Voltage Asymmetry of Spin-Transfer Torques

Abstract

We present a Non-Equilibrium Green's Function based model for spin torque transfer (STT) devices which provides quantitative agreement with experimentally measured (1) differential resistances, (2) Magnetoresistance (MR), (3) In-plane torque and (4) out-of-plane torque over a range of bias voltages, using a single set of three adjustable parameters. We believe this is the first theoretical model that is able to cover this diverse range of experiments and a key aspect of our model is the inclusion of multiple transverse modes. We also provide a simple explanation for the asymmetric bias dependence of the in-plane torque, based on the polarization of the two contacts in energy range of transport.

Full text

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IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 2, MARCH 2012 261
Voltage Asymmetry of Spin-Transfer Torques
Deepanjan Datta, Behtash Behin-Aein, Supriyo Datta, and Sayeef Salahuddin
Abstract—Experimentally, it is seen that the free magnetic layer
ofaspintorquetransfer(STT)deviceexperiencesalargerin-plane
torque when a negative (rather than positive) voltage is applied to
the fixed layer. This is surprising because magnets do not have
any intrinsic asymmetry. In this paper, we 1) provide a simple
physical explanation, based on the polarization of fixed layer in the
energy range of transport; 2) extend it to explain the asymmetric
bias dependence of out-of-plane torque as observed in some of the
experiments; and 3) propose an asymmetric STT structure that
can lead to a significant difference in the in-plane torques exerted
ontwocontacts,eveniftheyareidentical.Thiseffect3hasnotbeen
observedtoourknowledgeandifdemonstratedcanfindimportant
applications.
Index Terms—Asymmetry, in-plane torque, nonequilibrium
Green’s function (NEGF), out-of-plane torque, spin polarization,
spin torque transfer (STT).
I. INTRODUCTION
S
polarized electrons have generated much interest due to their
abilitytowriteinformationwithoutanyexternalmagneticfields
[1]–[11]. The bias behavior of spin torque applied to magnetic
tunnel junctions (MTJs) is critical for applications including
high-densitymagneticrandomaccessmemorydevices.Thespin
torque can be decomposed into in-plane (spin-transfer) (τ?) and
out-of-plane (field-like) (τ⊥) components. Experimentally, it is
seen that the free layer experiences a larger τ?when negative
(ratherthanpositive)voltageisappliedtothefixedlayer[9]–[12]
(seealsoFig.1).Thisseemssurprisingbecausetheferromagnets
do not have any obvious asymmetry that could cause the effect
and there is no consensus regarding the underlying reason [5],
[15]–[18]. Moreover, while most of the experiments [8]–[12]
present symmetric τ⊥(V ), Petit et al. [13] and Oh et al. [14]
have observed asymmetries in this component as well.
Different reasons have been proposed to explain the bias
asymmetry of spin torque [5], [15]–[18]. Recent calculations
predicted the bias asymmetry of τ?in terms of parallel and an-
tiparallel spin currents [15]–[17]. Slonczewski also predicted
PIN torque transfer (STT) devices (see Fig. 1) that can
switch magnetization of a ferromagnetic layer using spin-
ManuscriptreceivedApril5,2011;acceptedJuly6,2011.Dateofpublication
July29,2011;dateofcurrentversionMarch9,2012.ThisworkattheUniversity
of California, Berkeley, was supported in part by National Science Foundation.
The review of this paper was arranged by Associate Editor R. Lake.
D. Datta, B. Behin-Aein, and S. Datta are with the Department of Electrical
and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA
(e-mail: ddatta@purdue.edu; behinb@purdue.edu; datta@purdue.edu).
S. Salahuddin is with the Department of Electrical Engineering and Com-
puter Sciences, University of California, Berkeley, CA 94720 USA (e-mail:
sayeef@eecs.berkeley.edu).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TNANO.2011.2163147
Fig. 1.
bias voltages with respect to free FM layer. Left (fixed) ferromagnetic layer is
alongˆ
M, while the right (free) layer is along ˆ m, separated by an insulating
layer. We see that in-plane torque exerted on the free FM layer at positive bias is
larger than the torque at negative bias, i.e., |? τ?,m(Vb> 0)| > |? τ?,m(Vb< 0)|.
Schematic of symmetric trilayer layer MTJ device under two opposite
the asymmetry of τ?in terms of voltage dependence of the spin
polarization PCof the fixed ferromagnetic layer [5]. Moreover,
some theoretical calculations showed asymmetric behavior in
τ⊥with voltage [17], [18]. In this paper, we explain the bias
asymmetry of both τ?and τ⊥solely based on the PC of the
“fixed ferromagnet/insulator” interface in the energy range of
interestwithoutinvokinganydetailedferromagneticbandstruc-
ture. Then, using a simple example of parabolic ferromagnetic
band structure, we show that PC(E), which is larger below the
Fermi level than above it, clearly shows the correct nature of
asymmetry of τ?(V ). This should be true for any complicated
PC(E) as long as it generally remains larger below the Fermi
level than above it. Opposite trend would have been observed
if a material has larger PC above Fermi energy than below it.
Furthermore, if a material has a constant PC(E), this should
result in no asymmetries in τ?(V ). Invoking the same functional
dependence of PC(E), we will also explain the experimental
bias dependence of τ⊥(V ). In the experiments in [8]–[12], same
ferromagnetic electrodes were used, which cause no change in
the τ⊥(V ) when the voltage polarity is changed as τ⊥is pro-
portional to product of the interface spin polarizations of the
ferromagnetic electrodes. However, in [13], [14], different fer-
romagnetic electrodes were used. The asymmetry in τ⊥(V ) can
be explained assuming that in the latter experiments, the free
layer’s spin polarization PCwas constant in the energy range of
interest. This causes τ⊥to be dependent only on the interface
spin polarization of the fixed layer (which is also the case for
τ?(V )), causing an asymmetry in τ⊥(V ) similar to τ?(V ).
Our analysis is based on nonequilibrium (or Keldysh)
Green’s function formalism (NEGF) for quantum transport (see
Appendix I for details of the model) [19], [20]. In a recent pa-
per, we have shown the quantitative agreement of this model
with the diverse experimental aspects of STT devices namely
1536-125X/$26.00 © 2011 IEEE

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262 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 2, MARCH 2012
differential resistances, tunnel magnetoresistance (TMR), τ?
and τ⊥ for several experiments (also see Figs. A2–A4 in
Appendix I) [21]. It is important to note that while the quan-
titative agreement with the experiments depends on the details
of the model and the parameters used, as we will show, it is
the qualitative nature of PC(E) that determines the asymmetry
of torques irrespective of the quantitative details of the ferro-
magnetic band structure. Based on our understanding of torque
asymmetry, we also present a vision for designing STT devices,
which can be useful for nonreciprocal switching and also pro-
vides a new route of designing spin-based circuits.
II. SPIN CURRENT?ISAND SPIN TORQUE ? τ
IN A TRILAYER DEVICE
Fig. 1 shows the schematic of a trilayer STT device that con-
sists of a fixed magnet with magnetization alongˆ
magnet with magnetization along ˆ m, separated by an insulator
layer. The spin torque, in an STT device, can be explained in
termsofspincurrent?IS(seeAppendixIfordetails),whichisthe
rate of change of the spin angular momentum of the conduction
electrons.?ISin a trilayer device can be expressed as
M and a free
?IS(E) = IS,mˆ m + IS,Mˆ
M + IS,⊥(ˆ
M × ˆ m)
(1)
as long as ˆ m andˆ
be zero if they were). The spin-transfer (? τ?,m) and field-like
(? τ⊥,m) components of the torque exerted on the free ferromag-
netic layer are, respectively, given by [21], [22]
M are not collinear (the torque would anyway
? τ?,m(Vb) =
?
?
?
?
dE(?IS,?− (?IS,?· ˆ m)ˆ m)
=
dE IS,M(E)(ˆ m ׈
M × ˆ m)
(2a)
? τ⊥,m(Vb) =
dE(?IS,⊥− (?IS,⊥· ˆ m)ˆ m)
=
dE IS,⊥(E)(ˆ
M × ˆ m).
(2b)
Note that?IS,?is the sum of the first two terms on the right-hand
side of (1), while IS,⊥represent the third term. It is evident
that ˆ τ?,mand ? τ⊥,m at every energy are proportional to IS,M
and IS ,⊥at that energy, respectively. The overall torque can
be found by integrating these values over the energy range of
transport which is a few kBT above and below the Fermi levels
of the contacts. To understand the torque asymmetry, we are
interested in the energy dependence of IS,M and IS ,⊥in this
energyrange.AfterpresentingthedependenceofIS,MandIS ,⊥
on the interface spin polarizations of the fixed (PCM) and free
(PCm)magneticlayers,wearguethatthefunctionaldependence
ofPCMandPCmonenergyistheresponsiblemechanismforthe
observed nonlinearity in torques. Furthermore, by invoking the
right general functional from of PC(E), we are able to explain
the correct symmetries and asymmetries observed in ˆ τ?,mand
? τ⊥,m.
III. ENERGY DEPENDENCE OF PCAND NONLINEAR TORQUES
Spin polarization PC of a “ferromagnet/insulator” interface
is defined as (G↑− G↓)/(G↑+ G↓) [22], where G↑,↓= (q2/h)
?T↑,↓(k?) are the tunneling conductances of up (↑) and down
is the transverse component of the wave vector. Note that PCis
calculated with our quantum transport model [19]–[21], but us-
ing a ferromagnet/insulator/nonmagnetic metal (NM) junction
so that both up and down spin electrons can propagate through
the NM region with equal probability. It can be shown that
the following relations are approximately true for the structures
considered in this paper (see Appendix II for details):
(↓) spin electrons, T↑,↓are the transmission functions, and k?
IS,M = PCM¯T0(fL− fR)
IS,⊥= PCMPCm¯Teff(fL+ fR)
where PCM and PCm are the spin polarization of the “fixed-
layer (ˆ
M)/insulator” and “free-layer (ˆ m)/insulator” interfaces,
respectively, along the spin quantization axes of the magnets.
Note that¯To represents the transmission of the channel with
unpolarizedcontacts.¯Teffisaneffectivetransmissionassociated
with the out-of-plane spin current (IS ,⊥) (see Appendix II).
Note that unlike¯To,¯Teffexists only when the contacts are spin
polarized.
In general, for the tunneling devices,¯Tois reasonably uni-
form over the energy range where fL–fR is nonzero, causing
the charge current to be antisymmetric with respect to voltage
making the resistance R(V ) symmetric in nature (see Fig. A2 in
Appendix I). Now, if PCM is constant around E = Ef, then
the magnitude of the in-plane torque [see (2a)] is rewritten
as τ?(V ) =?dE ISM(E) = PCM
the experimentally observed asymmetric τ?(V ) also shown in
Fig. 2(a). As a result, asymmetric τ?(V ) is a clear manifestation
of energy dependence of PCMaround Fermi energy.
Such energy dependence is also evident in the behavior of
τ⊥(V ). Equation (3b) states that IS ,⊥and, hence, τ⊥,mdepend
on the product of the spin polarizations (PCMand PCm) of both
ferromagnetic interfaces and should be symmetric with voltage
for symmetric devices which is also observed experimentally.
However, if we assume PCM and PCm to be constant with
energy, the magnitude of the out-of-plane torque [see (2b)] is
rewritten as τ⊥(V ) = PCMPCm
proportional to |V | and contradicts the experimentally observed
τ⊥(V ) ∝ V2as seen in [8]–[12]. This also suggests that the bias
dependence of τ⊥(V ) cannot be explained without invoking the
energy dependence of Pc.
(3a)
(3b)
?dE¯T0(fL− fR), which
should be perfectly antisymmetric in V , and this contradicts
?dE¯Teff(fL+ fR) which is
A. τ?,m(Vb> 0) ?= τ?,m(Vb< 0)
Consider case (A) in Fig. 2 when a positive bias Vb> 0 is
applied to the free layer ˆ m with respect to fixed magnetic layer
ˆ
M.Thefree-layersub-bandsareshifteddownbyanamountqVb
with respect to that of the fixed layer. Since tunneling occurs
in the energy range a few kBT above and below the Fermi
levels μL and μR (Vb > 0), we are interested in PCM(E) in
this energy range. But when the voltage polarity is reversed,

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DATTA et al.: VOLTAGE ASYMMETRY OF SPIN-TRANSFER TORQUES263
Fig. 2.
device. We assume the fixed magnet is along ˆ z-direction and the free magnet
is along ˆ x-direction. The parameters used for NEGF simulation: Lox= 1nm,
Ef= 2.25 eV, Δ = 2.15 eV, Ub= 1 eV, m∗
0.2. (b) Energy versus PC is drawn for the ferromagnetic contacts showing
relative contributions of DOS of majority and minority carriers available for
tunneling, when a voltage is applied to the fixed magnetˆ
of the fixed magnet is compared for voltages with same magnitude but of
opposite polarities. Since PCM (E > Ef) < PCM (E < Ef) around Ef,
torque exerted on the free layer |? τ?,m(Vb> 0)| > |? τ?,m(Vb< 0)|.
(a) Asymmetric behavior of τ?versus voltage Vbfor a symmetric STT
FM/mo= 0.2, and m∗
ox/mo=
M. Polarization PCM
the free-layer sub-bands are shifted up by same amount qVb
(see case (B) in Fig. 2) and we should consider PCM(E) in
this energy range. Now, we will compare PCM(E) for the two
polarities of Vb. PC(E) is drawn for the fixed layer showing
relative contributions of density of states (DOS) of the majority
and minority carrier modes available for tunneling (see Fig. 2).
Note that the bias asymmetry of? τ?,m(V ) [see (2a)] arises from
the energy dependence of PCM, specifically because
PCM(E < Ef) > PCM(E > Ef).
(4)
This also suggests if a material has constant PC(E), it will not
showtorqueasymmetry(asshowninsomeAbinitiocalculations
[23]) or, if PC(E) is higher above the Fermi energy than below,
it will show the opposite trend with voltage. Indeed, one could
argue that the fact that experimentally measured torques are
asymmetric shows that PCM(E) must be decreasing function of
E around E = Ef. Therefore, at each energy E around Ef,
IS,M (E > Ef) < IS,M (E < Ef) and when integrated over
energy, we get
??? τ?,m(Vb> 0)??>??? τ?,m(Vb< 0)??
(5)
Fig. 3.
different values of asymmetry energy parameter δ = EC,free− EC,fixed= 0
(solid), and 1 (dashed) eV. We assume the fixed magnet is along ˆ z-direction and
the free magnet is along ˆ x-direction. Slight asymmetry between the bottom of
the conduction bands of FM electrodes drastically changes τ⊥(V ) from pure
quadraticdependenceforcompletesymmetricdevices(δ = 0)toapproximately
linear dependence for asymmetric devices around zero bias (see inset). This
figure reconciles the apparently contradictory bias behavior of τ⊥, [8]–[13].
(b)EnergyversusPCisdrawnfortheferromagneticcontactsoftheAsymmetric
MTJ device. Since PCm is constant in the energy range of transport, we get
τ⊥∼ PCM.
as observed in all experiments [9]–[12]. This also explains the
observation that critical voltage for spin torque induced switch-
ing from AP to P configuration (positive bias in our convention)
is often lower compared to P to AP switching.
One may argue that (4) is valid only for simpler free-electron
models where the FM DOS is a monotonic function of energy
(as the three dimensional DOS is proportional to√E). On the
other hand, in the models involving complex band structures,
DOS may not be a monotonic function of energy. However, our
argument of the asymmetry is solely based on the decreasing
nature of PCM(E) around E = Ef, not on the DOS of fixed
layer. For example, if a material has a complex band structure
instead of the parabolic one (used to calculate torques), it will
stillshow|τ?,m(Vb>0)|>|τ?,m(Vb<0)|providedPCM(E >
Ef) < PCM(E < Ef). This is a clear testable prediction that
can be proved or disproved by experiments.
(a) Oscillatory behavior of τ⊥as a function of bias voltage for two
B. τ⊥,m(Vb> 0) ?= τ⊥,m(Vb< 0)
As discussed in Section I, most of the experiments [8]–[12]
demonstrated symmetric field-like component, i.e., τ⊥(V ) ∝

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264IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 2, MARCH 2012
Fig. 4.
can comprise of metal or, semiconductor having very low spin polarization at the FM/NM interface. Though for the symmetric STT devices, |? τ?,m(Vb> 0)| =
|? τ?,M(Vb< 0)|, for the proposed device |? τ?,m(V )| ? |? τ?,M(V )| irrespective of voltage polarity. We assume the left magnet is along ˆ z-direction and the right
magnet is along ˆ x-direction. The parameters used for NEGF simulation is same as in Fig. 2(a) with LNM= 5nm and EC NM= Ef− 0.5 eV. (b) Energy versus
PC is drawn for the ferromagnetic contacts of the proposed device. Since PCM(E < Ef) > PCM(E < Ef) ? PCm(E) around Ef, spin torque exerted on
the ferromagnetic contacts |? τ?,m(Vb> 0)| > |? τ?,m(Vb< 0)| ? |? τ?,M(V )|.
(a) Proposed device consists of a symmetric STT device with a nonmagnetic layer inserted between the oxide and right FM layer. The nonmagnetic layer
V2. However, Petit et al. [13] and Oh et al. [14] observed
asymmetries in τ⊥(V ) around zero bias. Recent calculations
predicted an oscillatory bias dependence of τ⊥which can be
tunedbyconductionbandasymmetryofFMcontacts[17],[18].
Our model shows similar oscillatory behavior of τ⊥(V ) [see
Fig. 3(a)]. Moreover, similar to τ?(V ), we explain the origin of
asymmetries in τ⊥(V ) in terms of energy dependence of inter-
face spin polarization of fixed layer, i.e., PCM. As derived in
Appendix II (and stated earlier), the magnitude of IS ,⊥and,
hence, τ⊥depends on the product of the spin polarizations of
both ferromagnetic interfaces, i.e.,
τ⊥∝ PCMPCm.
(6)
This remains unaffected when the voltage polarity is reversed
for a device with symmetric magnetic electrodes as shown in
Fig. 3(a) (solid curve). However, for asymmetric devices [17],
[18] with free-layer, PCmbeing constant around E = Ef, it is
only PCM(E) that determines the bias behavior of τ⊥(V ) and
likewiseτ?(V ),anasymmetryisobservedinthebiasdependence
of τ⊥(V ) [see the dashed curve in Fig. 3(a)]. This reconciles
the apparently contradictory experimental results observed in
[8]–[14].
IV. PROPOSED ASYMMETRIC DEVICE
AND NONRECIPROCAL TORQUE
So far, we have explained voltage asymmetry of τ?and τ⊥in
terms of the polarization of the interface spin polarization PCM
of the fixed layer in the energy range of interest. It is interest-
ing to note that there is an interesting symmetry |τ?,m(Vb>
0)| = |τ?,M(Vb< 0)| as evident from Fig. 2(a). However, this is
just a consequence of the structural symmetry. To illustrate this
point, consider a nonsymmetric structure, for which we show
that |τ?,m| ? |τ?,M| (see Fig. 4). Of course, this is just a model
calculation intended to illustrate an interesting point that may
perhaps find applications.The proposed device has thepotential
of switching in nonreciprocal manner, directed from the left to
the right magnet, and not the other way around (no feedback).
The most surprising and unusual feature of Fig. 4 is “nonre-
ciprocity” of torque, i.e., |? τ?,m(Vb> 0)| > |? τ?,M(Vb< 0)| and
vice versa. The concept underlying the nonreciprocity of the
proposed device is illustrated [see Fig. 4(b)] as follows: when a
negativevoltageVb<0isappliedtoleftmagnet,thetorque? τ?,m
exertedontheothercontactwhichiflargeenough(duetohigher
spin polarization at the ferromagnet/insulator interface) will flip
the magnetization of right magnet. But if the same voltage Vb<
0 is applied to the right magnet, due to poor spin injection at
the FM/NM interface (i.e., ohmic contact) torque exerted on

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DATTA et al.: VOLTAGE ASYMMETRY OF SPIN-TRANSFER TORQUES265
the left magnet ? τ?,Mis extremely low and is unable to flip the
magnetization of the left magnet. This concept of nonreciprocal
switching, however, is yet to be demonstrated experimentally.
Our objective is to motivate experimental investigation of the
proposed concept in a suitably designed spin torque structure.
V. CONCLUSION
In conclusion, we have shown that the bias dependence of
both τ?and τ⊥(and their symmetry/asymmetry) can simply
be understood from the energy dependence of the tunneling
spin polarization of the “ferromagnet/insulator” interface in the
energy range of interest without invoking any detailed ferro-
magnetic band structure. Based on the polarization difference
of the ferromagnetic contacts, we also predict that an appropri-
ate designed asymmetric STT structure can lead to a significant
difference in the magnitude of torques exerted on two contacts,
even if they are identical. However, this effect has not been
observed and if demonstrated can find new route of designing
spin-based circuits [24].
APPENDIX I
DETAILS OF THE NEGF MODEL
Fig.A1(a)showstheschematicofanMTJdevicethatconsists
ofafixedmagnetwithmagnetizationalongˆ
with magnetization along ˆ m, separated by an insulator layer, θ
beingtheanglebetweenthem(θ=cos−1(ˆ m.ˆ
along the direction perpendicular to the both fixed and free thin-
film planes. Our NEGF model for spin transport is based on
single-band effective mass Hamiltonian [H] described by the
parameters: 1) equilibrium Fermi level Ef; 2) spin splitting Δ;
3) barrier height of the insulator Ub; 4) effective masses for
electrons inside FM contacts (m∗
5) effective mass for electrons inside insulator m∗
NEGF-based transport model for MTJ devices with a single-
band effective mass Hamiltonian [H] and self-energy ΣL,R,
which are used to calculate Green’s function G(E), electron
correlation function Gn, charge and spin current densities J
and JS, respectively.
M andafreemagnet
M)).Currentflows
FM,↑= m∗
FM,↓= m∗
FM); and
ox.We use an
A. Hamiltonian [H] and Self-Energy Matrix ΣL,R
Here, we will present the device Hamiltonian H and self-
energy matrices ΣL,Rfor each transverse mode with wave vec-
tor k?through the device. We assume that the fixed magnetˆ
and the soft magnet ˆ m are both in the ˆ x − ˆ z plane [see Fig.
A1(a)]. Hamiltonian [H] for each transverse mode is given by
(note: ? σ represent the vector of the Pauli spin matrices) [21]
Left Contact:
M
HL(i,j,k?)
⎧
⎪
=
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎨
?
−tFMI,
0,
αFM(k?) +qV
2
?
I +
?
I −? σ ·ˆ
2
M
?
Δ,i = j
if i and j are nearest neighbors
otherwise.
(A1.1)
Fig. A1.
H and self-energy matrices ΣL,Rwhose anti-Hermitian components ΓL ,R=
i(ΣL ,R−Σ†
tFMand toxrepresent the coupling parameter between each lattice site given
by tFM= ¯ h2/2m∗
in discrete representation. αFM= 2tFM+ ¯ h2k2
k2
represent transverse energy of a mode with wave vector k?inside FM and
insulator regions, respectively. m∗
inside FM and insulator region, respectively, JSpin Leftand JSpin Rightare
current densities going into and coming out of the soft ferromagnetic region.
(a) The device region is modeled using appropriate Hamiltonian
L,R)arebroadeningmatricesduetocontactsL andR,respectively.
FMa2, tox= ¯ h2/2m∗
oxa2, a being uniform lattice spacing
?/2m∗
?/2m∗
FM, αox= 2tox+ ¯ h2
FMand ¯ h2k2
?/2m∗
ox, and αint= αFM+ αox, where, ¯ h2k2
?/2m∗
ox
FMand m∗
oxare effective masses of electron
Left Interface:
Hinterface(i,i,k?)
?
Channel:
=
αint(k?) +Ub
2
+qV
2
?
I +
?
I −? σ ·ˆ
4
M
?
Δ. (A1.2)
Hchannel(i,j,k?)
⎧
⎪
Right Interface:
=
⎪
⎪
⎪
⎪
⎪
⎩
⎨
?
−toxI,
0,
αox(k?) + qV
?1
2−
i
N + 1
??
I,i = j
if i and j are nearest neighbors
otherwise.
(A1.3)
Hinterface(i,i,k?)
?
Right Contact:
=
αint(k?) +Ub
2
−qV
2
?
I +
?
I −? σ ·ˆ
4
M
?
Δ. (A1.4)
HR(i,j,k?)
⎧
⎪
=
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎨
?
−tFMI,
0,
αFM(k?) −qV
2
?
I +
?
I −? σ ·ˆ
2
M
?
Δ,i = j
if i and j are nearest neighbors
otherwise
(A1.5)

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266IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 2, MARCH 2012
where αFM(k?) = 2tFM+ ¯ h2k2
¯ h2k2
¯ h2/2m∗
masses of electron inside FM and insulator region, respectively,
aistheuniformlatticespacing,andI isthe2×2identitymatrix.
N is the total number of atomic sites (excluding the interface
points) inside the insulator. V is the applied voltage difference
between two FM contacts which is assumed to drop linearly
inside the insulator. Ubis the barrier height of the insulator.
To write the self-energy matrices ΣL,R we note that their
nonzero elements are the left (L) or, right (R) lattice points
respectively and these are given by a 2 × 2 matrix of the form
?
?/2m∗
FM, αox(k?) = 2tox+
?/2m∗
ox, αint(k?) = 0.5 × (αFM(k?) + αox(k?)), tFM=
FMa2, tox= ¯ h2/2m∗
oxa2; m∗
FMand m∗
oxare effective
j(i,i,k?) =
?−tFMexp(ik↑
ja)0
0
−tFMexp(ik↓
ja)
?
,
j = L,R
(A2)
where k↑,↓
j
are, respectively, given by
E = E↑,↓
C±qV
2
+
¯ h2k2
2m∗
?
FM
+ 2tFM(1 − cos(k↑,↓
ja)).
(A3)
(Use + qV/2 for the contact L, and − qV/2 for the contact
R) and E is the energy. The form given previously for ΣL,R
is appropriate, if the easy axis of the magnet is used as the
quantization axis for the spin. If the easy axis does not lie along
ˆ z, then a unitary transformation is needed to transform this
quantization axis to ˆ z [25].
Note that we view Ub, m∗
account for a wide variety of factors including imperfection at
ferromagnet/insulator interfaces. On the other hand, Efand Δ
are the material parameters. Efis chosen as 2.25V [11], which
is believed to be typical for CoFeB alloys used in experiments.
ThespinsplittingΔof3dstatesoftransitionmetalsiscorrelated
linearly with the spin angular moment mS [26]. Based on the
experimental studies, mS/n3d= 0.7 (for Co60Fe20B20alloys)
[27], where n3dthe number of d holes (∼3.4 for bulk Fe, but
not known for CoFeB alloys). Therefore, Δ is calculated to be
2.38 eV. We use a slightly lower value of Δ = 2.15 eV, which
is justified due to presence of surface imperfection and other
defectsattheFM/insulatorinterface.m∗
not reported in the literature and is chosen to obtain quantitative
agreement for R(V ) at different relative angular alignments of
ferromagnetic contacts. In practice, the values of m∗
be very different and might depend on the growth of the crystal
insidethelattice.Amongtheotherparameters,wechoosebarrier
height Ub∼ 0.7 − 1 eV and effective mass of electrons inside
insulator m∗
mass.TherangeofthevaluesofUbismotivatedbythepresence
of oxygen vacancy defects that is expected to reduce the barrier
height of ideal MgO tunnel barrier (3.7 eV) [28].
Note that that in our model, we assume periodic boundary
conditions in the transverse direction so that all allowed trans-
versewavevectorsk?(ortransversemodes)[seeFig.A1(b)]are
decoupled like individual 1-D wires. The details of the Green’s
function G(E) and spin-dependent correlation function Gn(E)
that are used to calculate charge and spin current densities be-
FM, and m∗
oxas parameters that
FMoftheCoFeBalloyis
FMcould
ox∼ 0.2 − 0.3 mo, where mois the free electron
Fig. A1.
described by split bands of up and down spins (Δ), while the oxide only
introduces a barrier Ub. Different bands shown with different colors (e.g., blue,
red, green) correspond to various transverse wave vectors k?, which we assume
tobeconservedforduringtransport.Ef,Δ,andUbaredefinedw.r.t.thebottom
of the conduction band of the ferromagnet.
(b) Energy-band diagram of the trilayer device. The magnets are
TABLE I
PARAMETERS Ef, Δ, m∗
AVAILABLE EXPERIMENTAL R(V ), TMR, AND TORQUE
FM, Ub, AND m∗
oxUSED TO MODEL TWO SETS OF
tween different lattice sites are summarized in Appendix I-B.
We sum over the results obtained from all transverse modes to
getthefinalresultrelevanttorealisticexperimentaldeviceshav-
ing sizable cross sections. Note that in our model, we assume
that the wave vector k?for each transverse mode is conserved
throughout the device. The transverse energies ΔEFM,t and
ΔEox,t in the FM region and the oxide (see Fig. A1(b)] are
different due to difference in the effective masses. In any practi-
cal tunnel-junction structure, the barrier is unlikely to be defect
free over the junction area because of surface imperfections and
lattice mismatch between Fe and MgO [29]. Such effects could
lead to nonconservation of k?, but are outside the scope of this
paper.
B. Green’s Function, Charge, and Spin Current Density
In previous section, we showed the device Hamiltonian [H]
and self-energy matrices ΣL,R for the device. Once [H] and
ΣL,R are known, all quantities of interest including Green’s
function G(E) can be calculated from the following set of equa-
tions [19], [20]:
Green’s Function: G(E) = [EI − H − ΣL− ΣR]−1
Spectral Density Function: A(E) = i?G − G†?
Electron density Function: Gn(E) = G
??in
L+
?in
R
?
G†

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DATTA et al.: VOLTAGE ASYMMETRY OF SPIN-TRANSFER TORQUES267
In-Scattering Function:
?in
L,R(E) = ΓL,R(E)fL,R(E)
??
Broadening Matrix: ΓL,R(E) = i
L,R−
?†
L,R
?
(A4)
where Gn(≡−iG<) refers to spin-dependent correlation func-
tionwhosediagonalelementsareelectrondensity.Aisthespec-
tral function, whose diagonal elements are local DOS, Σin
the in-scattering function representing the rate at which elec-
trons come in to the device from contacts and fL,R are the
Fermi functions in contacts L and R, respectively. Once we
know Gn, the charge and spin current density between lattice
sites are calculated as follows.
Current Operator Between Two Lattice Points j and j ∓ 1:
Iop,j,j±1=i
¯ h
Charge Current Density:
?
Spin Current Density:
?
The spin torque exerted on the free magnetic layer will be the
difference between spin current going into and coming out of
the soft magnet in the absence of scattering. We define the spin
torque as the divergence of the spin current as [30]
?
=
dS(?JSpin Left−?JSpin Right) =
L,Ris
?Hj,j±1Gn
j±1,j− Gn†
j,j±1H†
j±1,j
?
(A5)
Jj,j±1=
dE Real[Trace(Iop)]
(A6.1)
JS,j,j±1=
dE Real[Trace(? σ Iop)].
(A6.2)
? τ = −
dV [?∇ ·?JS] =
?
dS
?
−
?
dl[?∇ ·?JS]
?
?
?
dS?JSpin FM (A7)
where S and V are the area of cross section and the volume
of magnet, respectively. μB is the Bohr Magneton (i.e., mag-
netic moment per unit volume) and JSis the spin current den-
sity between two lattice points. In our calculation, however,
?dS?JSpin FMis replaced by?dS?JSpin Leftbased on follow-
1) Spin-polarized electrons become completely polarized
along the free-layer magnetization after traversing a few
monolayer inside the magnet.
2) Spin torque acting on the free layer has components only
transverse to the free-layer magnetization ˆ m. Therefore,
the torque exerted on the soft ferromagnet is rewritten as
?
= ˆ m ×
?
where?IS=?dS?JSpin Left. The in-plane (? τ?,m) and out-
ing two assumptions.
? τm=
dS?JSpin Left−
?
?
ˆ m ·
?
dS?JSpin Left
?
ˆ m
dS?JSpin Left
???
?IS
׈ m
(A8)
of-plane (? τ⊥,m) components of spin torque are calculated
as
? τ?,m=
?IS· (ˆ m ׈
(1 − (ˆ m ·ˆ
?IS· (ˆ
(1 − (ˆ m ·ˆ
M × ˆ m)
M)2)
(A9.1)
? τ⊥,m=
M × ˆ m)
M)2).
(A9.2)
C. Results
Our model successfully describes reported differential resis-
tances, TMR, and its roll-off with voltage (see Fig. A2). Note
that we obtain close agreement of R(V ) and TMR with the
measured data from two different experimental groups with a
small adjustment of Ub and m∗
in the left panel of Fig. A2) are the results of direct experi-
mental measurements. However, the reported values of τ?and
τ⊥(or, ∂τ?/∂V and ∂τ⊥/∂V ) are not measured directly (see
Figs. A3 and A4), rather derived from the measured RF volt-
age output (in the spin-transfer-driven ferromagnetic resonance
experiments) using a specific theoretical model with its own
built-in assumptions. Fig. A3 shows comparison between ex-
perimental [9] and calculated torkances, ∂τ?/∂V and ∂τ⊥/∂V ,
respectively, at θ = 58◦and 131◦. With regards to the absolute
magnitudes of ∂τ?/∂V , our simulation shows that near V =
0, (∂τ?/∂V )/sin θ is 0.11(¯ h/2q) kΩ−1, which is in agreement
with the reported data, i.e., (0.1(¯ h/2q) kΩ−1) and also with the
Ab initio study (0.14(¯ h/2q) kΩ−1) [23]. In contrast to the
∂τ?/∂V , the bias dependence of the out-of-plane component
∂τ⊥/∂V , shows ∂τ⊥/∂V ∝ V (normalized with respect to sin
θ), so that τ⊥= (A0+ A1V2) sin θ. Also, it is evident that A1
is completely independent of θ for the tunneling devices. Note
that the equilibrium part of τ⊥, i.e., A0is not reported in the
literature. It is important to note that A0is comparable to A1V2
at lower voltages and can manifests itself as an exchange field
introducing an asymmetry in the R–H loops. The measured data
reported by Kubota et al. [11] and corresponding calculation
are shown in the Fig. A4. Encouraging agreement of our theory
with both experiments for the magnitudes of τ?and τ⊥sup-
ports the validity of our model. Note that the bias dependence
of τ⊥(V ) shows different bias curvatures in [9] and [11], the
origin being the difference in the sign convention employed for
the direction of both components of the spin transfer torque in
the experiments.
ox. These quantities (presented
APPENDIX II
A. Justification of (3a)
In general, it is convenient to define a current operator for
contact j (j = L, R) (see [20, p. 317])
Iop,j= i
??GΓj− ΓjG†?fj−
Starting from (A1) and using properties of “Trace” of a ma-
trix, we show the charge current and the in-plane spin current,
??
jGn− Gn?†
j
??
.
(A1)

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268IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 2, MARCH 2012
Fig. A2.
between reported [8] and calculated R(V ) at four different relative angular positions (θ = 0◦, 52◦, 71◦, and 180◦) of fixed- and free-layer magnetizations. Inset of
upper right panel shows bias dependence of TMR. The zero voltage TMR (300K) is calculated to be 156% (indicated by “O”), which is very close to the reported
value of 154%. We also calculate the expected roll-off of TMR (43% at 0.54V; indicated by “×”). Lower panel shows the comparison between reported [11] and
calculated R(V ) for three different relative angular positions (θ = 0◦, 137◦, and 180◦). Zero voltage TMR is calculated to be 154%, which is exactly same as
measured from experiment.
Experiment and numerical simulation of bias dependence of differential resistance R(V ) for different θ. Upper panel shows the shows comparison
?Is,?can be written as (S?=˜I ⊗? σ.ˆ
IC(E) = Real?Trace?ΓLGΓRG†??(fL− fR)
M)
(A2.1)
IS,?(E) = Real?Trace?ΓLGΓRG†S?
where ΓL,Rare the anti-Hermitian components of self-energy
matrices?
spin components
??(fL− fR) (A2.2)
L,Rdue to the fixed (L) and free (R) ferromagnetic
contacts, respectively, which can be decoupled into spatial and
ΓL= γL⊗ (I + PCM? σ.ˆ
ΓR= γR⊗ (I + PCm? σ.ˆ m)
M)
(A3.1)
(A3.2)
γL,R being N × N spin-independent matrices in real space
with N lattice points,˜I and I being N × N and 2 × 2 identity
matrices respectively in real and spin space.
Starting with?
of the self-energy matrix?
j= ΓH
j− (i/2)Γj and G = GH− (i/2)A
(i.e., separating the Hermitian and anti-Hermitian components
jand the Green’s function G), it is
straightforward to show that
Iop,j= 0.5 × [Γj(Afj− Gn) + (Afj− Gn)Γj]
+ i[(GHΓj− ΓjGH)fj− (ΓH
= IAH
op,j
jGn− GnΓH
j)]
op,j+ IH
(A4)
A( = i (G − G†)) being the total spectral function and Gn
the spin correlation function (Gn= ALfL+ ARfR). AL,Rare
thepartialspectralfunctionsdueto“Left”and“Right”contacts,
respectively (AL,R= GΓL,RG†, A = AL+ AR).
Now,weevaluate the“Trace”ofIop,LmultipliedbythePauli
spin matrix along the direction of the magnetizationˆ
fixed FM layer
M of the
Trace[IAH
op,LS?]
= 0.5Trace[(AfL− Gn)(ΓLS?+ S?ΓL)]
= 0.5(fL− fR)
?
= Real[Trace(ΓLGΓRG†S?)](fL− fR)
×Trace
GΓRG†
?(γL⊗ (I + PCM? σ ·ˆ
M))(˜I ⊗? σ.ˆ
M)
M))
+(˜I ⊗? σ ·ˆ
M)(γL⊗ (I + PCM? σ.ˆ
??
(A5.1)

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DATTA et al.: VOLTAGE ASYMMETRY OF SPIN-TRANSFER TORQUES269
Fig. A3.
numerical simulation, we see that both the torkances are proportional to sin θ, so when these quantities are normalized with respect to sin θ, they should fall onto
single curves (see right panel of the figure). It also confirms the angular dependence of the torkances.
Measured [9] and numerical simulation of bias dependence of differential torkances ∂τ?/∂V and ∂τ⊥/∂V for two different values of θ. From our
Fig. A4.
orientation of fixed- and free-layer magnetization. As shown in the bottom panel of the figure, there is slight discrepancy in the magnitude of the measured and
calculated τ⊥.
Measured [11] and numerical simulation of bias dependence of spin-transfer (τ?) and field-like (τ⊥) components of spin torque for perpendicular

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270IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 2, MARCH 2012
Trace[IH
op,LS?]
= Trace[iGH(ΓLS?− S?ΓL)]fL
− Trace[iGn(ΓH
?
?
= 0.
LS?− S?ΓH
?(γL⊗ (I + PCM? σ.ˆ
?(γH
L)]
= Trace
iGH
M))(˜I ⊗? σ.ˆ
M)
−(˜I ⊗? σ ·ˆ
L⊗ (I + PCM? σ.ˆ
−(˜I ⊗? σ.ˆ
M)(γL⊗ (I + PCM? σ.ˆ
M))(˜I ⊗? σ.ˆ
M)(γH
M))
??
fL
− Trace
iGn
M)
L⊗ (I + PCM? σ.ˆ
M))
??
(A5.2)
The in-plane spin current IS,?is calculated from (A5.1). Now,
if Green’s function G(E) is spin independent, i.e., G(E) = (g ⊗
I), then it follows from (A5.1) that
Charge Current:
IC(E) = (fL− fR)Real[Trace(γLgγRg†)]
× Real[Trace([I + PCM? σ.ˆ
= (fL− fR)¯T0(1 + PCMPCm ˆ m.ˆ
In-Plane Spin Current:
M][(I + PCm? σ.ˆ m)])]
M)
(A6)
IS,?(E) = (fL− fR)Real[Trace(γLgγRg†)]
× Real[Trace([I + PCM? σ.ˆ
= (fL− fR)¯T0(PCMˆ
where¯T0= Real[Trace(γLgγRg†)] thus proving (3a) of our pa-
per. It is interesting to see that¯To can be directly calculated
using nonmagnetic contacts instead of ferromagnetic contacts
(i.e., by putting spin splitting Δ = 0). However, it should be
noted that G(E) = g ⊗ I is not generally true since G(E) =
[EI − H −?
G(E) = g ⊗ I + gCM⊗? σ.ˆ
with g, gCM, and gCmbeing N × N spin-independent matrices
in real space, where g ∼ [E − H0− Ub− (γH
(γH
PCm [γH
asymmetric devices, we can show g ? gCM, gCmas follows.
1) Symmetric MTJ device: H0+ Ub? γH
oflargebarrierheightUb),g ?gCM,gCm,andPCM,PCm
< 1. Therefore, G∼= [(E − H0− Ub) ⊗ I]−1= g ⊗ I.
2) Proposed FM/insulator/NM/FM: Left interface to be tun-
neling and right interface ohmic γL? γR(since γL,R∝
GL,Rand GL? GR, where GL,Rare conductance ma-
trices of the contacts), γH
(∼0) in the energy range of interest. Therefore, G∼=
[(E − H0− Ub− γH
B. Justification of (3b)
M][(I + PCm? σ.ˆ m)])? σ?]
M + PCm ˆ m)
(A7)
L−?
R]−1and?
L,Rare spin dependent like
ΓL,R. G(E) in general, can be written as
M + gCm⊗? σ.ˆ m
(A8)
L+ (i/2) γL) −
R+ (i/2) γR)], gCM ∼ PCM [γH
R+ (i/2) γR]. However, in MTJ as well as proposed
L+ (i/2) γL], and gCm∼
L,R, γL,R(because
L? γH
Rand PCM ? PCm
R− (i/2)γH
R) ⊗ I]−1 ∼= g ⊗ I.
StartingfromtheexpressionofcurrentoperatorIop,j,wewill
show that out-of-plane torque τ⊥can be written as
? τ⊥=
?
dE PCMPCm¯Teff(fL+ fR)(ˆ m ׈
M)
(B1)
with¯Teff= Real[Trace(γHgγg†)] thus proving the fact that
IS ,⊥ is proportional to the product of the spin polarizations
PCMand PCmof the ferromagnetic interfaces.
To derive the expression for IS ,⊥due to “Left” contact, we
evaluatethe“Trace”ofIop,LmultipliedbythePaulispinmatrix
along the direction of magnetization perpendicular to the FM
layers, i.e., S⊥=˜I ⊗? σ.(ˆ
Trace[IAH
M × ˆ m)
op,LS⊥]
= 0.5Trace[(AfL− Gn)(ΓLS⊥+ S⊥ΓL)]
⎡
⎣GΓRG†
= 0.5(fL− fR)Real[Trace(γH
?(I + PCm? σ.ˆ m)(I + PCM? σ.ˆ
= 0.
= 0.5(fL−fR)Trace
⎢⎢⎢
⎧
⎪
Lg γRg†)]
⎪
⎪
⎪
⎪
⎪
⎩
⎨
(γL⊗ (I + PCM? σ.ˆ
(˜I ⊗? σ · (ˆ
+(˜I ⊗? σ · (ˆ
(γL⊗ (I + PCM? σ ·ˆ
M))
M × ˆ m))
M × ˆ m))
M))
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎭
⎬
⎤
⎦
⎥⎥⎥
× Trace
M)? σ · (ˆ
M × ˆ m)
+(I + PCm? σ.ˆ m)? σ · (ˆ
M × ˆ m)(I + PCM? σ ·ˆ
M)
(B2)
?
Note that we assume G(E) = g ⊗ I to arrive at (B2). For the
remaining part of the current operator, i.e., Iop,LH
Trace[IH
op,LS⊥] = Trace[iGH(ΓLS⊥− S⊥ΓL)]fL
− Trace[iGn(ΓH
LS⊥− S⊥ΓH
L)]
where
Trace[iGH(ΓLS⊥− S⊥ΓL)]fL
= Trace
iGH
?
×
?(γL⊗ (I + PCM? σ ·ˆ
M))(˜I ⊗? σ · (ˆ
M × ˆ m))
−(˜I ⊗? σ · (ˆ
M × ˆ m))(γL⊗ (I + PCM? σ ·ˆ
M))
??
fL
= 0
(B3.1)
Trace[iGn(ΓH
LS⊥− S⊥ΓH
LS⊥− S⊥ΓH
LS⊥− S⊥ΓH
LgγLg†)]
?(I + PCM? σ ·ˆ
+ fRReal[Trace(γH
?(I + PCm? σ.ˆ m)(I + PCM? σ ·ˆ
= 2PCMPCmReal[Trace(γH
L)]
= Trace[iAL(ΓH
L)]fL
+ Trace[iAR(ΓH
L)]fR
= fLReal[Trace(γH
× Trace
M)(I + PCM? σ.ˆ
M)? σ · (ˆ
Lg γRg†)]
M)? σ · (ˆ
M × ˆ m)
−(I + PCM? σ.ˆ
M × ˆ m)(I + PCM? σ.ˆ
M)
?
× Trace
M)? σ · (ˆ
M × ˆ m)
−(I + PCm? σ.ˆ m)? σ · (ˆ
M × ˆ m)(I + PCM? σ.ˆ
Lg γRg†)]fR(1 − (ˆ m.ˆ
M)
?
M)2).
(B3.2)
Note that likewise Γj(j = L, R), ΓH
of self-energy matrices?
j(i.e., the Hermitian part
j) can be also decoupled into spatial

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DATTA et al.: VOLTAGE ASYMMETRY OF SPIN-TRANSFER TORQUES271
and spin components
ΓH
L= γH
ΓH
L⊗ (I + PCM? σ.ˆ
R= γH
M)
(B4.1)
R⊗ (I + PCm? σ.ˆ m)
(B4.2)
γH
Note that we use the following identities to derive (B3.2) (note:
εijkis the Levi–Civita antisymmetric tensor):
1) (? σ.? a)(? σ.?j) = δajI + iεajk(? σ.?k);
2) (? σ.? a)(? σ.?j) − (? σ.?j)(? σ.? a) = 2iεajk(? σ.?k);
3) (? σ.?i)(? σ.? a)(? σ.?j)=iεajkI+δaj(? σ ·?i)−δji(? σ.? a)+δai(? σ.?j).
Similarly for the “Right” contact, we get
L,Rbeing N × N spin-independent matrices in real space.
Trace[IAH
op,RS⊥] = 0
op,RS⊥] = −2PCMPCm
× Real[Trace(γH
Therefore, the out-of-plane spin current IS ,⊥due to both the
contacts will be
(B5.1)
Trace[IH
RgγLg†)]fL(1 − (ˆ m.ˆ
M)2).
(B5.2)
IS,⊥=
?IS.(ˆ
(1 − (ˆ m.ˆ
M × ˆ m)
M)2)
= PCMPCm[¯T1fL+¯T2fR]
(B6)
wherethecoefficientsaregivenby¯T1= Real[Trace(γH
and¯T2= Real[Trace(γH
1) Symmetric MTJ device:
RgγLg†)]
LgγRg†)].
γL= γR= γ
(B7.1)
γH
L= γH
R= γH
(B7.2)
PCM(E) = PCm(E).
(B7.3)
Therefore, (B6) becomes
IS,⊥= PCMPCm¯Teff(fL+ fR)
assuming¯T1≈¯T2≈¯Teff= Real[Trace(γHgγg†)], thus prov-
ing (3b) of our paper. Note that fL+ fRis always symmetric
with energy.
2) Asymmetric MTJ device:
(B8)
γL= γR= γ
(B9.1)
γH
L= γH
R= γH
(B9.2)
PCM(E) ?= PCm(E)
(B9.3)
with constant PCm around E = Ef. Therefore, τ⊥ ∼ PCM
and likewise τ?(V ), an asymmetry is introduced in the bias
dependence of τ⊥(V ). The change in sign of the bias curvature
of τ⊥around V = 0 is a manifestation of subtracting the offset,
i.e., τ⊥(V = 0) from τ⊥(V ).
ACKNOWLEDGMENT
The authors acknowledge the support of the Materials, Struc-
ture, and Devices Focus Center, one of the six research centers
funded under the Focus Center Research Program, a Semicon-
ductor Research Corporation Entity.
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Authors’ photographs and biographies not available at the time of publication.