basics of spintronics and magneto-sensoricsjanutka/teaching_pliki/lec5_basics_of... · 2019. 5....
TRANSCRIPT
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Basics of spintronicsand
magneto-sensorics
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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
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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
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Anisotropic magneto-resistance (AMR)
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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:
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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)
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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
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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
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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
=>
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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.
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Giant magneto-resistance (GMR)
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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)
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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
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Baibich et al. (Fert group) 1988: GMR in Fe/Cr multilayers
Max[∆R/R*100%]≈100% ! = > a giant effect
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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
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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.
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Tunneling magneto-resistance (TMR)
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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:
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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
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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
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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.
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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
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Example: Fe/MgO/Fe MTJ that allows for TMR of 480%
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Example: Parkin et al. (1999) obseved asymmetric TMR in exchange biased MTJ of MnFe/Co/AlO/Ni40Fe60
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Spin-transfer torque
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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
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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
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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 → ,
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Applications of spin-transfer torque
Current-driven motion of domain wall
Current-driven resistivity oscillations (GHz generation)
Machine learning
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Magnetizaton switching (in STT-MRAM)
Applications of spin-transfer torque
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Giant magnetoimpedance
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GMI before (a) and after (b) glass removal
The skin depth depends on ac frequency, conductivity and, via the transversepermeability, on the external field
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Electronic compass
Motion sensor
Non-invasive crack detectionPreassure and strain sensorsBrain activity sensorsand so on
Main advantage of GMI: high sensitivity to the field