Local density of states as a probe for tunneling magnetoresistance eect application to ferrimagnetic tunnel junctions Katsuhiro Tanaka1Takuya Nomoto1and Ryotaro Arita1 2

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Local density of states as a probe for tunneling magnetoresistance eﬀect: application

to ferrimagnetic tunnel junctions

Katsuhiro Tanaka,1Takuya Nomoto,1and Ryotaro Arita1, 2

1Research Center for Advanced Science and Technology,

University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8904, Japan

2Center for Emergent Matter Science, RIKEN, Wako, Saitama 351-0198, Japan

(Dated: October 5, 2022)

We investigate the tunneling magnetoresistance (TMR) eﬀect using the lattice models which

describe the magnetic tunnel junctions (MTJ). First, taking a conventional ferromagnetic MTJ as

an example, we show that the product of the local density of states (LDOS) at the center of the

barrier traces the TMR eﬀect qualitatively. The LDOS inside the barrier has the information on

the electrodes and the electron tunneling through the barrier, which enables us to easily evaluate

the tunneling conductance more precisely than the conventional Julliere’s picture. We then apply

this method to the MTJs with collinear ferrimagnets and antiferromagnets. We ﬁnd that the

TMR eﬀect in the ferrimagnetic and antiferromagnetic MTJs changes depending on the interfacial

magnetic structures originating from the sublattice structure, which can also be captured by the

LDOS. Our ﬁndings will reduce the computational cost for the qualitative evaluation of the TMR

eﬀect, and be useful for a broader search for the materials which work as the TMR devices showing

high performance.

I. INTRODUCTION

Utilizing the close connection between the spin and

charge degrees of freedom of electrons in solids, spintron-

ics has developed various phenomena that are novel from

the viewpoint of fundamental physics and promising

for industrial use [1–5]. Among those, the tunneling

magnetoresistance (TMR) eﬀect [6, 7] is one of the

representative phenomena in its wide application [8–12].

The TMR eﬀect is observed in the magnetic tunnel junc-

tion (MTJ), which consists of two magnetic electrodes

and the insulating barrier in between. The electrons

can tunnel through the MTJ as a quantum mechanical

current, and the tunneling resistances become diﬀerent

when the magnetic moments of the two electrodes

align parallelly or antiparallelly. The set of these two

alignments with diﬀerent tunneling resistances corre-

sponds to a bit taking a binary 0 or 1, which has been

utilized to the magnetic head and the magnetic random

access memory devices for the storages and the readout.

As well as the theoretical approaches [13–16], large

TMR ratios have been experimentally observed in the

MTJs such as the Fe(Co)/Al2O3/Fe(Co) [17, 18],

Fe(Co)(001)/MgO(001)/Fe(Co) [19, 20] and

CoFeB/MgO/CoFeB systems [21, 22]. Ferromag-

netic Heusler compounds have also been utilized as the

electrodes thanks to their half-metalicity [23–26].

While the main target of the spintronics was ferromag-

nets, recent spintronics has been extended to antiferro-

magnets and ferrimagnets owing to their superiorities to

ferromagnets; the smaller stray ﬁeld and the faster spin

dynamics [27–35]. The antiferromagnetic version of the

spintronic phenomena, e.g., the giant magnetoresistance

eﬀect [36–38] and the anomalous Hall eﬀect [39–42], has

been developed. Along with these advances, the TMR

eﬀect using antiferromagnets has also been intensively

investigated [43–48]. While most of the studies have

been theoretical attempts, experiments have also been

developed; the TMR eﬀect is observed in the MTJ whose

two electrodes are the ferromagnet and the ferrimagnetic

Heusler compound [44]. However, for more practical ap-

plication of the MTJs with antiferromagnets and ferri-

magnets to the devices, we should search for materials

constructing the MTJs which show a large TMR ratio,

and handy methods for the search are required.

In this paper, we examine the TMR eﬀect using the

lattice models mimicking the MTJs whose electrodes are

made of collinear ferrimagnets, including the antiferro-

magnets. Motivated by the studies indicating that the

interfacial electronic structures aﬀect the TMR eﬀect

and they can be probed by the local density of states

(LDOS) [49–53], we particularly focus on the LDOS to

analyze the TMR eﬀect. We ﬁnd that the product of the

LDOS at the center of the barrier usually reproduces the

transmission properties qualitatively in the ferrimagnetic

MTJs as well as the ferromagnetic ones. The LDOS has

the information both on the magnetic properties of elec-

trodes and on the tunneling electrons. Besides, from the

physics point of view, we show that multiple conﬁgura-

tions can be realized in the ferrimagnetic MTJs due to the

sublattice structure for each of the parallel and antipar-

allel magnetic conﬁgurations. The resultant TMR eﬀect

changes depending on the conﬁgurations, which suggests

that the magnetic conﬁgurations should be carefully ex-

amined when we deal with the ferrimagnetic MTJs.

Considering the above qualitative estimation in terms

of the LDOS, we present a hierarchy for evaluating the

TMR eﬀect in Fig. 1. To quantitatively estimate the

TMR eﬀect, we have to calculate the conductance itself

through the Landauer–B¨uttiker formula [54–57]. Tech-

nically, this method can be applied to any system and

gives us highly accurate results, whereas its numerical

cost is often expensive, particularly in calculating from

arXiv:2210.01441v1 [cond-mat.mes-hall] 4 Oct 2022

Local density of states

Appearance:

bulk density of states / spin configuration

Transport calculation:

Landauer–Büttiker formula

Coverage / Computational cost

High

Low

FIG. 1. Hierarchy for the evaluation of the tunneling mag-

netoresistance eﬀect. The upper has a broader coverage and

gives more quantitative results, with the higher calculation

cost. Symbols in the right side, G(E), Nσ(E;x, y), and

Nσ(E), denote the conductance, the local density of states,

and the density of states of the bulk, respectively, which are

the quantities obtained in each calculation.

ﬁrst-principles. By contrast, the Julliere’s picture, which

claims that the density of states of the bulk electrodes

determines the eﬃciency of the MTJ [6, 58], is a sim-

ple and convenient picture to predict the TMR eﬀect.

However, the picture is valid only for limited cases, and

currently it is found that the electronic states of the

tunneling electrons are signiﬁcant rather than those of

the bulk electrodes. Our results can be placed between

these two methods. Calculating the LDOS of the MTJs

is much easier than the Landauer–B¨uttiker calculation.

Additionally, the estimation from the LDOS can cover a

broader range of the MTJs with higher reliability than

the prediction from the electronic structures of the bulk

electrodes.

The remainder of this paper is organized as follows. In

Sec. II, we introduce the model describing the MTJ. We

simulate the TMR eﬀect using the ferromagnetic elec-

trodes in Sec. III, and see how the LDOS works for pre-

dicting the tunneling conductance. In Sec. IV, we calcu-

late the TMR eﬀect in the ferrimagnetic and antiferro-

magnetic MTJs applying the prediction from the LDOS.

We discuss in Sec. V the hierarchy shown in Fig. 1 and the

correspondence between our models and the MTJ with

real materials. Section VI is devoted to the summary and

perspective of this study.

II. MODEL AND METHOD

We construct the two-dimensional square lattice MTJ

using two semi-inﬁnite lattices which work as electrodes

and the barrier in between, which is schematically shown

in Fig. 2. We treat the tight-binding Hamiltonian with

the s–dcoupling on this system, which is given as

H=H0+Ht+Hs−d,(1)

H0=X

εini,(2)

Ht=−tX

hi,ji,σ c†

i,σcj,σ + h.c.,(3)

Hs–d=−JX

i∈electrode

(si·σ)αβ c†

i,αci,β .(4)

Here, c†

i,σ (ci,σ) is the creation (annihilation) opera-

tor of an electron with spin-σon the i-th site, and

ni=Pσc†

i,σci,σ is the number operator. The on-site

potential is denoted as εi, and the electron hopping be-

tween two sites is written as t. The summation in Htis

taken over the neighboring two sites, which is expressed

by hi, ji. The eﬀect of the magnetism of the electrodes

is introduced by the s–dcoupling, Hs–d, where the local-

ized spin moment, si, and the conducting electrons cou-

ple each other with the magnetic interaction constant, J.

The spin degrees of freedom of the conducting electrons

are expressed by σ, which is the vector representation of

the 2 ×2 Pauli matrices. We take the two-dimensional

cartesian coordinate, (x, y), for the MTJ, where the x-

axis is parallel to the conducting path which inﬁnitely

extends, and the y-axis is perpendicular to the conduct-

ing path (see Fig. 2(a)). The width of the barrier in

the x-direction is L, and the width of the MTJ in the

y-direction is W. The lattice constant is taken to be

unity. We hereafter impose the open boundary condition

on the y-direction, while we conﬁrmed that the periodic

boundary condition does not change the overall results.

The calculations of the transmissions are performed

using the kwant package [59], in which the quantum

transport properties are computed based on the scatter-

ing theory and the Landauer–B¨uttiker formula.

III. TUNNELING MAGNETORESISTANCE

EFFECT WITH FERROMAGNETIC

ELECTRODES

Let us ﬁrst recall the conventional TMR, namely the

TMR using the ferromagnetic electrodes as shown in

Fig. 2(a). We set the localized spin moments as si=

t001for all sites in the left electrode. For the

sites in the right electrodes, we set si=t001and

t0 0 −1for the parallel and antiparallel conﬁgura-

tions, respectively. The schematics of these two conﬁg-

urations are shown in Figs. 3(a) and 3(b). We set the

hopping tas a unit, t= 1. The on-site potential is set

as εi= 0 for the electrodes, and εi= 10 for the barrier

region. The system size is set as L= 8 and W= 160.

electrode

(ferrimagnetic)

electrode

(ferrimagnetic)

barrier

(nonmagnetic)

(a)

electrode

(ferromagnetic)

electrode

(ferromagnetic)

barrier

(nonmagnetic)

(b)

FIG. 2. Schematics of the two-dimensional magnetic tun-

nel junctions (MTJ) used in our calculations. (a) MTJ with

the ferromagnetic electrodes. (b) MTJ with the ferrimagnetic

electrodes. Arrows represent the localized spin moments on

the electrodes.

A. Bulk properties

Before discussing the properties of the tunneling con-

ductance, we see the properties of the bulk ferromagnetic

metals used as the electrodes, namely, the energy bands

and the density of states (DOS). The DOS is given as

Nσ(E) = RBZ dkδ(E−Ek,σ), where Ek,σ is the energy

band with spin-σ. For the energy bands of the bulk elec-

trodes, we consider the two-dimensional square lattice

described by Hgiven in Eq. (1). The energy bands are

found to be E±

k=−2t(cos kx+ cos ky)±J. The DOS of

the ferromagnet is shown in Fig. 3(c) at J= 1, where the

right-hand side and the left-hand side are the ones with

the up-spin and down-spin, respectively. By introducing

a ﬁnite J, the energy band with up-spin gains the energy

−J, while that with down-spin shifts by +J.

B. Transmission and local density of states

In Fig. 3(d), we show the TMR ratio deﬁned as

rt=TP−TAP

TP+TAP

,(5)

on the plane of Jand E. Here, TP/AP denotes the trans-

mission for the parallel/antiparallel conﬁguration. We

note that the deﬁnition above is slightly diﬀerent from

the conventional optimistic/pessimistic ones for the rea-

son on normalization. At J= 0, the whole system is

nonmagnetic and has the degeneracy on the spin degrees

of freedom, and thus TP=TAP holds at each energy.

Namely, rtis zero. When we introduce a ﬁnite magnetic

interaction J, the degeneracy is lifted, and TPstarts to

take a larger value than TAP; a ﬁnite TMR ratio is ob-

served. Due to the asymmetric structure of the barrier in

the energy, rtis also asymmetric with respect to E= 0.

The J-dependence of TPand TAP is shown in Figs. 3(e)

and 3(f). At E=−2, both of TPand TAP decrease when

Jincreases, and TAP reaches zero at J= 2 [60]. When

E= 0, TPincreases with J, while TAP decreases to zero.

To better understand the transmission properties, we

examine the LDOS in addition to the bulk DOS. In the

naive Julliere’s picture, the product of the bulk DOS,

g(E), deﬁned as

g(E) = X

NL,σ(E)NR,σ(E),(6)

describes the transmission [6, 58], where NL/R,σ (E) is

the bulk DOS with spin-σ,Nσ(E), of the left/right elec-

trodes. However, g(E) does not consider the barrier, and

thus this picture holds only for the limited cases. In-

stead, the LDOS has been utilized to capture the details

on the MTJs. In particular, it has been proposed that

the transmission can be described using the LDOS at the

interfaces of the MTJ. In fact, when the potential of the

barrier is high enough, the conductance derived by the

Kubo formula [14] is proportional to the product of the

LDOS at the left and right interfaces, and to the expo-

nential function, e−κL, representing the decay inside the

barrier [49, 50]. Here, κstands for the decaying property,

namely, 1/κ means the spin diﬀusion length [61]. Hence,

we consider the product of the LDOS at the interfaces,

gi(E) given as

gi(E) = 1

σNσE; 1,W

2NσE;L, W

2

+NσE; 1,W

2+ 1NσE;L, W

2+ 1,(7)

where Nσ(E;x, y) (1 ≤x≤L, 1 ≤y≤W) is the LDOS

of the barrier at (x, y). Since we impose the open bound-

ary condition in the y-direction, we take the average over

the sites at y=W/2 and W/2 + 1 in Eqs. (7) and (8) to

reduce the eﬀects of the oscillation due to the boundary,

and we implicitly assume that the spins do not ﬂip in-

side the barrier [62]. However, in general, it is not easy

to precisely evaluate the exponent κ; we should need the

transmission coeﬃcients [63] in addition to ﬁnding the

electronic structures.

Alternatively, we here utilize the LDOS at the center of

the barrier. Expecting that it contains both the details

of the MTJ and the decay, we consider the product as

follows;

gc(E) = 1

σNσE;L

2,W

2NσE;L

2+ 1,W

2

+NσE;L

2,W

2+ 1NσE;L

2+ 1,W

2+ 1.

(8)

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Localdensityofstatesasaprobefortunnelingmagnetoresistanceeect:applicationtoferrimagnetictunneljunctionsKatsuhiroTanaka,1TakuyaNomoto,1andRyotaroArita1,21ResearchCenterforAdvancedScienceandTechnology,UniversityofTokyo,Komaba,Meguro-ku,Tokyo153-8904,Japan2CenterforEmergentMatterScience,RIKEN,Wako,Sai...

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Local density of states as a probe for tunneling magnetoresistance eect application to ferrimagnetic tunnel junctions Katsuhiro Tanaka1Takuya Nomoto1and Ryotaro Arita1 2

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