Observation of the possible chiral edge mode in Bi1-xSbx

After the classification of topological states of matter has been clarified for non-interacting electron systems, the theoretical connection between gapless boundary modes and nontrivial bulk topological structures, and their evolutions as a function of dimensions are now well understood. However, such dimensional hierarchy has not been well established experimentally although some indirect evidences were reported, for example, such as the half-quantized Hall conductance via quantum Hall effect and extrapolation in the quantum-oscillation measurement. In this paper, we report the appearance of the possible chiral edge mode from the surface state of topological insulators under magnetic fields, confirming the dimensional hierarchy in three dimensional topological insulators. Applying laser pulses to the surface state of Bi1-xSbx, we find that the sign of voltage relaxation in one edge becomes opposite to that in the other edge only when magnetic fields are applied to the topological insulating phase. We show that this sign difference originates from the chirality of edge states, based on coupled time-dependent Poisson and Boltzmann equations.

Chiral edge states are boundary states emerging in two-dimensional (2D) electron systems and reflect its bulk topology. These exotic states have been observed in quantum Hall systems [1][2][3][4], graphene [5,6], HgTe [7,8], surface of three-dimensional (3D) topological insulators (TI) [9,10,11], and quantum anomalous Hall systems [12,13]. In fact, prerequisite for observing the edge states has been thought to be to apply high perpendicular magnetic field B so that well-separated Landau levels are established. When the Fermi energy EF is placed in between the bulk Landau gap, as a consequence of band bending, there are states crossing it at the edges. In contrast to usual quantum Hall systems, 2D Dirac electron systems exhibits half-integer quantum Hall effect, as first observed in graphene [5,6] and later in HeTe [7,8]. case, interestingly, a hierarchy structure is observable, starting from the bulk 3D TI to one-dimensional (1D) edge state through topological 2D surface state. Topological 2D surface state is a boundary state of the bulk 3D TI and 1D edge state occurs as a boundary state of the topological 2D surface state when gap opens at the Dirac node and EF of the 2D surface state is placed at the node.
In general, in the topological state of matter [14][15][16][17][18], there exist structures called bulk-edge correspondence between gapless boundary modes and nontrivial bulk topological structures, and their evolutions as a function of dimensions. This dimensional hierarchy shows how gapless boundary states evolve as dimensional reduction is performed. Theoretically, the existence of gapless surface states in three-dimensional topological insulators is attributed to the topological B E   ⋅ term with an angle coefficient π θ = [19][20][21][22][23]. For the two-dimensional surface state, this term develops into the Chern-Simons term when time reversal symmetry is broken by the applied magnetic fields [19][20][21][22][23]. The half-quantized Hall conductance is a fingerprint of this dimensional hierarchy that starts from topological insulators in 3D [24]. Unfortunately, the half-quantized Hall conductance could be seen only by extrapolation in the quantum-oscillation measurement [25][26][27]. Unambiguous evidence is to observe the single chiral edge mode more directly, responsible for the half-quantized Hall conductance. This was achieved only very recently in 3D TIs [9,10,11].
In this paper, we try to understand the bulk-boundary correspondence in Bi1-xSbx by performing specially designed transient experiments, which leads to the surface Dirac electrons with a gap. According to the dimensional hierarchy, the single chiral edge mode should appear at least in the region of weak magnetic fields, which results from the single Dirac cone topologically protected by the 2 Z index for the surface mode. It is a main point of this study to observe such a chiral edge state by applying laser pulses to the surface state and scrutinizing the relaxation of voltage drop in each onedimensional edge. We find that the sign of voltage drop in one side, saying To detect such a chiral edge mode, we devise measurements at T = 4.2 K of transient thermoelectric voltages induced by the Nd-YAG pulsed laser with wavelength of 1064 nm [28] in a magnetic field. The energy density on the sample is ~ 15 mJ/cm 2 . We call this experiment M1 hereafter, where the sample is covered half to block the laser light [ Fig. 1(b)]. The metal mask was manually made by Al foils covered with Kapton tape. The Kapton tape was used for electrical insulation. The experimental setup and dimension of the samples in the M1 experiment are illustrated schematically in Fig. 1(b). Typical thickness of the sample is around 0.5 mm. The exposure area and mask size are 3 mm 2 and 6 mm 2 , respectively. The laser pulse instantaneously causes the distribution of electrons to deviate from equilibrium. In a metal, non-equilibrium temperatures of electrons are produced and electrons are equilibrated with phonons within few picoseconds through electron-phonon coupling [29,30]. In the case of a semiconductor, the laser pulse generates electron-hole pairs also within few picoseconds [31].
These processes eventually increase the temperature of an entire system. The increase of temperature is expected to be about 10 K, evaluated from the delivered energy from the laser pulse and the specific heat of the BiSb alloy. Thus, the laser irradiation causes the temperature of the system higher in the exposed part of the sample at the very initial stage. In our experiments, after the laser irradiation, we investigate the equilibrating process by measuring the temporal evolution of the voltage differences In this experiment, we used Bi1-xSbx single crystals with x = 0.0 %, 3.0 %, 11.6 %, and 21.5 %.
We selected these samples taking account for the phase diagram of the BiSb alloy and their gap values [32]. It is well-known that the topological phase transition from a band insulator to a topological insulator occurs across x ~ 3 -4 % in Bi1-xSbx [32,33]. In this special composition, this system evolves into a Weyl metal state under the magnetic field, and indeed negative longitudinal magnetoresistance [34] and violation of Ohm's law [35] originated in chiral anomaly of the Weyl metallic state were observed. However, the system in this composition is still metallic due to contributions of the hole band around the T point in the reciprocal space. A full insulating state appears above x ~ 7 % when the band at T sinks below the Fermi level. In this respect one may regard Bi and Bi1-xSbx with x = 3.0% as band "insulators" while Bi1-xSbx with x = 11.6% and 21.5% as topological insulators. The temperature dependence of resistivity informs us that our samples for x < 3.0 % are metallic with residual resistivity at T = 4.2 K less than 0.03 mΩcm while those with x > 7.0 % are weakly insulating with residual resistivity of 0.7 -2.1 mΩcm. Fig. 2 in the M1 configuration for Bi1-xSbx single crystals with x = 0.0 %, 3.0 %, and Fig. 3 for x = 11.6 %, and 21.5 %.
Several important points are clearly noticed in these figures. First, the transient voltage signals rise with a characteristic time less than 0.05 µsec. The signals reach the maximum and decay with characteristic time dependence. We will discuss the characteristic time scales in detail later. In fact, the time scales of the rise and decay are much larger than the time constant of electronic processes measured by the femto-second experiments, which is time scale of only few picoseconds [29][30][31]. The maximum signal is proportional to the magnetic field in the low-field region, but it is saturated or slightly decreases at higher magnetic fields around 1-4 T. The polarity of the signal is determined by the sign of the applied magnetic field and thus, the sign of the voltage signal changes when the magnetic field is reversed. The most interesting observation is the sign of under a given magnetic field. It turns out to be the same in the band insulating region but it is opposite in the topological insulating region. In the case of Bi, Fig. 2 ] for the x = 11.6 % sample and Fig. 3 ] for the x = 21.5 % sample. As the Bi and Bi1-xSbx with x = 3.0 % possess quite different band structure, the peculiar band structure is not a key factor to give rise to the same sign in the band insulating region.
The observation in the band insulating region indicates that the transient current flows from the region at high temperature to that at low temperature as expected. However, the sign difference in the topological insulating region implies that the transient current circulates in either clockwise or counter-clockwise, depending on magnetic fields [ Fig. 1(a)] in contrast to the case of the band insulator.
We attribute the origin of this sign difference to the possible appearance of chiral edge modes on the surface of a topological insulator under magnetic fields. To further support this scenario, we devised another experiment, which was also performed at T = 4.2 K with a different mask configuration as shown in Fig. 1(c). This configuration, which we call the M2 or differential, cancels non-circulating currents, and thus it maximizes the circulating ones. In this setup, all voltage contacts are screened by two masks while the middle of the sample is open. The schematic diagram of experimental setup are presented, along with the sample dimensions and exposure area in Fig. 1(c). In case that there are no 1D chiral edge modes and the sample-mask-contact configuration is perfectly symmetric, the voltage 3 (5) and 4 (6) should be exactly same, giving the zero-voltage difference between 3 (5) and 4 (6). Because the real experimental setup is not perfectly symmetric, a small signal due to experimental asymmetry may appear.
However, its magnitude is expected to be smaller than the corresponding signal in the M1 configuration.
for Bi1-xSbx with x = 0 % and 3 % is much reduced on the whole compared to M1, whose maximum is only 1/4 of that in M1. In contrast, for topological insulating samples becomes more enhanced overall, compared to the case of M1. The magnitude of the peak signal is two times larger than that in M1. In addition, the magnitude of ( ) , and the relaxation dynamics of the chiral mode is described by the Boltzmann equation, for an appropriate gauge choice, where τ is full width at half maximum of the laser pulse and ξ is a length scale related with the laser spot size.

Solving
the Boltzmann equation, we obtain , where is the density of surface electrons and is an inverse temperature. We point out that the sign difference appears in this expression due to the chirality. Inserting this distribution function into the Maxwell equation, we reach the following expression , where C is a positive constant that results from the velocity integration.
As this is a nonlinear equation, it is not easy to solve. A usual way is to linearize this equation, expanding the sinh-term up to the first order in the electric potential. Then, an asymptotic solution is given by limit. The first term represents an offset associated with the boundary condition, which is not important here. The second term describes the relaxation dynamics of voltage drop. A crucial point is that the relaxation time scale is purely given by the pulse width of the right hand side in the Maxwell equation. This is a crucial result for the 1D chiral fermion, directly originating from the absence of the dissipation term on the right hand side of the Boltzmann equation. It is also clear that the sign difference in the voltage drop results from the sign difference of the velocity of the chiral mode on each side.
In Fig. 6, we present the analysis of ( )  samples, we suspect that the chiral edge mode in Bi1-xSbx with x = 21.5 % is more strongly coupled to bulk electronic degrees of freedom than the case of Bi1-xSbx with x=11.6 %. We speculate that stronger couplings between the chiral edge mode and bulk degrees of freedom in Bi1-xSbx with x = 21.5 % may originate from larger localization length into the bulk from the edge than that of Bi1-xSbx with x = 11.6 %, implying that Bi1-xSbx with x=11.6 % has stronger topological properties than Bi1-xSbx with x = 21.5 %.
In Fig. 6(d), we analyse the scaled , where the exponential Boltzmann factor is "linearized". Although it is extremely difficult to solve this nonlinear differential equation, one can expect that the relaxation time becomes enlarged because of scattering processes, i.e., The inset of Fig. 6(d) implies three steps of relaxation processes, which correspond to three different time scales; τ estimated above, the oscillation period, and the period related with damping of this oscillation. Similar experiments on several n-and p-doped semiconductors without magnetic fields also confirmed existence of three steps of relaxation processes; the carrier generation and recombination, the diffusion of carriers, and the diffusion of thermal flux or phonons along the temperature gradient [36][37][38]. Even in the presence of external magnetic fields, these three processes will still be main mechanisms for the relaxation processes. The carrier mobility would be affected by the cyclotron motion under magnetic fields. We leave sincere analysis for Bi as an interesting future study. As the analysis in Fig. 6(d) is also equally well applied to Bi1-xSbx with x = 3 %, three relaxation processes are expected to exist in this sample. It is also noted that the peak signal at B = 4 T in the M1 experiment of Bi is much larger than that of other compositions. As the M1 experiment is a dynamic version of thermoelectric measurement, this peak signal is proportional to thermoelectric coefficient. This implies that Bi has larger thermoelectric coefficient than Bi1-xSbx with x = 3 %. Therefore, by accurately interpreting the time scales discussed above, the M1 experiment can provide important clues to understand and optimize thermoelectric performance of any thermoelectric material including Bi and BiSb alloy, particularly in the band "insulating" region.
Finally, we emphasize that the rather precise scaling behaviour from 0.4 T to 4 T implies the existence of only one single chiral-edge mode. This is consistent with the expectation that one single chiral-edge mode should exist on the surface state in the weak-field limit. On the other hands, strong magnetic fields are expected to lead to a series of Landau levels on the surface state. In fact, such Landau levels on the surface of 3D TI were directly measured in the dual-gated BiSbTeSe2 devices [11]. The