Entanglement and manipulation of the magnetic and spin–orbit order in multiferroic Rashba semiconductors

Entanglement of the spin–orbit and magnetic order in multiferroic materials bears a strong potential for engineering novel electronic and spintronic devices. Here, we explore the electron and spin structure of ferroelectric α-GeTe thin films doped with ferromagnetic Mn impurities to achieve its multiferroic functionality. We use bulk-sensitive soft-X-ray angle-resolved photoemission spectroscopy (SX-ARPES) to follow hybridization of the GeTe valence band with the Mn dopants. We observe a gradual opening of the Zeeman gap in the bulk Rashba bands around the Dirac point with increase of the Mn concentration, indicative of the ferromagnetic order, at persistent Rashba splitting. Furthermore, subtle details regarding the spin–orbit and magnetic order entanglement are deduced from spin-resolved ARPES measurements. We identify antiparallel orientation of the ferroelectric and ferromagnetic polarization, and altering of the Rashba-type spin helicity by magnetic switching. Our experimental results are supported by first-principles calculations of the electron and spin structure.

Supplementary Table 1: ARPES Rashba-Zeeman-gas best-fit parameters.   Supplementary Note 1 Models for combined magnetic and spin-orbit order.

Effect of ferromagnetic order on a Rashba gas.
Here we describe the simplified two-dimensional Rashba-Zeeman-gas model discussed in the main text. We consider the momentum-dependent expectation values of the spin components interacting with ferromagnetic order defined by the unit vectorm = (m x , m y , m z ) and expressed by means of the Zeeman splitting ∆ Z in energy . The Hamiltonian is given by where E 0 is the band bottom, m * is the effective mass, α the Rashba parameter, ∆ Z the Zeeman gap, σ are the Pauli matrices and µ is the chemical potential. The energy eigenvalues are given by Supplementary Figure 1 summarizes the resulting dispersions of the spin-polarized bands for different orientations of the magnetic field B along the {xyz} axes. Only for magnetization along the surface normal (z-axis) the degeneracy is lifted for both k x and k y momenta by opening the Zeeman gap. For completeness Supplementary Table 1 summarizes all parameters used in the simplified Rashba-Zeemangas fit of the experimental data in Fig. 3 of the main text.

Dirac fermion with Rashba-like splitting and magnetic order.
It has been realized some time ago that the band dispersion of IV-VI semiconductors don't follow the typical free-electron like parabolic dispersion and that the bands can be better reproduced with a massive Dirac fermion model. The basic band dispersion in this case can be described by Here m D is the Dirac mass and represents the curvature for small k-values, v is the band velocity and represents the steepness of the bands for larger k-values, and E 0 is a band offset.
To this model the Rashba-like spin splitting and Zeeman gap opening can be added in a perturbative manner whereby it should be realised that, in order to keep a constant momentum splitting as a function of binding energy, the Rashba-like term should be of lower order as the main dispersive term. In the following we only consider magnetic order along the surface normal. This leads to the following phenomenological expression Here α D is the Rashba-like parameter in units of eVÅ 1 2 . It should be noted that the obtained values can only indirectly be compared to the Rashba parameters (in eVÅ) which are obtained for parabolic band dispersions. The Zeeman gap E Z is directly proportional to the ∆ D energy term according to Using this model the quality of the fit in Fig. 3 of the main text improves especially for larger momentum values. It should be noted that in this model the unit of the Rashba-like parameter α D changes and therefore can no longer be directly compared to free electron approximation model discussed in the main text. The fitted results from both models are displayed in Supplementary Figure 2 for both ARPES data and multiple scattering theory. Both α R and α D show a similar dependency on Mn doping and also the Zeeman gap shows identical values for both models. This further reinforces the validity of the used models. The error bars for α R (±7%, see Fig.3 of main text) and α D (±4%) were obtained from varying the fit parameters (α R and m * for the free electron approximation; and α D , m d and v for the massive fermion model). By incorporating third, and higher order corrections in k the model can be further refined. However, in this case it is preferential to directly refer to the ab-initio calculations based on multiple scattering theory seen in Fig. 3. Finally, Supplementary Here we illustrate the use of resonant photoemission to reveal the changes in the electronic structure of GeTe caused by the Mn doping. In this type of experiment one resonantly enhances the photoemission signal from certain elements by tuning the photon energy onto the corresponding peaks in the X-ray absorption spectrum (XAS). This yields elemental and chemical state resolution of the valence states (1, 2). In Fig. 2 of the main text, we could see that tuning the photon energy onto the main peak of the Mn L-edge XAS resonantly enhanced the Mn 3d related angle-integrated photoemission signal through the whole width of the valence band. We will now relate this resonant enhancement to particular bands resolved in electron momentum. Supplementary Figure  This difference reflects different strengths of the hybridization of these bands with the Mn states, depending on the particular character of the corresponding wavefunctions. The observed hybridization demonstrates integration of Mn states into the GeTe host states, consistent with the high solubility of Mn atoms in the GeTe lattice (3,4). Of importance are two experimental observations: • As seen in both difference spectra in Supplementary Figure 3c,d, the bands around the Zeeman gap are hybridized with Mn.
• The resonant image B in panel (b) and the difference image in (c), confirm the absence of any impurity states in the vicinity of E F which might otherwise have interfered with the RZ-splitting in this region.
With a small increase of photon energy staying within the main absorption peak, the experimental band structure image C in Supplementary Figure 3a stays essentially the same as the resonant one except for a smaller Mn 3d intensity enhancement. The image D in Supplementary Figure 3b, taken above the resonance, is identical to the pre-resonant one (image A in Fig. 3b) apart from changes in k z and a weak afterglow of the impurity state.
Supplementary Note 3 Experimental alignment of normal emission. Supplementary Note 4 Additional SARPES data.

SARPES 3D-vectorial peak-fitting analysis.
The COPHEE experimental station at the Swiss Light Source is a unique facility for SARPES experiments with a 3D Mott polarimeter (5,6). Combined with an angle-resolving photoelectron spectrometer it produces complete data sets consisting of photoemission intensities as well as spin polarization curves for three orthogonal vector components. SARPES data in the main text show the populations of electrons with momentum along A-Z-A and U-Z-U having their spin parallel (up) or antiparallel (down) to the local momentum-dependent spin quantization axis. To ensure equivalent measurement conditions, data were taken by tilting the sample [the τ direction seen in Supplementary Figure 6c] with Z-A or Z-U oriented perpendicular to the scattering plane. In a well established fitting routine (7) the photoemission spectrum is first dissected into individual peaks and background. Supplementary Figure 5(a,b) illustrate how the background subtracted total intensity momentum-distribution curves (MDC) are fitted with Voigt functions in both A-Z-A and U-Z-U directions. Supplementary Figure 5(c) shows ARPES data along A-Z-A measured at hν=22 eV, the dashed rectangle with red line in panel indicate the energy broadening of 60 meV and energy setting for α-GeTe and Ge 0.87 Mn 0.13 Te SARPES measurements. The polarization curves are modeled until the best fit is reached by simultaneously fitting the MDC intensity and the polarizations P x , P y and P z . The definition of the local {xyz} axes and polar representation of the 3D spin vectors is seen in panel (d). A spin polarization vector is assigned to each peak-fit. Their lengths correspond to the degree of polarization, their in-plane angle is defined by the experimental geometry sketched in Supplementary Figure 6c. SARPES data were measured by rotating the sample azimuth φ such that Z-U or Z-A directions were aligned perpendicular to the scattering plane. In the sample coordinate system the x (y) axis was oriented along Z-U (Z-A) direction.
The orange lines inside the polarization panels P x , P y and P z compare the output from the 3D-fit with measured data summarized in Supplementary Figure 5e for α-GeTe(111) and Fig. 5f for Ge 0.87 Mn 0.13 Te, respectively. From the total intensities and the polarization data we generate the spin-resolved populations I x , I y and I z along the three coordinate axes as a function of electron momenta. Consistently with the bulk Rashba initial states for α-GeTe (8), the Ge 0.87 Mn 0.13 Te in-plane P x,y polarization feature canted spin helicity with nearly factor two higher P y than P x along A-Z-A. Thus, similar to α-GeTe, in Ge 0.87 Mn 0.13 Te the in-plane spin texture is preserved. Their difference in P z is to be attributed to the spin reorientation discussed in next section, combined with α-GeTeP z -warping between the two equivalent Z-A and Z-A directions in the bulk Brillouin zone (Supplementary Figure 6c), as seen in Fig. 4a of main text.

Bulk states spin texture above and below the Zeeman gap.
To elucidate the spin texture above and below the Zeeman gap, and, to assess the overall in-plane P x,y spin texture of the Rashba-split bulk states, Supplementary Figure 6 visualize energy-resolved spin polarization for four finite momenta (≈ ±0.1Å −1 ) along U-Z-U and A-Z-A, denoted with A-D labels. For simplicity we ignore the canted spin and focus on maximum spin polarizations which are P x along U-Z-U (Supplementary Figure 6a) and P y along A-Z-A (Fig. 6b).
Step-by-step the spin polarization in panels (a-b) above (≈0.1 eV binding energy) and below the Zeeman gap (≈0.3 eV binding energy) are indicated with violet arrows. All the spinors summarized in panel (d) show the opposite spin texture of the Ge 0.87 Mn 0.13 Te bulk states just above and below the Zeeman gap, as also sketched in Fig. 1g of main text.
The band anisotropy between Z-U and Z-A induces a rather strong hexagonal warping which is responsible for the out-of-plane P z spin polarization component (9)(10)(11). As mentioned in the main text, the symmetry breaking due to ferromagnetic order in the out-of-plane direction reorient the spins in the P z -direction for Ge 1− As for the switching methodology under B-field, our results indicate that within the Ge 0.87 Mn 0.13 Te surface electronic structure there are Rashba-type states with strong surface localization resilient to switch. On the other hand the states which hybridize with, or originate from bulk Rashba bands, do switch. Our conjecture is that a similar behavior will be also reflected in the E-field control of the spin texture. We note that due to the remanent magnetic field of the experimental station we cannot guarantee that the effective magnetic field M inside the sample was the same when switching the B-field between ±700 Gauss. This could be the reason for the difference in polarization amplitudes seen in Supplementary Figure 8a and Fig. 4d of main text.