Intrinsic electron-glassiness in strongly-localized Be films
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Intrinsic electron-glassiness in strongly-localized Be films
Z. Ovadyahu , X. M. Xiong and P. W. Adams Racah Institute of Physics, The Hebrew University, Jerusalem,91904, Israel, Department of Physics and Astronomy,Louisiana State University, Baton Rouge, Louisiana 70803.
Abstract
We present results of out–of-equilibrium transport measurements made on strongly-localized Beryllium films and demonstratethat these films exhibit all the earmarks of intrinsic electron-glasses. These include slow (logarithmic) relaxation, memory effects,and more importantly, the observation of a memory dip that has a characteristic width compatible with the carrier-concentrationof beryllium. The latter is an empirical signature of the electron-glass. Comparing various non-equilibrium attributes of theberyllium films with other systems that exhibit intrinsic electron-glasses behavior reveals that high carrier-concentration is theironly common feature rather than the specifics of the disorder that rendered them insulating. It is suggested that this shouldbe taken as an important hint for any theory that attempts to account for the surprisingly slow relaxation times observed inthese systems.
PACS numbers: 72.20.Ee 72.20.Ht 72.70.+m
INTRODUCTION
The interplay between static disorder and Coulombinteractions may precipitate a glassy state in an An-derson insulator. This ‘electron-glass’ scenario was dis-cussed in several papers [1–4]. In theory, this propertyis generic to all degenerate Fermi systems with local-ized states interacting via a Coulomb potential. Exper-imental evidence for these glassy effects, however, hasbeen somewhat scarce, presumably due to specific ma-terial requirements. It turns out that only systems withrelatively high carrier-concentration n exhibit relaxationtimes that can be conveniently monitored by transportmeasurements. Conductance relaxations that persist formany seconds, and memory effects characteristic of in-trinsic [5] electron-glass, seem to be peculiar to systemswith n > cm − [5]. A prominent group of materi-als that exhibit electron-glass behavior with long relax-ation times are granular metals; Al [6], Bi [7], Pb [7],Ni [8], and Au [9], all having high carrier-concentration n cm − . Hitherto, the only non-granular systems that exhib-ited intrinsic electron-glass behavior were crystalline andamorphous indium-oxide films (In O − x and In x O re-spectively) [5], which are ionic compounds.In this work we report on the low temperature trans-port properties of strongly-localized Be films, and demon-strate that they exhibit intrinsic glassy effects. Thisis the first non-granular mono-atomic system to showthese effects. These include logarithmic relaxation ofthe out-of-equilibrium conductance and, more impor-tantly, a memory-dip that has all the earmarks of in-trinsic electron-glass. Although beryllium, like a typicalmetal, has a Fermi energy E F of few electron-volts, it hasan unusually low density of states at E F [10]. A signa-ture of the Be peculiar density of states, namely, at theFermi energy, the density-of-states decreases with energy, is actually observed in our field effect measurements aswill be demonstrated below. The low density of statesof beryllium is presumably the main reason why stronglocalization is achievable in this material just by makingthe sample thin enough (yet still physically continuous). EXPERIMENTAL
Samples used in the experiments reported here were18 ± µ m glass slides. These were silver-painted on theirbackside so as to form a gate for the field effect mea-surements. The samples were typically 400 µ m wide and600 µ m long strips and had sheet resistance R (cid:3) in therange 23kΩ to 120kΩ at 295K and 100kΩ-160MΩ at ≈ ≈ ∝ exp[-( T T ) / ] with T in the range 100-900K. Fig-ure 1 illustrates this behavior for two of the samples thatwere used in this study.The different values of R (cid:3) in the studied series ofsamples were obtained by a judicious oxidation of theBe films in an oxygen-enriched chamber. The changeof the samples resistance was constantly monitored dur-ing the oxidation process. At the range of sample thick-ness d ≈
18 ˚A, the room-temperature sheet-resistance ofthe samples is ≈ (cid:3) at 4K. The sheet-resistance of sam-ples much thicker than 18 ˚A may not reach the quantumresistance at 4K (recall that R (cid:3) > ~ /e is a pre-requisitefor electron-glass behavior [5]). For example, in a previ-ous study [12] a beryllium film with a nominal thicknessTypeset by REVTEX 1 .15 0.20 0.25 0.30 0.35 0.40 0.45 0.5010 T =175K T =906K R (cid:1) ( k (cid:0) ) T -1/2 (K -1/2 ) FIG. 1: Resistance versus temperature plots for two of theberyllium films used in this study. d of ≈
20 ˚A, only slightly thicker than the films studiedhere, exhibited G(T) ∝ exp[-( T T ) / ] with T =1.6K. Thisshould be compared with T & , and the as-sociated larger localization length in this case means thatstrongly-localized behavior is attained only at tempera-tures that are well below the range covered here. Suchsamples are not included in our present study where oneof our goals is to compare the results with previouslystudied electron-glasses, which were measured at or near4K.The conductivity of the samples was measured usinga two-terminal ac technique employing a 1211-ITHACOcurrent pre-amplifier and a PAR-124A lock-in amplifier.Measurements reported below were performed with thesamples immersed in liquid helium at T=4.1K main-tained by a 100 liters storage-dewar, which allowed longterm measurements of samples as well as a convenientway to maintain a stable temperature bath. Unless oth-erwise indicated, the ac voltage bias was small enoughto ensure linear response conditions (judged by Ohm’slaw being obeyed within the experimental error). Fullerdetails of measurements techniques are given elsewhere[13].A variety of techniques were employed to character-ize the films microstructure. Fig.2 shows a Transmis-sion Electron Microscope (TEM) micrograph of a Be filmprepared in the same way and with similar thickness asthe samples used for the transport studies. The physicalcontinuity of the film is evident in the figure. On care- FIG. 2: Transmission Electron Microscope micrograph of ≈ < > planes of the hexagonal BeO (reflected as satel-lites in the Fourier transform plate). ful examination, the micrograph shows occasional fringesthat indicate the presence of small crystals. These wereidentified as BeO by direct imaging and further, by theirdiffraction pattern (interestingly, the diffraction patternof Be, being light on electrons, was presumably too weakto register a clear pattern over the background set bythe amorphous carbon support-film). The BeO crystalswere clearly observable in TEM dark-field imaging offtheir < > diffraction line. This enabled an estimate oftheir crystallographic size and partial volume in the sam-ples. Randomly distributed BeO crystallites of sizes upto 50-70˚A were observed in these dark-field images. Weestimate that less than ≈
10% of the film area is occupiedby fully oxidized Be crystallites, and therefore transportpresumably occur through non-oxidized Be phase. Yet,the insulating BeO crystallites, somewhat restrict thevolume available for conductivity (much like punchingholes in the film would). A result of this geometrically-constrained structure is that the transport properties ofthe films show some mesoscopic effects that one usuallyencounters in smaller systems measured at similar tem-peratures and comparable degree of disorder [14, 19].The physical continuity of the Be phase in the filmwas ascertained by performing local electron-energy-loss-spectroscopy (EELS) on the parts of the film that werenot occupied by BeO crystals. The EELS spectra takenfrom theses areas was consistent with that of metallic Be.A slight shift of energy, +3% of the peaks position inthe spectra was detected, possibly due to strain. Thepresence of free Be in the samples was also confirmedby X-ray Photoemission Spectroscopy, which was carriedout on the actual samples that were used for the trans-port measurements.2 R (cid:2) =40M (cid:3) (c) t (s) R (cid:4) =6.4M (cid:5) (b) G ( a r b . un i t s ) R (cid:6) =11M (cid:7) (a) FIG. 3: Non-equilibrium transport behavior of typical Befilms under different protocols; (a) After a quench-cool fromT ≈ ac from the (Ohmic) 20mV ac bias.(c) Same sample as (b) after the 20mV ac has been restored.Each graph is labeled with the average R (cid:3) of a sample underthe measurements conditions. RESULTS AND DISCUSSION
We turn now to the non-equilibrium transport proper-ties of the films. The first signature of glassy behavior inthese films is encountered upon quench-cooling the sam-ple to 4K; after an initial fast drop (reflecting the changein temperature), the conductance G keeps on decreasingslowly (logarithmically) long after the sample has reachedthe bath temperature. A typical quench-cooling proto-col is shown in Fig.3a. The figure also illustrates theslow conductance excitation process upon ‘stressing’ [15]the film with a non-Ohmic source-drain voltage Fig.3b.The ensuing relaxation of G after the source-drain volt-age was set back to its Ohmic value is shown in Fig.3c.These excitation-relaxation curves are clearly similar tothose previously observed in glassy In O − x samples [15]A controlled way to take the system out of equilib-rium is a change of the potential difference between thesample and a near-by gate. This technique has beenwidely used in the study of several electron-glasses [5–7, 13]. Among other things, it may be used to estimatea typical relaxation time τ under a given set of condi-tions [16]. An example of such protocol is illustrated inFig.4. In this protocol, one uses the conductance relax-ation law ∆G(t/t )= ∆G(1 sec) − a · log(t/t ) where t (cid:8) time since V g =105 V is established (cid:9) G / G ( % ) V g =105 VV g =0 V G ( a r b . un i t s ) t (s) FIG. 4: A gate-excitation protocol for a specific Be sample(R (cid:3) =40MΩ) . The insert depicts the characteristic log(t) de-pendence of the EG from which the typical relaxation time τ isestimated using the baseline conductance G (105V) (markedhere by the dashed line).measured independently. is the experimental resolution time, and the equilibriumvalue of G at V g =105V to extract the value of τ definedby G( τ ) ≡ G(1 sec) . A characteristic feature that is believed to be commonto all intrinsic electron-glasses is a memory-dip; this is acusp-like minimum in G(V g ) centered at the gate voltageV g where the system was allowed to equilibrate [5–7, 17].A conspicuous memory-dip was consistently observed inall our Be films. Fig.5 shows this feature for two Besamples in the studied series. For both samples, G(V g )scans were taken after a ≃
24 hour equilibration underV g =0 volt. Note first that the memory-dips have thesame shape and width independent of R (cid:3) and indepen-dent of wether the G(V g ) scans were taken by sweepingV g through the equilibrium-V g or symmetrically aroundit [13] (c.f., the lower graph of Fig.5).In previously studied electron-glasses the width of thememory-dip was found to systematically depend on thecarrier-concentration n of the system [17]. On the ba-sis of the G(V g ) data, we have estimated the typicalwidth in our Be films in the same manner as was donein [17]. This involves several stages; First the change ofcharge ∆Q associated with the cusp-width is estimatedfrom ∆V g by taking heed of the sample-gate capacitance.The relevant energy is then calculated using the beryl-lium ( ∂ n/ ∂µ ) E F and the screening length. This proce-dure gave the energy-width Γ ∗ (as defined in [17]) as ≈ n (Fig.4 of [17]) such Γ ∗ corresponds to n of or-der 10 -10 cm − . This is consistent with our Hall ef-3 .5027.5047.5067.5087.510 (cid:10) ±105V R (cid:11) =0.32M (cid:12) R (cid:13) =150M (cid:14) G ( a r b . un i t s ) -200 -100 0 100 2005.3555.3705.385 V g (V) (cid:15) +198V (cid:16) -198V-198V (cid:17) +198V FIG. 5: The conductance versus gate voltage for two Be films(labeled by their R (cid:3) ) illustrating the memory-dip structure.After the sample was allowed to equilibrate under V g =0 for ≈ g =0 in each volt-age direction (symmetrical G(V g ) scans) or, from -198V to+198V. Note that the structure is skewed due to the con-tribution of the anti-symmetric (equilibrium) field-effect (seetext). fect measurements that gave n ∼ =7-8 · cm − as wellas with the concentration predicted by band-structurecalculations ( n = 0.016 state/atom [18] tantamount to n ≃ . · cm − ). This correlation between the widthof the memory-dip and the carrier concentration of thematerial is an important empirical test for the intrinsicnature of the electron-glass [5].The G(V g ) traces (Fig.5) reveal some mesoscopic fluc-tuations (reproducible with V g scans) superimposed onthe memory-dip (note e.g., the modulation of G(V g )around V g ≈ − (cid:3) . Interestingly, ∆G/G(R (cid:3) )for Be is almost identical to that measured under thesame conditions in other electron-glasses. In Fig.6 wecompare the results of the current study with some olddata [19] taken on In O − x films exhibiting quite simi-lar behavior. A similar agreement is observed betweenthat data of Fig.6 and the results obtained on granularaluminum films (c.f., Fig.17 of [20]).On the other hand, the anti-symmetric part (c.f.,Fig.6) of the G(V g ) for the beryllium samples has the -1 Beryllium In O (cid:18) G / G ( % ) R (cid:19) (k (cid:20) ) -0.3-0.2-0.110203040 FIG. 6: The relative magnitude of the memory-dip versusR (cid:3) for the studied Be samples. The data of [15] are shownfor comparison. Insert: The slope (relative change of G per400V of V g ) of the anti-symmetric part of G(V g ) comparedwith typical results for In O − x samples. opposite slope to that observed in In O − x and In x O[5, 13]. The sign of this slope is controlled by the energyderivative of the thermodynamic density of states ∂ n/ ∂µ at the Fermi level; ∂ G(V g )/ ∂ V g ∝ ∂∂E ( ∂ n/ ∂µ ) E F . Theanti-symmetric part of ∂ G(V g )/ ∂ V g we observe in all ourBe films (represented by the dashed curves in Fig.5) isconsistent with the negative slope of ∂ n/ ∂µ | E F found intheoretical calculations for the Be band-structure [21].These calculations assumed an ideal Be crystal whichmight not be relevant for the disordered structure. It isnot uncommon however that band-structure features cal-culated for the perfect crystal persist in the disorderedmaterial (as actually observed for some optical proper-ties of Be [22]). The magnitude of the slope depends alsoon the film resistance as is shown for both In O − x andBe in the insert to Fig.6. Being a low density system, ∂∂E ( ∂ n/ ∂µ ) E F in In O − x is much larger than in metals,which in turn makes ∂ G(V g )/ ∂ V g larger.It is interesting to note that the currently known elec-tronic systems that exhibit intrinsic glassiness (with as-sociated long relaxation times) are quite diverse in mostother aspects. For example, in terms of microstruc-ture, there are in this group representatives of all typesof disordered structures; poly-crystalline (In O − x , andBe), granular or discontinuous (Al, Pb, Au, Ni), andamorphous (In x O, Bi). Most of these systems con-tain oxide, whether as an intrinsic part of the material4In O − x , In x O), or to stabilize a granular structure (e.g.,Al). However, the discontinuous Ni films, being pre-pared and measured under high-vacuum conditions [8],is oxygen-free and show the same effects as the otherintrinsic electron-glasses [8]. In some of these systemsthere may be local order due to superconductivity (Pb,Bi, the high n version of In x O, Be, Al) or magnetism(Ni) at the temperatures of the experiments but not inothers. Finally, all these systems obey some form of ac-tivated conductivity, G(T) ∝ exp[-( T T ) α ] with 0.3 <α< α is singled out in the group.In other words, the conductivity versus temperature lawG(T) exhibited by these systems is not due to a specifichopping mechanism. It does not set this group apartfrom other hopping systems that do not exhibit long re-laxation times. Indeed, a G(T) law that resembles isobserved in many disordered semiconductors while theirrelaxation times are very short [23] (presumably due totheir low n [5]).In fact, the only common feature of the materials thatshow long relaxation times appears to be their relativelyhigh carrier-concentration (in addition of course to be-ing strongly-localized thus exhibiting hopping conductiv-ity). The common, out-of-equilibrium features that allthese electron-glasses exhibit are suggestive of a genericmechanism. 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