state of holes (valance band) | $D_v(E) = \dfrac{8\sqrt{2}\pi m_h^{*3/2}}{h^3}\sqrt{E_v-E}$ |
| Fermi-Dirac distribution for electrons | $f(E) = \dfrac{1}{1+\exp(\frac{E-E_F}{k_BT})}$ |
| Fermi-Dirac distribution for holes | $1-f(E)$ |
| Boltzmann approximation | $E-E_F \gg k_BT$ |
| Effective density of state of electrons in 3D | $N_c^{\text{3D}} = N_{e, \text{eff}} = 2 \left( \dfrac{2\pi m_e^* k_BT}{h^2} \right)^{3/2}$ |
| Effective density of state of holes in 3D | $N_v^{\text{3D}} = N_{h, \text{eff}} = 2 \left( \dfrac{2\pi m_h^* k_BT}{h^2} \right)^{3/2}$ |
| Effective density of state in 2D (use appropriate mass) |
$N^{\text{2D}} = \dfrac{m^* k_B T}{\pi\hbar^2}$ |
| Effective density of state in 1D (use appropriate mass) |
$N^{\text{1D}} = \sqrt{\dfrac{m^* k_B T}{2\pi\hbar^2}}$ |
| Effective density of state in 0D (use appropriate mass) |
$N^{\text{0D}} = 2$ |
| Electron (carrier) density | $n_e(E) \approx N_{e, \text{eff}}\exp\left(-\dfrac{E_c-E_F}{k_BT}\right)$ |
| Hole (carrier) density | $n_h(E) \approx N_{h, \text{eff}}\exp\left(-\dfrac{E_v-E_F}{k_BT}\right)$ |
| Intrinsic carrier density | $n_i = \sqrt{N_cN_v}\exp\left(-\dfrac{E_g}{2k_BT}\right)$ |
| Exciton separation distance | $a_{\text{ex}} = a_0\dfrac{\varepsilon}{m_{\text{ex}}/m_e}$ |
Semiconductor doping
| Description | Equations |
|---|---|
| Variables | oc - open circuit bi - built-in sc - short circuit |
| Solar cell conductivity $\mu$ - electron mobility |
$K \propto \mu n_i$ |
| Open-circuit voltage (p-n solar cells) $N_a$ - donor atom concentration $\Delta n$ - excess electrons generated by photons (photocurrent) $q=e$ - elementary charge |
$\begin{aligned}V_{\text{oc}} &= \dfrac{E_{\mathrm{HL}}}{q} = \dfrac{E_{F_{n}} - E_{F_{p}}}{q} \cr &= \dfrac{k_BT}{q}\ln\left(\dfrac{(N_a+\Delta n)\Delta n}{n_i^2}\right) \cr &\approx \dfrac{k_BT}{q}\ln\left(\dfrac{I_{ph}}{I_0}\right)\end{aligned}$ |
| Built-in voltage (p-n diode) | $\begin{aligned}V_{\text{bi}} &= \dfrac{E_{F_{n}} - E_{F_{p}}}{q} \cr &=\dfrac{k_BT}{q}\ln\left(\dfrac{N_aN_d}{n_i^2}\right)\end{aligned}$ |
| Relationship between open circuit and built-in voltages | $V_{\text{oc}} < V_{\text{bi}} \newline V_{\text{bi}} = V_{\text{oc}} + \dfrac{2k_BT}{q}$ |
| Depletion width | $w = \sqrt{\dfrac{2\varepsilon_r \varepsilon_0 (N_a+N_d)}{qN_aN_d}V_{\text{eff}}}$ |
| Fill factor | $\mathrm{FF} = \dfrac{P_{\text{max}}}{V_{\text{oc}}I_{\text{sc}}}$ |
| Efficiency | $\eta = \dfrac{P_{\text{max}}}{P_{\text{in}}} = \dfrac{(\mathrm{FF})V_{\text{oc}}I_{\text{sc}}}{P_{\text{in}}}$ |
Molecular Modes and Energetic Properties
Energy: Electronic $\gg$ Vibrational $\gg$ Rotational $\gg$ Thermal noise ($k_BT$) $\gg$ Translational
Vibration modes of diatomic molecules
| Description | Equations |
|---|---|
| Spring potential | $V(x) = \frac{1}{2}k_s x^2$ |
| Frequency of quantum harmonic oscillator $m_1 \gg m_2 = m$ |
$\nu = \dfrac{1}{2\pi}\sqrt{\dfrac{k_s}{m}}$ |
| Energy levels of quantum harmonic oscillators | $E_n = (n + \frac{1}{2})h\nu \newline n = 0, 1, 2, …$ |
| Zero-point energy | $E_0 = \frac{1}{2}h\nu$ |
| Equidistant energy levels | $\Delta E_{\text{vib}} = h\nu_{\text{vib}}$ |
| Reduced mass | $\mu = \dfrac{m_1m_2}{m_1+m_2}$ |
| Frequency of quantum harmonic oscillator $m_1 ~ m_2$ |
$\nu = c\tilde{\nu} = \dfrac{1}{2\pi}\sqrt{\dfrac{k_s}{\mu}}$ |
| Vibrational wave number of quantum harmonic oscillator | $\tilde{\nu} = \dfrac{1}{\lambda} = \dfrac{1}{2\pi c}\sqrt{\dfrac{k_s}{\mu}}$ |
| Wave number of quantum harmonic oscillator | $k_\lambda = \dfrac{2\pi}{\lambda} = \dfrac{1}{c}\sqrt{\dfrac{k_s}{\mu}}$ |
| Dissociation energy $D_e$ and actual dissociation energy $D_0$ |
$D_0 = D_e + \frac{1}{2}h\nu$ |
| Morse potential $r_e$ - equilibrium position, bond length $a$ - inverse width of Morse potential |
$V(r) = D_e (1 - e^{-a(r-r_e)})^2 \newline a = \sqrt{\dfrac{k_s}{2D_e}} = \omega\sqrt{\dfrac{\mu}{2D_e}}$ |
| Vibrational temperature | $\Theta_{\text{vib}} = \dfrac{h\nu}{k_B}$ |
| Vibrational energy | $E_{\text{vib}} = k_B\Theta_{\text{vib}}$ |
Rotational modes of diatomic molecules
| Description | Equations |
|---|---|
| Angular velocity (frequency) | $\omega = 2\pi\nu$ |
| Linear velocity | $v_i = r_i\omega$ |
| Moment of inertia | $I = m_1r_1^2 + m_2r_2^2 = \mu R^2$ |
| Kinetic energy of rigid rotor | $E_k = \frac{1}{2}I\omega^2$ |
| Rotational constant | $B = \dfrac{\hbar^2}{2I} \newline \tilde{B} = \dfrac{B}{hc} = \dfrac{h}{8\pi^2 cI} \ [\mathrm{cm^{-1}}]$ |
| Rotational temperature | $\Theta_{\text{rot}} = \dfrac{B}{k_B} = \dfrac{\hbar^2}{2Ik_B}$ |
| Rotational energy | $E_{\text{rot}} = k_B\Theta_{\text{rot}}$ |
| Energy of rigid rotor | $\begin{aligned}E_J &= J(J+1)\dfrac{\hbar^2}{2I} \cr &= J(J+1)B \cr &= J(J+1)k_B\Theta_{\text{rot}}\end{aligned}$ |
| Probability of being in a particular rotational energy state | $f_J = (2J+1)\exp(-J(J+1)\Theta_{\text{rot}}/T)$ |
| Energy for absorption | $\Delta E_{\text{rot}}^{J\to J-1} = 2B(J+1)$ |
| Energy for emission | $\Delta E_{\text{rot}}^{J\to J+1} = 2BJ$ |
| Separation between transitions | $\Delta(\Delta E_{\text{rot}}^{J\to J-1}) = \Delta(\Delta E_{\text{rot}}^{J\to J+1}) = 2B$ |
| Non-degenerate translational energy level in 1D | $E_n^{\text{1D}} = \dfrac{n^2h^2}{2mL^2}$ |
| Non-degenerate translational energy level in 3D | $E_n^{\text{3D}} = \dfrac{h^2}{8m}\left(\dfrac{n_x^2}{L_x^2} + \dfrac{n_y^2}{L_y^2} + \dfrac{n_z^2}{L_z^2}\right)$ |
Polyatomic molecules
| Degree of freedom | Linear molecule with $n$ atoms |
Nonlinear molecule with $n$ atoms |
|---|---|---|
| Translational | $3$ | $3$ |
| Rotational | $2$ | $3$ |
| Vibrational | $3n-5$ | $3n-6$ |
| Description | Equations |
|---|---|
| Intensity of IR vibrational signal | $I_{\text{IR}} \propto \left(\dfrac{du_D}{d\xi}\right)^2$ |
| Rotational motion energy | $E_J = \dfrac{J(J+1)\hbar^2}{2I} \newline J = 0, 1, 2, …$ |
| Degeneracy of rotational motion energy | $g_J = 2J+1$ |
| Moment of inertia | $I = \sum\limits_{i=1}^n m_i (x_i - x_{cm})^2$ |
Lattice vibration and phonon
| Description | Equations |
|---|---|
| Harmonic potential | $V(r_i-r_j) = \sum\limits_{i, j} \frac{1}{2}m\omega^2(r_i-r_j)$ |
| Energy of phonon (lattice vibration) in 3D | $E_{n_x, n_y, n_z}^{\text{3D}} = (n_x + n_y + n_z + \frac{3}{2})\hbar\omega \newline n = 0, 1, 2, …$ |
| Debye frequency Upper limit of dispersion frequency $n$ - atom number density |
$\omega_D = 6\pi^2 n c_{\text{sound}}^3$ |