# Magnetic surface reconstruction in the van-der-Waals antiferromagnet Fe1+xTe

C. Trainer  School of Physics and Astronomy, University of St. Andrews, St.
Andrews KY16 9SS, UK  M. Songvilay  School of Physics and Astronomy,
University of Edinburgh, Edinburgh EH9 3JZ, UK  N. Qureshi  Institut Laue-
Langevin, 71 avenue des Martyrs, CS20156, 38042 Grenoble Cedex 9, France  A.
Stunault  Institut Laue-Langevin, 71 avenue des Martyrs, CS20156, 38042
Grenoble Cedex 9, France  C. M. Yim  School of Physics and Astronomy,
University of St. Andrews, St. Andrews KY16 9SS, UK  E. E. Rodriguez
Department of Chemistry and Biochemistry, University of Maryland, College
Park,Maryland 20742, USA  C. Heil  Institute of Theoretical and Computational
Physics, Graz University of Technology, NAWI Graz, 8010 Graz, Austria  V.
Tsurkan  Center for Electronic Correlations and Magnetism, Experimental
Physics V, University of Augsburg, D-86159 Augsburg, Germany  Institute of
Applied Physics, Academy of Sciences of Moldova, MD 2028 Chisinau, Republic of
Moldova  M. A. Green  School of Physical Sciences, University of Kent,
Canterbury, CT2 7NH, UK  A. Loidl  Center for Electronic Correlations and
Magnetism, Experimental Physics V, University of Augsburg, D-86159 Augsburg,
Germany  P. Wahl  School of Physics and Astronomy, University of St. Andrews,
St. Andrews KY16 9SS, UK  C. Stock  School of Physics and Astronomy,
University of Edinburgh, Edinburgh EH9 3JZ, UK

(August 27, 2024)

###### Abstract

Fe1+xTe is a two dimensional van der Waals antiferromagnet that becomes
superconducting on anion substitution on the Te site. The properties of the
parent phase of Fe1+xTe are sensitive to the amount of interstitial iron
situated between the iron-tellurium layers. Fe1+xTe displays collinear
magnetic order coexisting with low temperature metallic resistivity for small
concentrations of interstitial iron $x$ and helical magnetic order for large
values of $x$. While this phase diagram has been established through
scattering [see for example E. E. Rodriguez et al. Phys. Rev. B ${\bf{84}}$,
064403 (2011) and S. Rößler et al. Phys. Rev. B ${\bf{84}}$, 174506 (2011)],
recent scanning tunnelling microscopy measurements [C. Trainer et al. Sci.
Adv. ${\bf{5}}$, eaav3478 (2019)] have observed a different magnetic structure
for small interstitial iron concentrations $x$ with a significant canting of
the magnetic moments along the crystallographic $c$ axis of $\theta$=28 $\pm$
3∘. In this paper, we revisit the magnetic structure of Fe1.09Te using
spherical neutron polarimetry and scanning tunnelling microscopy to search for
this canting in the bulk phase and compare surface and bulk magnetism. The
results show that the bulk magnetic structure of Fe1.09Te is consistent with
collinear in-plane order ($\theta=0$ with an error of $\sim$ 5∘). Comparison
with scanning tunnelling microscopy on a series of Fe1+xTe samples reveals
that the surface exhibits a magnetic surface reconstruction with a canting
angle of the spins of $\theta=29.8^{\circ}$. We suggest that this is a
consequence of structural relaxation of the surface layer resulting in an out-
of-plane magnetocrystalline anisotropy. The magnetism in Fe1+xTe displays
different properties at the surface when the symmetry constraints of the bulk
are removed.

##  I Introduction

Van der Waals forces differ from ionic and covalent bonding in terms of the
strength and also the range of forces. Mayor (2016) Two dimensional materials
that are based on sheets weakly held together by van der Waals forces have
recently been studied in the context of graphene Neto  _et al._ (2009), and
also for the investigation of two dimensional ferromagnetism Miller (2017).
Two dimensional magnetic van der Waals crystals have also been of interest in
the context of iron based superconductivity Ishida  _et al._ (2009); Kamihara
_et al._ (2008); Johnston (2010); Paglione and Greene (2010); Dai (2015);
Inosov (2016); Stock and McCabe (2016); Dai  _et al._ (2012); Lumsden and
Christianson (2010) with arguably the structurally simplest such
superconductor being the monolayer compound Hsu  _et al._ (2008)
Fe1+xTe1-y(Se,S)y Wen  _et al._ (2011) consisting of weakly bonded iron
chalcogenide sheets. Under anion substitution, an optimal superconducting
transition temperature of $\sim 14\mathrm{K}$ has been reported in
Fe1+xTe0.5Se0.5 Sales  _et al._ (2009) and $\sim 10\mathrm{K}$ in FeTe1-xSx
Mizuguchi  _et al._ (2009). In this paper we investigate the difference
between the bulk and surface magnetic structures in the non-superconducting
parent compound Fe1+xTe through the use of spherical neutron polarimetry and
spin polarized scanning tunnelling microscopy in vector magnetic fields.

The single layered chalcogenide Fe1+xTe1-yQy (where $Q$=Se or S) has been
important in the study of iron based superconductors owing to its relatively
simple single layer structure and because it is highly electronically
localized Yin  _et al._ (2011); Rößler  _et al._ (2010); Si and Abrahams
(2008); Si (2009) in comparison to other iron based systems. This is a
property that is also reflected in the oxyselenides. McCabe  _et al._ (2014);
Zhao  _et al._ (2013); Zhu  _et al._ (2010); Freelon  _et al._ (2015) The
electronic and magnetic properties of Fe1+xTe1-yQy can be tuned via two
variables - the parameter $x$ determines the amount of interstitial iron
located between the weakly bonded FeTe layers and disordered throughout the
crystal, and $y$ is the amount of anion substitution and provides a chemical
route towards superconductivity. It should be noted that while the
interstitial iron is disordered introducing magnetic clusters Thampy  _et al._
(2012), the electronic properties have been found to be homogeneous, He  _et
al._ (2011); Xu  _et al._ (2018) and recently discussed in the context of
device fabrication Zalic  _et al._ (2019). The sensitivity of the properties
to stoichiometry is also reflected in Fe1+δSe. McQueen  _et al._ (2009) There
have been several studies that have shown $x$ and $y$ to be correlated and
hence both influence the superconductivity. Rodriguez  _et al._ (2011a); Sun
_et al._ (2016); Babkevich  _et al._ (2010); Bendele  _et al._ (2010) In
particular, the tetrahedral bond angles Huang  _et al._ (2010); Xu  _et al._
(2016, 2010); Wen  _et al._ (2012) are altered with interstitial iron
concentration $x$ along with tuning the material across several magnetic and
structural phase transitions. It is this interplay between the structural,
magnetic, electronic and superconducting properties which makes this material
exciting, with the expectation that understanding the relation between these
phases leads to an understanding of superconductivity.

Fe1+xTe has been found to display two spatially long-range correlated magnetic
phases as a function of iron concentration $x$ separated by a region of
spatially short-ranged magnetic order Bao  _et al._ (2009); Koz  _et al._
(2013); Wen  _et al._ (2009). For small interstitial iron concentrations
$x\leq 0.12$, the magnetic structure is collinear Li  _et al._ (2009);
Zaliznyak  _et al._ (2012) below a temperature where a structural transition
occurs from a tetragonal ($P4/nmm$) to monoclinic ($P2_{1}/m$) unit cell.
Electronically, this also marks a transition from a resistivity which is
“semi/poor”-metallic to being metallic in character at low temperatures. Chen
_et al._ (2009) Despite the metallic character, the low energy spin
fluctuations are consistent with localized transverse spin-waves Stock  _et
al._ (2017); Song  _et al._ (2018). The second magnetic phase at
concentrations $x>0.12$ is helical in nature combined with a
“semi/poor”-metallic behavior at all temperatures. Rößler  _et al._ (2011) The
two disparate magnetic phases induced with the variable $x$ are separated by a
collinear spin density wave Materne  _et al._ (2015); Parshall  _et al._
(2012) located near a Lifshitz point Rodriguez  _et al._ (2013). As well as
the tuning of the magnetic and crystallographic structures with interstitial
iron concentration, the magnetic excitations also display a large dependence
on $x$ Stock  _et al._ (2011, 2012), even at high energy transfers Lumsden
_et al._ (2010); Stock  _et al._ (2014); Zaliznyak  _et al._ (2011); Xu  _et
al._ (2016); Lipscombe  _et al._ (2011).

We will focus on the low interstitial iron concentrations in this study. The
magnetic structure for low interstitial iron concentrations is termed a
“double-stripe” structure and has been investigated extensively with both
unpolarized and uniaxial polarized neutron scattering. This collinear magnetic
phase has magnetic moments aligned along the crystallographic $b$ axis and
magnetic Bragg peaks in the neutron cross section at $\vec{Q}$=$({1\over
2},0,{1\over 2})$ or denoted as $(\pi,0)$. Being metallic at low temperatures
and easily cleavable offers the opportunity to apply surface techniques for
investigating the electronic and magnetic properties.

In particular, spin-polarized scanning tunnelling microscopy has been used to
manipulate the excess iron concentration at the surface layer. Enayat  _et
al._ (2014); Singh  _et al._ (2015) This work has demonstrated that the
surface magnetic structure faithfully follows the bulk magnetic phase diagram
as a function of interstitial iron in terms of both the crystallographic and
magnetic structures giving consistent results with neutron scattering on the
interstitial iron concentration where the magnetic and crystallographic
structures change. However, an important difference was observed for the
magnetic structure in the collinear “double-stripe” phase for small values of
$x$. Tunneling measurements show a periodicity consistent with the stripe
phase reported based on neutron scattering Sugimoto  _et al._ (2013), however
recent measurements in vector magnetic fields have observed a significant out
of plane canting, along the crystallographic $c$-axis, of the magnetic moment
of $\theta\sim 28^{\circ}$ where neutron scattering reports the moments to be
entirely in the $ab$ plane ($\theta=0$). Hänke  _et al._ (2017); Trainer  _et
al._ (2019) Interestingly, such a magnetic structure is consistent with early
studies on Fe1.12Te,Fruchart  _et al._ (1975) however given more recent work
it is possible that this concentration was at the boundary between collinear
and helical magnetism possibly complicating the interpretation.

Figure 1: Comparison of magnetic order from neutron scattering and STM. The
axial magnetic structure with the moments aligned along the crystallographic
$b$ axis ($\theta$=0) contrasts with the canted structure measured with
tunnelling measurements where the moments are canted out of the plane by
$\theta$=28 $\pm$ 3∘.

Based on the discrepancy between current neutron diffraction results, scanning
tunneling microscopy, and older diffraction work, we revisit this problem
applying spherical neutron polarimetry to determine the out of plane angle due
to any canting of the spins in bulk single crystals of Fe1+xTe. This study
focuses on the iron deficient portion of the phase diagram which exhibits
collinear order because this is where the differences between neutron
scattering and STM are most prominent.

In this paper, we compare spin polarized scanning tunnelling microscopy
measurements of the magnetism on the surface with a study of the bulk magnetic
structure. The two different magnetic structures that will be compared in this
paper are illustrated in Fig. 1. This paper is divided into five sections
including this introduction. We first present the results from spin polarized
scanning tunnelling microscopy of the canting angle in the surface layer. We
then investigate the canting angle in the bulk from spherical neutron
polarimetry and analyze the results in terms of a possible canting in the
bulk. We finally compare these results and discuss the differences and
possible origins, including dipolar and anisotropy terms in the magnetic
Hamiltonian. Through this comparison we find that the surface layer of the two
dimensional van der Waals Fe1+xTe magnet exhibits a magnetic surface
reconstruction.

##  II Spin Polarized STM measurement

Figure 2: (a) - Spin polarized STM image of the surface of an
$\mathrm{Fe}_{1.1}\mathrm{Te}$ sample with a bias voltage of $100\mathrm{mV}$
and a set point current of $50\mathrm{pA}$. Recorded with an applied magnetic
field of $\mathrm{2T}$ applied out of the plane of the image along the sample
$c$ axis. (b) As (a) but with the direction of the applied magnetic field
reversed. (c) Half the difference of the images shown in (a) and (b), directly
proportional to the sample magnetization along the $c$ axis. Inset - The
Fourier transform of (c) showing the magnetic ordering vector ($q_{a}$). (d)
The average of the images (a) and (b), directly proportional to the non spin
polarized component of the tunneling current. Inset - the Fourier transform of
(d) showing the atomic peaks due to the $\mathrm{Te}$ lattice. (e) The out of
plane canting angle ($\theta$) measured from images recorded with three
orthogonal directions of applied field. (f) Histograms of the absolute value
of the measured out of plane canting angle for data sets of three dimensional
spin polarization data recorded on different samples of $\mathrm{FeTe}$. The
red data points corresponds to the data shown in (e). Solid lines represent
fits of a Gaussian function plus a linear background.

We first discuss spin polarized STM measurements of Fe1+xTe probing the
magnetic structure at the surface.

###  II.1 Experimental Details

Spin polarized STM measurements were conducted on samples of Fe1+xTe with
excess iron concentrations $x$ ranging from $5\%$ to $11.5\%$. The
measurements were performed using a home-built cryogenic STM operating at a
base temperature of $2\mathrm{K}$ and mounted in a Vector magnet that is
capable of applying a field of up to $5\mathrm{T}$ in any direction relative
to the sample Trainer  _et al._ (2017). Atomically clean surfaces for STM
measurement were prepared by cleaving the samples in-situ at a temperature of
$\sim 20\mathrm{K}$ White  _et al._ (2011). Magnetic tips were created by
collecting excess Fe atoms from the sample surface Enayat  _et al._ (2014);
Trainer  _et al._ (2019); Singh  _et al._ (2015). In this way, the tunneling
current between the STM tip and sample becomes sensitive to the relative angle
between the magnetization of the tip and the sample. The tunneling current
($I_{\mathrm{SP}}$) due to the spin polarization of the tip
($P_{\mathrm{tip}}$) and the sample ($P_{\mathrm{sample}}$) can be expressed
asWiesendanger (2009)

$I_{\mathrm{SP}}=I_{0}\left[1+P_{\mathrm{tip}}P_{\mathrm{sample}}\cos(\phi)\right],$
(1)

where $\phi$ is the angle between the tip and sample magnetizations. Figure
2(a) shows a typical spin polarized STM topographic image of the iron
telluride surface (Fe1.1Te). The excess iron atoms are seen in the STM images
as bright protrusions on the surface. The bi-collinear antiferromagnetic order
is imaged as a stripe like modulation running parallel to the sample $b$ axis
with a wavevector $(q_{a})$ along the sample $a$ axis. The imaged wavelength
and direction of this ordering is in excellent agreement with that determined
from neutron scatteringBao  _et al._ (2009); Rodriguez  _et al._ (2011b). For
ferromagnetic tips, the magnetization of the tip is found to follow the
direction of an applied magnetic field Enayat  _et al._ (2014); Trainer  _et
al._ (2019); Hänke  _et al._ (2017) therefore imaging the surface with the tip
polarized by a field applied $180\degree$ to the original field orientation
results in a $\pi$ phase reversal of the imaged magnetic order. This reversal
of the imaged magnetic order can be seen by comparing images recorded with
opposite applied field orientations shown in Figs. 2(a) and (b).

###  II.2 STM Results

It is possible, through Eqn. 1, to directly measure the sample’s surface spin
polarization from the spin polarized STM images. This is done by taking the
difference of the images recorded with oppositely polarized tips which is
proportional to $2P_{\mathrm{tip}}P_{\mathrm{sample}}\mathrm{cos}(\phi)$. Fig.
2(c) shows such a difference image showing the component of the sample’s
magnetic order that is parallel to the crystal $c$ axis. The sum of the two
images recorded with oppositely polarized tips resembles the topography that
would be recorded if the sample was imaged with a non-spin polarized tip (Fig.
2(d)).

By recording spin-polarized STM images with magnetic field applied in three
orthogonal spatial directions it is possible to determine the precise
orientation of the sample’s spin structure at the surface Zhang  _et al._
(2016); Trainer  _et al._ (2019). The out of plane canting angle ($\theta$) of
the surface spins resulting from this measurement is shown in Fig. 2(e). Clear
canting of the spins away from the $ab$ plane can be observed. By plotting the
absolute value of this angle as a histogram, shown in Fig. 2(f), a clear peak
at $\sim 30\degree$ can be observed. This measurement has been repeated for a
sample of Fe1.05Te, the results of which are also shown in Fig. 2(f). We
determine the out of plane canting of the surface spins by fitting a Gaussian
distribution plus a linear background to the data. By combining the fits to
both data sets we obtain an average out of plane canting angle of $30.3\pm
2.0^{\circ}$. This substantial canting of the surface spins is seen across
multiple samples and has been observed in previous STM studies on this
compoundHänke  _et al._ (2017); Trainer  _et al._ (2019).

sample # | $x(\%)$ | $\mathrm{I}(q_{a})||b$ | $\mathrm{I}(q_{a})||c$ | $\theta(\degree)$
---|---|---|---|---
1 | 5 | 0.1221 | 0.0550 | 24.2527
2 | 10 | 0.2246 | 0.1570 | 34.9522
2 | 10 | 0.1542 | 0.1721 | 48.1350
2 | 10 | 0.2719 | 0.0186 | 3.9146
2 | 10 | 0.2243 | 0.1761 | 38.1379
3 | 11.5 | 0.2248 | 0.1275 | 29.5626
Table 1: Results from $6$ different independent SP-STM measurements on
different samples of Fe1+xTe with different magnetic tips. The table shows the
intensity of the imaged magnetic order when the tip spin is parallel to the
crystal $b$ axis and when the tip spin is parallel to the crystal $c$ axis and
the resulting canting angle of the surface spins.

We have also conducted further studies on other samples of Fe1+xTe, where
measurements were only recorded with the tip polarized along the crystal $b$
and $c$ axes. The data is shown in Table 1. The intensity of the magnetic peak
$\mathrm{I}(q_{a})$ for field applied along the crystallographic $b$ and $c$
axes respectively and the corresponding canting angle ($\theta$) are shown.
The average out-of-plane canting angle obtained from these values is $29.8\pm
13.7\degree$. The error bar here contains contributions from variations in the
magnetic properties of the STM tips, differences between samples and the
alignment of the magnetic field plane with the crystallographic axis of the
sample. This is the magnetic structure shown in Fig. 1 $(b)$.

Figure 3: A color schematic of the polarization matrix for the spin models
under consideration for the $Q$-vectors $Q=(1/2,0,1/2)$ and $Q=(1/2,0,3/2)$:
(a) for a single domain of the double-stripe order with canted spins with
$\theta=28^{\circ}$, (b) for multiple domains of the same order as in (a)
using the symmetry relations in table LABEL:table_pos and (c) for the axial
order with spins pointing along $b$. $X-$, $Y-$ and $Z-$ are the spin
components along the three spatial directions, color encodes the polarization.

##  III Neutron Scattering Experiments

Having discussed the canted magnetic structure at the surface, we now apply
neutron scattering to study the bulk magnetism. Neutron scattering, unlike
x-rays or photon based measurements, is a bulk measurement of materials owing
to the interaction between neutrons and matter being mediated by nuclear
interactions. For example, for single crystalline Fe1.09Te with a neutron
wavelength of $\lambda=1.1\mathrm{\AA}$ the $1/e$ scattering length for the
sum of absorption and incoherent cross sections is $\sim 2\mathrm{cm}$. The
results presented in this section are therefore a measure of the bulk averaged
response. Neutrons carry a magnetic moment making them sensitive to the
localized magnetic moments in materials.

###  III.1 Experimental Details

To investigate the polarization matrix of the magnetic order in Fe1+xTe
sensitive to the orientation of the local iron moments, we used the CRYOPAD
(Cryogenic Polarization Analysis Device) developed at the ILL Tasset  _et al._
(1999); Brown  _et al._ (1993). Unlike conventional polarization measurements
which involve studying spin flip scattering along a particular
crystallographic axis, CRYOPAD allows all components of the polarization
matrix to be studied governed by the Blume-Maleev equations. Blume (1963);
Maleev  _et al._ (1963) Single crystals of Fe1.09Te were synthesized by the
Bridgemann method Rodriguez  _et al._ (2013, 2011b). All measurements
discussed here were done at a base temperature of $2\mathrm{K}$ using the IN20
spectrometer with the sample aligned such that Bragg peaks of the form (H 0 L)
lay within the horizontal scattering plane. The structural and magnetic
transition in this material occurs at $\sim 60\mathrm{K}$ resulting in
structural domains present at low temperatures. Fobes  _et al._ (2014) The
possible symmetry operations resulting from these domains are displayed in
Table LABEL:table_pos and discussed below.

$\vec{Q}$ | $P_{ij}^{\mathrm{measured}}$ |  $P_{ij}^{\mathrm{axial}}$ |  $P_{ij}^{\mathrm{canted}}$ $(\theta=28^{\circ})$
---|---|---|---
$(\frac{1}{2},0,\frac{1}{2})$ | $\left(\begin{array}[]{ccc}-0.896(3)&0.064(4)&-0.002(4)\\\ -0.086(3)&-0.879(3)&-0.018(4)\\\ -0.049(4)&-0.032(4)&0.882(3)\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-0.768&0\\\ 0&0&0.768\end{array}\right)$
$(\frac{3}{2},0,\frac{3}{2})$ | $\left(\begin{array}[]{ccc}-0.889(9)&0.048(10)&-0.025(11)\\\ -0.068(10)&-0.887(9)&-0.010(15)\\\ -0.058(11)&-0.004(11)&0.874(9)\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-0.768&0\\\ 0&0&0.768\end{array}\right)$
$(\frac{3}{2},0,\frac{1}{2})$ | $\left(\begin{array}[]{ccc}-0.887(9)&0.042(10)&-0.039(11)\\\ -0.071(10)&-0.877(9)&-0.016(15)\\\ -0.042(11)&-0.020(10)&0.887(9)\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-0.636&0\\\ 0&0&0.636\end{array}\right)$
$(\frac{1}{2},0,\frac{3}{2})$ | $\left(\begin{array}[]{ccc}-0.887(3)&0.068(4)&-0.003(4)\\\ -0.060(3)&-0.880(3)&-0.016(4)\\\ -0.048(4)&-0.031(4)&0.884(4)\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-0.946&0\\\ 0&0&0.946\end{array}\right)$
$(\frac{1}{2},0,\frac{5}{2})$ | $\left(\begin{array}[]{ccc}-0.894(7)&0.059(8)&-0.032(8)\\\ -0.055(8)&-0.883(7)&-0.013(8)\\\ -0.068(8)&-0.022(8)&0.882(7)\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-0.979&0\\\ 0&0&0.979\end{array}\right)$
$(\frac{5}{2},0,\frac{1}{2})$ | $\left(\begin{array}[]{ccc}-0.846(15)&0.029(17)&-0.007(18)\\\ -0.10(2)&-0.884(15)&-0.003(18)\\\ -0.050(18)&-0.043(18)&0.859(16)\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\ 0&0&1\end{array}\right)$ | $\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-0.618&0\\\ 0&0&0.618\end{array}\right)$
Table 2: A list of the experimentally measured polarization matrices at $T=2\mathrm{K}$ measured on IN20. The calculated matrix elements Qureshi (2019), assuming 100 % beam polarization, are shown for the axial spin structure and the canted magnetic structure with $\theta=28^{\circ}$ for comparison. Table 3: Low temperature structural domains considered here for the magnetic structural analysis. 1 | x | y | z
---|---|---|---
2 | x | y | -z
3 | -x | -y | z
4 | -x | -y | -z

Spherical neutron polarimetry is sensitive to the direction of the ordered
magnetic moment, spin chirality, and coupling between nuclear and magnetic
cross sections Blume (1963). In the case of structural domains that exist at
low temperature (which average out the off-diagonal elements) from the
structural transition (Table LABEL:table_pos) and in the absence of spin
chirality and coupling to a nuclear cross section, the polarization matrix
measured with spherical neutron polarimetry becomes diagonal and takes the
following form,

$P_{ij}=\begin{pmatrix}-1&0&0\\\
0&{{|M_{\perp,y}|^{2}-|M_{\perp,z}|^{2}}\over|\vec{M}_{\perp}|^{2}}&0\\\
0&0&-{{|M_{\perp,y}|^{2}-|M_{\perp,z}|^{2}}\over|\vec{M}_{\perp}|^{2}}\end{pmatrix}$

where $\vec{M}_{\perp}\equiv\vec{Q}\times\vec{M}\times\vec{Q}$. Here
$\vec{Q}\equiv\vec{k}_{i}-\vec{k}_{f}$ is the momentum transfer and $\vec{M}$
is the magnetic moment direction. The matrix element $P_{xx}\equiv P_{11}$ is
strictly $=-1$ and deviations from this are a measure of the neutron beam
polarization. It is important to note that the polarization matrix does not
provide information on the magnitude of $\vec{M}$, but only the direction.

In this experiment, the polarization matrix was measured at six magnetic Bragg
peaks at $T=2\mathrm{K}$. The full experimental polarization matrices
$P_{ij}^{\mathrm{measured}}$ for these Bragg peaks are shown in Table 2. The
calculated matrices $P_{ij}^{\mathrm{axial}}$ and $P_{ij}^{\mathrm{canted}}$
shown in the table are discussed below in the context of our comparison with
tunnelling and previous neutron results.

###  III.2 Neutron scattering results

Figure 1 illustrates the two magnetic structures that we will compare the
polarized neutron scattering results to in this section. The reported
structure based on neutron diffraction on single crystals and also powders
suggests that the structure is collinear with the moments aligned along the
crystallographic $b$ axis (Fig. 1 (a)). The structure is often referred to as
a “double-stripe” magnetic structure. This is contrasted to a recent magnetic
structure reported using scanning tunnelling microscopy (Fig. 1 (b)). The
magnetic structure obtained from STM has the magnetic moments collinear but
canted along the crystallographic $c$ axis by an angle of $\theta=29.8\pm
13.7^{\circ}$. For the purposes of this section we refer to the neutron
scattering structure which is aligned along the $b$ axis as “axial” and the
structure reported by spin polarized tunnelling microscopy as “canted”. We now
discuss the application of neutron spherical polarimetry to revisit the bulk
magnetic structure of collinear Fe1+xTe.

Figure 4: A summary of the spherical polarimetry data from IN20 measured for
Fe1.09Te and compared against calculations. (a,b) histograms of the
polarization matrix elements $P_{ij}$ for calculations based on the canted and
axial magnetic structures respectively. (c) the same histogram for the
measured matrix elements. (d,e) plots of the calculated polarization matrix
elements as a function of measured values for both the canted and axial
magnetic structures. Figure 5: A parameterization of the goodness of fit
($\chi^{2}$) to the data as a function of canting angle $\theta$ as defined in
the text. A neutron beam polarization of $0.88$ was taken for the analysis.
(a) $\chi^{2}$ over the full range of canted angles from $0-180^{\circ}$. (b)
$\chi^{2}$ in a narrow range of angles from $0-5^{\circ}$.

While the application of unpolarized neutron powder diffraction and also
uniaxial polarized neutrons maybe arguably ambiguous in determining canting of
the magnetic moments owing to the number of accessible peaks and statistics
for low interstitial iron concentrations, spherical neutron polarimetry is
very sensitive to this canting. We illustrate this in Fig. 3 which displays a
color representation of the calculated polarization matrices at the magnetic
momentum positions $\vec{Q}$=($\frac{1}{2}$, 0, $\frac{3}{2}$) and
($\frac{1}{2}$, 0, $\frac{1}{2}$). Three different models are presented. Panel
$(a)$ displays a calculation based on the canted model proposed by tunnelling
measurements for a single structural and magnetic domain crystal. This
calculation shows a non zero off-diagonal values for the matrix elements for
the $P_{yz}$ and $P_{zy}$ positions. However, Fe1+xTe undergoes a structural
distortion from a tetragonal to a monoclinic unit cell that is coincident with
magnetic ordering. The four domains are related by symmetry as displayed in
Table LABEL:table_pos. The corresponding matrix including the effects of
domains is diagonal and is illustrated in Fig. 3(b). The magnitudes of the
matrix elements $|P_{yy}|\neq|P_{zz}|$. This contrasts with the case where the
magnetic moments point within the $ab$ plane as termed “axial” in this paper
and schematically shown in Fig. 3(c) where $|P_{yy}|=|P_{zz}|$.

Figure 4 illustrates a comparison of our results to the predicted matrix from
both magnetic structures. Figs. 4(a, b) illustrate histograms of the
calculated polarization matrix elements for the both the canted (tunnelling,
Fig. 4(a) and axial (neutron, Fig. 4(b) magnetic structures. The canted
magnetic structure results in polarization matrix elements for a range of
values ranging from $-1\rightarrow 1$. The largest number of matrix elements
appear at $0$ resulting from the averaging over domains meaning that all off-
diagonal matrix elements are calculated to be $0$ (Fig. 3). The axial magnetic
structure, in contrast only displays three matrix elements ($[-1,0,1]$)
distinguishing it from the canted magnetic structure.

A histogram of the experimentally measured polarization matrix elements is
plotted in Figs. 4(c) and shows a distribution of measured elements centered
around three values, in qualitative agreement with the axial (neutron)
magnetic structure. We note that there is a large distribution around
$P_{ij}=0$ and is further shown in the table displayed above (Table 2 in the
experimental section). The origin of this error results from the incomplete
polarization of the beam and also due to small misalignments ($\sim
1-2^{\circ}$, see the appendix of Ref. Giles-Donovan  _et al._ , 2020 for an
analysis of the errors) of the sample with respect to the beam polarization.
Based on the comparison between the Figs. 4 (a-c), the neutron data is
consistent with the axial magnetic structure rather than the prediction of a
broad spread of matrix elements which would result from a canted magnetic
structure. This is further illustrated in Figs. 4 (d,e) which shows the
calculated matrix elements as a function of the measured matrix elements. The
spread of the data from a single straight line is a measure of the “goodness
of fit”. The canted magnetic structure in panel (c) clearly provides a much
poorer description of the data over the axial one displayed in panel (d).

Figure 5 shows a plot of $\chi^{2}$ (a measure of the goodness of fit) as a
function of canting angle $\theta$ quantifying the sensitivity of our
measurement using spherical polarimetry and also establishing a measure of the
errorbar in our experiment. For this figure, we have defined $\chi^{2}$ in
terms of the measured and calculated polarization matrix elements $P_{i}$ by

$\chi^{2}\equiv\sum_{i,j}\sum_{\alpha}(P^{\mathrm{measured},\alpha}_{ij}-P^{\mathrm{calculated},\alpha}_{ij})^{2},$
(2)

where the summation index $\alpha$ is taken over all Bragg peak peaks and the
index $ij$ are the matrix elements probed in this experiment. The plot of
$\chi^{2}$ as a function of canting angle for the 6 Bragg peaks (Table 2)
studied on IN20 shows a broad minimum near $\theta\sim 0^{\circ}$ and a
distinct maximum with $\theta=90^{\circ}$ when the moments are pointing along
the crystallographic $c$ axis. The vertical red line in Fig. 5(a) is the
canted $\theta=28\pm 3^{\circ}$ proposed by tunnelling measurements. The
$\chi^{2}(\theta)$ curve clearly shows that our spherical neutron polarimetry
data is inconsistent with a canted structure with a broad minimum observed
near $\theta=0^{\circ}$ which is the axial structure found previously in
powders and single crystal unpolarized neutron measurements. The nature of the
broad minimum in the $\chi^{2}$ surface indicates an underlying errorbar in
the magnetic structure measured here of $\sim$ $\pm 5^{\circ}$. The neutron
scattering data shows that the magnetic structure in iron deficient Fe1.09Te
is inconsistent with the magnitude of the canted magnetic structure reported
for the single layer limit in tunnelling measurements.

##  IV Discussion

The comparison of the spherical neutron polarimetry with spin-polarized STM
shows clearly that for Fe1+xTe, a surface magnetic reconstruction forms where
the spins on the iron site tilt out of the $ab$-plane. In the following we
will discuss possible mechanisms leading to this reconstruction.

###  IV.1 Surface relaxation – Density functional theory (DFT) calculations

To attempt to explain the difference between the measured magnetic structure
of the surface and the bulk we have performed DFT calculations on an FeTe slab
where the surface was allowed to relax from the lattice positions in the bulk.
First-principles calculations were performed using the Quantum Espresso
Giannozzi  _et al._ (2009) code. We employed optimized norm-conserving
Vanderbilt pseudopotentials Hamann (2013) with the Perdew-Burke-Ernzerhof
exchange-correlation functional in the generalized gradient approximation
Perdew  _et al._ (1996). For the calculation, four layers of FeTe, without
excess iron atoms ($x=0$), with a vacuum region of $13\mathrm{\AA}$ in the $z$
direction, bicollinear magnetic order along $x$, and ferromagnetic order along
$z$ were considered. The $z$-length of the unit cell was kept fixed during
variable cell relaxation runs and spin-polarization taken into account. We
chose a kinetic energy cutoff for the plane waves of $80\mathrm{Ry}$, a
Methfessel-Paxton Methfessel and Paxton (1989) smearing of $0.02\mathrm{Ry}$,
and a $8\times 16\times 1$ Monkhorst-Pack Monkhorst and Pack (1976) k-mesh.
Details of the crystal structure and atomic positions were taken from
experiment Li  _et al._ (2009) and then geometrically relaxed. The surface
layer was found to relax away from the bulk layer to such an extent that the
$c$ axis parameter of the surface layer changes from $6.2\mathrm{\AA}$ to
$6.8\mathrm{\AA}$. The lateral position of the surface layer also change from
that of the bulk with the Fe atoms of the surface layer displaced by up to
$10\mathrm{pm}$ from the unrelaxed position. Furthermore slight changes in the
length of the $\mathrm{Fe}-\mathrm{Te}$ bonds of up to $3\mathrm{pm}$ were
observed. This reconstructed structure is illustrated schematically in Fig.
6(a).

Figure 6: (a) Structural model of the surface layer relaxed in DFT
calculations. The relaxation of the top surface layer can be seen. (b)
Calculation of the energy per Fe atom due to the dipole interaction as a
function of out of plane canting angle $\theta$ of a bi-layer of
$\mathrm{FeTe}$ as described in the text. The maximum energy per Fe atom due
to the dipole interaction is found from this calculation to be $3.3\mathrm{\mu
eV}$. The energy minimum is found at $\theta=\pm 90^{\circ}$, i.e. out of the
surface plane.

###  IV.2 Magnetic dipole interactions

As a potential explanation for the canting of the spins in the surface layer,
we have considered the magnetic dipole interaction between the Fe atoms. We
have constructed a numerical model of the surface of Fe1+xTe to calculate the
preferred spin orientation of the surface Fe atoms given dipolar interactions
with the layer below. The model consists of a bulk layer of $101\times 101$ Fe
atoms with a surface layer of $41\times 41$ Fe atoms. The spins of each Fe
atom were fixed into the bicollinear AFM order with their in-plane component
fixed to point along the crystallographic $b$ axis. The magnetic moment of
each Fe atom was taken to be $2\mu_{\mathrm{B}}$ Rodriguez  _et al._ (2011b).
The energy of the interaction of the dipoles in the system is then determined
by numerically summing over all the spins in the system. The equation for this
process is:

$E=\sum_{i\neq
j}\frac{\mu_{0}}{4\pi|\bm{r}_{ij}|^{2}}\left[\bm{\mu}_{i}\cdot\bm{\mu}_{j}–\frac{3}{|\bm{r}_{ij}|^{2}}(\bm{\mu}_{i}\cdot\bm{r}_{ij})(\bm{\mu}_{j}\cdot\bm{r}_{ij})\right]$
(3)

where the indices $i$ and $j$ indicate the different $\mathrm{Fe}$ positions
in both layers. By varying the canting angle $\theta$ of the surface spins we
determined how the energy of the dipole interaction varies as a function of
$\theta$, this is shown in Fig. 6(b). We determine from this analysis that the
dipole interaction in Fe1+xTe would favor aligning the Fe spins along the
crystallographic $c$ axis, a result that is in good agreement with the effects
of magnetic dipole interactions in bulk crystals Johnston (2016). Indeed if
one were to consider only dipole interactions in addition to the AFM ordering
in the bulk then the magnetic moments of the Fe would align with the sample
$c$ axis. The fact that this is not observed in neutron scattering
measurements leads us to the conclusion that a substantial magnetic anisotropy
from the crystalline electric field keeps the Fe spins pointing in the $ab$
plane. We note that the tendency of the magnetic dipole interaction to favour
out-of-plane order is stronger for the surface layer than it is in the bulk.
The energy scale associated with the dipolar interaction per Fe spin is
$\sim\mu$eV.

###  IV.3 Magnetocrystalline anisotropy

The energy scale of the dipolar interaction is extremely small in comparison
to the measured $\sim$ 5 meV anisotropy gap found with neutron inelastic
scatteringStock  _et al._ (2011). Both, the direction and magnitude of
magnetic dipole interactions suggest that these are not sufficiently strong to
explain the out-of-plane tilting of the magnetization in the surface layer.
This leaves the magnetocrystalline anisotropy resulting from crystalline
electric field effects Turner  _et al._ (2009); Yosida (1996) as a possible
origin for the magnetic surface reconstruction. In bulk Fe1+xTe, the magnetic
anisotropy results in an in-plane orientation of the spinsEnayat  _et al._
(2014). At the surface, the broken symmetry resulting from the loss of a
mirror plane and structural relaxation of the surface layer discussed above
imply that the magnetic anisotropy can differ significantly.

##  V Conclusions

The spherical neutron polarimetry results show a distinct difference between
the bulk magnetic structure in Fe1.09Te measured with neutron scattering and
the canted magnetic structure reported with tunnelling measurements in the
single layer limit. This illustrates a difference between bulk and surface
magnetism in this Van der Waals magnet. It should be noted that the cases of
tunnelling from a surface and neutron scattering from the bulk are not
studying the exact same situation. In the bulk neutron response, each magnetic
Fe1+xTe layer effectively represents a mirror plane, this is not the case of a
hard surface as is the situation in tunnelling. Therefore, from a symmetry
perspective, there is no constraint forcing both situations to be identical.

The magnetic moments in Fe1.09Te interact through either effects of bonding
(including possible itinerant interactions such as RKKY exchange) or dipolar
interactions. For interactions within the $ab$ plane of Fe1.09Te these should
be dominated by the effects of bonding which result in strong dispersion of
the magnetic excitations along these directions Stock  _et al._ (2014). The
situation along the $c$-axis is less clear as the FeTe layers are only weakly
bonded through Van der Waals forces. However, dipolar forces which decay
$\sim{1\over r^{3}}$ are still present in the magnetic Hamiltonian and these
could be strongly influential to the magnetic correlations along the
crystallographic $c$-axis. Magnetic neutron inelastic scattering have indeed
found the weak $c$-axis correlations Xu  _et al._ (2017); Stock  _et al._
(2011) occuring without the presence of strong bonding and only Van der Waals
forces.

Another effect not directly tied to the crystalline electric field effects
discussed above that may be the origin of the difference between surface
(tunnelling) and bulk (neutron) responses is interstitial iron. Previous
neutron scattering results have shown a strong connection between the magnetic
correlations and the interstitial iron concentration $x$. Leiner  _et al._
(2014); Stock  _et al._ (2011) With increasing interstitial iron
concentration, the crystallographic $c$-axis decreases which could in turn
increase the importance of the dipolar terms in the magnetic Hamiltonian. The
interstitial sites may also be magnetic and this could influence the structure
in the FeTe plane.

We note that differences in the magnetic structure and periodicity between
tunnelling and bulk neutrons scattering have been reported before. Comparative
measurements done in superconducting La2-xSrxCuO4 Christensen  _et al._ (2004)
with tunnelling and neutron scattering have observed different wavevectors,
however a similar response in the dynamics. The role of dipolar and
crystalline electric field terms in the magnetic Hamiltonian may be an issue
that needs to be considered in all magnetic layered and two dimensional
structures.

While further calculations will be required to ultimately understand the
difference in magnetic structures observed on the surface and the bulk, our
study illustrates the sensitivity and difference between the magnetism in the
Fe1.09Te Van der Waals magnet between the bulk and the surface. This has been
established through a comparison between spherical polarimetry to determine
that the bulk magnetic structure of iron deficient Fe1.09Te has $\theta$=0
$\pm$ 5∘, while using spin-polarized STM to characterize the surface magnetic
order. We suggest the difference between magnetic structures found between
scanning tunnelling microscopy and neutron scattering originates from the
relaxation of the surface layer and the corresponding changes in
magnetocrystalline anisotropy.

We acknowledge financial support from the EPSRC (EP/R031924/1 and
EP/R032130/1) and NIST Center for Neutron Research. C.H. acknowledges support
by the Austrian Science Fund (FWF) Project No. P32144-N36 and the VSC4 of the
Vienna University of Technology.

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