Spintronics

Spintronics studies how spin angular momentum and magnetic order are generated, transported, detected, and used to control electronic or magnetic states. Its central objects are not isolated spins but nonequilibrium spin accumulations, spin-polarized charge currents, pure spin currents, interfacial torques, and magnetization-dependent electrical signals.

A useful device separates four functions:

1. generation, by a ferromagnetic contact, optical orientation, the spin Hall effect, or interfacial spin–momentum locking;
2. transport, by itinerant electrons, holes, magnons, or collective magnetic order;
3. manipulation, by precession, exchange fields, spin–orbit fields, spin-transfer torque, or spin–orbit torque;
4. detection, through a ferromagnetic analyzer, inverse charge–spin conversion, magnetoresistance, optical rotation, or magnetic resonance.

Those stages are linked by interfaces. A large bulk polarization or spin Hall conductivity does not guarantee a large device signal if the interface absorbs, depolarizes, reflects, or shunts the relevant current.

Required background. Density and Current Operators supplies spin-current and torque definitions, while Spin–Orbit Coupling in Solids supplies charge–spin conversion mechanisms.

Helpful background. Ferromagnetism provides injection and magnetoresistive context, while Boltzmann Transport provides the semiclassical transport closure.

Spin density, spin current and transport conventions

Spin-transport literature uses several quantities all called “spin current.” They are not numerically interchangeable.

Let $e>0$ denote the elementary charge, so an electron has charge $-e$. The labels $\uparrow,\downarrow$ refer to physical electron spin $\pm\hbar/2$ along the chosen quantization axis. A spin-resolved conventional charge-current density has units $\mathrm{A\,m^{-2}}$. Define

$$\mathbf j_c = \mathbf j_\uparrow + \mathbf j_\downarrow , \qquad \mathbf j_s^{(q)} = \mathbf j_\uparrow - \mathbf j_\downarrow .$$

The superscript $(q)$ marks the charge-equivalent normalization. Its current polarization is

$$P_j = \frac{ \mathbf j_s^{(q)}\cdot\hat{\mathbf n} }{ \mathbf j_c\cdot\hat{\mathbf n} } = \frac{ j_\uparrow-j_\downarrow }{ j_\uparrow+j_\downarrow },$$

for transport along $\hat{\mathbf n}$. This scalar description is adequate only when a common spin axis remains meaningful. For codirected channel currents, $|P_j|\le1$. With opposite flows, that bound need not hold. A pure spin current has $\mathbf j_c=0$ but nonzero spin flux, so this ratio is then undefined.

The angular-momentum-current tensor $Q_i^{\,a}$ describes flow along spatial direction $i$ of spin angular momentum polarized along $a$. Its units are angular momentum per area per time. We define

$$j_{s,i}^{\,a} \equiv \frac{2e}{\hbar} Q_i^{\,a},$$

so $j_{s,i}^{\,a}$ also has units $\mathrm{A\,m^{-2}}$. The inverse relation is

$$Q_i^{\,a} = \frac{\hbar}{2e} j_{s,i}^{\,a}.$$

For electron transport with quantization axis $z$, a particle-number flux $\mathcal J_{\uparrow,\downarrow,i}$ carries charge current $j_{\uparrow,\downarrow,i}=-e\mathcal J_{\uparrow,\downarrow,i}$. Consequently,

$$Q_i^{\,z} =-\frac{\hbar}{2e}\,j_{s,i}^{(q)}, \qquad j_{s,i}^{\,z}=-j_{s,i}^{(q)}.$$

Thus the two charge-equivalent normalizations above differ by a sign, even though they share units. We retain $\mathbf j_s^{(q)}$ for two-current injection and $j_{s,i}^{\,a}=2eQ_i^{\,a}/\hbar$ for angular-momentum conversion. Some authors call $Q_i^{\,a}$ the spin current; others use a charge-equivalent quantity and suppress its conversion factor. Every quoted spin Hall angle, interface conductance, or torque efficiency should therefore state both normalization and sign.

For collinear transport, the spin accumulation is the splitting of spin-resolved electrochemical potentials,

$$\mu_s \equiv \mu_\uparrow-\mu_\downarrow .$$

For noncollinear transport it becomes a vector $\boldsymbol{\mu}_s$ in spin space. It has units of energy, not magnetization. A spin density and a spin accumulation are related through the appropriate density of states and susceptibility; they are not identical variables.

Below, $\mathbf m=\mathbf M/M_s$ is the unit magnetization, $\hat{\mathbf p}$ is an injected physical-spin direction and $t_F$ is the active magnetic-layer thickness. The electron flow is opposite the conventional current; neither direction alone specifies the spin polarization.

Spin continuity and diffusion

For spin density $s^a$, a local balance equation has the form

$$\frac{\partial s^a}{\partial t} + \partial_i Q_i^{\,a} = \mathcal T^a - \frac{s^a-s_{\mathrm{eq}}^a}{\tau_{\mathrm{sf}}},$$

where $\mathcal T^a$ is a coherent torque density and the final term represents a simple spin-flip relaxation model. When the Hamiltonian does not conserve $s^a$, current and torque can be repartitioned by torque-dipole terms. The measurable content lies in the complete balance law and boundary conditions, not in a current symbol alone.

In a diffusive nonmagnetic conductor, for small departures from equilibrium, isotropic relaxation and a locally linear density–accumulation relation, a common phenomenological equation is

$$\frac{\partial\boldsymbol{\mu}_s}{\partial t} = D_s\nabla^2\boldsymbol{\mu}_s - \frac{\boldsymbol{\mu}_s}{\tau_s} + \boldsymbol{\Omega}_L \times \boldsymbol{\mu}_s + \mathbf S,$$

where $D_s$ is a spin-diffusion coefficient, $\boldsymbol{\Omega}_L$ is the Larmor-frequency vector, and $\mathbf S$ is a source. Without precession or a source, each steady component obeys

$$\nabla^2\mu_s = \frac{\mu_s}{\lambda_s^2}, \qquad \lambda_s = \sqrt{D_s\tau_s}.$$

In one dimension,

$$\mu_s(x) = A e^{-x/\lambda_s} + B e^{x/\lambda_s}.$$

Boundary conditions discard the exponentially growing branch in a semi-infinite lead. The diffusion length is therefore not the mean free path: it combines many momentum-scattering events with a slower loss of spin memory.

Define the electron electrochemical voltage by $\zeta_\sigma=-\mu_\sigma/e$. Then conventional current flows down a voltage gradient, $\mathbf j_\sigma=-\sigma_\sigma\boldsymbol\nabla\zeta_\sigma$, and

$$\zeta_s = \zeta_\uparrow-\zeta_\downarrow .$$

For equal spin-channel conductivities in a nonmagnetic metal,

$$\mathbf j_s^{(q)} = - \frac{\sigma}{2} \boldsymbol{\nabla}\zeta_s .$$

Equivalently, $\mathbf j_s^{(q)}=(\sigma/2e)\boldsymbol\nabla\mu_s$. Specifying the voltage conversion removes the apparent sign ambiguity.

Spin injection across an interface

Consider a ferromagnet F, a nonmagnetic channel N, and an interface c. In a one-dimensional collinear diffusion model it is useful to compare spin-resistance-area products,

$$r_N = \rho_N\lambda_N,$$

and

$$r_F = \frac{ \rho_F\lambda_F }{ 1-P_F^2 }.$$

Here $P_F=(\sigma_\uparrow-\sigma_\downarrow)/(\sigma_\uparrow+\sigma_\downarrow)$ is the bulk conductivity polarization. Both $r_N$ and $r_F$ have units $\Omega\,\mathrm{m^2}$. For positive spin-channel contact conductances per area $\Sigma_\uparrow$ and $\Sigma_\downarrow$, define

$$\begin{aligned} \Sigma&=\Sigma_\uparrow+\Sigma_\downarrow, \\ P_c&=\frac{\Sigma_\uparrow-\Sigma_\downarrow}{\Sigma}, \\ r_c&=\frac{\Sigma}{4\Sigma_\uparrow\Sigma_\downarrow} =\frac{1}{\Sigma(1-P_c^2)}. \end{aligned}$$

The effective contact spin resistance-area product $r_c$ differs from the bare parallel-channel electrical resistance-area product $1/\Sigma$. In the linear, one-dimensional model with semi-infinite F and N regions, a collinear spin-conserving contact, and no interfacial spin relaxation, Rashba (2000) obtains the injected conventional-current polarization

$$P_{\mathrm{inj}} = \frac{ P_F r_F + P_c r_c }{ r_F+r_N+r_c }.$$

Here $P_{\mathrm{inj}}=(j_\uparrow-j_\downarrow)/j_c$ at the interface. These definitions fix the factors in the displayed model; a measured contact voltage can also contain spin-accumulation contributions. The limiting cases are

$$\begin{aligned} r_N\gg r_F,r_c &\Longrightarrow |P_{\mathrm{inj}}|\ll1, \\ r_c\gg r_F,r_N &\Longrightarrow P_{\mathrm{inj}}\to P_c. \end{aligned}$$

The first limit is the conductivity-mismatch problem for a transparent ferromagnet–semiconductor contact: the channel’s large spin resistance suppresses injection. A spin-selective tunnel barrier can raise $r_c$ and restore polarization. An arbitrarily resistive barrier is not automatically optimal, because it reduces charge current, signal bandwidth, and impedance matching.

Real interfaces add:

• spin-dependent interface transmission;
• spin-memory loss from interfacial disorder and spin–orbit coupling;
• interface states and Fermi-level pinning;
• current crowding and nonuniform contact geometry;
• magnon-assisted and inelastic channels;
• proximity magnetism and intermixing.

Bulk polarization, contact polarization, and the polarization delivered to the channel are therefore distinct experimental quantities.

Nonlocal detection and Hanle precession

In a lateral nonlocal geometry, one ferromagnet injects current into a nonmagnetic channel while a distant ferromagnet detects voltage in a nominally open circuit. This separation strongly suppresses ordinary charge-voltage backgrounds.

Take an effectively infinite one-dimensional channel, a point injector at $x=0$ and a detector at $L>0$. Both contacts are sufficiently resistive that spin absorption can be neglected. If the total injected conventional-current difference is $P_{\mathrm{inj}}I$, symmetry sends half into each side of the channel. With cross-sectional area $A_N$,

$$\begin{aligned} R_N&=\frac{\rho_N\lambda_N}{A_N},\\ \zeta_s(x)&=P_{\mathrm{inj}}I R_N e^{-|x|/\lambda_N}. \end{aligned}$$

Indeed, integrating the spin-current balance across the injector gives $I_s^{(q)}(0^+)-I_s^{(q)}(0^-)=P_{\mathrm{inj}}I$, while $I_s^{(q)}=-(\sigma A_N/2)\partial_x\zeta_s$. The detector reads a conductance-weighted average of the two channel voltages. Relative to the local charge voltage, its open-circuit signal is $V_{\mathrm{NL}}=P_{\mathrm{det}}\zeta_s(L)/2$. Therefore

$$R_{\mathrm{NL}}\equiv\frac{V_{\mathrm{NL}}}{I} =\frac{P_{\mathrm{inj}}P_{\mathrm{det}}R_N}{2} e^{-L/\lambda_N}.$$

The polarizations are signed along one common physical-spin axis; reversing one contact reverses this spin signal. For equal polarization magnitudes in the parallel and antiparallel states, the full difference is $\Delta R_{\mathrm{NL}}=|P_{\mathrm{inj}}P_{\mathrm{det}}|R_Ne^{-L/\lambda_N}$. Confusing this difference with either signed level produces a factor-of-two error. A semi-infinite channel accepting all injected spin current on one side doubles the level amplitude. Finite contacts and spin absorption require a new boundary-value solution (Takahashi and Maekawa, 2003).

A transverse field rotates the diffusing spins. Define the one-dimensional diffusion kernel

$$K(L,t)=\frac{e^{-L^2/(4D_s t)}}{\sqrt{4\pi D_s t}}.$$

For injection along one spin axis, the ideal Hanle response contains

$$R_{\mathrm{NL}}(B)\propto\int_0^\infty K(L,t)e^{-t/\tau_s} \times\cos(\Omega_Lt)\,\mathrm dt.$$

The kernel sums over all travel times, the exponential accounts for spin relaxation and the cosine records precession. For this ideal infinite channel, introduce

$$\ell=\frac{L}{\lambda_s},\qquad b=\Omega_L\tau_s,$$

and $z=\sqrt{1-i b}$, using the square root with positive real part. The integral gives

$$\frac{R_{\mathrm{NL}}(B)}{R_{\mathrm{NL}}(0)} =\operatorname{Re}\!\left[\frac{e^{-\ell(z-1)}}{z}\right].$$

Different travel times precess by different angles, so this is generally not a Lorentzian; at sufficient separation the detected projection can even reverse sign. The parameter $b$ becomes order one when

$$|\Omega_L|\tau_s \sim 1, \qquad \boldsymbol{\Omega}_L = \frac{g\mu_B}{\hbar}\mathbf B$$

for an isotropic $g$ factor. For $g=2$ and $\tau_s=1\,\mathrm{ns}$ this corresponds to about $5.7\,\mathrm{mT}$, but the line width also depends on $L/\lambda_s$. Reliable extraction of $D_s$ and $\tau_s$ may require contact-induced relaxation, drift, anisotropic $g$ tensors, stray fields, nuclear fields, and magnetic switching to be modeled rather than absorbed into a single Lorentzian.

Top: the two sides share the injected spin current, giving $\zeta_s/(P_{\mathrm{inj}}I R_N)=e^{-|x|/\lambda_N}$ in the stated infinite-channel model. Bottom: exact diffusion–precession signals, each divided by its own zero-field value, for $L/\lambda_s=1$ and $3$. The more distant detector samples a different travel-time distribution; the sign reversal reflects precession of the measured spin component, not a negative diffusion coefficient.

Giant magnetoresistance

Giant magnetoresistance (GMR) is the dependence of a metallic multilayer’s resistance on the relative orientation of neighboring magnetic layers. It is a nonequilibrium transport effect, not simply the equilibrium energy that favors one alignment.

The Mott two-current picture begins with weak spin-flip scattering: majority and minority carriers conduct approximately in parallel and experience different resistances. For two identical ferromagnetic layers, let $R_\uparrow$ and $R_\downarrow$ be the resistance of one layer for spin quantized along its magnetization. In the parallel state,

$$R_P = \frac{ 2R_\uparrow R_\downarrow }{ R_\uparrow+R_\downarrow }.$$

In the antiparallel state, each channel encounters one majority-like and one minority-like layer, so

$$R_{\mathrm{AP}} = \frac{ R_\uparrow+R_\downarrow }{ 2 }.$$

Thus

$$\frac{ R_{\mathrm{AP}}-R_P }{ R_P } = \frac{ (R_\uparrow-R_\downarrow)^2 }{ 4R_\uparrow R_\downarrow } \ge 0.$$

Defining

$$\beta = \frac{ R_\downarrow-R_\uparrow }{ R_\downarrow+R_\uparrow },$$

the same result becomes

$$\frac{\Delta R}{R_P} = \frac{\beta^2}{1-\beta^2}.$$

This symmetric resistor network explains positive GMR. Different layers or interfaces can have opposite spin asymmetries and can produce inverse GMR; the sign is not a general theorem for every spin valve. Current-in-plane (CIP) and current-perpendicular-to-plane (CPP) geometries sample bulk and interface scattering differently. In CPP structures, the Valet–Fert theory solves spin-dependent diffusion with bulk asymmetries, interface resistances, finite spin-diffusion lengths, and spin accumulation. Layer thicknesses much larger than $\lambda_s$ do not preserve spin contrast throughout their full volume. This loss concerns spin memory; the classical diffusion model does not require electronic phase coherence.

Magnetoresistance mechanisms

Effect	State variable controlling resistance	Typical structure
Anisotropic magnetoresistance (AMR)	angle between current and one magnetic order parameter	single magnetic conductor
GMR	relative orientation of metallic magnetic layers	metallic spin valve or multilayer
Tunneling magnetoresistance (TMR)	relative orientation across a tunnel barrier	magnetic tunnel junction
Spin Hall magnetoresistance	spin reflection and absorption at a spin–orbit-metal/magnet interface	bilayer
Colossal magnetoresistance (CMR)	coupled magnetic, electronic, and structural state	correlated magnetic oxide

For an ideal spin-conserving tunnel junction, Jullière’s model gives

$$\mathrm{TMR} \equiv \frac{ R_{\mathrm{AP}}-R_P }{ R_P } \simeq \frac{ 2P_1P_2 }{ 1-P_1P_2 },$$

where $P_1$ and $P_2$ are tunneling density-of-states polarizations. Crystalline symmetry filtering, barrier defects, bias, temperature, magnons, and interface resonances can invalidate this simple density-of-states formula.

Charge–spin conversion

Spin–orbit coupling permits reciprocal conversion between longitudinal charge flow and transverse spin flow. To make signs explicit, suppose a charge current flows along $+\hat{\mathbf x}$, spin angular momentum flows along $+\hat{\mathbf z}$, and its polarization is $+\hat{\mathbf y}$. We define the sign of the spin Hall angle by

$$j_{s,z}^{\,y} = \theta_{\mathrm{SH}} j_{c,x}.$$

Changing the interface normal or choosing electron flow instead of conventional current changes the verbal handedness. The tensor relation, axes, and sign convention should accompany every number.

The intrinsic Berry-curvature contribution, skew scattering, side jump, and the conventional spin-current operator are developed in Spin–Orbit Coupling in Solids. At the device level, a measured conversion also depends on layer thickness $t$, spin-diffusion length $\lambda_s$, resistivity $\rho$, effective spin-mixing conductance $g_{\mathrm{eff}}^{\uparrow\downarrow}$, interface transparency and current shunting.

A fitted torque efficiency is therefore not automatically the intrinsic bulk spin Hall angle.

The inverse spin Hall effect converts an injected spin flux into a transverse charge signal. Onsager reciprocity constrains direct and inverse linear-response coefficients after all time-reversal-odd variables, especially magnetization, are transformed consistently. It does not excuse ignoring open-circuit boundary conditions or sign conventions.

At a two-dimensional inversion-asymmetric interface, the Edelstein effect produces a nonequilibrium spin accumulation from an in-plane electric field. The inverse Edelstein response converts a nonequilibrium interfacial spin density into a two-dimensional charge current. When the input is an injected spin flux, spin relaxation and escape determine the resulting density and the measured conversion length. This interfacial spin density is not the same object as a bulk transverse spin Hall current, even though both can generate similar torques.

Spin pumping and interfacial mixing

A precessing magnet can emit spin angular momentum into an adjacent conductor. For a slowly varying unit magnetization, the pumped angular-momentum current per area is

$$\mathbf Q_s^{\mathrm{pump}} = \frac{\hbar}{4\pi} \left[ g_r^{\uparrow\downarrow} \mathbf m\times\dot{\mathbf m} - g_i^{\uparrow\downarrow} \dot{\mathbf m} \right].$$

The spatial flow direction here is outward from the ferromagnet; the vector components specify spin polarization. The real and imaginary parts $g_r^{\uparrow\downarrow}$ and $g_i^{\uparrow\downarrow}$ use the dimensionless-channel convention per area, with units $\mathrm{m^{-2}}$, not electrical conductance per area. A receiving conductor can develop spin accumulation and send angular momentum back, reducing the net exported flux (Tserkovnyak et al., 2002).

Neglecting the imaginary part or its small gyromagnetic-ratio correction, the net interfacial angular-momentum loss adds Gilbert damping:

$$\Delta\alpha = \frac{ |\gamma|\hbar }{ 4\pi M_s t_F } g_{\mathrm{eff}}^{\uparrow\downarrow}.$$

Here $g_{\mathrm{eff}}^{\uparrow\downarrow}$ is real and $|\gamma|$ is in $\mathrm{s^{-1}T^{-1}}$. With interfacial spin-memory loss, this damping parameter measures total angular momentum lost by the magnet; it is not necessarily the conductance for spin reaching the bulk receiving layer. Inverse spin Hall voltage or a nonlocal detector samples that transmitted part. Ultrafast terahertz emission can also detect spin flux, but the slow-precession formula above does not by itself describe femtosecond nonequilibrium transport. A linewidth increase alone is not unique proof of spin pumping: two-magnon scattering, radiative damping, eddy currents, inhomogeneous broadening, and magnetic proximity effects require controls.

Spin pumping and spin-transfer torque are reciprocal interfacial processes. One exports angular momentum from a moving magnet; the other absorbs angular momentum from an incident nonequilibrium spin distribution.

Spin-transfer and spin–orbit torques

The Landau–Lifshitz–Gilbert equation with current-induced torques can be written

$$\dot{\mathbf m} ={}-|\gamma|\mu_0\mathbf m\times\mathbf H_{\mathrm{eff}} +\alpha\mathbf m\times\dot{\mathbf m} +\boldsymbol\tau_{\mathrm{STT}}+\boldsymbol\tau_{\mathrm{SOT}}.$$

For a spin flux polarized along $\hat{\mathbf p}$ and absorbed by a uniform magnetic layer, the leading angular structures are

$$\boldsymbol{\tau}_{\mathrm{DL}} = \tau_{\mathrm{DL}} \mathbf m\times \left( \hat{\mathbf p}\times\mathbf m \right),$$

and

$$\boldsymbol{\tau}_{\mathrm{FL}} = \tau_{\mathrm{FL}} \mathbf m\times\hat{\mathbf p}.$$

The first is called damping-like because its symmetry can oppose or reinforce Gilbert damping for a suitable trajectory. The second has the symmetry of precession about an effective field. The names describe vector form, not a unique microscopic origin.

Define a signed magnetization-torque efficiency $\xi_{\mathrm{DL}}$ through

$$\tau_{\mathrm{DL}} = \frac{ |\gamma|\hbar }{ 2eM_s t_F } \xi_{\mathrm{DL}}j_c,$$

The efficiency includes conversion, transmission, and the sign appropriate to this equation for magnetization. For example, if the transverse angular momentum absorbed into an electron-spin ferromagnet per area and time is $(\hbar/2e)\eta j_c\,\mathbf m\times(\hat{\mathbf p}\times\mathbf m)$, then $\mathbf M=-|\gamma|\mathbf S_{\mathrm{ang}}$, with $\mathbf S_{\mathrm{ang}}$ the angular-momentum density, gives $\xi_{\mathrm{DL}}=-\eta$. A signed magnetization-torque efficiency must therefore not be identified silently with the angular-flux conversion ratio.

For a spin valve, $\hat{\mathbf p}$ is the physical injected-spin direction set by a polarizing magnetic layer, and the torque is usually called spin-transfer torque. In a spin–orbit bilayer, $\hat{\mathbf p}$ can be generated by the spin Hall or Edelstein effect and the torque is usually called spin–orbit torque.

In a smoothly varying magnetic texture, current within the magnet also produces adiabatic and nonadiabatic torques:

$$\boldsymbol\tau_{\mathrm{texture}} ={}-(\mathbf u\cdot\boldsymbol\nabla)\mathbf m +\beta_{\mathrm{na}}\mathbf m\times \left[(\mathbf u\cdot\boldsymbol\nabla)\mathbf m\right].$$

The drift parameter $\mathbf u$ is proportional to current polarization, and $\beta_{\mathrm{na}}$ is a nonadiabatic coefficient. This continuum form assumes a texture smooth on microscopic transport scales.

For a collinear polarizer and a nearly circular small-amplitude mode about a uniform equilibrium, the linear antidamping threshold follows by balancing spin torque against damping. With $H_{\mathrm{eff}}>0$ the mode field scale,

$$|j_c^{\mathrm{crit}}| \sim \frac{ 2e }{ \hbar } \frac{ \alpha M_s t_F }{ |\xi_{\mathrm{DL}}| } \mu_0H_{\mathrm{eff}}.$$

The required polarity supplies antidamping for the chosen state. Crossing this threshold can produce sustained oscillation or reversal; it does not guarantee switching. In particular, a perpendicular magnet with in-plane spin polarization generally has a different spin–orbit-torque switching problem, including symmetry breaking and possible static loss of stability. Its threshold need not scale as $\alpha$. Nonuniform reversal, pulse length, heating and the desired error probability further affect actual write currents.

Materials and devices

Different devices need different combinations of transport length, readout contrast and write efficiency:

Platform	Main function	Useful attributes	Frequent limitation
Metallic ferromagnet/nonmagnet	injection, GMR, spin torque	high current density, mature interfaces	short $\lambda_s$, Joule heating
Magnetic tunnel junction	TMR readout, STT switching	large resistance contrast, nonvolatility	barrier defects, write stress, stochastic switching
Heavy spin–orbit metal or semimetal	spin Hall generation and inverse detection	efficient charge–spin conversion	shunting, resistive loss, interface uncertainty
Semiconductor or two-dimensional channel	gate control, long-distance transport	tunable density and spin–orbit coupling	contact mismatch, disorder, environmental sensitivity
Magnetic insulator	magnon transport, spin pumping	negligible electronic charge current in magnet	interface conversion and magnon relaxation
Antiferromagnet or ferrimagnet	fast dynamics, compensated or low-moment operation	weak stray fields, rich torque symmetry	readout, domain control, material variability

Read and write architectures

Metallic spin valves and GMR multilayers enabled compact magnetic-field sensors and hard-disk read heads. A magnetic tunnel junction uses TMR to distinguish a free layer from a reference layer. In spin-transfer-torque MRAM, the current used to read the tunnel junction also passes through it during writing. In many spin–orbit-torque designs, a lateral current in an adjacent layer writes a magnetic element while the tunnel junction provides a separate read path.

The magnetic energy barrier must satisfy

$$\Delta \equiv \frac{E_b}{k_BT}$$

large enough for retention, while write torque must still cross the barrier quickly and reliably. Magnetic Anisotropy explains this barrier and its thermally activated crossing. Lowering $E_b$ reduces write cost but also retention; increasing damping can accelerate settling while raising antidamping current.

“Low power” is not a material property. Energy per successful operation must include line resistance, pulse duration, capacitive loading, write errors, standby leakage, peripheral circuitry, and the cost of maintaining the required field or symmetry breaking.

Interpreting a measured spin signal

Electrical spin signals must be compared with ordinary charge transport and the magnetic state. Reversing one injector or detector tests the predicted spin parity, while a transverse-field Hanle scan tests precession. Neither check alone excludes stray-field Hall signals, thermoelectric backgrounds or contact-induced relaxation. Geometry and contact absorption belong in the transport fit.

For torque measurements, spin-torque ferromagnetic resonance and harmonic Hall analysis separate angular responses only within a model. Oersted fields, anomalous Nernst voltages, spin Seebeck response, planar Hall mixing and current crowding can imitate some torque symmetries. Independent resistivity, magnetic thickness, damping and anisotropy measurements reduce the ambiguity. Pumping and reciprocal detection provide complementary checks, with interfacial spin loss treated explicitly.

Exercises

1. Diffusion from a narrow injector

A one-dimensional nonmagnetic channel receives a localized steady spin source at $x=0$. Away from the source,

$$\frac{\mathrm d^2\mu_s}{\mathrm dx^2} = \frac{\mu_s}{\lambda_s^2},$$

and $\mu_s$ must remain finite as $x\to\pm\infty$. Find the profile if $\mu_s(0)=\mu_0$. What detector spacing reduces the signal to $1/e$ of its interfacial value?

Solution

On the right, finiteness removes the growing exponential; on the left it removes the exponential that grows toward negative infinity. Continuity at the injector gives

$$\mu_s(x) = \mu_0 e^{-|x|/\lambda_s}.$$

The ratio at a detector a distance $L$ away is

$$\frac{\mu_s(L)}{\mu_s(0)} = e^{-L/\lambda_s}.$$

It equals $1/e$ at

$$L=\lambda_s.$$

The derivative jumps at $x=0$ because the idealized injector is a delta-function source. A finite-width injector smooths that cusp.

2. Two-current GMR

Derive the parallel and antiparallel resistances of the symmetric two-layer model and show that $R_{\mathrm{AP}}\ge R_P$.

Solution

In the parallel state, the two channel resistances are $2R_\uparrow$ and $2R_\downarrow$ in parallel:

$$\frac{1}{R_P} = \frac{1}{2R_\uparrow} + \frac{1}{2R_\downarrow}.$$

Therefore

$$R_P = \frac{ 2R_\uparrow R_\downarrow }{ R_\uparrow+R_\downarrow }.$$

In the antiparallel state, each channel has series resistance $R_\uparrow+R_\downarrow$, and the two equal channels are in parallel:

$$R_{\mathrm{AP}} = \frac{ R_\uparrow+R_\downarrow }{ 2 }.$$

Their difference is

$$R_{\mathrm{AP}}-R_P = \frac{ (R_\uparrow-R_\downarrow)^2 }{ 2(R_\uparrow+R_\downarrow) } \ge0.$$

Equality holds only when the two spin channels scatter identically. The proof applies to the stated resistor model, not automatically to every nonlinear or quantum-coherent device.

3. Charge-equivalent and angular-momentum currents

A bilayer carries $j_{c,x}=8.0\times10^{10}\,\mathrm{A\,m^{-2}}$ and has $\theta_{\mathrm{SH}}=0.12$ in the convention $j_{s,z}^{\,y}=\theta_{\mathrm{SH}}j_{c,x}$. Find $j_{s,z}^{\,y}$ and $Q_z^{\,y}$.

Solution

The charge-equivalent spin-current density is

$$\begin{aligned} j_{s,z}^{\,y} &= (0.12) (8.0\times10^{10}) \\ &= 9.6\times10^9\, \mathrm{A\,m^{-2}}. \end{aligned}$$

The angular-momentum-current density is

$$Q_z^{\,y} = \frac{\hbar}{2e} j_{s,z}^{\,y}.$$

Using $\hbar/(2e)\simeq3.29\times10^{-16}\,\mathrm{J\,s\,C^{-1}}$,

$$Q_z^{\,y} \simeq 3.16\times10^{-6}\, \mathrm{J\,m^{-2}}.$$

Since $\mathrm J$ equals angular momentum per unit time, this is equivalently angular momentum per area per time. The numerical values differ by the dimensional factor $\hbar/2e$; neither should be quoted without units. For physical electron-spin channels quantized along $y$, the corresponding conventional-current difference is $j_\uparrow-j_\downarrow=-9.6\times10^9\,\mathrm{A\,m^{-2}}$, with the opposite sign.

4. Antidamping threshold and retention

Take $\hat{\mathbf p}=\hat{\mathbf z}$, $\mathbf H_{\mathrm{eff}}=H_0\hat{\mathbf z}$, $H_0>0$, and retain a constant damping-like coefficient $\tau$. Linearize the Gilbert equation about $\mathbf m=\hat{\mathbf z}$. Which sign and magnitude of $\tau$ destabilize this state? How does the result relate to reducing the anisotropy of a perpendicular uniaxial memory element?

Solution

For $\mathbf v=(m_x,m_y)^{\mathsf T}$, let $J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$ and $\omega_0=|\gamma|\mu_0H_0$. The linearized equation is

$$(I-\alpha J)\dot{\mathbf v}=(\omega_0J-\tau I)\mathbf v.$$

Its growth rates are

$$\lambda_\pm= \frac{-\tau-\alpha\omega_0\pm i(\omega_0-\alpha\tau)}{1+\alpha^2}.$$

Positive $\tau$ stabilizes this particular state; negative $\tau<-\alpha\omega_0$ makes the real part positive. Substitution of the defined torque efficiency yields the displayed current threshold. This is a local stability calculation, not a proof of deterministic reversal.

For an ideal uniaxial macrospin with $\mu_0H_0=2K_{\mathrm{eff}}/M_s$, the threshold magnitude is proportional to $\alpha K_{\mathrm{eff}}t_F/|\xi_{\mathrm{DL}}|$. Reducing $K_{\mathrm{eff}}$ lowers this threshold but also lowers $E_b=K_{\mathrm{eff}}V$ and exponentially shortens thermal retention. A noncollinear torque geometry or nonuniform reversal requires a different stability calculation.

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