In a ferromagnetic (FM) metal or semiconductor, the spontaneous magnetization of the sample generates an anomalous contribution to the transverse channel of transport phenomena. The most well-known example is the anomalous Hall effect (AHE), in which an extra Hall voltage proportional to the magnetization is present even in zero magnetic field [1, 2]. The AHE has attracted tremendous attention in the past few decades due to its complex origin and potential applications in spintronics. Three distinct mechanisms have been proposed for the AHE. The two extrinsic ones are related to impurities, which cause side jumps or skew scatterings of itinerant electrons by spin–orbit coupling [2–5]. The intrinsic mechanism ascribes the AHE to the Berry-phase curvature of the occupied electronic states, which reveals the consequence of momentum space topology on transport properties [2, 6–8]. In general the three mechanisms take effect simultaneously in a given material, making it difficult to clarify their individual contributions [9]. Recently, the quantum AHE (QAHE) has been observed in a magnetic topological insulator (TI), in which a quantized Hall resistance Ryx=h/e2 exists even in zero magnetic field [10–13]. This finding not only completed the decade-long search for the last member of the quantum Hall trio [14], but also unambiguously proved the intrinsic mechanism of the AHE due to the Berry-phase curvature [15].
Besides the anomalous electrical transport studied in magnetic TIs, another type of magnetism-induced transport phenomenon, which is much less explored but equally interesting, is the anomalous thermoelectric or thermomagnetic effect. In a conductive material, a temperature gradient in x direction 
generates a longitudinal electric field (Ex). The ratio 
is called the thermopower or the Seebeck coefficient. In a perpendicular magnetic field Bz, the moving charge carriers experience a Lorentz force perpendicular to their velocity and Bz, hence generating a y-direction electric field Ey. This transverse thermoelectric effect is called the Nernst effect [16, 17], and an anomalous Nernst effect (ANE), namely the thermoelectric counterpart of the AHE, is available in a FM material [9, 18]. The ANE offers a great platform for exploring the intriguing physics of spin–orbit coupling. The unique electronic structures of a FM TI, such as the nontrivial bulk band structure, the ambi-polar Dirac-like surface states and the chiral edge states [19–21] may also bring exotic phenomena in ANE, thus creating novel applications.
In this Letter, we report electromagnetic and thermomagnetic transport studies of a magnetic TI thin film, Cr0.15(Bi0.1Sb0.9)1.85Te3. We found that the temperature and gate voltage dependence of the AHE exhibits the typical behavior of a QAHE insulator. The ANE, however, shows a sign reversal when the Fermi level (EF) is tuned across the charge neutrality point (CNP) of the surface Dirac cone. We show that the ambi-polar behavior of the ANE can be explained by the semiclassical Mott relation, in conjunction with the ambi-polar Dirac band structure.
The Cr0.15(Bi0.1Sb0.9)1.85Te3 TI thin film with thickness of 6 quintuple layer (QL) is grown on insulating SrTiO3(111) substrate by using state-of-the-art molecular beam epitaxy. The sample growth condition is the same as that for the magnetic TI film in which the QAHE was first discovered [10]. The schematic setup of the transport measurements is shown in figure 1(a), in which the electromagnetic and thermomagnetic properties can be measured on the same sample. The magnetic TI thin film is manually scratched into a Hall-bar configuration and the electrodes are made by mechanically pressing small pieces of indium ingots onto the TI film, and then connected to gold leads. One end of the substrate is connected to a copper heat sink and a thin-film heater is mounted on the other end to produce a steady and uniform temperature gradient through the TI film. Two fine-gauge thermocouples (type E, CHROMEGA/constantan) are connected in subtractive series and thermally anchored to the substrate to monitor the temperature difference across the sample. The carrier density of the magnetic TI thin film is controlled by the applied gate voltage with the SrTiO3 substrate acting as the gate dielectrics. The direction of the applied magnetic field is perpendicular to the film plane. The electromagnetic properties are measured in an isothermal condition using the delta mode of the Keithley 6221 current source plus the 2182 A nanovoltmeter. The thermomagnetic measurements are carried out in high-vacuum condition with the pressure lower than 1×10−6 mbar. The dc voltages of the Nernst signals are recorded by Keithley 2182 A nanovolmeters.
We first measure the AHE of the 6 QL Cr0.15(Bi0.1Sb0.9)1.85Te3 thin film under various gate voltages (Vg) at the base temperature T=1.5 K. As shown in figure 1(b), the magnetic field (μ0H) dependences of the anomalous Hall resistance show square-shaped hysteresis loops with a coercive field (Hc) around 0.1 T. With increasing Vg, the anomalous Hall resistance first increases, reaches a maximum value near Vg=70 V, and then decreases again. Figure 1(c) displays the Vg dependence of 
(the Hall resistance at μ0H=0 T), which reaches a maximum value of 17.4 kΩ at Vg=70 V. This gate voltage thus corresponds to the situation where the EF lies at the CNP within the energy gap at the surface-state Dirac cone opened by the FM order [10]. Both the Vg dependence and the maximum 
value are in excellent agreement with that of the original QAHE sample measured at T=1.5 K (figure 4(A) in [10]). These results indicate that the magnetic TI film studied here has high quality and can achieve the QAHE at lower temperature.
To investigate the anomalous thermomagnetic transport, we measure the Nernst signal (defined as 
) in this magnetic TI. Figures 2(a) and (b) display the Hall effect and the Nernst effect traces measured at different temperatures from 5 to 40 K, respectively. At T=40 K, both Syx and ρyx traces are linear to magnetic field, which is the typical behavior of the ordinary transverse transport of charge carriers. Pronounced curvature develops at T=30 K due to the magnetic field alignment of the local moments, which is characteristic of a paramagnetic metal [22]. For T<15 K, the sample enters the FM phase and a square-shaped hysteresis loop starts to appear. Both transport effects exhibit the same direction of the hysteresis loop (anti-clockwise loop as indicated by the magenta arrows), and nearly identical coercivity at each temperature. The strong resemblance between the temperature dependence of the AHE and ANE in this magnetic TI suggests that the two anomalous transport effects share the same physical origin.
In contrast to the similar temperature dependence, the gate voltage dependence of the two anomalous transport effects reveal something very puzzling. Figures 3(a) and (b) display the magnetic field traces of ρyx and Syx measured under various Vg at T=5 K. The CNP corresponds to Vg between 55 and 70 V, when the anomalous ρyx reaches the maximum level. For Vg=−20, 0 and 20 V, EF lies below the Dirac point so that the hole-type surface state carriers dominate the transport process. In this regime the two anomalous transport effects are highly analogous, both showing the hysteretic behavior with the same direction, and the gradual increase of anomalous signal with Vg. With further increase of Vg towards the CNP, the anomalous ρyx keeps increasing, whereas the anomalous Syx starts to decrease. For Vg>70 V, the EF lies above the Dirac point, so that the electron-type surface state carriers dominate the transport process, and the anomalous ρyx starts to decrease with increasing Vg. In this regime the anomalous Nernst signal first decreases to almost zero at Vg=100 V, and more surprisingly, above that the anomalous Syx changes its sign from positive to negative. The direction of the ANE loops changes from anticlockwise to clockwise (as indicated by the magenta and blue arrows, respectively). Therefore, the ANE of the magnetic TI exhibit a highly unusual ambi-polar behavior when the EF is tuned across the CNP of the surface Dirac cone.
The sharp contrast between the Vg dependence of AHE and ANE reveals that in the magnetic TI, the anomalous transport responses to an electric field and a temperature gradient are considerably different. Below we will show that the Mott relation, in conjunction with the ambi-polar electronic band structure of the topological surface states, give a natural explanation for the ambi-polar ANE observed here in the QAHE regime.
The Mott relation was originally derived from the Boltzmann equation for diffusive transport coefficients, in which the thermopower can be expressed as 
where e, kB, and σ are the electron charge, the Boltzmann constant and electrical conductivity, respectively [23]. The basic idea is that the thermopower is proportional to the energy derivative of the electrical conductivity evaluated at EF. In the presence of a magnetic field, S and σ have the tensor form including both x and y components. In this case the Nernst signal Syx can be expressed as
In our experiment the position of EF can be tuned by Vg, therefore the energy derivative of the conductivity with respect to EF can be calculated as 
From the experimentally measured ρyx and ρxx values at varied gate voltages, we can calculate the Vg dependent σxx and σxy, as summarized in figure 4(a) for μ0H=0.2 T and T=5 K. Moreover, 
can be estimated by using a simple Dirac-like surface band structure with linear E–k dispersion (the contribution of the bulk carriers and the small gap of the surface band opened at the Dirac point by the magnetism can be ignored). For a TI thin film, 
where vF is the Dirac Fermion velocity and is ∼105 m s−1 in this system [24]. Meanwhile, n2D can be related to Vg as: 
where Cg and VD are the capacitance per unit area and the gate voltage at the Dirac point. Cg can be calculated as 
where 
and 
are the dielectric constant and the thickness of the SrTiO3 substrate. Here, 
is about 0.25 mm in our experiment and 
is estimated from [25, 26]. Therefore, we obtain the relation between EF and Vg: 
and thus 
Substituting it into equation (1), we obtain the anomalous Syx value calculated from the Mott relation, 
which is plotted as blue solid line in figure 4(b). As can be seen clearly, 
agrees well with the measured Syx at μ0H=0.2 T and T=5 K (the red solid symbols). Most importantly, the anomalous Syx value changes sign from positive to negative with increasing Vg, consistent with the sign reversal observed experimentally. Note that the Mott relation only works for continuous σ(E) at low temperatures. The small quantitative deviation between theory and experiment may be caused by the discontinuity of σ(E) around band edges and the contribution from the bulk carriers at finite temperature, which also causes a reduction of AHE from the quantized level [10].
In conclusion, we observe a sign reversal of the ANE in a magnetic TI when the Fermi level is tuned across the CNP by an external gate voltage. We show that this ambi-polar behavior of the ANE can be explained semi-quantitatively by using the Mott relation between electrical and thermal transport coefficients, in conjunction with the ambi-polar Dirac-like band structure of the topological surface states, thus is unique to magnetic TIs. These results shed important new lights on the anomalous thermomagnetic transport properties and potential applications of magnetic TIs.
This work was supported by the National Natural Science Foundation of China and the Ministry of Science and Technology of China. YX acknowledges support from Tsinghua University Initiative Scientific Research Program and the National Thousand-Young-Talents Program.
