Electronic transport that exploits the spin degree of freedom in addition to charge. Foundational: Mott 1936 two-current model of spin-polarised conduction in ferromagnets; Julliere 1975 tunneling-magnetoresistance ratio between parallel…
spintronics
GMR: ρ_AP − ρ_P = 4β²ρ*λ_sf/L-dependent — parallel/antiparallel resistance difference
Baibich-Fert 1988 / Grünberg 1989 GMR (2007 Nobel): Fe/Cr multilayers show a ~50% resistance drop when external field aligns neighbouring…
Spin-transfer torque: J_s > J_c ⇒ precessional switching of free-layer magnetisation
Berger 1996 / Slonczewski 1996 spin-transfer torque: a spin-polarised current passing through a ferromagnetic free layer transfers angular…
Spin-Hall effect: J_c → J_s transverse — intrinsic + skew + side-jump mechanisms
Dyakonov-Perel 1971 / Hirsch 1999 spin-Hall effect: charge current in a material with spin-orbit coupling generates a transverse spin…
Julliere TMR ratio: (R_AP − R_P)/R_P = 2 P₁P₂ / (1 − P₁P₂)
Julliere 1975 tunneling-magnetoresistance ratio for a magnetic tunnel junction (MTJ) with two ferromagnetic electrodes separated by an…
Spin-Hall angle: θ_SH = σ_SH / σ_xx — charge-to-spin-current conversion efficiency
Spin-Hall angle θ_SH is a dimensionless material figure of merit for charge-to-spin-current conversion: θ_SH = σ_SH / σ_xx, i.e. the ratio…
Mott two-current model: ρ_P ~ parallel of (ρ_↑ ∥ ρ_↓), both low; ρ_AP ~ series
Mott 1936 / 1964 two-current model: in a ferromagnet, electrons with spin-majority and spin-minority at E_F scatter through independent…
Heisenberg exchange H = −J·Σ S_i·S_j on spin-commutator algebra with Jacobi identity
Heisenberg exchange Hamiltonian on the spin-commutator Lie algebra. Setup: localised spin-s operators S_i^a (a = x, y, z) on lattice sites…
SU(2) spin-rotation: R(n̂, θ) = exp(−iθ·n̂·σ/2) = cos(θ/2)·I − i·sin(θ/2)·n̂·σ
SU(2) spin-1/2 rotation operator. Setup: Pauli matrices σ^x, σ^y, σ^z with [σ^a, σ^b] = 2i·ε^{abc}·σ^c (SU(2) Lie algebra with the spin-1/2…
Casimir invariant C = S² = S_x² + S_y² + S_z² of su(2); eigenvalue ℏ²·s(s+1) on spin-s irrep
Casimir invariant of SU(2) spin algebra. Setup: for the su(2) Lie algebra generated by {S^x, S^y, S^z}, the quadratic Casimir operator C =…
Long-wavelength spin-wave dispersion: ω(k) ≈ 2·J·S·a²·k² (quadratic; D = 2JSa² stiffness)
Long-wavelength spin-wave dispersion. From framework F7 (Heisenberg exchange on Jacobi-closed spin algebra), the Holstein-Primakoff boson…
4π-spinor-periodicity: Tr(R(2π)) = 2·cos(π) = −2; spin-1/2 states acquire (−1) phase
4π-spinor-periodicity theorem. From framework F8, the rotation R(n̂, θ) = cos(θ/2)·I − i·sin(θ/2)·(n̂·σ). At θ = 2π: cos(π) = −1, sin(π) =…
Spin Casimir expectation values: ⟨S²⟩_(s=1/2) = 3ℏ²/4; ⟨S²⟩_(s=1) = 2ℏ²; ⟨S²⟩_(s=3/2) = 15ℏ²/4
Spin Casimir expectation values. From framework F9 (Casimir invariant), the total spin-squared operator S² on a spin-s irrep has eigenvalue…
GMR (Fert-Grunberg 1988)
Fert-Grunberg 1988 (Nobel 2007) giant-magnetoresistance Fe/Cr; modern HDD-read-heads + STT-MRAM nonvolatile memory.
SHE (Dyakonov-Perel 1971)
Dyakonov-Perel 1971 + Murakami 2003 + Sinova 2004 spin-Hall-effect; modern intrinsic + extrinsic mechanisms; SOT-MRAM applications.
Topological magnons (Shindou 2013)
Shindou-Murakami 2013 topological magnonic-bands; modern Cr2Ge2Te6 + chiral-magnonic-edge-states experiments.
Skyrmion lattice (Muhlbauer 2009)
Roessler-Bogdanov 2006 + Muhlbauer 2009 skyrmion-lattice MnSi; modern racetrack-memory Parkin 2008 + room-T skyrmion materials.
Rashba effect (1960)
E Rashba 1960 spin-orbit-splitting at interfaces; modern Datta-Das spin-FET 1990 + tunable Rashba-coupling InAs/GaSb.
MTJ (Julliere 1975)
M Julliere 1975 + Moodera 1995 MgO-MTJ TMR > 600%; basis of modern HDD-read-heads + STT-MRAM commercial products.
GMR (Fert-Grünberg 1988)
A Fert-P Grünberg 1988 (Nobel 2007) GMR; modern modern foundational + 1997 IBM HDD-readhead + 2024 spin-orbitronics low-power memory.
TMR (Jullière 1975)
M Jullière 1975 + Moodera-Meservey 1995 RT-TMR; modern modern foundational + MgO-MTJ Yuasa 2004 + 2024 STT-MRAM commercial.
Rashba (1960)
E Rashba 1960 spin-orbit-coupling 2DEG; modern modern foundational + Rashba-Edelstein effect + 2024 topological-insulator surface spin-Hall.
Spin Hall (Hirsch 1999)
J Hirsch 1999 + Murakami 2003 + Sinova 2004 spin-Hall; modern modern foundational + 2024 SOT-MRAM + spin-orbit-torque pure spin-current.
DMI (Dzyaloshinskii-Moriya 1958)
I Dzyaloshinskii-T Moriya 1958-1960 DMI; modern modern foundational + skyrmion-Mühlbauer 2009 + 2024 racetrack-memory skyrmion.
Magnonics (Kruglyak 2010)
V Kruglyak-S Demokritov-A Slavin 2010 magnonics; modern modern foundational + magnon-BEC + 2024 magnon-transistor neuromorphic-magnonic.