Post on 26-Feb-2021
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