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PHYSICAL REVIEW B 83, 024407 (2011)
Magnetism of 3dtransition-metal monolayers on Rh(100)
A. Al-Zubi,*G. Bihlmayer,†and S. Bl¨
ugel
Institut f¨
ur Festk¨
orperforschung and Institute for Advanced Simulation, Forschungszentrum J¨
ulich and
JARA, D-52425 J¨
ulich, Germany
(Received 2 November 2010; published 24 January 2011)
Weemploy the full-potential linearized augmented plane-wave method to report a systematic density-functional
theory study of the magnetic properties of the 3dtransition-metal (V, Cr, Mn, Fe, Co, and Ni) monolayers deposited
on the Rh(100) substrate. We find that all monolayer films are magnetic. The size of the local magnetic moments
across the transition-metal series follows Hund’s rule with a maximum magnetic moment of 3.77μBfor Mn.
The largest induced magnetic moment of about 0.46μBwas found for Rh atoms adjacent to the Co film. When
relaxations are included, we predict a ferromagnetic (FM) ground state for V, Co, and Ni, while Cr, Mn, and Fe
favor a c(2 ×2)antiferromagnetic (AFM) state, a checkerboard arrangement of up and down magnetic moments.
The magnetic anisotropy energies of these ultrathin magnetic films are calculated for the FM and AFM states.
With the exception of Cr, the easy axis of the magnetization is predicted to be in the film plane. Rough estimates
of the ordering temperatures are given. To gain an understanding of the c(2 ×2) AFM state of Fe/Rh(100), we
analyze this result with respect to the trends of the magnetic order of 3dmonolayers on other 4dsubstrates, such
as Pd(100) and Ag(100).
DOI: 10.1103/PhysRevB.83.024407 PACS number(s): 75.70.Ak, 71.15.Mb, 73.20.−r
I. INTRODUCTION
During the past two decades, theoretical and experimental
investigations were performed to understand the magnetism
of ultrathin magnetic films on (100) oriented nonmagnetic
substrates. Mainly the weakly interacting coinage metals (Cu,
Ag, Au) and some transition metals (TMs), for example,
Pd, have been chosen as substrates in order to minimize the
interaction between monolayer and substrate.1,2Cu with an
experimental lattice constant of a0=3.61 ˚
A turned out to be
an ideal template for fcc bulk TMs, while Ag (a0=4.09 ˚
A)
and Au (a0=4.08 ˚
A) are templates to grow bcc metals.
Employing density functional theory (DFT), some general
trends were identified for 3dmonolayers (MLs) on these
substrates: (i) The magnetic moments of the monolayers are
considerably enhanced as compared to the equivalent bulk
systems and, (ii) similar to the bulk cases, Fe, Co, and Ni
are ferromagnetic (FM) on these substrates, while V, Cr,
and Mn prefer a c(2 ×2) antiferromagnetic (AFM) structure
(i.e., a checkerboard arrangement of antiparallel magnetic
moments,3,4) a magnetic structure that cannot be derived from
respective bulk phases. Experimentally, Ortega and Himpsel
studied 3dmonolayers on Ag(100) and confirmed the theoreti-
cal predictions, most notably the magnetism of V on Ag(100).5
More recently, the field has taken a different turn, moving
away from monolayers on weakly interacting substrates to
those on 4dand 5dtransition-metal ones. This is motivated by
a couple of unexpected findings, which include the prediction
of a ferromagnetic phase for the prototype antiferromagnet Cr,6
the prediction6–9and experimental verification of the c(2 ×2)
AFM phase for Fe,10 the prediction of c(2 ×2) AFM Co,6
and the discovery of a homochiral cycloidal magnetic phase
for Mn (Refs. 11,12), all as monolayers on W(100). Signs of
magnetic frustration have been reported from calculations of
Fe/Mn alloys on W(001) (Ref. 13) and also Fe/Ir(001) was
found to be close to a transition from FM to AFM coupling,
depending on the relaxation of the overlayer.14 Experimentally
and theoretically, a complex long-ranged magnetic structure
of an Fe monolayer on Ir(111) has been observed.15
These findings motivate a more systematic investigation
of monolayers on 4dand 5dsubstrates in general. In this
paper we investigate 3dmonolayers on the 4dTM substrate
Rh(100). Rh has a large Stoner enhanced susceptibility, as
shown for Rh films on Fe (Ref. 16), and FeRh is known to form
ordered alloys in the cesium chloride (CsCl-type) structure
with subtle magnetic properties.17–19 The lattice constant of
Rh (a0=3.80 ˚
A) is in between those of Cu and Ag and thus
Rh serves as a potential substrate to grow artificial phases
of 3dtransition-metal films such as fcc Fe stabilized under
tensile strain or bcc Co under compressive strain. The Rh(001)
substrate provides favorable growth conditions for transition-
metal films despite a large lattice mismatch of fcc Fe or Co and
bcc Fe with Rh of about 6%, 8%, and −7%, respectively. For
example, no notable intermixing has been encountered at the
interface of Fe/Rh(100) during growth of Fe films.20 Epitaxial,
pseudomorphic layer-by-layer growth of one and two layers
of Co on Rh(100) was reported by Begley et al.,21 and several
groups 20,22–24 have been able to grow pseudomorphically even
thicker films of face-centered tetragonal Fe on Rh(100). Recent
reports show that FeCo alloys can be grown on Rh(100) where
the composition of the alloy can be used to tailor the magnetic
anisotropy of the system.25 Moreover, the same group used
Rh(100) spacers to achieve a FM or AFM interlayer exchange
coupling depending on the Rh thickness.26
Hayashi et al.23,24 concluded on the basis of soft x-ray
magnetic circular dichroism experiments measured at room
temperature that a monolayer and a bilayer of Fe are not
ferromagnetic and interpreted them as magnetically dead,
caused by the large strain exerted in the interface of the thin
film and the substrate. Hwang et al.27 found experimentally
a suppression of the ferromagnetic order of Fe overlayers on
the Rh(100) surface, and they as well as Spisak and Hafner28
predicted a c(2 ×2) AFM order for one ML Fe on Rh(100) on
the basis of DFT calculations.
We study the chemical trend of the interlayer relaxation,
the magnetic moments and structure, the magnetic anisotropy
energy (MAE), and the magnetization direction across the
024407-1
1098-0121/2011/83(2)/024407(7) ©2011 American Physical Society
A. AL-ZUBI, G. BIHLMAYER, AND S. BL ¨
UGEL PHYSICAL REVIEW B 83, 024407 (2011)
transition-metal series of 3dMLs on Rh(100). We calculate
the spin-orbit and dipole-dipole contribution to the MAE and
together with the energy difference between the FM and AFM
states we estimate the N´
eel and Curie temperatures of uniaxial
antiferromagnets and ferromagnets, respectively. The results
on the electronic and magnetic properties are compared to
previous theoretical studies on 4dmetal substrates such as
Pd(100) (Ref. 3) and Ag(100).4Since the dband of the sub-
strate is filled by one additional electron when comparing Rh
(a0=3.80 ˚
A) to Pd (a0=3.89 ˚
A) and Pd to Ag (a0=4.09 ˚
A),
and the lattice constant increases by about 2.5% each, the
comparison of the results of the 3dTM MLs on these
substrates allows an efficient analysis of the role of different
in-plane lattice constants, the hybridization between 3dand
4dorbitals, and the charge transfer across the 3d/4dinterface
for the magnetic properties in these films. In particular, from a
comparison of the non-, ferro-, and antiferromagnetic densities
of states (DOS) of Fe on Rh, Ag, or Pd, we elucidate the
mechanisms that drive the Fe ML into the antiferromagnetic
state on Rh. Moreover, we compare this system with Cr on Rh
to explore the trends through the TM series and to understand
the role of hybridization on the magnetic order.
II. METHOD
We determined the structural, electronic, and magnetic
properties of 3dTM monolayers on Rh(100) by performing
first-principles calculations using the full-potential linearized
augmented plane-wave (FLAPW) method29 in film geometry
as implemented in the FLEUR code.30 The generalized-gradient
approximation of Perdew, Burke, and Ernzerhof was applied,31
leading to a Rh bulk lattice constant of 3.819 ˚
A, which is only
0.4% larger than the experimental lattice constant of 3.804 ˚
A.32
The film was modeled by a symmetric seven-layer Rh(100)
slab covered by a single 3dmonolayer on each side, using
the calculated Rh in-plane lattice constant. Relaxations were
considered for the topmost two layers, that is, the 3dML and
the interface layer Rh(I).
Both the FM and the c(2 ×2) AFM configuration were
relaxed. We used about 120 LAPW basis functions per atom
with a muffin-tin radius of 1.22 ˚
Aforthe3dmonolayer atoms
and 1.28 ˚
A for the Rh atoms. The irreducible part of the two-
dimensional Brillouin zone (I2DBZ) was sampled with 78 k
points to determine the relaxations and the energy differences
between the different magnetic configurations with the same
c(2 ×2) unit cell. For the calculations of the MAE, spin-orbit
coupling (SOC) was included in the Hamiltonian and 1024 k
points were used in the full 2DBZ. Tests with 4096 kpoints
for selected systems confirmed convergence with respect to
this parameter. For these calculations the force theorem 33 was
employed, but the potential was converged including SOC,
although with less dense sampling of the reciprocal space.
III. RESULTS
A. Relaxations and magnetic moments
We calculated the relaxations between the layers iand j,
dij , defined as
dij =dij −d0
d0
,(1)
where dij is the spacing between the layers iand j, and d0is
the ideal bulk interlayer distance of the substrate.
The relaxations of the interlayer spacing between the 3d
monolayer and the topmost substrate layer, d12, and the first
Rh interlayer spacing, d23, are presented in Fig. 1for both
FM and AFM configurations. As a global trend we find that,
as a function of the 3dband filling, the interlayer relaxation
between the first and second Rh layer is reduced from a 7%
tetragonal expansion for V at the beginning of the series to the
almost ideal bulk interlayer spacing for the Ni monolayer at
the end of the series.
Analyzing d12, we notice that for the FM configuration
the smallest inward relaxation of the 3dmonolayers occurs
for Mn and Fe on Rh(100), where the magnetovolume effect is
strongest; that is, the large magnetic moments (see Fig. 2)of
these TMs compensate the strong inward relaxation—caused
by the larger Rh lattice constant—most efficiently. Of course,
there are also other factors controlling d12; for example, for
V the relaxation is smaller than for Cr, although the magnetic
moment of the latter is much larger than that of vanadium.
Here also the fact that the bulk lattice constant of V is 5%
larger than that of Cr has to be considered.
Comparing our results to data existing in the literature,
we notice that the relaxations in Fig. 1for Fe/Rh(100) are
stronger than in the calculation of Hwang et al.,27 which
reported −2.8% and −9.4% contraction of the first interlayer
distance for FM and AFM order, respectively. Since they used
a methodology similar to ours, the difference has to originate
from the relaxations of the deeper layers, which were ignored
in Ref. 27. Indeed, in another ab initio calculation for this
surface, including multilayer relaxations, values for d12 of
−7.2% and −12.6% were obtained for FM and AFM order,
respectively,28 in good agreement with the data from Fig. 1.
Also, the values for d23 agree within 0.2%. Experimentally,
low-energy electron diffraction measurements by Begley and
collaborators 34 reported a relaxation of d12 =−8.4±1.6%.
Taking temperature effects and uncertainties in the determina-
tion of the exact Fe coverage into account, the agreement is
VCr Mn Fe Co Ni
-15
-10
-5
0
5
10
Δd12 (%) Δd23 (%)
Rh(I)
3d monolayer
FM
AFM
FIG. 1. (Color online) Relaxations of the first and second inter-
layer spacing, d12 and d23, respectively, for 3dTM monolayers
on Rh(100) for the FM and the c(2 ×2) AFM configuration. The
corresponding changes are given with respect to the substrates’ bulk
interlayer spacing, which is 1.91 ˚
A.
024407-2
MAGNETISM OF 3dTRANSITION-METAL MONOLAYERS ... PHYSICAL REVIEW B 83, 024407 (2011)
0
1
2
3
4
magnetic moment (μB)
Ag(001)
Pd(001)
Rh(001)
VCr Mn Fe Co Ni
0.0
0.2
0.4
1 ML 3d on
Rh(001)
Ag(001)
Pd(001)
Rh(I)
FM
AFM
FIG. 2. (Color online) Magnetic moments of (top) 3dTM
monolayers on Ag, Pd, and Rh(100) surfaces and (bottom) the
interface Rh moments. The TM moments are denoted by solid (open)
symbols for the FM (AFM) solutions. For the FM case, the magnetic
moment of the interface Rh atoms is given by solid (green) squares.
Thedataforthe3dTMs on Ag(100) and Pd(100) are taken from
Refs. 3and 4.
quite reasonable. The same authors reported slightly smaller
relaxations of −6.8±1.6% for Co on the Rh(100) surface,21
while our calculations indicate even stronger relaxations for
the Co films, quite independent of the magnetic order.
With the exception of Ni, in all cases the equilibrium
distance between the interface Rh layer and the bulk Rh
underneath, d23, increases with respect to the substrates’
bulk interlayer spacing. Moving from left to right through
the periodic table, we find that d23 decreases, support-
ing the interpretation that the d-band filling controls these
relaxations.35,36 This highlights the importance of multilayer
relaxations for the early transition monolayers. The same
trend can be seen for the monolayers with c(2 ×2) AFM
configuration: while d12 is also influenced by the magnetic
moment of the 3dmonolayer, d23 depends almost solely on
the d-band filling.
In Fig. 2the magnetic moments of the 3dTM monolayers
on Rh(100) are compared with the Ag(100) and Pd(100)
substrates for the FM and AFM structures at the relaxed
interlayer distances. The magnetic moments on Rh(100) are
smaller than those on Ag(100) or Pd(100) for the early TMs,
whereas they are quite similar for the late TMs. This can be
understood on the basis of the observation that the overlap of
the dwave functions with the substrate is larger for the early
TMs than for the late TMs. Therefore, the dependence of the
TM magnetic moments on the chosen substrate is largest at the
beginning of the TM series. The largest moment, in all cases,
is found for Mn. The V and Ni magnetic moments vanish for
the c(2 ×2) AFM structure.
For FM order of the 3dmetal, the induced magnetic moment
of the Rh interface layer couples antiferromagnetically with
Cr and ferromagnetically with Mn, Fe, Co, and Ni, whereas
almost no moment is induced by V. The largest induced mag-
netic moment is caused by the Co monolayer. The large value
of 0.46μBon the Rh atom agrees with the values found for
Co/Rh(100) multilayers.37 However, in these experiments even
larger Rh moments were reported for Fe/Rh(100) multilayers.
In fact, our value of 0.27μBon Rh for Fe/Rh(100) agrees nicely
with previous theoretical results;28 the reasons for the deviation
from experiments might come from the multilayer structure.
For Fe, Co, and Ni layers, the induced moments in the deeper
Rh layers show an oscillatory AFM and FM coupling that
is qualitatively in line with the observed interlayer exchange
coupling.26 The induced moments are much larger for Rh than
for substrates like Ag(100), highlighting the importance of the
substrate for magnetic properties like the magnetic anisotropy,
which is discussed below. Of course the AFM ordered TM
films induce no magnetic moment in the interface Rh layer,
and the induced moments in the deeper layers are considerably
smaller and coupled antiferromagnetically to the nearest 3d
monolayer atoms.
B. Magnetic order
The total energy difference E =EAFM −EFM between
the c(2 ×2) AFM and the FM configuration is plotted in
Fig. 3for the 3dTM monolayers on different substrates. For
the Rh(100) substrate, we found a FM ground state for V,
Co, and Ni, while it is c(2 ×2) AFM for Cr, Mn, and Fe.
The data for Ag(100) and Pd(100) are taken from Refs. 3
and 4. For V and Ni, we see that the energy differences
are small: 4 and 15 meV, respectively. Experimentally, the
case of a single monolayer V on Ag(100) was discussed
controversially, claiming ferromagnetic order,38 or the absence
of ferromagnetic order,39 as well as evidence for antiferromag-
netic order.5Therefore, we checked E carefully as a function
of several computational parameters, like k-point sampling of
the temperature broadening at the Fermi level. Since several
studies reported on a field-induced or field-assisted FM-to-
AFM transition in FeRh films,40 we also checked the stability
of our results with respect to the influence of external electric
fields. For the Fe monolayer on Rh(100), no significant change
VCr Mn Fe Co Ni
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
E ( eV / 3d-atom )
Ag(001)
Pd(001)
Rh(001)r
Rh(001)ur
AFM
FM
Δ
FIG. 3. (Color online) The magnetic order of 3dTMs on Ag, Pd,
and Rh(100): positive E =EAFM −EFM indicates a FM ground
state, while negative values denote AFM order. The data for 3dTM
on Ag(100) and Pd(100) were taken from Refs. 3and 4. The open
squares connected with the dashed line represent the values of E
for the unrelaxed 3dTMs on Rh(100).
024407-3
A. AL-ZUBI, G. BIHLMAYER, AND S. BL ¨
UGEL PHYSICAL REVIEW B 83, 024407 (2011)
TABLE I. Differences of the total energy, E, of the AFM and
FM states for Fe monolayers on different substrates, where negative
energies indicate that the AFM state is more stable. Additionally, we
give the magnetic moments of Fe in the FM and the AFM state.
E (meV) MFM (μB)MAFM (μB)
Fe/Cu(100) 340 2.61 2.35
Fe/Ag(100) 210 3.01 3.06
Fe/Pd(100) 180 3.19 3.20
Fe/Rh(100) −60 3.01 2.93
Fe/W(100) −170 2.05 2.67
aThe Cu(100), Ag(100), Pd(100), and W(100) data are taken from
Refs. 42,4,41,and6, respectively.
of the results was observed in the presence of experimentally
obtainable external electric fields. From Fig. 3we observe for
V and Cr a trend toward FM order as we change the substrate
from Ag or Pd to Rh, while an increasing tendency toward
AFM is observed for Mn and the late TMs.
To highlight the influence of relaxations for the magnetic
ground state, we also show in Fig. 3the predicted E for
the unrelaxed structures: It can be noticed that, for all TMs
where the relaxation for FM and AFM structures are very
similar (V, Mn, Co, and Ni; see Fig. 1), relaxation effects on
E are rather small and do not change the predicted magnetic
ground state. On the other hand, for Co and Fe we observe
strong modifications of E due to relaxation: In Fe, where the
AFM structure shows a stronger inward relaxation than the FM
structure, the ground state changes from FM toward AFM. This
strong dependence of the relaxation on the magnetic structure
was already noticed in Ref. 27 and a similar trend was observed
for Fe on Ir(001).14 This is not unexpected, since Rh and Ir are
in the same row of the periodic table. For Cr the opposite trend
in relaxation is observed and the AFM structure is destabilized.
The magnetic structure that leads to the most favorable band
alignment between the TM and the substrate for bonding is
energetically most stable. We discuss the relations between
electronic and magnetic structure in more detail in Sec. IV.
For a weakly interacting substrate, like Ag(100), the trends
of E can be understood on the basis of the densities of
states of the TM monolayers.41 As we move on to more
strongly interacting substrates like Pd or Rh, this sinus-like
curve in Fig. 3is shifted gradually to the right; that is,
hybridization with the substrate effectively lowers the d-band
filling in the TM film. This trend is even more pronounced
for strongly interacting early TM substrates, like W(100),
causing a complete reversal of the magnetic trends.6A
compilation of results on different substrates is presented in
Table I.
C. Magnetic anisotropy
Apart from the magnetic order, the magnetic anisotropy
is one of the most important quantities characterizing thin
magnetic films because it not only determines the magneti-
zation direction (in plane or out of plane) but it also leads
to the stabilization of the magnetic order against thermal
fluctuations in two-dimensional systems and determines as
such the critical temperatures, the N´
eel temperature in case
of the c(2 ×2) antiferromagnets, or Curie temperature for
the ferromagnets. In principle, two terms contribute to the
MAE: the dipole-dipole interaction, leading in a ferromagnetic
film to the magnetic shape anisotropy (MSA) and the spin-
orbit coupling term giving the dominant contribution to
the magnetocrystalline anisotropy (MCA). Since in ultrathin
magnetic films the MCA is typically larger than the MSA,
we first discuss the effects of spin-orbit coupling and add
the dipole-dipole contributions at the end of this discussion.
All values given here refer to the energy difference between
in-plane (010) and out-of-plane (100) magnetization, positive
values indicating that the latter is more stable.
To investigate the influence of the substrate on the mag-
netic film, we compare our results to 3dtransition-metal
unsupported monolayers (UMLs) or monolayers on weakly
interacting substrates. These show on the right-hand side of
the periodic table (Co, Ni) a tendency toward strong in-plane
magnetization that changes to stable out-of-plane orientation
at the transition from Co to Fe. Then the values of the
anisotropy get smaller from Mn to V and can show small
oscillations (also in sign), depending on the chosen substrate
or lattice constant.43 This tendency can also be seen for UMLs
at the Rh(100) in-plane lattice constant in Fig. 4. Since the
magnetic anisotropy is a small quantity, typically two orders
of magnitude smaller than the energy differences between
different magnetic structures (cf. Fig. 3), one might expect
that hybridization effects allow these values to be manipulated
considerably. As can be seen from Fig. 4, this is indeed that
case, although the overall trend—as described above—remains
visible.
Focusing on the case of Fe to discuss the effect of the
substrate on the MCA, we first notice that both FM and AFM
ordered UMLs have large out-of-plane anisotropies. Also, for
Fe at the W(100) lattice constant, huge MCA values of 3.2
and 1.0 meV were reported for the AFM and FM structures,
respectively.10 On the smaller lattice constant of Rh(100) these
values are smaller and the effect of hybridization with the
VCr Mn Fe Co Ni
-3
-2
-1
0
1
2
MCA [meV/3d-atom]
3d/Rh(001)
3d contr.
UML
VCr Mn Fe Co Ni
-3
-2
-1
0
1
2
AFM
FM
FIG. 4. (Color online) Spin-orbit contribution to the magne-
tocrystalline anisotropy (MCA) for one ML 3d/Rh(100) in (left)
FM and (right) AFM order. Crosses indicate the anisotropy of
the unsupported monolayers, while the stars represent only the 3d
contribution. Positive values correspond to an out-of-plane easy axis
and negative values to an in-plane magnetization. The values for the
magnetic ground state (FM or AFM) are shown on white background.
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MAGNETISM OF 3dTRANSITION-METAL MONOLAYERS ... PHYSICAL REVIEW B 83, 024407 (2011)
substrate is quite different: While the value of about 0.8meV
remains almost unchanged for FM order, in the AFM state the
easy axis changes and the MCA drops to −0.2 meV due to the
Rh substrate. On W(100), the opposite trend was observed:
In the FM case the easy axis switched to in plane, while the
AFM ordered film kept its out-of-plane easy axis also on the
substrate.
In principle, two mechanisms for a reorientation of the
easy axis due to the substrate are conceivable: Either the
charge transfer changes the electronic properties of the TM
monolayer or the SOC contribution on Rh and the induced
Rh moments are responsible for the altered anisotropy. To
distinguish between these effects we can exclude the influence
of spin-orbit coupling in the substrate by setting the SOC
matrix elements to zero in the Rh atomic spheres. The
so-obtained MCA can then be attributed solely to the 3d
TM atoms and is indicated by stars in Fig. 4. For Fe and
the elements to the left of Fe, the charge transfer leads to
a reduction of the MCA with respect to the unsupported
monolayers and eventually to in-plane polarization, while for
Co and Ni the opposite effect is observed. Considering further
the effect of the induced Rh moments (noticing that only in the
FM case moments are induced in the substrate) leads—with the
exception of Cr—to an increase of the MCA. Also, with respect
to the induced Rh moments, Cr is exceptional, as it induces a
moment opposite to the TM layer, but it is not obvious whether
there is a connection to the magnetic anisotropy. For the AFM
structures, there are no induced moments, so the SOC effect on
the Rh atoms can only come from small changes due to SOC
in the band structure. We see that for Fe and Co this effect is
quite substantial.
Finally, we note that the balance of the above-mentioned
effects is quite sensitively influenced by the relaxation of the
magnetic monolayers. For example, for an unrelaxed FM film
of Fe on Rh, we observe just a small tendency toward an out-
of-plane easy axis that is further enhanced by the relaxation of
the film. The more Fe approaches the substrate, the stronger the
induced magnetic moments in Rh and the more important the
influence of the substrate. Additionally, the more pronounced
crystal field effect from the substrate on the Fe ML tends to
stabilize the out-of-plane anisotropy.44
Since the ground state of Cr, Mn, and Fe is AFM and
for V, Co, and Ni it is FM, we observe that a (small)
out-of-plane macroscopic moment is only obtained for the
V overlayer. Up to now, our arguments were only based on the
SOC contribution to the magnetocrystalline anisotropy, but
the MAE also contains the dipole-dipole interaction, which
we neglected so far. Since this interaction scales with the
square of the magnetic moments, we expect it to be im-
portant in the middle of the 3dseries, where films with an
antiferromagnetic ground state are found. For FM films, the
dipole-dipole interaction can be approximated in a continuum
model, giving rise to the shape anisotropy which is for a thin
film with an atomic volume Vand magnetic moments m(in
μB) given as
Kshape =−2π1
2c2
m2
V,(2)
where atomic (Hartree) units are used and the negative sign
implies that for FM films the dipole-dipole interaction favors
an in-plane easy axis. In AFM films there is no contribution
from the dipole-dipole interaction in the continuum model and
the dipole sum has to be carried out explicitly, where it is found
that the out-of-plane direction is favored.45 In our calculations
we included also the induced magnetic moments of the
substrate. Since the V moment is rather small, the dipole-dipole
interaction is also tiny and the MSA adds for this FM film only
−2μeV to the MCA. For AFM structures, the dipole-dipole
contribution is positive, adding for Cr and Fe about 0.10 meV
and for Mn 0.15 meV to the MCA, which makes the MAE
for Fe and Mn on Rh(100) small, but still negative. In the FM
Co and Ni overlayers, the MSA further stabilizes the in-plane
orientation of the magnetization.
In two-dimensional systems, the magnetic anisotropy is
of particular importance for the stabilization of the magnetic
order at finite temperatures. For example, in a system with
uniaxial anisotropy it was shown that the ordering temperature
in two dimensions, T2, can be derived by a renormaliza-
tion of the ordering temperature in three dimensions, T3,
according to46
T2=2T3
ln(π2J/K),(3)
where Jis the nearest-neighbor exchange coupling as defined
in a Heisenberg model and Kis the uniaxial anisotropy. For
a square lattice with out-of-plane anisotropy, this model can
be used to estimate the ordering temperature from the energy
difference between FM and AFM structures (Fig. 3) and the
anisotropy (Fig. 4). Since there are only two systems with
out-of-plane easy axis in the ground state, V and Cr, we can
focus on these two. However, for V the exchange coupling,
J, and the anisotropy, K, is tiny and the Curie temperature is
only 3 K. For Cr, we estimate a value of 295 K, so that the
chances are good that this structure can be directly observed
experimentally with a spatially nonaveraging technique.
In in-plane magnetized films, the temperature dependence
of the magnetization depends on both the MAE as discussed
above and the magnetic anisotropy within the plane.47 Al-
though these in-plane anisotropies are tiny (about 2 μeV per
atom for the Fe overlayer), they are essential in stabilizing
the magnetic order at low temperatures, where these values
can be used to describe the temperature dependence of
the magnetization within the random phase approximation.
Unfortunately, the extension toward T2is problematic in this
case.48
IV. DISCUSSION
To analyze the role of the hybridization between the Fe ML
and the Ag, Pd, or Rh substrate, the local density of states
(LDOS) for the nonmagnetic (NM) and the FM configuration
of one ML Fe on different substrates are shown in Fig. 5.
We see from the NM LDOS that there is no overlap of the
Ag 4dband with the Fe DOS, which is pinned at the Fermi
level due to its incomplete 3dband filling. Therefore, the
Fe bandwidth is small and (according to the Stoner model)
the high DOS at the Fermi level favors ferromagnetism. In
contrast there is a strong 3d-4dhybridization between the Fe
ML and the the Rh(100) substrate. The broadening of the Fe d
band reduces the DOS at the Fermi level, leading finally to an
024407-5
A. AL-ZUBI, G. BIHLMAYER, AND S. BL ¨
UGEL PHYSICAL REVIEW B 83, 024407 (2011)
-6 -4 -2 0 2 4
2
4
6
8NM
-6 -4 -2
-2
0
2
4FM
-6 -4 -2 0 2 4
2
4
6
LDOS (states/eV)
-6 -4 -2
-2
0
2
-6 -4 -2 0 2 4
E-EF [eV]
0
2
4
6
-6 -4 -2 0 2 4
E-EF [eV]
-4
-2
0
2
Pd(I)
Ag(I)
spin +
spin +
spin -
spin -
Rh(I)
Fe
spin +
spin -
Fe
Ag(I)
Pd(I)
Rh(I)
Fe
Fe
FIG. 5. (Color online) Local densities of states (LDOS) for the Fe
atoms and the substrate atoms at the interface (I) for the nonmagnetic
(NM) and FM configurations of Fe on Ag(100), Pd(100), and Rh(100)
(top to bottom). The LDOS of the 3dmetal is drawn with thick (red)
lines, and the substrates’ LDOS is shown with thin lines.
antiferromagnetic ground state. The case of Fe/Pd(100) is in
between these extremes, but still a FM order is obtained as a
ground state. From the FM DOS it can be seen that a small FM
moment can be induced in the Pd interface that is absent in the
case of Fe/Ag(100).
Coming back to the previous discussion of the effects
of bonding (relaxation) on the magnetic structure, we now
compare Fe on Rh(100) to Cr/Rh(100). The fact that the latter
has a strong tendency toward a c(2 ×2) AFM ground state
can be explained from the NM LDOS (top of Fig. 6): We
see that the Fermi level falls in a minimum of the Cr LDOS,
favoring the AFM ground state even more than in the case of
Fe (see Fig. 3). A comparison of the magnetic DOS shows
that in Fe/Rh(100) the minority DOS is pinned at the Fermi
level, while in the Cr case the position of the majority dband
is determined by the band filling and the whole exchange-split
d-like DOS is energetically at a much higher position. This
modifies the hybridization to the substrate, as can be seen in
the relaxations: While Fe and Cr have very similar bulk lattice
constants, the inward relaxation of FM Cr on Rh(100) is much
bigger than that of FM Fe/Rh(100) (see Fig. 1). In the case
of the AFM solutions, the trend is exactly opposite; that is,
Fe relaxes much more than Cr. In Cr in the AFM state, the
dbandwidth is much smaller than in the FM state, leading to
a decreased hybridization with the substrate. In contrast, the
AFM Fe DOS can be seen to be stronger hybridized to the
-6 -4 -2
2
4
Fe/Rh(001)
-6 -4 -2 0 2 4
2
4
Cr/Rh(001)
-6 -4 -2
-4
-2
0
2
-6 -4 -2 0 2 4
-4
-2
0
2
LDOS (states/eV)
-6 -4 -2 0 2 4
E-EF [eV]
-6
-4
-2
0
2
-6 -4 -2 0 2 4
E-EF [eV]
-6
-4
-2
0
2
FM FM
AFM
AFM
spin +spin +
spin + spin +
spin -
spin -
spin -
spin -
Rh(I) Rh(I)
Cr Fe
NM
NM
FIG. 6. (Color online) LDOS for the (top) NM, (middle) FM,
and (bottom) AFM configurations of (left) Cr and (right) Fe on the
Rh(100) surface. The LDOS of the 3dmetal is drawn with thick lines,
and the substrates’ LDOS is shown with thin (green) lines.
Rh, so that an increase in hybridization (stronger relaxation)
is connected to a stabilization of the AFM structure (Fig. 3),
as opposed to the observations for the Cr ML. We should note
here that in all these cases the magnetic moments (and the
exchange splitting) are approximately the same (Fig. 2), so
magnetovolume effects should be comparable.
V. SUMMARY
Monolayers of 3dtransition metals were studied on top of
a Rh(100) surface including relaxations of the topmost layers.
As in the previously studied 3dTM series on Ag(100) and
Pd(100), the trend of the magnetic moments follows Hund’s
first rule with the largest moments on Mn. Opposed to that,
the largest induced magnetic moments in Rh are obtained at
the end of the TM series. The magnetic order was found to
be FM for V, Co, and Ni overlayers, whereas Cr, Mn, and
Fe favor an AFM ground state. Depending on the substrate,
Fe can be tuned from an AFM state on Rh and W to FM
coupling on Pd and Ag(100). A similar behavior was reported
recently for Fe on W(100) and Ta(100) surfaces: Complex
magnetic structures can be expected when the nearest-neighbor
exchange coupling is small, that is, when E =EAFM −EFM
is small.49 Also, frustration effects, like on Rh(111) surfaces,
could lead to interesting magnetic structures.50 From the
LDOS it can be seen how hybridization affects the magnetic
order in these systems: Compared to Ag or Pd(100) substrates,
024407-6
MAGNETISM OF 3dTRANSITION-METAL MONOLAYERS ... PHYSICAL REVIEW B 83, 024407 (2011)
the broadening of the dband is enhanced, leading to a tendency
in favor of AFM ordering for Fe but FM for Cr. Calculations of
the magnetic anisotropy showed that only V and Cr monolayers
on Rh(100) have an out-of-plane easy axis in their ground state,
while the magnetization of Mn, Fe, Co, and Ni is oriented
in plane.
ACKNOWLEDGMENTS
We acknowledge valuable discussions with J. Kudrnovsk´
y
and A. Lehnert, and the financial support of the ESF EURO-
CORES Programme SONS under Contract No. ERAS-CT-
2003-980409.
*Present Address: Max-Planck-Institut f¨
ur Eisenforschung,
D¨
usseldorf, Germany.
†g.bihlmayer@fz-juelich.de
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