Quantized conductance doubling and hard gap in a two-dimensional semiconductor-superconductor heterostructure

Abstract

Coupling a two-dimensional (2D) semiconductor heterostructure to a superconductor opens new research and technology opportunities, including fundamental problems in mesoscopic superconductivity, scalable superconducting electronics, and new topological states of matter. One route towards topological matter is by coupling a 2D electron gas with strong spin-orbit interaction to an s-wave superconductor. Previous efforts along these lines have been adversely affected by interface disorder and unstable gating. Here we show measurements on a gateable InGaAs/InAs 2DEG with patterned epitaxial Al, yielding devices with atomically pristine interfaces between semiconductor and superconductor. Using surface gates to form a quantum point contact (QPC), we find a hard superconducting gap in the tunnelling regime. When the QPC is in the open regime, we observe a first conductance plateau at 4e² /h, consistent with theory. The hard-gap semiconductor-superconductor system demonstrated here is amenable to top-down processing and provides a new avenue towards low-dissipation electronics and topological quantum systems.

ARTICLE

Received 6 Mar 2016 | Accepted 8 Aug 2016 | Published 29 Sep 2016

M. Kjaergaard1, F. Nichele1, H.J. Suominen1, M.P. Nowak2,3,4, M. Wimmer2,3, A.R. Akhmerov2, J.A. Folk5,6,

K. Flensberg1, J. Shabani7,w, C.J. Palmstrm7 & C.M. Marcus1

Coupling a two-dimensional (2D) semiconductor heterostructure to a superconductor opens new research and technology opportunities, including fundamental problems in mesoscopic superconductivity, scalable superconducting electronics, and new topological states of matter. One route towards topological matter is by coupling a 2D electron gas with strong spinorbit interaction to an s-wave superconductor. Previous efforts along these lines have been adversely affected by interface disorder and unstable gating. Here we show measurements on a gateable InGaAs/InAs 2DEG with patterned epitaxial Al, yielding devices with atomically pristine interfaces between semiconductor and superconductor. Using surface gates to form a quantum point contact (QPC), we nd a hard superconducting gap in the tunnelling regime. When the QPC is in the open regime, we observe a rst conductance plateau at 4e2/h,

consistent with theory. The hard-gap semiconductorsuperconductor system demonstrated here is amenable to top-down processing and provides a new avenue towards low-dissipation electronics and topological quantum systems.

1 Center for Quantum Devices and Station Q Copenhagen, Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark.

2 Kavli Institute of Nanoscience, Delft University of Technology, PO Box 4056, 2600 GA Delft, The Netherlands. 3 QuTech, Delft University of Technology, PO Box 4056, 2600 GA Delft, The Netherlands. 4 AGH University of Science and Technology, Faculty of Physics and Applied Computer Science, Mickiewicza 30, 30-059 Krakw, Poland. 5 Department of Physics and Astronomy, University of British Columbia, Vancouver, British Columbia,Canada V6T1Z1. 6 Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia, Canada V6T1Z4. 7 California NanoSystems Institute, University of California, Santa Barbara, California 93106, USA. w Present address: Physics Department, City College of the City University of

New York, New York 10031, USA. Correspondence and requests for materials should be addressed to C.M.M. (email: mailto:[email protected]

Web End [email protected] ).

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Quantized conductance doubling and hard gap in a two-dimensional semiconductorsuperconductor heterostructure

ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12841

Recent work on semiconductor nanowires has offered evidence for the existence of Majorana zero modes, a signature of topological superconductivity13. A

characteristic of the rst studies in this area was signicant subgap tunnelling conductance (a so-called soft gap), attributed to disorder at the semiconductorsuperconductor (SmS) interface4,5. In nanowires, the soft-gap problem was recently resolved by growing Al epitaxially on InAs nanowires, yielding greatly reduced subgap conductance6,7. Studies of SmS systems based on top-down processed gateable two-dimensional electron gases (2DEGs) coupled to superconductors have not explicitly addressed the soft-gap issue yet8,9. However experiments on such systems have demonstrated other theoretical predictions, such as quantization of critical current911, the retro-reection property of Andreev scattering12, and spectroscopy of a gate-dened quantum dot with superconducting leads13,14, which do not require a hard proximity-induced gap in the semiconductor.

The two main results we present in this paper are both consequences of the transparent epitaxial SmS interface and overcome the soft gap problem for 2D electron gases. The rst is a doubling of the the lowest quantized conductance plateau, from 2e2/h in the normal state to 4e2/h in the superconducting state, as predicted theoretically15. The second is a strong suppression of conductance for voltages smaller than the superconducting gap when the quantum point contact (QPC) is in the tunnelling regimethat is, the detection of a hard superconducting gap in a proximitized 2DEG. Conductance doubling arises from Andreev reection transferring charge 2e into the superconductor16. The hard gap reects the absence of electronic states below the superconducting gap in the semiconductor. Using gate voltage to control the QPC, we measure conductance across the transition from weak tunnelling to the open-channel regime and nd good (but not perfect) agreement with the theory of a normal-QPC-superconductor structure15.

ResultsProperties of the 2DEG and the superconducting Al lm. The starting material is an undoped InAs/InGaAs heterostructure with epitaxial Al as a top layer, grown by molecular beam epitaxy17. A cross-sectional TEM showing a sharp epitaxial SmS interface is shown in Fig. 1a. In the devices reported here, the thickness of the InGaAs barrier was b 10 nm, and the Al lm

thickness was a 10 nm. A Hall ball fabricated on the same

wafer with the Al removed (see Methods) gave density n 3 1012 cm 2 and mobility m 104 cm2 V 1 s 1, yielding

a mean free path leB230 nm. In a similar wafer, weak anti-localization analysis gave a spinorbit length lsoB45 nm (ref. 17).

The Al lm has a critical temperature Tc 1.56 K, corresponding

to a gap D0 235 meV, enhanced from the bulk value of Al, and

consistent with other measurements on Al lms of similar thickness18. The in-plane critical eld of the Al lm is Bc 1.65 T

(ref. 17).

Quantized conductance doubling. A scanning electron micro-graph of Device 1 is shown in Fig. 1b. The conductance of the QPC is tuned by negative voltages applied to the gates. The QPC is located B150 nm in front of the region where the Al lm has not been removed. Figure 2 shows conductance traces for two lithographically similar QPCs. In the superconducting state, both devices show increased conductance at the plateau of the QPC and suppressed conductance below GB0.8G0, where G0 2e2/h,

relative to the normal state. This behaviour is the hallmark of Andreev reection being the dominant conduction mechanism through the QPC15,19. Raising the temperature above the critical temperature of the Al lm, applying an out-of-plane magnetic eld, or applying a bias larger than the gap, all bring the lowest plateau back to 2e2/h (Fig. 2). The dip structure at the transition between conductance plateaus was also observed in a similar experiment on nanowires20, and is presumably caused by mode mixing due to disorder, leading to a reduction in transparency of the already open rst channel. A constant contact resistance RcB1 kO has been subtracted in each viewgraph, a value chosen to move the rst plateau in the normal state to G0.

Hard superconducting gap. By further depleting the electron gas in the constriction, the device is operated as a tunnel probe of the local density of states in the InAs 2DEG. This technique has been applied to studying subgap properties of semiconductor nanowires coupled to superconductors13,6,21,22. In Fig. 3a, the

a b

Device 2

Vsd (mV) B

(mT) 0

0

a b

Al oxide

InGaAs (barrier)

InGaAs (barrier)

InAlAs

Al

a

G (2e2 /h)

G (2e2 /h)

b

InAs

z [100]

Figure 2 | Quantized conductance in the Andreev quantum point contact. (a) Differential conductance, G, as a function of gate voltage Vg at zero bias (black line), at source-drain bias larger than the gap (red line), and at elevated temperature (green line). At zero bias and base temperature, the rst conductance plateau is at 4e2/h, double the value at higher temperature or bias. (b) The differential conductance in a second, lithographically identical device at zero bias (black line), at source-drain bias larger than the gap (red line), and in a magnetic eld applied perpendicular to the plane of the chip (blue line).

y [011]

Aluminum

InGaAs/InAs

Ti/Au

y [011]

Figure 1 | Epitaxial aluminium on InGaAs/InAs and device layout.(a) Cross-sectional transmission electron micrograph of epitaxial Al on InGaAs/InAs. On the wafer imaged here, the height of the InGaAs barrier is b 5 nm and Al lm thickness aB5 nm. Scale bar, 5 nm. (b) False-colour

scanning electron micrograph of Device 1 (see main text for details). Scale bar, 1 mm.

6

5

4

3

2

1

0 8 7 6 5 Vg (V)

6

5

4

3

2

1

09 8 7 6

Vg (V)

x [011]

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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12841 ARTICLE

a c

G (2e2/h)

0

1

2

2

1.0

1

0.5

V sd(mV)

0.0

2*/e

0.5

101

9.0 8.8 8.6 8.4 8.2 Vg (V)

G (V sd= 0mV)(2e2 /h)

1.0

b

102

2

1

0

Vg (V)

10

5

0

G (2e2 /h)

G (10 22e 2/h)

102

104

1.0 0.5 0.0 0.5 1.0 Vsd (mV)

102 101 1

G (|Vsd| > 0.8mV)(2e2/h)

Figure 3 | Transition from 4e2/h conductance to hard superconducting gap. (a) Differential conductance, G, in Device 1 as a function of gate voltage Vg and source-drain voltage bias Vsd. (b) Vertical cuts in a in the tunnelling (red line) and one-channel (blue line) regime. Supplementary Figure 1 shows data from a lithographically similar device on a wafer with no InGaAs barrier (that is, b 0 nm) between the top layer Al and the InAs 2DEG. (c) Differential

conductance at zero source-drain voltage, G(Vsd 0 mV), versus averaged differential conductance at nite source-drain voltage, G(|Vsd|40.8 mV). Red

and blue circles indicate data corresponding to cuts in b. Green line is the theoretically predicted conductance in an Andreev enhanced QPC (equation (1) with no tting parameters).

QPC voltage is decreased to gradually transition from the one-channel regime, where the zero bias conductance is 4e2/h, to the tunnelling regime, where conductance is strongly suppressed for |Vsd|o190 mV. From these measurements, the gap in the density of states of the InAs due to the proximity to the Al is estimated to be D B190 meV (measured peak-to-peak). The value of D is similar, but not identical, to the gap in the Al lm as estimated from Tc, as discussed above.

In the case of perfect Andreev reection from the superconductor/semiconductor interface, the conductance of one channel through a constriction proximal to the interface is given by

Gns 2G0

G2nn

2G0 Gnn

2

; 1

where Gns is the conductance when the lm is superconducting, and Gnn is the conductance in the normal state15. In Fig. 3c, the prediction in equation (1) with no free parameters (green line) and experimental data are shown. Here, Gnn is the average

conductance for |Vsd|40.8 mV, justied by the equality of applying a bias and raising the temperature above Tc, as shown in Fig. 2a. Equation (1) is consistent with the data over two orders of magnitude in Gns, indicating that the zero bias conductance up to 4e2/h is well described by the prediction of perfect Andreev reection of a single QPC mode. Equation (1) represents the only

quantitative theory of the relation between subgap conductance and normal state conductance (that is, the hard gap) of which we are aware, and the agreement between equation (1) and the experiment in Fig. 3c leads to the designation of a hard gap in this superconductor2DEG system. However, the systematic deviation between data and prediction in Fig. 3c for Gnso10 2 2e2/h could be a manifestation of a small remnant

non-zero normal scattering probability.The shapes of the conductance curves at eVsdtD in the

tunnelling regime (red line in Fig. 3b) are smeared relative to the conventional BardeenCooperSchrieffer (BCS) density of states of a superconductor. This could be due to broadening of the BCS coherence peaks in the disordered superconducting lm formed in the 2DEG under the Al23, a weak coupling between Al and 2DEG5 or the layout of the tunnel probe relative to the proximitized 2DEG2426.

Temperature dependence of the density of states. The temperature dependence of the conductance in the Andreev QPC is different in the one-channel and in the tunnel regime (Fig. 4). The one-channel regime (Fig. 4a,b) has a pronounced kink at T Tc, presumably associated with the sudden onset of Andreev

enhanced subgap conductance. In contrast, the temperature dependence in the tunnel regime (Fig. 4c,d) is smeared close to Tc due to thermally excited quasiparticles.

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a

c

0.15

0.10

0.05

0.000.5 0.0 0.5

One-channel regime

T (K)0.1

Tunnelling regime

2

1

1.0 0.5 0.0 0.5 1.0

T (K)0.10.6

G (2e2/h)G (2e2/h)

Vsd (mV) Vsd (mV)

G (2e2/h)

G (2e2/h)

b

2

1

0.06

0.04

0.02

0.00 3.0

2.5

2.0

1.5

1.0

0.5

0.0

T (K)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

T (K)

One-channel regime d

Figure 4 | Temperature dependence of the enhanced subgap conductance and the hard superconducting gap. (a) Differential conductance, G, as a function of source-drain bias voltage, Vsd, at ve temperatures in the one-channel regime. See Supplementary Fig. 2a for similar data measured on a wafer with no InGaAs barrier between the top layer Al and the InAs 2DEG. (b) Temperature dependence at zero bias (corresponding to cut along the dashed, grey line in a in the one-channel regime. (c) Similar measurement to a but in the tunnelling regime. (d) As in b for two different values of gate voltage, Vg, both in the tunnelling regime. Insets in b and d show results from numerical simulations (see Supplementary Figs 35 for more details on numerical results).

The temperature dependence is simulated (insets in Fig. 4) by calculating G

R

dEG E

@f@E where f is the Fermi function

that accounts for thermal broadening. The conductance G E

is

calculated by combining scattering matrices of electrons and holes in the normal region and Andreev reection at the superconductor interface (details of the simulation are given in Methods). The scattering matrices are calculated using the numerical package Kwant27, and the simulation are performed using the device geometry from the micrograph in Fig. 1b. The temperature dependence of the gap is modeled with

D T D

1 T=Tc

q

2

, and the Andreev reection amplitude is taken from ref. 15. The simulation shows good quantitative agreement with the data.

Magnetic eld dependence of the density of states. To drive a superconductor/semiconductor device into a topological regime, one requirement is gmBB4D , while the native superconductor retains its gap. Figure 5 shows the in-plane magnetic eld dependence of D , from which an approximate critical eld

B c 300 mT is extracted. A rough estimate of the g-factor can be

inferred by assuming the critical B c results from Zeeman energy surpassing the induced superconducting gap, that is gmBB cD ,

which yields gB10, similar to the g-factor in bulk InAs. In Fig. 5d, the zero-bias conductance is shown for the two different in-plane directions, and the slight direction dependence of B c could be due to an anisotropic g-factor in the InAs crystal lattice.

The induced gap in the 2DEG disappears at in-plane magnetic elds signicantly smaller than the critical eld of the Al lm itself. The 2DEG has a strong spinorbit interaction (lsoB45 nm),

which, taken together with the intimate coupling to the superconductor, makes this material system a feasible candidate to realize topological superconducting devices. By using top-down fabrication techniques and the electrostatic gating demonstrated here, effective one-dimensional systems can be produced, in which an in-plane magnetic eld can close the induced superconducting gap to reach a topological phase.

In conclusion, we observe quantization doubling through a QPC proximal to a superconductor/semiconductor interface, conrming a long-standing theoretical prediction15. Operated as a gate-tunable tunnel probe of the local density of states, the QPC shows a hard superconducting gap induced in the 2DEG. The magnetic eld dependence of the induced gap compares favourably with the critical eld of the superconducting lm, opening possibilities to pursue topological states of matter in one-dimensional structures fabricated from epitaxial Al/2D InAs material.

Methods

Fabrication and measurement setup. Ohmic contacts to the InAs electron gas are formed directly by the epitaxial Al. Mesa structures are patterned by standard IIIV chemical etching techniques. The aluminium is etched using commercial Transene Aluminum Etch D. Subsequent to the selective Al etch, an insulating 40 nm Al2O3 layer is deposited using atomic layer deposition and metallic gates (5 nm Ti/50 nm Au) are evaporated onto the device. The

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a

c

0.15

0.21

0.39

By (T)

By (T)

0.0

0.1

0.2

0.4

0.0

0.06

G (2e2 /h)

G (2e2 /h)

G (102 2e2 /h)

G (102 2e2 /h)

8

6

4

2

2

1

1.0 1.0

0.5 0.0 0.5 Vsd (mV)

0 0.5 0.0 0.5 Vsd (mV)

b

d

2

1

3

2

1

0 0.6

0.4

0.2

0.0

0.0

0.3

0.6

0.9

B (T) B (T)

Figure 5 | In-plane magnetic eld of the enhanced subgap conductance and the hard superconducting gap. (a) Differential conductance, G, as a function of source-drain bias, Vsd, at several in-plane magnetic elds applied along the point contact constriction. (b) Zero-bias conductance as a function of the in-plane magnetic eld, By. (c) Similar measurement to a but in the tunnelling regime. Supplementary Fig. 2b shows data on a lithographically similar device fabricated on a wafer with no InGaAs barrier between the top layer Al and InAs 2DEG (that is, b 0 nm). (d) As in b but in the tunnelling regime, for both

in-plane directions (By is along and Bx is perpendicular to the constriction).

measurements were performed in a dilution refrigerator with a base-mixing chamber temperature TmcB30 mK, using four-terminal lock-in techniques and

DC measurements.

Measurement details. The data in Fig. 3 is measured in a DC setup, incrementing the voltage in steps of size 3 mV. The data are smoothed over 10 steps and the derivative is calculated numerically to obtain the differential conductance.

A constant contact resistance Rc 800 O is subtracted from the data, moving the

conductance at Vg 8.2 V for Vsd40.8 mV to 2e2/h. The four-terminal

resistance of the device is Rd 400 O with Vg 0 V. The difference between Rc and

Rd is most likely dominated by the change of resistivity near the gated region, when the gate is turned on, as well as the distance from the voltage probe to the QPC region. The voltage probes are located B15 mm away from the QPC and the gates overlap the mesa over an area B1.6 mm2. The normal state conductance is calculated as the average of G(Vsd) for Vsd in the range (0.8 mV, 1 mV). The analysis is largely unaffected by changing the averaging window for values |Vsd|40.6 mV. The cuts in Fig. 3b are taken by averaging over a 12 mV (30 mV)

window in Vg for the one-channel (tunnelling) regime. Finally, each datapoint in Fig. 3c is calculated as the average over a 10 mV range in Vg.

Model for numerical simulations. We calculate the conductance of the junction in two steps. First, we determine the scattering properties of the normal region which we assume is a 1.1 mm wide channel of length L, where we have taken dimensions from SEM in Fig. 1b. It is described by the spinless Hamiltonian,

H

2k2

2m VQPC x; y

W x X

d

arctan W x X

d

g S y; W x X g S y; W x X

g S y; W x X g S y; W x X ;

3

where

g u; v

1

2p arctan

uv dR

; 4

p . The potential landscape of the simulation is shown in Supplementary Fig. 3.

We include disorder29 by adding a random on-site energy Vd(x, y) distributed uniformly between Ud/2 and Ud/2 where

Ud m

and R

u2 v

2 d2

6l3F

p3Dx2le

s : 5

Due to limitation of the computational mesh resolution we exclude the disorder from the vicinity of the QPC and take Uda0 only for x4700 nm.

We calculate the scattering matrix of the normal part of the junction for a particle at the energy e as

SN e

r e

t e

t0 e r0 e

Vd x; y

m: 2

We model the QPC as two rectangular gates located at X 400 nm, with the

width 2W, separated by the length 2S and located at the distance d above

the 2DEG (see Supplementary Fig. 3 for illustration of W and S). We calculate the potential generated by the QPC electrodes, VQPC(x, y), for the gate voltage Vg as

follows28

VQPC x; y

eVg

1p arctan

; 6

using Kwant package27 and discretizing the Hamiltonian in equation (2) on a mesh with the spacing Dx Dy 3 nm. The quantities r(e) and t(e) denote reection and

transmission submatrices for a time-reversal symmetric system. In the second step, we combine the scattering matrices calculated for e and e (that correspond to

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms12841

electron and hole, respectively) with the matrix that accounts for the Andreev reection at the superconductor interface

SA rA

0 rp rp 0

; rp

eif1 0

0 e if1

; 7

where

rA

e D T

sign e D T

s

e2

D T

2

1 : 8

The latter equation describes the Andreev reection amplitude15 including the

temperature-dependent pairing potential D T

D

1 T=Tc

q

2

. Finally, we

calculate the conductance according to

Gns E

Z

deG e

@f E; e

@e

; 9

where f stands for the Fermi function

f E; e

1

e e E

=k T 1

; 10

and where G e

N re e

k k2. N is the number of modes in the normal

lead. The quasielectron and quasihole reection matrices are given by:

re e r e

t0 e

rAr0 e

rA

1

k k2 rh e

1 r0 e

rAr0 e

rA

t e

; 11

1

rh e t0 e

rA

1 r0 e

rAr0 e

rA

t e

: 12

Additionally, the normal-state conductance is given by Gnn t e0

k k2. Results of

the simulations are shown in Supplementary Figs 35.

Data availability. All data presented in the main paper and supplement, as well as code used to generate simulations are available from the authors upon request.

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Acknowledgements

Research support by Microsoft Project Q, the Danish National Research Foundation. C.M.M. acknowledges support from the Villum Foundation. F.N. acknowledges support from a Marie Curie Fellowship (no. 659653). M.P.N. acknowledges support from ERC Synergy Grant. A.A. is supported by an ERC Starting Grant. M.W. and A.A. are supported by the Foundation for Fundamental Research on Matter (FOM) and the Netherlands Organization for Scientic Research (NWO/OCW) as part of the Frontiers of Nanoscience program. We are indebted to S. Kraemer for the TEM analysis, performed at the UCSB MRL Shared Experimental Facilities (NSF DMR 1121053),a member of the NSF-funded Materials Research Facilities Network.

Author contributions

M.K., F.N., H.J.S. and C.M. conceived the experiment. M.K., F.N. and H.J.S. designed, fabricated and measured the devices and wrote the manuscript, with comments from all other authors. J.A.F. and K.F. provided input on interpretations. M.P.N., M.W and A.R.A. developed theory and code for the simulations. The wafer was grown byJ.S. and C.J.P.

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How to cite this article: Kjaergaard, M. et al. Quantized conductance doubling and hard gap in a two-dimensional semiconductorsuperconductor heterostructure. Nat. Commun. 7:12841 doi: 10.1038/ncomms12841 (2016).

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