ARTICLES

Sci. Rep. Kanazawa Univ.
Vol. 62, pp. 23–35, 2018

Perpendicular and in-plane magnetic anisotropies
in Cr-buffered Fe(x)/MgO interface
from first-principles approach
Nurul I KHSAN1,2∗, Tomosato K ANAGAWA1 , Indra PARDEDE1,3 , Daiki YOSHIKAWA1 ,and Tatsuki O DA1,4
1 Graduate School of Natural Science and Technology, Kanazawa University,

Kanazawa, 920-1192, Japan
2 School of Computing, Telkom University, Bandung 40257, Indonesia
3 Department of Physics, Institut Teknologi Sumatera, Lampung 35365, Indonesia
4 Institute of Science and Engineering, Kanazawa University, Kanazawa, 920-1192, Japan(Received June 25, 2018 and accepted in revised from August 23, 2018)

Abstract Electronic structure and magnetic anisotropy energy (MAE) of Cr-buffered
Fe/MgO interface were investigated in terms of thickness dependence by means of a fully
relativistic density functional approach. The electronic structure indicated that the interface
state gets occupied unlike a typical rigid band picture as the number of Fe layers decreases,
leading large perpendicular anisotropies in the oscillating behavior for its thickness dependence. The maximum of the MAE originating from spin-orbit interaction (SOI) reaches
2mJ/m2 . It was found that the saddle point nature and dispersionless at the Fermi level may
be an origin of its large perpendicular magnetic anisotropy. To evaluate the MAE originating from SOI, the in-plane shape magnetic anisotropy originating from magnetic dipole
interaction (MDI) was estimated. The calculated MAE from MDI required a suppression of
the calculated total magnetization for realizing a fair agreement with an experimental MAE.
With using a rescaled magnetization it was found that a possible perpendicular anisotropy
may appear in the thickness range of experimental data.
Keywords.

1

magnetic anisotropy, magnetic thin film, Fe/MgO interface, MRAM

Introduction

Metal-insulator interface has been attractive in material science for the last decade. The interfaceof magnetic-oxide, such as Fe/MgO family, holds an important role in the development of magnetoresistive random access memory (MRAM) [1–4]. In the early development stage the modulationof the ferromagnetic free layer is driven by the spin transfer torque mechanism [5, 6]. The challenge is not only the magnetization control, but also large tunnel magnetoresistance [7] and low
damping constant [8]. These three are required to realize MRAM applications.
∗ Corresponding author

Email: ikhsan@cphys.s.kanazawa-u.ac.jp

24

Perpendicular and in-plane magnetic anisotropies in Cr-buffered Fe(x)/MgO interface

Cr

Fe

O

Mg

Figure 1: Schematic diagram of the slab systems; Cr(6ML)/Fe(xML)/MgO(5ML) for x = 5.

Subsequent advancement of the magnetization control by electric field effect on this interface
is discovered [9]. Later, it can be controlled by spin-orbit torque (SOT) [10], and the combinationof voltage control and spin-Hall effect of the underlayer, which is also known as voltage-controlspintronics-memory (VoCSM) [11]. Even now the evolution of read-write mechanism still develops, but the fundamental object remains the same Fe/MgO interface. The unique characteristics,
like perpendicular magnetic anisotropy (PMA) and spin transfer (voltage, spin torque or VoCSM)
controlled magnetic junction, are promising for information technology device, especially for highdensity and low power consumption data storage [12]. Recently, the unique spin structure with
voltage controlled Dzyaloshinskii-Moriya interaction was also discovered in this interface [13].
Such multi-functional interface has a lot of interesting physical properties and potential applications for spintronics or magnonic devices [14].
In the works for the development of the devices, the family of Fe/MgO interfaces has been used
as a kernel technological element. They have shown a strong perpendicular magnetic anisotropy
for the thin Fe without any heavy element [7, 15]. The multi-functional properties, mentioned in
the previous paragraphs, are mostly originated from the electronic structure. In such system, theinterface state has been discussed in the several works [16, 17]. These states are consequences ofthe band formation consisting of non-bonding orbitals on the interface. Although such characterhas been observed as interesting characteristics, detail dependence of electronic structure has notbeen investigated.
The recent improvement of computational performance allows us to estimate the magnetic
anisotropy or its EF effect precisely and numerically [18–20]. Such improvement contributes not
only to physical and qualitative explanations in the property of magnetic anisotropy, but also tosemi-quantitative agreements. In particular, the slope in the EF variation has been proved to have arealistic meaning, when compared with the experimental results [21]. On the magnetic anisotropy
energy (MAE), the comparison with experiment has a distance from explaining the experimental
measurements with a quantitative agreement. The experimental progress on the interface magnetic
anisotropy in the thin films gives us a fascinating opportunity on a direct comparison between thetheoretical and experimental approaches.
This work was devoted to the discussions on electronic, magnetic, and structural properties ofFe(x ML)/MgO(001), as a reinvestigation in the viewpoint of two dimensional electronic structure.
We obtained remarkable Fe-thickness dependences of MAE, implying a picture of non-rigid bandfilling in the interface states. These results are discussed in terms of electronic band theory, compared with the available experimental data. In the comparison, we also discuss the computational
accuracy on MAE within the current state-of-the-art for density functional theory (DFT).

2

Model and method

In order to investigate the interface of Fe/MgO, we consider a slab system, vacuum (0.79 nm)/Cr(wML)/Fe (x ML)/MgO (5 ML)/vacuum (0.79 nm) (ML=atomic monolayer), in the computation

N. I KHSAN, T. K ANAGAWA, I. PARDEDE, D. YOSHIKAWA, and T. O DA

25

MAE(SOI) (mJ/m2 )

1.4
1.2
1
0.8
0.6
2

3

4

5 6 7
t Cr (ML)810Figure 2: Thickness dependence of Cr layer on MAE in Cr/Fe(3ML)/MgO.

(Fig. 1). The atoms are specified by number as Cr(1), Cr(2), ..., Fe (1), Fe (2), ..., O (1), ..., Mg(1), ... etc. from the Fe/MgO interface. At this interface, an Fe atom was placed just next to the O
atom due to its stability, and in the Cr and Fe layers the body-centered layer-stacking sequence wasused [22, 23]. The in-plane lattice constant extracted from bulk Cr was employed. The thicknessof Fe thin layer (x) was varied from 1 ML to 10 ML.
We carried out first-principles density functional calculations, which employ fully relativistic (with spin-orbit interaction) and scalar-relativistic (without spin-orbit interaction) ultrasoft
pseudopotentials and planewave basis [24, 25], by using the generalized gradient approximation [26]. In these calculations, the smearing of Fermi-Dirac function with 315 K was introduced as in the previous works [27, 28]. As the result, the total energy Etot , which has a variational property in the occupation number as well as in the wavefunction and the atomic coordinate. The MAE originating from spin-orbit interaction (SOI) was estimated from the totalenergy difference between the different magnetization directions [100] (x-axis) and [001] (z-axis),
MAE(SOI) = E[100] − E[001], where [001] specifies the direction of film thickness (right-handside in Fig. 1). We used the 32 × 32 × 1 mesh of k point sampling [29] in MAE estimation [30].
Using the scalar-relativistic level computation, in which a k-mesh of 24 × 24 × 1 is used, we
induced structural relaxation while keeping both the in-plane lattice constant and the atomic coordinates of O(3). The MAE from the shape anisotropy, MAE(MDI), was estimated using themagnetostatic dipole interaction (MDI) and assuming the atomic magnetic moments obtained by
the density functional calculation [31]. In our work, the thickness of Cr layer was set to 6 ML
(w = 6). Fig. 2 shows the thickness dependence of MAE for the Fe 3ML (x = 3) system with thein-plane lattice constant extracted from Fe bcc bulk which is very similar to that of Cr bulk. In
Fig. 2 the MAE is almost independent of Cr thickness within the accuracy of 0.1mJ/m2 for a given
Fe thickness. Note that the contribution of Cr surface with vacuum to the MAE(SOI) is negligible.
This was confirmed in a free standing Cr layer.

3

Results

3.1

MAE(SOI)

We report the thickness dependence of MAE(SOI) in Fig. 3. It indicates an oscillating perpendicular anisotropy with respect to Fe thickness (x), and the maximum of 2.0 mJ/m2 at Fe 2ML(x = 2)26

Perpendicular and in-plane magnetic anisotropies in Cr-buffered Fe(x)/MgO interface

2.2
2
1.8

MAE (mJ/m2 )

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

2

4

6
t Fe (ML)810Figure 3: Thickness dependence of the magnetic anisotropy energy (MAE) from spin-orbit interaction in
Cr/Fe(xML)/MgO. The bullet indicates the experimental value (ref. [4]), where 1ML thickness is assumed
to be 0.142 nm.

and other maximal values at 5ML(x = 5) and 7ML(x = 7). The behavior shows that an odd-evenalternating oscillation at the thicker systems (4ML-10ML). For the thinner systems, the amplitude of oscillation is largely enhanced since large changes are expected in the electronic structurearound the Fe/MgO interface. Hallal et al. also reported an oscillating MAE as a function of
Fe thickness [20]. In their system, the thickness varies from 5ML to 13ML in a multi-layer ofFe/MgO with a double interface, while our slab model has the single Fe/MgO interface and more
realistic in accordance with the experimental measurement [4].

3.2

Fe 2ML case

The maximum MAE at Fe 2ML is much larger than the previous theoretical and experimental
values in the Fe/MgO interface family, comparable to the interface contribution extracted from theextrapolation fitting in the experiment [32]. At Fe 5ML our value agrees well with the experimental
value [4].
Figure 4 shows the projected density of states (PDOS) at the interface in x = 2 and band
dispersions with 3d orbital components in the vicinity of the Fermi level (EF ). As shown, the EF
is located between the two peaks of PDOS consisting of 3d orbitals. These electronic states are
relatively localized since the main orbital components are made of the non-bonding 3d-orbitals.
Indeed, the states form a flat band around the k-point k1 = π/a(1/2, 1/2) (see Fig. 4) and a saddle
point near k1 in two-dimensional Brillouin zone (2DBZ). The band flatness appears along X̄–Ȳ
line in 2DBZ. These features are remarkably observed for both the occupied and unoccupied bands
in Fig. 4(c). There is a saddle point nature around k1 in 2DBZ (not exact of saddle point). AlongX̄–Ȳ, there is a maximum at k1 in the 3d-orbital band just above EF , and simultaneously alongΓ̄–M̄ a minimum near k1 .
This feature is the origins of sharp PDOS peaks in the interface states, appearing more orless in the Fe/MgO and its family systems. However, for realizing such features, there may be a
combination of two conditions. The one is an appropriate orbital hybridization between 1st Fe and2nd Fe layers. This keeps splitting the mixed eigenstates of dxz and dyz components at M̄ point to
the lower and higher eigen energies, while in the Fe 1ML system, those stays remain on or aroundEF [33]. The 2nd condition is also an orbital hybridization between Fe d3z2 −r2 and O pz . This keeps

27

N. I KHSAN, T. K ANAGAWA, I. PARDEDE, D. YOSHIKAWA, and T. O DA(b)total 3ddxz+yz

1

1

0

0
dxy

−1

−1

1

1

0

0
dx2−y2

−1

−1

1

1

0

0
d3z 2−r2

−1
−3

−2

Ȳ

M̄

Γ̄

X̄

k1

0.6 (c) x =2
0.4
Energy(eV)

1.5
1
0.5
0
−0.5
−1
−1.5

Number of electrons/atom

DOS(States/eV/cell)(a)x = 2
3
2
1
0
−1
−2
−3

0.2
0
−0.2
−0.4

−1
−1

0
1
Energy(eV)

2

3

−0.6

Γ

M

X

ΓX
YΓ
M
Wave vectors k

X

ΓX

Y

Figure 4: Electronic structures at the Fermi level in Cr/Fe(2ML)/MgO without including spin-orbit interaction, (a) projected densities of states (PDOS)(form the top) 3d total dxz+yz , dxy , dx2 −y2 ,d3z2 −r2 components,(b) selected k-point path, and (c) band dispersion for the Fe 3d orbitals at the interface Fe(1). The colors ofgrey, orange (yellow), blue, green, and red are used for indicating the components of total 3d orbital, dxz+yz ,dxy , dx2 −y2 , and d3z2 −r2 , respectively. The light blue curve in (a) indicates integrated number of electrons ofeach orbital component.

the d3z2 −r2 away from EF , not disturbing the localized states of non-bonding dxy , dx2 −y2 , dxz , anddyz at EF . The latter has been well known as one of important origins for realizing perpendicularanisotropy [15]. This is because the orbital d3z2 −r2 always contributes only to in-plane magnetic
anisotropy, assuming that the contribution from the majority spin state can be neglected due to alarge exchange splitting [33]. The MAE maximum in Fe 2ML is obtained as the consequences ofthe origins discussed above. Note that there are vertical couplings of SOI around k1 area in 2DBZ,
which contribute to perpendicular anisotropy; couplings of dxy –dx2 −y2 and dxz –dyz .

3.3

Fe 5ML caseIn x = 5, such dispersionless states is more clearly observed just above EF (0.1 eV in Fig. 5(b)), theband changes upward along Γ̄–M̄ and slightly downward along X̄–Ȳ. In the more real-life scenario
of x = 5 configuration in Fig. 5, the formation of interface state is also observed, but unfortunately
the vertical coupling of dxy –dx2 −y2 and dxz –dyz at k1 is already shifted away from the Fermi level tohigher energy. Therefore we have lower value in the magnetic anisotropy energy, but we still havethe coupling of dxz –dyz orbital at middle point of X̄–Γ̄ and Γ̄–M̄ which is contribute significantly
to the perpendicular magnetic anisotropy.

28

Perpendicular and in-plane magnetic anisotropies in Cr-buffered Fe(x)/MgO interface

total 3ddxz+yz

1

1

0

0
dxy

−1

−1

1

1

0

0
dx2−y2−1

1

00d3z 2−r2
−3

−2

0.4
0.2
0
−0.2
−0.4

−1

1

−1

0.6 (b) x =5

Energy(eV)

1.5
1
0.5
0
−0.5
−1
−1.5

Number of electrons/atom

DOS(States/eV/cell)(a)x = 5
3
2
1
0
−1
−2
−3

−0.6
Γ

M

X

ΓX
YΓ
M
Wave vectors k

X

ΓX

Y

−1
−1

0
1
Energy(eV)23Figure 5: Electronic structures at the Fermi level in Cr/Fe(5ML)/MgO without including spin-orbit interaction, (a) partial density of states (PDOS), (b) band dispersion for the Fe 3d orbitals at the interface Fe(1).
The colors of grey, orange (yellow), blue, green, and red are used for indicating the components of total 3dorbital, dxz+yz , dxy , dx2 −y2 , and d3z2 −r2 , respectively. The light blue line in (a) indicates integrated numberof electrons of each orbital component.
0.08

total 3ddxz+yzdxydx2 −y2d3z 2 −r2

Number of electrons

0.06
0.04
0.02
0
−0.02
−0.04
−0.06

2

3

4

5
6
7
tFe (ML)8910Figure 6: Thickness dependence of the number of 3d electron on the interface Fe. The relative values withrespect to those of Fe 10ML are shown. The colors of grey, orange (yellow), blue, green, and red are used
for total 3d orbital, dxz+yz , dxy , dx2 −y2 , and d3z2 −r2 , respectively.

3.4

Thickness dependences of interface electronic structure and number of electrons

Thickness dependence in MAE has a relationship with the number of electrons (NOE) in the 3dorbital of minority spin state on the interface Fe. In the large variation range of MAE (< Fe 4ML),
the NOE decreases as thickness, and in the odd-even alternating range (≥ Fe 4ML) its NOE does
not synchronize so much with the MAE, as shown in Fig. 6. For the former, the number of Fe

29

N. I KHSAN, T. K ANAGAWA, I. PARDEDE, D. YOSHIKAWA, and T. O DA(a)x = 13(b)x = 2(c)x = 3(d)x = 4(e)x = 5(f)x = 6(g)x = 9dxz+yz2

DOS (States/eV/cell)

1

0

dxy
1

0.5

0

dx2 −y2
10.5

0−0.3 0 0.3−0.3 0 0.3−0.3 0 0.3−0.3 0 0.3

Energy (eV)−0.3 0 0.3−0.3 0 0.3−0.3 0 0.3

Figure 7: Projected density of states (PDOS) for the interface state in Cr/Fe(xML)/MgO; (a)–(g) for x =1 − 6 and 9, respectively. The top, middle, bottom rows in the panels show the components of dxz+yz , dxy ,and dx2 −y2 .

works as hole doping, while in the latter the MAE may be influenced by the details of electronic
structures at the interface. Interestingly, note that the NOEs of dxy and dx2 −y2 show an alternating
behavior in their relative values (See Fig. 6). Indeed, the alternating nature appears at the dxycomponent in the band dispersions (not shown) and PDOS(see Fig. 7).
The number of 3d electrons at the interface Fe is summarized in Fig. 6. The total number of 3delectron decreases as the thickness increases until Fe 4ML. This decrease is in accordance with therelationship of electronegativity between Fe and Cr. The electrons on Fe of larger electronegativity
gradually increases as the thickness of Fe increases from Fe 4ML.
Figure 7 shows the series of PDOS for interface states. Except for x = 1, the sharp peaks
appear, implying the localized nature of wave functions. It is worthy to notice that the series of
x = 2 − 5 do not show a simple rigid band filling. The shape of dxy component changes while thatof dx2 −y2 is kept without any large change. In details, the peaks of dxy and dx2 −y2 are located justbelow and above the EF level, respectively, in Fe 2ML, and as the thickness the peaks of dxz+yz
moves to higher energies with that of dx2 −y2 . Additionally, it is interesting to see the PDOS peak
sharpen in the thicker systems (see the case of x = 9). The origin of interface states energy shift
is speculated as an orbital hybridization with the Cr underlayer. At the Cr/Fe interface, the 3dorbital components at EF was shifted due to the hybridization to a higher energy for Cr and to alower energy for Fe. This realizes the electron transfer from Cr to Fe atoms, showing a property ofsmaller electronegativity for Cr. Consequently, the 3d orbital on the Fe of Fe/MgO interface canget electrons as the thickness of Fe layer decreases. Indeed, it was observed clearly in the Fe layer
with small x’s as a vicinity effect of Cr. Interestingly, the similar energy lowering of IS occurs as
the decrease of in-plane lattice constant [34], inducing the modulations in MAE.

30

Perpendicular and in-plane magnetic anisotropies in Cr-buffered Fe(x)/MgO interface

3

Magnetic Moment (µB)

2.8
2.6
2.4
2.2
Fe(1)
Fe(2)
Fe(3)
Fe(4)
Fe(5)
Fe(6)
Fe(7)
Fe(8)
Fe(9)
Fe(10)

2
1.8
1.6
1.4

1

2

3

4

5
6
t Fe (ML)

7

8

9

10

Figure 8: Atomic magnetic moment of Fe in the ferromagnet layer, Fe(1) are located at the interface of
MgO.
2.5
2

Magnetic Moment (µB)

1.5
1
Cr(1)
Cr(2)
Cr(3)
Cr(4)
Cr(5)
Cr(6)

0.5
0
−0.5
−1
−1.5
−2

1

2

3

4

5
6
t Fe (ML)

7

8

9

10

Figure 9: Atomic magnetic moment of Cr in the ferromagnet layer, Cr(1) are located at the interface with
Fe.

3.5

Variation of the atomic magnetic moment

We report the results of atomic magnetic moments obtained by integrating the spin density in theatomic sphere with the radius (Cr: 0.90 Å, Fe: 0.90 Å), as shown in Fig. 8 and Fig. 9. As generaltrends the Fe magnetic moment is enhanced (2.8 µB ) and reduced (2.3 µB ) at the Fe/MgO andCr/Fe interfaces, respectively, compared with those of the inside layers (2.5 µB ). In the Cr layer,
the magnetic moments couples also antiferromagnetically between the neighboring layers. These
configurations are similar to the previous theoretical work on the Fe(001)/MgO/Cr/Fe magnetic
tunnel junctions [35]. Note that the magnetic alignment related with the Cr atoms has not been
known in the thin film materials (devices). To bulk Cr, an incommensurate antiferromagnetic spindensity wave has been known for a long time [36] and the interface with Fe may possibly show anoncollinear magnetic configuration [37].
As shown in Fig. 9, the magnetic moment on Cr(6) is larger than those of the other Cr MLs,
because this atom faces to vacuum and the electron wave function is more localized near thevacuum. This feature contributes to larger magnetic moment.

Total Magnetic Moment (µB )N. I KHSAN, T. K ANAGAWA, I. PARDEDE, D. YOSHIKAWA, and T. O DA

25

31

8
6

20

4
2

15

0

1

2

3

10
Fe onlyplus Cr
experiment

5
0

2

4

6
tFe (ML)8

10Figure 10: Thickness dependence of magnetizations, compared with the available experimental data(ref.
[32]). Plus symbol indicates total magnetizations contributed from Fe atom only, solid box includes thecontribution of Cr, bullet indicates experimental data.
0

MAE(MDI) (mJ/m2 )

−0.5
−1
−1.5
−2
−2.5
2

4

6
t Fe (ML)810Figure 11: Theoretical MAE(MDI) (yellow triangles), normalized MAE(MDI) (blue reverse triangles), andexperimental MAE(SA) (red dots), the dashed lines are only guide.

3.6

Total magnetizations

In the theoretical approach we obtained the magnetic dead layer (MDL) with 0.053 nm thick at
most, while the experiment clearly showed the MDL with 0.1 nm thick [32]. The theoretical magnetization is larger than the experimental one by about 24 % at the Fe 5 ML, as shown in Fig. 10.
These differences could be ascribed to an alloying effect at the Cr/Fe interface in the experiment.
This allows to draw a suppression of the in-plane anisotropy from MDI contribution, or alternatively an enhancement in the strength of interface perpendicular anisotropy. The nearest neighborpairs of Cr–Cr and Fe–Cr tend to enhance the antiferromagnetic coupling between the magnetic
moments, and possibly inducing a noncollinear magnetic structures due to a competition in magnetic couplings [37]. Associated with such complexity in magnetic configuration, the MAE(SA)modulates due to the change in MDI.

32

Perpendicular and in-plane magnetic anisotropies in Cr-buffered Fe(x)/MgO interface

2.5
2

MAE (mJ/m2 )

1.5
1
0.5
0
−0.5
−1
−1.5
−2
2

4

6
t Fe (ML)810Figure 12: Total magnetic anisotropy energy (MAE), compared with the experimental data (bullets)(ref.
[32]). The open yellow triangle symbol specifies the data estimated theoretically; MAE(SOI)+MAE(MDI),
the reverse blue triangle symbol represents the MAE(SOI+MDI) after the 26% reduction of theoreticalatomic moment, and the red cross symbol represents the sum of the theoretical MAE(SOI) and the shapeanisotropy MAE(SA) estimated using the experimental magnetization.

3.7

Shape anisotropy

The shape anisotropy is realistically important for determining the whole magnetic anisotropy.
When we compare our MAE(SOI) with those from measurements, the shape anisotropy needs tobe estimated precisely. Here, we estimated the shape anisotropy energy, MAE(SA), in two ways;
from the MDI [31] using the theoretical atomic magnetic moments and the estimation (−µ0 Ms2 /2)
using the experimental saturation magnetization (Ms ) [32]. In the former, all the Cr layer were also
taken into account. This shape anisotropy contribute to the in-plane component, large negativevalue means more reduction to the MAE(SOI). So we could observe that the calculation frompure dipole-dipole interaction [31] provide much reduction compared to the one extracted fromexperimental ones [32]. Just as described in Fig. 10, the estimation of total magnetization are
overestimate experimental value, therefore we consider the case of normalized ones. Later we canuse this value as the corrected factor of the MAE(MDI).

3.8

Total MAE

Figure 12 reports the total MAE, namely summation of MAE(SOI) and MAE(SA), in comparison
with the available experimental data. This figure shows that our estimation is much reduced due
to large in-plane contribution from the MDI. As shown in Fig. 11, the experimental MAE(SA) is
much reduced in absolute, compared with the theoretical one. This is an origin why we have somuch reduced value. By considering the reduced MDI (rMDI), we could obtain that the MAE(SOI+rMDI) describe in the blue reverse triangle. In this result we could observe an agreementwith the experimental data [32].

4Discussions

The obtained MAE(SOI) result can be compared to Bruno relation [38] in order to confirm thevalidness of our calculation results. This relation can describe MAE in the ferromagnetic transition
metals like Fe, Co, Ni. In Fig. 13, we can observe that in even number of ferromagnet layers theN. I KHSAN, T. K ANAGAWA, I. PARDEDE, D. YOSHIKAWA, and T. O DA

2.2

33

MAE(SOI)
MAE(Bruno’s)
MAE(Bruno’s Fe(1))
MAE(Bruno’s Fe(2))

21.8

MAE (mJ/m2 )

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2

1.5

2

2.5

3

3.5
t Fe (ML)

4

4.5

5

5.5

Figure 13: The estimation from Bruno’s relation compared to the calculation of MAE from SOI. Blacksolid square represent MAE from SOI, red cross symbol are MAE calculated from Bruno relation, blue starsymbol are the contribution from Fe(1) at the interface, and orange empty square symbol are contribution
from Fe(2).

MAE(SOI) is close to the value estimated by assuming 62 meV to the spin-orbit coupling constantin the Bruno relation. This much underestimates the MAE for the odd number cases. This relation
may not be filled for the complex system and the system with a large spin orbit coupling constant.
In another way, this model is too simple for the cases of odd ML. Still, we could expect thesame out-of-plane tendency for these numbers of ML in Fig. 13. We could consider that the main
contribution of magnetic anisotropy energy comes from the first layer of interface iron with the
MgO. The contribution of nearest neighbor is a constant around 0.2 mJ/m2 , and the contribution
from next nearest neighbor is almost zero. Therefore, the latter is not included in Fig. 13.
In the case of consideration when we decrease the thickness of Fe layer, for example become
2ML, the ferromagnetic state may become unstable against thermal disturbance. This could bedenied in a discussion of Cr–Fe magnetic interaction. The Fe at the Cr/Fe interface has a strong
antiferromagnetic interaction of Heisenberg type. This stabilizes the ferromagnetic state of Fe thinlayer. The previous works reported large exchange interactions of J = 69 meV and |J|=59, 20 meV
for Fe-Fe and Cr-Fe nearest neighbors, [37, 39, 40] respectively. These values could be enough tomaintain the ferromagnetic states at room temperatures. However, practically, associated with theexistence of magnetic dead layer in such system, [32] Fe atom may diffuse into the Cr layer during
the fabricating process at high temperatures. Indeed, when exchanging the Fe and Cr atoms at the
Cr/Fe interface in Fe 4ML, the total energy does not become so higher (50 meV/in-plane Fe).
As shown in Figs. 4 and 7 the interface state discussed here is presented just as some sharp
peaks in this work. The main body of states is on EF or just above EF by less than 0.2 eV. Theseenergy levels are located at the same energy range in which the enhanced magnetic tunnelingspectra appears [4]. The interface states in the present work may have the potential to make a state
resonating with the conduction electron through the s−d interaction at the interface [41]. Indeed,
the enhanced spectra in the spin-dependent tunnel conductance has been observed, implying some
unknown/non-resolved mechanism behind [2, 16].
The Fe layer, forming a quantum well structure, is terminated at the Fe/MgO and Cr/Fe interfaces. At the edges, the amplitudes of wave function have a feature of odd-even alternating
behavior in the one dimensional model chain of which the atomic site is connected with electron transfer integral. The dxy component at the Fe/MgO interface is largely affected due to its
non-bonding nature, and can be sensitive to external perturbation from the other edge of Cr/Fe

34

Perpendicular and in-plane magnetic anisotropies in Cr-buffered Fe(x)/MgO interface

interface. The electron transfer between the dxy orbitals of neighboring Fe MLs is proportional
to Txy,xy (k) ∼ eik·R {3t(ddσ ) + 2t(ddπ) + 4t(ddδ )}/9, and is not negligible because the absoluteof transfer integral can reach to about 0.1 eV [33]. Consequently, as an edge effect of the model
chain, some non-negligible changes of dxy are supposed to appear at the interface states.

5

Summary

We performed first principles electronic structure calculations for the interface systems with Cr
underlayer of Fe/MgO interface and estimated the MAE originating from the SOI and MDI. The
exotic oscillating behavior was observed in the MAE from SOI. We found that the DFT approach
can describe the MAE with good accuracy, compared with the experimental data. Our calculationshows that the MAE(SOI) in 5ML system with 1.5 mJ/m2 is comparable to the experimental ones.
The series of total MAE’s indicated perpendicular and in-plane magnetic anisotropies in the Crunderlayer Fe/MgO. The reduction by the in-plane shape anisotropy energy was considered as thecorrection factor to the pure DFT calculation approach. By introducing the rescaled magnetization
similar to the experimental one, the thickness range of perpendicular magnetic anisotropy was
found to correspond to experimental result. From the electronic structure, the dispersionless andsaddle point natures appearing in the band dispersions around k1 were found to play the importantrole in the perpendicular anisotropy. In the Fe 2ML, particularly, the flat bands consisting of dxz ,dyz , dxy , dx2 −y2 are located just below and above the Fermi level. These bands contribute to thelarge MAE(SOI) of 2 mJ/m2 . The change of interface states was discussed as the origin of change
in the perpendicular magnetic anisotropy. The detail electronic and magnetic properties of the Cr
buffered Fe/MgO slab systems might be useful in future materials design for such multi-functional
interfaces.

Acknowledgement
The first-principles calculations were performed using the facilities of the Supercomputer Center,
Institute for Solid State Physics, the University of Tokyo, Japan. This research partly used computational resources of the K computer and other computers of the HPCI system provided by the
AICS and Information Technology Center of Nagoya University through the HPCI System Research Project (Project ID:hp160227, hp160107, hp170168). This work was partly supported by
the ImPACT Program of Council for Science, Technology and Innovation (Cabinet Office, Japan
Government), and by the Computational Materials Science Initiative (CMSI), Japan. The authors(N.I and I.P.) acknowledges Japanese Government (MEXT) Scholarship in the Program for theDevelopment of Global Human Resources for Kanazawa University.

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