We investigate charge pumping in carbon nanotube quantum dots driven by the
electric field of a surface acoustic wave. We find that at small driving
amplitudes, the pumped current reverses polarity as the conductance is tuned
through a Coulomb blockade peak using a gate electrode. We study the behavior
as a function of wave amplitude, frequency and direction and develop a model in
which our results can be understood as resulting from adiabatic charge
redistribution between the leads and quantum dots on the nanotube

Anderson, D.

Buitelaar, M. R.

Cobden, D. H.

Jones, G. A. C.

Kashcheyevs, V.

Leek, P. J.

Smith, C. G.

Talyanskii, V. I.

Wei, J.

English

arXiv

We investigate charge pumping in carbon nanotube quantum dots driven by the electric field of a surface acoustic wave. We find that, at small driving amplitudes, the pumped current reverses polarity as the conductance is tuned through a Coulomb blockade peak using a gate electrode. We study the behavior as a function of wave amplitude, frequency, and direction and develop a model in which our results can be understood as resulting from adiabatic charge redistribution between the leads and quantum dots on the nanotube

Buitelaar, MR

Kashcheyevs, V

Leek, PJ

Talyanskii, VI

Smith, CG

Anderson, D

Jones, GA

Wei, J

Cobden, DH

Oxford University Research Archive (ORA)

Physical Review Letters

Adiabatic charge pumping in carbon nanotube quantum dots.

ORA - Oxford University Research Archive

M. R. Buitelaar

V. Kashcheyevs

P. J. Leek

V. I. Talyanskii

C. G. Smith

D. Anderson

G. A. C. Jones

J. Wei

D. H. Cobden

Crossref

Adiabatic Charge Pumping in Carbon Nanotube Quantum Dots

Oxford University Research Archive

UCL Discovery

Adiabatic Charge Pumping in Carbon Nanotube Quantum DotsM.R. Buitelaar,1 V. Kashcheyevs,2,3 P. J. Leek,1 V. I. Talyanskii,1 C.G. Smith,1 D. Anderson,1G.A. C. Jones,1 J. Wei,4 and D.H. Cobden41Cavendish Laboratory, University of Cambridge, Cambridge, CB3 0HE, United Kingdom2Institute for Solid State Physics, University of Latvia, Riga, LV-1063, Latvia3Faculty of Physics and Mathematics, University of Latvia, Riga, LV-1002, Latvia4Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA(Received 21 April 2008; published 17 September 2008)We investigate charge pumping in carbon nanotube quantum dots driven by the electric field of asurface acoustic wave. We find that, at small driving amplitudes, the pumped current reverses polarity asthe conductance is tuned through a Coulomb blockade peak using a gate electrode. We study the behavioras a function of wave amplitude, frequency, and direction and develop a model in which our results can beunderstood as resulting from adiabatic charge redistribution between the leads and quantum dots on thenanotube.DOI: 10.1103/PhysRevLett.101.126803 PACS numbers: 85.35.Kt, 73.23.Hk, 73.63.Fg, 73.63.KvThe dynamics of charge transport in quantum dots is ofconsiderable fundamental interest and has applications infields as diverse as electrical metrology and quantum in-formation processing. A sensitive probe of the response ofa quantum dot to variation of external parameters (such as agate voltage) is the dc charge pumping current generated inthe absence of an applied bias [1]. If this variation is slowcompared with other characteristic energy scales of thesystem, the pumping is adiabatic and does not depend onthe detailed time evolution over the pumping cycle. Inother words, it is of a geometric nature. An example isthe situation in which two parameters "1 and "2 of aquantum dot are periodically varied such that they followa closed path in ð"1; "2Þ parameter space. In that case, thecurrent depends only on the integral of some responsefunction Rð"1; "2Þ over the area A enclosed by the pumpingtrajectory.For open, noninteracting electron systems, this is ex-pressed by the Brouwer formula in which the responsefunction is related to the scattering matrix of the system[2]. In interacting systems, correlations add complexityand modify the predictions. Recent theoretical work hasconsidered pumping in quantum dots with weak interac-tions [3,4], in the Kondo regime [5], in the Coulombblockade regime [6], and for superconducting leads [7,8].Experimentally, however, the rich variety of predictedphenomena has remained largely unexplored [9,10]. Apromising model system to test the various theories ofcharge pumping is carbon nanotubes, since all of thetransport regimes mentioned above have already beenrealized in conventional nanotube devices [11–13].In this Letter, we pursue the potential of carbon nano-tube quantum dots and demonstrate charge pumping up tofrequencies of 2.6 GHz. More specifically, we show that, inthe Coulomb blockade regime, the pumping can be de-scribed geometrically, using a response function that isbased on a simple two-level system to approximate theCoulomb blockade behavior of tunnel-coupled quantumdots.While charge pumping could, in principle, be realizedby modulating side or top gates that are capacitivelycoupled to the nanotubes [14], we adopt a different ap-proach in which the electric field of a surface acousticwave (SAW) pumps charge through the device [15–18].To this end, an individual nanotube is contacted on apiezoelectric quartz substrate by palladium source anddrain electrodes that are separated by 5 m. A side gateelectrode is used to vary the electrostatic potential in thenanotube. Several millimeters beyond each contact are twoSAW transducers having resonant frequencies fSAW ofabout 2.6 GHz and 544 MHz [see the insets in Fig. 1(c)].An important feature of the SAW charge pump is that itavoids direct capacitive coupling between the high-frequency and device electrodes and enables a clear dis-tinction between the signal due to the SAW (occurring onlyat fSAW) and rectified currents from radiated fields (pos-sible at all frequencies). Details of sample fabrication andtransducer operation were described in Ref. [16].Figure 1(a) shows the dc transport properties of thenanotube at temperature T ¼ 5 K. The approximate peri-odicity and large charging energy UC  10–15 meV ob-served indicate that the nanotube is divided into two (ormore) sections such that the conductance is dominated byCoulomb blockade of a single electron puddle that is muchsmaller than the 5 m source-drain separation. This alsofollows from the suppression of the linear-response con-ductance and additional features [arrows in the inset inFig. 1(a)] observed in the differential conductance whichare tentatively interpreted as signatures of the chargingenergy of a second (larger) electron puddle in series; see,e.g., Ref. [19]. Figure 1(c) shows the induced current ISAWin the presence of a SAW field at2:6 GHz in the absenceof a source-drain bias voltage. For a SAW velocity vSAW 3200 m=s on quartz, this corresponds to a SAW wave-PRL 101, 126803 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending19 SEPTEMBER 20080031-9007=08=101(12)=126803(4) 126803-1  2008 The American Physical Societylength SAW ¼ vSAW=fSAW  1:2 m [16]. When a low-power PSAW is applied to the SAW transducers, a dccurrent is induced whose direction alternates as a sensitivefunction of gate voltage Vg. The peak-and-dip features inthe current are clearly correlated with the Coulomb block-ade peaks, and the current changes polarity on reversal ofthe SAW direction.These features were studied in more detail as a functionof SAW amplitude and for both available SAW frequen-cies. Figure 2(a) shows ISAW as a function of PSAW. Thepeak-and-dip features increase in magnitude and moveoutwards as PSAW is increased. When we plot the deriva-tive of the current with respect to the SAW amplitudeVSAW / P1=2SAW, as in Fig. 2(c), it becomes apparent thatthe dominant features move essentially linearly with VSAW.At the highest applied SAW powers (up to 20 dBm or100 mW), the features merge into a background of currentin the direction of the SAW across the whole gate voltagerange [15].All of the observed behavior can be explained by amodel in which disorder in the nanotube causes it tocontain two or more localized electron puddles in series.For small SAW amplitudes (i.e., smaller than the disorderpotential), the SAW modulates the energies of the states inthese puddles, and adiabatic charge redistribution betweenthe puddles and the leads gives rise to a dc current. We firstdiscuss a minimal model in which there are two puddles,each acting as a quantum dot with a single spinless level[see Fig. 3(a)], before introducing asymmetries in thetunnel couplings and multiple levels in the dots to betterreflect the likely experimental situation. Such a simpletwo-level system provides a useful approximation to theCoulomb blockade physics of tunnel-coupled double dots;see e.g., Ref. [20]. The only parameters that enter themodel are the tunnel couplings L and R of the left andright dots to left and right leads, respectively, VSAW, thephase difference  of the SAW between the two dots, andthe tunnel coupling  between them. The instantaneouseffective Hamiltonian of this double-dot system can bewritten asH d ¼ "1  iL=2 =2=2 "2  iR=2 : (1)For any periodic time dependence of the dot energies "1and "2, the adiabatic current [2,21–23] from left to rightcan be calculated as a surface integral [2] I ¼efRA Rð"1; "2Þd"1d"2 over the area A enclosed parametri-cally by the pumping trajectory in the ð"1; "2Þ plane.The response function R can be obtained from Eq. (1)(see, e.g., Ref. [23]) and in the zero-temperature limit takesa straightforward form:-1.65 -1.60 -1.55 -1.50-15-10-50-0.2 0.0 0.2I (nA)P SAW (dBm)-0.20.00.2I (nA)-0.20.00.2-1 10Vg (V)I (ef)P SAW (a.u.)02040-0.2 0.0 0.2I (ef)Vg (V)-1.8 -1.6 -1.4 -1.2V SAW/Vmax0.51(a)(b) (c)ε0 (ΓL)f = 2607 MHz-4 40ε0 (ΓL)FIG. 2 (color online). (a) Color scale representation of theSAW-induced current as a function of PSAW and Vg, showingsign reversal as Vg crosses a Coulomb blockade peak. The linetrace is taken at PSAW ¼ 8 dBm. (b) Calculated current usingthe model explained in the main text. (c) Derivative of the SAW-induced current with respect to the SAW amplitude as a functionof Vg and (normalized) VSAW. The maximum SAW amplitude isestimated to be150 mV [16]. Inset: dI=dVSAW as calculated inthe model, showing the same linear peak splitting.0.10.0-0.1-0.2-0.3-1.2-1.4-1.6 -1.3-1.5-1.70.0-2.0-4.0Vsd = -2.5 mV15-1500.2Vg (V)I (nA)V sd(mV)-10 0 10I (nA)-0.4(b)(c)(a)544 MHz 2.6 GHz-1.05 -1.00 -0.95freq (MHz)2575 2600 2625 Vg = -20-4I SAW (nA)-1.65 VVsdVgCNTARight moving SAW Left moving SAWSAWI SAW (nA)FIG. 1 (color online). (a) Color scale plot of the current as afunction of Vsd and Vg. Inset: dI=dVsd around a Coulomb peak(dark ¼ more conductive). (b) Current trace at Vsd ¼ 2:5 mVshowing Coulomb blockade oscillations. (c) ISAW as a functionof Vg. In red is the current produced by the left-moving SAW at2602 MHz with PSAW ¼ 15 dBm. In blue is the currentproduced by the right-moving SAW at 2607 MHz with PSAW ¼10 dBm. Right inset: Simplified schematic of the device. Leftinset: Photograph of the transducers. Top inset: Frequency de-pendence of the current at Vg ¼ 1:65 V and PSAW ¼ 15 dBm.A peak is observed at the transducer resonant frequency only,demonstrating that the observed current features do not resultfrom directly radiated fields or from photon-assisted tunneling.PRL 101, 126803 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending19 SEPTEMBER 2008126803-2Rð"1; "2Þ ¼ ð32=Þ2LRð"1R þ "2LÞ½2LR2 þ 4"212R þ 4"222L þ ð2  4"1"2Þ2 þ 2L2R2:The response function is shown graphically in Figs. 3(b)and 3(c) for different parameter values. It exhibits onepronounced minimum (blue) and one maximum (red).For the pumping trajectory, we assume the SAW periodi-cally modulates the levels according to"1 ¼ =2þ 1eVg þ eVSAW cosð2ftÞ; (2)"2 ¼ þ=2þ 2eVg þ eVSAW cosð2ftþÞ: (3)Here 1 and 2 are the coupling efficiencies of the gate tothe two dots and  is a level offset parameter. The effect ofthe SAW on the tunnel barriers is thought to be negligiblefor the small SAW amplitudes considered here. For thecase of symmetric coupling L ¼ R [Fig. 3(b)], the peaksof negative and positive pumping current correspond toelliptical trajectories around the two triple points in thestability diagram of the double dot [20,24]. In obviousnotation we may represent these as the ð0; 0Þ ! ð0; 1Þ !ð1; 0Þ ! ð0; 0Þ electron and ð1; 1Þ ! ð1; 0Þ ! ð0; 1Þ !ð1; 1Þ hole cycles, respectively.One property of the pumping current that follows im-mediately is that it changes polarity when the direction ofthe pumping contour reverses, i.e., when the SAW direc-tion is reversed. Another is that, for small SAWamplitudes,I / V2SAW sin [2], because in this limit Rð"1; "2Þ is ap-proximately constant within the area A ¼ e2V2SAW sinenclosed by the trajectories.As Vg is varied, the center of the pumping trajectoryfollows a straight line along the direction "0 which, for thesimple case 1 ¼ 2 and  ¼ 0, is the solid diagonal inFigs. 3(b)–3(d). In this case, the current traces IðVgÞ exhibitsymmetric sign-reversing features. If the dots are single-level and completely symmetric, i.e., L ¼ R and  ¼ 0,then the current traces for different PSAW are dependentonly on the ratio =. This simple situation alreadymatches the experimental data quite well, as shown inFig. 2(b), taking = ¼ 0:5. Figure 3 also illustrates theeffects of asymmetries. A nonzero level offset  shifts thediagonal line [see, e.g., the dotted line in Fig. 3(b)], whileasymmetric coupling to the gate 1  2 changes theslope (dashed line). The result is an asymmetry betweenthe positive and negative current peaks which is mostpronounced in the weak pumping regime. For largerSAW powers, such that eVSAW  , the difference be-tween positive and negative peaks begins to average out.In the limit of strong pumping, the pumping trajectoriesencompass both positive and negative parts of Rð"1; "2Þwhose contributions to the pumping tend to cancel. Thisexplains the experimental observation [Fig. 2(c)] that theopposite-sign peak pairs in the pumping current versus Vgmove apart linearly in VSAW: The peak current occurswhere the integral of R is maximal as a function of "0,and this happens at j"0j  eVSAW.The qualitative behavior of the model is quite insensitiveto details such as asymmetry in the tunnel couplings or amultiplicity of levels in the dots. In particular, the experi-mental situation in which the conductance is dominated bya single quantum dot can be modeled by taking a singlelevel on the right dot and multiple levels on the left, asshown in Fig. 3(d). The sign-reversing nature of theISAW-Vg traces is still a robust feature although signaturesof extra levels may appear for certain pumping trajectories.Finally, we examine whether the experimental effects ofchanging SAW are consistent with the model. Figure 4(a)shows ISAW at fSAW ¼ 2607 MHz in the vicinity of aCoulomb peak. A slight asymmetry between positive andnegative peaks is seen for small PSAW. In the model, this isreproduced [see Fig. 4(b)] by, for instance, assuming an-0.2 0.20R (ΓL-2)-0.2 0.20R (ΓL-2)-2 20R (ΓL-2)ΓL:ΓR:∆ = 1:0.1:0.5 ΓL:ΓR:∆ = 1:1:0.5Source DrainΓL ΓR∆δVSAWVSAW(c)(b)(a)(d)− ε1 (ΓL)0 1-10 1-10 1-101-101-1φ = π/2φ = π/10δε 0(0,1)ΓL:ΓR:∆ = 1:1:0.5(0,0) (1,0)(1,1)δ− ε 2 (ΓL)FIG. 3 (color online). (a) Schematic of a double dot in whichthe single-electron states are periodically modulated by the SAWwith phase difference . (b) Color scale representation of thefunction Rð"1; "2Þ for symmetric coupling to the leads. TheSAW-induced current is proportional to the integral of thisfunction over the area traversed in ð"1; "2Þ space. For  ¼=2, the trajectories are circles with diameters proportional toVSAW; for ¼ =10, they are narrow ellipses. The ordered pairsðn;mÞ indicate the electron occupancy of the quantum dots.(c) Same as panel (b) for asymmetric coupling to the leads.(d) Calculation of Rð"1; "2Þ with 3 levels on the left dot taken tohave spacing equal to 2 and identical couplings to the level inthe other dot.PRL 101, 126803 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending19 SEPTEMBER 2008126803-3asymmetric coupling to the leads and a small offset ,corresponding to the situation depicted in Fig. 3(c). Otherkinds of asymmetry such as 1  2 have a similar result.Figure 4(c) shows the current at fSAW ¼ 544 MHz, usingthe second pair of transducers. The current is now nearlyalways of negative polarity. Figure 4(d) shows the corre-sponding model prediction, which agrees very well. Herethe effect of decreasing fSAW is just to reduce the phasedifference between the dots (! 0 when SAW becomesmuch larger than the device dimensions). This reduces thewidth and area of the elliptical pumping trajectory. Theresult can be understood qualitatively from Fig. 3(c) notingthat the level offset  shifts the line followed by the ellipsecenter towards the negative (blue) regions of Rð"1; "2Þ. Forlarge SAW powers and circular pumping trajectories, theasymmetry in the pumped current is still relatively smallsince the enclosed area includes a domain where Rð"1; "2Þis positive [illustrated by the circles in Fig. 3(c)]. However,after decreasing PSAW (smaller radius) or fSAW (narrowerellipse), the integral includes only regions of negativeRð"1; "2Þ, enhancing the asymmetry. Note that the en-hancement of the asymmetry in ISAW when decreasingSAW power or frequency is expected whenever the centerof the pumping trajectory follows a line in ½"1; "2 parame-ter space that is asymmetric with respect to the minima andmaxima of Rð"1; "2Þ, irrespective of the precise form ofthis function. The results in Fig. 4 are therefore consistentwith the model also when in the experiment Rð"1; "2Þ issignificantly more complex than the model situationsshown in Fig. 3. To completely describe the experiment(the model actually underestimates the amount of pumpedcurrent and does not account for the fine structure observedat 544 MHz) would require a better knowledge of themicroscopic details of the device than we have at present.We thank A. Aharony, O. Entin-Wohlman, andL. Levitov for discussions. M. R. B. has been supportedby the UK QIP IRC (GR/S82176/01). V.K. has beensupported by Latvian Council of Science, EuropeanSocial Fund, and German–Israeli Project Cooperation(DIP).[1] D. J. Thouless, Phys. Rev. B 27, 6083 (1983).[2] P.W. Brouwer, Phys. Rev. B 58, R10 135 (1998).[3] I. L. Aleiner and A.V. Andreev, Phys. Rev. Lett. 81, 1286(1998).[4] P.W. Brouwer, A. Lamacraft, and K. Flensberg, Phys. Rev.B 72, 075316 (2005).[5] T. Aono, Phys. Rev. Lett. 93, 116601 (2004).[6] J. Splettstoesser, M. Governale, J. Ko¨nig, and R. Fazio,Phys. Rev. Lett. 95, 246803 (2005).[7] M. Blaauboer, Phys. Rev. B 65, 235318 (2002).[8] J. Splettstoesser, M. Governale, J. Ko¨nig, F. Taddei, and R.Fazio, Phys. Rev. B 75, 235302 (2007).[9] H. Pothier, P. Lafarge, C. Urbina, D. Esteve, and M.H.Devoret, Europhys. Lett. 17, 249 (1992).[10] M. Switkes et al., Science 283, 1905 (1999).[11] J. Nyga˚rd, D.H. Cobden, and P. E. Lindelof, Nature(London) 408, 342 (2000).[12] M. Bockrath et al., Nature (London) 397, 598 (1999).[13] M. R. Buitelaar et al., Phys. Rev. Lett. 91, 057005 (2003).[14] Y. Wei, J. Wang, H. Guo, and C. Roland, Phys. Rev. B 64,115321 (2001).[15] P. J. Leek et al., Phys. Rev. Lett. 95, 256802 (2005).[16] M. R. Buitelaar et al., Semicond. Sci. Technol. 21, S69(2006).[17] J. Ebbecke, C. J. Strobl, and A. Wixforth, Phys. Rev. B 70,233401 (2004).[18] Y.-S. Shin et al., Phys. Rev. B 74, 195415 (2006).[19] J. Park and P. L. McEuen, Appl. Phys. Lett. 79, 1363(2001).[20] W.G. van der Wiel et al., Rev. Mod. Phys. 75, 1 (2002).[21] M. Bu¨ttiker, H. Thomas, and A. Preˆtre, Z. Phys. B 94, 133(1994).[22] O. Entin-Wohlman, A. Aharony, and Y. Levinson, Phys.Rev. B 65, 195411 (2002).[23] V. Kashcheyevs, A. Aharony, and O. Entin-Wohlman,Phys. Rev. B 69, 195301 (2004).[24] W. J.M. Naber, T. Fujisawa, H.W. Liu, and W.G. van derWiel, Phys. Rev. Lett. 96, 136807 (2006).[25] The correspondence between  and SAW is uniquelydefined only when SAW is larger than twice the distancebetween the centers of the two dots.-15-5-1.00 -0.950-10105Vg (V)f = 2607 MHzφ = π/100.20.0-0.2255075-1.5 0 1.50.10.0-0.1-0.25 0.250I (nA) -0.5 0.50f = 544 MHz-0.10 0I (nA)P SAW (dBm)I (nA)-0.30 0I (ef)(a) (b)(c) (d)P (a.u.)I (ef)ε0 (ΓL)-1.00 -0.950.10.0-0.1I (nA)P SAW (dBm)-1.5 0 1.50.50.0-0.5I (ef)φ = π/2255075P (a.u.)I (ef)FIG. 4 (color online). (a) Color scale representation of theSAW-induced current as a function of SAW power and Vg forfSAW ¼ 2607 MHz. (b) Calculated SAW current for a doublequantum dot model with an asymmetry between the tunnelcouplings L and R, as discussed in the text. The parametersare L:R:: ¼ 1:0:0:1:0:5:0:3, and the phase difference  ¼=2. (c) SAW-induced current for the same resonance peak as inpanel (a) for fSAW ¼ 544 MHz. (d) Calculated SAW current asin panel (b) but for a phase difference  ¼ =10, correspondingto the fivefold increase of SAW [25].PRL 101, 126803 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending19 SEPTEMBER 2008126803-4