Recently [1], it was found that a two-dimensional topological insulator is implemented not only in HgTe quantum wells but also in CdHgTe quantum wells. Thus, it was experimentally demonstrated that the state of the two-dimensional topological insulator can be controlled not only by changing the thickness of the HgTe film but also by using a film with a different composition of the initial materials. In particular, the addition of a small amount of Cd makes it possible to rearrange the energy spectrum of the quantum well in such a way that the critical thickness of the quantum well corresponding to the transition from the direct to inverted spectrum increases, and thus the state of the topological insulator can be implemented at large thicknesses, as also shown in [1]. However, the experimental samples in [1] had macroscopic (over 100 μm) dimensions and thus demonstrated only diffusion transport.

In this work, we study mesoscopic samples based on the system described above, implement the quasi-ballistic transport regime in them, and analyze the behavior of mesoscopic conductance fluctuations caused by the interaction of bulk and edge states. The experimental samples studied in this work were micron-sized Hall bars (their topology and dimensions are shown in Fig. 1a), which were fabricated on the basis of a CdHgTe quantum well 11.5 nm thick with the (013) orientation and were equipped with metal TiAu gates. The measurements were carried out in the temperature range of 0.08–10 K in magnetic fields up to 2 T using a standard lock-in detection technique at frequencies of 1–12 Hz and at measuring currents through the sample of 0.1–100 nA, depending on the character of the measurements. Figure 1c shows typical dependences of the local (\({{R}_{{\text{L}}}}\)) and nonlocal (\({{R}_{{{\text{NL}}}}}\)) resistances measured in the shortest part of the Hall bar (see Fig. 1b), at \(T = 0.85\) K. These curves have a similar asymmetric shape, a maximum corresponding to the position of the Fermi level in the energy gap and the charge neutrality point, and a rapid decrease when the Fermi level is shifted either to the conduction band or to the valence band. The local resistance is 27 kΩ at the maximum (i.e., twice the resistance in the ballistic transport regime) and the nonlocal resistance is 20 kΩ. In general, the observed picture is characteristic of two-dimensional topological insulators in HgTe quantum wells in the quasi-ballistic transport regime [2, 3]. The inset of Fig. 1c shows the temperature dependence of the maximum local resistance RL(Vg) at Vg = –2 V. We note that the dominant edge transport regime begins at \(T < 1\) K. The estimate of the activation energy based on the data from the inset of Fig. 1c gives about 1 meV for the gap, which agrees with the value found in the analysis of edge transport in macroscopic samples [1].

Fig. 1.
figure 1

(Color online) (a) Layout of the 9-pin Hall bars under study coated with a metal gate. The distance between the nearest potentiometric contacts along the edge of the sample located under the gate is 5 µm. (b) (Left) Local and (right) nonlocal 4-point resistance measurement geometry. The measurements were carried out in the part of the Hall bar with the smallest distance between the contacts. (с) Local RL and nonlocal RNL resistances versus the gate voltage Vg at a temperature of \(T = 0.85\) K. The inset: the temperature dependence of the maximum of the local resistance \(R_{{\text{L}}}^{{{\text{max}}}}({{V}_{{\text{g}}}})\) at Vg = –2 V.

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A more careful analysis of the gate dependences reveals mesoscopic fluctuations of the local and nonlocal conductivities in the samples under study with amplitudes comparable for both types of resistances. This demonstrates the key role of edge transport in the formation of these fluctuations. Figures 2a and 2b show families of dependences \({{R}_{{\text{L}}}}(B)\) and RNL(B) at different gate voltages corresponding to the dominance of edge transport, i.e., at \(T = 80\) mK. It can be clearly seen that they are close to periodic oscillations with an average period of about 0.15 T. We note that similar oscillations with a much longer period (0.5 T) and smaller amplitude were earlier observed in the local transport regime in two-dimensional topological insulators based on HgTe quantum wells [4]. In this work, these oscillations were interpreted as Aharonov–Bohm oscillations associated with the quantization of the magnetic flux through closed trajectories of two types: (i) trajectories formed as a result of the interaction with bulk states, which are puddles of bulk charge carriers caused by fluctuations in the impurity potential [5], and (ii) trajectories caused by the appearance of closed loops of edge states under the action of roughness at the edge of the quantum well. Unfortunately, oscillations in nonlocal geometry were not measured in the experiments presented in [4]. Thus, strictly speaking, it was not unambiguously proven that the Aharonov–Bohm oscillations found in [4] are due to edge transport.

Fig. 2.
figure 2

(Color online) (a, b) Magnetic-field dependences of the (a) local \({{R}_{{\text{L}}}}\) and (b) nonlocal RNL resistances measured at the minimum temperature and various gate voltages Vg near the charge neutrality point. (c, d) Autocorrelation functions of normalized fluctuations of local and nonlocal resistances obtained from curves shown in panels (a) and (b), respectively. Each autocorrelation function can be represented as a periodic function and an exponentially decaying envelope, which is typical of autocorrelation functions close to periodic fluctuations. In this case, it clearly shows that the amplitude of oscillations of autocorrelation functions reaches a maximum near Vg = –2 V, while the damping decrement reaches a minimum. An increase in the amplitude is also observed with the transition to the conduction band (upper two lines in panel (c)), but this is due to the formation of Shubnikov–de Haas oscillations in the local resistance, and in this case is an artifact.

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Our experimental data shown in Fig. 2 directly indicate that the observed Aharonov–Bohm oscillations reflect the response of the edge states. We compare the behaviors of fluctuations \(\Delta R\) of local and nonlocal resistances. The period and amplitude of these fluctuations are not necessarily determined from the calculated Fourier spectra of almost periodic fluctuations and their analysis. A more adequate method is based on the calculation and analysis of autocorrelation functions of normalized local and nonlocal resistance fluctuations. To this end, we used the formula \(\Delta R = (R - \langle R\rangle ){\text{/}}R(B = 0)\), where \(\langle R\rangle \) means a monotonic component obtained from the experimental data using a low-pass filter with a cutoff frequency of 4 T–1, i.e., all oscillations with a period longer than 0.25 T were averaged. Next, autocorrelation functions presented in Figs. 2c and 2d were calculated from the normalized fluctuations thus obtained in the magnetic field range of 0–1.5 T. All autocorrelation dependences obtained are typical of almost periodic fluctuations and can be described by the function \(A\cos (\omega \Delta B)\exp ( - k\Delta B)\), where A is the fluctuation amplitude, ω is the characteristic frequency, and k is the damping decrement indicating the degree of randomness (dispersion) of the analyzed fluctuations in amplitude and period. The analysis of the entire family of functions indicates that (i) the oscillation period is almost independent of the gate voltage and is \( \approx {\kern 1pt} 0.15\) T for both the local and nonlocal resistances and (ii) the amplitude of normalized oscillations has an explicit maximum near the charge neutrality point. The second increase in the amplitude is also observed when the gate voltage is moved to the conduction band. This phenomenon is caused by the formation of Shubnikov–de Haas oscillations on the dependences \({{R}_{{\text{L}}}}(B)\), and therefore, it is an artifact within the scope of this work.

Since the amplitudes and phases are the same for local and nonlocal responses, it is reasonable to assume that the discussed oscillations are precisely described by the Aharonov–Bohm effect at the edge current states of a two-dimensional topological insulator, as predicted in [5]. A small damping decrement of the autocorrelation functions indicates that the areas of the interference loops on which oscillations are formed are approximately the same. In turn, this indicates that the characteristic size of the edge roughness of the quantum well under study is not described by the expected random distribution, but has a certain value. This size can be estimated as \(d = \sqrt {\frac{4}{\pi }\frac{h}{{e\Delta B}}} = \) 190 nm from a fluctuation period of 0.15 T, and the contour perimeter L is thus \( \approx {\kern 1pt} 600\) nm.

The detection of almost periodic fluctuations in the magnetic field may seem a random result associated with a particular sample and the formation of a single closed loop in it. However, the available data indicate that this is not the case. First, oscillations are observed in all parts of the sample (from 3 to 30 μm in length), which explicitly indicates the presence of many interference contours, and all of them are characterized by a similar period in the magnetic field, which again indicates their characteristic size of \( \approx {\kern 1pt} 200\) nm. Second, similar oscillations (but with a much smaller amplitude) were also observed in macroscopic samples [1]. We assumed that the formation of similar-sized closed loops in this structure was due to the small energy gap (on the order of 1 meV [1]) in the presence of electrostatic potential inhomogeneities.

Figure 3a shows the temperature dependence of the rms oscillations \(\Delta {{R}_{{\text{L}}}}\) of the local resistance. It can be seen that this dependence is almost exponential with saturation at \(T < 0.5\) K. This fact is another confirmation of a non-Gaussian size distribution of the discussed roughnesses. The properties of oscillations described above suggest that the temperature dependence of their amplitude is described by the simple expression

$$\Delta R = A\exp ( - L{\text{/}}{{L}_{\phi }}),$$
(1)

where L is the contour perimeter and \({{L}_{\phi }}\) is the phase coherence length.

Fig. 3.
figure 3

(Color online) (a) Temperature dependence of rms oscillations of the local resistance \(\Delta {{R}_{{\text{L}}}}\) at the charge neutrality point, i.e., at Vg = –2.2 V. (b) Temperature dependence of the phase coherence length \({{L}_{\phi }}\) calculated from the data shown in panel (a) using Eq. (1). The dependence was fitted by the formula \({{L}_{\phi }} = A{\text{/}}{{T}^{\alpha }}\) with the fitting parameter \(\alpha = 0.495\).

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Using Eq. (1) and assuming that \({{L}_{\phi }} = L\) at the minimum temperature, one can determine \({{L}_{\phi }}\) and its temperature dependence. The result is shown in Fig. 3b. It indicates that the phase coherence length increases as \({{L}_{\phi }} \sim {{T}^{{ - \alpha }}}\), where \(\alpha = 0.5 \pm 0.05\), from 100 to 600 nm with decreasing temperature from 8 K to 80 mK. Such a temperature dependence qualitatively corresponds to the theory for a one-dimensional diffusion metal [6]. Let us discuss the results. As noted above, in the structures under study with a two-dimensional topological insulator, a quasi-ballistic regime is implemented. The ballistic path length \({{l}_{{\text{B}}}}\) along the edge is estimated at about 3 µm. This value is an order of magnitude larger than the phase coherence length (see Fig. 3b). In the samples under study, the magnetoresistance is relatively low and is no higher than a few tens of percent (see Fig. 2a), so the ratio \({{L}_{\phi }} \ll {{l}_{{\text{B}}}}\) holds even in a magnetic field. Apparently, this fact made the observation of Aharonov–Bohm oscillations possible. Note that a slight increase in the resistance of the sample in a weak magnetic field (up to 0.3 T) contradicts the results obtained in [1, 7], where it was shown that a weak magnetic field leads to an exponential increase in the resistance of the sample with a proportional decrease in \({{l}_{{\text{B}}}}\) and transition to the relation \({{l}_{{\text{B}}}} \ll {{l}_{\phi }}\). In our samples, the physical picture is opposite, which is confirmed by direct measurements of \({{L}_{\phi }}\).

To summarize, quantum interference transport in a two-dimensional topological insulator in a quantum well based on the ternary compound CdHgTe has been experimentally studied in detail. Aharonov–Bohm oscillations on closed trajectories by one-dimensional helicoidal edge states have been detected, and furthermore, the phase coherence length for the motion of charge carriers along these states has been determined for the first time.