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Tunable Dirac interface states in topological superlattices

Relativistic Dirac fermions are ubiquitous in condensed matter physics. Their mass is proportional to the material energy gap and the ability to control and tune the mass has become an essential tool to engineer quantum phenomena that mimic high energy particles and provide novel device functionalities. In topological insulator thin films, new states of matter can be generated by hybridizing the massless Dirac states that occur at material surfaces. In this work, we experimentally and theoretically introduce a platform where this hybridization can be continuously tuned: the Pb1-xSnxSe topological superlattice. In this system, topological Dirac states occur at the interfaces between a topological crystalline insulator Pb1-xSnxSe and a trivial insulator, realized in the form of topological quantum wells (TQW) epitaxially stacked on top of each other. Using magnetooptical transmission spectroscopy on high quality MBE grown Pb1-xSnxSe superlattices, we show that the penetration depth of the TQW interface states and therefore their Dirac mass is continuously tunable with temperature. This presents a new pathway to engineer the Dirac mass of topological systems and paves the way towards the realization of emergent quantum states of matter using Pb1-xSnxSe topological superlattices.


Introduction
Engineered quantum effects in heterostructures [1] of condensed-matter are behind several revolutionizing ideas and advances in fundamental and applied physics. Examples are the quantum spin Hall [2] and fractal quantum Hall effects [3] in semiconductor and graphene heterostructures, as well as band engineered quantum cascade lasers [4] and entangled photon sources. [5] With the emergence of topological phases of matter and the discovery of topological insulators, [6] [7] a new horizon has been opened to develop heterostructures combining materials of alternating band topology. In topological insulators, Dirac cones emerge at the surfaces of an insulator hosting a bulk band inversion. In a superlattice of alternating topological and trivial materials, these states appear at the topological material's interfaces, and their mutual coupling and hybridization can lead to novel quantum phases such as Weyl fermions [8] and Su-Schriefer-Heeger (SSH) excitations. [9] A natural question is whether and how one can engineer the hybridization between the topological interface states and tune the emergent nontrivial topological phases using external knobs.
A highly attractive material system to achieve this purpose are the topological crystalline insulators (TCI) such as Pb1-xSnxSe and Pb1-xSnxTe. [10] - [16] Topological phase transitions can be induced in TCIs by tailoring the crystal structure and breaking crystalline symmetries. [11] [14] [17] - [20] Pb1-xSnxSe and Pb1-xSnxTe host four Dirac cones per surface Brillouin zone ( Fig. 1(a)). [21] [22] [23] For example, their characteristics -the effective electron mass and energy gap -are highly sensitive to strain [24] [25] and lattice distortions [19] and can be controlled by electric [18] and magnetic fields. [26] This has led to novel theoretical proposals such as the electrical-field-tuned quantum spin Hall state, [18] [27] and the helical interfacial flat band state that appears upon straining those materials. [25] The latter is particularly interesting as it is predicted to occur in heterostructures of TCIs separated by normal insulating barriers, [25] and yields an emergent superconducting phase with an enhanced critical temperature. [28] [29] Heterostructures of TCIs are therefore an exciting platform to engineer tunable topological phases.
Up to now, however, most studies on TCIs dealt with bulk single crystals and thick epilayers, [13] [22] [30] that required surface sensitive and local probes to thoroughly characterize the topological states under ultra-high vacuum conditions far from practical device applications. In particular, these techniques are not sensitive to states occurring at buried interfaces in superlattices. This fact impeded the prospects of realizing novel states of matter in these systems.
In this work, we unravel the topological interface states (TIS) formed in epitaxial TCI superlattices consisting of multiple Pb1-xSnxSe topological quantum wells (TQW) separated by normal insulator (NI) Pb1-yEuySe barriers ( Fig. 1(b,c)). Using magneto-infrared spectroscopy to probe the Landau level structure of the quantum confined states formed at the buried interfaces, we reveal the behavior of the TIS as a function of temperature. Detailed theoretical analysis of the experimental data reveals that the penetration depth, the hybridization strength and the Dirac mass of the TIS can be effectively tuned not only by varying the quantum well thickness but also by continuously changing temperature ( Fig. 1(d)). Our results thus establishes Pb1-xSnxSe topological quantum wells (TQWs) as a viable platform that hosts tunable topological Dirac states at buried interfaces. Our findings are of key importance for the realization of new emergent states of matter in TI and TCI heterostructures. [25] [9] FIG 1. (a) Identical Dirac cones on the (111)-surfaces of bulk Pb0.75Sn0.25Se. (b) Topological quantum-well Pb0.75Sn0.25Se/Pb1-yEuySe superlattices of alternating TCI/NI layers studied in this work. The band alignment and band profiles are shown in (c). L6 ± denote the conduction and valence band extrema at the L-points in the rocksalt Brillouin zone of Pb0.75Sn0.25Se and Pb1-yEuySe. εA (<0 at 4.2K) is the L6 ± energy separation in the band-inverted Pb0.75Sn0.25Se quantum wells (A) and εB (>0) that of the normal insulator Pb1-yEuySe barriers (B). V is the conduction band offset and d the well thickness. (d) Sketch of the evolution of the wave function probability density and Dirac cones of the topological interface state as a function of temperature across the topological phase transition T*, illustrating the tunability of the penetration depth and hybridization gap of the TIS.

Experimental method
Magneto-optical spectroscopy experiments [31] [32] are performed on coherent TCI superlattices of Pb0.75Sn0.25Se/Pb1-yEuySe grown by molecular beam epitaxy on (111) BaF2 substrates (see Appendix A). The Sn content x = 0.25 of the TQW was chosen to be above the TCI phase transition that occurs at x = 0.16 at T=4.2K. [32] Infrared spectroscopy is performed at different temperatures and magnetic fields on two TQW samples. The first -TQW 36 -has a quantum and barrier thickness of 36nm and 26 periods. The second -TQW 24 -has 24nm quantum wells, 120nm barriers and 16 periods.
Spectra were recorded in an optical cryostat equipped with a superconducting magnet providing magnetic fields up to 15T and temperatures down to 4.2K. The probe is coupled to a FTIR spectrometer that operates in the far-IR and mid-IR to cover a range extending from roughly 1meV to 1eV. The signal was either recovered at the exit of the probe and measured using an external HgCdTe detector cooled to 77K or detected inside the probe using a composite Si bolometer mounted below the sample and cooled to 4.2K. The external detection setup was used to perform temperature dependent measurements between 4.2K and 200K. The detection cutoff in this case is close to 80meV. All the measurements are performed in the Faraday geometry with B || [111] direction, the growth axis. It is emphasized that due to the large penetration depth of the infrared photons, the entire heterostructure is probed.

A. Magnetooptical specotroscopy of topological interface states
Figure 2(a) shows relative transmission spectra acquired at different magnetic fields for sample TQW-36 at 4.2K. The spectra exhibit pronounced minima due to interband Landau level (LL) transitions that are observed at magnetic fields down to 2T. Their strength significantly gains in amplitude and strongly blueshifts with the applied magnetic field. It is therefore obvious that they originate from Landau quantization. The large number of transitions evidences the high mobility and low carrier density of the TQW structures resulting from the pseudomorphic heteroepitaxial growth and effective control of the carrier concentration, [32] with a Fermi energy less than 40meV above the mid-gap. Corresponding spectroscopic data for TQW-24 are shown in the supplemental material. [33] From the magneto-optical spectra, we construct fan charts of the Landau level transitions in Fig. 2(b) to extract the band structure parameters. As a first step, we identify the transitions of the TIS (E1 and H1) marked by the red arrows in Fig. 2(a) and represented in (b) by red points. A second set of weaker transitions also appears (black arrows in Fig. 2(a)) and is assigned to transitions between the second subbands E2-H2 and hybrid ones between TIS and E3 (see Fig. 2(c)). These are represented by the open black and purple circles in the fan chart. Using the k·p envelope function approach developed in Refs.
[1], [34], [35] and detailed in appendix B, we compute the quantum confined states and magneto-optical transitions, taking into account the opposite topological character of the TQWs and NI barriers ( Fig. 1(c)). The fits represented by the solid lines in Fig. 2(b) evidence an excellent agreement with the experimental data. This yields the material band structure parameters listed in Table I. Table I. Structure and band parameters of the TCI/NI TQW-36 and TQW-24 investigated in this work. d and dbarrier are the Pb1-xSnxSe and Pb1-yEuySe thicknesses and x and y their composition as obtained by x-ray diffraction (see appendix 1). The bulk band gaps εA and εB of the TQW and barriers as well as the Dirac velocity vc derived from Landau level spectroscopy and k·p analysis are also listed (see supplement S2 and S3). [ The E(k) dispersion of the TQW states and their Landau levels versus magnetic field derived from the k·p fits are presented in Fig   From the measured energy gap Δ and Dirac velocity , [32] , [42] we can compute the Dirac mass of the TIS: Indeed the mass acquisition due to hybridization is extremely small since the QW is relatively large (36nm), evidencing nearly-massless Dirac states at the TCI/NI interface. It is important to note at this point, that while for SnTe and Pb1-xSnxTe (111) two types of Dirac cones have been observed at the surface, [43] in the case of Pb1-xSnxSe, the bulk Fermi surface is nearly-isotropic for x≈0.25 as reported previously. [32] [44] Consequently, although Dirac cones occur at two different crystalline symmetric points of the Brillouin zone (Γ ̅ and M ̅ ) as shown in Fig. 1(a), their velocity and their shape are identical. The measured Dirac mass thus corresponds to all 4 Dirac valleys of Pb1-xSnxSe.
A remarkable feature of our result is that the amplitude of the magneto-optical transitions from the Dirac interface states is larger than those stemming from the trivial QW subbands. This means that in heterostructures the TIS transitions dominate the signal, contrary to the case of bulk material. We will next reveal a peculiarity of this TIS of Pb1-xSnxSe -its temperature tunability.
B. Tuning the topological interface states with temperature For topological Pb1-xSnxSe one can tune the magnitude of the bulk energy gap εA of the system by changing the temperature. [45] [46] Hence, one can vary the penetration depth λ of the TIS into the Pb1-xSnxSe well according to (see appendix 2): The experimental results shown in Fig. 3 for TQW-36 demonstrate how we achieve a tuning of and a controlled suppression of TIS via temperature by tailoring λ for a fixed QW thickness. Remarkably, the sample mobility is sufficiently high to resolve the LLs at temperatures as high as 200K as evidenced in Fig.  3(a). Figure 3(b-e) show the resulting Landau level fan charts at different temperatures T = 60K, 80K, 120K and 200K; the results for TQW-24 are shown in the supplement. [33] The whole experimental data set is fit by the k·p model that takes into account the variation of εA versus T obtained from a bulk-like reference sample. [33] The model parameters are shown in table II. The excellent fit to the data is shown by the solid lines in Fig. 3(b-e).
From the theoretical analysis of the experiments, we extract the temperature dependent Dirac gap Δ(T) (arrows in Fig. 3(b-e)) and the corresponding Dirac mass mD(T). They are plotted in Fig. 3(f) for both TQW samples. A monotonic increase of Δ and mD is seen with increasing temperature, whereas the bulk gap εA changes from negative to positive at 110 K. Evidently, for the thinner TQW-24, Δ is systematically larger, which unambiguously confirms the quantum confinement and hybridization related nature of the gap, and the topological character of the TQW states.

C. Discussion
Two regimes (i) and (ii) emerge from Fig. 3(f) when the Dirac gap Δ is compared to the bulk gap εA. Regime (i) occurs when Δ<|εA| and εA < 0, i.e., when the Dirac gap of the TIS is smaller than the bulk gap that is also inverted. According to Fig. 3(f) this applies when T is below the critical temperature T*=70 K for TQW-36, and for T<T*=50K for TQW-24. In the second regime (ii), Δ>|εA |. To shed light on the nature of the Dirac states in these regimes, we evaluate the probability density obtained by the k·p calculations based on the fit to the data. Figure 4(a) and 4(b) display the results for TQW-36 at T=4.2K and 60K (<T*) for regime (i). The band alignment diagram and E(k) dispersions reveal that the E1 and H1 TQW states (red lines) lie within the Pb1-xSnxSe bulk gap (Δ<|εA|). As a result, the probability density χ of the Dirac states is peaked at the interfaces (red solid lines in Fig. 4(a,b)). Its decays towards the center of the quantum well is governed by λ given by Eq. (1). Approaching the critical temperature T* where Δ→|εA|, the TIS penetrates deeper into the QW (see Fig. 4(b) at 60K). As a result, the coupling strength and hybridization gap Δ of the TIS increases (see Fig. 3(f)), even though Pb1-xSnxSe is still in the non-trivial TCI state with εA<0. Therefore, the observed increase of Δ is accompanied by an increasing penetration λ of the TIS as shown in Fig. 4(e). Through this relation, a tuning of both λ and mD is achieved for the TIS.
A key feature of this regime is that as long as Δ <|εA|, the ground state remains peaked at the interface and is of topological origin. This suggests that topological characteristics such as spin-momentum locking are still partially preserved even when Δ≠0. Previous ARPES measurements on Bi2Se3 films in fact observed a progressive suppression of the spin-polarization of the surface states upon gapping, [40] suggesting that upon increasing top-bottom surface hybridization the loss of spin-momentum locking is only partial as long as the states remain pinned to the surface or interface.
Further increasing the temperature leads to a transition into regime (ii) where Δ >|εA| (with either εA>0 or εA<0). As shown by Fig. 4(c,d) for TQW-36 at T=80K and T=200K (>T*), the E1 and H1 energy levels move, respectively, above and below the bulk TCI band edges. As revealed by the computed probability density χ displayed in Fig. 4(c,d), the wave function of E1/H1 is no longer peaked at the interfaces. Here, Δ is simply the confinement induced gap between the first E1/H1 QW subbands. Note that the interfacial nature of E1 and H1 remains unchanged when εA changes sign (at 110K), as it is entirely determined by the value of Δ relative to |εA|. Lastly, we compute the continuous change of λ versus T, from Eq. (1) and the variation of |εA| and Δ with temperature in regime (i). The results are displayed in Fig. 4(e), demonstrating the tunability of the phenomenon. Since the unique characteristics of the TIS such as their spin-momentum locking and scattering properties are linked to λ, Fig. 4(e) reveals how these can be continuously tuned in TCI TQWs. It is important to highlight here that the bulk Dirac cone of Hg1-xCdxTe has been also shown to be tunable with temperature. [49] However, in the case of HgTe, the 2D topological states and the heavy-hole band are known to hybridize. [50] [51] [52] Therefore, the main strength of Pb1-xSnxSe, is that it hosts nearly-ideal Dirac cones with easily tunable characteristics.

IV. Conclusion
Overall, we have realized a TQW heterostructures based on heteroepitaxial Pb1-xSnxSe/Pb1-xEuxSe superlattices and revealed the topological nature of the buried TIS using magnetooptical Landau level spectroscopy. While previous works on topological thin films relied on surface sensitive probes that could not directly access the changing hybridization strength at subsurface topological interfaces, our optical technique is able to directly obtain this information via the changing energy gap of the interfacial Dirac states. It is thus important to recognize the importance of using such a technique to probe buried TIS in topological heterostructures. To this end, our results provide detailed phase diagrams (Figs. 3(f) and 4(f)) that demonstrate how the Dirac mass can be tuned by changing TQW thickness and more interestingly by varying temperature. While this is a remarkable feature of the IV-VI TCI QW systems our conclusions are of general interest to other TI systems. Keeping in mind the recently proposed analogy between the TQW and the 1D SSH model, [9] in that the hybridization of TIS in TQW can be related to hoping amplitude between lattice sites, the dynamics of the SSH model are temperature tunable in such a system.
Lastly, we highlight that our realization of high-quality superlattices and our successful observation of the TIS at TQW interfaces is a major and essential step towards engineering interfacial quantum phenomena predicted in TCIs. In fact, heterostructures of IV-VI materials with engineered lattice-mismatch, [53] [24] are predicted to host pseudo-Landau levels, helical flat-band states and an interfacial superconductor [25] [29] phase with an enhanced critical temperature. [28] Its possible coexistence with the TIS observed here in band gap engineered Pb1-xSnxSe or Pb1-xSnxTe TQW structures will provide a new platform to realize an intrinsic topological superconductor [54] for Majorana-based systems. We therefore motivate further work on strain and quantum engineering of TQW as a platform for emergent many-body topological states.

Appendix A: Sample design and growth.
Epitaxial growth of topological Pb1−xSnxSe/Pb1-yEuySe superlattices was performed using molecular beam epitaxy (MBE) in ultra-high vacuum of 5×10 −10 mbar at a substrate temperature of 380°C on BaF2 (111) using PbSe, SnSe, Eu, Te and Bi2Se3 effusion sources. The composition of the TQW Pb1−xSnxSe was set to x=25% to achieve band inversion, with a corresponding negative band gap εA = -40 meV at 4.2K as checked on bulk-like reference layers. [33] Pb1-yEuySe with x = 0.04 -0.055 was chosen as barrier material. [36] Incorporation of Eu increases the band gap to a value of 270 -314 meV (at 4.2K), as shown in the supplement. [33] This provides a stronger confinement of the TQW levels and pushes the absorption edge of the barriers well above the TQW LL transitions, while retaining a good lattice matching (0.6%). The thermal expansion coefficient of the well and the barrier are similar, [55] so the mismatch can be assumed constant with temperature.
To obtain a low carrier concentration of the TQWs, n-type doping using Bi was employed to compensate the native p-type character of Pb1−xSnxSe (see Ref. [32]) due to doubly charged cation vacancies. Superlattices were grown with the number of periods N > 16 to obtain a sufficiently high absorption of the topological interface states in the infrared. Pseudomorphic 2D growth is achieved as checked by in-situ reflection high-energy electron diffraction. This yields a very high structural quality of the superlattices and interfaces as demonstrated by the x-ray diffraction spectra of Fig. 5(a,d), exhibiting a large number > 10 of sharp superlattice satellite peaks. The x-ray reciprocal space maps recorded around the symmetric (222) (Fig. 5(b,e)) and asymmetric (153) (Fig. 5(c,f)) Bragg reflection show that all satellite peaks are perfectly aligned along the [111] the growth direction, evidencing that the whole stack is fully pseudomorphic with coherent interfaces. Additionally, the observation of high order satellite peaks rules out any significant intermixing at the Pb1−xSnxSe/ Pb1−yEuySe interfaces. This is further confirmed by previous secondary ion mass spectroscopy measurements on similar superlattices shown in ref. [56]. More evidence of the excellent quality of the samples is provided in the supplementary material (S4) [33] where we estimate the sample mobility by analyzing the linewidth [57] of magnetooptical transitions. Mobilities of 15000cm 2 /Vs and 24000cm 2 /Vs are respectively found for TQW-24 and TQW-36. the Kane k·p matrix element. The band anisotropy is neglected as it is known to be very small (K≈1) in Pb0.75Sn0.25Se, and the four L-valleys are assumed equivalent. [32] In the Dirac formalism, we introduce = 0 , the Dirac velocity.
V is assumed to be the same for the conduction and valence band, [36] yielding a symmetric confinement potential. This assumption holds for the low-energy subbands of interest to this work (H1, E1, H2, E2,...). Crystalline inversion symmetry is assumed to be preserved. In zone A of Fig. 1(c), the zero energy is taken at the L6 + band edge, so that the L6band edge is at −| | at 4.2K. In zone B of Fig. 1(c), the L6band edge ends up being at − | A| and the L6 + band edge at -. At = = 0, the problem is reduced to a two eigenvalues system: In this case, the change of topology at the interfaces causes the wave vector to be imaginary = , therefore yielding 1 (1) = ℎ( ) for -|εA|<E<0. The probability density of the first subband ( ) = | 1 (1) | 2 is therefore peaked at the interface. It is the topological interface state (TIS) of the QW. Its penetration depth into the well is therefore given by:

= 1
The TIS energy eigenvalues can be expressed as The expression for the penetration depth is therefore reduced to: = 2ℏ √ 2 − ∆ 2 ( ) is shown in the inset of Fig. 2(c) for TQW-36. In general, the method used here is equivalent to a Ben-Daniel Duke model where the non-parabolicity ('Dirac-ness') of the energy bands is accounted for. This method is discussed thoroughly in Ref.
[1], [34] and [35]. It is similar to the methods used in previous works to compute the effect of confinement of the topological surface states in TIs and TCIs. [27] , [47] , [48] Appendix C: Derivation of the k-dispersion and Landau levels.
We derive an effective Hamiltonian using the , =0 solutions found by solving the transcendental equations. The effective Hamiltonian is expressed in the orthonormal (Ψn ⃗⃗⃗⃗ , φn ⃗⃗⃗⃗ ) 0< < basis. The , -dependent part of the 4x4 . Hamiltonian (Eq. 2) is regarded as a perturbation. We can write it as: The eigenvalue problem corresponding to the effective B-dependent Hamiltonian is numerically solved for 0≤B≤15T, and -1≤N≤10.
Appendix D: Anti-crossing of Landau levels. An anti-crossing between , and ±1, ±1 Landau levels (i: band index, N: LL index) is computed using the conventional approach of solving the eigenvalue problem corresponding to the interaction of the two levels according to: Here is magnitude of the avoided crossing. It is taken to be equal to 5meV for the -1 E1 0 E2 anticrossing shown in Fig. 2(b,d). decreases significantly with increasing N. This type of anticrossing is characteristic of the Landau levels of non-parabolic semiconductors with band minima at the L-points. [59]