Spin memory of the topological material under strong disorder

Robustness to disorder - the defining property of any topological state - has been mostly tested in low-disorder translationally-invariant materials systems where the protecting underlying symmetry, such as time reversal, is preserved. The ultimate disorder limits to topological protection are still unknown, however, a number of theories predict that even in the amorphous state a quantized conductance might yet reemerge. Here we report a directly detected robust spin response in structurally disordered thin films of the topological material Sb2Te3 free of extrinsic magnetic dopants, which we controllably tune from a strong (amorphous) to a weak crystalline) disorder state. The magnetic signal onsets at a surprisingly high temperature (~ 200 K) and eventually ceases within the crystalline state. We demonstrate that in a strongly disordered state disorder-induced spin correlations dominate the transport of charge - they engender a spin memory phenomenon, generated by the nonequilibrium charge currents controlled by localized spins. The negative magnetoresistance (MR) in the extensive spin-memory phase space is isotropic. Within the crystalline state, it transitions into a positive MR corresponding to the weak antilocalization (WAL) quantum interference effect, with a 2D scaling characteristic of the topological state. Our findings demonstrate that these nonequilibrium currents set a disorder threshold to the topological state; they lay out a path to tunable spin-dependent charge transport and point to new possibilities of spin control by disorder engineering of topological materials

Robustness to disorder -the defining property of any topological state -has been mostly tested in low-disorder translationally-invariant materials systems where the protecting underlying symmetry, such as time reversal, is preserved. The ultimate disorder limits to topological protection are still unknown, however a number of theories predict that even in the amorphous state a quantized conductance might yet reemerge. Here we report a directly detected robust spin response in structurally disordered thin films of the topological material Sb2Te3 free of extrinsic magnetic dopants, which we controllably tune from a strong (amorphous) to a weak (crystalline) disorder state. The magnetic signal onsets at a surprisingly high temperature (∼ 200 K) and eventually ceases within the crystalline state. We demonstrate that in a strongly disordered state disorder-induced spin correlations dominate the transport of charge -they engender a spin memory phenomenon, generated by the nonequilibrium charge currents controlled by localized spins. The negative magnetoresistance (MR) in the extensive spin-memory phase space is isotropic. Within the crystalline state, it transitions into a positive MR corresponding to the weak antilocalization (WAL) quantum interference effect, with a 2D scaling characteristic of the topological state. Our findings demonstrate that these nonequilibrium currents set a disorder threshold to the topological state; they lay out a path to tunable spin-dependent charge transport and point to new possibilities of spin control by disorder engineering of topological materials Electronic disorder [1] and elementary excitations in quantum condensed matter are fundamentally linked and it is well established that spatially fluctuating potentials tend to promote decoherence and localization of fermions, i.e. formation of Anderson insulators [2]. The interplay of interactions and disorder often leads to new quantum behaviors; disorder typically boosts interparticle correlations both in charge and in spin channels, and that could either aid or suppress the motion of charge [3]. Spin effects related to disorder are particularly important when spin-orbit coupling (SOC) is strong [4], and when spindependent charge transport can be electrically manipulated for uses, e.g. in spin-based electronics [5].
Strong SOC is a hallmark of three-dimensional (3D) topological insulators [6], where 2D gapless spin-polarized Dirac surface states are robust against backscattering. Most topological materials are known to contain a natural population of charged defects [7] that do not cause a destruction of the topological Dirac states [8]; indeed, they can be compensated [9] as long as Dirac mass [10] or puddle [11] disorders do not enter. Under weak disorder, a coherent interference of electron waves survives disorder averaging [12], and strong SOC enhances conductivity by a weak antilocalization (WAL) correction related to the topological π-Berry phase [6] when magnetic impurities are absent. The 2D WAL channels can be outnumbered by the weak localization (WL) channels [13] -this is a precursor of Anderson localization [2], which occurs at strong disorder. Under strong disorder, theory and numerical simulations [14][15][16][17] predict an emergence of a new topological state, dubbed 'topological Anderson insulator', in which conductance G 0 = 2e 2 /h is quantized. Indeed, recent theoretical demonstrations of topological phases in amorphous systems [18,19] point to promising new possibilities in engineered random landscapes. Strong disorder, however, is not trivial to install, quantify and control, and topological matter under such conditions has not yet been experimentally tested.
Here we implement an extensive range of site disorder -from amorphous to crystalline state -in Sb 2 Te 3 , the material which is a known 2 nd generation topological insulator [20][21][22], and report that under strong structural and electronic disorder conditions dynamic spin correlations dominate charge transport over a surprisingly large range of magnetic fields. These correlations imprint spin memory on the electrons hopping via localized spin sites. Predicted to be small [23] and thus practically unobservable in the conventional materials, the effect found here is large; it persists over a surprisingly wide range of disorder and well within the disordered crystalline topological phase, as long as variable range electron hopping (VRH) [24] is at play. It eventually transitions into the characteristic 2D WAL regime when the gapless surface channels are reestablished. The spin memory uncovered in this work is not an orbital effect; it is distinct from the weak localization interference effect (WL) observed in the magnetically doped topological insulators [25]. It originates from the presence of disorder-induced localized spins [3] and, as witnessed by the characteristically non-analytic arXiv:1911.00070v1 [cond-mat.mtrl-sci] 31 Oct 2019 negative magnetoresistance (neg-MR), is governed by the distribution of very large spin g-factors that widens with decreasing localization length ξ in a way akin to an assembly of quantum wells [26].
The experiments were performed on thin (∼ 20 − 50 nm) films of Sb 2 Te 3 , in which extreme positional disorder (amorphous state) is possible to obtain (SI, Section A). Sb 2 Te 3 is a well known phase-change material [27] (SI, Fig. S1), that undergoes amorphous-to-crystalline transformation with the concurrent orders-of-magnitude resistive drop, and hence a huge range of disorder could be controllably explored.
Let us recall that in the presence of disorder a finite population of singly occupied states below the Fermi energy E F has been discussed as long as 20 years back by Sir Neville Mott [3]. Magnetic response from randomly localized spins in such state was expected to be weak and, as far as we know, has never been experimentally demonstrated. So our first surprising finding was a very robust magnetic signal from the disordered Sb 2 Te 3 films directly detected using a custom-designed µHall sensors (Figs. 1A,B), with the thin film flakes mechanically exfoliated from their substrates and transferred onto the active sensor area (Materials and Methods, SI, Fig. S2). Our films do obey Anderson scaling (see Fig. 1C), and we surmise that here the observed effective moment per atom is significantly amplified by the large effective Landé g-factor [4] in Sb 2 Te 3 , where SOC is strong [20]. The detected signal depends on the magnetic history (field-cooling vs. zero-field cooling), which, together with slow magnetic relaxation (Fig. S2C) reflects a glassy nature of the localized state. We emphasize that magnetic signal crucially depends on the level of disorder; indeed, it becomes barely detectable in the crystalline phase as disorder in the same film is reduced by thermal annealing (Fig. 1B).
Our second key finding is shown in Fig. 1D. The change in the longitudinal magnetoresistance ∆R xx at low magnetic fields is with a remarkable fidelity impervious to the tilt of magnetic field. It is relatively large and negative (i.e. charge transport becomes less dissipative) up to a field H max (Figs. 1D and S3). This field isotropy of the low-field dissipation 'quench' naturally suggests a spin-dominated mechanism rather than orbital effect, and so the pertinent question to ask is how such behavior can proceed in a system where electronic states are localized and the extrinsic spinfull impurities are absent (Table S1).
With strong disorder charge transport is a complex electron hopping process that at low temperatures proceeds via quantum tunneling between localized states assisted by phonons [24,30]. While considerations of magnetotransport have mainly focused on the orbital effects, a recently proposed idea [23] takes note of putative nonequilibrium spin correlations in the localized regime created by the flowing current when electron hopping times τ are short relative to the spin relaxation times τ s . These time scales determine the magnetic field range over which spin-correlation-driven neg-MR (positive magnetoconductance) ought to be present. The idea is illustrated in Fig. 1E. When the current is injected, an electron with spin S attempting to hop to an available empty site can do it in two ways: directly or via an intermediate site occupied by a localized spin s. It may take several attempts for the indirect hops to succeed and the return probability will depend on history, i.e. on whether the tunneling electron can form a triplet or a singlet state with s. For example, in the absence of disorder a triplet state would remain so in the presence of applied magnetic field and no reduction of magnetoresistance (increase of magnetoconductance) is expected. Under strong disorder (such as shown in Fig. 1C), however, spin g-factors will be spatially random so that localized spins at different sites will precess incoherently and spin correlations will be destroyed by the field. Accordingly, in a simple model [23] the change in magnetoconductance ∆G xx arising from such spin correlations should follow not a power law [31] but a unique non-analytic form: , d s = 4/3 is the spectral dimension of the percolation cluster [32] (which is the relevant dimension in the hopping process), l is index of the diffusing spin, Γ(− ds 2 ) is the gamma function ∼ = −4, and H = µ B τ ∆g is the limiting magnetic field range set by the hopping rate 1/τ and the disorder-induced spread ∆g of spin g-factors.
The strongly localizing behavior we observe in the longitudinal resistance R xx ( Fig. 2A) at low temperatures (below ∼ 10 K) follows variable range hopping law Fig. 2B. The E-S energy scale T 0 characteristic of the hopping process (Fig. S1B) is tracked on decreasing disorder by a well controlled thermal annealing schedule (Materials and Methods); it is inversely proportional to the electron localization length [24] ξ, which we will show controls the g-factor distribution width ∆g. We remark that in this regime (at low T ) the detected magnetic moment appears 'flat' in temperature (Fig. 1B).
In the variable range hopping (VRH) regime, the fit of conductance to Eq. (1) for two states of disorder is illustrated in Fig. 2C. As seen in the figure, at low magnetic fields the characteristic non-analytic behavior is accurately followed; here the ratio of hopping time to spin relaxation time τ /τ s and the hopping field scale H were used as fitting parameters (Materials and Methods). The fits at different disorder levels controlled by the anneals at different temperatures T a are shown in Fig. S4. The ratio τ /τ s strongly depends on the level of disorder (Fig. 2D), with the hopping and spin relaxation rates, 1/τ ∝ H and 1/τ s ∝ H s , in close correspondence with the disorder dependence of R xx all the way through crystallization transition (see Fig. 5 below). The hopping time τ can be independently extracted from the E-S energy (Fig. S5A), and, as expected for the hopping conductivity τ increases exponentially on decreasing temperature (Fig. S5B). This allows us to consistently obtain the evolution of spinrelaxation time τ s (Fig. 2E) and ∆g (Fig. 3A) with decreasing disorder (increasing T a ), and hence that of the low-field spin-relaxation scale H s = µ B τs∆g (Fig. S6); H s marks a crossover from the concave-up field shape associated with τ s to concave-down behavior at higher fields, see Fig. 2C. An intriguing question arises as to what controls the unexpectedly large (Tesla-range) field scale where negative magnetoresistance, the hallmark of dynamic spin memory, is found. For the material systems with small spin-orbit coupling where g-factor ∼2, the expected field range would be in the 10 −4 − 10 −5 Tesla range. In theory [23], this field scale is obtained from the competition of the magnetic energy of the spins, gµ B H, and either thermal energy or the exchange energy J between neighboring spins -it ought to be well below the competing effects. Here, however, with large effective g value [33], gµ B H/k B ∼ 20 K is comparable to the spin-memory range. This brings us back to disorder-induced spin correlations. Fig. 1B shows that under extreme disorder the onset of magnetic response is abrupt and at a remarkably high temperature T s ∼ 200 K. While the details of spin correlations in this Anderson-like-localized glassy state clearly deserve further experimental and theoretical studies, a rough estimate of J ∼ k B T s /z ≈ 70 K, using local coordination number [27] z ∼ 3 expected in Sb 2 Te 3 , implies that here short range interactions between localized spins play the key role (see SI, Section A).
Let us now consider g-factor fluctuations in a strongly disordered state. The g distribution width naturally arising from our magnetoconductance data (Fig. 3A) within the model considered above is spectacularly wide at the highest level of disorder, ∆g 40; indeed, it exceeds the effective g value of ∼ 30 obtained e.g. directly in the same topological insulator family from the electron spin resonance (ESR) experiments [33]. Such large g spread is uncommon but not unprecedented. Giant fluctuations of g-factors have been reported in e.g. InAs nanowires where ∆g > |g ef f | and the effective factor |g ef f | is also large [29]. Furthermore, in semiconducting quantum wells |g ef f | has been known to increase roughly linearly [26] with quantum confinement energy E Q as |g ef f | g 0 + βE Q , where β is a material-specific constant. Here we propose that in the strongly localized Anderson-like state, quantum confinement is enforced by the wells constrained by the localization length ξ (Fig. 3B). In this view, a simple particle-in-a-box approximation gives E Q ∝ ξ −2 that indeed fully scales with ∆g (Fig. 3A). The expected linear |g ef f | vs. E Q would then set |g ef f | ∼ ∆g −α with α ∼ 0.1, with |g ef f | approaching the ESR-determined value in the unconfined state, see Fig. 3C.
Our experiments reveal that positive ∆G xx (H) (neg-MR), evolves progressively with decreasing disorder; it persists over a spectacularly large disorder range, all the way through the crystallization process and beyond (Fig. 4A). The large limiting field range set by H max at strong disorder falls with increasing T a (Fig. 4B) to pinch off ∆G xx (H) eventually to null. A clear visual of the field-disorder phase space is shown in Fig. 4C, where the strength of disorder is represented by E Q . H − E Q diagram shows that spin-memory region is restricted by H Ta follows the confinement energy EQ(Ta). EQ is modeled by a particle-in-a-box and calculated using ξ as a box size.
In the amorphous state in the conduction tail states effective mass is strongly enhanced [28]. Inset: P (g) calculated in the strong SOC regime [29] for ∆g = 27. (B), A model of disorder landscape within the Anderson bandwidth W riding on a long-range smooth potential [30]. (C), Effective g-factor, |g ef f |, increases with EQ. We note that the sign of g for many semiconductors, particularly with strong SOC, is negative [4].
to relatively low fields, but when the system is less localized the 'envelope' of the spin-memory space switches to H max . The temperature range over which spin memory is evident is set by the VRH process (Fig. 2B), with ∆G xx (H) well described by Eq. (1). Above ∼ 10 K , outside the VRH region, ∆G xx (H) becomes nearly 'flat' (Fig. 4D); there both τ s (Fig. S7A) and |g ef f | (see Fig. S7B) appear to saturate, and spin-memory phenomenon is not expected. The typical field scale associated with the E-S energy T 0 ∼ 10 − 30 K is in the 1-2 T range, in close correspondence with H at low T . As before, H max H , and in the low-temperature localized state H max becomes the limiting crossover field (Fig. 4E). Above T a 180 • C (E Q ≈ 60 meV), in the strongly disordered crystalline state spin memory phenomenon is not detectable. The exit from the spin-memory state at ∼ = 180 • C is clearly evidenced by a transition from the negative MR (neg-MR) to a positive MR (pos-MR)'cusp' characteristic of the WAL state (Fig. 5A). The WAL cusp scales with the transverse component of applied magnetic field H ⊥ = Hcosθ (Fig. 5B), consistent with the 2D (orbital) character expected in a topological insulator [9] under weak disorder. This 2D scaling should be contrasted with isotropic (3D) scaling of the neg-MR peak in the spin-memory state. Thus, unlike the WL-WAL transition driven by magnetic impurities [25], the transition from spin-memory onto a WAL state is also a 3D-2D dimensionality transition at which the electron system rapidly delocalizes. The change of the localization length ξ with decreasing disorder is smooth at the transition to WAL (Fig. 5C) , with electron mean free path only limited by the film thickness and grain size in the crystalline state at high T a (Fig. S8). We remark again that WAL onsets at 180 • C, way above the crystallization transition at 140 • C (see inset in Fig. 5C), pointing to a disorder threshold for the topological state.
To summarize, we directly detect spin response of the topological material Sb 2 Te 3 under strong structural disorder, which we can control and tune by thermal annealing from amorphus to crystalline state. Under strong disorder, the system develops spin-correlations that drive the spin-memory phenomenon controlling the transport of charge. Both in magnetic field and in disorder strength, the parameter space where spin memory exists is unexpectedly broad -it persists well into a disordered crystalline state where, eventually, a 2D WAL quantum interference correction is recovered. While a simple phenomenology we used captures the key features of transport under strong disorder, this regime is undoubtedly complex and theory has yet to provide a full understanding of charge transport under such conditions, particularly when spin-orbit coupling is strong. A new perspective revealed by our findings is that spinmemory effect sets a disorder threshold at which topological protection of the gapless surface states is reclaimed. Save for the edge currents, the control of spin-dependent transport arising from nonequilibrium correlations can, in principle, be achieved by using electrostatic gating [34], and by modifying correlations with an addition of localized spins using currently practiced semiconductor doping techniques.
We wish to acknowledge Roland Winkler for his key insights regarding g-factors. We thank Igor Aleiner and Boris Spivak for their useful comments. We are grateful to Andy Kellock for the RBS and PIXE analysis of the films. This work was supported by the NSF grants DMR-1312483-MWN, DMR-1420634, and HRD-1547830 (L.K.-E.).

APPENDIX
Film growth and structural characterization. Films of Sb 2 Te 3 with thicknesses ranging from 20 to 100 nm were sputter-deposited at room temperature in Ar gas at 4 mTorr and a flow of 46 sccm from a nominally stoichiometric target using 15 W DC power on Si 3 N 4 (100 nm)/Si substrates. The stoichiometry was confirmed by Rutherford Backscattering (RBS) and particle induced X-ray emission (PIXE). RBS data were collected at NEC 3UH Pelletron using a Si surface barrier detector with He + ions at 2.3 MeV. PIXE data was collected using a Si-Li detector with H + ions at 1 MeV. Elemental analysis was done at Evans Analytical Group. X-ray diffraction characterization was performed using Bruker D8 Discover system with the da Vinci configuration using a monochromated beam (λ Cu = 1.5418Å) and a scintillator detector with analyzer crystal (HR-XRD). The film morphology was characterized using the FEI Titan Themis 200 transmission electron microscope (TEM), 200 kV, with TEM resolution 0.9Å and 4k × 4k Ceta 16M CMOS camera. Disorder was characterized by Raman spectra using 633 nm linearly polarized excitation in a backscattering configuration [35], with power kept below 2 mW to avoid heating effects.
Magnetic measurements were performed using customdesigned on-chip µ-Hall sensors based on In 0.15 Ga 0.85 As heterostructures (SI). To measure magnetization, ∼50 micron in lateral size thin Sb 2 Te 3 film samples were exfoliated from their substrates and placed directly on the SiO 2 -passivated sensor using PDMS. At each temperature an empty twin sensor was used for background subtraction. Transport measurements were performed in a 14 Tesla Quantum Design PPMS system in 1 mTorr of He gas on many film samples, each subjected to the same annealing protocol used to tune the level of disorder. Lithographically patterned structures combining both Hall bar and van der Pauw electrical contact configurations with Ti/Au metallurgy were used ( Fig. 2A). Measurements were performed on as-deposited films and on the same shows a distinct kink at crystallization and a smooth transition to a 2D WAL state. T0 was obtained from fitting to the E-S VRH formula [30]. Inset: Crystallization at Ta ∼ = 140 • C (yellow dash) is clearly seen as sharp lines in the Raman spectra vs. Ta. Transition to WAL at 180 • C is indicated by red dash.
films after each 5 min annealing step in a box furnace in flowing nitrogen in the temperature range across crystallization at T a ∼ 140 • C. We used a numerical Monte Carlo technique to fit our transport data to Eq. (1).