Discussion and Summary

Chapter 4: Electrical transport studies of Magnetic

4.4 Discussion and Summary

photo-carriers in the MTI, which were theoretically predicted to causes structural distortions that suppressed the FM coupling between neighboring Cr ions, leading to suppression in |Rxy| as observed experimentally. However, detailed studies of field sweeps at different temperatures both with and without light will be necessary to fully understand the interplay of photo-induced carrier densities and spin alignment by circularly polarized light.

massive Dirac fermions of MTIs, in contrast to the WAL behavior of massless Dirac fermions in the surface state of pure TIs [92]. Both (0+6)-10% and (5+6)-10% binary TI can be studied in the future to confirm the origin of the weak anti-localization and weak localization on binary TI.

Table 4.4 Summary of electrical transport results for ternary TI/MTI and MTI samples

Note: Measurements of the (0+6)-10% ternary MTI and (3+6)-10% ternary TI/MTI under circularly polarized light were made by Adrian Llanos.

On the other hand, ternary TI/MTI bilayer samples exhibited ferromagnetic hysteresis loops with negative coercive fields in the Rxy vs. B isotherms for 13 K < T < 25 K, whereas no hysteresis loops were found in the Rxx vs. B isotherms over this temperature range. Below 13 K, both Rxy vs. B and Rxx vs. B isotherms revealed standard ferromagnetic hysteresis loops.

Additionally, we found that the slope of d|Rxy|/d|B| in the Rxy vs. B isotherms changed signs around 13 K in the high-field limit where magnetization saturated, implying sign-change in the majority carriers around T = 13 K for (3+6)-10% ternary TI/MTI.

For bilayer (1+6)-10% ternary TI/MTI, Rxx vs. B only exhibited weak localization (WL) behavior, similar to the monolayer (0+6)-10% ternary MTI. Interestingly, WL of 6-QL monolayer ternary MTI was stronger than the WL behavior of (1+6)-10% ternary TI/MTI, consistent with the dominating bulk contributions from the gapped ternary MTI. In contrast, for (3+6)-10% ternary TI/MTI, the feature of WAL appeared at all temperatures while below 13 K, the WL feature showed up, implying the competition between WL and WAL behavior.

Further studies of the field sweeps for (1+6)-10% ternary TI/MTI at higher temperatures to see if WAL may appear when the surface gap closes will help correlate the physical origin of the WL/WAL competition to the surface/bulk states contributions.

The appearance of spontaneous magnetization with a preferred direction in zero fields implied broken symmetry between a positive and a negative magnetic field of the same magnitude below 𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}. This asymmetry was also observed when circularly polarized light was applied to the sample. We conjectured that the appearance of a preferred magnetization direction might be attributed to the asymmetric interfaces of the TI and MTI layers, leading to Rashba-like effects in these strong spin-orbit coupled materials.

One noteworthy point was that our transport measurements on different pieces of bilayer ternary TI/MTI samples often revealed slight differences. We attribute these slight differences found in the transport properties to slight variations in either the Fermi level or the electronic band structures of the ternary TIs. Specifically, we consider the evolution of the surface and bulk band structures of the ternary TIs as a function of temperature, as schematically shown in Figure 4.20 for the (3+6)-10% ternary TI/MTI bilayer sample, and in Figure 4.21 for the (1+6)-10% ternary TI/MTI bilayer sample.

The schematic band structures of (𝐵𝐵𝑖𝑖,𝑆𝑆𝑆𝑆)₂𝑇𝑇𝑒𝑒₃ shown in both Figure 4.20 and Figure 4.21 are based on the assumption that the bandstructure is between that of 𝐵𝐵𝑖𝑖2𝑇𝑇𝑒𝑒3 and 𝑆𝑆𝑆𝑆2𝑇𝑇𝑒𝑒3. If we further assume that for the (3+6)-10% ternary TI/MTI sample the Fermi level EF is located above the Dirac point and slightly below the bulk valance band (Figure 4.20), and that for the (1+6)-10% ternary TI/MTI sample the Fermi level EF is located below both the

surface state Dirac point and the bulk valance band (Figure 4.21), we are able to consistently account for all phenomena seen in our experiments.

Figure 4.20: Schematic plots for the evolution of band structures for a ternary TI (3+6)-10% at different temperature regions. Here the bandstructure of the surface state is represented by red lines, and that of the gapped bulk conduction and valence bands are represented by the blue line. (a) 𝑇𝑇>

𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}~25𝐾𝐾 where the surface state is gapless, with massless surface electrons being the majority carriers and bulk holes being the minor carriers. (b) 𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}>𝑇𝑇>𝑇𝑇_𝑥𝑥1~ 13 𝐾𝐾, where a small gap opens up so that the surface electrons become massive and bulk holes remain invariant. (c) 𝑇𝑇=𝑇𝑇_𝑥𝑥1, 𝑇𝑇_𝑥𝑥1 is the charge neutrality temperature where the electron and hole densities become comparable.

(d) 𝑇𝑇<𝑇𝑇_𝑥𝑥1 , where the bulk holes remain the majority carriers while the surface contribution decreases.

Let's first consider the (3+6)-10% ternary TI/MTI sample with a bandstructure shown in Figure 4.20. When 𝑇𝑇 >𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}~ 25 𝐾𝐾, the surface gap is closed so that the massless surface electrons are the majority carriers while the massive bulk holes are the minority carriers. For temperatures slightly below 𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}, a surface gap opens up due to the appearance of a finite magnetization. However, the surface gap is smaller than the indirect bulk gap, and the majority carriers are still surface electrons, although the electrons become massive. When 𝑇𝑇= 𝑇𝑇𝑥𝑥1 ~ 13 𝐾𝐾, the surface electron density becomes comparable to that of the bulk holes so that it is' a charge neutrality point. The hysteresis loop at this temperature, therefore, looks paramagnetic. At = 𝑇𝑇_𝑥𝑥2 ~ 7 𝐾𝐾, the bulk holes become the majority carriers, and the surface gap becomes comparable to the indirect bulk gap. For 𝑇𝑇< 𝑇𝑇_𝑥𝑥2 ~ 7 𝐾𝐾, the surface gap becomes larger than the indirect bulk gap so that the surface bands no longer contribute much

to the transport properties, and the bulk holes remain the majority carriers. In this limit, Rxx increases suddenly, and photo-induced excitations begin to affect Rxy.

For Rxx vs. B isotherms, the gapless surface electrons contribute to a V-shaped AWL behavior due to the prohibition of direct backscattering in the surface state of TIs. On the other hand, bulk hole carriers contribute to a W-shaped isotherm because of WL behavior in the low field limit and classical magnetoresistance in the high-field limit. At T = 13 K, both features show up in the Rxx vs. B curve. The competition between WL and WAL behaviors is consistent with our conjectures of two components of carriers. Thus, all experimental phenomena associated with the (3+6)-10% ternary TI/MTI sample can be consistently explained by the bandstructure shown in Figure 4.20.

Figure 4.21: Schematic plots for the evolution of band structures for a ternary TI (1+6)-10% at different temperature regions. Here the bandstructure of the surface state is represented by red lines, and that of the gapped bulk conduction and valence bands are represented by the blue line. (a) 𝑇𝑇>

𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}~30𝐾𝐾 where the surface state is gapless, with both surface massless holes bulk holes as carriers. (b) 𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}>𝑇𝑇>𝑇𝑇𝑥𝑥~ 10 𝐾𝐾, where a small gap opens up so that the surface holes become massive and bulk holes remain invariant. (c) 𝑇𝑇<𝑇𝑇𝑥𝑥~10𝐾𝐾, 𝑇𝑇𝑥𝑥1, the majority carriers are dominated by bulk holes while surface carriers diminish.

Next, we consider the (1+6)-10% ternary TI/MTI sample with a bandstructure shown in Figure 4.21. For 𝑇𝑇 >𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏} ~ 30 K, the surface gap is closed so that both the surface-state massless Dirac holes and bulk holes contribute to the sample conduction (Figure 4.21(a)).

For 𝑇𝑇𝑥𝑥~10 K <𝑇𝑇< 𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}, a small gap opens up in the surface state, and the corresponding carriers include both the surface-state massive Dirac holes and bulk holes (Figure 4.21(b)).

The coercive fields are very small due to small ferromagnetic domains, and even become slightly negative for 12 K < T ≤ 15 K, which may be attributed to the Rashba effect. Below Tx ~ 10 K, the surface gap becomes sufficiently large so that the dominant carriers are bulk holes (Figure 4.21(c)). The bulk ferromagnetism is well stabilized with large ferromagnetic domains so that the coercive fields increase rapidly with decreasing temperature. Moreover, for all temperatures, the carriers remain the same sign so that d|Rxy|/d|B| in the Rxy vs.

B isotherms of the (1+6)-10% ternary TI/MTI sample is always positive up to 1 T, as shown in Figure 4.7, which is in contrast to the sign-changing behavior found in the (3+6)-10%

ternary TI/MTI sample.

Figure 4.22: Schematic plots for the evolution of band structures for a binary TI (1+6)-10% at different temperature regions. Here the bandstructure of the surface state is represented by red lines, and that of the gapped bulk conduction and valence bands are represented by the blue line. (a) 𝑇𝑇>

𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}~30𝐾𝐾 where the surface state is gapless, the majority carriers are massless Dirac electrons.

(b) 𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}>𝑇𝑇>𝑇𝑇𝑥𝑥1, where a small gap opens up so that the surface electrons become massive but still majority carriers (c) 𝑇𝑇𝑥𝑥1>𝑇𝑇>𝑇𝑇𝑥𝑥2, the Fermi level is within the surface gap. Majority carriers are from thermally activated surface states.(d) 𝑇𝑇 <𝑇𝑇_𝑥𝑥2, the surface gap is greater than the indirect bulk gap. Majority carriers are from thermally activated bulk states.

The aforementioned discussions suggest that by accounting for the slight bandstructure variations in the ternary MTI systems as shown in Figure 4.20 and Figure 4.21, we are able to consistently explain varying experimental findings in both the (1+6)-10% ternary TI/MTI and the (3+6)-10% ternary TI/MTI samples. In this context, it is worth considering whether the bandstructure of binary MTIs may be responsible for the complete absence of hysteresis

loops in the Rxy-vs.-B and Rxx-vs.-B isotherms at 𝑇𝑇<𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}. As shown in Figure 4.22(a), the Dirac point of the surface state in Bi2Se3 is significantly far from all bulk conduction and valence bands, which is in contrast to the bandstructure of ternary MTIs. Therefore, for 𝑇𝑇 >

𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏} ~ 30 K and the Fermi level slightly above the Dirac point, the surface state is gapless, and the dominant carriers are surface-state massless Dirac electrons. For 𝑇𝑇< 𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}, a surface gap opens up and the majority carriers are massive Dirac electrons when the surface gap is small so that the Fermi level still intersects the top surface state (Figure 4.22(b)), thermally activated massive Dirac electrons and holes when the surface gap becomes sufficiently large so that the Fermi level falls within the surface gap (Figure 4.22(c)), and thermally activated bulk electrons and holes when the surface gap becomes larger than the indirect bulk gap (Figure 4.22(d)). The surface gap evolution depicted from Figures 4.22(b) to 4.22(d) at 𝑇𝑇< 𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏}can be realized by either decreasing the temperature in a constant field or by increasing the magnetic field at a constant temperature, and in the latter case the gapped surface state naturally leads to WL behavior in the Rxx-vs.-B isotherms for small fields, as shown in Figure 4.4. Overall, we conjecture that the absence of intrinsic bulk carriers in the ferromagnetic state of the binary MTI cannot support sufficiently large magnetic domains to yield discernible coercive fields below 𝑇𝑇_𝐶𝐶^{𝑏𝑏𝑢𝑢𝑣𝑣𝑏𝑏} so that no hysteresis loops could be resolved in either Rxy-vs.-B or Rxx-vs.-B isotherms. Thus, details of the bandstructure and the Fermi level location in the MTIs appear to play critical roles in determining the electrical transport properties of the MTIs.

In conclusion, our electrical transport studies of bilayer TI/MTI samples have revealed many interesting properties associated with the interplay of magnetism with the surface and bulk states of these topological materials. In particular, we note that the appearance of QAHE in ternary MTIs only at extremely low temperature (~ 30 mK) and the rapid decrease of the AHE with increasing temperature may be attributed to the finite contributions from bulk states because of the bandstructure effects (Figures 4.20 and 4.21) [93]. Promising enhancement of the AHE induced by circularly polarized light has also been observed.

Further exploration of photo-induced effects on controlling the magnetism of MTIs based on the findings derived from this work will likely yield exciting new results.