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arXiv:0804.2557v5 [cond-mat.mes-hall] 24 Feb 2009

Single-band tight-binding parameters for Fe-MgO-Fe magnetic heterostructures

Tehseen Z. Raza

School of Electrical and Computer Engineering and NSF Network for

Computational Nanotechnology, Purdue University, West Lafayette, IN 47907

Hassan Raza

School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853

We present a computationally efficient transferable single-band tight-binding model (SBTB) for

spin polarized transport in heterostructures with an effort to capture the band structure effects.

As an example, we apply it to study transport through Fe-MgO-Fe(100) magnetic tunnel junction

devices. We propose a novel approach to extract suitable tight-binding parameters for a material by

using the energy resolved transmission as the benchmark, which inherently has the bandstructure

effects over the two dimensional transverse Brillouin zone. The SBTB parameters for each of the four

symmetry bands for bcc Fe(100) are first proposed which are complemented with the transferable

tight-binding parameters for the MgO tunnel barrier for the ∆1 and ∆5 bands. The non-equilibrium

Green’s function formalism is then used to calculate the transport. Features like I-V characteristics,

voltage dependence and the barrier width dependence of the tunnel magnetoresistance ratio are

captured quantitatively and the trends match well with the ones observed by ab initio methods.

PACS numbers: 72.25.-b, 85.75.-d, 75.47.-m, 75.47.Jn, 85.35.-p

I.INTRODUCTION

Transport across multilayered heterostructures is a

problem of great interest both for its intrinsic physics

as well as its device applications [1]. Very often each of

the component materials has been studied extensively on

its own, but it is difficult to make use of this knowledge

because various studies employ different models and it is

difficult to combine their results. As a result, the only vi-

able approach is to start anew for each heterostructure.

The objective of this paper is to present a scheme for

extracting suitable single-band tight-binding (SBTB) pa-

rameters for each of the component materials, thus trans-

lating the results of earlier studies all into one common

SBTB platform which can then be used in a standard

non-equilibrium Green’s function (NEGF) based model

for quantum transport. We illustrate our approach with

an example of great current interest, namely an Fe-MgO-

Fe magnetic tunnel junction (MTJ) device. We use the

principles described in this paper to obtain the SBTB

parameters for bcc Fe(100) from the extended H¨ uckel

parameters [2] and those for MgO from ab initio mod-

els [3, 4, 5]. Using these SBTB parameters, extracted

from different sources, in an NEGF model for transport

we obtain I-V characteristics, voltage dependence of tun-

nel magnetoresistance (TMR) ratios and barrier width

dependence of TMR in good agreement with published

first-principles results for the same structure.

Ab initio modeling of materials is at an advanced stage

[6]. Coupled with the quantum transport, these models

have been successfully applied to nanoscale systems and

heterostructures. However, such methods are resource

intensive, and simplified models that capture the essen-

tial physics due to the underlying electronic structure

effects are desirable. Since the seminal work of study-

ing material properties using tight binding parameters

0.8

1

4-Layer

0.4

?m2)

y (A/?

nsity

nt den

urren

Cu

TMRT

0

0

2x 102

-6

1

(SBTB) JP

(SBTB) JAP

(ab initio) J

(ab-initio) JAP

12-Layer

R

TMR

1

00

(ab-initio) JP

SBTB SBTB

ab-initio

001122

0

001122

Vbias

FIG. 1: Total current densities for the parallel and the anti-

parallel configurations and the tunnel magneto resistance ra-

tio (TMR) for a 4-layer (top) and 12-layer (bottom) device.

SBTB transferable parameters are optimized for the MgO

tunnel barrier to match the current levels from the ab initio

calculations [3, 4, 5]. The bias dependence of TMR is also cap-

tured well within this simple single-band tight-binding model.

by Slater and Koster [7], there has been a motivation

to develop computationally efficient yet accurate models

to capture the underlying physical mechanisms and the

bandstructure effects. Such simple methods may or may

not capture all the intricate details present in the more

sophisticated models, e.g. ab initio, semi-empirical tight-

binding [8, 9] and empirical tight-binding methods, but

they do provide a platform for large scale calculations.

The simplest method currently available in this context

is an effective mass model. Although very successful for

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2

matching experiments, it is well known that it does not

capture the electronic structure effects. We find the same

for bcc Fe(100) because the band dispersions do not re-

main parabolic over the two dimensional (2D) transverse

Brillouin zone (BZ). Furthermore, the energy bands have

finite width, whereas bandwidth is infinite in continuum

effective mass models and it depends on the lattice spac-

ing in discrete effective mass models. Moreover, the var-

ious band edges shift over the 2D BZ and this shift is

not necessarily dictated by the transverse mode energy

?2k2

t/2mt, as prescribed by an effective mass model. Our

objective in this paper is to propose a computationally

efficient method for heterostructures with an effort to in-

corporate the physics of electronic structure effects over

the 2D transverse BZ in the transport quantities. Mo-

tivated by this, we propose a single-band tight-binding

(SBTB) model for transport through heterostructures

and outline the procedure for extracting suitable tight-

binding parameters. The computational complexity of

this model is the same as that of an effective mass model.

Recently, Fe-MgO-Fe heterostructures have appeared

as the most noteworthy example in spintronics [10].

These devices have emerged as one of the candidates for

random access memory applications. The prediction of

high tunnel magneto resistance (TMR) ratio for crys-

talline MgO barrier of over 1000% [11, 12] was followed

by observations of about 200% TMR ratios at room tem-

perature in Fe-MgO-Fe and CoFe-MgO-CoFe MTJ de-

vices [13, 14]. Since then, there has been an increased

effort to integrate them into practical devices. Although,

ab initio [3, 4, 5, 11, 15, 16] and empirical tight bind-

ing [12] studies have been reported, their computational

complexity limit their use for rapid device prototyping.

This method not only allows simulating device character-

istics efficiently but it also gives an inherently simple and

intuitive understanding for the underlying device physics.

A comparison between the SBTB and ab initio models is

shown in Fig. 1. For low bias, it matches well with Ref.

[5]. At high bias, our model predicts TMR roll-off and

ultimately becoming negative, which was later observed

in Ref. [17].

Our approach is to present SBTB parameters for differ-

ent materials independently and then couple them across

the interface. Instead of fitting the parabolic bands to the

bandstructure to extract effective masses, we propose to

match the energy resolved transmission. The energy re-

solved transmission plots reflect the band structure ef-

fects for the majority and the minority spin bands over

the transverse BZ and we propose to capture this effect in

an averagemanner with the SBTB parameters. As shown

in Fig. 3, the transmission is strongly dependent on the

transverse BZ, and it is imperative to incorporate these

effects. Thus, we first extract the SBTB parameters for

bcc Fe(100) from the energy resolved transmission plots

calculated using a semi-empirical atomistic method based

on extended H¨ uckel theory (EHT). Next, we propose the

barrier parameters, for tunneling through the MgO re-

gion for the ∆1 and ∆5 bands by comparing with the

55

0.4

yz

kk

a

?

??

0

yz

kk

??

1.4

yz

kk

a

?

??

2

yz

kk

a

?

??

11

?

)

y (eV)

EnergyE

00

2'

?

5

?

2

?

-5

[100] direction

FIG. 2: Band structure for the majority spin over the two

dimensional transverse Brillouin zone computed using EHT.

In general, the bands are not parabolic and hence an effective

mass description is not valid.

TABLE I: SBTB parameters for the majority(↑) and the

minority(↓) spin bands of bcc Fe(100). The band offsets (Ebo)

for ∆1, ∆2, ∆′

states of bands are also given [20].

2and ∆5 bands are shown. Orbital symmetry

Band Symmetry

to(eV )

Ebo(eV )

↑↓↑↓

∆1

4s,4pz,3dz2

3dxy

2.52.5 -11

∆2′

0.20.2 -1.50.4

∆5

4px,4py,3dxz,3dyz

11-3.5 -2.0

∆2

3dx2−y2

-0.2-0.35-2.1-0.8

TABLE II: SBTB parameters for MgO. Ubis the MgO barrier

height, to is the hopping parameter for ∆1 and ∆5 bands.

to(eV ) Ub(eV )

∆1 band0.642.8

∆5 band0.644.5

I-V characteristics through Fe-MgO-Fe calculated using

ab initio methods [3, 4, 5]. Only ∆1 and ∆5bands are

considered and ∆2and ∆2′ bands are ignored due to their

large decay rates [11].

This paper is divided into four sections. In Sec. II,

we discuss the theoretical approach and the assumptions

made. In Sec. III, we discuss the results. Finally, in Sec.

IV, we provide the conclusions.

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3

4

22

MAJORITY SPIN

33

1

?/a)

ky(?

k

2

0

1

-1

00

-2

-2

-2

-1

-1

0

0

1

1

2

2

kz(?/a)

MINORITY SPIN

ky(?/a)

kz(?/a)

FIG. 3: Equilibrium transmission over the two dimensional

transverse BZ for Ef = 0. k||= (kx,ky) resolved transmission

for the majority and the minority spin. The BZ is shown by

the dotted line. The transmission has units of 1019m−2and

k has units of m−2.

II.THEORETICAL MODEL AND

ASSUMPTIONS

To calculate the transmission for homogeneous materi-

als using the SBTB method, for each band we start with

the following Hamiltonian [18],

HSBTB=

?

Ebo+ 2to

−to

for i = j

for |i − j| = 1

(1)

where Ebo is the band offset and to is the hopping pa-

rameter. This results in cosine dispersion as:

ǫ(k) = Ebo+ 2to[1 − cos(ka)](2)

which gives a bandwidth of 4to. Here a is the lattice spac-

ing and k is the wave vector in the transport direction,

where a = 4.2˚ A and 2.86˚ A for MgO and Fe respectively.

This dispersion is exactly the same as that of an effective

mass Hamiltonian discretized using the finite-difference

method. However, in SBTB, to is a constant whereas

in effective mass approach it is inversely proportional to

the lattice spacing. In SBTB method, each lattice point

corresponds to a unit cell. Therefore, corresponding to

a ten layer (five unit cells) device considered to calcu-

late transmission using the EHT calculations, there are

five lattice points in the SBTB model. The extracted

SBTB parameters (Ebo and to) for various bcc Fe(100)

symmetry bands are shown in Table I.

For heterostructures, the above Hamiltonian is modi-

fied depending on the device material. e.g., in our system

the device is made up of a tunnel barrier, hence ULis the

Laplace potential linearly dropped across the insulator

region, which is added to the MgO Hamiltonian matrix

as follows:

HSBTB=

?

Ebo+ 2to+ UL(i,j) for i = j

−to

for |i − j| = 1

(3)

For MgO, the hopping parameters (to) and the band off-

sets (Ebo= Ef+Ub) are given in Table II. The method of

extracting these band parameters will be explained later

in this section.

At the heterostructure interface, the off-diagonal ele-

ments are taken such that the resulting Hamiltonian is

Hermitian to ensure that the energy eigenvalues are real

and current is conserved [19]. Additionally, the effect of

different interface structures may be incorporated by in-

troducing additional SBTB parameter for the interface,

and is left for future work.

We then use the equilibrium part of NEGF [19] to cal-

culate the transmission (ˆT), which gives unity transmis-

sion in the bandwidth region due to the one-dimensional

nature of the transport. Finally, the transmission per

unit area is calculated analytically by summing over the

2D transverse BZ as:

TSBTB=

1

4π2

?(π

a,π

a)

(−π

a,−π

a)

dk||ˆT(El) =

1

a2ˆT (4)

where a = 2.86˚ A is the cubic Fe lattice constant. This

equation is the key equation for the SBTB method, where

the energy resolved transmission can be captured aver-

agely by a suitable band width for a band specified by a

single hopping parameter toand the band offset Ebo.

We now present the method of extracting the suitable

band parameters for bcc Fe(100) and MgO. Due to the 1D

nature of the transport for the system under study, the

transmission is unity within the bandwidth. Therefore,

instead of matching the fine details in the transmission as

shown in Fig. 3 due to the varying band offsets and band-

widths over the 2D transverse BZ, we capture the trans-

mission in an average way by estimating the bandwidths

and band offsets for the various symmetry bands. Un-

like an effective mass model with infinite bandwidth, the

finite bandwidth in SBTB model is important to match

the transmission. Since this bandwidth equals 4to (see

eq. 2), the hopping parameters toand band offsets Ebo

can then be calculated.

The transmission calculated for bcc Fe(100) using the

EHT method serves as a benchmark for the SBTB cal-

culations. Adapted from Ref. [20] and also shown in

Table II, ∆2 and ∆2′ bands have d-type orbital sym-

metry. At k||= 0, the ∆5 band is doubly degenerate.

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4

5

MAJORITY

MINORITY

EHT

SBTB

5

MAJORITY

MINORITY

EHT

SBTB

?

?

55

??

0

rgy (eV)

Ener

0

rgy (eV)

Ener

1

024024

-5

Transmission (1019m-2)

024024

-5

Transmission (1019m-2)

(a)(b)

5

MAJORITY

MINORITY

EHT

SBTB

5

MAJORITY

MINORITY

EHT

SBTB

22

??

2'2'

??

0

rgy (eV)

Ener

0

rgy (eV)

Ener

024024

-5

Transmission (1019m-2)

024024

-5

Transmission (1019m-2)

(c)(d)

FIG. 4: The proposed individual band contribution to the en-

ergy resolved transmission per unit area using SBTB. Trans-

mission per unit area through bulk Fe using EHT (solid line)

and for ∆1, ∆2, ∆2′ and ∆5 bands using SBTB (dashed line)

for the majority (left) and the minority (right) spin. The

SBTB transferable parameters for these bands are estimated

from their finite bandwidths in the energy resolved transmis-

sion and are shown in Table I.

However for k||?= 0, this degeneracy is lifted. It is well

established that the bands with d-type orbital symme-

try are localized in energy and have smaller bandwidths

due to reduced hybridization. We therefore attribute the

two peaks in transmission due to the ∆2′ and ∆2 sym-

metry bands superimposed on the ∆1and ∆5bands as

shown in Fig. 4. We estimate the average bandwidths

and band offsets from the above-mentioned analysis and

subsequently extract to. The transmissions for ∆1, ∆2,

∆2′ and ∆5 bands are shown in Fig. 4. More sophis-

ticated methods may be used for estimating the band-

widths, which is left for future work.

For MgO parameterization, current through Fe-MgO-

Fe is compared against ab initio studies [3, 4, 5].

SBTB, the current density in P and AP configuration for

each spin orientation is given as:

In

J =e

h

?

dElTSBTB[f1− f2](5)

The ∆1 band current density in AP configuration is

nearly zero due to its half-metallic nature, thus the total

AP current density is dominated by the ∆5 band. We

thus, systematically extract the parameters for ∆5band

and then for the ∆1band by matching with the AP and

the P current density respectively as given by the ab ini-

tio calculations [3, 4, 5]. to is taken the same for both

the bands and Ub for the ∆1 and the ∆5 band is used

as a fitting parameter. Hence, in the SBTB model for

Fe-MgO-Fe MTJ there are three fitting parameters, Ub

for ∆1and ∆5bands and to. It is worthwhile to mention

here that unlike the Wentzel-Kramers-Brillouin (WKB)

approximation, where only the product of effective mass

Energy (eV)

MAJORITY SPINMINORITY SPIN

EHT

SBTB

Transmission (1019m-2)

Energy window

Integrated Transmission

(1020m-2)

EHT

SBTB

(a)

(b)

FIG. 5: Comparison of the energy resolved transmission and

the integrated transmission through bulk Fe using SBTB and

EHT for the majority and the minority spin. (a) The energy

resolved transmissions using the two methods match qualita-

tively. The transmissions are not matched below about -3eV

and -2eV for the majority and the minority spin respectively.

(b) The reference for integration is taken as Ef = 0.

m∗and Ubmatters, we find that in SBTB, only the re-

ported values of Uband to gave quantitative agreement

with the current levels for various barrier widths.

III.DISCUSSION OF RESULTS

In Fig. 2, the bandstructure of bcc Fe calculated us-

ing EHT is shown in the [100] direction for the majority

spin for different values of k||. In general, over the 2D

transverse BZ, these bands do not remain parabolic and

hence it is not possible to represent these band disper-

sions by suitable effective masses. Furthermore, they also

have different band edge Ebo. We find a similar trend for

the minority spin bands (not shown). Fig 3 is a plot of

k||resolved equilibrium transmission at the Fermi energy

using EHT for the majority and the minority spin. Here

the peaks in transmission are not only present at the Γ

point, but are dispersed throughout the BZ. Therefore,

it is imperative to consider the transverse BZ while cal-

culating the transport. This is also evident in the rich

transmission features as shown in Fig. 5(a) for the ma-

jority and the minority spins using EHT. These features

in transmission are due to varying band offsets and band-

widths for various k||over the transverse BZ.

In Fig. 5(a), the energy resolved transmission (given

by Eq. 4) calculated using SBTB is compared with one

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5

0.6

)

?m2)

y (A/

ensity

nt de

urren

Cu

JJP

JAP?

JP

JJAP

1

?

?

0.4

JAP

1

5

?

?

5

0.2

0

00.511.52

Vbias

FIG. 6: Current density for the ∆1 and ∆5 bands for a 4-

layer Fe-MgO-Fe MTJ device. Current densities for parallel

and anti-parallel configurations using the SBTB method. The

parameters for MgO are shown in Table II and for Fe in Table

I. The low bias transport in the P configuration is dominated

by the ∆1 band and in the AP configuration it is dominated

by the ∆5 band.

calculated using EHT. In Fig. 5(b), a comparison of in-

tegrated transmission using SBTB and EHT is shown

for the majority and the minority spins. Even though

the energy resolved transmission using SBTB matches

qualitatively with EHT calculation, the integrated trans-

mission for various energy windows match quantitatively,

and this is the most relevant quantity for the quantum

transport.

We also compare the low bias conductance for bcc

Fe(100) calculated using SBTB and EHT methods. The

low bias conductance (Go) at finite temperature is given

as [21]:

Go=

2q2

hkT

?µo+10kT

µo−10kT

dET(E)

e(E−µo)/kT

[1 + e(E−µo)/kT]2

(6)

where µois the contact Fermi energy. We obtain Goas

2.16×1015Sm−2and 2.23×1015Sm−2for the majority

and 1.08 × 1015Sm−2and 1.14 × 1015Sm−2for the

minority spin using SBTB and EHT respectively, which

match reasonably with each other.

In Fig. 6, the calculated current densities for ∆1and

∆5bands using the SBTB parameters are shown for a 4-

layer Fe-MgO-Fe device. These parameters are transfer-

able for various barrier widths and a quantitative agree-

ment in current densities and TMR ratios is obtained

with ab initio calculations [3, 4, 5] as shown in Fig. 1. JP

is dominated by the ∆1band current density at low bias

due to a lower potential barrier seen by the ∆1band. On

the other hand, the total AP current density is dominated

by the ∆5band. After a certain voltage, there is a sharp

increase in AP current of ∆1band due to its half-metallic

nature. Due to this increase, there is a sharp roll-off in

TMR, which ultimately becomes negative. These high-

bias predictions have also been reproduced in Ref. [17].

The TMR calculated using the SBTB method is shown

Fig. 1. TMR values match well with the ones obtained

using the ab initio model in Refs. [3, 4, 5]. To the best

of our knowledge, such a bias dependence and quantita-

tive agreement has not yet been captured within a simple

model whose computational efficiency is on the order of

an effective mass model.

IV.CONCLUSIONS

We have presented a single-band tight-binding model

for studying heterostructures with Fe-MgO-Fe(100) mag-

netic tunnel junction devices as an example due to their

technological importance. We have tried to capture the

band structure effects by using band parameters for var-

ious symmetry bands in the contacts and the barrier re-

gion. Features in TMR which are manifestation of the

electronic structure of material were captured quantita-

tively within this simple model, whose computational

complexity is on the order of an effective mass model.

Within the same set of parameters, I-V characteristics,

voltage dependence of TMR and thickness dependence of

TMR were captured quantitatively.

Acknowledgments

We thank S. Datta for very useful discussions. The

authors are also thankful to Dr. Heiliger for useful dis-

cussions and for sharing their data in electronic format.

T. Z. Raza acknowledges support by the MARCO Fo-

cus Center for Materials, Structure and Devices.

Raza is thankful to National Science Foundation (NSF)

and to Nanoelectronics Research Institute (NRI) through

Center for Nanoscale Systems (CNS) at Cornell Univer-

sity. We thank nanohub.org and NSF sponsored Network

for Computational Nanotechnology (NCN) for computa-

tional resources. T. Z. Raza is also thankful to E. C. Kan

for providing office space and physical resources.

H.

Appendix

For bcc Fe, EHT transferable parameters are adapted

from Ref.[2] and are also given in Table III. These

parameters are benchmarked against Slater Koster pa-

rameters [7, 22]. EHT is a semi-empirical tight-binding

method with non-orthogonal Slater type orbitals basis

set. This method has been used for large systems due

to its computational simplicity [23]. The overlap matrix

elements are calculated as Sij =< i|j >. The diagonal

elements of Hamiltonian (H) are the ionization energies

of the corresponding orbitals. The off-diagonal elements

are constructed using the overlap matrix (S) as: