# Experimental Observation of Temperature-Driven Topological Phase Transition in HgTe/CdHgTe Quantum Wells

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## Abstract

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_{c}).

## 1. Introduction

_{0.7}Hg

_{0.3}Te QW grown on CdTe buffer the band structure inverted if QW thickness d exceeds the critical value d

_{c}= 6.3 nm, whereas for d < d

_{c}the band structure is normal [1]. At the critical thickness, the band gap closes, establishing single-valley 2D massless Dirac fermions [3].

_{c}.

_{c}, above which an inverted band ordering is transformed into a normal one [2]. The latter can be conditionally interpreted as negative values for B

_{c}. Thus, B

_{c}= 0 corresponds to a topological phase transition between the BI and TI phases. As well as the band ordering, a critical magnetic field also depends on temperature and pressure, and therefore, can be varied by tuning these external parameters [6].

_{c}(above 200 K) the gapless state could not be directly observed in the samples studied. Ikonnikov et al. [10] have reported on magnetospectroscopy of HgTe quantum wells in magnetic fields up to 45 T in the temperature range from 4.2 K to 185 K. They show that although their samples are TI only at low temperatures, the signature of the TI phase persists in optical transitions at high temperatures and high magnetic fields. In our previous works we have reported on the observation of temperature driven topological phase transition using far-infrared magnetoabsorption spectroscopy [11] and magnetotransport [12,13]. In the present work we analyze these two methods to make a conclusion about their applicability for the TI-BI phase transition investigation.

## 2. Results

_{c}is high enough so that α and β transitions are not forbidden by the Pauli principle and the crossing point does not fall in the restrahlen bands (see Figure 2f). Consequently, in the vicinity of the conditions corresponding to the phase transition when B

_{c}is close to zero no manifestation of the LL anticrossing can be observed (see Figure 2g,h).

_{c}are affected by the aforementioned α-α′ and β-β′ anticrossing [8,15]. Hence at these conditions the behavior of the α and β lines predicted by a simple axial model cannot be observed due to interference with low symmetry effects around B

_{c}and the limitation of the maximal magnetic field. At critical temperature the crossing of α and β lines was observed (Figure 2h). Above the critical temperature we can clearly see that β line (that has higher slope) lies above α line as expected for BI state (Figure 2i,j).

_{g}, allows one to perform a magnetotransport mapping of the structure of LLs of the sample. As it was proposed in the pioneer paper by Büttner et al. [3], the peaks of ∂σ

_{xy}/∂V

_{g}curves give the precise positions of crossings between the Fermi level and the LLs. Therefore plotting ∂σ

_{xy}/∂V

_{g}for each magnetic field value makes it possible to reveal the dispersion of the LLs (see Figure 4).

_{c}as the position of the crossing of zero-mode LLs or their linear extrapolation. Consequently, the value of B

_{c}can be negative in the normal semiconductor phase. It is necessary to note that the extrapolation we use is much more accurate than the one used in magnetooptical experiments. The measurements have shown that when the Fermi level is located within the conduction or the valence band the value of the 2D carrier concentration depends linearly on the gate voltage V

_{g}[16]. Consequently, as far as the LL degeneracy rate is also proportional to the 2D carrier concentration, all LLs in Figure 4 are exactly linear everywhere except the range where the Fermi level lies in the gap (−2V < V

_{g}< −1.5V). Thus, the method used in work [13] allows precise measurement of both positive and negative B

_{c}values (see Figure 5) and is a reliable instrument to probe a temperature-induced phase transition between the BI and TI phases.

## 3. Conclusions

_{c}). Magnetooptical experiments give only a qualitative picture and any estimations of the energy gap based on these measurements are inaccurate.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Landau levels of HgTe/Cd

_{0.7}Hg

_{0.3}Te QW calculated within axial approximation for: (

**a**) band insulator (d = 5.5 nm); (

**b**) zero-gap semiconductor (d = 6.3 nm); (

**c**) topological insulator (d = 7.4 nm). Zero-mode LLs are shown with bold lines. LL indices are shown with both numbers on the lines and line colors. We use the same calculation method and LL indices notation of as in [8,9].

**Figure 2.**Fan chart of inter-LL transitions in 6-nm wide HgTe/Cd

_{0.62}Hg

_{0.38}Te QW (

**a**–

**e**) and 8-nm wide width HgTe/Cd

_{0.8}Hg

_{0.2}Te QW (

**f**–

**j**) from work [11]. The calculated transitions are shown in solid lines while the experimental data are represented by the open symbols in the same colors as the theoretical curves. The estimated values of the energy gap are shown by the horizontal arrows. Shaded areas indicate the GaAs and HgCdTe reststrahlen bands.

**Figure 4.**Colormap of the experimental Landau level fan charts of derivative ∂σ

_{xy}/∂V

_{g}in 6.5-nm wide HgTe/Cd

_{0.65}Hg

_{0.35}Te QW as a function of both magnetic field and gate voltage from work [13]; brighter color represents higher value of the derivative; dark color represents a lower value. The white curves present positions of σ

_{xy}= (2n + 1)e

^{2}/(2h). The blue dashed curves fit the experimental behavior of the zero-mode LLs based on σ

_{xy}values at high magnetic fields.

**Figure 5.**Theoretical (black curve) and experimental (open symbols) values for critical magnetic field as a function of temperature for 6.5 nm HgTe/Cd

_{0.65}Hg

_{0.35}Te QW from work [13].

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**MDPI and ACS Style**

Zholudev, M.S.; Kadykov, A.M.; Fadeev, M.A.; Marcinkiewicz, M.; Ruffenach, S.; Consejo, C.; Knap, W.; Torres, J.; Morozov, S.V.; Gavrilenko, V.I.;
et al. Experimental Observation of Temperature-Driven Topological Phase Transition in HgTe/CdHgTe Quantum Wells. *Condens. Matter* **2019**, *4*, 27.
https://doi.org/10.3390/condmat4010027

**AMA Style**

Zholudev MS, Kadykov AM, Fadeev MA, Marcinkiewicz M, Ruffenach S, Consejo C, Knap W, Torres J, Morozov SV, Gavrilenko VI,
et al. Experimental Observation of Temperature-Driven Topological Phase Transition in HgTe/CdHgTe Quantum Wells. *Condensed Matter*. 2019; 4(1):27.
https://doi.org/10.3390/condmat4010027

**Chicago/Turabian Style**

Zholudev, Maksim S., Aleksandr M. Kadykov, Mikhail A. Fadeev, Michal Marcinkiewicz, Sandra Ruffenach, Christophe Consejo, Wojciech Knap, Jeremie Torres, Sergey V. Morozov, Vladimir I. Gavrilenko,
and et al. 2019. "Experimental Observation of Temperature-Driven Topological Phase Transition in HgTe/CdHgTe Quantum Wells" *Condensed Matter* 4, no. 1: 27.
https://doi.org/10.3390/condmat4010027