Spin-orbitronics

Spin-Hall effect and spin torque

Spin-Hall-related mechanisms of relaxation of the electronsa) extrinsic spin-Hall effects (originally studied with ragard to explain the AHE in ferromagnets)

(i) Spin Skew (Mott) Scattering (Smith 1958): the SOC due to the field of (heavy-metal) impurities splits the electronbeam, due to the relation

It is independent of the impurity concentration and it does not influence the relaxation time τ1 (inducedby the electric field, τ0 is an intrinsic relaxation time).

(ii) Side-Jump Scattering (Berger 1970)That scattering mechanism is similar in principle to the Stern-Gerlach experiment, and it relatesto usual relaxation time of the medium (in absence of the electric field) and it is dependent on the impurityconcentration

b) Intrinsic spin-Hall effect: arises from the SO-lifted degeneracy of the electronic bands of different spins

=>

Stern-Gerlach experimenthttps://www.youtube.com/watch?v=rg4Fnag4V-E

Schematic view of the Skin Skew Scattering (a),

Side Jump Scattering (b),

Intrinsic SH effect (c)

Phenomenological description of SHE

by befinition of SH angle(direct SHE)

Symmetry analysis leads to

D denotes the diffusioncoefficient

(inverse SHE)

Rashba Hamiltonian and spin galvanic effect

- spin galvanic effect

- inverse spin galvanic (Edelstein) effect

=>

=>

=>

- Inverse spin galvanic effect

- spin galvanic effect

Theory of intrinsic sH effect in the Rashba system

Starting with the Rashba Hamiltonian ,

we deternime the spin dynamics with the Bloch equation (of a two level medium) .Here n denotes the Bloch vector that is equivalent to the spin-½ director (versor),and the coupling „constant” ∆ is defined by the Hamiltonian: and let

Let ∆=(∆1,∆2,0), n=(n1,n2,nz), p(t=0)=(0,p2,0), thus ∆(t=0)=(∆1,0,0), whilen(t=0)=(1,0,0) for the majority spin, and assume ∆1(t)≈const. We obtain

Up to the leading order in the slow-time dependences: , thus,

Utilizing , we arrive at for the majority spin. For the minority spin one has to changethe sign:

The sH-current density reads

When both (minority and majority) bands are occupied, thenand (in general case ).

Inverse spin-Hall effect in ferromagnetic metals

sHE:

IsHE:

In (d), the currentis self-polarizing

In (c), the charge distributionis asymmetric

Spin-Hall magnetoresistance

- non-crystalline AMR

- spin-Hall MR

(a) SHE, (b) SHE (upon switching the charge current off)(c) ISHE(d)-(e) SHE due to the absorption of spin-up carriers

by ferromagnet and ISHE due to their reflection.In (d), absorption is mninimized, thus, the conductivity is maximized.In (e), absorption is maximized, thus, the conductivity is minimized,

(f) Resulting spin accumulation in normal-metal layer

Spin-orbit torque

Spin-Hall spin-transfer torque (SH-STT)The Slonczewski-like STT is induced by spin-Hall current that flows perpendicular to the layer

Ηere, η=ϵ/P defines the so called injection efficiency

Rashba spin-transfer torque (Rashba STT)is due to the current parallel to the layer (non-adiabatic-like STT)

->

In fact, both SOTs can be generated by the Rashba coupling due to the spin galvanic effect (SH-STT) or inverse spin galanic effect (Rashba STT)

Spin-orbitronics of perpendicular-magnetic-anisotropy (PMA) layers

Perpendicular magnetic anisotropy

In the picture: dopant vs. interface induced Dzyaloshinskii-Moriya interaction.

PMA is observed in:

1) Ferromagnetic multilayers (FM/Pt, FM/Pd, FM/Ru, FM/Ta, FM/Au). A strong hybridization

of 3d and 5d orbitals at the interface (DM interaction) enhances the SOC in the ferromagnet of TM.

Especial case is Co/Ni due to a high magnetization at the interface

2) Crystalline alloys (FePt, FePd, CoPt, MnGa, MnAl in L10-symmetry structure,

Mn-based Heusler alloys: e.g. MnAlGa, MnCoGa) deposited on substrate of relevant symmetry.

2) Amorphous RE-TM alloys: GdFeCo, TbFeCo, etc.

3) CoFeB-oxide multilayers (CoFeB-MgO). A hybridization of 3d-orbitals of Fe with 2p-orbitals of O.

PMA is especially useful for data storage, alowing for increasing bit density compared to utiliznig

in-plane magnetization

Dzyaloshinskii-Moriya interaction at the microscopic level (bulk or inter-layer)

It leads to the additional term in the micromagnetic Hamiltonian that breaks

the space-reversal symmetry.

For bulk DMI

For interfacial DMI

(because of the direction of D12 vector, see figure)

Magnetic bubbles and Skyrmions

Note on a „sigma-model” with Skyrme-Fadeev term (1961)

The Lagrangian of density

is conserved under the operations of group. For , the relativistic

problem reduces to the stationary Skyrme-Fadeev sigma-model:

that is a 3D generalization of 2D problem of Belavin-Polyakov.

Skyrmions are unstable excitations of that system for l=0 (Belavin-Polyakov state),

while l>0 results in the stabilization.

vortex (of LMA structure, stabilized by the shape anisotropy of a dot)

Bloch-like skyrmion

Neel-like skyrmion

of large PMA

= magnetic bubble - DW-like ansatz

Neel-like skyrmion

of weak PMA - Belavin-Polyakov ansatz

Let R denotes the skyrmion radius, and D the exchange length of the PMA anisotropy

Neel-like state is preferable by Dzyaloshinskii-Moriya interaction of

The DM energy of Bloch-like skyrmion is zero, while EDM~ᵡp for Neel-like skyrmion

(thus, it introduces chirality-spiltting for any given p).

Magnetic skyrmions are stabilized by perpendicular magnetic anisotropy (PMA), thus,

they are metastable

(a)-(b) Combined absorption-induced SHE and refelction-induced ISHE, In (a), strong absorption-weak reflection,In (b), weak absorption-strong reflection

(c) SHE-ISHE in PMA bilayer; middle absorption and middle reflection

Skyrmion motion can be driven with current via STT or via spin-Hall effect (SHE) when

the magnetic layer is deposited on a non-magnetic metal. The Thiele equation of skyrmion

leads to the longitudinal velocity

While perpendicular velocity is nonzero, in a skyrmion racetrack, it is suppressed

by the interaction with the closer nanostripe edge

We see the STT-induced motion to be similar to that of DW in the viscous regime (below

the Walker-like breakdown)

Let us take a better insight into the spin-Hall driving when looking at the motion of

chiral (Dzyaloshinskii-Moriya) Neel-like DW in 1D system with PMA

In narrow nanostripe, the ordering in y-z plane is favorable,

thus, Neel DW is described with a single dynamical parameter

Current-driven motion of DWs using the spin-Hall effect

The spin-Hall torque of the constant depends on:

a current-density JSE that is a percentage of the current density in the bulk of the substrate,

the thickness of the magnetic layer t

a spin-Hall angle qSH that is defined via th ratio of the transverse „spin” current to the

lngitudinal „chargé” current

In figure, the top layer is added in order

to exclude the interfacial (Rasba or DM-type)

spin-orbit effects, while cross-section asymmetry

results in non-vanishing spin-Hall effect

In DW systems, l corresponds to the DW width and in the DW area

Notice: there is no viscous regime of motion of the Neel-like chiral DW,

(the Walker-breakdown field and current are zero).

Current-driven motion of DWs in layers with PMA

2008

V=100m/s

2011

2013

V=350m/s

(viscous motion)

2014

V=750m/s

(viscous motion)

a) longitudinal writing

b) perpendicular writing

c) perpendicular writing

with nonmagnetic underlayer

(spin-Hall, Rasha,

or Dzyaloshinskii-Moriya torques)

d) perpendicular writing

in synthetic antiferromagnet racetrack

(reducing the stray field HK to zero)

DW-racetrack-design evolution

Since 2013, skyrmion racetracks

are investigated in parallel

Spin transistors and spin injection

Datta, Das 1990

Semiconductor-based spin transistor

In (a), analyzer and polarizer are both rotated by 45o from Y or Z

The output power is

In (b), the Rashba couplingsplits the electron beam of

inducing

Correct quantitative description requires inclusion of the transverse quantization of the electrons

Spin injection at ferromagnet/2D-electron-gas (F/2DEG) interface (of semiconductor-based spin transistor)

Johnson 2002: the electrical voltage shifts the Fermi level of 2DEG due to the Rashbacoupling and the shift is different for the two spin subbands (splittedby Rashba Hamiltonian). The Fermi-energy splitting is due to different lengthsof the maximum wavevectors of the spin-up and spin-down carriers.

This rises a difficulty with quantitative description of the spin-transistor efficiency

=>

Problem: the spin relaxation (change of the angle ϴk)of the electron at the Fe-GaAs interfaceis almost complete at the distance of about 50µm.

All-metal three-terminal device (F/N/F trilayer)

Johnson 1993

Spin injection at N-F interfaces; according to Johnson and Silsbee (1988), the spin current can be tunedby the resistance (impedance)

Spin injection for paralel and antiparallel magnetization orientations in F1 and F2

Hence, the resistence (voltage) causes a difficulty with correct working

All-metal spin-transistor

Johnson 1994

Efficient manipulation of spin injection with a voltage requires using an additional non-magnetic lead

Magnetic field effect transistor (MFET)

Spin-polarized solar battery: circular polarization of light (via filteringthe solar photons) causes the spin polarizationof the photovoltaic electron-hole pairs and result in the spin-polrization of the photocurrentof electrons (hole spin in III-V semiconductors

is relaxed very fast)

Idea of MFET: the width of the depletion layer of a p-n junction can be tunned with the magnetic field (normal to the junction)instead of using electric field, provided g-factorsof the electrons and holes are high (the Zeeman energyis high). Large values of g are obtainable via doping III-V semiconductors with Mn (p-doped or n-dopeddiluted magnetic semiconductors)

In the bottom picture, a scheme of the measurement of the junction efficiency with an electrode of a given widthbetween p and n sectors

Disadvantage of metal-based spin transistors: they do not offer large amplification of the spin current. Unlike in semicoducting devices, draining a small numer of carriers from the transistor base does not result in a hightransmission between the emiter and colector

Spin-orbit effect on the band structure of bulk semiconductors of Zinc-blende (III-V) or diamond (Si, Ge) structure

Optical selection rules. The creation probability of the heavy-hole transition if three times as large as that of the light-hole transition

In the absence of the inversion symmetry, the Rashba or Dresselhaus Hamiltoniansplits the conduction band as well

With the inverson-symmetry brokenvia z-axis confinement:

=>

Spin injection into semiconductors: major mechnisms of the spin relaxation and dephasing

D’yakonov-Perel’ mechanism: in non-centrosymmetric systems, for non-zero k, the effective field

due to e.g. Rashba or Dresselhaus drives the carrier-spin precession.Independently the carrier-momentum undergoes a collision-induced relaxation. Usually, (high-enough temperatures); (the spin-rotation period is large compared to the momentum-relaxation time τp).

The relevant spin-relaxation time is evaluated with

Note: D-P mechanism dominates the spin relaxation in n-doped quantum wells.

Elliott-Yafet mechanism: the spin-flip scattering of the electron due to a k-dependent, SO-induced admixtureof the valence-band states to the conduction-band wave function(similar to the admixture of localized d-states in AMR systems).The Bloch states are no longer spin eigenstates, thus, any proces of the scatteringby impurities or phonons can be related to flpping the spin in the average.

Note: E-Y mechanism dominates the spin relaxation in narrow gap semiconductors.