As the electron passes through the electric field of the nucleus, a magnetic field is produced in the reference frame of the electron.

Mott polarimeter

Based on the well-known connectionbetween the direction (cw or ccw) of circularly polarized lightand the spin polarization (up or down) of the excited electrons,

ink was finally found in the form of magneto-chiraldichroism, that is, a difference in the magnetic optical activity ofthe two enantiomers of a chiral medium.

c

G

polarization P.

/H-TaS 2 CMIS (c) and a S-MBA/H-TaS 2 CMIS (d) m

c,d

Magnetic-field-dependent

, Magnetic-field-dependent tunnelling current m

ceeding 60%,

a robust tunnellingmagnetoresistance >300%

unique vdW gaps,

Therecently emerged 2DAC

mCP-AFM

low spin selectivity or limited stability, andhave difficulties in forming robust spintronic devices5–8.

d or exchange interaction with magnetic atoms,thus preserving the time-reversal symmetry

relative to the square root of frequency and its value is usually on the order of nV/√Hz

When working with lock-in amplifiers, the input bandwidth is usually small, so the shot noise does not affect the measurements as much.

codimension

I is not a regularvalue!

{β1, β2 , . . . , βn } of V∗ is said to be dual to the basis {b1, b2, . . . , bn } ofV

φ∗ d  dφ∗

J ∑I(ψ−1i ◦ ψj )∗ ( fI det(D(ψ−1i ◦ ψj )I,J )).

ψ−1i

ψi (Ui )

Superachromatic Wave Plates

each consists of three quartz and three magnesium fluoride (MgF2) plates that are optically cemented to maximize transmission and carefully aligned to minimize the wavelength dependence of the retardance.

We do not recommend disturbing this retaining ring, as it is likely to affect the optical alignment of the fast axis of the wave plate.

Zero-Order Achromatic Wave Plates

Mounted Achromatic

LCP zero-order wave plates produce a smaller decrease in retardance at larger AOIs.

J = L + S is what is conserved, so the spin-orbit coupling should take the form L · S

c is the specific heat of a quantum oscillator calculated earlier with naturalfrequency vsq,

vs

ΘD

this can be justified by considering that the lattice has acharacteristic length scale

all three acoustic modes

singular value.

fibre of φ

does notrefer to local parametrizations.

it turns outthat practically all abstract manifolds can be embedded into a vector space

nverse of ψ, which is a map ψ−1 : ψ(U) → U, is continuous

(ii) Dψ(t) is one-to-one for all t ∈ U;

Changing the order of integration (see Remark 5.3

α be a k − 1-

tj ,

formal linear combination ∑pq1 aq {xq }, which represents a distribution of pointcharges, and the linear combination of vectors ∑pq1 aq xq,

guage of linear algebra, the k-chains form a vector space with a basisconsisting of the k-cubes

formal linear combination

f (t1 , t2) dt2 dt1  − f (t1, t2) dt1 dt2. How can this besquared with formula (5.1)?

p∗ (h) det(Dp) ds1 ds2 · · · dsk

almost completelyunaffected

Although the image c(R) may look very different from the blockR,

, k-forms can beintegrated over k-dimensional parametrized regions

Thenwe know that α is exact.

orms this means that α  dg,

α  F · dx,

o by the substitution formula, Theorem B.9,we have ∫c◦p α  ± ∫c α, where the + occurs if p′ > 0 and the − if p′ < 0

g

Differentiationand integration are related

lmost completely

formula is seldom used to calculate pullbacks in practice

dφi1

curl(F) · dx  ∗dα.

div(F)  ∗d∗α.

ector-valued 1-form

exterior differentiation

By convention, forms of negative degree are0

exterior differential calculus

c′(0)  0,which does not span a line.

parametrizations to give a formal definition of the notion of amanifold in Chapter 6.

closed square is not a manifold, because the corners are not smooth.1

affine subspaces

Pulling back forms is nicely compatible with theother operations that we learned about (except the Hodge star).

(α, β)  (∗α, ∗β)

d(αβ)  (dα)β + (−1)k α dβ for all k-forms α and l-forms β

for any increasing multi-index I, Ic denotes the complementary increasingmulti-index, which consists of all numbers between 1 and n that do not occur in I

dα ∑Id fI dxI .

d( f g)  f dg + g d f

s a graded algebra.

representative linear interferometric techniques

markedly increasing the writing speed of SOT magneticrandom-access memory devices.

coherentmagnetization switching

ultrafast heating model a

ncubation delay

~9-picosecond electrical pulse

the worse the reversing of a filter will be; hence, inverting a filter is not always a good solution as the error amplifies. Deconvolution offers a solution to this problem.

∂ fi∂x j ∂ fj∂xi

(−1)k

( fI dgJ + gJ d fI )

Thereforeαn+1  0for any form α on Rn of positive degree.

the increasing multi-indices of degree k

To avoid such ambiguities it is good practice tostate explicitly the domain of definition when writing a differential form

Single prism

The use of transfer matrices in this manner parallels the 2×2 matrices describing electronic two-port networks, particularly various so-called ABCD matrices

ray transfer matrix (RTM) M,

n sin θ,

s beam can be propagated through an optical system with a given ray transfer matrix by using the equation

using transfer matrices of higher dimensionality, that is 3×3, 4×4, and 6×6, are also used in optical analysis.

Writing the wave equations in the light-cone coordinates returns this equation without utilizing any approximation.[18]

e ν(x, y)e i(βν z−iωt)

And in a lossless waveguide, power conservationrequires

Special numerical methods which exploit the structure of the oscillation are required, an example of which is Ooura's method for Fourier integrals[6] This method attempts to evaluate the integrand at locations which asymptotically approach the zeros of the oscillation (either the sine or cosine), quickly reducing the magnitude of positive and negative terms which are summed.

2

By this mode of argument itbecomes transparently clear how the a that enters into the prefactor comes toacquire its (otherwise perplexing) absolute value braces

2x

of two Lorentz oscillators, 𝑈(𝑄,! , 𝑄! )

The dot product of a scalar with anything is zero.

wedge product is p∧q = ½[(pq)−(pq)*]. This is pure imaginary, and constitutes the high-grade piece of the ordinary product. This has norm |p||q|sin(θ) where θ is the angle between the two vectors, which agrees with the ideas in reference 5.

The second term in on the RHS of equation 61 is the wedge product V∧W. The Trace of the first term is the dot product.

A×B·C = A∧B∧C

grade-flipping does not change the cardinality of the basis.

That is to say, it will have the largest possible grade, namely grade=d.

A ⌞ B:=⟨A B⟩r−s    (right contraction)     (51a)         (forward contraction) A ⌟ B:=⟨A B⟩s−r

A •H B:=⟨A B⟩|s−r|    provided r>0 and s>0     (51e)         (Hestenes inner product)

1s 1v         D=1     1s 2v 1b        D=2    1s 3v 3b 1t       D=3   1s 4v 6b 4t 1q      D=4

e earlier definition of dot product (equation 7) and the definition of components (equation 45).

There is no advantage in imagining some super-vector that has the γi vectors as its components.

γi γj = − γj γi      for all i ≠ j

In Minkowski spacetime, any such basis will have the following properties:

:= γ1γ2             B := γ2γ3            then    AB := γ1γ3

sC = s·C                 = s∧C

It is easy to calculate the geometric product: AB = 1 + γ1γ4 + γ2γ3 +  γ1 γ2 γ3 γ4

We make a point of keeping things non-chiral, to the extent possible, because it tells us something about the symmetry of the fundamental laws of physics, as discussed in reference 3.

The paintbrush picture is a little dodgy in the case where C is a scalar

Suppose P, Q, and R are grade=1 vectors. Then P ∧ (Q·R) is simple. It’s just a vector parallel to P, magnified by the scalar quantity (Q · R).

|p−q| to p+q inclusive (counting by twos)

2.15  More About the Geometric Product

w define the wedge product between any blade and any blade.

P∧Q∧R := 1  6 (PQR + QRP + RPQ − RQP − QPR − PRQ)

P∧(Q∧R) := P∧Q∧R

blade is defined to be any scalar, any vector, or the wedge product of any number of vectors.

vector,   bivector,   trivector, ⋯,  multivector

vector,   bivector,   trivector, ⋯,  blade

quantity γ0 γ1 + γ2 γ3

It is necessarily either a blade or the sum of blades, all of the same grade.

PQ = −QP = P∧Q     iff P ⊥ Q

grade≤1

A bivector can be visualized as a patch of flat surface, which has area and orientation

We add bivectors edge-to-edge,

The eigenvalues of Z are x ± iy an

we are motivatedtherefore to introduce the operation

the fol-lowing symmetries:

The statevariables are divided into two groups.

and it would haveled to ~xxfs ¼ 0

free energy being a perfect differential, and be-cause the constitutive relations (47) do not involveconvolution integrals

prospects of obser-ving a magnetic monopole are rather remote [38, 39],for now it is appropriate to regard the Tellegen med-ium as chimerical

Hence, non-zero va-lues of BðwÞ of actual materials are not know

h magnetoelectric tensors arecommonplace. Typically, such properties are exhibitedat low frequencies and low temperatures.

The constitutive functions must be characterized aspiecewise uniform

Physically, this constraint arises from the follow-ing two considerations:

generalized duality transformation

reciprocity constraint. But it is not, because itdoes not impose any transpose-symmetry requirementson ~ee, ~aa, ~bb and ~nn

D x; tð Þ ¼ e0 ~EE x; tð Þ þ ~PP x; tð Þ ; ð15Þ~HH x; tð Þ ¼ m
10 ~BB x; tð Þ
~MM x; tð Þ

Both ~PP x; tð Þ and ~MM x; tð Þ are nonunique to the extentthat they can be replaced by ~PP x; tð Þ
r  ~AA x; tð Þ and~MM x; tð Þ þ ð@=@tÞ ~AA x; tð Þ, respectively, in (12) and (13)without affecting the left sides of either equation.

r  ~MM x; tð Þ

Actually, only spatialaveraging is necessary [6], because it implies temporalaveraging due to the finite magnitude of the universalmaximum speed e0m0ð Þ

he constitutive parameters satisfy the On-sager relations

deviation
S of theentropy from its equilibrium value as the quadratic expres-sion [7]

ndicates averaging over time t

variables are supposed to be odd with respect to are-versal of velocities of the microscopic particles constitutingthe linear medium; in other words

. In crystals and molecular systems,magnetoelectricity – a phenomenon with local coexistence of electric and magnetic moments – takesplace when space inversion is locally broken [8].

The cross-polarization matrices in constitutive relations of bianisotropic media are, in fact,pseudotensors

e cross-polarization coupling vanishes in the long-wavelength limit

the trailing-shield head invented by Michael Mallary. This head offered higher field gradients

control the level of exchange-coupling between grains

oxide-segregant exchange-break between grains

uniaxial anisotropy constant Ku, which in turn is higher for a material with a higher magnetic coercivity

Perpendicular recording uses higher coercivity materials because the head's write field penetrates the medium more efficiently in the perpendicular geometry.

superparamagnetic limit

Hard disk technology with longitudinal recording has an estimated limit of 100 to 200 gigabit per square inch (16 to 31 Gb/cm2) due to the superparamagnetic effect, though this estimate is constantly changing.

19

We measure thesaturation magnetization of our Py layers using vibrating sample magnetometry

Surface magneto-optic Kerr effect

Comparison of themeasured out-of-plane Oersted field with a finite-elementcalculation of th

As expected, we find that theout-of-plane Oersted field is antisymmetric about the centerof the wire, and the equivalent spin-orbit field is approxi-mately constant across the wire width

expected change in the Kerr rotation signal isDwðmÞ ¼ aPolarDhM þ bQuadratic cos 2/polD/M: (5)A derivation of this result using a Jones-matrix calculation isgiven in the supplementary material

quarter wave plate (labeled QWP-1) to compensate a slightbirefringence of the beam splitter

mode-locked Ti:Sapphire laser working at 780 nm centerwavelength.

(4)

Ms is much larger than any of the other field terms

Permalloy, the in-plane anisotropy is negligible

This is replaced by a quarterwave plate QWP-2 for detecting the current-induced in-plane magnetizationrotation.

esulting from two orthogonal effec-tive-magnetic-field components h y0 and hz

We initially align themagnetization along the x0 direction using an external field

in-plane AC current, Iac cos xt, at 1013 Hz with Iac ¼ 10 mA

MOKE is analogous to the anomalous Hall effect (/ m z),while the quadratic MOKE is the analog of the planar Halleffect (/ m x m y).

intimate connection between the permittivity tensor andthe conductivity tensor in electromagnetism.

quadratic MOKE changes the polarization fromcircular to slightly elliptical (see supplementary material).

rotation of the polarization angle due to the mag-netization can be written as 26wðmÞ ¼ aPolarmz þ bQuadraticmxmy þ    ;

We verify the accu-racy of this method for measuring spin-orbit-induced torquesby studying a series of Pt/Permalloy (Ni81Fe19 ¼ Py) bilayers

sensitiv-ity comparable to the techniques based on magnetotranspor

Recently, Montazeri et al. demonstrated the existenceof a quadratic MOKE response in the setup with normal lightincidence.

nance (ST-FMR)6 can be used for metallic magnets with eitherperpendicular or in-plane anisotrop

econd-harmonic Halleffect measurements work well for measuring torques actingon a metallic magnetic layer with perpendicular magneticanisotropy, but for magnets with in-plane anisotropy, the needto separate out thermally induced signals makes this techniquemore difficult to apply.

nvolve current-induced torques arising from spin-orbitinteractions, either in heavy-metal (HM)/ferromagnet (FM)bilayers5,6,11,12 or topological insulator (TI)/FM bilayers

inding values in excellent agreement with spin-torque ferromagnetic resonance measurements

voids interference when sub-sequently overlapping forward- and backward-generatedpair contributions.