PHYSICAL REVIEW B
VOLUME 60, NUMBER 1
1 JULY 1999-I
Tetragonal equilibrium states of iron
P. M. Marcus and V. L. Moruzzi
IBM Research Division, T.J. Watson Research Center, Yorktown Heights, New York 10598
S.-L. Qiu
Department of Physics, Alloy Research Center, Florida Atlantic University, Boca Raton, Florida 33431-0991
~Received 6 January 1999!
First-principles total-energy calculations on tetragonal Fe show that the ferromagnetic and antiferromagnetic
phases have tetragonal equilibrium states with c/a.1, the fcc value. The bulk layers of an epitaxial film of Fe
on Cu~001!, which are almost fcc with the Cu lattice constant, are shown to be stable in the antiferromagnetic
phase, but inherently unstable in the ferromagnetic phase. The structure of tetragonal equilibrium antiferromagnetic Fe is estimated to be a53.47 Å, c53.75 Å. @S0163-1829~99!04125-9#
I. INTRODUCTION
The electronic structure of bulk Fe is especially interesting because of the rich variety of states it exhibits. Thus, Fe
exists in both bcc and fcc structures and has many magnetic
phases. This work examines the behavior of Fe in the more
general case of tetragonal atomic structure and looks for
metastable equilibrium tetragonal states of Fe in various
magnetic phases. The results show the existence of noncubic
tetragonal equilibrium states in both the ferromagnetic ~FM!
and antiferromagnetic ~AF! phases with comparable energies.
First-principles calculations of fcc Fe based on the KohnSham equations1 indicate that the lowest energy state is nonmagnetic ~NM! with lattice constant a53.44 Å ~WignerSeitz radius r WS52.54 a.u.). The FM and AF energy and
moment curves of Ref. 1, Fig. 2, which include the type-I AF
phase @alternating moments on ~001! planes#, have been verified many times.2–4 Figure 1 is a calculation using the augmented spherical wave method of Ref. 1, but now based on a
four-atom cell ~two adjacent bcc cells!, which gives the calculation greater stability, especially near phase transitions.
The second-order phase transition to the AF phase from the
NM phase is found to occur at a slightly expanded r WS
~0.4%! than the NM minimum, as shown by the position of
the abrupt rise of the local moment of the AF phase. Figure
1 applies to the bulk layers of Fe films of 5–10 atomic layers
grown in coherent epitaxy on Cu~001! at room temperature,
since the bulk layers are very nearly fcc at the Cu lattice
constant5 a53.61 Å, r WS52.67 a.u. with c/a50.98.
Although Fig. 1 shows the NM phase has the minimum
energy state of fcc Fe at a53.44 Å, it is wrong to think of
this state as the lowest equilibrium state of Fe near the fcc
structure. A simple symmetry argument based on Fig. 1
shows that the energy E of the AF phase can be reduced
below the fcc NM minimum by tetragonal deformation. The
minimum energy NM and AF states of fcc Fe are nearly
degenerate, but the E of the AF state will vary linearly with
tetragonal deformation @ DE}D(c/a) # since the AF state
does not have cubic symmetry, whereas the NM state does
have cubic symmetry, hence will be flat under tetragonal
deformation @ DE}„D(c/a)…2 # . Then in one direction of te0163-1829/99/60~1!/369~4!/$15.00
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tragonal deformation the E of the AF state E AF will go below
the E of the NM state E NM.
Later it is shown that increase of c/a by 8% lowers E AF
by about 4 mRy/atom, whereas E NM has a minimum at c/a
51 and actually increases when c/a changes from 1. Hence
epitaxial tensile strain exerted by Cu~001! on the Fe film
strains the tetragonal equilibrium state nearly into constrained fcc Fe at the Cu lattice constant and the fcc calculation of Fig. 1 applies. Thus at r WS52.67 a.u. Fig. 1 shows
the AF phase below the FM phase by about 3 mRy/atom and
below the NM phase by about 4 mRy/atom, in agreement
FIG. 1. Energies (E2E 0 ) mRy per atom relative to the energy
minimum and magnetic moments per atom in Bohr magnetons m B
of the nonmagnetic ~NM!, high-spin ferromagnetic ~FM!, and type-I
antiferromagnetic ~AF! phases of fcc Fe as functions of WignerSeitz radius r WS . Calculated with a four-atom cell, the augmented
spherical wave method and the fixed spin-moment procedure from
the spin-polarized Kohn-Sham equations.
369
©1999 The American Physical Society
370
P. M. MARCUS, V. L. MORUZZI, AND S.-L. QIU
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FIG. 3. Relative energy per atom (E2E 0 ) of Fe relative to FM
energy at the experimental volume V511.53 Å 3 as a function of
c/a, where a5a 8 & is the fct basal plane lattice constant. The NM
~h!, FM ~circle!, and AF ~3! points are from the first-principles
calculations of Peng and Jansen ~Ref. 11!. The curves are interpolated between the calculated points to show the two minima in the
AF and FM curves, particularly the tetragonal minima at 1.08 ~AF!
and 1.20 ~FM! that give equilibrium states.
FIG. 2. ~a! Constant energy contours ~small dashes!, two energy
minima ~•! and saddle point ~3! of vanadium V on the tetragonal
plane, coordinates c/a 8 and V/V 0 , where a 8 is the body-centeredtetragonal basal plane lattice constant and V 0 is the bcc reference
volume 12.57 Å3. The dashed lines bound the region of inherent
instability and the full line is the epitaxial Bain path ~EBP! of states
produced by isotropic epitaxial strain on the equilibrium states at
the two minima. ~b! Energy per atom E(c/a) along the EBP ~full
line! and E(c/a) at volume V51.04V 0 ~dashed line!. The plots
show that c/a 8 is the same at the minima and saddle points of the
two curves and E is slightly greater at the minima of the dashed
curve.
of FM Fe, but without the complications of separating the
other phases that occur in Fe.
Figure 2~a! shows contour lines ~short dashes! on which
E(a,c) is constant in the tetragonal plane which has coordinates c/a 8 and V/V 0 ; a 8 and c are tetragonal lattice constants @a 8 is the lattice constant of the square cross section of
a body-centered-tetragonal ~bct! description, and a5a 8 & is
the lattice constant for a face-centered-tetragonal ~fct! description#. Figure 2 uses the bct description; Figs. 3 and 4 use
the fct description. V is the volume per atom5c/a 8 2 /2
5ca 2 /4, and V 0 is the reference volume—for vanadium V 0
is the bcc value 12.57 Å3.
The contour lines in Fig. 2~a! show there are two minima
with measurements which show6 the bulk of the film to be
AF.
Section II uses the theory of tetragonal structures in Refs.
7 and 8 to deduce the metastable tetragonal equilibrium
structures of both AF and FM Fe from first-principles calculations and elastic and structural measurements. In Sec. III
the stability of the AF phase and instability of the FM phase
of Fe on Cu~001! are shown. Problems that need further
study are indicated.
II. DETERMINATION OF TETRAGONAL EQUILIBRIUM
STATES
Finding the tetragonal equilibrium states of Fe requires
knowledge of some features of tetragonal structure which are
discussed in detail in Ref. 7. Figure 2 illustrates the characteristic behavior of tetragonal states by showing the behavior
of nonmagnetic tetragonal vanadium. Figure 2~a! is adapted
from Ref. 7 ~Fig. 1! by the addition of the energy curves in
Fig. 2~b!; both figures are based on the first-principles calculations of tetragonal states in Ref. 8. The tetragonal behavior
of V is described here because it is very close to the behavior
FIG. 4. Part of the tetragonal plane of Fe showing the LEED
point ~3!, the EBP in linear elastic approximation, slope 21.92 Å3,
~full line! and tetragonal equilibrium state of AF Fe at c/a51.08,
V511.34 Å 3 ~circle!. Assuming the FM phase has the same equilibrium V at c/a51.20 ~h! and that the FM EBP has the same
slope as the AF EBP, then the FM EBP is the dashed line through
the FM equilibrium point. The FM EBP intersects the line on the
tetragonal plane on which a53.61 Å ~slope a 3 /4511.76 Å 3 ) at the
point c/a51, V511.76 Å 3 ~n!, hence the FM Fe phase on Cu~001!
is fcc with the Cu lattice constant.
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TETRAGONAL EQUILIBRIUM STATES OF IRON
of E ~•! and a saddle point ~3!. These are general features of
tetragonal structures.8,9 The minima are the equilibrium
states of vanadium V; the lower minimum is at the bcc state,
hence the ground state is bcc. The higher minimum is a
noncubic equilibrium state at c/a 8 51.78 which has actually
been observed when stabilized by coherent epitaxy of V on
Ni~001! and on Cu~001!.10 Around the minima are states of
constrained stability, such as those constrained by epitaxy.
The solid line which passes through the minima and the
saddle point is the epitaxial Bain path ~EBP!, which gives all
the constrained states of V produced by isotropic epitaxial
~two-dimensional! strain on the equilibrium states. A point
on the EBP for Fe will describe the structure of the bulk
layers of an Fe film epitaxial on Cu~001!. However, as will
be shown, the magnetic phase must be specified.
Reference 7 shows that the region between the two lines
of long dashes in Fig. 2~a!, in which the saddle point is
central, contains inherently unstable tetragonal states. Such
states cannot be stabilized by applied stresses, in contrast to
the constrained states around the minima mentioned above,
which are outside this unstable region. This behavior of tetragonal states will be used later to determine the stabilities
of the magnetic states of Fe on Cu~001!. Reference 7 also
gives a formula for the slope of the EBP, which will be used
later; at cubic points it is
S
dV
d ~ c/a !
D
52
EBP
2 ~ 122 n ! V
,
~ 11 n !~ c/a !
~1!
where n is the Poisson ratio of the cubic material ~for stresses
along a cubic axis!.
In Fig. 2~b! the solid curve is the energy per atom along
the EBP, E EBP(c/a), and the dashed curve is the energy per
atom at a constant V51.04V 0 as a function of c/a, E V (c/a).
This V is about 2% higher than the equilibrium V at c/a 8
51.78. Determination of the Fe tetragonal equilibrium states
requires the feature shown in Fig. 2~b!, namely that the c/a
values of the minima and of the saddle points of the two
curves coincide. This correspondence of the c/a values is
due to c/a and V being orthogonal coordinates at cubic stationary points and nearly orthogonal at noncubic stationary
points @Ref. 7, Eq. ~7!#. Also note that the E values at the
minima of E V (c/a) are only slightly higher than at the
minima of E EBP(c/a), since E is flat around the minima.
The available information on the energies of tetragonal Fe
from first principles consists of the work of Peng and
Jansen,11 who calculated some values of E V (c/a) for the AF,
FM, and NM phases of E. They used the full-potential linearized plane wave method including relativistic corrections
and solved the spin-polarized Kohn-Sham equations with the
von Barth–Hedin exchange correlation interaction, but without nonlocal corrections like the generalized gradient approximation ~GGA!. Also they used the experimental volume per atom of the bulk of the Fe film on Cu~001!, V
511.53 Å 3 . Their results, replotted from Table I of Ref. 11,
FM
are shown in Fig. 3. The E AF
V (c/a) and E V (c/a) curves are
notably different. Both show the characteristic two minima
of tetragonal structures. The FM curve resembles E V (c/a)
for vanadium with the ground state at the bcc structure and
the second minimum at large c/a 8 51.70 (c/a51.20). The
fragment of E NM
V (c/a) is enough to show a minimum at
371
c/a51, the fcc point, as anticipated in Sec. I. The AF curve
has a minimum at c/a51.08, also as anticipated in Sec. I.
Thus these constant-volume energy curves locate an equilibrium state of the AF phase at c/a51.08, c/a 8 51.53, and of
the FM phase at c/a51.20, c/a 8 51.70. However the values
of V, hence of a and c separately, of the equilibrium states
are not yet known.
To find V for the equilibrium state of the AF phase we use
the structure measured by quantitative low-energy electron
diffraction ~LEED! analysis5 of Fe on Cu~001!, which gives
the bulk structure of the film as a53.61 Å, c53.54 Å,
thereby giving a point at c/a50.98, V511.53 Å 3 on the
EBP of AF Fe. Then we use Eq. ~1! at the fcc point to find
the slope of the EBP as 21.92 Å3, on using the measured
Poisson ratio12 of g-Fe n 50.44 and V511.50 Å 3 ~slightly
smaller at c/a51 compared to V at c/a50.98) to draw the
EBP shown in Fig. 4. At c/a51.08 the EBP has V
511.34 Å 3 , hence a53.47 Å, c53.75 Å at the tetragonal
AF equilibrium point.
An estimate of a and c for the tetragonal FM equilibrium
state, where no LEED structure is available, can be made
with the plausible assumption that the equilibrium value of V
is the same as that for the AF phase, 11.34 Å3. Then with
c/a51.20 the FM equilibrium tetragonal lattice constants are
a53.35 Å, c54.02 Å. For both the AF and FM phases we
have used the minima of E V (c/a) at a V 2% larger than the
equilibrium V’s to find the c/a of the equilibrium states, the
same volume difference as was illustrated for V in Fig. 2~b!.
III. DISCUSSION
Section II shows that enough information is now available
to make a reasonable estimate of the structures of the metastable tetragonal equilibrium states of both the AF and FM
phases of Fe. First-principles total-energy calculations were
combined with measured epitaxial film lattice constants and
measured elastic constants of g-Fe. To check these structures
in a consistent calculation free of the various assumptions
used, requires direct evaluation from first principles of the
EBP’s of Fe in the AF, FM, and NM phases, as was done for
FM Co in Ref. 13.
The AF phase with equilibrium in-plane lattice constant
a53.47 Å requires a 4% isotropic epitaxial tensile strain to
be grown epitaxially on Cu~001!, producing a nearly fcc Fe
lattice at the Cu lattice constant, a53.61 Å, r WS52.67 a.u.
Hence, the fcc calculation of Fig. 1 applied to the bulk of the
film on Cu~001! shows that the AF phase is lowest in energy,
consistent with observations on the films.6
The existence of a nearly fcc Fe lattice at the Cu lattice
constant a Cu is thus a happy coincidental result of a Cu
53.61 Å. An exact fcc Fe lattice would be produced on a
substrate with a53.58 Å, since the AF EBP of Fe in Fig. 4
shows V511.50 Å 3 5a 3 /4 at c/a51. Note that diamond has
a53.57 Å and epitaxial Fe films have been shown to grow
well on diamond~001!.14 However no magnetic measurements were made on the films,
The FM equilibrium phase at a53.35 Å requires a 7.8%
tensile strain to be epitaxial on Cu~001!. The EBP of FM Fe
is drawn as a straight line in Fig. 4 ~long dashes! through the
point c/a51.20, V511.34 Å 3 assuming the same slope as
the AF phase, i.e., assuming the same Poisson ratio. At c/a
372
P. M. MARCUS, V. L. MORUZZI, AND S.-L. QIU
51 the FM phase then has V511.75 Å 3 , which gives a5c
53.61 Å, hence the FM phase is even closer to fcc than the
AF phase.
Although both phases are seen to be fcc on Cu~001!, Fig.
3 shows an important difference between the AF and FM
films on Cu~001!. Namely, the FM phase has cubic symmetry at the fcc point ~in a collinear description!. At a structure
with cubic symmetry the energy has an extremum, hence
change of the energy is second order, i.e., flat, with respect to
deformations of the structure. Therefore E FM
V (c/a) by symmetry is flat at the fcc point in a collinear description. Moreover, the flat shown in Fig. 3 at the fcc point is actually a
maximum at c/a51. Then, as noted in Sec. II, the E EBP also
has a maximum at c/a51 and E FM(a,c) has a saddle point
at c/a51. This means that the FM phase of Fe on Cu~001! is
in the middle of the unstable region and is inherently unstable.
However, the AF phase, which has ~001! planes with alternating moments, does not have cubic symmetry at the fcc
point, since the @001# direction is then different from the
@100# and @010# directions. Hence, the AF EBP varies linearly at c/a51 ~and also at 0.98!, and the saddle point is far
removed at c/a50.71. Thus, the AF phase of Fe on Cu~001!
is a constrained stable state. Note also in Fig. 3 that the AF
energy at c/a50.98 is about 4 mRy/atom above the minimum E at c/a51.08, as was noted in Sec. I.
This difference in the stability of the AF phase compared
to the FM phase of Fe films on Cu~001! is consistent with the
observation that AF films grow easily to 10 layers, whereas
FM films do not grow thicker than 4 layers, which then have
no bulk layers.15 If a substrate could be found with a closer
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PRB 60
to 3.35 Å, the FM phase might be stabilized, although Fig. 3
shows its minimum energy is 3 mRy/atom higher than the
AF minimum. When Fe is grown epitaxially on substrates
with c/a sufficiently below 1 to be outside the unstable region, the FM state is a constrained stable state of bcc Fe and
is more stable than the AF state, as is shown in Fig. 3. This
strained state of bcc Fe was found to occur for Fe/Rh~001!
~Ref. 16! (c/a50.82), for Fe/Cu3Fe(001) ~Ref. 17! (c/a
50.72), for Fe/Ag~001! ~Ref. 18! (c/a50.70), and for Fe/
Pd~001! ~Ref. 19! (c/a50.79).
Recent work has shown that fcc Fe at the Cu lattice constant has additional magnetic phases with still lower energies
than the AF phase shown in Fig. 1. Thus a noncollinear
spin-spiral magnetic phase has been found,4 which branches
off below the AF phase. Also, an AF phase with a longer
period ~moment sequence ↑↑↓↓ on four successive layers!
has been found.20 It remains to be seen whether tetragonal
deformation of these phases brings them below the energy
minimum at c/a51.08 of the simple ~type-I! AF phase considered here.
ACKNOWLEDGMENTS
P.M.M. thanks M. Scheffler of the Fritz-Haber Institute,
Berlin, Germany and J. Kirschner of the Mikrostrukturphysik
Institute, Halle, Germany for hospitality during the writing
of this paper. P.M.M. and V.L.M. thank IBM for providing
facilities. S.-L.Q. is grateful to NSF for research funds ~Contract Nos. DMR-9120120 and 9500654!. Some of the calculations were carried out using the computational resources
provided by the MDRCF, which is funded jointly by the NSF
and FAU.
14 051 ~1998!; ~V/Cu! Y. Tian, F. Jona, and P. M. Marcus ~unpublished!.
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