Conductance through a helical state in an InSb nanowire

7.7 MB·

by  Jakob Kammhuber

Conductance through a helical state in an InSb nanowire
J. Kammhuber,1 M.C. Cassidy,1 F. Pei,1 M.P. Nowak,1,2 A. Vuik,1 D. Car,1,3
S.R. Plissard,4 E.P.A.M. Bakkers,1,3 M. Wimmer,1 and L.P. Kouwenhoven1,∗
1QuTech and Kavli Insitute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
2Current adress: Faculty of Physics and Applied Computer Science,
AGH University of Science and Technology, al. A.Mickiewicza 30, 30-059 Kraków, Poland
3Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
4CNRS-Laboratoire d’Analyse et d’Architecture des Systemes (LAAS),
Université de Toulouse, 7 avenue du colonel Roche, F-31400 Toulouse, France
7
1
0 The motion of an electron and its spin are gen- mode of 2D quantum spin hall topological insulators,11,12
2 erally not coupled. However in a one dimensional and in quantum wires created in GaAs cleaved edge over-
(1D) material with strong spin-orbit interaction growth samples.13 They have also been predicted to exist
n
a (SOI) a helical state may emerge at finite mag- incarbonnanotubesunderastrongappliedelectricfield,14
J netic fields,1,2 where electrons of opposite spin RKKY systems,15 and in InAs and InSb semiconducting
4 will have opposite momentum. The existence of nanowires where they are essential for the formation of
2 this helical state has applications for spin filter- Majorana zero modes. Although the signatures of Ma-
ing and Cooper pair splitter devices3,4 and is an joranas have been observed in nanowire-superconductor
l] essentialingredientforrealizingtopologicallypro- hybrid devices,16,17 explicit demonstration of the heli-
al tected quantum computing using Majorana zero cal state in these nanowires has remained elusive. The
h modes.5–7 Here we report electrical conductance measurement is expected to show a distinct experimental
- measurements of a quantum point contact (QPC) signature of the helical state - a return to 1e2/h conduc-
s
e formed in an indium antimonide (InSb) nanowire tance at the 2e2/h plateau in increasing magnetic field as
m as a function of magnetic field. At magnetic fields different portions of the band dispersion are probed.1,2,18
exceeding 3T, the 2e2/h plateau shows a reen- While ballistic transport through nanowire QPCs is now
.
at trant conductance feature towards 1e2/h which in- standard,19,20 numerical simulations have shown that the
m creases linearly in width with magnetic field be- visibility of this experimental signature critically depends
fore enveloping the 1e2/h plateau. Rotating the on the exact combination of geometrical and physical
-
d external magnetic field either parallel or perpen- device parameters.18
n dicular to the spin orbit field allows us to clearly Here we observe a clear signature of transport through
o
attribute this experimental signature to SOI. We ahelicalstateinaQPCformedinanInSbnanowirewhen
c
[ compare our observations with a model of a QPC the magnetic field has a component perpendicular to the
incorporatingSOIandextractaspinorbitenergy spin-orbit field. We show that the state evolves under
1 of ∼ 6.5meV, which is significantly stronger than rotationoftheexternalmagneticfield,disappearingwhen
v
8 the SO energy obtained by other methods. the magnetic field is aligned with BSO. By comparing
Spin-orbit interaction is a relativistic effect where a our data to a theoretical model, we extract a spin orbit
7
8 charged particle moving in an electric field E with mo- energy E = 6.5meV, significantly stronger than that
SO
6 mentumkandvelocityv =k/m ,experiencesaneffective measured in InSb nanowires by other techniques.
0
1.0 TmhaegnmetaigcnfieetilcdmBoSmOe=nt(o−f1t/hme0ecl)ekct×roEn sipninit,sµre=stefSr/amm0e,. devFiicgeu.rAen1aInsShbownsanaoswcihreemisadtiecpiomsiategde oonf aatdyepgiecnaelrQatPelCy
0 interacts with this effective magnetic field, resulting in doped silicon wafer covered with a thin (20nm) SiN di-
7 a spin-orbit Hamiltonian HSO = −µ.BSO that couples electric. The QPC is formed in the nanowire channel in
1 the spin to the orbital motion and electric field. In crys-
a region defined by the source and drain contacts spaced
:
v talline materials, the electric field arises from a symmetry ∼ 325nm apart. The chemical potential µ in the QPC
Xi breaking that is either intrinsic to the underlying crystal channel, which sets the subband occupation, is controlled
lattice in which the carriers move, known as the Dressel-
by applying a voltage to the gate V . The electric field in
ar haus SOI,8 or an artificially induced asymmetry in the the nanowire, E, generated by thegbackgate and the sub-
confinement potential due to an applied electric field, or
strate that the nanowire lies on, both induce a structural
Rashba SOI.9 Wurtzite and certain zincblende nanowires inversion asymmetry that results in a finite Rashba spin
possess a finite Dresselhaus SOI, and so the SOI is a com-
orbit field. As the wire is translationally invariant along
binationofboththeRashbaandDresselhauscomponents.
its length, the spin orbit field, B , is perpendicular to
For zincblende nanowires grown along the [111] growth SO
both the electric field and the wire axis. The effective
direction the crystal lattice is inversion symmetric, and
channel length, L ∼245nm, as well as the shape of
so only a Rashba component to the spin-orbit interaction QPC
the onset potential λ ∼ 80nm are set by electrostatics
is thought to remain.10 which are influenced by both the thickness of the dielec-
Helical states have been shown to emerge in the edge tric and the amount of electric field screening provided
2
by the metallic contacts to the nanowire (Fig 1b). Here field,beforefullyenvelopingthe1·G plateauformagnetic
0
we report measurements from one device. Data from fieldslargerthanaround5.5T. Linetracescorresponding
an additional device that shows the same effect, as well to the colored arrows in Fig 2b are shown in Fig 2d. The
as control devices of different channel lengths and onset feature is robust at higher temperatures up to 1K, as
potentials, is provided in the Supplementary Information. well across multiple thermal cycles (see Supplementary
The energy-momentum diagrams in Fig 1c-e show the Information). Using the 1D nanowire model with θ =17◦
we find that the helical gap feature vanishes into a con-
dispersion from the 1D nanowire model of Refs. 1 and 2
including both SOI with strength α and Zeeman splitting tinuous 0.5·G0 plateau when EZ >2.4ESO. Using the
E =gµ B, whereg istheg-factor, µ theBohrmagne- extracted g-factor g = 38 of our device (see Fig 3 and
Z B B Supplementary Information) we find a lower bound for
ton and B the magnetic field strength. These dispersion
relations explain how the helical gap can be detected: the spin-orbit energy ESO =5.5meV, corresponding to a
Without magnetic field, the SOI causes the first two spin spin-orbit length lSO =1/kSO ≈22nm. For a second de-
degeneratesub-bandstobeshiftedlaterallyinmomentum vice, we extract a similar value ESO =5.2meV. Recently
it has been highlighted that the visibility of the helical
space by ±k =m∗α/(cid:126)2 with m∗ the effective electron
SO gap feature depends crucially on the shape of the QPC
mass, as electrons with opposite spins carry opposite mo-
potential.18 To verify that our observation is compati-
mentum,asshowninFig1c. Thecorrespondingspin-orbit
ble with SOI in this respect, we perform self-consistent
energy is given by E = (cid:126)2k2 /2m∗. However, here
SO SO simulations of the Poisson equation in Thomas-Fermi
Kramers degeneracy is preserved and hence the plateaus
approximation for our device geometry. The resulting
in conductance occur at integer values of G = 2e2/h,
0 electrostatic potential is then mapped to an effective 1D
as for a system without SOI. Applying a magnetic field
QPC potential for a quantum transport simulation using
perpendicular to BSO the spin bands hybridize and a
parameters for InSb (for details, see Supplementary In-
helical gap, of size E opens as shown in Fig 1d. When
Z formation). These numerical simulations, shown in Fig
the chemical potential µ is tuned by the external gate
voltage, it first aligns with the bottom of both bands 2c, fit best for lSO =20nm (ESO =6.5meV) and agree
well with the experimental observation, corroborating our
resulting in conductance at 1·G before reducing from
0 interpretation of the re-entrant conductance feature as
1·G to 0.5·G when µ is positioned inside the gap.
0 0 the helical gap.
This conductance reduction with a width scaling linearly
with increasing Zeeman energy, is a hallmark of trans- Voltage bias spectroscopy, as shown in Fig 3a confirms
port through a helical state. When the magnetic field is that this state evolves as a constant energy feature. By
orientated at an angle θ to B , the size of the helical analyzingthevoltagebiasspectroscopydataatarangeof
SO
gap decreases as it is governed by the component of the magneticfields,wequantifythedevelopmentoftheinitial
magnetic field perpendicular to B , as shown in Fig 1e. 0.5·G plateau, as well as the reentrant conductance
SO 0
Additionally, the two sub-band bottoms also experience feature (Fig 3b). From the evolution of the width of the
a spin splitting giving rise to an additional Zeeman gap. first 0.5·G plateau, we can calculate the g-factor of the
0
For a general angle θ, the QPC conductance thus first first sub-band g = 38±1. This number is consistent
rises from 0 to 0.5·G , then to 1·G , before dropping with the recent experiments, which reported g factors of
0 0
to 0.5·G again. The helical gap thus takes the form of 35−50.21,22 ComparingtheslopesoftheZeemangapand
0
a re-entrant 0.5·G conductance feature. By comparing thehelicalgapE /E ≈tanθprovidesanalternativeway
0 h Z
to a 1D nanowire model, we can extract both the size todeterminetheoffsetangleθ. Wefindθ =13◦±2◦which
of the helical gap E ≈ E sinθ and the Zeeman is in reasonable agreement with the angle determined by
helical Z
shift E ≈ E cosθ (see Supplementary Informa- magnetic field rotation.
Zeeman Z
tion). This angle dependency is a unique feature of SOI
To confirm that the reentrant conductance feature
and can be used as a decisive test for its presence in the
agrees with spin orbit theory, we rotate the magnetic
experimental data.
field in the plane of the substrate at a constant magni-
Figure 2 shows the differential conductance dI/dV of tude B = 3.3T, as shown in Fig 4a. When the field is
our device at zero source-drain bias as a function of gate rotated towards being parallel to B O, the conductance
S
and magnetic field. Here the magnetic field B is offset dip closes, while when it is rotated away from B , the
SO
at a small angle θ = 17◦ from B (see Fig 2a). We dip increases in width and depth. In contrast, when the
SO
determine that our device has this orientation from the magnetic field is rotated the same amount around the
angle-dependence of the magnetic field, by clearly resolv- y-z plane, which is largely perpendicular to B , there
SO
ing the 1·G plateau before the re-entrant conductance is little change in the reentrant conductance feature, as
0
feature, which is reduced at larger angles (see Supplemen- shown in Fig 4b. Figure 4c shows the result of rotating
tary Information). For low magnetic fields, we observe through a larger angle in the x-y plane shows this feature
conductance plateaus quantized in steps of 0.5·G , as clearly evolves with what is expected for spin orbit. Our
0
typical for a QPC in a spin polarizing B-field with or numerical simulations in Figure 4d agree well with the
without SOI. Above B =3T, the 1·G plateau shows a observed experimental data. The small difference in the
0
conductance dip to 0.5·G . This reentrant conductance angleevolutionbetweenthenumericalsimulationsandex-
0
feature evolves continuously as a function of magnetic perimental data can be attributed to imperfect alignment
3
of the substrate with the x-y plane. region,andexistsatzeromagneticfield,unlikethefeature
inourdata. Additionally,aKondoresonanceshouldscale
The extracted SO energy of 6.5meV is significantly
larger than that obtained via other techniques, such as with Vsd =±gµBB/e as a function of external magnetic
field, decreasing instead of increasing the width of the
weak anti localization (WAL) measurements,23 and quan-
region of suppressed conductance. Given the g factor
tum dot spectroscopy.22 This is not entirely unexpected,
measured in InSb quantum dots, and its variation with
due to the differing geometry for this device and differ-
ent conductance regime it is operated in. Quantum dot the angle of applied magnetic field g =35−50,22 we can
exclude both these effects. Similarly the Fano effect and
measurements require strong confinement, and so the
disorder can also induce a conductance dip, but these
Rashba SOI is modified by the local electrostatic gates
effects should not increase linearly with magnetic field.
used to define the quantum dot. Weak anti-localization
The 0.7 anomaly occurs at the beginning of the plateau,
measurements are performed in an open conductance
and numerical studies have shown it does not drastically
regime, however they assume transport through a diffu-
sive, rather than a ballistic channel. Neither of these affect the observation of the helical gap.26 In conclusion,
measurements explicitly probe the spin orbit interaction we have observed a return to 1e2/h conductance at the
where exactly one mode is transmitting in the nanowire, 2e2/hplateauinaQPCinanInSbnanowire. Thecontin-
uousevolutioninincreasingmagneticfieldandthestrong
the ideal regime for Majoranas, and so the spin orbit
angle dependence in magnetic field rotations agree with a
parameters extracted from QPC measurements offer a
SOI related origin of this feature and distinguish it from
more accurate measurement of the SOI experienced by
Fabry-Perot oscillations and other g-factor related phe-
theMajoranazeromode. Also,theSOIinananowirecan
nomena. Additional confirmation is given by numerical
bedifferentforeverysubband,anditisexpectedthatthe
simulations of an emerging helical gap in InSb nanowires.
lowest mode has the strongest spin-orbit due to a smaller
confinement energy.10 Additionally, the finite diameter of Theextractedspinorbitenergyof6.5meV issignificantly
largerthanwhathasbeenfoundbyothertechniques,and
the nanowire, together with impurities within the InSb
more accurately represents the true spin orbit energy in
crystal lattice both break the internal symmetry of the
the first conduction mode. Such a large spin orbit energy
crystal lattice and may contribute a non-zero Dresselhaus
reduces the requirements on nanowire disorder for reach-
component to the spin orbit energy that has not been
previously considered. While high quality quantized con- ing the topological regime,27 and offers promise for using
InSb nanowires for the creation of topologically protected
ductance measurements have been previously achieved in
quantum computing devices.
shortchanneldevices19 (L∼150nm),thechannellengths
required for observing the helical gap are at the experi-
mental limit of observable conductance quantization. As
shown in the Supplementary Information, small changes METHODS
in the QPC channel length, spin-orbit strength or the
QPC potential profile are enough to obscure the helical Device Fabrication
gap, particularly for wires with weaker SOI. We have
fabricated and measured a range of QPCs with different The InSb nanowires were grown using the metalor-
length and potential profiles, and only two devices of ganicvaporphaseepitaxy(MOVPE)process.28TheInSb
L∼300nm showed unambiguous signatures of a helical nanowires were deposited using a deterministic deposi-
gap. tion method on a degenerately doped silicon wafer. The
Several phenomena have been reported to result in wafercoveredwith20nmoflowstressLPCVDSiNwhich
anomalous conductance features in a device such as this. is used as a high quality dielectric. Electrical contacts
OscillationsinconductanceduetoFabry-Perotresonances (Cr/Au, 10nm/110nm) defined using ebeam lithography
are a common feature in clean QPCs. Typically the first werethenevaporatedattheendsofthewire. Beforeevap-
oscillationatthefrontofeachplateauisthestrongestand orationthewirewasexposedtoanammoniumpolysulfide
theoscillationsmonotonicallydecreaseinstrengthfurther surface treatment and short helium ion etch to remove
along each plateau.18,23 In our second device, we clearly the surface oxide and to dope the nanowire underneath
observe Fabry-Perot conductance oscillations at the be- the contacts.19
ginning of each plateau, however these oscillations are
significantly weaker than the subsequent conductance dip.
Furthermore we observe Fabry-Perot oscillations at each Measurements
conductance plateau, while the reentrant conductance
feature is only present at the 1·G plateau. Addition- Measurements are performed in a dilution refrigerator
0
ally, the width of the Fabry-Perot oscillations does not withbasetemperature∼20mKfittedwitha3-axisvector
changewithincreasingmagneticfield,unliketheobserved magnet, which allowed for the external magnetic field to
re-entrant conductance feature. A local quantum dot in be rotated in-situ. The sample is mounted with the
the Coulomb or Kondo regimes can lead to conductance substrate in the x-y plane with the wire orientated at a
suppression, which increases in magnetic field.25 However small offset angle θ =17◦ from the x-axis. We measure
both effects should be stronger in the lower conductance the differential conductance G=dI/dV using standard
4
lock-in techniques with an excitation voltage of 60µV Numerical transport simulations
and frequency f = 83Hz. Additional resistances due
to filtering are subtracted to give the true conductance We use the method of finite differences to discretize
through the device. the one-dimensional nanowire model of Ref 2. In order
to obtain a one-dimensional QPC potential, we solve
the Poisson equation self-consistently for the full three-
dimensional device structure treating the charge density
in the nanowire in Thomas-Fermi approximation. To this
end, we use a finite element method, using the software
FEniCS.29 The resulting three-dimensional potential is
then projected onto the lowest nanowire subband and
interpolated using the QPC potential model of Ref18.
Transport in the resulting tight-binding model is calcu-
lated using the software Kwant.30
∗ [email protected] electric fields. Phys. Rev. Lett. 106, 1–4 (2011).
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ACKNOWLEDGMENTS
We gratefully acknowledge D. Xu, Ö. Gül, S. Goswami,
D. van Woerkom and R.N. Schouten for their technical
assistance and helpful discussions. This work has been
supported by funding from the Netherlands Foundation
for Fundamental Research on Matter (FOM), the Nether-
lands Organization for Scientific Research (NWO/OCW),
the Office of Naval Research, Microsoft Corporation Sta-
tion Q, the European Research Council (ERC) and an
EU Marie-Curie ITN.
AUTHOR CONTRIBUTIONS
J.K and F.P. fabricated the samples, J.K. M.C.C. and
F.P. performed the measurements with input from M.W..
M.W., M.N. and A.V. developed the theoretical model
and performed simulations. D.C., S.R.P. and E.P.A.M.B.
grew the InSb nanowires. All authors discussed the data
and contributed to the manuscript.
2
a c B = 0 T
E(k) μ
A
Si Cr/Au InSb E V E
3N SO
4
B n++ Si k Bso -k +k k 0.5 1 G (2e2/h)
Z SO SO
BY ECDd B B
Vg SO
B
X E(k) μ
b E
Z
ɛ- ɛ+
k G (2e2/h)
μ 0.5 1
e B θ B
SO
L λ V (x)
QPC g E(k) μ
~E sinθ
Z
~E cosθ
Z
k G (2e2/h)
0.5 1
FIG. 1. The helical gap in a 1D nanowire device. a, An InSb nanowire device with a Rashba spin-orbit field B
SO
perpendicular to the wave vector k and the electric field E. A voltage is sourced to one contact, and the resulting conductance
measured from the second contact. The chemical potential in the wire, µ, is tuned with a global backgate V . b, The QPC
g
channel of length L is defined by the two contacts. The shape of the onset with a lengthscale λ is set by the dielectric and
screening of the electric field from the metallic contacts resulting in an effective QPC length L =L−2λ. c, The energy
QPC
dispersion of the first two subbands for a system with SOI at external magnetic field B =0. The SOI causes subbands to shift
by k in momentum space, as electrons with opposite spins carry opposite momentum. When the electrochemical potential µ
SO
inthewireistunedconductanceplateauswilloccuratintegervaluesofG . d,AtfinitemagneticfieldB perpendiculartoB ,
0 SO
the spin polarized bands hybridize opening a helical gap of size E (green). In this region the conductance reduces from 1·G
Z 0
to 0.5·G when µ is positioned inside the gap. e, When the magnetic field is orientated at an angle θ to B , the size of the
0 SO
helical gap decreases to only include the component of the magnetic field perpendicular to B . For all angles the reentrant
SO
conductance feature at 0.5·G in the 1·G plateau will scale linearly with Zeeman energy.
0 0
3)
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G
B=3T B=4.5T B=5.5T
0.2 0.4 0.2 0.4 0.2 0.4
V (V) V (V) V (V)
g g g
FIG. 2. Magnetic field dependence of the helical gap. a, The nanowire lies in the x-y plane at an angle θ=17◦ relative
to the external magnetic field. b, Differential conductance dI/dV at zero source-drain bias as a function of back gate voltage
and external magnetic field. At low magnetic fields conductance plateaus at multiples of 0.5·G are visible. Above B =3T, a
0
reentrantconductancefeatureat0.5·G appearsinthe1·G plateau. ThefeatureevolveslinearlywithZeemanenergyindicated
0 0
by dashed green lines. c, Numerical simulations of the differential conductance as a function of the potential E and external
a
magnetic field for L=325nm, θ=17◦ and l =20nm (See Supplementary Information for a more detailed description of the
SO
model). In the numerical simulations, the conductance plateaus have a different slope compared to the experimental data as
the calculations neglect screening by charges in the wire. d, Line traces of the conductance map in b. As the helical gap is
independent of disorder or interference effects, these and other anomalous conductance features average out in a 2D colorplot
improving the visibility of the helical gap in b compared to the individual traces in d.
4
a b
G
B=4T
5 (2e2/h) E
2 20 Subband
)
V
m 0.5 1 1.5
(0

Vsd V)
e E
0 m Zeeman
-5 (
10
E
)
h
2/ 1 E
e helical
2
( V = 0mV

G sd 0
0
0.25 0.5 2 4 6
V (V) B (T)
g
FIG. 3. Voltage bias spectroscopy of the helical gap. a, Conductance measurement as a function of QPC gate and
source-drain bias voltage at B = 4T. The observed helical gap (green) is a stable feature in voltage bias. Dotted lines are
drawn as guide to the eye indicating the plateau edges. b. Evolution of the energy levels extracted from scans similar to a, at
increasing magnetic field. Fits with intercept fixed at zero (dotted lines) give the g-factor of the first subband and the offset
angle via g=1/(µ cosθ)·dE/dB and E /E ≈tanθ. We find g=38±1 and θ=13◦±2◦. Individual scans are
B helical Zeeman
included in the Supplementary Information.
5
SN01_79 proctriton (B-angle phi (degree)) 3.3Tesase----d012344321 .........d0000000000a.t0a 1 (powSe0Nr.0-1x-11_z-718e p-0rBo2cS00triton (B-angle phi (degree)) 3.3TeO;.so2sθfefs----dy012344321e .........dt0000000000-ad.-t05aa0 c1003 (θ (°)0. p2e/h)3(-oB0wG;121pe0)ro1.713-1w-1e-1Bre-0=--.1043-21x-0.023e;.o2--1Tf0fzs.25e0 t°-;pd-s05ac.0lc5a0a03l0. e3(-nB 0dG;ape)otaw-e1S0r2-0.N-69.1400-116e_;-s7G(0c292a 0 elp02;es02r00001111 co/.a5ha........c02580258xltriton (B-angle phi (degree)) 3.3Te)ee ssdbprocessed data 1 (power--1-1e-020;offset--5000-0;power--1-1e-020;scale data-12906;scale axes-0.03-1)s-a0eta----.d0123443210- 1.........30d00000000020.-a691.t00a)6 1;s c(aploew00001111 aS........e02580258x0Ner.0-1s-processed data 1 (power--1-1e-020;offset--5000-0;power--1-1e-020;scale data-12906;scale axes-0.03-1)1-10x_-.701z93e -p-1Ф0r)o2cB00triton (B-angle phi (degree)) 3.3Te;S.so2Osfefs----d012344321e y.........dt0000000000-ad.-t05aa0 c1003 Ф (°)(20.e/h) p3(-oB0wG;pe0-)ro1.440-1-w1e-1rBe-0--.1=042-3100-e.;.4yo32-f0.fT-5s2ez°0t-;d- s05apc.0c5a003ll0. ea3(-B 0dG;npa)otaew-e10r2-0.-69.140-16e;-s0G(c22a0 el02;es0200001111 c/.a5ah........02580258xle)e dsprocessed data 1 (power--1-1e-020;offset--5000-0;power--1-1e-020;scale data-12906;scale axes-0.03-1)a-0ta.-01032.-6910)6;scale00001111 a........02580258xesprocessed data 1 (power--1-1e-020;offset--5000-0;power--1-1e-020;scale data-12906;scale axes-0.03-1)-0.03-1)
2 2
G ( 12.5° G ( 4.5°
0 0
0.2 0.4 0.2 0.4
V (V) V (V)
g g
c d
G G
35 B = 3.6T (2e2/h) 35 (2e2/h)
2 1
30 30
2
0 37° 0
25 25
)
(°)20 0 θ (°20

θ )2
h 17°
/
15 2e 15
2
(

G
10 0 10
2
0°
5 5
0
0 0 30 20 10
0.2 0.4 0.2 0.4 E (meV)
a
V (V) V (V)
g g
FIG. 4. Angle dependence of the helical gap. a, Rotation of the magnetic field at B =3.3T in the x-y plane parallel to
thesubstrateshowsstrongangledependenceofthehelicalgap. TheconductancedipcloseswhenB isrotatedtowardsB and
SO
opens when B is rotated away from B . b, Rotation of the magnetic field at B =3.3T in the y-z plane, mostly perpendicular
SO
to B . While the angle range is identical to a there is little change in the conductance dip. c, Rotation of the magnetic field
SO
at B =3.6T in the x-y plane over a large angle range. The conductance dip disappears when B is parallel to B which gives
SO
the exact offset angle between B and B , θ=17◦. d, Numerical simulations of the differential conductance in a magnetic
SO Z
field rotated along θ in the x-y plane with L=325nm and l =20nm.
SO
6
Supplementary Information for: Conductance through a helical state in an InSb
nanowire
J. Kammhuber,1 M. C. Cassidy,1 F. Pei,1 M. P. Nowak,1,2 A. Vuik,1 D. Car,1,3
S. R. Plissard,4 E. P. A. M. Bakkers,1,3 M. Wimmer,1 Leo P. Kouwenhoven1,∗
1QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands
2 Current adress: Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, al.
A.Mickiewicza 30, 30-059 Kraków, Poland
3Department of Applied Physics, Eindhoven University of Technology, 5600 MB Eindhoven, The Netherlands
4CNRS-Laboratoire d’Analyse et d’Architecture des Systemes (LAAS), Université de Toulouse, 7 avenue du colonel Roche,
F-31400 Toulouse, France
I. NUMERICAL SIMULATIONS OF THE CONDUCTANCE THROUGH A HELICAL STATES
A. Poisson calculations in a 3D nanowire device
Observing the helical gap in a semiconducting nanowire crucially depends on the smoothness of
the electrostatic potential profile between the two contactsS1. When the potential profile changes
too abruptly, it forms a tunnel barrier which suppresses conductance well below quantized values,
thereby masking features of the helical gap. On the other hand, if the potential varies on a length
scale much larger than the characteristic spin-orbit coupling length l , transmission through the
SO
‘internal state’ (the smaller-momentum state of the two right-moving states in the bottom of the
lower band) is suppressed. This reduces the first 2e2/h plateau in the conductance to a 1e2/h plateau,
thereby concealing again the helical gap.
Because of the crucial role of the electrostatic potential, we perform realistic Poisson calculations
to compute the potential φ(~r) in the nanowire (with ~r = (x,y,z)), solving the Poisson equation of
the general form
ρ(~r)
∇2φ(~r) = − , (S1)
(cid:15)
with (cid:15) the dielectric permittivity and ρ the charge density. For the charge density ρ, we apply the
Thomas-Fermi approximationS2
e 2m∗eφ(~r)!3/2
ρ(~r) = , (S2)
3π2(cid:15) (cid:126)2
where m∗ is the effective mass of InSb.
For a given charge density ρ, we solve Eq. S1 numerically for the potential using the finite element
package FEniCSS3. We model the two normal contacts as metals with a fixed potential V = 0.22 V,
N
assuming a small work function difference between the nanowire and the normal contacts. The back
gate is modeled as a fixed potential V along the bottom surface of the dielectric layer. We use the
G
dielectric permittivities for InSb and SiN in the wire and the dielectric layer respectively. The FEM
mesh, with its dimensions and boundary conditions, is depicted in Fig. S1a.
We apply the Anderson mixing schemeS4 to solve the nonlinear equation formed by Eqs. S1 and S2
self-consistently. An example of a self-consistent Poisson potential with Thomas-Fermi density is
plotted in Fig. S1b.
B. Conductance calculations in a 1D model with a projected potential barrier
To apply the 3D Poisson potential in a simple 1D nanowire model, we convert the three-dimensional
ˆ
potential φ(x,y,z) to a one-dimensional effective potential barrier φ(x) by projecting φ on the