Controlling the conductance and noise of driven carbon-based Fabry-Perot devices
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Controlling the conductance and noise of driven carbon-based Fabry-P´erot devices
Luis E. F. Foa Torres and Gianaurelio Cuniberti Institute for Materials Science and Max Bergmann Center of Biomaterials,Dresden University of Technology, D-01062 Dresden, Germany (Dated: October 28, 2018)We report on ac transport through carbon nanotube Fabry-Perot devices. We show that tuning theintensity of the ac gating induces an alternation of suppression and partial revival of the conductanceinterference pattern. For frequencies matching integer multiples of the level spacing of the system∆ the conductance remains irresponsive to the external field. In contrast, the noise in the low biasvoltage limit behaves as in the static case only when the frequency matches an even multiple of thelevel spacing, thereby highlighting its phase sensitivity in a manifestation of the wagon-wheel effectin the quantum domain . Achieving control of the electrical response of nanome-ter scale devices by means of external fields is a main goalof nanoelectronics. This quest for control is usually pur-sued using static electric or magnetic fields, whereas theuse of time-dependent excitations [1, 2] remains muchless explored. Besides offering captivating phenomena[3, 4, 5, 6], such as the generation of a dc current at zerobias voltage [3, 6, 7], understanding the influence of acfields becomes necessary if nanoscale devices are going tobe integrated in everyday electronics.Carbon based materials including fullerenes, graphiticsystems and polymers, are promising building blocks forthese devices. Among them, carbon nanotubes [8] standout due to their extraordinary mechanical and electricalproperties [8]. Applications include transistors, sensorsand interconnects [9]. As compared to other molecularsystems, single walled carbon nanotubes offer the uniqueopportunity of achieving almost ballistic transport. In-deed, thanks to the low resistance of the nanotube-electrodes contacts, Fabry-Perot (FP) conductance oscil-lations were experimentally observed [10] and the currentnoise measured [11]. Although some experimental [12]and theoretical [13, 14, 15, 16, 17] studies on the effectsof ac fields on nanotubes are available, the effect of acfields on the conductance and noise in this Fabry-Perotregime is unknown: Is it possible to tune the interferencepattern through ac fields? Can ac fields help us to unveilphase sensitive information encoded in the current noise?In this Letter, we answer the above questions by mod-elling the electronic transport through driven carbonnanotube FP resonators. We show that by tuning thefrequency and intensity of a harmonic gate (see scheme inFig. 1) one can control not only the conductance but alsothe current noise [18]. Moreover, we show that the cur-rent noise carries phase sensitive information not avail-able in the static case.
Tight-binding model and Floquet solution.
Several ap-proaches can be used to describe time-dependent trans-port. They include: the Keldysh formalism [19, 20, 21],schemes that use density functional theory [22], the equa-tion of motion method [23], and schemes that exploit thetime-periodicity of the Hamiltonian through Floquet the-
FIG. 1: (color online) Top, scheme of the device consideredin the text, a CNT connected to two electrodes and an acgate. The panels marked with a, b, c and d, are the Fabry-Perot conductance interference patterns (as a function of biasand gate voltages) observed with no ac gating (panel a) ismodified when different different driving frequencies and am-plitudes are applied: b (suppression), c (revival and phaseinversion) and d (robustness). White and dark blue corre-spond to maximum and minimum conductances respectively. ory [24, 25]. Here, as a general framework we use the Flo-quet scheme [25] combined with the use of Floquet-Greenfunctions [26]. Within this formalism, the dc componentof the time dependent current as well as the dc conduc-tance (called simply conductance hereafter), can be fullywritten in terms of the Green’s functions for the system
FIG. 2: (color online) Half amplitude of the FP conductanceoscillations as a function of the ac field intensity. These resultsare for a metallic zig-zag nanotube of length L = 440 nm and γ t = 0 . γ , at zero bias voltage and temperature. [2, 26]. The current noise can be obtained from the corre-lation function S ( t, t ′ ) = h [∆ I ( t )∆ I ( t ′ ) + ∆ I ( t ′ )∆ I ( t )] i , ∆ I ( t ) = I ( t ) − h I ( t ) i being the current fluctuation op-erator. The noise strength can be characterized by thezero frequency component of this correlation function av-eraged over a driving period, ¯ S, which can be casted ina convenient way within this formalism [2]. Further sim-plifications can be achieved by using the broad-band ap-proximation and an homogenous gating of the tube [2].For simplicity we consider an infinite CNT describedthrough a π -orbitals Hamiltonian [8] H e = P i E i c + i c i − P h i,j i [ γ i,j c + i c j + H . c . ], where c + i and c i are the creationand annihilation operators for electrons at site i , E i arethe site energies and γ i,j are nearest-neighbors carbon-carbon hoppings. To model the FP interferometer, a cen-tral part of length L (the “sample”) is connected to therest of the tube through matrix elements γ t smaller thanthe hoppings in the rest of the tube which are taken tobe equal to γ = 2 . L can be used to tune thelevel spacing ∆ ∝ /L . For the case of uniform gatingof the sample, it is modeled as an additional on site en-ergy E j ∈ CNT = eV g + eV ac cos(Ω t ). This non-interactingmodel is justified when screening by a metallic substrateor by the surrounding gate lessens electron-electron inter-actions. When these interactions come into play effectsbeyond our present scope may emerge [16]. Suppression, revival and robustness of FP oscillationsunder AC gating.
In the following we consider a uniformgating of the tube, the same effects are expected for an acbias or illumination with radiation of wavelength largerthan the device length as in Ref. [27] where frequen-cies of up to 3THz were achieved. At low to moderatefrequencies ( ~ Ω < ∆), our main observation is that theamplitude of the FP conductance oscillations is reducedand can even be suppressed by tuning the intensity of thefield. This is clearly shown in Fig. 2 (solid line) wherethe half amplitude of the conductance oscillations (com- FIG. 3: (color online) a) Contour plot showing the half am-plitude of the FP oscillations as a function of V ac and Ω atzero temperature and bias voltage. b) Same contour plot forthe current noise ¯ S . White in both figures is for maximumvalues of the half amplitude and noise respectively. puted at zero temperature) is shown as a function of theac field intensity V ac . Different curves correspond to dif-ferent frequencies. Interestingly, for certain frequencies,the FP pattern is completely suppressed by tuning theintensity of the ac field, an effect which survives in theadiabatic limit.Figure 1, shows the FP patterns obtained at the pointsmarked on the curves in Fig. 2 . There, by comparisonwith the static case (a), one can observe the suppression(b) and subsequent revival with a phase inversion (c)of the FP oscillations as V ac increases. On the otherhand, panel d shows a situation in which the frequency istuned to meet the wagon-wheel or stroboscopic condition ( ~ Ω ≈ n ∆ , n integer). The overall dependence of the FPamplitude on both ~ Ω and V ac is better captured by thecontour plot in Fig. 3-a. White corresponds to maximumvalues of the half amplitude and black to zero values.In general, although the field produces no appreciablechange in the conductance whenever ~ Ω ≈ n ∆, a staticbehavior in the noise requires a more stringent conditionas we will see later.To rationalize these features let us first consider theadiabatic limit. In this limit, the period of the ac os-cillation is long enough such that at each instant oftime the system can be considered as static with an ap-plied field which coincides with its instantaneous value.Within this approximation, the conductance is given by G ad = G avg + A × J (2 πeV ac / ∆) × cos(2 πeV g / ∆), where A and G avg are the half amplitude and average value of theconductance in the static situation (vanishing ac field).We can see that the amplitude is modulated by a factor J (2 πeV ac / ∆). Thus, the FP interference is destroyedwhenever 2 πeV ac / ∆ is a root of J . This relation wouldallow the experimental determination of the effective am-plitude of the ac gating.Deviations from the adiabatic result are due when thecondition ~ Ω ≪ ∆ is not fulfilled. In this case, by assum-ing the broad-band approximation, the dc conductancecan be written in the Tien-Gordon [28] form: G ( ε F , V g ) = P n | a n | G static ( ε F + n ~ Ω , V g ) , where a n = J n ( eV ac / ~ Ω)represents the probability amplitude of emitting (absorb-ing) n photons. Thus, the averaging of the conductancetakes place only among energies differing in discrete stepsthereby rendering the smoothing of the oscillations lesseffective. A higher intensity of the field is thereforeneeded to suppress the conductance oscillations. Indeed,whenever ~ Ω is commensurate with ∆ the contrast of theFP pattern is insensitive to the ac field intensity. Thedash-dotted line in Fig. 2 shows the amplitude vs. V ac for a frequency close to such a condition.Length dependence: In the infinite tube limit (∆ → ∼ max( eV ac , N ~ Ω), where N is the typical number of photons) is small enough(max( eV ac , N ~ Ω) ≪ ε F ), thereby giving generality to ourresults for systems other than nanotubes. The reservoirsare assumed to be in thermal equilibrium and current-induced heating is assumed not to compromise the tubestability. The typical current-induced heat flow is ofabout 50nW, much below the experimental limits re-ported in [29]. AC effects on the current noise and quantum wagon-wheel effect.
The current noise as characterized by thedimensionless Fano factor (not shown here) is affected bythe ac field only at high frequencies and driving ampli-tudes ( ~ Ω , V ac ∼ ∆). More interesting is the low bias,low temperature limit. For a static Hamiltonian, thenoise power S vanishes in this limit as no fluctuationsremain in the electron distributions. This is in strikingcontrast with the situation in which an ac field is appliedto the conductor where there is always a non-zero currentnoise. This is because even at zero temperature and bias,the ac field introduces probabilistic scattering processes(photon-assisted transitions) which add uncertainty tothe effective electron distributions, thereby giving a non-vanishing contribution to the current noise [2]. In Fig.3-b a contour plot of the current noise S as a functionof the driving frequency and ac intensity is presented.The color scale ranges from 6 × − A / Hz (black) to2 × − A / Hz (white). The thermal contribution tothe noise at 800mK is estimated to be 6 × − A / Hzand would therefore not be noticeable in this color scale.Figure 3-b shows that, in contrast to what is observedfor the conductance, whenever the wagon-wheel or stro-boscopic condition is attained S does not behave as inthe static case (i. e. different from zero). Indeed, thereis a suppression of S only for ~ Ω commensurate with twice ∆. This is due to the fact that the noise under acconditions is sensitive to the phase of the transmission amplitude which changes only by π (and not 2 π ) fromone resonance to the next one. In between these minimathere are local maxima whose intensity is proportional to V ac . A similar situation was reported for a double quan-tum dot [30], and for a driven system composed of twobarriers of varying strength and a uniform varying po-tential in between [31]. The noise suppression observedhere can be understood by using a simplified expressionfor the noise as done before for the conductance. Themain conclusion is that when ~ Ω = 2 n ∆ ( n integer), thenoise behaves as in the static case. This effect can bevisualized as a manifestation of the wagon-wheel effectin the quantum domain where a static behavior in thephase sensitive noise requires a doubling of the strobo-scopic frequency. Conclusions.
In summary, we have analyzed the ef-fects of ac gating on the conductance and noise of FPnanotube-based resonators. It was shown that the acfield can be used to tune the conductance and noise ofthe device. Suppression of the FP conductance oscilla-tions can be achieved even in the adiabatic limit by tun-ing the driving amplitude. In contrast, when the driv-ing frequency matches (a multiple of) the level spacing(wagon-wheel condition), the conductance coincides withthe one of the static system. In contrast, the noise coin-cides with the static one only when ~ Ω is commensuratewith twice ∆ (quantum wagon-wheel condition), there-fore highlighting its phase sensitivity. Although here weconsidered only nanotubes, our main results are expectedto be valid for more general FP resonators.We thank S. Kohler and M. Moskalets for useful com-ments and M. del Valle for discussions. This work wassupported by the Alexander von Humboldt Foundationand by the EU project CARDEQ under contract No.IST-021285-2. Computing time was provided by ZIH-TUD. [1] G. Platero and R. Aguado, Phys. Rep. , 1 (2004).[2] S. Kohler, J. Lehmann, and P. H¨anggi, Phys. Rep. ,379 (2005).[3] D. J. Thouless, Phys. Rev. B , 6083 (1983).[4] F. Grossmann, T. Dittrich, P. Jung, and P. H¨anggi, Phys.Rev. Lett. , 516 (1991).[5] M. Wagner, Phys. Rev. A , 798 (1995).[6] B. L. Altshuler and L. I. Glazman, Science , 1864(1999); M. Switkes, C. M. Marcus, K. Campman, and A.C. Gossard, Science ,1905 (1999).[7] B. Kaestner, V. Kashcheyevs, S. Amakawa, M. D. Blu-menthal, L. Li, T. J. B. Janssen, G. Hein, K. Pierz, T.Weimann, U. Siegner, and H. W. Schumacher, Phys. Rev.B , 153301 (2008); A. Fujiwara, K. Nishiguchi, Y. Ono,Appl. Phys. Lett. , 042102 (2008); B. Kaestner, C. Le-icht, V. Kashcheyevs, K. Pierz, U. Siegner, and H. W.Schumacher, ibid. , 012106 (2009).[8] J.-C. Charlier, X. Blase, and S. Roche, Rev. Mod. Phys. , 677 (2007); R. Saito, G. Dresselhaus, and M. S. Dres-selhaus, Physical Properties of Carbon Nanotubes (Impe-rial College Press, London, 1998).[9] J. C. Coiffic, M. Fayolle, S. Maitrejean, L. E. F. FoaTorres, and H. Le Poche, Appl. Phys. Lett. , 252107(2007).[10] W. Liang, M. Bockrath, D. Bozovic, J. H. Hafner, M.Tinkham, and H. Park, Nature , 665 (2001).[11] F. Wu, P. Queipo, A. Nasibulin, T. Tsuneta, T. H. Wang,E. Kauppinen, and P. J. Hakonen, Phys. Rev. Lett. ,156803 (2007); L. G. Herrmann, T. Delattre, P. Morfin,J.-M. Berroir, B. Placais, D. C. Glattli, and T. Kontos, ibid. , 156804 (2007); Na Young Kim, P. Recher, W.D. Oliver, Y. Yamamoto, J. Kong, and H. Dai, ibid. ,036802 (2007).[12] J. Kim, H. So, N. Kim, J. Kim, K. Kang, Phys. Rev.B , 153402 (2004); Z. Yu and P. Burke, Nano Lett. , 1403 (2005); C. Meyer, J. Elzerman, and L. Kouwen-hoven, Nano Lett. , 295 (2007); P. J. Leek, M. R. Buite-laar, V. I. Talyanskii, C. G. Smith, D. Anderson, G. A.C. Jones, J. Wei, and D. H. Cobden, Phys. Rev. Lett. , 256802 (2005).[13] C. Roland, M. Buongiorno Nardelli, J. Wang, and H.Guo, Phys. Rev. Lett. , 2921 (2000).[14] P. A. Orellana and M. Pacheco, Phys. Rev. B , 115427(2007).[15] C. Wuerstle, J. Ebbecke, M. E. Regler, and A. Wixforth,New J. of Phys. , 73 (2007).[16] M. Guigou, A. Popoff, T. Martin, and A. Crepieux, Phys.Rev. B , 045104 (2007). [17] L. Oroszlany, V. Zolyomi, and C. J. Lambert,arXiv:0902.0753 (unpublished).[18] Ya. M. Blanter and M. B¨uttiker, Phys. Rep. , 1(2000).[19] H. M. Pastawski, Phys. Rev. B , 4053 (1992).[20] A.-P. Jauho, N. S. Wingreen, and Y. Meir, Phys. Rev.B , 5528 (1994); C. A. Stafford and N. S. Wingreen,Phys. Rev. Lett. , 1916 (1996).[21] L. Arrachea, Phys. Rev. B , 125349 (2005).[22] S. Kurth, G. Stefanucci, C. Almbladh, A. Rubio, and E.K. U. Gross, Phys. Rev. B , 035308 (2005).[23] A. Agarwal and D. Sen, J. Phys. Cond. Matt. , 046205(2007).[24] M. Moskalets and M. B¨uttiker, Phys. Rev. B , 205320(2002).[25] S. Camalet, J. Lehmann, S. Kohler, and P. H¨anggi, Phys.Rev. Lett. , 210602 (2003).[26] L. E. F. Foa Torres, Phys. Rev. B , 245339 (2005).[27] B. J. Keay, S. Zeuner, S. J. Allen, Jr., K. D. Maranowski,A. C. Gossard, U. Bhattacharya, and M. J. W. Rodwell,Phys. Rev. Lett. , 4102 (1995).[28] P. K. Tien and J. R. Gordon, Phys. Rev. , 647 (1963).[29] L. Shi, J. Zhou, P. Kim, A. Bachtold, A. Majumdar, andP. L. McEuen, arXiv:0904.3284v1 (unpublished).[30] M. Strass, P. H¨anggi, and S. Kohler, Phys. Rev. Lett. ,130601 (2005).[31] M. Moskalets and M. B¨uttiker, Phys. Rev. B78