Effects of resonant interface states on tunneling magnetoresistance
O. Wunnicke, N. Papanikolaou, R. Zeller, and P. H. Dederichs
Institut fu
¨
r Festko
¨
rperforschung, Forschungszentrum Ju
¨
lich, D-52425 Ju
¨
lich, Germany
V. Drchal and J. Kudrnovsky
´
Institute of Physics, Academy of Sciences of the Czech Republic, CZ-18040 Praha 8, Czech Republic
共Received 2 October 2001; published 22 January 2002兲
Based on model and ab initio calculations we discuss the effect of resonant interface states on the conduc-
tance of epitaxial tunnel junctions. In particular we show that the ‘‘hot spots’’ found by several groups in ab
initio calculations of symmetrical barriers of the k
储
-resolved conductance can be explained by the formation of
bonding and antibonding hybrids between the interface states on both sides of the barrier. If the resonance
condition for these hybrid states is met, the electron tunnels through the barrier without attenuation. Even when
both hybrid states move together and form a single resonance, strongly enhanced transmission is still observed.
The effect explains why, for intermediate barrier thicknesses, the tunneling conductance can be dominated by
interface states, although hot spots only occur in a tiny fraction of the surface Brillouin zone.
DOI: 10.1103/PhysRevB.65.064425 PACS number共s兲: 72.25.⫺b, 73.20.⫺r, 73.40.Rw, 73.40.Gk
The tunneling magnetoresistance 共TMR兲 of magnetic tun-
nel junctions consisting of ferromagnet
兩
insulator
兩
ferro-
magnet layers has attracted a strong scientific interest, partly
due to their potential application as magnetic random access
memories. Miyazaki and Tezuka
1
and Moodera et al.
2
were
able to obtain TMR ratios up to 20% in room-temperature
experiments and recently room-temperature values of more
than 50% were reported by various groups. The understand-
ing of the TMR and of the electronic structure has not pro-
gressed equally quickly. Model calculations
3,4
have shed
light on various aspects of the effect, but only recently have
ab initio calculations of the electronic structure and the spin-
dependent transport been reported.
5–8
In this paper, we will consider the tunneling through epi-
taxial junctions, which are characterized by two-dimensional
periodicity. Here recent ab initio calculations of the
k
储
-resolved conductance show a very interesting phenom-
enon: for certain discrete k
储
values ‘‘hot spots’’ or ‘‘spikes’’
appear in the transmitted intensity, showing that electrons
with such k
储
values can apparently tunnel through the junc-
tion with no or very little attenuation while all other states
are very strongly damped.
9–11
This effect occurs only in the
minority band of the ferromagnet and only for ferromagnetic
coupling. If present, it can dominate the tunnel characteris-
tics for intermediate thicknesses. For large thicknesses, in the
asymptotic limit, the behavior is determined by the complex
band structure of the insulator,
12
i.e., by those metal-induced-
gap states, which have the smallest imaginary part of the
perpendicular component k
z
of the Bloch vector. An example
for such hot spots is given in Fig. 1, showing the results of
ab initio Korringa-Kohn-Rostoker calculations for a junction
consisting of two fcc Co共001兲 half-crystals separated by 4
monolayers 共ML兲 of vacuum. The results are based on
density-functional theory in the local-density approximation
and the Landauer formula for the conductance. We have cho-
sen a vacuum layer as the simplest model of an insulating
barrier. Quite similar effects are also found in the
calculations
9–11
for insulating barriers.
The k
储
-resolved conductance is plotted in the two-
dimensional 共001兲 Brillouin zone at the Fermi level, for the
case of ferromagnetic-moment alignment of the two Co half
crystals in Fig. 1共a兲 for the majority electrons and in Fig.
1共b兲 for the minority ones. Figure 1共c兲 gives the conductance
for the case of antiparallel alignment. The majority conduc-
tance shows a smooth peak at the ⌫
¯
point. This is the ‘‘nor-
mal’’ behavior expected for a potential barrier, since elec-
trons with perpendicular incidence experience the smallest
decay in the vacuum region. As explained in terms of the
complex band structure,
12
this behavior is also expected for
most insulators. In contrast to this the minority conductance
is dominated by four double peaks in the ⌫
¯
–X
¯
directions
with extremely high intensity, compared to which the contri-
butions from other peaks seem to be negligible. The structure
of these double peaks will be discussed later. For the case of
antiparallel alignment 关Fig. 1共c兲兴 the k
储
-resolved conduc-
tance shows both features, i.e., a smooth peak at the ⌫
¯
point
with similar intensity as the majority conductance 关Fig. 1共a兲兴
and four double peaks at the same k
储
values in the ⌫
¯
–X
¯
directions. However, compared to the minority case 关Fig.
1共b兲兴, the intensity of these hot spots is reduced by more than
two orders of magnitude so that they are of minor impor-
tance. While it is tempting to attribute the hot spots to nu-
merical problems in the complicated evaluation of the con-
ductance, several groups have demonstrated recently that
these spots are connected with the occurrence of interface
states in the minority band.
9–11
Yet details of the mechanism,
in particular, why these hot spots are practically not damped,
are still not properly understood,
13
as we believe.
Here we present a detailed investigation of the properties
and the occurrence of such hot spots. As we will show, these
spectacular spikes only occur for symmetrical or nearly sym-
metrical barriers and are due to the formation of bonding and
antibonding hybrids of the interface states on both sides of
the barrier. As long as the bonding-antibonding splitting of
these two hybrids is larger than the genuine width of the
interface resonance the particle can tunnel through the barrier
PHYSICAL REVIEW B, VOLUME 65, 064425
0163-1829/2002/65共6兲/064425共6兲/$20.00 ©2002 The American Physical Society65 064425-1
without attenuation, i.e., the barrier becomes fully transpar-
ent. On the other hand for larger thicknesses both hybrid
levels fuse together into a single resonance, full transmission
is no longer possible, and attenuation sets in. However the
hybridization effect is still important and the peaks dominate
the behavior for an intermediate thickness region, before for
even large thicknesses the normal resonance behavior as in
Fig. 1共c兲 for the antiparallel alignment is obtained. We will
discuss this effect by two methods, first by a simple analyti-
cal model and second by ab initio calculations for the
Co
兩
vacuum
兩
Co junction using the tight-binding 共TB兲 linear
muffin-tin orbital 共TBLMTO兲 method.
In the model calculation we assume a barrier with a con-
stant potential of height V
B
and thickness D. The potential
V(x,y,z) in the two half crystals is independent of z, the
direction normal to the barrier. For the in-plane x and y de-
pendencies we assume a weak potential corrugation in the x
direction described by, e.g., the Fourier coefficients V
G
x
and
V
⫺ G
x
for the smallest reciprocal-lattice vectors in the x di-
rection. The eigenfunctions are then calculated in the nearly-
free-electron approximation. In the following we discuss
only the case of perpendicular incidence k
储
⫽ 0. In the unper-
turbed case, i.e., V
G
x
⫽ 0, we obtain an incident wave e
ik
z
z
with energy E⫽ k
z
2
and a second wave e
ik
z
z
cos(G
x
x) with
energy E⫽ k
z
2
⫹ G
x
2
. For V
G
x
⫽0, but small, the new eigen-
functions, i.e., the Bloch waves, still have the same dominat-
ing plane-wave character, either e
ik
z
z
or e
ik
z
z
cos(G
x
x), but
they also have a small admixture of the second plane-wave
component, so that the energy dispersion is only slightly
changed. In addition we introduce two attractive
␦
functions
⫺

1
␦
(z⫹ D/2) and ⫺

2
␦
(z⫺ D/2) at the interfaces on both
sides of the barrier, which allow us to introduce interface
states. The solutions in the two half crystals and in the barrier
are matched at the interfaces z⫽⫺D/2 and z⫽⫹D/2. If the

1
共or

2
) value is sufficiently large, it introduces at the left
共or right兲 interface an interface state for an energy below the
bottom of each band. The interface state below the lower
band (k
z
2
) is localized, while the one below the higher band
(k
z
2
⫹ G
x
2
) is resonant, since it hybridizes weakly with the
Bloch waves of the lower band (k
z
2
).
We consider now a Bloch wave k⫽ (0,0,k
z
) incident on
the barrier. If the barrier is sufficiently high, the transition
probability
兩
t
兩
2
is relatively small, as is shown in Fig. 2共a兲
for energies away from the energy of the resonant interface
states.
In the vicinity of the resonance, we observe anomalies.
The dashed curve gives the results for a single resonance on
one side of the interface, obtained by allowing only

1
or

2
to be nonzero. At the resonance energy we obtain then a
maximum in the transmission, thus an enhanced tunneling
probability which is due to the enhancement of the incoming
wave function by the large amplitude of the resonant inter-
face state. The following zero of
兩
t
兩
2
, an antiresonance, can
be explained by the Fano effect.
14
While this behavior is as
expected, a dramatic effect occurs in the case of two degen-
erate interface states on both sides of the barrier, created by
two equal
␦
potentials

1
⫽

2
. We observe two resonance
peaks, both showing full transmission,
兩
t
兩
2
⫽ 1, of the inci-
dent wave. The peaks are symmetrically situated at lower
and higher energies compared with the energy of the single
resonance, and both are accompanied by an antiresonance.
The calculation of the wave functions 关Fig. 2共b兲兴 shows that
for the lower peak both interface states form a bonding hy-
brid 共solid line兲 and for the higher peak an antibonding hy-
brid 共dashed line兲. Thus, for these resonances the total wave
function is nearly symmetrical or antisymmetrical with re-
spect to the center of the barrier, which directly explains the
FIG. 1. k
储
-resolved ab initio conductance plot in the SBZ for a
Co
兩
4 vac
兩
Co tunnel junction. The figure shows the conductance for
parallel aligned moments in the majority band 共a兲 and the minority
band 共b兲 as well as the conductance for antiparallel aligned mo-
ments 共c兲 of the Co half crystals.
O. WUNNICKE et al. PHYSICAL REVIEW B 65 064425
064425-2
full transmission without any attenuation since by entering
these states the particle comes to the other side of the barrier
without tunneling. Note that the total wave function is not
fully symmetrical or antisymmetrical, since the incident
wave, incident from the left or right, breaks the symmetry of
the system. Thus if the bonding-antibonding splitting ⌬ is
larger than the natural resonance width ⌫, ⌬⭓⌫, full trans-
mission will occur, while when both resonances fuse to-
gether, ⌬Ⰶ ⌫, the normal resonance behavior is observed.
This can also be understood in a time-dependent picture. If
the lifetime t
R
⫽ ប/⌫ of the resonance is much larger than the
hopping time t
H
⫽ ប
/⌬, which the particle needs to coher-
ently hop between the interface states on both sides of the
barrier, then during the lifetime of the resonance, bonding
and antibonding states can be formed, allowing the particle
to fully penetrate the barrier via these hybrids. If t
R
Ⰶ t
H
or
⌬Ⰶ ⌫ this channel is no longer open and the incident wave is
attenuated.
This effect shows up equally dramatically in Fig. 3, where
the transmission probability
兩
t
兩
2
at the resonance peak is
plotted versus the barrier thickness D. In the case of no in-
terface states (

1
⫽

2
⫽ 0) the normal exponential depen-
dence on the barrier thickness is obtained. For a single inter-
face state on one side of the barrier, the tunneling is
somewhat enhanced, but basically the same attenuation is
observed. On the other hand, for the symmetrical barrier with
degenerate interface states, full transmission is obtained for
small and intermediate thicknesses and the exponential decay
is strongly delayed. For very large thicknesses one finds in
all three cases the same exponential decrease, i.e., with the
same decay length, but with somewhat different amplitudes,
being largest in the two-resonance case.
More details about the tunneling behavior can be under-
stood from a phase-shift analysis. In the one-dimensional
case, as well as in the pseudo-one-dimensional case consid-
ered here 共i.e., for a given k
储
component兲, the scattering can
be described by two phase shifts
␦
S
(E) and
␦
A
(E). Here
␦
S
共or
␦
A
, respectively兲 refers to the phase shift, when two
waves with equal amplitudes 共or opposite amplitudes兲 are
incident from the left and right sides on a symmetrical bar-
rier, so that the total wave function is symmetrical 共or anti-
symmetrical兲 with respect to reflection around the center of
the barrier. With these phase shifts the transmission coeffi-
cient t(E) can be written as
15,16
t⫽ cos(
␦
S
⫺
␦
A
)e
i(
␦
S
⫹
␦
A
)
.In
the off-resonance energy region, the difference
␦
S
⫺
␦
A
is
slightly larger than ⫺
/2 共mod 2
), so that the transmission
兩
t
兩
2
Ⰶ 1. Near the bonding resonance, the phase shift
␦
S
(E)
quickly increases by
, so that
␦
S
⫺
␦
A
first crosses the reso-
nance value 0 共mod 2
) with
兩
t
兩
2
⫽ 1, and then the value
/2, for which
兩
t
兩
2
⫽ 0 共antiresonance兲. In the vicinity of the
antibonding resonance,
␦
A
(E) increases equally quickly by
, so that the difference
␦
S
⫺
␦
A
first crosses the value
/2
for the antiresonance and then the resonance value 0 of the
antibonding resonance. This is exactly the behavior of the
transmission probability seen in Fig. 2共a兲. When the bonding
and antibonding peaks move together, the ‘‘jumps’’ of the
phase shifts
␦
S
(E) and
␦
A
(E) start to overlap and partially
compensate each other in the difference
␦
S
⫺
␦
A
. Therefore
first one loses the crossing of the value
/2, i.e., the antireso-
nances disappear, while the curve
␦
S
⫺
␦
A
still crosses the
value 0 twice, so that the bonding and antibonding reso-
nances still show full transmission with
兩
t
兩
2
⫽ 1. However,
when the two resonances further move together, then the
maximum of the curve
␦
S
(E)⫺
␦
A
(E) becomes smaller than
0, and at this point attenuation sets in. This discussion ex-
plains the behavior seen in Fig. 3, showing that for the sym-
metrical barrier one obtains at the resonance energies full
transmission up to a critical thickness.
The transition region of the two-interface-state case in
Fig. 3, where the transmission
兩
t
兩
2
is no longer 1, but still
much larger than in the two other cases, is particularly inter-
esting and important for understanding the conductance re-
sults of Fig. 1. In this region we have ⌬ⱗ ⌫, so that the
splitting ⌬ is smaller than the resonance width ⌫ and the
bonding and antibonding peaks have moved together already
exhibiting a single resonance peak. Nevertheless also in this
case the hybridization effect is still important. This is illus-
FIG. 2. Results of the analytical model. In plot 共a兲 the transmis-
sion
兩
t
兩
2
for a 4 a.u. thick barrier with one interface state 共dashed
line兲 and for a symmetrical barrier with two interface states 共solid
line兲 is shown. In plot 共b兲 the real parts of the wave functions at the
bonding 共solid line兲 and antibonding 共dashed line兲 peaks are plotted
over the distance perpendicular to the interface plane (z axes兲. The
interface planes at z⫽⫾2 a.u. are indicated by two thin lines.
FIG. 3. Thickness dependence of the maximum transmission
兩
t
兩
2
at the resonance energies for no, one, and two interface states is
shown.
EFFECTS OF RESONANT INTERFACE STATES ON . . . PHYSICAL REVIEW B 65 064425
064425-3
trated in Fig. 4 for a symmetrical barrier with 8 a.u. thickness
and two interface states. For the energy at the maximum of
the transmission curve the wave function 关Fig. 4共b兲兴 is
clearly asymmetrical, with the real part 共solid line兲 being
bondinglike and the imaginary part 共dashed line兲 anti-
bondinglike. This can be explained by the fact that for t
R
ⱗ t
H
, when the resonance lifetime is shorter than the hop-
ping time, a certain percentage of electrons can still hop
during the lifetime of the resonance to the interface on the
other side, in this way avoiding the tunneling and enhancing
the transmission probability. The importance of this effect
cannot be seen well in Fig. 3, since in this thickness region
all three transmissions are small, so that important differ-
ences cannot be observed, but show up, e.g., in a semiloga-
rithmic plot.
In the following we come back to the conductances of the
Co
兩
vacuum
兩
Co 共001兲 barrier shown in Fig. 1 and present a
detailed study of the resonance effects with very high k
储
resolution. Contrary to the previous model case in the trans-
port calculation we vary the k
储
vector in the surface Brillouin
zone 共SBZ兲 and fix the energy at the Fermi level E
F
.We
have calculated the transmission T(k
储
,E
F
) by the Landauer
formalism using the ab initio TB-LMTO method, for which
we refer to Ref. 17. To resolve the spiky structures of the hot
spots a very large number 共980 700兲 of k
储
points in the irre-
ducible part of the SBZ has been used. For the following
results two fcc Co 共001兲 half crystals have been considered,
being separated by three layers of vacuum 共3 vac兲. For all k
储
in the SBZ, all eigenstates, either delocalized or localized,
have been evaluated and analyzed by the transfer-matrix
method 共for a recent application to multilayers, see Ref. 18兲.
Here we consider both the parallel alignment of the Co mo-
ments, in particular the conductance in the minority band, as
well as the conductance for the antiparallel alignment.
First we note the important similarities between the above
model results and the resonance effects in Fig. 1共b兲 and Fig.
1共c兲. In the case of parallel alignment, i.e., for a symmetrical
barrier with two degenerate surface states, we find extremely
high transmission peaks of nearly 1, while for the antiparallel
alignment where for both spin polarizations only one inter-
face state on one or the other side of the barrier exists, the
resonance peaks are reduced by more than two orders of
magnitude. This is fully in line with the model results shown
in Fig. 2–4.
Let us discuss in more detail the structure of the four
dominating double spots in Figs. 1共b兲 and 1共c兲, which occur
along the ⌫
¯
–X
¯
line, i.e., the diagonal k
x
⫽ k
y
. The isointen-
FIG. 4. Results of the analytical model. In plot 共a兲 the transmis-
sion
兩
t
兩
2
for a 8 a.u. barrier for a symmetrical barrier with two
interface states is shown. In plot 共b兲 the real part 共solid line兲 and the
imaginary part 共dashed line兲 of the wave function at the resonance
energy are plotted over the distance perpendicular of the interface
plane (z axes兲. The interface planes at z⫽⫾4 a.u. are indicated by
two thin lines.
FIG. 5. Enlarged ab initio transmission plot of a Co
兩
3 vac
兩
Co
tunnel junctions for 共a兲 parallel aligned moments in the minority
band and for 共b兲 antiparallel aligned moments of the Co half spaces.
Darker areas correspond to higher transmission probabilities. The
reciprocal-lattice vectors are given in units of 1/a, where a is the
lattice constant of the fcc Co. Note that the intensity scales in 共a兲
and 共b兲 are different by a factor 10.
O. WUNNICKE et al. PHYSICAL REVIEW B 65 064425
064425-4
sity contours of the transmission are shown in Figs. 5共a兲 and
5共b兲, where darker areas denote higher intensities.
For symmetry reasons the contours show mirror symme-
try with respect to the ⌫
¯
–X
¯
line. For the symmetrical case
关Fig. 5共a兲兴 all the isointensity lines merge together in two
points on this line k
x
⫾
⫽ k
y
⫾
⫽ (2.485 172⫾ 0.000 167)1/a (a
is the lattice constant of the fcc Co兲, while for the antiparallel
case in Fig. 5共b兲 all lines merge in a single point at k
x
⫽ k
y
⫽ 2.485 172)1/a. As indicated by the vanishing linewidth at
these points, these are localized states which do not contrib-
ute to the current. At these points they coexist with two in-
cident states with full transmission. As the above discussion
suggests, in the symmetrical case of Fig. 5共a兲 the wave func-
tions of the two split localized states are bonding and anti-
bonding combinations of the surface states of the two Co
half crystals. In fact in both cases the localization is enforced
by symmetry: The localized state is symmetric with respect
to a reflection at the ⌫
¯
–X
¯
axis while all propagating states
for this k
储
point are antisymmetric. However, for a small
deviation from the diagonal, other states are intermixed,
leading to the loss of orthogonality and to a transition from a
localized to a resonant state with a small, but finite half-
width. Note the large intensity difference between both con-
ductances, which is directly evident even though the inten-
sity scale for the antiparallel case is blown up by a factor 10.
For the parallel alignment the darkest contour area, indicat-
ing a transmission larger than 0.999, contains a line with full
transmission 1. With increasing deviation from the diagonal
the linewidth increases and the two resonances join together
into a single resonance, so that for larger distances the trans-
mission decreases.
For the total conductance one has to integrate the
k
储
-resolved conductances over the SBZ. It is directly evident
from Fig. 5 共Ref. 19兲 that only the hot spots for parallel
alignment will give an important contribution. However the
important contribution does not arise from the highest peaks
with full transmission, since in Fig. 5共a兲 these represent only
a line, being of measure zero for the integration. The big
contributions arise from the whole area around this line,
where the transmission is smaller than 1, but is still strongly
enhanced by the partial hopping effect explained above 共see
Fig. 4兲. Of course, both effects are directly connected, since
this strong enhancement only occurs in the vicinity of the
bonding-antibonding resonances. To obtain a more quantita-
tive feeling about the importance of hot spots, we present in
Table I the relative areas of the SBZ, for which the transmis-
sion lies between 0.1 and 1, and the relative contributions of
these areas to the total conductance for four different junc-
tions with 3 ML of vacuum.
The areas present only a tiny fraction of the SBZ, which
makes reliable numerical calculations very complicated. In
the case of antiparallel alignment, the Co
兩
3 vac
兩
Co barrier
still has hot spots, as seen from Fig. 1共c兲 or Fig. 5共b兲, but
with the chosen criterion they are of zero importance. The
same is true for the stronger asymmetric barrier
Co
兩
3 vac
兩
5Cu
兩
Co even for the parallel coupling and the
minority band. On the other hand for the barrier
Co
兩
3 vac
兩
5Co
兩
Cu, which from a physical point of view is
only slightly asymmetrical, hot spots are still important, even
though full transmission cannot occur. Finally for the sym-
metrical junction Cu
兩
5Co
兩
3 vac
兩
5Co
兩
Cu the hot spots
are, due to quantum well effects in the Co layers, even more
important than for the Co half space junctions.
As our calculations show, high-symmetry lines in the SBZ
are favorable for hot spots, since for symmetry reasons lo-
calized states are more likely to exist on these lines, so that
for nearby k
储
values resonance effects will occur. But hot
spots can also occur elsewhere in the SBZ.
To our best knowledge, in all calculations hot spots are
only observed in the minority band. The most likely reason
for this is that the occurrence is limited to a multiband Fermi
surface. As our analytical model showed, one needs a second
band to provide the coupling of the interface state to the
conducting ones, since otherwise the interface state would
remain localized.
As we have demonstrated above, large effects from inter-
face resonances in general require symmetrical or nearly
symmetrical barriers, since a one-sided resonance is less ef-
fective for the tunneling process. It is the formation of bond-
ing and antibonding states that leads to full transmission at
the resonance peaks and at the same time to strongly en-
hanced tunneling due to the partial hopping effect in the
vicinity of these resonances. Therefore we believe that inter-
face roughness as well as a finite bias voltage can substan-
tially reduce the importance of hot spots. However, this is a
quantitative question since a fully symmetrical barrier is not
required and reliable calculations would be highly desirable.
To summarize, we evaluated by model and ab initio cal-
culations the effects of interface states on the conductance of
magnetic tunnel junctions. The hot spots found by different
groups in ab initio calculations can be explained by the
formation of bonding and antibonding hybrids between
the interface states on both sides of the barrier. If the
resonance condition for these hybrid resonances is met
or nearly met, the electron can tunnel through the barrier
without or with only little attenuation. The effect explains
why the tunneling conductance can be dominated for
intermediate barrier thicknesses by interface states, although
these hot spots occur only in a tiny fraction of the surface
Brillouin zone. Surface states on one side of the interface
TABLE I. Contributions to the conductance in the SBZ with a
transmission probability between 0.1⬍ T(k
储
,E
F
)⭐1 for different
systems 共first column兲. The second column denotes whether the two
Co half spaces are aligned antiparallel or parallel, in which case
only the spin minority contributions are given. The relative area of
the SBZ and the relative contribution to the total conductance are
shown in the third and fourth columns.
System Area 共%兲
Contri-
bution 共%兲
Co
兩
3 vac
兩
Co antiparallel 0 0
Co
兩
3 vac
兩
Co minority 0.0014 24.1
Co
兩
3 vac
兩
5Co
兩
Cu minority 0.0003 5.2
Co
兩
3 vac
兩
3Cu
兩
Co minority 0 0
Cu
兩
5Co
兩
3 vac
兩
5Co
兩
Cu minority 0.0471 64.9
EFFECTS OF RESONANT INTERFACE STATES ON . . . PHYSICAL REVIEW B 65 064425
064425-5
