Quantum Hall effect originated from helical edge states in Cd$_3$As$_2$

The recent experimental observations of the quantum Hall effect in 3D topological semimetals have attracted great attention, but there are still debates on its origin. We systematically study the dependence of the quantum Hall effect in topological semimetals on the thickness, Fermi energy, and growth direction, taking into account the contributions from the Fermi-arc surface states, confinement-induced bulk subbands, and helical side-surface edge states. In particular, we focus on the intensively studied Dirac semimetal Cd$_{3}$As$_{2}$ and its slabs grown along experimentally accessible directions, including [001], [110], and [112]. We reveal an ignored mechanism from the Zeeman splitting of the helical edge states, which along with Fermi-arc 3D quantum Hall effect, may give a non-monotonic dependence of the Hall conductance plateaus on the magnetic field in the most experimentally studied [112] direction slab. Our results will be insightful for exploring the quantum Hall effects beyond two dimensions.

The Hall conductance for a slab of thickness L can be found as σ H = σL, where the Hall conductivity can be found from the Kubo formula [32] where e is the elementary charge, is the reduced Planck constant, V eff is the volume of the slab, |Ψ δ is the eigenstate of energy E δ for H in the y -direction magnetic field and open boundaries at y = ±L/2, v x and v z are the velocity operators, f (x) is the Fermi distribution.
The [001] slab -For a Cd 3 As 2 slab grown along the [001] direction, the bulk spectrum is quantized into discrete gapped subbands (See Sec. SI of [33]) because of the quantum confinement effect [34][35][36]. The spectrum opens a gap, which decays with increasing L (probably with an oscillation as well). The effective Hamiltonian H nn for each subband n(= 1, 2, . . .) is equivalent to a quantum spin Hall insulator [37][38][39] characterized by the spin Chern number [40][41][42] are the valence-band Chern numbers of the spin-up and spin-down blocks of the n-th subband. Each Chern number C ↑/↓ n represents a chiral edge state circulating around the side surfaces [ Fig. 1(a)]. The total spin Chern number C s = n C n s is equal to the number of pairs of helical edge states. As shown in Fig. 1(b), the oscillatory decay of the band gap with increasing L is always accompanied by the variation of the spin Chern number C s at each dip. In the Dirac semimetal Na 3 Bi, a topological phase transition to the quantum spin Hall state has been  observed [43].
However, the spin Chern number is not measurable because the measurable Hall conductance is associated with the total Chern number σ H = e 2 h n,s=↑,↓ C s n , which is zero in the absence of the magnetic field because of timereversal symmetry. A magnetic field can break timereversal symmetry as well as the balance between C ↑ n and C ↓ n , leading to measurable Hall conductance σ H whose magnitude increases with increasing magnetic field, as shown in Figs. 1(c-d). We also plot the Hall conductance σ ↑,↓ H for the spin-up and spin-down blocks of the Hamiltonian [ Fig. 1(c)], which confirms that the non-zero quantum Hall conductance is originated from the fieldinduced imbalance between counter-propagating chiral edge states. Also, Fig. 1(d) shows that the Hall conductance approaches zero for thinner slabs because of the mixing of counter-propagating chiral edge states. This mechanism due to the splitting of helical edge states was previously ignored and could benefit the further experimental explorations.
The  [23,44] arising from the Fermi-arc surface states [25,27]. On each of the top and bottom surfaces of a Weyl semimetal, there are topologically-protected sur- , to support a "3D" quantum Hall effect [23]. A Dirac semimetal can be regarded as two time-reversed Weyl semimetals to host two copies of the Fermi-arc quantum Hall effect. In addition, in a Dirac semimetal, the Fermi-arc surface states and their time-reversal partners on a single surface can form a 2D electron gas [45], to support a quantum Hall effect as well. For both cases, the Hall conductance plateaus are supposed to decrease with increasing magnetic field, much like those in conventional 2D electron gases [1], where the magnetic field presses the occupation of electrons to lower Landau levels, as shown in Figure 2(d) for different slab thickness L.
Moreover, Fig. 2(d) shows that the width of the quantized plateaus is stable for thicker slabs (L > 50 nm), while show obvious variations for ultrathin slabs (L < 20 nm) with decreasing thickness. This can be understood using Figs. 2(b-c), where the area of the Fermi loop S converges to a constant for thick slabs but decreases exponentially with decreasing L, due to the hybridization of the opposite surfaces in ultrathin slabs. According to the Lifshitz-Onsager relation, S determines the plateau width of the Hall conductance, which explains the quantized pattern in Fig. 2(d). Moreover, in Dirac semimetals E F shifts away from E w as the Zeeman effect splits E w (See Sec. SIII of [33]), leading to a systematic shift of the Hall plateaus with increasing thickness [ Fig. 2(d)].
The slab, that is, the Hall plateaus decrease with increasing magnetic field, which indicates the quantized con- The angle dependence - Figure 4(a) illustrates the dependence on θ, the angle between the line connecting the Dirac nodes and the k x -k z plane. For example, the Dirac nodes are located on the k x -k z plane when θ = 0 [ Figs. 2(a-b)]. Figure 4(b) shows that S decreases and C s increases with increasing θ, indicating the competition between the Fermi-arc surface states and sidesurface helical edge states. Figure 4(c) shows σ H as a function of θ and 1/B for L = 100 nm. For θ = 0 ([110] direction) and θ = π/2 ([001] direction), the quantized conductance is only contributed by the Fermi-arc surface states and imbalanced helical edge states, respectively. For other θ, the quantized Hall conductance originates from both the Fermi-arc surface states and the helical edge states. Furthermore, σ H as a function of (E F − E w , θ) [ Fig. 4(d)] shows that the quantum Hall effect may be due to the confinement-induced bulk subbands, giving another origin for the experimentally observed quantum Hall effect in Cd 3 As 2 [20,21]. In experiments, it may be difficult to distinguish whether the thickness dependent conductance plateaus are consequences of the bulk subbands or the Fermi-arc surface states. The non-monotonic dependence of σ H on the magnetic field may play another significant role to detect the side-surface helical edge states or Fermi-arc 3D quantum Hall effect in a [112] Dirac semimetal slab.
Discussion -Above, we have shown that the Hall plateaus can be attributed to the Fermi-arc surface states, confinement-induced bulk subbands, and helical side-surface edge states. It may be challenging to distinguish them in a standard Hall-bar device. The helical side-surface edge states can be identified through the nonlocal measurements [46]. Furthermore, in Sec. SIV of [33], we propose two different local transport devices, which may help revealing the mechanism of the observed quantum Hall effects in experiments.