topograph - kickoff · topograph engineering topological superconductivity in graphene bme-eötvös...

TopoGraph Engineering topological superconductivity in graphene

TopoGraph Engineering topological superconductivity in graphene

BME-Eötvös (Hungary) CSIC (Spain)

TUDelft (Netherlands)

Péter Makk (coordinator)

Szabolcs Csonka Rakyta Péter

Andor Kormányos Srijit Goswami Attila Geresdi

Pablo San-Jose Ramón Aguado

9

OverviewThe road to Quantum Computation

Cold atoms

NV centers in diamond Spin Qubits

Superconducting Qubits

Topological Qubits (Majorana) Atomic chains

Semiconducting nanowires Epitaxial semiconductors

2D crystals?3

Quantum ManifestoA New Era of Technology Draft - March 2016

8

2015 2035

ATOMIC

QUANTUM

CLOCK QUANTUM

SENSORQUANTUM

INTERNET

UNIVERSAL

QUANTUM

COMPUTERQUANTUM

SIMULATOR

INTERCITY

QUANTUM

LINK

Quantum Technologies Timeline

1B

4C

1C

1D 1E

3F2E

1F

1A

2A

3A

4A

4B

2B

3D3E

2C

2D

3C

4E4D 4F

3B

1. Communication

A Core technology of quantum repeaters

B Secure point-to-point quantum links

C Quantum networks between distant cities

D Quantum credit cards

E Quantum repeaters with cryptography and eavesdropping detection

F Secure Europe-wide internet merging quantum and classical communication

2. Simulators

A Simulator of motion of electrons in materials

B New algorithms for quantum simulators and networks

C Development and design of new complex materials

D Versatile simulator of quantum magnetism and electricity

E Simulators of quantum dynamics and chemical reaction mechanisms to support drug design

3. Sensors

A Quantum sensors for niche applications (incl. gravity and magnetic sensors for health care, geosurvey and security)

B More precise atomic clocks for time stamping of high-frequency financial transactions

C Quantum sensors for larger volume applications including automotive, construction

D Handheld quantum navigation devices

E Gravity imaging devices based on gravity sensors

F Integrate quantum sensors with consumer applications including mobile devices

4. Computers

A Operation of a logical qubit protected by error correction or topologically

B New algorithms for quantum computers

C Small quantum processor executing technologically relevant algorithms

D Solving chemistry and materials science problems with special purpose quantum computer > 100 physical qubit

E Integration of quantum circuit and cryogenic classical control hardware

F General purpose quantum computers exceed computational power of classical computers

5 – 10 years

0 – 5 years

> 10 years

9

Goals"To develop the basic components for graphene-based topological qubits"

Topological superconductivity and Majoranas in graphene

4

Majorana Zero Modes in Graphene

P. San-Jose,1 J. L. Lado,2 R. Aguado,1 F. Guinea,3,4 and J. Fernández-Rossier2,51Instituto deCienciadeMaterialesdeMadrid,ConsejoSuperiorde InvestigacionesCientíficas (ICMM-CSIC),

Sor Juana Inés de la Cruz 3, 28049 Madrid, Spain2International Iberian Nanotechnology Laboratory (INL), Avenida Mestre José Veiga,

4715-330 Braga, Portugal3Instituto Madrileño de Estudios Avanzados en Nanociencia (IMDEA-Nanociencia), 28049 Madrid, Spain4Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom

5Departamento de Fisica Aplicada, Universidad de Alicante, 03690 Alicante, Spain(Received 27 July 2015; published 15 December 2015)

A clear demonstration of topological superconductivity (TS) and Majorana zero modes remains oneof the major pending goals in the field of topological materials. One common strategy to generate TS isthrough the coupling of an s-wave superconductor to a helical half-metallic system. Numerous proposalsfor the latter have been put forward in the literature, most of them based on semiconductors or topologicalinsulators with strong spin-orbit coupling. Here, we demonstrate an alternative approach for the creation ofTS in graphene-superconductor junctions without the need for spin-orbit coupling. Our prediction stemsfrom the helicity of graphene’s zero-Landau-level edge states in the presence of interactions and from thepossibility, experimentally demonstrated, of tuning their magnetic properties with in-plane magnetic fields.We show how canted antiferromagnetic ordering in the graphene bulk close to neutrality induces TS alongthe junction and gives rise to isolated, topologically protected Majorana bound states at either end. We alsodiscuss possible strategies to detect their presence in graphene Josephson junctions through Fraunhoferpattern anomalies and Andreev spectroscopy. The latter, in particular, exhibits strong unambiguoussignatures of the presence of the Majorana states in the form of universal zero-bias anomalies. Remarkableprogress has recently been reported in the fabrication of the proposed type of junctions, which offers apromising outlook for Majorana physics in graphene systems.

DOI: 10.1103/PhysRevX.5.041042 Subject Areas: Graphene, Superconductivity,Topological Insulators

I. INTRODUCTION

The realization of topological superconductivity (TS), anovel electronic phase characterized by Majorana excita-tions, has become a major goal in modern condensed-matter research. Despite promising experimental progress[1–13] on a number of appealing implementations [14–17],a conclusive proof of TS remains an open challenge. Here,we report on a new approach to obtain TS and Majoranastates in graphene-superconductor junctions. Key to ourproposal is the interaction-induced magnetic ordering ofgraphene’s zero Landau level (ZLL). Coupling this uniquestate to a conventional superconductor gives rise to noveledge states whose properties depend on the type ofmagnetic order. In particular, the canted antiferromagneticphase is a natural host for Majorana bound states. Ourproposal combines effects that were recently demonstratedexperimentally (tunable spin ordering of the ZLL [18,19]

and ballistic [20,21] graphene-superconductor junctionsof high transparency [21] operating in the quantum Hallregime [20]) and is thus ready to be tested.While intrinsic TS is rare, it can be synthesized effec-

tively through the coupling of a conventional s-wavesuperconductor (SC) and tailored electronic gases withspin-momentum locking. Using this recipe, it has beenpredicted that Majorana excitations should emerge whenone induces superconductivity onto topological insulators[14] or semiconductors with strong spin-orbit coupling[15]. Particularly attractive are implementations of one-dimensional TS using either semiconducting nanowires[16,17] or edge states in two-dimensional quantum spinHall (QSH) insulators since the main ingredients arealready in hand. These ideas have spurred a great dealof experimental activity [1–13]. Despite this progress,however, an unambiguous demonstration of TS is, argu-ably, still missing. Important limitations of these systemsinclude disorder, bulk leakage, or the imperfect proximityeffect (the so-called soft gap problem). Thus, it is worth-while to explore alternative materials.One particularly interesting option is graphene [22],

which exhibits very large mobilities even in ambient

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 3.0 License. Further distri-bution of this work must maintain attribution to the author(s) andthe published article’s title, journal citation, and DOI.

PHYSICAL REVIEW X 5, 041042 (2015)

2160-3308=15=5(4)=041042(15) 041042-1 Published by the American Physical Society

conditions, and where a ballistic proximity effect hasrecently been demonstrated [21]. Graphene was the firstmaterial in which a topological insulating phase wasproposed [23] in the presence of a finite intrinsic spin-orbit coupling. Kane and Mele showed that a gap opens atthe Dirac point, and a single helical edge mode with a spin-locked–to–propagation direction develops at each edge. Insuch a QSH regime, gapping the edge states throughproximity to a conventional superconductor gives rise toa one-dimensional TS along the interface [14]. Graphene’snegligible spin-orbit coupling, however, has proved to be afundamental roadblock in this program.In this work, we present a simple mechanism to realize

the above situation in graphene without recourse to spin-orbit coupling. We consider a graphene ribbon in thequantum Hall (QH) regime, in which, unlike in the QSHcase, time-reversal symmetry is broken by a strong mag-netic flux (Appendix A). In contrast to conventional two-dimensional electron gases, graphene develops a zeroLandau level at the Dirac point, which has been shownin pristine samples to become split because of electronicinteractions [24–29]. Experimental evidence [19] pointstowards spontaneous antiferromagnetic ordering [30–32],although other broken symmetries have been discussed[18]. In this work, we consider all possible magnetic orders.Figure 1 summarizes the different possibilities, rangingfrom ferromagnetic (F) to antiferromagnetic (AF) ordering[33], including canted AF which may be controlled by an

external in-plane Zeeman field as argued in Ref. [19]. Thedifferent orders are parametrized by the angle θ between thespin orientation of the ZLL in the two graphene sublattices,so that θ ¼ 0 for F and θ ¼ π for AF.While θ is considered, in our model, as an externally

tunable parameter as in Ref. [19], we have checked that amean-field calculation in a honeycomb Hubbard modelunder an in-plane Zeeman field (Appendix A) yields thesame bulk and edge phenomenology presented in this work[31]. Corrections beyond the mean field and the Hubbardmodel have been explored theoretically in the past and havepredicted the formation of a Luttinger liquid domain on aninfinite vacuum edge [34]. The corresponding excitationdensity resembles the noninteracting edge for F order,rather than the AF case. The interacting problem at a highlytransparent superconducting contact remains an openproblem. We conjecture that, given their robust topologicalorigin, the Majorana phenomena described here at a mean-field level would survive in the Luttinger regime, at leastwithin a limited range of parameters, and with power-lawcorrections to the transport results. These issues, however,remain beyond the scope of this work.

II. TOPOLOGICAL SUPERCONDUCTIVITYIN QUANTUM HALL GRAPHENE

An infinite ferromagnetically ordered (θ ¼ 0) ribbon invacuum has a QH mean-field band structure [35] as shown

(a) (d)

(e) (f) (h) (i)

(g)

(K)

(j)

(b) (c)

FIG. 1. Sketch, band structure, and phase diagram of a 350-nm-wide graphene sample (Fermi velocity vF ¼ 106m=s) in the quantumHall regime (out-of-plane field Bz ¼ 1 T) in various configurations. The chemical potential (dotted line) is tuned within the gapΔZLL ≈ 20 meV of the zero Landau level, which has ferromagnetic (a–c), antiferromagnetic (d–f), or canted antiferromagnetic ordering(g–i) due to electronic interactions. The band structures under each sketch correspond to an infinite graphene ribbon surrounded byvacuum (vac/Gr/vac panels) or coupled to a superconductor of gap ΔSC (chosen to be large, around 7 meV, for visibility) along the topedge as in the sketches (vac/Gr/SC panels, Nambu bands). Bands in red correspond to eigenstates localized at the top edge of the ribbon.The zero-energy local density of states is shown in blue in each sketch. (j) Low-energy band structure of states along a graphene-superconductor interface folded onto the Γ point, in the gapless (left panel, zoom of panel f), nontrivially gapped (middle panel, cantedorder), and trivially gapped (right panel, with strong intervalley scattering) phases. (k) Phase diagram of said interface, computed fromits low-energy effective Hamiltonian (see text), as a function of magnetic angle θ and intervalley coupling w. The three possible phasesare shown, bounded by threshold values w0, wins, and θins (see main text).

P. SAN-JOSE et al. PHYS. REV. X 5, 041042 (2015)

041042-2

9

GoalsRequirements:

High-mobility, low-disorder crystals: encapsulation Induced superconductivity: clean lateral contacts

5

FLAG-ERA JTC 2017 Full Proposal

[TopoGraph] page 5 of 25

Figure 1: Device structures. a) Van der Waals heterostructures based on graphene allow engineering materials properties on the atomic scale. By coupling it to SC electrodes (marked by red) topological superconducting state can be formed. b) An hBN/twisted bilayer graphene (TBLG)/hBN heterostructures coupled to SC electrodes, with graphite bottom-gate (contacted with a gold electrode). hBN is blue, graphite and graphene is gray, and gold electrodes are yellow. c) Side view of the TBLG device with graphite bottomgate and a topgate. d) Side view of another device with WSe2 substrate (orange), which will be used to induce SOI in graphene.

1.2 State of the art The field of topological superconductivity, while relatively new, has seen tremendous progress in recent years [1]. This includes experimental advances with both intrinsic topological superconductors and engineered topological superconductors by means of the proximity effect. We here follow the latter route for which experimental breakthroughs have come about through a combination of realistic theoretical proposals, advances in materials growth, improved device fabrication technologies and optimized measurement protocols. Key material systems include nanowires with large spin-orbit coupling, 2D TIs, and ferromagnetic atomic chains – all interfaced with a superconductor. After the initial spectroscopic signatures of MZMs in nanowires [3] there is growing evidence for their existence, especially in wires with MBE grown epitaxial aluminium [8]. Studies of the AC Josephson effect in HgTe quantum wells have demonstrated the existence of a 4p periodic component in the current-phase relation (CPR) [9], a necessary requirement. Finally, spatially resolved STM studies have shown an enhanced density of states at the ends of magnetic chains, consistent with the presence of MZMs [10]. Although these material systems are quite promising several technical problems persist, like remaining bulk conductance in TIs, disorder in nanowires or interfacing these materials with SC contacts is also challenging. Compared to these materials graphene has several advantages. It is a 2D material, proximity effects can change its properties on the atomic scale, it has high mobility and can be easily interfaced with SC materials. While graphene was historically the first proposed 2D TI system [11], realizing a topological superconductor in graphene has been, until recently, unrealistic. This situation has changed significantly in the past 2-3 years, mainly due to the ability to create extremely clean graphene devices based on van der Waals heterostructures [12], which have shown evidence of helical edge modes, akin to quantum spin Hall (QSH) state [6,7]. Furthermore, combining such heterostructures with superconductors via transparent interfaces have led to the observation of supercurrents and signatures of CAR in the quantum Hall regime, and intriguing edge modes [13] in gapped graphene

9

GoalsRequirements:

Quantum Spin Hall effect (QSHE) With spin-orbit coupling (SOC): induced by substrate

Without SOC: bilayer + interactions + Quantum Hall

6

4

F Finocchiaro et al

screened interactions, in the form of a purely on-site Hubbard term. The Hamiltonian then becomes = +H H H0 int where = ∑ ↑ ↓H U n ni i iint . To compute

the effects of Hint we employ a standard mean field decoupling, so that Hint is approximated by a self-

consistent = ∑ + +↑ ↓ ↑ ↓H U n n n n EMFi i i i iint U[ ] ,

with EU an unimportant constant5.In monolayer graphene, at half filling and for high

enough perpendicular magnetic fields, interactions are able to break the SU(4) = SU(2) × SU(2) spin and sublattice symmetry of the ZLL [20, 41–51]. Within the above Hubbard model, the symmetry breaking happens already at the self-consistent mean field level, which at zero filling ν = 0 predicts a sublattice-splitted SU(2)-symmetric ZLL with a gap ∆AF and characterized by an antiferromagnetically (AF) ordered ground state [20, 22, 41, 42, 47], consistent with experiments [23]. Figure 1(f) shows the ν = 0 Hubbard spectrum for the monolayer at small energies, with the AF ground state polarization depicted in figure 1(i). This AF ground state, with opposite spin polarization on different A/B sublattices, hosts no protected edge states, and is thus fully insulating. As the filling factor is increased to ν = 1,

the remaining SU(2) symmetry of the split ZLL level is broken, see figure 1(g). A small ν = 1 gap ∆FI opens, and the corresponding bulk ground state becomes fer-rimagnetic (FI), with uncompensated spin polarization between A/B sublattices, see figure 1(j). Protected edge states emerge within the FI gap, which, remarkably, are perfectly spin-polarized (red and blue in figure 1(g) denote opposite spin polarization = ±s 1z⟨ ⟩ ). The spin polarization of these edge states stems entirely from interactions, as no Zeeman coupling is included here.

As in the non-interacting case, the spectrum of the TwBG nanoribbon exhibits similar phenomenology as the two decoupled layers. This is true as long as ℓ ℓ>B M. Otherwise the Landau level broadening δZLL may exceed the relevant gap, be it ∆AF or ∆FI, and the bulk of the system then becomes metallic, as shown in figure 1(h).

2.3. Pseudo-QSH under an interlayer biasWe emphasise once more, in relation to the possibility of a pseudo-QSH phase (with helical counterpropagating edge states of opposite spins), the fact that within the ν = 1 gapped phase, each layer hosts one spin-polarized state at a given edge6, as sketched in panel (2) of figure 2(e), without the need of a Zeeman field. This is one of the key ingredients that allow the bilayer to be

Figure 2. (a) Pseudo-QSH bandstructure of two decoupled graphene monolayer with armchair edges, at fillings ν = 1 and ν = −1. τ = ±1 stands for the top and bottom layer respectively. (b) Pseudo-QSH bandstructure for a TwBG nanoribbon at the same filling, induced by a bias Vb between the two layers. Colors in the right (left) panel indicate spin (layer) polarization. Note that the ribbon has two edges, so that there are two states per edge. (c) Spin- and layer-resolved spatial density of the edge states along a TwBG semi-infinite plane in the pseudo-QSH regime. (d) Phase diagram of the two-terminal conductance G in units of e2/h for fixed values of B = 4 T, U = 1.5t and U2 = 0.28U, as a function of the total gating Vg and interlayer bias Vb. (e) Sketch of the edge states configuration in the four FI-ordered phases marked as (1–4) in (d), with red and blue denoting opposite spin polarizations.

6 The precise statement is that the total number of right-moving minus left-moving states per edge and layer is one.

5 Note that Fock terms are not required in this decoupling, as H0 is SU(2) symmetric. The Hartree decoupling then merely amounts to a specific choice of magnetization direction for symmetry breaking.

2D Mater. 4 (2017) 025027

same devices (Fig. 2a). A simple explanation for the difference is thatthe counterpropagatingmodes of the (±1,∓1) states have opposite spinpolarizations, which are the expected Zeeman plus exchange-driven

ground states for monolayer graphene at ν = ±1 (refs 20,33). Whenthe spin wavefunctions on each layer are orthogonal, interlayer tunnel-ling processes are forbidden and the edge states are protected from

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Figure 2 | Transport in graphene electron–hole bilayers. a, Cartoons depicting edge state configurations with νtop = −νbottom. b, Measured two-probeconductance G for νtot = 0 as a function of displacement field, D , at B = 4 T. The (−1,+1) state is conductive while the (−2,+2) state is insulating. c, Magneticfield dependence of νtot = 0 line (white dashed line shows the location of the line trace in b). d, Two-probe conductance map, G , as function of νtot and D .Conductance is given by νtot(e

2/h) for all configurations except for the (±1,∓1) states (indicated by the dashed white circles). Contact resistances have beensubtracted from the positive and negatives sides of the data to fit the νtot = ±1 plateaus. e, Schematic map of possible filling factor combinations.

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Figure 3 | Non-local measurements of helical edge states. a, Optical image of four-probe device. Scale bar, 5 µm. Graphite leads are highlighted in orangeand the gold top-gate that covers the device is highlighted in yellow. b, Schematics of different measurement configurations for the four-probe device.c, Non-local resistance, RNL, as a function of νtot and displacement field, D , measured at B = 4T. Dashed white circles highlight the (±1,∓1) states that exhibita strong non-local signal, indicating transport through highly conductive counterpropagating edge modes. Axis ranges are identical to those in Fig. 2d. In the(0,0) insulating state, RNL fluctuates strongly due to low current signals near the noise limit (bright white features). d, RNL (black line, left axis) compared totwo-probe resistance, Rtwo-probe, (grey line, right axis) of a constant D line cut through the (+1,−1) state (horizontal dashed line in Fig. 3c). RNL is near zerowhen Rtwo-probe exhibits a conductance plateau, since the voltage drop along a chiral edge state is zero. During plateau transitions, the bulk becomesconductive, resulting in a small peak that is suppressed by the non-local geometry of the measurement. e, Magnetic field dependence of RNL and Rxx in the(+1,−1) state. Plotted points are determined by averaging the resistance over the region in the νtot and D map corresponding to the (+1,−1) state (error barsrepresent one standard deviation from the mean).

NATURE NANOTECHNOLOGY DOI: 10.1038/NNANO.2016.214 LETTERS

NATURE NANOTECHNOLOGY | ADVANCE ONLINE PUBLICATION | www.nature.com/naturenanotechnology 3

© 2016 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

4

F Finocchiaro et al

screened interactions, in the form of a purely on-site Hubbard term. The Hamiltonian then becomes = +H H H0 int where = ∑ ↑ ↓H U n ni i iint . To compute

the effects of Hint we employ a standard mean field decoupling, so that Hint is approximated by a self-

consistent = ∑ + +↑ ↓ ↑ ↓H U n n n n EMFi i i i iint U[ ] ,

with EU an unimportant constant5.In monolayer graphene, at half filling and for high

enough perpendicular magnetic fields, interactions are able to break the SU(4) = SU(2) × SU(2) spin and sublattice symmetry of the ZLL [20, 41–51]. Within the above Hubbard model, the symmetry breaking happens already at the self-consistent mean field level, which at zero filling ν = 0 predicts a sublattice-splitted SU(2)-symmetric ZLL with a gap ∆AF and characterized by an antiferromagnetically (AF) ordered ground state [20, 22, 41, 42, 47], consistent with experiments [23]. Figure 1(f) shows the ν = 0 Hubbard spectrum for the monolayer at small energies, with the AF ground state polarization depicted in figure 1(i). This AF ground state, with opposite spin polarization on different A/B sublattices, hosts no protected edge states, and is thus fully insulating. As the filling factor is increased to ν = 1,

the remaining SU(2) symmetry of the split ZLL level is broken, see figure 1(g). A small ν = 1 gap ∆FI opens, and the corresponding bulk ground state becomes fer-rimagnetic (FI), with uncompensated spin polarization between A/B sublattices, see figure 1(j). Protected edge states emerge within the FI gap, which, remarkably, are perfectly spin-polarized (red and blue in figure 1(g) denote opposite spin polarization = ±s 1z⟨ ⟩ ). The spin polarization of these edge states stems entirely from interactions, as no Zeeman coupling is included here.

As in the non-interacting case, the spectrum of the TwBG nanoribbon exhibits similar phenomenology as the two decoupled layers. This is true as long as ℓ ℓ>B M. Otherwise the Landau level broadening δZLL may exceed the relevant gap, be it ∆AF or ∆FI, and the bulk of the system then becomes metallic, as shown in figure 1(h).

2.3. Pseudo-QSH under an interlayer biasWe emphasise once more, in relation to the possibility of a pseudo-QSH phase (with helical counterpropagating edge states of opposite spins), the fact that within the ν = 1 gapped phase, each layer hosts one spin-polarized state at a given edge6, as sketched in panel (2) of figure 2(e), without the need of a Zeeman field. This is one of the key ingredients that allow the bilayer to be

Figure 2. (a) Pseudo-QSH bandstructure of two decoupled graphene monolayer with armchair edges, at fillings ν = 1 and ν = −1. τ = ±1 stands for the top and bottom layer respectively. (b) Pseudo-QSH bandstructure for a TwBG nanoribbon at the same filling, induced by a bias Vb between the two layers. Colors in the right (left) panel indicate spin (layer) polarization. Note that the ribbon has two edges, so that there are two states per edge. (c) Spin- and layer-resolved spatial density of the edge states along a TwBG semi-infinite plane in the pseudo-QSH regime. (d) Phase diagram of the two-terminal conductance G in units of e2/h for fixed values of B = 4 T, U = 1.5t and U2 = 0.28U, as a function of the total gating Vg and interlayer bias Vb. (e) Sketch of the edge states configuration in the four FI-ordered phases marked as (1–4) in (d), with red and blue denoting opposite spin polarizations.

6 The precise statement is that the total number of right-moving minus left-moving states per edge and layer is one.

5 Note that Fock terms are not required in this decoupling, as H0 is SU(2) symmetric. The Hartree decoupling then merely amounts to a specific choice of magnetization direction for symmetry breaking.

2D Mater. 4 (2017) 025027

9

MethodsFabrication

Dry pickup method for encapsulation (hBN, TMDC) Edge contact method (MoRe and NbTiN)

Measurement Local tunnelling spectroscopy Current-phase relations in SQUID geometry Fraunhofer patterns Microwave spectroscopy

Simulation MathQ / QBox.jl codes (CSIC) EQuUs codes (BME)

7



filling factors, a QSH state is formed and coupling to SC contacts should enable the formation on MBSs. Here, and in the previous method challenges can also lie in controlling the doping close to the superconducting contacts, which might dope the graphene or screen the electric fields. This could lead to changes of the local filling factor close to the control; therefore, separate gates controlling the doping close to the contacts might be needed. All of the above proposals allow to generate MZMs, however to probe them current-phase relation, Fraunhofer and local bias spectroscopy measurements will be needed [16].

Figure 2: Different methods for probing the topological states in graphene. Red colour marks SC

contacts. a) Tunnelling spectroscopy using local probes separated from graphene with a thin hBN

barrier. b) Tunnelling spectroscopy using side coupled QPCs. c) CPR measurement of a graphene with

SOI using SQUID-type measurements. d) Similar CPR measurements using radiation detection

techniques.

We will address the CPR with several complementary techniques to provide a compelling evidence for the topological phase transition. It has been predicted, that a 4p-periodic supercurrent is the hallmark of topological superconductivity [1,9]. A direct measurement of the CPR can be achieved by embedding the Josephson junction (JJ) into a SQUID, where the other arm is a topologically trivial junction of much higher critical current (Fig. 2c). Here, by measuring the critical current of the SQUID as a function of the applied flux, the CPR is directly mapped. We have recently used this method to measure the CPR of graphene using a second graphene JJ with much higher critical current [15]. Another possibility is to utilize the AC Josephson effect, since there is a universal relation between the applied bias voltage and emitted photon frequency of the JJ. The topological phase transition manifests itself as a factor of two decrease in frequency, which is then measured by conventional microwave spectroscopy or on-chip detection techniques [17] (Fig. 2d). Finally, the transition can be also monitored using Shapiro steps measurements. We have experience with all these techniques at zero magnetic field, but high magnetic fields impose new challenges, since conventional JJs cannot be used. Therefore, we will use graphene JJs at high doping as auxiliary junctions, which still posses decent supercurrent at high filling factors. The direct measurement of the Andreev bound state (ABS) spectra is another method to observe the topological phase transition. One approach we will pursue is to use our expertise in hBN as a tunnel barrier [19] and fabricate local probes to directly measure local DOS in graphene (Fig. 2a). This can be combined with small point like contacts as we have recently shown [20]: above graphene a thin hBN is placed as a barrier, and then a thicker one, with predefined hole, which can make the contact








techniques.









techniques.









techniques.




Figure 1: Device structures. a) Van der Waals heterostructures based on graphene allow engineering materials properties on the atomic scale. By coupling it to SC electrodes (marked by red) topological superconducting state can be formed. b) An hBN/twisted bilayer graphene (TBLG)/hBN heterostructures coupled to SC electrodes, with graphite bottom-gate (contacted with a gold electrode). hBN is blue, graphite and graphene is gray, and gold electrodes are yellow. c) Side view of the TBLG device with graphite bottomgate and a topgate. d) Side view of another device with WSe2 substrate (orange), which will be used to induce SOI in graphene.

1.2 State of the art The field of topological superconductivity, while relatively new, has seen tremendous progress in recent years [1]. This includes experimental advances with both intrinsic topological superconductors and engineered topological superconductors by means of the proximity effect. We here follow the latter route for which experimental breakthroughs have come about through a combination of realistic theoretical proposals, advances in materials growth, improved device fabrication technologies and optimized measurement protocols. Key material systems include nanowires with large spin-orbit coupling, 2D TIs, and ferromagnetic atomic chains – all interfaced with a superconductor. After the initial spectroscopic signatures of MZMs in nanowires [3] there is growing evidence for their existence, especially in wires with MBE grown epitaxial aluminium [8]. Studies of the AC Josephson effect in HgTe quantum wells have demonstrated the existence of a 4p periodic component in the current-phase relation (CPR) [9], a necessary requirement. Finally, spatially resolved STM studies have shown an enhanced density of states at the ends of magnetic chains, consistent with the presence of MZMs [10]. Although these material systems are quite promising several technical problems persist, like remaining bulk conductance in TIs, disorder in nanowires or interfacing these materials with SC contacts is also challenging. Compared to these materials graphene has several advantages. It is a 2D material, proximity effects can change its properties on the atomic scale, it has high mobility and can be easily interfaced with SC materials. While graphene was historically the first proposed 2D TI system [11], realizing a topological superconductor in graphene has been, until recently, unrealistic. This situation has changed significantly in the past 2-3 years, mainly due to the ability to create extremely clean graphene devices based on van der Waals heterostructures [12], which have shown evidence of helical edge modes, akin to quantum spin Hall (QSH) state [6,7]. Furthermore, combining such heterostructures with superconductors via transparent interfaces have led to the observation of supercurrents and signatures of CAR in the quantum Hall regime, and intriguing edge modes [13] in gapped graphene

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Work PackagesWP1: Design and optimisation of basic components

Characterise substrate-induced SOC (WSe2) Optimise SC contacts Optimise geometries

WP2: engineer QSHE+SC, with SOC induced by substrate Topological phase diagram (CPR, WAL, Shapiro, microwave, tunnel…)

WP3: engineer pseudo-QSH+SC on bilayer graphene in the QH regime Topological phase diagram (CPR, WAL, Shapiro, microwave, tunnel…)

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(Copenhagen), Silvano de Franceschi (Grenoble). Interactions with this extensive scientific network will certainly be valuable in finding solutions to important problems, which arise during the course of the project. List of milestones:

No of Milestone

Delivery month

WP involved

Title

M1 12 1 QSH state with SC contacts observed M2 24 1 Benchmark of new codes for critical currents M3 30 2,3 Demonstration of MBSs from ABS measurements M4 36 2,3 Demonstration of MBSs using current-phase relation

measurements Risk analysis:

Risk description Likelihood Impact Mitigation plan Vortices appearing in the QH limit low WP1 Try another SC material, different

contact geometry (holes in the SC films as nucleation sites, etc)

Proposed complex-frequency algorithms for critical currents are not more efficient than conventional ones

low WP1 Optimize conventional algorithms by treating subgap and above-gap supercurrents independently

The SOI is not sufficiently strong compared to disorder scale

medium WP1/2 Try to minimize disorder using graphite gates, or try other SOI materials (e.g. using the result of iSpintext network)

SC contacts change the local filling factor close to the contact

medium WP1/3 Design and fabricate local gates to tune the regions close to the contacts

Complex devices structures in WP2/WP3 are not developed in time

high WP2/3 Due to the parallel run of WP2 and WP3 the other WP can still progress on.

Regular CPR measured in the topological regime due to QP poisoning

high WP2/3 Try different graphene-SC interface preparation and fabrication quasiparticle traps and engineer faster CPR measurements.

2.3 Consortium

2.3.1 Description of the consortium Topograph is set up from 2 leading experimental groups and 2 theory groups (forming 3 nodes together) with complementary expertise and tasks necessary to carry out the outlined program. They are experts in the field of graphene nanodevices, SC hybrid structures, Majorana physics and spintronics. Due to the small size of the group and due to their different expertiese we expect intensive, a live collaboration between the nodes. We would also mention, that although industrial partners are not members of the consortium, TUD works in close collaboration with leading industrial players in quantum computation (in particular Microsoft Research). Below, detailed descriptions of the nodes are given.

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Resources

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Total person/month 198

Total budget 560 K€

Team size 8 + 3 in 3 nodes

Project duration 36 months

Project start 1st May 2018

Project type Experiment + Theory

topograph - kickoff · topograph engineering topological superconductivity in graphene bme-eötvös...

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