Long-range influence of manipulating disordered-insulators locally
aa r X i v : . [ c ond - m a t . d i s - nn ] A p r Long-range influence of manipulating disordered-insulators locally
Z. Ovadyahu
Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
Abstract
Localization of wavefunctions is arguably the most familiar effect of disorder in quantum systems. It has been recentlyargued [V. Khemani, R. Nandkishore, and S. L. Sondhi, Nature Physics, , 560 (2015)] that, contrary to naive expectation,manipulation of a localized-site in the disordered medium may produce a disturbance over a length-scale much larger thanthe localization-length, ξ . Here we report on the observation of this nonlocal phenomenon in electronic transport experiment.Being a wave property, visibility of this effect hinges upon quantum-coherence, and its spatial-scale may be ultimately limitedby the phase-coherent length of the disordered insulator. Evidence for quantum coherence in the Anderson-insulating phasemay be obtained from magneto-resistance measurements which however are useful mainly in thin-films. The technique usedin this work offers an empirical method to measure this fundamental aspect of Anderson-insulators even in relatively thicksamples. INTRODUCTION
Disorder may lead to a variety of non-trivial phenom-ena in both classical and quantum systems. The mostfamiliar of these phenomena is Anderson localization[1]. This phenomenon has been established in electronictransport [2], propagation of light [3] and sound waves[4], and in disordered Bose-Einstein condensates [5].Localization of wavefunctions may seem a way to allowmanipulation of a particular site in a solid while partsof the system that are remote from it are unaffected.This expectation has been recently questioned; Khemani,Nandkishore, and Sondhi (KNS) [6] shown that adiabati-cally changing the potential on a local site will produce aneffect over a distance that may exceed ξ by a considerablemargin. This long-range effect may have important con-sequences for quantum-computing manipulations and forfundamental issues such as the orthogonality-catastrophe[6, 7].In this work we describe a method that allows observa-tion of the KNS effect in an electronic system and showresults that demonstrate the quantum nature of the phe-nomenon. EXPERIMENTALSample preparation
The samples used in this study were amorphousindium-oxide (In x O) made by e-gun evaporation of99.999% pure In O onto room-temperature Si-wafersin a partial pressure of 1.3x10 -4 mBar of O and a rateof 0.3 ± N of the samples, measured by the Hall-Effect at room-temperatures, was N ≈ (1 ± cm -3 .Using free-electron formula, this carrier-concentration isassociated with ∂ n/ ∂µ ≈ erg -1 cm -3 . The Si wafers(boron-doped with bulk resistivity ρ ≤ -3 Ωcm) wereemployed as the gate-electrode in the field-effect experi- ments. A thermally-grown SiO layer, 2 µ m thick, servedas the spacer between the sample and the conductingSi:B substrate. Films thickness was measured in-situ bya quartz-crystal monitor calibrated against X-ray reflec-tometry. Samples geometry was defined by the use ofstainless-steel mask during deposition into rectangularstrips 0.8 ± ± Measurement techniques
Conductivity of the samples was measured using a two-terminal ac technique employing a 1211 ITHACO currentpreamplifier and a PAR 124A lock-in amplifier using fre-quencies of 30-75Hz depending on the RC of the sample-gate structure. R is the source-drain resistance and C isthe capacitance between the sample and the gate (C inour samples was typically ∼ =10 -10 F and R for the samplesstudied in this work ranged between 1.5-20MΩ). Exceptwhen otherwise noted, the ac voltage bias in conductiv-ity measurements was small enough to ensure near-ohmicconditions. Most measurements were performed with thesamples immersed in liquid helium at T ≈ σ that for 4 < T <
50K wasof the Mott form [8]; σ ( T ) ≈ exp h − ( T /T ) i (1)as illustrated in Fig.1 for two typical samples. This al-lowed an estimate of the localization-length ξ through [8]:k B T ≈ ( ξ ∂ n/ ∂µ ) -1 where ∂ n/ ∂µ is the thermodynamicdensity-of-states. With ∂ n/ ∂µ ≈ erg -1 cm -3 , the ξ val-ues for the samples reported below ranged between 2.7nmto 3.3nm. These ξ values are close to the inter-carrier dis-tance N -1/3 of this version of In x O as may be expected forTypeset by REVTEX 1 .35 0.40 0.45 0.50 0.55 0.60 0.65 0.7010 -2 -1 s ( W - c m - ) T -1/4 (K -1/4 ) T »3160K T »3250K FIG. 1: Conductivity versus temperature for two In x O sam-ples from the same preparation batch with thickness d=82nm. These exhibit Mott variable-range-hopping yielding sim-ilar activation energies T and localization lengths ξ ≈ samples that are far from the metal-insulating transitionwhich applies to all our studied samples. This makes theestimate for ξ , based on the σ (T) data, a plausible value.Taking the sample far from equilibrium to study itsthermalization dynamics is accomplished in this workby exposing the sample to an AlGaAs diode oper-ating at ≈ ± µ m mounted on the sample-stage ≈ RESULTS AND DISCUSSION
In a field-effect experiment, the charge δ Q added to thesample when the gate-voltage is changed by δ V g , residesin a thin layer of thickness λ ≈ (4 π e ∂ n/ ∂µ ) -1/2 at theinterface between the sample and the spacer [10, 11] (Thedielecric constant of the material κ is of the order of 10).The thickness of this layer in In x O is λ ≈ δ Q to the system dueto δ V g had an effect extending over length-scales muchlonger than both λ and ξ .This observation may be inferred from G(V g ) plots,taken at different times t Q , while the sample is relaxing t Q (s)
28 420 750 7500 G ( x - - ) d=45 nmd=82 nm (a) t Q (s)
20 140 500 1400 d=65 nm (b)(c) t Q (s)
20 140 300 1460 V g (V) t Q (s)
20 80 420 7200 d=145 nm (d)
FIG. 2: Results of the G(t,V g ) protocol performed on In x Osamples with similar composition (carrier-concentration of ≈ cm -3 ) but different thickness d. Each G(V g ) plot wereobtained with the same sweep-rate ∂ Vg/ ∂ t=0.5V/s. Bathtemperature T=4.1K after being quench-cooled from an excited state; Considerthe G(t,V g ) plots for the samples in Fig.2. The protocolused throughout the series of measurements shown in thiscomposite figure was as follows. The sample, immersedin liquid He at T=4.11K, was exposed for 3 seconds toinfra-red source (light-emitting-diode at 0.82 micron ra-diation) taking it from equilibrium. G(V g ) scans werethen taken with constant ∂ V g / ∂ t starting from V g =0,at which the gate-voltage was kept between subsequentscans. These are labeled in the graphs by the time t Q thatelapsed since turning off the infra-red source (and theonset of relaxation towards restoring equilibrium underV g =0). Each of these G(V g ) plots reflects the energy de-pendence of ∂ n/ ∂µ modulated by a ”memory-dip” whichresults from the interplay between disorder and Coulombinteraction [12].Note first the difference between the thinnest andthickest samples in the series, Fig.2a and 2d respectively.2n the former, the G(V g ) plots taken at different timestend to merge for V g ≥
10V while, for the 145 nm sample,they tend to become parallel.A simple explanation to the results exhibited by thesample in Fig.2d is that the added charge only affectsthe part of the sample that is close to the spacer-interfacewhile the rest of it is unaffected. In this case the sampleis effectively composed of two conductors in parallel; onewhere the G(t,V g ) curves are like the pattern exhibitedby the sample in Fig.2a, and another for which G(t) justmonotonically decreases after the quantum-quench, in-dependent of δ V g . Superimposing these two componentsqualitatively reproduces the G(t,V g ) curves exhibited bythe 145 nm sample in Fig.2d.It is important to understand the different roles playedby the infrared exposure versus the gate-sweeps in theseexperiments. Changing the gate-voltage or exposing thesample to infrared will take the system from equilibrium.However, these agents do not play a symmetric role inthe protocol; the infrared exposure is a one-shot eventdriving the system far from equilibrium. Sweeping thegate is used to take a snapshot of how far the system ison its relaxation trail. This is done intermittently as timeprogresses and yields a certain swing δ G(t) reflecting thedevelopment of a memory dip. This δ G may be thencompared with the background conductance-value thatis going down with time due to the original excitationby the infrared source (which, as alluded to in sectionIII, affects the entire thickness of the sample). The formof the observed G(t,V g ) plots will tell whether or notthe gate-sweeping affects the entire sample volume (as inFig.2a, 2b and 2c) or only part of it (as in Fig.2d).The G(t,V g ) curves pattern characteristic of a thinsample has been first observed in [13] on crystallineindium-oxide and later in [14] on a different ver-sion of In x O than the one used here (namely, with N ≈ cm -3 ).To account for the behavior of the three thinner sam-ples is a more challenging task; apparently in these in-stances the disturbance caused by the added charge ex-tends throughout their entire thickness - over a length-scale of d which, for the 82nm sample, is 25 to 30 timeslarger than the localization-length ξ . It is hard to see howsuch a long-range effect is possible unless wavefunction-overlap that are L ≫ ξ apart is much better than might beexpected from exponential decay. The Coulomb interac-tion due to δ Q over this length-scale, even if unscreened,is too weak relative to the local disorder to affect G(V g )during the time V g is swept.High transmission-channels through disordered mediawould offer an explanation for the long-range effect.These resonant channels are theoretically possible butexponentially rare [15–17]. By contrast, the scenario pro-posed by KNS creates such resonant channels in the dis-ordered system with high probability by using a time-dependent adiabatic process [6]. Adapted for our geome- try, quasi-extended states are parametrically formed per-pendicular to the film plane by slowly varying the localpotential V at the interface layer. As will be now shown,this scenario accounts for all aspects of the experimentalresults.Let us first look at r zd , the extent of the ‘zone-of-disturbance’ expected of the KNS-produced resonances[6]: r zd ≈ ξ · ln (cid:18) W ∂ V/ ∂ t · ~ (cid:19) (2)With the value of the quenched-disorder in our sam-ples [18] W ≈ ∂ V/ ∂ t ≈ ξ ≈ zd ≈ zd ≈ g ) curves converge at high gate-voltages which im-plies r zd ≥ d. In addition, the mean values that δ V attainsin the V g -interval used in the experiments, covers theenergy separation δ E ≈ ( ∂ n /∂µ · L ) -1 for states that areapart by any L & ξ . This secures ample ‘tuning-margin’for creating the quasi-extended states by the KNS sce-nario.A fundamental requirement on the KNS mechanism isthat phase-coherence must be preserved throughout thespatial-scale in question. This requirement follows fromthe quantum-mechanical nature of the process. In otherwords, the range of disturbance may be r zd in Eq.1 only when L φ , the phase-coherent-length in the medium obeysL φ > r zd .Evidence for phase-coherence in Anderson-localizedfilms over scales of tens of several nano-meters has beenreported. This evidence is based on two phenomena,both strictly requiring phase-coherence: orbital magneto-conductance [20, 21], and Andreev tunneling [22]. Thelatter, performed on In x O films of similar composition asused in the current work, demonstrated that a coherence-length of ≃ ≈
4K is realizable in this system.A further test of the role of quantum-coherence in thenonlocal effect discussed here is to see how the G(t,V g )plots change when dephasing is judiciously introduced.Once the dephasing-rate is large enough to cause L φ < d,the resulting G(t,V g ) plots should revert from the ‘con-verging’ pattern to that resembling the results in Fig.1d.To implement this test in a controlled way, one needsa dephasing agent that can be turned on and off atwill. An effective and easy to control mechanism for de-phasing Anderson-insulators is using a non-ohmic fieldin the transport measurement [20]. This has beendemonstrated in magneto-conductance measurements onstrongly-localized indium-oxide films [20]. This techniquewas applied on three different In x O samples and the re-sults corroborate the expected behavior caused by theextra dephasing. Figure 3 illustrates the results of one ofthese experiments:3 .44.54.64.74.84.96.76.86.97.07.17.2 0 5 10 15 209.69.79.89.910.010.1 t Q (s)
20 80 200 450 1300 V SD =0.1V (a) t Q (s)
19 82 202 452 1232 V SD =5V (b) G ( x - - ) V g (V) t Q (s)
20 80 200 440 1520 V SD =10V (c) FIG. 3: Results of the G(t,V g ) protocol applied on a single65nm thick In x O sample under different source-drain fields(distance between the source-drain contacts is 1mm). Plate(a) shows the results of the protocol taken under linear-response conditions while in plates (b), and (c) the G(t,V g )plots were taken using non-ohmic voltages in the measurementcausing the somewhat enhanced conductance. Bath temper-ature T=4.1K Figure 3a shows a set of G(t,V g ) curves taken in linear-response. These ‘converging’ plots are consistent withr zd > d. Using non-ohmic V SD for measuring G(V g ) onthe same sample produced however, different results; theG(t,V g ) curves (Fig.3b and 3c) resemble the pattern ob-tained for the thick sample in Fig.2d where presumablythe range of δ V g is smaller than the sample thickness.Another indication that, under the higher V SD con-ditions, part of the sample is not affected by the gate-voltage is shown in Fig.4. This figure compares the rela-tive magnitude of the memory-dips taken under the samefields used in Fig.3a, 3b, and 3c. The figure shows alarge reduction in the memory-dips magnitude for thetwo non-ohmic V SD used relative to the linear responseplot. The reduced range of disturbance implied by thedata in Fig.3b, 3c and 4 is consistent with the dephasingeffect of non-ohmic fields causing L φ to become the short-est scale. Similar behavior was observed on two othersamples with d=65nm and d=82nm upon application of -20 -15 -10 -5 0 5 10 15 201.001.051.101.15 V g (V) G ( V g ) / G ( ) V SD FIG. 4: The memory-dips taken under the same source-drainvoltages and sweep-rates as the data in Fig.2. These weretaken, in each case, after the sample was allowed to relax atV g =0V for 24 hours. The bath temperature was T=4.1K. R ( M ) V SD (mV) FIG. 5: Sample resistance as function of the source-drain volt-age V SD . Plots are given for three sample labeled by theirthickness. The nominal field used for ohmic-regime measure-ment was typically F=1V · m -1 . The source-drain separationfor all these samples was 1mm.The upper plot is taken on thesame sample as in Fig.3 above. non-ohmic fields.For further discussion of the results of the non-ohmicfields, we show in Fig.5 resistance versus source-drainvoltage V SD plots for three of the samples used in thisstudy.Note first that the increase in the overall conductanceof the sample used in Fig.3 under V SD of 5V and 10V (bya factor of ≈ ≃ g ) plots;when measured in linear-response, samples with the samed, but conductance that differed by as much as order ofmagnitude, still exhibited the same converging G(t,V g )curves. Secondly, in terms of dephasing the effect of anon-ohmic field act in the same direction as higher sam-ple temperature [20]. The increase of effective tempera-ture ∆T due to the applied source-drain field F SD may4e roughly estimated as ∆T ≃ e ξ F SD /k B which, for theF ≈ -1 used in Fig.3c is tantamount to ∆T ≈ φ in an Anderson-insulator may be ≥ ≈ φ of this order of magnitude is typical of diffu-sive samples at this temperature [23]. This may conflictwith common intuition expecting disorder to decreasetransport-related spatial-scales. However the dependenceof L φ on disorder is not clear even in diffusive systems de-spite extensive studies [24], let alone in the more intricateAnderson-insulating phase where this issue has barelybeen studied. On the basis of current knowledge it is notimpossible that L φ in the insulating phase be as largeas in the metallic phase. In terms of mechanisms, theinsulating phase may even have an advantage; electron-electron inelastic scattering that, at low temperatures,is the main source of dephasing in diffusive systems, issuppressed in the insulating phase. This has been antici-pated on theoretical grounds [24], and was shown exper-imentally [25]. Moreover, the electron-phonon inelasticrate is likely also suppressed due to the reduced overlapbetween the initial and final electronic states involved inthe inelastic event. Therefore dephasing due to inelas-tic scattering may actually be weakened in the strongly-localized regime. On the other hand, once interactionsare turned on, a potential source for dephasing appearsthat may not have existed in the weak-disorder regime -spin-flips. This mechanism may become important oncethe on-site Coulomb repulsion is strong enough to pre-cipitate a finite density of singly-occupied states at theFermi-energy [26]. These singly-occupied sites act like lo-cal magnetic impurities and may contribute to dephasing[27]. This potential source of dephasing may be the rea-son for the paucity of experiments reporting on quantum-interference effects in Anderson insulators in systems thatdo exhibit such effects in their diffusive regime. Evidencefor quantum-coherent effects is usually based on obser-vation of anisotropic magneto-conductance. This tech-nique however becomes ineffective for films thicker thanfew tens of a nanometer [21], a weakness not shared byour protocol.In sum, we demonstrated the existence of a nonlocaleffect in strongly disordered Anderson-insulators extend-ing over surprisingly long spatial-scales. It was shownthat this effect is consistent with the mechanism proposedby KNS. The study also revealed that this spatial-scaleis limited by the phase-coherent length of the medium.Therefore the KNS effect is expected to be considerablyweakened by temperature while being only logarithmi-cally sensitive to the rate-dependence of a local poten-tial change. 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