# Two-dimensional magnetic interactions in LaFeAsO

M. Ramazanoglu  Ames Laboratory, Ames, IA, 50011, USA  Department of Physics
and Astronomy, Iowa State University, Ames, IA, 50011, USA  J. Lamsal  Ames
Laboratory, Ames, IA, 50011, USA  Department of Physics and Astronomy, Iowa
State University, Ames, IA, 50011, USA  G. S. Tucker  Ames Laboratory, Ames,
IA, 50011, USA  Department of Physics and Astronomy, Iowa State University,
Ames, IA, 50011, USA  J.-Q. Yan  Oak Ridge National Laboratory, Oak Ridge, TN,
37831, USA  S. Calder  Oak Ridge National Laboratory, Oak Ridge, TN, 37831,
USA  T. Guidi  ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot,
Oxon OX11 OQX, United Kingdom  T. Perring  ISIS Facility, Rutherford Appleton
Laboratory, Chilton, Didcot, Oxon OX11 OQX, United Kingdom  R. W. McCallum
Ames Laboratory, Ames, IA, 50011, USA  Department of Physics and Astronomy,
Iowa State University, Ames, IA, 50011, USA  T. A. Lograsso  Ames Laboratory,
Ames, IA, 50011, USA  Department of Physics and Astronomy, Iowa State
University, Ames, IA, 50011, USA  A. Kreyssig  Ames Laboratory, Ames, IA,
50011, USA  Department of Physics and Astronomy, Iowa State University, Ames,
IA, 50011, USA  A. I. Goldman  Ames Laboratory, Ames, IA, 50011, USA
Department of Physics and Astronomy, Iowa State University, Ames, IA, 50011,
USA  R. J. McQueeney  Ames Laboratory, Ames, IA, 50011, USA  Department of
Physics and Astronomy, Iowa State University, Ames, IA, 50011, USA

(March 15,2013)

###### Abstract

Inelastic neutron scattering measurements demonstrate that the magnetic
interactions in antiferromagnetic LaFeAsO are two-dimensional. Spin wave
velocities within the Fe layer and the magnitude of the spin gap are similar
to the AFe2As2 based materials. However, the ratio of interlayer and
intralayer exchange is found to be less than $\sim 10^{-4}$ in LaFeAsO, very
similar to the cuprates, and $\sim$ 100 times smaller than that found in
AFe2As2 compounds. The results suggest that the effective dimensionality of
the magnetic system is highly variable in the parent compounds of the iron
arsenides and weak 3-D interactions may limit the maximum attainable
superconducting $T_{c}$.

###### pacs:

75.25.-j, 61.05.fg

The discovery of high-temperature superconductivity in the iron arsenide
compounds Kamihara08  immediately led to comparisons to the copper oxide
superconductors. Both systems possess layered crystal structures, suggesting
that two-dimensional (2-D) behavior may be a shared feature amongst the high-
temperature superconductors. In particular, the enhanced spin fluctuations
that arise from reduced dimensionality is regarded as a critical element of
high-temperature superconductivity. In the case of the copper oxide materials,
the magnetism occurs within square copper oxide sheets that are weakly coupled
to each other due to separation by ionic layers (such as BaO or LaO). The
parent La2CuO4 compound, for example, is an insulator with a strongly
anisotropic resistivity measured within ($\rho_{ab}$) and perpendicular
($\rho_{c}$) to the Cu layers ($\rho_{c}/\rho_{ab}\approx$ 500 at high
temperatures).Preyer89  The magnetic excitations measured with inelastic
neutron scattering (INS) are well-understood within a 2-D Heisenberg model as
the ratio of interlayer to intralayer exchange is very small
($J_{c}/J_{ab}\approx 10^{-4}-10^{-5}$).Endoh88 ; Keimer92

In the iron arsenides, magnetism also occurs in separated square FeAs layers,
but the dimensionality of the magnetic interactions is debated. Measurements
of the anisotropic properties have been mainly performed on the AFe2As2 (122)
system ($A=$ Ca, Sr, Ba), where large single-crystals are available. In the
122 materials, the FeAs layers are separated by an alkali-earth metal layer
(with Fe-Fe layer separation of 5.5 - 6.5 Å from Ca - Ba, respectively).
Transport properties in parent 122 compounds display only a weak anisotropy
(for example, $\rho_{c}/\rho_{ab}\approx 1-3$ Tanatar09 ) in contradiction to
the strongly 2-D transport properties observed in the cuprates. Angle-resolved
photoemission (ARPES) experiments on the 122 compounds indicate a significant
variation of the Fermi surface geometry along the $c$-axis that is also
consistent with a 3-D system.Heimes ; Kim  Finally, INS measurements with $A=$
Ca,Diallo08 ; Diallo09 ; Zhao09 , Sr Zhao08 , and Ba Sato09 ; Harriger11 ,
indicate a fairly substantial interlayer magnetic exchange interaction
($J_{c}/J_{ab}\approx 2-6\%$) which supports three-dimensional (3-D)
magnetism.

Very little is known about the magnetic interactions in the $R$FeAsO (1111)
family of superconducting materials that currently claim the largest
superconducting transition temperature of $T_{c}^{max}\approx$ 55 K (whereas
$T_{c}^{max}\approx$ 40 K for the 122 compounds).Johnston10  Based on a larger
interlayer spacing ($\approx$ 8.7 Å), 1111 compounds are expected to be closer
to the 2-D limit than the 122 compounds. The recent availability of large
single-crystals of LaFeAsO Yan09  have enabled measurements of the anisotropic
resistivity ($\rho_{c}/\rho_{ab}\approx$ 2-20)Jesche12 , which is similar to
the 122 compounds. However, ARPES measurements of the effective dimensionality
of the electronic system are inconclusive due to the presence of surface
states.Liu10  In this Letter, we use INS measurements of the spin-wave
spectrum in the parent LaFeAsO compound to show that magnetic exchange
coupling is 2-D, despite the inference of only weak anisotropy from bulk
measurements, with a ratio of exchange interactions comparable to the cuprates
($J_{c}/J_{ab}<10^{-4}$). This result provides evidence that the magnetism can
vary from 2-D to weakly 3-D in different iron arsenide compounds, with
possible implications for the maximum achievable $T_{c}$.Monthoux

The sample used for INS experiments consists of dozens of small single-
crystals of LaFeAsO with a total mass of approximately 600 mg that are co-
aligned to within $\sim$2 degrees. Details of crystal growth and
characterization are described elsewhere.Yan09  Previous neutron and x-ray
scattering measurements show that the crystals undergo a tetragonal-
orthorhombic structural phase transition at $T_{\mbox{\scriptsize S}}=155$ K,
followed by stripe antiferromagnetic ordering transition at
$T_{\mbox{\scriptsize N}}=140$ K.Yan09  The wavevector of the stripe AFM
ordered state is $\mathbf{Q}_{AFM}=$ (1/2,1/2,1/2)T when indexed with
reference to the high-temperature $P4/nmm$ tetragonal structure. In this
paper, the scattering data is presented with respect to low-tempereture
orthorhombic $Cmma$ unit cell (in other words; $\mathbf{Q}_{AFM}=$ (1,0,1/2)O)
where we define
$\mathbf{Q}=(H,K,L)=\frac{2\pi}{a}H\hat{\imath}+\frac{2\pi}{b}K\hat{\jmath}+\frac{2\pi}{c}L\hat{k}$
and the lattice constants are $a\approx b=$ 5.68 Å and $c=8.75$ Å. INS
measurements were performed on the MERLIN spectrometer at the ISIS Neutron
Scattering Facility at Rutherford-Appleton Laboratory and the HB3 spectrometer
at the High Flux Isotope Reactor at Oak Ridge National Laboratory. For these
measurements, the samples were mounted in the $(H,0,L)$ scattering plane.

For subsequent discussion, both the MERLIN and HB3 data are described using a
model of damped Heisenberg spin waves with nearest ($J_{1a}$,$J_{1b}$) and
next-nearest ($J_{2}$) interactions within the Fe layer, and an interlayer
exchange ($J_{c}$). We also include a single-ion anisotropy energy ($D$) to
account for an observed spin gap. Within linear spin-wave theory, the
dispersion is given by
$\hbar\omega(\textbf{q})=\sqrt{A_{\textbf{q}}^{2}-B_{\textbf{q}}^{2}}$ with
$A_{\textbf{q}}=2S[D+2J_{2}+J_{1a}+J_{c}+J_{1b}(\cos(\pi K)-1)]$ and
$B_{\textbf{q}}=2S[J_{1a}\cos(\pi H)+2J_{2}\cos(\pi H)\cos(\pi
K)+J_{c}\cos(2\pi L)]$. The magnetic susceptibility can be written as a damped
simple harmonic oscillator (with damping parameter $\Gamma$),

$\chi^{{}^{\prime\prime}}(\textbf{q},E)=\frac{\chi_{0}\Gamma
E}{[E^{2}-(\hbar\omega(\textbf{q}))^{2}]^{2}+\Gamma^{2}E^{2}}.$ (1)

and the INS intensity in arbitrary units is
$f^{2}(Q)\chi^{{}^{\prime\prime}}(\textbf{Q}-\textbf{Q}_{AFM},E)(1-e^{-E/kT})^{-1}$
where $f(Q)$ is the magnetic form factor of the Fe2+ ion and
$\textbf{q}=\textbf{Q}-\textbf{Q}_{AFM}$ is the reduced wavevector within the
magnetic Brillouin zone.

MERLIN measurements were performed with the incident neutron beam oriented
along $L$ and an incident energy $E_{i}=$ 150 meV. The spectrum of spin
fluctuations were measured deep in the stripe AFM ordered state at $T=$ 5 K
and are shown in Figs. 1 (a) and (b) for the longitudinal ($H$) and transverse
($K$) directions relative to $\textbf{Q}_{AFM}$ (see Fig. 1(c) for reference).
In these spectra, an assumed isotropic and energy-dependent non-magnetic
background signal was estimated by summing data at all scattering angles after
masking the INS signal near the magnetic zone centers. The spectrum below 100
meV consists of steep spin waves concentrated close to $\textbf{Q}_{AFM}$ and
all symmetrically equivalent wavevectors in the twinned orthorhombic structure
[see Fig. 1(c)]. The signal above $\approx$ 100 meV becomes too weak to
observe. The MERLIN data is therefore best understood in the small-$q$ limit
with spin waves described by an anisotropic linear dispersion relation,

$\hbar\omega(\textbf{q})=\sqrt{\Delta_{0}^{2}+v_{a}^{2}q_{x}^{2}+v_{b}^{2}q_{y}^{2}+v_{c}^{2}q_{z}^{2}}.$
(2)

The spin gap and spin wave velocities are given by
$\Delta_{0}=2S\sqrt{2DJ_{+}}$, $v_{a}=aSJ_{+}$, $v_{b}=bS\sqrt{J_{+}J_{-}}$,
$v_{c}=cS\sqrt{J_{c}J_{+}}$, respectively, where $J_{+}=2J_{2}+J_{1a}+J_{c}$
and $J_{-}=2J_{2}-J_{1b}$.

The anisotropy of the dispersion within the Fe layer appears as the elliptical
shape of the neutron intensity in Fig. 1(c). Estimates of the longitudinal and
transverse spin wave velocities (based on fits discussed below) are $v_{a}=$
555 $\pm$ 100 meV Å and $v_{b}=$ 420 $\pm$ 55meV Å respectively. The in-plane
spin wave velocities in LaFeAsO are comparable, though slightly larger, than
the 122 materials, in agreement with first-principles electronic band
structure calculations.Han09  The longitudinal velocity exceeds the transverse
velocity ($v_{a}>v_{b}$) and the anisotropy of the spin excitations within the
Fe layer is defined as
$\eta=(v_{a}^{2}-v_{b}^{2})/(v_{a}^{2}+v_{b}^{2})$.Tucker12  The value of
$\eta=$ 0.25 for LaFeAsO is similar to the values found in the parent 122
compounds where $\eta=$ 0.2-0.4.Tucker12

Figure 1: (a) Longitudinal and (b) transverse cuts through the inelastic
neutron scattering spectrum of LaFeAsO at T=5 K as measured on the MERLIN
spectrometer with $E_{i}=$150 meV after background substruction (see text).
(c) Data averaged over an energy transfer range from 35-75 meV showing
anisotropic spin fluctuations in the $H-K$ plane centered at
$\textbf{Q}_{AFM}$. (d) Energy spectra at different average $K$ values along
the transverse direction; 0, 0.1, and 0.15. Lines shown are fits described in
the text. In these panels, the $L$ component of the wavevector varies with the
in-plane wavevector and energy trasfer because of the fixed crystal
orientation with respect to the incident beam direction.

MERLIN measurements cannot ascertain any substantial interlayer exchange
interactions, which should appear as $L$-dependent oscillations in the
intensity. In time-of-flight INS experiments with the $c$-axis fixed along the
incident beam, $L(E)$ is a function of the energy transfer $E$. So, the
absence of substantial $E$-dependent intensity oscillations in Figs. 1(a),
(c), and (d) indicates 2-D magnetism.

In order to verify this 2-D behavior, we performed measurements of the low-
energy spin excitations at $T=$ 5 K on the same sample using the HB3
spectrometer. Figure 2(a) shows that a substantial spin gap is observed at
$\textbf{Q}_{AFM}=(1,0,1/2)$. The gap shows an onset of $\approx$ 5 meV and a
peak at $\approx$ 11 meV and is comparable to the spin gaps observed in
$A$Fe2As2 Diallo08 ; Zhao08 ; Sato09 ; Park12  and NaFeAs Park12  compounds.
Low-energy INS measurements on polycrystalline LaFeAsO observe a similar sized
spin gap and report a 2D-like response.Ishikado09

The difference between the spin gaps at the magnetic zone center
$\textbf{Q}_{AFM}$ ($\Delta_{0}$) and the magnetic zone boundary point at
$\textbf{Q}_{ZB}=(1,0,1)$ ($\textbf{q}_{ZB}=(0,0,1/2)$) provides a direct
measurement of $J_{c}$. Using the Heisenberg model, the difference in spin
gaps is

$[\hbar\omega(\textbf{q}_{ZB})]^{2}-\Delta_{0}^{2}=16S^{2}J_{c}J_{+}.$ (3)

A comparison of Figs. 2(a) and 2(b) shows that the magnitude of the spin gap
is similar at $\textbf{Q}_{ZB}$, thereby providing very strong confirmation
that $J_{c}$ must be weak. Figs. 2(c) and (d) show an absence of any
$L-$dependent sinusoidal modulations of the intensity along $(1,0,L)$ both at
the gap onset at 5.5 meV and the peak at 10 meV. Finally, we show in Fig.3
that a longitudinal cut at the gap onset of 5.5 meV reveals weak intensity at
both $\textbf{Q}_{AFM}$ and $\textbf{Q}_{ZB}$, which again confirms that the
two spin gaps are the same. The HB3 data shown in Figs.2 and 3 was fit using
Eqns. (1) and (2) after convolution with the instrumental resolution using the
RESLIB programReslib . These data were used to obtain values for the low
energy damping and the spin gaps at $\textbf{Q}_{AFM}$ and $\textbf{Q}_{ZB}$,
yielding $\Gamma=8\pm 1$ meV, $\Delta_{0}=11.6\pm 0.5$ meV, and
$\hbar\omega(\textbf{q}_{ZB})=11.2\pm 0.5$ meV, respectively. All of these
data and subsequent fits give substantive proof that no observable dispersion
exists along $L$ and, therefore, magnetic interactions in LaFeAsO are 2-D in
nature.

Figure 2: Measurements of the low energy spin excitations in LaFeAsO at $T=$5
K as measured on HB3. Energy dependence of the magnetic scattering at (a)
$\textbf{Q}_{AFM}=$ (1,0,1/2) and (b) magnetic zone boundary position
$\textbf{Q}_{ZB}=(1,0,1)$. Constant energy scans depicting the $L$ dependence
of the magnetic scattering along (1,0,$L$) (c) at gap onset at 5.5 meV, and
(d) above the gap at 10 meV. Lines are fits to a damped spin wave model as
described in the text. Insets and panels (c) and (d) show the raw data (green
squares) and background scans (crystal rotated from nominal Q by 20 degrees,
black diamonds) that were used to estimate the magnetic scattering (red
symbols) in (a)-(d). The arrows in these panels are indicating the onset value
of the enegrgy gap. Figure 3: Longitudinal cuts of the low energy spin
excitations in LaFeAsO at $T=$ 5 K as measured on HB3. Data are shown (a) at
the onset of the gap at $E=$ 5.5 meV and $\textbf{Q}_{AFM}=(1,0,1/2)$, (b) at
5.5 meV and $\textbf{Q}_{ZB}=(1,0,1)$, (c) at 10 meV and $\textbf{Q}_{AFM}$,
and (d) at 10 meV and $\textbf{Q}_{ZB}$. Lines show the fits to the damped
spin wave model,described in the text, with only a free amplitude and
background. The rest of the parameters are fixed to the values given in Table
1. The quality of the agreement between data and the model is also a
confirmation of the results.

For the MERLIN data, the TOBYFIT program Tobyfit  was used to fit the
corrected data to the 2-D Heisenberg model after convolution with the
instrumental resolution and accounting for orthorhombic twinning. The values
of $\Delta_{0}$ and $\Gamma$ used in the MERLIN fits are fixed to the values
determined by the HB3 data. The corrected data has been symmetrized by
averaging all four equivalent quadrants of reciprocal space and subtracting an
estimate of the non-magnetic and background scattering. The main MERLIN
fitting results are displayed in Figs. 1 and 4 as a series of longitudinal and
transverse cuts through $\textbf{Q}_{AFM}$ at different energy transfers from
$E=$15 to 75 meV. At low energies, cuts through the steep magnetic excitations
consist of a single sharp peak centered at $\textbf{Q}_{AFM}$ due to
resolution limitations (the resolution width is indicated by the horizontal
line in Fig. 4). Above $E\sim$55 meV, the peak splitting from counter
propagating spin wave modes can be resolved and is more pronounced in
transverse cuts where the spin wave velocity is lower. The two curves in Fig.
4 represent a global fit to all cuts shown (blue line) as well as local fits
to each cut (red line) with both procedures yielding similar values for the
fitting parameters. Without the ability to observe the spin wave dispersion at
the magnetic zone boundary positions, such as $\textbf{q}=(0,1,0)$
[$\textbf{Q}=(1,1,0)$], the fits are not sensitive to the difference between
nearest-neighbor exchange constants, $S(J_{1a}-J_{1b})$. The full set of
fitting parameters listed Table 1 represent both the HB3 and MERLIN data quite
well.

Table 1: Parameters obtained from fitting the $J_{1a}-J_{1b}-J_{2}$ spin wave
model with damping and single-ion anisotropy. The damping factor ($\Gamma$),
energy gap ($\Delta_{0}$), and exchange energies $<SJ_{\pm}>$ are in meV while
spin-wave velocities $v_{a\above 0.0ptb}$are in meV Å.

HB3 |  MERLIN Local |  MERLIN Global
---|---|---
$\Gamma=8\pm$1 |  $<SJ_{+}>=102\pm$20 |  $<SJ_{+}>=93\pm$15
$\Delta_{0}=11.2\pm$.6 |  $<SJ_{-}>=59\pm$7 |  $<SJ_{-}>=51\pm$5

$<v_{a}>=555\pm$100
---
$<v_{b}>=420\pm$55

Figure 4: Longitudinal (left) and transverse (right) constant energy cuts of
the magnetic scattering in LaFeAsO up to 75 meV as measured on MERLIN. Global
(blue line) and local (red line) fits to the damped spin wave model described
in the text are shown. The shoulders in the tails of transverse $E=$75 meV cut
fit is due to the effect of the twinning of orthorhombic cyrstal, which is
included in the model calculations. For these cuts the L component of the
wavevecotor is function a of the in-plane momentum vectors and the energy
transfer, as explained earlier.

In summary, the details of the spin wave spectrum and magnetic energy scales
of LaFeAsO are similar in many ways to the 122 compounds. The energy scale for
exchange interactions within the Fe layer and their average in-plane
anisotropy are nearly equivalent. This is in accordance with ab initio
calculations of the spin excitation spectrum.Han09  The measurements must be
extended up to higher energies in order to determine whether any substantial
difference exists between $J_{1a}$ and $J_{1b}$. At lower energies, we find
that the magnitude of the spin gap is also similar to the 122 compounds. The
common energy scale of the 122 and NaFeAs spin gaps was recently discussed in
Park12 , as it does not follow from the expectations of simple single-ion
anisotropy due to substantial differences in the magnitude of the ordered
moments in the two systems. The similar spin gap observed in LaFeAsO, along
with its relatively small ordered moment (0.4 $\mu_{B}$),Cruz08  would seem to
add some strength to this argument.

The most important difference in the 122 and 1111 compounds is the interlayer
exchange. In our measurements, the zone center and (0,0,1/2) zone boundary
spin gaps in LaFeAsO are equal within error. Considering the error bars may
allow a 1 meV difference in spin gaps, we can estimate an upper limit for the
exchange anisotropy (based on Eq. 3) of $J_{c}/J_{+}<10^{-4}$ which is similar
to the cuprates and places LaFeAsO strongly in the 2-D limit. In comparison,
$J_{c}/J_{+}=2-6\%$ is $\approx$ 100 times larger for the parent 122
compounds.Diallo08 ; Diallo09 ; Zhao09 ; Zhao08 ; Sato09 ; Harriger11  The 2-D
antiferromagnetism found in the 1111 compounds may be responsible for some
enhancement of ($T_{c}^{max}\approx$ 55 K) and weak 3-D magnetic interactions
present in the 122 family compounds may present a limitation to higher
superconducting transition temperatures ($T_{c}^{max}\approx$ 40 K).

RJM would like to thank D. C. Johnston and V. Antropov for useful discussions.
The work at Ames Laboratory was supported by the U.S. Department of Energy,
Office of Basic Energy Science, Division of Materials Sciences and Engineering
under Contract No. DE-AC02-07CH11358. Work at Oak Ridge National Laboratory is
supported by U.S. Department of Energy, Office of Basic Energy Sciences,
Scientific User Facilities Division.

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