Magnetic surface reconstruction in the van-der-Waals antiferromagnet Fe_{1+x}Te
C. Trainer, M. Songvilay, N. Qureshi, A. Stunault, C. M. Yim, E. E. Rodriguez, C. Heil, V. Tsurkan, M. A. Green, A. Loidl, P. Wahl, C. Stock
MMagnetic surface reconstruction in the van-der-Waals antiferromagnet Fe x Te C. Trainer, M. Songvilay, N. Qureshi, A. Stunault, C. M. Yim, E. E. Rodriguez, C. Heil, V. Tsurkan,
6, 7
M. A. Green, A. Loidl, P. Wahl, and C. Stock School of Physics and Astronomy, University of St. Andrews, St. Andrews KY16 9SS, UK School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK Institut Laue-Langevin, 71 avenue des Martyrs, CS20156, 38042 Grenoble Cedex 9, France Department of Chemistry and Biochemistry, University of Maryland, College Park,Maryland 20742, USA Institute of Theoretical and Computational Physics,Graz University of Technology, NAWI Graz, 8010 Graz, Austria 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 School of Physical Sciences, University of Kent, Canterbury, CT2 7NH, UK (Dated: January 5, 2021)Fe x Te is a two dimensional van der Waals antiferromagnet that becomes superconducting onanion substitution on the Te site. The properties of the parent phase of Fe x Te are sensitiveto the amount of interstitial iron situated between the iron-tellurium layers. Fe x Te displayscollinear magnetic order coexisting with low temperature metallic resistivity for small concentrationsof interstitial iron x and helical magnetic order for large values of x . While this phase diagram hasbeen established through scattering [see for example E. E. Rodriguez et al. Phys. Rev. B , 064403(2011) and S. R¨oßler et al. Phys. Rev. B , 174506 (2011)], recent scanning tunnelling microscopymeasurements [C. Trainer et al. Sci. Adv. , eaav3478 (2019)] have observed a different magneticstructure for small interstitial iron concentrations x with a significant canting of the magneticmoments along the crystallographic c axis of θ =28 ± ◦ . In this paper, we revisit the magneticstructure of Fe . Te using spherical neutron polarimetry and scanning tunnelling microscopy tosearch for this canting in the bulk phase and compare surface and bulk magnetism. The resultsshow that the bulk magnetic structure of Fe . Te is consistent with collinear in-plane order ( θ = 0with an error of ∼ ◦ ). Comparison with scanning tunnelling microscopy on a series of Fe x Tesamples reveals that the surface exhibits a magnetic surface reconstruction with a canting angleof the spins of θ = 29 . ◦ . We suggest that this is a consequence of structural relaxation of thesurface layer resulting in an out-of-plane magnetocrystalline anisotropy. The magnetism in Fe x Tedisplays different properties at the surface when the symmetry constraints of the bulk are removed.
I. INTRODUCTION
Van der Waals forces differ from ionic and covalentbonding in terms of the strength and also the range offorces. Two dimensional materials that are based onsheets weakly held together by van der Waals forceshave recently been studied in the context of graphene ,and also for the investigation of two dimensional fer-romagnetism . Two dimensional magnetic van derWaals crystals have also been of interest in the con-text of iron based superconductivity with arguablythe structurally simplest such superconductor being themonolayer compound Fe x Te − y (Se,S) y consistingof weakly bonded iron chalcogenide sheets. Under anionsubstitution, an optimal superconducting transition tem-perature of ∼
14K has been reported in Fe x Te . Se . and ∼
10K in FeTe − x S x . In this paper we investi-gate the difference between the bulk and surface magneticstructures in the non-superconducting parent compoundFe x Te through the use of spherical neutron polarime-try and spin polarized scanning tunnelling microscopy invector magnetic fields.The single layered chalcogenide Fe x Te − y Q y (where Q =Se or S) has been important in the study of ironbased superconductors owing to its relatively simple sin- gle layer structure and because it is highly electronicallylocalized in comparison to other iron based systems.This is a property that is also reflected in the oxyse-lenides. The electronic and magnetic properties ofFe x Te − y Q y can be tuned via two variables - the pa-rameter x determines the amount of interstitial iron lo-cated between the weakly bonded FeTe layers and dis-ordered throughout the crystal, and y is the amount ofanion substitution and provides a chemical route towardssuperconductivity. It should be noted that while the in-terstitial iron is disordered introducing magnetic clus-ters , the electronic properties have been found to behomogeneous, and recently discussed in the contextof device fabrication . The sensitivity of the proper-ties to stoichiometry is also reflected in Fe δ Se. Therehave been several studies that have shown x and y to becorrelated and hence both influence the superconductiv-ity. In particular, the tetrahedral bond angles are altered with interstitial iron concentration x alongwith tuning the material across several magnetic andstructural phase transitions. It is this interplay betweenthe structural, magnetic, electronic and superconduct-ing properties which makes this material exciting, withthe expectation that understanding the relation betweenthese phases leads to an understanding of superconduc- a r X i v : . [ c ond - m a t . s t r- e l ] J a n tivity.Fe x Te has been found to display two spatially long-range correlated magnetic phases as a function of ironconcentration x separated by a region of spatially short-ranged magnetic order . For small interstitial ironconcentrations x ≤ .
12, the magnetic structure iscollinear below a temperature where a structuraltransition occurs from a tetragonal ( P /nmm ) to mon-oclinic ( P /m ) unit cell. Electronically, this also marksa transition from a resistivity which is “semi/poor”-metallic to being metallic in character at low temper-atures. Despite the metallic character, the low energyspin fluctuations are consistent with localized transversespin-waves . The second magnetic phase at concen-trations x > .
12 is helical in nature combined witha “semi/poor”-metallic behavior at all temperatures. The two disparate magnetic phases induced with the vari-able x are separated by a collinear spin density wave located near a Lifshitz point . As well as the tuning ofthe magnetic and crystallographic structures with inter-stitial iron concentration, the magnetic excitations alsodisplay a large dependence on x , even at high energytransfers .We will focus on the low interstitial iron concentrationsin this study. The magnetic structure for low interstitialiron concentrations is termed a “double-stripe” structureand has been investigated extensively with both unpo-larized and uniaxial polarized neutron scattering. Thiscollinear magnetic phase has magnetic moments alignedalong the crystallographic b axis and magnetic Braggpeaks in the neutron cross section at (cid:126)Q =( , , ) or de-noted as ( π, This work hasdemonstrated that the surface magnetic structure faith-fully follows the bulk magnetic phase diagram as a func-tion of interstitial iron in terms of both the crystallo-graphic and magnetic structures giving consistent resultswith neutron scattering on the interstitial iron concentra-tion where the magnetic and crystallographic structureschange. However, an important difference was observedfor the magnetic structure in the collinear “double-stripe” phase for small values of x . Tunneling measure-ments show a periodicity consistent with the stripe phasereported based on neutron scattering , however recentmeasurements in vector magnetic fields have observeda significant out of plane canting, along the crystallo-graphic c -axis, of the magnetic moment of θ ∼ ◦ whereneutron scattering reports the moments to be entirely inthe ab plane ( θ = 0). Interestingly, such a magneticstructure is consistent with early studies on Fe . Te, however given more recent work it is possible that thisconcentration was at the boundary between collinear andhelical magnetism possibly complicating the interpreta- b a c a b c a b c b a c b a c b a c Fe Te (axial/neutron) Fe
Te (canted/tunnelling) aab bc
FIG. 1. Comparison of magnetic order from neutron scatter-ing and STM. The axial magnetic structure with the momentsaligned along the crystallographic b axis ( θ =0) contrasts withthe canted structure measured with tunnelling measurementswhere the moments are canted out of the plane by θ =28 ± ◦ . tion.Based on the discrepancy between current neutrondiffraction results, scanning tunneling microscopy, andolder diffraction work, we revisit this problem applyingspherical neutron polarimetry to determine the out ofplane angle due to any canting of the spins in bulk sin-gle crystals of Fe x Te. This study focuses on the irondeficient portion of the phase diagram which exhibitscollinear order because this is where the differences be-tween neutron scattering and STM are most prominent.In this paper, we compare spin polarized scanning tun-nelling microscopy measurements of the magnetism onthe surface with a study of the bulk magnetic structure.The two different magnetic structures that will be com-pared in this paper are illustrated in Fig. 1. This paperis divided into five sections including this introduction.We first present the results from spin polarized scanningtunnelling microscopy of the canting angle in the surfacelayer. We then investigate the canting angle in the bulkfrom spherical neutron polarimetry and analyze the re-sults in terms of a possible canting in the bulk. We finallycompare these results and discuss the differences and pos-sible origins, including dipolar and anisotropy terms inthe magnetic Hamiltonian. Through this comparison wefind that the surface layer of the two dimensional vander Waals Fe x Te magnet exhibits a magnetic surfacereconstruction.
II. SPIN POLARIZED STM MEASUREMENT
We first discuss spin polarized STM measurements ofFe x Te probing the magnetic structure at the surface.
A. Experimental Details
Spin polarized STM measurements were conducted onsamples of Fe x Te with excess iron concentrations x ranging from 5% to 11 . . Atomically clean sur-faces for STM measurement were prepared by cleavingthe samples in-situ at a temperature of ∼ . Mag-netic tips were created by collecting excess Fe atoms fromthe sample surface . In this way, the tunneling cur-rent between the STM tip and sample becomes sensitiveto the relative angle between the magnetization of thetip and the sample. The tunneling current ( I SP ) due tothe spin polarization of the tip ( P tip ) and the sample( P sample ) can be expressed as I SP = I [1 + P tip P sample cos( φ )] , (1)where φ is the angle between the tip and sample magneti-zations. Figure 2(a) shows a typical spin polarized STMtopographic image of the iron telluride surface (Fe . Te).The excess iron atoms are seen in the STM images asbright protrusions on the surface. The bi-collinear anti-ferromagnetic order is imaged as a stripe like modulationrunning parallel to the sample b axis with a wavevector( q a ) along the sample a axis. The imaged wavelengthand direction of this ordering is in excellent agreementwith that determined from neutron scattering . Forferromagnetic tips, the magnetization of the tip is foundto follow the direction of an applied magnetic field therefore imaging the surface with the tip polarized bya field applied 180 ◦ to the original field orientation re-sults in a π phase reversal of the imaged magnetic order.This reversal of the imaged magnetic order can be seenby comparing images recorded with opposite applied fieldorientations shown in Figs. 2(a) and (b). B. STM Results
It is possible, through Eqn. 1, to directly measure thesample’s surface spin polarization from the spin polarizedSTM images. This is done by taking the difference of theimages recorded with oppositely polarized tips which isproportional to 2 P tip P sample cos( φ ). Fig. 2(c) shows sucha difference image showing the component of the sample’smagnetic order that is parallel to the crystal c axis. Thesum of the two images recorded with oppositely polarized sample x (%) I( q a ) || b I( q a ) || c θ ( ◦ )1 5 0.1221 0.0550 24.25272 10 0.2246 0.1570 34.95222 10 0.1542 0.1721 48.13502 10 0.2719 0.0186 3.91462 10 0.2243 0.1761 38.13793 11.5 0.2248 0.1275 29.5626TABLE I. Results from 6 different independent SP-STM mea-surements on different samples of Fe x Te with different mag-netic tips. The table shows the intensity of the imaged mag-netic order when the tip spin is parallel to the crystal b axisand when the tip spin is parallel to the crystal c axis and theresulting canting angle of the surface spins. tips resembles the topography that would be recordedif the sample was imaged with a non-spin polarized tip(Fig. 2(d)).By recording spin-polarized STM images with mag-netic field applied in three orthogonal spatial directionsit is possible to determine the precise orientation of thesample’s spin structure at the surface . The out ofplane canting angle ( θ ) of the surface spins resulting fromthis measurement is shown in Fig. 2(e). Clear cantingof the spins away from the ab plane can be observed.By plotting the absolute value of this angle as a his-togram, shown in Fig. 2(f), a clear peak at ∼ ◦ canbe observed. This measurement has been repeated for asample of Fe . Te, the results of which are also shownin Fig. 2(f). We determine the out of plane canting ofthe surface spins by fitting a Gaussian distribution plus alinear background to the data. By combining the fits toboth data sets we obtain an average out of plane cantingangle of 30 . ± . ◦ . This substantial canting of the sur-face spins is seen across multiple samples and has beenobserved in previous STM studies on this compound .We have also conducted further studies on other sam-ples of Fe x Te, where measurements were only recordedwith the tip polarized along the crystal b and c axes. Thedata is shown in Table I. The intensity of the magneticpeak I( q a ) for field applied along the crystallographic b and c axes respectively and the corresponding cantingangle ( θ ) are shown. The average out-of-plane cantingangle obtained from these values is 29 . ± . ◦ . Theerror bar here contains contributions from variations inthe magnetic properties of the STM tips, differences be-tween samples and the alignment of the magnetic fieldplane with the crystallographic axis of the sample. Thisis the magnetic structure shown in Fig. 1 ( b ). III. NEUTRON SCATTERING EXPERIMENTS
Having discussed the canted magnetic structure atthe surface, we now apply neutron scattering to studythe bulk magnetism. Neutron scattering, unlike x-raysor photon based measurements, is a bulk measurement Δ h (pm) -40 40 h (pm) (a)(b) (c)(d) q a q Te -90 90 (°) (e)(f) | | ( ° ) o cc u r en c e s Fe Tefit - 30.8 ° Fe Tefit - 29.6 ° FIG. 2. (a) - Spin polarized STM image of the surface of an Fe . Te sample with a bias voltage of 100mV and a set point currentof 50pA. Recorded with an applied magnetic field of 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 magneticordering vector ( q a ). (d) The average of the images (a) and (b), directly proportional to the non spin polarized componentof the tunneling current. Inset - the Fourier transform of (d) showing the atomic peaks due to the Te lattice. (e) The out ofplane canting angle ( θ ) measured from images recorded with three orthogonal directions of applied field. (f) Histograms of theabsolute value of the measured out of plane canting angle for data sets of three dimensional spin polarization data recorded ondifferent samples of FeTe. The red data points corresponds to the data shown in (e). Solid lines represent fits of a Gaussianfunction plus a linear background. of materials owing to the interaction between neutronsand matter being mediated by nuclear interactions. Forexample, for single crystalline Fe . Te with a neutronwavelength of λ = 1 . /e scattering length for thesum of absorption and incoherent cross sections is ∼ A. Experimental Details
To investigate the polarization matrix of the magneticorder in Fe x Te sensitive to the orientation of the lo-cal iron moments, we used the CRYOPAD (CryogenicPolarization Analysis Device) developed at the ILL .Unlike conventional polarization measurements which in- volve studying spin flip scattering along a particular crys-tallographic axis, CRYOPAD allows all components ofthe polarization matrix to be studied governed by theBlume-Maleev equations.
Single crystals of Fe . Tewere synthesized by the Bridgemann method . Allmeasurements discussed here were done at a base temper-ature of 2K using the IN20 spectrometer with the samplealigned such that Bragg peaks of the form (H 0 L) laywithin the horizontal scattering plane. The structuraland magnetic transition in this material occurs at ∼ The possible symmetry operations resulting fromthese domains are displayed in Table III and discussedbelow.Spherical neutron polarimetry is sensitive to the direc-tion of the ordered magnetic moment, spin chirality, andcoupling between nuclear and magnetic cross sections .In the case of structural domains that exist at low tem- (cid:126)Q P measured ij P axial ij P canted ij ( θ = 28 ◦ )( , , ) − . . − . − . − . − . − . − . . − − − − .
768 00 0 0 . ( , , ) − . . − . − . − . − . − . − . . − − − − .
768 00 0 0 . ( , , ) − . . − . − . − . − . − . − . . − − − − .
636 00 0 0 . ( , , ) − . . − . − . − . − . − . − . . − − − − .
946 00 0 0 . ( , , ) − . . − . − . − . − . − . − . . − − − − .
979 00 0 0 . ( , , ) − . . − . − . − . − . − . − . . − − − − .
618 00 0 0 . TABLE II. A list of the experimentally measured polarization matrices at T = 2K measured on IN20. The calculated matrixelements , assuming 100 % beam polarization, are shown for the axial spin structure and the canted magnetic structure with θ = 28 ◦ for comparison.TABLE III. Low temperature structural domains consideredhere for the magnetic structural analysis.1 x y z2 x y -z3 -x -y z4 -x -y -z perature (which average out the off-diagonal elements)from the structural transition (Table III) and in the ab-sence of spin chirality and coupling to a nuclear crosssection, the polarization matrix measured with spheri-cal neutron polarimetry becomes diagonal and takes thefollowing form, P ij = − | M ⊥ ,y | −| M ⊥ ,z | | (cid:126)M ⊥ |
00 0 − | M ⊥ ,y | −| M ⊥ ,z | | (cid:126)M ⊥ | where (cid:126)M ⊥ ≡ (cid:126)Q × (cid:126)M × (cid:126)Q . Here (cid:126)Q ≡ (cid:126)k i − (cid:126)k f is the momen-tum transfer and (cid:126)M is the magnetic moment direction.The matrix element P xx ≡ P is strictly = − (cid:126)M , but only the direction.In this experiment, the polarization matrix was mea-sured at six magnetic Bragg peaks at T = 2K. The full experimental polarization matrices P measured ij for theseBragg peaks are shown in Table II. The calculated ma-trices P axial ij and P canted ij shown in the table are discussedbelow in the context of our comparison with tunnellingand previous neutron results. B. Neutron scattering results
Figure 1 illustrates the two magnetic structures thatwe will compare the polarized neutron scattering resultsto in this section. The reported structure based on neu-tron diffraction on single crystals and also powders sug-gests that the structure is collinear with the momentsaligned along the crystallographic b axis (Fig. 1 (a)). Thestructure is often referred to as a “double-stripe” mag-netic structure. This is contrasted to a recent magneticstructure reported using scanning tunnelling microscopy(Fig. 1 (b)). The magnetic structure obtained from STMhas the magnetic moments collinear but canted along thecrystallographic c axis by an angle of θ = 29 . ± . ◦ .For the purposes of this section we refer to the neutronscattering structure which is aligned along the b axis as“axial” and the structure reported by spin polarized tun-nelling microscopy as “canted”. We now discuss the ap-plication of neutron spherical polarimetry to revisit thebulk magnetic structure of collinear Fe x Te.While the application of unpolarized neutron powderdiffraction and also uniaxial polarized neutrons maybearguably ambiguous in determining canting of the mag-netic moments owing to the number of accessible peaks a) Canted - Single domain
X Y ZXYZ b) Canted - multi-domain
X Y ZXYZ c) Axial
X Y ZXYZ X Y ZXYZ X Y ZXYZ-1 -0.5 0 0.5 1X Y ZXYZ
Q=(1/2, 0, 1/2)Q=(1/2, 0, 3/2) P ij FIG. 3. A color schematic of the polarization matrix forthe spin models under consideration for the Q -vectors Q =(1 / , , /
2) and Q = (1 / , , / θ = 28 ◦ ,(b) for multiple domains of the same order as in (a) usingthe symmetry relations in table III and (c) for the axial orderwith spins pointing along b . X − , Y − and Z − are the spincomponents along the three spatial directions, color encodesthe polarization. and statistics for low interstitial iron concentrations,spherical neutron polarimetry is very sensitive to thiscanting. We illustrate this in Fig. 3 which displays acolor representation of the calculated polarization matri-ces at the magnetic momentum positions (cid:126)Q =( , 0, )and ( , 0, ). Three different models are presented.Panel ( a ) displays a calculation based on the cantedmodel proposed by tunnelling measurements for a single structural and magnetic domain crystal. This calculationshows a non zero off-diagonal values for the matrix ele-ments for the P yz and P zy positions. However, Fe x Teundergoes a structural distortion from a tetragonal to amonoclinic unit cell that is coincident with magnetic or-dering. The four domains are related by symmetry asdisplayed in Table III. The corresponding matrix includ-ing the effects of domains is diagonal and is illustratedin Fig. 3(b). The magnitudes of the matrix elements | P yy | (cid:54) = | P zz | . This contrasts with the case where themagnetic 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 FIG. 4. A summary of the spherical polarimetry data fromIN20 measured for Fe . Te and compared against calcula-tions. (a,b) histograms of the polarization matrix elements P ij for calculations based on the canted and axial magneticstructures respectively. (c) the same histogram for the mea-sured matrix elements. (d,e) plots of the calculated polar-ization matrix elements as a function of measured values forboth the canted and axial magnetic structures. predicted matrix from both magnetic structures. Figs.4(a, b) illustrate histograms of the calculated polariza-tion matrix elements for the both the canted (tunnelling,Fig. 4(a) and axial (neutron, Fig. 4(b) magnetic struc-tures. The canted magnetic structure results in polariza-tion matrix elements for a range of values ranging from − →
1. The largest number of matrix elements appearat 0 resulting from the averaging over domains meaningthat all off-diagonal matrix elements are calculated to be0 (Fig. 3). The axial magnetic structure, in contrast onlydisplays three matrix elements ([ − , , P ij = 0 and is further shown in thetable displayed above (Table II in the experimental sec-tion). The origin of this error results from the incompletepolarization of the beam and also due to small misalign-ments ( ∼ − ◦ , see the appendix of Ref. 73 for an ° Fe Te, T= 2 K tunnelling(28 3 ° ) a) b) FIG. 5. A parameterization of the goodness of fit ( χ ) to thedata as a function of canting angle θ as defined in the text. Aneutron beam polarization of 0 .
88 was taken for the analysis.(a) χ over the full range of canted angles from 0 − ◦ . (b) χ in a narrow range of angles from 0 − ◦ . analysis of the errors) of the sample with respect to thebeam polarization. Based on the comparison betweenthe Figs. 4 (a-c), the neutron data is consistent with theaxial magnetic structure rather than the prediction of abroad spread of matrix elements which would result froma canted magnetic structure. This is further illustratedin Figs. 4 (d,e) which shows the calculated matrix el-ements as a function of the measured matrix elements.The spread of the data from a single straight line is ameasure of the “goodness of fit”. The canted magneticstructure in panel (c) clearly provides a much poorer de-scription of the data over the axial one displayed in panel(d).Figure 5 shows a plot of χ (a measure of the goodnessof fit) as a function of canting angle θ quantifying thesensitivity of our measurement using spherical polarime-try and also establishing a measure of the errorbar in ourexperiment. For this figure, we have defined χ in termsof the measured and calculated polarization matrix ele-ments P i by χ ≡ (cid:88) i,j (cid:88) α ( P measured ,αij − P calculated ,αij ) , (2)where the summation index α is taken over all Bragg peak peaks and the index ij are the matrix elements probedin this experiment. The plot of χ as a function of cant-ing angle for the 6 Bragg peaks (Table II) studied onIN20 shows a broad minimum near θ ∼ ◦ and a distinctmaximum with θ = 90 ◦ when the moments are point-ing along the crystallographic c axis. The vertical redline in Fig. 5(a) is the canted θ = 28 ± ◦ proposed bytunnelling measurements. The χ ( θ ) curve clearly showsthat our spherical neutron polarimetry data is inconsis-tent with a canted structure with a broad minimum ob-served near θ = 0 ◦ which is the axial structure foundpreviously in powders and single crystal unpolarized neu-tron measurements. The nature of the broad minimumin the χ surface indicates an underlying errorbar in themagnetic structure measured here of ∼ ± ◦ . The neu-tron scattering data shows that the magnetic structure iniron deficient Fe . Te is inconsistent with the magnitudeof the canted magnetic structure reported for the singlelayer limit in tunnelling measurements.
IV. DISCUSSION
The comparison of the spherical neutron polarimetrywith spin-polarized STM shows clearly that for Fe x Te,a surface magnetic reconstruction forms where the spinson the iron site tilt out of the ab -plane. In the follow-ing we will discuss possible mechanisms leading to thisreconstruction. A. Surface relaxation – Density functional theory(DFT) calculations
To attempt to explain the difference between the mea-sured magnetic structure of the surface and the bulk wehave performed DFT calculations on an FeTe slab wherethe surface was allowed to relax from the lattice positionsin the bulk. First-principles calculations were performedusing the Quantum
Espresso code. We employed op-timized norm-conserving Vanderbilt pseudopotentials with the Perdew-Burke-Ernzerhof exchange-correlationfunctional in the generalized gradient approximation .For the calculation, four layers of FeTe, without excessiron atoms ( x = 0), with a vacuum region of 13˚A in the z direction, bicollinear magnetic order along x , and fer-romagnetic order along z were considered. The z -lengthof the unit cell was kept fixed during variable cell re-laxation runs and spin-polarization taken into account.We chose a kinetic energy cutoff for the plane waves of80Ry, a Methfessel-Paxton smearing of 0 . × × k -mesh. Details of thecrystal structure and atomic positions were taken fromexperiment and then geometrically relaxed. The sur-face layer was found to relax away from the bulk layer tosuch an extent that the c axis parameter of the surfacelayer changes from 6 . . bulk surface(a) (b) ( ° ) E / F e ( e V ) FIG. 6. (a) Structural model of the surface layer relaxed inDFT calculations. The relaxation of the top surface layer canbe seen. (b) Calculation of the energy per Fe atom due to thedipole interaction as a function of out of plane canting angle θ of a bi-layer of FeTe as described in the text. The maximumenergy per Fe atom due to the dipole interaction is found fromthis calculation to be 3 . µ eV. The energy minimum is foundat θ = ± ◦ , i.e. out of the surface plane. with the Fe atoms of the surface layer displaced by upto 10pm from the unrelaxed position. Furthermore slightchanges in the length of the Fe − Te bonds of up to 3pmwere observed. This reconstructed structure is illustratedschematically in Fig. 6(a).
B. Magnetic dipole interactions
As a potential explanation for the canting of the spinsin the surface layer, we have considered the magneticdipole interaction between the Fe atoms. We have con-structed a numerical model of the surface of Fe x Te tocalculate the preferred spin orientation of the surface Featoms given dipolar interactions with the layer below.The model consists of a bulk layer of 101 ×
101 Fe atomswith a surface layer of 41 ×
41 Fe atoms. The spins ofeach Fe atom were fixed into the bicollinear AFM orderwith their in-plane component fixed to point along thecrystallographic b axis. The magnetic moment of eachFe atom was taken to be 2 µ B65 . The energy of the in-teraction of the dipoles in the system is then determinedby numerically summing over all the spins in the system.The equation for this process is: E = (cid:88) i (cid:54) = j µ π | r ij | (cid:20) µ i · µ j – 3 | r ij | ( µ i · r ij )( µ j · r ij ) (cid:21) (3)where the indices i and j indicate the different Fe posi-tions in both layers. By varying the canting angle θ ofthe surface spins we determined how the energy of thedipole interaction varies as a function of θ , this is shownin Fig. 6(b). We determine from this analysis that thedipole interaction in Fe x Te would favor aligning the Fespins along the crystallographic c axis, a result that is in good agreement with the effects of magnetic dipole inter-actions in bulk crystals . Indeed if one were to consideronly dipole interactions in addition to the AFM orderingin the bulk then the magnetic moments of the Fe wouldalign with the sample c axis. The fact that this is not ob-served in neutron scattering measurements leads us to theconclusion that a substantial magnetic anisotropy fromthe crystalline electric field keeps the Fe spins pointing inthe ab plane. We note that the tendency of the magneticdipole interaction to favour out-of-plane order is strongerfor the surface layer than it is in the bulk. The energyscale associated with the dipolar interaction per Fe spinis ∼ µ eV. C. Magnetocrystalline anisotropy
The energy scale of the dipolar interaction is extremelysmall in comparison to the measured ∼ . Both, thedirection and magnitude of magnetic dipole interactionssuggest that these are not sufficiently strong to explainthe out-of-plane tilting of the magnetization in the sur-face layer. This leaves the magnetocrystalline anisotropyresulting from crystalline electric field effects as apossible origin for the magnetic surface reconstruction.In bulk Fe x Te, the magnetic anisotropy results in anin-plane orientation of the spins . At the surface, thebroken symmetry resulting from the loss of a mirror planeand structural relaxation of the surface layer discussedabove imply that the magnetic anisotropy can differ sig-nificantly. V. CONCLUSIONS
The spherical neutron polarimetry results show a dis-tinct difference between the bulk magnetic structurein Fe . Te measured with neutron scattering and thecanted magnetic structure reported with tunnelling mea-surements in the single layer limit. This illustrates a dif-ference between bulk and surface magnetism in this Vander Waals magnet. It should be noted that the casesof tunnelling from a surface and neutron scattering fromthe bulk are not studying the exact same situation. Inthe bulk neutron response, each magnetic Fe x Te layereffectively represents a mirror plane, this is not the caseof a hard surface as is the situation in tunnelling. There-fore, from a symmetry perspective, there is no constraintforcing both situations to be identical.The magnetic moments in Fe . Te interact through ei-ther effects of bonding (including possible itinerant inter-actions such as RKKY exchange) or dipolar interactions.For interactions within the ab plane of Fe . Te theseshould be dominated by the effects of bonding which re-sult in strong dispersion of the magnetic excitations alongthese directions . The situation along the c -axis is lessclear as the FeTe layers are only weakly bonded throughVan der Waals forces. However, dipolar forces which de-cay ∼ r are still present in the magnetic Hamiltonianand these could be strongly influential to the magneticcorrelations along the crystallographic c -axis. Magneticneutron inelastic scattering have indeed found the weak c -axis correlations occuring without the presence ofstrong bonding and only Van der Waals forces.Another effect not directly tied to the crystalline elec-tric field effects discussed above that may be the ori-gin of the difference between surface (tunnelling) andbulk (neutron) responses is interstitial iron. Previousneutron scattering results have shown a strong connec-tion between the magnetic correlations and the intersti-tial iron concentration x . With increasing interstitialiron concentration, the crystallographic c -axis decreaseswhich could in turn increase the importance of the dipo-lar terms in the magnetic Hamiltonian. The interstitialsites may also be magnetic and this could influence thestructure in the FeTe plane.We note that differences in the magnetic structure andperiodicity between tunnelling and bulk neutrons scatter-ing have been reported before. Comparative measure-ments done in superconducting La − x Sr x CuO withtunnelling and neutron scattering have observed differ- ent wavevectors, however a similar response in the dy-namics. The role of dipolar and crystalline electric fieldterms in the magnetic Hamiltonian may be an issue thatneeds to be considered in all magnetic layered and twodimensional structures.While further calculations will be required to ulti-mately understand the difference in magnetic structuresobserved on the surface and the bulk, our study illus-trates the sensitivity and difference between the mag-netism in the Fe . Te Van der Waals magnet between thebulk and the surface. This has been established througha comparison between spherical polarimetry to determinethat the bulk magnetic structure of iron deficient Fe . Tehas θ =0 ± ◦ , while using spin-polarized STM to charac-terize the surface magnetic order. We suggest the differ-ence between magnetic structures found between scan-ning tunnelling microscopy and neutron scattering orig-inates from the relaxation of the surface layer and thecorresponding changes in magnetocrystalline anisotropy.We acknowledge financial support from the EPSRC(EP/R031924/1 and EP/R032130/1) and NIST Centerfor Neutron Research. C.H. acknowledges support by theAustrian Science Fund (FWF) Project No. P32144-N36and the VSC4 of the Vienna University of Technology. L. Mayor, Phys. World , 28 (2016). A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov,and A. K. Geim, Rev. Mod. Phys. , 109 (2009). J. L. Miller, Physics Today , 16 (2017). K. Ishida, Y. Nakai, and H. Hosono, J. Phys. Soc. Jpn. , 062001 (2009). Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono,J. Am. Chem. Soc. , 3296 (2008). D. C. Johnston, Adv. Phys. , 803 (2010). J. Paglione and R. Greene, Nat. Phys. , 645 (2010). P. Dai, Rev. Mod. Phys. , 855 (2015). D. S. Inosov, C. R. Physique , 60 (2016). C. Stock and E. E. McCabe, J. Phys. Condens. Matter ,453001 (2016). P. Dai, J. Hu, and E. Dagotto, Nature Phys. , 709 (2012). M. Lumsden and A. D. Christianson, J. Phys. Condens.Matter , 203203 (2010). F. C. Hsu, J. Y. Luo, T. K. Chen, T. W. Huang, P. M.Wu, Y. C. Lee, Y. L. Huang, Y. Y. Chu, D. C. Yan, andM. K. Wu, Proc. Natl. Acad. Sci. USA , 14262 (2008). J. S. Wen, G. Xu, G. D. Gu, J. M. Tranquada, and R. J.Birgeneau, Rep. Prog. Phys. , 124503 (2011). B. C. Sales, A. S. Sefat, M. A. McGuire, R. Y. Jin, D. Man-drus, and Y. Mozharivskyj, Phys. Rev. B , 094521(2009). Y. Mizuguchi, F. Tomioka, S. Tsuda, T. Yamaguchi, andT. Takano, Appl. Phys. Lett. , 012503 (2009). Z. P. Yin, K. Haule, and G. Kotliar, Nat. Mater. , 932(2011). S. R¨oßler, D. Cherian, S. Harikrishnan, H. L. Bhat, S. Eliz-abeth, J. A. Mydosh, L. H. Tjeng, F. Steglich, andS. Wirth, Phys. Rev. B , 144523 (2010). Q. Si and E. Abrahams, Phys. Rev. Lett. , 076401 (2008). Q. Si, Nat. Phys. , 629 (2009). E. E. McCabe, C. Stock, E. E. Rodriguez, A. S. Wills,J. W. Taylor, and J. S. O. Evans, Phys. Rev. B ,100402(R) (2014). L. L. Zhao, S. Wu, J. K. Wang, J. P. Hodges, C. Broholm,and E. Morosan, Phys. Rev. B , 020406(R) (2013). J.-X. Zhu, R. Yu, H. Wang, L. L. Zhao, M. D. Jones, J. Dai,E. Abrahams, E. Morosan, M. Fang, and Q. Si, Phys. Rev.Lett. , 216405 (2010). B. Freelon, Y. H. Liu, J.-L. Chen, L. Craco, M. S. Laad,S. Leoni, J. Chen, L. Tao, H. Wang, R. Flauca, Z. Yamani,M. Fang, C. Chang, J.-H. Guo, and Z. Hussain, Phys. Rev.B , 155139 (2015). V. Thampy, J. Kang, J. A. Rodriguez-Rivera, W. Bao,A. T. Savici, J. Hu, T. J. Liu, B. Qian, D. Fobes, Z. Q.Mao, C. B. Fu, W. C. Chen, Q. Ye, R. W. Erwin, T. R.Gentile, Z. Tesanovic, and C. Broholm, Phys. Rev. Lett. , 107002 (2012). X. He, G. Li, J. Zhang, A. B. Karki, R. Jin, B. C. Sales,A. S. Sefat, M. A. McGuire, D. Mandrus, and E. W.Plummer, Phys. Rev. B , 220502(R) (2011). Z. Xu, J. A. Schneeloch, M. Yi, Y. Zhao, M. Matsuda,D. M. Pajerowski, S. Chi, R. J. Birgeneau, G. Gu, J. M.Tranquada, and G. Xu, Phys. Rev. B , 214511 (2018). A. Zalic, S. Simon, S. Remennik, A. Vakahi, G. D. Gu,and H. Steinberg, Phys. Rev. B , 064517 (2019). T. M. McQueen, Q. Huang, V. Ksenofontov, C. Felser,Q. Xu, H. Zandbergen, Y. S. Hor, J. Allred, A. J. Williams,D. Qu, J. Checkelsky, N. P. Ong, and R. J. Cava, Phys.Rev. B , 014522 (2009). E. E. Rodriguez, C. Stock, P. Y. Hsieh, N. P. B.dn J. Paglione, and M. A. Green, Chem. Sci. , 1782 (2011). Y. Sun, T. Yamada, S. Pyon, and T. Tamegai, Sci. Rep. , 32290 (2016). P. Babkevich, M. Bendele, A. T. Boothroyd, K. Conder,S. N. Gvasaliya, R. Khasanov, E. Pomjakushina, andB. Roessli, J. Phys. Condens. Matt. , 142202 (2010). M. Bendele, P. Babkevich, S. Katrych, S. N. Gvasaliya,E. Pomjakushina, K. Conder, B. Roessli, A. T. Boothroyd,R. Khasanov, and H. Keller, Phys. Rev. B , 212504(2010). S. X. Huang, C. L. Chien, V. Thampy, and C. Broholm,Phys. Rev. Lett. , 217002 (2010). Z. Xu, J. A. Schneeloch, J. Wen, E. S. Boˇzin, G. E.Granroth, B. L. Winn, M. Feygenson, R. J. Birgeneau,G. Gu, I. A. Zaliznyak, J. M. Tranquada, and G. Xu,Phys. Rev. B , 104517 (2016). Z. Xu, J. Wen, G. Xu, Q. Jie, Z. Lin, Q. Li, S. Chi, D. K.Singh, G. Gu, and J. M. Tranquada, Phys. Rev. B ,104525 (2010). J. Wen, Z. Xu, G. Xu, M. D. Lumsden, P. N. Valdivia,E. Bourret-Courchesne, G. Gu, D.-H. Lee, J. M. Tran-quada, and R. J. Birgeneau, Phys. Rev. B , 024401(2012). W. Bao, Y. Qiu, Q. Huang, M. A. Green, P. Zajdel, M. R.Fitzsimmons, M. Zhernenkov, S. Chang, M. Fang, B. Qian,E. K. Vehstedt, J. Yang, H. M. Pham, L. Spinu, and Z. Q.Mao, Phys. Rev. Lett. , 247001 (2009). C. Koz, S. R¨oßler, A. A. Tsirlin, S. Wirth, and U. Schwarz,Phys. Rev. B , 094509 (2013). J. Wen, G. Xu, Z. Xu, Z. W. Lin, Q. Li, W. Ratcliff, G. Gu,and J. M. Tranquada, Phys. Rev. B , 104506 (2009). S. Li, C. de la Cruz, Q. Huang, Y. Chen, J. W. Lynn,J. Hu, Y.-L. Huang, F.-C. Hsu, K.-W. Yeh, M.-K. Wu,and P. Dai, Phys. Rev. B , 054503 (2009). I. A. Zaliznyak, Z. J. Xu, J. S. Wen, J. M. Tranquada,G. D. Gu, V. Solovyov, V. N. Glazkov, A. I. Zheludev,V. O. Garlea, and M. B. Stone, Phys. Rev. B , 085105(2012). G. F. Chen, Z. G. Chen, J. Dong, W. Z. Hu, G. Li, X. D.Zhang, P. Zheng, J. L. Luo, and N. L. Wang, Phys. Rev.B , 140509(R) (2009). C. Stock, E. E. Rodriguez, P. Bourges, R. A. Ewings,H. Cao, S. Chi, J. A. Rodriguez-Rivera, and M. A. Green,Phys. Rev. B , 144407 (2017). Y. Song, X. Lu, L.-P. Regnault, Y. Su, H.-H. Lai, W.-J.Hu, Q. Si, and P. Dai, Phys. Rev. B , 024519 (2018). S. R¨oßler, D. Cherian, W. Lorenz, M. Doerr, C. Koz,C. Curfs, Y. Prots, U. K. R¨oßler, U. Schwarz, S. Eliza-beth, and S. Wirth, Phys. Rev. B , 174506 (2011). P. Materne, C. Koz, U. K. R¨oßler, M. Doerr, T. Goltz,H. H. Klauss, U. Schwarz, S. Wirth, and S. R¨oßler, Phys.Rev. Lett. , 177203 (2015). D. Parshall, G. Chen, L. Pintschovius, D. Lamago, T. Wolf,L. Radzihovsky, and D. Reznik, Phys. Rev. B ,140515(R) (2012). E. E. Rodriguez, D. A. Sokolov, C. Stock, M. A. Green,O. Sobolev, J. A. Rodriguez-Rivera, H. Cao, andA. Daoud-Aladine, Phys. Rev. B , 165110 (2013). C. Stock, E. E. Rodriguez, M. A. Green, P. Zavalij, andJ. A. Rodriguez-Rivera, Phys. Rev. B , 045124 (2011). C. Stock, E. E. Rodriguez, and M. A. Green, Phys. Rev.B , 094507 (2012). M. Lumsden, A. D. Christianson, E. A. Goremychkin,S. E. Nagler, H. A. Mook, M. B. Stone, D. L. Abernathy, T. Guidi, G. J. MacDougall, C. de al. Cruz, A. S. Sefat,M. A. McGuire, B. C. Sales, and D. Mandrus, Nat. Phys. , 182 (2010). C. Stock, E. E. Rodriguez, O. Sobolev, J. A. Rodriguez-Rivera, R. A. Ewings, J. W. Taylor, A. D. Christianson,and M. A. Green, Phys. Rev. B , 121113(R) (2014). I. A. Zaliznyak, Z. Xu, J. M. Tranquada, G. Gu, A. M.Tsvelik, and M. B. Stone, Phys. Rev. Lett. , 216403(2011). O. J. Lipscombe, G. F. Chen, C. Fang, T. G. Perring, D. L.Abernathy, A. D. Christianson, T. Egami, N. Wang, J. Hu,and P. Dai, Phys. Rev. Lett. , 057004 (2011). M. Enayat, Z. X. Sun, U. R. Singh, R. Aluru, S. Schmaus,A. Yaresko, Y. Liu, V. Tsurkan, A. Loidl, J. Deisenhofer,and P. Wahl, Science , 653 (2014). U. R. Singh, R. Aluru, Y. Liu, C. Lin, and P. Wahl, Phys.Rev. B , 161111(R) (2015). A. Sugimoto, R. Ukita, and T. Ekino, Phys. Procedia ,85 (2013). T. H¨anke, U. R. Singh, L. Cornils, S. Manna, A. Kamla-pure, M. Bremholm, E. M. J. Hedegaard, B. B. Iversen,P. Hofmann, J. Hu, Z. Mao, J. Wiebe, and R. Wiesendan-ger, Nat. Commun. , 13939 (2017). C. Trainer, C. M. Yim, C. Heil, F. Giustino, D. Croitori,V. Tsurkan, A. Loidl, E. E. Rodriguez, C. Stock, andP. Wahl, Sci. Adv. , eaav3478 (2019). D. Fruchart, P. Convert, P. Wolfers, R. Madar, J. P. Sen-ateur, and R. Fruchart, Mater. Res. Bull. , 169 (1975). C. Trainer, C. M. Yim, M. McLaren, and P. Wahl, Rev.Sci. Instrum. , 093705 (2017). S. C. White, U. R. Singh, and P. Wahl, Rev. Sci. Instrum. , 113708 (2011). R. Wiesendanger, Rev. Mod. Phys. , 1495 (2009). E. E. Rodriguez, C. Stock, P. Zajdel, K. L. Krycka, C. F.Majkrzak, P. Zavalij, and M. A. Green, Phys. Rev. B ,064403 (2011). K. F. Zhang, X. Zhang, F. Yang, Y. R. Song, X. Chen,C. Liu, D. Qian, W. Luo, C. L. Gao, and J. F. Jia, Appl.Phys. Lett. , 061601 (2016). F. Tasset, P. J. Brown, E. Lelievre-Berna, T. Roberts,S. Pujol, J. Allibon, and E. Bourgeat-Lami, Physica (Am-sterdam) , 69 (1999). P. J. Brown, J. B. Forsyth, and F. Tasset, Proc. R. Soc.A , 147 (1993). M. Blume, Phys. Rev. , 1670 (1963). S. V. Maleev, V. G. Baryakhtar, and R. A. Suris, Sov.Phys. - Solid State , 2533 (1963). D. Fobes, I. A. Zaliznyak, Z. Xu, R. Zhong, G. Gu, J. M.Tranquada, L. Harriger, D. Singh, V. O. Garlea, M. Lums-den, and B. Winn, Phys. Rev. Lett. , 187202 (2014). N. Qureshi, J. Appl. Cryst. , 175 (2019). N. Giles-Donovan, N. Qureshi, R. D. Johnson, L. Y. Zhang,S.-W. Cheong, S. Cochran, and C. Stock, Phys. Rev. B , 024414 (2020). P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car,C. Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni,I. Dabo, A. D. Corso, S. de Gironcoli, S. Fabris, G. Fratesi,R. Gebauer, U. Gerstmann, C. Gougoussis, A. Kokalj,M. Lazzeri, L. Martin-Samos, N. Marzari, F. Mauri,R. Mazzarello, S. Paolini, A. Pasquarello, L. Paulatto,C. Sbraccia, S. Scandolo, G. Sclauzero, A. P. Seitsonen,A. Smogunov, P. Umari, and R. M. Wentzcovitch, J. Phys.Condens. Matter , 395502 (2009). D. R. Hamann, Phys. Rev. B , 085117 (2013). J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.Lett. , 3865 (1996). M. Methfessel and A. T. Paxton, Phys. Rev. B , 3616(1989). H. J. Monkhorst and J. D. Pack, Phys. Rev. B , 5188(1976). D. C. Johnston, Phys. Rev. B , 014421 (2016). A. M. Turner, F. Wang, and A. Vishwanath, Phys. Rev.B , 224504 (2009). K. Yosida,
Theory of Magnetism (Springer-Verlag, NewYork, 1996). Z. Xu, J. A. Schneeloch, J. Wen, B. L. Winn, G. E. Granroth, Y. Zhao, G. Gu, I. Zaliznyak, J. M. Tranquada,R. J. Birgeneau, and G. Xu, Phys. Rev. B , 134505(2017). J. Leiner, V. Thampy, A. D. Christianson, D. L. Aber-nathy, M. B. Stone, M. D. Lumsden, A. S. Sefat, B. C.Sales, J. Hu, Z. Mao, W. Bao, and C. Broholm, Phys.Rev. B , 100501(R) (2014). N. B. Christensen, D. F. McMorrow, H. M. Rønnow,B. Lake, S. M. Hayden, G. Aeppli, T. G. Perring,M. Mangkorntong, M. Nohara, and H. Takagi, Phys. Rev.Lett.93