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Breakdown of the interlayer coherence in twisted bilayer graphene

Youngwook Kim,1 Hoyeol Yun,2 Seung-Geol Nam,1 Minhyeok Son,3 Dong Su Lee,4 Dong Chul

Kim,5 S. Seo,6 Hee Cheul Choi,3 Hu-Jong Lee,1 Sang Wook Lee,2, ∗ and Jun Sung Kim1, †

1Department of Physics, Pohang University of Science and Technology, Pohang 790-784, Korea2Division of Quantum Phases and Devices, School of Physics, Konkuk University, Seoul, 143-701, Korea

3Department of Chemistry and Division of Advanced Materials Science,

Pohang University of Science and Technology, Pohang 790-784, Korea4Institute of Advanced Composite Materials, Korea Institute of Science and Technology, Wanju-gun 565-902, Korea

5Department of Electronics and Telecommunications,

Norwegian University of Science and Technology, NO-7491 Trondheim, Norway6Department of Physics, Sejong University, Seoul, 143-747, Korea

(Dated: November 5, 2018)

Coherent motion of the electrons in the Bloch states is one of the fundamental concepts of thecharge conduction in solid state physics. In layered materials, however, such a condition often breaksdown for the interlayer conduction, when the interlayer coupling is significantly reduced by e.g. largeinterlayer separation. We report that complete suppression of coherent conduction is realized evenin an atomic length scale of layer separation in twisted bilayer graphene. The interlayer resistivityof twisted bilayer graphene is much higher than the c-axis resistivity of Bernal-stacked graphite,and exhibits strong dependence on temperature as well as on external electric fields. These resultssuggest that the graphene layers are significantly decoupled by rotation and incoherent conductionis a main transport channel between the layers of twisted bilayer graphene.

PACS numbers: 71.20.-b, 71.20.Ps, 71.18.+y, 72.20.My

In many layered systems, the interlayer coupling isone of the key parameters for altering their electronicproperties [1–6]. When a thick insulating block is in-serted between the metallic layers, the interlayer cou-pling can be significantly reduced, leading to breakdownof the interlayer coherence, as nicely demonstrated intwo dimensional electron gas (2DEG) in semiconduc-tor superlattice[6]. Such an ”interlayer version” of theMott-Ioffe-Regel limit is realized when the layer sep-aration exceeds the mean free path across the layers,which is evidenced by qualitatively different temperaturedependences of the intralayer (metallic) and the inter-layer (semiconducting) resistivities. The intriguing in-terlayer conduction has been observed in various sys-tems including high-Tc cuprates [1], organic crystals [2],dichalcogenides [3], graphite [4, 5] and semiconductor su-perlattices [6], but its underlying mechanism related tothe interlayer incoherence has been under debate in lastdecades [7].

Graphene, an ideal 2DEG system, also exhibits richelectronic properties depending on how it is stacked ontop of another graphene layer [8, 9]. While bilayergraphene in Bernal stacking has massive charge carrierswith a zero band gap, twisted bilayer graphene with arandom orientation of the layers has a massless electronicdispersion similar to that of monolayer graphene [10].Twisted bilayer graphene is of particular interest be-cause several intriguing properties such as renormaliza-tion of Fermi velocity [10, 11], van Hove singularities[12],and electronic localization [11, 13] were recently discov-ered. Experimental studies including angle-resolved pho-toemission spectroscopy [14], scanning tunneling spec-

troscopy [12], Raman spectroscopy [15, 16], and in-planetransport [17, 18], suggest that the layers are decoupledin twisted bilayer graphene, and the misoriented layersare often considered as being electrically isolated. How-ever, it is still not clear what sense the layers are decou-pled on an atomic length scale of the layer separation,and how strong the interlayer coupling is in twisted bi-layer graphene [9, 19].

In this Letter, we present experimental evidence forcomplete suppression of the interlayer coherence betweenin twisted bilayer graphene. We found that the interlayerresistivity of twisted bilayer graphene is much higherthan that of Bernal-stacked graphite by at least 4 ordersof magnitude. In contrast to the almost temperature-independent intralayer resistivity, the interlayer resis-tivity exhibits strong negative temperature dependence.These results demonstrate that incoherent tunneling isa main conduction channel between the layers, due tothe momentum-mismatch of the 2 dimensional Fermi sur-faces from each layer. We thus suggest that the twistedbilayer graphene provides a rare example of the layeredsystems showing the interlayer incoherence even with asmall layer separation of ∼ 4 A.

In order to fabricate the graphene cross junction weemployed a mechanical transfer process [20]. Twisted bi-layer graphene is formed in the overlapped region of twomonolayer graphene layers, as shown in Fig. 1(a). Dur-ing the process, the surfaces of both the lower and theupper graphene layers are kept free of contact to chemicalsolvents until they overlap with each other. This is cru-cial for ensuring tight contact between the layers withoutany chemical adsorbates in-between.

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FIG. 1: (color online) (a) Optical image of a cross junctionof two monolayer graphene sheets (A and B) forming twistedbilayer graphene in the overlapped region (C). (b) AFM imagewith a scale bar of 500 nm. (c) The height profile along theline indicated in (b). (d) The height histogram of the boxedregion in (b). The data is fitted by the sum (black) of twoGaussian functions (red), yielding d ≈ 0.4 nm as indicated bythe arrow. (h) Raman spectra taken from the monolayer (Aand B) and the twisted bilayer (C) regions. The inset showsthe Raman map of the relative frequency shift of the G bandand the 2D band in the twisted bilayer region with respect tothose of the monolayer region.

Before discussing the interlayer transport properties oftwisted bilayer graphene, we confirm that in the over-lapped region, two graphene layers are tightly stacked.Figure 1 shows the optical and the atomic force mi-croscopy (AFM) images of the twisted bilayer graphenesample that is mainly discussed in this manuscript. Theseparation between the layers is d ≈ 0.4 nm, determinedby the line-cut height profile [Fig. 1(c)] and the heighthistogram over the step from the bottom to the top lay-ers [Fig. 1(d)]. This is very close to the layer spacing ofd ∼ 0.34 nm in Bernal-stacked graphite.

Twisted stacking of the graphene layers in the over-lapped region is also demonstrated by Raman spec-troscopy. According to recent studies [15, 16], the doubleresonance peak (2D) of twisted bilayer graphene resem-bles a single Lorentzian peak rather than multiple peaksas found in Bernal-stacked bilayer graphene. Further-more, the Fermi velocity of twisted bilayer graphene isknown to be reduced as compared to the monolayer in-ducing a blue-shift of the 2D peak. Both features arefound in our sample when we compare the spectra of

the overlapped region (C) with those of the bottom (A)and the top (B) layers. The Raman 2D peak at C is blue-shifted (2682 cm−1) as compared to the peaks at A and B(2672 cm−1). By contrast, the Raman G peaks appear atthe same position of ∼ 1582 cm−1 for the twisted bilayerand the monolayer regions [Fig. 1(e)]. These observationsconfirm that the graphene layers are indeed coupled athigh energy levels in the overlapped region. Accordingto Ref. [16], a blue-shift of the 2D peak by ∼ 10 cm−1

with an enhancement of its intensity corresponds to thetwist angle of 15o-30o.

The interlayer transport measurements in the twistedlayers were done using four branches of monolayers aselectrical leads. For both bottom and top monolayers,the intralayer resistivity as a function of back-gate volt-age (Vg) shows a narrow peak at Vg = 2 V without sig-nificant temperature dependence [21]. For the interlayertransport, we apply the current between each branch ofthe bottom and the top layers and measure the interlayer

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FIG. 2: (color online) (a) Schematic diagram of a twistedbilayer graphene device. (b) Optical image of a twisted bi-layer graphene device (S1) with a scale bar of 10 µm. (c)Current-voltage characteristics of the interlayer conductionat various temperatures from 2 K to 280 K (using the samecolor code in (d)) (d) The interlayer resistivity of twisted bi-layer graphene (S1) with variation of the gate voltages (Vg)at different temperatures. (e) The temperature dependenceof the normalized interlayer and the intralayer resistivity atVg = VN . (f) Temperature dependence of the ρinter(T ) forseveral twisted bilayer graphene devices (S1-S4) at Vg = VN .The crossover temperature T ∗ from the T -independent to thestrong T -dependent regimes is indicated by the arrows.

3

voltage drop using the other two branches as illustratedin Fig. 2(a). In total, four devices of the twisted bilayergraphene were studied (Table 1), and all devices showqualitatively similar behaviors of the interlayer conduc-tion. We present the results obtained from S1 [Fig. 2(b)]unless otherwise noted. Figure 2(c) displays the current-voltage (I-V ) characteristics through the layers at thecharge neutral point with a back-gate voltage of Vg=VN

in the temperature range from 2 K to 280 K. The I-Vcharacteristics exhibit the ohmic behaviors, and it is sus-tained down to 2 K.

Despite the linear I-V characteristics the interlayer re-sistivity (ρinter) is very large. Figure 2(d) shows the in-terlayer resistivity ρinter as a function of Vg at varioustemperatures. Here we define ρinter ≡ Rinter ·A/d, whereA is the area of the overlapped region, d is the interlayerdistance, and Rinter is the interlayer resistance[20]. At280 K, ρinter∼ 2000 Ωcm, which is at least four ordersof magnitude higher than the reported values of the c-axis resistivity ρc for graphite. Typically ρc ranges from∼ 10−4 Ωcm for single crystalline graphite [4] to ∼10−1

Ωcm for highly oriented pyrolytic graphite (HOPG) [4, 5].

The interlayer resistivity shows a strong temperaturedependence as shown in Fig. 2(e). ρinter(T ) at chargeneutral point (Vg = VN ) increases with lowering tempera-ture, and it saturates at ∼ 7800 Ωcm below the crossovertemperature T ∗ ∼ 90 K. The similar temperature de-pendence was observed in other samples [Fig. 2(f)].The negative temperature dependence (dρinter/dT <0) is distinct from the intralayer resistance measuredfor the bottom and the top monolayers, which is al-most temperature-independent [Fig. 2(e)]. More impor-tantly, this behavior contrasts to the metallic behavior,dρc/dT > 0 found in single crystalline Bernal-stackedgraphites [4]. These results provide experimental evi-dence for the incoherent interlayer conduction betweenthe layers [22].

Recent theoretical studies on twisted bilayer graphenesuggested that Dirac cones from the two monolayers aremisoriented as shown in Fig. 3(a) [19]. When the twistangle is away from the commensurate angle, the overlapof the FS’s from each layer does not occur, and the energydifference between states with the same extended mo-

TABLE I: The characteristics of twisted bilayer graphenesamples. The interlayer resistivity (ρinter), the interlayerequilibration rate (τ−1

RC) at 2 K are listed together with thecrossover temperature (T ∗) for Vg = VN . The relative shiftof 2D Raman band (∆2D) of the twist bilayer graphene withrespect to that of monolayer graphene is also listed.

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S1 7.80 5.54×108 ∼ 90 ≈ 8S2 3.98 1.11×109 ∼ 30 ≈ 10S3 10.00 4.33×108 ∼ 42S4 5.24 8.25×108 ∼ 40

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FIG. 3: (color online) (a) The momentum mismatch of twosets of Dirac cones (red and blue) from different layers intwisted bilayer graphene. (b) The neighboring Dirac conesoverlaps at high energies. Thermally-excited tunneling be-tween two FS’s of each layer is indicated by the arrow. (c)Phonon-assisted tunneling with a momentum exchange of qenhances the overlap between the FS’s. (d) The interlayerresistivity (ρinter) of twisted bilayer graphene (S1) at dif-ferent Vg’s as a function of temperature. The solid linesare the fit of the metal-insulator-metal junction model whoseschematic illustrations are shown in the inset (see the text).(e) The Vg dependence of the conductivity σ(0) at zero tem-perature limit. The open (solid) symbols represent the caseof hole (electron) carriers. The height U (f) and the width w

(g) of an effective barrier estimated from the fitting are shownas a function of Vg. The solid line is the guide-to-eyes. Thecrossover temperature T ∗ (f) and the resistivity ratio ρinter(2K)/ρinter(280 K) (g) are also plotted. The interlayer separa-tion d = 0.4 nm is indicated by the dashed line in (g).

mentum is typically much larger than the Fermi energy.In this case, the interlayer tunneling of charge carrierswith conservation of their in-plane momentum is signifi-cantly suppressed. The interlayer equilibriation time e.g.the RC time τRC for the interlayer conduction is expectedto be 10−6 - 10−12 sec for n = 5 × 1012 cm−2 at zero tem-perature limit. Using the measured σ(2 K) at Vg = 30V and the interlayer dielectric constant ǫGG ∼ 2.6ǫ0 [18],we estimate τRC ∼ 2 × 10−10 sec, that is well within therange of theoretical estimates [19].

For quantitative analysis, we employ a model of metal-insulator-metal junction [23]. The current through a po-

4

tential barrier is given by

I ∼

∫T (E)Db(E)Dt(E + eV )[fb(E)− ft(E + eV )]dE,

(1)where Db,t(E) and fb,t(E) are the density of states(DOS), and the Fermi-Dirac distribution function for thebottom and the top monolayers, respectively. In themodel, the conductivity saturates at a zero temperaturelimit σ(0) ∼ Db(EF )Dt(EF )T (0). Assuming no signif-icant Vg-dependence of T (0), σ(0) ∼ |Vg − VN |, whichroughly agrees with our experiments [Fig. 3 (e)]. Thefinite and relatively large σ(0) near the charge neutralpoint is due to the electron-hole puddles [24], where thecharge density inhomogeneity of n0 ∼ 3× 1011 cm−2 areexpected [25].

We now discuss the temperature dependence of theinterlayer resistivity, which is determined by tunnelingprobability between the two monolayers. Since the inter-layer tunneling occurs when the Fermi circles intersectin the extended Brillouin zone[19], it is allowed for ex-cited carriers at higher energies where the two neighbor-ing Dirac cones overlap each other as shown in Fig. 3(b).Thus we can consider an effective barrier with its heightU corresponding to the energies required for interlayertunneling between two misoriented FS’s [27]. The tem-perature dependence of ρinter(T ) is well reproduced bythe fit using Eq. (1) as shown in Fig. 3(e). Here we as-sumed a square barrier with a height of U and a widthof w [Fig. 3(e)] using the Wentzel-Kramers-Brillouin ap-proximation. For the DOS of monolayer graphene, thephenomenological DOS is used with a broadening factorΓ = 60 meV [20, 25, 26].

The resulting barrier height U is reduced at high|Vg−VN |’s showing electron-hole symmetry, which resem-bles the behaviors of T ∗ [Fig. 3(f)]. This is indeed whatis expected since thermal energy needed for the interlayertunneling decreases when the Fermi circles become largerwith increase of carrier density. The barrier width w isfound to be 1.0-1.3 nm with a moderate Vg dependencesimilar to that of ρinter(2 K)/ρinter(280 K) [Fig. 3(g)].Note that the estimated w is about 3 times larger thanthe interlayer separation d ≈ 0.4 nm measured by theAFM [Fig. 1]. We tested various cases with differ-ent magnitude and shape of U , but w > d ≈ 0.4 nmis needed in order to explain the observed strong tem-perature dependence, i.e. a large value for the ratioρinter(2 K)/ρinter(280 K).

We attribute this discrepancy to the interlayer tunnel-ing assisted by phonon at high temperatures [7]. In thiscase, the carriers are scattered by acoustic phonon modeswithin the phonon sphere of diameter q ≈ kBT/~vs wherevs is sound velocity. In particular, in graphene q can bemuch larger than the size of Fermi circle at moderate tem-peratures [28], and thus phonon-assisted tunneling sig-nificantly enhances the overlap between the misoriented

FS’s as illustrated in Fig. 3(c). With lowering tempera-ture, therefore, not only the thermally-excited tunnelingbut also the phonon-assisted tunneling are suppressed,and the effect of the momentum mismatch of the FS’sbecome pronounced. This results in the stronger temper-ature dependence of ρinter(T ), leading to more significantsuppression of interlayer tunneling than expected in themetal-insulator-metal junction model. Further studiesare highly desirable for understanding the incoherent in-terlayer tunneling process assisted by phonon and otherbosons, as discussed recently [7]. Our findings clearlymanifest that the momentum-mismatch of the FS’s dueto misorientation is essential for breakdown of the inter-layer coherence in twisted bilayer graphene.

In conclusion, using graphene cross junctions we in-vestigate the interlayer transport properties of twistedbilayer graphene. The large magnitude of the resistiv-ity and its strong dependence on temperature and alsoon external electric fields reveal that the main interlayerconduction mechanism is incoherent tunneling due to themomentum-mismatch of FS’s. Thus the twisted bilayergraphene is a rare example of the layered systems wherethe interlayer incoherence is induced by misorientation ofthe layers even with an atomic scale of layer separation.We envisage that the interlayer conduction can be mod-ified by the insertion of other nanosheets using multiplestacking [29] or chemical intercalation [30]. Therefore,graphene cross junctions with further functionalizationcan provide a new platform incorporating vertical trans-port of graphene layers for possible electronic applica-tions.

The authors thank G. T. Kim, M. Jonson, and E.E. B. Campbell for fruitful discussion. This work wassupported through BSR (2009-0076700, 2011-0021207 ),EPB Center (2010-0001779), SRC (2011-0030046), andWCU (R31-2008-000-10057-0), CRC (2010K000981) pro-grams by the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science andTechnology.

∗ Electronic address: leesw@konkuk.ac.kr† Electronic address: js.kim@postech.ac.kr

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[20] See Supplemental Material [URL will be inserted by pub-lisher] for additional information regarding the devicefabrication and the detailed analysis on the results.

[21] The mobilities of electron (hole) carriers are as high as13400 (11100) cm2/Vs and 8960 (8820) cm2/Vs for thetop and bottom layers, respectively.

[22] In HOPG, similar semiconducting temperature depen-dence of ρc has been observed at high temperatures, butit is followed by the metallic behavior at low tempera-tures [4, 5]. This can be due to the heterogeneous con-duction of Bernal-stacked domains with stacking faultsinduced by the in-plane mosaicity. The twisted bilayergraphene, therefore, is well in the incoherent regime,which is an extreme opposite to Bernal-stacked graphitevia HOPG.

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[26] E. H. Hwang and S. Das Sarma, Phys. Rev. B 82,081409(R) (2010).

[27] Since the momemtum-mismatch between the pair ofneighboring Dirac cones are different, U should be con-sidered as an effective value averaged over the extendedBrillouin zone.

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