AB INITIO STUDY OF IRON AND
Cr/Fe(001)
H. C. HERPER∗, E. HOFFMANN and P. ENTEL
Theoretical Low-Temperature Physics,
Gerhard Mercator University, 47048 Duisburg, Germany
(Received ...)
Abstract
In the present paper interfacial mixing of a thin chromium overlayer on bcc iron is studied. The calculations are performed in the
framework of a pseudo-potential technique using the generalized gradient approximation for the exchange-correlation functional. Although
Fe and Cr do not alloy in the bulk system at low temperatures, strong
intermixing effects have been observed with Auger spectroscopy, if Cr
is epitaxially grown on bcc Fe(001). Besides these structural effects
we discuss the magnetic structure of the interface. It can be shown
that the results are in good agreement with the experimental findings
of Pfandzelter (Pfandzelter, Igel and Winter, 1996), provided we allow for lattice relaxation. Additionally, the structural and magnetic
properties of iron investigated by the pseudo-potential method are
compared to our former full potential results.
Keywords: Magnetic multilayers, Mixing effects, Density-functional theory
1. INTRODUCTION
Hetero-epitaxial growth processes of metals on metals, oxides or semiconductors are a great technologically challenge, because the occurrence of islands and interdiffusion often suppresses the layer-by-layer growth. Today,
there are still many open questions concerning the interface structure and
the penetration depth of the surface atoms, because Auger spectroscopy and
other surface sensitive methods do not give detailed information about the
structure of more than two layers. Therefore, they give no evidence of the
alloying effects at deeper layers. A metal on metal system of special interest
∗
Corresponding author, Tel:
heike@thp.uni-duisburg.de
+49-203-379-3564, fax:
1
+49-203-379-3665, email:
is chromium on bcc iron. Here the effects of interface alloying are discussed
for a Cr monolayer on bcc Fe(001) and are compared to the results from
Auger spectroscopy (Pfandzelter, Igel and Winter, 1996). Measurements of
bulk Fe-Cr samples have shown that a miscibility gap exists, which stretches
nearly over all concentrations (Kubaschewski, 1982). Only two stable phases
have been found. A fcc phase on the iron rich side and a σ-phase around 45
at.% Cr exist. Both phases vanish with decreasing temperature. However,
intermixing effects have been observed, if Cr is deposited on an Fe substrate.
The magnetic properties also exhibit interesting features. An odd number
of Cr layers is expected to favor ferromagnetic (FM) coupling, but instead of
that antiferromagnetic (AFM) spin structures are observed (Unguris, Celotta
and Pierce, 1992), due to alloying effects at the interface. In addition, it is
known that Cr itself has an incommensurate spin density wave (Fawcett,
1988), but if it is deposited on a surface the magnetic behavior changes, and
collinear spin structures are favored.
The magnetic and structural properties of the Cr-Fe interface are investigated within an ab initio pseudo-potential method. First, we check the
accuracy of the pseudo potentials, because it is well known that the treatment of the magnetovolume effects of iron is complex. The present results
for bulk iron are compared to the experimental findings, as well as to our
former full potential results (Herper, Hoffmann and Entel, 1999).
2. BULK PROPERTIES
The magnetic and structural properties of iron are investigated by using
two different pseudo potentials. The calculations are performed using the
VASP code (Kresse and Hafner, 1996; Kresse and Furtmüller, 1996) and the
generalized gradient approximation (GGA) (Perdew and Wang, 1992). Both
methods employ ultra soft Vanderbilt pseudo potentials, but the description
of the 3p semi-core electrons is different. In the first potential (PS1) the
3p orbitals are part of the pseudo core and possible dispersion effects are
neglected. In the second a pseudo potential (PS2) is used, which does not
include the semi-core states. In this case the 3p states are treated selfconsistently. The results for the magnetic and structural properties of iron are
summarized in Fig. 1. The choice of the pseudo potential has no significant
influence on the properties of bcc iron, which can be seen from Table 1.
Even for the simpler PS1 potential there are only small deviations from
the experimental values and from the full potential results. The calculated
bulk modulus is about 5.8% smaller than the experimental value, whereas the
volume is slightly overestimated. These deviations decrease if the PS2 potential is used. This has also been found by Moroni and coworkers (Moroni,
Kresse, Hafner and Furthmüller, 1997). Nevertheless, both pseudo potentials
are at least sufficient for the description of the ground state properties of bcc
iron. This does not hold for fcc iron, which is known to show magneto-volume
2
3
3
B)
2
2
M(
1
1
PS1 (4s 3d)
PS2 (3p 3d 4s)
0
0
fcc AFM-I
Etot (mRy/atom)
24
fcc AFM-I
24
fcc NM
16
16
8
8
fcc NM
fcc FM
fcc FM
bcc FM
bcc FM
0
0
64
72
80
88
64
V/atom (a.u.)
72
80
88
V/atom (a.u.)
Figure 1: Calculated phase diagram of iron. Two different pseudo potentials
are used. The PS1 (left panel) includes the 3p electrons and in the PS2 (right
panel) the 3p electrons are treated as valence states. The FM bcc phase is
marked through dotted lines. Full lines indicate fcc states.
instabilities leading to the anti-Invar effect. In a previous work we have shown
that the anti-Invar effect can be well understood from full-potential calculations (Herper, Hoffmann and Entel, 1999). However, the PS1 fails for γ-Fe,
because fcc Fe would have a FM ground state, which is obviously not the
case. The AFM state with a smaller volume has a higher energy and can
only be occupied at higher temperatures. This situation would be expected
for a typical Invar system like Fe65 Ni35 , but not for an anti-Invar system. If
the PS2 is used instead of PS1 the AFM state is more stable than all other
fcc states, which is in accordance with the experimental results. This was
not clear in a previous work (Moroni, Kresse, Hafner and Furthmüller, 1997)
where the AFM and FM high-spin states of γ-Fe are nearly degenerated.
The difference can be accounted for on the basis that we use a higher plane
wave cut-off, therefore, a larger number of plane waves, which enhances the
accuracy. In the present work the self-consistent treatment of the 3p orbitals
3
bcc Fe (FM)
bcc Cr (AFM)
PS1
PS2
FLAPW
EXPT.
V
79.33
78.36
77.22
78.94
B
162
166
174
172
V
82.00
—
80.30
81.10
B
130
—
173
160
Table 1: Calculated bulk moduli B (in GPa) and ground state volumes V in
(a.u.) for FM bcc Fe. The pseudo-potential calculations are performed by
using the VASP program (Kresse and Hafner, 1996; Kresse and Furtmüller,
1996). The full potential results are taken from Herper et al. (1999). All
calculations are done within the GGA (Perdew and Wang, 1992). Experimental data for iron are from Acet et al. (1994). The Cr data are taken from
Donohue (1982); Guillermet and Grimvall (1989).
is obviously an improvement, but the description of the two FM states is still
problematic. In contrast to what is expected from the full potential calculations, the sequence of the high-spin and low-spin state is wrong, because the
low-spin state has a higher energy. Summarizing we can state that the PS2
obviously works better for fcc Fe, but unfortunately it is rather time consuming and offers no real advantage over the full potential. As mentioned above,
the ground state properties of bcc iron come out sufficiently well for both
pseudo potentials. Therefore, the PS1 can be used for the investigations of
a Cr overlayer on bcc Fe(001).
In the following part we discuss several aspects of the Fe-Cr bulk system
in order to make sure that the interfacial effects discussed later on are no
artefacts of the bulk properties. Therefore, bcc Fe100−x Crx compounds with x
varying in steps of 25% are examined in view of their miscibility and magnetic
properties. All compounds prefer more or less an antiparallel alignment of
the Fe and Cr spins, see Fig. 2). On the average the iron moment is enhanced
compared to pure bcc iron as long as the compounds contain less than 50 at.%
Fe. The Cr moment breaks down for FeCr, but with increasing Cr content
it increases again. Therefore, the net moment of the unit cell decreases from
pure Fe to FeCr3 .
In addition, the total energies of the compounds have been used to calculate the mixing energy em at T = 0 K
Em = EFe−Cr − [(100 − x)EFe + xECr ]
(1)
with EFe and ECr being the total energies of FM bcc iron and AFM bcc
chromium. The mixing energy provides useful information about ordering
and disordering trends in the system. The mixing energies for the Fe-Cr
system are always positive, see Fig. 2. This means that no cubic Fe-Cr
compounds exist at T = 0 K. Structures other than cubic have not been
4
3
Mixing energy (mRy/atom)
Fe100-xCrx
M(
B/atom)
2
1
0
-1
-2
0
20
12
8
4
0
40 60 80 100
x (at.% Cr)
Fe100-xCrx
16
0
20
40
60
80 100
x (at.% Cr)
Figure 2: Distribution of the average magnetic moments versus the chromium
amount in bcc Fe100−x Crx compounds (left panel). Open symbols correspond
to the magnetic moment of the Fe atoms and filled squares mark the Cr
moments. The full circles indicate the net moment per unit cell. The mixing
energy for bcc Fe100−x Crx compounds versus the Cr amount is shown in the
right panel.
investigated. The observed decomposition agrees qualitatively with the experimental findings (Kubaschewski, 1982; Pepperhoff and Acet, 2000). Experimentally the decomposition starts below 800 K. Therefore, calculated
absolute values are by a factor of 3 higher than the measured values (Pepperhoff and Acet, 2000), but nevertheless, these simple calculations reveal
the right chemical trend.
3. Cr ON BCC Fe(001)
It has been already mentioned in the beginning that no bulk Fe-Cr alloys exist
at low temperatures. Therefore, one could suspect that Cr atoms deposited
on a bcc Fe substrate remain on the surface, which obviously is not the
case. Numerous experiments (Pfandzelter, Igel and Winter, 1996; Davies,
Stroscio, Pierce and Celotta, 1996) show that the Cr atoms penetrate into
the substrate. Some theoretical work has also been done in this field (Turek,
Weinberger, Freyss, Stoeffler et al., 1998; Freyss, Stoeffler and Dreyssé, 1997).
However, there are still a number of questions concerning the penetration
depth, interface alloying as well as the magnetic behavior. In order to study
the mixing effects at the Fe-Cr interface and the related magnetic effects, we
have performed supercell calculations for one Cr monolayer on a bcc Fe(001)
substrate. All calculations are done with the VASP program employing the
5
Cr
vacuum
Fe
vacuum
11 ML
7 ML
Figure 3: Periodic supercells for 1 ML Cr on bcc Fe(001). The present systems consist of 9 ML Fe/2 ML Cr and 5 ML Fe/2 ML Cr, whereas the second
Cr layer is added for symmetry reasons. The atoms are covered with 11 ML
and 3 ML of vacuum, respectively.
GGA. We use the PS1 potential, which is discussed in Section . We have
used bcc supercells in slab geometry with one monolayer (ML) Cr on an
Fe substrate. For symmetry reasons a second monolayer has been added to
the bottom of the cell, which can be seen in Fig. 3. In order to study the
influence of the system size and to rule out side effects from the periodic
structure, we use two different types of super cells. They consist of 7 and 11
layers of atoms covered with 3 and 11 layers of vacuum, respectively. The
initial lattice constant is chosen as a = 5.41 a.u., which corresponds to the
calculated value for bcc FM iron (Table 1).
Starting from a perfect Cr monolayer on the Fe substrate for the two
different unit cells, we change the Cr amount on the surface. In order to
examine the interface structure, some of the Cr atoms are interchanged with
the Fe atoms from the first layer below the surface, until the Cr layer is completely covered with an Fe monolayer. Thereby, we change the Cr amount in
steps of 25 at.%. It should be mentioned that all investigated structures are
ordered surface compounds, and no disorder effects are taken into account.
All calculations have been performed twice using the 7 ML (+3 ML vacuum)
and the larger 11 ML (+ 11 ML vacuum) supercells. The results show no significant differences for the total energy or the magnetic moments. There are
only small deviations of about 1.5 mRy in the absolute values of the total
energy, and the magnetic moments are identical for both systems. Only the
size of the relaxation effects seems to depend on the size of the unit cell,
which will be discussed further below. However, the results for the concentration dependence show a good agreement with the experimental findings
(Pfandzelter, Igel and Winter, 1996), see Fig. 4. The multilayer structure
6
7 ML, ideal positions
7 ML, relaxed
11 ML, ideal positions
11 ML, relaxed
Energy (mRy/interface)
16
8
0
1 ML Cr on Fe(001)
-8
0
20
40
60
x (at.% Cr in top layer)
80
100
Figure 4: Total energy per interface versus the Cr amount of the top layer.
Both results, without lattice relaxation (dotted lines, open symbols) and after
the relaxation (full lines, filled symbols) are given.
is most stable, if half of the Cr atoms are located in the 1st Fe layer below
the surface. This is connected with an energy gain of about 2.8 mRy per
interface relative to the energy of the ideal Cr monolayer (100 at.% Cr on
the surface). The gain is somewhat smaller for the smaller supercell, but
it shows the same trend. Further penetration of Cr atoms into the 1st Fe
layer leads to a strong increase of the total energy. Therefore, such configurations are less stable and cannot be expected at low temperatures. This
corresponds well to the experimental results from Auger spectroscopy. After growing a single monolayer Cr on a bcc Fe substrate the remaining Cr
amount on the surface is 45 at.%. The rest of the Cr atoms move into the 2nd
layer. Deeper layers solely consist of Fe atoms (Pfandzelter, Igel and Winter,
1996). This is in nearly perfect agreement with the present results, whereas
the Cr atoms are not allowed to interchange with the Fe atoms from deeper
layers. It should be emphasized that the agreement with the experimental
findings is connected with the allowance of lattice relaxation. If the atoms
are kept fixed on the positions of the ideal lattice no interface mixing would
occur and most of the Cr atoms would remain on the surface (Fig. 4). The
configuration with 50 at.% Cr on the surface would then have 2 mRy higher
energy relative to the energy of the perfect Cr overlayer.
In the following we have investigated lattice relaxation effects in detail for
the considered surface compounds. The layer-resolved results for the relative
movements ∆a/a from the ideal positions are given in Fig. 5. The presented
7
Layer
M(
2
3
4
1
10
2
5
0
100% Cr
2
3
4
5
1
10
2
5
a/a (%)
0
2
3
4
5
6
0
-5
-2
75% Cr
2
3
4
5
-10
6
1
2
3
4
20
2
a/a (%)
B/atom)
M(
6
75% Cr
0
-2
50% Cr
-4
1
2
3
4
5
5
6
50% Cr
10
0
-10
6
1
4
10
2
5
2
3
4
5
6
25% Cr
a/a (%)
B/atom)
5
-10
6
4
M(
4
0
4
1
0
0
-5
-2
25% Cr
-4
1
2
3
4
5
-10
6
1
4
10
2
5
2
3
4
5
6
0% Cr
a/a (%)
B/atom)
3
-5
-2
-4
M(
2
100% Cr
1
B/atom)
6
4
-4
M(
Layer
5
a/a (%)
B/atom)
1
0
0
Cr, 7 ML
Fe, 7 ML
Cr, 11 ML
Fe, 11 ML
-5
-2
0% Cr
-4
1
2
3
4
5
-10
6
1
Layer
2
3
4
5
6
Layer
Figure 5: Magnetic moment (left column) and relative lattice relaxation
(right column) versus the number of the layer. The numbers in the boxes
express the relative Cr content on the surface.
8
values are averages over all Fe or Cr atoms per layer. As expected, the
relaxation effects for the deeper layers are rather small, except for the totally
covered Cr layer (0 at.% on the surface). In the latter case the relative lattice
relaxations are of about 2%. This confirms out concentration dependent total
energy results, because this state has a much higher energy and tries to lower
it by strong movements of all atoms, compare Fig. 4. In all other cases only
the first three layers show considerable lattice relaxation effects. Especially
large relaxation effects are observed for the surface atoms of the system with
50 at.% Cr on top. The Cr atoms move out of the multilayer, whereas the Fe
atoms move inwards. This means that the Cr atoms on the surface and in the
2nd layer try to repel each other, which supports the experimental findings
that in Cr surface alloys nearest neighbor occupation is suppressed (Davies,
Stroscio, Pierce and Celotta, 1996). In view of this large repulsion it can
be understood that the calculations without lattice relaxation do not give
the correct structure. In this case the ideal Cr monolayer will be preferred,
because this structure shows the smallest relaxation effects, see Fig. 5.
The interchange of Fe and Cr atoms is also connected with strong changes
of the magnetic structure. It should be mentioned that the Fe moments of
the deeper layers are slightly higher than the bulk value of 2.2 µB , which may
be an artifact of the periodic structure. The chromium atoms on the surface
are always antiparallel aligned to the Cr atoms in the 2nd layer, whereas
the Fe atoms are ferromagnetically ordered, which is expected for this lattice
constant (Fig. 1). In accordance with the experimental findings we observe an
increase of the surface magnetic moments as compared to the bulk moments.
The magnetic moments for the Cr atoms amount to 3.0 − 3.3 µB depending
on the concentration of Cr on the surface. These values are in agreement
with the results from in situ measurements (Turtur and Bayreuther, 1994).
In accordance with photoemission experiments (Hillebrecht, Roth, Jungblut,
Kisker et al., 1992), we observe an antiparallel alignment of the Cr monolayer
on the Fe substrate. This holds even if the Cr monolayer is completely covered
with a monolayer Fe (Fig. 5). If the relative Cr content in the 2nd layer
becomes larger than 25 at.%, the Cr moments in this layer align antiparallel
to the bulk magnetic moments, whereas the Cr atoms on the surface accept
a parallel alignment to the magnetic moments of the Fe atoms. The AFM
coupling of the Cr atoms seems to be stronger than the coupling between
the Fe and Cr atoms. The structures with incomplete Cr layers can be
understood as a simple model for non-layer-by-layer growth. This means
that a new Cr layer, which starts growing on an incomplete Cr layer, is
always antiparallel aligned to the Cr atoms in the former layer. One should
keep in mind that the total number of atoms corresponds to one monolayer.
It has to be investigated, whether this behavior changes with increasing the
number of Cr layers. Finally, it should be mentioned that the influence of
the lattice relaxation on the magnetic structure is less important. Apart
from the absolute value of the moments, the same magnetic structures have
already been found when the atoms are kept fixed.
9
4. SUMMARY
In the first part we have studied the phase diagram of iron within a pseudopotential method using two different pseudo potentials. It has been shown
that the semi-core states of Fe have to be calculated self-consistently in order
to describe the magneto-volume effects of fcc Fe. For bcc Fe, and therefore
for the present multilayer calculations, it is sufficient to treat the 3p states
as core states. Secondly, we have discussed the intermixing effects as well as
the magnetic structure of one monolayer Cr on bcc Fe(001) using the same
method. The penetration of the Cr atoms in the substrate has been changed
in steps of 25 at.% Cr. From our results we can conclude that 50% of the
Cr atoms penetrate into the 2nd layer. This is connected with large lattice
relaxation effects. Therefore, this is not observed if the atoms are kept fixed.
In this case all Cr atoms would remain on the surface.
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11