supporting information anomalous zero-bias conductance
TRANSCRIPT
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:[email protected]
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.
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