with thesame value of differential resistance

Differentialresistance

an angle β (45◦–135◦) relative to the exchangebias of the SAF to produce a nonzero offset angle θ

<5 × 10−14 F,

[Ta(3)/CuN(41.8)/Ta(3)/CuN(41.8)/Ta(3)/Ru(3.1)

synthetic antiferromagnet (SAF

CoFe(0.5)/CoFeB(3.4)

After subtractingthe much weakened background,

Geometry of the MTJ devices

Pulse generator ΙΙ(20¬30 ns

Pulse generator Ι(5¬10 ns)

chiral anomaly

t frustrated ferromagnetic or antiferro-magnetic order, 1

SOTs generated by the anomalous Hall effect inFM/NM/FM multilayers were predicted 13 and experimentallyrealized14

s [7]∆HX = −2 D0 ± Dπ/21 − 2∆HY = −2 Dπ/2 ± D01 − 2(5)(

RXY = RPHEmX mY

ξDL is ex-pected to be approximately independent of tCo

amplitude of the planar-Hall voltageoscillations is proportional to H2

μ0Meff from the first-harmonic Hall sig-nal as a function of in-plane magnetic field swept alongφH = 0 or π/2

provides resultsfor the SOTs in better agreement with HH measurementson samples with in-plane anisotropy

ing-angle sputtering results in relatively high resitivities forthe heavy-metal films: 1.5 μΩ m for Pt and 3.8 μΩ m for Ta atroom temperature.

he efficiency of these spin−orbittorques can be maximized if the magnetic layer is as thin aspossible while maintaining a low magnetic damping, if thesaturation magnetization is minimized,

This means that, to get a whole resonance lineshape, ω0ought to be on the order of GHz.

We like the units where γ ≈ 28 GHz/T,

which show data taken at 4 GHz, which is very far away from the FMR frequencyof this sample (in these field ranges

A hysteresis loop with external field applied out of theplane of the sample, results in: RAHE = 4 Ω

Shunt Through Method:

ρxx = ρ0 + ρ φ,SMRxy m2y

ferrimagnet (Tm 3 Fe 5 O12 = TmIG)

ystems with strong spin-orbit coupling, anE or T can also drive analogous transverse-flowing spincurrents, yielding spin Hall effects (SHE

dissipationless spin current

ny HQ Graphene have led to the advancements such asthe installation of the first commercial automated 2D transfer system inside a glovebox in PI’slab(Fig. 4(c))38, 3

“dry-transfer”

highest SOT efficiency to date17, 20, 21.

advantageous reductions in magnetic damping

Born out of a Nobelwinning physics discovery,

Joule heating by the injected current asthe sole source of a thermal gradient and assume ∇T ∝ I 2R s ,where Rs is the sample resistance.

The ISHE converts a spin currentinto an electromotive force ESHE by means of spin–orbit scattering

Refractive index also varies with wavelength of the light as given by Cauchy's equation: The most general form of Cauchy's equation is n ( λ ) = A + B λ 2 + C λ 4 + ⋯ , {\displaystyle n(\lambda )=A+{\frac {B}{\lambda ^{2}}}+{\frac {C}{\lambda ^{4}}}+\cdots ,} where n is the refractive index, λ is the wavelength

This isbased on the long-held assumption that a SOT’s effect on the magnetization isindistinguishable from that of a magnetic field.

ly quasi-TEmodes, but the design process easily extends to quasi-TM modes.

Major dielectricfunctions only can be obtained for materials with cubic, hex-agonal, trigonal, tetragonal, and orthorhombic crystal systems[139]. Such functions can no longer be meaningfully definedfor materials with monoclinic and triclinic crystal systems;

A necessary condition for the underlyingstructure to represent an orthogonal system of electric suscep-tibilities, A must be wavelength independent.

A rotation ARφ;θ;ψ to diagonalize ε can always be found forsymmetric tensors

only distin-guishable from χB
0 at B ≠ 0

symmetry breaking in space such asby chiral arrangement of matter produce nonsymmetric contri-butions

which is capable of deter-mining the normalized upper 3 × 3 block of the sample Muellermatrix [22].

a truly 3D phenomenon and dispenses withthe requirement of an ideal sheet with infinitesimal small thick-ness.

optical Hall effect extends the elec-trical Hall effect to a truly 3D phenomenon

ωyωz

We are generally not concernedwith the particulars of the resonance,

C current is now at a much smaller frequency (usually a few kHz), sonow the magnetization is perturbed quasistatically and not driven into resonant

δ ˆm is very small so its higher-order terms are negligible

µ0Meff = µ0Ms − 2K⊥/Ms;

α d ˆmdt +

H + µ0Meff( ˆm · ˆZ)ˆZ

τ = τDL + τz

The torques produced by a poly-crystalline thin film must obey Rashba symmetry [18].

endogenous parameter

esonant linewidth on direct current (dc)

by extrapolation of the range of the magnetic field

Pt/permalloy (Py) a

ental method used to quantify SOTs inheterostructures that have in-plane magnetic anisotropy[2,5–7].

residual signals

n at the modulation frequency due to heating.

urrent-induced excitations of a small volume ofmagnetic material with magnetic damping much larger

onreciprocal phase shift in reflection from anuntwinned single crystal of YBa2Cu307_5, below 50 K (top) and at room temperature(bottom)

~ 1 mW

ultimate limit on our sensitivity will be set by the shotnoise of the laser.

nti-reflection coated source

determine that the optical Kerr effectwas the primary source of error with this laser, producing on offset of ~ 70 \irad,depending on the coupling efficiency in the loop

A variant of magneto-optic spectroscopy, X-raycircular dichroism, is an important probe for measuring the orbital contribution tothe overall magnetic moment of materials.

he nonlinear magneto-optic Kerr ef-fect(NLMOKE)

thecontrast mechanism is well understood and linear in the magnetization, unlike thescanned probe microscopes,

eflection ampUtude|r*pp| may be related to elUpticity, and the phase of r ^ is related to rotation.

lliptically polarized with the planeof polarization in the plane of incidence.

En and Es, must be out of phase

the component paraUel to the surface, ks, is real.

for the Kerr effects, with definitions of the coordinate systemsused for the magnetization and the incident and reflected electric fields.

and an analysis basedon the Fresnel equations is no longer valid

i xx along z and i± = ixx ± ie ^

nonlocal response to understand a number of opticaleffects, such as optical activity and the anomalous skin effect.

ctij(u) = since j t = ™ aij(u)Aj

<£»•

eignekets satisfy |m) =|m), a

no overall rotation

/ f extW = - ^ j ' A ext(t),

symmetry of the tensors 6y and /Zy

Magneto-optics only probes an optical penetration depth into the sample, typically ioo A

We used the firstconfiguration for most of the experiments because it allowed us to monitor the power

h usually it is in the former position

A = 670 nm [9]. Thisinstrument suffered from a number of difficulties associated with imperfections in thebulk electro-optic modulator (EOM) and with the laser diode source

Thermal sources, which have very short coherence lengths, proveto be very effective in this regard

Faraday rotation in the fiber loop

The 840 nm interferometer has orders ofmagnitude lower amplitude modulation than the 670 nm interferometer,

placing the sample a sixthof the way around the loop, and modulating at three times the proper frequency(3u>m) [9]

In the normal-incidence Kerrconfiguration, the sample is capable of backscattering a considerable amount of light,and this effect must be addressed.

if we were to rely solely on this method of polarization controlthe tolerance on the extinction coefficient would be insurmountably high: for phasesensitivity of 1 /zrad, the polarization extinction must be better than 120 dB

of other noise sources much earlier than any fundamentalshot noise limit

21.7 nrad/v/mW Hz

if we do not go to thetrouble of specially preparing “sqeezed” photon number states

for example, if we choose to interfereorthogonal linear polarization states, for example by rotating one of the fibers inFig. 2.2 by ninety degrees, we will be able to measure the ellipticity epp = ^m (6ptp).This will come at a considerable cost in signal strength, however, because nearly allof the fight will be in the wrong polarization state when it arrives at the polarizer(see Fig. 2.1).

corresponding to the real andimaginary parts of when we are at normal incidence

We set the linear polarization state in one fiber end to be parallel to the optical table,

we must maintain a finiteloop length of about 30 m in order to keep the modulation frequency in the MHzrange.

e Sagnac interferometer to magneto-optic measurements ofmaterials began in 1989

parasitic effects due to imperfections in the phase modulator will besignificantly reduced near this frequency [10].

They showed that proper preparation of the polarization states travelingin the interferometer allowed them to measure magnetic fields via the Faraday effectof the fiber [6, 7

conventional superconductors break only gauge symmetry,

rows correspond to irreducible representations

ZG
◁ G

such, All Irreps of any Abelian Group, e.g., SO(2), are 1-dimensional

1d subspace

{ρg ·⃗v, ∀⃗v ∈ V (i)} = V (i)

Vector Space V ∼= Fd, an

Aut(H) < Sym(H)

x ⋄ y−1

be a subset of its actions

all (continuous) Permutations

homomorphism that defines how a group acts-on aset, and the set of actions on the Se

antisymmetric outer product

to avoid the formation ofvortices in the SC state.

e ac magnetic field HiSHE generatedvia the iSHE induced charge current JiSHEq is inductively

suppression of the dampinglike torque generated in the Ptlayer by the inverse spin Hall effect, which can be understood by the changes in spin current transport in thesuperconducting NbN layer.

supercurrent is induced along the x axis by enforcing the pairpotential to have a constant phase in a small region close toeach of the two boundaries along x

pair potential i = |i| exp(iφi).

With respect to thisreference frame, a rotation of the axes by π/2 degrees aboutthe z axis leads to a sign change of ˜αD , while a rotation of π/4degrees changes the tensor to ηso = ˜αD σz.

m = −γ m × [Heff + Hso ] + αGm × ˙

textured ferromagnets

as been observed in systems with broken spatialinversion symmetry and strong spin-orbit coupling (SOC)

he maximum achievable spin-orbit torque field is estimated to be on the order of 0.16 mT

such that the latter induces a Group Action on theVector Space, V , such that the latter induces a Group Action on the Vector Space, V ,

group G acts-on itself by conjugation

The set of all (continuous) Permutations of the Set, X = Rd, form a Group,Symc(Rd), the Symmetric Group of the set

G −→ Sym(X)

Group Action on the underlying Vector Space,

Rep of a Group, g ∈ G, on Vector Space

b) HomF(V, W )G

L ∈ HomF(V, W )

simple proportion (gain or attenuation), an integral (low-pass filter) and/or derivative (high-pass filter).

signal of strength M2

A general structure shouldbe regarded as chiral (Fig. 1e), as long as the total twist is non-zero atany point.

The structural chirality is not equivalent to thebreaking of inversion

bsence of any M, there can be an AFM CD unrelatedto M but proportional to the AFM order L in 𝒫𝒫𝒫𝒫-symmetric AFMs asreported in Cr2O3 (

ferromagnetic spintronics

ptical control of chirality andmagnetization,

tead a clean signal of a precessing spin-waveexcitation.

high field regime, the spin-transfer effectcannot produce a full reversal of the thin-layer moment

which involvedcontinuous ferromagnetic layers

pin-polarized electrons flowing from the thinCo layer to the thick layer can switch the moment of thethin layer antiparallel to the thick-layer moment,

the concept of reciprocity to understand Kerr effect, the useof fluctuation–dissipation theorem, which is another principalresult of linear response theory (that is, the study of irreversiblethermodynamics associated with linear processes that are referredto the equilibrium state of a system) yields an identical result

Reciprocity then requiresthat

Onsager reciprocity relations and fluctuation–dissipationtheory rely on the assumption that near equilibrium, macroscopicresponse and decay process occur in the same manner as thedecay of equilibrium fluctuations

ε ω ε ω′ = ′r r r r( , ; ) ( , ;

V r r( , ; )

ielectric function

spin–orbit coupling and exchange splitting

he fiber is highly birefringent and the diode light source has8-μm coherence length, only light that couples, after reflectingfrom the sample, between different axes in the fiber willtraverse optical path lengths that differ by less than a coherencelength and interfere coherently at the polarizer

tements of the symmetries of the electromagneticfield and its measurement entail that the reflection amplitudesbe considered quantum mechanically

polarization state ± near the source, if not as a plane wave.

gen-eral optically active

the reciprocity theorem

becomes larger than two

m through long range dipole-dipole couplin

readout of magnon density and propagation using photons of visible ligh

Samples swept with circularly polarized beams

OKE response measures primarily the TM sublattice.

strongertemperature dependence and the T

This results in a narrow composition rangewhere the two sublattice

lting in a net magnetization equal to zero

ces. In the case of thelight RE (4f electrons < 7) the two sublattices are exchange-coupledferromagnetically whereas for heavy RE (4f electrons ≥ 7) the twosublattices are exchange-coupled antiferromagnetically, forming aferrimagnet.

describes deterministic magnetization reversal ofthe material under the beam with no external magnetic field.

f GdFeCo (ref. 3), TbCo

0 fJ is expected to be sufficient

exhibiting magnetization compensation (andtherefore angular momentum compensation)

0.4 nm Irinterlayers

ferromagnetic sample

magnetization switching usingfemto- or picosecond pulsed lasers3

challenge present theories of AO-HDS

all-optical helicity-dependent switching (AO-HDS)

manipulating magnetic systems without applied magnetic fields have attracted growing attention

On the other hand,SOT is orthogonal to the magnetization of the free layer, which isexpected to provide “instant-on” switching torque.

examine the nature of a ∧ b, consider the formula ( a ⋅ b ) 2 − ( a ∧ b ) 2 = a 2 b 2 , {\displaystyle (\mathbf {a} \cdot \mathbf {b} )^{2}-(\mathbf {a} \wedge \mathbf {b} )^{2}=\mathbf {a} ^{2}\mathbf {b} ^{2},}

here exist explicit cut-and-dried algorithms for calculating the Hodge dual of B, especially if B is known in terms of components in some basis. See the discussion in reference 13, or see the actual code in reference 14.

dot-multiplying by a vector lowers the grade by 1

As another example, in d=3, it converts a vector to a certain “corresponding” pseudoscalar. Meanwhile, in d=4, it converts a vector into the “corresponding” pseudovector (not pseudoscalar)

Spinors (aka Pauli spin matrices)

The matrix representation is particularly efficient if you have one particular rotation and wish to apply it repeatedly, using it to rotate a large number of vectors. The advantage is most conspicuous in four or more dimensions. See section 7.

the standard Schroedinger wave function is a solution the Schroe-dinger equation (107)

definea rotor D by assuming that it satisfies the equationdDdt = 12( emc iB)D

We can generalize (96) to arbitrary states by interpretingρE = 〈∂t ψ iσ3¯hψ†

For a stationary solution with B × s = 0,

we can writeψ = ρ 12 U, where U U † = 1

where ψ′ = ψC† is the wave function relative to the alternative quantizationaxis σ′3. The matrix analog of this transformation is a change in matrix repre-sentation for the column spinor Ψ

i is now the unit pseudoscalar

Now write Ψ in the formΨ = ψu,

with complete generality that ψ in (80) is areal even multivector. Now we can reinterpret the σk in ψ as vectors in GAinstead of matrices. Thus, we have established a one-to-one

where HS is the Schroedinger hamiltonian

a column matrix Ψ

σ · a σ · b = a · b I + iσ · (a × b).