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

Abstract

A striking prediction in topological insulators is the appearance of the quantized Hall resistance when the surface states are magnetized. The surface Dirac states become gapped everywhere on the surface, but chiral edge states remain on the edges. In an applied current, the edge states produce a quantized Hall resistance that equals the Chern number C = ±1 (in natural units), even in zero magnetic field. This quantum anomalous Hall effect was observed by Chang et al. With reversal of the magnetic field, the system is trapped in a metastable state because of magnetic anisotropy. We investigate how the system escapes the metastable state at low temperatures (10 to 200 mK). When the dissipation (measured by the longitudinal resistance) is ultralow, we find that the system escapes by making a few very rapid transitions, as detected by large jumps in the Hall and longitudinal resistances. Using the field at which the initial jump occurs to estimate the escape rate, we find that raising the temperature strongly suppresses the rate. From a detailed map of the resistance versus gate voltage and temperature, we show that dissipation strongly affects the escape rate. We compare the observations with dissipative quantum tunneling predictions. In the ultralow dissipation regime, two temperature scales (T1 ~ 70 mK and T2 ~ 145 mK) exist, between which jumps can be observed. The jumps display a spatial correlation that extends over a large fraction of the sample.

Keywords
  • Topological insulators
  • quantum anomalous hall effect
  • metastability
  • dissipative quantum tunneling
  • chern number
  • chiral edge states
  • dissipationless transport
  • quantization of hall resistance

INTRODUCTION

In the semiconductors known as topological insulators (exemplified by Bi2Se3, Bi2Te3, and Bi2Te2Se) (14), strong spin-orbit interaction, together with band inversion, leads to electronic surface states that display a Dirac-like linear dispersion. On each surface, the Dirac states have only one spin degree of freedom, with the spin locked transverse to the momentum. The node of the Dirac cone is protected against gap formation if time-reversal symmetry (TRS) prevails (this constrains the nodes to be pinned at time-reversal invariant momenta).

The unusual properties of the surface Dirac states have been intensively investigated by angle-resolved photoemission spectroscopy, scanning tunneling microscopy, and transport experiments. A natural question is what happens when TRS is broken by inducing a magnetization at the surface? Theory posits that breaking of TRS opens a gap at the Dirac node everywhere on the surface (involving the surface “bulk” state as opposed to the edge states) (14). However, a conducting edge state remains, which runs around the perimeter of the sample. The edge state is chiral (it is either right- or left-moving depending on the magnetization vector) and dissipationless (backscattering of the electrons is prohibited). As in the quantum Hall effect (QHE), the edge state displays a Hall resistance that is rigorously quantized (Ryx = h/e2, where h is Planck’s constant and e is the electron charge), whereas the longitudinal resistance Rxx vanishes (15). However, the quantization of Ryx occurs even if the external magnetic field H is 0. This phenomenon is known as the quantum anomalous Hall (QAH) effect. The quantization of Ryx in zero field was first established experimentally by Chang et al. (6) using ultrathin films of Cr-doped (Bi,Sb)2Te3. Similar results in related topological insulator films were subsequently reported by several groups (713). The topological nature of the Hall quantization produces a hysteretic loop in Ryx that reflects the magnetic hysteresis when H (the applied magnetic field) is slowly cycled beyond the coercive field Hc. If the system is prepared with magnetization M || H || −z, the chiral edge modes lead to Ryx = −h/e2 = −25.812 kilohms. This state is characterized by a Chern number C = −1. As H changes sign, the system becomes metastable; to exit the metastable state, it undergoes a very sharp transition to the state with C = 1 and Ryx = +h/e2. Here, we focus on the nature of this transition at millikelvin temperatures T. In conventional magnets, the hysteretic transition reflects the gradual diffusive motion of domain walls. Surprisingly, the transition actually proceeds by large, discrete jumps in Ryx. We find that slightly increasing T suppresses the jump probability. A detailed investigation of how the jump probability varies with H and T suggests that the jumps reflect quantum tunneling events in the presence of dissipation.

RESULTS

Characterization and benchmarks

In the experiment, we used samples of Cr-doped (Bi,Sb)2Te3 grown on a SrTiO3 substrate by molecular beam epitaxy (MBE) to a thickness of 10 nm and cut into Hall bars (0.5 mm × 1 mm) [see Materials and Methods and the study of Kandala et al. (11)]. As grown, the surface states are n-doped, with the chemical potential lying high above the surface Dirac node. A large, negative back-gate voltage of Vg = −80 V is required to lower μ to the surface Dirac node. Within the optimal gate window −120 V < Vg < −80 V (the limits are nominal; they change by ±10% in successive experiments), the Hall resistance Ryx attains near-ideal quantization at a temperature of T = 10 mK. Figure 1A shows that, at Hc ~ 0.14 T, Ryx undergoes narrow transitions between −1 and +1, with narrow peaks in the resistance Rxx of the width ΔH of ~20 mT (hereafter, we quote resistances in units of h/e2).

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.

An important question is the stability of the quantized values Ryx = ±1. If the magnetization is actually not spontaneous, Ryx should relax away from the quantized values when H is returned to 0, whereas a truly magnetized ground state with C = ±1 should display no sign of relaxation. To test this, we prepared the state in positive field H || z. We then returned H → 0 and monitored Ryx for a period of 8 to 9 hours. As shown in Fig. 1B, we do not detect any sign of relaxation at 49 and 72 mK for Vg = −80 V. To us, this shows that, at the Hall plateaus, the system is in the true ground state with C = ±1. Moreover, it remains stable in the limit H → 0 for at least 8 to 9 hours.

The expanded scale in Fig. 1C shows that Ryx deviates from 1.0000 by about four parts in 104 within the optimal gating window. Figure 1D shows in expanded scale the behavior of Rxx away from the peaks. Significantly, Rxx at H = 0 falls to values <3 × 10−4 at 10 mK, consistent with ultralow dissipation. These benchmarks are comparable to the best achieved to date (9, 10).

To provide a broader view of the transport behavior at 10 mK, we show the curves of Ryx and Rxx versus H in Fig. 2 , with Vg set at values from 0 to −120 V. The Hall curves Ryx (Fig. 2A) remain very close to 1.0000 in the optimal gate window but start to fall monotonically as |Vg| decreases, reaching 0.03 when Vg = 0 (surface states strongly n-doped). Throughout the gate interval, −70 to 0 V, the sign of the n-type carriers imparts a gentle slope to Ryx for fields |H| > Hc. Turning to the longitudinal resistance, we show in semilog scale the curves of Rxx versus H for the same gate voltage values (Fig. 2B). As noted above, Rxx falls to values as low as 3 × 10−4 (in terms of h/e2) within the field window |H| < Hc (see curve at Vg = −90 V). The small narrow spikes at H = ±10 mT are caused by the quenching of superconductivity in the indium contacts (in an experiment in which indium is replaced by gold in all contacts, we find that the spikes are completely absent). When |H| is increased beyond Hc, the dissipation increases exponentially. As we tune Vg outside the optimal gate window, the curves of Rxx also increase exponentially, eventually reaching ~0.2 when Vg approaches 0. The pattern of an exponential increase in dissipation when either Vg is tuned outside its optimal window or |H| is increased above the coercive field implies the existence of a small energy gap that “protects” the dissipationless state with accurately quantized Ryx. We estimate the activation gap below (see section Activation across a small gap).

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.

Jumps in Rxx and Ryx

The main results (the observation of large jumps in Rxx and Ryx) are described in this subsection. At each T, we prepare the system with Chern number C = −1 (H || −z). To investigate the transition C: −1 → 1, we scan H very slowly (1 to 10 mT/min) across Hc (left to right in all figures). Starting at our lowest T (10 to 50 mK), Ryx displays a nominally smooth profile as it changes from −1 to 1 (Fig. 3A). The smooth variation, which has a profile that is T-independent in width up to 580 mK, is likely associated with conventional domain wall motion.

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.

In the interval of 50 to 65 mK, small precursor steps become discernible above an onset field Hs. At the characteristic temperature T1 of ~65 mK, we observe the appearance of remarkably large jumps with magnitudes of ΔRyx ~ 1. Within the interval T1T2 = 145 mK, the jumps appear in a stochastic yet systematic pattern, as shown in Fig. 3B. Their occurrence is independent of the adopted sweep rate (1 to 10 mT/min). We focus on the first jump, which is always the largest (in over 50 scans recorded). As T rises above T1, its magnitude ΔRyx initially increases (curves at 83→131 mK). Above 138 mK, ΔRyx rapidly diminishes to become unresolved. As shown in Fig. 3C, all curves above T2 are smooth and devoid of jumps. Although jumps are not observed at 10 mK in the ultralow dissipation state, they reappear when we tune Vg to exit the optimal window. In Fig. 3D, Ryx at −90 V is smooth, but jumps appear when Vg is raised above −40 V. The jumps are reproducibly observed even after repeatedly warming to 300 K. They are distinct from magnetization jumps observed in manganites below 4.7 K (when charge-ordered insulating regions coexist with ferromagnetic regions) (14). The jumps in manganites appear at characteristic fields that are nearly T-independent, although they are strongly sensitive to field-cooling conditions. The jumps here are qualitatively different.

The jumps are also prominent in the dissipative channel (curves of Rxx). At low T (10 to 50 mK), Rxx exhibits a nominally smooth, asymmetric profile (Fig. 4A). Above T1, large jumps appear at the leading (left) edge, whereas the trailing edge remains smooth, as the width increases to 38 mT. Figure 4B follows these curves above T2 to 216 mK.

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.

As noted for Ryx, the escape field HJ(T) of the initial jump increases with T. However, in the window 125 to 138 mK, the jumps now have opposite signs (positive at the leading edge). Jumps observed in Ryx and Rxx are closely similar when H is swept in the opposite direction (figs. S2 and S3).

Figure 4 (C and D) shows how the two important quantities (the jump magnitude and the initial field) vary with T. As noted, the jump magnitude ΔRyx is initially small within the precursor interval but increases rapidly at T1 to saturate at a plateau value of ~1 (Fig. 4C). As T is further raised above T2, ΔRyx collapses to a value below our resolution. Above T2 (up to 500 mK), the Hall curves vary smoothly throughout the hysteresis loop. In contrast to the nonmonotonicity of ΔRyx(T), the escape field HJ(T) increases steeply from T1 to T2 (solid symbols in Fig. 4D). It is interesting that HJ is still rising with steepening slope when the jump magnitude falls below resolution above T2. For comparison, we also plot the gentle decrease of Hc and Hs (open symbols) over the same interval of T (obtained from curves in fig. S4).

Spatial correlations

An interesting way to study the jumps is provided by simultaneous measurements of Rij across well-separated pairs of voltage contacts. The resistance Rij,mn is defined as Vmn/Iij, where Vmn is the voltage measured across contacts (m, n) with the current Iij injected at the source contact i and drained at j (see inset, Fig. 5A). A typical pattern is that a jump in Ryx measured across one pair of contacts barely affects the signal across a different pair, but at a slightly larger H, the reverse holds.

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.

Figure 5 (A and B) shows traces of R14,53 and R14,62 versus H at T = 105 and 120 mK, respectively. In both panels, the initial jump (at H = 0.129 T) is large across the downstream Hall pair (3, 5), whereas the upstream Hall pair (2, 6) is barely affected. At a larger H (0.133 T), the second jump is prominent in the upstream pair but much smaller in the downstream pair. Simultaneously, the resistance R14,23 in the right pair (2, 3) first increases and then decreases, whereas in the left pair (5, 6), R14,65 shows the opposite sequence (Fig. 5, C and D).

Despite the stochastic nature of the jumps, the overall pattern is systematic and reproducible, especially for the first two jumps in a trace. In Fig. 6, we plot four successive scans of the traces taken successively, with T fixed at 86 mK and Vg = −80 V. The sample is prepared in the same way in each scan, with the sweep rate fixed at 1 mT/min. Figure 6A reports the Hall resistances (light curves for the upstream pair and bold curves for the downstream pair). In Fig. 6B, we show the corresponding longitudinal resistance curves (bold curves for the left pair and light curves for the right pair). The first two jumps are reproducible, although the jump magnitudes and HJ can vary by 10%. The variations from one jump to the next shown here are typical of the reproducibility across the whole temperature range investigated (10 to 580 mK).

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.

How fast are the jumps? To minimize extraneous perturbations, all the traces shown above were taken at very slow field sweep rates (typically 1 mT/min). We have attempted to estimate the duration of the first jump Δt by minimizing the capacitive loading in the measurement circuitry in the dilution refrigerator. To this end, we sequentially removed several stages of the filter elements in the circuit (which increased the noise in the traces) while slowing the field scan to 0.1 mT/min. The final trace in this process is measured with only the low-temperature filter present. The noisy trace (Fig. 7) shows that Δt is shorter than 1 ms, the time constant of the remaining filter.

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.

Activation across a small gap

Extending the measurements over a broad range of Vg and T, we find that the ultralow dissipation state is protected by a minigap ΔR ~ 190 mK. Setting Vg within the optimal gating window (−110 to −75 V) places μ inside ΔR (these numbers can shift by 10% between runs because of charge trapping at the interface). Once μ exits the minigap, dissipation rises exponentially.

Figure 8A shows the changes in Ryx and Rxx as Vg is varied from −120 to 0 V , with T fixed at 10 mK and H fixed at ~0 T (solid triangles) or 1 T (solid circles). The Hall curves (red and magenta symbols) show that, as |Vg| is decreased, Ryx stays close to the quantum value h/e2 (the optimal gating window) until |Vg| decreases below 70 V in zero field. Thereafter, Ryx decreases exponentially to 0.03 as Vg→0. [The lower limit of the optimal window (−70 V in Fig. 8A) varies from run to run because of hystereses caused by charge trapping in the SrTiO3 substrate.] The resistance Rxx at H = 0 (blue triangles) increases exponentially with a lower threshold than Ryx. A finite field (1 T; black circles) lowers the threshold significantly. We identify the exponential changes in both Ryx and Rxx with raising of μ above the upper limit of the gap ΔR by gating.

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).

The magnitude of the minigap ΔR is determined by detailed measurement of the conductance σxx versus T. From the semilog plot of σxx versus 1/T in Fig. 8B, we infer that ΔR ~ 190 mK for H = 33 mT (black squares). At 0.5 T, the equivalent plot gives a slightly smaller ΔR (155 mK), consistent with the steeper increase in the curve with black symbols in Fig. 8A. Both plots imply that the minigap decreases noticeably with increasing H. This is also consistent with the gradual increase of the background dissipation discussed in conjunction with Fig. 2B.

The changes to the transport curves induced by varying either Vg or T provide firm evidence that the ultralow dissipation state in the QAH state is protected by a minigap ΔR ~190 mK. An exponential increase in the dissipation is observed when T is increased or when we move μ out of the optimal window by gating. The size of the gap is also decreased by increasing H beyond 0.5 T. The gradual increase of the background with H, together with the negative slope of Ryx outside the coercive field window, suggests that the bulk carrier population (n-type) is gradually increased by H when μ falls outside the optimal window. To summarize the results, we schematically plot the behavior of the curves ΔR, T1, and T2 in the Vg-T plane (Fig. 9).

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).

DISCUSSION

Metastability and the effect of T

The reversal of H from the direction in which the system is prepared (H || −z) traps the system in a metastable state (spin ↓) until escape occurs near the coercive field. We investigated the nature of the transition in the magnetic topological insulator at millikelvin temperatures. In a conventional ferromagnet, escape from the metastable state corresponds to gradual motion of domain walls separating up and down domains, which allows the spin-↑ domain to expand. This leads to the familiar smooth hysteretic M-H loop. However, here, the transition that involves both a change of the Chern number (C = −1→1) and the average M displays a rich array of additional features, most notably the large jumps in Ryx and Rxx in the interval T1 < T < T2 (Figs. 3 and 4). The jumps occur on time scales shorter than 1 ms. The first jump involves reversal of M over roughly half the sample. (As mentioned, we identify the smooth T-independent variation of Ryx above T2 with conventional domain wall motion. Below T2, the smooth variation persists as a background channel between the jumps.)

In two recent investigations (12, 13) of the transition C = −1→1, the variation of Ryx is much broader than the ones observed here. Instead of jumps in Ryx, a shoulder at C = 0 (zero-Hall conductivity feature) is observed. Compared to the present sample (thickness, 10 nm), the samples used are in the ultrathin limit [5 nm in the study of Kou et al. (12) and 6 nm in the study of Feng et al. (13)], where hybridization between the states on the two surfaces is expected. Both the presence of zero-Hall shoulder and the absence of jumps suggest that these ultrathin samples are in a distinctly different regime of QAH behavior from the present sample. The degree of disorder appears to be higher in the ultrathin regime. For example, in the zero-Hall sample (13), the dissipation remains high at 50 mK (Rxx ~ 4 at H = 0), whereas accurate Hall quantization is not attained (Ryx ~ 0.9).

We start by applying the standard Cahn-Hilliard nucleation theory to the expansion of a bubble of the true ground state (↑) within a sample in the metastable state (↓). The free energy of a two-dimensional (2D) bubble of radius R isEmbedded Image(1)where ΔF > 0 is the free energy gain in the bulk of the bubble. The surface tension σ, which expresses the cost of the wall, is proportional to the exchange energy. When R exceeds the critical radius Rc = σ/ΔF, the bubble expands without limit (if dissipation is absent), happily paying the cost of the wall to gain bulk energy. Conversely, if R < Rc, the bubble shrinks to 0. At the critical radius, Fb has the critical value Fbc = πσ2F. For a purely classical (thermally activated) escape process, the lifetime of the metastable state isEmbedded Image(2)where 1/τ0 is an attempt frequency. Clearly, if σ and ΔF are both T-independent (which we argue is the case here at all T < 0.5 K), the lifetime of the metastable state decreases exponentially with increasing T.

Trapping of the system in a metastable state can be represented as a particle located in a potential well U(M), where M is a collective coordinate representing the local magnetization M(r) (see sketch in Fig. 4C, inset). After the system is prepared with H || −z at temperature T ~100 mK, it remains trapped for extremely long times if H is tuned to a positive, but small, value 0 < H < Hs(T) (~0.122 T at 70 mK). As shown in Fig. 1B, the stability for H = 0 exceeds 8 hours at 10 mK. Incrementing H above Hs leads to a lowering of the barrier and a decrease in tE. When H is equal to HJ(T), we infer that tE (T) falls inside the observation time window (1 to 3 min), so the first jump is observed. By measuring how HJ(T) varies with T, we can deduce the effect of T on the escape rate.

The curve of HJ(T) (Fig. 4D) shows that HJ increases steeply with T. This trend is counterintuitive if we view the escape as a classical process. In such “over the barrier” processes, raising T invariably speeds up the escape event. Quite generally, in the Langer formulation (15), an escape occurs when a particular spin configuration creates a saddle point in the potential landscape U(M(r)) that breaches the barrier. As the probability for this to occur rises exponentially with T, raising T should lead to a sharp reduction of tE (and hence a smaller HJ). This is opposite to the observed trend of HJ(T). Contrary to the general expectation that “escape” should become easier as more spin configurations become accessible at higher T, we find that escape is more difficult.

Next, we consider a hybrid classical quantum process in which escape results from thermally assisted tunneling. Absorption of bosons (phonons or spin waves) from the bath brings the system’s energy closer to the top of the barrier, thereby enhancing the tunneling rate. Clearly, raising T increases the phonon or spin-wave population, leading to an enhanced escape rate, again in conflict with HJ(T).

Dissipative quantum tunneling

One way to obtain an escape rate that increases with decreasing T is dissipative quantum tunneling. Here, dissipation as monitored by Rxx plays an important role.

The quantum tunneling of a magnetization configuration out of a metastable state is closely related to the decay of the false vacuum investigated by Coleman (16, 17) and Callan and Coleman (16, 17). Through an instanton process (see section S3), the system tunnels “under the barrier” to form a small bubble of the true ground state. Thereafter, the bubble expands rapidly without limit (if dissipation is absent) in the classical Cahn-Hilliard process.

To incorporate dissipation, Caldeira and Leggett (18, 19) considered a single particle of mass m in the tilted double-well potential U(q) with dynamics described by the classical equation of motionEmbedded ImageDissipation—introduced by a linear coupling to many classical oscillators—appears as the damping coefficient η(ω), which may depend on frequency ω. They find that increasing the dissipation strongly suppresses the tunneling probability Γ. In effect, each sampling (measurement) of the particle’s position by an oscillator collapses its wave function; this resets the tunneling process. Whereas η by itself decreases Γ, raising T has the opposite effect (20). The opposing effects of dissipation and T have been investigated experimentally (21).

Within the optimal gating window in our experiment, Rxx is thermally activated with an energy gap ΔR. Hence, the dissipation increases faster than T (by a factor of 17 versus 4 between 50 and 200 mK). We reason that Γ should decrease, consistent with the observed trend in HJ.

At very low T, changes caused by increasing η compete with purely statistical effects due to T. Following Grabert et al. (20), the tunneling rate Γ(T,η) at low T may be expressed asEmbedded Image(3)where B(η,0) is the η-dependent exponent at T = 0. The T-dependent exponent is given by A(T,η) = f(η)T2 (for ohmic dissipation). Although the analytic expression for B(η,0) is not known, numerical integration (22) shows that, at very large η, B(η,0) is asymptotically linear in η, but it approaches a constant as η → 0. Because they are additive and have opposite signs, B(η,0) and A(T,η) compete in the exponent of Γ(T,η). Unfortunately, theory provides only limited guidance to their relative magnitudes. We reason that at low T and small η, A(T,η) is very small compared with B(η,0) [Grabert et al. (20) show that A(T,η) vanishes exponentially like exp(−ħω0/kBT) in the limit η → 0, where ω0 is the characteristic frequency at the well minimum]. Hence, B(η,0) dominates the exponent in Γ(T,η), that is, Γ(T,η) is an overall decreasing function of η, with negligible dependence on T (except insofar as raising T in our situation causes η to increase exponentially). As we raise T between T1 and T2, we expect Γ(T,η) to decrease, consistent with the trend in HJ(T). Tests of macroscopic quantum tunneling have been carried out in experiments involving Josephson junctions (21, 23, 24), molecular magnets (2527), and domain wall dynamics (2830).

A new feature in the present experiment is that both the jump magnitude ΔRyx and its spatial correlation also provide information on the rapid wall expansion that occurs after tunneling. Because dissipation diverts some of the energy gain, a large dissipation will severely restrict the final bubble size, overdamping the expansion (31, 32). To us, this provides a persuasive explanation of the behavior of ΔRyx(T) observed above 120 mK (Fig. 4C). The magnitude ΔRyx provides a measure of the size of the spin-up regions after the expansion corresponding to the first jump. In the interval 80 mK < T <120 mK, the expansion of the true ground state extends to roughly half the sample. As T rises above 120 mK, dissipation increasingly limits the expansion. Above T2, the final radius (as measured by the jump size) falls below resolution. Even if tunneling persists (as implied by the continued increase in HJ near T2), the final bubbles are far too small to be detected by Ryx. Raising T in this interval opens a dissipative channel that sharply limits the size of the true ground state bubbles. [Because the activation gap protecting the quantized state is similar in magnitude (ΔR ~ 190 mK; section Activation across a small gap), we propose that the bulk electrons are the source of dissipation limiting the wall expansion.]

Above T2, the smooth variations of Ryx versus H (Fig. 3C) imply a gradual increase of the spin-up domains associated with domain wall motion as in conventional magnets. At all T investigated, this smooth evolution is present as a background process that exists in parallel with the jumps of interest.

At lower T, a major puzzle uncovered is the steady decrease of ΔRyx to very small steps when T decreases below 70 to 10 mK for Vg = −80 V (Figs. 3B and 4C). The nonmonotonic profile of ΔRyx versus T implies that a second process that suppresses the tunneling rate becomes increasingly important at very low T. We emphasize that this suppression is confined to the ultralow dissipation state (the region −120 < Vg < −80 V in Fig. 9). Once dissipation is activated by gating, the jumps reappear at 10 mK. As shown in Fig. 3D, the curve at Vg = −90 V does not show jumps, whereas jumps reappear in the curves when Vg is changed to (−40, −10) V (all curves are at 10 mK).

It is instructive to view this pattern in the Vg-T plane (Fig. 9). The reappearance of the jumps for Vg > −40 V implies that the boundary T1 decreases rapidly from 70 to <10 mK once Vg leaves the optimal gating window. Hence, a large gap ΔR (which strongly suppresses activation of bulk carriers) favors a large T1, whereas a small ΔR decreases T1 to below 10 mK. The pattern implies that the absence of jumps below T1 is intrinsic to the ultralow dissipation state.

The observation suggests the following (tentative) picture. Because zero dissipation leads to a steep increase in the correlation length ξ of the spins in the limit T → 0, we reason that the tunneling rate Γ will again be suppressed if Ns (the number of spins that tunnel coherently) increases to macroscopic values (for example, the size of the whole sample). In the Wentzel-Kramers-Brillouin (WKB) approximation, where Γ ~ exp(−B/ħ) with Embedded Image, with q as a collective coordinate, a large Ns or large mass m leads to strong suppression of Γ.

When μ lies within the minigap, the bulk electrons become negligible in the limit T → 0 so that ξ extends over macroscopic regions and Γ becomes very small. Increasing the dissipation by changing Vg to (−40, −10) V serves to reduce the size of the coherent region. Hence, jumps can reappear at 10 mK. Although this argument is speculative, the experiment potentially provides a way to measure ξ and its effect on Γ in the limit T → 0.

Nucleation in superfluid helium

Another known example in which the escape rate increases with decreasing T is the metastable A phase of superfluid 3He produced by supercooling below the AB transition. If 3He is cooled from the normal state to the superfluid state at a finite pressure P (say, 30 bar), it first undergoes a second-order transition to the anisotropic A phase at the critical temperature (Tc ~ 2.3 mK). At a slightly lower T (~2 mK), a first-order transition to the isotropic B phase occurs. Because of the first-order nature of the AB transition, the A phase is metastable as a supercooled liquid until it reaches ~0.7 Tc at ~1.4 mK when bubbles of the B phase expand to fill the whole sample.

The Cahn-Hilliard theory (Eq. 1) predicts that, in 3D, Rc = 2σABF and Fbc = (2π/3)R3ΔF, where σAB is the surface tension and FB is the condensation energy of the B phase. Using known quantities, one calculates Fbc to be extremely large (106→107 kBT at 0.7Tc). The predicted lifetime of the A phase τA ought to be many times the age of the universe. The observed nucleation temperature posed a deep puzzle until Leggett proposed the theory (dubbed “baked Alaska”) (33) that nucleation is triggered by cosmic rays. The strongly stochastic and often irreproducible nature of the nucleation process made experimental tests difficult. Subsequently, it was shown that τA is reduced by factors of 10 to 100 if the superfluid is exposed to gamma rays from a 60Co source, in strong confirmation of the cosmic ray theory (34). Experimentally, τA(T) is observed to decrease steeply with decreasing T with the empirical form τA ~ exp(a(Rc/R0)n), with n = 3 to 5 (a and R0 are fit parameters).

Despite the similarity of the lifetime trend versus T, the two experiments are quite different. In 3He, the extremely large barrier Fbc precludes thermal activation as a relevant factor at 1 mK. The T dependence of τA arises because, in the vicinity of the Tc curve, the critical divergence of σAB(T) leads to a divergent increase in Rc (hence, of Fbc) as T approaches Tc from below. The primary effect of decreasing T is to strongly decrease the barrier height Fbc. Clearly, this depends on being close to the second-order transition curve Tc (to have a strongly T-dependent Rc). In addition, we need an intervening first-order (AB) transition for supercooling to occur.

In the QAH system, extensive measurements of Rij versus both H and T [reported here and in previous studies (610)] have found no evidence for thermodynamic phase transitions (first or second order) below 4 K (the critical temperature for the ferromagnetic transition is in the range of 15 to 30 K). The barrier height in the metastable state (which depends on Hc and the exchange energy) is controlled by H but is essentially T-independent below 0.5 K. Because the knob (H) that changes the barrier height is independent of the knob that controls the dissipation (T), we can infer that the observed trend of HJ(T) reflects the effect of dissipation on the escape rate. Both reasons persuade us that the 3He supercooling theory is not applicable here.

Multiple paths and network model

The pattern of correlation between Hall signals observed across pairs of Hall contacts separated by 100 μm (or more) suggests that the jumps strongly perturb the magnetization pattern. As an illustration, we start with a one-wall model that allows R14;mn to be calculated for assumed positions of the wall using the Büttiker formalism (see section S4). We calculate that the downstream Hall pair R14;53 undergoes a jump but not the upstream R14;62 (fig. S5). R14;23 jumps up, whereas R14;65 jumps down, in agreement with Fig. 5 (C and D). Next, we assume that the wall is further displaced to lie between the pair (2, 6) and contact 1 at the second jump. The calculated upstream Hall pair undergoes a jump, and the resistance pairs now reverse their jump directions, consistent with observation. However, the single-wall model is inadequate in many ways. Its biggest difficulty is that the predicted jumps ΔRyx are always 2 (in units of h/e2), whereas the observed magnitudes are roughly 1 or much smaller. This flaw appears to be intrinsic to the one-wall assumption.

We generalize to a multidomain picture in which tunneling starts from many seeds distributed across the sample. A natural starting point is the closely related plateau transition in the conventional integer QHE (IQHE), which has received considerable attention (see section S5) (3537). As H is varied, the chemical potential μ crosses the center of a Landau level broadened by the disorder potential V(x). The current is diverted from the dissipationless edge modes to a very small subset of extended but dissipative bulk states at the band center (states away are Anderson-localized). In the bulk, the centers of the cyclotron orbits describe chiral trajectories that track the energy contours of V(x). Tunneling occurs between chiral trajectories that get close to each other (this leads to scattering). Hence, the transition of Ryx from one Hall plateau to the next occurs concurrently with the appearance of dissipation (finite σxx).

The network model studied by Chalker and collaborators (section S5) (35, 36) provides an elegant way to treat the flow of current through multiply connected paths. The sample is modeled as a regular lattice of corner sharing loops defined by directed links. At the shared corners (the nodes), incoming currents are scattered to one of the two outgoing links, with amplitudes parametrized by β (eqs. S10 and S11 in section S5). They find that the localization length ξ(β) diverges as |β − βc|ν, with ν = 2.5, where β = βc at maximal dissipation (where σxx peaks).

For the Chern transition in the QAH problem (eq. S13 in section S5), Wang et al. (38) have established the existence of three types of domains with Chern number C = 0, ±1. Identifying the Chern number domain walls with the energy contours in the IQHE problem, Wang et al. map the Chern transition to the network model of Chalker and colleagues and obtain a divergence in ξ with the same exponent ν. Wang et al. infer that σij obeys the two-parameter scaling relationship (39). Evidence for this scaling in the QAH problem has been obtained by plotting σxx versus σxy in the vicinity of the Chern number transition (7, 8). Evidence for the ν = 0 QAH state has also been reported by Yoshimi et al. (40).

We discuss our results in this context. Just before the first jump in Ryx occurs, the sample is predominantly in the C = −1 state (because Rxx is finite and very small, small regions with C = 0 exist). After the first jump, which involves (by assumption) expansion from multiple seeds spread over the sample, the C= 0 regions define a multiply connected network of conducting paths that extend across the width of the sample. The observation that ΔRyx is close to 1 (but never 2) suggests to us that the converted regions have C = 0 [this provides support for the three-domain proposition of Wang et al. (38)]. In this network, connectivity between the chiral modes is mediated by tunneling at the nodes as in the network model (the microscopic tunneling between chiral modes occurring at the nodes is to be distinguished from the macroscopic quantum tunneling that triggers the jumps). The bulk conduction is dominated by a few backbone paths (as in percolation networks) that extend across the sample width to give ΔRyx ~ 1. We expect the sudden appearance of a network of regions with C = 0 to strongly perturb the system away from the two-parameter scaling behavior. This is borne out in an interesting way.

In Fig. 9B, we show plots of σxx versus σxy at selected T for Vg fixed at −80 V (within the optimal gating window). Above T2 (curves at 142 and 152 mK), the measured σij trace out a nearly ideal semicircle (the dome). Within the interval (T1, T2), the first jump and subsequent ones kick the orbit high above the dome—the orbit returns only when the transition is nearly complete (σxye2/h). A notable pattern is that the perturbed orbits extend above the dome (the exception is a small sliver inside the dome for prejump segments at T < 100 mK). This rich pattern suggests that the two-parameter scaling behavior is valid above T2, but below T2, the jumps take the system high above the scaling dome.

Although two-parameter scaling cannot be used to obtain σxy when jumps occur, it seems possible to calculate σxy and σxx numerically from large networks, as carried out for ξ by Chalker and colleagues (35, 36). Calculations clarifying the conditions under which ΔRyx is smaller than 1 will add valuable insight into what happens at the jumps.

Outlook and conclusions

The central features uncovered are the large jumps in the Hall resistance of magnitude ΔRyx ~ h/e2 and in Rxx when the QAH system is trapped in the metastable state with C = −1, within the interval T1 < T < T2. A key finding is that the escape rate is sharply decreased as T increases. From the thermal activation of Rxx, we have inferred that the increase in dissipation overwhelms any purely thermal effect associated with increasing T; the increased dissipation then suppresses the tunneling probability Γ. Hence, the Caldeira-Leggett theory seems to be the most appealing explanation for the jumps.

Because the transitions involve tunneling of a field ϕ(x, t) with many degrees of freedom (as opposed to one collective degree of freedom), the present experiment uncovers additional features that invite further investigation. The most interesting seems to be the existence of a temperature scale T1 below which jumps are not observed within the experimental time scale, which we tentatively identify as yet another effect of dissipation. As shown in Fig. 9, the ultralow dissipation state is protected by a small energy gap ΔR from thermal excitation of bulk electrons. The latter opens up a dissipative channel that strongly affects the tunneling process. Above T2, the dissipation strongly limits the expansion of the bubble after tunneling occurs.

A recent scanning nano-squid investigation of the magnetization on closely similar QAH films at T down to 300 mK (41) has not detected the bubble expansion corresponding to our jumps. However, 300 mK is twice our limit T2 for resolving the jumps. Extending the squid experiments to below 100 mK (where ΔRyx ~ 1) will allow imaging of the macroscopic bubbles corresponding to regions transformed by the first jump in Rxy. We anticipate that the static size of the spin-up regions will increase rapidly to macroscopic lengths (100 μm) as T is lowered from T2 to 100 mK. A related experiment revealing the coincidence of superparamagnetism and quantization in the QAH problem was recently reported (42). Investigation of the correlations between jumps can be feasibly extended to a large number (~20) of contacts to provide a more detailed map of the spatial displacement of the wall.

The role of dissipation in macroscopic quantum tunneling (for example, in determining Ns) remains a challenge for both theory and experiment. The QAH system, in which the chemical potential, dissipation, field strength, and temperature can be varied independently, provides a powerful system to explore these issues quantitatively.

MATERIALS AND METHODS

Samples were grown by MBE using thermally heated single elemental (at least 5N purity) sources. The SrTiO3 substrates were annealed in an oxygen environment between 925° and 965°C for about 2.5 hours and were checked after annealing with atomic force microscopy for an atomically ordered surface with a root-mean-square roughness of <0.3 nm before growth. Substrates were indium-mounted on the MBE sample holder and outgassed at 500°C for an hour in the MBE chamber before growth. Cr-doped (Bi,Sb)2Te3 was grown at 240°C as measured by a pyrometer, with a growth rate of about 0.5 quintuple layer per minute. After growth, the sample was allowed to cool to room temperature and was capped with about 3 nm of Al. The sample was then exposed to atmosphere to allow the Al capping layer to oxidize, forming Al2O3. Finally, the sample was heated in an argon environment to melt the indium and free the sample from its substrate. The remaining indium on the back of the substrate was then used for the back-gate contact. For further details, see the work of Kandala et al. (11).

In the experiment, the Hall bar (width, 0.5 mm; length, 1 mm) was made by scratching the thin film with a sharp needle controlled by programmable stepper motors. The ohmic contact was made by firmly pressing tiny pieces of indium onto the contact pads. The pressing action was believed to help the indium penetrate through the Al2O3 capping layer and form a good contact with the underlying Cr-doped (Bi,Sb)2Te3 layer. The measurements were performed with the sample immersed in the 3He-4He mixture of a top-loading wet dilution refrigerator (Oxford Instruments TLM400). To filter out room temperature radiation, three filter stages were added to the circuit. At the room temperature end, an LC π filter provided more than 80-dB attenuation above 100 MHz. Two cryogenic filters were installed in the mixing chamber. One was a powder filter that filters out radiation above 1 GHz, whereas the second was an RC filter with a cutoff frequency of 300 kHz. The overall time constant of the whole circuit was ~10 ms. Nearly all the results reported were taken using a Keithley 6221 precision current source and a Keithley 2182 nanovoltmeter operating in delta mode (with an overall time constant of ~1 s). The relative accuracy of the resistance measurements was confirmed to be better than 0.1% using a 100-ppm precision resistor. The exceptions (Figs. 2D, 4, 5, and 6) were performed using a Stanford Research SR830 lock-in amplifier at 2-Hz frequency, with the signal preamplified by a Stanford Research SR560 and/or PAR113 preamplifiers (time constant set at 2 s in Figs. 4, 5 and 6 and 5 s in Fig. 2D). The excitation current was 1 nA in both methods. For the high-speed measurements of the jump duration (Fig. 7), we applied a dc of 2.5 nA. The room temperature filter was removed. The voltage signal was amplified by the SR560 preamplifier and then digitized using a Zurich Instrument HF2 lock-in amplifier at a sampling rate of 6.4 kilo samples per second. For these measurements, the field sweep rate was reduced to 0.1 mT/min.

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.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.

REFERENCES AND NOTES

Acknowledgments: We thank D. Goldhaber-Gordon, J. Wang, Y. Wang, and S.-C. Zhang for valuable discussions. Funding: We acknowledge support from the Defense Advanced Research Projects Agency/Space and Naval Warfare (N66001-11-1-4110) and a Multidisciplinary University Research Initiative award for topological insulators (ARO W911NF-12-1-0461). N.P.O. acknowledges support from the Gordon and Betty Moore Foundation EPiQS Initiative through grant GBMF4539. Author contributions: M.L., N.S., A.Y., and N.P.O. designed the experiment. A.R.R. and N.S. grew the samples. M.L., W.W., and A.K. performed the measurements. M.L., W.W., J.L., A.Y., and N.P.O. carried out the analysis and modeling of the results. M.L. and N.P.O. wrote the manuscript with inputs from all authors. Competing interests: The authors declare that they have no competing interests. Data and materials availability: The raw data for the curves in all figures are available in Dryad Correspondence, and requests for materials should be addressed to M.L. (minhaol{at}princeton.edu) and N.P.O. (npo{at}princeton.edu).
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