Magnetism in Semiconducting Molybdenum Dichalcogenides

Transition metal dichalcogenides (TMDs) are interesting for understanding fundamental physics of two-dimensional materials (2D) as well as for many emerging technologies, including spin electronics. Here, we report the discovery of long-range magnetic order below TM = 40 K and 100 K in bulk semiconducting TMDs 2H-MoTe2 and 2H-MoSe2, respectively, by means of muon spin-rotation (muSR), scanning tunneling microscopy (STM), as well as density functional theory (DFT) calculations. The muon spin rotation measurements show the presence of a large and homogeneous internal magnetic fields at low temperatures in both compounds indicative of long-range magnetic order. DFT calculations show that this magnetism is promoted by the presence of defects in the crystal. The STM measurements show that the vast majority of defects in these materials are metal vacancies and chalcogen-metal antisites which are randomly distributed in the lattice at the sub-percent level. DFT indicates that the antisite defects are magnetic with a magnetic moment in the range of 0.9-2.8 mu_B. Further, we find that the magnetic order stabilized in 2H-MoTe2 and 2H-MoSe2 is highly sensitive to hydrostatic pressure. These observations establish 2H-MoTe2 and 2H-MoSe2 as a new class of magnetic semiconductors and opens a path to studying the interplay of 2D physics and magnetism in these interesting semiconductors.


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Fig. S1. ZF μSR time spectra and temperature-dependent parameters for MoSe 2 . Zero-field µSR time spectra for the polycrystalline (Sample D) sample of MoSe 2 , recorded for various temperatures in the range between 10 K and 300 K, are  shown in Fig. S1. Below T M 100 K, in addition to the strongly relaxed signal, a spontaneous muon-spin precession with a well-defined frequency is observed, which is clearly visible in the raw data. The inset of Fig. S1 shows the temperature dependence of the local magnetic field µ 0 H int at the muon site for both single crystalline (Sample E) and polycrystalline (Sample D) samples of MoSe 2 . There is a smooth increase of µ 0 H int below T M 100 K, reaching the saturated value of µ 0 H int = 310 mT at low temperatures. Observation of the spontaneous muon-spin precession indicates the occurrence of long range static magnetic order in semiconducting 2H-MoSe 2 , similar as for 2H-MoTe 2 . However, the magnetic ordering temperature T M 100 K as well as the internal field µ 0 H int 310 mT in 2H-MoSe 2 is higher as compared to the ones T M 40 K and µ 0 H int 200 mT, observed in 2H-MoTe 2 . This difference might be related to the different magnetic structures in these two samples 2H-MoSe 2 than in 2H-MoTe 2 . On the other hand, the fraction of the strongly damped signal is higher in 2H-MoSe 2 than in 2H-MoTe 2 .
Zero-field µSR measurements reveal that the homogeneous internal magnetic fields in 2H-MoTe 2 and 2H-MoSe 2 appear below T M 40 K and 100 K, respectively. However, the weak-TF µSR experiments show that the short-range (inhomogeneous) magnetism appear at much higher temperatures. Figure S2 displays the fraction of the low-frequency oscillations V osc , determined from the weak-TF µSR data, as a function of temperature in 2H-MoTe 2 and 2H-MoSe 2 . In MoTe 2 , at 450 K, V osc exhibits nearly a maximum value and decreases with decreasing temperature and tends to saturate below 300 K. This ∼ 30 % reduction of V osc can arise for few different reasons and it is most likely not magnetic in origin. However, there is an additional decrease of V osc starting below T onset M ∼ 180 K, which is due to the appearance of the inhomogeneous short-range magnetism and V osc continues to reduce until it reaches the minimum value V osc 0.1 at the long-range magnetic ordering temperature T M 40 K, below which zero-field µSR shows a well defined uniform internal magnetic fields. This implies that, the long range magnetic order is achieved only below T M 40 K, while the short range magnetism appear at higher temperature T onset M ∼ 180 K. The same is also observed in MoSe 2 (see Fig. S2), i.e., the short range magnetism appear below T onset M ∼ 250 K, while the long-range order is achieved below T M 100 K. Such a big difference between the short range and long range magnetic ordering temperatures was also observed in antiferromagnetic EuTiO 3 [48]. The polycrystalline samples of 2H-MoTe 2 and 2H-MoSe 2 have been studied by means of the Electron Spin Resonance (ESR) technique with the emphasis on checking for the paramagnetic impurity phases. ESR experiments were performed with a Bruker EMX spectrometer at X-band frequencies (ν = 9.4 GHz) equipped with a continuous He gas-flow cryostat in the temperature range 10 < T < 300 K. In Figure S3a and b the ESR spectra of 2H-MoTe 2 and 2H-MoSe 2 , respectively, are shown for various temperatures. It is remarkable that no trace of ESR signal was found in the sample 2H-MoTe 2 (see Fig. S3a) down to the lowest temperature, indicating the absence of any even small amount of paramagnetic impurities such as Fe or Ni. This provides strong support to the intrinsic nature of magnetic order in 2H-MoTe 2 , observed by µSR. We  found a small ESR signal in 2H-MoSe 2 with the g-factor of 4 (See Fig. S3b), which indicates the presence of small amount of paramagnetic Fe-impurities. However, this tiny amount of paramagnetic Fe-impurities would not lead to the observed long-range magnetic order in 2H-MoSe 2 . ESR experiments give evidence that the magnetic order in 2H-MoTe 2 and 2H-MoSe 2 is not related to the presence of an impurity phase but it is an intrinsic state.
Pair Distribution Function (PDF) Structure Confirmation of 2H-MoTe2 and 2H-MoSe2 Total scattering X-ray measurements were performed at the National Synchrotron Light Source II (XPD, 28-ID-2), Brookhaven National Laboratory. Finely ground powders of 2H-MoTe 2 and 2H-MoSe 2 were sealed in polyimide capillaries and diffraction patterns were collected in a Debye-Scherrer geometry with an X-ray energy of 66.479 keV (λ = 0.1865Å) using a large-area 2D Perkin Elmer detector. The detector was mounted with a sample-to-detector distance of 204.14 mm. The samples were measured at 90 K and 300 K using an Oxford CS-700 cryostream. The experimental geometry, 2θ range, and detector misorientations were calibrated by measuring a crystalline nickel powder directly prior to data collection at each temperature point, with the experimental geometry parameters refined using the PyFAI program [ ]. Standardized corrections are then made to the data to obtain the total scattering structure function, F (Q), which is then Fourier transformed to obtain the PDF, using PDFgetX3 [ ] within the xPDFsuite [51]. The maximum range of data used in the Fourier transform (Q max , where Q = 4π sin θ/λ is the magnitude of the momentum transfer on scattering) was chosen to be 25Å −1 to give the best tradeoff between statistical noise and real-space resolution. PDFgui was used to construct unit cells from reference structures, carry out structure refinements, and determine the agreement between calculated PDFs and data, quantified by the residual, R w [52].
Average structure PDF refinements for 2H-MoTe 2 and 2H-MoSe 2 are performed over a wide r-range from 1.5 < r < 50Å. Lattice constants and atomic displacement parameters were constrained by hexagonal symmetry. The results of the PDF analysis are summarized in Figure S4. For both temperatures T = 90 K and 300 K, the PDF is in good agreement with the 2H polytype (SG:P 6 3 /mmc) reported by D. Puotinen and R.E. Newnham [ ]. No evidence of structural distortions or segregation, originating from the dilute concentration of intrinsic defects was found, in line with the observation of a spatially homogeneous distribution of defects from STM. The best PDF fit for 2H-MoTe 2 at 300 K, yields refined lattice parameters  of a = b = 3.5186Å, and c = 13.9631Å. For 2H-MoSe 2 at 300 K, a = b = 3.2875Å and c = 12.9255Å.

High pressure µSR data for MoTe2
The results of the high pressure weak-TF µSR experiments are summarised in Figs. S5(a-d). Namely, we plot the temperature dependence of V osc for various hydrostatic pressures. A strong reduction of T M as well as of the corresponding magnetic fraction is observed under pressure. For the highest applied pressure of p = 1.6 GPa the T M decreases by ∼ 30 K and the corresponding fraction by ∼ 50 %. Note that the pressure dependence of the high temperature magnetic transition was not measured due to technical issues related to the high pressure µSR technique.

Magnetization measurements of MoTe2 and MoSe2
We carried out magnetization experiments on both MoTe 2 and MoSe 2 and measured the temperature dependence of the macroscopic magnetic moment under zero-field cooled (ZFC) (sample was cooled down to the base-T in zero magnetic field and the measurements were done upon warming) as well as field-cooled (FC) conditions (the sample was cooled down to the base-T in an applied magnetic field and the measurements were done upon warming). Figures S6a and b show the moment for MoSe 2 and MoTe 2 , respectively, recorded in an applied field of 10 mT. A large difference between ZFC and FC magnetic moments are seen for both samples. The samples show a combination of a temperature-independent diamagnetism, small van Vleck-type paramagnetism (which is determined by the energy separation of bonding and anti-bonding states E a -E b . E a -E b is proportional to the band gap E g ) and a ferromagnetic contribution that onsets near 230 K and 180 K for MoSe 2 and MoTe 2 , respectively. The difference between the FC and ZFC response suggests that different ferromagnetic domains tend to cancel out (anti-align) after ZFC. If we FC, then the domains align. The onset temperatures of hysteresis 230 K and 180 K for MoSe 2 and MoTe 2 , respectively, are close to the temperature T onset M below which µSR experiments show the appearance of inhomogeneous magnetism (see Figure S2). This means that below T onset M some small ferromagnetic domains (droplets of magnetic order) form, which produces inhomogeneous magnetic fields and would result the absence of coherent oscillations in the µSR signal. Instead, the various magnetic fields produced in the sample will give rise to a strong damping of the muon asymmetry, which we clearly observe at T onset M = 180 K in MoTe 2 and T onset M = 250 K in MoSe 2 . µSR observes homogeneous magnetism below T M 40 K and 100 K for MoTe 2 and MoSe 2 , respectively, and the anomalies (such as an additional increase of the moment and of the difference) at around these temperatures can also be seen in magnetization data ( Figures S6a and b). Thus, the µSR data for oscillations and damping and the SQUID data for temperature and field dependence of the magnetization are completely consistent with each other.
For the sample MoSe 2 the magnetic contribution dominates over the diamagnetism, gives rise to total positive moment. In contrary, for MoTe 2 diamagnetism dominates over the magnetic contribution. Assuming that the core diamagnetism is nearly the same in these materials, this difference might be related to the stronger magnetism (bigger moments) in MoSe 2 than in MoTe 2 . This is consistent with µSR experiments, showing factor of two higher magnetic ordering temperature and factor of 1.5 higher internal magnetic field in MoSe 2 as compared to those in MoTe 2 .
The field dependence of the magnetic moment for MoSe 2 and MoTe 2 , recorded at three different temperatures are shown in Fig. S7a and b, respectively. Large hysteresis loop is observed for both samples at the base temperature, confirming the presence of ferromagnetism in these semiconductors. The magnitude of the loop decreases with increasing temperature and fully closes at high temperatures. This is consistent with the temperature dependent magnetisation data which shows no difference between FC and ZFC response at temperatures above 230 K and 180 K for MoSe 2 and MoTe 2 , respectively. The coercive field, estimated at T = 5 K, is 300 G and 400 G for MoSe 2 and MoTe 2 , respectively.  The Hubbard U value used in our simulations is in the range of 0.5 to 4.0 eV to account for strong on-site interactions at the defect as mentioned explicitly in the Methods. We calculated the local magnetization of the antisite defect as a function of U . The strong U dependence of moment is found as shown in Figure S8b. We have also explicitly calculated the magnitude of the Hubbard U using linear response theory as included in Figure  S8a. The magnitude obtained is of U LR = 2.72 eV for the antisite defect (this value is marked as the dashed line in Figure S8b), which is within the limit of initial range calculated the magnetic properties of the defects. .(a) Occupation number versus rigid potential shifts α for antisite defects for the bare, non-interacting potential χ0 and the interacting potential χ.From the angular coefficients of both curves we can extract the optimum ULR for our system, ULR = χ0 −1 -χ −1 [ ].
(b) Variation of the local magnetization at the defect antisite versus U . At U = 0, no magnetic moments are observed as the defect shows a symmetric configuration at the Mo-Mo bonds. At U > 0.5 eV, this symmetry is broken and the defect develops an appreciable magnetic moment that increases with U as a result of the increased localization of the bands.
versus Hubbard U