7. Tunneling magnetoresistancehebergement.u-psud.fr/joovon/teaching/old/2010-2011/part... · 2013....

7. Tunneling magnetoresistance

Tuesday, 25 January 2011

Review of multilayer transport

2

Recall that giant magnetoresistance can be understood (phenomenologically) in terms of an equivalent circuit model

Spin-up and spin-down electrons form two independent conduction channels (“two-current” model)

ρ↑

ρ↑ρ↓

ρ↓ρ↑ ρ↑

ρ↓ ρ↓

What happens if the metallic spacer layer separating the two ferromagnetic layers is replaced by an insulating layer?

Transport occurs through quantum tunneling


Quantum-mechanical tunneling

3

In quantum physics, a particle can exist in or tunnel through a region where E < V0

V(x)E

V0E− 22m

∂2

∂x2+ V (x)

|ψ(x) = E|ψ(x)

1D Schrödinger equation

− 22m

∂2

∂x2|ψ(x) = E|ψ(x)

− 22m

∂2

∂x2|ψ(x) = (E − V0)|ψ(x)

E − V0 = −∆E < 0

|ψ(x) = eikx−ωtk = ±

2mE

2

|ψ(x)b = eqx−iωtq = ±

2m∆E

2

Outside the barrier region, we have propagating plane waves

Inside the barrier region, we have evanescent waves

Plane waves

Evanescent waves



4

Need to ensure continuity of wavefunction and its derivative at the interfaces

Aeqx +Be−qx

r: reflection coefficientt: transmission coefficient

Tunneling probability is given by probability of finding the particle in region 2

P = ψ∗2ψ2 = t∗t = |t|2

Probability decreases exponentially as a function of the barrier width L

eikx + re−ikx

teikx

P 16k2q2

(k2 + q2)2e−2qL



5

What is the typical “penetration depth” into the barrier? q = ±

2m∆E

2

∆E = 1 eV

m = m0 = 9.1× 10−31 kg

1

q 0.2 nm

Therefore, to observe tunneling effects, one requires barrier widths to be on the nanometer scale.

STMmagnetic tunnel

junction

insulator/tunnel barrier


General barrier form, WKB

6

For an energy barrier with an arbitrary form, the WKB approximation is a useful technique for obtaining approximate solutions to the Schrödinger equation

V(x)E

V0

L

− 22m

∂2

∂x2+ V (x)

|ψ = E|ψ

∂2

∂x2|ψ = 2m

2 (V (x)− E)|ψ = q(x)2|ψ

|ψ = exp

± x

0dx q(x)

The WKB approximation involves assuming

∂2ψ(x)

∂x= ±∂q(x)

∂xψ(x) + q(x)2ψ(x)

This works well if the barrier is smooth

q(x) =

2m(V (x)− E)

2q(x)2 ∂q

∂x


Bias voltages

7

We have used a scattering argument to derive transmission (and reflection) probabilities

But in the absence of bias voltages, no current flows

EF

A bias voltage must be applied in order to create an asymmetry in the tunneling current

Je(V = 0) = 0

eV EF

EF


Tunnelling current

8

In the WKB approximation, the tunneling probability can be expressed in the general form

P (E) = t∗t = exp

−2

x

0dx q(x)

dN

dt=

4πm2

3

E

0dEx P (Ex)

∞

0dE||f(E)

The number of tunneling electrons per unit time f(E) =1

e(E−EF )/kBT + 1

Je = e

dN1→2

dt− dN2→1

dt

=

4eπm2

3

E

0dEx P (E)

∞

0dE|| [f(E)− f(E + eV )]

The net tunneling current is the difference between right-going and left-going currents

Fermi-Dirac distribution

eV EF

EF


Magnetic tunnel junctions

9

Recall: In ferromagnetic metals, the spin-up and spin-down electrons do not have the same Fermi surface e.g. asymmetry leads to different scattering rates at interfaces

In magnetic tunnel junctions, the metallic spacer layer as present in a GMR multilayer structure is replaced by a thin insulating layer

Transport through this insulating layer is by quantum tunneling

Tunneling for spin-up and spin-down electrons is also asymmetric!

metallic ferromagnet

insulator (e.g. oxides)

metallic ferromagnet

EFEF


Tunneling magnetoresistance (TMR)

10

Similarly to all metallic case, transport through insulator layer is spin-dependent

i.e. spin-up and spin-down electrons do not see the same barrier height

Tunneling magnetoresistance (TMR) arises from tunnelling resistance that depends on relative magnetization orientation (e.g., of a trilayer system)

LETTERS

nature materials | VOL 3 | DECEMBER 2004 | www.nature.com/naturematerials 869

by using molecular beam epitaxy (MBE) and microfabrication techniques (see Methods). Cross-sectional transmission electron microscope (TEM; Hitachi H-9000NAR) images of an MTJ with tMgO = 1.8 nm are shown in Fig. 1. Single-crystal lattices can be identifi ed in the images. The lattice image for MgO(001) (Fig. 1b) illustrates that the lattice spacing is elongated along the [001] axis by 5% and is compressed along the [100] axis by 1.2% compared with the lattice of bulk MgO. Although the MgO lattice is compressed along the [100] axis to match the Fe lattice, the in-plane lattice constant of MgO is still 2.5% larger than that of bulk Fe. This lattice mismatch is relaxed by dislocations formed at the interfaces (see Fig. 1b). More dislocations are observed at the lower interface than at the upper interface. This is because the lattice of the top Fe electrode is expanded by 1.9% along the [110] axis to match the MgO lattice.

The magnetoresistance at bias voltages up to 1,300 mV was measured at 293 K and 20 K by using the d.c. four-probe method. The bias direction was defi ned with respect to the top Fe electrode. Typical magnetoresistance curves for the Fe/MgO/Fe/IrMn MTJ at 293 K and 20 K are shown in Fig. 2a. At 293 K the MTJ has an MR ratio of 180%, which is more than twice the highest room-temperature MR ratio reported to date13. Resistance of the MTJ for a 1 × 1 µm area (resistance–area product RA) is plotted as a function of tMgO in Fig. 2b. Its exponential increase as a function of tMgO is typical of ideal tunnel junctions14. According to the Wenzel–Kramer–Brillouin (WKB) approximation, the slope of the log(RA) versus tMgO plot corresponds to 4π(2mϕ)1/2/h, where m, ϕ and h are, respectively, the electron mass, the potential barrier height (energy difference between the Fermi level and the bottom of the conduction band in the tunnel barrier), and Planck’s constant15. The slope yields a barrier height ϕ of 0.39 eV. Simmons’ equations for I–V characteristics15 yield ϕ = 0.37–0.40 eV. The barrier height of our MTJs is considerably lower than the values in the literature9,10, which should be due to the oxygen vacancy defects in MgO (see Methods). Oxygen vacancies in MgO can form charge-neutral gap states (F-centres) about 1.2 eV below the bottom of conduction band16, which raises the Fermi level above the vacancy states and makes the barrier height lower than 1.2 eV. It should be noted that the barrier height of an ideal MgO tunnel barrier9 (3.7 eV) is too high for the device applications. It should also be noted that for an Al–O tunnel barrier, a lower barrier height yields a lower MR ratio17. It is surprising that in Fe/MgO/Fe MTJs, there is an enormous TMR effect despite the low barrier height. This is very favourable in applications because both a low RA and a high MR ratio can be achieved. Fe/MgO/Fe MTJs with RA values ranging from 300 to 10,000 Ω µm2, which are desirable for MRAMs, have huge MR ratios over 150% at room temperature.

The dependence of the MR ratio on tMgO gives valuable information on the physical mechanism of the TMR effect. According to theoretical calculations7,8, the MR ratio increases with increasing tMgO. This can be understood as follows. When the tunnel barrier is thick, the tunnelling current is dominated by electrons with momentum vectors normal to the barrier, because tunnelling probability decreases rapidly when the momentum vectors deviate from the barrier-normal direction.

a

b

c

–200 1000–100 200

H (Oe)

25

20

15

10

5

T = 20 K

T = 293 K

Parallel

Antiparallel

107

106

105

104

103

102

101

1.51.0 2.0 2.5 3.0tMgO (nm)

200

150

50

0

100

250

MR

ratio

(%)

1.51.0 2.0 2.5 3.0tMgO (nm)

T = 20 K

T = 293 K

0.30 nm

µm2

)Ω

RA

( k

µm2

)Ω

RA

(

Figure 2 Tunnel magnetoresistance of Fe(001)/MgO(001)/Fe(001) junctions. a, Magnetoresistance curves (measured at a bias voltage of 10 mV) at T = 293 K and 20 K (MgO thickness tMgO = 2.3 nm). The resistance–area product RA plotted here is the tunnel resistance for a 1 × 1 µm area. Arrows indicate magnetization confi gurations of the top and bottom Fe electrodes. The MR ratio is 180% at 293 K and 247% at 20 K. b, RA at T = 20 K (measured at a bias voltage of 10 mV) versus tMgO. Open and fi lled circles represent parallel and antiparallel magnetic confi gurations. The scale of the vertical axis is logarithmic. c, MR ratio at T = 293 K and 20 K (measured at a bias voltage of 10 mV) versus tMgO.

nmat1257-print.indd 869nmat1257-print.indd 869 10/11/04 11:17:29 am10/11/04 11:17:29 am

© 2004 Nature Publishing Group

© 2004 Nature Publishing Group

LETTER TO THE EDITOR

L232 T. Miyazaki, N. Tezuka /Journal of Magnetism and Magnetic Materials 139 (1995) L231-L234

(B) at the Fermi surface. Slonczewski [9,10] also discussed the magnetic tunneling effect (he named first this effect magnetic-tunneling-valve effect) in the light of band theory and showed how the expres- sion for conductance ratio of Maekawa et al. may be modified by consideration of wave-function matching at the interfaces between sublayers. He also demonstrated the dependence of the tunneling conductance on the angle between the magnetizations of two magnetic layers. Eq. (1) shows that the relative conductance ratio (magnetoresistance ratio) is proportional to the product of spin polarization of the tunneling electrodes. The spin polarization of electrons tunneling for Fe, Co, Ni, and Gd films was obtained experimentally by Tedrow and Meserbey [11]. The result suggests that the spin polarization of Fe, Co and Ni is proportional to their magnetic moments. Therefore, it is expected that a magnetic tunneling junction with a large magnetoresistance ratio can be formed by using a material with a large intensity of magnetization as magnetic electrodes.

The iron electrodes and aluminum were prepared by electron beam evaporation and by rf sputtering, respectively. First, a 1000 A thick Fe layer was evaporated in a form of 1 ! 15 mm z onto a glass substrate kept at 200°C in order to reduce the magnetic anisotropy and/or coercive force. The pressure during evaporation was approximately 1 ! 10 -6 Torr. Then, a 55 A thick A1 layer was sputtered onto the center of the Fe film in a circular form with a 5 mm diameter. The system base pressure was lower than 1 X 10 -6 Torr and the Ar sputtering gas pressure was 1.5 mTorr. Typical growth rate was 5.6 A / s . The aluminum was oxidized in air at room temperature for 24 h. On the A120 3 layer, a 1000 thick Fe layer was formed in the rectangular shape of 1 ! 15 mm 2 perpendicular to the long axis of the first Fe film. In this case the substrate was kept at room temperature. In this way we can vary the induced magnetic anisotropy and/or coercive force for both iron electrodes. The experimental evidence of the different magnetic anisotropy will be shown in Fig. 2 and the magnetization curve explained. The magnetoresistance was measured by a dc four probe method in magnetic fields up to 200 Oe in the temperature range 4.2-300 K. The magnetization hysteresis curve was measured by a vibrating sample magnetometer.

0.008 I I I

20 i . ' ' 4

• . . " " i ! ' . . ~ • q "L

0.007 ' ~ 10 = i "

i i

i i

F . , . . i L

. . ~ . . : . . . . .. . . . . . . . . . : . ~

0.006 l i i I i -100 -50 0 50 100

H ( O e ) Fig. 1. Resistance as a function of the magnetic field for 1000 ,~ Fe/AI203/1000 ,~ Fe junction.

Fig. 1 shows the dependence of the resistance on the intensity of magnetic field at room temperature. The resistance increases sharply at +20 Oe and decreases gradually up to about +52 Oe, followed by a rapid decrease with further increase of the magnetic field.

The magnetoresistance ratio A R / R s is approximately 18%, where AR is the resistance change from the antiparaUel to parallel magnetization and R s is the resistance at saturated magnetization. The value is one order higher than that reported previ- ously for other magnetic tunneling junctions. Here it should be noted that other samples prepared at the

-100 - 5 0 ~ I f

. . . .

M(Arb. Units)

...M I 5 0 1 0 0

H(Oe)

Fig. 2. Hysteresis curve corresponding to the magnetoresistance curve in Fig. 1.

Fe/Al2O3/Fe Fe/MgO/Fe

P

AP

P

AP

P

From Miyazaki et al., J. Magn. Magn. Mater 139, L231 (1995) From Yuasa et al., Nat. Mater. 3, 868 (2004)


Jullière model

11

1. Assume no spin flips when electrons tunnel through barrier

Two independent conduction channels for spin-up and spin-down

Tunneling of spin state from first film into second film is determined by unfilled states of same spin in second film

2. Assume conductance G for a spin channel is given by the product of the effective density of states n of the two ferromagnetic electrodes

M. Jullière, Phys. Lett. 54A, 225 (1975)

Gσ = nσLn

σR G = G↑ +G↓

P =n↑ − n↓

n↑ + n↓

TMR ≡ RAP −RP

RP=

2PLPR

1− PLPR

conductances

spin polarization


Spin polarization of 3d metals

12

Large spin-polarization is needed for large TMR

Topical Review

Figure 1. Spin splitting of the density of states (ρ) in a ferromagnetdue to the exchange field.

from a ferromagnet to a paramagnet by tunnelling throughan insulator, this current is also polarized due directly to thedensity of states asymmetry. FM elements may thus be used asspin-polarized current sources in spin-electronic circuits. Mostspin-electronic phenomena are based on either one or both ofthese asymmetries (whose common origin is the band structuresplitting) prevailing in the relevant physical system [1].

2.1. Spin asymmetry: density of states asymmetry versusmobility asymmetry

In fact, the two asymmetries often compete with one anotherin spin electronics. The Fermi surface in most FM materialscontains components which have both s- and d-character.The s-like effective masses are small compared to the d-likemasses and so any current that flows is primarily mediated bys-electrons. However, the d-electrons are significantly split bythe exchange interaction and as a result present very differentdensities of states into which the s-electrons may be scattered.Thus, from figure 1, the down s-channel (whose spin type hasa large d density of states at the Fermi energy) suffers themost scattering and hence has lower mobility than the others-channel: this latter consequently carries most of the current.

Thus in a system with s- and d-like character at the Fermisurface, the tendency is for the current to be carried by theminority carriers (where ‘minority’ is taken to mean thosewith the lower density of states at the Fermi energy, and thisconvention will be used throughout this paper) whereas in ahalf-metallic ferromagnet (see next section), the current mayonly be carried by majority carriers. This conflict between thetwo types of asymmetry is one reason why spin-tunnel devices(section 6) have an advantage over their competitors sincethey exploit only the density of states asymmetry; hence, themobility asymmetry has no chance to compete and reduce theoverall device performance. This has direct relevance to thequestion of spin injection into semiconductors.

2.2. The half-metallic ferromagnet

In the extreme limit of spin asymmetry lies the half-metallicferromagnet [2] in which the band structure splitting is suchthat only one spin channel has available states at the Fermisurface and hence all current must be carried by these so-calledmajority spins. Practical examples include chromium dioxide

Figure 2. Schematics of the difference in the densities of statesbetween a ferromagnet and a half metallic ferromagnet.

Figure 3. Illustration of the spin accumulation at theferromagnet/paramagnet interface.

(CrO2), lanthanum strontium manganite (La0.7Sr0.3MnO3)and some Heusler alloys. In reality, obtaining half-metallicspin-electronic behaviour is fraught with problems mainly todo with the interfaces. Conversely, some materials whosebulk electrical conduction deploys both spin channels may,due to hybridization, form half-metallic interfaces with othermaterials.

2.3. Spin injection across an interface: spin accumulation

Now that we have considered the basic principles behindthe origin of spin asymmetry, we can briefly consider animportant phenomenon which lies at the heart of early spin-electronic devices. Providing one carrier spin type is dominantin the electrical transport of a ferromagnet, when a currentis passed from this ferromagnet to a PM metal such assilver or aluminium, it brings with it a net injection of spinangular momentum and hence also of magnetization [3]. Themagnetization which builds up in the new material is knownas a spin accumulation (figure 3). Its size is determined by theequilibrium between the net spin-injection rate at the interfaceand the spin-flipping rate in the body of the paramagnet. Itfollows that the spin accumulation decays exponentially awayfrom the interface on a length scale called the ‘spin diffusionlength’.

R122

Assuming 50% spin polarization, Jullière’s model predicts TMR of ~67%

TMR ≡ RAP −RP

RP=

2PLPR

1− PLPR


Crystalline barriers

13

Free electron models, such as the Jullière model, are OK for amorphous barriers, such as aluminium oxide, as all electronic effects related to crystal symmetry are smeared out.

These models give a qualitative account of the dependence of the TMR on the spin polarization of the ferromagnetic electrodes.

However, these do not work for crystalline barriers, for which band structure effects need to be taken into account

!"#$%&'( !"#$%&'()*+)(,*+-.$+*/("0))))))))1223)45.$,&6+)7/8$$')$+)#69+&-*(:;)<*:*($6.6)

)%*#+!,#!-.(./.0.!&1!.2#!,.3#!,#!4332#!56(7%87!9::;

427<=!>?.7.!&1!.2#!4332%&@!56(7%87!A&11&B7=!9::;

!"#$%&'()&*$$%+),*$(&'-$.)/".%0)-$)!#1)&*$$%+)/"22'%2.

C47D@&3<7%1&@=!E<)&"!.F<B36<?7=!G*H!3<2(8B%71.22%'&CI3<'!.''&.2%'*=!B&8B(71.22%J.1%<'!<K!E<)&"!KB<F!16&!G*H!%'1&BK.8&7!.'@!%F3B<L&F&'1!%'!G*H!8B(71.22%'%1(!0%16!MN::O!P88!1&Q1?B&


Fe/MgO/Fe

14

High quality crystalline Fe/MgO/Fe represents a candidate system for spin filtering based on band symmetry

High TMR ratios observed in these systems (up to 600% at 300 K)

!

Fe !"1

Fe #"2

MgO

MgO


MgO

15

Sci. Technol. Adv. Mater. 9 (2008) 014106 Topical Review

Figure 12. Absolute square of the wavefunction for incident electrons of different symmetries as a function of position. The two top panelsdescribe tunneling for parallel alignment of the magnetic moments of the electrodes. The upper left panel describes k = 0 contributions toG↑↑. The upper right panel describes k = 0 contributions to G↓↓. Similarly, the bottom two panels describe contributions to G↑↓ and G↓↑.

valence bands as well as a conduction band. Electrons can alsotunnel through the barrier via the valence bands; this wouldbe described as ‘hole’ tunneling because it would describea process in which a valence electron makes an upwardtransition from the top of the valence band to the Fermi energyof the right electrode simultaneously with the transition ofan electron at the Fermi energy of the left electrode makinga downward transition that fills the hole. The situation iseven more interesting, because as we shall show, the redvalence band (labeled 1) is really the same band as the redconduction band (also labeled 1).

The first-principles based layer KKR approach was usedto calculate kz as a function of E as shown in figure 4. Justas for the simple barrier model, energies exist for which nostates with real kz(E, k) exist. However, for these energies,states exist which are imaginary or complex. In fact all of thepropagating Bloch states continue as evanescent states whenthey are no longer valid propagating states. In particular, forMgO, the valence band with 1 symmetry continues throughthe gap region as an evanescent state for which kz is imaginaryand re-emerges at the top of the gap as the conduction band.An excellent approximation for this1 state in the vicinity ofthe gap is

1k2z

= −h2

2m∗v(E − Ev)

+h2

2m∗c(E − EC)

,

where Ev and m∗v represent the energy and effective mass at

the top of the valence band while Ec and m∗c represent the

analogous quantities at the bottom of the conduction band. ForMgO, m∗

v and m∗c seem to be the same so that the interpolation

formula for the evanescent band in the gap can besimplified to

h2k2z2m∗ = (E − Ev)(E − Ec)

(Ec − Ev).

The minimum value of k2z is [k2z ]min = −m∗(Ec−Ev)2h2 and occurs

at mid-gap, E = (Ec + Ev)/2. For comparison, the simplebarrier model would give k2z = −me(Ec−Ev)

h2 at mid-gap.The continuation of the bands into the gap region as

evanescent states can be understood on very elementary termsfor MgO. A minimal model for the electronic structure in thevicinity of the gap would include the Oxygen p-states and theMg-s state. The tight-binding Hamiltonian matrix for kx = 0and ky = 0, can be written as

H(kz) =

Es w[exp(ikza/2) − exp(−ikza/2)]w[exp(−ikza/2) − exp(ikza/2)] Ep

.

The diagonal elements of the matrix represent the atomicenergies of the Mg-s and oxygen-pz states (possibly modifiedby terms independent of kz). The off diagonal matrix elementsrepresent interactions between the Mg-s and nearest neighbor

9

Taken from W. H. Butler, Sci. Technol. Adv. Mater. 9, 014106 (2008)


8. Spin transfer torques


GMR: Analogy with polarized light

17

polarizing filter measurement filter

I1 ≈ 50%

I2 ≈ 0%

incident beamof unpolarized

light

GMR parallel state

GMR antiparallel

state

We saw in the last lecture that giant magnetoresistance (GMR) can be well-described by the “two-current” model.

In fact, there is a nice analogy between spin transport and polarized light


Non-collinear transport

18

polarizing filter measurement filter

incident beamof unpolarized

light

Coherent superposition ofboth polarization states

f(k, r) →

[

f↑↑ f↑↓f↓↑ f↓↓

]

What happens if the two magnetizations are not collinear?

If we go back to the transport theory of last week’s lecture, we can describe this by generalizing the spin-dependent Boltzmann equation to use 2x2 matrices

Off-diagonal elements represent coherences between the two spin states.


Non-collinear transport

19

coherence length?

Ferromagnet 1 Ferromagnet 2

However, these coherences cannot be long-lived. Consider two semi-infinite ferromagnets:

Eventually, the transmitted electrons must participate to the two-current conduction in the second ferromagnet, relative to the local quantization axis.

What is the typical coherence length?


Single electron at N/F interface

20

M. D. Stiles and A. Zangwill, Phys. Rev. B 66, 014407 (2002)

FN

quantization axis

εF =h

2k2F

2mεF =

h2(k↑,↓

F)2

2m∓

∆

2

k↓F

< k↑F

Consider the following quantum-mechanical scattering problem. Imagine an incident free electron in the normal metal layer arriving at the normal metal (N)/ferromagnet (F) interface.

Because the bands in the ferromagnet are spin-split, there is a spin-dependent step potential at the interface.

F

∆k↓k↑



21

ψin = eikxxeik||·r||

(

cos(θ/2)e−iφ/2| ↑ 〉sin(θ/2)eiφ/2| ↓ 〉

)

incident wavefunction

ψref = e−ikxxeik||·r||

(

R↑ cos(θ/2)e−iφ/2| ↑ 〉R↓ sin(θ/2)eiφ/2| ↓ 〉

)

transmitted wavefunction

reflected wavefunction

ψtr = eik||·r||

(

T↑eik↑xx cos(θ/2)e−iφ/2| ↑ 〉

T↓eik↓xx sin(θ/2)eiφ/2| ↓ 〉

)

Assume constant effective mass

Apply usual quantum mechanical matching conditions across interface to obtain reflection and transmission coefficients



22

ψiσ(r)

occupied single-particle wavefunction

state i spin σ

n(r) =∑

iσ

ψ∗

iσ(r)ψiσ(r) particle density

j(r) = Re∑

iσ

ψ∗

iσ(r) v ψiσ(r) particle current density

m(r) =∑

iσ

ψ∗

iσ(r) s ψiσ(r) spin density

↔

Q(r) = Re∑

iσσ′

ψ∗

iσ(r) s ⊗ v ψiσ′(r) spin current density

Some useful quantities for basic particle transport:

Some useful quantities for spin transport:

ψ = (ψ↑,ψ↓)

s = (σx,σy,σz)

Pauli matrices



23

Qzx

|Rσ|2

+kσ

x

kx

|Tσ|2

= 1

jinx

+ jrefx

= jtrx

Qinzx

+ Qrefzx

= Qtrzx

reflection and transmission probabilities

particle current

longitudinal spin current

∇ ·!j +∂n

∂t= 0

What is conserved?



24

Qxx

Qyx

Qin⊥x

+ Qref⊥x

!= Qtr⊥x

continuity equationfor spin

externaltorque

spin accumulation

∇ · !Q +∂ !m

∂t= −

δ !m

τ↑↓+ !next

What is not conserved?

There is a discontinuity of the transverse spin current at the interface

From the continuity equation of spin flow, this discontinuity must appear as a source (or sink) term

Gauss’ theorem: transverse spin current must be absorbed at the interface of the ferromagnet

From conservation of spin-angular momentum, the missing transverse spin current must be transferred to the ferromagnet


Mechanisms of spin-transfer

25

– Spin precession

∆k = k↓F− k

↑F

spatial precession frequencies

– Spin filtering spin-dependent reflection and transmission probabilities

Qrefxx = −

!

4vx sin(θ)ReR∗

↑R↓eiφ

Qtrxx(r) =

!

4

v↑x + v↓x2

sin(θ)ReT ∗↑ T↓e

iφe−i(k↑x−k↓

x)x

– Spin rotation

R∗↑R↓e

iφ= |R∗

↑R↓|ei(φ+δφ)


Spin transfer torques

26

The transverse spin current is absorbed at the interface of ferromagnet 2.

What happens if ferromagnet 2 is very thin?

The torque due to absorption of transverse spin current can affect the magnetization dynamics.

This is often referred to as spin-transfer torque.

Ferromagnet 1 Ferromagnet 2

L. Berger, Phys. Rev. B 54, 9353 (1996)J. C. Slonczewski, J. .Magn. Magn. Mater. 159, L1 (1996)



27

[

∂s2

∂t

]

st

∝ s1⊥ = s2 × (s1 × s2)

Transfer of transverse spin-angular momentum leads to a new term for magnetization dynamics



28

Gilbert damping Spin-transfer torque

The usual equations of motion is supplemented with extra term

Depending on current direction and M1, the spin-transfer torque can lead to negative damping.

d !M2

dt= −|γ| !M2 × !Heff

2 +α

M0

!M2 ×d !M2

dt+ g(θ)

ωt

M20

!M2 × ( !M1 × !M2)

g(θ)

ωt =gµB

M0

1

d2

Je

e

angular dependence of spin-transfer

rate of spin-transfer

M2M1


Angular dependence of spin torques

29

The angular dependence of spin-transfer is sensitive to transport details

g(θ)

e.g. Co/Cu/Co


Spin torques a nanoscale effect

30

Why didn’t we see this effect before??

Spin-transfer torque is proportional to current density, not current e.g. for a 1 mm2 wire, would need 400 000 A !!!

––

Just like giant magnetoresistance, spin-transfer torques only appear (in the practical sense) on the nanoscale

GMR: (1988) needs nm thicknesses Spin-momentum transfer: needs nm thicknesses AND (1996) nm lateral dimensions


Consequences on magnetization dynamics

31

Spin-transfer torques can lead to magnetization reversal, even in the absence of an applied field

Nanopillar structureCross-section 60 ×180 nm2

F. J. Albert et al., Phys. Rev. Lett. 89, 226802 (2002)

Applied currents switch magnetization back and forth, depending on sign


7. Tunneling magnetoresistancehebergement.u-psud.fr/joovon/teaching/old/2010-2011/part... · 2013....

Documents

Transcript of 7. Tunneling magnetoresistancehebergement.u-psud.fr/joovon/teaching/old/2010-2011/part... · 2013....