Sample Size Effects on the Transport Characteristics of Mesoscopic Graphite Samples
J. Barzola-Quiquia, J.-L. Yao, P. Rödiger, K. Schindler, P. Esquinazi
aa r X i v : . [ c ond - m a t . m t r l - s c i ] S e p physica status solidi, 1 November 2018 Sample Size Effects on the TransportCharacteristics of MesoscopicGraphite Samples
J. Barzola-Quiquia, J.-L. Yao, P. R ¨odiger, K. Schindler, P. Esquinazi * Division of Superconductivity and Magnetism, Universit¨at Leipzig, Linn´estraße 5, D-04103 Leipzig, GermanyReceived XXXX, revised XXXX, accepted XXXXPublished online XXXX
PACS ∗ Corresponding author: e-mail [email protected] , Phone +49-341-9732751, Fax +49-341-9732769
In this work we investigated correlations between theinternal microstructure and sample size (lateral as wellas thickness) of mesoscopic, tens of nanometer thickgraphite (multigraphene) samples and the temperature ( T ) and field ( B ) dependence of their electrical resis-tivity ρ ( T, B ) . Low energy transmission electron mi-croscopy reveals that the original highly oriented py-rolytic graphite material – from which the multigraphenesamples were obtained by exfoliation – is composed of astack of ∼ nm thick and micrometer long crystallineregions separated by interfaces running parallel to thegraphene planes. We found a qualitative and quantita-tive change in the behavior of ρ ( T, B ) upon thickness ofthe multigraphene samples, indicating that their internalmicrostructure is important. The overall results indicate that the metallic-like behav-ior of ρ ( T ) at zero field measured for bulk graphitesamples is not intrinsic of ideal graphite. The resultssuggest that the interfaces between crystalline regionsmay be responsible for the superconducting-like prop-erties observed in graphite. Our transport measure-ments also show that reducing the sample lateral sizeas well as the length between voltage electrodes de-creases the magnetoresistance, in agreement with re-cently published results. The magnetoresistance of themultigraphene samples shows a scaling of the form( ( R ( B ) − R (0)) /R (0)) /T α = f ( B/T ) with a sampledependent exponent α ∼ , which applies in the wholetemperature 2 K ≤ T ≤ K and magnetic field range B ≤ T. Copyright line will be provided by the publisher
Ideal graphite consists on layers ofhoneycomb lattices of carbon atoms, characterized by twonon-equivalent sites, A and B, in Bernal stacking config-uration (ABABAB . . . ). Although there is not yet consenton the interlayer cohesive or binding energy per carbonatom between graphene layers in ideal graphite [1], severalexperimental facts suggest that this coupling is not largerthan ∼ . eV [2]. The huge anisotropy in the resistivity(ratio of the in-plane divided out-of-plane resistivities) ∼ ρ a /ρ c & at low and room temperatures,respectively, in good quality samples indicate the quasitwo-dimensionality of the transport in the graphene layersof the graphite structure [2]. On the other hand, the effectsof the real microstructure, the size and quality of the crys-talline regions and their interfaces or even the influence ofthe overall sample size on the transport properties in single graphene, tens of graphene layers (multigraphene) as wellas in bulk graphite samples are still not well studied.A quick comparison between the published tem-perature ( T ) dependence of the resistivity ρ ( T ) at zeromagnetic field ( B ) suggests that ρ ( T ) behaves differentlyin graphene and in high quality graphite. Whereas highlyoriented pyrolytic graphite (HOPG) with narrow rockingcurve widths shows a ρ ( T ) that decreases decreasing T . K, ρ ( T ) of single graphene layers appearsto steadily increase decreasing T , keeping the electrondensity n low enough (see for example [3] and referencesinside). Although the influence of the substrates on theelectronic transport of graphene is significant and can-not be neglected [4], the overall published experimentaldata do suggest that the ubiquitous decrease of ρ ( T ) (at T . K) appears only in good quality HOPG samples
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Barzola-Quiquia et al.: Sample Size Effects [5]. It is interesting to note that it remains still unclear,which is the absolute resistivity (parallel and perpendic-ular to the graphene planes) as well as the temperaturedependence of defect free, ideal graphite. This is a basicquestion that still remains fully open due to the sensitivityof the graphite structure to defects. The results presentedin these studies suggest that the resistivity of ideal graphiteis not metal-like, i.e. ρ ( T ) decreases with T in all the T − range.Our work provides experimental hints on the cor-relations between the internal structure of the samplesand their transport properties. The results are not onlyimportant to understand ideal graphite/graphene propertiesbut also to understand the role of defects and/or inter-faces within the graphite structure. Recently done highresolution magnetoresistance measurements on multi-graphene samples show anomalous hysteresis loops belowa “critical” temperature that indicates the existence ofsuperconducting grains with high critical temperature em-bedded in a semiconducting matrix [6]. Those results arethe last ones of a series of experimental hints suggestingthe existence of granular superconductivity in graphite (fora short review see Ref. [7]). The data presented in thisstudy provide a hint where these superconducting regionsmight be located. Size effects
Recently published experimental workon highly oriented pyrolytic graphite (HOPG) showed thatthe change of the electrical resistance with magnetic field,i.e. the ordinary magnetoresistance (MR), decreases withthe sample size even for samples hundreds of micrometerlarge [8]. This effect was ascribed to the large carrier meanfree path ℓ as well as the large Fermi (or de Broglie) wave-length λ F in graphite. Recently, Garc´ıa et al. [9] reportedthe development of an experimental method and its theoret-ical basis to obtain without free parameters these two trans-port properties based on the measurement of the resistancethrough micro-constrictions on a ∼ µ m thick HOPGsample. In that work micrometer large values for ℓ and λ F at T ≤ K were obtained in agreement with the expec-tations from the magnetoresistance results[8]. The carrierswith the largest mean free path have ℓ ∼ µ m appear tobe limited by the crystallite size in the used high qualityHOPG sample [9].In the last 50 years, the transport properties of graphitehave been interpreted in terms of two- (three-) bandBoltzmann-Drude approach[1]. However, the use of thestandard approaches to understand the electrical propertiesof graphite is doubtful since due to the large carrier meanfree path ballistic instead of diffusive transport should beapplied. The large Fermi wavelength (due to the smallcarrier density of the carriers λ F ∼ µ m) implies thatdiffraction effects within the sample may play also arole. Moreover, the differences in the transport propertiesbetween apparently similar samples together with elec-tron force microscopy (EFM) results obtained on HOPGsamples [10] provide further evidence that HOPG samples should be considered as a non-uniform electronic system.This non-uniformity is not an intrinsic property but itdepends on parameters like the defect density or inter-faces within the measured sample region and thereforewe expect that the electrical resistance will depend on themeasured sample size. The overall results presented inthis study especially the sample size effects confirm oncemore the inadequacy of the standard models to understandthe transport properties of graphite. All these results aswell as the possibility that granular superconductivitycould influence the transport of HOPG cast doubts on theapplicability of the theoretical descriptions as has beendone in the literature up to date to understand the transportproperties of graphite. Single graphene, multigraphene and the role ofthe microstructure
Nowadays there is a general interestof the solid state community on the transport properties ofsingle graphite layers dubbed graphene [11]. As in the veryfirst transport experiments done in graphene [12,13], thesesamples are usually fixed on substrates. In general neitherthe possible variations of the electrical potential at the sur-face of the substrates as well as their shape variations northe influence of the environment were considered impor-tant issues that may influence the transport properties ofthe graphene samples. Recent experimental evidence ob-tained in suspended graphene samples appears to confirmthe detrimental effect of the substrates on the mobility ofgraphene [14,15]. The lowest achievable electronic den-sity ( n & cm − ) in free standing as well as fixed-on-substrate graphene samples is still far away from theDirac point, due to defects in the graphene structure, intrin-sic bending or due to the influence of the substrate itself.This restriction limits the Fermi wavelength, λ F , and there-fore the largest achievable mobility ( µ = ( e/h ) ℓλ F ) sincethe carrier mean free path ℓ cannot be larger than the sizeof the graphene samples. In fact, recently published work[4] showed that the minimum conductivity is governed notby the physics of the Dirac point singularity but rather bycarrier-density inhomogeneities induced by the potential ofcharged impurities that may come from the substrate.One possibility to overcome these limitations is to useseveral nanometers thick multigraphene samples, whichare much less sensitive to the environment as well as to thesubstrates. In high quality graphite samples the graphenelayers inside are of larger perfection than single graphenelayers. Therefore it should be possible to obtain theirintrinsic transport properties without external influenceother than those coming from the lattice defects. A directproof of the perfection of the graphene layers in highquality graphite is given by the very low carrier densitymeasured at low temperatures, which is nearly two ordersof magnitude smaller than the minimum obtained forsuspended graphene samples [9]. In this work we studied the temperature and magneticfield response of the electrical resistivity of multigraphenesamples with thickness between 10 nm and 20 µ m. Fur- Copyright line will be provided by the publisher ss header will be provided by the publisher 3 ther characterization of the microstructure was done with alow voltage transmission electron microscope (TEM) thatallowed us to observe some details of the internal structureof the samples as the typical distance between the inter-phases separating crystalline regions. Micro-Raman stud-ies on some of the nanometer thick samples were also per-formed. We provide evidence for two size effects on thetransport properties of multigraphene samples. One is re-lated to the thickness and correlates to the internal mi-crostructure of the samples. The second size effect dealswith the influence of the sample lateral size on the transportproperties. The paper has four more sections. Section 2 ex-plains some details of the experimental methods we used.Section 3 describes the measured samples and their char-acteristics and includes the TEM and Raman results. Sec-tion 4 shows the main transport results. This section is di-vided in three subsections where we describe the differentbehavior of the transport properties upon the sample size.The main conclusions are given in Sec. 5
In order to carry out a system-atic study we have performed measurements in differenttens of nanometer thick multigraphene samples obtainedby exfoliation from the same highly oriented pyrolyticsample with a mosaicity of . ◦ ± . ◦ (HOPG(0.4)). Onepart of the original HOPG sample was left with a thicknessof ∼ µ m, which transport properties resemble theusual behavior observed in high-quality HOPG of similarcharacteristics [5].The initial HOPG material of dimensions × × . mm was glued on a substrate using GE 7031 varnish.We used a simple technique to produce the multigraphenefilms, which consists in a very carefully mechanical pressand rubbing the initial material on a previously cleanedsubstrate. As substrate we used p-doped Si with a 150 nmSiN layer on top. This substrate helps to select the multi-graphene films because – in comparison with Si substrateswith a top layer of SiO – the SiN layer provides a highercolor contrast allowing us to use optical microscopy toselect the film. After the rubbing process we put threetimes the substrate containing the multigraphene filmsin a ultrasonic bath during 2 min using high concentrateacetone. This process cleans and helps to select only thegood adhered multigraphene films on the substrate. Afterthis process we used optical microscopy and later scanningelectron microscopy (SEM) to select and mark the positionof the films. For the production of the electrical contactswe used conventional electron lithography process. Af-terwards the contacts were done by thermal deposition ofPd (99,95%) in high vacuum conditions. We have usedPd because it does not show any Schottky barrier whenused with carbon. Measurements of the resistance of thePd-electrodes alone showed negligible magnetoresistance.For the transport measurements the sample was gluedon a chip carrier. The contacts from the chip carrier to the electrodes on the sample substrate were done using a µ m gold wire fixed with silver paste.The advantage of using HOPG of good quality isthat in these samples and due to the perfection of thegraphene layers and low coupling between them, a lowtwo-dimensional carrier density × cm − . n . cm − in the temperature range 10 K . T . Kis obtained [9]. The carrier density values obtained inRef. [9] are smaller than in typical few layer graphene(FLG) samples probably due to lattice defects generatedby the used method to produce them and/or surface doping[14,15].
Transport measurements
Low-noise four-wires(two for the input currents and two for the voltage mea-surement) resistance measurements have been performedby AC technique (Linear Research LR-700 Bridge with8 channels LR-720 multiplexer) with ppm resolutionand in some cases also with a DC technique (Keithley2182 with 2001 Nanovoltmeter and Keithley 6221 currentsource). The temperature stability achieved was ∼ . mKand the magnetic field, always applied normal to thegraphene planes, was measured by a Hall sensor – justbefore and after measuring the resistance – and locatedat the same sample holder inside a superconducting-coilmagneto-cryostat. We used current amplitudes between and µ A. The measurement of the resistance at differ-ent positions of the same sample indicate that the contactresistance contributions in the absolute value as well asin the temperature and magnetic field measurements arenegligible, as expected due to the four-wire configurationused. The magnetoresistance measurements were donealways with the magnetic field applied perpendicular tothe graphene planes, i.e. parallel to the samples c-axis.
Transmission Electron Microscopy
The images ofthe internal structure of the HOPG sample were obtainedusing a Nova NanoLab dual beam microscope from theFEI company (Eindhoven). A HOPG lamellae was pre-pared for transmission electron microscopy (TEM) usingthe in-situ lift out method of the microscope. The TEMlamellae of HOPG was cut perpendicular to the graphenelayers. Therefore, the electron diffraction provided infor-mation on the crystalline regions and their defective partsparallel to the graphene layers. After final thinning, thesample was left on a TEM grid. A solid-state scanningtransmission electron microscopy (STEM) detector forhigh-resolution analysis of thinned samples was used. Thevoltage applied to the electron column was 18 kV and thecurrents used were between 38 to 140 pA.
Micro-Raman Spectroscopy
Raman spectra ofmultigraphene samples were obtained at room temperatureand ambient pressure with a Dilor XY 800 spectrometer at514.53 nm wavelength (Green) and a µ m spot diameter.The incident power was varied between 0.5 to 3 mW tocheck for possible sample damage or laser induced heatingeffects. No damage and significant spectral change wasobserved in this range of incident power. Copyright line will be provided by the publisher
Barzola-Quiquia et al.: Sample Size Effects
Table 1 shows the sam-ple dimensions and names. The error bars in the thicknessare the maximum estimated ones, taking into account themaximum error in the measurement and calibration of theoptical and/or atomic force microscope (AFM) as well asthe irregularities in the samples borders. The error in theabsolute value of the resistivity takes those errors into ac-count as well as errors in the width and length. Optical pic-tures of some of the measured samples are included in thefigures below.
Figure 1 shows the bright field (a) and dark field (b) detailsobtained with the low-voltage STEM. Figure 1(c) shows ablow out of a detail of (a). The different gray colors indi-cate crystalline regions with slightly different orientations.The images indicate that the average thickness of the crys-talline regions is ± nm. One can also resolve theinterfaces perpendicular to the c-axis of the layers and be-tween the regions as well as the end parts of the crystallineregions along the graphene layers direction, see Fig. 1(c).Electron back scattering diffraction measurements done onsimilar HOPG samples indicate that the typical size of thesingle crystalline regions (on the (a,b) plane) ranges be-tween 1 to ∼ µ m [9]. If the interface between the crys-talline regions as well as the defects in the crystalline lat-tice have some influence on the transport properties wewould expect to see a change in the behavior of the trans-port properties between samples of thickness of the orderor less than the average thickness of the crystalline regions. The Raman spectrum ofgraphite [16], a few layers and single graphene [17,18]have been thoroughly measured and discussed in recentpublished studies and review. In this study we are inter-ested in three Raman maxima, namely the G-peak around1582 cm − due to a Raman active in-plane optical phononE g , its neighbor peak, the D-line around 1350 cm − ,which is very sensitive to the amount of lattice disorder,and the D’-line[17] (or 2D peak[18]) at ∼ cm − andits splitting in two or more maxima upon the number ofgraphene layers the sample has [17].Figure 2 shows the Raman spectra of samples L5 (up-per picture) and L2A (bottom) around the D’-line. Thesplitting into two peaks of the D’-line is in agreement withrecently published results [17]. The main central narrow Table 1
Samples’ names, resistivity and size.
Name resistivity thickness length width ρ (4 K )[ µΩ cm]HOPG ± ± µ m 4.4 mm 1.1 mmL5 ±
15 12 ± nm µ m µ mL8A ±
100 13 ± nm µ m µ mL2A ±
33 20 ± nm µ m µ mL8B ±
25 45 ± nm µ m µ mL7 ± ± nm µ m µ m peak is at ± cm − whereas the value of the split-ting is ± cm − in very good agreement with that ob-served for HOPG bulk. Taking into account that both sam-ples L5 and L2A have a thickness between 10 and 20 nm,this difference in thickness does not affect the Raman D’-line. Taking this line as reference one would conclude thatboth samples are identical.However, according to literature the structural qualityof the samples can be resolved by investigating the D-lineat ∼ cm − . We note that also the edges of the sam-ples as well as the borderlines between regions of differ-ent thicknesses may contribute to the D-band signal. TheRaman spectra between 1300 and 1700 cm − have beenmeasured at different positions of sample L5 and in sam-ple L2A. The results are shown in Fig. 3 for the two sam-ples. The broad D-line at ∼ cm − is clearly seen insample L5 (at the same position as in Fig. 2) but it is com-pletely absent in sample L2A. We checked that a similarcurve is observed at different positions of the sample L5.The small peak at ∼ cm − in sample L2A is due tothe substrate. The G line at ≃ cm − is observed forboth samples, see Fig. 3. From these results we would con-clude that sample L5 has more disorder or that the bordersor edges have a larger influence to the Raman spectra thanin sample L2A. From literature[19] we would expect thatthis disorder should have some influence on the transportproperties of graphene as well as in a multigraphene sam-ple. Figure 4 showsthe resistivity at 4 K of the six measured samples vs. theirthickness. It is clearly seen that the resistivity decreasesincreasing the sample thickness. The average change inresistivity between ∼ nm to µ m thick samples isabout two orders of magnitude, far beyond geometrical er-rors. A similar behavior was observed recently in multi-graphene samples obtained with a different, micromechan-ical method[20]. Because in that work no explicit absolutevalues of the resistivity at 4 K were given, we estimate ittaking the in that work given mobility, assuming that thecarrier density does not depend on the thickness and fixingarbitrarily the value of µΩ cm for the 12 nm thick sam-ple reported in Ref. [20]. The open circles shown in Fig. 4are the data points from Ref. [20]. A reasonable agreementbetween the two independently obtained measurements isobtained that speaks for the reproducibility of the observeddependence.The authors in Ref. [20] suggested that the decreaseof mobility µ (i.e. an increase in the resistivity at con-stant carrier density) decreasing sample thickness providesan evidence for boundary scattering. Taking into accountthe fact that one graphene layer shows finite mobility [14,15], boundary scattering is certainly not the correct expla-nation for the observed behavior. A possible explanation Copyright line will be provided by the publisher ss header will be provided by the publisher 5
Figure 1
Transmission Electron Microscopy pictures taken parallel to the graphene layers of the HOPG lamelle. Thec-axis is perpendicular to the clearly observable stripes of different gray colors, each representing a crystalline region witha slightly different orientation. The arrows in (a) and (b) indicate 400 nm length scale and in (c) 100 nm.for the observed trend is that the larger the thickness thelarger is the amount of defects and interfaces in the sam-ple, see Fig. 1, that produces the decrease in the resistiv-ity. Since HOPG is a highly anisotropic material with hugeanisotropy in the resistivity, it appears reasonable to as-sume that certain kind of lattice defects (vacancies, dis-locations, etc.) may produce a sort of short circuits be-tween layers, changing the dimensionality of the carriertransport and decreasing the resistivity. It appears unlikely,however, that randomly distributed point-like lattice de-fects can be the reason for the observed behavior. The re-sults suggest the existence of a kind of thickness thresholdaround ∼ nm for the muligraphene samples obtainedfrom HOPG(0.4) bulk graphite, see Fig. 5.Regarding the two samples characterized with Ramanwe note that the sample L5, which shows a D-line (seeFig. 3) presumably due to the contribution of lattice dis-order, has a lower resistivity than sample L2A that showsno D-line. In Ref. [19] was shown that extended defects ingraphene can lead to self doping. Moreover, the presence of such defects can still lead to long carrier mean free pathand a decrease in the resistivity. Our results speak for anon-simple influence of defects in graphite.Figure 5 shows the normalized resistivity as a functionof temperature at zero magnetic field for all measuredsamples. The overall results agree with those obtained formultigraphene samples in Ref. [20]. However, we note thefollowing, interesting details:- There is an apparent difference in the T − dependencebetween thick and thinner samples. The samples thickerthan ∼ nm show a rather metallic behavior whereasthinner samples a semiconducting like, see Fig. 5.- There is basically no difference in the T − dependencebetween sample L5 and L2A, with exception of the regionat T < K where R decreases decreasing T in thethicker sample L2A. This fact indicates that the disorderthe Raman D-line indicates does not affect strongly the T -dependence of the resistivity.- The overall results indicate that the maximum at ∼ Kin the resistivity observed in sample L2A may have
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Barzola-Quiquia et al.: Sample Size Effects
Figure 2
Raman spectra around the D’-line for samples L5(upper picture) and L2A (bottom picture). The continuouslines in the upper picture represent the Lorentzian peaksused to fit the data. The insets show optical microscopepictures of the samples with the array of Pd electrodes.the same origin as the decrease in the resistivity below ∼ K observed in thicker samples, see Fig. 5. Tak-ing into account the TEM results, see Fig. 1, we wouldconclude that this metallic-like behavior is not intrinsic toideal, defect-free graphite or multigraphene but it is dueto the influence of the interfaces inside the samples withlarge enough thickness.The obtained results suggest that the true T − dependenceof the resistivity in an ideal, defect-free multigraphenesample should be semiconducting-like, i.e. it should in-crease decreasing temperature. This dependence is actuallythe one expected for an ideal semimetal with zero or smallenergy gap, in agreement with the large decrease in carrierdensity decreasing temperature recently obtained in HOPGsamples [9]. The MR was measured for a few sam-ples. Here we concentrate ourselves on two samples L5 and L7, see Fig. 5, which show a clear difference in the T − dependence of the resistivity. The results for thesetwo samples are typical ones. The MR depends mostlyon the behavior of the resistivity, i.e. whether it shows asemiconducting behavior like in samples L2, L2A (above50 K) and L8A, or a metallic one, like in samples HOPG,L8B and L7.Figure 6 shows the MR vs. magnetic field defined asMR = ( R ( B ) − R (0)) /R (0) at different constant temper-atures for sample L5. Above a field of ∼ T the MR in-creases with temperature. At lower fields it remains nearly T − independent. This increase of MR with T , althoughanomalous, it is actually what one expects if the resistancedecreases with T , as is the case for sample L5. Note, how-ever, that the MR is much weaker than the one measuredin HOPG or Kish graphite bulk samples where the MR islarger than 1000 % at fields above 0.5 T and at low temper-atures [5,21,22].A qualitative and quantitative different behavior of theMR is obtained for the metallic-like sample L7, see Fig. 7.Here the MR decreases with temperature and it is a factorof ∼ larger than in the thinner sample L5. Because bothsamples have similar lateral sizes the difference in behaviorin the MR should be related to the difference in thickness,i.e. in the number of interface regions within the sample -1 )Sample L2A Sample L5 Raman shift (cm -1 ) R a m an i n t en s i t y ( a . u . ) Figure 3
Raman spectra around the D- and G-line (1350cm − and 1582 cm − )) for samples L5 (upper picture) andL2A (bottom picture). Copyright line will be provided by the publisher ss header will be provided by the publisher 7
10 100 1000 10000110100 R e s i s t i v i t y ( (cid:181) c m ) Sample Thickness d(nm)
Figure 4
Absolute resistivity at 4 K vs. sample thickness d for the six measured samples of this work ( (cid:4) ). The datapoints ( (cid:13) ) are taken from Ref. [20] as explained in thetext. L5 L2A L8A L8B L7 HOPG R ( T ) / R ( K ) Temperature T(K)
Figure 5
Normalized resistance vs. temperature at zero ap-plied field for all the samples measured in this work.that also may influence the resistivity. We show below thatthe MR can be related to the resistivity of each sample.Figure 8 shows the MR for both samples L5 and L7 at6 T field at different temperatures as a function of the re-sistivity at the same temperatures. This figure shows thatthe decrease or increase of the MR with T is related to theresistivity value at zero field. This result appears to be com-patible with the semi classical picture for the magnetoresis-tance, i.e. the longer the relaxation time – or the smaller the Applied Field B(T) ( R ( B )- R ( )) / R ( ) Sample L5
Figure 6
The magnetoresistance at different constant tem-peratures for sample L5. The selected voltage electrodeswere the two with the largest distance, near the current in-put electrodes, see Fig. 2.resistivity – the larger can be the effect of the field on theresistance.As shown in Fig. 8 the MR of multigraphene samplesas well as in HOPG bulk samples [23] does not follow thewell known Kohler’s rule. This rule describes the MR as afunctional scaling with the ratio
B/ρ , i.e. in lowest orderthe MR ∝ ( B/ρ (0)) . There are at least three differentreasons for the failing of the Kohler rule in graphite,namely:- One reason is related to the huge electron mean freepath ℓ ( T ) in graphite, which is much larger than the ra-dius of curvature of an electron orbit under a magneticfield r c = m ∗ v F /eB , with m ∗ . . m the effectiveelectron mass, v F ∼ m/s the Fermi velocity and e theelectron charge. For a field of 1 T, for example, we have r c ∼ . µ m whereas ℓ > . µ m for T <
K at zerofield [9].- Other reason is that the semiclassical picture of the MRbreaks down when the Fermi wavelength λ F gets largerthan r c , i.e. what is the meaning of the classical cyclotronradius r c ∼ . µ m at B ∼ T when the wavelength ofthe electrons λ F & . µ m below 200 K [9].- A third effect might be related to the contribution ofthe sample internal structure and interfaces. These defectsmay not only influence the dimensionality of the transportin graphite (short circuiting the graphene planes) but also Copyright line will be provided by the publisher
Barzola-Quiquia et al.: Sample Size Effects -8 -6 -4 -2 0 2 4 6 8020406080100120140160 ( R ( B )- R ( )) / R ( ) Applied Field (T)
4K 100K 150K 220KSample L7
Figure 7
The magnetoresistance at different constant tem-peratures for sample L7. The selected voltage electrodeswere those of the d3 configuration, see inset in Fig. 10.might be the origin for localized, granular superconductingregions. Recently done study of the behavior of the MR ofmultigraphene samples suggests the existence of granularsuperconductivity [6]. The main experimental evidencecomes from the anomalous irreversible behavior of theMR, which appears compatible with Josephson-coupledsuperconducting grains [6].We note that Bi, as graphite, has a low density, loweffective mass of carriers and huge values of the electronmean free path [24,25]. It shows a very similar magneticfield induced metal-insulator transition and also has a verylow resistivity [21]. We may speculate that a similar su-perconductivity phenomenon may play a role. In fact re-cently published work found that crystalline interfaces inBi bicrystals of inclination type show superconductivity upto 21 K [26,27]. A similar situation may occur at the inter-faces between crystalline regions in oriented graphite, seeFig. 1.An important point we would like to stress here isthat the peculiar behavior of the MR in HOPG samples[5] is observed only for fields applied perpendicular tothe graphene layers. For fields parallel to the graphenelayers there is no MR, i.e. the measured very weak MRcan be explained by the misalignment of the field with
30 40 50 60 70 801.01.21.41.61.82.02.2 20406080100120 [ ( , T )-( , T ) ]/ ( , T ) Resistivity (0,T) ((cid:181) cm) L5 L7
Figure 8
The magnetoresistance at 6 T field vs. the resis-tivity at zero field and at different constant temperaturesfor samples L5 and L7 (right y − axis).respect to the graphene planes [28]. This speaks for a hugeanisotropy of the assumed superconducting region. There-fore the available experimental data suggest the interfaceregions as possible candidates where this superconductiv-ity might be located, see Fig. 1. Within the same schema itappears clear that the decrease or even the level-off of theresistivity decreasing temperature, see Fig. 5, would not beintrinsic but due to the influence of the interface regions.In case we have non-percolative superconducting ar-rays of grains, one would expect to have a relatively largerincrease in the resistance with magnetic field mostly in thetemperature region where the influence of the coupling be-tween superconducting grains at no applied field is observ-able, i.e. in a region where either the resistance decreasesdecreasing temperature or it levels off below a certain tem-perature as for samples L5 and L8A below 25 K or be-low 30 K for sample L2A, see Fig. 5. This is expected forgranular superconductors due to a coherent charge transferof fluctuating Cooper pairs between the grains [29]. Thissensitive change of the temperature dependence of the re-sistance under an applied magnetic field has been alreadyseen in Ref. [30] for bulk HOPG samples. The possibil-ity of superconductivity in graphite has been discussed inrecent years, see Refs. [7,6] for further reading.On the other hand, in a multigraphene sample we cer-tainly have other defects that would not trigger local super-conductivity but increase the carrier density, decreasing λ F and therefore for a large enough sample (see section 4.3)the MR should increase. Simultaneously, reducing λ F theSchubnikov-de Haas (SdH) oscillations in the MR shouldbe recovered, as seen in sample L7 at low temperatures andhigh fields, see Fig. 7. This is similar to the effect observed Copyright line will be provided by the publisher ss header will be provided by the publisher 9
B(T)/T (T/K) ( R ( B )- R ( )) / ( R ( ) T ) ( K - ) L5; = 1.3 4 K 20 K 100 K 270 K
L7; = 0.94 K100 K 220 K
Figure 9
The magnetoresistance divided by the tempera-ture to the exponent α vs. the ratio of the applied field totemperature B/T for samples L5 and L7. The scaling isachieved for both samples using two different exponents α = 0 . . for sample L7 (L5).in graphene layers by applying a large enough bias voltage,increasing the carrier density, decreasing λ F and recover-ing the SdH oscillations [12].Finally, we would like to remark an interesting aspectof the MR related to its field and temperature depen-dence. The field dependence of the MR is quasi-linear infield at low enough temperatures and high enough fields ( B > . T), see for example the MR for sample L5 inFig. 6. This behavior is still not well understood; possibleexplanations are based on Landau level quantization of theDirac fermions [31] or due to the circulations of quantum,inhomogeneous current paths at the borders of graphiteplatelets creating a Hall-like voltage contribution to theMR [32]. As mentioned above, neither the MR of themultigraphene samples studied in this work nor the one inbulk graphite follows the classical Kohler scaling. We havefound however, that the MR data for the multigraphenesamples L5 and L7 show an impressive scaling when ( R ( B ) − R (0) / ( R (0) T α ) is plotted vs. the ratio B/T , ascan be seen in Fig. 9 for α = 1 . and . , respectively.This kind of universal scaling of the magnetoresis-tance has been observed previously in different materialsnear a metal-insulator transition, like in metallic Si:B( α = 1 / ) [33], icosahedral AlPdRe ( α ≃ . ) [34] or inintercalated amorphous carbon ( α = 1 / ) [35]. Whereasthe scaling with α = 1 / has a physical interpretationbased on electron-electron interactions [33], deviationsfrom this value remain unexplained. The scaling obtainedfor samples L5 and L7 is indeed extraordinary since it covers several orders of magnitude in both scaled axes andappears to be unique in the literature. Future studies shouldclarify whether this scaling is related to the influence ofsuperconductivity in the MR. As has been shown in recent work on HOPGbulk samples[8] the lateral size of the sample has aninfluence on the MR. This can be qualitatively under-stood if we take into account that: (a) The de-Brogliewavelength for massless Dirac fermions λ D ≃ hv F /E F ( v F ≃ m/s is the Fermi velocity and a typical Fermienergy k B E F . K) or for massive carriers witheffective mass m ⋆ . . m , λ m = h/ √ m ⋆ E F , as wellas the Fermi wavelength λ F ∼ (2 π/n ) / can be of theorder of microns or larger due to the low density of Diracand massive fermions. (b) Due to the extraordinary largevalues of the mean free path of the carriers in bulk HOPG( . ◦ ) ℓ [ µ m ] ∼ ((10 µ m ) − + (6 × T − ) − ) − [9],at low enough temperatures we expect to see a decreaseof the MR reducing the lateral sample size or the distancebetween the voltage electrodes. Figure 10
The magnetoresistance of sample L7 at 4 K vs.applied field, obtained at three different voltage pairs (d1 tod3) as shown in the optical microscope picture in the inset.Figure 10 shows the MR of sample L7 measured atthree different voltage electrode positions d1, d2 and d3,see inset. As expected, the larger the distance between volt-age electrodes the larger the MR. Note that a clear effect is
Copyright line will be provided by the publisher observed changing the distance from ∼ to 16 µ m indi-cating that the mean free path should be of this order, inagreement with recently published results [9]. -8 -6 -4 -2 0 2 4 6 8
2K 4K 10K 25K 50K 75K 100K
Applied Field (T) ( R ( B )- R ( )) / R ( ) Figure 11
Magnetoresistance for a ± µ m thick HOPGsample at different temperatures with a constriction. In themiddle of the HOPG sample and between the voltage andcurrent electrodes a constriction was prepared by focussedGa + -ion beam. The inset shows a scanning electron micro-scope image of the constriction.The lateral size effect on the MR is also nicely observedin bulk HOPG samples with a constriction. Using the Ga + -ion beam of the DBM we have prepared a ∼ µ m wideconstriction at the middle and between the sample elec-trodes, see inset in Fig. 11 where the results for the MR atdifferent temperatures are shown. The MR of the HOPGsample with a constriction is about two orders of magni-tude smaller than in the original sample. Leaving by sidethe influence of the SdH oscillations we note that the MR at B ≤ T remains nearly T -independent from 4 K to 25 Kand decreases at higher temperatures. We note also that atzero field the ratio between the resistivities at 50 K and4 K ρ (4) /ρ (50) ∼ . for the sample with constriction,see Fig. 11, whereas it is ∼ . for the original sample, seeFig. 5. Therefore, the behavior of the MR can be under-stood assuming that the mean free path remains larger thanthe width W of the constriction at T < K and is smallerat higher T . This lateral size effect on the MR indicatesthat the effective mean free path of the carriers responsible for the MR should be of the order of µ m at ∼ K, inexcellent agreement with recently reported values for sim-ilar HOPG samples [9]. In case granular superconductiv-ity is confirmed in oriented graphite, future studies shouldclarify to which extent this phenomenon contributes to theobserved lateral size effect in the MR.
The data obtained in this investigationimply that multigraphene samples show two size effects.The smaller the thickness of the multigraphene samplethe larger is the resistivity. The correlation between thethickness dependence of the resistivity and the microstruc-ture of highly oriented pyrolytic graphite suggests that theinterfaces between crystalline regions and parallel to thegraphene layers could be the regions where granular super-conductivity is located. The differences of the temperatureas well as the magnetic field dependence of the resistivityof different multigraphene samples suggest that thoseinterfaces play a main role. The available data indicate thatthe intrinsic T -dependence of the resistivity of ideal multi-graphene or graphite would be semiconducting-like. Thiswould imply that interpretations based on the metal-liketransport properties of graphite as well as multigraphenesamples should not be related to the properties of an idealgraphite structure. The lateral size dependence of thetransport properties, specially in the magnetoresistance,can be understood taking into account the large effectivemean free path of the carriers. Acknowledgements
It is a pleasure to thank Dr. S. Reyn-tjens from the FEI company in Eindhoven for providing us withthe TEM images of the HOPG sample. Fruitful discussionswith N. Garc´ıa are gratefully acknowledge. We gratefully thankU. Teschner and W. Grill for the Raman measurements. Thiswork was supported by the DFG under DFG ES 86/16-1. J.-L.Yao acknowledges the support of the A. v. H. Foundation and J.B-Q. the support of the EU project “Ferrocarbon”.
References [1] B. T. Kelly, Physics of Graphite (London: Applied SciencePublishers, 1981).[2] Y. Kopelevich and P. Esquinazi, Adv. Mater. (Weinheim,Ger.) , 4559 (2007).[3] Y. W. Tan, Y. Zhang, H. Stormer, and P. Kim, Eur. Phys. J.Special Topics , 15–18 (2007).[4] J. H. Chen, C. Janga, S. Adam, M. S. Fuhrer, E. D.Williams, and M. Ishigam, Nature Physics , 377–381(2008).[5] Y. Kopelevich, P. Esquinazi, J. H. S. Torres, R. R. da Silva,and H. Kempa, Advances in Solid State Physics, B. Kramer(Ed.), Vol. 43, (Springer-Verlag Berlin, 2003), pp. 207–222.[6] P. Esquinazi, N. Garc´ıa, J. Barzola-Quiquia, M. Mu˜noz,P. R¨odiger, K. Schindler, J. L. Yao, and M. Ziese,arXiv:0711.3542.[7] Y. Kopelevich and P. Esquinazi, J. Low Temp. Phys. ,629–639 (2007), and refs. therein. Copyright line will be provided by the publisher ss header will be provided by the publisher 11 [8] J. C. Gonz´alez, M. Mu˜noz, N. Garc´ıa, J. Barzola-Quiquia,D. Spoddig, K. Schindler, and P. Esquinazi, Phys. Rev.Lett. , 216601 (2007).[9] N. Garc´ıa, P. Esquinazi, J. Barzola-Quiquia, B. Ming, andD. Spoddig, Phys. Rev. B , 035413 (2008).[10] Y. Lu, M. Mu˜noz, C. S. Steplecaru, C. Hao, M. Bai,N. Garc´ıa, K. Schindler, and P. Esquinazi, Phys. Rev. Lett. , 076805 (2006), see also the comment by S. Sadewasserand Th. Glatzel, Phys. Rev. lett. , 269701 (2007) and thereply by Lu et al., idem , 269702 (2007); D. Martinez-Martin and J. Gomez-Herrero, arXiv:0708.2994 (unpub-lished); R. Proksch, Appl. Phys. Lett. , 113121 (2006).[11] M. I. Katsnelson, Materialstoday , 20 (2007).[12] K. S. Novoselov, A. K. Geim, S. V. Morozov, S. V.Dubonos, Y. Zhang, and D. Jiang, Nature , 197 (2005).[13] Y. Zhang, Y. W. Tan, H. St¨ormer, and P. Kim, Nature ,201 (2005).[14] K. I. Bolotin, K. J. Sikes, Z. Jiang, M. Klima, G. Fuden-berg, J. Hone, P. Kim, and H. L. Stormer, Solid State Com-mun. , 351 (2008).[15] X. Du, I. Skachko, A. Barker, and E. Y. Andrei,arXiv:0802.2933.[16] S. Reich and C. Thomsen, Philos. Trans. R. Soc. London,Ser. A , 2271 (2004).[17] D. Graf, F. Molitor, K. Ensslin, C. Stampfer, A. Jungen,C. Hierold, and L. Wirtz, Nano Lett. , 238–242 (2006).[18] A. C. Ferrari, J. C. Meyer, V. Scardaci, C. Casiraghi,M. Lazzeri, F. Mauri, S. Piscanec, D. Jiang, K. S.Novoselov, S. Roth, and A. K. Geim, Phys. Rev. Lett. ,187401 (2006).[19] N. M. R. Peres, F. Guinea, and A. H. C. Neto, Phys. Rev. B , 125411 (2006).[20] Y. Zhang, J. P. Small, W. V. Pontius, and P. Kim, Appl.Phys. Lett. , 073104 (2005).[21] X. Du, S. W. Tsai, D. L. Maslov, and A. F. Hebard, Phys.Rev. Lett. , 166601 (2005).[22] T. Tokumoto, E. Jobiliong, E. Choi, Y. Oshima, andJ. Brooks, Solid State Commun. , 599 (2004).[23] H. Kempa, P. Esquinazi, and Y. Kopelevich, Phys. Rev. B , 241101(R) (2002).[24] A. N. Friedman, Phys. Rev. , 553 (1967).[25] V. F. Gantmakher and Y. S. Leonovov, JETP Letters , 162(1968).[26] F. Muntyanua, A. Gilewski, K. Nenkov, J. Warchulska, andA. Zaleski, Phys. Rev. B , 132507 (2006).[27] F. Muntyanua, A. Gilewski, K. Nenkov, A. Zaleski, andV. Chistol, Solid State Commun. , 183–185 (2008).[28] H. Kempa, H. C. Semmelhack, P. Esquinazi, andY. Kopelevich, Solid State Commun. , 1–5 (2003).[29] I. V. Lerner, A. A. Varlamov, and V. M. Vinokur, Phys. Rev.Lett. , 117003 (2008).[30] Y. Kopelevich, V. Lemanov, S. Moehlecke, and J. Torres,Phys. Solid State , 1959–1962 (1999).[31] A. A. Abrikosov, Europhys. Lett. , 789 (2000).[32] H. Kempa, P. Esquinazi, and Y. Kopelevich, Solid StateCommunication , 118–122 (2006).[33] S. Bogdanovich, P. Dai, M. P. Sarachik, and V. Dobrosavl-jevic, Phys. Rev. Lett. , 2543–2546 (1995).[34] V. Srinivas, M. Rodmar, R. K¨onig, S. J. Poon, and O. Rapp,Phys. Rev. B , 094206–1–8 (2002). [35] L. Kumari and S. V. Subramanyam, Mater. Sci. and Eng. B , 48–53 (2006)., 48–53 (2006).