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
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
Transport across multilayered heterostructures is a
problem of great interest both for its intrinsic physics
as well as its device applications . 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  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
. 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
(ab initio) J
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 , 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
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
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 .
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  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.
. At high bias, our model predicts TMR roll-off and
ultimately becoming negative, which was later observed
in Ref. .
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
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 .
2and ∆5 bands are shown. Orbital symmetry
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 )
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 .
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.
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
To calculate the transmission for homogeneous materi-
als using the SBTB method, for each band we start with
the following Hamiltonian ,
for i = j
for |i − j| = 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
Ebo+ 2to+ UL(i,j) for i = j
for |i − j| = 1
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 . 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  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:
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.  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.
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:
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
MAJORITY SPINMINORITY SPIN
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  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
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
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
[1 + e(E−µo)/kT]2
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. .
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.
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.
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.
For bcc Fe, EHT transferable parameters are adapted
from Ref. 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 . 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:
TABLE III: EHT parameters for Fe majority and minority
spin adapted from Ref. . KEHT = 2.3.
Orbital Eon−site(eV )
ξ1 (˚ A−1) ξ2 (˚ A−1)
-11.9179 0.2396 0.84341.48913.3483
-10.36970.3229 0.77961.4891 3.3483
2KSij(Hii+ Hjj) (7)
The real space Hamiltonian H(? r) and the overlap S(? r)
matrices are transformed into Fourier (?k) space as :
Hmnei?k.(? rm−? rn)
Smnei?k.(? rm−? rn)
where?k = (?kx,?ky,?kz).
center unit cell and m represents the neighboring unit
cells. The band structure for the majority spin channel
is shown in Fig. 2 for various transverse wave vectors.
Next, the equilibrium transmission is calculated for a ten
layer device (five unit cells) with an infinite cross-section.
For calculating the transmission, we use the equilibrium
scheme of NEGF formalism . This calculation was
also checked by a more simple numerical calculation by
counting the independent propagating modes at a partic-
ular energy for bcc Fe. The infinite device cross-section
area allows to transform the real space Hamiltonian and
overlap matrices in the transverse direction to reciprocal
space. Thus, for each transverse reciprocal?k||= (?ky,?kz),
we have a one dimensional (1D) lattice. Finally, the en-
ergy resolved transmission per unit area is obtained by
The integer n represents the
and is shown in Fig. 4 for the majority and the minority
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