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Dirac Landau Level Spectroscopy in Pb_{1−x}Sn_{x}Se and Pb_{1−x}Sn_{x}Te across the Topological Phase Transition: A Review

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## Abstract

**:**

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te. More recent works also took interest in studying the topological phase transition from trivial to non-trivial topology that occurs in such materials as a function of increasing Sn content. A peculiar property of these materials is the fact that their bulk bands disperse following a massive Dirac dispersion that is linear at low energies above the energy gap. This makes Pb

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te ideal platforms to simultaneously study 3D and 2D Dirac physics. In this review, we will go over infrared magneto-optical studies of the Landau level dispersion of Pb

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te for both the bulk and surface bands and summarize work that has been done on this matter. We will review recent work on probing the topological phase transition in TCI. We will finally present our views on prospects and open questions that have yet to be addressed in magneto-optical spectroscopy studies on Pb

_{1-x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te.

## 1. Introduction

_{2}TIs [1]. Contrary to TIs protected by time-reversal symmetry (TRS) [2,3,4], a TCI material is protected by mirror symmetry or reflection symmetry present in the crystal [1,2,3,4,5]. Note that crystal symmetries can be broken by material surfaces. As a consequence, topological surface states of a TCI can exist only on some surfaces of the crystal, which is not the case in TI materials. The key role of mirror symmetry present in TCIs makes the study of different surface orientations very attractive.

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te (Figure 1a), a band inversion (Figure 1b) occurs as a function of increasing Sn content at four equivalent L-points in the Brillouin zone. In the trivial phase, ${L}_{6}^{-}$ is the lowest conduction band and ${L}_{6}^{+}$ is the highest valence band that form an energy gap E

_{g}of the system. The far-bands are distant and not shown in Figure 1b. As shown in Figure 1b, in the trivial phase, the energy gap (E

_{g}> 0) initially decreases with increasing Sn composition, then closes at x = x

_{c}, and finally re-opens as x > x

_{c}(E

_{g}< 0) in the topological non-trivial phase (inverted regime) [8,9,10]. The band inversion results in the emergence of the topological surface states (TSS) in the non-trivial regime. This leads to an even Z

_{2}topological invariant and implies a trivial character when TRS is considered. However, it was shown that as long as TRS is preserved, the crystalline mirror symmetry with respect to the {110} crystallographic planes of the rocksalt crystal leads to topologically protected band crossings at an even number (2 or 4) of points on the surface Brillouin zone [6]. These points correspond to where the bulk L-points project on each respective surface. Therefore, the (001) surface of Pb

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te ends up having four Dirac cones along the $\overline{\Gamma}$-$\overline{X}$ linecuts of the surface Brillouin zone (Figure 1c) [6]. The (111) surface has one Dirac cone at the $\overline{\Gamma}$-point and three at the $\overline{M}$-points (Figure 1d) [11,12]. This class of topological materials is referred to as topological crystalline insulators. It was first evidenced using angle-resolved photoemission spectroscopy (ARPES) in Pb

_{1−x}Sn

_{x}Se [13], then subsequently in Pb

_{1−x}Sn

_{x}Te [14] and SnTe [15].

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te can be controlled via the chemical composition, both systems offer an ideal platform to study the physics of the topological phase transition that leads to the TCI states. More specifically, the dynamics of the bulk band structure has generated great interest in being examined in light of the current understanding of 3D and 2D condensed matter physics and its subtle interplay with band topology. A powerful tool used to perform such band structure investigation is, and has always been, magneto-optical Landau level spectroscopy. It is a bulk sensitive tool that allows one to get accurate information about the bulk, without undermining the detection of topological surface state related features. This technique finally provides quantitative band structure information that can be useful to future studies on magnetotransport, quantum electronic devices, and infrared detectors.

**k.p**perturbation theory used to study the lead salt compounds is equivalent to the Bernevig–Hughes–Zhang model Hamiltonian for TIs. These two models will then be shown to be the 3D Dirac Hamiltonian that we can use to identify the topological nature of Dirac fermions in Pb

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te. We will also revisit the problem of Landau quantization of the bulk carriers in Pb

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te, highlighting the Dirac character of energy bands in the vicinity of the topological phase transition. We will present a detailed discussion on how a Dirac fermion spectrum is obtained, based on previous findings that use a six-band perturbative

**k.p**approach developed by Mitchell and Wallis in 1966 for lead salts [16]. In Section 3, we will summarize recent results on Landau level IR magneto-optical spectroscopy measurements of the bulk band structure of Pb

_{1−x}Sn

_{x}Se [17] and Pb

_{1−x}Sn

_{x}Te [18]. We will then discuss the recent observation of the ground state cyclotron resonance of topological surface states at high magnetic fields in Section 4 [18]. In Section 5, we will review recent progress on studies of the behavior of the bulk bands through the topological phase transition [17]. In Section 6, we will summarize future perspectives on bulk and surface Landau level studies in IV–VI materials and present challenges that need to be addressed in the future. Section 7 presents final conclusions.

## 2. Hamiltonian for IV–VI Semiconductors

#### 2.1. Fermi Surface Anisotropy in Pb_{1−x}Sn_{x}Se and Pb_{1−x}Sn_{x}Te

^{2},or, equivalently, the

**k.p**matrix element anisotropy factor K = (P

_{⏊}/P

_{ǁ})

^{2}[8], where P

_{⏊}and P

_{ǁ}are, respectively, transverse and longitudinal momentum matrix elements.

#### 2.2. Band Structure of the Longitudinal Valley in IV–VI Semiconductors

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te TCIs. In 2006, Bernevig, Hughes and Zhang (BHZ) proposed an explicit model Hamiltonian to describe the quantum spin Hall effect (QSHE) that was theoretically predicted to be realized in HgTe/CdTe quantum wells, known as the first 2D TI [20]. For 3D TIs in the Bi

_{2}Te

_{3}family with a single Dirac cone on the surface, such materials can be similarly described by the model Hamiltonian developed by Zhang et al. in 2009 [21,22]. Here, we show that the

**k.p**perturbation theory used to study TCI materials [8,23,24,25] is equivalent to the BHZ Hamiltonian for TIs. This model is a 3D Dirac Hamiltonian and reliably describes relativistic-like Dirac fermions in Pb

_{1-x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te.

**k.p**model including the lowest conduction and the highest valence levels (${L}_{6}^{\pm}$), for

**z**//[111], reads:

_{g}is the band gap, P

_{⏊}and P

_{ǁ}are, respectively, transverse and longitudinal momentum matrix elements, k

_{±}= k

_{x}± ik

_{y}, and m

_{0}is the electron rest mass.

_{g}/2, ${v}_{c}$ = P

_{⏊}/m

_{0}and ${v}_{c}^{\prime}$ = P

_{ǁ}/m

_{0}. Here, ${v}_{c}$ is the velocity perpendicular to the z-axis and ${v}_{c}^{\prime}$ is the velocity parallel to the z-axis. We thus get a massive Dirac Hamiltonian with uniaxial anisotropy along the z-direction written as:

_{⏊}

^{2}= k

_{x}

^{2}+ k

_{y}

^{2}and the ± signs refer, respectively, to the energy of the conduction ${E}^{c}(\overrightarrow{k})$ and valence ${E}^{v}(\overrightarrow{k})$ bands. In this two-band model, the Dirac transverse mass is given by m = Δ/v

_{c}

^{2}.

^{2}-approximation [8,16,23,24], the diagonal terms of Equation (2) are changed and replaced by:

_{1}= ћv

_{c}

^{’}, A

_{2}= ћv

_{c}, B

_{1}= $-\frac{{\u045b}^{2}}{2\tilde{m}}$ and B

_{2}= $-\frac{{\u045b}^{2}}{2\tilde{\mu}}$. This Hamiltonian is nothing but the 3D Dirac Hamiltonian with uniaxial anisotropy along the z-direction and $\overrightarrow{k}$-dependent mass terms.

^{4}terms, the dispersion relation of the conduction and valence bands is given by:

- If Δ > 0, the material is trivial;
- If Δ < 0, the material is topological.

#### 2.3. Landau Levels of the Longitudinal Valley

**k.p**approach, where the ${L}_{6}^{\pm}$ bands are exactly accounted for and the effect of four far-bands (two conduction and two valence bands) is included perturbatively in k

^{2}-approximation. At k

_{z}= 0, the Landau level energies of the conduction (c) and valence (v) bands are expressed as:

_{B}= eħ/2m

_{0}is the Bohr magneton, and B is the applied magnetic field. The cyclotron frequencies of the conduction (ω

_{c}) and valence (ω

_{v}) bands are defined, respectively, as ${\omega}_{c}=\omega +{\tilde{\omega}}_{c}$ and ${\omega}_{v}=-\omega +{\tilde{\omega}}_{v}$, where ω = eB/m is the cyclotron frequency in the two-band model ($\frac{1}{m}=\frac{{v}_{c}^{2}}{\Delta}$). ${\tilde{\omega}}_{c}$ and ${\tilde{\omega}}_{v}$ are the far-band contributions. Since the far-bands are nearly equally distant from the conduction and valence bands (${L}_{6}^{\pm}$), we assume that these two bands are symmetric. The far-band contributions are thus given by ${\tilde{\omega}}_{c}=\frac{eB}{\tilde{m}}=\tilde{\omega}$ and ${\tilde{\omega}}_{v}=-\frac{eB}{\tilde{m}}=-\tilde{\omega}$. Similarly, the g factors of the conduction (g

_{c}) and valence (g

_{v}) bands are given by ${g}_{c}=g+{\tilde{g}}_{c}$ and ${g}_{v}=-g+{\tilde{g}}_{v}$, where the tilted terms represent the far-band contributions. The far-band cyclotron energy contribution is assumed to be equal to the effective spin splitting. We thus get ${\tilde{g}}_{c}{\mu}_{B}B=-\u045b\tilde{\omega}$ and ${\tilde{g}}_{v}{\mu}_{B}B=\u045b\tilde{\omega}$.

^{−}) Landau level, which is non-degenerate in spin. Everything finally reduces to:

_{z}= 0 where the joint density of states is optimal, simply read:

_{z}= 0, similarly yield:

#### 2.4. Velocity as a Function of θ

## 3. Magneto-Optical Landau Level Spectroscopy of Bulk Massive Dirac Fermions in (111) Pb_{1−x}Sn_{x}Se and Pb_{1−x}Sn_{x}Te

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te have recently been studied in detail using infrared (IR) magneto-optical spectroscopy in the Faraday geometry [17,18]. Far- and mid-infrared (FIR and MIR) magneto-optical absorption measurements were performed. A 30–700 cm

^{−1}range is covered in FIR and a 700–7500 cm

^{−1}is covered in MIR. Large samples grown on cleaved (111) BaF

_{2}substrates by means of molecular beam epitaxy (MBE) were used. The applied magnetic field was oriented along the [111] direction (growth axis) and perpendicular to the sample surface. An Si composite bolometer cooled down to 4.5K was used to detect the transmitted signal. Transmission spectra were acquired and analyzed by a Bruker Fourier transform infrared (FTIR) spectrometer (Bruker, Germany). The relative transmission at a given magnetic field B is defined to be the normalization of the sample transmission T(B) by the zero-field transmission T(0).

_{1−x}Sn

_{x}Se. The bulk band parameters can be precisely extracted as long as the sample has a high enough mobility (>5000 cm

^{2}/Vs) and low carrier density (<5 × 10

^{18}cm

^{−3}). Results from our work on Pb

_{1−x}Sn

_{x}Se (x = 0.14) in Reference [17] are shown in Figure 4. Figure 4a,b, respectively, show the relative transmission amplitude from transitions observed between 11T and 15T in the MIR range and between 15 T and 17 T for the FIR range. We know a priori from transport measurements that this sample is slightly p-type having a Fermi level intersecting the bulk valence band. In Figure 4a, the transition occurring at the lowest energy corresponds to the first interband 1

^{v}− 0

^{c}transition (Figure 4c). The second interband transition at slightly higher energy is a 2

^{v}− 1

^{c}transition (Figure 4a,c). Notice that this transition seems split, as a result of the bulk valley degeneracy. In the FIR range (Figure 4b), the first interband transition is again reproduced. The ground state intraband cyclotron resonance (1

^{v}− 0

^{v}) can also be observed and is labeled CR. The energy of the transmission minima is extracted then plotted versus field in Figure 4d. Open red and full black circles denote, respectively, transitions attributed to the oblique and longitudinal bulk valleys.

_{1−x}Sn

_{x}Se. Here, L and O stand, respectively, for longitudinal and oblique valleys. The reststrahlen band of BaF

_{2}substrate is between 22 and 55 meV. In this region, no transition can be observed. The velocity of massive Dirac fermions can then be measured with a precision that is better than 2%, and the band gap can be determined, typically, with an uncertainty of 5 meV. Additionally, transitions pertaining to Landau levels from different valleys can be well differentiated.

_{1−x}Sn

_{x}Te, confirming the Dirac nature of the bulk bands. They are also reported in References [17,18]. Samples having x between 0 and 0.56 have been measured. The band inversion is seen to occur at about x = 0.4, in agreement with the experimental literature. The velocity is higher than that measured for Pb

_{1−x}Sn

_{x}Se for the longitudinal valley.

## 4. The Landau Levels of Massless Dirac Surface States in (111) Pb_{1−x}Sn_{x}Se and Pb_{1−x}Sn_{x}Te

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te are four-fold degenerate for both materials, with one massless Dirac cone occurring at the $\overline{\Gamma}$-point of the 2D Brillouin zone and three occurring at the $\overline{M}$-points. In both cases, the Landau levels are given by:

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te, has been observed recently.

_{1−x}Sn

_{x}Te having x = 0.46 [18] and x = 0.56 has been reported. The velocity obtained from the CR-TSS in that case was almost equal to the velocity of the bulk longitudinal valley. The transition was thus associated with the $\overline{\Gamma}$-point Dirac cone. The $\overline{M}$-point Dirac cone in (111) Pb

_{1−x}Sn

_{x}Te is expected to yield a similar transition with a lower velocity likely equal to that of the oblique valleys [11,12,32,33]. It has not been observed in (111) samples grown on BaF

_{2}, since it likely occurs in the reststrahlen band of the BaF

_{2}substrate.

_{1−x}Sn

_{x}Se (x = 0.19) that is n-type and has a Fermi level intersecting the bulk conduction band. They are shown in detail in Figure 5 as adapted from Reference [17]. Figure 5a,b, respectively, show the relative transmission amplitude from transitions observed between 11T and 15T in the MIR range and between 15T and 17T for the FIR range. In Figure 5a, interband transitions from bulk Landau levels are shown; the transition occurring at the lowest energy corresponds to the first interband 0

^{v}− 1

^{c}transition (see Figure 5c). The second interband transition at slightly higher energy is 1

^{v}− 2

^{c}(Figure 5a,c). In the FIR range (Figure 5b), the first interband transition is again reproduced. At low energies, a CR from a N = 0 − N = 1 transition of massless TSS is observed (see Figure 5c). The energy of the transmission minima is extracted and then plotted versus the field in Figure 5d. Open red and full black circles denote, respectively, transitions attributed to the oblique and longitudinal bulk valleys. Blue points refer to the CR-TSS. All bulk transitions agree with Equations (13) and (14). The CR-TSS agrees quite well with Equation (17) for ${v}_{D}$ equal to that of the bulk 4.7 × 10

^{5}m/s (solid blue line). This additional transition is also not seen in the trivial x = 0.14 sample, consolidating the fact that it is due to the presence of TSS. In Pb

_{1−x}Sn

_{x}Se, in contrast with Pb

_{1−x}Sn

_{x}Te, the CR-TSS is likely due to all four Dirac cones (at $\overline{\Gamma}$- and $\overline{M}$-points), since they are almost all identical and all have velocities that are almost equal to 4.7 × 10

^{5}m/s. This is mainly the reason why the CR-TSS in Pb

_{1-x}Sn

_{x}Se is significantly stronger than that in Pb

_{1−x}Sn

_{x}Te. This agrees with ARPES studies of Pb

_{1−x}Sn

_{x}Se [34].

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te, the additional transition associated with TSS occurs at energies where no transitions are expected from the bulk and lies above the ground state cyclotron resonance of the bulk bands. It additionally satisfies Equation (17), and is seen to disappear across the topological phase transition in magneto-optical spectroscopy. These three arguments are evidence that the additional transition that is observed is indeed the TSS ground cyclotron resonance.

## 5. IR Magneto-Optical Determination of the Topological Index

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te using magneto-optical absorption experiments. Looking back at the expression of the bulk Landau levels as a function of magnetic field, we have noticed that the velocity given by Equation (11) can be related to the relative sign of the energy gap, or equivalently to the topological index [35,36]. If the critical velocity ${v}_{c}$ is known, or can be estimated, one can extract the sign of the topological index by measuring:

## 6. Future Perspectives

#### 6.1. The Nature of the Topological Phase Transition in IV–VI Topological Materials

_{1−x}Sn

_{x}Se attempted to quantify the bulk band gap and have challenged this claim [37]. In magneto-optical Landau level spectroscopy, the determination of the band gap is precise to about 2–5 meV in samples having good enough transport parameters. Up until now, no composition of Pb

_{1−x}Sn

_{x}Se has been seen to exhibit a gapless 3D Dirac state. In Pb

_{1−x}Sn

_{x}Te, matters are more complicated. We have reported a thorough quantification of the bulk band gap near the topological phase transition, and have not yet observed a bulk gap closure [17]. Pb

_{1−x}Sn

_{x}Te samples having high Sn content are, however, Bi-doped (0.01%–0.2%) [38]. The Bi content is likely to have some impacts on the band gap. However, it is expected to be minimal for such concentrations. The energy gap that we observe in Pb

_{1−x}Sn

_{x}Te is never smaller than 30 meV. Overall, it is unclear whether the gapless 3D Dirac state is fundamentally unfavorable or suppressed by alloy disorder.

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te. The findings typically agree with ARPES results from Reference [30]. Further work needs to be done to determine whether the behavior is fundamentally discontinuous or suppressed by disorder.

#### 6.2. Surface and Bulk State Landau Levels at High Magnetic Fields

_{1−x}Sn

_{x}Te, the $\overline{M}$-Dirac cones were shown to be highly anisotropic and asymmetric. It is thus expected to behave like the non-ideal Dirac cones of Bi-based TIs. The Landau level dispersion of non-ideal Dirac cones was discussed under the scope of the Berry phase by McKenzie and Wright [42], and Taskin and Ando [43]. It was shown that a strong Zeeman term will coexist with the cyclotron term of Dirac fermions, and will yield a field-dispersing zeroth Landau level. This has not yet been observed in Landau level spectroscopy on TCI materials. The difficulty lies mainly in the fact that the $\overline{M}$-Dirac cones may be shifted with respect to the $\overline{\Gamma}$-Dirac cone and may have Fermi levels that are farther from the Dirac point [27]. Low-index Landau levels are hence challenging to observe. Intraband transitions involving $\overline{M}$-Dirac cones are thus expected to lie at relatively low energies and might require high magnetic fields to yield transitions in the IR range (>1 THz).

## 7. Conclusions

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te, and presented a number of challenging questions that need to be addressed. Finally, we have presented two predictions about unconventional effects stemming from the dispersion of Landau levels and their valley degeneracy that remain untested. Magneto-optical Landau level absorption is the ideal tool to put those predictions to the test. However, the conditions required to observe these effects are not easy to reach. Pb

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te have shown great promise so far in exhibiting topological surface state signatures in Landau level spectroscopy, owing to their low carrier density and high mobility. They are, therefore, systems that deserve great attention and may soon lead to significant achievements in the field of topological matter.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) rocksalt (space group Fm$\overline{3}$m) crystal structure of Pb

_{1−x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te alloys; (

**b**) schematic illustration of a topological phase transition in Pb

_{1-x}Sn

_{x}Se and Pb

_{1−x}Sn

_{x}Te systems, from topological trivial regime to topological non-trivial regime, occurring at a critical Sn concentration x

_{c}; (

**c**) (001)- and (

**d**) (111)-oriented 3D bulk Brillouin zones with respective 2D surface Brillouin zones shown on top. The Dirac cones corresponding to the (001) (

**c**) and (111) (

**d**) surfaces are also shown on each respective surface Brillouin zone.

**Figure 2.**Illustration of the Fermi surface geometry. (

**a**) Longitudinal bulk valley of IV–VI semiconductors is an ellipsoid with its major axis oriented parallel to the [111] direction. 2a is the minor axis of the ellipsoid and 2b is its major axis. A magnetic field will yield cyclotron motion on a plane perpendicular to the field. The section of the Fermi ellipsoid by this plane yields a circular 2D Fermi surface shown below the ellipsoid; (

**b**) In the case of a [111] oblique valley, the ellipsoid is tilted by 70.5° with respect to the [111] direction and direction of the applied field. The 2D section is thus an ellipse; (

**c**) In the case of (001)-oriented crystals with a magnetic field oriented parallel to the [001] direction, the ellipsoid is tilted by 53°, and the section also yields an ellipse.

**Figure 3.**The equivalence between the Landau levels in the six-band

**k.p**model used by Mitchell and Wallis and in the massive Dirac model that includes the far-band correction terms. The arrows show the interband transitions of the same energy in the Faraday geometry where the selection rules are ΔN = ±1 and Δσ = ±1.

**Figure 4.**Infrared (IR) magneto-optical absorption measurement in Pb

_{1−x}Sn

_{x}Se for x = 0.14. (

**a**) Mid-IR and (

**b**) Far-IR relative transmissions measured at high magnetic fields; (

**c**) Landau levels dispersion and examples of Landau level interband transitions occurring in this system; and (

**d**) Landau level transition diagram. Transitions in the bulk oblique and longitudinal valleys are represented by empty red and full black circles, respectively. All curve fits (red for oblique and black for longitudinal valleys) are obtained from a massive Dirac fermion model that includes far-band contributions. The green rectangle is the BaF

_{2}substrate reststrahlen band where no transition can be observed.

**Figure 5.**IR magneto-optical absorption measurement in Pb

_{1−x}Sn

_{x}Se for x = 0.19. (

**a**) MIR and (

**b**) FIR relative transmissions measured at high magnetic fields; (

**c**) examples of Landau level interband transitions and the ground state cyclotron resonance of the topological surface states (CR-TSS); and (

**d**) Landau level transition diagram. Transitions in the bulk oblique and longitudinal valleys are represented by empty red and full black circles, respectively. Blue circles denote the CR-TSS. Curve fits for bulk band transitions (red for oblique and black for longitudinal valleys) are obtained from a massive Dirac fermion model that includes far-band contributions. The blue solid line is obtained from a massless Dirac model. The green rectangle is the BaF

_{2}substrate reststrahlen band where no transition can be observed.

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Phuphachong, T.; Assaf, B.A.; Volobuev, V.V.; Bauer, G.; Springholz, G.; De Vaulchier, L.-A.; Guldner, Y. Dirac Landau Level Spectroscopy in Pb_{1−x}Sn_{x}Se and Pb_{1−x}Sn_{x}Te across the Topological Phase Transition: A Review. *Crystals* **2017**, *7*, 29.
https://doi.org/10.3390/cryst7010029

**AMA Style**

Phuphachong T, Assaf BA, Volobuev VV, Bauer G, Springholz G, De Vaulchier L-A, Guldner Y. Dirac Landau Level Spectroscopy in Pb_{1−x}Sn_{x}Se and Pb_{1−x}Sn_{x}Te across the Topological Phase Transition: A Review. *Crystals*. 2017; 7(1):29.
https://doi.org/10.3390/cryst7010029

**Chicago/Turabian Style**

Phuphachong, Thanyanan, Badih A. Assaf, Valentine V. Volobuev, Günther Bauer, Gunther Springholz, Louis-Anne De Vaulchier, and Yves Guldner. 2017. "Dirac Landau Level Spectroscopy in Pb_{1−x}Sn_{x}Se and Pb_{1−x}Sn_{x}Te across the Topological Phase Transition: A Review" *Crystals* 7, no. 1: 29.
https://doi.org/10.3390/cryst7010029