Current without bias and diode effect in shuttling transport of nanoshafts
K. Morawetz, S. Gemming, R. Luschtinetz, L. M. Eng, G. Seifert, A. Kenfack
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Current without bias and diode effect in shuttling transport of nanoshafts
K. Morawetz , , S. Gemming , R. Luschtinetz , L. M. Eng , G. Seifert , A. Kenfack Forschungszentrum Dresden-Rossendorf, PF 51 01 19, 01314 Dresden, Germany Max-Planck-Institute for the Physics of Complex Systems, N¨othnitzer Str. 38, 01187 Dresden, Germany Institute of Physical Chemistry and Electrochemistry, TU Dresden, 01062 Dresden, Germany and Institute of Applied Photophysics, TU Dresden, 01062 Dresden, Germany
A row of parallely ordered and coupled molecular nanoshafts is shown to develop a shuttlingtransport of charges at finite temperature. The appearance of a current without applying an externalbias voltage is reported as well as a natural diode effect allowing unidirectional charge transportalong one field direction while blocking the opposite direction. The zero-bias voltage current appearsabove a threshold of initial thermal and/or dislocation energy.
PACS numbers: 73.63.Fg, 73.23.-b,85.85.+j,87.15.hj,05.60.Cd
Organic field effect transistors (OFETs) based on dif-ferent polymers [1, 2, 3] attract an increased interest dueto numerous potential applications as flexible and lowcost storage and microelectronic devices. For achievingexcellent electric properties such as high charge carriermobilities and low resistive losses, which are requiredfor technologically attractive device applications, a highstructural ordering of the semiconductor molecular mate-rial is necessary [4, 5]. Oligothiophenes and their deriva-tives can be regarded as one of the most promising sys-tems for building such self-organized structures acrossmultiple length scales due to the variety of intra- and in-termolecular interactions, which originates from the po-larizability of the sulfur electrons and the aromatic π -electron system [6, 7, 8]. Just recently an OFET struc-ture has been built from ultra-thin self-assembly filmsmade up from oligothiophenes, which are arranged in ahigh-order lamellar stacking perpendicular to the sub-strate surface [3].In general, the charge transport is largest in the di-rection perpendicular to the plane of the thiophene rings[9, 10]. This finding addresses already the basic differ-ence between electronic transport in organic conductorsand in classical semiconductors. Though the band gapbetween the lowest unoccupied and the highest occupiedlevel of about 3 eV in quarterthiophene suggests an anal-ogy to the conduction and valence band in semiconduc-tors, there are crucial differences. While in conventionalsemiconductors the transport is due to delocalized statesand limited by the scattering, the transport in moleculesis due to localized states dominated by hopping [11].In this paper we suggest a new mechanism of chargetransport which is possible for flexible molecular tubes,shafts or any elastically deformable assembly which couldprovide an alternative explanation of transport proper-ties in nanoshaft-based OFETs. The performance ofOFETs based on oligothiophene [3] shows characteris-tic features, e.g. the current starts at a certain thresholdof gate voltage and reaches a saturation value for certaindrain voltages. We will show in the following that for aregular arrangement of long elastic molecules a shuttling transport can be established, which can model these fea-tures. In particular, we will describe a shuttling effectof nanoshafts leading to a diode behavior as well as toa directed current without external bias voltage. Thisresembles the ratchet effect described as Brownian mo-tors [12, 13, 14] also realizable with oscillating laser fields[15, 16]. The thermal noise plays an important role foractivating such motors to overcome a certain barrier [17].In the effect presented here the activating threshold isgiven by an initial kinetic or potential energy necessaryfor the first nanoshaft to reach the contacts.Shuttling in single-dot devices had been already inthe center of interest [18, 19]. Besides promising ap-plications for nano-electromechanical devices the cou-pling of quantized transport with micromechanical can-tilevers [20] bridges the classical mechanical and thequantum physics in an exciting way. Our model, ana-lytically solvable, demonstrates that a similar shuttlingmode as known from single-dot transport by mechani-cal cantilevers is achievable by a row of elastically cou-pled molecular tubes. The model is based on the coupledspring chain. Despite their simplicity, such models sur-prise with the ability to describe unknown phenomena ofclassical quantization phenomena in velocities [21]. Themodel of freely rotating linear chains has been investi-gated also in view of quantization [22] and has been ap-plied to entanglement problems [23]. We note that theshuttling transport described here should not be mixedup with the so called ‘chain shuttling’ in polymerizationreactions, where a growing polymer chain is transferredbetween different metal catalysts [24, 25].The elasticity of polymers with respect to the bendingand rigidity has been studied by molecular dynamic sim-ulations [26]. The model of hard-sphere chains is usedto describe the stiffness and diffusion in polymers [27].These elastic properties play an important role in themesoscale modeling of carbon nanotubes [28]. Such car-bon nanotubes on glassy carbon electrodes can enhancethe current response by a factor of 1000 as reported in[29]. Therefore it is of high actual interest to reinves-tigate the transport in nanotubes. We will concentrate FIG. 1: Snapshots of time evolution of the chain of shuttling nanoshafts (left) with positive charge (green) and negative charges(red) or no charge (grey). The kinetic energy, currents on both sides and the recombination rate are plotted from left to right. in this paper on charge transport perpendicular to thedirection of nanoshafts.Specifically, we consider N nanoshafts of length L par-allely oriented in the y -direction. The shafts are standingperpendicular on the substrate, and are periodically ar-ranged along the x -direction at a certain distance fromeach other (see Fig. 1). We expect them to couple elas-tically by their top ends. Then the shafts bend in xdirection when applying the tangential force F with thedisplacement x ( z ) = F z E I (cid:16) L − z (cid:17) = 32 x ( L ) (cid:16) zL (cid:17) (cid:16) − z L (cid:17) (1)where E is the Young modulus and I = R z dA is thearea moment of inertia being I = πr / r or I = ab /
12 fornanoshafts with rectangular cross section of sides a and b . The maximal displacement on the top ends z = L is x ≡ x ( l ) = F L / E I from which the spring con-stant k = F/x = 3 E I/L is given provided we knowthe Young modulus. In other words we have a coupledlinear chain of top ends obeying the differential equations d dt x i = ω ( x i − − x i + x i +1 ) with ω = k/m given bythe mass of the nanoshafts. The nanoshafts are locatedin-between two solid contacts such that the coupled equa-tions are bounded which can be formally expressed by x = x N +1 = 0. Additionally each nanoshaft can carrya charge q i which can be exposed to a time-dependentexternal field E ( t ). The total coupled equation systemfor the time-dependence of the nanoshaft top ends reads d x i ( t ) dt = ω [ x i − ( t ) − x i ( t ) + x i +1 ( t )] + a i ( t ) (2)with ω = 3 E I/mL and a i ( t ) = q i E ( t ) /m . For simplic-ity we will use the time in units of 1 /ω further on. Forquarterthiophene we have a typical value of k = 1N/m and a mass of m = 5 . × − g which leads to the scale ω = 1 . l = 0 . mω l /q =6 . × V/m again given explicitly for quarterthiopheneand the charge current in units of qω = 21 . q = e . The typical energyunits are mω l / . φ nν = p / ( N + 1) sin [ nνπ/ ( N + 1)], x i ( t ) = N X n =1 φ ni (cid:20) ( c n cos ω n t + d n sin ω n t )+ t Z dt ′ sin ω n ( t − t ′ ) ω n N X m =1 a m ( t ′ ) φ nm (3)with the eigenfrequencies ω n = 2(1 − cos [ nπ/ ( N + 1)])and the initial condition determining c n = P ν φ nν x ν (0)and d n = P ν φ nν ˙ x ν (0) /ω n .Now we proceed by considering the nanoshafts betweentwo oppositely charged plates, q sides , modeling the con-tacts. Each nanoshaft can carry a negative charge q i < q i > q sides − q i . Each time such an event happens the timeevolution according to the analytical solution (3) restartswith the new initial conditions and new charge distribu-tion. In this way we use the speed of analytical solutiontogether with the nonlinear process of recharging.For exploratory reasons we restrict ourselves to 11nanoshafts of length 20 l and have arranged them at adistance of 3 l . These are typical parameters of quar-terthiophene used in recent experiments [3]. First wecharge all shafts equivalently and start the process ofshuttling by bending the most left tube to the left con-tact, allowing to transfer the first charge to the left lead. FIG. 2: The time evolution for the recombination rate(above), the total kinetic energy (middle) and the total cur-rent (below) for 3 different applied fields. The time unites arein ω = 3 EI/mL and the length units are l . In figure 1 we have plotted different snapshots of thetime evolution in units of 1 /ω . We see that the systemstarts to shuttle and to transport positive charges fromthe left to the right and negative charges from the right tothe left side. This leads to steps in the current counted ascharges delivered on the corresponding side per elapsedtime. The recombination rate is increasing with time;complete movies can be found in [30].The snapshots in figure 1 are actually taken from a runwithout applying any external field. The gates serveshere merely as a reservoir of charges, not as capacitorplates. The astonishing observation is now that even forsuch an unbiased case a finite total current is developing.This fact is demonstrated in figure 2 where the time evo-lution of the recombination rate, the kinetic energy, andthe total current which is the sum of right and left cur-rents, are plotted for different external fields. The totalcurrent and the recombination rate reach saturation withincreasing time while the kinetic energy remains constanton average. If one applies an electric field of E = 0 . E = − .
2, the sys-tem is accelerated and delivering more and more charges, limited finally only by the length of the tubes. In otherwords we observe a pure diode effect just due to shut-tling of coupled chains of nanoshafts. The total currentreached after long times for different applied fields is de-picted in figure 3 and illustrates this diode effect as aquite systematic one.One may argue that this diode effect as well as theobservation of a current without external bias is due tothe initial charging of the nanoshafts. In the figure 3we have plotted the finally reached total current versusthe applied field for two cases, the initially charged caseconsidered so far, and the case where initially all shiftsare uncharged. We see that identical final currents areobtained without external bias for both cases and thediode effect is present, as well. -0.4 -0.2 0 0.2 0.4
E [m ω l /q] J t o t [ q ω ] completely charged initially: q i =1uncharged initially: q i =0 FIG. 3: The total current versus applied electric field for twocases, initially uncharged tubes and initially charged tubes.
Now we want to return to the puzzling observation thata charge transport occurs even without external bias. Tounderstand the conditions under which such an effect oc-curs we have to clarify the dependencies on the initialkinetic and potential energy and the geometric config-urations. The length L of the nanoshafts determinesonly the time scale ω and scales out here. Thereforethe distance between the nanoshafts ∆ x in the row isthe only geometric parameter left. Further, the initialkinetic and potential energy we determine by the elonga-tion x and speed v of the leftmost nanoshaft while allother nanoshafts are initially not displaced and at rest.In figure 4 we plot the dependence of the reached cur-rent on the initial displacement and the initial velocity.Using dimensionless units for the velocity v ∗ = v/ωl and the displacement x ∗ = x /l the initial energy is E tot = mω l ( v ∗ + x ∗ ) / l . We see in the right plot of figure 4 that the cur-rent is increasing quadratically with the initial velocity,i.e. the current is proportional to the initially depositedkinetic energy. This initial kinetic energy can be equiva-lently realized by thermal motion and characterized by atemperature. In the simulation described so far we bendthe leftmost nanoshaft sufficiently close to the electrodesuch that the shuttling can happen even without initialkinetic energy. If we bend it less we obtain a thresholdin kinetic energy analogously to the thermal threshold inBrownian motors [17]. The dependence of the current onthe initial position is seen in the left plot of figure 4. Wehave chosen an initial velocity below the threshold whichshows that a threshold in the position or potential energyhas to be overcome in order to create the shuttling cur-rent. We note that the shuttling transport develops onlyif we start with asymmetric initial conditions. Otherwisecounter-oscillations of the left and right half of shaftsblock any transport. Therefore the symmetry breakingis due to initial conditions and this model illustrates anelectronic ratchet effect with thresholds in kinetic or po-tential energy corresponding to a high system tempera-ture or initially elongation being mω ∆ x / -3 -2 -1 0 1 2 3 x / ∆ x v [ ω l ]0.050.10.15x =-1.5 ∆ xv =1.5 ω ∆ x J t o t [ q / ω ] FIG. 4: The total current without voltage bias versus initialdisplacement (left) and versus initial velocity (right) of themost left nanoshaft.
In summary we have found that a chain of perpendic-ularly arranged coupled and chargeable nanoshafts showa shuttling transport of charges. As a surprising effectit turns out that a finite current is established alreadywithout external bias only due to the initial asymmet-ric deformation of the nanoshafts. The effect observedhere is reminiscent of the ratchet effect. Even in the ab-sence of a net macroscopic force or noise a current can begenerated. This resembles a lot the prominent motor pro-teins found inside cells. There, proteins such as Mysosinand Kynesin use chemical energy which is gained fromhydrolysis of ATP into ADP to move along asymmetricpathways transporting vesicles inside cells, contractingmuscles and being important in the process of cell divi-sion [13, 14].This work was supported by DFG Priority Program1157 via GE1202/06 and the BMBF and by EuropeanESF program NES. [1] T. Yasuda and T. Tsutsui, Chem. Phys. Lett. , 395(2005).[2] T. Koyanagi, M. Muratsubaki, Y. Hosoi, T. Shibata,K. Tsutsui, Y. Wada, and Y. Furukawa, Chem. Lett. ,20 (2006).[3] K. Haubner, E. Jaehne, H. J. P. Adler, D. Koehler,C. Loppacher, L. M. Eng, J. Grenzer, A. Herasimovich,and S. Scheinert, phys. stat. sol. (a) (2008).[4] C. D. Dimitrakopoulos and P. R. L. Melenfant, Adv.Mater. , 99 (2002).[5] F. Garnier, Acc. Chem. Res. , 209 (1999).[6] M. Melucci, M. Gazzano, G. Barbarella, M. Cavallini,F. Biscarini, P. Maccagnani, and P. Ostoja, J. Am.Chem. Soc. , 10266 (2003).[7] E. A. Marseglia, F. Grepioni, E. Tedesco, and D. Braga,Mol. Cryst. Liq. Cryst. , 137 (2000).[8] G. Barabrella, M. Zambianchi, A. Bongini, and L. An-tolini, Adv. Mater. , 834 (1993).[9] A. Facchetti, Mater Today , 28 (2007).[10] A. Yassar, F. Demanze, A. Jaafari, M. E. Idrissi, andC. Coupry, Adv. Funct. Mater. , 699 (2002).[11] G. Horowitz, Adv. Matter. , 365 (1998).[12] P. H¨anggi and R. Bartussek, in Noninear Physics ofComplex Systems: Current Status and Future Trends ,edited by J. Parisi, S. C. M¨uller, and W. Zimmermann(Springer-Verlag, New York, 1996), p. 294.[13] F. J¨ulicher, A. Adjari, and J. Prost, Rev. Mod. Phys. , 917 (1997).[14] R. D. Astumian and P. H¨anggi, Physics Today
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