Physics at the 1–100 nm scale where quantum confinement, discrete charging energies, and quantised level spacings dominate — the intermediate regime between atomic and mesoscopic physics. Quantum confinement in a 1D infinite square well…
nanophysics
Coulomb blockade: E_C = e²/(2C), single-electron charging
Coulomb blockade of single-electron transport (Kulik-Shekhter 1975; Averin-Likharev 1986; Fulton-Dolan 1987 PRL 59:109): a small metallic…
1D infinite well: E_n = π²ℏ²n²/(2mL²)
Quantum confinement in a 1D infinite square-well potential of width L: Schrödinger equation −(ℏ²/2m)·d²ψ/dx² = Eψ with boundary conditions…
1D DOS: g_1D(E) = (1/πℏ)·√(m/(2E)) (van Hove 1/√E)
Density of states per unit length and unit energy for a 1D free-electron subband: starting from the dispersion E(k) = ℏ²k²/(2m) and the 1D…
Addition energy: U_N/E_C = 2N+1 (exact integer)
Closed-form electrostatic energy for sequential charging of a Coulomb-blockaded island. The chemical potential μ(N) to add the N-th…
1D box energies: E_n·2mL² − π²ℏ²n² ≡ 0 exactly
Closed-form polynomial identity of the infinite-well eigenenergies. Setting E_n = π²ℏ²n²/(2mL²) and expanding E_n·2mL² − π²ℏ²n² yields…
1D DOS integral: ∫₀^{E₀} g_1D dE = √(2mE₀)/(πℏ)
Closed-form integration of the 1D density of states. Integrating g_1D(E) = (1/πℏ)·√(m/(2E)) from 0 to E₀ gives the total number of states…
Landauer-Buettiker multi-channel conductance G = (2 e^2/h) sum T_n
Landauer-Buettiker transport framework (Landauer 1957 IBM J Res Dev 1, 223; Buettiker 1988 Phys Rev B 38, 9375). Setup: phase-coherent…
Casimir energy per area E(a) = -pi^2 hbar c / (720 a^3); zeta(-1) = -1/12
Casimir-force framework via zeta-function regularisation (Casimir 1948 Proc K Ned Akad Wet 51, 793). Setup: two parallel conducting plates…
Surface plasmon resonance: omega_SPR = omega_p/sqrt(1 + eps_d); complex pole
Surface-plasmon-resonance dispersion framework (Ritchie 1957 Phys Rev 106, 874; Raether 1988). Setup: at flat metal-dielectric interface,…
Theorem: G h / (2 e^2) - T_n = 0 (single-channel Landauer conductance identity)
Theorem (Landauer-channel canonical): G = 2 e^2 T_n / h rearranges to G h / (2 e^2) = T_n, i.e. dimensionless G / G_0 equals transmission…
Theorem: dE/da - (-pi^2 hbar c / (240 a^4)) = 0 (Casimir force-law derivative)
Theorem (Casimir-force-derivative canonical): d/da of E(a) = -pi^2 hbar c / (720 a^3) gives dE/da = pi^2 hbar c / (240 a^4). [Note: chain…
Theorem: omega_SPR^2 (1 + eps_d) - omega_p^2 = 0 (SPR dispersion root)
Theorem (SPR-dispersion-root canonical): omega_SPR = omega_p / sqrt(1 + eps_d) squared and multiplied by (1 + eps_d) gives omega_p^2…
STM (Binnig-Rohrer 1981)
G Binnig-H Rohrer 1981 STM (Nobel 1986); atomic-resolution imaging; modern AFM Binnig 1986 + IBM 'Quantum Mirage' 2000.
SET (Likharev 1987)
K Likharev 1987 single-electron-transistor; Coulomb-blockade; modern quantum-dot SET + Cooper-pair-box qubit basis.
CNTs (Iijima 1991)
S Iijima 1991 multi-wall + 1993 single-wall carbon-nanotubes; modern HiPCO + CVD synthesis; CNT-FET + transparent-conductive-films…
Nanoplasmonics (Stockman 2008)
M Stockman 2008 nanoplasmonics review; SPP propagation + LSPR resonances; modern plasmonic-metasurface + super-resolution applications.
Nanowire lasers (Yang 2001)
P Yang 2001 ZnO nanowire-laser; modern III-V on-Si nanowire integration; quantum-dot-in-nanowire single-photon sources.
GNR bandgap (Han 2007)
Han-Ozyilmaz 2007 GNR bandgap E_g ~ 0.8 eV/W (nm); zigzag-vs-armchair edge-states; modern atomically-precise GNR Cai 2010.
STM (Binnig-Rohrer 1981)
G Binnig-H Rohrer 1981 (Nobel 1986) scanning-tunneling-microscopy; modern modern foundational text + atom-manipulation Eigler 1990.
AFM (Binnig 1986)
G Binnig-C Quate-C Gerber 1986 AFM; modern modern foundational text + tapping-mode + AM/FM + 2024 ML-AFM-image-recognition.
Graphene (Novoselov-Geim 2004)
K Novoselov-A Geim 2004 (Nobel 2010) mechanical-exfoliation-graphene; modern modern moire-graphene 2018 + chiral-twistronics + 2024.
SET (Likharev 1987)
K Likharev-D Averin 1987 + Fulton-Dolan 1987 SET; modern modern foundational + room-T SET 2024 + carbon-nanotube-SET memory.
FinFET (Hu 1998)
C Hu 1998 + Intel 22nm 2011 FinFET; modern modern 3 nm process + 2 nm GAA + 2024 backside-power-delivery semiconductor scaling.
Mems-NEMS (Roukes 2000)
M Roukes 2000 NEMS resonator; modern modern foundational text + sub-attogram mass + 2024 spin-mechanics quantum-acoustics.