Structural properties of the heterostructures
The Pb1 − xSnxSe/EuSe superlattices oriented in the (111) direction are grown on BaF2 (111) substrates by MBE. First, a thick buffer layer of insulating Pb0.84Eu0.16Se (400-500nm) is grown to reduce the lattice mismatch between the Pb1 − xSnxSe wells (a = 6.107–6.088Å) and the substrate (a = 6.196Å). Above the buffer layer we grow a periodic stack of Pb1 − xSnxSe/EuSe multiquantum wells. We study 5 samples listed in Table 1. Two samples N1 and N2 are dedicated to neutron reflectivity measurements. we focus on samples S1 and N1 in the manuscript.
X-ray diffraction (XRD) carried out at T = 10K, allow us to extract the strain state of S1 at temperatures of interest. In Fig. 1(c), a reciprocal space map (RSM) taken along (444) at T = 10K shows the Bragg peaks of the substrate, the (Pb,Eu)Se buffer layer and the Pb1 − xSnxSe well for sample S1. Periodic peaks from the superlattice structure are also resolved indicating a highly coherent heterostructure. The patterns allow us to extract the superlattice period for each sample. This is consistently checked with transmission electron microscopy (TEM) measurements and X-ray reflectometry (XRR) (see supplementary section 1) to extract the well and EuSe thickness separately. The structural properties of the samples are shown in Table 1. In Fig. 1(d), an RSM along (513) allows us to compute the in-plane and out-of-plane lattice constant and extract information about the strain. The superlattice is under tensile strain \(({ϵ}_{\left|\right|}\approx +0.4\%)\) in the in-plane \(\)direction and compressive strain in the [111] direction.
Transmission electron microscopy images confirm the observations of XRD. Figure 1(e) shows a well-aligned stack of Pb1 − xSnxSe and EuSe probed using TEM in sample S1. The inset of this image displays an elemental mapping of the Eu atom using energy dispersive X-ray spectroscopy that demonstrates the periodic repetition of the EuSe layers. A zoom-in figure (Fig. 1(f)) shows an interface between the well and the barrier demonstrating a short-range roughness of about a monolayer. We highlight that samples with thicker EuSe layers (> 3nm) yielded rougher interfaces, therefore we restrict this analysis to thinner layers. Additional XRD, XRR and TEM measurements are included in supplementary section 1. The large gap of EuSe (> 1eV) prohibits interactions between neighboring quantum wells, so the heterostructure can be safely considered as a multi-quantum well (see supplementary section 2).32
Table 1
Sample list. The Sn concentration is determined using energy dispersive spectroscopy performed during the TEM measurements and X-ray diffraction on controlled bulk samples. The well and barrier thickness are extracted from a period determined by X-ray diffraction of the superlattice and cross-section TEM measurements. The uncertainty accounts for the interface roughness. X-ray diffraction patterns for all samples are shown in Supplementary section 1.
Sample | Sn concentration x | Pb1 − xSnxSe Well thickness(nm) | EuSe Barrier thickness(nm) | No. of periods |
S1 | 0.28 by EDX | 22(± 1) | 1.41 ± 0.35 (4 ± 1ML) | 10 |
S2 | 0.34 by XRD | 25(± 2) | 2.4 | 5 |
S3 | 0.14 by XRD | 32 | 6 | 5 |
N1 | 0.10 by XRD | 48 | 11 | 5 |
N2 | 0.22 by XRD | 35 | 4 ± 2, varies by layer | 5 |
Magnetooptical Measurements
We employ magneto-optical infrared spectroscopy to probe the energy gap of the TIS in proximity to EuSe. Experiments are performed in applied magnetic fields up to 17.5T using transmission and reflection measurements (Fig. 2(a)). The transmission spectra revealed a very weak signal between 25 and 72 meV corresponding to the reststrahlen band of the BaF2 substrate. Therefore, we employed the reflection geometry to study the infrared signal from the heterostructure in this spectral range. In the presence of a magnetic field, infrared excites transitions between various Landau levels (LLs) as illustrated in Fig. 2(b). The minima in the normalized transmission spectra T(B)/T(B = 0) allow us to extract the energy of these transitions.
The experimental spectra obtained using transmission measurements are shown in Fig. 2(c) for S1. Owing to the high mobility of the TIS in Pb1 − xSnxSe, LL transitions that shift to higher energy as the field increases can be observed at fields as low as 3T. Some are marked with blue and gray dots to highlight the field dependence. Color coding will become clear later.
To obtain information about the transitions occurring in the reststrahlen of the substrate, we carried out reflectivity measurements in the far-infrared. The relative reflectance is shown in Fig. 2(d). Below 75meV, reflectance is strong, and a prominent minimum marked with a black square is observed. Other field-independent modulations of the reflectivity are ignored, since they cannot be related to LLs and are of no interest for our analysis.
We identify all transitions observed as minima in Fig. 2(c,d) by points presented in Fig. 3(a). Those transitions are shown in the empirical LL graphs plotted in Fig. 3(b). By modelling these LL transitions using a massive Dirac Hamiltonian30 we can determine their origin. The transitions marked in blue in Fig. 2(c) and Fig. 3(a) are due to interband transitions between the conduction and valence LLs of the TIS. The intraband band transition between the 0th and the 1st LL is shown by the black square. Those marked in gray are associated with interband transitions from the LLs of trivial quantum well subbands. We make this attribution following analysis using the semi-empirical model discussed next.
Figure 3(a) shows a curve fit carried out using the following relation which describes interband magnetooptical transitions for massive Dirac fermions given dipole selection rules 4041 (see supplementary section 2):
$${ϵ}_{n}^{E,i}-{ϵ}_{n\pm 1}^{H,i}=\sqrt{{{\Delta }}_{\text{i}}^{2}+2e{v}^{2}{\hslash }^{2}nB}+\sqrt{{{\Delta }}_{\text{i}}^{2}+2e{v}^{2}{\hslash }^{2}(n\pm 1)B}$$
Here \({{\Delta }}_{i}\) the band edge position of each subband, i is the subband index (i = 1 for the TIS), E/H denote the conduction and valence subbands respectively, \(v\) is the band velocity taken to be the same for all subbands, \(n\) is the Landau index, and B is the magnetic field. \({ϵ}_{n}^{E,i}\) thus denotes the energy of the nth LL of the ith electron subband. A k.p formalism using the envelope function scheme allows us to derive the band dispersion to show that it indeed satisfies this quasi-ideal massive Dirac model (see supplementary material).30,42 The solid blue lines in Fig. 3(a) represent transitions obtained from Eq. (1) for the TIS and the gray lines represent those from trivial QW subbands. Their extrapolation to B = 0 yields the gap between each pair of conduction and valence subbands with the same index i (i.e.\({2{\Delta }}_{i}\) the gap between Ei and Hi). \({2{\Delta }}_{1}\) is ETIS. It is zero for a thick quantum well and if time-reversal symmetry is preserved. The corresponding LLs for all subbands up to i = 3 are shown in Fig. 3(b) with the transitions shown as arrows.
The cyclotron resonance (CR) corresponds to the intraband transition between the valence n = 0 and n = 1 levels of the TIS, whose energy is given by:
$${ϵ}_{CR} =\sqrt{{{\Delta }}_{1}^{2}+2e{v}^{2}{\hslash }^{2}nB}-{{\Delta }}_{1}$$
This transition is shown as the black line in Fig. 3(a) and the black arrow in Fig. 3(b). The fit to the interband transitions and the CR in Fig. 3(a) yield the following gaps: the TIS gap \({E}_{TIS}=2{{\Delta }}_{1}=14\pm 6meV\), the gap between E2 and H2 \(2{{\Delta }}_{2}=68\pm 2meV\), and between E3 and H3 \(2{{\Delta }}_{2}=204\pm 2meV.\) The Fermi energy can also be determined from to be 40meV from the field at which the 3–4 transition disappears as the n=3 LL crosses the Fermi level (Fig. 3(a,b)). This experimental measurement allows us to reconstruct the band dispersion (in the QW plane) shown in Fig. 3(c). Despite the presence of the EuSe layer, the TIS gap cannot exceed 20meV within one standard deviation. The fit also yields the velocity \(v=4.3\pm 0.05\times {10}^{5}m/s\). Since we observe several transitions, our uncertainty on \(v\) and \({{\Delta }}_{i}\) is very small. Data taken on sample S2 is presented in the supplement and yields similar but less precise results as its mobility was lower.
The experimental results for both samples S1 and S2 are shown in Table 2. The extracted energy gaps are compared to a theoretical calculation carried out to determine the impact of quantum confinment and strain on the trivial and topological states (supplementary sections 2 and 3). It utilizes the band alignment shown in Fig. 3(c), and it is further discussed in the supplement. The resulting gaps are also shown in Table 2. The TIS gap from the experiment is only slightly larger than the theoretical value calculated without including magnetic exchange. The calculated value after the inclusion of the impact of strain is within experimental error. Strain reduces the topological bulk gap of the Pb0.72Sn0.28Se, enhancing the hybridization between top and bottom TISs.46,47,48 For this reason, it is found to enhance ETIS.
Table 2
Experimental and theoretical energy gaps for S1 and S2. The calculation utilizes an envelope function scheme implemented using the band alignment shown in the inset of Fig. 3(c) and discussed in the supplementary section 2. Magnetic exchange is neglected in these calculations. The measurements for S2 are shown in supplementary section 4.
Sample ID | ETIS = \({2{\Delta }}_{1}\)(meV) | E2-H2 gap\({2{\Delta }}_{2}\) | E3-H3 gap\({2{\Delta }}_{3}\) |
S1 – Exp. | 14 ± 6 | 136(± 4) meV | 206 meV |
S1 – Cal. | 7 | 129 meV | 204 meV |
S1 – Cal. with strain | 10 | 122 meV | 198 meV |
S2 – Exp. | 10 ± 10meV | N/A | N/A |
S2 – Calc. | 2meV | 141meV | 201meV |
Temperature dependence.
Additional temperature dependent magnetooptical spectroscopy measurements are carried out to further corroborate the origin of the energy gap. Spectra taken at various temperatures for B = 13T are shown in Fig. 4(a). The transition involving the N = 1H and N = 2E transition of the interface states is seen to vary in energy versus temperature, starting at 64K. At 1.6, 4.2 and 6K, the magnetooptical spectra are nearly-identical. The energy dependence at high temperatures is analyzed by fitting to the model discussed above, to extract ETIS. The fan-charts shown in Fig. 4(b-e) demonstrate the changing energy gap, extracted by extrapolating the Landau levels to zero field using the massive Dirac model. Both ETIS and \({2{\Delta }}_{2}\) are extract and plotted in Fig. 4(e). ETIS is constant at the lowest temperatures 1.6K to 6K, where we expect the magnetic ordering transition of EuSe to take place. It increases between 64K and 120K. This increase can be fully understood as the result of the increasing penetration depth of surface states as a function of temperature,30caused by the decreasing bulk energy gap of Pb1 − xSnxSe.35,43 When this penetration depth increases, the hybridization between the top and bottom surfaces is enhanced and causes an enhanced gapping of their Dirac spectrum. These measurements suggest that the magnetic gap induced by proximity with EuSe is much smaller than the confinement gap at low temperatures.
Magnetic measurements.
We have lastly performed SQUID magnetization measurements to identify the magnetic phases of our EuSe barriers. Figure 5(a) shows the magnetic moment versus temperature with B = 100G applied parallel to the sample surface. The curve consists of two main contributions: the paramagnetic signal (PM) of the Pb1 − xEuxSe buffer layer, which increases as the temperature decreases following the Curie law and the antiferromagnetic signal from the EuSe barriers, which yields a peak at the Néel temperature. All curves shown in Fig. 5(a) exhibit such a peak, indicating AFM order from the EuSe barriers. The extracted Néel temperature TN is plotted in Fig. 5(b). It increases with Sn concentration. This is likely due to the strain caused by a change in the lattice parameter of the Pb1 − xSnxSe layer with increasing x.44 The field-dependent magnetization was also measured and shown in Fig. 5(c) for the three samples considered here at 2K. Based on the estimated thickness we have calculated the average magnetic moment \({\overline{m}}_{Eu}\) per Eu2+ in Fig. 5(c) after the correction for diamagnetism of the BaF2 substrate and the PM of the buffer layer (see supplementary section 5). The fine structure of the magnetization versus magnetic field – reflecting transitions from an AFM ground state to an FiM phase and finally to a saturated state – is only visible for samples with a thick EuSe layer (\(\ge 4nm\)). But, the magnetic moment saturates to a value close to the expected saturation from Eu2+ ions, 7\({\mu }_{B}/E{u}^{2+}.\) Thus, for magnetic fields at which optical measurements are carried out, the moment of the EuSe layers can be considered saturated, and the extrapolation of the LLs to B = 0 should yield ETIS in the presence of a ferromagnet in proximity.
To probe the depth profile of the magnetism at buried interfaces, we carried out PNR measurements on two samples (N1 and N2) at Oak Ridge National Lab. Data for N1 is shown in Fig. 5 and N2 is discussed in supplementary section 6. The PNR measurements yield the spin resolved reflectivities R+ and R− as a function of the wavevector transfer Q at a fixed magnetic field (Fig. 5(d)). Superscripts plus (or minus) denote measurements with neutrons with spin parallel (or antiparallel) to the direction of the applied magnetic field. The depth profiles of the nuclear and magnetic scattering length densities (NSLD and MSLD) correspond to the depth profile of the chemical and in-plane magnetization vector distributions, respectively. Both are extracted using the fit shown in Fig. 5(d) and plotted in Fig. 5(e). Figure 5(f) shows the spin asymmetry ratio \(SA = ({R}_{+} - {R}_{-})/({R}_{+} + {R}_{-})\)obtained from the experimental and fitted curves at ± 4.85T. The SA signal evidences the presence of a depth dependent magnetic moment. The simulated depth profile in Fig. 5(e) and the SA in Fig. 5(f) confirm that we successfully obtain the intended periodic repetition of magnetic layers in proximity to every Pb1 − xSnxSe layer. The peak MSLD signal is nearly constant in each of the 7 layers and converts to a value close to 5.5\({\mu }_{B}/E{u}^{2+}\) slightly lower than the value recovered from magnetometry. Despite the interface being atomically sharp at the nm scale, some roughness can be resolved in TEM measurements at the \(\mu m\) scale. This can cause a slight reduction of the magnetic moment probed by PNR.