Table Of ContentSpin-orbit interaction in chiral carbon nanotubes probed in pulsed magnetic fields
S. H. Jhang,1 M. Marganska,2 Y. Skourski,3 D. Preusche,1 B. Witkamp,4
M. Grifoni,2 H. van der Zant,4 J. Wosnitza,3 and C. Strunk1,∗
1Institute of Experimental and Applied Physics, University of Regensburg, 93040 Regensburg, Germany
2Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany
3Dresden High Magnetic Field Laboratory, Forschungszentrum Dresden-Rossendorf, 01314 Dresden, Germany
4Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
The magneto-conductance of an open carbon nanotube (CNT)-quantum wire was measured in
pulsed magnetic fields. At low temperatures we find a peculiar split magneto-conductance peak
closetothechargeneutralitypoint. Ouranalysisofthedatarevealsthatthissplittingisintimately
0
connected to the spin-orbit interaction and the tube chirality. Band structure calculations suggest
1
0 that the current in the peak regions is highly spin-polarized, which calls for application in future
2 CNT-based spintronic devices.
n
PACSnumbers: 73.63.Fg,75.47.-m,73.23.Ad,85.75.-d
a
J
6 A source of spin-polarized electrons is one of the im- (See EPAPS for further experimental details).
2
portantbuildingblocksofafuturespin-basedelectronics Figure 1a shows the magneto-conductance G(B ) of a
(cid:107)
[1]. Very high degrees of polarization can potentially be small-bandgapCNTdevicelocatedneartheCNP(diam-
]
l achievedbyexploitingspin-orbitinteraction(SOI)[2,3]. eterd∼1.5nm). At82K,theconductanceGofthetube
l
a Based on the low atomic number Z = 6 of carbon the initially increases to reach a maximum at B = 5.9 T,
h 0
spin-orbit interaction in carbon nanotubes (CNTs) was then it exponentially drops to zero at higher fields due
-
s mostly believed to be very weak, until a recent exper- to the Aharonov-Bohm (AB) effect [5, 9]. Interestingly,
e
iment [4] has demonstrated the effect of spin-orbit in- when the device was cooled down to 4.2 K, the conduc-
m
teraction in clean CNT quantum dots. Evidence for the tance maximum G at B was split into two distinct
max 0
.
t spin-orbitsplittinginsimplemagneto-conductance(MC) peaks at magnetic fields B = 3.1 T and B = 11.1 T.
a 1 2
measurements has not yet been reported. We note that these peaks are symmetric with respect to
m
InthisLetter,wepresentMCdataforthecomplemen- the conductance dip at B∗ ≈ 7 T, and G(B∗) is similar
- 0 0
d tary situation of an open CNT-quantum wire obtained in magnitude to G(B(cid:107) = 0). The key to the explanation
n in pulsed magnetic fields. In a parallel magnetic field of the data lies in the magnetic field dependence of the
o B , a small band-gap CNT evolves via a metallic state one-dimensional band structure.
c (cid:107)
into a semiconducting one, resulting in a typical peak AspecificCNTisuniquelylabeledbythechiralindices
[
in the MC [5, 6]. In one of our tubes, however, we ob- (n,m), which define the chiral angle θ and the quantized
1 served a splitting of this MC-peak into two peaks at low values of the transversal wavevector k⊥ [10]. The values
v
7 temperature. RecordingMC-tracesatdifferentVg shows of k⊥, combined with the graphene dispersion cones, de-
6 that the splitting vanishes when moving away from the termine the quasi-onedimensional band structure of the
7 charge neutrality point (CNP). A thorough comparison CNTs. AgivenCNTismetallicifthelinesofallowedk⊥
4 to band structure calculations reveals that the splitting crosstheDiracpointsK,K(cid:48);otherwiseitissemiconduct-
1. is explained by the SOI, which becomes strong for small ing. For nominally metallic CNTs (n−m = 3l, with l
0 tube diameters. An interesting implication of our analy- an integer), the dispersion relation E(k(cid:107)) near the Dirac
0 sis is the prediction of a highly spin-polarized current in points reads [4, 11–14]:
1
: the peak regions. (cid:113) (cid:16)g (cid:17)
v Theexperimentshavebeenperformedondevicesmade E(k(cid:107)) = ±¯hvF k(cid:107)2+k⊥2 + 2µBB(cid:107)+τεSO σ,
i
X of individual CNTs prepared on Si/SiO2/Si3N4 sub- k = k +k0 +k , (1)
⊥ AB ⊥ SO
strates. The heavily p-doped Si was used as a back
r
a gate and the thickness of the insulating layer was 350 where k is the wave vector parallel to the tube axis, v
(cid:107) F
nm. CNTs were grown by means of a chemical vapor theFermivelocity, gµ B σ beingtheZeemantermwith
2 B (cid:107)
deposition method [7] and Pd (50 nm) electrodes were σ =±1forspinparallel/antiparalleltothetubeaxis,and
defined on top of the tubes by e-beam lithography. In τ = ±1 for the K and K(cid:48) Dirac points. The transver-
order to exclude strain effects on the band structure [8], sal wave vector k contains three distinct contributions,
⊥
onlystraightandlong(∼50µm)CNTswereselectedfor which are discussed below.
devices and the distance between two Pd electrodes was The Aharonov-Bohm flux φ = B πd2/4 results in
AB (cid:107)
∼ 500 nm. The dc magneto-conductance was studied in a shift k = (2/d)(φ /φ ) of k , where φ = h/e is
AB AB 0 ⊥ 0
pulsed magnetic fields of up to 60 T, applied parallel to the flux quantum. Therefore, one can convert a metallic
the tubeaxis. The accuracy of the alignment was ∼±5◦ CNTintoasemiconductingone, orviceversa, bytuning
2
(a) 2.0 presence of an AB flux φSO ≈ 10−3φ0 [4, 11], and pro-
B 110000 duces a spin-orbit energy splitting ∆SO = 2h¯vF |kSO|.
0 82 K For a CNT with d ∼ 1 nm, φSO corresponds to (cid:39) 5T,
while φ is (cid:39)5000T.
1.5 S) 0
82 K μ10 Incontrast, thetermwithεSO =−δ cos(3θ)/d, added
)h) (G to the root in the Eq. 1 (like the Zeeman term), solely
/ 4.2 K
2 shifts the energy but not k , leading to an asymmetric
(eG ( 11.00 B B 11 spin-orbitenergysplittingfo⊥rthehole(∆SO+2εSO)and
1 2
00 2200 4400 the electron band (∆ −2ε ) of chiral metallic tubes
BB SO SO
0 B(T) [13, 14]. As ε contains the factor cos(3θ), it is small
0.5 SO
for near armchair tubes. The parameter δ ranges from
Vg= + 5.6 V 0.3-0.7 nm meV [13, 14].
∆ B 4.2 K
The resulting evolution of the band structure in mag-
0.0
netic field is visualized in Fig. 1b. At zero-field the
0 20 BB || ||((TT)) 40 60 band gap Eg0 = Ecurv − ∆SO is reduced by the SOI.
(b) With the application of B(cid:107), the two spin sub-bands sep-
BB == 00 BB == BB BB == BB BB == BB BB >> BB arated by the SOI cross the corner point of the Bril-
|| || 1 || 0 || 2 || 2
louin zone (either at K or K(cid:48)), thus explaining two sub-
EE
curv ∆ K K K K sequent MC-peaks at B1 and B2. In between, a conduc-
SO tance dip appears at B∗ when the spin sub-bands are
0
located symmetrically around the corner point. If the
K B
|| Zeeman-like terms in Eq. 1 are neglected the energy gap
has a local maximum at k = −k0 corresponding to
AB ⊥
FIG. 1: (a) Magneto-conductance (MC) of a small-bandgap E (B∗) ≈ ∆ . The distance between the two peaks,
g 0 SO
CNTdevicenearthechargeneutralitypoint(CNP)measured ∆B = (4/πd2)∆φ , is determined by ∆φ = 2 φ
AB AB SO
at 82 K and 4.2 K. The inset shows the MC in a semi-log (the factor 2 comes from σ = ±1). For the observed
scale. The observed double-peak in G(B ) defines the char-
(cid:107) values of ∆B = 8 T, and d=1.5nm we find
acteristic fields B∗,B and B . (b) One of the Dirac cones
0 1 2
near the K-points, intersected by lines of allowed k⊥ values πd2∆B
for a small-bandgap CNT with spin-orbit interaction (SOI). φ = ≈1.7 ×10−3φ . (3)
SO 8 0
Spin-up(blue)andspin-down(red)bandsdisplayacurvature
inducedbandgapE andareseparatedby∆ ,duetothe
curv SO For a conservative confidence interval of ±0.5 nm for d
SOI.WithincreasingB ,spin-splitsub-bandsshiftduetothe
ABeffectandcrossthe(cid:107)K(K(cid:48))point,successivelyclosingthe determined with an atomic force microscope one obtains
energy gap at B1 and B2. For simplicity, the Zeeman term 0.76 < 103φSO/φ0 < 3 compatible with previous studies
andε ,anadditionalZeeman-liketerminducedbytheSOI, [4, 11, 20]. Eqs. (1), (2) and (3) result in the energy
SO
are neglected. However, B and B are not affected by those splitting ∆ at B =0 (assuming k =0)
1 2 SO (cid:107) (cid:107)
terms (see EPAPS).
4h¯v φ
∆ = F SO ≈2.5±0.8meV. (4)
SO d φ
0
theallowedvaluesofk withamagneticfieldparallelto
⊥
the tube axis [5, 9, 15–17]. This value corresponds to ∼ 30 K and explains the dis-
In addition, curvature [18] affects the allowed val- appearance of the double-peak structure and the single
ues of k⊥ and induces small band gaps in nominally conductance maximum at B(cid:107) = B0 (cid:39) B0∗ for the 82 K
metallic CNTs. The curvature-induced shift [18, 19] trace of Fig. 1. Because ∆SO is inversely proportional to
k0 =−τa cos(3θ)/(2d)2 of the allowed k-states results the diameter, it becomes large for small-diameter tubes
⊥ 0
in a band gap E = 2¯hv |k0| at B = 0, where a is [22].
curv F ⊥ (cid:107) 0
the C−C bond length. With further increase of φAB the energy gap Eg lin-
Asecondconsequenceofcurvatureisaspin-dependent early opens again as both orbital sub-bands gradually
shift move away from the corner points of the Brillouin zone.
The exponential decrease of G at high fields (the inset
k =−σ(2/d)(φ /φ ), (2) of Fig. 1a) is thus explained by charge carriers thermally
SO SO 0
activated over the magnetic-field-induced band gap, as
ofk bythespinorbitinteraction[4,11–14,20,21]which described by previous authors [5, 23]. Since the inset in
⊥
removesthefour-foldspinandK,K(cid:48)-degeneracyinfavor Fig. 1a suggests that the conductance depends exponen-
of two Kramers doublets corresponding to parallel and tially on the band gaps E0 and E (B∗), they dominate
g g 0
antiparallel alignment of orbital and spin magnetic mo- theconductanceatB =0andB =B∗. Hence,theap-
(cid:107) (cid:107) 0
ments. This SOI-induced shift in k is equivalent to the proximate equality G(0) ≈ G(B∗) inferred from Fig. 1a
⊥ 0
3
((12,9)) ((17,,2))
(a) (c)
K
K
SO (b) K
2ε K
V = 0 V -O
g S
∆
KK
-5.3 V O K θ = 5.5°
θ = 25.3° +2εS B2 K (near zigzag)
((nneeaarr aarrmmcchhaaiirr)) ∆SO BB11 K
B (T) B (T) B (T)
|| || ||
FIG.2: (a)Calculateddensityofstatesinaparallelmagneticfieldforthe(12,9)CNT.(b)Zoomintotheareaboundedbythe
grayboxclearlyshowsthatthebandgapisclosedatB andB ingoodagreementwiththepeakpositionsobservedinFig1a.
1 2
Whitearrowsindicatethespin-polarizationofthebandsnearthecrossingpointsB andB . (c)DOScalculatedforthe(17,2)
1 2
CNT. Due to the larger curvature-induced band gap of the (17,2) tube with θ close to 0◦, the Zeeman-energy splitting at B
0
(gµ B ≈3.7meV) is larger than the spin-orbit energy splitting (∆ ≈2.33meV). Hence, the SOI-induced peak splitting is
B 0 SO
pronounced for the CNTs with chiral angles close to 30◦. Orbital (K and K(cid:48)) and spin states (white arrows) are indicated.
suggests the following relation between the two energy man splitting. The DOS calculated for the (12,9) tube
scales E and ∆ : explains the evolution of the magneto-conductance very
curv SO
well. The band gap is closed at B by the spin-down
E0 E −∆ 1
g = curv SO ≈1. (5) andsubsequentlyatB2 bythespin-upsub-band,invery
Eg(B0∗) ∆SO good agreement with the observed double-peaks at the
CNP. The calculated energy gaps at zero-field and at
Wenowturntothediscussionoftheeffectoftubechi-
B∗ agree with Eqs. (4) and (5). On the other hand,
rality. The strong dependence of B and B∗ on the chi- 0
0 0 the DOS calculated for the (17,2) CNT predicts a sig-
rality can be used to identify the chiral indices of small-
nificantly reduced intermediate gap region in Fig. 2c, in
bandgap CNTs [5]. Out of 53 small-bandgap CNTs with
spite of almost the same d and ∆ , when compared
d=1.5±0.5nm,onlyfivetubes[(12,9),(13,10),(16,10), SO
with the (12,9) tube. The (17,2) tube has a much larger
(18,9) and (19,10)] display values of B ≈5-8 T compat-
0 curvature-induced gap E ≈ 18.5 meV, resulting in
ible with our data, while B can take much larger values curv
0 a much higher B ≈ 31.6 T. Because the spin-orbit gap
for other CNTs, e.g., the (12,3) and (17,2) tubes. 0
competeswiththeZeemansplitting,thepeaksplittingin
TableIlistsvaluesofφ and∆ forthesechiralities,
SO SO
calculated from Eqs. (3) and (4) and the observed ∆B
= 8 T. The φ of the (12,9) tube is closest to φ ≈
SO SO
10−3φ , predicted in Ref. 11 and measured in Ref. 4.
0 TABLE I: Determination of the chirality from G(B ) at the
WhenwefurthertakeintoaccounttheconditionE ≈ (cid:107)
curv chargeneutralitypoint. φ wascalculatedfromEq.(3)with
SO
2∆SO for the CNT measured (Eq. (5)), we realize that the measured value of ∆B = 8 T and used as input for the
the (12,9) and (18,9) tubes satisfy this constraint best. evaluation of ∆ using Eq. (4).
SO
Taking the (12,9) tube with the chiral angle θ =25.3◦
(n,m) d θ E B ,B∗ φ ∆
(close to the armchair configuration) as the most proba- curv 0 0 SO SO
ble candidate, we calculated the density of states (DOS) ( nm ) ( ◦ ) ( meV ) ( T ) ( 10−3φ0 ) ( meV )
inaparallelmagneticfield. Forcomparison,weshowthe (12,9) 1.43 25.3 4.6 7.8 1.54 2.36
DOSofa(17,2)tube, whichhasalmostthesamediame- (13,10) 1.56 25.7 3.5 5.5 1.84 2.58
ter but a very different chiral angle θ =5.5◦ close to the (16,10) 1.78 22.4 4.7 6.4 2.39 2.94
zigzag-configuration. (18,9) 1.86 19.1 6.0 7.8 2.63 3.08
From Fig. 2, it becomes apparent that in an applied (19,10) 2.00 19.8 4.9 5.9 3.02 3.30
magnetic field the band edges change with four distinct
(12,3) 1.08 10.9 28.1 63.1 0.88 1.78
slopes away from the two Kramers doublets both in the
(17,2) 1.42 5.5 18.5 31.6 1.51 2.33
electron and hole bands, reflecting the orbital and Zee-
4
4 the conductance kinks observed at 8 and 24 T for the
((aa)) ) 4 ((bb))
h BB == 00 G(B ) curve at V = -5.3 V in Fig 3a, may be explained
33 - 5.3 V 2G (e/G 20 -10 0 |1|0 20 +66..44 VV 1 Kby↑(cid:48),tKh(cid:107)e↓sautb(cid:39)se2q4ueTntignlothsseoDfOKS↓(cid:48).Hsuobw-ebvaenrd,sasatthBec(cid:39)alc8uTlatainodn
neglects quantum interference effects in the Fabry-Perot
V ((V))
gg )
h) - 4.6 V 2/h regime, such as the AB beating effect [24], we cannot
2e/(e 2 0 V 0(eG ecxonpdecutcttaonecxepwlaitinhinallouferatsuimrepsleofmtohdeeml.easured magneto-
G +7 V 1
++ 44.5599 VV In conclusion, we have investigated single walled car-
1
bon nanotubes up to very high magnetic fields. The
++ 33.55 VV 00 1100 2200 00 magneto-conductance of a quasi-metallic tube shows a
Vg = + 5.6 V B (T) peculiar double peak, which can be explained in terms
00 ||
of spin split conduction bands, separated by a strong
0 10 20 30
B (T) spin-orbit interaction, which exceeds the Zeeman split-
||
ting. Our finding may open the path towards the appli-
FIG. 3: (a) G(B ) traces at 4.2 K for the hole side of the cation of CNTs as highly efficient ballistic spin filters.
(cid:107)
CNP.AsthebandgapgrowswithB(cid:107) athighfields,thegate ThisresearchwasfundedbytheDeutscheForschungs-
characteristic G(V ) exhibits the behavior of a p-type CNT
g gemeinschaftwithinGK1570andSFB689andpartially
field-effect-transistor with on-off conductance ratio of several
supportedbyEuroMagNETundertheEUcontractRII3-
orders of magnitude. The insert shows G(V ) at B = 0.
g (cid:107)
CT-2004-506239.
Each colored dot corresponds to the same colored trace of
G(B ). The black solid line in the insert is a fit to exper-
(cid:107)
imental points (see error bar) given as a guide line to the
eye. (b) G(B ) curves for the electron side of the CNP. The
(cid:107)
double-peakstructureisobservedonlywhentheFermienergy
is tuned close to the CNP at Vg (cid:39)+6V. ∗ e-mail:[email protected]
[1] I. Zutic, J. Fabian, and S. D. Sarma, Rev. Mod. Phys.
76, 323 (2004).
G(B ) is most pronounced for near armchair tubes with [2] Y. Kato et al., Science 306, 1910 (2004).
(cid:107)
chiral angles close to 30◦. [3] J.Wunderlichetal.,Phys.Rev.Lett.94,047204(2005).
[4] F. Kuemmeth et al., Nature 452, 448 (2008).
So far, all our discussion focused on the vicinity of the
[5] G. Fedorov et al., Nano Lett. 7, 960 (2007).
charge neutrality point. As a crucial test of our analy-
[6] T. Nakanishi and T. Ando, J. Phy. Soc. Jpn. 74, 3027
sis, we traced the evolution of the G(B ) curves for var-
(cid:107) (2005).
ious values of gate voltage (Vg). In Fig. 3a, magneto- [7] J. Kong et al., Nature 395, 878 (1998).
conductance traces at 4.2 K are displayed for the hole [8] E. D. Minot et al., Phys. Rev. Lett. 90, 156401 (2003).
side of the CNP. Deeply inside the hole band the con- [9] H.AjikiandT.Ando,J.Phy.Soc.Jpn.62,1255(1993).
ductance exceeds 3e2/h, close to the theoretical limit [10] See, e.g., J.-C. Charlier, X. Blase, and S. Roche, Rev.
Mod. Phys. 79, 677 (2007) and the references therein.
of 4e2/h. This shows that our device is in the ballis-
[11] T. Ando, J. Phy. Soc. Jpn. 69, 1757 (2000).
tic regime, where the conductance is determined by the
[12] D. V. Bulaev, B. Trauzettel, and D. Loss, Phys. Rev. B
numberofavailablesub-bandswithanaveragetransmis-
77, 235301 (2008).
sion probability of ∼ 0.8. At B(cid:107) = 0, the hole conduc- [13] W.Izumida,K.Sato,andR.Saito,J.Phy.Soc.Jpn.78,
tanceisaround3e2/handdiminishesdownto∼0.5e2/h 074707 (2009).
asE istunedtowardstheCNP(V∗ ∼+6V).Whilethe [14] J.-S. Jeong and H.-W. Lee, Phys. Rev. B 80, 075409
F g
magneto-conductance is initially positive at low fields, it (2009).
[15] S. Zaric et al., Science 304, 1129 (2004).
becomes negative at high fields (B (cid:29) B ) for all gate
(cid:107) 0 [16] U. C. Coskun et al., Science 304, 1132 (2004).
voltages, indicating the growth of E due to the AB ef-
g [17] B. Lassagne et al., Phys. Rev. Lett. 98, 176802 (2007).
fect. At B > 30 T, the gate characteristic G(V ) ex-
(cid:107) g [18] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 78, 1932
hibitsthebehaviorofap-typeCNTfield-effect-transistor (1997).
withanon-offconductanceratio>103. Thedouble-peak [19] A. Kleiner and S. Eggert, Phys. Rev. B 64, 113402
structure is pronounced only in the vicinity of the CNP (2001).
(+5.6 V ≤ V∗ ≤ +6.4 V).Twoadditional G(B ) curves, [20] D.Huertas-Hernando,F.Guinea,andA.Brataas,Phys.
g (cid:107) Rev. B 74, 155426 (2006).
presentedinFig.3b,showthatthetwopeaksmergeagain
[21] L. Chico, M. P. Lopez-Sancho´, and M. C. Mun˜oz, Phys.
into one as E is shifted to the electron side across the
F Rev. B 79, 235423 (2009).
CNP.
[22] Therefore, ∆ is larger for our nanotube, compared to
SO
ApproximatingthebackgatecouplingtotheFermien- the value observed for the tube (d∼ 5 nm) in Ref. 4.
ergy shift as ∆E ≈ 0.7×10−3∆V , the DOS calcula- [23] E. D. Minot et al., Nature 428, 536 (2004).
F g
tion matches the gate voltages in Fig. 3. For instance, [24] J. Cao et al., Phys. Rev. Lett. 93, 216803 (2004).
