Tunable few-electron double quantum dots and Klein tunnelling in ultra-clean carbon nanotubes
TTunable few-electron double quantum dots and Klein tunnelling inultra-clean carbon nanotubes
G. A. Steele , G. Gotz , & L. P. Kouwenhoven Kavli Institute of NanoScience, Delft University of Technology, PO Box 5046, 2600 GA, Delft, The Nether-lands.
Quantum dots defined in carbon nanotubes are a platform for both basic scientific studies[1, 2,3, 4, 5] and research into new device applications[6]. In particular, they have unique propertiesthat make them attractive for studying the coherent properties of single electron spins[7, 8, 9,10, 11]. To perform such experiments it is necessary to confine a single electron in a quantumdot with highly tunable barriers[1], but disorder has until now prevented tunable nanotube-based quantum-dot devices from reaching the single-electron regime[2, 3, 4, 5]. Here, we uselocal gate voltages applied to an ultra-clean suspended nanotube to confine a single electronin both a single quantum dot and, for the first time, in a tunable double quantum dot. Thistunability is limited by a novel type of tunnelling that is analogous to that in the Klein paradoxof relativistic quantum mechanics.
Single spins in carbon nanotube quantum dots are expected to be very stable against both relaxationand decoherence[11]. Nuclear spins, the principal source of spin decoherence in GaAs[7, 8], can be completelyeliminated and, furthermore, a strong spin-orbit interaction recently discovered in carbon nanotubes[9] en-ables all-electrical spin manipulation[10, 11], while preserving long spin relaxation and decoherence times[11].Electron spins in carbon nanotube quantum dots are therefore attractive for implementation of a quantumbit (qubit) based on spin for applications in quantum-information processing[6]. In double quantum dotsystems, precise control of the tunnel coupling between the two quantum dots, and between the quantumdots and the leads attached to them, is critically important for spin readout schemes[12, 13, 1], and also toprevent loss of spin and phase information through exchange of an electron with the leads.Double quantum dots can also be used to explore novel quantum tunnelling phenomena. In Kleintunnelling[14, 15, 16], for example, an electron tunnels with a high probability through a long and tall1 a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l otential energy barrier when the height of the barrier is made comparable to twice the rest mass of theelectron. It is not feasible to create such a barrier for free electrons due to the enormous electric fieldsrequired, but the low effective rest mass of the electrons in small bandgap nanotubes makes the observationof such Klein tunnelling in nanotube devices possible[16].By depositing metallic gates isolated by a dielectric layer on top of a nanotube, several groups havedemonstrated tunable double quantum dots in nanotubes lying on a substrate[2, 3, 4, 5]. A disadvantage ofthis technique is that nanotubes in these devices suffer from significant disorder induced by the substrate andby the chemical processing required to fabricate the device. As the electron density is reduced, this randompotential dominates and breaks the segment of nanotube into multiple disorder-induced “intrinsic” quantumdots before reaching the few-electron regime.Wet etching of the device after fabrication to remove the substrate-induced disorder has been usedpreviously to obtain single electron quantum dots in carbon nanotubes[17, 18], although experience hasshown that the yield of such devices is quite low. Recently, a new fabrication method has been developed forproducing ultra-clean quantum dots in suspended carbon nanotubes with a high yield in which all chemicalprocessing is done before nanotube growth[19]. Studying single quantum dots in these devices has uncoverednew carbon nanotube physics, including a strong spin orbit interaction due to the nanotube curvature[9] andevidence of Wigner crystallization of electrons at low density[20]. While devices fabricated in this way areextremely clean, they have some significant limitations: in particular, the confinement is produced only bySchottky barriers, which cannot be easily tuned in-situ. Furthermore, due to an insufficient number of localgates, it has not been possible to create a tunable double quantum dot in these ultra clean devices.In order to overcome these limitations, we have developed a new method of integrating multiple localgates with the ultra clean fabrication. A schematic of the device is shown in Figure 1. As described in theMethods section, we grow a carbon nanotube over gates that are patterned in a thin doped silicon layer. Ourcurrent design provides three independent gates, although fabrication can easily be modified to include ascalable number of gates inside the trench (see Supplementary Information). In this letter, we use these threegates in two different ways. In device D1, with L = 1.5 µ m, the gates are used to define a single electron andsingle hole quantum dot where electrons and holes are confined by tunable pn -junctions instead of Schottky2ontacts. In device D2, with L = 300 nm, we rely on tunnel barriers from the Schottky contacts, but nowuse the three gates to create a tunable single electron and single hole double quantum dot.In all previous measurements of quantum dots in carbon nanotubes containing a single electron, carrierswere confined by Schottky barriers formed at the metal contacts[9, 17], or by potentials defined from trappedoxide charges[18]. In figure 2, we demonstrate a single electron quantum dot defined only by gate voltages.We begin by applying a negative voltage to the splitgates, creating a p-type nanotube source and drainon top of the oxide. Sweeping the backgate voltage V BG , shown in figure 2a, the current initially showsweak modulations from resonances in the leads when the suspended segment is p-type ( ppp configuration),and is then completely suppressed as the suspended segment is depleted ( pip configuration). As we sweepfurther, we form a pnp quantum dot showing clean Coulomb blockade, where single electrons in the suspendedsegment are confined by pn junctions to the leads. Figure 2c shows a stability diagram as a function of bothbackgate and bias voltage, demonstrating that we have reached the single electron regime. As the confinementpotential and doping profile are determined by our local gates, we can also confine single holes in an npn configuration in the same device simply by inverting the gate voltages, shown in figure 2d. In figure 2e weshow the current as a function of the backgate voltage and the voltage on the splitgates. In the left of theplot, the leads are doped p-type, and a positive backgate induces a single electron pnp quantum dot. In theright of the plot, the leads are doped n-type and a negative backgate induces a single hole npn quantum dot.By adjusting the splitgate voltages, the pn junction width, and thus the tunnel barriers, can be tuned whilekeeping the electron number fixed (see figure 2f).In device D2, we use the gates in our design for a different purpose: here, we rely on less transparentSchottky contacts as incoming and outgoing tunneling barriers, and now use the backgate and the twosplitgates as three independent local gates to create a double quantum dot potential in the nanotube with atunable interdot coupling. Figure 3 shows the current through the device as a function of the two splitgatevoltages. In the lower left and upper right regions of the plots, the two splitgates dope the two segments ofthe nanotube with carriers of opposite sign, resulting in a pn double quantum dot with an interdot barrierformed from a pn junction. In the upper left (bottom right) corner, the two splitgates dope both sides ofthe nanotube p-type (n-type). In figure 3a, V BG is set to ground, which gives a potential in the middle3f the nanotube that is attractive for holes but repulsive for electrons. We consequently observe single dotbehaviour for the first hole and weakly coupled double dot behaviour for the first electron. In figure 3b,we apply a positive backgate voltage, V BG = 250 mV. The potential in the middle of the nanotube is nowrepulsive for holes: the first hole enters a weakly coupled double dot, while electrons fill a mostly single dotpotential. (At some gate voltages, the presence of the oxide creates a non uniform potential which resultsin strongly coupled double dot instead of purely single dot behaviour. See section S1 of the SupplementaryInformation for further discussion.) By changing V BG , we can continuously tune the interdot coupling in thefew-electron and few-hole regime from weakly coupled double dot to single dot behaviour.In figure 4, we investigate the tunable inderdot coupling in our double quantum dot more detail bystudying current at the (0,1e) ↔ (1e,0) triple point transition of a weakly coupled double quantum dot. Ina weakly coupled double quantum dot, current can only flow at specific values of the gate voltages, knownas triple points, where the levels in the two dots are aligned, allowing an electron to tunnel from one dot tothe other[21]. In figures 4a to c, V BG is made more negative, creating a larger barrier for electron tunnellingbetween the dots, suppressing the current at the triple point. However, as we sweep V BG further, shown infigures 4d and e, the current increases again, despite creating an even larger barrier for electron tunnelling.The explanation of this curious increase of the current is a novel tunnelling process analogous to thetunnelling paradox in high energy physics proposed by Klein[14, 15, 16]. Specifically, we will define Kleintunnelling as any enhancement of the tunnelling of an electron through a barrier due the so-called negativeenergy solutions (positron states) that arise in relativistic quantum mechanics (see Supplementary Informa-tion for further discussion). In figure 4, the enhancement of the interdot coupling we observe at large tunnelbarrier heights is an example of Klein tunnelling in a carbon nanotube, where now the valance band of thenanotube plays the role of the negative energy solutions in relativistic quantum mechanics. What is uniqueabout the data in figure 4 is that we have created a direct implementation of Klein’s gedanken experiment inour double quantum dot device, where we are able to tune continuously from the normal tunnelling regimeto the Klein tunnelling regime simply by changing the barrier height with a gate voltage. We have alsoobserved Klein tunnelling for holes (see Supplementary Information). In figure 4, what we observe is a kindof “virtual” Klein tunnelling, where the electron virtually occupies a state in the empty valance band in order4o tunnel from the left to the right dot, similar to a cotunnelling process[22]. In addition to our observationsin a double quantum dot, the npn data in figure 2 can be though of as a type of Klein tunnelling in a differentregime, where the valance band is now occupied with holes, and where Klein tunnelling occurs by the electronsequentially tunneling across the two pn -junctions. This also emphasizes the close relation between Kleintunnelling in high energy physics and interband tunnelling phenomena in semiconductor physics, such asZener tunnelling in insulators[23] and direct interband tunnelling in an Esaki diode[24].Analyzing the current at the (0,1e) ↔ (1e,0) transition quantitatively using the result from Stoof andNazarov[25, 26], we calculate the tunnel rates Γ L and Γ R of the barriers to the leads, and the interdot tunnelcoupling t c , shown in figure 4h. At these gate voltages, we are in the limit of weak interdot coupling: t c ∼ µ V << Γ L , Γ R ∼ t c , is decreased from an initial value of 9 µ V to a minimumof 3 µ V as a function of backgate, before the onset of Klein tunnelling results in an increase up to 9 µ V aswe approach gate voltages where an npn triple dot is formed. Γ L and Γ R are found to be independent of thebackgate voltage, indicating that the backgate is not influencing the Schottky barrier transparency.Finally, we comment that although we are in the appropriate double quantum dot coupling regime, wehave not found evidence of spin blockade at any of the expected transitions[27]. (A parallel magnetic field of1.5T was applied to ensure that the nanotube valley degeneracy was lifted). One possible explanation for thisis a singlet-triplet splitting in the (0,2e) state that is much smaller than the 3 mV single particle spacing weobserve in the single electron quantum dot. This could be an indication of Wigner crystal formation[20]: in aWigner crystal, the electron wavefunction overlap is very small, and consequently the single-triplet splitting isstrongly suppressed. This possibility will be investigated further using devices with more gates, which couldallow us to probe the Wigner crystallization transition by tuning the quantum dot confinement potential.We have presented a new technique for confining single electrons and single holes in ultra-clean carbonnanotubes. By eliminating disorder and incorporating local gates, a new level of control over single electronconfinement has been achieved, allowing us to observe a novel type of tunnelling in a single electron carbonnanotube device. While our motivation for such a device comes from the spin physics of carbon nanotubes[28],the fabrication itself could have a much broader use in carbon nanotube applications, such as electricallydoped pn junctions for carbon nanotube optical emission[29], where low disorder and multiple gates for5lectrical control of pn junctions could allow the development of new types of optically active devices. Methods
Fabrication begins with a p++ Si wafer with 285 nm of thermal silicon oxide. On top of this, a 50 nmthick n++ polysilicon gate layer is deposited, followed by a 200 nm LPCVD-TEOS oxide layer. Usingelectron-beam lithography and dry etching, a trench of approximately 300 nm deep is etched, forming thetwo splitgates from the n++ Si gate layer. A 5/25 nm W/Pt layer is deposited to serve as source and draincontacts, and nanotubes are then grown from patterned Mo/Fe catalyst[30]. In about half of the devices,a single carbon nanotube is suspended across the trench making electrical contact to the source and drain.Transport through the devices is characterized at room temperature, and selected devices are cooled to < µ m, W = 300 nm and bandgap E g = 60 mV, and D2 with L = 300 nm, W = 500 nm and E g =25 mV, where bandgaps are determined by subtracting the charging energy from the size of the empty dotCoulomb diamond. References [1] Hanson, R., Kouwenhoven, L. P., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K. Spins in few-electronquantum dots.
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Acknowledgments
It is a pleasure to acknowledge P. L. McEuen for the suggestion of using pn junctions as tunable barriers,as well as D. Loss, T. Balder, I. T. Vink, R. N. Schouten, L. M. K. Vandersypen, and M. H. M. van Weertfor useful discussions and suggestions. Supported by the Dutch Organization for Fundamental Research onMatter (FOM), the Netherlands Organization for Scientific Research (NWO), and the Japan Science andTechnology Agency International Cooperative Research Project (JST-ICORP). Additional Information μ m ab Si Backgate285 nm200 nm
Si Gate
LPCVD SiO Thermal SiO LL W V SG2 V sd V SG1
CNT
W/Pt V BG W/PtSiO /SG1SiO /SG2W/PtSiO /BG Figure S1 : Integrating local gates with ultra-clean carbon nanotubes. a , A schematic of the device. Apredefined trench is etched to create two splitgates from a 50 nm thick n++ polysilicon gate layer between twosilicon oxide layers. A Pt metal layer is deposited to act at as source and drain contacts, and a nanotube isthen grown from patterned catalyst. Device D1 has L = 1.5 µ m, W = 300 nm, and D2 has L = 300 nm, W =500 nm. b , In a subset of devices, a single nanotube bridges the trench, contacting the metal source and drainelectrodes, as shown in this colourised SEM micrograph. The micrograph shows an example of a device with thesame dimensions as device D1. -200200 -150 ppp pip pnp -200150 -8 -6 -4 -2 0-50-2502550 -150 nin nnnnpn a C u rr e n t ( n A ) V SG1/2 = -270 mV dc fe C u rr e n t ( n A ) C u rr e n t ( n A ) -1 0 1 2840-4-8 ppp nnn (c) (d) C u rr e n t ( n A ) C u rr e n t ( n A ) ppp pnppip
1e 2e V SG1/2 = -70 mV n p n
Conduction BandValance Band pnp
HolesElectrons0 V BG (V)V BG (V) V BG (V)V SG1 = V
SG2 (V) V B G ( V ) V S D ( m V ) V S D ( m V ) V BG (V) Figure S2 : Gate defined single-electron and single-hole quantum dots. a , Coulomb peaks of a pnp quantumdot in device D1 taken at a V
SG1 = V
SG2 = -50 mV and V sd = 1 mV. The splitgates are used to dope the NTsource and drain leads with holes. As V BG is swept from negative to positive voltages, the suspended segmentis depleted giving a pip configuration, followed by a pnp configuration as single electrons are filled in an n-typequantum dot, as illustrated in the energy diagrams in b . c , Stability diagram of the pnp dot: the charging energy ofthe first electron E ∼ meV is remarkably large due to the weak capacitive coupling of the suspended segmentto the gates and the metal source drain layers. d , The potential landscape in the device can be completelycontrolled by the gate voltages: by reversing the gate voltages, single holes are confined in a npn configuration. e , A 2D plot showing backgate sweeps at different splitgate voltages and V SD = 10 mV. The two splitgates areset to the same voltage. The stability diagrams in c and d are taken at V SG1 / values indicated by the arrows.(Resonances from residual disorder in the long NT leads can be seen as oscillations as a function of V SG1 / in the ppp and nnn configurations.) f , Using the splitgates, we can tune the width of the pn junction depletion region,and hence the tunnel barriers: at V SG1 = V
SG2 = -70 mV, the potential from the splitgates is shallow, giving awide depletion region and a current of 0.5 nA for the first electron Coulomb peak at V SD = 10 mV. At V SG1 =V SG2 = -270 mV, the potential across the pn junction is steeper, now giving a narrower depletion region and acurrent of 13 nA for the first electron. (The V SG1 / = -270 mV trace has been offset in V BG and in current.)
00 200 300 400500400300200100 200 300 400500400300200 ) A p ( t n e rr u C (0,0) (1e) (1e,1h) (2e,1h) (3e,1h)(0,1h)(1h,0)(1h,1h)(2h,1h) (1h,1e)(1h,2e)(1h,3e) (0,2e)(0,3e) (2e,0) (3e,0) (4e,0)(5e) (6e) (7e)(0,0) (1e,0) (2e,0)(1e,1e)(0,1e) (1e,2e) (2e,1e)(2e,2e)(1e,2h)(1h)(0,2h)(0,3h)(2h,1e)(3h,1e) (2h,0)(5h)(6h) (4h) (2e,1h)(1e,1h) ( h , e ) (4e) a b V BG = 0 mV V BG = 250 mV (3h,0) (3h) V SG1 (mV) V S G ( m V ) V SG1 (mV) V S G ( m V ) Figure S3 : A tunable double quantum dot in the few-electron and few-hole regime.
Current as a functionof the two splitgate voltages at V SD = 0.5 mV for device D2. In device D2, electrons are confined in the nanotubeby Schottky barriers at the metal contacts, with a potential that is tunable using the three gates. Electron andhole occupation numbers are determined from the transition to a pn double quantum dot, as described in theSupplementary Information. a , V BG = 0. At this voltage, a barrier for electrons is induced in the middle ofthe device. Electrons are added to a weakly coupled double dot potential, while holes are added to a singledot potential. b , V BG = 250 mV. A more positive V BG creates a double dot potential for holes and a singledot potential for electrons. The interdot coupling for both the electron and the hole double dot can be tunedcontinuously using the backgate voltage. .01.71248138 ) A p ( t n e rr u C V BG (mV) Г R , L ) V m ( t c ( μ ) V V BG = 125 mV V BG = 101 mV V BG = 27 mV V BG = -21 mV V BG = -45 mV (0,1e)(1e,0) NormalTunnellingKleinTunnelling ) A p ( t n e rr u C Energy Detuning
0 10 20 V BG = -45 - 21 3 27 52 76 101 a b cd e fg h NormalKlein (a)(b)(c)(d)(e)
Figure S4 : Klein tunnelling in a single electron double quantum dot.
Current at the (1e,0) ↔ (0,1e) triplepoint for a single electron double quantum dot at V SD = 5 mV. (Note that the interdot capacitance E interc ∼ . mV is much smaller than the bias, and thus the triple point bias triangles for the electron and hole cycle[21]strongly overlap.) Transitions to the excited state of the outgoing dot are visible as lines in the triangle runningparallel to the baseline give a quantized level spacing of 3 mV, consistent with a dot length of ∼
500 nm. In a through c , the backgate is made more negative, creating a larger barrier for electron tunnelling. As a result, thecurrent through the double dot is decreased. In d and e , however, the current begins to increase again despitea larger barrier for electron tunnelling. f , This increase in current results from tunnelling of an electron belowthe barrier through a virtual state in the valence band, analogous to Klein tunnelling in high energy physics. g ,Line cuts of the triple point data in a - e showing the current for the ground state baseline transition at differentbackgate voltages. The line cuts are taken along the dashed line in e . The x-axis shows the distance along this lineconverted into the energy detuning between the left and right dot ground state levels. For the rightmost traces,interdot tunnel coupling is mediated by normal electron tunnelling, while for the leftmost traces, Klein processesprovide the interdot tunnel coupling. h , Parameters from a fit to the Stoof-Nazarov equation. The interdot tunnelcoupling initially decreases as the barrier height increases (V BG = 125 to 27 mV), and then increases due to theonset of Klein tunnelling as the barrier height becomes comparable to the bandgap (V BG = 27 to -45 mV). unable few-electron double quantum dots and Klein tunnelling inultra-clean carbon nanotubes: Supplementary Information G. A. Steele , G. Gotz , & L. P. Kouwenhoven Kavli Institute of NanoScience, Delft University of Technology, PO Box 5046, 2600 GA, Delft, The Nether-lands.
S1 Determining electron numbers
Absolute electron numbers in the device are identified by the transition from a nn or a pp single dot to a pn or np double dot, as shown in figure S1. For example, at V SG1 = −
250 mV and V
SG2 ∼
260 mV, we remove thelast hole from the right side of the nanotube, (p,p) → (p,0). As we sweep V SG2 further, at V
SG2 ∼
380 mV,we fill an electron into the right dot. Here we see an abrupt transition from single dot behaviour to doubledot behaviour, signaling the transition to a (p,n) double dot. This transition allows us to clearly identify theelectron numbers in the device. The electron number assignment was also confirmed by large bias Coulombdiamond measurements such as those shown in figure 2 of the main text.At V
SG1 ∼
500 mV, the device suffers from a “switch” in gate voltage: this switch, which appeared onthe third cooldown of the device, is likely due to a charge trap in the oxide. Aside from this, the deviceis extremely stable. It is also very robust with respect to thermal cycling: after 2 thermal cycles includingexposure to air, the barrier transparencies were unchanged and the position of the first Coulomb peak movedby less than 50 mV in gate space.Note also that although the backgate voltage used in figure S1 should result in single dot behaviourfor holes, the data show some bending of the Coulomb peak trajectories along the (p,p) to (0,p) and (p,0)transitions, indicating a strongly tunnel coupled double dot type of behaviour. It is also visible along the(n,n) to (n,0) and (0,n) transitions in figure 3(b) of the main text, and at higher electron numbers in figureS1. This results from a somewhat non uniform potential induced by the presence of the oxide under partof the tube, likely due to a combination of trapped charges in the oxide and the abrupt change in dielectricconstant. 1 a r X i v : . [ c ond - m a t . m e s - h a ll ] J u l
250 0 500 7507505000-250 V S G ( m V ) V SG1 (mV) (n,n)(0,n)(p,n)(p,n)(p,0)(p,p) (p,p) (0,p) (n,p) (n,p)(n,0)(n,n) (0,0) C u rr e n t ( p A ) Figure S1 : A 2D splitgate sweep over a larger range used to determine electron numbers from the transition toa pn double quantum dot. Data is taken at V BG = 50 mV and V SD = 0.5 mV. ) A p ( t n e rr u C Energy Detuning (meV) Γ out = 0.4 mV t c = 3.3 μ VV BG = 52 mV C u rr e n t ( p A ) Energy Detunig (meV) V SD = 5 mV Γ out = 0.5 mV t c = 3.6 μ V V BG = 52 mVV SD = -5 mV a b Figure S2 : Fit of (0,1e) ↔ (1e,0) transition at V BG = 52 mV to the Stoof-Nazarov theoretical result for a ,positive and b , negative bias. A detuning independent inelastic contribution to the current of 350 fA is clearlyvisible in the reverse bias trace. This inelastic current is also present in a , but is more difficult to identify due toa nearby excited state of the outgoing dot in forward bias. S2 Stoof-Nazarov Equation
To analyze the data quantitatively, we fit the current at the ground state to ground state transition alongthe baseline of the triple point bias triangle as a function of energy detuning (cid:15) to the expression from Stoofand Nazarov[1, 2]. By performing such an analysis, we are able to isolate the contribution of the middletunnel barrier from the measurement of the current through the double quantum dot. For a interdot tunnelcoupling t c and tunnel rates Γ L,R to the left and right leads, the elastic current in a double quantum dot isgiven by: I el ( (cid:15) ) = et c Γ R t c (2 + Γ R / Γ L ) + Γ R / (cid:15)/h ) (1)In the limit of weak interdot tunnel coupling, t c << Γ L , Γ R , this reduces to a simple Lorentzian line shapeof the form: I el ( (cid:15) ) = 4 et c / Γ R (cid:15)/ Γ R h ) (2)A fit of the data to equation 2 for a single electron double dot is shown in figure S2. The fit was performed for (cid:15) < (cid:15) >
0, the fit deviates from the Lorentzian3ineshape due to inelastic processes[3].
S3 Relativistic tunnelling through a barrier and the Klein Paradox
Consider an electron of energy E and momentum ¯ hk incident on a square barrier of height V as shown infigure S3. We are interested in the probability that the electron is transmitted to x > L using the Diracequation. The solutions of the Dirac equation have two branches[4]: a set of positive energy solutions with E >
E <
0. The two branches are separated by an energygap 2 mc . The vacuum state is interpreted as having the negative energy solutions filled with electrons (the“Dirac sea”), and a hole in the Dirac sea is then interpreted as a positron. For a barrier height that is smallcompared to 2 mc , shown in figure S3(a), the Dirac equation gives a wavefunction that decays exponentiallyinside the barrier: for an incident energy E (cid:28) V , the probability of the electron tunnelling to the region x > L is small. This is also what is predicted by the non-relativistic Schroedinger equation.However, if the barrier height becomes very large, so that V is comparable to 2 mc , the negative energysolutions of the Dirac equation strongly modify the tunnelling process. In particular, Klein noticed that for V ∼ mc , as shown in S3(b) and S3(c), an electron moving at non-relativistic speeds incident on the barrierat position x = 0 can tunnel to x > L on the other side of the barrier with nearly unity probability. In thecontext of the non-relativistic Schroedinger equation, such a high tunnelling probability would be completelyunexpected, hence the idea of such tunnelling as a paradox.The tunnelling enhancement can be divided in to two regimes, illustrated in figures S3(b) and S3(c).We will refer to the first, illustrated in figure S3(b), as the (Klein) Tunnelling regime[5]. Here, the electronpropagates inside the barrier as an evanescent wave, but the transmission probability can be very high sincethe decay length is significantly longer that that from the Schroedinger equation due to the negative energysolutions. We will refer to the second regime, shown in figure S3(c), as the (propagating) Klein regime. Here,the wavefunction in the barrier is oscillatory in nature and does not decay. Both cases are examples of whatwe will call non-classical “Klein Tunnelling” in which the electron emerges at x > L with a much higherprobability than that predicted by the Schroedinger equation.The electronic spectrum of a carbon nanotube at low energies is also given by a Dirac equation that4 b c Normal Tunnelling (Klein) TunnellingRegime Klein (propagating)Regime
Figure S3 : Relativistic tunnelling through a barrier. Positive energy solutions of the Dirac equation are separatedfrom the negative energy solutions by a an energy gap mc . We consider the probability that an electron incidenton a barrier of height V at x = 0 with energy E is transmitted to the region x > L . a, For V (cid:28) mc , thewavefunction inside the barrier decays exponentially with a decay length κ = p m ( V − E ) / ¯ h , as predictedby the non-relativistic Schroedinger equation. We refer to this as the “Normal” tunnelling regime. b, For V slightly less than mc , the wavefunction also decays exponentially inside the barrier. However, due to thenearby negative energy solutions of the Dirac equation, the decay length is now much longer, given by κ = p m (2 mc − V + E ) / ¯ h , and the transmission probability is much higher than that predicted by the Schroedingerequation. We refer to this as the (Klein) Tunnelling regime. c, For
V > mc , the electron now propagates insidethe barrier without decaying by occupying a negative energy solution of the Dirac equation. Inside the barrier, thewavefunction is a plane wave e ik x with energy E = V − mc − E . We refer to this as the Klein (propagating)regime.
5s the same as that for normal electrons[6, 7], but with 2 mc replaced by the bandgap Eg , and the speedof light c replaced by the Fermi velocity of graphene v F ( ∼ × m/s). Free electrons in the Diracequation correspond to electrons in the conduction band of the nanotube, and positrons in the Dirac equationcorrespond to holes in the valance band. Thus, it should be possible to observe phenomena analogous to thetwo Klein tunnelling regimes of S3(b) and S3(c) in a carbon nanotube device.In figure 2 of the main text, we present data demonstrating a single hole npn and single electron pnp quantum dot. The current that we observe at the Coulomb peaks can be considered as an example of the(propagating) Klein regime illustrated in figure S3(c), where the potential barrier from our gate voltagesis larger than the bandgap. Figure 2(d) from the pnp configuration corresponds to the (propagating) Kleinregime for positrons, and figure 2(c) from the npn configuration corresponds to the same regime for electrons.In figure 4 of the main text, we show an example of the (Klein) tunnelling regime of figure S3(b). Inthe data, we observe a continuous transition from the normal tunnelling regime to that where the negativeenergy solutions of the Dirac equation provide an enhancement of electron tunnelling, as in the original Klein gedanken experiment. We also note that the unusual tunnelling process shown in figure S3(b), where thedecay length in the barrier is increased due to the negative energy states, has recently been proposed as amechanism of generating exchange coupling between two distant quantum dots in graphene nanoribbons[7].Our experiment demonstrates explicitly such tunnelling in a carbon nanotube. S4 Klein tunnelling for a single hole double quantum dot
In figure S4(a), we show Klein tunnelling for a single hole double quantum dot. Qualitatively, the processis the same as that for the single electron double dot. In the single hole double dot, the tunnel rates to theleads are smaller by a factor of 2-3 compared to the single electron double dot: in the device, the Fermi levelpinning at the metal contacts is such that electrons see a smaller Schottky barrier. This can also be seenin figure S1, where the electron single dot peaks show more current and broadening than those of the holesingle dot. 6 V BG = 176 mV V BG = 203 mV V BG = 256 mVV BG = 283 mV V BG = 336 mV V BG = 363 mVV SD = 3 mV C u rr e n t ( p A )
200 300 BG (mV) Г R , L ) V m ( t c ( μ ) V a bc NormalTunnellingKleinTunnelling
Figure S4 : a,b Klein tunnelling at the (0,1h) ↔ (1h,0) single hole double dot transition and c , parameters froma fit to the Stoof-Nazarov expression. S5 Extending fabrication to include more gates
In the device studied in this paper, the incoming and outgoing barriers of the double dot were formed bySchottky barriers at the metal contacts. Because we have only three gates, for the double dot we couldnot tune the incoming and outgoing barriers independently of the electron number. As a result, we havesignificant lifetime broadening of the energy levels in the double dot configuration.To overcome this, we can easily extend the fabrication to include more gates inside the trench, as shownschematically in figure S5(a). As an example of this, a test structure using this type of fabrication is shownin figure S5(b).In addition to tuning Γ
L,R independent of electron number, these gates could also be used to tune thesingle dot confinement potential. If Wigner crystallization is indeed responsible for the lack of spin blockadein our current device, we should be able to observe a transition to a finite singlet-triplet splitting by tuningthe confinement energy using these extra gates. 7
00 nm a b
Figure S5 : a , Schematic of device with more gates in the trench and b , an SEM micrograph of a test device witha suspended nanotube. References [1] Stoof, T. H. & Nazarov, Y. Time-dependent resonant tunneling via two discrete states.
Phys. Rev. B ,1050–1053 (1996).[2] Fujisawa, T., van der Wiel, W. G. & Kouwenhoven, L. P. Inelastic tunneling in a double quantum dotcoupled to a bosonic environment. Physica E , 413–419 (2000).[3] Fujisawa, T. et al. Spontaneous emission spectrum in double quantum dot devices.
Science , 932–935(1998).[4] See, for example, Perkins, D. H.,
Introduction to High Energy Physics (Addison-Wesley, Menlo Park,1987).[5] Bernardini, A. E. Relativistic tunneling and accelerated transmission.
J. Phys. A , 215302+ (2008).[6] Bulaev, D. V., Trauzettel, B. & Loss, D. Spin-orbit interaction and anomalous spin relaxation in carbonnanotube quantum dots. Phys. Rev. B , 235301 (2008).[7] Trauzettel, B., Bulaev, D. V., Loss, D. & Burkard, G. Spin qubits in graphene quantum dots. NaturePhys.3