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Oscillating Magnetoresistance in Graphene p-n Junctionsat Intermediate Magnetic FieldsDOI:10.1021/acs.nanolett.6b05318

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Citation for published version (APA):Overweg, H., Eggimann, H., Liu, M. H., Varlet, A., Eich, M., Simonet, P., Lee, Y., Watanabe, K., Taniguchi, T.,Richter, K., Fal'ko, V., Ensslin, K., & Ihn, T. (2017). Oscillating Magnetoresistance in Graphene p-n Junctions atIntermediate Magnetic Fields. Nano Letters, 17(5), 2852-2857. https://doi.org/10.1021/acs.nanolett.6b05318

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Download date:16. Aug. 2021

Oscillating magnetoresistance in graphene p-n junctions at intermediate magneticfields

Hiske Overweg, Hannah Eggimann, Anastasia Varlet, Marius Eich,

Pauline Simonet, Yongjin Lee, Klaus Ensslin, and Thomas IhnSolid State Physics Laboratory, ETH Zurich, CH-8093 Zurich, Switzerland

Ming-Hao LiuInstitut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany and

Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan

Kenji Watanabe and Takashi TaniguchiAdvanced Materials Laboratory, National Institute for Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan

Klaus RichterInstitut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany

Vladimir I. Fal’koNational Graphene Institute, University of Manchester, Manchester M13 9PL, UK

(Dated: April 6, 2017)

We report on the observation of magnetoresistance oscillations in graphene p-n junctions. Theoscillations have been observed for six samples, consisting of single-layer and bilayer graphene, andpersist up to temperatures of 30 K, where standard Shubnikov-de Haas oscillations are no longerdiscernible. The oscillatory magnetoresistance can be reproduced by tight-binding simulations. Weattribute this phenomenon to the modulated densities of states in the n- and p- regions.

p-n junctions are among the basic building blocks ofany electronic circuit. The ambipolar nature of grapheneprovides a flexible way to induce p-n junctions by elec-trostatic gating. This offers the opportunity to tune thecharge carrier densities in the n- and p-doped regionsindependently. The potential gradient across a p-n in-terface depends on the thickness of the involved insula-tors and can also be modified by appropriate gate volt-ages. Due to the high electronic quality of present daygraphene devices a number of transport phenomena inpnp or npn junctions have been reported, such as ballisticFabry-Perot oscillations1–3 and so-called snake states4,5,both of which depend on characteristic length scales ofthe sample.

Here we report on the discovery of yet another kindof oscillation, which does not depend on any such lengthscale. The oscillations occur in the bipolar regime, inthe magnetic field range where Shubnikov-de Haas os-cillations are observed in the unipolar regime. Thesenovel oscillations in the bipolar regime are governed bythe unique condition that the distance between two resis-tance minima (or maxima) in gate voltage space is givenby a constant filling factor difference of ∆ν = 8. Thefeatures are remarkably robust: they occur in sampleswith one and two p-n interfaces; in single and bilayergraphene; up to temperatures of 30 K (where Shubnikov-de Haas oscillations have long disappeared); over a largedensity range; for interface lengths ranging from 1 µmto 3 µm and in both pnp and npn regimes. The oscil-lations have been observed in a magnetic field range ofB = 0.4 T up to B = 1.4 T. Their periodicity does not

sample name A B C D E F

sample width W (µm) 1.3 1.4 1.1 0.9 3 1.2

sample length L (µm) 3.0 1.4 1.0 2.3 3 2.8

top gate length LTG (µm) 1.1 0.7 0.55 1.2 1.0 1.0

distance to top gate (nm) 23 44 28 57 35 25

number of graphene layers 2 1 2 2 2 2

junction type npn pn npn pn npn npn

TABLE I. Characteristics of samples A-F

match the periodicity of the aforementioned snake states.In this paper we address this phenomenon and suggest amodel which can qualitatively explain the oscillations.

Measurements were performed on six samples in to-tal, which all consist of a graphene flake encapsulatedbetween two hexagonal boron nitride (h-BN) flakes on aSi/SiO2 substrate. They all show similar behavior. Thispaper focuses on measurements performed on one sam-ple (sample A), with the device geometry sketched inFig. 1a. Specifications of the other five samples are sum-marized in table I. The bilayer graphene (BLG) flake wasencapsulated with the dry transfer technique described inRef. 6. A top gate was evaporated on the middle part ofthe sample, which divides the device into two outer re-gions, only gated by the back gate (single-gated regions),and the dual-gated middle region. The other five sampleswere made with the more recent van der Waals pick-uptechnique.7 Unless stated otherwise, the measurementswere performed at 1.7 K. An AC voltage bias of 50 µVwas applied symmetrically between the Ohmic contacts

2

(‘source’ and ‘drain’ in Fig. 1a, inner contacts in Fig. 1b)and the current between the same contacts was measured.The transconductance dG/dVTG was measured by apply-ing an AC modulation voltage of 20 mV to the top gate.

Figure 1c shows the conductance as a function of topgate voltage VTG and back gate voltage VBG. Chargeneutrality of the single-gated regions shows up as a hori-zontal line of low conductance and is marked by a whiteline. The diagonal line of low conductance correspondsto charge neutrality of the dual-gated region. The slopeof this line is given by the capacitance ratio of the topand back gate. Together these lines divide the map intofour regions with different combinations of carrier types:two with the same polarities in the single- and dual-gatedregions (pp’p and nn’n) and two with different polarities(npn and pnp). The conductance in the latter regionsshows a modulation which is more clearly visible in thetransconductance (see Fig. 2a). The oscillatory conduc-tance is caused by Fabry-Perot interference of charge car-riers travelling back and forth in the region of the sampleunderneath the top gate. Their periodicity yields a cav-ity length LTG = 1.1 µm, which is in agreement with thelithographic length of the top gate. The Fabry-Perot os-cillations were studied in more detail in Ref. 3, which re-vealed the ballistic nature of transport in the dual-gatedregion.

The Fabry-Perot oscillations disappear in a magneticfield of B & 100 mT (see Fig. 2b-d). Yet at magneticfields of B = 0.4 T a new oscillatory pattern appears inthe npn and pnp regime. This can be seen in the conduc-tance and transconductance maps recorded at B = 0.5 T,shown in Figs. 3a,b,d,e. The oscillations follow neitherthe horizontal slope of features taking place in the single-gated region, nor the diagonal slope of the dual-gatedregion. They are therefore expected to occur at the in-terface between the p- and n-doped regions. This wasconfirmed by measurements on sample D, which had twocontacts in the single-gated region and two contacts inthe dual-gated region. For this sample, only the conduc-tance along paths involving the interface shows oscilla-tions (see Supporting Information).

On top of this novel oscillatory pattern the transcon-ductance of sample A in Fig. 3b(e) shows faint diagonallines in the nn’n(pp’p) regime, which are Shubnikov-deHaas oscillations in the dual-gated region. The occur-rence of Shubnikov-de Haas oscillations shows that inthis moderate magnetic field regime the Landau levelsare broadened by disorder on the scale of their spacing,resulting in a modulation of the density of states.

Using a plate capacitor model described in the supple-mental material of Ref. 3, the gate voltage axes can beconverted into density and filling factor axes, νX withX = SG,DG for the single- and dual-gated regions, re-spectively. The result of this transformation is shownin Fig. 3c. The oscillatory pattern has a slope of one,i.e. it follows lines of constant filling factor difference∆ν = νSG − νDG. It appears that the oscillations can be

(a) LLTG

W

top gatehBN tophBN bottom

source drain

SiO2

Si back gategraphene

←

(b)

BLG

←

5 µm

BNT

BNB

0 120G(e2/h)

-6 -4 -2 0 2 4 6V TG (V)

-40

-20

0

20

40

VB

G(V

)

(c)

nn’nnpn

pp’p pnp

−→nDG−→nSG

FIG. 1. Characterization of the device. (a) Schematic of thedevice: a bilayer graphene flake is encapsulated between h-BN layers. It is contacted by Au contacts and a Au top gateis patterned on top, which defines the dual-gated region. (b)Optical microscope image of the sample. The four contacts,of which only the inner ones were used, appear orange. Thetop gate is outlined by a red curve. (c) Conductance of thesample at B = 0 T, T = 1.7 K. Four regions of differentpolarities are indicated. A zoom of the transconductance inthe boxed region with a solid line is shown in Fig. 2. Thedashed (dotted) box indicates the gate voltage range in whichFigs. 3a,b (d,e) were measured.

described by:

G = 〈G〉+A cos(2π∆ν

8) (1)

where A is the amplitude of the oscillations, which ison the order of 4 % of the background conductance 〈G〉at T = 1.7 K. The distance between one conductancemaximum and the next can therefore be bridged by ei-ther changing the filling factor in one region by 8, or bychanging the filling factor in both regions oppositely by4. It should be noted that Eq. (1) can be used to describethe oscillations in all six samples, regardless of the num-ber of graphene layers and the sample width (see table Iand the Supporting Information).

The oscillations persist in magnetic fields up to B =1 T for sample A and the periodicity scales with ∆ν for

3

-15

-10

-5

VB

G(V

)

(a)

B = 0.0 T

(b)

B = 0.1 T

4.5 5 5.5 6V TG (V)

-15

-10

-5

VB

G(V

)

(c)

B = 0.2 T

4.5 5 5.5 6V TG (V)

(d)

B = 0.3 T

-12 12dG/dV TG (e2/h V−1)

FIG. 2. Disappearance of Fabry-Perot oscillations with in-creasing magnetic field. The measurement was taken in theboxed region with solid lines in Fig. 1c. At B = 0 T (a) thetransconductance shows clear Fabry-Perot oscillations. Theydisappear in a magnetic field of B & 0.1 T (b–d).

the entire magnetic field range. In higher magnetic fieldsthe conductance is dominated by quantum Hall edgechannels and takes on values below e2/h in the npn andpnp regimes, in agreement with observations by Amet etal.8. Other works report on the (partial) equilibration ofedge channels8–13 and shot noise14,15 in p-n junctions inthe quantum Hall regime.

The oscillatory conductance is quite robust againsttemperature changes. Figure 4a,b show the decay of theamplitude as a function of temperature T . The oscilla-tory conductance in the pnp and npn regime disappearat a temperature around T = 30 K. As can be seen inFig. 4c, at T = 10 K the oscillations are still clearlypresent, while the Shubnikov-de Haas oscillations in thepp’p regime have already faded out. The persistence upto T = 30 K indicates that the studied phenomenon doesnot require phase coherence on the scale of the devicesize. The phase coherence length at T = 1.7 K is esti-mated to be on the order of the device size, but it fallsoff with 1/T .16

The above discussed oscillations can be reproduced bytransport calculations for an ideal SLG p-n junction atan intermediate magnetic field B, based on the scalabletight-binding model17. The ideal junction is modeled byconnecting two semi-infinite graphene ribbons (orientedalong armchair) with their carrier densities given by nLin the far left and nR in the far right. A simple hyperbolic

0 40G (e2/h)

-6 -5 -4 -3 -2 -1V TG (V)

10

20

30

40

VB

G(V

) nn’n

npn

(a)-12 12

dG/dV TG (e2/h V−1)

-6 -5 -4 -3 -2 -1V TG (V)

T = 1.7 K

SdHW

/ Rc =

7

W/ R

c =11

(b)

120 150 180 210νDG

-100

-80

-60

ν SG

slope

=1

∆ν

=22

4

∆ν

=23

2

∆ν

=24

0

∆ν

=24

8

∆ν

=25

6

∆ν

=26

4

∆ν

=27

2

∆ν

=28

0

∆ν

=28

8

1.2 1.6 2.0 2.4 2.8nDG (× 1012 cm−2)

-1.4

-1.2

-1.0

-0.8

n SG

(×10

12cm−

2)

-12 12dG/dV TG (e2/h V−1)(c)

0 40G (e2/h)

0 1 2 3 4 5 6V TG (V)

0

-10

-20

-30

VB

G(V

)

pp’p

pnp

(d)-12 12

dG/dV TG (e2/h V−1)

0 1 2 3 4 5 6V TG (V)

SdH

T = 1.7 K

W/ R

c =7

W/ R

c =11

(e)

FIG. 3. Magnetotransport at B = 0.5 T. (a) Conductance ofthe sample at 0.5 T, showing an oscillatory pattern in the npnregime. The measurement was taken in the dashed boxed re-gion of Fig. 1c. (b) The oscillatory pattern in the npn regimeis more clearly visible in the transconductance. Green dashedlines indicate the pattern expected for snake states. In thenn’n regime some faint lines can be distinguished, followingthe slope of the charge neutrality line of the dual gated region.These are Shubnikov-de Haas oscillations. (c) Transconduc-tance at B = 0.5 T in the pnp regime as a function of chargecarrier density (and filling factor) in the single- and dual-gated region. The oscillatory pattern follows the indicatedline of slope one and can therefore be described by lines ofconstant filling factor difference ∆ν = νDG − νSG. (d),(e)Same as (a),(b), but with opposite charge carrier polarities.The oscillations are essentially particle-hole symmetric.

4

2 27T (K)

4.5 5.0 5.5 6.0V TG (V)

-0.04

-0.02

0.0

0.02

0.04(G

-〈G〉)/〈G〉

(a)

0 10 20T (K)

0

0.02

0.04

A/〈G〉

(b)

-4 4dG/dV TG (e2/h V−1)

0 2 4 6V TG (V)

-30

-20

-10

0

VB

G(V

)

(c)

T = 10 K

no SdH

pp’p

pnp

FIG. 4. Temperature dependence. (a) Oscillatory part of theconductance as a function of top gate voltage and temperaturemeasured along the line cut indicated by the black line inFig. 4c. (b) Amplitude A of the oscillatory conductance asa function of temperature. The oscillations disappear aroundT = 30 K. (c) Transconductance at T = 10 K, B = 0.5 T inthe pnp regime. Whereas Shubnikov-de Haas oscillations inthe pp’p regime have faded out, the oscillatory pattern in thepnp regime persists.

tangent function with smoothness 50 nm bridging nL andnR is considered; see the inset of Fig. 5a for an example.To cover the density range up to ±3× 1012 cm−2 corre-sponding to a maximal Fermi energy of Emax ≈ 0.2 eV,the scaling factor sf = 10 is chosen because it fulfillsthe scaling criterion17 sf 3t0π/Emax ≈ 141 very well;here t0 ≈ 3 eV is the hopping energy of the unscaledgraphene lattice. Note that the following simulationsconsider W = 1 µm for the width of the graphene ribbon,but simulations based on a different width show an iden-tical oscillation behavior (see Supporting Information fordetails), confirming its width-independent nature as al-ready concluded from our measurements.

The transmission function T (nR, nL) across the idealp-n junction at B = 0.5 T is shown in Fig. 5a, where fineoscillations along symmetric bipolar axis (marked by theblue arrows) from np to pn through the global charge

neutrality point can be seen. Two regions marked by thewhite dashed boxes in Fig. 5a are zoomed-in and shown inFigs. 5b and d for a closer look and comparison with themeasurements of sample B and E (Figs. 5c and e, respec-tively). Despite certain phase shifts (observed in Figs.5b, d, and e) that are beyond the scope of the presentstudy, good agreement between our transport simulationand experiment showing the oscillation period well ful-filling Eq. (1) can be seen.

Other works4,5 report on the formation of so calledsnake states along p-n interfaces in graphene. Snakestates result in a minimum in the conductance wheneverthe sample width W and the cyclotron radius Rc satisfyW/Rc = 4m − 1 with m a positive integer. In the den-sity range of Fig. 3b,e this would lead to two resonancesat most (indicated by green dashed lines in Fig. 3b,e),which is far less than the observed number of resonances.On top of that, snake states are inconsistent with theobserved absence of a dependence on sample width. Fur-thermore, the tight-binding simulation also confirms thatthe observed effect is independent of the sample widthand cannot be suppressed by introducing strong latticedefects in the vicinity of the p-n junction (see SupportingInformation). We therefore rule out snaking trajectoriesas a possible cause of the observed oscillations.

Another process which could give rise to oscillations ina graphene p-n junction in a magnetic field is the inter-ference of charge carriers which are partly reflected andpartly transmitted at the interface. When the charge car-rier densities are equal on both sides of the interface, elec-trons and holes will have equal cyclotron radii and there-fore the paths of transmitted and reflected charge carrierswill form closed loops. For the case of equal density, thismodel predicts the right periodicity of the oscillations.18

Experimentally, however, the measured oscillations arestill visible when the densities on both sides of the p-ninterface are quite different: at the point (VBG,VTG) =(12,-6) V for example (see Fig. 3b), the cyclotron radii onthe p and n side are respectively 0.36 µm and 0.16 µm.The path lengths hence differ by 2∆Rc = 0.40 µm, whichis more than seven times the Fermi wavelength (0.02 µmand 0.05 µm). It seems unlikely that interference betweencharge carriers on skipping orbits can still occur in thisdensity regime. On top of that, the tight-binding simu-lations show that the oscillatory pattern is still presentwhen introducing large-area lattice defects in the vicinityof the p-n junction, which destroy the skipping trajecto-ries (see Supporting Information). The observed robust-ness against temperature changes is in contradiction withthis model as well. Thus, the observed oscillations cannotbe ascribed to interference of charge carriers on cyclotronorbits at the p-n interface.

Since the oscillations occur in both single-layer and bi-layer graphene, we exclude an explanation that relies onspecific details of the dispersion relation. In the mag-netic field range where the oscillations are observed, thesample width is comparable to the classical cyclotron di-ameter. This excludes explanations based on classical

5

b

d

-3 -2 -1 0 1 2 3

nR

(1012 cm -2)

-3

-2

-1

0

1

2

3n

L (

10

12 c

m-2

) 20

40

60

T

-200 0 200

x (nm)

-1012

n(x

)

-180 -160 -140 -120ν

R

140

160

180

νL

25

30

35

40

45

T

-180 -160 -140 -120ν

DG

140

160

180

νS

G

-12

0

12G'

-140 -120 -100 -80 -60 -40ν

R

60

80

100

120

10

20

30

T

-140 -120 -100 -80 -60 -40ν

DG

60

80

100

120

-12

0

12G'

(a)

(b)

Theory

(c)

Experiment

(d)

Theory

(e)

Experiment

FIG. 5. (a) Transmission T as a function of the carrier densities on the left, nL, and right, nR, for an ideal SLG p-n junctionat a perpendicular magnetic field B = 0.5 T based on a tight-binding transport calculation (color range restricted for clarity).Oscillations occur in the vicinity of the symmetric bipolar axis marked by blue arrows. Inset: an example of the consideredcarrier density profile corresponding to the white cross. White dashed boxes correspond to the density regions shown in panels(b) and (d), where the carrier density values are transformed in filling factors. (c)/(e) Transconductance G′ measured forsample B/E shown with the same filling factor range as (b)/(d).

electron flow following skipping orbit-like motion alongedges.

A mechanism which may cause the oscillations involvesthe alignment of the density of states (DOS) around theFermi energy. Diagrams of the DOS in the single- anddual-gated regions are sketched in Figs. 6a-c. Figure 6dshows a zoom in the map of the oscillatory transconduc-tance of Fig. 3c. At point a in this zoom the filling factorin the dual-gated regime is νDG = 180 and νSG = -92in the single-gated region. Because of the fourfold de-generacy of the Landau levels, Landau level numbers areN = 45 and N = -23 respectively, as shown in the DOSdiagram of Fig. 6a. When following the oscillatory pat-tern from point a to point b, the two combs of DOS re-main aligned with one another and only the Fermi levelchanges. This in contrast to what happens when movingfrom point a to point c: the DOSs shift with respect toone another and the transconductance oscillates. It couldtherefore be the case that the alignment of the DOS af-fects the conductance of the p-n interface in a way sim-ilar to the magneto-intersubband oscillations (MISO) ofa two-dimensional electron gas (2DEG):19,20 the occupa-tion of two energy subbands of a 2DEG can lead to en-hanced scattering between the subbands when the DOSsof the subbands are aligned. Although the p- and n- re-gions are spatially separated in the case of graphene p-njunctions, a similar enhancement of the coupling at theinterface may be observed. In the pp’p and nn’n regimethe interfaces are much more transparent (see conduc-tance in Fig. 4a,d and Ref. 21), therefore the interface

plays a negligible role in the total conductance. This ex-plains why the oscillations are only visible in presence ofa p-n interface. As the two outer regions of the samplehave the same density up to an insignificant difference inresidual doping, the two interfaces contribute in a similarway. The number of interfaces can at most influence thevisibility of the oscillations. In practice we find that thevisibility is however mostly influenced by sample quality.Just as for the oscillations we report on, MISO persist upto relatively high temperatures. The spacing predictedby this model lacks a factor of two compared to the ex-periment, however: it would predict the argument of thecosine of Eq. (1) to be 2π∆ν/4. Further investigationis needed to explain this discrepancy between the MISOmodel on the one hand and the experimental data andthe tight-binding simulations on the other hand.

In conclusion, we have observed oscillations in the con-ductance of six graphene p-n junctions in the magneticfield range of B = 0.4− 1.5 T. The oscillations are inde-pendent of sample width and can be described by the fill-ing factor difference between the single- and dual-gatedregions. The oscillations are quite robust against tem-perature changes: they fade out only in the range ofT = 20 − 40 K, whereas Shubnikov-de Haas oscillationsdecay below T = 10 K. The oscillations can be well re-produced by tight-binding transport calculations consid-ering an ideal p-n junction at a constant magnetic field.Up to a factor of two, the oscillatory pattern can be ex-plained by considering the density of states alignment ofthe single- and dual-gated regions.

6

DOS DG

(a)

EF

E

N = 44

N = 45

N = 46

N = 47

DOS SG

N = -25

N = -24

N = -23

N = -22

DOS DG

(b)

EF

E

N = 45

N = 46

N = 47

N = 48

DOS SG

N = -24

N = -23

N = -22

N = -21

DOS DG

(c)

N = 42

N = 43

N = 44

N = 45

EF

E

DOS SG

N = -25

N = -24

N = -23

N = -22-12 12

dG/dVTG (e2/h V−1)(d)

168 172 176 180 184 188νDG

-88

-92

ν SG

a

b

c

FIG. 6. (a-c) Schematics of the densities of states as a functionof energy for the single- and dual-gated regions of the sam-ple at positions a-c of the measurement (d) Zoom in Fig. 3c.The alignment of the densities of states can lead to an oscil-latory pattern with the right slope: along the line from pointa to point b the two combs of densities of states stay aligned,whereas the combs shift with respect to one another whenmoving from point a to point c.

ACKNOWLEDGEMENT

We thank Peter Makk and Francois Peeters for fruit-ful discussions. We acknowledge financial support fromthe European Graphene Flagship and the Swiss NationalScience Foundation via NCCR Quantum Science andTechnology. M.-H.L. and K.R. acknowledge financialsupport by the Deutsche Forschungsgemeinschaft withinSFB 689.

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21 P. Rickhaus, M.-H. Liu, P. Makk, R. Maurand, S. Hess,S. Zihlmann, M. Weiss, K. Richter, and C. Schonenberger,Nano Letters 9, 5819 (2015).

Supporting Information forOscillating Magnetoresistance in Graphene p-n Junctions at Intermediate Magnetic Felds

Hiske Overweg,1 Hannah Eggimann,1 Ming-Hao Liu,2, 3 Anastasia Varlet,1 Marius Eich,1 Pauline Simonet,1 YongjinLee,1 Kenji Watanabe,4 Takashi Taniguchi,4 Klaus Richter,2 Vladimir I. Fal’ko,5 Klaus Ensslin,1 and Thomas Ihn1

1Solid State Physics Laboratory, ETH Zurich, CH-8093 Zurich, Switzerland∗2Institut fur Theoretische Physik, Universitat Regensburg, D-93040 Regensburg, Germany

3Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan4National Institute for Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan

5National Graphene Institute, University of Manchester, Manchester M13 9PL, UK

TIGHT BINDING SIMULATIONS

Overview

In the main text, we have shown a transmission map asa function of left and right carrier density, calculated usingthe real-space Green’s function method based on the scalabletight-binding model [1]. The considered graphene ribbon ofwidth W = 1 µm is subject to a model density function de-scribing an ideal pn junction with smoothness 50nm. The fullmap is repeated here in Fig. S1(a), with a white box markingthe region plotted in Fig. S1(b).

In this Supporting Information, we show more numericalresults in order to demonstrate that the observed oscillationis independent of the smoothness of the pn junction and thewidth of the graphene ribbon, and is not related the currentalong the pn junction. Instead, the oscillation is shown by thelast numerical test to be closely related to the Landau levelsaway from the pn junction.

For quantitative and systematic comparisons, we will focuson the density range shown in Fig. S1(b) and the line cut onit along the dashed line shown in Fig. S1(c). All calculationsshown in the following consider the same density range andresolution as Figs. S1(b) and (c), which can be regarded as thereference panels of this Supporting Information. In particu-lar, the line cut of Fig. S1(c) will be repeatedly shown in thefollowing results.

Smoothness dependence

Figure S2(a) presents the transmission map with smooth-ness of 25nm, showing a similar pattern seen in Fig. S1(b)where the junction smoothness is 50nm. A more quantitativecomparison is shown in Fig. S2(b) for the line cuts of the twocases. Despite a slightly higher T obtained for the sharperjunction due to the Klein collimation [2], i.e., the sharper thepn junction, the wider the finite transmission probability ofthe angle distribution and hence the conductance, the gen-eral trend of the oscillation is shown to be independent of thesmoothness.

In the rest of the numerical results, the smoothness will befixed to 50nm.

Width dependence

Figure S3 presents the transmission map based on agraphene ribbon with W = 0.9 µm shown in its panel (a) andW = 0.8 µm shown in its panel (b). Comparing the line cutsin Fig. S3(c), along with the reference line of Fig. S1(c) forthe case of W = 1 µm, the feature of the oscillation is clearlyshown to be width-independent. On the other hand, the oscil-lation amplitude decreasing with the reduced graphene widthimplies that the oscillation may be closely related to the Lan-dau levels in the bulk away from the pn junction, since thewider the graphene ribbon the better the Landau levels candevelop.

In the rest of the numerical results, the graphene width willbe fixed as W = 1 µm.

Strong lattice defects

Next we show that the oscillation is not related to the cur-rent along the pn junction. To this end, we consider large-arealattice defects located in the vicinity of the pn junction. Thebasic idea is that if the oscillation came from any interferencedue to the current along the pn junction, such as the snakestate [3], a large-area lattice defect at the pn interface or inthe vicinity of it would act as a strong scatterer, destroying the

-2 0 2

nR

(1012 cm-2)

-3

-2

-1

0

1

2

3

nL (

10

12 c

m-2

)

204060

T

-2.5 -2 -1.5 -1

nR

(1012 cm-2)

1.5

2

2.5

nL (

10

12 c

m-2

)

10

20

30

40

50

T

1.5 2 2.5

nL (1012 cm-2)

20

40

60

T

(a)

(b)

(c)

FIG. S1. (a) Transmission map T (nR,nL) same as Fig. 5(a) in themain text (smoothness 50nm and graphene width W = 1 µm); whitedashed box marks the region shown in (b), where a black dashed lineindicates the line cut of T (nL) at fixed nR ≈−2×1012 cm−2 shownin (c).

2

-2.5 -2 -1.5 -1

nR

(1012 cm-2)

1.5

2

2.5n

L (

10

12 c

m-2

)

20

40

60

T

1.5 2 2.5

nL (1012 cm-2)

20

40

60

T

(a) (b)

FIG. S2. (a) Transmission map T (nR,nL) similar to Fig. S1(b) butwith smoothness 25nm of the pn junction. The red dashed line indi-cates the line cut shown as a red line in (b), where the black line isthe reference line identical to Fig. S1(c) for the case with smoothness50nm.

-2.5 -2 -1.5 -1

nR

(1012 cm-2)

1.5

2

2.5

nL (

10

12 c

m-2

)

20 40 60

T

-2.5 -2 -1.5 -1

nR

(1012 cm-2)

20 40 60

T

1.5 2 2.5

nL (1012 cm-2)

20

40

60

T

(a) (b)

(c)

FIG. S3. Transmission maps T (nR,nL) similar to Fig. S1(b) withthe same smoothness of 50nm but with (a) W = 0.9 µm and (b) W =0.8 µm. Line cuts along the red/blue dashed line marked in (a)/(b) arecompared in (c) together with the reference line (black) of Fig. S1(c)for the case of W = 1 µm.

interference and hence suppressing the oscillation. Contrarily,if the oscillation survives the introduced large defects, the cur-rent along the pn junction will then be ruled out from possibleorigins of the oscillation.

We first consider a 50× 400nm2 defect in Figs. S4(a) and(b); the defect is placed in front of the pn junction (at a dis-tance 150nm) in the former, and exactly on the pn junction inthe latter. Despite an additional modulating pattern observedin Fig. S4(a), the fine oscillation patterns remain visible inboth cases. By increasing the defect area to 300× 300nm2,the transmission map shown in Fig. S4(c) still exhibits thesame oscillation pattern. A quantitative comparison of the linecuts summarized in Fig. S4(d) together with the reference linefrom Fig. S1(c) clearly shows that the oscillations observedin Figs. S4(a)–(c) belong to the same type as all those shownpreviously.

The fact that the strong defect introduced in the vicinity ofthe pn junction cannot suppress the oscillation clearly indi-cates that any possible interference effect due to the current

-2.5 -2 -1.5 -1

nR

(1012 cm-2)

1.5

2

2.5

nL (

10

12 c

m-2

)

15 20 25 30

T

-2.5 -2 -1.5 -1

nR

(1012 cm-2)

10 15 20 25

T

-2.5 -2 -1.5 -1

nR

(1012 cm-2)

15 20 25 30

T

1.5 2 2.5

nL (1012 cm-2)

0

50

T

(a) (b) (c)

(d)

FIG. S4. (a)–(c) Transmission maps T (nR,nL) similar to Fig. S1(b)with the same smoothness of 50nm and width W = 1 µm, but witha large-area defect represented by the black rectangle shown in theindividual inset to the right of each panel, where the color back-ground depicts the y-independent model function n(x,y) describingthe density variation of the pn junction. The size of the defect is50×400nm2 in (a,b) and 300×300nm2 in (c). Line cuts along thered/blue/purple dashed line marked in (a)/(b)/(c) are compared in (d)together with the reference line (black) of Fig. S1(c) for the casewithout the defect.

along the pn junction cannot be the origin causing the oscilla-tion. Instead, the oscillation seems to depend only on the Lan-dau levels that are well developed in the semi-infinite leads.

Fixed leads

So far, all the presented calculations are based on an infinitegraphene ribbon with a pn junction in the middle, as describedin the main text. Technically, this is achieved in numerics byconsidering a scattering region of size L×W attached to twoleads from the left and right, both floating with the densityprofiles at the attaching edge of the scattering region. As longas L is much longer than the smoothness of pn junction (L =400nm has been adopted in all the presented calculations),the density values at the left and right edges of the scatteringregion will saturate to a constant, and the entire open quantumsystem of the finite-size scattering region attached to the twofloating semi-infinite leads will resemble an ideal pn junctionin the middle of an infinitely long graphene ribbon, exhibitingan L-independent transmission behavior.

As a final and conclusive numerical test, we now fix theFermi energies in the two semi-infinite leads at 0.1eV, andconsider the same range and parameters as the reference panelof Fig. S1(b). The calculated transmission map is shownin Fig. S5(a), which no longer exhibits the fine oscillation.The line cuts of fixed leads vs. floating leads compared inFig. S5(b) clearly show that the oscillation completely van-ishes in the present case of fixed leads.

The vanishing oscillation is consistent with what we havespeculated from the previously shown tests that the oscillationoriginates from the resonance between Landau levels well de-

3

-2.5 -2 -1.5 -1

nR

(1012 cm-2)

1.5

2

2.5n

L (

10

12 c

m-2

)

6

8

10

12

14

16

T

1.5 2 2.5

nL (1012 cm-2)

0

20

40

60

T

(a) (b)

FIG. S5. (a) Transmission map T (nR,nL) with the same range andparameters considered in Fig. S1(b) but with the two leads fixed atenergy E = 0.1eV. The red dashed line indicates the line cut shownas a red line in (b), where the black line is the reference line identicalto Fig. S1(c) for the case with floating leads.

veloped in the far left and far right in the semi-infinite leads.The present case shown in Fig. S5 considers fixed Fermi ener-gies in the leads that no longer float with the densities nR andnL. Together with the fact that the length L = 400nm W ofthe scattering region is too short for the Landau levels to form,the vanishing oscillation is therefore reasonably expected. Byincreasing the length of the scattering region to at least L≈W ,revival of the oscillation is expected for the case of fixed leads.

Note that the situation of fixed leads is actually closer to theexperiment, because the densities in graphene regions close tothe contacts are rather pinned by the contact doping. However,the samples in our experiments (summarized in Table I in themain text) are long enough (several microns in all samples)for the Landau levels to develop well (with level spacing notfar enough compared to disorder broadening in the magneticfield range we focus on) due to their cleanness and thereforeexhibit the oscillation. Our numerical results based on floatingleads correspond to the ideal case of infinitely long samplesand therefore exhibit optimized oscillation.

MAGNETORESISTANCE OSCILLATIONS IN SAMPLESB-F

Figures S6-S10 show magnetoresistance oscillations ofsamples B-F, which look similar to the ones observed in sam-ple A (see Fig. 3 of main text). The periodicity of the oscilla-tions is the same for all samples.

4-terminal measurements in sample D

The device layout of sample D is schematically shown inFig. S11. A DC voltage of 100mV was applied to the samplewith a R = 10 MΩ resistor in series. This led to a constantcurrent of I = 10 nA flowing from contact 1 to contact 4. The

voltage drop between contact pairs (1,2), (2,3) and (3,4) wheremeasured. To calculate the conductance a contact resistancewas subtracted where appropriate. As is shown in Fig. S11b-d, the oscillatory magnetoresistance is only observed when ap-n interface is present (i.e. in between contacts (2,3)).

-12 -6 0 6 12dG/dV TG (e2/h V−1)

-10 -8 -6 -4V TG (V)

0

10

20

30

40

VB

G(V

)

sample BB = 0.5 T

-200 -150 -100 -50νDG

120

160

200

ν SG

∆ν

=-3

32

∆ν

=-3

16

∆ν

=-3

00

∆ν

=-2

84

∆ν

=-2

68

∆ν

=-2

52

-2.5 -2.0 -1.5 -1.0 -0.5nDG (× 1012 cm−2)

1.6

2.0

2.4

n SG

(×10

12cm−

2)

(a)

(b)

FIG. S6. (a) Transconductance dG/dVTG of sample B at B = 0.5 T.(b) Transconductance as a function of filling factor in the single anddouble gated region.

∗ overwegh@phys.ethz.ch[1] M.-H. Liu, P. Rickhaus, P. Makk, E. Tovari, R. Maurand,

F. Tkatschenko, M. Weiss, C. Schonenberger, and K. Richter,Phys. Rev. Lett. 114, 036601 (2015).

[2] V. V. Cheianov and V. I. Fal’ko, Phys. Rev. B 74, 041403 (2006).[3] P. Rickhaus, P. Makk, M.-H. Liu, E. Tovari, M. Weiss, R. Mau-

rand, K. Richter, and C. Schonenberger, Nat. Commun. 6, 6470(2015).

4

-12 -6 0 6 12dG/dV TG (e2/h V−1)

-10 -8 -6 -4 -2 0V TG (V)

0

20

40

60V

BG

(V)

sample CB = 0.8 T

-200 -150 -100 -50νDG

100

120

140

ν SG

∆ν

=-3

36

∆ν

=-3

20

∆ν

=-3

04

∆ν

=-2

88

∆ν

=-2

72

∆ν

=-2

56

∆ν

=-2

40

∆ν

=-2

24

∆ν

=-2

08

∆ν

=-1

92

∆ν

=-1

76

-4.0 -3.0 -2.0 -1.0nDG (× 1012 cm−2)

2.0

2.4

2.8

n SG

(×10

12cm−

2)

(a)

(b)

FIG. S7. a) Transconductance dG/dVTG of sample C at B =0.8 T. (b) Transconductance as a function of filling factor in thesingle and double gated region.

-100 -50 0 50 100dG/dV TG (e2/h V−1)

-6 -5 -4 -3 -2 -1 0V TG (V)

0

5

10

15

20

25

VB

G(V

)

sample DB = 0.4 T

-200 -150 -100 -50 0νDG

40

80

120

160

ν SG

∆ν

=-2

40

∆ν

=-2

24

∆ν

=-2

08

∆ν

=-1

92

∆ν

=-1

76

∆ν

=-1

60

∆ν

=-1

44

∆ν

=-1

36

-1.5 -1.0 -0.5 0.0nDG (× 1012 cm−2)

0.4

0.8

1.2

1.6

n SG

(×10

12cm−

2)

(a)

(b)

FIG. S8. (a) Transconductance dG/dVTG of sample D at B =0.4 T. (b) Transconductance as a function of filling factor in thesingle and double gated region.

5

-12 -6 0 6 12dG/dV TG (e2/h V−1)

-6 -4 -2 0V TG (V)

0

1

2

3

4V

BG

(V)

sample EB = 0.5 T

-150 -100 -50νDG

40

80

120

ν SG

∆ν

=-2

32

∆ν

=-2

16

∆ν

=-2

00

∆ν

=-1

84

∆ν

=-1

68

∆ν

=-1

52

∆ν

=-1

36

∆ν

=-1

20

∆ν

=-1

04-2.0 -1.5 -1.0 -0.5

nDG (× 1012 cm−2)

0.8

1.2

1.6

n SG

(×10

12cm−

2)

(a)

(b)

FIG. S9. (a) Transconductance dG/dVTG of sample E at B =0.5 T. (b) Transconductance as a function of filling factor in thesingle and double gated region.

-12 -6 0 6 12dG/dV TG (e2/h V−1)

-6 -5 -4 -3 -2V TG (V)

0

20

40

60

VB

G(V

)

sample FB = 0.8 T

-150 -100νDG

40

80

120

ν SG

∆ν

= -248

∆ν

= -232

∆ν

= -216

∆ν

= -200

∆ν

= -184

∆ν

= -168 ∆

ν= -15

2

∆ν

= -136

-3.0 -2.5 -2.0 -1.5nDG (× 1012 cm−2)

1.0

1.5

2.0

2.5

n SG

(×10

12cm−

2)

(a)

(b)

FIG. S10. (a) Transconductance dG/dVTG of sample F at B =0.8 T. (b) Transconductance as a function of filling factor in thesingle and double gated region.

6

1

2 3

4TG

0

5

10

15

20

25

VB

G(V

)

-100 0 100dG23/dV TG (e2/h V−1)

-6 -4 -2 0V TG (V)

0

5

10

15

20

25

VB

G(V

)

-100 0 100dG12/dV TG (e2/h V−1)

-6 -4 -2 0V TG (V)

-100 0 100dG34/dV TG (e2/h V−1)

(a)

(b)

(c)

(d)

FIG. S11. (a) Schematic drawing of sample D. Contacts are labelled1-4. (b) Transconductance between contacts (1,2) at B = 0.4 T. (c)Transconductance between contacts (2,3) showing an oscillatory pat-tern in the p-n regime. (d) Transconductance between contacts (3,4).