Supporting Information

Anomalous zero-bias conductance peak in a Nb-InSb nanowire-Nb hybrid

device

M. T. Deng,1 C. L. Yu,1 G. Y. Huang,1 M. Larsson,1 P. Caroff,2 and H. Q. Xu1, 3,∗

1Division of Solid State Physics, Lund University, Box 118, S-221 00 Lund, Sweden

2I.E.M.N., UMR CNRS 8520, Avenue Poincare,

BP 60069, F-59652 Villeneuve d’Ascq, France

3Department of Electronics and Key Laboratory for the Physics and Chemistry of Nanodevices,

Peking University, Beijing 100871, China

(Dated: November 21, 2012)

Abstract

In this Supporting Information, we provide the details of device fabrication,more measurement results

for the device presented in the main paper, and the measurement results fora normal metal-InSb nanowire-

superconductor hybrid device. In addition, we provide the results of measurements for a Nb-based thin film

stripe with the same thickness as we used in the fabrication of superconductor-InSb nanowire hybrid devices.

A theoretical model that describes a one-dimensional superconductor-semiconductor nanowire quantum

dot-superconductor system is also provided. The results of this simple model provide a physical insight into

the transport signatures of Majorana fermions in superconductor-semiconductor nanowire-superconductor

hybrid devices.

∗Corresponding author:hongqi.xu@ftf.lth.se

1

I. DEVICE FABRICATION

Our superconductor-semiconductor nanowire hybrid devices are fabricated from high crys-

talline quality, zinc blende InSb segments of epitaxially grown InAs/InSb heterostructure

nanowires. The heterostructure nanowires are grown on InAs(111)B substrates at 450oC by

metal-organic vapor-phase epitaxy in a two-stage process using aerosol Au particles as initial

seeds. Growth of InAs stems first replaces direct nucleationby a wire-on-wire growth process,

effectively favoring a high yield of epitaxial top InSb nanowire segments. Figure 1(a) show a

scanning electron microscope (SEM) image of as grown InAs/InSb heterostructure nanowires.

Contrary to most other III-V nanowires, the InSb nanowire segments are free of any extended

structural defects and do not show tapering. For further details about the growth, structural, and

basic field effect transistor properties of the InAs/InSb heterostructure nanowires, see Refs. [1–4]

and the references therein.

The grown InAs/InSb heterostructure nanowires are transferred to degenerately doped, n-type

Si substrate, capped with a 100 nm thick SiO2 layer, with predefined Ti/Au bonding pads and

300 nm500 nm

(a) (b)

Etched In

Sb

segment

InSb

InAs

FIG. 1: (a) SEM image of InAs/InSb heterostructure nanowires grown by metal-organic vapor phase epitaxy

on an InAs(111)B substrate. The lower thinner segments are InAs nanowire parts and the upper thicker

segments are InSb nanowire parts. The image is recorded with a 30o tilt of the substrate from the horizontal

position and the scale bar is not compensated for the tilt. (b) SEM image of a segment of an InSb nanowire

with a selectively etched part. The InSb nanowire segment was transferred to a Si/SiO2 substrate and was

then selectively etched in a (NH4)2Sx solution. The diameter of the nanowire is roughly 15 nm smaller after

etching.

2

markers. Using an optical microscope, the positions of the wires relative to the metal markers

are recorded. Then, two 470 nm wide Nb-based superconductorcontacts with a separation of

100−150 nm are defined on the InSb segment of each selected InAs/InSb heterostructure nanowire

using electron beam lithography, sputtering and lift-off techniques. For our superconductor-InSb

nanowire quantum dot-superconductor hybrid devices, the Nb-based superconducting contacts

consisting of both Ti/Nb/Ti (3 nm/80 nm/5 nm) trilayers and Ti/Nb/Al (3 nm/80 nm/5 nm) tri-

layers are employed. The 3 nm Ti bottom layers in both cases serve as adhesion layers, whereas

the top 5 nm Ti layers or the top 5 nm Al layers are used for protecting the Nb from oxidation. In

our normal metal-InSb nanowire quantum dot-Nb hybrid devices, the superconductor contacts are

made of Ti/Nb/Ti (3 nm/80 nm/5 nm) trilayers in the same procedure as for superconductor-InSb

nanowire quantum dot-superconductor devices, while the normal metal contacts are defined by

thermal evaporation of Ti/Au (3 nm/80 nm) layers and lift-off process. It is however important

to note that an oxygen plasma treatment has been performed prior to the metal deposition in the

fabrication of the contacts in order to remove resist residues. To remove the native oxide layers on

the InSb nanowires, we have also performed 60 seconds of wet etching/passivation in a (NH4)2Sx

solution. The diameters of the etched nanowires are roughly15 nm smaller than the diameters

of these nanowires before etching, see the SEM image shown inFig.1(b) for an example. How-

ever, we do not use Ar plasma milling prior to metal sputtering or evaporation in order to avoid

crystalline damage and unintentional Ar ion implantation [5]. Finally, we note that in addition to

the two superconducting contacts, there is a Ti/Au metal layer on the back side of each substrate

which has been employed as a global back gate.

II. ELECTRICAL CHARACTERIZATION OF THE DEVICE REPORTED IN THE MAIN AR-

TICLE

The device reported in the main article is a Nb-InSb nanowirequantum dot-Nb hybrid device,

see Figs. 1(a) and 1(b) in the main article. Some results of the measurements for the device have

already been reported in the main article. Here, we provide more measurement results for the

device at more positive back gate voltages, where the transport through the nanowire show Fabry-

Perot interference-like patterns [3, 6]. Figure 2(a) shows the measurements of the differential

conductance as a function ofVsd andVbg (charge stability diagram) atB = 0 T over a large range

of Vbg. As can be seen in Fig. 2(a), whenVbg > −4 V, a chess board pattern, i.e., characteristics of

3

Vsd

(m

V)

4-8 -2 0 2-6 -4-10-12V

bg (V)

-6

-4

-2

0

2

4

6

dIs

d /d

Vs

d (e2/h

)

0

3

2

1

(a)

4-2 0 2-6 -4-10-12

Vbg

(V)

(e) (f) (g)

Rn (k

Ω)0.8

0.4

0.0

1.2

20

16

12

8

I C (n

A)

0.3

0.2

0.1

0.0

80

60

40

20

0.3

0.4

0.2

0.1

10

18

26

Vbg

= 2.75 V

Vsd

(mV)

dI s

d /d

Vs

d (e

2/h

)

43210-1-2-3-4

2.6

2.4

2.2

2.0

2ΔInSb

(b)

Isd

(nA)0.0 1.0 2.0-1.0-2.0

Vsd

(μ

V)

20

10

0

-10

-20

-0.5 V

-1.1 V

Vbg

= -3.6 V

-1.5 -0.5 0.5 1.5

(c)

4-8 -2 0 2-6 -4-10-12V

bg (V)

I sd (n

A)

-1.5

-1.0

-0.5

0

0.5

1.0

1.5

(d)

dV

sd

/dIs

d (kΩ)

dV

sd

/dIs

d (kΩ)

0300

050

dV

sd

/dIsd (k

Ω)

0

50

25

FIG. 2: (a) Differential conductance on a color scale as a function of source-drain bias voltageVsd and

back gate voltageVbg (charge stability diagram) measured for the device reported in the main article at

B = 0 T. The measurements show a Fabry-Perot interference-like pattern atVbg > −4 V and a quasi-particle

Coulomb blockade diamond structure atVbg < −9 V. (b) Differential conductance as a function ofVsd

measured for the device atVbg = 2.75 V, i.e., along the dashed line in (a). The two conductance peaks

indicated by the arrows can be attributed to the first-order multiple Andreev reflections associated with the

proximity effect induced superconductor energy gap∆InS b of the InSb nanowire. (c) Source-drain voltage

Vsd as a function of source-drain currentIsd measured for the device atVbg = −0.5, −1.1 and−3.6 V. The

red and black curves are recorded in the upward and the downward current sweeping direction, respectively.

(d) Differential resistance on a color scale as a function ofIsd andVbg measured for the device atB = 0

T. Here, the critical supercurrent shows a clearVbg dependence. (e)-(g) show the critical supercurrentIc

(red curve) and the normal state resistanceRN (black curve) as a function ofVbg measured for the device at

B = 0 T. It is generally seen that the smallerRN is, the largerIc is.

Fabry-Perot-like interference, is visible in the measured differential conductance. The differential

conductance reaches more than 3e2/h in the gate voltage region ofVbg > −4 V. This observation

is consistent with the results obtained previously in InSb nanowire devices with normal metal

4

contacts [3]. The Fabry-Perot-like interference pattern suggests a long quasi-particle coherence

length in the nanowire and the quasi-particle transport is ballistic or near ballistic between the two

contacts.

In Fig. 2(a), we also see that there are two stripes of high conductance located symmetrically

around zero bias voltage. To see these features of high differential conductance, we show in

Fig. 2(b) a measured differential conductance trace taken atVbg = 2.75 V, i.e., along the dashed

line in Fig. 2(a). The trace clearly shows two conductance peaks. As indicated in the figure, the

two conductance peaks can be attributed to the first order multiple Andreev reflections associated

with the proximity effect induced superconductor energy gap∆InS bof the InSb nanowire. Unlike in

the region ofVbg < −10 V, multiple Andreev reflection features associated with the superconductor

energy gap of the Nb contacts are not clearly visible in the figure in the back gate voltage region

of Vbg > −4 V.

The device reported in the main article is also characterized by current biased measurements.

Figure 2(c) shows the measured source-drain voltage of the device at a temperature of 25 mK as a

function of applied source-drain currentIsd at three different voltagesVbg applied to the back gate.

A zero resistance branch is clearly seen in each measured curve, which indicates the presence of

a dissipationless Josephson supercurrent in the junction.The Josephson junction switches to a

dissipative transport branch when the applied current is larger than a critical valueIc. The upward

current sweeping trace (red curve) and downward sweeping trace (black curve) have different

switching points, i.e., the device shows a hysteretic behavior. This hysteretic behavior has also

been seen in Josephson junctions made from semiconductor nanowires and superconducting Al [7–

9], and could be the result of phase instability typically found in a capacitively and resistively

shunted Josephson junction or simply due to a heating effect. The supercurrentIc is related to the

resistance of the junction in the normal state [7–9] and can thus be tuned in our device by applying

a voltageVbg to the back gate. For example, in Fig. 2c, we see thatIc is 1.1 nA atVbg = −3.6 V but

it is only 0.2 nA atVbg = −1.1 V. The tunability of the critical supercurrentIc can be visualized

more clearly in Fig. 2(d) where the differential resistance is plotted as a function ofIsd andVbg. We

note that theVbg axis in this figure has the same scale as in Fig. 2(a) and in the two gate voltage

regions where there are no plotted data, the measurements ofthe differential conductance were

not performed. In Fig. 2(d) the critical supercurrentIc is characterized by the width of the low

differential conductance region and it is clearly seen thatIc is a function ofVbg.

Figures 2(e), 2(f) and 2(g) show the measured critical current Ic and the normal state resistance

5

(b) Vbg

= -3.6 V

Isd

(nA)Isd

(nA)

0.0 1.0 2.0-1.0-2.0

Vs

d (

μV

) 5

10

0

-10

-5

-1.5 -0.5 0.5 1.5

25 mK300 mK700 mK1000 mK

B (T

)

0

-2

-2 -1 0 1 2

-1

2

1

dVsd

/dIsd

(kΩ)0 2010 30

Vbg

= - 3.6 V

(a)

FIG. 3: (a) Differential resistance on a color scale measured for the device reported in the main article

at Vbg = −3.6 V as a function of the applied source-drain currentIsd and the magnetic fieldB applied

perpendicularly to the substrate. The supercurrent is seen to persist as the magnetic field goes up toB ∼ 2

T. (b) Source-drain voltageVsd measured for the device reported in the main article as a function of applied

source-drain currentIsd at Vbg = −3.6 V andB = 0 T and at four different temperaturesT = 25, 300, 700,

and 1000 mK. A supercurrent continues to be visible when the temperature goes up toT = 700 mK, but it

disappears atT ∼ 1 K.

Rn as a function ofVbg. The resistanceRn is deduced from the differential resistance atIsd = 1.5

nA in Fig. 2(d). At 1.5 nA and beyond theVsd − Isd characteristics show approximately straight

lines [cf. Fig. 2(c)]. Generally, a smallIc is observed at a region of gate voltage for whichRn is

large. However, the measuredIcRn product in our device is not a constant, but varies from 2µV

to 12µV. These values are overall much smaller than the expected value of IcRn ∼ ∆InS b/e= 0.25

mV or ∆Nb/e = 1.5 mV for an ideal Josephson junction embedding a short, diffusive normal

conductor. Such reduced experimental values have also beenobserved in semiconductor nanowire

based Josephson junctions made with superconducting Al contacts [7, 9] and can typically be

attributed to premature switching due to thermal activation in a capacitively and resistively shunted

junction and to finite transparencies at the superconductor-nanowire interfaces [10].

Figure 3(a) shows the measured differential resistance of our hybrid device as a function of the

applied source-drain currentIsd and the magnetic fieldB applied perpendicular to the substrate

at Vbg = −3.6 V. It is generally seen that the supercurrentIc decreases as the magnetic fieldB

increases and disappears after the magnetic field becomes higher than a critical valueBc. In an Al

based Josephson junction made from an InSb nanowire, the value ofBc is generally found to be a

6

few 10 mT [9]. We find that, with Nb contacts,Bc is much larger and can reach a few T (see below

for a further discussion about the superconducting properties of Nb thin film contacts). Figure 3(b)

displays the source-drain voltageVsd as a function of source-drain currentIsd measured at different

temperatures for the device atVbg = −3.6 V andB = 0 T. The measurements show the tempera-

ture dependence of the Josephson supercurrentIc in our device. As the temperature increases,Ic

decreases gradually. Eventually,Ic disappears at the temperatures higher than a critical valueof

Tc ∼ 1 K.

III. ZERO-BIAS CONDUCTANCE PEAK STRUCTURE OBSERVED IN THE MEASURE-

MENTS OF A Au-InSb NANOWIRE QUANTUM DOT-Nb DEVICE

As we mentioned in the main article, we also fabricated and measured a normal metal-InSb

nanowire quantum dot-superconductor device as shown schematically in Fig. 4(a). The device

was again made from the InSb nanowire segment of an InAs/InSb heterostructure nanowire. How-

ever, only one of its two contacts was made from superconducting Nb (in a 3 nm Ti/80 nm Nb/5

nm Ti triple layer form) and the other one was made from normalAu (in a 4 nm Ti/80 nm Au

double layer form). Figure 4(b) shows an SEM image of the fabricated device, where the spacing

between the two contacts is about 210 nm. Similar devices were studied in Ref. [11] where the

observation of zero-bias conductance peaks were reported.An InSb nanowire quantum dot was

formed in the junction between the two contacts in our device, which differs from the devices stud-

ied in Ref. [11]. Figure 4(c) shows the differential conductance of the device measured atB = 0 T

andT = 25 mK as a function ofVbg andVsd (charge stability diagram). Due to the relatively large

size of the quantum dot, the addition energyEadd is found to be only about 1.5− 2 meV, which is

significantly smaller than that in the Nb-InSb nanowire quantum dot-Nb devices reported above.

At small source-drain bias voltages, we see the existence ofa gap of low conductance over the en-

tire measured back gate voltages in the figure. This low conductance gap arises from the proximity

effect induced superconductor energy gap of the InSb nanowire.Figure 4(d) shows a trace of the

differential conductance measured for the device atVbg = 0.835 V. The low conductance gap in the

low bias voltage region can be easily identified. At the edgesof the gap, two conductance peaks

appear located symmetrically aroundVsd = 0 V. These peaks arise from tunneling through the sin-

gularity points in the quasi-particles density of states inthe superconductor InSb nanowire and the

distance between the two peaks is given by 2∆InS b/e, where∆InS b is the proximity effect induced

7

500 nm

Ti/Nb/Ti

Ti/Au

InSb

(b)

0.95

0.90

0.85

0.80

0.75

Vsd

(mV)0.0 1.00.5-0.5-1.0

Vb

g (V

)

0.0 0.3 0.6

dIsd

/dVsd

(e2/h)

(c)

B= 0 T

Vsd

(mV)0.0

0.02

0.04

0.08

0.12

0.14

0.06

0.10

0.60.40.2-0.2-0.6 -0.4

dI s

d /d

Vs

d (e

2/h

) (d) Vbg

= 0.835 V

ΔInSb Δ

InSb

(a)InSb InAs

FIG. 4: (a) Schematic layout of a normal metal-InSb nanowire quantum dot-superconductor device. (b)

SEM image of the fabricated Au-InSb nanowire quantum dot-Nb device with the measurements presented

in this and the next figure. (c) Differential conductance on a color scale as a function ofVsd andVbg (charge

stability diagram) measured atB = 0 T. Here, no zero-bias conductance peak is visible. (d) Trace of the

differential conductance of the device atVbg = 0.835 V. The two conductance peaks indicated by the two

black arrows can be attributed to quasi-particle tunneling through singularities in the density of states in the

superconducting InSb nanowire. The two conductance peaks indicatedby red arrows are most likely caused

by the Andreev bond states in the junction region.

superconductor energy gap in the InSb nanowire. From the measurements, we can deduce a value

of ∆InS b∼ 0.27 meV. We also note that the measurements also show two weak subgap differential

conductance peaks as indicated by red arrows in Fig. 4(d). These two subgap peaks are most likely

due to Andreev bound states located symmetrically around zero energy in the dot [12, 13].

No zero-bias conductance peak can be seen in Fig. 4(c) where the measurements were made for

8

Vsd

(mV)0.0 1.00.5-0.5-1.0

0.95

0.90

0.85

0.80

0.75

Vb

g (V

)

0.0 0.5 1.0

dIsd

/dVsd

(e2/h)

B= 1.25 T

(a) (b)

Vsd

(mV)

dI s

d /d

Vsd (

e2/h

)

0.00 0.50-0.50 -0.25 0.250.02

0.06

0.10

0.14 Vbg

= 0.90 V

(c)

Vsd

(mV)

dI s

d /d

Vsd (e

2/h

)

0.00 0.50-0.50 -0.25 0.25

0.5

0.4

0.6

0.7V

bg= 0.84 V

(d)

Vsd

(mV)

dI s

d /d

Vs

d (e

2/h

)

0.00 0.50-0.50 -0.25 0.250.06

0.10

0.14V

bg= 0.78 V

Vsd

(mV)0.0 0.40.2-0.2-0.4

0.0 0.5 1.0

dIsd

/dVsd

(e2/h)

B (T

)

0.0

0.8

1.2

1.6

0.4

(e)

Vbg

= 0.835 V

Vsd

(mV)0.0 0.40.2-0.2-0.4

dIsd

/dVsd

(e2/h)

B (T

)

0.50

1.25

1.50

2.00

1.00

(g)

Vbg

= 0.81 V

10-2.0 10-1.0

B (T

)

dI s

d /d

Vs

d (e

2/h

)

Vsd

(mV)0.0 0.20.1-0.1-0.2

0.0

2.0

2.5

3.5

3.0

1.5

1.0

0.5

0.0

1.4

Vbg

= 0.835 V(f)

FIG. 5: (a) Differential conductance on a color scale as a function ofVsd andVbg (charge stability diagram)

measured atB = 1.25 T for the Au-InSb nanowire quantum dot-Nb device shown in Fig. 4(b). A weak

zero-bias conductance peak is visible over the measuredVbg region in the figure. (b)-(d) Traces of the

differential conductance of the device taken from (a) at three different values ofVbg indicated by dashed

line in (a). A zero-bias conductance peak is clearly seen in each trace. (e) Differential conductance on a

color scale as a function ofVsd andB measured for the device atVbg = 0.835 V. A zero-bias conductance

peak is present at magnetic fieldsB ∼ 1.2 − 1.5 T. (f) The same measurements as in (e), but the traces

of the differential conductance measured at selected magnetic fields in the region ofB ∼ 0 − 1.4 T. The

measured curves are successively offset by 0.06e2/h for clarity. (g) The same measurements as in (e) but

for Vbg = 0.81 V. Superimposed on the figure are the traces of the differential conductance taken at different

B values indicated by horizontal lines in the figure. A zero-bias conductance peak is clearly seen in the

magnetic field region betweenB ∼ 0.9− 1.8 T.

the Au-InSb nanowire quantum dot-Nb device at zero magneticfield. To search for zero-bias con-

ductance peaks, the transport signature of Majorana fermions in the superconductor Nb-covered

InSb nanowire, we drove the InSb nanowire to a nontrivial topological superconductor phase by

9

applying a magnetic field perpendicular to the substrate andthus to the nanowire. Figure 5(a)

shows the charge stability diagram measured for the device at B = 1.25 T. A weak zero-bias con-

ductance peak is visible in the whole gate voltage range. Fora better visualization of the zero-bias

conductance peak, Figs. 5(b) to 5(d) show three traces of thedifferential conductance measured at

back gate voltagesVbg = 0.9, 0.84 and 0.78 V, i.e., along the three dashed lines in Fig. 5(a). A

zero-bias conductance peak is clearly seen in each of the three traces regardless of differences in

the back ground conductance.

Figure 5(e) shows the differential conductance measured for the device at a fixed back gate

voltageVbg = 0.835 V as a function ofVsd andB, while Fig. 5(f) shows the corresponding line

plots. In these measurements that no zero-bias conductancepeak feature are found at magnetic

fieldsB < 0.8 T. However, atB ∼ 0.8 T, two weak conductance peaks appear in the close vicinity

of Vsd = 0 V. As the magnetic field is increased further, the two conductance peaks gradually merge

into a single zero-bias conductance peak atB ∼ 1.2 T. The zero-bias conductance peak remains

visible until B ∼ 1.5 T. As in Ref. [11], we can attribute the zero-bias conductance peak to the

transport through the Majorana fermion states in the InSb nanowire covered by the superconductor

Nb contact. Figure 5(g) shows the same measurements of differential conductance as a function

of Vsd andB but for the device atVbg = 0.81 V. In this figure, a few line plots of the differential

conductance taken at different back gate voltages, as indicated by horizontal solid lines are shown.

Here, again, we see no zero-bias conductance peak at low magnetic fields, but it is clearly visible

at magnetic fieldsB ∼ 0.9− 1.8 T as a result of the transport through the Majorana fermion states

in the Nb-covered InSb nanowire.

IV. SUPERCONDUCTING PROPERTIES OF THE TI/NB/TI TRILAYER

In order to separate the intrinsic superconducting properties of the Nb contacts from the trans-

port properties of the Nb/InSb nanowire/Nb Josephson junction, we have fabricated a device con-

sisting of a 120µm long and 400 nm wide Ti/Nb/Ti (3 nm/80 nm/5 nm) thin film connected to

Au contacts, see the schematic in Fig. 6(a). Magnetotransport measurements of the device were

performed in a3He cryostat with a base temperature of 300 mK using a current bias setup. Fig-

ure 6(b) shows the differential resistancedVsd/dIsd of the device measured as a function of applied

bias currentIsd and external magnetic fieldB. The bias current is swept from negative to positive

values. A zoom-in section of the measurements in a high magnetic field region is shown in the

10

(a)

Isd (mA)

T (

K)

5

9

7

-0.3 0.30

010dV/dI (kΩ)

(d)

T (K)4 9 14

500

0

(e)

L = 120 μm

W = 400 nm

Ti/Nb/Ti (3 nm/80 nm/5 nm)

Au Au

Isd (mA)

B (

T)

-0.3 0.300

2

4

6

010dV/dI (kΩ)

(b)

B (T)

dV

/dI

(Ω)

0 5 10

400

200

0

(c)

Isd (mA)

B (

T)

-0.2 -0.1 0.20.105

6

7

8

500600dV/dI (Ω)

dV

/dI

(Ω)

FIG. 6: (a) Schematic layout of the Ti/Nb/Ti thin film device. (b) Differential resistance measured as a

function of source-drain currentIsd and magnetic fieldB applied perpendicular to the plane of the Nb thin

film. The bias current is swept from negative to positive values. A zoom-insection of the measurements in

a high magnetic field region is shown in the inset of (c). (c) Zero-bias differential resistance as a function of

magnetic field. (d) Differential resistance as a function of source-drain currentIsd and temperatureT. The

bias current is swept from negative to positive values. (e) Zero-biasdifferential resistance as a function of

temperatureT.

inset of (c). The dark blue region seen in Fig. 6(b) corresponds to the region where the thin film is

in the superconducting state and a supercurrent flows through the film. The peak in the differential

resistance at positive and negative bias currents signals atransition between the superconducting

state and the normal state at the value of the critical supercurrent. In the slightly lighter blue region

of Fig. 6(b), the thin film is in the normal, resistive state.

We can estimate the resistivityρ = 1.44× 10−7 Ωm from the normal state resistance and the

dimensions of the Nb layer. This value is used together with the standard valuen = 5.56× 1022

cm−3 of the electron density in Nb to calculate the electron mean free pathl. We obtain a value of

l = 6 nm. Using the electron density of states at the Fermi energyof Nb, N(ǫ f ) = 9.8×1046 J−1m−3,

11

the diffusion constantD = 2.7× 10−3 m2/s can be determined from the relation 1/ρ = N(ǫ f )e2D.

Based on the diffusion constant and the superconducting energy gap∆Nb = 1.5 meV determined

from the Andreev reflection measurements in the main articleand this Supporting Information,

we can estimate the superconducting coherence lengthξ0 =√~D/∆Nb = 35 nm and the London

penetration depthλL =√

~ρ/(µ0π∆Nb) = 130 nm [14]. As expected for a type II superconductor

like Nb, we find thatλL > ξ0.

For finite external magnetic fields the superconducting energy gap∆Nb will decrease, leading

to a smaller critical supercurrent compared to the zero magnetic field case. Type II supercon-

ductors are associated with two critical fieldsBc1 andBc2, whereBc1 is the critical field at which

magnetic flux starts to penetrate into the superconductor and Bc2 is the critical field at which the

superconductivity and, therefore, also the supercurrent are completely suppressed. From Fig. 6(c),

which shows the zero-bias differential resistance as a function of magnetic field, we see a reduced

resistance persisting up to fields of∼ 6 to 7 T. Based on this and the measurements shown in the

inset of Fig. 6(c), we can deduce that the upper critical fieldis in a range ofBc2 ∼ 6 − 7 T. As-

suming that the kinks and associated broadening of the superconducting to normal state transition,

see the white arrows in Fig. 6(b), are caused by flux penetration into the superconductor, we get

Bc1 ∼ 2.5 T. We cannot, however, exclude the possibility thatBc1 is smaller than 2.5 T based on

our measurements. The upper limit ofBc1, on the other hand, can be determined from the increase

in zero-bias resistance at∼ 4 T seen in Fig. 6(c).

From the measurements of the temperature dependence of the critical current shown in Fig. 6(d)

and of the zero-bias differential resistance as a function of temperatureT shown in Fig. 6(e), we

can deduce the critical temperatureTc ∼ 8− 9 K for our Nb film, which is very close to the bulk

value of NbTc ∼ 9.25 K [15].

V. THEORETICAL MODEL AND SIMULATION RESULTS

According to the experimental setup as sketched in Fig. 7(a), we establish a semiconductor

nanowire model which consists of three segments: two proximity effect induced superconducting

segments located under two superconductor contacts and a middle normal semiconductor junction

segment. The entire nanowire has a strong spin-orbit interaction (SOI) and the normal semicon-

ductor segment is connected to the two superconducting segments via tunnel couplings in order to

take into account the effect of the formation of the quantum dot between the two superconductor

12

contacts. Furthermore, we assume that a magnetic field is applied perpendicularly to the nanowire

as indicated in Fig. 7(a). Our semiconductor nanowire modelresembles the superconductor-

semiconductor-superconductor heterostructure system ofLutchynet al. [16], but with two major

differences. (i) The phase difference of the superconducting order parameters of the left and right

segment is unknown and is taken to be a tunable constant in thecalculations presented below. We

concentrate on the energy spectrum rather than the Josephson current. (ii) The couplings between

the normal semiconductor segment and the two superconducting segments are set to be equal but

tunable, i.e., we consider the simplest system of a symmetric double barrier type.

The Bogoliubov-de Gennes (BdG) Hamiltonian of our model system, modified from Ref. [16],

is constructed by assuming that the nanowire is oriented along thex axis, the SOI is of the Rashba

type, and the electric and magnetic field are applied along thezdirection. Under the Landau gauge

and using the Nambu spinors [un↑,un↓, vn↓,−vn↑] with unσ andvnσ being the particle and hole wave

function at thenth site, the Hamiltonian can be written in a one-dimensionaltight-binding form as

(HBdG)n,m = [(2t − µ)τz+ Vzσz+ ∆n(τx cosφn − τy sinφn)]δn,m

+(−tn + iαnσy)τzδn,m−1 + (−tn − iαnσy)τzδn,m+1.

In the above equation, indicesm andn run through the lattice sites in the model. Site depen-

dent parameterstn are the hopping integrals withtn = Γ at the two interfaces between the semi-

conductor and two superconductive segments andtn = t otherwise. ∆n = |∆∗| in the left and

right superconducting segments, and 0 in the middle normal semiconductor segment, where∆∗

is the proximity effect induced superconductor energy gap in the two superconductor contacted

nanowire segments.φn = φ in the right superconductive segment and 0 elsewhere, afterremoving

an unimportant global phase, whereφ is the phase difference of the two superconductive nanowire

segments.µ is the chemical potential of the system and is set to zero in the calculations presented

here.αn = tnα0 is the Rashba SOI energy withα0 being the dimensionless parameter describing

the Rashba SOI strength.Vz =12gµBB is the Zeeman energy, whereg the effectiveg factor,µB the

Bohr magneton, andB the magnetic field. Finally,σi andτi are Pauli matrices which operate on

spin space and particle-hole space, respectively.

The model above has been employed to explore the evolutions of the energy spectrum with in-

creasing couplingΓ and phase differenceφ. A special emphasis is placed on the properties of the

zero energy Majorana fermion states atµ = 0. Figures 7(b)-7(d) show the results of the calcula-

tions for the system, modeled by 901 lattice sites with the middle normal semiconductor segment

13

(a) B

NW

SC

SC

Γ= 0.0 t, ϕ= 0(b)

0.0

0.1

0.2

0.3

ǀΨǀ2

lx

0

800

200400

600E (t)

0.050.10

0.00-0.05-0.10

Γ= 0.2 t, ϕ= 0

(c)

E (t)lx

0

800

200400

600

0.050.10

0.00-0.05-0.10

0.0

0.1

0.2

0.3

ǀΨǀ2

Γ= 0.6 t, ϕ= 0(d)

E (t)lx

0

800

200400

600

0.050.10

0.00-0.05-0.10

0.0

0.1

0.2

0.3

ǀΨǀ2

Γ= 0.3 t, ϕ= π/3(f)

0.0

0.1

0.2

0.3

ǀΨǀ2

lx

0

800

200400

600E (t)

0.05

0.00

-0.05

Γ= 0.3 t, ϕ= π/2(g)

0.0

0.1

0.2

0.3

ǀΨǀ2

lx

0

800

200400

600E (t)

0.05

0.00

-0.05

Γ= 0.3 t, ϕ= π(h)

0.0

0.1

0.2

0.3

ǀΨǀ2

lx

0

800

200400

600E (t)

0.05

0.00

-0.05

Γ= 0.3 t(i)

ϕ

0.00

0.04

-0.04

0.02

-0.02

E (t)

2πππ/2 3π/20

ϕ= 0(e)

0.0Γ (t)

1.0

0.00

0.06

-0.06

0.04

-0.04

0.2 0.4 0.6 0.8

0.02

-0.02

E (t)

FIG. 7: (a) Schematic of the model system consisting of a semiconductor nanowire (NW), with a strong

Rashba SOI, and two superconductor (SC) contacts (grey regions).The nanowire is placed along thex

axis. Due to the proximity effect, the nanowire segments covered by the superconductor contacts arein

a superconducting state with a superconductor energy gap∆∗. These two superconducting segments are

connected to the normal semiconductor segment via tunnel couplingsΓ. The magnetic fieldB is applied

perpendicularly to the nanowire along thez axis. (b) to (d) Wave function probability distributions along

the nanowire for the first few low energy states calculated forΓ = 0, 0.2t, and 0.6t at φ = 0, |∆∗| = 0.2t,

Vz = 0.5t, andα0 = 1. (e) Energies of the first few low energy states calculated for the system as a function

of Γ. (f) to (h) The same as in (b) to (d) but forΓ = 0.3t atφ = π/3, π/2, andπ, respectively. (i) Energies of

the first few low energy states calculated for the system as a function ofφ.

represented by 42 sites, for|∆∗| = 0.2t, Vz = 0.5t, andα0 = 1, at different coupingsΓ and different

phase differencesφ. In these figures, the wave function probability distributions are plotted for a

few low energy states. AtΓ = 0, we find two pairs of zero energy Majorana fermion states with

each pair being localized at the two ends of a proximity effect induced superconducting nanowire

segment. AsΓ becomes finite, the two Majorana fermion states near the middle normal semicon-

14

ductor segment interact, leading to the creation of a pair ofnormal fermion states–a quasi-particle

state and a quasi-hole state–with energies located symmetrically around zero [17–22]. However,

the other two Majorana fermion states seen atΓ = 0 remain intact and the entire nanowire behaves

as a nontrivial topological superconductor system. AtΓ = 0.6t, the two normal fermion states are

seen to move further apart in energy, while the two Majorana fermion states remain unchanged.

A summary of the results of the calculation is shown in Fig. 7(e) where the continuous energy

evolutions of a few low energy states with increasingΓ are presented.

Figure 7(f)-7(h) show the results of the calculations for the same system withΓ = 0.3t but at

different phase differences. It is seen that asφ is increased, the pair of the normal fermion states

created as the annihilation of the two inner Majorana fermion states move towards zero energy.

This pair of normal fermion states become zero energy statesexactly atφ = π and then move

apart in energy again asφ is increased further. However, the other two Majorana fermion states

localized at the two outer ends of two superconductor nanowire segments always remain at zero

energy with increasingφ, i.e., the energy positions of these two Majorana states areindependent

of the phase differenceφ. Figure 7(i) shows a summary of the results of the calculations, where

the continuous energy evolutions of the four lowest energy states with increasingφ are presented.

In conclusion for this section, when two proximity effect induced superconducting semicon-

ductor nanowires in a nontrivial topological superconductor phase are brought to be coherently

coupled via a normal semiconductor nanowire segment, the two Majorana fermion states local-

ized near the normal semiconductor nanowire segment can be annihilated to create a pair of nor-

mal fermion states–a quasi-particle fermion state and a quasi-hole fermion state. Then, the entire

nanowire will turn into a nontrivial topological superconductor system with a pair of Majorana

fermions localized at the two ends of the entire nanowire. Cooper pairs can transport through the

semiconductor nanowire via the pair of Majorana fermions [23], leading to an enhancement in the

zero-bias conductance, as observed in the experiment reported in the main article.

[1] Caroff, P. et al. High-quality InAs/InSb nanowire heterostructures grown by metal-organic vapor-phase

epitaxy.Small4, 878-882 (2008).

[2] Nilsson, H. A., Caroff, P., Thelander, C., Larsson, M., Wagner, J. B., Wernersson, L.-E., Samuelson,

L. & Xu, H. Q. Giant, level-dependent g factors in InSb nanowire quantum dots.Nano Letters9,

15

3151-3156 (2009).

[3] Nilsson, H.A., et al. InSb Nanowire Field-Effect Transistors and Quantum-Dot Devices.IEEE journal

of selected topics in quantum electronics17, 907-914 (2011).

[4] Plissard, S. R.,Slapak, D. R., Verheijen, M. A., Hocevar, M.; Immink, G.W. G., van Weperen, I.,

Nadj-Perge, S., Frolov, S. M., Kouwenhoven, L. P. & Bakkers, E. P.A. M. From InSb Nanowires to

Nanocubes: Looking for the Sweet Spot.Nano Letters12 1794-1798 (2012).

[5] Magnee, P. H. C. et al. Influence of low energy Ar sputtering on the electronicproperties of InAs based

quantum well structures.Applied Physics Letters67, 3569-3571 (1995).

[6] Kretinin, A.V., et al. Multimode Fabry-Perot conductance oscillations in suspended stacking-faults-

free InAs nanowires.Nano letters10, 3439-45 (2010).

[7] Doh, Y.-J., van Dam, J. A., Roest, A. L., Bakkers, E. P. A. M., Kouwenhoven, L. P. & De Franceschi,

S.Science309, 272-275 (2005).

[8] Xiang, J., Vidan, A., Tinkham, M., Westervelt, R. M. & Lieber, C. M. Ge/Si nanowire mesoscopic

Josephson junctionsNature Nanotech.1, 208-213 (2006).

[9] Nilsson, H. A., Samuelsson, P., Caroff, P. & Xu, H. Q. Supercurrent and multiple Andreev reflections

in an InSb nanowire Josephson junction.Nano Letters12, 228-233 (2012).

[10] Tinkham, M. Introduction to Superconductivity (Dover, New York,1996).

[11] Mourik, V. et al. Signatures of Majorana fermions in hybrid superconductor-semiconductor nanowire

devices.Science336, 1003-1007 (2012).

[12] Pillet, J.-D. et al. Andreev bound states in supercurrent-carryingcarbon nanotubes revealed.Nature

Physics6, 965-969 (2010).

[13] Dirks, T. et al. Transport through Andreev bound states in a graphene quantum dot.Nature Physics7,

386-390 (2011).

[14] Anthore, A., Pothier, H., & Esteve, D. Density of states in a superconductor carrying a supercurrent.

Physical review Letters90, 127001 (2003).

[15] Finnemore, D.K., Stromberg, T.F.,&Swenson, C.A. Superconducting properties of high-purity Nio-

bium.Physical review149, 231-243 (1966).

[16] Lutchyn, R., Sau, J. & Das Sarma, S. Majorana Fermions and a Topological Phase Transition in

Semiconductor-Superconductor Heterostructures.Physical Review Letters105, 077001 (2010).

[17] Kitaev, A.Y. Unpaired Majorana fermions in quantum wires.Physics-Uspekhi131, 130-136 (2001).

[18] Oreg, Y., Refael, G. & von Oppen, F. Helical Liquids and MajoranaBound States in Quantum Wires.

16

Physical Review Letters105, 177002 (2010).

[19] Jiang, L. et al. Unconventional Josephson Signatures of Majorana Bound States.Physical Review

Letters107, 236401 (2011).

[20] Alicea, J. et al. Non-Abelian statistics and topological quantum information processing in 1D wire

networks.Nature Physics7, 412 (2011).

[21] Alicea, J., New directions in the pursuit of Majorana fermions in solid state systems.arXiv:1202.1293

(2012).

[22] Sau, J. D., Clarke, D. J. & Tewari, S. Controlling non-Abelian statistics of Majorana fermions in

semiconductor nanowires.Physical Review B84, 094505 (2011).

[23] Wu, B. H. & Cao, J. C. Tunneling transport through superconducting wires with Majorana bound

states.Physical Review B85, 085415 (2012).

17