Thickness dependence of the quantum Hall effect in films of the three-dimensional Dirac semimetal Cd₃As₂ Open Access

Three-dimensional Dirac semimetals, such as Cd₃As₂,^1–4 can possess topologically nontrivial surface Fermi arcs, which are thought to be protected^1,5–7 even in the case of a band gap opening in the bulk states. Compared to Weyl semimetals, for which surface Fermi arcs have been observed in angle-resolved photoemission spectroscopy,^8,9 their experimental detection has remained more elusive. Magnetotransport measurements should, in principle, allow for their observation. For example, these states can form two copies of “Weyl orbits” that allow for quantum oscillations, when top and bottom surfaces connect at the Dirac node projections of the surface Brillouin zone for sufficiently thin samples.^10–12 Signatures of such orbits have been observed in quantum oscillations from lamellae cut from bulk crystals of Cd₃As₂ (thickness ≥150 nm),¹³ and hints of Hall plateaus have recently been observed in platelets of similar thicknesses.¹⁴ In both studies, however, the bulk states still contributed significantly to the transport, which can complicate interpretation.

Recently, we have reported the quantum Hall effect for thin (20 nm–30 nm) epitaxial Cd₃As₂ films.¹⁵ In these very thin films, an energy gap exists for the bulk electronic states, consistent with expectations from theory.^1,11,16 At low temperatures, all mobile carriers were shown to reside in two-dimensional states without any parasitic conduction from the bulk.

While the observation of the quantum Hall effect is definitive evidence of two-dimensional carriers, additional studies can provide more information about their nature and origin. For example, by varying the film thickness, the relative contributions of quantum-confined bulk states and surface states to the transport may be determined. Furthermore, films grown on different substrates may provide insights into the robustness of surface states with respect to microscopic parameters, such as interfacial charge transfer. In this study, we investigate the quantum Hall effect of Cd₃As₂ films grown epitaxially on CdTe substrates, as a function of the film thickness. The results are consistent with the presence of robust, gapless surface states, as expected from theoretical predictions.

Cd₃As₂ films were grown by molecular beam epitaxy (MBE) on (111)CdTe substrates. Growth conditions were similar to those used in our previous studies, in which Cd₃As₂ films were grown on (111) GaAs substrates with epitaxial GaSb buffer layers.^15,17,18 More details about the MBE procedures can be found in the supplementary material. Cd₃As₂ has a tetragonal unit cell (I4₁/acd space group¹⁹), while the substrates have the cubic zincblende structure. On both types of substrates, the film/substrate orientation is (112)Cd₃As₂||(111)zincblende. The lattice mismatch between CdTe and Cd₃As₂ is 2.3%, which is slightly smaller than that for GaSb (−3.5%). X-ray diffraction (XRD) confirmed single-phase, epitaxial films (see Fig. 1). Miscut (4°) CdTe substrate surfaces were used to avoid twinning of the Cd₃As₂ films and to promote two-dimensional film growth (see the supplementary material). Furthermore, while films on (111) GaSb buffers showed a small (<1°) misalignment between the film and substrate surface planes, they were parallel for the CdTe substrates, likely because of the smaller lattice mismatch. The thickness of the Cd₃As₂ films was varied between 10 nm and 350 nm. Magnetoresistances were measured as a function of the temperature in a Quantum Design Dynacool cryostat with magnetic fields (B) up to 9 T and in a Triton Oxford Instruments dilution refrigerator at 40 mK in magnetic fields of up to 14 T. A small systematic offset in the values obtained from the electronics used for measuring resistances in the Oxford Instruments refrigerator was corrected using reference measurements performed in the Dynacool instrument. Transport properties were measured using Hall bar devices of dimensions 100 μm × 100 μm (width × spacing between the contact arms), except for the 350-nm-thick film, which was measured in the van der Pauw configuration. Hall mobilities and carrier densities were extracted from the low field (−0.5 T to +0.5 T) data. All films were n-type.

Figure 2 shows the longitudinal magnetoresistance (R_xx) and Hall resistance (R_xy) of a 10-nm-thick film measured at 40 mK. Well-developed Hall plateaus, which coincide with an almost vanishing R_xx at high B, can be seen. Minima in R_xx coincide with the quantum Hall plateaus in R_xy. The sequence of the plateaus, R_xy = $h/νe2$⁠, where h is Planck’s constant, e is the elementary charge, and ν is the filling factor, is ν = 2, 4, 6, 8, indicating two-fold degeneracy. The lowest Landau level is reached at about 10 T without lifting the degeneracy. Figures 3(a) and 3(b) show R_xx normalized to the zero-field value and R_xy, respectively, for films with different thickness, measured at 2 K. Shubnikov-de Haas oscillations can be detected for all films, and plateaus in R_xy develop for thicknesses below 70 nm. Sheet carrier densities at 2 K, extracted from the Shubnikov-de Haas oscillations and low-field Hall measurements, respectively, are shown in Fig. 3(c) for the films with different thicknesses. While the carrier concentration determined from the quantum oscillations is nearly independent of the thickness, the Hall carrier density increases with the thickness. For the thinnest films, the two carrier densities match. The positions of the peaks in the Shubnikov-de Haas oscillations are plotted in a fan diagram [Fig. 3(d)] and follow a straight line.

The relative amplitude of the Shubnikov-de Haas oscillations (Fig. 4), $ΔRxxR0$⁠, where R₀ is the resistance at zero field, as a function of the temperature (T) can be used to extract an effective mass, m*, and quantum mobility, $μq=m*e/τq$⁠, where τ_q is the quantum scattering time, according to²⁰ 

Here, k_B is Boltzmann’s constant, $ℏ$ is reduced Planck’s constant, and ω_c is the cyclotron frequency. The results from the fits to the data in Fig. 4 (solid lines) are shown in Table I, along with the low-field Hall mobilities, μ_H. In general, μ_q < μ_H is expected,²¹ even in the case of topologically protected states, because the protection is only against backscattering, not impurity scattering.²² 

We next discuss the results. For thick films, both bulk and surface state carriers contribute to the Hall effect, resulting in a larger sheet carrier density. The effect of quantum confinement is to produce a bulk gap, which is expected to increase with decreasing film thickness. Theoretical predictions indicate that bulk states in films as thick as 90 nm show an energy gap, while the surface states remain gapless to much lower thicknesses.¹⁶ The decrease in μ_H with decreasing thickness (see Table I) indicates a higher mobility of the (residual) bulk carriers, whose contribution to μ_H diminishes at low thicknesses. At low temperatures, sufficiently thin films have a bulk gap that is large enough that the bulk carriers freeze out and do not contribute to the transport. At this point, only the carriers in the two-dimensional states contribute to transport, which give rise to the Shubnikov-de Haas oscillations and the quantum Hall effect. The quantum Hall effect is cleanest (vanishing R_xx at the plateaus) for the thinnest films, as expected because of the absence of the bulk contribution. The fact that the quantum Hall effect in thin Cd₃As₂ films is independent of the substrate (CdTe in this study and GaSb in our prior study¹⁵) shows that it is intrinsic to Cd₃As₂ and relatively independent of microscopic details, such as band alignments (charge transfer) with the substrate.

Interestingly, the carrier density derived from the quantum oscillations is nearly independent of the film thickness, indicating that the two-dimensional states, which in the thinnest sample give rise to the quantum Hall effect, are already present even in thicker films. This finding supports the notion that the quantum oscillations and quantum Hall effect arise from the intrinsic surface states predicted by theory for three-dimensional Dirac semimetals. If the quantum oscillations were due to quantum confined bulk states, one would expect a pronounced thickness dependence. The extracted values for m* and μ_q (Table I) show some thickness dependence, most notably around 60–70 nm, which coincides with the thickness where the bulk band gap appears. Further studies are needed to understand this; for example, it may be due to a change in the connectivity between bulk and surface states.

The degeneracy of the quantum Hall plateaus is consistent with theoretical expectations. The generic surface states in Dirac semimetals, such as Cd₃As₂, are double Fermi arcs that connect the projections of the Dirac nodes on the surface, which are connected near the Dirac nodes through the bulk.¹⁰ This allows for two sets of “Weyl orbits” in a quantizing magnetic field. Under certain conditions, surface states may become disconnected from the projection of the bulk nodes, forming orbits derived from two copies of topological insulator-like Dirac cones.²³ Either scenario could explain the two-fold degeneracy of the quantum Hall effect. We note that this implies that any spin degeneracy is already lifted at low fields, as is perhaps expected, given the large g-factor.^24,25 Interestingly, in our prior studies of films on GaSb,¹⁵ the two-fold degeneracy was lifted in high magnetic fields, resulting in the appearance of a ν = 1 plateau. Microscopic parameters, such as a substrate miscut, can lift valley degeneracy in two-dimensional electron gases in semiconductors, which leads to Landau level splitting in a magnetic field (see, e.g., Ref. 26). Furthermore, films on GaSb have higher mobilities¹⁵ and are thus likely to exhibit reduced disorder broadening of the Landau levels, making it easier to resolve the splitting. Finally, we briefly note that while the phase shift in fan diagrams, such as shown in Fig. 3(d), can provide information about the Berry phase,^27,28 this shift also depends on the number of Dirac nodes enclosed by the electron orbits. Without further understanding of these orbits, the value of the intercept in the fan diagram therefore does not permit any conclusions about the topological nature of the two-dimensional states in these films.

In summary, by analyzing the thickness dependence of the quantum transport of epitaxial Cd₃As₂ films grown on CdTe, we have shown that the observation of an even-integer quantum Hall effect is consistent with theoretical predictions of robust, gapless surface states. In particular, two-dimensional states were found to give rise to quantum oscillations at all thicknesses investigated here and the quantum Hall effect emerges once bulk conduction is eliminated by reducing the film thickness and the opening of a bulk band gap. The results, along with our previous study,¹⁵ open up numerous opportunities to study their unique properties.

See supplementary material for details about the MBE growth, film microstructure, and additional transport data.

The authors thank Omor Shoron, Leon Balents, and Jim Allen for discussions. They also gratefully acknowledge support through the Vannevar Bush Faculty Fellowship program by the U.S. Department of Defense (Grant No. N00014-16-1-2814). Partial support was also provided by a grant from the U.S. Army Research Office (Grant No. W911NF-16-1-0280). The dilution fridge used in the measurements was funded through the Major Research Instrumentation program of the U.S. National Science Foundation (Award No. DMR 1531389). This work made use of the MRL Shared Experimental Facilities, which are supported by the MRSEC Program of the U.S. National Science Foundation under Award No. DMR 1720256.