# Impact of Spin-Orbit Coupling on Quantum Transport in Magnetic Tunnel
Junction with an anti-ferromagnetic Capping Layer

Shradha Chandrashekhar Koli  School of Integrated Circuit Science and
Engineering, MIIT Key Laboratory of Spintronics, Beihang University, Beijing
100191, China  Bertrand Dupé  Fonds de la Recherche Scientifique (FNRS),
B-1000 Bruxelles, Belgium  Nanomat/Q-mat/CESAM, Université de Liège, B-4000
Sart Tilman, Belgium  Patrick M. Buhl  Johannes-Gutenberg-Universität Mainz,
Institut für Physik, 55128 Mainz, Germany  Hangyu Zhou  School of Integrated
Circuit Science and Engineering, MIIT Key Laboratory of Spintronics, Beihang
University, Beijing 100191, China  Jiaqi Zhou  School of Integrated Circuit
Science and Engineering, MIIT Key Laboratory of Spintronics, Beihang
University, Beijing 100191, China  Weisheng Zhao  School of Integrated Circuit
Science and Engineering, MIIT Key Laboratory of Spintronics, Beihang
University, Beijing 100191, China

(August 27, 2024)

###### Abstract

Using first-principles calculations, we explore the effect of non-magnetic and
magnetic capping-layers, in particular the anti-ferromagnetic heavy-metal,
IrMn, on the tunneling magnetoresistance (TMR) in magnetic tunnel junctions
(MTJs). We conduct a comparative study of the different capping-layer by
employing conventionally used non-magnetic heavy-metal capping layers in
symmetric-MTJs X/FeCo/MgO/FeCo/X where X=Ta, W, Mo and IrMn. Firstly, a
relatively large TMR is obtained in the IrMn-IrMn MTJ, without SOC, which can
be attributed to both the reduced lattice-mismatch at the IrMn/FeCo interface
resulting in a large spin-polarization and the anti-ferromagnetic order of
IrMn that reflects spins at IrMn/FeCo interface resulting into the non-
identical signatures of both the anti-parallel conduction channels. When SOC
is switched-on, a gigantic enhancement in the TMR is reported which is
attributed to the spin-mixing of $\Delta_{1}$ and $\Delta_{5}$ states and the
additional increase in the degeneracy of the out-of-plane orbitals levels, _p_
z, _d_ ${}_{z}^{2}$ and _d_ xz. Finally, we conclude that the capping-layer-
induced crystallographic modifications within the FeCo-electrodes also plays a
decisive role in enhancing the TMR.

##  I Introduction

Capping layers are an essential part in magnetic tunnel junctions (MTJs). They
do not only offer a protection against oxydation that degrade the device but
also can be used to enhance their magnetic and transport properties. Since
this effect was discovered on tunnelling magneto-resistance (TMR)[1] and
perpendicular magnetic anisotropy (PMA) [2], tremendous efforts have been made
to explore this effect both theoretically and experimentally.

Ta, W and Mo have been comprehensively studied both as buffer- and capping-
layers alongside FeCo(B)-electrode MTJs [3, 4, 5]. For over a decade Ta has
been a generic choice in FeCo(B)-based perpendicular-magnetic tunnel junctions
(_p_ MTJ) [2]. But due to its thermal instability above 300∘ C [Ref], the
utilization of Ta in industrial FeCo(B)/MgO microelectronic devices becomes
challenging as the required processing temperature is more than 400∘ C.
Whereas, W and Mo provided a robust solution to the thermal incompatibility of
Ta along with increased PMA and TMR in FeCo(B)/MgO devices [Ref].

Among Ta, W, Mo and many others non-magnetic (NM) heavy-metals (HM), W appears
as a prominent member of NM-HM group as it provides a large PMA ($\sim 2.0$
erg/cm2) and a TMR in FeCo(B)/MgO-based MTJs even above 400∘ C [6, 7]. Due to
its large spin-orbit coupling (SOC) constant [Ref], W also became an
attractive choice for spin-orbit torque (SOT)-based FeCo(B)/MgO devices [Ref].
Then, a wide range of _d_ -non-magnetic heavy-metal capping layers with strong
SOC and large SOT-switching efficiency was explored to enhance the PMA and the
TMR in FeCo(B)/MgO MTJ. In 2013, Ir was found to drastically enhance both the
PMA and the TMR in SOT-_p_ MTJ [8]. A PMA as large as 4.17 erg/cm2 was
obtained in Ir/CoFe/MgO structure [9], coming from its large SOC and the
induced magnetic moments at the Fe-interface. The 5 _d_ -3 _d_ hybridization
between Ir and a FM metal is at the origin of the enhanced PMA and TMR [Ref].
With respect to this, we envision that using a capping layer composed of _d_
-group bi-metals could definitely have a positive impact on PMA and TMR in
CoFe(B)/MgO-based SOT-_p_ MTJs.

So far capping layers have been chosen due to their strong SOC that could
affect the transport properties of the associated device. To the best of our
knowledge, the addition of magnetic elements in binary alloys containing heavy
elements such as W, Ta and Ir have been widely overlooked. Such composition
would ensure a strong PMA as well as a strong modification of transport
properties depending on their magnetic configurations. Especially, alloys
composed of 3 _d_ transition metal and 5 _d_ heavy-metals (M-HM) have been
widely overlooked both theoretically and experimentally.

In 2005, K. Tsunekawa et al. investigated the effect of capping layers on the
TMR in CoFeB/MgO/CoFeB/HM MTJ with various mono-metallic (HM=Fe, Cu, Al, Ru,
Mg, Ta, Ir) and bi-metallic (HM=NiFe, CoFe, PtMn, IrMn) materials [1]. More
than 100% TMR was obtained in MTJs with mono-metallic capping layers such as
Ta, Ru, Mg and bi-metallic capping layer like PtMn and IrMn. The cause of
increase in TMR was attributed to the reactive interface between
Ferromagnetic/Heavy-Metal i.e. interfacial hybridization. As evident from
Fig.1 in Ref[1], the increase of TMR due to PtMn and IrMn capping layers was
relatively giant, although the detailed analysis of the origin of this
increase was not clear. In Ref[10], we speculate that the increase in the TMR
could be: (a) strong spin-orbit coupling of Ir-atoms and (b) anti-
ferromagnetic ordering of Mn-atoms with large spin magnetic moments.

Since, IrMn is comparatively more suitable in devices with PMA [11] and is
also employed in IrMn/FeCo(B)/MgO tri-layered SOT-structures [Ref], we choose
IrMn over PtMn as a magnetic capping layer in our _p_ MTJ. Due to the strong
spin-orbit coupling (SOC) of Ir and the large magnetic moments on Mn [12],
IrMn serves as a promising candidate in spin-valve structures and in SOT-pMTJ
devices [13, 12, 11].

For achieving high TMR, maintaining the (001) face-centered cubic (fcc)
crystallography of MgO-barrier (for coherent-tunneling) and crystallization of
body-centered cubic (bcc) FeCo(B)-electrode in (001)-direction (for dominating
presence of $\Delta_{1}$ state) is of paramount importance [Ref]. It was also
demonstrated that the crystallography of heavy-metal adjacent to the
FeCo(B)-electrode and the hybridization of HM/FeCo(B) interface also plays a
vital role in achieving high PMA/TMR [Ref]. Sometimes, the crystallographic
changes in FeCo(B) caused by the adjacent heavy-metal also has a sever impact
on TMR [Ref]. But, to our knowledge, this impact has not been widely studied.
Bcc Ta [Ref], W [Ref] and Mo [Ref] along (001)-direction had been employed in
FeCo(001)/MgO(001) structures as buffer/capping layers. Whereas, IrMn is well
known to grow in the (111)-direction with (001)-FeCo(B) [Ref]. But successful
growth of IrMn (001) [Ref] in FeCo(B)(001)/MgO(001) [ref], is also achievable.
Therefore, we enforce the (001)-crystallographic direction for all the layers
(capping, electrode and barrier) in our MTJ structures X/FeCo/MgO/FeCo/X
(X=Ta,W,Mo and IrMn).

To have better insight into the influence of SOC in IrMn on the tunneling
magnetoresistance, we first study the influence of anti-ferromagnetic capping
IrMn on TMR, without SOC. This helps to determine the spin-decomposed dominant
Bloch-state symmetries in the capping layer and the ferromagnetic electrode
and understand its contribution to the tunneling trough MgO-barrier. Whereas,
when SOC is switch on the understanding of the spin-mixed contribution of
Bloch-states becomes complex. The comparative and systematic study with other
capping layers like Ta, W and Mo, with and without including SOC will deepen
the understanding of influence of anti-ferromagnetic capping layer IrMn and
will emphasize on the impact of its large spin-orbit coupling.

This paper is decomposed as follows: Secrtions I describes …., section 2 (keep
that brief)

##  II Methods

###  II.1 Crystal structure of IrMn

To study the impact of an anti-ferromagnetic capping layer, an ordered CuAu-I-
type face-centered tertragonal (FCT) structure of L10-IrMn [14]
(????[patent]????) is considered. It has the equiatomic composition of the
alternatively arranged two Ir- and two Mn-atoms as shown in Fig.1 panel (a).
Spins on Mn-atoms are anti-ferromagnetically aligned in
$\mathbb{k}$-direction. The in-plane lattice constant of L10-IrMn is
$\mathbb{i}$=$\mathbb{j}$=3.855 Åand $\mathbb{k}$=3.643 Å[13]–[15]. In view of
maintaining a relatively smaller lattice-mismatch of FeCo/MgO/FeCo junction,
the L10 lattice of IrMn is rotated by 45∘ in-plane, obtaining a lattice
constant of $\mathbb{i}$=$\mathbb{j}$=$\mathbb{i}/\sqrt{2}$. Unlike the
typical ($001$)[$100$] orientations of L10-IrMn, ($001$)[$110$] (45∘) is
composed of only one atom in each plane. Therefore, the magnetic spin-
alignment on Mn-atoms is modified along ($001$) direction creating a stable
AFM alignment [Ref]. Fig.1 panel (b) shows the unit cell of modified magnetic
structure of 45∘ rotated L10-IrMn. This 45∘ rotated geometry, minimizes the
lattice-mismatch between IrMn/FeCo to -2.46% which is relatively lesser than
+4.7% of mismatch obtained for FeCo/MgO and, furthermore, is incomparably
lesser than that of Ta/FeCo, W/FeCo and Mo/FeCo structures.

With respect to previous studies [16, 17, 18, 19, 20], we intend to employ
five monolayers (ML) of capping layer forming a
symmetric-X(5)/FeCo(5)/MgO(5)/FeCo(5)/X(5) MTJ. The numbers in the parentheses
represent the number of monolayers employed in the MTJ structure. To confirm
if the modified spin-alignment on Mn-atoms can retain the stability of an AF
ordering in IrMn (45∘), a collinear magnetic relaxation was performed using
the Vienna Ab Initio Simulation Package (VASP) [21].

###  II.2 DFT calculations

###  II.3 Structural relaxation

The total free energies were computed to confirm that the energetically
favored interface-termination is Fe-Ir over Co-Ir in
IrMn($001$)[$110$]/FeCo($001$)[$100$] bilayer as in panel (c) of Fig.1.

Figure 1: Schematic illustration of energetically favorable crystallographic
structures. (a) L10-IrMn unit cell with Mn1 (up) and Mn2 (down) representing
anti-ferromagnetic alignment of spins on Mn-atoms.(b) The B2 structure
implemented in our calculation is highlighted using dotted lines. (c) An in-
plane crystallographic relationship of IrMn($001$)/FeCo($001$) with (d) an
interfacial atomic arrangement along the direction of transport. The direction
of transport is along the Z-axis. (d) two-probe device model of the
CoFe/MgO/CoFe MTJ-stack with same capping layer (X=Ta,W,Mo,IrMn) on both the
sides. A collinear magnetic spin-alignment, normal to the interface, is
considered.

The FeCo-electrode is considered epitaxial with respect to the 001 growth
direction of IrMn. The stability of both the Fe- and Co- terminated interfaces
were tested with respect to both the IrMn capping layer and the MgO-barrier.
The binding energy, Eb (eV.Å-1) is computed for various interface combinations
in structurally relaxed IrMn/FeCo-bilayer and IrMn/FeCo/Mgo-trilayer. The in-
plane lattice constant is fixed to the lattice constant of FeCo-electrode
(2.83 Å) in all structures. The binding energy Eb is defined as [22],

$\displaystyle
E_{b}=E_{\mathrm{IrMn}}+E_{\mathrm{FeCo}}+E_{\mathrm{MgO}}-E_{\mathrm{IrMn/FeCo/MgO}}$
(1)

Here, EIrMn/FeCo/MgO is the total free energy of IrMn/FeCo/MgO (with Fe-
termination) trilayer structure. EIrMn, EFeCo, and EMgO are corresponding to
the total energies of IrMn, FeCo and MgO, calculated using similar parameters
and atomic positions from the trilayer structure. The obtained total free
energies, Efree and the calculated binding energies, Eb are tabulated in
Table1. The obtained Efree for FeCo/MgO bi-layer is larger than that of
CoFe/MgO indicating that the Fe-atoms prefer to accumulate at the MgO
interface which is consistent with the experimental work [Ref]. In contrary,
the obtained Eb for CoFe/MgO bi-layer is larger than that of FeCo/MgO which
signifies that Co-terminated interfaces are energetically stable due to their
stronger bonding with interfacial Oxygen-atoms from MgO [22]. This argument is
additionally supported by calculating interfacial bond-length at FeCo/MgO and
CoFe/MgO interface in respective bi-layers. The Fe-O interface bond length of
2.23Å[22, 23] and Co-O of 2.15Å[22] signifies stronger bonding between Co-O at
CoFe/MgO interface over Fe-O bonding at FeCo/MgO interface. Hence, Co-
terminated interfaces are mostly preferred over Fe-terminated interfaces in
FeCo/MgO/FeCo junctions for TMR.

On the other hand, in both the bilayer and trilayer IrMn structures, Fe-
terminated interfaces showed relatively larger Efree and Eb as compared to
that of Co-terminated interfaces. This indicates that: (i) the Fe-atoms from
FeCo-electrode prefer to reside at the interface than in the bulk and (ii) Ir-
Fe termination at IrMn/FeCo interface is more stable than that of Co-Ir
termination. This could be because of: (a) strong spin-orbit coupling in Ir-
atoms results into stronger interfacial coupling between Ir-Fe termination
[Ref] and (b) IrMn-Fe favours stable grownt in (001)-direction whereas IrMn-Co
favours (111)-direction [Ref]. Therefore, IrMn(001)/FeCo(001)/MgO(001)
structure with Fe-termination is considered. Besides, Fe-Ir interface helps to
increase the interfacial PMA and TMR [24]

Table 1: The total free energy of fully-relaxed structures and the Binding energy as defined by Eq.1. FeCo and CoFe refer to Fe-terminated and Co-terminated interface, respectively. Structure |  $E_{free}$ (eV) |  $E_{b}$(eVÅ-1)
---|---|---
FeCo/MgO |  -95.243 |  0.568[22]
CoFe/MgO |  -94.265 |  0.69[22]
IrMn/FeCo |  -81.506 |  2.550
IrMn/CoFe |  -80.247 |  2.311
IrMn/FeCo/MgO |  -140.314 |  3.105
IrMn/CoFe/MgO |  -139.195 |  3.008

###  II.4 Transport calculations

As the heavy-metal is deposited over a FM-electrode (capping layer) and below
a FM-electrode (seed layer) influence differently [Ref], the complete
supercell of current-perpendicular-to-plane (CPP) like spin-valve MTJ model
was structurally relaxed. The supercell composition is
X(5)/FeCo(5)/MgO(5)/FeCo(5)/X(5) with X=Ta,W,Mo,IrMn as capping layers. The
electronic structure is obtained with the projector augmented wave (PAW)[25]
Pseudopotentials as implemented in Vienna Ab Initio Simulation Package
(VASP)[21]. The exchange-correlation energy was treated within the generalized
gradient approximation (GGA) using Perdew-Burke-Ernzerhof (PBE) functionals
[26, 27]. The convergence threshold for total atomic energies and Hellmann-
Feynman forces are set to 10-6 eV and 0.01 eV.Å-1, respectively. Kohn-Sham
orbitals were expanded using the plane-wave basis set with kinetic energy
cutoff of 550 eV. The sampling of the Brillouin zone is obtained with a
$\Gamma$-centered Monkhorst-Pack scheme[28] of $25\times 25\times 1$ k-points.
This ensures sufficient convergence of the forces. The illustration of the
atomic structures were obtained using VESTA[29] software.

The spin-dependent quantum transport properties of electronically stable MTJs
are calculated via the Keldysh-nonequilibrium-Green’s-function formalism
combined with real-space density functional theory (NEGF-DFT) as implemented
in Nanodcal[30, 30] package. The device model in Nanodcal is divided into
three sections: the central scattering (CS) region and the two leads (capping
layers). The CS region includes five monolayers of the capping layer as shown
in Fig.1 panel (e). The non-magnetic capping layers, Ta, W and Mo, with
lattice constants 3.317Å, 3.187Åand 3.168Årespectively, are chosen. Self-
consistent calculations are performed on CS region with the double plus
polarized (DZP) atomic orbitals basis. This basis is used to expand all the
physical quantities with electron kinetic energy cut-off of 5000 eV. The same
PBE-GGA functional was used in both structural relaxation and the quantum
transport calculations. SOC was also included self-consistently.

The spin-polarized conductance Gσ is given by the Landauer-Büttiker formula
as,

$\displaystyle
G_{\sigma}=\frac{e^{2}}{h}\sum_{k_{\parallel}}T_{\sigma}(k_{\parallel},E_{F}),$
(2)

where $T_{\sigma}(k_{\parallel},E_{F})$ is the transmission coefficient at
Fermi Energy $E_{F}$ with spin $\sigma=(\uparrow,\downarrow)$ and an in-plane
Bloch wave vector $k_{\parallel}=(k_{x},k_{y})$. $e$ is the charge of the
electron and $h$ is the Plank’s constant. For self-consistent calculations,
the Hamiltonian matrix and density matrix are converged down to less than 10-4
Hartree with a $k_{\parallel}$-mesh sampling of $10\times 10\times 1$ for
central scattering region and $10\times 10\times 100$ for the leads. The
transmission coefficient and conductance were evaluated using denser
$k_{\parallel}$-mesh of $300\times 300\times 1$ for all spin-channels. The TMR
ratio is defined as

$\displaystyle TMR=\frac{G_{P}-G_{AP}}{G_{AP}},$ (3)

where $G_{P}=G_{P}^{\uparrow\uparrow}+G_{P}^{\downarrow\downarrow}$ is the
conductance in the parallel configuration (PC) of spins in FeCo-electrodes.
$G_{P}^{\uparrow\uparrow}$ and $G_{P}^{\downarrow\downarrow}$ define the
conductance when a spin travels from a majority to a majority channel and from
a minority to minority channel, respectively. Similarly,
$G_{AP}=G_{AP}^{\uparrow\downarrow}+G_{AP}^{\downarrow\uparrow}$ is the
conductance in the anti-parallel configuration (APC) with
$G_{AP}^{\uparrow\downarrow}$ and $G_{AP}^{\downarrow\uparrow}$ being the
conductance of majority-to-minority and minority-to majority states channel,
respectively. This implies that the effective spin-polarization (SP) is
defined as

$\displaystyle
SP=\frac{G_{P}^{\uparrow\uparrow}-G_{P}^{\downarrow\downarrow}}{G_{P}^{\uparrow\uparrow}+G_{P}^{\downarrow\downarrow}}.$
(4)

Here, the magnetic moments in the MTJs are collinear.

##  III Results and Discussion

Firstly, we present the effect of different capping layers on the transport
properties in X/FeCo/MgO/FeCo/X (X=Ta, W, Mo and IrMn) MTJs. Since the TMR in
Eq.4 is defined as the ratio of conductance in two different spin channels, we
start by elucidating the spin-dependent conductance (Fig.2) in Ta-Ta, W-W, Mo-
Mo and IrMn-IrMn MTJs. For better understanding of the contribution of unmixed
spins conductance (without SOC) are shown in Fig.2(a), for the parallel(P) and
anti-parallel(AP) configurations. In the MTJs with non-magnetic capping layers
(Ta, W, Mo), the trend followed by the TMR,
$G_{P}^{\uparrow\uparrow}(\mathrm{Ta})<G_{P}^{\uparrow\uparrow}(\mathrm{W})<G_{P}^{\uparrow\uparrow}(\mathrm{Mo})$,
is in good agreement with the previous studies [31, 18].

In the parallel configuration, the conductance obtained for the majority-spin
channel (G${}_{P}^{\uparrow\uparrow}$) is very small in the Ta-Ta MTJ. The
lattice-mismatch of Ta/FeCo is very large as compared to that of FeCo/MgO.
This lattice-mismatch exerts an in-plane compressive strain on Ta and pushes
the Ta atoms in z-direction. This increases the inter-atomic distance between
Ta-Fe at the interface ([32]) forming a discontinued interface between FeCo
and Ta. As a consequence, contribution from the $\Delta_{1}$ state is reduced
[33]. This causes the $\Delta_{2}$, $\Delta_{2}^{\prime}$ and most
importantly, the $\Delta_{5}$ state to contribute, leading to a relatively
larger minority-spin channel conductance (G${}_{P}^{\downarrow\downarrow}$)
[33, 34].

The lattice-mismatch of W/FeCo is considerably smaller than that of Ta/FeCo.
This drastically reduces the inter-atomic distance between W-Fe at the
interface. Also, W capping helps the growth both, the FeCo(B)-electrodes and
the MgO-barrier [35]. This maintains the coexistence of $\Delta_{1}$ and
$\Delta_{5}$ states in the FeCo-electrode (left) [34] and offers a pathway for
coherent-tunneling of $\Delta_{1}$ and $\Delta_{5}$ states through the MgO-
barrier, in the majority-spin channel, increasing
G${}_{P}^{\uparrow\uparrow}$. As a result, the dominance of $\Delta_{5}$ state
in minority-spin channel is reduced leading to the decreased
G${}_{P}^{\downarrow\downarrow}$, compared to the Ta-Ta MTJ.

The lattice-mismatch and the inter-atomic distance between Mo-Fe is, only,
slightly reduced in Mo-Mo MTJ, owing to its almost similar lattice constant as
of W. (This is related to the interface formation enthalpy of Mo-Fe). Unlike
W, Mo severely alters the bcc symmetry of the FeCo-electrodes in ($001$)
direction [5, 36, 4]. This reduces the contribution of the $\Delta_{1}$ state
present at the Fermi energy in ($001$) bcc FeCo [33] forcing other Bloch
states, probably, the $\Delta_{5}$ state to dominate in the majority-spin
channel [34]. The reduced lattice-mismatch provides efficient tunneling of the
$\Delta_{5}$ state, further increasing the G${}_{P}^{\uparrow\uparrow}$.
Since, the minority-states are already dictated by the $\Delta_{5}$ state,
G${}_{P}^{\downarrow\downarrow}$ also increase than in W-W MTJ. The same
analogy can be used to explain the $\Delta_{5}$ increase of
G${}_{AP}^{\uparrow\downarrow}$ and G${}_{AP}^{\downarrow\uparrow}$ in the
antiparallel configuration.

With the similar understanding, we proceed with the analysis of the
conductance in IrMn-IrMn MTJ. The 45∘ in-plane rotation of IrMn lattice
reduces the lattice-mismatch (-2.46 %) of IrMn/FeCo. The inter-atomic distance
between Ir-Fe slightly increases (2.19 Å) than in W-W (2.15 Å) and Mo-Mo (2.14
Å) MTJs. This increase could be a compromise of bonding interaction between
Ir-Fe on one side and Ir-Mn on the other side. Furthermore, this relatively
reduced the lattice-mismatch and the inter-atomic distance at the interfaces
and the atomic density in IrMn-IrMn MTJ [33]. This should increase the
magnitude of $\Delta_{1}$ Bloch states in FeCo-electrode and the conductance
in majority-spin channel. On the contrary, the conductance
G${}_{P}{\uparrow\uparrow}$ and G${}_{P}{\downarrow\downarrow}$ decreases as
compared to that of Mo-Mo MTJ. Also, the conductance in anti-parallel
configuration (G${}_{AP}^{\uparrow\downarrow}$,
G${}_{AP}^{\downarrow\uparrow}$ has also decreased. Importantly, unlike other
MTJs, the G${}_{AP}^{\uparrow\downarrow}$ and G${}_{AP}^{\downarrow\uparrow}$
show non-identical behaviour, signifying the different characters of majority-
and minority-spins. This reveals the influence of anti-ferromagnetism in IrMn
on the adjacent FeCo-electrodes. In the anti-parallel configuration,, the
spin-alignment of only one of the FeCo-electrode (free-layer) has been flipped
by $90^{\circ}$, keeping the spin-alignment in both the IrMn capping layers
unchanged.

The TMR can be quantitatively understood by analyzing the spin-polarization
(SP) [37] of the tunneling electrons which can be expressed in terms of
conductance [38] as defined in Eq.4. The magnitude of the spin-polarization
(Fig.2(b)) in Ta-Ta, W-W and Mo-Mo MTJs is in correspondence with the trend
followed by the parallel conductance in Fig.2(a). Whereas, the spin-
polarization in IrMn-IrMn MTJ shows a different signature than that of the
obtained conductance. This could be the effect of the lattice-mismatch-induced
strain on the atoms of the capping layers [39]. Due to the large lattice-
mismatch, atoms of the capping layers (Ta, W, Mo) shrink to reach the in-plane
lattice constant of 2.83 Å. While the atoms shrink in x-direction they are
forced to expand in the z-direction. Therefore, the compressive strain in
x-direction now acts as the tensile strain in the z-direction, elongating the
capping layers and increasing the inter-atomic interfacial distance. As the
electrons traverse in the z-direction, the strain acting in the z-direction
shows strong impact on the spin-polarization. Cai, Y. et al. [40] demonstrated
that the spin-polarization responds differently to the compressive and the
tensile strain. The spin-polarization is inversely proportional to the tensile
strain acting on the atoms. This tensile strain enhances the contribution of
interfacial states leading to the reduction in the spin-polarization, and, in
turn, the TMR [40].

In Ta-Ta MTJ, due to the larger lattice-mismatch, the tensile strain is
extremely larger than that of W-W and Mo-Mo MTJ. This tensile strain immensely
increases the contribution of the interfacial states resulting into a negative
spin-polarization. As the contribution of interfacial states is predominant in
minority-spins channel [34], the conductance G${}_{P}{\downarrow\downarrow}$
supersedes G${}_{P}{\uparrow\uparrow}$ in Ta-Ta MTJ (see Fig.2(a)). Similarly,
the increase in the magnitude of spin-polarization in W-W and Mo-Mo MTJs is
the result of the corresponding decrease in the lattice-mismatch.

In case of IrMn-IrMn MTJ, the in-plane lattice constant of BCT-IrMn (2.73 Å)
is smaller than that of FeCo (2.83 Å). Therefore, unlike in Ta-Ta, W-W and Mo-
Mo MTJs, the atoms expand laterally. As a result, the atoms shrink in
z-direction causing compressive strain on the atoms. This preserves strong
interfacial coupling between Ir-Fe at the interface, resulting into a high
magnitude of spin-polarization. However, the obtained high spin-polarization
does not correspond with the trend followed by conduction in Fig.2(a). This
occurs because the anti-ferromagnetic ordering of the spins on Mn-atoms in
IrMn probably reflect the spins at IrMn/FeCo interface. This is also justified
by the non-identical conductance obtained in anti-parallel configuration.

As the anomalous correlation between the conductance and the spin-polarization
failed to explain the relative decrease in G${}_{P}^{\uparrow\uparrow}$ and
G${}_{P}^{\downarrow\downarrow}$ in IrMn-IrMn MTJ and as IrMn being a bi-metal
with strong spin-orbit coupling (SOC), we now present the analysis of
conductance including SOC, in Fig.2(c). Importantly, SOC is inevitably present
in the real systems. In Fig.2(c), the parallel conductance GP
(G${}_{P}^{\uparrow\uparrow}$+G${}_{P}^{\downarrow\downarrow}$) and the anti-
parallel conductance GAP
(G${}_{AP}^{\uparrow\downarrow}$+G${}_{AP}^{\downarrow\uparrow}$) indicate the
contribution of spin-mixed conduction channels, as the spins cannot be
decomposed when SOC is switched-on [41]. Meaning that the
majority(minority)-to-majority(minority) channel in parallel configuration and
majority(minority)-to-minority(majority) channel in the anti-parallel
configuration are indistinguishable.

In Fig.2(c), the trend followed by GP for Ta-Ta, W-W and Mo-Mo MTJ is in
direct correspondence with the trend followed by G${}_{P}^{\uparrow\uparrow}$
in Fig.2(a). Conversely, the GP for IrMn-IrMn MTJ increases with SOC showing
complete opposite behaviour than that of G${}_{P}^{\uparrow\uparrow}$.
Whereas, the trend observed in GAP and
G${}_{AP}^{\uparrow\downarrow}$/G${}_{AP}^{\downarrow\uparrow}$ remains mostly
identical. Except that the GAP for the strong SOC capping layers, W and IrMn,
slightly decreases that in Fig.2(a).

When SOC is switched-on, the contribution of the out-of-plane orbital
characters (_p_ z,_d_ ${}_{z}^{2}$,_d_ xz) [41] is enhanced due to the
additional increase in the degenerated energy levels. Thus, the contribution
of $\Delta_{1}$ and $\Delta_{5}$ Bloch states is enhanced in the majority-
spins channel and only the contribution of $\Delta_{5}$ state is enhanced in
the minority-spins channel. Since IrMn is a bi-metallic alloy with 5 _d_ Ir-
atoms and 3 _d_ Mn-atoms, the effect of the SOC-induced degenerated energy
levels is expected to increase twice than that of the mono-metallic capping
layers, Ta, W and Mo. Therefore, the relatively larger GP is observed in IrMn-
IrMn MTJ, when SOC is switched on.

Typically in the anti-parallel configuration only $\Delta_{5}$ state
contributes to the tunneling conductance [34]. In Ta-Ta and Mo-Mo MTJs, the
$\Delta_{5}$ state governs the majority-spins channel as well as minority-
spins channel. Therefore, in anti-parallel configuration, both the majority-
to-minority and the minority-to-majority channels provide easy pathway to the
$\Delta_{5}$ state. Furthermore, the magnitude of this efficiently tunneling
$\Delta_{5}$ state is increased due to SOC and hence, the GAP increases for
Ta-Ta and Mo-Mo MTJ, in Fig.2(c).

In Fig.2(d) we report on the tunneling magnetoresistance (TMR) including the
SOC contribution. The TMR without SOC in Ta-Ta, W-W and Mo-Mo MTJs shows the
same trend as the spin-polarization (Fig.2(b)) and the spin-dependent
conductance (Fig.2(a)). Whereas, in IrMn-IrMn MTJ, the TMR corresponds with
spin-polarization but not with the obtained conductance. This is attributed to
the non-identical anti-parallel conductance in IrMn-IrMn MTJ. In
G${}_{AP}^{\downarrow\uparrow}$, the tunneling of electrons from minority-to-
majority spins channel is almost forbidden. Hence, the electrons experience
large resistance resulting into a smaller conductance. This small anti-
parallel conductance results into a large TMR ratio [34].

When SOC is switched-on the trend followed by the TMR in Fig.2(d) is strongly
transformed. Firstly, the magnitude of the TMR is enhanced as compared with
that of the non-SOC TMR. This can be attributed to the SOC-induced increase in
the degeneracy of the band levels.

In Ta-Ta MTJ, the TMR remains unexpectedly low due to the large contribution
of the interfacial states in the minority-spins channel, due to the
tremendously large lattice-mismatch. In W-W MTJ, the $\Delta_{1}$ and
$\Delta_{5}$ states contribute equally as in FeCo/MgO/FeCo MTJ as explained by
Butler et al. [34]. On the contrary, in Mo-Mo MTJ, only $\Delta_{5}$ state
contributes in the minority-spins channel resulting into a lower TMR than that
of W-W MTJ. Precisely, the SOC-induced impact on the $\Delta_{1}$ state with
two out-of-plane orbital characters, _p_ z and _d_ ${}_{z}^{2}$, is larger
that that on the $\Delta_{5}$ state with only one out-of-plane orbital
character, _d_ xz. Therefore, the TMR in W-W is larger than the one of Mo-Mo
MTJ when SOC is switched-on. Similar to the W-W MTJ, in IrMn-IrMn MTJ, the
increase of the SOC-induced TMR can be attributed to the combined contribution
of $\Delta_{1}$ and $\Delta_{5}$ states with SOC-induced increase in
degeneracy in _p_ z, _d_ ${}_{z}^{2}$ and _d_ xz bands. The relative increase
of TMR in IrMn-IrMn MTJ compared to the W-W MTJ is due to: (a) bi-metallic
capping layer with two different _d_ -group metals and (b) the large spin-
polarization within the Mn-atoms [42].

Figure 2: Spin-dependent transport properties in FeCo/MgO/FeCo MTJ with
different capping layers. Plots of (a) conductance in parallel (P) and anti-
parallel (AP) spin configuration without SOC.(b) Spin-polarization of MTJ-
stacks without SOC, using Eq.4. (c) SOC-induced conductance in P and AP spin
configuration and (d) The Normalized TMR ratio as a function of SOC. The
values of conductance are in the units of $e^{2}h^{-1}$. Conductance with SOC
is not spin-decomposed. The solid lines are guide to the eye, only.

To gain further insight into the behavior of the conductance when SOC is
switched-on, we analyze the transmission coefficient as a function of
transverse momentum $k_{\parallel}(k_{x},k_{y})$ at Fermi energy. The
transmission spectra shown in Fig.3 is plotted in the two-dimensional
Brillouin zone (2DBZ) along ($001$)-plane. For parallel configuration,
major(minor) contribution to the conductance appears at the $\Gamma$-point
($k_{\parallel}=0,0$) from the $\Delta_{1}$($\Delta_{5}$) state symmetries
propagating perpendicular to the interface. The rate of decay of these
symmetries within the barrier defines the probability of transmission and the
shape of the transmission spectrum.

In Ta-Ta MTJ, the moderate transmission at the $\Gamma$-point, in Fig.3(a), is
due to the low contribution of $\Delta_{1}$ state and the high transmission
ring (red) around the $\Gamma$-point is due to the dominating $\Delta_{5}$
state which corresponds to the negative spin-polarization in Fig.2(b). The
effect of the increased interfacial states due to the large lattice-mismatch
can be seen as the sharp 100% transmission peaks at ($k_{\parallel}\neq 0.0$)
in Fig.3(b). Therefore, the conductance of the minority-spin channel with the
dominating $\Delta_{5}$ state is larger than that of the majority-spin channel
conductance. These points are represented as interfacial resonant states (IRS)
giving rise to a resonant tunneling mechanism in the minority-state channel.
The cumulative effect of the $\Delta_{1}$ and the $\Delta_{5}$ states due to
the SOC-induced mixing of spins is evident from the relatively high
transmission at $\Gamma$-point ($k_{\parallel}=0,0$) in Fig.3(c), unlike in
Fig.3(a).

In case of W-W MTJ, W exhibits a strong interfacial PMA [43] with _d_
${}_{z}^{2}$ orbital character in W, overlapped with the _d_ ${}_{z}^{2}$ and
_d_ xz orbitals of the Fe-atom in the FeCo-electrode. This strong PMA leads to
an increase in TMR [41]. This is evident from the strong concentrated peaks at
and around the $\Gamma$-point ($k_{\parallel}=0,0$) governed by the
$\Delta_{1}$ and the $\Delta_{5}$ states, respectively (see Fig.3(d)). Since
the tunneling is governed by the $\Delta_{1}$ state in the majority-spin
channel and due to the positive spin-polarization in W-W MTJ, the influence of
the resonant states appears to be auxiliary but not dominating as in the Ta-Ta
MTJ (see Fig.3(e)). Similarly, almost 100% propagation of the SOC-induced
$\Delta_{1}$ state at the $\Gamma$-point is confirmed in Fig.3(f). The broader
peak at and around the $\Gamma$-point is the result of the SOC-induced spin-
mixed transmission of $\Delta_{1}$ and $\Delta_{5}$ states.

Orbital hybridization of Mo(001)[110]/FeCo(001)[100] is different as compared
with that of the Ta/FeCo and W/FeCo interface, causing crystallographic
modifications in adjacent FeCo-Electrode [5, 36, 4]. Therefore, the
$\Delta_{1}$ state along ($001$) cannot be present in FeCo-electrode (left) at
the Fermi energy ($k_{\parallel}=0,0$). As a consequence, the $\Delta_{1}$
state will not dominate the the tunneling at the $\Gamma$-point in the
FeCo/MgO/FeCo-junction. In the parallel configuration ($\uparrow\uparrow$),
for Mo-Mo MTJ, the incident wave travelling from the left Mo capping layer and
the left FeCo-electrode will now travel entirely with $\Delta_{5}$ symmetries.
This $\Delta_{5}$ wave decays rapidly within the MgO-barrier [34]. After
tunneling through the MgO-barrier, this $\Delta_{5}$ wave tries to regain its
Mo-induced magnitude in the right FeCo-electrode in the majority-spin channel.
Therefore, the decay in FeCo-electrode (right) is relatively lower than that
of the MgO barrier. Similarly, as the $\Delta_{5}$ state is already dominating
in the right Mo capping layer, the propagating wave now transmits steadily
with $\Delta_{5}$ state. Therefore, successful transmission of $\Delta_{5}$
states occurs in the absence of $\Delta_{1}$ state along the
($001$)-direction. This is the origin of the zero-transmission hole (blue
circles) at $k_{\parallel}=0,0$ and relatively high-transmission around
$k_{\parallel}=0,0$ in Fig.3(g). Even then, a large conductance value is
obtained in the parallel configuration in Fig.2(a). This signifies that an
analysis of the conductance at $k_{\parallel}=0,0$ is imperative but not
sufficient to explain the dominant character of the large TMR ratio, because,
other contribution become equally crucial [34].

In the minority-spin channel ($\downarrow\downarrow$), the $\Delta_{5}$ state
is typically considered as dominant in FeCo-electrode, irrespective of the Mo-
induced crystallographic modifications. So, the incident wave now propagates
with a relatively higher magnitude of the $\Delta_{5}$ state than in the
majority-spin channel. This is reflected in the $\Delta_{5}$-dominated
minority-spin channel with stronger and wider resonant peaks at
$k_{\parallel}(k_{x},k_{y})$ in Fig.3(h). Relatively higher amplitude of
transmission around $k_{\parallel}=0,0$ are the cause of larger parallel
conductance in Fig.2(a) and the sharp but broader resonant peaks at
$k_{\parallel}(k_{x},k_{y})$ are the cause of large spin-polarization in
Fig.2(b). This leads to a comparatively larger TMR in Mo-Mo MTJ as compared
with that of the Ta-Ta and W-W MTJs.

When SOC is switched on, degenerated bands with, both, $\Delta_{1}$ and
$\Delta_{5}$ symmetries are lifted. Since, Mo is a non-SOC metal, SOC-induced
mixing of $\Delta_{1}$ and $\Delta_{5}$ states does not occur, but they co-
exist in the process of tunneling due to SOC. Mo-induced $\Delta_{5}$ state
remains dominant in the left FeCo-electrode with the increased SOC-induced
magnitude while the $\Delta_{1}$ becomes a secondary contribution. Within the
MgO-barrier, the $\Delta_{5}$ state decays rapidly while $\Delta_{1}$ state
decays slowly but still remains a secondary contribution. After tunneling
through the barrier, the $\Delta_{1}$ state becomes slightly dominant in the
majority-spin channel of the right FeCo-electrode. While $\Delta_{1}$ is
dominating in the right FeCo-electrode, $\Delta_{5}$ dominates in the right Mo
capping layer. Therefore, the transmission peak with distinct contributions of
the $\Delta_{1}$ state at $\Gamma$-point and the $\Delta_{5}$ state around the
$\Gamma$-point is observed in Fig.3(i). The reduction in TMR due to SOC could
be attributed to: (a) minor transmission of $\Delta_{1}$ state at
$\Gamma$-point and (b) dominance of different Bloch symmetries in both the
FeCo-electrodes resulting into larger GAP .

Figure 3: Transmission spectrum at the Fermi energy $E_{F}$ in 2D Brillouin
zone. Spin- and $k_{\parallel}$-resolved transmission probabilities for
(a)-(c) Ta-Ta, (d)-(f) W-W and (g)-(i) Mo-Mo MTJs are shown. $\Gamma$, X and M
in (a) indicate the high-symmetry points in ($001$)-plane along which the
spectrum is calculated. IRS in (b) indicates transmission due to interfacial
resonant states corresponding to minority-spin states. Spins are not
decomposed for transmission with SOC in (c), (f) and (i). $\Delta_{1}$ in (f)
and $\Delta_{5}$ in (g) indicate the highest and lowest probabilities of
transmission at $\Gamma$-point ($k_{\parallel}=(0,0)$). The bottom bar
indicates color coding for transmission coefficient plotted on a percentage
scale.

Unlike in the case of Mo, IrMn helps FeCo preserve its growth symmetry along
the ($001$)-direction [44, 45]. When SOC is not included, this leads to a
relatively stronger contribution of $\Delta_{1}$ state at the $\Gamma$-point
(see Fig.4(a)) in the majority-spin channel ($\uparrow\uparrow$) in the
parallel configuration. Due to the stronger impact of $\Delta_{1}$ states than
that of $\Delta_{5}$ state along ($001$)-direction, sharp but narrow
transmission peaks are observed in the minority-spin channel
($\downarrow\downarrow$) at $k_{\parallel}(k_{x},k_{y})$ (see Fig.4(b)). This
corresponds with the smaller lattice-mismatch-induced higher magnitude of the
spin-polarization in Fig.2(b). When SOC is switched-on, due to the strong SOC
in IrMn, indistinguishable mixing of $\Delta_{1}$ and $\Delta_{5}$ state
occurs. This is reflected in nearly the same magnitude (100%) and the broader
transmission peak at $k_{\parallel}=0,0$ ($\Delta_{1}$) and around
$k_{\parallel}=0,0$ ($\Delta_{5}$) in Fig.4(c). The influence of the anti-
ferromagnetic ordering of IrMn on the FeCo-electrode is also revealed from the
dissimilar transmission spectra of majority-to-minority Fig.4(d) and minority-
to-majority Fig.4(e) spins channel in the anti-parallel configuration.
Corresponding to the indistinguishable contribution of $\Delta_{1}$ and
$\Delta_{5}$ state, when SOC is included, both, the wave-function symmetries
decay with similar magnitude after propagating through the MgO-barrier. Hence,
relatively reduced distribution of SOC-induced transmission peak is obtained
in Fig.4(f).

Figure 4: Spin- and $k_{\parallel}$-resolved transmission probabilities across
IrMn-IrMn MTJ at the Fermi energy in 2D Brillouin zone. Transmission without
SOC and with SOC are shown for (a)-(c) parallel configuration and (d)-(f)
anti-parallel configuration. Spins are not decomposed for transmission with
SOC in (c) and (f). $\Gamma$, X and M in (a) indicate the high-symmetry points
in ($001$)-plane along which the spectrum is calculated. IRS in (b) indicates
transmission due to interfacial resonant states corresponding to minority-spin
states. The bottom bar indicates color coding for transmission coefficient
plotted on a percentage scale.

To further understand the SOC-induced influence of the IrMn capping layers on
the adjacent FeCo-electrodes, we study the scattering states ($|\Psi_{s}(z)|$)
at the Fermi energy (EF), in the parallel and anti-parallel configuration.
With the real-space analysis of the scattering state, one can understand the
path taken by the tunneling electron in different conduction channels. To
better understand and visualize the contribution of the Bloch symmetries, the
spin-decomposed distribution of SOC-induced scattering state is plotted in
Fig.5. At $\Gamma$-point ($k_{\parallel}=0,0$), the right incident spin-wave
traveling from the left IrMn capping layer enters the left FeCo-electrode and
tunnels through the MgO-barrier with an exponential decay in the PC
$\uparrow\uparrow$/PC $\downarrow\downarrow$ (Fig.(5(a)) indicating a direct
tunneling mechanism. As confirmed in Fig.(5(a), this scattering wave PC
$\uparrow\uparrow$ is composed of _d_ ${}_{z^{2}}$ orbitals of Ir, Mn (Fe, Co)
and _p_ z orbitals of O in the MgO-barrier. The high amplitude scattering
states in the left IrMn and the left FeCo-electrode and the extremely delayed
decay of these orbitals reveal the dominant contribution of $\Delta_{1}$ state
in the majority-spin channel of IrMn-IrMn. In PC $\downarrow\downarrow$
(Fig.5(a)), the wave is composed of only _d_ xz orbitals in IrMn (FeCo) layer.
The interfacial orbital hybridization at Fe-O interface is also observed.
Besides, the amplitude of scattering states, before and after propagating
through the MgO-barrier, is entirely identical, indication that the scattering
wave is governed by $\Delta_{5}$ state and that it is entirely transmitted.
This further reveals the influence of the SOC-induced co-existence of the
$\Delta_{1}$ and $\Delta_{5}$ states in transmission spectrum in Fig.4(c)
resulting into tremendously large TMR. Similar to the direct tunneling, the
transmission at IRS ($k_{\parallel}=(0.75,0.63)$) in Fig.4(b) is also
enhanced. The amplitude of the scattering wave is much larger around the MgO-
barrier in PC $\downarrow\downarrow$ (Fig.5(b)).This negligible attenuation of
scattering wave is caused by the resonant states (IRS) localized at the
interface. These states travel with the the help of evanescent states in the
MgO-barrier and hence the amplitude of the scattering wave within the MgO-
barrier considerably increases in PC $\downarrow\downarrow$. These resonant
states do not contribute to the conductance in majority-spin channel
($\downarrow\downarrow$) when SOC is not included. However, as spin-mixed
contribution from $\Delta_{1}$ and $\Delta_{5}$ state is obtained when SOC is
included, IRS equally contributes to the tunneling conductance. Hence, the
larger G$\\_P$ (Fig.2(c)) than that of G${}_{P}^{\uparrow\uparrow}$ is
achieved (Fig.2(a)). As observed in Fig.5(a,b) the contribution from the FeCo-
electrodes (Left/Right) and the IrMn capping (Left/Right) to the scattering
wave is precisely of the same magnitude, which indicates the significant
impact of an AFM-capping layer on an adjacent FM-electrode. This is also
evident from the unequal amplitudes of the scattering waves in
$\uparrow\downarrow$ and $\downarrow\uparrow$ of anti-parallel configuration
(APC) in Fig.5(c). Despite rotating the spins of only right FeCo-elctrode in
anti-parallel configuration, the spin-alignment of the right IrMn capping
layer also appears to have changed. This results in relatively lower anti-
parallel conductance (Fig.2(c)) leading to an enhanced SOC-induced TMR.

Figure 5: Real-space distribution of spin-decomposed scattering states
$|\Psi_{s}(z)|$ across the IrMn-IrMn MTJ, under equilibrium, at the Fermi
energy. The scattering states are calculated at different $k_{\parallel}$
points: (a),(c) $k_{\parallel}=(0,0)$ and (b) $k_{\parallel}=(0.75,0.63)$ The
values of $k_{\parallel}$ are given in the units of $2\pi/a$. In (a)
PC$\uparrow\uparrow$ resembles $\Gamma$-point in Fig.4(a) indicating ’direct
tunneling’ mechanism. While in (b), $PC\downarrow\downarrow$ indicates
’Resonant tunneling’ governed by IRS as shown in Fig.4(b). (c) shows
asymmetrical traversing of spin-wave in anti-parallel configuration at
$\Gamma$-point. The dashed vertical lines mark, roughly, the interfaces. The
bottom bar represents color coding for logarithmic scaled $|\Psi_{s}(z)|$ when
the spin-wave travels along the direction of transport (z-direction).

##  IV Conclusion

In conclusion, we demonstrate the effect of employing an anti-ferromagnetic
capping layer on the TMR in X/FeCo/MgO/FeCo/X symmetric-MTJ where X represents
Ta, W, Mo or IrMn.

## Acknowledgement

B. Dupé and P. Buhl acknowledge funding by the DFG under Grant No. DU
1489/3-1.

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