Gate-dependent spin-orbit coupling in multi-electron car - Niels Bohr ...

Gate-dependent spin-orbit coupling in multi-electron carbon
nanotubes
T. S. Jespersen 1∗† , K. Grove-Rasmussen 1,2∗† , J. Paaske 1 , K. Muraki 2 ,
T. Fujisawa 3 , J. Nygård 1 , K. Flensberg 1 .
1 Niels Bohr Institute & Nano-Science Center, University of Copenhagen,
Universitetsparken 5, DK-2100 Copenhagen, Denmark
2 NTT Basic Research Laboratories, NTT Corporation,
3-1 Morinosato-Wakamiya, Atsugi 243-0198, Japan
3 Research Center for Low Temperature Physics, Tokyo Institute of Technology,
Ookayama, Meguro, Tokyo 152-8551, Japan
∗ To whom correspondence should be addressed; E-mail: tsand@fys.ku.dk, k grove@fys.ku.dk
† Equal contribution
Understanding how the orbital motion of electrons is coupled to the spin degree
of freedom in nanoscale systems is central for applications in spin-based electronics
and quantum computation. We demonstrate this coupling of spin and orbit in
a carbon nanotube quantum dot in the general multi-electron regime in presence
of finite disorder. Further, we find a strong systematic dependence of the spinorbit
coupling on the electron occupation of the quantum dot. This dependence,
which even includes a sign change is not demonstrated in any other system and
follows from the curvature-induced spin-orbit split Dirac-spectrum of the underlying
graphene lattice. Our findings unambiguously show that the spin-orbit
coupling is a general property of nanotube quantum dots which provide a unique
platform for the study of spin-orbit effects and their applications.
1

The interaction of the spin of electrons with their orbital motion has become a focus of attention
in quantum dot research. On the one hand, this spin-orbit interaction (SOI) provides a route
for spin decoherence, which is unwanted for purposes of quantum computation 1–3 . On the other
hand, if properly controlled, the SOI can be utilized as a means of electrically manipulating the
spin degree of freedom 4–7 .
In this context, carbon nanotubes provide a number of attractive features, including large
confinement energies, nearly nuclear-spin-free environment, and, most importantly, the details of
the energy level structure is theoretically well understood and modeled, as well as experimentally
highly reproducible. Remarkably, the SOI in nanotubes was largely overlooked in the first two
decades of nanotube research and was only recently demonstrated by Kuemmeth et al. for the special
case of a single carrier in ultra-clean CNT quantum dots 3, 8, 9 . Except for these reports, the SOI
in nanotubes is experimentally unexplored. Theoretically, the focus has exclusively been on the
SOI-modified band structure of disorder-free nanotubes 10–14 . Therefore, two important questions
remain: how the effective SOI depends on electron filling and how it appears in the general case of
quantum dots subject to disorder. Here we answer these two questions.
Firstly, by low-temperature electron transport we demonstrate the presence of a significant SOI
in a disordered CNT quantum dot holding hundreds of electrons. We identify and analyze the
role of SOI in the energy spectrum for one, two, and three electrons in the four-fold degenerate
CNT electronic shells, thus describing shells at any electron filling. By rotating the sample, we
present for the first time spectroscopy of the same charge-states for magnetic fields both parallel
and perpendicular to the nanotube axis, thus controlling the coupling to the orbital magnetic
moment. Remarkably, a single-electron model taking into account both SOI and disorder quantitatively
describes all essential details of the multi-electron quantum dot spectra. Secondly, by
changing the electron occupancy we are able to tune the effective SOI and even reverse its sign
in accordance with the expected curvature-induced spin-orbit splitting of the underlying graphene
Dirac spectrum 1, 11–14 . Such systematic dependence has not been demonstrated in any other ma-
2

terial system and may enable a new range of spin-orbit related applications. This microscopic
understanding and detailed modeling is in stark contrasts to situations encountered in alternative
strong-SOI quantum-dot materials, such as InAs or InSb nanowires, where the effective SOI arises
from bulk crystal effects combined with unknown contributions from surface effects, strain and
crystal defects 15 . These systems often exhibit semi-random fluctuations of, e.g., the g-factor as
single electrons are added 16, 17 . Thus, beyond fundamental interest and the prospect of realizing recent
proposals of SOI-induced spin control in CNTs 1, 18 , our findings pave the way for new designs
of experiments utilizing the SOI in quantum dots.
Zero-field splitting of the four-fold degeneracy
Our experimental setup is presented in Fig. 1a. We fabricate devices of single-wall CNTs on highly
doped Si substrates capped with an insulating layer of SiO 2 (see Methods section). The size of the
quantum dots is defined by the contact separation (400 nm) and the electrical properties are investigated
in a voltage biased two-terminal configuration applying a voltage V sd between source-drain
contacts and measuring the resulting current I. The differential conductance dI/dV sd is measured
by standard lock-in techniques. When biased with a voltage V g , the Si substrate acts as an electrostatic
gate controlling the electron occupancy of the dot. The devices are measured at T = 100 mK
in a 3 He/ 4 He dilution refrigerator, fitted with a piezo-rotator allowing in-plane rotations of the device
in magnetic fields up to 9 T.
Figure 1b shows a typical measurement of dI/dV sd vs. V sd and V g in the multi-electron
regime of a small-band-gap semiconducting nanotube. The pattern of diamond-shaped regions of
low conductance is expected for a quantum dot in the Coulomb blockade regime and within each
diamond the quantum dot hosts a fixed number of electrons N, increasing one-by-one with increasing
V g . The energy E add , required for adding a single electron appears as the diamond heights
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and has been extracted in Fig. 1c. The four-electron periodicity clearly observed in Fig. 1b and c
reflects the near four-fold degeneracy in the nanotube energy spectrum 19, 20 ; one factor of 2 from
the intrinsic spin (↑, ↓) and one factor of 2 from the so-called isospin (K, K ′ ) that stems from the
rotational symmetry of the nanotube - electrons orbit the CNT in a clockwise or anticlockwise
direction. As is generally observed 19–23 , the addition energy for the second electron in each quartet
(yellow in Fig. 1c) exceeds those for one and three. This was previously interpreted as a result
of disorder-induced coupling ∆ KK ′
of the clockwise and anticlockwise states 21, 24 that splits the
spectrum into two spin-degenerate pairs of bonding/antibonding states separated by ∆ KK ′. As
mentioned, Kuemmeth et al. showed recently that for the first electron in an ultra-clean suspended
nanotube quantum dot, the splitting was instead dominated by the spin-orbit coupling. The first
question we address here is whether SOI also appears in the many-electron regime and how it may
be modified or masked by disorder.
Modeling spin-orbit coupling and disorder
Performing level spectroscopy with a magnetic field B applied either parallel (B ‖ ) or perpendicular
(B ⊥ ) to the nanotube axis proves to be a powerful tool to analyze the separate contributions from
disorder and spin-orbit coupling. This is illustrated in Figs. 2a-d, which show calculated singleparticle
energy level spectra for four limiting combinations of ∆ KK ′
and the effective spin-orbit
coupling ∆ SO (all limits are relevant for nanotube devices depending on the degree of disorder
and CNT structure 13, 14 ; details of the model are provided in the Supplementary Information). In
all cases a parallel field separates the four states into pairs of increasing (K-like states) and decreasing
(K ′ -like states) energies. The magnitude of the shift is given by the orbital g-factor g orb
reflecting the coupling of B ‖ to the orbital magnetic moment caused by motion around the CNT 25 .
Further, each pair exhibits a smaller internal splitting due to the Zeeman effect. Figure 2b shows
4

the disorder induced coupling of K and K ′ states resulting in an avoided crossing at B ‖ = 0 and
the zero-field splitting discussed above. In the opposite limit with SOI only (Fig. 2c), the zerofield
spectrum is also split into two doublets, but the field dependence is markedly different and
no avoided crossing appears. In the simplest picture, this behavior originates from coupling of the
electron spin to an effective magnetic field B SO = −(v × E)/c 2 experienced by the electron as
it moves with velocity v in an electric field E. Here the speed of light, c, reflects the relativistic
origin of the effect. In nanotubes, the curvature of the graphene lattice generates an effective radial
electric field, and since the velocity is mainly circumferential (and opposite for K and K ′ ), B SO
polarizes the spins along the nanotube axis and favors parallel alignment of the spin and orbital
angular momentum. Thus, even in the absence of disorder, the spectrum splits into two Kramers
doublets (K ↓, K ′ ↑) and (K ↑, K ′ ↓) separated by ∆ SO . Interestingly, since a perpendicular field
does not couple K and K ′ the doublets do not split along B ⊥ on the figure. As a consequence, the
g-factor, when measured in a perpendicular field, will vary from zero when ∆ SO ≫ ∆ KK ′
(Fig.
2c) to 2 in the opposite limit (Fig. 2b).
The final case, including both disorder and SOI, is of particular importance for the present
study, and the calculated spectrum is displayed in Fig. 2d for ∆ KK ′
> ∆ SO . Importantly, the
effects of SOI are not masked despite the dominating disorder: For parallel field, SOI remains responsible
for an asymmetric splitting of the Kramers doublets (α, β) vs. (δ, γ), and the appearance
of an additional degeneracy in the spectrum at finite field (δ and γ states). In a perpendicular field,
the effect of SOI is to suppress the Zeeman splitting of the two doublets and since the eigenstates
of the SOI have spins along the nanotube axis it couples the states with spins polarized along B ⊥
resulting in the avoided crossing indicated on the figure.
5

Spin-orbit interaction revealed by spectroscopy
With this in mind we now focus on the quartet with 4N 0 ≈ 180 electrons highlighted in Fig. 1b
and expanded in Fig. 3a. In order to investigate the level structure we perform cotunneling spectroscopy,
as illustrated in the schematic Fig. 3b 26 : In Coulomb blockade, whenever e|V sd | matches
the energy of a transition from the ground state α to an excited state (β, γ, δ), inelastic cotunnel
processes, that leave the quantum dot in the excited state, become available for transport. This
significantly increases the current and gives rise to steps in the conductance. These appear as gateindependent
features in Fig. 3a (arrows) and are clearly seen in the inset showing a trace through
the center of the one-electron (4N 0 + 1) diamond along the dashed line. Thus following the magnetic
field dependence of this trace, as shown in Fig. 3c, maps out the level structure. The energies
of the excitations are given by the inflection points of the curve (i.e. peaks/dips of d 2 I/dV 2
sd) 27 and
the level evolution is therefore directly evident in Fig. 3d-i, which show color maps of the second
derivative vs. V sd and B ⊥ , B ‖ for V g positioned in the center of the one, two, and three electron
charge states. As explained below, the SOI is clearly expressed in all three spectra.
Consider first the one-electron case: In a parallel field (Fig. 3d), the asymmetric splitting of
the two doublets is evident (black vs. green arrows) and applying the field perpendicularly (Panel
g), the SOI is directly expressed as the avoided crossing indicated on the figure. The measurement
is in near-perfect agreement with the single-particle excitation spectrum calculated by subtracting
the energies of Fig. 3b and shown by the solid lines. The calculation depends on only three parameters:
∆ SO = 0.15 meV set directly by the avoided crossing, ∆ KK ′
= 0.45 meV determined
from the zero-field splitting of the doublets (see Fig. 2d), and g orb = 5.7 set by the slopes of the
excitation lines from α to γ, δ in Panel d.
Consider now the role of SOI for the doubly occupied CNT quartet. This situation is of particular
importance for quantum computation as a paradigm for preparation of entangled states and
a fundamental part of Pauli blockade in double quantum dots 28 . Figures 3e,h show the measured
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spectra in parallel and perpendicular fields. The model perfectly describes the measurement and
now contains no free parameters since these are fixed by one-electron measurement. Six states are
expected: the ground state singlet-like state ˜S 0 formed by the two electrons occupying the lowenergy
Kramers doublet 29 , three triplet-like ˜T − , ˜T 0 , ˜T + and a singlet-like state ˜S 1 , which all use
one state from each doublet and the singlet-like ˜S 2 with both electrons occupying the high-energy
doublet. The ground state ˜S 0 does not appear directly in the measurement, but sets the origin for
the cotunneling excitations. The excitation to the high-energy ˜S 2 (dashed line) is absent in the
experiment, since it cannot be reached by promoting only a single electron from ˜S 0 . In Fig. 3h
excitations to ˜T − and ˜T + are clearly observed, while ˜S 1 and ˜T 0 merge into a single high-intensity
peak showing that any exchange splitting J is below the spectroscopic line width ≈ 100µeV 23 . In
other quartets an exchange splitting is indeed observed (see Supplementary Information). In the
two-electron spectra the SOI is directly expressed as the avoided crossing at B ⊥ ≈ 4.5 T accompanying
the ˜S 0 ↔ ˜T − ground state transition 15, 30 . In quartets of yet stronger tunnel-coupling it is
replaced by a singlet-triplet Kondo resonance 31 (see Supplementary Information).
Finally, the spectrum of three electrons in the four-electron shell is equivalent to that of a
single hole in a full shell; at low fields the δ-state becomes the ground-state and γ the first excited
state, while α and β then constitute the excited doublet. As seen by comparing Fig. 2b and 2d
SOI breaks the intra-shell electron-hole symmetry of the nanotube spectrum. This is evident in
the experiment when comparing Fig. 3d and 3f: In 3f, increasing B ‖ , the lowest excited state γ,
barely separates from the ground state δ and at B ‖ = 1.1 T they cross again, causing a ground state
transition. At the crossing point, the spin-degenerate ground-state results in a zero-bias Kondo
peak (see inset) 32 . Interestingly, this degeneracy also forms the qubit proposed in Ref. 1. For the
B ⊥ -dependence the one- and three-electron cases remain identical and Fig. 3i exhibits again the
SOI-induced avoided crossings.
7

Gate-dependent spin-orbit coupling
Having established the presence of SOI in the general many-electron disordered quantum dot, we
now focus on the dependence of ∆ SO on the quantum dot occupation. To this end, we have repeated
the spectroscopy of Fig. 3 for a large number of CNT quartets and in each case extracted
∆ SO by fitting to the single-particle model (all underlying data are presented in the Supplementary
Information). Figure 4a shows the result as a function of V g . An overall decrease of ∆ SO is observed
as electrons are added to the conduction band and, interestingly, a negative value is found
in the valence band (i.e. SOI favoring anti-parallel rather than parallel spin and orbital angular
momentum, thus effectively interchanging the one and three-electron spectra).
The magnitude of ∆ SO is given by the spin-orbit splitting of the underlying graphene band
structure as we will now discuss.
For flat graphene this splitting is very weak (∆ graphene
SO
1 µeV) 12 as it is second-order in the already weak atomic SOI of carbon ∆ C SO ∼ 8 meV. In
nanotubes, however, the curvature induces a coupling between the π- and σ-bands and generates
a curvature-induced spin-orbit splitting, which is first order in the atomic SOI and thus greatly
enhances ∆ SO . Around a Dirac-point of the Brillouin zone (e.g. K) the graphene band structure
appears as on Fig. 4b 11–14 : The spin-up and spin-down Dirac cones are split by SOI both in energy
and along k ⊥ , the momentum in the circumferential direction of the CNT. The schematic also
highlights the CNT band structure (Fig. 4a upper inset) obtained by imposing periodic boundary
conditions on k ⊥ . In a finite-length CNT quantum dot also the wavevector along the nanotube axis,
k ‖ , is quantized, and letting ɛ N denote the energy of the N th longitudinal mode the effective SOI
for a small-gap CNT becomes
∆ SO,± = E↑ K − E↓ K (1)
√
√
= 2∆ 0 SO ± (∆ g + ∆ 1 SO) 2 + ɛ 2 N ∓ (∆ g − ∆ 1 SO) 2 + ɛ 2 N.
Here the upper(lower) sign refers to the conduction(valence) band, ∆ g is the curvature induced
energy gap 33, 34 , and the two terms ∆ 0 SO and ∆ 1 SO are the band structure spin-orbit parameters due to
8
∼

curvature 11–14 . The separate contributions of the two terms are illustrated in the lower inset to Fig.
4a. ∆ 1 SO was found already in the work of Ando 10 and accounts for the k ⊥ -separation of the Diraccones
in Fig. 4b. For the CNT band structure this term acts equivalently to an Aharonov-Bohm flux
from a parallel spin-dependent magnetic field, which changes the quantization conditions in the k ⊥ -
direction. Characteristically, its contribution to ∆ SO decreases with the number of electrons in the
dot (ɛ N ) and reverses sign for the valence band. This contrasts the ɛ N -independent contribution
from the recently predicted ∆ 0 SO-term 12–14 , which acts as an effective valley-dependent Zeeman
term and accounts for the energy splitting of the Dirac-cones in Fig. 4b. For the nanotube studied
in Kuemmeth et al. 8 ∆ SO has the same sign for electrons and holes, i.e., |∆ 0 SO| > |∆ 1 SO|. In our
case, the negative values measured in the valence band demonstrate the opposite limit |∆ 0 SO| <
|∆ 1 SO|. Thus, the measured gate-dependence of ∆ SO agrees with the spin-orbit splitting of the
graphene Dirac spectrum caused by the curvature of the nanotube, and fitting to Eq. 1 (Fig. 4a,
dashed line) yields ∆ 0 SO = 10 ± 10 µeV and ∆ 1 SO = 220 ± 25 µeV 35 . Band structure models
relate these parameters to the structure of the nanotube 12–14 : ∆ 0 SO = λ 0 ∆ C SO∆ g D and ∆ 1 SO =
λ 1 ∆ C SO/D, where D is the nanotube diameter and λ 1,2 constants that depend on the CNT class
(semiconducting, small-band-gap). Typically CVD-grown single wall CNTs have diameters in
the range 1-3 nm, while obtaining D from the measured values of g 36 orb gives D ≈ 6 nm. Thus
taking D = 1-6 nm we estimate λ 1 = 0.03-0.17 nm and λ 0 = 0.7-4 · 10 −6 (nm · meV) −1 . While
λ 1 is consistent with the value quoted in the theoretical literature (0.095 nm) 14 , the calculated λ 0
value −4·10 −3 (nm · meV) −1 does not match the experiment and similar deviations appear 14 when
comparing the theory to the SOI values measured in Ref. 8. The origin of this discrepancy remains
unknown, and further work on SOI in nanotubes with known chirality is needed to make further
progress.
9

Methods
The devices are made on a highly doped Silicon wafer terminated by 500 nm of SiO 2 . Alignment
marks (Cr, 70 nm) are defined by electron beam lithography prior to deposition of catalyst
islands made of Iron nitrate (Fe(NO 3 ) 3 ), Molybdenum acetate and Alumina support particles 37 .
The sample is then transferred to a furnace, where single wall carbon nanotubes are grown by
chemical vapor deposition at 850-900 ◦ C in an atmosphere of hydrogen, argon and methane gases.
Pairs of electrodes consisting of Au/Pd (40/10 nm) spaced by 400 nm are fabricated alongside
the catalyst islands by standard electron beam lithography techniques.
Finally, bonding pads
(Au/Cr 150/10 nm) are made by optical lithography and the devices are screened by room- and
low-temperature measurements.
We measured the sample in an Oxford dilution refrigerator fitted with an Attocube ANRv51
piezo rotator which allows high precision in-plane rotation of the sample in large magnetic fields.
The rotator provides resistive feedback of the actual position measured by lock-in techniques. For
electrical filtering, room-temperature π-filters and low-temperature Thermocoax are used. The
base temperature of the modified refrigerator is around 100 mK. The CNT measurement setup
consists of a National Instrument digital to analog card, custom made optically coupled amplifiers,
a DL Instruments 1211 current to voltage amplifier and a Princeton Applied Research 5210 Lockin
amplifier. Standard dc and lock-in techniques have been used to measure current and differential
conductance dI/dV sd while d 2 I/dV 2
sd is obtained numerically.
10

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29. The states are not the conventional spin singlets and triplets as the they are modified by SOI
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˜T − and ˜S 2
↔ ˜T + avoided crossings occur simultaneously and as ˜T − to ˜T + excitations are
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Acknowledgements
We thank P. E. Lindelof, J. Mygind, H.I. Jørgensen, C.M. Marcus, and F. Kuemmeth for discussions
and experimental support. T.S.J. acknowledges the Carlsberg Foundation and Lundbeck
Foundation for financial support. K.G.R. K.F. J.N. acknowledges The Danish Research Council
and University of Copenhagen Center of Excellence.
Author contributions
T.S.J. and K.G.R. performed the measurements, analyzed the data and wrote the paper. T.S.J.
designed the rotating sample stage. K.G.R. made the sample. K.M., T.F. and J.N. participated in
14

discussions and writing the paper. J.P. and K.F. developed the theory and guided the experiment.
15

Figure 1
Four-fold periodic nanotube spectrum. a, Schematic illustration of the device
and setup. CNT quantum dots are measured at T = 100 mK in a standard two-terminal
configuration in a cryostat modified to enable measurements in a high magnetic field at
arbitrary in-plane angles θ to the CNT axis. b, Typical measurement of the differential
conductance dI/dV sd vs. source-drain bias V sd and gate voltage V g for a multi-electron
CNT quantum dot. c, Addition energy as a function of V g . In b and c the characteristic
filling of four-electron shells is clearly seen.
Figure 2
Role of spin-orbit interaction and disorder for the nanotube energy spectrum.
Calculated single-particle energy spectrum as a function of magnetic field applied perpendicular
(B ⊥ ) and parallel (B ‖ ) to the CNT axis in the limiting cases of neither SOI nor
disorder a, disorder alone b, SOI alone c, and the two combined ∆ KK ′ > ∆ SO > 0 d.
Depending on the CNT type, electron filling and degree of disorder, all four situations can
occur.
Figure 3
Spin-orbit interaction in a disordered multi-electron nanotube quantum dot.
a, Measurement of dI/dV sd vs. V sd and V g corresponding to the consecutive addition of
four electrons to an empty shell (indicated on Fig. 1b). A strong tunnel coupling results
in significant cotunneling which is evident as horizontal lines truncating the diamonds
(arrows). The black trace shows a cut along the dashed line. b, Schematic illustration
of the relevant inelastic cotunneling processes. c, Traces along the dashed line in a for
various B ‖ (red: B = 0, scale-bar: 0.1e 2 /h). d-f, The second derivative d 2 I/dV 2
sd along the
center of the N 0 + 1, N 0 + 2 and N 0 + 3 diamonds, respectively, as a function of a parallel
magnetic field. Peaks/dips appear at inflection points of the differential conductance and
thus correspond to the energy difference between ground and excited states. In f the inset
shows dI/dV sd vs. −0.3 < V sd < 0.3 mV and B ‖ = 0; 0.55; 1.1; 1.65 T (arrows) illustrating
16

the splitting and SOI-induced reappearance of a zero-bias Kondo resonance. g-i, As d-f
but measured as a function of B ⊥ . The effective spin-orbit coupling appears directly as
the avoided crossings indicated by ∆ SO . In d-i the black lines results from the singleparticle
model with parameters ∆ SO = 0.15 meV, ∆ KK ′
= 0.45 meV, and g orb = 11.4. The
dashed lines in e,h correspond to the excitations to the two-electron singlet-like ˜S 2 state
which cannot be reached by promoting a single electron from the ground state ( ˜S 0 ) and
therefore expected to be absent in the measurement.
Figure 4
Tuning ∆ SO in accordance with the curvature-induced spin-orbit splitting of the
nanotube Dirac-spectrum. a, Measured effective spin-orbit coupling strength as a function
of V g extracted from spectroscopy measurements like in Fig. 3, repeated for multiple
shells. The dashed line is a fit to the theory. Lower inset: Expected dependence of ∆ SO
on ɛ N highlighting the two SOI-contributions ∆ 0 SO and ∆ 1 SO. b, Graphene dispersion-cones
around one K-point of the graphene Brillouin zone. Due to SOI the spin-up (blue) and
spin-down (red) Dirac cones are split in both the vertical (E) and k ⊥ -direction. The cut
shows the resulting CNT band structure also shown in the upper inset in a.
17

Figure 1
a
b
c
V sd (mV)
E add (meV)
2
0
-2
8
3
Fig. 3
-8
4N0~180
+24
-4
+4 +12 +16 +20
+8
+28
5.5 V g (V)
6.5
+32
1
0
dI/dV sd (e 2 /h)

Figure 2
a
∆ KK' = 0
∆ SO = 0
E K' b ∆ KK' > 0
∆ SO = 0
K'
E
A
A
∆ KK'
K'
K
K
K
B
B
B
B ||
B
B ||
c
∆ KK' = 0
∆ SO > 0
E
∆ SO
K'
K'
K
d
∆ KK' > ∆ SO
∆ SO > 0
E
2 2
∆ KK' + ∆ SO
∆ β
SO
δ
γ
K
α
B
B ||
B
B ||

Figure 3
a
V sd (mV)
2
0
-2
3
dI/dV sd (e 2 /h)
0 0.75
4N 0
+1 +2 +3
5.6 V g (V) 5.7
b
B
∆ SO
α
B || = -3T
B ||
-3 3
2∆ SO
d 4N 0 + 1 e 4N 0 +2 f
4N 0 +3
E
δ
β γ
c
V sd (mV)
B || = 3T
1
V sd (mV)
0
0
V sd (mV)
-3
-3 B || (T)
3 -3 B || (T) 3
1.5
g
h
~
S 2
V sd (mV)
∆
0
SO
~
T +
~
T 0
~
T -
~
S 1
-3 B || (T)
3
i
∆ SO
-1
35
-27
d 2 2
I/dV sd (mS/V)
-1.5
-7 B (T)
7 -7 B (T)
7 -7 B (T)
7

Figure 4
a
0.3
E
Fig 3
b
E
∆ SO
∆ SO (meV)
0.0
k ||
∆ SO
k
k ||
Band gap
2∆ 0
2∆ 1
-0.3
-4 0 4 8
V g (V)
ε n

Supplementary Information
Gate-dependent spin-orbit coupling in
multi-electron carbon nanotubes
T. S. Jespersen, K. Grove-Rasmussen, J. Paaske,
K. Muraki, T. Fujisawa, J. Nygård, and K. Flensberg
Abstract
We here present the theory and additional experimental data supporting
the conclusions in the article. First, the carbon nanotube dispersion
relation including spin-orbit interaction is reviewed starting from a modified
Dirac Hamiltonian, and the single-particle model used in the fitting
procedure is explained. Second, tunneling spectroscopy data and the corresponding
analysis used to extract the gate-dependent spin-orbit coupling
strength are presented.
Contents
1 Carbon nanotube quantum dot with spin-orbit interaction and
iso-spin mixing 2
1.1 The spectrum for parallel magnetic field . . . . . . . . . . . . . . 3
1.2 Including disorder . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Two-electron spectrum . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Supplementary data 4
2.1 Conduction band . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Sequential tunneling spectroscopy . . . . . . . . . . . . . 5
2.1.2 Cotunneling spectroscopy . . . . . . . . . . . . . . . . . . 7
2.2 Valence band . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
i

1 Carbon nanotube quantum dot with spin-orbit
interaction and iso-spin mixing
In this section, we discuss the theory for the spectrum of a carbon nanotube
quantum dot and how it is modified by spin-orbit coupling and iso-spin (valley)
mixing. The theory is used to fit the spectroscopic data and to extract the
values of the spin-orbit coupling and KK ′ iso-spin mixing terms. Starting from
a clean, disorder-free nanotube (i.e. no valley mixing), the Hamiltonian near
the K and K ′ points is [1, 2]
H 0 = v ( τp x σ 1 + p y σ 2
)
+ ∆g σ 1 + τs y σ 1 ∆ 1 SO + τs y ∆ 0 SO + V (y), (1)
where σ i , i = 1, 2, 3 are the Pauli matrices in the A-B graphene sub-lattice
space, v is the Fermi velocity, and τ = ±1 the iso-spin index. Here p y is the
momentum along the tube (coordinate y), while p x is in the circumferential
direction (coordinate x). The spin is described by the Pauli matrix s y with
eigenvalues s = ±1 for eigenstates with spins aligned along the nanotube axis,
and V (y) is the confining potential along the tube. The curvature induced mass
term ∆ g depends on the diameter D and the chiral angle η (defined to be 0
for zigzag tube), see below. The spin-orbit interaction part of the Hamiltonian
has two types of curvature spin-orbit terms: one diagonal and one non-diagonal
in A-B space. The strength of these are ∆ 0 SO and ∆1 SO , respectively. The
curvature induced spin-orbit interaction is proportional to 1/D and the intraatomic
spin-orbit interaction V so ≃ 6-8 meV [2, 3], while the diagonal part also
depends on the chirality of the tube. Bandstructure calculations [2] give that
∆ 1 SO = α 1
V so
D , α 1 ≈ 0.095 nm (2)
∆ 0 SO = α 2
V so
D cos (3η) , α 2 ≈ −0.090 nm (3)
∆ g = β D 2 cos (3η) , β ≈ 24 meV· nm2 (4)
These values are results of a tight-binding calculation, with intra atomic spinorbit
interactions only, and should therefore be taken with some caution. In
our experiment, we cannot determine D and η independently, however, using
an estimate for the diameter, one can compare with the measured value of ∆ 1 SO
and the ratio ∆ 0 SO /∆ g as is done in the article.
The eigenstates of H 0 are of the form
ψ τ = e iKτ ·r e ik xx ϕ τs (y). (5)
An external magnetic field B causing an Aharonov-Bohm flux Φ through a
cross-section of the nanotube shifts the wave number by k Φ = 2Φ/DΦ 0 =
eBD/4, with Φ 0 = h/e. By imposing periodic boundary conditions along
the circumferential direction the wave vector in the x-direction becomes k x =
k Φ + 2ν/3D, where ν is 0 or ±1 depending on the type of tube. For small
bandgap tubes, likely to be the situation in the experiment, one has ν = 0.
ii

The longitudinal part of the envelope wave function ϕ τs (y) depends on the
confining potential. Let us assume that there is a large region in the middle of
the dot with a constant potential. In this region the wavefunction is a superposition
of the two states ϕ τs
± = A ± exp(±iky τs y), ky
τs > 0 [4]. The constants
A ± are determined by the boundary conditions at the ends of dot y = 0 and
L. For a sharp termination of the potential, we have A + ≈ −A − and hence
ky τs ≈ Nπ/L, N = 1, 2, 3, ...
Choosing the spin quantization axis to be along the tube axis the Hamiltonian
H 0 is then written as
(
0 v(τk
H 0 =
x − ik τs
v(τk x + iky τs ) 0
and the eigenenergies are
E τs = sτ∆ 0 SO ±
y )
) (
+
sτ∆ 0 SO
sτ∆ 1 SO + ∆ g
sτ∆ 1 SO + ∆ g
sτ∆ 0 SO
)
,
(6)
√
(∆ g + τ∆ Φ + τs∆ 1 SO )2 + ϵ N 2
, (7)
with ϵ N = vNπ/L and the Aharonov-Bohm energy given as
∆ Φ = vk Φ = evDB
4
= mω cvD
. (8)
4
Finally, in addition to the orbital coupling, the magnetic field also couples to
the spin degree of freedom by the Zeeman term
H Z = − 1 2 µ sB · s, (9)
where µ s = −gµ B and s i are the Pauli matrices in spin space. Using the above
solutions in absence of H Z as basis states, the Hamiltonian (for each isospin) is
(
E1,τ − 1
H 0 + H Z =
2 µ 1
sB ∥ 2 µ )
sB ⊥
1
2 µ sB ⊥ E −1,τ + 1 2 µ , (10)
sB ∥
which is readily diagonalized.
1.1 The spectrum for parallel magnetic field
For the magnetic field parallel to the tube axis the energy of the four states
in the quantum dot is to linear order in B given by
where
and
∆ SO = 2∆ 0 SO ±
E τs = E 0 + τs ∆ SO
2
+ (τg orb,τs + 1 2 gs)µ BB ∥ , (11)
√
√
(∆ g + ∆ 1 SO )2 + ϵ 2 N ∓ (∆ g − ∆ 1 SO )2 + ϵ 2 N , (12)
g orb,sτ = evD
( )
∆g + τs∆ 1 SO
√
≈ evD ∆
√ g
. (13)
2µ B
(∆ g + τs∆ 1 SO )2 + ϵ
2 2µ B N
∆ 2 g + ϵ
2 N
Here the upper(lower) sign corresponds to the conduction (valence) band.
iii

1.2 Including disorder
Since isospin mixing is an experimentally very important effect, this must
be included in the modelling of the quantum dot spectrum to correctly describe
the data. The microscopic nature of the sources to KK ′ mixing is in general
unknown, but several factors can contribute to the mixing, e.g., couplings to the
substrate which breaks rotational invariance, a random distribution of defects
in the tube, or end effects for non-adiabatic confining potentials. In our basis
for the clean wire, these effects result in matrix elements ∆ KK ′ = ⟨sτ|H KK ′|s¯τ⟩
(notice that spin is conserved, assuming non-magnetic symmetry breaking only).
Therefore, with KK ′ mixing included, the Hamiltonian within a single nanotube
shell in a magnetic field having an angle θ relative to the tube axis is (using the
basis (K ↑ y , K ′ ↓ y , K ↓ y , K ′ ↑ y ))
⎛
H = E 0 + ⎜
⎝
+ 1 2 gµ BB ⎜
⎝
∆ KK ′ 0 0 E −1,1
⎛
cos θ 0 sin θ 0
⎞
⎞
E 1,1 0 0 ∆ KK ′
0 E −1,−1 ∆ KK ′ 0
⎟
0 ∆ KK ′ E 1,−1 0 ⎠
0 − cos θ 0 sin θ
sin θ − cos θ 0
0 sin θ 0 cos θ
⎟
⎠ . (14)
The energies in the diagonal E τ,s are taken as the expressions (11) and (13).
After diagonalizing H we can thus fit the experimentally observed spectrum,
using ∆ SO , ∆ KK ′ and g orb as fitting parameters.
1.3 Two-electron spectrum
In absence of exchange coupling the two-electron spectrum is simply given
by filling up the one-electron states found above. We classify these according
to their spectroscopic nature, i.e., how they split in a magnetic field and denote
them therefore as singlet- and triplet-like states. They are not singlets and
triplets in the usual sense of spin singlets and triplets.
2 Supplementary data
A central point of our work is that of Fig. 4a in the article, which shows the
gate voltage-dependence of the effective spin-orbit coupling strength ∆ SO . Each
data point in the figure is obtained by fitting the single-particle energy spectrum
to magnetic field dependent tunneling spectroscopy measurements. This section
presents the analysis of the underlying data. We start by discussing results from
the conduction band where both sequential- and cotunneling spectroscopy have
been performed (Sec. 2.1). Subsequently, we present data from the valence band
where a negative value of ∆ SO is found by cotunneling spectroscopy (Sec. 2.2).
iv

2.1 Conduction band
The spin-orbit parameter ∆ SO has been extracted from 14 shells in the conduction
band labeled A-N with shell A(N) corresponding to the lowest(highest)
positive gate voltage. Shell E at V g ≈ 5.6 − 5.7 V corresponds to the quartet for
which the cotunneling spectroscopy was analyzed in Fig. 3 of the article. For
lower gate voltages, the lead-nanotube Schottky barrier is larger, significantly
decreasing the cotunneling current and for shells A and B cotunneling spectroscopy
is no-longer feasible. Instead we show in section 2.1.1 how sequential
tunneling spectroscopy also yield the effective spin-orbit coupling strength in
complete accordance with the single-particle model. The additional 12 shells
(C-N) in the conduction band allow cotunnelling spectroscopy as presented in
section 2.1.2.
2.1.1 Sequential tunneling spectroscopy
The electron filling of quantum states for smaller gate voltages than shown
in Fig. 1b in the article is analyzed in Fig. S1a-b. They show the stability diagram
(S1b) with relatively weak lead-dot tunnel-coupling and the corresponding
addition energies (S1a) extracted from the widths of the diamonds (and scaled
by the gate coupling factor). The large addition energy for every fourth electron
(red data points in Fig. S1a) clearly shows the regular four-electron shell filling,
which persists over many electronic shells. A zoom of the two shells labeled
A and B is shown in Fig. S1c. Due to the weak coupling, gate-independent
cotunneling steps are not observed. Instead, lines parallel to the diamond edges
representing sequential transport through excited states of the charge-state at
which they terminate are well resolved and can thus be used for spectroscopy.
The linewidth ≈ 200-300 µeV is determined by the tunnel coupling of the quantum
dot to the leads and sets the energy resolution of the measurement. Figure
S1d shows the transconductance dI/dV g vs. V sd and energy (V g ) around the
4N 1 ∼ 120 electron charge state with N 1 completely filled four-electron shells
as marked in Fig. S1c. The Coulomb diamonds are clearly observed and at the
upper (lower) crossing point one electron is added (removed) to the next empty
(last filled) quartet. Note, that by measuring transconductance (rather than
dI/dV sd ), the lines acquire an additional sign reflecting levels entering(positive
dI/dV g ) or leaving(negative) the available bias window [5]. This feature eases
the identification of the relevant levels in Fig. S1e, where traces along the black
line at fixed bias V sd = 4 mV are measured while varying B ∥ . Consider first
the spectrum of the 4N 1 + 1 charge state (shell B). Increasing B ∥ the doubly
degenerate ground-state splits into two states (α,β) while the excited states
(γ, δ) steeply separates from the ground-state but are only discernably split at
B ∥ ≈ 3 T as emphasized in the inset. As discussed in the article, this asymmetry
is a direct consequence of SOI (see Fig. 2d in the article) and the measurements
are accurately reproduced by the calculation overlaid on the B ∥ > 0 part of Fig.
S1e with ∆ SO = 0.2 meV, ∆ KK ′ = 0.58 meV, and g orb = 7.8. Importantly, even
though the SOI is masked in the low field regime by the dominating disorder
v

E add
(meV)
V sd (mV) V sd (mV)
Energy (meV)
15
10
6
0
-6
10
5
0
-5
-10
10
0
a
-12
b
c
-8
4.0 4.1 V gate (V) 4.2 4.3
4N 1 + 1
(α,β)
(γ,δ)
4N 1
4.0
-4
d
4N 1 ~120
+4
+8
+12
+16
A
α
w
δ
α β γ δ
w
γ
β
e
4N 1
+20
+24
4.5 B 5.0
f
+28
g
0.1
0
0.05
0
dI/dV sd (e 2 /h)
dI/dV sd (e 2 /h)
(α,β)
(γ,δ)
4N-1e
-10 1 - 1 θ = 0 θ= π/2
-5 0 5 -5 0 5 -6 0 6
V sd (mV)
B || (T)
B (T)
B = 3.25 T
0 -π/2 π/2
θ (rad.)
3
-3
dI/dV gate (mS)
Figure S1: a, Addition energy as a function of V gate extracted from the stability diagram
in b revealing consecutive four-electron shell filling for a large number of shells. c, Charge
stability diagram of the two shells (A & B) in which the sequential tunneling analysis is made
for 3 and 1 electrons in the two shells, respectively. d, Transconductance dI/dV g vs. V g and
V sd around a full electron shell 4N 1 ≈ 120. Upper(lower) arrow indicates the first excited
state of the first(last) electron in an otherwise empty(filled) shell. e, Transconductance along
the dashed line in d at V sd = 4 mV measured as a function of parallel magnetic field. Inset
shows a cut along the vertical line emphasizing the asymmetric splitting of the two doublets;
(α, β) vs. (γ, δ) which is the consequence of SOI. Labels α, β, γ, δ refer to Fig. 2d in the
article. f, As d but applying the magnetic field perpendicular to the axis; the doublets split
equally. g, Measurements in a constant magnetic field B = 3.25 T while varying the angle
θ between B and the CNT axis. In e-g the solid lines are a single fit to the theory with a
total of only four parameters ∆ SO = 0.2 meV, g orb = 7.8 and ∆ KK ′ = 0.58(0.75) meV for
the 4N 1 + 1(4N 1 − 1) charge state. The theory has been overlaid only for B > 0 not to mask
the data. Dashed lines in e and g are excitations belonging to the 4N 1 + 1 charge state also
anticipated by the model with a level-spacing of 6 meV.
vi

V sd (mV)
V sd (mV)
V sd (mV)
4
2
0
-2
-4
5.25
6
b
3
0
-3
-6 0
5.8 6.0 6.2 6.4 6.6 6.8 7.0 7.2 7.4 7.5
6
3
0
-3
a
c
-6
0
7.4 7.6 7.8 8.0 8.2 8.4
Fig. 2, Shell E
0
5.3 5.4 5.5 5.6
5.7
5.8 5.9
L2
G2
M2
C1 D1 D2 E1 E2 E3 F1 F2 F3
H2
I2
V gate (V)
dI/dV sd
(e 2 /h)
2
4
2
0
-2
J2
d
K2
N2
dI/dV sd
(e 2 /h)
1
dI/dV sd
(e 2 /h)
2
dI/dV sd
(e 2 /h)
1
-4 0
9.2 9.3 9.4
Figure S2: a-d, Charge stability diagrams at higher gate voltages with the CNT quantum
dot being in the cotunneling regime. Arrows point to charge states where cotunneling spectroscopy
have been measured (see Figs. S3 and S4). The shell labeled E is analyzed in details
in Fig. 3 of the article.
and the limited spectroscopic resolution, it clearly stands out in the high field
regime as the asymmetry of the splitting of the two doublets (α, β) vs. (γ, δ).
The corresponding evolution for the 4N 1 −1 charge state is similar (shell A), and
is fitted with the same values for ∆ SO and g orb and taking ∆ KK ′ = 0.75 meV.
Figure S1f shows the corresponding measurement in a perpendicular field
B ⊥ . The level structure is in complete agreement with the model which does
not contain any free parameters. Since ∆ SO is below the spectroscopic resolution
the avoided crossing cannot be clearly discerned in the measurement
(unlike the case for the cotunneling measurements in Fig. 3 in the article). For
completeness we show in Fig. S1g the spectrum for fixed B = 3.25 T while continuously
varying the angle θ between the nanotube and the field. Here the
presence of SOI is responsible for the angle dependence of the internal doublet
splittings (arrows), and again the model is in near-perfect agreement with the
data without any free parameters.
2.1.2 Cotunneling spectroscopy
For higher gate voltages, as already mentioned, the tunnel coupling to the
electrodes increases, and allows for inelastic cotunneling spectroscopy. As shown
in Fig. 3 in the article, the spin-orbit coupling can in this case be directly readoff
by avoided level crossings in perpendicular magnetic field for 1, 2, and 3
vii

electrons. Moreover, the spin-orbit coupling breaks the intra-shell electron-hole
symmetry, which can be probed by comparing the 1 and 3 electron cases in
parallel field. Figures S2a-d show charge stability diagrams in the cotunneling
regime, where arrows point to charge states for which field-dependent cotunneling
spectroscopy has been performed. In Fig. S2b-d, the level broadening
due to electrode-dot coupling is even stronger and the spin-orbit coupling is
extracted from the avoided crossing (of magnitude 2∆ SO ) between the twoelectron
singlet-like ˜S 0 and triplet-like ˜T − states in perpendicular field.
Figure S3 shows cotunneling spectroscopy plots (d 2 I/dVsd 2 versus gate and
magnetic field) of shells C, D and F in the cotunneling regime marked in Fig.
S2a. The overall features reproduce Fig. 3 in the article and show excellent
agreement with the model (black lines). Parameters used in the fits are all
summarized in Table S1. We now comment on the details observed in each
shell, and on how the spin-orbit strength and other parameters are found.
Shell C is analyzed in Figs. S3a-b, which show cotunneling spectroscopy for 1
electron in perpendicular and parallel magnetic fields. The spin-orbit coupling
is directly estimated in Fig. S3a from the avoided crossing at B = ±4.5 T
yielding ∆ SO ≃ 140 µeV. Furthermore, the orbital coupling ∆ KK ′ ≃ 495 µeV
and g orb ≃ 5.7 are chosen to produce the theoretical predicted excitations, which
perfectly match the observed threshold of inelastic cotunneling (peaks and dips).
Measurements of shell D is displayed in Figs. S3c-g which show a different
behavior than shell C, because the orbital coupling is smaller than the spinorbit
coupling. The spin-orbit coupling for this shell is extracted in Fig. S3d
from the avoided crossing between ˜S 0 and ˜T − states in case of the two-electron
cotunneling spectroscopy in perpendicular magnetic field. The theory matches
the experimental data except in Fig. S3d, where some small deviations are
observed at high fields. All other figures show good agreement between theory
and experiment, most remarkably in Fig. S3g, where the sample is rotated to an
angle of 22.5 degrees between the nanotube axis and the magnetic field. Note,
that the theory lines in Fig. S3e-g at negative biases are only plotted for low
fields to clearly reveal the underlying data.
Shell F is presented in Figs. S3h-m, which show the magnetic field dependence
in parallel and perpendicular magnetic fields for fillings 1-3 (as in Fig.
3 in the article). The overall conductance in this shell is higher than for the
shells analyzed above. Moreover, the zero-field threshold of inelastic cotunneling
is seen to vary with filling, which is not fully accounted for in our model.
The clear avoided crossing for 2 electrons in perpendicular magnetic field (Fig.
S3i), and the breaking of electron-hole symmetry in parallel magnetic field for
electron fillings 1 and 3 (Figs. S3k,m) unambiguously reveal the spin-orbit interaction.
Interestingly, for 2 electrons, four (not three as in Fig. 3 in the article)
excitation lines are seen to emerge from the zero field peak at V sd = 0.25 meV
in Fig. S3i. Such behavior is indeed expected in the presence of finite exchange
interaction (J > 0), which splits the ˜S 1 and ˜T states. The deviation between
experiment and theory at large B || in Fig. S3l may be attributed to avoided
crossings with states from the next shell. The fitting parameters are chosen to
mainly match the inelastic cotunneling thresholds of Figs. S3h (not S3j) and
viii

V sd (mV)
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
a
C1
-6 -4 -2 0 2 4
6
2
b C1 ||
1
0
-1
-2
-3 -2 -1 0 1 2
3
2
-2
dI 2 /dVsd 2 (mS/V)
V sd (mV)
V sd (mV)
V sd (mV)
V sd (mV)
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
3
2
1
0
-1
-2
-3
c
D1
0 1 2 3
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
0 -1 -2 -3 -4 -5 -6 -7 0 1 2 3 4 5 6 7
3
2
e D1 || f D2 g
2
||
D2 22.5deg
1
1
0
0
-1
-2
-3
d
0 1 2 3
1.5
2
h F1 i F2
1.5 j F3
1.0
1
1.0
0.5
0.5
0.0
0
0.0
-0.5
-0.5
-1
-1.0
-1.0
-1.5
-2
-1.5
0 1 2 3 4 5 6 7 -6 -4 -2 0 2 4 6 0 1 2 3 4 5 6 7
3
3
2
20
2 k F1 ||
2 l F2 || m F3 ||
1
1
1
0
0
0
-1
-1
-1
-2
-2
-3
-3
0 1 2 3 -3 -2 -1 0 1 2 3 0 1 2 3
B (T) B (T) B (T)
D2
-1
-2
-2
0 -1 -2 -3 -4 -5 -6 -7
dI 2 /dVsd 2 (mS/V)
5
-5
dI 2 /dVsd 2 (mS/V)
-20
Figure S3: Cotunneling spectroscopy d 2 I/dVsd 2 versus bias and magnetic field for shell C
(a,b), D (c-g) and F (h-m). The labels indicate shell and electron filling in correspondence
with Fig. S2a as well as orientation of the field relative to the carbon nanotube axis. Data are
acquired at gate voltages corresponding to the center of the respective Coulomb diamond. The
lines display the excitation spectrum, (i.e., threshold of inelastic cotunneling) predicted by
the single- and two-particle model having four independent parameters, the orbital coupling
(disorder) parameter ∆ KK ′, the spin-orbit coupling ∆ SO , the orbital g-factor g orb and the
exchange interaction J. Parameters are summarized in Table S1.
ix

V sd (mV) V sd (mV)
V sd (mV)
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
1.5
g
1.0
0.5
0.0
-0.5
a G2 b H2 c
d J2 e K2 f
M2
-1.0
-1.5
-20 20
-100
0 1 2 3 4 5 6 70 1 2 3 4 5 6 7
B (T) B (T)
Λ
h
N2
2∆ SO
100
dI 2 /dV sd
2 (mS/V)
I2
L2
0 1 2 3 4 5 6 7
B (T)
Figure S4: a-h, Cotunneling spectroscopy dI 2 /dVsd 2 versus bias and perpendicular magnetic
field at half filling for shells G-N (see stability diagrams in Fig. S2b-d). The spin-orbit coupling
∆ SO is estimated from the avoided crossing between the singlet-like ˜S 0 and the triplet-like
˜T − states as shown in Panel b. The zero field energy difference Λ between the ground and
excited Kramers doublets allows the calculation of the orbital coupling parameter ∆ KK ′ . The
lines represent the calculated spectrum of the two-particle model using the two parameters
∆ SO and ∆ KK ′ as well as a finite exchange J. The deviations from the model is attributed
to g-factor renormalization caused by Kondo correlation which are important in the strongly
coupled regime (see text). Note, that the color-scale bar for shell N is different than for shells
G-M.
S3i. A more detailed analysis of CNT two-electron physics including exchange,
disorder and spin-orbit interaction will be presented elsewhere.
For shells G-N (Fig. S2) the lead-dot coupling is even stronger, leading to
pronounced Kondo physics for 1 and 3 electron fillings and Kondo enhanced
inelastic cotunneling at half filling [6]. Figure S4 shows two-electron cotunneling
spectroscopy for perpendicular magnetic field. The orbital coupling ∆ KK ′ is
found from the zero-field splitting Λ between the two Kramers doublets (green
arrows in Panel b), and the spin-orbit coupling appears as the zero bias avoided
crossing between the ˜S and ˜T states (black arrows in Panel b). Given these
two parameters for each shell the two-electron excitation spectrum is calculated
as shown by the black lines. Interestingly, for shells H-M, the ˜S- ˜T avoided
crossings occur at somewhat higher magnetic fields than predicted. This is
consistent with correlation induced renormalization (lowering) of the g-factor
[7]. As the coupling is decreased in case of shell N, the correspondence between
the model and the experimental threshold of cotunneling is improved confirming
x

that the effect is related to the coupling and not the electron filling. The g-
factor renormalization is an interesting topic for detailed study and the good
understanding of the single-particle spectrum of single wall carbon nanotube
electronic shells constitutes an optimal starting point for further explorations.
xi

V sd (mV)
V sd (mV) V sd (mV)
V sd (mV)
V sd (mV)
4
2
0
-2
-4
1.5
1.0
0.5
0.0
-0.5
-1.0
1
0
-1
-2
-3
-3 -2 -1 0 1 2 3 0 1 2 3 -3 -2 -1 0 1 2
1.5
1.0
h W1 i W2 j W3
0.5
0.0
-0.5
-1.0
-1.5
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
3
k W1 l m
150
2
|| W2 || W3 ||
1
0
-1
-2
a
-2.8 -2.7
V gate (V)
-2.6 -2.5
-1.5
0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
3
2
e W1 2
|| f W2 || g W3 ||
-3
-3 -2 -1 0 1 2
B (T)
W1
W2
W3
b c d
W1 W2 W3
3 0 1 2
B (T)
3 -3 -2 -1 0 1 2
B (T)
3
3
dI 2 /dV sd
2 (mS/V)
0
dI 2 /dV sd
2 (mS/V)
-150
Figure S5: a, Stability diagram showing four-electron shell structure in the valence band,
where the arrows indicate electron filling 1-3 in shell W. We here consider electron filling in
the valence band shell instead of hole filling to make easier comparison to the measurements
analyzed in the conduction band. Measured dI/dV sd versus bias and perpendicular/parallel
(b-d)/(e-g) magnetic field for electron filling 1, 2 and 3. Numerically obtained d 2 I/dVsd
2
versus bias and perpendicular/parallel (h-j)/(k-m) magnetic field. The green arrows in e, k,
g, m point to the threshold of inelastic cotunneling involving the ground- and first excited
states. The blue arrows indicate the magnetic field for which this threshold is zero. This occurs
for B || ≃ 1 T and B || = 0 T for fillings 1 and 3, respectively, thereby revealing a negative spinorbit
coupling. The parameters used in the calculated spectrum (lines in (h-m)) are given in
Table S1.
xii

2.2 Valence band
For negative gate voltages transport is carried by holes in the valence band
of the small-band-gap semiconducting carbon nanotube. The coupling to the
electrodes is significantly stronger than for the conduction band and Kondo
physics dominates as seen in Fig. S5a. Cotunneling spectroscopy in perpendicular
and parallel magnetic field is measured for 1, 2 and 3 electrons in one of
the weakest coupled shell W (see arrows). For completeness, both differential
conductance dI/dV sd (S5b-g) and its derivative d 2 I/dVsd 2 (S5h-m) are shown.
Due to significant level broadening of the cotunneling features, the (avoided)
crossing in perpendicular magnetic field for two electrons results in a singlettriplet
Kondo resonance as seen in Figs. S5c,i [8], and the spin-orbit coupling
cannot be directly extracted as the case for weaker coupling (Fig. S4). Instead,
we focus on the parallel magnetic field dependence for electron filling 1 and 3.
In presence of orbital coupling (∆ KK ′ > 0) and positive spin-orbit coupling
(∆ SO > 0) as discussed in the article, the lines corresponding to the lowest excitation
cross zero bias only at B = 0 for electron filling 1, while a crossing also
occurs at finite parallel magnetic fields for filling 3 (see Fig. 3d,f in the article
and Fig. S3k,m). Similar behavior is seen for shell W in Fig. S5k, m where the
one and three electron behavior now is interchanged: the cotunneling threshold
corresponding to the lowest excitation (green arrows in Fig. S5e, k) cross at
finite fields for filling 1 (blue arrow). This shows that the spin-orbit coupling
∆ SO is negative thus favoring antiparallel rather than parallel orientation of the
orbital and spin angular momentum (see Fig. S6 for the effect of negative ∆ SO
on the level structure). The theoretically expected cotunneling thresholds are
indicated by the black lines in Fig. S4 with the values of the orbital coupling
and the spin-orbit coupling listed in Table S1. As discussed in the article, a
negative spin-orbit coupling is indeed consistent with the expectation from the
Dirac band structure, and our findings illustrate a different regime than previously
reported, which found a positive spin-orbit coupling for the (first) carriers
a b
∆ E
SO > 0 ∆ SO < 0
∆ KK' > ∆ SO
∆ KK' > |∆ SO |
B
B ||
B
E
B ||
Figure S6: Energy spectrum of quartet with positive (a) and negative (b) spin-orbit coupling
resulting in an interchange of the two Kramers doublets (red arrows).
xiii

V sd (mV)
0.75
0
-0.75
1.5
a Z1 Z3 Y1 Y3
B ||=0T
-3.30 -3.25 -3.20 -3.15 -3.10 -3.05
V gate (V)
b Y1 || c Y3 ||
dI/dV sd (e 2 /h)
1.1
0
V sd (mV)
0
-1.5
1.5
d Z1 || e Z3 ||
dI 2 /dV sd
2 (mS/V)
50
V sd (mV)
0
-50
V sd (mV)
-1.5
-3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3
f B || (T)
B || (T)
1.0
Z1 Z3 Y1 Y3 X1 X3
0.5
0
-0.5
~200µV
~290µV
0.8
B ||=2T
-1.0
-3.30 -3.25 -3.20 -3.15 -3.10 -3.05 -3.00 -2.95 0
V gate (V)
dI/dV sd (e 2 /h)
Figure S7: a, Stability diagram of shells Y and Z in the valence band from a different
cooldown of the same sample. b-e, Cotunneling spectroscopy d 2 I/dVsd 2 versus V sd and parallel
magnetic field of shell Y and Z showing the breaking of electron-hole symmetry reflecting a
negative spin-orbit coupling strength, i.e., the lowest excitation lines cross at respectively
finite and zero field for b,d and c,e. The black lines are the expected excitations given the
parameters in Table S1. (f) Stability diagram at B || = 2 T showing shells X, Y, and Z. The
spin-orbit interaction is revealed by a difference in splitting of the Kondo resonance for 1 and
3 electrons in a shell (yellow and blue arrows). A negative spin-orbit interaction of similar
strength is therefore inferred also for shell X.
in both the conduction and valence band [3].
The analysis is repeated for shells Y and Z at larger negative gate voltages
xiv

presented in Fig. S7a, where the asymmetry between the 1 and 3 electron behavior
is observed in Figs. S7b-e. By comparing to theory (black lines) we find
a spin-orbit coupling strength ∼ −75 µV for both shells. Finally, we show that
the spin-obit coupling also is observed in stability diagrams as an asymmetric
splitting of the 1 and 3 electron Kondo resonances in parallel magnetic fields.
This is illustrated in Fig. S7f showing a stability diagram of the three shells X,
Y and Z at B || = 2 T, where an asymmetric splitting of the Kramers doublets
for filling 1 and 3 is clearly revealed. The alternating behavior of small and large
splitting is highlighted by pairs of yellow and blue arrows. In the case of shell
Y the difference in splittings is ∼ 90 µV, consistent with the spin-orbit coupling
strength deduced above. We therefore conclude that also shell X has a negative
spin-orbit coupling in the same order as shell Y and Z.
References
[1] J. Jeong and H. Lee, Phys. Rev. B 80 (2009).
[2] W. Izumida, K. Sato, and R. Saito, J. Phys. Soc. Jpn. 78, 074707 (2009).
[3] F. Kuemmeth, et al., Nature 452, 448 (2008).
[4] D. V. Bulaev, B. Trauzettel, and D. Loss, Phys. Rev. B 77, 235301 (2008).
[5] L. Kouwenhoven, D. Austing, and S. Tarucha, Rep. Prog. Phys. 64, 701
(2001).
[6] J. Paaske, et al., Nature Physics 2, 460 (2006).
[7] A. C. Hewson, J. Bauer, and W. Koller, Phys. Rev. B 73, 045117 (2006).
[8] J. Nygård, D. H. Cobden, and P. E. Lindelof, Nature 408, 342 (2000).
xv

Shell # V gate (V) Λ(µeV) ∆ SO (µeV) ∆ KK ′(µeV) J(µeV) g orb
A 4.12 775 200 750 - 7.8
B 4.25 615 200 580 - 7.8
C 5.40 515 140 495 - 5.7
D 5.55 200 185 75 0 5.5
E (Fig. 3) 5.68 475 150 450 0 5.7
F 5.78 560 100 550 450 5.4
G 6.04 195 115 155 - -
H 6.30 315 65 310 - -
I 6.57 190 95 165 - -
J 7.09 245 125 210 100 -
K 7.36 390 60 385 - -
L 7.56 435 60 430 - -
M 7.69 410 60 405 - -
N 9.34 330 65 325 - -
W -2.67 225 -100 200 0 ≃ 4.4
X -2.98 - ≃ −75 - - -
Y -3.14 ≃ 605 ≃ −75 ≃ 600 - ≃ 5.8
Z -3.25 ≃ 705 ≃ −75 ≃ 700 - ≃ 5.8
Table S1: Parameters extracted for each measured shell in the conduction (A-N) and valence
(W-Z) band. Shells A and B are the only shells analyzed in the sequential tunneling regime
(Fig. S1), while C-F are in the weak cotunneling regime (Fig. S3 and Fig. 3). As the levelbroadening
increases even further (G-N), the spin-orbit coupling ∆ SO and the energy splitting
between the two Kramers doublets Λ is estimated at half filling by the zero-bias avoided
crossing at finite magnetic field and the zero-field inelastic cotunneling
√
threshold, respectively
(see Fig. S4b). The orbital coupling is related as ∆ KK ′ = Λ 2 − ∆ 2 SO , and in some cases
a ˜S 0 - ˜T splitting caused by exchange interaction J can be observed (J = 0 means that the
exchange interaction is smaller than the spectroscopic line width in the shell). In the valence
band, the spin-orbit coupling is observed by the breaking of electron-hole symmetry for 1 and
3 electrons in a parallel magnetic field (S5 and S7). The uncertainty of the spin-orbit coupling
strength is shown in Fig. 4 in the article. The dashes indicate that measurement have not
been made to extract the corresponding parameter. Note, that the decreasing values of g orb
versus gate voltage (filling) in the conduction band is consistent with Eq. (13). As expected
the orbital coupling ∆ KK ′ shows no gate voltage correlation.
xvi