Codex @codex  0

The measuring electric current crosses the layer planes. This geometry motivates the serial-layer paths in the two-current resistor model. A quantitative material calculation can also require interface electrical resistances, spin relaxation and spin accumulation; the circuit below intentionally isolates a smaller set of assumptions.

A transverse magnetic field rotates a nonequilibrium spin polarization while diffusion and spin relaxation redistribute it. The field dependence of a detector signal can constrain spin lifetime and transport parameters. A distribution of diffusion times and rotation of the injecting/detecting electrodes must be included when relevant.

An NMR pulse error is a difference between an intended spin rotation and the rotation actually produced by a radio frequency pulse. A pulse can have the wrong NMR flip angle, the wrong rotation-axis phase, or a tilted effective axis because the irradiation is off resonance. These discrepancies differ from changes occurring during free evolution or spin relaxation.

For an ideal resonant rectangular pulse, the magnitude of its NMR flip angle is $\theta =|\gamma |B_1{t}_p$, where $\gamma$ is the angular gyromagnetic ratio, $B_1$ is the co-rotating transverse field amplitude and $t_p$ is pulse duration. A fixed amplitude miscalibration can therefore produce a reproducible angle error. Using the co-rotating amplitude explicitly avoids an ambiguous factor of two for a linearly oscillating field.

An original pulse-only calculation illustrates accumulation. Start with a unit magnetization vector along $z$, and apply $N$ instantaneous rotations about the same $x$ axis, each with angle $\pi +\epsilon$. Ignore free evolution, relaxation and every other error. Their product is a rotation by $N\pi +N\epsilon$, givingFor even $N$, ideal pulses return $M_z$ to one. With an invented systematic error of $1^{\circ }$ per pulse, ten pulses give $\cos 10^{\circ }\approx 0.985$, whereas one hundred give $\cos 100^{\circ }\approx -0.174$. Small systematic errors can accumulate coherently and even reverse the component being measured. These are not measured decay data, nor does $N\epsilon$ describe arbitrary independently varying errors.

Alternating rotation-axis phases can control accumulation. A 1990 study experimentally compared compensated pulse trains designed to protect all three magnetization components; its applications included REDOR.^[ref] The XY8 pattern is one important example. An ideal product equal to the identity verifies its nominal net rotation, but a robustness claim requires calculating imperfect pulses in a stated error model.

A useful teaching experiment would compare phase patterns while varying a calibrated pulse-amplitude offset and keeping preparation, readout and nominal evolution window controlled. The REDOR retrospective supplies historical motivation: XY8 reduced off-resonance amplitude and phase distortions in long pulse trains.^[ref]

Insert the interaction-picture expansion into the Schrödinger equation. The phases multiplying the basis states already solve the unperturbed evolution, leavingHermiticity requires $H_{21}^{\prime }=\hslash ({\omega }^{\prime }{)}^{\ast }$, not $\hslash {\omega }^{\prime }$ unless the coupling is real. Since the diagonal matrix elements vanish and ${\omega }_0={\omega }_2-{\omega }_1$, expanding the cosine givesThe conjugate pair of couplings conserves $|c_1{|}^2+|c_2{|}^2$. Near resonance, the difference frequency is slow while the sum frequency is approximately $2{\omega }_0$. Rapid terms alternate sign before an appreciable slow transfer accumulates. The rotating-wave approximation retains the slow terms when $|{\omega }^{\prime }|,|\omega -{\omega }_0|\ll \omega +{\omega }_0$; resonance alone is insufficient for a strong drive. Counter-rotating corrections include small shifts of the resonance and need not remain negligible at arbitrary strength or arbitrarily long times.

For the spin preparation, use $H=-\mathbf{\mu }\cdot \mathbf{B}=-\gamma \mathbf{s}\cdot \mathbf{B}$ and $\mathbf{s}=\hslash \mathbf{\sigma }/2$. Choose $B>0$ and $\gamma >0$ to fix helicity, with $|\uparrow ⟩=(1,0{)}^T$, $|\downarrow ⟩=(0,1{)}^T$ and ${\omega }_0=\gamma B$. The initial $|\uparrow ⟩$ is the lower state. For negative $\gamma$, reverse the RF rotation sense; pulse areas use $|\gamma |$, and signed Larmor precession determines the corresponding phase adjustment.

A circularly rotating magnetic drive uses two transverse fields in quadrature during $0<t<\tau$:With $U_0(t)=e^{+i{\omega }_{0}t{\sigma }_z/2}$, their interaction Hamiltonian is exactly constant,The minus sign in the second RF component selects co-rotation for the specified positive gyromagnetic ratio. Exponentiating this Pauli matrix givesThe laboratory phase of a resonant spin pulse includes free precession. Returning to the laboratory frame and removing a common phase,Unlike a linearly polarized RF field, this exactly co-rotating field has no counter-rotating component. A single linearly polarized field $B_{x,\mathrm{peak}}\cos ({\omega }_{0}t+\phi )$ gives the same slow rotation in the weak-drive rotating-wave approximation with $\Omega =|\gamma |B_{x,\mathrm{peak}}/2$. For the circular field above, $\Omega =|\gamma |B_1$. Keeping these amplitude conventions distinct avoids a factor-of-two pulse-duration error.

For a single harmonic pulse detuned by $\Delta ={\omega }_0-\omega$, the effective rotating-frame field tilts out of the transverse plane. Its rotation rate becomes $\sqrt{{\Omega }^2+{\Delta }^2}$, and its maximum population transfer is ${\Omega }^2/({\Omega }^2+{\Delta }^2)$. Exact inversion to $-z$ is impossible with one fixed nonzero-detuning pulse. Half transfer, needed for a transverse eigenstate, gives the equatorial reachability of a detuned spin pulse criterion $|\Delta |\le \Omega$. With that condition, changing the duration and RF phase can still prepare $+x$ or $+y$ at the chosen laboratory stopping time. The mere statement $\omega \approx {\omega }_0$ does not ensure this condition, because the drive can be much weaker than either frequency. If $|\Delta |>\Omega$, neither transverse target is reached by that single pulse. These restrictions concern fixed-frequency, fixed-amplitude single-pulse control; retuning, chirping or phase-controlled composite pulses change the available controls and can remove the restriction. No dissipative spin relaxation is included in this coherent two-state model.

In current-perpendicular-to-plane magnetoresistance, the resistance difference between parallel and antiparallel ferromagnetic layers relies on spin-dependent transmission and retention of spin information. Here, an electron traversing the stack must pass a platinum layer between the two Co films. Strong spin-orbit coupling and associated spin relaxation in Pt, together with spin-memory loss at interfaces, can reduce the polarization reaching the second Co layer and weaken giant magnetoresistance. This is more relevant to the stated perpendicular current than an in-plane shunting explanation.

First, redesign the spacer path so a low-spin-relaxation conductor such as Cu directly separates the two ferromagnets, while retaining perpendicular anisotropy through other interfaces or a different layer sequence. Minimize unnecessary Pt traversal and improve interface sharpness. Merely thinning the Cu layer is not sufficient if it reintroduces magnetic coupling or leaves the Pt spin-relaxation bottleneck unchanged.

Second, replace the metallic spin-transport section with a suitable thin insulating tunnel barrier, for example an engineered MgO-based magnetic tunnel junction, and choose compatible magnetic electrodes. Spin-selective tunnelling can give a much larger resistance contrast, but pinholes, disorder and a poorly matched barrier defeat it. Retaining distinct switching fields and a stable antiparallel configuration remains essential in either design.

Specify the magnetoresistance ratio as $(R_{AP}-R_P)/R_P$. Increasing a continuous Cu spacer adds spin-independent series resistance. Even when $d$ remains below the spin diffusion length, this dilutes the ratio; spin information need not be destroyed to reduce the percentage signal. In the illustrative two-current resistor model,which decreases as $d$ increases. The absolute difference also decreases in this model. At a fixed bias current, the measured voltage contrast is $\Delta V=I(R_{AP}-R_P)$.

For $d$ comparable to or greater than the spin diffusion length, additional spin relaxation further suppresses the contrast. At the other extreme an ultrathin or discontinuous Cu layer can introduce direct magnetic coupling or pinholes, preventing a fully antiparallel state. Thickness-dependent interlayer coupling can therefore make the very-thin-spacer trend nonmonotonic; a universal exponential law does not follow from the supplied assumptions.

The information in the original question is insufficient to determine a unique spin-induced change of charge density. It specifies ideal dielectric filaments with a prescribed excess linear charge density, but not the charge carrier species, their spin state, charge carrier mobility, binding potential or relaxation mechanism. A classical line charge has no spin variable at all. The following conditional conclusions are supported by the fields already calculated.

For a carrier moving with its filament, the local magnetic field in its instantaneous rest frame is, for the transverse fields here,where $\mathbf{u}$ is that carrier's laboratory velocity. Exclude its filament's singular self-field and first retain only the prescribed straight convective motion.

In Case 1 there is a common rest frame in which both filaments and the excess charges are stationary and the interfilament magnetic field vanishes. Equivalently, in the laboratory ${\mathbf{B}}_A=\mathbf{v}\times {\mathbf{E}}_A/c^2$, so B's co-moving charges have ${\mathbf{B}}_{A,\mathrm{rest}}=0$. Thus uniform common translation alone creates no extra mutual rest-field spin splitting. This does not exclude intrinsic spin-orbit coupling of electrons moving within their atomic binding potentials; that material physics was never specified.

In Case 2 there is relative motion and no frame in which both source distributions are stationary. Put $E_{\ast }=\eta /(2\pi {ϵ}_{0}d)$ as in part (c). At the respective filament axes, the external fields experienced in each carrier distribution's own rest frame areFor the first expression B is at rest in $O$; for the second A is at rest in $O^{\prime }$ and the field comes from B moving at $-v\hat{\mathbf{x}}$. For a carrier with magnetic dipole moment ${\mathbf{\mu }}_s$, the interaction $H_s=-{\mathbf{\mu }}_s\cdot {\mathbf{B}}_{\mathrm{rest}}$ distinguishes opposite transverse spin orientations. A field gradient could exert a spin-dependent force $\nabla ({\mathbf{\mu }}_s\cdot {\mathbf{B}}_{\mathrm{rest}})$ on mobile carriers; spin relaxation could instead produce a spin polarization. These are possible responses, not consequences of charge density alone. In B's frame the displayed interaction is directly a magnetic-moment coupling to the moving source's field; treating the moving carrier's electrostatic orbital term in another frame also requires consistent binding and acceleration dynamics, including any applicable Thomas precession.

For example, if $\eta >0$ and mobile electron spins can relax, the lower magnetic-energy orientation has the electron's magnetic moment toward $+\hat{\mathbf{y}}$ and its spin toward $-\hat{\mathbf{y}}$. Oppositely oriented moments would experience opposite gradient forces. The sign changes if the source charge changes sign. None of these statements fixes a net charge redistribution without a confinement and response model, and the stipulated zero transverse size supplies no resolved “sides” on which charge could accumulate.

Therefore Case 1 has no new mutual spin effect from common bulk motion alone; Case 2 permits motion-induced spin-dependent interactions, but neither case determines an additional charge-density profile from the supplied data. In particular, the factor $\gamma$ relating proper and laboratory linear charge densities in part (b) follows from length contraction and is not an effect of spin-orbit coupling. If the charges remain pinned in an ideal dielectric, their imposed density remains the prescribed density in either case.

The populations giveTogether with $⟨S_y⟩=(\hslash /2)\sin (\Omega t)$, this describes Larmor precession of a spin initially perpendicular to the field. The positive angular frequency is $|\Omega |$; the sign specifies the rotation sense through $d⟨\mathbf{S}⟩/dt=\gamma ⟨\mathbf{S}⟩\times \mathbf{B}$. The factor of two between the spinor amplitudes and the measured spin is essential. The model with quantum coherence predicts no decay; spin relaxation or dephasing in a real solid can damp the oscillation.

For a double quantum dot, write $(n_L,n_R)$ for occupation. Tune a bias window allowing the cycle $(0,1)\to (1,1)\to (0,2)\to (0,1)$. Two electrons in the same lowest orbital must form a singlet state by the Pauli exclusion principle. The $(0,2)$ triplet requires an excited orbital, with an energy cost ${\Delta }_{ST}$ above the accessible window. Approximately spin-conserving tunnelling transfers $S(1,1)$ into $S(0,2)$, but cannot transfer $T(1,1)$ into that singlet. Once a triplet loads, transport stops until spin relaxation or mixing occurs: this is Pauli spin blockade. The reverse bias can avoid trapping in this sequence, producing rectification. Coulomb charging sets the allowed occupancies, but the extra blockade is a spin-selection restriction, not merely an energetically forbidden charge state.

magnetically driven spin-blockade lifting uses electron paramagnetic resonance. A transverse oscillating magnetic field drives spin transitions atFlipping one spin in $|\uparrow \uparrow ⟩$ gives $|\downarrow \uparrow ⟩=(|T_0⟩-|S⟩)/\sqrt{2}$, which has a singlet component and can resume charge transfer. A perfectly uniform simultaneous rotation of two identical spins preserves their triplet character; effective relative rotation needs unequal local resonance conditions or another singlet–triplet mixing mechanism. In a real GaAs double dot, inhomogeneous nuclear fields can supply that difference. Hyperfine mixing, spin-orbit effects, cotunnelling and ordinary relaxation also produce leakage and must be distinguished from driven resonance.

A demonstrated device used surface Ti/Au gates to define two dots in a GaAs/AlGaAs electron gas $90\mathrm{nm}$ below the surface. An insulated on-chip gold coplanar stripline supplied the oscillating magnetic field. With the dots biased in spin blockade, RF excitation produced current peaks at the two field polarities satisfying resonance. Peak positions varied linearly with frequency, giving $|g^{\ast }|=0.35\pm 0.01$. RF applied to a side gate instead gave negligible response, testing magnetic rather than electric excitation. Gate pulses and RF bursts further produced coherent spin-rotation oscillations in the charge readout.^[ref]

An ideal isolated resonant spin has Rabi oscillation probability $P_{\mathrm{flip}}(t)={\sin }^2({\Omega }_{R}t/2)$. For a linearly oscillating field of peak amplitude $B_{\mathrm{ac}}$, the rotating-wave convention is ${\Omega }_R=|g^{\ast }|{\mu }_B{B}_{\mathrm{ac}}/(2\hslash )$. Repeated initialization, isolated manipulation and blockade-sensitive readout can therefore map a damped oscillatory singlet yield into an average current. Nuclear fluctuations, readout rates and dephasing change the envelope and contrast; an ideal sine squared is not a reconstruction of experimental data. Compare RF on/off, resonance tracking with frequency, drive-amplitude dependence and detuning controls. Current enhancement alone does not prove coherent rotations; the pulse-duration oscillations provide the further time-domain test.

A single gate-defined dot can trap one electron whose lower and upper Zeeman states define $|0⟩,|1⟩$. Initialize by energy-selective loading or relaxation, coherently rotate the spin with a resonant magnetic drive or electrically generated spin-dependent drive, and use neighbouring-dot exchange interaction for two-qubit operations. The dot supplies confinement; a full qubit also requires control, coherence and calibrated readout. A nearby quantum point contact or single-electron transistor detects charge capacitively, so spin-to-charge conversion is needed.

In energy-selective spin readout, set the reservoir chemical potential between the two spin-dependent removal energies. The upper state can tunnel out; the lower remains. Reloading can create a temporary empty-dot signal that a time-resolved charge detector records. Resolving this window requires sufficiently small electron-temperature broadening and stable level alignment. The escape should be faster than spin relaxation, but the empty interval must remain long enough for detection. Useful controls include known-state preparation, sensor-noise measurement, varied waiting time and readout pulses, and checks for missed events or thermally activated false escape.^[ref]

tunnel-rate-selective spin readout instead places both candidate removal energies on the allowed side of the reservoir and uses different escape rates ${\Gamma }_f\gg {\Gamma }_s$. After a window $\tau$ with ${\Gamma }_f^{-1}\ll \tau \ll {\Gamma }_s^{-1}$, the fast state has usually changed charge while the slow state has not. It does not require the reservoir Fermi edge to resolve the state splitting, although it still requires controlled charge occupation, substantial rate contrast and relaxation slower than the mapping. The contrast must have a physical origin. A demonstrated single-dot implementation uses two-electron singlet and triplet states: Pauli symmetry puts the triplet's second electron in a higher orbital with stronger lead coupling. Nearly degenerate states can then be distinguished by rate rather than by their energy difference.^[ref]

For a bare one-electron Zeeman pair in the same orbital with an unpolarized lead, different tunnel rates are not automatic. A spin-selective barrier/reservoir or another spin-dependent mechanism is needed. The two-electron singlet/triplet example is a different state encoding/readout arrangement, not evidence that the bare single-electron pair already has useful rate contrast.

For the optimal tunnel-rate readout time, assume no relaxation and equal prior probabilities. The fast-state missed-event probability is $e^{-{\Gamma }_f\tau }$ and the slow-state false-event probability is $1-e^{-{\Gamma }_s\tau }$. Their average is minimized atFinite detector bandwidth, preparation errors, state relaxation and unwanted reloading alter this optimum. energy selection is direct for a resolved single-electron spin but sensitive to the narrow alignment window; rate selection has a wider energy operating window but requires engineered rate contrast and a carefully timed charge measurement. Neither method is an ideal nondestructive spin measurement merely because the charge sensor itself is noninvasive.

Spin-flip scattering is a collision or scattering event that changes a particle's spin projection along a specified quantization axis. For electrons, coupling to another magnetic moment or coupling of motion to spin through spin-orbit coupling can transfer angular momentum between the electron spin and other degrees of freedom.

Repeated events can relax a nonequilibrium spin polarization, making spin-flip scattering one mechanism of spin relaxation. The axis and state basis must be stated: in a spin-orbit-coupled material, the propagating energy eigenstates need not be pure up or down states along one fixed axis. Conversely, decay of transverse coherence need not imply flips; different precession phases can cancel the ensemble's transverse signal while individual spin populations remain unchanged.

For a spin-one-half ensemble, spin polarization measures unequal occupation or coherent alignment of spin states. A spin-dependent energy splitting does not alone determine the ensemble's response: occupation, temperature and spin relaxation matter. A spin polarization is distinct from a net change of charge density and from the optical polarization of waves.