ARTICLES

Sci. Rep. Kanazawa Univ.
Vol. 62, pp. 37–49, 2018

Fermi level smearing effect in the perpendicular interface
Fe/MgO magnetic anisotropy
Nurul I KHSAN1,2∗, Indra PARDEDE1,3 , Tomosato K ANAGAWA1 , Daiki YOSHIKAWA1 ,and Tatsuki O DA1,4
1 Graduate School of Natural Science and Technology, Kanazawa University, Kakuma, Kanazawa,

Ishikawa 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, Kakuma, Kanazawa, Ishikawa 920-1192,

Japan(Received June 25, 2018 and accepted in revised form August 28, 2018)

Abstract Effects of the Fermi level smearing introducing temperature dependence ofenergy-band occupation numbers were investigated in terms of magnetic anisotropy by using a density functional theory. In a perpendicular magnetization film Fe(0.7nm)/MgO, themagnetocrystalline anisotropy shows a decrease with respect to the smearing temperature;
0.4mJ/m2 from 11K to 527K. This reduction is not negligible and is expected to partiallycompensated by a reduction in the shape magnetic anisotropy, assuming reasonable valuesof the Curie temperature and saturated magnetic moment. The resulting temperature dependences in the total perpendicular magnetic anisotropy energy is in agreement with those ofavailable experimental data semi-quantitatively.
Keywords. Magnetocrystalline anisotropy, Shape anisotropy, First-principles electronic
structure calculation, Density functional theory

1

Introduction

Perpendicular magnetic anisotropy (PMA) has an important role for designs of better devices ofspin transfer torque recording magnetoresistive random access memory (STT-MRAM) [1, 2]. In
the approaches of both theory and experiment, a lot of progress has been made for developing functionals or improving performances [3–12]. The properties of temperature dependence in PMA are
highly requested for designing ferromagnetic magnetic materials. Recently, the experimental workby Xiang et al. showed a temperature effect of perpendicular magnetic anisotropy. This effect is areduction of 0.15 mJ/m2 in the case that the temperature decreases form 300K to 10K [10]. This
work motivates us to investigate temperature effects by theoretical approach. Up to now, there
are many works of theoretical approach for the systems of localized magnetic moments (single
ion-magnetic anisotropies) [13–15]. In these approaches, the magnetic anisotropy energy (K) was∗ Corresponding author

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

38

Fermi level smearing effect in the perpendicular interface Fe/MgO magnetic anisotropy

treated as a cubic polynomial function of magnetization (M). For the metallic epitaxial films, asquare function was employed at the low temperature [16,17]. The theoretical approaches showedthat K is proportional to M 2 in case of L10 -ordered FePt [18, 19]. The recent density functional
approach to disordered magnetic bulk alloys explains an anomalous temperature dependence ofmagnetocrystalline anisotropy [20].
For the design of materials in emerging nanoscale memory and logic device, Alzate et al.
showed that in the system MgO/CoFeB/Ta-based MTJ [21], M as temperature dependence fitted
well with the Blöch law (T 3/2 ) [22]. In addition to this, K as temperature dependence fitted wellwith a power law of M 2 or similar one. Wen et al. also showed a similar behavior of temperaturedependence on M and K in Ru/Co2 FeAl/MgO-based MTJ [23]. For the thinner films of several
iron monolayers, the temperature dependence of M was implied to change from T 3/2 to T 2 [24].
In the materials for perpendicular-MRAM devices, the property of PMA is mainly ascribed tospin-orbit coupling (SOC) in the metallic electronic structure. The magnetocrystalline anisotropy
for metal has been estimated successfully since the 1980’s using density functional approaches.
The PMA of thin films has qualitatively or semi-quantitatively been explained. This is a consequence of the fact that the PMA from SOC overcomes the magnetic shape anisotropy (SA) whichfavors in-plane magnetization. The latter contribution to magnetic anisotropy has been investigated for a long time, including its temperature-dependent property. However, the temperaturedependence of the magnetic anisotropy caused by metallic energy bands has not been investigatedvery well, particularly for thin film systems.
In the present work, we investigated smearing effects on the Fermi level in terms of themagnetic anisotropy energy of the metal slab system for magnetic devices, by means of a firstprinciples calculation. These effects are expected to contribute to a temperature dependence in themagnetic anisotropy caused by SOC. In combination with SA analyses, a saturating behavior in K
at room temperatures may be comprehended, compared with the available experimental data [10].

2

Method2.1

General theory

Finite temperature effects can be taken into account in the band-energy by the Mermin’s approach [25]. In such kind of approaches a set of temperature-dependent electron occupations is
introduced. Using an analogy of quasi-one-particles, the total Hamiltonian of system H can bedivided into two parts; H = H0 + ∆H, where H0 and ∆H represent quasi-one-electron’s part andthe other part, respectively. Schrödinger equation for H0 is represented as follow: H0 ϕi = εi ϕi ,where εi and ϕi are the eigenvalue and eigenwavefunction. ∆H contains all interactions between
quasi-one-electrons. The free energy of system at the temperature T is expressed as follows:
Ftot =∑ fi (εi − µ) − kB T S + ∆F + µNe ,

(2.1)

i

∆F = −kB T log Tr [exp{−∆H/kB T }] ,(2.2)

where Ne and µ are the number of electrons and chemical potential, fi is the occupation numberof band energy, defined as fi = 1/[exp{(εi − µ)/kB T } + 1]. Note that ∑i fi = Ne . The second
term in Eq. (2.1) represents the entropic energy of non-interacting one-particles and S = ∑i si =∑i y( fi ), where y(x) = −{x log x + (1 − x) log(1 − x)}. The third term ∆F in Eq. (2.1) represents
the contributions from all the other interactions between electrons which are not included in thefirst (one-particle) term. Ferromagnetism is stabilized by an exchange magnetic field (molecularNurul I KHSAN, Indra PARDEDE, Tomosato K ANAGAWA, Daiki YOSHIKAWA, and Tatsuki O DA

(a)

39

(b)
1.7
Dense k mesh
Sparse k mesh

[𝟎𝟎𝟏]

ESM

1.6

MAE (mJ/m2)

vacuum

Fe
Cr
Mg
O

1.5
1.4
1.3
1.2

vacuum
[𝟏𝟎𝟎]

0

100

200

300

400

500

Temperature (K)Figure 1: (a) Computational slab system equipped with an effective screening medium (ESM), and (b)Fermi level smearing effect in magnetic anisotropy energy (MAE) with respect to temperature. The symbolsof triangle and circle specify the data by dense and sparse meshes, respectively.

field) arising from the other magnetic moments. This effect is taken into account partially in εi and∆F. The magnetic dipole-dipole interaction between electrons, contributing to the SA, is includedin ∆F.
In magnetic metals, the magnetic anisotropy largely depends on the details of electronic structures at the Fermi level. The theory of force theorem [26] clearly indicates that the one-particle
energy εi plays an important role in the estimation of magnetic anisotropy. As also indicated at
the first term in Eq. (2.1) the anisotropy energy may vary with the electron occupation fi whichsmears the Fermi level as temperature increases. This smearing effect becomes large in case of
the electronic structure which has a set of flat bands (spiky in the one-particle density of states)
at the Fermi level. In particular, such feature can be found in low dimensional systems. In thedemonstration shown later, the system has fine electronic structures in the energy width with several times of kB Tr (Tr : room temperature) at the Fermi level. The approach along the free energyof Eq. (2.1) needs an appropriate additional treatment to fully include the spin fluctuations of bothtransverse and longitudinal components when discussing the temperature dependence of magnetization [18, 19, 27].

2.2

Practical approach

The magnetic anisotropy also depends on the magnetic moment. As temperature increases, themagnetic moment decreases, and thus the anisotropy energy often decreases. The main part ofsuch reduction may be realized by the contribution of SA. The temperature dependence of themagnetic moment originates from the spin fluctuations. In order to evaluate it, one can take amethod based on the microscopic electronic structure [18,19]. However, for focusing the smearing
effect at the Fermi level and for simplicity, this work employs a well-known sophisticated modelas the temperature-dependent magnetization M(T ), as follows:
M(T ) = M0 y(T /T ∗ ),(2.3)

where y is a given function, M0 = M(0), and T ∗ is a sophisticated parameter, such as Curie temperature (Tc ). T ∗ is used as a sort of fitting parameters. In this work, due to the thinner magnetic

40

Fermi level smearing effect in the perpendicular interface Fe/MgO magnetic anisotropy

slab, we employed y(T /T ∗ ) = 1 − (T /T ∗ )2 [24].
The magnetic anisotropy energy is presented as
K = Kb + Ksa ,(2.4)

where Kb is from the band energy and Ksa from the SA. Kb is expressed as the energy differenceof free energy Fb between the different magnetization directions, such as [100](x-direction) and[001](z-direction) [28]. The Fb is given as follows [29, 30] :Fb =∑ fnk (εnk − µ) − kB T S + Ed + µNe ,

(2.5)

nk

where Ed is the double counting term in the total energy [26]. Using Eq. (2.5),[100]

Kb = Fb

[001]

− Fb

.

(2.6)

Ksa is expressed as
int
Ksa = −µ0 M 2 /2Ω + ∆Ksa
,

(2.7)

where Ω and µ0 are the volume of magnetic slab and permeability of vacuum, respectively, and
int is the interface contribution which is not included in the 1st term. ∆K int originates from∆Ksa
saboth the discreetness of stacking atomic layers [31] and the deviation from spherical atomic spin
moment density at the interface magnetic atoms [32]. These are due to the shape of magnetization
distribution, reducing the in-plane SA in ferromagnetic Fe layers.

2.3

Computational system

In the demonstration, we used the slab system, vacuum(0.79nm)/Cr(6ML)/Fe (5ML)/MgO(5ML)/vacuum(0.79nm) (ML=atomic monolayer) [see Fig. 1 (a)]. At the Fe/MgO interface the 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 was used. The in-plane lattice constant extracted frombulk Cr was employed. The effective screened medium (ESM) was introduced as an ideal metal.
It is for avoiding an artificial electric field caused by the periodic boundary condition along the[001] direction [33]. The density functional calculation employs a fully relativistic (with spinorbit interaction) ultrasoft pseudopotentials and planewave basis [34], by using the generalized
gradient approximation [35]. We used a 32 × 32 × 1 (sparse) mesh for the k point sampling in
the estimations of Kb . At the low temperatures, unfortunately, the sparse mesh mentioned above
cannot give any convergence in the self-consistent calculation, and a 64 × 64 × 1 (dense) mesh
was also introduced. The difference from the sparse mesh appeared at the low temperatures, suchas less than or equal to 100-200 K, and becomes small at 300 K. The dense mesh requires a large
amount of computational source so that the number of calculated temperature points were limited
to a few.

3

Result and Discussion

3.1

Magnetic anisotropy from spin-orbit interaction

Figure 1(b) shows Kb per unit area as a function of T . These values are positive, contributing toa perpendicular magnetic anisotropy as expected in a family of Fe/MgO interfaces [2, 3, 36], andNurul I KHSAN, Indra PARDEDE, Tomosato K ANAGAWA, Daiki YOSHIKAWA, and Tatsuki O DA(a) Band dispersion curves41(b) PDOS

0.3

316 K
53 K

Energy(eV)

0.2
0.1
0

95 (meV)
-0.1

85
75

-0.2

0
-0.3

200

400 (K)

_

_

_

__

_

_

_

Γ

M

X

Γ Γ

M

X

Γ 0 1 2 3 4 5 6 7

k wave vector

States/eV/cellFigure 2: (a) Band dispersion curves (left and center) and (b) partial density of states (right) for theminority-spin-state 3d orbital on the interface Fe in the [001] magnetization system determined withthe temperature-dependent Fermi level smearings of 316 K (full curves and symbols) and 52.8 K (broken curves). The symbols specify the angular orbital components projected on the interface Fe atoms;
dxy , dx2 −y2 , d3z2 −r2 (left) in blue, green, and red bullets, respectively, and dxz , dyz (center) in orange and yellow. The Fermi levels are adjusted to zero in the vertical axis with the horizontal full line, and the Fermi
energy (chemical potential µ) decreases by 0.11 eV as temperature. The inset figure in (b) shows the typical
data of εnk − µ with respect to the temperature at k = 0.33 × Γ̄ M̄ (vertical red arrow).

similar to the data (1.5 mJ/m2 ) from the experimental measurement at the room temperature for
the Fe thickness (tFe ) of 0.7nm [12]. Such positive contribution may be attributed to the SOCs
between the orbital components of dxy and dx2 −y2 , or dxz and dyz in the respective occupied andunoccupied states [37]. The smearing effect decreases Kb monotonically by 0.38 mJ/m2 from 10.5
K to 527 K. This decreasing quantity is not negligible, implying one of important ingredients forthe temperature dependence of magnetic anisotropy.

3.2

Electronic structure

Kb decreasing with temperature is a consequence of electronic structure. To confirm the variation
property, in Fig. 2 we show the band dispersion curves and the partial density of states (PDOS)
for two different temperatures (52.8 K and 316 K). The eigenvalue with respect to the chemical
potential (εnk − µ) is increased roughly by 0.01 eV in almost all the Brillouin zone. In particular,
focusing the unoccupied states dominated by 3d orbitals at around 0.09 eV, the eigenvalue tends to0 − µ + αT 2 , where ε 0 and µ are the eigenvalue and chemical potential,
behave (εnk − µ) = εnk00nk
respectively, extrapolated to 0 K, and α is a positive constant (typical temperature dependenceof εnk − µ is shown in the inset of Fig. 2). Consequently, the quantity of 1/(εnk − µ) decreases
as temperature, implying a decrease in Kb considering the second-order perturbation formula for0 − µ ) is muchspin-orbit interaction [37]. In the present case, the α is small so that αT 2 /(εnk0smaller than the unity, showing a gradual temperature dependence in Kb like a linear, instead of
a complicated dependence. Quantitatively the decreasing rate by the Fermi level smearing should

42

Fermi level smearing effect in the perpendicular interface Fe/MgO magnetic anisotropy
0.8
Dense : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K
int: ∆Κ sa /Α

MAE (mJ/m2)

0.7

Sparse : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α
Exp.

0.6

0.5

0.4
0

100

200

300

400

500

Temperature (K)Figure 3: Total magnetic anisotropy energy (K) per unit area with respect to temperature. The symbols
int = 0 and those ofof triangle, bullet, circle, diamond indicate the data of T ∗ = 900, 1100, 1300K for ∆Ksa
int /A = 0.09mJ/m2 , where A is the interface area. The empty squarecross and plus for T ∗ = 1300K and ∆Ksa
symbols specify the experimental data extracted from reference [10].

be canceled out partially by the other contributions so as to showing a gradual decline such as
observed in the experimental measurement [10].

3.3

Including shape magnetic anisotropy

In order to validate the smearing effects obtained above comparing them with the experimental
results, we had performed a simulation on K using the parameters of M0 and T ∗ . In this simulation we used the Fe layer thickness (tFe = 0.710nm). This tFe is comparable to those obtained by
the first-principles calculation. As a result the M0 can be fixed to the value which reproduces theexperimental magnetization at 300 K (M(300K) = 1.83 T [38]) [11]. We also found that, comparing the results of the T ∗ s of 800, 900, 1000, 1100, 1200, 1300, and 1400K, the T ∗ s between
1200K and 1300K give a reasonable fitting to the temperature dependence of available experimenint . In Fig. 3, the total K per area
tal data [10] except a temperature-independent value implying ∆Ksa
int = 0.
are plotted as temperature for the parameters of T ∗ = 900K, 1100K, and 1300K, where ∆Ksa∗
At the lower T s, the in-plane SA becomes larger and the difference from the experimental dataalso becomes large at the low temperatures.
int . This plot implies that,In Fig. 3, the total K per unit area is also shown for a non-zero ∆Ksa
int /A = 0.09mJ/m2 ), K becomes close to the experassuming the interface contribution of SA (∆Ksa
int is not so far from a realistic contribution,imental results. This quantitative assumption in ∆Ksa
because the quadrupole atomic spin density of prolate type at the interface can reduce the in-plane
SA by an energy comparable to that in the free-standing Fe 1ML (0.10mJ/m2 ) [32]. Note that suchcontribution does not depend on the total magnetization. On the fitting to another experimental
int /A = 0.35mJ/m2 , T ∗ = 1400K) prodata of M(300K) = 2.09T [12], the set of parameters (∆Ksa
int is not too
vides a reasonable temperature dependence in K (see Appendix). This parameter of ∆Ksa
large, because the parameter originates from both interfaces in the Fe layer. Further investigations
int are required for analyzing real magnetic interfaces.
on the origin of ∆Ksa
Our analysis on the temperature dependence of K can predict a behavior at the higher temperatures. As shown in Fig. 3, our result indicates that it increases as temperature increases afterNurul I KHSAN, Indra PARDEDE, Tomosato K ANAGAWA, Daiki YOSHIKAWA, and Tatsuki O DA

43

some temperature. Note that such behavior is a consequence that as temperature increaces, theperpendicular Kb decreases and the in-plane SA (|Ksa |) also decreases more rapidly with a growthof spin fluctuations. The behavior that the SA works as an enhancement of PMA or as a suppressing origin in the reduction of PMA at higher temperatures, may be one of important generalinterests in the thin ferromagnetic materials. Note that at further higher temperatures near Tc the
Mermin’s approach is no longer useful and should be corrected in accordance with existing large
spin-fluctuations [27]. In a further general treatment on magnetic anisotropy, temperature effects
originating from a magnetoelastic anisotropy should be considered.
In our analysis, the parameter of T ∗ is relatively flexible, compared with the other parameterof M0 . T ∗ around 1300K seems to be suitable to explain the available experimental data. If
T ∗ is regarded as Tc , this Tc (= 1300K) is comparable to those of bcc Fe (Tc = 1043K). Whenconsidering a Tc formula in the approach of localized spin moment, Tc is proportional to both theexchange coupling constant and the square of magnetization. Roughly speaking, the effective Tc
may increase, because the magnetic moments on Fe atom tend to be enhanced at the interfaces.
Further information on Tc should be required in the computational approach as well as in theapproaches of phenomenological theory and experimental measurement.
Our present approach to a temperature dependent MAE may indicate a correspondence with anavailable experimental data. However, it is still unclear that the spin fluctuation at finite temperatures is fully considered. This is because the electronic structure calculation, which is a basis of
the MAE originating from the spin-orbit interaction, does not include effects of finite temperaturein the atomic spin configuration, while the spin configuration at finite temperatures should be an
ensemble of various directions for atomic magnetic moments. The last picture is a reason why thestrength of total magnetization tends to decrease as temperature increases. An improved approach
to MAE estimation is required for a computational material design of magnetic materials.

4

Summary

We performed the first-principles calculations on the MAE from the band energy contribution using the temperature-dependent Fermi level smearing in the Fe(5ML)/MgO slab. The contributionof the MAE decreases by 0.4 mJ/m2 as temperature from 10.5K to 527K. When employing thesimple formula for the SA and assuming the experimental saturated magnetization with the appropriate temperature dependence for ultra-thin films, the total MAE shows a nearly flat part aroundthe room temperature. This theoretical analysis may predict an increase of the perpendicular total
MAE at higher temperatures. Such increase may be a possible general consequence of the balance
between a large perpendicular band energy contribution and small in-plane shape anisotropy. The
present work provides a new pathway to understand origins of a temperature dependence in MAE.

44

Fermi level smearing effect in the perpendicular interface Fe/MgO magnetic anisotropy

1

M/M0

0.8
0.6
M(T) = M(0)[1-(T/T *)3/2]
M(T) = M(0)[1-(T/T *)2]

0.4

M(T) = M(0)[1-(T/T *)2]1/2
1/2
M(T) = Ps(T/T *)M(0)[1-(T/T *)]

0.2
0
0

0.2

0.4

0.6

0.8

1

T/T *

Figure 4: Reduced magnetization, M/M0 as a function of reduced temperature T /T ∗ .

AAppendix: Simulation on shape anisotropy

The reduced magnetization, M(T )/M(0), is expressed as a function of the reduced temperature,
T /T ∗ . In the simulations on SA, we tried to choose the following functions [22, 24, 39, 40] :" 2 #
T
M(T ) = M(0) 1 −
,
(A.1)
T∗
"
 3/2 #T,(A.2)
M(T ) = M(0) 1 −
T∗
s

 2
T
,
M(T ) = M(0)  1 −
(A.3)
T∗
"s
 
 #
T
T
M(T ) = PS
M(0)
1−
,
(A.4)
∗
T
T∗
where
PS (T /T ∗ ) =
1 + 0.410721(T /T ∗ ) + 1.65224(T /T ∗ )2 − 5.60792(T /T ∗ )3 + 9.67475(T /T ∗ )4
−9.30572(T /T ∗ )5 + 4.74404(T /T ∗ )6 − 1.00258(T /T ∗ )7 .
(A.5)

These functions are shown in Fig. 4. Using the assumed function of M(T ), the Total MAE (K)as a function of temperature is fitted to an experimental data [10]. M(0) was determined so as
to reproduce an experimentally measured magnetization at 300K; M(300K) = 1.83T [11] and2.09T [12] were used in Figs. 5 and 6, respectively.
The total magnetic anisotropy for each M(T ) function is presented in Fig. 5, compared with theexperimental data extracted from Xiang et al. [10]. The functions of Eqs. (A.1), (A.2), (A.3), and(A.4) are used in Figs. 5 (a), (b), (c), and (d), respectively, with the experimental magnetization[M(300K)=1.83T] extracted from Ref. [11]. Various T ∗ s were used from 800 to 1400K. In Fig. 5,
the resulting data are presented only for 900, 1100, and 1300K. The choice of the first M(T ) function (which is proportional to T 2 ) could reproduce the tendency of experimental data, especiallyfor the case of T ∗ =1300K. The flat part of total MAE is remarkably reproduced when the temperature increases from 200K to 300K. As shown in Fig. 5 (a), the choice of lower T ∗ gives anNurul I KHSAN, Indra PARDEDE, Tomosato K ANAGAWA, Daiki YOSHIKAWA, and Tatsuki O DA
1Sparse : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α
Exp.

MAE (mJ/m2)

0.8

(b)

Dense : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ intsa /Α

1
Sparse : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α
Exp.

0.8

MAE (mJ/m2)

(a)

0.6

0.4

45

Dense : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α

0.6

0.4

0.2
0.2
0
0

100

200

300

400

500

0

100

Temperature (K)1Sparse : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α
Exp.

MAE (mJ/m2)

0.8

300

400

500

Temperature (K)(d)

Dense : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ intsa /Α

1
Sparse : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α
Exp.

0.8

MAE (mJ/m2)

(c)

200

0.6

0.4

0.2

Dense : Τ * = 1300 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α

0.6

0.4

0.2

0

0
0

100

200

300

Temperature (K)

400

500

0

100

200

300

400

500

Temperature (K)Figure 5: Total magnetic anisotropy energy (K) per unit area with respect to temperatures; (a), (b), (c),and (d) for the functions in Eqs. (A.1), (A.2), (A.3) and (A.4), respectively. The symbols of triangle, bullet,
int = 0 and those of cross and plus for T ∗circle, diamond indicate the data of T ∗ = 900, 1100, 1300K for ∆Ksa
int
2
= 1300K and ∆Ksa /A = 0.09mJ/m , where A is the interface area. The empty square symbols specify theexperimental data extracted from Ref. [10]. The experimental magnetization [M(300K)=1.83T] extracted
from Ref. [11].

increasing tendency of total MAE as the temperature increases. These features are not favorable,compared with the experimental tendency. When we choose the second M(T ) function presented
at Eq. (A.2), the total MAE at low temperature is largely underestimated and increases linearly as
the temperature increases. The total MAE from the third M(T ) function gives a reasonable comparison with the experimental data. However, this choice of function and set of parameters tends
to show a different trend at the flat part around 200K and 300K of experimental data. The choiceof the fourth M(T ) function gives an overestimation at low temperatures, compared with the experimental data and also gives a linear decreasing tendency as the temperature increases. At hightemperatures it provides a lower total MAE, compared to other three functions. This M(T ) function cannot also reproduce the behavior around 200K and 300K in the experimental data. Fromthe simulation presented in Fig. 5, the first M(T ) function (A.1) with T ∗ of 1300K may provide
an acceptable result reproducing the experimental data.
Next, another experimental magnetization is considered. The total magnetization of M(300K)=
2.09T is extracted from Koo et al. [12]. The results of simulation are presented in Fig. 6, as forthe magnetization of 1.83T. This total magnetization largely underestimates all of our data in the
total MAE. The best result is provided by the first M(T ) function with T ∗ =1400K. This result

46
1Sparse : Τ * = 1400 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α
Exp.

MAE (mJ/m2)

0.8

(b)

Dense : Τ * = 1400 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ intsa /Α

Sparse : Τ * = 1400 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α
Exp.

1
0.8

MAE (mJ/m2)(a)Fermi level smearing effect in the perpendicular interface Fe/MgO magnetic anisotropy

0.6

0.4

Dense : Τ * = 1400 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α

0.6
0.4
0.2
0

0.2

−0.2
0
0

100

200

300

400

500

0

100

Temperature (K)1Sparse : Τ * = 1400 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α
Exp.

MAE (mJ/m2)

0.8

300

400

500

Temperature (K)(d)

Dense : Τ * = 1400 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ intsa /Α

1
Sparse : Τ * = 1400 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α
Exp.

0.8

MAE (mJ/m2)

(c)

200

0.6

0.4

0.2

Dense : Τ * = 1400 K: Τ * = 1100 K: Τ * = 900 K: ∆Κ int
sa /Α

0.6

0.4

0.2
0

100

200

300

Temperature (K)

400

500

0

100

200

300

400

500

Temperature (K)Figure 6: Same quantities (K per unit area) as in Fig. 5 with the different parameters. The symbols of
int = 0 and those crosstriangle, bullet, circle, diamond indicate the data of T ∗ = 900, 1100, 1400K for ∆Ksa∗
int
2
and plus for T = 1400K and ∆Ksa /A = 0.35mJ/m . The experimental magnetization [M(300K)=2.09T]
extracted from Ref. [12].
may reproduce the behavior around 200K and 300K in the experimental data when we choose
int /A = 0.35mJ/m2 as the interface anisotropy. In the simulations of Fig. 5 and Fig. 6, we∆Ksa
int /A = 0.09mJ/m2 and ∆K int /A = 0.35mJ/m2 , respectively. Although the fabriintroduced ∆Ksa
sa
cated systems are different between Nozaki et al. [11] and Koo et al. [12], those magnetic slabs
int /A should be explained with a reasonable origin
are similar to each other. The difference in ∆Ksa
in future.

Nurul I KHSAN, Indra PARDEDE, Tomosato K ANAGAWA, Daiki YOSHIKAWA, and Tatsuki O DA

B

47

Appendix: Magnetic moments of density functional approach

The magnetic moments obtained by the density functional approach are displayed for typical temperatures of 53K (sparse mesh) and 316K (sparse mesh) in Table 1 and Table 2. These data were
estimated from the spin density by integrating it in the atomic sphere with the radius (Fe: 1.32Å,
Cr: 0.90Å).
Table 1: Spin magnetic moments in µB on Cr and Fe atoms. The temperature specifies the value used in the
Fermi level smearing. [001](z-direction) and [100](x-direction) specify the direction of total magnetization.

Temperature
53K [001]
53K [100]316K [001]316K [100]Total
12.7964
12.7964
12.7844
12.7848Fe(1)2.84812.84992.84202.8428Fe(2)2.53112.52952.52862.5286Fe(3)2.59842.59752.59602.5957Fe(4)2.49862.49922.50082.5007Fe(5)2.32022.32032.31702.3170Temperature
53K [001]
53K [100]316K [001]316K [100]Cr(1)-1.0678-1.0674-1.0689-1.0689Cr(2)0.99990.99990.999
0.9989Cr(3)-1.0902-1.0903-1.0905-1.0906Cr(4)1.22471.22451.22091.2208Cr(5)-1.4434-1.4433-1.4421-1.4421Cr(6)2.33292.33302.33092.3308

Table 2: Orbital magnetic moments in µB on Cr and Fe atoms. The temperature specifies the value used in
the Fermi level smearing.

Temperature
53K [001]
53K [100]316K [001]316K [100]Total
0.290610.25549
0.28550.2572Fe(1)0.08476
0.06251
0.08330.0649Fe(2)0.05967
0.048220.05740.0478Fe(3)0.050760.049440.04980.0492Fe(4)0.053010.05068
0.05240.0505Fe(5)0.042410.044640.04270.0448Temperature
53K [001]
53K [100]316K [001]316K [100]Cr(1)0.009960.011820.00980.0121Cr(2)-0.00612-0.00607
0.00620.0062Cr(3)0.00935
0.008880.00940.0091Cr(4)-0.00746-0.00911
0.00750.0090Cr(5)0.01050.009310.01050.0093Cr(6)-0.01491-0.02182
0.01500.0211

48

C

Fermi level smearing effect in the perpendicular interface Fe/MgO magnetic anisotropy

Appendix: MAE from Bruno’s relation

One can estimate MAEs from the orbital magnetic moments by using Bruno’s formula [41][001]

MAE = ξ

mo

[100]

− mo
4µB

,

(C.1)
[001]

[100]

where ξ is the spin-orbit coupling constant (Fe: 51 meV/atom) and mo(mo ) is the orbital
magnetic moment for the [001] ([100]) magnetization direction. The MAE was estimated foreach atom and displayed with the total for the typical temperature in Table 3. The totals areunderestimated compared with the respective values indicated in Fig. 1(b) in the main text. These
underestimation may be attributed to the details of 3d-orbital hybridization on Fe atom around the
Fermi level. Interestingly, the difference between 53K and 316K on the total MAE is very similar
to the corresponding value indicated in Fig. 1(b) in the main text.
Table 3: Magnetic anisotropy energy (mJ/m2 ) of Bruno’s formula estimated from the orbital moments
shown in Table. 2.

Temperature
MAE(53K)
MAE(316K)
difference

Total
0.87
0.70
0.17

Fe(1)
0.55
0.45
0.10

Fe(2)
0.28
0.24
0.04

Fe(3)
0.03
0.01
0.02

Fe(4)
0.06
0.05
0.01

Fe(5)
-0.06
-0.05
-0.01

Acknowledgements
The first-principles calculations were performed using the facilities of the Supercomputer Center, Institute for Solid State Physics, the University of Tokyo, Japan. This work was partly supported by Grant-in-Aid for Scientific Research from JSPS/MEXT (Grant Nos. 15K05165 and
18K04923). This research partly used computational resources of the K computer and othercomputers of the HPCI system provided by the AICS and Information Technology Center ofNagoya University through the HPCI System Research Project (Project ID:hp160107, hp160227,
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 the Development of Global Human Resources for
Kanazawa University.

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