Research ArticleQuantum Mechanics

Large discrete jumps observed in the transition between Chern states in a ferromagnetic topological insulator

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Science Advances  29 Jul 2016:
Vol. 2, no. 7, e1600167
DOI: 10.1126/sciadv.1600167
  • Fig. 1 Quantization of the Hall resistance in a 10-nm-thick film of Cr-doped (Bi,Sb)2Te3 at millikelvin temperatures.

    (A) Hall plateaus in the Hall resistance Ryx at 10 mK (blue curves). The transition between Chern states at the coercive field Hc (~0.14 T) is nearly vertical. The longitudinal resistance Rxx (red curves) displays ultralow dissipation, apart from the sharp peaks at Hc. (B) Testing of the stability of the Chern state Ryx = h/e2 over an extended period (8 to 9 hours), with H fixed at 0 starting at the time indicated by vertical arrows. The two traces at 49 mK (gray trace) and 72 mK (green) are shown displaced for clarity. (C) The expanded scale shows that Ryx deviates from h/e2 by a few parts in 104 in the optimal gating window 110 < Vg < −90. (D) The expanded scale shows that Rxx drops to <3 × 10−4 h/e2 near H = 0. Rxx is thermally activated with a gap ΔR ~ 190 mK.

  • Fig. 2 The dependences of Ryx and Rxx on gate voltage Vg at T = 10 mK.

    (A) Traces of Ryx versus H at selected Vg from 0 to −120 V. The negative slopes at large H show that the bulk carriers that appear when |Vg| < 40 V are n-type. (B) Plot of Rxx for the same values of Vg in semilog scale. The small anomalies at H = ±10 mT are caused by the quenching of superconductivity in the In contacts. The coercive field Hc determined by the sharp peaks in Rxx is nearly independent of Vg.

  • Fig. 3 Jumps in Ryx at the transition between Chern states.

    (A) Traces of Ryx at selected T (10→70 mK) as H is slowly swept (10 mT/min) past Hc, as indicated by the arrows. The transition between Chern states C = ±1 is nominally smooth below 40 mK. Starting near T1 ~ 65 mK, small vertical jumps appear near the start of the transition. (B) Ryx versus H in the important interval 70 mK ≤ T ≤ 145 mK, in which large jumps are clustered. Starting at 83 mK, the system “escapes” the C = 1 state by a very large initial jump, followed by a cascade of smaller ones. Focusing on the initial jump, we see that field HJ(T) that triggers the jump (kink feature) increases steadily as T rises to 142 mK. Above 131 mK, the jump magnitude ΔRyx sharply decreases, becoming unresolved above T2 ~ 145 mK. (C) From T2 to 580 mK, Ryx changes smoothly over the (now broadened) transition. (D) Schematic of how the Ryx curves change with Vg at 10 mK. At optimal gating (Vg = −90 V), the transition is smooth, but as Vg is increased to −10 V, the jumps reappear, implying that a small amount of dissipation is necessary to seed the jump at the lowest T. The experimental time constants are 1 s in all panels, except in (D), where the constant is 5 s.

  • Fig. 4 Profiles of resistance Rxx versus H at selected T as H is slowly swept at 1 mT/min (arrows).

    (A) Below 50 mK, the profile is nominally smooth and asymmetric, whereas small vertical jumps appear near the leading edge as T rises above T1. (B) In the interval 83→216 mK, the smooth profile is dissected by large jumps ΔRxx that are positive (upward) near the leading edge and negative near the trailing edge. Above T2, the profile is again smooth but twice as broad as at 10 mK. (C) We plot the distribution of the initial jump magnitude ΔRyx versus T within the optimal gating window. Above T1, ΔRyx initially rises to a value of ~h/e2 and then falls steeply to 0 at T2. Jumps are not resolved above T2. The inset is a sketch of the system (red circle) trapped in a potential U(M). (D) Escape field HJ(T) plot of the initial jump versus T (solid symbols) inferred from Ryx and Rxx from both sweep-up and sweep-down runs. Below T1, HJ(T) terminates at the curve of Hs(T) (the onset field for small, precursor steps plotted as open triangles), which shares the same slope as the coercive field Hc(T) (open circles). Hc is defined by the maximum in Rxx (when the curve is smooth). Hc is undefined between 100 mK and T2, but it smoothly extends to measurements above T2.

  • Fig. 5 The jumps in Ryx (top) and Rxx (bottom) detected by simultaneous measurements of R14,mn, as labeled [I applied to contacts (1, 4) and voltage measured across (m, n)].

    The field sweep rate is 1 T/min. (A and B) At both 105 mK (A) and 120 mK (B), the initial jump strongly affects the downstream Hall signal (R14,53) but barely changes the upstream pair (R14,62). However, at the second jump, the upstream pair is strongly perturbed. The inset in (A) shows the contacts and the wall separating Chern state C = 1 (green) from +1 (yellow) after the first jump occurs. (C and D) In the resistance (bottom row), the first jump increases R14,23 but decreases R14,65 at both 105 mK (C) and 120 mK (D). The signs in both pairs become reversed at the second jump, also consistent with the upstream motion of the wall.

  • Fig. 6 Four traces of the resistances R14,mn measured in succession at 86 mK with Vg = −80 V.

    For each trace, the sample is prepared in the C = −1 state, and the transition is induced by sweeping H slowly through Hc at a rate of 1 mT/min while monitoring the resistances R14,mn. (A) Upstream Hall resistance R14,26 (light curves) and downstream Hall resistance R14,53 (bold). (B) Corresponding longitudinal left contact resistance R14,65 (bold curves) and right contact resistance R14,23 (light) plots. Although the jumps are stochastic, the overall pattern for the first two jumps is fairly reproducible from one run to the next.

  • Fig. 7 Time trace of a jump event observed in Rxx measured with the room temperature filters in the circuit removed (but not the low-temperature filters) at T = 117 mK and Vg = −85 V.

    The jump occurs on a time scale Δt that is shorter than 1 ms, the time constant of the remaining filter. The field sweep rate was 0.1 mT/min. a.u., arbitrary units.

  • Fig. 8 Gate voltage dependence of Rxx and Ryx and the activation gap at different magnetic fields.

    (A) Semilog plot of Ryx and Rxx as a function of Vg, with T fixed at 10 mK and H fixed at either 0 T (solid triangles) or 1 T (solid circles). As |Vg| decreases, the exponential increase in Rxx (blue and black symbols) implies that μ is being raised above the minigap ΔR. Comparison between Rxx at H = 0 T (blue) and 1 T (black) shows that a finite H decreases ΔR. Similarly, the curves for Ryx (red and magenta) show an exponential deviation from the quantum value h/e2 as |Vg| is decreased. The Hall curves persist longer at the quantized value than do the resistance curves. (B) The semilog plot of σxx versus 1/T with Vg = −80 V reveals an activated conductivity at both H= 0.5 T and 33 mT. The straight line fits yield values of ΔR ~190 mK (at H ≈ 0) and 155 mK (0.5 T).

  • Fig. 9 Crossover boundaries in the (Vg, T) plane (A) and scaling plot of σxx versus σxx (B).

    (A) The optimal gating window lies left of the vertical dashed line (−120 V < Vg < −80 V). The resistance gap ΔR, which protects the ultralow dissipation (ULD) state, is equal to ~190 mK within this window but falls rapidly outside the window. In the shaded region above ΔR, the system is dissipative (darker shade indicates larger Rxx). Jumps in Rxx and Ryx are observed between the two characteristic temperatures T1 and T2. The curve of T1 decreases steeply outside the window. In the ultralow dissipation region below T1, jumps are not observed on our experimental time scales. (B) The scaling plot describes a nearly ideal semicircle above T2 (curves at 142 and 152 mK). Within the interval (T1, T2), the appearance of jumps kicks the orbits high above the semicircle. This implies violation of the two-parameter scaling behavior (38).

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/7/e1600167/DC1

    section S1. Sample.

    section S2. Jumps in negative field.

    section S3. Tunneling out of metastable state.

    section S4. One-wall model.

    section S5. Multiple domains and the network model.

    fig. S1. Photo of the Hall bar used in the experiment.

    fig. S2. Curves of Ryx versus H for the transition between Chern states induced by sweeping H to negative values (arrow), with Vg fixed at −80 V.

    fig. S3. Field profiles of Rxx as H is decreased through −Hc at temperatures 10 to 83 mK (A) and 83 to 152 mK (B).

    fig. S4. The effect of increasing T on the curves of Ryx (A) and Rxx (B), with Vg fixed at −120 V (upper edge of the optimal gating window).

    fig. S5. Sketch of the device with a domain wall bisecting the film.

  • Supplementary Materials

    This PDF file includes:

    • section S1. Sample.
    • section S2. Jumps in negative field.
    • section S3. Tunneling out of metastable state.
    • section S4. One-wall model.
    • section S5. Multiple domains and the network model.
    • fig. S1. Photo of the Hall bar used in the experiment.
    • fig. S2. Curves of Ryx versus H for the transition between Chern states induced by sweeping H to negative values (arrow), with Vg fixed at −80 V.
    • fig. S3. Field profiles of Rxx as H is decreased through −Hc at temperatures 10 to 83 mK (A) and 83 to 152 mK (B).
    • fig. S4. The effect of increasing T on the curves of Ryx (A) and Rxx (B), with Vg fixed at −120 V (upper edge of the optimal gating window).
    • fig. S5. Sketch of the device with a domain wall bisecting the film.

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