Basics of spintronicsand

magneto-sensorics

Basics: Drude theory of conductivity and galvanomagnetic transport

Consider the damped electron motion under a constant driving force:

In the stationary state (dv/dt=0), , where µ is called the carrier mobility.

Thus, the current density takes the form . Via the Ohm’s law , one arrives at

the Drude formula

Let us extend the Drude model to the case of the magnetic-field presence:

In the stationary state; and

In the case of B=(0,0,Bz);

Defining the zero resistivity; , one rewrites the Ohm formula in the form

or

Consequences of the theory of galvanometric transport

(i) Hall effectIn the stationary state:

thus

(measuring; i, B, UH, we can determine the charge-carrier density)

=>

(ii) Lorentz magnetoresistance

=>

=> field-induced deflection does not influence the longitudinal conductivity

=> field-induced deflection influences the longitudinal resistivity

Note: Lorentz MR is low except in materials with compensated numer and mass of the electrons and holes, semimetals: Bi, InSb, …. In Bi, Lorentz MR is 18% at the field of 0.6T

Anisotropic magneto-resistance (AMR)

Since the spin current add up to the total current; j=j↑+j↓, the resistivities satisfy 1/ϱ =1/ϱ↑+1/ϱ↓.

The acceleration of the electrons can be expressed withthree relaxation times, where τ↑↓ relates to the spin-flip relaxation

With , (Drude formula) one writes

Here, ϱ’P↑,↓(T) are close to the temperature (phonon) corrections in pure (non-magnetic) metal ϱP↑,↓(T).

With , the total resistivity can be written in the form

In the simplest case ϱ’P↑,↓(T)=ϱP↑,↓(T), at low T, (the temperature corrections are small), we get

Basics: Two-current model of the resistivity of metallic ferromagnets

caused by defects caused by phonons caused by magnetism

Breaking the Mattheisen rule:

Idea of AMR in s-d systems

In 3d-systems spin-flip scattering is weak or completely forbidden (picture on the left).The spin-orbit (LS) coupling allows for the scattering of the 4s-electrons into the 3d-orbitals with or without flippingthe spin (picture on the right)

Theory of AMR in s-d systems (Campbell-Fert-Jaoul 1970)

Consider L-S coupling Hamiltonian as a perturbation to the exchange Hamiltonian,(A is small compared to the exchange field Hz

exch). The spin-↓ wavefunctions up to the second perturbation order are

where, the orientaion of d-orbitals correspond to

The perturbation parameter ϵ=A/Hzexch

The state can only be scattered into the d-state of m=0, while into the d-state of m=0 or m=±2.The resistivity , and ϴ denotes the angle between the incident wave (current

direction) and the magnetization.The resistance contributions

depend on the DOSs

In Ni (the s-d metal with a small spin polarization), the scattering is dominated by the spin-flip processes: s↑ → d↓.In half-metal ferromagnets, the scattering is domianted by the spin-conserving processes: s↑ → d↑.

Therefore the sign of the AMR ratio is positive for Ni (max[AMR]=2%), Co, and Fe while it is negative for half-metal ferromagnets Fe3O4, La0.7Sr0.3MnO3, La0.7Ca0.3MnO3

Especial case is Fe4N, that is a „strong ferromagnet” with negative spin polarization and negative AMR ratio.

In Co2FexMn1-xSi (xϵ[0;1]), one observes the half-metal (min[AMR=-0.4%)to „strong-ferromagnet” transition at x≈0.7, as seen from the plot

For Ni-Co and Ni-Fe alloys, one observes max[AMR]≈6% at 300K

Defining

we arrive at

Lorentz MR AMR Hall effect

n=M/|M|

ϱⱵ(B)=resistivity for j perpendicular to Mϱ‖(B)=resistivity for j paralel to MϱH(B)=Hall resistivity

In polycrystalline materials

Hall pseudovector

=>

Application of AMR to reading heads (barber-pole sensor)

Measurement of AMR:

Let φ be the angle between the magnetizationand the external field

When the field is of the Oersted type, then and

=>

Let (the Oersted field does not play any role now).

Then,

which allows distinguishing between up and down directionsof the magnetization.

Giant magneto-resistance (GMR)

In the picture: orange represents a magnetic reference layer, green – a magnetic free layer, grey – a nonmagnetic layer

Assume the layer thicknesses smaller than the electronic mean free path. Then

For arbitrary orientations of the layer magnetizations

Explanation: due to the stray fields in layers B and C the electrons are spin-polarized (a proximity effect). The uncompensation of the numer of spin-up and spin-down electrons results

in different relaxation times of both.

The quantitative theory of GMR in trilayers (based on the Bolzmann equation)has been developed by Camley and Barnaś (1989)

Grunberg et al. 1989: GMR in Fe/Cr/Fe trilayer. RKKY-like mechanism is responsible for the antiferromagneticcoupling of the Fe layers, while the ferromagnetic coupling is possible depending on the interlayer distance

(a quantitative theory:Bruno and Chappert 1992)

In the figure, AMR of single Fe layer is shownfor comparison

Baibich et al. (Fert group) 1988: GMR in Fe/Cr multilayers

Max[∆R/R*100%]≈100% ! = > a giant effect

Spin valves; GMR in systems with unidirectional anisotropy (a breakthrough in reading-head technology)

Exchange coupling (exchange bias) at the interface of the referencemagnetic layer to the antiferromagnet induces the unidirectionalanisotropy in the former layer. The result is a strongly asymmetric GMR

Dieny, Parkin, Gurney et al. 1991, IBM

Reversal in the pinned layer

Note: despite MR in multilayers is giant, in the spin-valve structure it is several percent only. However, the SV is easier to miniaturize than the AMR (barber-pole) sensors,enabling a technological breakthrough.

Tunneling magneto-resistance (TMR)

Basics: balistic transport of electrons

Consider the electrons travelling through a connection of the smaller length than the electronic mean free path, thus,they are not scattered. Such a transport is called ballistic. The conductance is given by the Landauer-Buttiker formula.The current intensity is a difference of the current from the left and the right leads

The Landauer-Buttiker formula: relates the current intensity to the chemical potentials of the leads.

The difference of them is a contact-like potential Energy

The conductance needs to be modified when there is N conduction bands (channels), with

If the ballistic limit is not achieved, the Landauer-Buttiker formula is modified via inclusion of the transmission coefficientsfor each conduction bands:

Tunneling through the barier

The transmission probability reads

In the limit of thick/high barier ,

Including the transverse motion in the wavefunctionone modifies the formulae ( ) with

In the picture: orange represents a magnetic reference layer, blue – an insulating layer. The current is perpendicularto the layers

Julliere model of TMR (1975): in the case of the same magnetization alignment,different tunneling probabilities for spin-up and spin-downelectrons follow from different concentrations of spin-up and spin-down carriers in itinerant ferromagnets

P - paralel alignmentAP - antiparalel alignment

PL, PR denote the spin polarizations in left and right leads

Notice: maximum TMR relates to the case of two half-metallic leads Equivalently:

Explanation: according to the modified Landauer-Buttikerformulae, both currents differ from each otherby a factor of the transiton coefficients

The transition probability reads

and the conductancetakes the for (Landauer formula):

The current intensity is calculated with the transition coefficients, via ;

where denotes the energetic DOS

In the limit of small ∆V:

Generalizing the tunneling problem, one assumes the electrons in the leads to be Bloch-like

Note: a more exact Berdeen approach to the conductance calulation includes the temperature distributionof electrons in both leads.

Example: Julliere (1975) observed TMR of 14% at 4.2K for MTJ of: Co/Ge(100Å)/Fe

Example: Moodera et al. (1995) observed TMR of 10% at 295K for MTJ of: CoFe/Al2O3/Co

Example: Fe/MgO/Fe MTJ that allows for TMR of 480%

Example: Parkin et al. (1999) obseved asymmetric TMR in exchange biased MTJ of MnFe/Co/AlO/Ni40Fe60

Spin-transfer torque

Consider the 5-layer system of normal metals and ferromagnets. Electrons flow from the layer A to the layer C in the picture.The effective potentials for spin-up (spin-down) electrons contains the Coulomb and Storner-exchange contributions.The spin momenta per unit area of the ferromagnetic layers are represented with , and a rotating frame

is defined by

Let ϴ denotes the angle between S1 and S2

The spin vector of the polarized electron incident from the layer B onto F2 is

and the wavevector of the polarized-up (-down) electron is

In a system of units that ћ2/2m=1, denoting the ortogonal to ξ component kp

within WKB approximation, on writes

The spinor wave function is written with

The particle fluxand the Pauli-spin flux(related to the continuity equations)are calculated for the flow between B and C as in the limit ofslowly-varying potential. These expressions describe the conical rotation of the electron spin about the magnetization of the F2 layer.

Slonczewski (1996) theory of STT in magnetic multilayers and Landau-Lifshitz-Gilbert-Slonczewski equation

The magnet F2 back-reacts to the spin rotation of a single electron in a way that the total angular momentum is conserved,thus,

Averaging ∆S2 over the direction of the electron motion, thus, over , one finds

In the ballistic limit (the multi-layer thickness is much smaller than the electronic mean free path);where denote the chargé (leftward) and

spin (rightward) current densities.Consider three classes of the electronic states: (i) Electron of any spin (ii) Electron of spin+ (iii) Electron of any spin is

is fully transmitted is transmitted is reflectedspin- is reflected

The total charge and spin current densities are . Hence,

Evaluating the ratio: , we notice that the Fermi vector Q is almost equal to the majority Fermi vector K+; and One arrives at where

The Landau-Lifshitz-Gilbert-Slonczewski equation of the magnetization in layer F2

The magnetization vector can be obtained from S2 via dividing by the thickness t of the F2 layer, and upon a generalization beyond the ballistic regime, the Landau-Lifshitz-Gilbert-Slonczewski equation takes the form

Here Λ>=1 is a measure of the magnetoresistance asymmetry. In the symmetric structure; Λ=1. The secondary(non-adiabatic) STT can be significant for the magnetic tunnel junctions (MTJs) geommetry.

Consider now the current flow in a non-uniformly magnetized ferromagnet. The LLG equation with a symmetric STTcan be applied to its magnetization dynamics with transforming βmp/t → ,

Applications of spin-transfer torque

Current-driven motion of domain wall

Current-driven resistivity oscillations (GHz generation)

Machine learning

Magnetizaton switching (in STT-MRAM)

Applications of spin-transfer torque

Giant magnetoimpedance

GMI before (a) and after (b) glass removal

The skin depth depends on ac frequency, conductivity and, via the transversepermeability, on the external field

Electronic compass

Motion sensor

Non-invasive crack detectionPreassure and strain sensorsBrain activity sensorsand so on

Main advantage of GMI: high sensitivity to the field