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Copyright © 2013 American Scientific Publishers All rights reserved Printed in the United States of America Journal of Nanoscience and Nanotechnology Vol. 13, 3889–3896, 2013 Temperature and Magnetic Field Effects on Electron Transport Through DNA Molecules in a Two-Dimensional Four-Channel System Yong S. Joe1 3 ∗ , Sun H. Lee2 , Eric R. Hedin3 , and Young D. Kim1 1 RESEARCH ARTICLE Department of Physics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130-701, Korea 2 Geophysical Institute, University of Alaska, Fairbanks, AK 99775, USA 3 Center for Computational Nanoscience, Department of Physics and Astronomy, Ball State University, Muncie, Indiana 47306, USA We utilize a two-dimensional four-channel DNA model, with a tight-binding (TB) Hamiltonian, and investigate the temperature and the magnetic field dependence of the transport behavior of a short DNA molecule. Random variation of the hopping integrals due to the thermal structural disorder, which partially destroy phase coherence of electrons and reduce quantum interference, leads to a reduction of the localization length and causes suppressed overall transmission. We also incorporate a variation of magnetic field flux density into the hopping integrals as a phase factor and observe Aharonov-Bohm (AB) oscillations in the transmission. It is shown that for non-zero magnetic flux, the transmission zero leaves the real-energy axis and moves up into the complex-energy Delivered to:the Kyung University plane. We also point out by thatPublishing the hydrogenTechnology bonds between base Hee pair with flux variations play a IP:periodicity 163.180.37.28 On: Mon, 2013 10:13:48 role to determine the of AB oscillations in 06 the May transmission. Copyright American Scientific Publishers Keywords: DNA Molecule, Electron Transport, Thermal Disorder, Aharonov-Bohm Oscillations. 1. INTRODUCTION Charge transport and electrical conduction through DNA sequences have acquired considerable attention in recent years. After the inter-base hybridization of -orbitals perpendicular to the planes of the stacked base-pairs in double-stranded (ds) DNA was found by Eley and Spivey,1 it was revealed that both positive charges (holes) and electrons would propagate through -stacks of DNA bases. Thus, the idea of using DNA as a component of future molecular electronic devices has been reported and is still being explored in nanotechnology and nanoelectronics.2–4 Charge transport measurements through DNA molecules have shown controversial results. The wide range of transport behaviors can be attributed to many experimental complications, such as contact between DNA molecules and electrodes, length and sequence of DNA, temperature, and humidity in each experiment. In order to understand the diverse features of the electrical transport properties of DNA molecules, many theoretical studies were ∗ Author to whom correspondence should be addressed. J. Nanosci. Nanotechnol. 2013, Vol. 13, No. 6 devoted to this topic using various models and techniques. For instance, one-dimensional (1D) and two-dimensional (2D) tight-binding (TB) models5 6 and density-functional methods7–10 have been employed. Klotsa et al.11 used two TB models of DNA, including a one-channel fishbone model and a two-channel ladder model, and obtained the electronic properties in terms of localization lengths. They showed that as backbone disorder increased, the localization lengths increased and thus, larger currents flow. The semiconductivity of DNA, using a ladder system which has two main chains with hopping between nearest-neighbor sites and inter-chains between the ds-DNA was investigated by Iguchi.12 He also suggested the backbone chains and hydrogen bonds contribute to the electronic properties of DNA. In particular, the temperature dependence of transport behavior of a short DNA molecule has been studied by Feng et al.13 taking into account Coulomb interaction of electrons and coupling between electrons within a twolevel system in the DNA molecule. In addition, the effect of the twist angle between neighboring base pairs due to thermal fluctuations is considered in the theoretical 1533-4880/2013/13/3889/008 doi:10.1166/jnn.2013.7206 3889 Joe et al. RESEARCH ARTICLE Temperature and Magnetic Field Effects on Electron Transport Through DNA Molecules investigations of charge transport through a model DNA sequence with a superconducting electrode.14 In these models, however, the possibility of electron transport along the sugar-phosphate backbone is ignored. Hence, it is necessary to have a more sophisticated depiction of DNA model which has four possible conduction channels for charge carrier propagation by incorporating intra-backbone couplings, hydrogen bonds, and a coupling between the base-pairs and backbones. The variation of the temperature in this advanced model induces structural disorder and ranFig. 1. A short poly(G)-poly(C) chess model for electronic transport domizes every hopping integral in the DNA molecule, such through a 2D four-channel DNA model. Circles and hexagons denote nucleobases with onsite potential energies xi = 775 eV (i = 1–5, guaas intra-backbone couplings and the coupling between the nine base), i = 887 eV (i = 6–10, cytosine base), = 85 eV (backbase-pairs and backbones. Furthermore, the effects of an bone), and 0 = 775 eV (metallic leads). Lines denote various hopping external magnetic field in this system should be considamplitudes: t0 = 1 eV (intra-lead coupling), tL12 = tR12 = 03 eV (couered to further examine general features of charge transport pling between the leads and the end bases), ti i+1 Ti i+1 = 02 eV (couthrough DNA for better understanding of environmental pling between the nearest neighbor bases), t = 03 eV (coupling between backbone and DNA base pairs), hi (hydrogen bonds), and Ba (intraparameters. backbone coupling). In this article, we consider a 2D, four-channel DNA model, which is more representative of the actual DNA and the yellow circles are the sites of the leads. DNA base molecule, with inhomogeneous hopping strengths between pairs have an energy given by the ionization potentials of base pair and backbone sites, the hydrogen bonds between respective bases, taken as G = 775 eV and C = 887 eV. base-pairs, and the intra-coupling along the backbone. These are interconnected and linked to the backbone, In this system, we study temperature-dependence of the leads, and nearest-neighbor nucleotides by – stacking charge transport properties. Since temperature causes therinteraction, and hydrogen bonds. Every line between sites mal fluctuations and other structural changes of the DNA, denotes coupling with a specified hopping amplitude. we incorporate this effect into the TB model Hamiltonian Hamiltonian in a 2D four-channel chess model through the hopping integrals and examine the transmisDelivered by Publishing TechnologyThe to:TB Kyung Hee University can be written as sion resonances, the localization length, current–voltage IP: 163.180.37.28 On: Mon, 06 May 2013 10:13:48 characteristics, and the differential Copyright conductanceAmerican at vari- Scientific Publishers (1) HTot = HLead + HDNA + HLead-DNA ous temperatures. In addition, we investigate the electronic properties of a short poly(G)-poly(C) DNA molecule in the Here, the Hamiltonian for a short poly(G)-poly(C) DNA presence of an external magnetic field. A magnetic field molecule is described by with flux density penetrating the center of the 2D DNA HDNA = i ci† ci + j di† di + a†i ai + bi† bi structure induces an Aharonov-Bohm (AB) phase differij ence between the electron wave functions of the upper and lower DNA strands and produces AB oscillations in − t ci† ai + t di† bi + hi ci† d + hc i the transmission due to quantum interference effects. It is shown that the periodicity of the AB oscillations is directly − ti i+1 ci† ci+1 + Ti i+1 di† di+1 proportional to the number of loops in the DNA molecules. i + Ba a†i ai+1 + Ba bi† bi+1 + hc 2. THEORETICAL MODEL AND CALCULATIONS We consider four possible conduction channels for charge carrier propagation by incorporating intra-backbone couplings and hydrogen bonds, shown schematically in Figure 1 which is called the “chess” model. Electron transport through the DNA molecule, connected between two semi-infinite electrodes, arises through four different channels which consist of -orbital overlapping between the nearest neighboring bases within the two main conduction chains and along the upper and lower backbones. In Figure 1, the individual upper purple circles (lower pink circles) represent DNA guanine (cytosine) bases, the green hexagons are sugar-phosphate backbone sites, 3890 (2) where ci† ci , di† di , a†i ai , and bi† bi are the creation (annihilation) operators at the i-th G/C base and the i-th upper and lower backbones, and ij is the onsite potential energy of DNA G(C) base and is backbone onsite energy. The couplings, ti i+1 Ti i+1 , are the hopping amplitudes between bases along the long axis and t is a hopping integral between each base and backbone. The intra-backbone couplings are denoted as Ba and each base-pair is coupled by the hydrogen bonds, hi . The DNA molecule is coupled to two semi-infinite metallic leads by the tunneling Hamiltonian HLeads-DNA = −tL1 l0† c1 − tL2 l0† d1 − tR1 l1† cN − tR2 l1† dN + hc (3) J. Nanosci. Nanotechnol. 13, 3889–3896, 2013 Joe et al. Temperature and Magnetic Field Effects on Electron Transport Through DNA Molecules where tL12 tR12 are the contact hopping strengths between the left (right) lead and the end DNA bases, and li† li is the creation (annihilation) operator at the i-th site of the leads. The leads themselves are modeled by another TB Hamiltonian as HLeads = 0 li† li − t0 li† li+1 + hc (4) t → t cos i i+1 , and hi → hi cos i i+1 . Using cos i i+1 1 − 2 /2 1 − kB T /2I2 × i , we can now form the final random hopping integrals as tT i i+1 → tT i i+1 1 − kB T /2I2 × i t → t 1 − kB T /2I2 × i (5) hi → hi 1 − kB T /2I × i where 0 is the lead onsite energy (0 = 775 eV) and where i is a random fluctuation factor. We note here that t0 is the intra-lead hopping amplitude. In our numerical the on-site energy fluctuations, which can be absorbed into calculations, we use re-scaled parameters and the electron the static energy disorder, are not taken into account. energy E, all of which are normalized with respect to the Figure 2 shows a contour plot of the transmission hopping integral of the leads, taken as t0 = 1 eV. as a function of both electron energy and temperature The TB approximation to the Schrödinger equation and two plots of the transmission coefficient as a funcfora system, depicted in Figure 1, can be written as tion of electron energy at T = 0 K and T = 300 K. − Vn m m +n n = E n . Here, the sum runs over the nearFour well-developed mini-bands with five peaks each are est neighbors of n, E is the electron energy, and n is merging together in Figure 2(b) due to the existence of the site energy. The parameters Vn m are overlap integrals intra-coupling along the backbones, the inclusion of the (or coupling parameters) involving the overlap of the single hydrogen bonds between the base pairs, and the coupling site, atomic-like wave functions from sites m and n with the between bases and backbone sites. As the temperature is single-site potential of site n. The electron energy window is 575 ≤ E ≤ 975, as set by the TB dispersion relation (E = −2t0 cos ka + 0 for the uniform leads. By applying the wave functions into the TB Schrödinger equations and solving the matrix equation for the linearized TB Hamiltonian, we obtain the transmission amplitude as a function of the incoming electron energy, E. desired transDelivered byThe Publishing Technology to: Kyung Hee University mission coefficient and the corresponding conductance areMon, 06 May 2013 10:13:48 IP: 163.180.37.28 On: obtained by taking the square of theCopyright transmission ampli- Scientific Publishers American tude, T = tE2 , and G = 2e2 /h T respectively. i i 2 3.1. Temperature Effects Charge transport in DNA is a complex phenomenon because the environment plays a significant role in determining the conductivity of DNA. Temperature is one of the important factors in experiments with biomaterials, since variation of the temperature induces structural disorder and fluctuations of the system. Here, we apply the variation of the temperature to the hopping integrals in terms of twist-angle fluctuations, and investigate the transport behavior for electrons through a short poly(G)-poly(C) chess DNA molecule in order to observe the effects of temperature. We introduce a relative twist angle i i+1 deviated from its equilibrium value between i and i + 1 that follows a Gaussian distribution with average twist angle, I I+1 = 0. In the meantime, its variance is taken according to the equipartition theorem, i2 i+1 = kB T /I2 , where I2 /kB = 250 K, T is the temperature in kelvins, I is the reduced moment of inertia for relative rotation of the two adjacent bases, and is the oscillator frequency of the mode.14–17 Then, the temperature-dependent hopping integrals can be obtained as tT i i+1 → tT i i+1 cos i i+1 , J. Nanosci. Nanotechnol. 13, 3889–3896, 2013 Fig. 2. (a) Contour plot of the transmission versus electron energy and temperature, and the transmission coefficient versus electron energy for two different temperatures, (b) T = 0 K and (c) T = 300 K with fixed parameters: hi = 05 eV, Ba = 02 eV, and random factors i = 05 −03 0 03 −05. 3891 RESEARCH ARTICLE 3. RESULTS AND DISCUSSION RESEARCH ARTICLE Temperature and Magnetic Field Effects on Electron Transport Through DNA Molecules Joe et al. increased, thermal fluctuations manifesting in the random T ∝ exp−L/, where L is the total length of the system. Notice that the magnitude of the localization length hopping amplitudes destroy the phase coherence of the at T = 0 K is approximately 10 times larger than that at electrons and reduce quantum interference. In Figure 2(c), T = 300 K. This clearly indicates that thermal structural the magnitude of the envelopes in the transmission specfluctuations localize the electronic wave functions, resulttrum, which initially have unit transmission, become suping in a temperature-dependent localization length. pressed and smear out below unity due to the decrease With the knowledge of the transmission T E, we in the number of transmitting states, while the resonance can evaluate the current–voltage (I–V ) characteristics by positions are shifted due to the phase changes of the applying the standard formalism based on the scattering electrons. theory of transport,25–28 A DNA double helix with disoriented base-pairs due to the thermal fluctuations can be viewed as a 2D disordered 2e system. In this system, the disorder leads to electronic I= dE T EfL E − fR E (6) h − localization. Hence, the thermal structural fluctuations will considerably limit electron transport through DNA and Here, fL/R E = 1/eE−L/R /kB T + 1 is the Fermi distrimake electron wave functions more localized. To address bution function, where kB is a Boltzmann constant and the effects of thermal structural fluctuations on electron L/R stands for the electrochemical potential of the left localization, we plot the localization length as a function (right) metal electrodes. The difference between these is of electron energy for two different temperatures, T = controlled by the applied source–drain bias voltage as 0 K and T = 300 K in Figure 3. Localization length is L − R = Vsd . Even if inelastic electron–phonon scatterinversely related to the Lyapunov coefficient N , which ing can occur in the presence of thermal vibrations of is widely used for a powerful proof-tool to sort out the the DNA molecule, inelastic scattering has a minor effect main features of complex localization patterns, and is on the conductance because the electron–phonon coupling calculated using transmission coefficients T E. Hence, is very weak.29 Hence, this expression takes into account the localization length can be written as E = N −1 , only the structural disorder due to thermal fluctuations and where N = − lnT E/2N and N is the number of the broadening of the Fermi function. The I–V characterbase-pairs.17–24 The asymptotic behavior of the localization istics are shown in Figure 4 for two different temperatures length indicates that theDelivered thermal fluctuations can partially T =to: 0 KKyung (solid Hee line) University and T = 300 K (dotted line). The by Publishing Technology destroy coherent charge transport and reduce the mean nonlinear I–V curves exhibit IP: 163.180.37.28 On: Mon, 06 May 2013 10:13:48 a current gap at low applied transmission coefficient. In other words, high temperature Copyright American Scientific bias, andPublishers the voltage threshold for current onset for both leads to the disorder of the system and a reduction of temperatures is about the same (Vsd = 32 volts). This indithe localization length and consequently a reduction of cates that the thermal structural disorder does not affect the electron conductance, according to the relationship, the voltage gap in the I–V characteristics. When Vsd > 32 volts, however, reduced current is observed at higher temperature, since the static distortion sites due to the thermal structural vibration and twist modes increase elastic scattering of electrons through the DNA molecule. As the temperature increases, the linear behavior of the current after the threshold voltage is changed to a step-like behavior, which is indicative of resonant conductance peaks in the differential I–V curve. The differential conductance Fig. 3. Localization lengths as a function of electron energy are plotted for two different temperatures (a) T = 0 K and (b) T = 300 K. 3892 Fig. 4. Current as a function of source–drain voltage for T = 0 K (solid line) and T = 300 K (dashed line). The corresponding differential conductance, dI/dV , versus applied voltage is shown in the inset. J. Nanosci. Nanotechnol. 13, 3889–3896, 2013 Joe et al. Temperature and Magnetic Field Effects on Electron Transport Through DNA Molecules dI/dV , shown in the inset of Figure 4, exhibits a wide and single resonance peak with a full-width half-maximum of ∼ 1.3 volts for T = 0 K and a triple-peak structure with reduced amplitudes for T = 300 K. We attribute these resonant peaks in the differential conductance, arising from the resonant tunneling through the resonant energy levels in the DNA molecule, to the re-arrangement of energy levels due to the thermal structural fluctuations. 3.2. Magnetic Flux Effects tT i i+1 → tT i i+1 e±i hi → hi e±i and t → t e±i (7) where = 2 /N 0 denotes the total phase shift with the number of sites N . Here, measures the total flux through the system in units of the flux quantum, 0 h/e, and the plus or minus signs in the exponential phase factors are applied when the electron moves in the counterclockwise or clockwise direction, respectively. First, we study magnetic flux dependence of the electron conduction through DNA molecules in the absence of hydrogen bonds. In Figure 5, we show contour plots of the transmission T E as a function of magnetic flux and electron energy for different numbers of base-pairs: (a) one base-pair and (c) five base-pairs. For a fixed incident energy (E = 8) the transmission spectra versus magnetic flux is also depicted in (b) one base-pair and (d) five Delivered by Publishing Technology to: Kyung Hee University IP: 163.180.37.28 On: Mon, 06 May 2013 10:13:48 Copyright American Scientific Publishers Fig. 5. Contour plots of the transmission as a function of electron energy (E and magnetic flux (/0 , without considering backbone effects or hydrogen bonds (hi = 0), for different numbers of base-pairs: (a) one base-pair and (c) five base-pairs. Transmission spectra versus magnetic flux for fixed incoming energy (E = 8) are shown for (b) one base-pair and (d) five base-pairs. As the number of base-pairs increases, the amplitude of the AB oscillations diminishes. However, from the enlarged plot of the transmission for five base-pairs, depicted in the inset of (d), it is clearly seen that the transmission has a small oscillatory, flux-dependent behavior with a periodicity of 0 . J. Nanosci. Nanotechnol. 13, 3889–3896, 2013 3893 RESEARCH ARTICLE In this section, we investigate the electronic properties of a short poly(G)-poly(C) DNA molecule in the presence of an external magnetic field. When a magnetic flux is present, there exist trajectories enclosing a finite flux, which affect the physical properties of the system. In order to focus on the AB effects in this system, we assume that our model in the absence of the backbone effect either has a single loop in the ds-DNA molecule without inter-base hydrogen bonds, or multiple loops with the inclusion of interbase hydrogen bonds. [In Figs. 7(a) and (c) notice that we color both the backbone sites and couplings between base pairs and backbone a light gray in order to indicate lack of consideration of the backbone effect]. A magnetic field flux density penetrating through the center of the 2D DNA structure induces AB phase difference between the electron wave functions of the upper and lower DNA strands and produces AB oscillations in the transmission T E as a function of magnetic flux. In order to observe the quantum interference through the double-helix DNA, the hopping integrals are modified by a multiplication of the Peierls gauge phase factor, defined as Temperature and Magnetic Field Effects on Electron Transport Through DNA Molecules the on-site energy of either the upper (guanine) or lower (cytosine) bases, giving less interaction between the two strands. We note here that the periodicity of the AB oscillations remains the same as 0 , regardless of the size of the loop, because there exists only a single loop in the DNA molecule in the absence of hydrogen bonds. Next, we examine the transmission characteristics as a function of electron energy for a fixed value of magnetic flux for two different numbers of base-pairs. In Figure 6, we show transmission resonances as a function of electron energy (left column) and contour plots of transmission in the complex-energy plane (right column) for different values of the magnetic flux and different number of base-pairs. In the case of one base-pair and /0 = 1 RESEARCH ARTICLE base-pairs in Figure 5. For a single base pair, AB oscillations with flux variation arising from quantum interference have a period of 0 and a noticeable amplitude of 0.4. In the case with five base-pairs, the amplitude of the AB oscillations in the transmission is shown to be negligible in Figure 5(d). However, from the enlarged plot depicted in the inset of (d), it is clearly seen that the transmission has a small oscillatory, flux-dependent transmission, in which the AB resonance oscillations are out of phase by 180 in comparison with Figure 5(b) [The locations of peaks are shifted by 0 /2]. The reduced amplitude of the AB oscillations in the case with 5 base-pairs is due to the transmission being restricted more to only one or the other strand at electron energy values in proximity to Joe et al. Delivered by Publishing Technology to: Kyung Hee University IP: 163.180.37.28 On: Mon, 06 May 2013 10:13:48 Copyright American Scientific Publishers Fig. 6. Transmission resonances as a function of electron energy: (a) /0 = 1, (c) /0 = 03 for one base-pair, and (e) /0 = 03 for five base-pairs. Contour plots of the transmission in the complex-energy plane: (b) /0 = 1, (d) /0 = 03 for one base-pair, and (f) /0 = 03 for five base-pairs. Two distinct BW resonances appear as peaks and poles at E ≈ 7.75 and E ≈ 8.87 in (a)–(d), but the transmission zero leaves the real-energy axis and moves up into the complex-energy plane in (d). 3894 J. Nanosci. Nanotechnol. 13, 3889–3896, 2013 Joe et al. Temperature and Magnetic Field Effects on Electron Transport Through DNA Molecules the two transmission bands have almost no overlap, which severely mitigates the amplitude of the AB oscillations, as noted in Figure 5(d). The five well-defined resonance peaks within each band are shown in Figure 6(f) as five distinct resonance poles in the complex-energy plane. Finally, we investigate the electron phase shift through a short DNA molecule with the inclusion of hydrogen bonds between the base pairs. The existence of hydrogen bonds generates many sub-rings, each enclosing magnetic flux. As the number of base-pairs in the DNA molecule changes, the number of loops within the DNA varies. For instance, a single base-pair generates two enclosed paths (2 loops) and five base-pairs produces six enclosed paths (6 loops) due to the hydrogen bonds hi i = 1 5 connecting bases across the long axis of DNA [see Figs. 7(a) and (c)]. The total transmission T E versus magnetic flux (/0 for one base-pair (2 loops) with hi = 05 and E = 75 is displayed in Figure 7(b) showing periodic AB oscillations with 20 periodicity. A contour plot of T as a function of E and /0 for five base-pairs with hi = 05 is Delivered by Publishing Technology to: Kyung Hee University IP: 163.180.37.28 On: Mon, 06 May 2013 10:13:48 Copyright American Scientific Publishers Fig. 7. Schematics of DNA molecules with magnetic flux for (a) one base-pair with 2 loops and (c) five base-pairs with 6 loops. (b) Transmission versus flux for one base-pair with hi = 05 and E = 75. In the case of five base-pairs, a contour plot of the transmission as a function of E and /0 is shown in (d), and the transmission versus flux for (e) hi = 05 and (f) hi = 09 with fixed E = 75. The periodicity of AB oscillations in terms of 0 is the same as the number of loops through the DNA. J. Nanosci. Nanotechnol. 13, 3889–3896, 2013 3895 RESEARCH ARTICLE (or = l 0 , where l = 0 1 2 ), two distinct BreitWigner (BW) resonances appear at E ≈ 7.75 and E ≈ 8.87 which correspond to the onsite energy values of guanine and cytosine, respectively [Fig. 6(a)]. When /0 = 03, the transmission minimum at E ≈ 8.3 is slightly shifted to higher energy, E ≈ 8.4, and no longer reaches to zero [Tmin E = 02 in Fig. 6(c)]; the two peak positions of the BW resonances remain the same, however. In other words, the transmission zero [ReE = 0 and ImE = 0 in Fig. 6(b)] leaves the real-energy axis and moves progressively up into the complex-energy plane as the magnitude of the flux increases from 0.0 to 0.5 [ReE = 0 and ImE = 08 for /0 = 03 in Fig. 6(d)]. For values of flux greater than 0.5, the transmission zero jumps to the negative half of the complex-energy plane and returns to the real-energy axis from below as /0 → 10. In the case of five base-pairs and /0 = 03, the transmission T of the structure exhibits weakly split groups of transmission resonances in each mini-band due to the inter-base-pair tunneling [Fig. 6(e)]. With five base-pairs, RESEARCH ARTICLE Temperature and Magnetic Field Effects on Electron Transport Through DNA Molecules shown in Figure 7(d), where pronounced oscillatory behavior as a function of flux variation is observed in the two transmission mini-bands centered within 72 < E < 80 and 87 < E < 94. We also plot the transmission versus flux for two different hydrogen bonds, hi = 05 and hi = 09 with a fixed E = 75 in Figures 7(e) and (f), respectively. In both cases (five base-pairs with 6 loops), the periodicity increases to 60 . For hi = 09, we observe a series of Fano-like resonance (a pair combination of a full transmission and a full reflection) in the transmission as a function of the modulated magnetic flux threading the DNA molecule. This indicates that careful fine-tuning of the magnetic flux allows for selective switching application in molecular electronic devices. It should be noted from Figure 7 that the periodicity of the AB oscillations is directly proportional to the number of loops in the DNA molecules. In other word, one base-pair with 2 loops has a periodicity of 20 and the case with five base-pairs and 6 loops has a periodicity of 60 in the transmission versus flux. In general, the periodicity of AB oscillations in the transmission is (q + 10 , where q is the number of base pair in DNA. Therefore, the length of DNA molecule plays a crucial role in determining the periodicity and patterns of AB oscillations in the transmission. Joe et al. pair in DNA. By examining the transmission resonance with a constant magnetic flux in the case of a single basepair system, we demonstrate that the transmission zero leaves the real-energy axis and moves up or down into the complex-energy plane for values of the magnetic flux not equal to an integer multiple of the flux quantum. Acknowledgment: One of the authors (Y. S. Joe) acknowledges support from both the International Scholar Program at Kyung Hee University and the Korea Research Foundation and Korean Federation of Science and Technology Societies Grant, funded by the Korean Government (MOEHRD, Basic Research Promotion Fund). References and Notes 1. D. D. Eley and D. I. Spivey, Trans. 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Soler, Molecular Phys. 101, 1587 (2003). short DNA molecule in a 2D four-channel DNA model, 10. J. X. Zhong, Nanotech. 2, 105 (2003). using a TB Hamiltonian. First, the structural disorder 11. D. Klotsa, R. A. Romer, and M. S. Turner, Biophys. J. 89, 2187 and fluctuations induced by the variation of temperature (2005). 12. K. Iguchi, J. Phys. Soc. Jpn. 70, 593 (2001). are incorporated into the random variation of hopping 13. J. F. Feng, X. S. Wu, S. J. Xiong, and S. S. Jiang, Solid State Comstrengths. Since the random hopping amplitudes destroy munications 139, 452 (2006). phase coherence of the electrons and reduce quantum inter14. W. Ren, J. Wang, Z. Ma, and H. Guo, Phys. Rev. B 72, 035456 ference, the transmission resonances are smeared out and (2005). suppressed below unity. This structural disorder of the 15. P. Tran, B. Alavi, and G. Gruner, Phys. Rev. Lett. 85, 1564 (2000). 16. S. Roche, Phys. Rev. Lett. 91, 108101 (2003). DNA molecule leads to a reduction of the localization 17. Z. G. Yu and X. Song, Phys. Rev. Lett. 86, 6018 (2001). length due to the decrease in the overlap of the electronic 18. E. Marcia, F. Triozon, and S. Roche, Phys. Rev. B 71, 113106 (2005). wave functions on adjacent sites, and changes the linear 19. A. Lagendijk, B. van Tiggelen, and D. S. Wiersma, Physics Today behavior of the current–voltage characteristics to a step62, 24 (2009). like behavior of the current after the threshold voltage. 20. M. Zwolak and M. D. Ventra, Nano Lett. 5, 421 (2005). 21. G. Xiong and X. R. Wang, Phys. Letts. A 344, 64 (2005). Second, the presence of magnetic flux through the DNA 22. H. Yamada, International Journal of Modern Phys. B 18, 1697 molecule induces a phase shift between the electron waves (2004). of the upper and lower DNA strands and produces AB 23. M. Storzer, P. Gross, C. Aegerter, and G. Maret, Phys. Rev. Lett. oscillations in the transmission. In the absence of hydrogen 96, 063904 (2006). bonds, variation of the magnetic field flux density, which 24. H. Yamada, Phys. Letts. A 332, 65 (2004). 25. R. Landauer, Philos. Mag. 21, 863 (1970). is incorporated into hopping integrals as a phase factor, 26. M. Buttiker, Phys. Rev. B 35, 4123 (1987). produces AB oscillations in the transmission whose peri27. D. K Ferry and S. M Goodnick, Transport in Nanostructure odicity is the elementary flux quantum, 0 , regardless of Cambridge University Press, New York (1997). the length of DNA molecule. In the presence of hydro28. S. Datta, Quantum Transport: Atom to Transistor (Cambridge Unigen bonds, on the other hand, the periodicity of the AB versity Press, New York, 2005). 29. M. Hjort and S. Stafstrom, Phys. Rev. Lett. 87, 228101 (2001). oscillations is (q + 10 , where q is the number of base Received: 12 September 2012. Accepted: 11 January 2013. 3896 J. Nanosci. Nanotechnol. 13, 3889–3896, 2013