Fabry-Perot interference, Kondo effect and Coulomb blockade in carbon nanotubes
aa r X i v : . [ c ond - m a t . m e s - h a ll ] N ov APS/123-QED l Fabry-Perot interference, Kondo effect and Coulomb blockade in carbon nanotubes
K. Grove-Rasmussen, ∗ H. I. Jørgensen, and P. E. Lindelof
Nano-Science Center, Niels Bohr Institute, University of Copenhagen, Denmark (Dated: June 14, 2013)High quality single wall carbon nanotube quantum dots have been made showing both metallicand semiconducting behavior. Some of the devices are identified as small band gap semiconduct-ing nanotubes with relatively high broad conductance oscillations for hole transport through thevalence band and low conductance Coulomb blockade oscillations for electron transport throughthe conduction band. The transition between these regimes illustrates that transport evolves frombeing wave-like transmission known as Fabry-Perot interference to single particle-like tunneling ofelectrons or holes. In the intermediate regime four Coulomb blockade peaks appear in each Fabry-Perot resonance, which is interpreted as entering the SU(4) Kondo regime. A bias shift of oppositepolarity for the Kondo resonances for one electron and one hole in a shell is in some cases observed.
PACS numbers:
INTRODUCTION
Single wall carbon nanotubes (SWCNT) are interest-ing objects for the study of low dimensional mesoscopicsystems, where the observed phenomena crucially de-pends on the coupling to the contacts. For good contact,the SWCNT acts as an electron wave guide creating res-onances at certain energies. Such systems are regardedas open quantum dots with the resonances correspond-ing to the broad energy levels of the quantum dot [1]. Inthe opposite limit of very low transparency, the electronsare forced to tunnel one by one due to Coulomb blockadeand the energy levels sharpens due to their longer lifetime [2, 3]. In this so-called closed quantum dot regimethe electron number on the SWCNT is well-defined ex-cept at charge-degeneracy points where single electrontunneling occurs. An intermediate regime also exists inwhich the electron number on the dot is still fixed butsignificant cotunneling is allowed. This leads to differ-ent kinds of Kondo effects related to the total excessspin [4, 5, 6] and/or the orbital degree of freedom onthe SWCNT quantum dot [7, 8]. We will in this paperexamine the transition between these regimes in smallband gap semiconducting nanotubes, where the couplingto the SWCNT is different for transport through the va-lence and conduction band [9]. High quality measure-ment is presented from each transparency regime show-ing that each Fabry-Perot oscillation develops into fourCoulomb blockade resonances with finite conductance inthe valleys for intermediate transparency (Kondo effect)and well known Coulomb diamond patterns with four-fold degenerate shell structure at lowest transparency[10].
EXPERIMENTAL METHODS
SWCNTs are grown from predefined catalyst islandsby chemical vapor deposition on a highly doped siliconsubstrate capped by a 500 nm silicon dioxide layer. Aftergrowth, pairs of electrodes are defined by electron beamlithography some microns away from the catalyst islandsin hope that one SWCNT bridges the gap between thetwo electrodes. In the case of the small band gap semi-conducting SWCNT the electrodes consist of Au/Pd bi-layers (40 nm/10 nm). Finally bonding pads of Au/Crare made by optical lithography connecting the electronbeam lithography defined electrodes. The devices areelectrically probed at room temperature with typical re-sistances in the range of 20-200 kΩ. These are cooled tolow temperatures where device characteristics clearly re-veals if only one SWCNT is in the gap. More details ondevice fabrication can be found in Refs. [11, 12]. Themeasurements presented in this article are mostly madeat 4 K in a DC-setup.
METALLIC AND SEMICONDUCTINGNANOTUBES
SWCNTs have the remarkable property that depend-ing on the exact arrangement of the carbon atoms theycan either be metals or semiconductors. Figure 1 showsthe gate (and temperature) dependence of the linear con-ductance for two different types of SWCNTs, which de-fines whether the nanotube is semiconducting or metallic.Figure 1(a) displays the behavior of a metallic SWCNTcharacterized by its relatively constant linear conduc-tance as a function of gate voltage close to room tem-perature. On the contrary the behavior shown in Fig.1(b) corresponds to a semiconducting SWCNT due to itsstrongly gate dependent linear conductance [13]. Thesemiconducting SWCNT has clearly high conductancefor negative gate voltages, which is due to hole trans-port through the valence band as indicated in the insetshowing the band structure (p-type). In the range of4 V < V gate <
10 V the chemical potential of the leads isaligned with the band gap, i.e., no conduction occurs. Athigher temperatures signs of electron transport throughthe conduction band is seen at high positive gate volt-ages due to thermal excitation of electrons into the con-duction band. The SWCNT shown in Fig. 1(b) has arelative large band gap, while conduction for small bandgap semiconductors reappears for positive gate voltage inthe gate range shown due to electron transport throughthe conduction band (see below). Note, that the resis-tance between the two devices differ by an order of mag-nitude due to different coupling to the leads. In generalthe coupling can to some extent be controlled by choiceof electrode material [14].Figure 1(a-b) also shows the temperature dependence,where the gate dependence of the linear conductancefor both devices evolves into regular oscillations at lowtemperature. These oscillations are a manifestation ofCoulomb blockade, which as mentioned above happensfor SWCNT weakly coupled to the leads. The regularityof the Coulomb oscillations indicates whether good qual-ity of the SWCNT has been obtained or if more than oneSWCNT is bridging the gap between the electrodes. Forhigh quality SWCNTs regular oscillations should persistthrough a gate region of typically | V gate | <
10 V as shownin (a). On the contrary if the SWCNT has defects onlysmall regions of gate voltage might have well behavedoscillations.
SMALL BAND GAP SEMICONDUCTINGNANOTUBES
We will now focus on small band gap semiconductingSWCNTs, where the band gap is so small that transportcan be tuned from hole transport through the valenceband to electron transport through the conduction bandby the back gate [9, 15, 16, 17]. Figure 2(a) shows thelinear conductance versus gate voltage at source-drainvoltages V sd ∼ e V for a small band gap SWCNTat T = 4 K (device 1). The nature of the SWCNT isidentified by the relatively high conductance region fornegative gate voltages (hole transport) in contrast to thelow conductance region for positive gate voltages (elec-tron transport). The broad oscillations for hole transportfor gate voltages between − i.e. , an openquantum dot [1]. In contrast for positive gate voltagesregular low conductance oscillations are observed due toCoulomb blockade. Figure 2(b) shows the high posi-tive gate region for electron transport where relativelyhigh conductance Coulomb blockade resonances are seen.They are spaced into four reflecting the spin and orbital -10 -5 0 5 100.00.51.01.52.00.000.050.100.15 (a) V gate [V] (b) k || E G [ e / h ] G [ e / h ] FIG. 1: Two different types of SWCNTs. (a) A metallicSWCNT identified by its weak gate voltage dependence athigh temperature ( T = 220 , , ,
10 K, black to blue). Atlow temperatures oscillations in the conductance versus gatevoltage are seen due to Coulomb Blockade. Inset: Band dia-gram of a metallic SWCNT (armchair). (b) A semiconductingSWCNT identified by its strong gate dependence of the lin-ear conductance. Oscillations of the conductance at low tem-perature in the p-type region is due to single hole transport(Coulomb blockade). Inset: Band diagram of a semiconduct-ing SWCNT with the electrochemical potential in the valenceband corresponding to the situation for V gate < T = 150 , , , , , , . degree of freedom. When this device is cooled to lowertemperature (50 mK) Kondo resonances are seen withinthe four-fold shell structure (not shown) in contrast tothe lower conducting region, e.g., gate voltages from 5 Vto 8 V, where only single electron tunneling is possible.The different transparency of the n- and p-type re-gions can be understood from Fig. 2(c). For negativegate-voltages (left figure) the bands are bending in sucha way that holes can tunnel from source into the va-lence band and out to drain. The Schottky barrier forhole transport is relatively small because the workfunc-tion of the Pd contacts is close to the valence band edgeleading to a relatively high conductance [9]. Transportcan be changed to electron transport through the con- FIG. 2: Measurements at 4 K of a small band gap semiconducting SWCNT (device 1). (a) Linear conductance versus gatevoltage. For hole transport through the valence band at negative gate voltage high conductance Fabry-Perot oscillations areobserved, while electron transport through the conduction band at positive gate voltage is dominated by Coulomb blockade dueto higher Schottky barriers. (b) Zoom at high positive gate voltages where the conductance increases and Coulomb blockaderesonances are spaced in four. (c) Schematic band diagrams of a small band gap single wall carbon nanotube contacted to leads,where the band bending is controlled by the gate voltage. The red/blue band is the conduction/valence band, respectively.Left: The condition for hole transport through the valence band. Holes tunnel into/out of the valence band through a relativesmall Schottky barrier. Right: Condition for electron transport through the conduction band, where electrons tunnel into/outof the conduction band through a larger Schottky barrier. Thus high conductance is observed through the valence band incontrast to low conductance through the conduction band, i.e. , Fabry-Perot versus Coulomb blockade regime. duction band (right figure) by applying positive voltageto the gate. The Schottky barrier is in this case signif-icantly larger leading to a low coupling of the SWCNTto the electrodes. Between these two transport regionsno transport is allowed because the chemical potential inthe leads is within the band gap of the semiconductingSWCNT.More information on the transport properties can berevealed by bias spectroscopy, where the differential con-ductance is measured as a function of gate and bias volt-ages. A bias spectroscopy plot of part of the Fabry-Perotregion is shown in Fig. 3(a). The low conductance regions(red areas) form a mesh due to interference of the holewaves reflected back and forth. Regarding the device asa quantum dot the white regions at zero bias correspondto being in resonance, while a red regions are off reso-nance. The level spacing can be extracted as indicatedin Fig. 3(b) by the black arrow ∆ E ∼ e V consistentwith the device being around L = ~ v F π ∆ E ∼
400 nm. Herewe have used a linear dispersion since we are far fromthe band gap. Furthermore, a four-fold degeneracy ofthe level is assumed, which is clearly revealed in the n- type region of the device (see below). The device hasa high asymmetric resistive coupling, because the con-ductance at resonance is lower than 4 e /h . The asym-metry Γ L / Γ R can be found from G res = e h L Γ R (Γ L +Γ R ) ,where G exp,res ∼ e /h is the conductance at resonanceyielding Γ L / Γ R = 0 .
17. The capacitive couplings to thesource and the drain electrodes are also slightly differentor/and the capacitive coupling to the gate is comparableto the source and drain capacitance because of the dif-ferent slopes of the low conductance lines (red). The ca-pacitances will be examined more closely in the Coulombblockade case below.Figure 3(b) shows a bias spectroscopy plot in the p-type region for another small band gap semiconduct-ing SWCNT with lower transparency (device 2). Signsof Fabry-Perot oscillations are still observed, but eachFabry-Perot resonance is split into four smaller peaks [9].These peaks are due to Coulomb blockade and single holetunneling, i.e. , the four-fold degenerate level becomes vis-ible due to quantization of the charge. The finite conduc-tance in the valleys between the peaks are indication of
FIG. 3: Measurements at 4 K of two different small band gap SWCNTs with Au/Pd contacts (device 1 and 2). (a) Biasspectroscopy plot of a small gate region of device 1 in the p-type gate region of Fig. 2(a) showing a Fabry-Perot interferencepattern, i.e., an open quantum dot. The level spacing is estimated to ∆ E ∼ e V. (b) Single hole transport is visible ontop of the broad Fabry-Perot resonances when the transparency is lowered (device 2, p-type) by four Coulomb blockade peakssplitting each resonance. The four peaks with finite conductance in the valleys is due to SU(4) Kondo effect. (c) For evenlower transparency of device 1 clear Coulomb blockade diamonds are seen. The four-fold degeneracy is clearly observed andthe charging energy and level spacing can be extracted as shown by the red and black arrows, respectively. The dashed redlines indicate the slope used to estimate the capacitances of the device in n-type region.
SU(4) Kondo effect, which will be treated in more detailbelow [7].Figure 3(c) shows a bias spectroscopy plot in the n-type region of device 1, where the transparency is re-duced even further. Clear Coulomb blockade diamondsand excited states are observed. The numbers indicatethe relative electron filling of the SWCNT quantum dotand a number dividable by four corresponds to a filledshell identified as three small diamonds followed by a bigger diamond. A shell thus consists of a four-fold de-generate level as expected due to orbital and spin de-grees of freedom. The charging energy can be extractedfrom the three small diamonds as half the source-drainheight U c ∼
11 m e V (left red arrow). The orbital split-ting and exchange energy are very small since the threeconsecutive small diamonds in one shell are almost equalin height, e.g., diamonds 5, 6 and 7. Every fourth di-amond is bigger (filled shell) since the addition of the
Dev1(cid:13)p-type(cid:13)Dev2(cid:13)p-type(cid:13)Dev1(cid:13)n-type(cid:13)Dev1(cid:13)p-type(cid:13)x5
FIG. 4: Linear conductance at 4 K, where the behavior evolvesfrom Fabry-Perot to Coulomb blockade resonances because ofthe decreasing coupling to the leads. The red curve showsbroad Fabry-Perot resonances from Fig. 3(a) with no singlehole charging effects, while the four-fold periodicity becomesvisible due to Coulomb blockade for the black curve from Fig.3(b). This structure is even more evident when the couplingdecreases further as shown for the n-type region of device1 (magenta and blue curves). All the curves are translatedalong the gate axis, but not scaled. first electron in a new shell requires both a chargingenergy and a level spacing energy. The additional di-amond height of the larger diamonds thus yields thelevel spacing ∆ E ∼ e V (right black arrow). This isconsistent with the level spacing identified from the ex-cited state lines shown for electron filling 7 (left blackarrow). The asymmetry of the diamonds are due tothe capacitive coupling of the source C s , drain C d andgate C g electrodes to the SWCNT. Estimating the slopes α s = 0 .
51 and α d = − .
02 of the two lines constitut-ing the diamond (dashed red lines) as well as the gatevoltage distance ∆ V g = 33 . C s = 4 . C d = 4 . C g = 4 . α s = C g / ( C g + C d ), α d = − C g /C s and ∆ V g = e/C g ,where C = C s + C d + C g is the total capacitance to thesurroundings, α s/d corresponds to aligning the electro-chemical potentials of the dot with the source/drain andan asymmetric biasing with the drain on ground is used[18]. The equal magnitude of the gate and source/draincapacitive coupling thus makes the diamond asymmetric.The transition between the different transparencyregimes is even more clearly revealed in the linear conduc-tance versus gate voltage shown in Fig. 4. All curves areextracted at zero bias from bias spectroscopy plots in thep- or n-type region of the two small band gap semicon-ducting SWCNTs presented above (device 1 and 2). Themost conducting device (red curve) shows broad Fabry- Perot oscillations with no sign of single hole transportconsistent with holes added continuously to the SWCNTdue to the relatively good coupling. When the couplingto the SWCNT weakens, the holes become more local-ized on the SWCNT and single hole transport is observed(black curve). Four Coulomb blockade resonances emergein each broad Fabry-Perot resonance consistent with eachresonance (level) being four-fold degenerate. For evenweaker coupling a closed quantum dot is formed (ma-genta and blue curves), where the measurement stemsfrom the n-type region of device 1. The curves are notscaled but only shifted along the gate axis. Similar ob-servations have been made by Cao et al. on very clean suspended small band gap semiconducting SWCNTs andalso in our measurement the Coulomb energy seem to di-minish as the transparency increases [9]. This is seen bythe distance between the peaks within one shell for themagenta curve is smaller than in the case of the blackcurve. Intermediate regime
We now want to examine the transport behavior inthe regime of intermediate transparency in more detail.Figure 5(a) shows a bias spectroscopy plot taken fromFig. 3(b) with the filling of only two shells each con-taining a four-fold degenerate level, i.e., addition of 8holes as the gate voltage becomes more negative. Thenumbers 0,..,4 show the number of holes in one of theshells. The Coulomb diamonds are still faintly visibledespite the relative high transparency, where the big di-amond(s) correspond to filled shells. The charging en-ergy and level spacing can be estimated as above yield-ing U c ∼ ∆ E ∼ . α d = − . andα s = 0 . V g = 27 mV) in the shell giving C s = 32 aF, C d = 27 aF and C g = 6 aF. In the case of the closedquantum dot of device 1, the gate capacitance is almostidentical ( ∼ V sd / ∆ V gate ∼ .
008 (reddashed line). Furthermore, the Coulomb blockade reso-nances for the hole transitions 0’-1’ (1’-2’) and 3’-4’ (2’-3’) are shifted equally in bias but with different polarity.Similarly, the Kondo peak for holes fillings 1’ and 3’ (i.e.,one electron) are shifted oppositely in bias ( ∼ ± . V gate ∼ . CONCLUSION
In conclusion measurements on very clean single wallcarbon nanotube quantum dots have been presented. Wefocused on small band gap semiconducting SWCNTs andthe transition from an open to a closed quantum dot(Fabry-Perot interference to Coulomb blockade). The ap-pearance of four peaks is observed in each Fabry-Perotresonance as the transparency is decreased, which is in-terpreted as entering the SU(4) Kondo regime. TheKondo resonances for one hole and one electron in a shellare in some cases shifted to opposite biases. At even lowertransparency clear Coulomb blockade shell structure witha four-fold degeneracy due to orbital and spin degrees offreedom is observed.
FIG. 5: (a-b) Bias spectroscopy plots at 4 K of smaller gate re-gions taken from Fig. 3(b) showing SU(4) Kondo effect. Num-bers indicate the number of holes in a shell. The Coulombdiamonds are shown by white dashed lines and the white dou-ble arrows reveal the level spacing and the charging energy.The red dashed lines show how the Coulomb and Kondo reso-nances are shifted in bias as holes are added to the shell, i.e.,no and finite slope for (a) and (b), respectively. The green,blue and red arrows point to bias cut for hole filling one, twoand three displayed in (c-d). Clear Kondo peaks are observed,while they are shifted to opposite bias for hole filling 1’ and3’ in (d).
ACKNOWLEGDEMENT
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