Spin transfer torques in magnetic tunnel junctions

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Spin transfer torques in magnetic tunnel junctions

Aurélien Manchon, Natalya Ryzhanova, Mairbek Chschiev, A. Vedyayev,

K.-J. Lee, Bernard Dieny

To cite this version:

Aurélien Manchon, Natalya Ryzhanova, Mairbek Chschiev, A. Vedyayev, K.-J. Lee, et al.. Spin

transfer torques in magnetic tunnel junctions. 2008. �hal-00258895�

hal-00258895, version 1 - 25 Feb 2008

SPINTEC, URA 2512 CEA/CNRS, CEA/Grenoble, 38054 Grenoble Cedex 9, Fran e

2

Department of Physi s, M. V. Lomonosov Mos ow State University, 119899 Mos ow, Russia

3

MINT Center, University of Alabama, P.O. Box 870209, Tus aloosa, Alabama, USA 4

Department of Materials S ien e and Engineering, Korea University, Seoul 136-713, Korea



(Dated: 26thFebruary2008) Abstra t

This hapter presentsareviewon spintransfertorque inmagneti tunneljun tions. Intherst part, we prop ose an overview of exp erimental and theoreti al studies addressing urrent-indu ed magnetizationex itationsinmagneti tunnel jun tions. Themostsigni ant results arepresented and the main observable hara teristi s are dis ussed. A des ription of the me hanism of spin transfer in ferromagnets is nally prop osed. In the se ond part, a quantum des ription of spin transp ort in magneti tunnel jun tions with amorphous barrier is develop ed. The role of spin-dep endent ree tionsas well as ele tronin iden e and spin-ltering by thebarrier are des rib ed. We showthattheseme hanismsgiverise tosp e i prop ertiesofspintransferin tunneljun tions, very dierent from the ase of metalli spin-valves. In the third part, the theoreti al observable featuresofspintransferinmagneti tunneljun tionsarederivedandthevalidityoftheseresultsis dis ussed and omparedtore entexp eriments. To on lude this hapter,westudytheme hanism of spin transfer in half-metalli tunnel jun tions, exp e ted to mimi MgO-based magneti tunnel jun tions.

PACSnumb ers:

Keywords: Spin Transfer Torque, Magneti tunnel jun tions, Tunnelling Magnetoresistan e, Current-indu edMagnetizationSwit hing

I. Intro du tion 4

I I. Overview of exp eriments and mo dels 5

A. Current-indu edmagnetizationswit hing 6

1. General prop erties 6

2. STTversusTMR 8

B. Current-indu edmagnetizationex itations 8

C. Originof spintransfertorque 10

1. Phenomenologi aldes ription 10

2. Spin transferin an arbitraryferromagnet 12

D. Theoriesofspin transferin magneti tunnel jun tions 14

I I I. Quantum origin of spin torque in magneti tunnel jun tions 16

A. Freeele tronmo del 16

B. Spin transp ortin aMTJ 19

C. In iden e sele tionin anamorphousbarrier 20

1. -sele tion due totunnelling 20

2. Spin sele tion due toferromagnets 21

D. Spin lteringin rystallinestru tures 22

E. Torquesand oupling 23

IV. Observable prop erties 25

A. Angulardep enden e 25

B. De aylength ofspin density 26

1. Ballisti interferen es 26

2. Spin s atteringme hanisms 28

3. RealFermi surfa es 29

C. Biasdep enden e 29

1. Freeele tron mo del 29

2. Cir uittheory 31

3. Asymmetri jun tion 33

4. Roleof magnonsemissions 33

D. Re ent exp erimental investigations 35

1. Radio-frequen ysignatureofspintorque 35

2. Thermallya tivatedphasediagrams 36

A knowledgments 44

REFERENCES 44

The study of the oupling b etween an ele tri al urrent and lo alized spins in transi-tion metals, leading to giant magnetoresistan e ee ts

1,2

, has renewed our knowledge of fundamentalele troni sand op ened wide elds ofresear hin this domain. The idea thata spin-p olarized urrentmayinturna tonthelo almagnetizationofsu haferromagnethave b een prop osed in the late 1970's by Berger

3

, when investigating the intera tion b etweena domainwalland an ele tri al urrent.

However, this torque - usually alled spin transfer torque (STT) - exerted by the spin-p olarized urrenton the lo al magnetizationrequires high urrent densitieswhi h an only b e rea hed in sub-mi roni devi es (nano-pillars, p oint onta ts or nano-wires). The de-velopmentof thin lm dep osition te hniques, as well as ele troni lithography in the early 1990's ledto the fabri ation ofspin-valve pillarswith dimensionsas smallas 100100 nm

2 . Spin-valves, rst studied by Dieny et al.

4

in 1991, onsist of two ferromagneti thin lay-ers (less than 10 nm-thi k), separated by a metalli (Cu, Al) or tunnelling (Al

2 O

3

, MgO, TaOx)spa er. One ofthe ferromagnetis pinnedby an antiferromagneti systemso that its magnetizationdire tionis onlyweaklyae ted by an external magneti eld.

The theoreti al demonstration of spin transfer torque in metalli spin valves (SVs) ten years ago

5,6

gavea new breathto giantmagnetoresistan erelatedstudies

7

,promising ex it-ing new appli ationsin non-volatile memorieste hnology

8

and radio-frequen yos illators

9

. A numb er of fundamental studies in metalli spin valves revealed the dierent prop er-ties of spin torque and led to a deep understanding of urrent-indu ed magnetization dynami s

10,11,12,13,14

. Parti ularly,several theoreti al studies des rib ed the stru ture of the torque in metalli magneti multilayersand showed the imp ortant role of averaging due to quantuminterferen es,spin diusionand spin a umulation

15,16,17

. Sin e the rst exp erimental eviden e of spin-dep endent tunnelling

18

, magneti tunnel jun tions (MTJs) have attra ted mu h attention b e ause of the p ossibility to obtain large tunnelling magnetoresistan e (TMR) at ro om temp erature

19

. The p ossibility to use MTJs as sensing elements in magnetoresistive heads, as non-volatile memory elements or in re-programmable logi gates has also stimulated a lot of te hnologi al developments aiming at the optimization of MTJs' transp ort prop erties and their implementation in sili on-based ir uitry

8,20

. Be auseof these appli ations, MTJs have b een intensively studied and the role of interfa es

21 , barrier 22 , disorder 23 and impurities 24

have b een addressed in many publi ations

25

. There enta hievementof urrent-indu edmagneti ex itationsandreversal inMTJs

26,27

has renewedthe already veryimp ortantinterestof thes ienti ommunityin MTJs.

The re entobservationofspin transfertorque inlowRA(resistan eareapro du t) MTJs using amorphous

26,27

or rystalline barriers

20,28

op ened new questions ab out the transp ort me hanism in MTJs with non ollinear magnetization orientations. As a matter of fa t, whereas the urrent-p erp endi ular-to-plane (CPP) transp ort inSVs is mostlydiusive and

governed by spin a umulation and relaxation phenomena , spin transp ort in magneti tunnel jun tions is mainly ballisti and governed by the oupling b etween spin-dep endent interfa ial densities of states: all the p otentialdrop o urs within the tunnel barrier. The hara teristi s of spin transfer torque are thus exp e ted to b e strongly dierent in MTJs omparedto SVs.

In this hapter, we prop ose a des ription of spin transfer torque in magneti tunnel jun tions, highlighting the dieren es with metalli spin valves. In se tion I I, an overview of the exp erimentson spin transfer torque is given as well as a des ription of the origin of STT in arbitraryferromagneti systems.

In se tion I I I, the quantum origin of spin transfer torque in MTJs is des rib ed using a simple free-ele tronapproa h. The sele tion of the in ident ele trons due to the tunnel barrier is depi ted and the relaxation of the transverse and longitudinal omp onents of the spin density (spin a umulation) is dis ussed. It is shown that these two ee ts may ontributeto a non negligibleeld-liketerm(also alled out-of-plane omp onent), ontrary to SVs where this termisnegligible.

In se tion IV, we present the angular and bias dep enden ies of the in-plane and out-of-plane omp onents of spin transfer torque. The imp ortant angular asymmetry usually observed in metalli systems disapp ears in magneti tunnel jun tions due to the redu ed inuen e of the longitudinal spin a umulation on the transverse spin urrent. Then, in agreementwithdierenttheoriesandveryre entexp eriments,weshowthat thebiasdep en-den iesof the two omp onentsof STT exhibitnon linear variations due to the sp e i non linear transp ort through the tunnel barrier. We also dis uss the existen e of other sour es whi h an stronglyae t thisbias dep enden e,su has the existen eof interfa ial asymme-try,in ompleteabsorption of the transverse omp onentof spin urrentor, mostimp ortant, emission ofspin wavesdue to hot ele trons.

Finally inse tionV,wepresenttheinuen eof in reasings-d ex hange ouplingon spin torque and esp e ially dis uss the ase of half metalli tunneljun tions, whi hmight mimi MgO-based MTJs. In half metalli ele tro des, the spin transfer exp onentially de ays near the interfa e stillgiving riseto anon zero torque on the lo almagnetization.

I I. OVERVIEW OF EXPERIMENTS AND MODELS

The observation of spin transfer torque in magneti tunnel jun tions is onlyvery re ent (2004) due to the di ulty to obtain high-quality low RA MTJs. As a matter of fa t, as we stressed out in the intro du tion, observing the magneti inuen e of spin transfer torquerequiresthe inje tionofhigh urrentdensitiesintheMTJs,oftheorderof10

7 A/ m

2 while onserving a high urrent p olarization. Redu ing the thi kness of the tunnel barrier generally leads to b oth the redu tion of TMR, as well as the app earan e of pinholes

29

ee tthrough MgO rystallinebarrier allowedto obtain lowresistan emagneti tunnel jun tions together with high urrent p olarization, thus fullling the requirements for the observation of STT in MTJs. Diao et al.

32

and Huai et al.

33

have ompared the urrent-indu ed magnetization reversal in MgO-based and AlOx-based MTJ and showed that the ee tive p olarization p of the interfa ial densities of states is signi antly higher in MgO-MTJ (p 46%) than in AlOx-MTJ (p 22%), due to spin-ltering ee ts in rystalline MgO barrier. Even if the existen e of su h interfa ialp olarization is questionable

34,35

, this estimationillustratesthe signi ant improvementa hievedwith MgO-based MTJs.

A. Current-indu ed magnetization swit hing

1. Generalproperties

As we stated inthe intro du tion, a magneti tunnel jun tion is a tunnelling spin valve, as displayed inFig. 1, omp osed of two ferromagneti ele tro des(CoFe, CoFeB) separated by a tunnelling barrier. One ferromagneti layer (referen e layer) is antiferromagneti ally oupled (usually through a thin Ru layer) to a so- alled"pinned layer". This pinned layer is magneti ally oupled to an antiferromagnet (IrMn, FeMn). This te hnique, known as syntheti antiferromagnet

36

,stronglystabilizesthereferen elayerwhileredu ingthedip olar eld emitted on the free layer. The free layer magnetization may then b e oriented by an external eld, while keepingthe magnetizationof the referen e layer ina xeddire tion.

Figure 1: S hemati s of a magneti tunnel jun tion. The bias voltage is dened p ositively when theele tronsowfromthereferen e layertowardthefreelayer.

Therstobservationof urrent-indu edmagnetizationswit hinginmagneti tunnel jun -tions has b een p erformed by Huai et al.

26

and Fu hs et al.

27

in AlOx-based low RA MTJ (RA<10:m

2

),in nano-pillar with ellipti shap e (120230 nm 2

in Ref.

26

suring resistan e lo ops as a fun tion of the external applied eld H and the applied bias voltageV,as displayedinFig. 2. Inthisgure,wemeasuredtheresistan eofaMgO-based MTJ, omp osedof CoFeBferromagneti ele tro des. Theresistan elo opas afun tionofthe external eld H for a xed applied bias voltage is givenin Fig. 2(a), while the resistan e lo op as a fun tion ofthe bias voltageV for a xedexternal eld is givenin Fig. 2(b).

Figure2: Resistan eofa CoFeB/MgO/CoFeBMTJs versus(a)theexternaleld(V=10mV)and (b)theappliedbiasvoltage(H=45Oe). ( )Tunnellingmagnetoresistan easafun tionofthebias voltage(H=45Oe). TMR=83:7%andA=50100nm

2 .

One observes sharp resistan e jumps in Fig. 2(b) for p ositive and negative bias whi h orresp ond to the swit hingof thefree layermagnetizationfromantiparallelto paralleland vi e-versa,resp e tively. In this jun tion, the riti al urrentneeded to swit hthe free layer magnetization is 510

6 A/ m

2

. The drop of resistan e as a fun tion of the bias voltage is asso iated withadropofTMR(seeFig. 2( )). Thisdrophas b eenattributed tospin-waves emissionsby hot ele trons

37

as wellas to the energy-dep enden eof the density of states at the jun tion interfa es. Note thatthis drop do es not exist inmetalli spin valvessin e only Fermiele tronssigni antly ontributeto the ele tri al urrentinmetals.

Sin e these rst observations, many eorts have b een arried out in order to obtain low riti al urrent magnetization swit hing in MTJs. Dieny et al.

38 , Fu hs et al. 39 and Huai et al. 40

prop osed dual typ e MTJs, in order to redu e the riti al swit hing urrent. These stru tures are of the typ e

39 CoFe 1 /AlOx/CoFe Fr ee /Cu/CoFe 2 , where CoFe 1 and CoFe 2 are antiparallel and the Cu/CoFe

2

interfa e is used to ree t the minority ele trons towards CoFe

Fr ee

in order to enhan e the spin transfer torque in this layer. With this s heme, riti al urrent weredividedby a fa tor 3.

Another metho d has b een prop osed by Inoku hi et al.

41

. By inserting a non magneti layermadeofZr,Hf,Rh,Ag,AuorVon thetopofthefreelayer,itisp ossibleto redu ethe riti al urrent by one order of magnitude and to rea h riti al urrent densities of 510

A. m .

2. STT versus TMR

Aninterestingp ointhas b eenunderlinedbyFu hsetal.

27

intheirpioneeringexp eriment, when observing urrent-indu ed magnetization swit hing at 77 K. As displayed on Fig. 3, themagnetizationofthefreelayer ouldb eswit hedfromantiparallel(bla kline)toparallel (red line) by applying an external urrent. The most interesting is that the magnetization swit hing o urred at abias voltage at whi hthe TMR was roughly zero, as shown by the arrows on Fig. 3.

Figure 3: Current-indu edmagnetization swit hingin AlOx-based MTJ,measured at77K. This swit hingisasso iated witha omplete quen hingof theTMR.From Ref.

27

.

This exp erimentdemonstratesthattheTMRde reasedo es notpreventthespintransfer. As a matter of fa t, whereas the p olarization of the olle ting ele tro de de reases when in reasing the bias voltage(due to energy-dep enden eof the interfa ialdensity of states as wellas magnon emission),the p olarization of the in identele tronsisonlyweaklyae ted. Consequently, a urrent-indu ed magnetization swit hing may o ur although the overall TMR is zero. In fa t, Levy and Fert

42

have shown that the ontribution of hot ele trons-indu ed spin-waveemission may play an imp ortantrole in su h systems.

B. Current-indu ed magnetization ex itations

Current-indu ed magnetization ex itationsare of great interest for appli ations, in par-ti ular ontrolling the noise sp e trumof read-head devi esor generatinghyp er-frequen ies.

di ult b e ause of the voltage limitation of the tunnel barrier whi h undergo ele tri al breakdown when submittedto bias voltage of typi ally1 V.

A rst study of the "spin-dio de ee t" was published by Tulapurkar et al.

43

, in 2005. The authors showed that the inje tion of a small radio-frequen y a - urrent into a MgO-based MTJ an generate a d -voltage a ross the devi e. This d -voltage app ears when the frequen y of the a - urrent is lose to the natural frequen y of FMRex itations. This resonan e an b e tuned by an external magneti eld. By this way,Tulapurkar et al. were the rst to observe a non negligible "ee tiveeld" term,b

j

, whi h was found to b e linear as a fun tion of the bias voltage. Re ent developmentsof this te hnique werea hieved by Kub ota etal.

44

. Theywillb e des rib ed inse tion IV. Another te hnique was prop osed by Sankeyet al.

45,46

. By studyingthe inuen eof spin transfer torque on the ferromagneti resonan e of the free layer, the authors were able to determinethe bias dep enden eof the spin transfer torque. These results willb e des rib ed in se tionIV.

Figure 4: Thermally a tivated FMR sp e tra of AlOx-based MTJ, as a fun tion of the inje ted urrent in paralleland antiparallel state. FromRef.

49

.

The inuen e of spin torque on thermally a tivated ferromagneti resonan e was also studied

47,48

. Petitetal.

49

havedemonstratedtheinuen eofspintransfertorqueonthermal noise inMTJs. Fig. 4 displays the thermallya tivated FMRsp e tra of aAlOx-based MTJ as a fun tion of the inje ted urrent. In parallel onguration, the amplitudeof the FMR p eak in reases as a fun tion of p ositive urrent and de reases when the inje ted urrent is negative(andinverselyinantiparallel onguration). On eagain,theauthors demonstrated the strong inuen eof the b

j

After this short overviewon previous relevant exp eriments,let us des rib e the physi al origin of spin transfer torque. To do so,wewillpro eed intwosteps: rstly,a phenomeno-logi al des ription of spin transfer will b e presented, using a simple on eptual s heme; se ondly, the expression of spin transfer torque in an arbitrary ferromagnet will derived fromquantum me hani al onsideration,justifying the phenomenologi alapproa h.

1. Phenomenologi aldes ription

The prin iple of spin transfer b etween two ferromagneti layers is sket hed on Fig. 5. Let us onsider an ele tri al urrent, spin-p olarized along the P dire tion (the ele tri al urrentmayb ep olarizedbyapreviousferromagneti layerforexample). Thisspin-p olarized urrent impinges on a N/F interfa e, where N is a normal metal (or a tunnel barrier) and F is a ferromagneti metalwhose magnetization M forms an angle  with P, so that P:M= os(6=0). Johnsonetal.

50

andVanSonetal.

51

showedthatanout-of-equilibrium magnetization(also alledspin a umulationindiusivesystems,orspin densityinballisti systems) app ears at this interfa e, due to the dierent spin-s attering rates in the N and F layers. In our system, sin e the impinging urrent is not p olarized following M, the rising out-of-equilibrium magnetization m p ossesses three omp onents. It an then exert a torque on the lo al magnetization M of the form T = J

sd =

B

Mm. Be ause of the fastangular pre ession ofthe ele tronsspin around Mand dueto therelaxationofthe spin a umulationm in the ferromagnet F,the transverse omp onent of the spin a umulation is qui klyabsorb ed lose to the N/F interfa e,on alength s ale 

J

,usually smallerthan 1 nmin metalli spin-valves

15,52

.

Another wayto understandspin transfer torque isto onsiderthatthe ele tri al urrent p ossesses an initial p olarization, des rib ed by the spin urrent J

s in

. One part of this im-pinging urrent is ree tedby the N/F interfa e,giving riseto a ree ted (ba kward) spin urrentJ

s r ef

. Inthe adiabati regime(theele tronspinpre essionisfast omparedto the lo- almagnetizationdynami s),afteralength

J

,itinerantele tronsarealignedalongthelo al magnetizationM and the transmitted spin urrentis then J

s tr ans

6=J s in

. The ree tedspin urrentJ

s r ef

b eing generally small,the net balan eof angular momentyieldsthe transverse omp onent of the in ident spin urrent: J

s in J s tr ans J s r ef =J s in ?

(note that transverse means transverse to M). Thus, the impinging ele trons lose the transverse omp onent of their magneti momentwhi h is transmitted to the lo alized ele trons, resp onsible for the lo almagnetization M. This spin transferis translated in atorque of the form:T= rJ

s . Stiles et al.

15

havedes rib ed the origin of spin transfer torque at aN/F interfa e,where Nisametal. Theauthors prop osed threeme hanismsgiving riseto spintransferinballisti systems. First,thespindep enden eoftheinterfa ialree tionand transmission o e ients

from left to right are qui kly reoriented (on a length  J

) when arriving in the right layer. The balan e b etween inwardand outward urrentsistransfertothelo al magnetization.

indu es a dis ontinuityof the spin urrentso that one part of the transverse omp onentof spin urrent is absorb ed at the interfa e. This dis ontinuity gives rise to a torque in the plane(P,M)whi htendsto alignPandM. Se ondly,the spin pre essionaround the lo al magnetization M, after averaging over the whole Fermi surfa e, gives rise to the omplete absorptionofthetransversespin urrentonalengths aleoftheorderof

J

=1nm. Finally, afterree tion by the interfa e,the ele tron spin formsan angle with both P and M. This spin rotationyieldsthe app earan eof another omp onentof the spintorque, p erp endi ular to the plane (P,M) and alled out-of-planetorque.

Thus,these three ontributionsgiveriseto atorque exertedby thespin a umulationon the lo al magnetization,writtenas:

T=a j M(MP)+b j MP (1) where a j and b j

are the in-plane and out-of-plane torque amplitudes. Note that in the rst theories of spin transfer torque by Slon zewski

5,53,54

and Berger

6,55

, the authors only deriveda

j

b e ause they onsidered that the ele tron spin remains inthe (P, M) plane, as orrob oratedbyab initio al ulations

15

. Thesetheoriesapplytometalli spinvalveswhere, duetothesmalllength

J

,spintransferisassumedtotakepla every losetotheinterfa e

56

. However,Edwards etal.

57

have derivedasizable out-of-plane torquein metalli spin-valves usingnonequilibriumGreen'sfun tionsandinterestingly,Zhangetal.

58

havedemonstrated thattakingintoa ountthespinpre essioninthetransp ortmo delsigni antlyenhan esthe b

j

term. In magneti tunneljun tions, b oth a j

and b j

termarise fromdierentme hanisms that willb e des rib ed inse tion I I I.

obtainsthe mo diedLLG equation,des ribing themagnetizationdynami softhefreelayer, submittedto b oth an external eld and a spin-p olarized ele tri al urrent:

M t = M(H e +b j P)+M M t a j M(MP) (2)

where is the gyromagneti ratio,  is the Gilb ert damping and H e

is the ee tiveeld, in ludingthe anisotropy eld,the demagnetizingeld and the external appliedeld. From Eq. 2, the out-of-plane torque a ts as an ee tive eld while the in-plane torque a ts as an ee tive (anti-)damping. As a fun tion of its sign, a

j

may ex ite or damp magneti ex itations in the magnetization M, whereas b

j

only ae ts the energy surfa e of the fer-romagneti layer. Dierent magneti b ehavior may b e observed: magnetization swit hing from a stable state to another, stabilization of magneti states at low energy minima, or magneti ex itations( oherentand in oherentpre essions).

2. Spin transfer in an arbitraryferromagnet

All along this se tion, we onsider the s-d mo del in whi h two p opulations of ele trons o exist: itinerantele trons(sp-typ eor itinerantd-typ e ele trons)andlo alizedele trons (d-typ emainly). Thelo alizedele tronsgiveriseto thelo almagnetizationoftheferromagnet. We also assume that the d lo al moments remain stationary. This mo del applies to the ele troni stru tureofferromagneti ele tro deswhose omp ositionslieonthe negativeslop e sideof the Slater-Néel-Pauling urve

59

(Ni, Co, NiFe,CoFe).

a. Itinerant ele trons dynami s Themotionofitinerantele tronsintheferromagneti materialsare representedbythe non-relativisti single ele tronHamiltonianin ludings d oupling: H = p 2 2m +U(r) J sd (:S d ) (3)

wherethe rstand se ondtermsare thekineti andp otentialenergies,whilethe thirdterm isthe s d ex hangeenergy,S

d

b eing the unit ve tor ofthe lo almagnetizationdue to the lo alized ele trons and J

sd

the s-d ex hange onstant. Let us dene the lo al spin density m(r;t) and the lo al spin urrent densityof itinerant ele tronsJ

s as m(r;t)=  (r;t) ~ 2  (r;t) (4) J s = ~ 2 2m Imf  (r;t)r r (r;t)g (5)

and the temp oral derivative ofthe spin density is:

d dt m(r;t)= ~ 2 f d dt   +   d dt g (6)

where = "

; #

is an arbitrary 2-dimension Hartree-Fo k wave fun tion. The two dimensionsreferto up (") and down (#)spin proje tionof the Hartree-Fo k wavefun tion. Fromthe time-dep endentS hro dinger equation i~d =dt=H ,weobtain the spin den-sity ontinuityequation:

dm dt = rJ s + 2J sd ~ S d m (7)

To orre tly des rib e the ferromagneti system under onsideration, one should add the intera tionsb etweenele tronsandlatti e,forexample. Indiusiveregime,one anintro du e a spin relaxationtermwhi hdep ends on the spin density

60 (m)= m  sf : dm dt = rJ s + 2J sd ~ S d m m  sf (8)

Eqs. 7 and 8 are of great imp ortan eto understand the role of spin transp ort in STT. One an see that the temp oral variation of the spin density (or spin a umulation) arises from the ontribution of three sour es: the spatial variation of spin urrent density, the torque exerted by the ba kground magnetization and a s attering sour e whi h a ts as a spin sink.

b. Lo alized ele trons dynami s The Hamiltonian of a single lo alized spin submitted to atimedep endent external eld and to an external urrent owis:

H = g B ~ S d :B 2J sd ~ S d :m= g B ~ S d :B eff (9)

where g is the Lande fa tor,  B

is the Bohr magnetron, S d

is the lo alized spin, B is the external magneti eld, m is the out-of-equilibrium spin density of the itinerant ele trons and B

eff

is the ee tive eld due to the ombination b etween the external eld and the itinerant ele tronspin density. ApplyingEhrenfest theorem

61 leads to d<S> dt = g B ~ <S>B eff (10)

where <> denotes averaging over allthe lo alizedstates, <S >=S d

. We an rewrite this equation as: dS d dt = g B ~ S d B 2J sd ~ S d m (11)

The rst termin ludes all the intera tions with magneti elds, likeexternal eld, magne-to rystalline anisotropy. The se ond term arises from the presen e of itinerant ele trons. In order to take into a ount the damping of the lo alized spin, one has to onsider a more ompleteHamiltonianthat in ludes many b o dy intera tions whi hleads to the usual Landau-Lifshitz-Gilb ert equation:

dS d dt = g B ~ S d B 2J sd ~ S d m+S d  dS d dt (12)

stru ture and setting g =2,and =2 B

=~, weobtain the mo died LLG equation:

dM dt = MH eff J sd  B Mm+M dM dt (13)

HereMisthe lo almagnetization,misthe out-of-equilibriumspina umulationor spin densityof itinerantele trons,and

H eff = H K M x M s e x + 2A ex M 2 s r 2 m 4M z e z +H ext e x (14) whereH K

isthe anisotropy eld, A ex

isthe ex hange onstant,and 4M z

isthe demagneti-zationeld. Thetermprop ortionalto J

sd

isatorqueexertedbythe spina umulationmon the lo almagnetizationM, similarto the one givenin Eq. 8. It isinterestingto note that onlythe transversespin a umulationmhas an inuen eonthe ba kgroundmagnetization state inthe formof a torque T along twoaxes:

T = J sd  B Mm= J sd  B [m x MP m y M(MP)℄ (15)

wherePistheunitve torparalleltothemagnetizationofthepinnedlayerandMistheunit ve tor parallel to the magnetizationof the free layer. The rst termin the right-hand-side of Eq. 15 is alled the eld-like term(or out-of-plane torque, or urrent-indu ed interlayer ex hange oupling) andthe se ondtermisthe usual Slon zewskiterm(orin-planetorque). The time s ale of itinerant spins dynami s is two orders of magnitude shorter than the time s ale of the ba kground magnetization dynami s. So one an onsider, in a rst ap-proximation,thattheitinerantspins an b edes rib edbythesteadystate equation(seeEqs.

7 and 8): rJ s (r;t)= 2J sd ~ mM (ballisti system) (16) rJ s (r;t)= 2J sd ~ mM+ m l sf (diusive system) (17)

Eqs. 16-17implythatthespatialtransferofspindensityp erunitoftimefromtheitinerant s-ele tronsto thelo alized d-ele trons (left-handsideterms)is equivalentto atorqueexerted by the transverse spin a umulation on the lo al magnetization (right-hand side terms), mo dulatedby the relaxationof the spin a umulationin diusive regime.

D. Theories of spin transfer in magneti tunnel jun tions

Slon zewski rst prop osed a free ele tron mo del of spin transp ort in a MTJ with an amorphous barrier

53

gle tingallnon-linear tunnelb ehaviors ( onsequently,the out-of-planetorque was found to b e zero). Ina twoband mo del,the torque was written as:

T = e 3 ( 4 k 2 " k 2 # )(k 2 " k 2 # ) 2 2 d( 2 +k 2 " ) 2 ( 2 +k 2 # ) 2 e 2d VM(MP) (18)

whereisthe barrierwaveve tor,k ";#

arethe Fermiwaveve torsformajorityandminority spins, d is the barrier thi kness and V, the bias voltagea ross the jun tion. Note that this mo delisrestri tedto re tangularbarrier,soverylowbiasvoltage. Morere ently, ombining BardeenTransferMatrixformalism(BTM)andhispreviousresults onthe relationb etween torquesand spin urrents

54

,the authorprop osed amoregeneralformulafor in-planetorque in magneti tunnel jun tions

35,62 : T = ~ 4 [ ++ + + + ℄m(mP) (19)   0 = 2eV ~ X p;q 2 p; ;q ; 0 (20) p; ;q ; 0 = ~ 2 2m Z dydz( p;  x  q ; 0  q ; 0  x p; ) (21)

where andaretheorbitalwavefun tionsforrightandleftinterfa e. Thisrelationstands forele tronswhoseenergy is lose to the Fermienergy. Theauthor underlined interestingly that Eq. 19 mayb e simplied ifthe integrals

  0

an b eseparated inthe form:

  0 /D L; D R; 0 (22) where D L(R);

is the density of states at the left (right) interfa e, for spin proje tion . In this ase, it is straightforward to see that the torque exerted on the right layer is redu ed to: T R = ~ 4 P L M(MP) (23) where P L

is the interfa ial p olarization of the density of states, as dened by Julliere

18

. This leads to abias asymmetryof the spin transfertorque, sin e the p olarizationP

L

isbias dep endent for only one dire tionof the applied voltage. The ondition of this separability hasb eendis ussedbySlon zewski

35

,Belas henkoetal.

63

andMathonetal.

64

. Theseauthors havesuggestedthatthephasede oheren es,indu edbydisorderinrealisti jun tions, ould redu e the p olarization fa tors to a pro du t b etween the interfa ial densities of states. It seemsthat this assumption is valid in magneti tunnel jun tions with not so thin barriers, esp e iallyin amorphousAlOx-based MTJs.

Theo donis etal.

65,66

re entlypresenteda tight-bindingmo del(TB)of MTJs,taking into a ount more realisti band stru tures than the usual free ele tron mo del. These studies showed that the in-plane torque should present an imp ortant bias asymmetry while the

on the bias voltage. This is in agreement with re ent studies of Wil zynski et al.

67

and Man hon etal.

68

,based on free ele tronmo del,as dis ussedin this hapter. The role of magnons have b een addressed by Levy et al.

42

and by Li et al.

69

. It was shown that magnons emission may strongly inuen e the bias dep enden e of spin transfer torque ontributing to mo dify the absorption length 

J

. This me hanismwill b e dis ussed in se tionIV.

Finally, note that all these theories assume amorphous barriers and a plane wave de-s ription of the transp ort, although most of the exp erimentsare arried out on rystalline MgO-based MTJs. A re ent publi ation from Heiligeret al.

70

addresses the hara teristi s of spin transfer torque in Fe/MgO/Fe rystalline jun tions. The dominant ontribution of

1

symmetrystrongly inuen esspin torque feature.

I I I. QUANTUM ORIGIN OF SPIN TORQUE IN MAGNETIC TUNNEL JUNC-TIONS

Wewillnow des rib ethe spin transp ort in magneti tunnel jun tions. Although mostof the exp erimentsare nowadays p erformedin rystallineMgO-based MTJ,one an get arst insight of TMR and spin torque by simply onsidering a free ele tron mo del of magneti tunnel jun tions.

We rst intro du e the free-ele tronmo del,and then depi t the spin transp ort in a MTJ with non- ollinear magnetization dire tions. Afterward, we will des rib e the role of the barrier on the spin transfer torque. Finally,the origin of the torques and oupling b etween the twoferromagneti layerswillb eexplained.

A. Free ele tron mo del

The basisofour al ulationisdepi tedinthe toppanelofFig. 6. Theout-of-equilibrium magneti tunneljun tionismo deledbya" ondu tor"(inthesensethatthetunnelbarrieris not innite)linkingtwomagneti reservoirs (F

L

and F R

) with non ollinearmagnetizations and with dierent hemi al p otentials 

L and  R 72 ( L > R ). A bias voltage V = ( L  R

)=e is applied a ross this " ondu tor". One has to onsider all ele trons with majority spins (solid arrows) and minority spins (dotted arrows), originated from left (rightward arrows) and right ele tro des (leftward arrows). In low bias limit (

L  

R

), the harge transp ort an b e approximately determined by the ele trons originated only from the left ele tro de withan energy b etweenE

F

and E F

eV.

In our ase (middle panel of Fig. 6), the magneti tunnel jun tion is omp osed of two ferromagneti layers, F

L

and F R

(made of the same material, for simpli ity), resp e tively onne ted to the left and right reservoirs and separated by an amorphous tunnel barrier.

Toppanel: spin-dep endentout-of-equilibriumtransp ortina ondu torlinkingtworeservoirsF L

and F

R

(whoseele tro hemi alp otentialsareresp e tively L

and R

)withnon ollinearmagnetization orientations. The solid arrows represent the majorityspins and the dotted arrows represent the minorityspins. Middle panel: MTJwith non ollinear magnetizationorientations. Bottompanel: Corresp ondingenergyproleoftheMTJ.Infree-ele tronapproximation,thelo aldensityofstates areparab oli formajority(solidline) andminority(dottedline)ele tronswithasplittingb etween thetwospinsub-bandsequals totheex hangeintera tionJ

sd .

Thex-axisisp erp endi ulartothe planeofthelayersand themagnetizationofF L isoriented following z: M L =M L z. The magnetizationM R of F R

isin the(x,z) plane and tilted from M

L

byanangle . Inthis onguration,thespindensityintheferromagneti layerp ossesses three omp onents : m=(m

x ;m y ;m z ). In F L

(we obtain the sameresults onsidering F R

), the transverse omp onents are m

x =<  x > and m y =<  y >,where  i

are the Pauli spin matri es and <> denotes averaging over orbital states and spin states, i.e. averaging over ele tronsenergyE,transversemomentumandspin states. The transversespin densityin the left layeris then givenby <

+ >=< x +i y > : m x +im y =<  + >=2< " # > (24)

In other words, the in-plane torque is given by the imaginary part of <  +

>, while the out-of-planetorque isgivenby itsreal part. One an understandthepro du t<

"

# >as a orrelationfun tionb etweenthe twoproje tionsofthe spinofthe impingingele trons. In ballisti regime,the spinof anele tronimpingingonaferromagnetwith aspinp olarization tiltedfromtheba kgroundmagnetizationpre essesaround this magnetization

15,66

. Lo ally, itstwo proje tions"and #following thequantizationaxis(denedby theba kground

mag-spin density m x

and m y

. Ifthe ele tronspin isfullyp olarized parallelor antiparallelto this magnetization, no pre ession o urs and its ontribution to the transverse spin density is zero.

We remind that we dened majority (minority) states as the spin proje tion parallel (antiparallel) to the magnetization of the left ele tro de. Therefore, <

"

#

> is the fra tion of ele trons whose spin is following x (real part) and y (imaginary part) in spin spa e.

In Keldysh out-of-equilibrium formalism

72,73

, the ondu tivity is al ulated onsidering the ontributionof the ele tronsoriginatingfrom theleft reservoirandfromtheright reser-voir(top panel of Fig. 6). The out-of-equilibrium Greenfun tion G(r;t;r

0 ;t

0

) (or Keldysh Green fun tion)is dened as a sup erp osition of these two ontributions:

G( r;t;r 0 ;t 0 )=f L L (r;t)  L (r 0 ;t 0 )+f R R (r;t)  R (r 0 ;t 0 ) (25) where L(R)

(r;t) are the ele tron wave fun tions originating fromthe left (right)reservoir at the lo ation r and timet and f

L(R)

are the Fermidistribution fun tions in the left and rightreservoirs.

Thus, the S hro dinger equationof the magneti tunneljun tion is:

H =  p 2 2m +U J sd (:S d )  " # ! =E " # ! (26)

where  the ve tor in Pauli matri es spa e :  =( x ; y ; z ) T

, E is the ele tron energy,U is the spin-indep endent p otentialalong the jun tion:

J sd (:S d )=J sd  z and U =E F for x<x 1 J sd (:S d )=0 and U(x)=U 0 x x 1 x 2 x 1 eV for x 1 <x<x 2 J sd (:S d )=J sd ( z os+ x sin ) and U =E F eV for x>x 2

We onsiderthat the p otentialdrop o urs essentiallywithin the barrier and weassume the bias voltage is low ompared to the barrier height (V << U=e). This allows to use WKB approximation to determinethe wave fun tions inside the barrier. Furthermore, the freeele tron approximation impliesparab oli disp ersion laws whi halso restri ts our study to lowbias voltage.

In the 2-dimensionalHartree-Fo k representation,spin-dep endent urrent and spin den-sity are denedusing the out-of-equilibriumlesser KeldyshGreen fun tion:

G +   0 (r;r 0 )= Z d  f L h  0 (") L (r 0 )  (") L (r)+  0 (#) L (r 0 )  (#) L (r) i +f R h  0 (") R (r 0 )  (") R (r)+  0 (#) R (r 0 )  (#) R (r) i (27)

where f L

= f (), f R

= f ( +eV), and f () is the Fermi distribution at 0 K. In-plane (a

j

M(MP)) and out-of-plane torques (b j

MP) an now b e determined from Eq.

24, whereas spin-dep endent ele tri al urrent densities are al ulated from the usual lo al denition: b j +ia j = J sd  B < + >=2 J sd  B a 3 0 (2) 2 Z Z G + "# (x;x;)dd (28) m z = J sd  B a 3 0 (2) 2 Z Z  G + "" (x;x;) G + ## (x;x;)  dd (29) J "(#) = ~e 4m e Z Z   x  x 0  G + ""(##) (x;x 0 ;)j x=x 0dd (30) J =J " +J # (31) G + "" (x;x;)and G + ##

(x;x;) are the energy-resolvedlo aldensity-of-states(LDOS) for up-and down-spins resp e tively,whereas

R G + "" (x;x;)dand R G + ##

(x;x;)d give the density of up- and down-ele trons at lo ation x along the stru ture.

To illustrate the ab ove al ulation, we use material parameters adapted to the ase of Co/Al

2 O

3

/Co stru ture: the Fermiwaveve torsformajorityand minorityspinsare resp e -tively k " F = 1:1 Å 1 , k # F = 0:6 Å 1

, the barrier height is U E F

= 1:6 eV, the ee tive ele tron mass within the insulator is m

eff =0.4

74

and the barrier thi kness is d=0.6 nm. These parametershaveb een ho osen to tthe exp erimentalI-V hara teristi s ofthe mag-neti tunneljun tionsstudiedinRef.

49

. Inallthisse tion,the magnetizationsforman angle of  =90

Æ

. We willjustifythis hoi e inthe following.

B. Spin transp ort in a MTJ

Althoughspin-dep endenttunnellingisawellknown pro ess,the des riptionwegivehere is of great imp ortan e to understand the sp e i hara teristi s of spin transfer torques in tunnellingtransp ort. Inthispart,wewill onsiderthelinearapproximationinwhi hthebias voltageV

b

is low enough so that the urrentis due to Fermiele tronsinje ted fromthe left ele tro de. When the ele tro desmagnetizationsare non ollinear,the ele trons are no more des rib ed as pure spin states, but as a mixing b etween majority and minority states. For example,let us onsiderone ele tronfromthe leftreservoir,initiallyinmajority spinstate, impingingontherightele tro de(seeFig. 7-step1). Therstree tion(step2)attheF

L =I interfa e do not intro du e any mixing sin e the insulator is non magneti . However, when (thetransmittedpartof)thisele tronisree tedortransmittedbythese ondinterfa eI=F

R (step3),theresultingstateintherightele tro deisamixingb etweenmajorityandminority states sin e the quantization axis in the right ele tro de is dierent from the quantization axis in the left ele tro de. Then, the transmitted spin is reoriented and pre esses (step 4) around the magnetization of the right ele tro de. Furthermore, the ree ted ele tron (step 5) is also in a mixed spin state and pre esses around the left ele tro de magnetization. In

ele tro des magnetizations. Step 1: the ele tron spin is p olarized along the magnetization of the left ele tro de. Step 2: After the rst ree tion/transmission by F

L

=I interfa e the ree ted and transmittedpartsremain in a purespin state. Step 3: The ree tion/transmissionby these ond interfa eI=F

R

reorientstheele tronspin. Step4and5: Thetransmittedandree tedspinspre ess aroundthelo al magnetization.

other words, after transp ort through the barrier, the ele tron spin is ree ted/transmitted with an angle. This reorientationgivesriseto spin transfer torque.

Note that there is no reason why the ele tron spin should remain in the plane of the ele tro desmagnetization. Wewillseethatafterthereorientation,theele tronspinp ossesses three omp onentsin spin spa e (and so two transverse omp onents).

C. In iden e sele tion in an amorphous barrier

1. -sele tion due totunnelling

It is well know that in non magneti tunnel jun tions, the transmission of an imping-ing ele trons dep endent on its in identdire tion. As a matter of fa t, the ee tive barrier thi kness involved in the tunnelling pro ess is larger for grazing in iden e than for nor-mal in iden e. The transmission o e ientde reasesexp onentially with the in-planewave ve tor , so that only ele trons whose wave ve tor is lose to the p erp endi ular in iden e signi antly ontribute to the tunnellingtransp ort.

Furthermore, inmagneti tunnel jun tions, the transmission o e ients also dep end on thespinproje tionoftheele trons,aswellasonthemagneti ongurationofthe ferromag-neti ele tro des. This"-sele tion"isillustratedinFig. 8(a). Asdis ussedpreviously,when the ele tro des magnetizationare non- ollinear,the spin of an impingingele tron,originally inapurespinstate, isreorientedafterree tionso thattheree tedstateisinamixedspin state. In our ase, only the ree tion o e ientsof the onservedspin part are rep orted in

fun tion of the in-plane waveve tor; (b) ree tion angles  (solid line) and  (dotted line) of an initially majority as a fun tion of . The applied bias voltageis V

b

= 0:1 V and  =90 Æ

. Insert: denitionof theree tionsangles.

Fig. 8(a).

Notethat onlyaverysmallpartof the inje tedp olarizedwaveisipp edduring the tun-nellingpro ess. However,thisdo esnotmeanthatspintransfertorqueissmallinMTJs,sin e only oherent mixedstates ontributeto the transversespin density,whi hisresp onsibleof the spin transfer torque.

2. Spin sele tion due to ferromagnets

Following the previous dis ussion ab out spin reorientation (see Fig. 7), it is p ossible to dedu e the angles at whi h the ele tron spin is ree ted by the barrier. We dene the azimuthalangle  andthe p olar angle  as indi atedinthe insert ofFig. 8(a).

Fig. 8(b) displaystheseanglesas afun tionofthein-planewaveve tor. Theazimuthal anglevariesb etween-64

Æ

to+77 Æ

whilethep olarangleremainsverysmall(lessthan0.2 Æ

, whi h means that the ele tron spin stays very lose to the quantization axis, as dis ussed ab ove). At=0:6 Å

1

( orresp onding to k # F

),=0whi hindi atesthat theee tivespin densitylies inthe plane ofthe magnetizations(M

L ;M

R

). Finally,the p olar angle do es not vary with the distan e, whi h means that the ree ted ele tron spin pre esses around O

z withasmallangle. A"bulk"spintransferresultsfromtheinterferen esofalltheree ted

in majorityspin state. Theparametersarethesameas in Fig. 8.

ele trons.

Thestrongdep enden eofasafun tionofthein-planewaveve tor, ombinedwiththe -sele tion losetothenormalin iden e(seeFig. 8(a)),impliesthat theee tivespinofthe transmitted ele trons p ossesses an imp ortant out-of-plane omp onent. In other words, the ee t of the spin-dep endent tunnelling is to strongly enhan e the out-of-plane omp onent of the spin torque, omparedto metalli spin valves. As a matter of fa t, inmetalli spin-valves,the whole Fermisurfa e ontributesto the spin transp ort so that the ee tiveangle  is verysmall

15

and orrelativelythe out-of-plane torque is negligible.

Fig. 9 shows the dep enden e of the angles as a fun tion of the s-d ex hange onstant J

sd

for p erp endi ular in iden e  = 0. Quite intuitively, the pre ession angle  in reases with J

sd

whereas the initial azimuthal angle  de reases in absolute value with J sd

. The spin-ltering ee t (the sele tion b etweenmajority and minority spin during the ree tion pro ess)in reaseswith J

sd

so thatthe ree tedspindire tiongets loser to theplane ofthe magnetizations.

D. Spin ltering in rystalline stru tures

Besides the two fundamental tunnelling sele tion me hanisms dis ussed ab ove, an ad-ditional spin ltering me hanism was prop osed by Butler et al.

30,31

whi h takes advantage from the ele troni stru ture of b oth ele tro de and insulator rystallinematerials ompris-ing MTJ. It is based on the fa t that only ele trons of ertain wave fun tion symmetries an easily propagate through the barrier. For instan e, in Fe(001) only the majority spin hannelhas ele troni states with

1

1 state inthe MgOgap with thesmallest de ayrate

30,31

. Asa result,Fe|MgO|Fe(001)tunnel jun tionhas averylarge ondu tan einparallelstatedueto fairlytransparent

1

majority hannelat k

jj

=0. Antiparallelmagnetizations onguration, on a ontrary,is low ondu -tivesin e the

1

symmetrystates do es not exist inthe minority bandstru ture around the Fermilevel

30,31

.

Spin transfer torque is nowadays usually observed in MgO-based rystalline jun tions, whereas only few theoreti al work has b een done on spin transfer in rystalline stru tures. The rst theoreti al studies of Heiliger et al.

70

on MgO-based MTJs indi ate a dominate ontributionof the

1

symmetryon spintransp ort whi hmay ae t theobservable hara -teristi s of STT,as dis ussedin se tion IV.

E. Torques and oupling

The me hanismswepreviouslydes rib edare at theoriginof spin-dep endentplane waves in the MTJ. The interferen esb etweenthese wavesgive rise to an out-of-equilibrium mag-netization m whi h ouplesthe ferromagneti ele tro des.

In thelinear regimeunder onsideration,thethree omp onentsofspin densityinthe left ele tro de an b e des rib edas follows:

m " xL +im " y L =A(V)sin  e i(k 1 +k 2 )(x x 1 ) r " 1 e i(k 1 k 2 )(x x 1 )  (32) m # xL +im # y L =A  (V)sin  e i(k 1 +k 2 )(x x 1 ) r # 1 e i(k 1 k 2 )(x x 1 )  (33) m " z L =B " (V) 1 k 1  r " 1 e 2ik1(x x1) +r " 1 e 2ik1(x x1)  (34) m # z L =B # (V)+ 1 k 2  r # 1 e 2ik 2 (x x 1 ) +r # 1 e 2ik 2 (x x 1 )  (35) whereA(V),B ";# (V) and r ";# 1

are o e ients dep endingon the jun tionparameters and on the bias voltage

17

and k 1;2

are the waveve tors ofmajorityand minorityspin, resp e tively. Considering m

"(#) +L

in Eqs. 32-35, two omp onents an b e distinguished : the rst one is prop ortional to e

i(k1+k2)(x x1)

, and due to the interferen e b etween the in ident wave with majority (resp. minority) spin and the ree ted wave with minority (resp. majority) spin; the se ond one is prop ortional to e

i(k 1 k 2 )(x x 1 )

and due to the interferen e b etween the ree ted waves with majority and minority spins. We note that the rst omp onents of m " +L and m # +L

are omplex onjugated so that their sumis real. Then, the interferen e b etween the in ident wave with majority spin and the ree ted wave with minority spin do es not ontribute to in-plane torque but only to out-of-plane torque. In-plane torque is then generated by the oherent interferen esb etweenree tedele tronswith opp osite spin proje tion (/e i(k 1 k 2 )(x x 1 ) ).

left ele tro de, as a fun tion of the distan e from theinterfa e. Top panel: m x

omp onent of spin density (solidline); thedashed lines are theenvelop es ofthe urve. Middle panel: m

y

omp onent of spindensity. Bottompanel: m

z

omp onent ofspin densitydue toinitially majority(solid line) andminority(dottedline) spinproje tion;thedashedlines arethemeanvaluesoftheos illations. The applied bias voltageis V

b

=0:1 V.The verti al line on the right isthe interfa eb etween the leftele tro de andthetunnel barrier.

Con erning m z L

, it is omp osed of one omp onentprop ortional to e 2ik 1 (x x 1 ) ,one om-p onentprop ortionalto e

2ik 2

(x x 1

)

and one onstantas afun tionofx. Thetwoformersare due to the interferen e b etween waves having the same spin proje tion but with opp osite propagation dire tionwhilethe latter isdue to interferen eb etweenwaveshavingthe same spin proje tion and the samepropagation dire tion.

Fig. 10 displays the details of the spin density omp onentsm x , m y et m z (des rib ed in Eq. 32) in the left ele tro de as a fun tion of x, when V

b

= 0:1 V. m x

p ossesses a quite omplexb ehavior with twop erio ds ofos illation (the dashedlinesshowthe envelop eof the urve),whereas m

y

isredu edto asingle os illation(Theos illation p erio dk 1

+k 2

vanishes whensummingthe ontributionof majorityandminorityspins);m

z

os illatesaroundmean values representedby horizontaldashed lines.

at zerobias

75,76

)isonlyprop ortionalto e i(k 1 +k 2 )(x x 1 )

. Butat nonzerobias, thedissipative part of the out-of-plane torque is prop ortional to b oth e

i(k 1 +k 2 )(x x 1 ) and e i(k 1 k 2 )(x x 1 ) .

IV. OBSERVABLE PROPERTIES

Up to now, in order to des rib e the quantum origin of spin torque in MTJ, we fo used on Fermiele tronsand lowbiasvoltage. Todepi ttheobservableprop ertiesofspintransfer torque in MTJ, we should take into a ount all the ele trons from the left and the right ele tro desso as to in lude non-linearpro esses.

A. Angular dep enden e

Fig. 11(a) shows the normalizedin-planeand out-of-plane omp onents,a nor m j and b nor m j , as a fun tion of the angle  b etweenthe ele tro des magnetizations,at V

b

=0 and V b

=0:1 V. The normalizedtorques are denedas:

T nor m

=T=T(90 Æ

)sin

It learly app ears that b oth omp onents are prop ortional to sin (the deviationfrom sin is smaller than 10

4

). This dep enden e is strongly dierent from what was predi ted in metalli spin valves

16,17,54

(see Fig. 11(b)) and has b eenattributed

35

to the single-ele tron natureof tunnelling.

As amatteroffa t,inmetalli spin-valves,thespin a umulation,duetospin-dep endent s attering at the interfa es, mo dies the p otential prole seen by the ele trons. This ee t isduetothe multi-ele tronsnatureofdiusivetransp ort, sin ethetransp ort ofoneele tron spin isae ted by the spin a umulationrising fromthe wholespin p olarized urrent. This spin a umulation strongly inuen es the angular dep enden e of the sta k resistan e and spin transfer torque

17

.

On the ontrary, in magneti tunnel jun tions, b e ause of the imp ortant height of the tunnel barrier ( 0:8 3:3 eV), all the p otential drop o urs inside the insulator and the spin a umulation(i.e. the feedba k of the urrent-indu edlongitudinalspin densityon the spin urrent)is negligible. In this ase, the angular dep enden e of torque is determinedby the angular dep enden eof the transmissionmatrix,as dis ussed inRef.

35

and yieldsa sine shap e. In the following,we willestimate the spin density for ==2.

Note that, at zero bias, the out-of-plane torque is still non-zero, ontrary to in-plane torque. The onservative part of the out-of-plane torque (interlayer ex hange oupling at zero bias) omes from the ontribution of ele trons lo ated under the Fermi level

75,76

. At zerobias, the urrentsfromleftand rightele tro desare equal,butthe ele tronpropagation still orresp onds to the s hemeshown inFig. 7: the mixingb etweenmajorityand minority states indu esa transverse omp onentinthe spin density.

(dottedline)in amagneti tunneljun tion;(b)Angulardep enden eof normalizedin-plane torque in ametalli spin-valve. From Ref.

17

.

B. De ay length of spin density

As dis ussedin se tionI IC2, spin transfer torque isestimatedfromthe transverse om-p onent of the spin density. This spin density (or spin a umulation in diusive systems) usuallyde aysduetoquantuminterferen esorspin-dep endents attering,sothatspintorque is generallyassumed to b e an interfa ialphenomenon.

1. Ballisti interferen es

In the present mo del, no spin-diusion is taken into a ount and the Fermi surfa e is assumed spheri al. Fig. 12 displays the two omp onents of transverse spin density as a fun tion of the lo ation in the left ele tro de. The interferen e pro ess b etween p olarized ele tronsyields a damp edos illation of the in-plane omp onentm

x

(giving riseto the out-of-plane torque) as presented in Fig. 12(a). We an distinguish two p erio ds of os illation T 1 = 2=  k " F k # F  and T 2 = 2=  k " F +k # F 

whereas at zero bias, only T 2

app ears (see inset of Fig. 12(a)). This an b e easily understo o d by onsidering ele trons from left and rightele tro des. The transversespin density in the left ele tro de due to ele tronsfrom the rightele tro de is:

m " +R =C " (V)sin e i(k 1 k 2 )(x x 1 ) (36) m # +R =C # (V)sin e i(k 1 k 2 )(x x 1 ) (37) where C ";#

(V) are o e ients dep ending on the jun tion parameters and on the bias voltage

17

. Itisnowp ossible toshowthatinthegeneralexpressionoftransversespin density

m + =m x +im y =m " +L +m # +L +m " +R +m # +R

the termsprop ortional to e

i(k1 k2)(x x1)

vanish at zero bias due to the an ellation of on-tribution of ele tronsfromthe left and rightreservoirs at zero biasvoltage(A(0)+A

 (0) =

density - inset: In-plane spin density at zero bias voltage; (b) Out-of-plane spin density. These quantitiesare al ulated atV

b =0:1V. C " (0) +C # (0)) so that m +

redu es to terms prop ortional to e i(k 1 +k 2 )(x x 1 )68 . F urther-more, these last terms only give a real omp onent sin e, as dis ussed ab ove, the majority and minority omp onentsof m

y

(giving riseto the in-planetorque) omp ensateea hother. Consequently, at zero bias, only the onservative part of the out-of-plane torque (zero bias interlayerex hange oupling) exists, due to the interferen eb etween in identand ree ted ele tronswithopp ositespinproje tion

75,76

. Butwhenthebiasvoltageisnonzero,the trans-p ort b e omes asymmetri and the terms prop ortional to e

i(k 1 k 2 )(x x 1 )

do not omp ensate ea h otheranymore whi hleads to twop erio ds of os illations as shown inFig. 12(a).

In-plane omp onent of spin transfer torque, prop ortional to m y

, exits only at non zero biasandp ossessesonlyonep erio dofos illationT

1

(seeFig. 12(b)). Itisworthytonotethat the transverse omp onentsof spin density is damp ed by 50% within the rst nanometers, and that the amplitude of the out-of-plane torque is of the same order than the in-plane torque. Thisde aylengthisverylarge omparedtoprevious theoreti alpredi tions

15,54

and exp erimentalinvestigations on SV

52

. As amatterof fa t, theballisti assumption holdsfor distan esmallerthanthe meanfreepath(5nminCo). In realisti devi es,spindiusion pro esses shouldin rease the de ay of the transverse omp onentsof spin density.

Finally, Fig. 13 shows the out-of-equilibrium longitudinal spin density
n dened as
n "(#) =n "(#) (V b =0:1) n "(#) (V b

=0).
nos illatesand asymptoti allyrea hesanonzero value. This meansthat when the biasvoltage isturnedon, anon equilibriumspin a umu-lationbuildsup. However,thisee tivespin a umulationisverysmall(
n

"
n

# 10

for majority(solid line) and minority (dottedline) ele tron spin proje tions. The bias voltageis V

b

=0:1 V.

ele tron/atom)and annot inuen espin urrentbuilding. Therefore,negle tingthe roleof longitudinal spin a umulation(spin density) inMTJ is justied.

2. Spin s attering me hanisms

Inrealmagneti tunneljun tions,oneshouldtakeintoa ountspin-ippro essesindu ed byspin-orbit ouplingaswellashotele trons-indu edspin-wavesemissionsthato urwithin the diusive ferromagneti ele tro des. Spin-orbit indu edspin-ip s attering (Elliott-Yafet me hanism

78,79

) as well as spin-wave s attering

80

lead to spin-diusion length, l sf

of 15-30 nminusualferromagneti ele tro des

81

. This spin-ipshouldin reasethespatial de ayrate of the spin density by a fa tor of e

l sf

x .

Spin-ip s attering by hot-ele trons indu ed spin wave is a spin-ip me hanism that sp e i ally o urs in magneti tunnel jun tions

37

. In tunnel jun tions, at non zero bias, spin-p olarizedele tronsfromtheleftele tro deimpingetothe rightele tro dewithanenergy higher than the lo al Fermi energy: they are alled "hot ele trons". These hot ele trons relaxtowardstheFermilevelbyinelasti s attering involvingphononandmagnon emission. Following the Fermi Golden rule, this spin-waves emission in reases with temp eratureand energy of thehot ele trons. Li etal.

69

have shown that the spin-diusion lengthdue to this me hanismis written: l sf /J F E F =J 2 sd V b (38) where J F

is the ferromagneti ex hange onstant and E F

the Fermi energy. The authors nd aspin-diusionlengthofab out 0.5-2nmforreasonable parameters. This demonstrates the essentialrole of magnon emissionsin magneti tunnel jun tions.

In order to more a urately des rib e spin-dep endent transp ort throughout rystalline barriers

30,31

(in parti ular MgO-based MTJs), the role of defaults in the barrier

71

, or inter-fa ial states ee ts, it is ne essary to go b eyond the free ele tron mo del and onsider the real band stru ture of the sta k.

Firstprin iplestudiesofrealisti Co/Cuinterfa es

82

(so,metalli spin-valves)showedthat the mismat h of the ele troni stru ture at the interfa e for spin down ele trons strongly redu es the transverse omp onentof spin density. As a matter of fa t, the spin-dep endent transmission at the interfa e b e omes more omplex. In parti ular, the ele tron phase distributionb e omes broadand asymmetri

15

. Thisleads to arapid interfa ialde ay ofthe transverse spin a umulationin metalli spin-valves. In MTJ, the non spheri al nature of the spin-dep endent Fermi surfa e

30,63,71

should also dramati ally alter the transverse spin density. This ould explain the fa t that the amplitudeof spin torque in the free-ele tron mo delwe prop osed istwoorders of magnitude higher than inexp eriments.

Heiligeretal.

70

re entlystudiedthespintransfertorqueinFe/MgO/Fe rystallinetunnel jun tion. The authors showedthatthe interfa ialspin density de ayisevenstronger inthis typ e of MTJ than inmetalli spin-valves. This de ay is attributedto the dominant ontri-bution of

1

ele tronsfor whi hFe b ehavesas a half-metal with resp e t to this symmetry. Spin transfer torque arising from the interferen es b etween majority (propagative states) and minority(evanes ent states) ele trons,islo alized lose to the MgO/Fe interfa e. This p oint willb e addressed inmoredetails in se tionV.

C. Bias dep enden e

1. Freeele tron model

The bias dep enden e of in-plane and out-of-plane torques in MTJ also presents strong dieren eswith metalli spin-valves. Werst al ulate thetotal spin torque exertedon the left ele tro de. Following the denition of Ref.

5

and Ref.

65

,the total torque is:

! T total = Z 1 x 1 rJ s dx=J s (x 1 ) (39)

Fig. 14 displays the total out-of-plane (a) and in-plane (b) torques as a fun tion of the applied bias voltage, for dierent values of the s-d ex hange parameter J

sd

. Consistently with Theo donis etal.

65

, the out-of-plane torque is quadrati whereas the in-plane torque is a ombinationb etween linearand quadrati bias dep enden e.

Finally,note that a hangeof sign of spin transfertorque at high p ositivebias voltage is exp e ted

65

oupling: J sd =0:38eV (op en ir les), J sd =0:76eV (lled ir les), J sd =1:62eV (op ensquares), J sd = 2:29 eV (op en triangles), J sd

= 2:97 eV (lled squares). Top inset: Bias dep enden e of STTforJ

sd

=1:62eV;thesolid line was al ulated followingtheusual wayandthesymb olswere al ulated using Eq. 46.

barrier height and high breakdown voltage (MgO seems a go o d andidate). Nevertheless, morete hnologi al developmentare neededto fabri atesu h jun tions.

However, Eq. 39 assumes that all the transverse spin density is relaxed within the free layer. Inother words,the initiallymisalignedin identele tronspin eventuallyalignson the lo almagnetizationwithin the freelayer. This assumptionseemsto b e valid, regarding the previousdis ussions. Nevertheless, onsideringweakspin-diusion pro essesas wellas non-half metalli jun tions(i.e. not likeFe/MgO/Fe), one may assumethat the ele tron spin is not fully aligned on the lo al magnetizationwhen leaving the free layer. This assumption may b e valid in magneti semi ondu tor-based tunnel jun tions, where the spin-diusion length is very large

83

. Fig. 15 displays the bias dep enden e of out-of-plane and in-plane torques for dierentintegration depths t (namely,dierentlayerthi knesses):

! T par tial = Z x 1 t x 1 rJ s dx=J s (x 1 ) J s (x 1 t) (40)

Thebiasdep enden e an hangedrasti allyandtheout-of-planetorque aneven hangeits sign(notethatthein-planetorquekeepsitsgeneralshap e). Thesedep enden iesarestrongly ae ted by the tunnel barrier hara teristi s and one should b e areful in the analysis of bias dep enden e.

sd

dierent values integrationdepth: t=0 Å(op ensquares),t=4 Å(lled triangles),t=10Å(lled ir les), t=1Å(op en ir les).

Figure 16: S hemati sof the ir uit mo delprop osedbySlon zewski

35,54

.

2. Cir uittheory

Slon zewski

35

prop osed a ir uit mo del to des rib e magneti tunnel jun tions in the general ase, without restri tion of the band stru ture of the ele tro des and of the barrier. Fig. 16 shows the s hemati sof this mo del. Theo donis etal.

65

havedemonstratedthat this mo delrepro du es wellthe bias dep enden eof the in-plane torque. Ifone onsiders the two pure spinstates inthe quanti ation axisof the leftele tro de j">

L

and j#> L

j"> L = os  2 j"> R +sin  2 j#> R (41) j#> L = sin  2 j"> R + os  2 j#> R (42)

where  is the angle b etween the magnetizations of the ele tro des. Then, the probability for an ele tron spin  in the left ele tro de to b e observed in a spin proje tion 

0 in the right ele tro de is P   0 =j <j 0 > j 2

. The asso iated resistan es indi atedon Fig. 16 are inverselyprop ortional to this probability,thus leadingto:

R   ( )=R  (0) os 2  2 (43) R  ( ) =R  ()sin 2  2 (44)

Usingthe expressionof in-planespin transfer torque derivedbySlon zewski

35 : a j =~(J " L J # L +(J # R J " R

) os )=2esin (45)

whereJ L(R)

 isthe urrentdensity ofthe spin proje tion  inL(R) ele tro de,we thennd:

a j = J s AP J s P 2 (46) where J s AP(P)

are the interfa ial spin urrent densities when the magnetizations are in an-tiparallel(parallel) onguration. Theo donisetal.

65

laimedthatthisrelationisindep endent of the ele troni stru ture or of the adopted des ription (free ele tron,tight-binding...). As amatteroffa t,theinsertofFig. 14 showsthe STT al ulatedusingEq. 39(solidline)and using Eq. 46 (symb ols), whi hare in verygo o d agreement. FromBrinkman'smo del

84

, the authors demonstrated that the omp onent T

jj

is the sup erp osition of a linear ontribution J

s P

and a quadrati ontribution J s AP

as afun tion of the bias voltage. As a matterof fa t,Brinkmanet al.

84

have showed,froma freeele tronmo del,that the urrentdensity owing a rossa non magneti tunnel jun tion whose barrier is asymmetri and submittedto abias V mayb e des rib ed by:

J(V)=f 1 (  )V f 2 (  )
V 2 +O (V 3 ) (47)  =( l + r )=2 (48)
= l  r (49) where  l and  r

are the barrier height at the left and right interfa es, measured from the b ottom of the ondu tion band. f

1

and f 2

are determined in Ref.

84

. In the ase of a magneti tunneljun tion, Eq. 47 apply to ea h spin proje tion. When the magnetizations are parallel,theMTJb ehaveslikeasymmetri tunneljun tionfor ea hspinproje tionand

  " 6=   # ,
 " =
 #

the MTJ b ehaveslike a asymmetri tunnel jun tion for ea h spin proje tion and   =   ,
 " =
 #

. The spin density isthen:

J s P =(f 1 (   " ) f 1 (   # ))V +O (V 3 ) (50) J s AP = 2f 2 (  )V 2 +O (V 3 ) (51)

Bythisway,Theo donis etal.

65

demonstratedthatthe generalformofthe Slon zewskiterm is a j = a 1 V +a 2 V 2 +O (V 3

). The balan e b etween the two bias dep enden ies, quadrati and linear, may b e mo diedby varying J

sd .

Notethatthe ir uitmo del annot des rib ethese ond omp onentb j

ofthespintransfer, sin eitmakestworestri tiveassumptions: i)duringthetransp ort, theele tronspinremains inthemagnetizationplane( =0-seeFig. 9)andii)thespin urrentis ompletelyabsorb ed attheinterfa e(nopre essionistakenintoa ount,sin etheele tronspinisinstantaneously reorientedalongthelo almagnetization). Thesetwohyp othesisignoretheee tswhi hgive riseto the out-of-plane torque

35

.

3. Asymmetri jun tion

Wil zynski et al.

67

re ently showed that the bias dep enden e of the torque is strongly ae tedbythesymmetryofthejun tion. Consideringtwodierentferromagneti ele tro des (dierent thi knessor dierent s-d ex hange oupling), the authors show that the bias de-p enden emayb e verydierent fromthe usual parab oli and se ond orderbias dep enden e depi tedin Fig. 14.

Slon zewski et al.

62

re ently prop osed a study of the inuen e of elasti and inelasti tunnelling in the spin transfer torque hara teristi s. This dis ussion is restri ted to the in-plane torque and the out-of-plane omp onent is predi ted to b e in the se ond order of bias voltage.

4. Role of magnons emissions

Magnons emission are also exp e ted to play an imp ortant role in spin-dep endent tun-nelling transp ort. As a matter of fa t, Zhang et al.

37

prop osed that impinging ele trons with energy higher than the Fermi level an emit spin waves by ipping their spin near the MTJ interfa e, leading to TMR drop as a fun tion of the applied bias voltage. Levy and Fert

42

re ently suggested that the partial dep olarization of spin- urrent by spin-waves emissionmaygiveriseto atorqueonthelo almagnetization,and onsequentlysigni antly ontributetospintransfertorque. Wegivehereasummaryofthepi tureprop osedinRef.

42

. The authors onsideredasystemsimilarto Slon zewski's

53

wherethe barrieris re tangu-larand submitted to lowbias voltage. In this ase, we saw that only in-planespin transfer

the in-plane torque p ossesses foursour es: T jj =(T elas +T int +T bulk tr ans +T bulk long )M(MP) (52)

where the four terms stand for the elasti torque (usual in-plane torque), the emission of interfa ial magnons and the emission of bulk magnons a ting on the transversal and longitudinal omp onentof the lo almagnetization.

a. Interfa ial magnons Magnons in general an only b e ex ited by ele trons whose energy is higher than the Fermi level and, their energy is ~!

l(r ) q

< eV. This leads to the formulationofthe torque due to interfa ial magnons ex itations,exertedon the l eft layer:

T int l /jt i j 2 sin V 2 f r N i l P r +N i r (P l os+F( ))g where N i l(r )

are the numb ers of spins p er unit area at the interfa e (in the left and right ele tro des,resp e tively),P

l(r )

aretheinterfa ialspinp olarizations, l(r )

are o e ientswhi h in ludematerialparametersandF( )isafun tionofthatwedonotdenehere(seeRef.

42

). This form is omplex and shows quadrati dep enden e as a fun tion of the bias voltage. Furthermore, the authors found that the torques indu ed by interfa ial magnon emission, appliedto leftandrightele tro de,areinopp ositedire tion(favorsparallelalignmentofthe left magnetizationand antiparallelalignmentof the rightmagnetization).

T int r = T int l (l !r ) (53)

To understand this ee t, Levy and Fert

42

give the following argument. The elasti spin urrent p olarization arises from the weighted ontribution of b oth left and right magneti ele tro des.

For the ele tro deat the higher ele tro hemi alp otential,left ele tro dehere, the authors found that the magnon emittedinthis ele tro de ausesthe p olarization to shift toward the p olarization of the right ele tro de, whi h ee tively is in the same dire tion than elasti torque.

However,for theele tro de with thelowerele tro hemi alp otential,rightele tro dehere, this reorientation of the p olarization redu es the ee t of the elasti term, reating an additivetorque in the opp osite dire tion.

b. Bulkmagnons Consideringtheele tronswhi hkepttheirspin losetotheinterfa e, one has to distinguish b etween two b ehaviors. Some of these ele trons are s attered with spin-ipinthebulk magneti leadwhereas othersare s attered without spin-ip. The spin-ip s atteredele trons ontributeto atransverse omp onentofthe spin urrent. Thisleads to the torques due to bulk magnon emission, exertedon the left and rightele tro des:

T bulktr ans l /V 3=2 jt b j 2 sin N b r [P l os+F 0 ( )℄ (54) T bulktr ans r /V 3=2 jt b m j 2 sin N b r (55)