Weak Localization and Transport Gap in Graphene Antidot Lattices
aa r X i v : . [ c ond - m a t . m e s - h a ll ] S e p Weak localization and transport gap in grapheneantidot lattices
J Eroms, D Weiss
Universit¨at Regensburg, Institut f¨ur Experimentelle und Angewandte Physik,D-93040 Regensburg, GermanyE-mail: [email protected]
PACS numbers: 73.23.-b,73.43.Qt,73.20.Fz
Submitted to:
New J. Phys.
Abstract.
We fabricated and measured antidot lattices in single layer graphene withlattice periods down to 90 nm. In large-period lattices, a well-defined quantum Halleffect is observed. Going to smaller antidot spacings the quantum Hall effect graduallydisappears, following a geometric size effect. Lattices with narrow constrictionsbetween the antidots behave as networks of nanoribbons, showing a high-resistancestate and a transport gap of a few mV around the Dirac point. We observe pronouncedweak localization in the magnetoresistance, indicating strong intervalley scattering atthe antidot edges. The area of phase-coherent paths is bounded by the unit cell sizeat low temperatures, so each unit cell of the lattice acts as a ballistic cavity.
Since its discovery in 2004 [1], graphene and its peculiar relativistic dispersionrelation have inspired experimentalists and theorists alike [2]. In the past few years,many fundamental effects have been demonstrated in graphene, often with specificmodifications due to the properties of Dirac fermions. For example, the quantum Halleffect in single layer graphene shows steps at values ( n + 1 /
2) 4 e /h (where n is aninteger), which is caused by the chiral motion of the carriers [3, 4]. Weak localizationwas found to be suppressed in some samples [5], and is generally more complex than indiffusive metals, since the strong isospin-orbit coupling leads to a destructive interferenceof time-reversed paths [6, 7]. The interplay of several scattering mechanisms then resultsin very rich physics, from weak localization to antilocalization, or complete suppressionof both. Electronic confinement in nanoribbons turns out to be linear in the inverseconstriction width, much stronger than in materials with a quadratic dispersion [8].In our study, we turn to a periodic nanoscale structure, namely antidot lattices [9].Here, a periodic lattice of holes is etched into a two-dimensional layer. Inhigh-mobility two-dimensional electron gases, pronounced oscillations appear in themagnetoresistance, which is a consequence of the the electron motion in the chaoticantidot billiard [10]. With the mobilities available in graphene layers on oxidizedsilicon substrates, clear ballistic effects of this kind are not yet feasible. However, since eak localization and transport gap in graphene antidot lattices R ( ) xx W R ( ) t W ( e / h ) xy s -6-10-14 sample E200sample E100 B = 10 TT = 1.6 K -2500 nm500 nm 6 102DP Figure 1.
Quantum Hall effect in single layer graphene antidot lattices. Upper panel:Lattice period a = 200 nm. A large number of oscillations is visible. Lower panel: a = 100 nm, 3-terminal resistance. Here, the quantum Hall effect can only be observedif a complete cyclotron orbit can pass between the antidots (sketch in lower rightinset). The scanning electron micrographs show the antidot pattern in samples E200and E100. DP marks the position of the Dirac point. the phase coherence length in graphene can approach 1 µ m at He temperatures, weexpect to see modifications of weak localization. Furthermore, Pedersen et al. recentlysuggested to use graphene-based antidot lattices to create a transport gap and confineelectrons in a tailor-made point defect in such a lattice [11]. As we will show, our antidotlattices show clear signatures of electron paths confined to phase-coherent unit cells ofthe lattice and exhibit a high-resistance state around the Dirac point.The graphene flakes were prepared by exfoliating natural graphite with the adhesivetape technique described in [1]. A highly doped Si-wafer with 300 nm of thermallygrown silicon dioxide was used as a substrate. Single layer flakes were identified by thecontrast of a grayscale image in the optical microscope [12]. We verified the reliability ofthis method with Raman spectroscopy and quantum Hall measurements. In total, fivesingle layer flakes (A to E) were selected. We patterned contacts with electron beamlithography using pre-defined markers, metal evaporation and lift-off. The metalizationwas either Ti/AuPd (flakes A to D) or Pd (flake E). The antidot lattices were preparedwith O reactive ion etching (flakes A to D) or Ar ion beam etching at 350 V (flake E).The sample names consist of the flake and the antidot lattice period a in nm. Table1 lists the lattice period a and the antidot diameter d of all the samples in this study. eak localization and transport gap in graphene antidot lattices Table 1.
Overview of the antidot samples and their geometry. Length and width referto the patterned region of the flake.Sample name lattice period a antidot diameter d length widthA140 140 nm 65 nm 2.2 µ m 1.3 µ mB140 140 nm 80 nm 3.4 µ m 2.5 µ mB200 200 nm 70 nm 3.2 µ m 2.5 µ mC200 200 nm 150 nm 5.0 µ m 1.7 µ mD90 90 nm 54 nm 5.1 µ m 2.9 µ mD100 100 nm 60 nm 5.1 µ m 2.9 µ mD110 110 nm 65 nm 5.0 µ m 2.9 µ mE100 100 nm 62 nm 4.0 µ m 4.0 µ mE200 200 nm 67 nm 4.0 µ m 4.0 µ mE400 400 nm 64 nm 4.0 µ m 4.0 µ m -60 -40 -20 0 20 40 600.010.11 20 K10 K7 K3 K1.6 K c ondu c t an c e ( e / h ) back gate voltage (V) Sample D90 T -1 (K -1 ) from IV R () W
300 pA ac
C200 R ( W ) V = 5.7 V g D90 T -1 (K -1 ) Figure 2.
Insulating behaviour of sample D90 (lattice period a = 90 nm) at varioustemperatures. The conductance at 1.6 K can be varied by 2 orders of magnitudein the accessible gate voltage range. Insets: the temperature dependence of samplesC200 (with very large antidots) and D90 at the Dirac point measured with an acbias of 300 pA and 1 nA, respectively. The zero bias data point of C200 at 1.6 K,taken from an I - V -curve, shows that the thermally activated behaviour continues tolow temperatures. The thin gray line corresponds to thermal activation with a 22 Kenergy gap. eak localization and transport gap in graphene antidot lattices d g ( e / h ) d g ( e / h ) d g ( e / h ) Figure 3.
Weak localization in samples E400, E200, and E100. Note the differentfield scale for sample E100. The thin gray line is a fit to equation 1, the dashed lineis a fit to the WL correction in a dirty semiconductor, with the same phase coherencelength. The graphs at 48 K were shifted vertically for clarity.
The length and width of the patterned region are also given.The measurements were performed in a He-cryostat at temperatures between 1.6 Kand 48 K using lock-in techniques with an ac bias current of 10 nA or lower. The carrierdensity was varied using the Si wafer as a back gate. Baking the sample in situ at 150 o Cfor several hours improved the homogeneity of the samples and moved the Dirac pointcloser to zero back gate voltage.Figure 1 shows a quantum Hall measurement at B = 10 T on flake E. Sample E200(and also E400, not shown here) shows clear Hall plateaus at quantized values ( n + ) e h and Shubnikov-de Haas oscillations in the longitudinal resistance, as expected for singlelayer graphene. While in E200 a large number of oscillations is visible, sample E100 ‡ only shows minima at filling factors ν = 2, 6, 10, and 14, i.e., at low carrier densities.Calculating the diameter of a cyclotron orbit 2 R C = 2¯ hk F /eB at the last visible fillingfactor, ν = 14 yields 2 R C = 40 nm, which corresponds to the lithographic width a − d of the constrictions between the antidots. At higher densities, the cyclotron orbits are ‡ During that measurement, only 3 contacts were working, so we could only measure the 3-terminalresistance. It always includes one contact resistance and either R xx or R , depending on the magneticfield direction and carrier polarity [13]. Here, at gate voltages below the Dirac point, the signal is R xx -like and we observe SdH-oscillations. The poor visibility of the ν = − B = 0 at the same back gate voltage. eak localization and transport gap in graphene antidot lattices a-d (nm) L ( n m ) f (a) T (K) L ( n m ) f (c) E400E100E200 T -0.5 L i ( n m ) (b) L i ( n m ) E400E100E200 (d) a-d (nm)
T (K)
T = 1.6 K T = 1.6 K
Figure 4. (a) and (b): Geometry dependence of the phase coherence length (a) andthe intervalley scattering length (b) from the WL fit for all samples in this study. Thedistance a − d refers to the width of the constrictions between the antidots. The dashedlines are guides to the eye. (c) and (d): Temperature dependence of L φ (c) and L i (d)of sample E100, E200, and E400. The error bars on the data of E400 in subfigure (d)give the range of L i when L ∗ is varied between 70 nm and 700 nm. larger and cannot move freely along the sample, therefore the quantum Hall effect issuppressed. In sample D90, with even smaller constrictions, only the plateau at ν = 2 isvisible (not shown). The mobility in our flakes was ranging between µ = 5000 cm /Vsand µ = 7000 cm /Vs. For example, at a carrier density of 10 cm − this leads to a meanfree path between 60 nm and 80 nm. Therefore, the mean free path was not sufficient toallow the observation of clear commensurability effects in the antidot lattices. Shen et al. recently claimed the observation of commensurability effects in graphene antidot latticeson SiC [14], although without an independent determination of the carrier density. eak localization and transport gap in graphene antidot lattices I - V -measurements on sample C200, we extract agap of around 6 mV along the whole antidot lattice § . The resistance at the Dirac pointis thermally activated with an activation gap of about 22 K · k B for samples C200 andD90. Since the lock-in measurement was performed with a finite ac current bias, theresistance apparently saturates once the sample voltage exceeds the size of the gap.Taking the true zero bias resistance from the full I - V -curve, we see that in fact thethermal activation continues down to 1.6 K (see left inset in figure 2).All our samples show very clear weak localization (WL) in the magnetoresistance(see figure 3). This is in strong contrast to the situation in unpatterned grapheneflakes of similar size, where conductance fluctuations often mask the localizationfeature completely and averaging over a range of gate voltages is necessary to unveilWL [7, 17]. More importantly, due to the strong isospin-orbit coupling in graphene theWL contribution is suppressed. WL only appears if the rate of intervalley scattering τ − (which breaks the isospin-orbit coupling) is higher than the phase-breaking rate τ − φ .Finally, the scattering rate τ − ∗ = τ − + τ − z + τ − w (where τ − w is due to trigonal warpingand τ − z is the intravalley scattering rate) also enters into the description of WL ingraphene [6]. The conductivity correction δg in diffusive, bulk graphene is governed bythe following equation: δg = e πh F BB φ ! − F BB φ + 2 B i ! − F BB φ + B ∗ !! (1)where F ( z ) = ln z + Ψ(0 . z − ), B φ, i , ∗ = ¯ hτ − φ, i , ∗ / eD and Ψ is the digammafunction [6]. D is the diffusion constant, and τ φ, i , ∗ are the respective scattering times.The corresponding lengths are L φ, i , ∗ = q Dτ φ, i , ∗ . The first term in equation 1 leads to apositive magnetoconductance and is exactly compensated by the second term, unless B i is large, i.e., strong intervalley scattering is present. The first term is also identical tothe magnetic field dependence of the WL in a dirty, two-dimensional semiconductor inthe limit where the mean free path is much shorter than the phase coherence length [18]. § We cannot determine if the voltage drop is distributed evenly between the antidot lattice cells, or ifthe voltage drops predominantly at the narrowest constriction. eak localization and transport gap in graphene antidot lattices L ∗ varying between 70 nmand 700 nm. The intervalley scattering length L i then simultaneously varies by about afactor of 1.7, where a shorter L ∗ leads to a longer L i or vice versa. The phase-coherencelength L φ , however, is always unambiguously defined, regardless of the values chosenfor L ∗ and L i , since it is taken from the positive magnetoconductance around B = 0.Figure 3 shows the conductivity correction δg (obtained by subtracting the zero-field conductance from the magnetoconductance) of the samples E400, E200, and E100at 1.6 K and 48 K and a voltage of 20 V applied to the back gate. Contrary tounpatterned graphene samples of mesoscopic size [7], in the antidot lattices the WLfeature is clearly visible without averaging over a range of gate voltages. We fittedthe data using equation 1, and also, for comparison, to the expression for a dirtysemiconductor. Both fits describe the positive magnetoconductance at low fields, givingthe same phase coherence length. Obviously, the theory of a diffusive semiconductorfails to reproduce the graphene data at higher fields, while the fit to equation 1 is verygood over the entire field range.Let us first consider intervalley scattering. Since a WL feature is visible, intervalleyscattering must be present in our samples. The etched boundaries of the antidotsare an obvious source of intervalley scattering. This can be seen by comparing thedata for different lattice periods at 1.6 K and 48 K and looking at the temperaturedependence of L φ (see figure 4). For samples E400 and E200, L φ decreases as T − . athigh temperatures. At 48 K, most electron paths will not reach an antidot edge withina phase-coherence length. Intervalley scattering is therefore not effective to make theWL correction visible. Indeed, the experimental data of E400 only shows a very faintWL feature at 48 K, in E200, WL at 48 K is visible but weak. Sample E100, where L φ allows to explore the entire unit cell even at 48 K, correspondingly shows a well definedWL feature at 48 K.In figures 4 b and 4 d we show the geometry and temperature dependence of L i as fitting parameter in equation 1. The error bars in the data for sample E400 infigure 4 d show the interdependence of L i and L ∗ . For example, the data point at T = 1 . L i is obtained by taking L ∗ between 200 nm and 700 nm, whereas L i = 160 nm is found by taking L ∗ = 70 nm.Comparing L i and the constriction width a − d (which is the shortest length scaleimposed by patterning) for all samples, we find that the intervalley scattering length L i from the diffusive fit is always much shorter than a − d . For example, for sampleE400 with a lattice period of 400 nm the diffusive theory yields L i ≈
100 nm. Wedo not expect that additional sources of intervalley scattering, other than the antidot eak localization and transport gap in graphene antidot lattices L i is an artefact of thediffusive theory, which assumes that the mean free path is much shorter than any otherlength scale of the system. In our samples, the mean free path is between 60 nm and80 nm, and the antidots, as sources of intervalley scattering, are introduced on a similarlength scale. In the diffusive theory, to get a clear WL feature, L i must be much shorterthan L φ , whereas in our situation each charge carrier hitting an antidot is scatteredbetween the valleys and WL is restored. A recent numerical study supports this view:In a tight-binding simulation a ballistic cavity with either abrupt scattering at the edges(strong intervalley scattering) or with mass confinement (no intervalley scattering) wasconsidered. The magnetoconductance only shows WL when intervalley scattering occursat the cavity boundary [19]. The WL correction to the magnetoconductance in a phase-coherent chaotic cavity is given by a Lorentzian [20], whose width corresponds to theflux through the cavity area. Within the experimental uncertainty, the measured datacan be fitted both with the ballistic and the diffusive theory. Fitting a Lorentzian (i.e.,the result of the ballistic theory) with the diffusive theory gives a phase coherence lengthof the order of the square root of the cavity area but the intervalley scattering length ismuch shorter than any of the cavity dimensions. We therefore conclude that the poorcorrespondence of the values for L i with any of the relevant lengths in our samples showsthat our samples are better described by the ballistic theory than the diffusive theory.Now we turn to the phase coherence length. In a truly ballistic sample, the WLcorrection should follow a Lorentzian [20] rather than the shape described by equation 1.However, since equation 1 contains 3 parameters, it is next to impossible to distinguishbetween both theories within the experimental uncertainty. Taking L φ = √ A , where A is the characteristic area from the Lorentzian fit, both theories give the same resultswhen fitting the experimental WL correction. In an unpatterned reference section onflake D, using equation 1, we determined L φ = 700 nm and L i = 550 nm at 1.6 K,i.e., larger than all the lattice periods we fabricated. In the antidot lattices, the phasecoherence length is limited by the lattice geometry as can be seen in figure 4 a. Moreprecisely, at 1.6 K, the lattice period sets a limit to the phase coherence length, whilein samples with a large antidot spacing the phase coherence length falls off as T − . at higher temperatures. Since the WL correction actually probes an area distributionof time-reversed paths we believe that the electron paths are confined to the ballisticunit cells of the antidot lattice, which are only weakly coupled by the nanoribbon-likeconstrictions. Alternatively, one might argue that the phase coherence is simply cutoff by scattering at the antidot edges. If that were the case, no closed coherent pathscould be formed, since the electrons would lose their phase memory when hitting anantidot, and WL would not be observed. The fact that we do observe a pronounced WLfeature shows that electrons experience intervalley scattering at the antidot edges, whichis essential to observing WL in graphene, but retain their phase, otherwise the closed eak localization and transport gap in graphene antidot lattices Acknowledgments
We would like to thank M. Hirmer, T. Korn and C. Sch¨uller for the Ramanmeasurements, and ˙I. Adagideli, C. Beenakker, J. Bundesmann, D. Horsell, E. Mucciolo,K. Richter, M. Wimmer and J. Wurm for fruitful discussions. [1] Novoselov K S, Geim A K, Morozov S V, Jiang D, Zhang Y, Dubonos S V, Grigorieva I V andFirsov A A 2004
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