Quantum Transport with Spin Dephasing: A Nonequilibrium Green's Function Approach

0.37 MB·English

by  Ahmet Ali Yanik

Quantum Transport with Spin Dephasing:
A Nonequlibrium Green’s Function Approach
Ahmet Ali Yanik∗
Department of Physics, and
Network for Computational Nanotechnology,
7
Purdue University, West Lafayette, IN, 47907, USA
0
0
2 Gerhard Klimeck
School of Electrical and Computer Engineering,
n
Network for Computational Nanotechnology,
a
Purdue University, West Lafayette, IN, 47907, USA and
J
Jet Propulsion Lab, Caltech, Pasadena, CA, 91109, USA
5
1
Supriyo Datta

] School of Electrical and Computer Engineering and
l Network for Computational Nanotechnology,
l
a Purdue University, West Lafayette, IN, 47907, USA
h
(Dated: February 6, 2008)
-
s
A quantum transport model incorporating spin scattering processes is presented using the non-
e
m equilibrium Green’s function (NEGF) formalism within the self-consistent Born approximation.
This model offers a unified approach by capturing the spin-flip scattering and the quantum effects
t. simultaneously. Anumericalimplementationofthemodelisillustratedformagnetictunneljunction
a deviceswithembeddedmagneticimpuritylayers. Theresultsarecomparedwithexperimentaldata,
m
revealingtheunderlyingphysicsofthecoherentandincoherenttransport regimes. Itisshown that
- small variations in magnetic impurity spin-states/concentrations could cause large deviations in
d junction magnetoresistances.
n
o
PACSnumbers: 72.10.-d,72.25.-b,72.25.Rb,71.70.Gm,73.43.Qt
c
[

4 I. INTRODUCTION Thisarticleisorganizedasfollows. Intheviewofped-
v agogicalclarity,aheurisicpresentationoftheNEGFfor-
7 malism with spin dephasing mechanisms is given Sec II
3 Quantum transport in spintronic devices is currently followed by a numerical implementation of the model in
0
a topic of great interest. Most of the theoretical work SecIII. Initially(SecIIIA)thedefinitionsofdevicechar-
5
reported so far has been based on the Landauer ap- acteristicsarepresentedforimpurity free MTJstogether
0
proach [1] assuming coherent transport, although a few withdeviceparametersbenchmarkedagainstexperimen-
6
0 authorshaveincludedincoherentprocessesthroughaver- talmeasurements. Howtoincorporatethespinexchange
/ aging over a large ensemble of disordered configurations scattering mechanisms into the electron transport calcu-
t
a [2, 3, 4]. However, it is not straightforward to include lations is shown (Sec IIIB), and the model is applied to
m dissipative interactions in such approaches. The non- MTJ devices with magnetic impurity layers (Sec IIIC).
- equlibriumGreen’sfunction(NEGF)formalismprovides Theoretical estimates and experimental measurements
d a natural framework for describing quantum transport are compared as well in this section (Sec IIIC), while
n
in the presence of incoherent and dissipative processes. a summary of the results is given in Sec IV.
o
Here, a numerical implementation of the NEGF formal-
c
: ism with spin-flip scattering is presented. For magnetic
v
tunneljunctions(MTJs)withembeddedmagneticimpu- II. MODEL DESCRIPTION
i
X ritylayers,thismodelisabletocaptureandexplainthree
distinctive experimental features reported in the litera-
r NEGF Method: The problem is partitioned into chan-
a ture [5, 6, 7, 8] regardingthe dependence of the junction
nel and contact regions as illustrated in Fig. 1 [9]. Com-
magnetoresistances(JMRs)on(1)barrierthickness,(2)
ponentsofthepartitioneddevicecanbeclassifiedinfour
barrierheightand(3)thenumberofmagneticimpurities.
categories:
The model is quite general and can be used to analyse
(i)Channel propertiesaredefinedbytheHamiltonian
anddesignavarietyofspintronicdevicesbeyondthe1-D
matrix [H] including the applied bias potential.
geometry explored in this article.
(ii)Contacts areincludedthroughself-energymatrices
[Σ ]/[Σ ] whose anti-hermitian component:
L R
∗Electronicaddress: [email protected] ΓL,R(E)=i ΣL,R(E)−Σ†L,R(E) , (1)
(cid:16) (cid:17)
2
Σin [Σout]. Broadening due to scattering is given by:
S S
(cid:2) (cid:3)(cid:14) Γ (E)= Σin(E)+Σout(E) , (3)
S S S
(cid:2) (cid:3)
from which the self-energy matrix is obtained through a
Hilbert transform:
Re Im
1 Γ (E′) Γ (E)
Σ (E)= S dE′−i S . (4)
S z2π E}′|−E { z }2| {
Z
FIG. 1: A schematic illustration of device partitioning in Eqs. (1-2) are the boundary conditions that drive the
NEGF formalism. Magnetization direction of the drain is coupled NEGF equations [Eqs. (3-8)], where Green’s
defined relative to thesource(∆θ=θR−θL). function is defined as:
G=[EI−H −U −Σ −Σ −Σ ]−1, (5)
L R S
describes the broadening due to the coupling to the con-
tact. The corresponding inscattering/outscattering ma- with the spectral function (analogous to density states):
trices are defined as:
A=i G−G† =Gn+Gp, (6)
Σin (E)=f (E−µ )Γ (E), (2a)
L,R 0 L,R L,R
where [Gn]/[Gp] refe(cid:2)r the ele(cid:3)ctron/hole correlation func-
tions (whosediagonalelementsaretheelectron/holeden-
Σout (E)=[1−f (E−µ )]Γ (E), (2b)
L,R 0 L,R L,R sity):
where f (E−µ ) = 1/1+exp[(E−µ )/k T] is
0 L,R L,R B Gn,p =G Σin,out+Σin,out+Σin,out G†. (7)
the Fermi function for the related contact. L R S
(iii) Electron-electron interactions are incorporated h i
throughthemeanfieldelectrostaticpotentialmatrix[U]. The in/out-scattering matrices Σin [Σout] are re-
S S
(iv) Incoherent scattering processes in the chan- latedtotheelectron/holecorrelationfunctions[Gn]/[Gp]
(cid:2) (cid:3)(cid:14)
nel region are described by in/out-scattering matrices through:
Σin,out (r,r′;E)= Dn,p (r,r′;~ω) Gn,p (r,r′;E∓~ω)d(~ω)
S;σiσj σiσj;σkσl sf σkσl
Z σXk,σlh i
+ Dn,p (r,r′;~ω) Gn,p (r,r′;E)d(~ω). (8)
σiσj;σkσl nsf σkσl
Z σXk,σlh i
Here the spin indices (σ ,σ ) refer to the (2x2) block di- croscopic spin-dephasing scattering mechanisms. These
k l
agonal elements of the on-site electron/hole correlation scatteringtensorscanbe obtainedfromthe spinscatter-
function which is related through the [Dn]/[Dp] tensors ing hamiltonian:
to the (σ ,σ ) spin components of the (2x2) block di-
i j
agonal of the in/out-scattering function. This term can
be interpreted as in the following. The first part de-
scribes the process of spin-flip transitions (subscript sf) HI(~r)= J ~r−R~j ~σ·S~j, (9)
due to the spin-exchange scatterings in the channel re- Rj
X (cid:16) (cid:17)
gion. The second part denotes the contributions of the
spin-conserving exchange scatterings (subscript nsf for where~r/R~ are the spatialcoordinates and~σ/S~ are the
j j
”no spin-flip”) in the channel region. Both of the con- spin operators for the channel electron/(j-th) magnetic-
tributing parts are previously shown by Appelbaum[10] impurity. For point like exchange scattering processes,
usingasimilartreatment. Herethe[Dn]/[Dp]arefourth- the scattering tensor for the first term in Eq. (8) cor-
order scattering tensors, describing the spatial corre- responding to the spin-flip transitions is given by (see
lation and the energy spectrum of the underlying mi- Appendix):
3
|σ σ i→ |↑↑i |↓↓i |↑↓i |↓↑i
k l
hσ σ |↓
i j
h↑↑| 0 F δ(ω∓ω ) 0 0
u,d q . (10)
h↓↓| F δ(ω±ω ) 0 0 0
[Dn,p(r,r′;~ω)] =δ(r−r′) J2N (ω ) d,u q
sf I q h↑↓|  0 0 0 0
ωq
P h↓↑|  0 0 0 0
 
Also, the no-spin flip component in Eq. (8) is given by:
|σ σi→ |↑↑i |↓↓i |↑↓i |↓↑i
k l
hσ σ |↓
i j
h↑↑| 1 0 0 0
. (11)
h↓↓| 0 1 0 0
[Dn,p(r,r′;~ω)] =δ(r−r′) 1J2N (ω )[δ(ω−ω )]
nsf 4 I q q h↑↓|  0 0 −1 0
ωq
P h↓↑|  0 0 0 −1
 
Current is calculated from the self-consistent solution the impurities have an internal degree of freedom, such
of the above equations for any terminal ”i”: as fluctuating internal spin states in the case of mag-
netic impurities. Electron scatterings from such impuri-
∞
ties randomize the electron spin states which can not be
q
Ii = h trace Σiin(E)A(E) −[Γi(E)Gn(E)] dE). simplyincorporatedthroughthedeviceHamiltonian. In-
Z stead scattering self energy matrices are needed for this
−∞ (cid:0)(cid:2) (cid:3) (cid:1)
(12) type of phase-relaxing scattering proccesses. An imple-
Thegeneralsolutionschemewithoutgoingintothede- mentation of this self-energy matrix treatment will be
tails can be summarized as follows . The matrices listed discussed in this part of the paper for electron-impurity
underthe categories(i) and(ii)arefixedatthe outsetof exchange scattering processes in MTJs. But first, we’ll
any calculations. While the [U], [Σin,out] and [Σ ] ma- discuss MTJ fundamentals and device characteristics in
S S
trices under the charging and scattering categories (iii) theabsenceofmagneticimpurities(thecoherent regime).
and (iv) depend on the correlation and spectral func-
tions requiring an iterative self-consisted solution of the
NEGF Equations [Eqs. (3-8)]. One important thing to A. MTJs: Coherent Regime
note is that for the numerical implementation presented
in Sec III we do not compute the charging potential [U] MTJs devices considered here consist of a tunnel-
self-consistently with the charge. The change in tunnel ing barrier (AlO ) sandwiched between two ferromag-
x
barriers is neglected and assumed not to influence the nets (Co) with different magnetic coercivities enabling
electrostatic potential. This allows one to focus on the independent manipulation of contact magnetization di-
dephasing due to the spin-flip interactions. rections (Fig. 2). Single band tight-binding approxi-
III. APPLICATION: MTJs WITH MAGNETIC
IMPURITIES
In the presence of ”rigid” scatterers such as impuri-
ties and defects, electron transport is considered coher-
ent as the phase relationships between different paths
are time independent. Accordingly, any static devia-
tions from perfect crystallinity leading to the scattering
of the electrons from one state to another can be incor-
porated into the transport problem through the device FIG.2: EnergybanddiagramforthemodelMTJsconsidered
Hamiltonian H. However the situtation is different when here.
4
mation is adopted [11] with an effective electron mass without any self-consistent solution requirements where
(m∗ = m ) in the tunneling region and the ferromag- H is Hamiltonian of the isolated system, and Σ /Σ
e L R
netic contacts. Accordingly for constant effective mass are the self-energies due to the source/drain contacts.
throughoutthedevice,transversemodescanbeincluded Here the matrices have twice the size of the channel re-
using 2-D integrated Fermi functions f (E −µ )= gion in the corresponding presentation due to the elec-
2D z L,R
m∗k T 2π–h2 ln[1+exp(µ −E /k T)] in Eq. (2) tron spin states. Accordingly, in real space representa-
B L,R z B
instead of numerically summing parallel k-components. tion for a discrete lattice whose points are located at
(cid:0) (cid:14) (cid:1)
The Green’s function of the device is simply defined as: x = ja, j being an integer (j = 1···N), the matrix
(E I−H −Σ −Σ ) can be expressed as:
z L R
G(E )=(E I −H −Σ −Σ )−1, (13)
z z L R
|1i |2i |N −1i |Ni
E I¯−α −Σ¯ β ··· ¯0 ¯0
h1| z 1 L
β+ E I¯−α ··· ¯0 ¯0
h2| z 2 , (14)
EzI−H −ΣL−ΣR =  ... ... ... ... ... 
hNhN−|1|  ¯0¯0 ¯0¯0 ······ EzI¯−β+αN−1 EzI¯−αβN −Σ¯R 
 
 
where α is a 2x2 on-site matrix: E = 2.2 eV and the exchange field ∆ = 1.45 eV [11].
n F
The tunneling regionpotentialbarrier[U ] is parame-
barr
α = Ec↑,n+2t+Un 0 , (15) terized within the band gaps quoted from the literature
n (cid:20) 0 Ec↓,n+2t+Un (cid:21) [12,13],whilethechargingpotential[U]isneglecteddue
to the pure tunneling nature of the transport.
and β = −tI¯ is a 2x2 site-coupling matrix with t = Coherent tunneling regime features are obtained by
~2 2ma2 and I¯ = ( 1 0). The left contact self-energy benchmarking the experimental measurements made in
0 1 impurity free tunneling oxide MTJs at small bias volt-
ma(cid:14)trix is nonzero only for the first 2x2 block: ages. Referring to I /I as the current values for the
F AF
parallel/antiparallelmagnetizations(∆θ =0/∆θ =π)of
the ferromagnetic contacts, the JMR is defined as:
Σ (1,1;E )=Σ¯ = −teikL↑a 0 , (16)
L z L " 0 −teikL↓a #
JMR=(I −I )/I . (19)
F AF F
whereE =E↑,↓+U +2t 1−cosk↑,↓a . Forthe right The dependence of the JMRs on the thickness and
z c L L
the height of the tunneling barriersis shown in Fig. 3(a)
contact only the last block(cid:16)is non-zero: (cid:17)
with an energy resolved analysis [Fig. 3(b)] for different
barrierthicknesses(0.7−1.4−2.1nm). JMRvaluesare
Σ (N,N,E )=Σ¯ =ℜ˜ −teikR↑a 0 ℜ˜†, (17) showntobeimprovingwithincreasingbarrierheightsfor
R z R " 0 −teikR↓a # allbarrierthicknesses,a theoreticallypredicted[7, 8, 14]
and experimentally observed [15, 16, 17, 18, 19, 20, 21,
where E =E↑,↓+U +2t 1−cosk↑,↓a with ℜ˜ being 22,23,24]featureinMTJs. Thebarrierheightsobtained
z c R R
here may differ from those reported in literature [15, 16,
the unitary rotation operato(cid:16)r defined as:(cid:17)
17, 18, 19, 20, 21, 22, 23, 24] based on empirical models
[25].
cos(∆θ/2) sin(∆θ/2)
ℜ˜(∆θ)= , (18) Experiments and theoretical calculations observe de-
−sin(∆θ/2) cos(∆θ/2)
(cid:20) (cid:21) terioration of JMRs with increasing barrier thicknesses
for two contacts with magnetization directions differing [Fig. 3(a)]. Whereas anenergy resolvedtheoreticalanal-
by an angle ∆θ. ysis shows that energy by energy junction magnetoresis-
A theoretical analysis ofMTJdevicesintheabsenceof tances defined as:
magneticimpuritylayersispresentedandcomparedwith
JMR(E )=(I (E )−I (E ))/I (E ), (20)
z F z AF z F z
theexperimentaldata[5,7]forvaryingtunnelingbarrier
heights and thicknesses. The parameters used here for remain unchanged [Fig. 3(b)]. This initially counter in-
the generic ferromagnetic contacts are the Fermi energy tuitiveobservationcanbeunderstoodbyconsideringthe
5
(a) (b) B. Adding Spin Exchange Scattering
JMR % −−−>123000 UUUbbbaaarrrrrr===521...346eeeVVV JMR Ratio −−−>00001.....24680 Ubarriettt===r=0121...741.6nnnemmmV 00001.....24680ωmalized (E) −−−>z ffroorSmtphinethmeeoxidcnheclaodnhegeveriecnsetcsacntotanetrusiirndegeroepfdrtohhceeersets.ueTsnnhaerrolieunggrhetstrphaoinsnsespilobarslet-
00 . 5 1 1.5 2 2.5 0 0 0.5 1 1.5 2 0 nor tic scattering process, the state of nothing else seem to
(cJMR % −−−>) 112205055 ThicknesNorm. JMR−−>s0 .(015n0timnhdiJc)e 2k−pNne−Ie n(s−eds5>Ve)n−t−> (dJMR Ratio −−−>)00001.....24680 EJJJz222 NNNEIII===n036eeeerVVVgy (eVt)= 0−.−7n−m> 00001.....24680ωormalized (E) −−−>z csiceltnihihhxbdcetaareoennirhur”ggenemideenard.efli.oanffrsTotmtWrhhapcaeeerhtscoiiaeooncftlmoencesrocmpsctheiorarensosakriginnseetunesctahretnetsenh”ytdihaismstoteteuhfppmrseurteto.rhaiicomteteeyfNspstsseouuhpvfrrieienirnttrocsyhstuocheaihnsnetpedilttmeriioennesrpsnlsgio,nutcagrtiaartihtelrlryieieoescqsuhtouaghaniinnhes--
0 0 0 n
0.4 0.8 1.2 0 0.5 1 1.5 2 the forces forcing the impurity spins into an unpolarized
Thickness (nm) −−−> E Energy (eV) −−−>
z
(%50 up, %50 down) spin distribution. These external
FIG. 3: For impurity free MTJs, (a)thickness dependence of forcesinacloselypackedimpuritylayercanbe magnetic
JMRs for different barrier heights are shown in comparison dipole-dipole interactions among the magnetic impuri-
with experimental measurements [15, 16, 17, 18, 19, 20, 21, ties or spin relaxation processes coupled with phononic
22,23,24]whilean (b)energyresolved analysis ofJMR(Ez) excitations. Nevertheless the physical origin of the equi-
(left-axis) andnormalized w(Ez) (right-axis) distributionsis libriumrestoringprocessesisnotofourinterest(atleast
also presented for a device with a tunneling barrier height of fromtunnelingelectron’spointofview)assumingequilib-
1.6eV.ForMTJswithimpuritylayers,(c)variationofJMRs
rium restoringprocesses are fast enough to maintain the
for varying barrier thicknesses and interactions strengths are
impurity spins in a thermal equilibrium state. Accord-
showntogetherwith(d)anenergyresolvedanalysis. Normal-
ingly, NEGF formalism incorporates spin dephasing ef-
ized JMRs are proven to be thickness independent as dis-
fectsoftheenvironmentintotheelectrontransportprob-
played in theinset.
lem through a boundary condition (the spin-scattering
self-energy). As discussed in Sec II, coupling between
redistribution of tunneling electron densities over ener- the number of available electrons/holes ([Gn]/[Gp]) at
gies with changing tunneling barrier thicknesses. Defin- a state and the in/out-flow ( ΣiSn [ΣoSut]) to/from that
ing w(E ) as a measure of the contributing weight of stateisrelatedthroughthefourthorderscatteringtensor
the JMRz(E ), one can show that experimentally mea- [Dn]/[Dp] in Eq. (8). (cid:2) (cid:3)(cid:14)
z
suredJMR is a weightedintegralofJMR(Ez)s overEz For the model systems considered here magnetic im-
energies: purityspinstatesaredegenerate(~ω =0)allowingonly
q
elastic spin-flip transitions. Spin-conserving scattering
processes are also neglected due to their minor effect on
JMR= ω(E )JMR(E )dE , (21) the JMRs. Accordingly, the spin scattering tensor rela-
z z z
Z tionship given in Eqs. (8,10 and 11) will simplify to:
where ω(E ) = I (E )/I is the energy resolved spin-
z F z F
continuum current component (weighting function). In
Eq. (21), independently from the barrier thicknesses
Σin,out(r,r;E)= Dn,p Gn,p (r,r;E). (22)
JMR(Ez) ratios are constant (solid line in Fig. 3(b)), S σiσj;σkσl sf σkσl
while the normalized w(E ) distributions shifts to- h i
z
wardshigher energieswith increasing barrierthicknesses
(dashed lines in Fig. 3(b)). Hence JMRs, an integral of [Dn]/[Dp] scattering tensors relate the electron/hole
themultiplicationofthew(E )distributionswiththeen- [Gn]/[Gp]correlationmatriceswiththe(2x2)blockdiag-
z
ergyresolvedJMR(E )s,decreaseswith increasingbar- onal elements of Σin [Σout] in/out-scattering matrices
z S S
rier thicknesses (Eq. (19)). of form:
(cid:2) (cid:3)(cid:14)
6
Σ¯in,out ¯0 ··· ¯0
S
1,1
 (cid:16) ¯0 (cid:17) Σ¯in,out ··· ¯0 
Σin,out = S 2,2 , (23)
S  ... (cid:16) ... (cid:17) ... ... 
 
 ¯0 ¯0 ··· Σ¯in,out 
 S 
 N,N 
 (cid:16) (cid:17) 
through the tensor relationship (shown below in matrix while preserving the normalized ω(E ) carrier distribu-
z
format) for the corresponding lattice site ”j” with mag- tions [Fig. 3(d)].
netic impurities: Further analysis shows that ”normalized” JMRs
deteriorates with increasing spin-dephasing strenghts
(J2N )independentlyfromthe tunnelingbarrierheights
Σin,out [Dn,p]sf Gn,p [Fig.4I(a)]. This generaltrendcanbe shownbymapping
S;↑↑ ↑↑
jj jj
(cid:16)Σin,out(cid:17)  0 Fu,d 0 0  (cid:16)Gn,p(cid:17)  the”normalized” JMRsintoasingle universal curve us-
S;↓↓ F 0 0 0 ↓↓ ingatunnelingbarrierheightdependentscalingconstant
jj =zJ2N d,u}| { jj ,
(cid:16)Σin,out(cid:17)  I 0 0 0 0 (cid:16)Gn,p(cid:17)  c(Ubarr) (inset in Fig. 4(a)).
 S;↑↓ jj  0 0 0 0  ↑↓ jj  This allows us to choose a particular value of barrier
    
(cid:16)Σin,out(cid:17)    (cid:16)Gn,p(cid:17)  height [U = 1.6eV] and adjust a single parameter J
 S;↓↑   ↓↑  barr
 jj   jj  to fit our NEGF calculations [Fig. 4(b-d)] with experi-
(cid:16) (cid:17)   (cid:16) ((cid:17)24)
mental measurements obtained from δ-doped MTJs [5].
where N is the number of magnetic impurities and
I
Submonolayerimpuritythicknessesgiveninthemeasure-
F /F represents fractions of spin-up/spin-down impu-
u d
mentsareconvertedintonumberofimpuritiesusingma-
rities for an uncorrelated ensemble (F +F =1).
u d
terial concentrations of Pd/Ni/Co impurities and device
This tensor relationship can be understood heuristi-
cross sections (6×10−4 cm2) [5].
cally from elementary arguments. The in/out-scattering
into spin-up component is proportionalto the density of Close fitting to the experimental data are observed at
the spin-down electrons/holes times the number of spin- 77K [Fig. 4(b-d)] using physically reasonable exchange
up impurities, N F : couplingconstantsofJ =2.63µeV/2.68µeV/1.41µeV for
I u
deviceswithPd/Ni/Coimpurities[26]. However,experi-
Σin,out =J2N F Gn,p . (25) mentallyobservedtemperaturedependenceofnormalized
S,↑↑ I u ↓↓
jj jj
(cid:16) (cid:17) (cid:16) (cid:17)
ptrooSnniesmn/tihloaislrelpysr,ottpihmoeretsiinot/nhoaeulttn-ouscmtahbteteedrreionnfgsistipynitnoo-fdtsohpweinns-pdiiomnw-punuprcietoliemecs--, (aR −−−>) 0.18 N. JMR −−>0.0150 c(Ubarr)J2NI5 (s eccVua) r lv−eed−> (bR −−−>) 0.18 Pd Impurity
N F : M M
I d ΣiSn,↓,o↓ut jj =J2NIFd Gn↑↑,p jj. (26) ormalized J000...246 UUUbbbaaarrrrrr===521...346eeeVVV ormalized J000...246 JJ7370K0K ==23..6032 µµeeVV
(cid:16) (cid:17) (cid:16) (cid:17) N 00 5 10 15 20 N 0 0 0.2 0.4 0.6 0.8
(c) J2NI (eV) −−−> (d) 10−12 NI −−−>
C. Incoherent Regime: Results −−> 1 Ni Impurity −−> 1 Co Impurity
− −
R 0.8 R 0.8
M M
inTthheeipnrceosheenrceentoftumnangenlientgicreigmimpuerditeivesiceischstaurdaicetderifsotricas alized J00..46 J77K =2.68 µeV alized J00..46 J77K =1.41 µeV
fixed barrier height U = 1.6 eV [Fig. 3 (c)] with orm0.2 J300K=3.00 µeV orm0.2 J300K=2.30 µeV
barr N 0 N 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
changing barrier thicknesses and electron-impurity spin 10−12 N −−−> 10−12 N −−−>
exchange interactions (J2N =0−6 eV). Nonlinear de- I I
I
creasing JMRs with increasing spin-exchange interac- FIG. 4: (a) Normalized JMRs deteriorates with increasing
tions are observed at all barrier thicknesses due to the spin-dephasing strengths independently from the tunneling
mixingofindependentspin-channels[5,6]whilethenor- barrier heights. This general trend can be scaled to a single
malized JMRs are proven to be thickness independent universal curve (inset). Experimentaldatatakenat77K and
(inset). This observation is attributed to the elastic na- 300K iscomparedwiththeoreticalanalysisinthepresenceof
ture of the spin exchange interactions yielding a total (b)Pd,(c)Niand(d)Comagneticimpuritieswithincreasing
drop in JMR(E ) values at all E energies in Eq. (19) impurity concentrations.
z z
7
JMRratioscannotbeaccountedforbyourmodelcalcu- IV. SUMMARY
lations. BroadeningsoftheelectrodeFermidistributions
due to changing temperatures from 77K to 300K seem
Summary: A NEGF based quantum transport model
to yield variations in normalized JMR ratios within a
incorporating spin-flip scattering processes within the
linewidth. As a result, different J exchange couplings
self-consistent Born approximation is presented. Spin-
are used in order to match the experimental data taken
flip scattering and quantum effects are simultaneously
at 300K.
captured. Spin scattering operators are derived for the
ForPd and Ni doped MTJs,relativelysmallvariations specific case of electron-impurity spin-exchange interac-
in J exchange couplings are needed (J /J =1.16 for tionsandtheformalismisappliedtospindependentelec-
300 77
Pd and J /J = 1.19 for Ni) in order to match the tron transport in MTJs with magnetic impurity layers.
300 77
experimentaldataat300K.Thesesmalltemperaturede- Thetheoryisbenchmarkedagainstexperimentaldatain-
pendencescouldbeduetothepresenceofsomesecondary volving both coherent and incoherent transportregimes.
mechanisms not included in our calculations. One such JMRsareshowntodecreasebothwithbarrierthickness
mechanismreportedinliteratureincludesthepresenceof and with spin-flip scattering but our unified treatment
impurity-assisted conductance contribution through the clearlybringsoutthedifferenceintheunderlyingphysics
defects (possibly created by the inclusion of magnetic im- [Fig. 3]. Our numerical results show that both barrier
purities within thebarrier) whichisknowntobestrongly height and the exchange interaction constant J can be
temperature dependent [27]. In fact, the contribution of subsumed into a single parameter that can explain a va-
the impurity-assistant conductance is proportional with riety of experiments [Fig. 4(a)]. Nonlinear dependence
the impurity concentrations in accordance with the ex- ofJMRratiosontheimpurityconcentrationsareshown
perimental measurements. [Fig. 4(b-d)] in difference with earlier linear estimates
from empirical models[5]. Accordingly, small differences
Another interesting feature observed in calculations in spin-states/concentrations of magnetic impurities are
[Fig. 4(b-c)] is the comparable J exchange couplings for shown to cause large deviations in JMRs. Interesting
thePdandNiimpuritiesattemperatures77Kand300K. similarities and differences among devices having Pd, Ni
Accordingly, a possible estimate of the impurity spin and Co impurities are pointed out, which could be sig-
states may be made by considering the most commonly natures of the spin states of oxidized Pd and Ni impuri-
encountered oxidation states of the Pd and Ni impuri- tiesandlow-spin/high-spinphasetransitionsforoxidized
ties. Closed-shell elemental Pd is only known to be in a cobalt impurities.
magnetic oxidized state of S=1 in octahedral oxygenco-
ordination according to the Hund’s rules [28]. Similarly,
we may attribute the comparable J couplings for Ni im-
purities due to the S=1 spin state of the Ni+2 which is
knowntobeafrequentlyobservedionizedstateinoxygen Acknowledgments
environment[29].
On the contrary, for Co doped MTJs, there’s a clear This work was supported by the MARCO focus cen-
distinction for normalized JMR ratios at different tem- ter for Materials, Structure and Devices and the NSF
peratures [Fig. 4(d)] which can not be justified by the Network for Computational Nanotechnology. Part of
presence of secondary mechanisms. Fitting these large this work was carried out at the Jet Propulsion Labo-
deviationsrequirelargevariationsinJ exchangecoupling ratory under a contract with the National Aeronautics
parameters [J /J = 1.73]. We propose this to be a and Space Administration NASA funded by ARDA.
300 77
resultofthermallydrivenlow-spin/high-spin phase tran-
sition [30, 31]. This is a credible argumentsince the oxi-
dationstateofthecobaltatomscanbeinCo+2(S =3/2,
high-spin)orCo+3 (S =0,low-spin)stateorpartiallyin
both of the states depending on the oxidation environ-
ment. Such thermally driven low-spin/high-spin phase
APPENDIX: SPIN DEPHASING SELF ENERGY
transitionsformetal-oxideshavebeenpredictedbytheo-
reticalcalculations and observedin experimentalstudies
[30,31]. Thesephasetransitionshavenotbeendiscussed In the following, the scattering tensors stated in
in MTJs community in connectionwith possible scatter- Eqs.(10-11)ofthemainpaperareobtainedstartingfrom
ing factors determining the temperature dependence of standardexpressionsobtainedintheself-consistentBorn
JMRs. Although from the avaliable experimental data approximationfromtheNEGFformalism. Herewestart
it’s not possible to make a decisive conclusion in this from the formulation in Ref. 9 (see Sections 10.4 and
direction, given the nonlinear dependence of JMRs on A.4)whichrepresentsageneralizationoftheearliertreat-
magnetic impurity states in our calculations, we believe ments [32, 33, 34, 35]. For spatially localized scatterers
that it’s important to point out this possibility here. we have:
8
H (t)=hσ |H (t)|σ i (A.2b)
I;σlσj l I j
Dn (t,t′)= τ (t)τ∗ (t′)
σiσj;σkσl * σisα;σksβ σjsα;σlsβ +
sXβ,sα
H† (t′)=hσ |H†(t′)|σ i (A.2c)
= hs |H (t)|s ihs |H† (t′)|s i (A.1a) I;σiσk i I k
* α I;σiσk β β I;σlσj α +
sXα,sβ
H† (t′)=hσ |H†(t′)|σ i (A.2d)
Dp (t,t′)= τ (t)τ∗ (t′) I;σlσj l I j
σiσj;σkσl * σlsβ;σjsα σksβ;σisα +
sXβ,sα
For an uncorrelated impurity spin ensemble with:
= hs |H (t)|s ihs |H† (t′)|s i (A.1b)
* β I;σlσj α α I;σiσk β +
sXα,sβ
ρ= w |s ihs |= w |s ihs | (A.3)
sα α α sβ β β
where|s iand|s iareimpurityspinsubspace statesand
α β Xsα Xsβ
H istheinteractionHamiltonian[Eq.(9)]definedwithin
I
the channel electron spin subspace as:
averaging in Eqs. (A.1a-A.1b) can be done through a
weightedsummation ofspin scattering rates of magnetic
H (t)=hσ |H (t)|σ i (A.2a) impurities:
I;σiσk i I k
Dn (t,t′) = w hs |H (t)|s i hs |H† (t′)|s i
σiσj;σkσl sβ α I;σiσk β β I;σlσj α
sXα,sβ
= hs |H (t)[ρ]H† (t′)|s i=tr ρH† (t′)H (t) (A.4a)
α I;σiσk I;σlσj α I;σlσj I;σiσk
Xsα (cid:16) (cid:17)
Dp (t,t′) = w hs |H (t)|s i hs |H† (t′)|s i
σiσj;σkσl sα β I;σlσj α α I;σiσk β
sXα,sβ
= hs |H (t)[ρ]H† (t′)|s i=tr ρH† (t′)H (t) (A.4b)
β I;σlσj I;σiσk β I;σiσk I;σlσj
Xsβ (cid:16) (cid:17)
The trace tr(ρA) for any operator A is independent of
representation. Accordingly, [Dn]/[Dp] scattering ten-
sorscanbeevaluatedusinganyconvenientbasisformag-
netic impurity spin states. a+(t)=σ+(t)= 0 eiωet , (A.6a)
Through Jordan-Wigner transformation, single spins 0 0
(cid:20) 0 0 (cid:21)
can be thought as an empty or singly occupied fermion a(t)=σ−(t)= . (A.6b)
e−iωet 0
state:
(cid:20) (cid:21)
|↑i≡a+|0i, (A.5a) For degenerate electron spin states (~ωe =0), there’s no
time dependence as such a†(t)→a† , a(t)→a.
Paulispinmatricesarerelatedtocreation/annihilation
|↓i≡|0i, (A.5b)
operatorsthroughσ =(a†+a)/2,σ =(a†+a)/2iand
x y
σ = a†a− 1/2. Accordingly, interaction Hamiltonian
withcreation/annihilation operatorsforthechannelelec- z
trons: H (~r,t)=Jδ ~r−R~ ~σ·S~(t) can be expressed as:
I
(cid:16) (cid:17)
9
1 1 1
H (~r,t)=Jδ ~r−R~ aS (t)+ a†S (t)+ a†a− S (t) , (A.7)
I 2 + 2 − 2 Z
(cid:16) (cid:17)(cid:20) (cid:18) (cid:19) (cid:21)
where S is the spin operator for the localized magnetic Substituting the interaction Hamiltonian from
impurity. Eq. (A.7) into Eqs. (A.4a-A.4b) will yield:
|σ σ i→ |↑↑i |↓↓i |↑↓i |↓↑i
k l
hσ σ |↓
i j
h↑↑| S (t′)S (t) S (t′)S (t) S (t′)S (t) S (t′)S (t)
Z Z + − + Z Z − , (A.8a)
h↓↓| S (t′)S (t) S (t′)S (t) −S (t′)S (t) −S (t′)S (t)
H† (t′)H (t) = − + Z Z Z + − Z
I;σlσj I;σiσk h↑↓|  S−(t′)SZ(t) −SZ(t′)S−(t) −SZ(t′)SZ(t) S−(t′)S−(t)
h↓↑| S (t′)S (t) −S (t′)S (t) S (t′)S (t) −S (t′)S (t)
 Z + + Z + + Z Z 
 
|σ σi→ |↑↑i |↓↓i |↑↓i |↓↑i
k l
hσ σ |↓
i j
h↑↑| S (t′)S (t) S (t′)S (t) S (t′)S (t) S (t′)S (t)
Z Z − + Z + − Z , (A.8b)
h↓↓| S (t′)S (t) S (t′)S (t) −S (t′)S (t) −S (t′)S (t)
H† (t′)H (t) = + − Z Z + Z Z −
I;σiσk I;σlσj h↑↓|  SZ(t′)S−(t) −S−(t′)SZ(t) −SZ(t′)SZ(t) S−(t′)S−(t) 
h↓↑| S (t′)S (t) −S (t′)S (t) S (t′)S (t) −S (t′)S (t)
 + Z Z + + + Z Z 
 
Thelocalizedmagneticimpurityspin-operatorscanbe with N being total number of impurities, the desired
I
written in its diagonalized impurity spin-subspace as: quantities [Dn]/[Dp] can be obtained by evaluating the
expectation values of the operators in Eqs. (A.8a) and
S =d† = 0 eiωqt (A.9a) (A.8b). Here the only non-zero elements are:
+ 0 0
(cid:20) (cid:21)
0 0
S =d= (A.9b)
− e−iωqt 0 1
(cid:20) (cid:21) tr(ρSZ(t′)SZ(t))= , (A.11a)
4
tr(ρS (t′)S (t))=F e−iωq(t−t′), (A.11b)
+ − u
1 1 1 0
S =d†d− = (A.9c)
Z 2 2 0 −1
(cid:20) (cid:21)
with ω = ∆E /~ where ∆E is the energy difference
q I I
between spin-up and spin-down states for the localized
magnetic impurities. tr(ρS (t′)S (t))=F eiωq(t−t′). (A.11c)
− + d
For a given impurity density matrix of the form
(Fu+Fd=1):
Finally, for a given impurity density matrix
F 0
ρ=N (ω ) u , (A.10) [Eq (A.10)], [Dn]/[Dp] tensors are obtained as:
I q 0 F
d
(cid:20) (cid:21)
|σ σ i→ |↑↑i |↓↓i |↑↓i |↓↑i
k l
hσ σ |↓
i j
h↑↑| 1/4 F e−iωq(t−t′) 0 0
u , (A.12a)
Dn(t,t′)= ωq J2NI(ωq) hh↓↑↓↓||  Fdeiω0q(t−t′) 10/4 −10/4 00 
P h↓↑|  0 0 0 −1/4 
 
 
10
|σ σ i→ |↑↑i |↓↓i |↑↓i |↓↑i
k l
hσ σ |↓
i j
h↑↑| 1/4 F eiωq(t−t′) 0 0
d , (A.12b)
Dp(t,t′)= ωq J2NI(ωq) hh↓↑↓↓||  Fue−iω0q(t−t′) 10/4 −10/4 00 
P h↓↑|  0 0 0 −1/4 
 
 
It’s convenient to work with the Fourier transformed Eqs. (10-11) [35]. For the calculations reported in this
functions as such (t−t′)→~ω: article, diagonal elements not leading to spin-dephasing
areomitteddue to theirnegligibleeffectonJMRratios.
e−iωq(t−t′)e−η|t−t′|/~ →δ(~ω−~ω ) (A.13) In this case [Dn]/[Dp] scattering tensors simplifies to a
q
form (Eq. (24)) which can be understood from simple
where η being a positive infinitesimal. With common-sense arguments (Eqs. (25) and (26)).
Fourier transformingEqs.(A.12a-A.12b) will simplify to
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