Concepts on Charge Transfer Through Naturally Vibrating DNA Molecule

Gene 509 (2012) 24–37

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Concepts on charge transfer through naturally vibrating DNA molecule S. Abdalla ⁎, F. Marzouki Physics Department, Faculty of Science, King Abdulaziz University at Jeddah, Saudi Arabia

a r t i c l e

i n f o

Article history: Accepted 30 July 2012 Available online 17 August 2012 Keywords: DNA molecule Electrical conduction Localized electrons Potential wells Relaxation times Electron density

a b s t r a c t Delocalization of charges thorough DNA occurs due to the natural and continuous movements of molecule which stimulates the charge transfer through the molecule. A model is presented showing that the mechanism of electrical conduction occurs mainly by thermally-activated drift motion of holes under control of the localized carriers; where electrons are localized in the conduction band. These localized (stationary-trapped) electrons control the movements of the positive charges and do not play an effective role in the electrical conduction itself. It is found that the localized charge-carriers in the bands have characteristic relaxation times at 5×10^−2 s, 1.94×10^−4 s, 5×10^−7 s, and 2×10^−11 s respectively which are corresponding to four intrinsic thermal activation energies 0.56 eV, 0.33 eV, 0.24 eV, and 0.05 eV respectively. The ac-conductivity of some published data are well fitted with the presented model and the total charge density in DNA molecule is calculated to be n= 1.88×10^19 cm^−3 at 300 K which is corresponding to a linear electron density n=8.66×10^3 cm^−1 at 300 K. The model shed light on the role of transfer and/or localization of charges through DNA which has multiple applications in medical, nano-technical, bio-sensing and different domains. So, repair DNA by adjusting the charge transport through the molecule is future challenges to new medical applications. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Every process of consuming bio-energy cannot entirely change to another form without some loss of heat which warms the body itself. Rattemeyer et al. (1981) and Ruth and Popp (1975) have defined the phenomenon of bio-photons (BPs) (or ultra weak photon emission of biological cells/systems). This says that all living systems can emit BPs in ubiquitous nature; for example, our DNA molecules can absorb light energy: part of this energy is stored in the molecules and the rest is emitted from them. The stored energy inside the molecules forces them to oscillate in a simple manner and one can call the DNA as “simple harmonic oscillator”. In principal, this oscillator should lose its energy with time, but by the continuous absorption of light energy, from body cells, can compensate any loss in the oscillator-energy and make an equilibrium state between absorbed and emitted energy which makes the molecule in permanent vibration. Thus, the DNA molecule acts as a permanent-resonator with a ubiquitous nature. This may confirm the fact that DNA is an organic-superconductor (Murakami, 1992). So, DNA is a wonderful superconductor that can perform its jobs at moderate Abbreviations: A, C, G and T, stand for DNA bases: adenine, cytosine, guanine and thymine, respectively; AEA, adiabatic electron affinity; Alpha, α correlation factor; BPs, phenomenon of bio-photons; CB, conduction band; DNA, deoxyribonucleic acid; exp (ΔE/kT), Boltzmann factor; FET, field effect transistor; h/2e, quantum wire limit of; HOMO, highest occupied molecular orbits; LUMO, lowest unoccupied molecular orbits; SWNT, single-walled carbon nano-tube; TSDC, thermally stimulated depolarization current; VB, valance band; ΔE, localization energy; ΔEi, potential barrier. ⁎ Corresponding author. Tel.: +966 582343822, +966 562010819 (Mobile). E-mail address: [email protected] (S. Abdalla). 0378-1119/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.gene.2012.07.082

temperatures (for example 37 °C for human cells). Also, super conductors have the ability to store light energy (Kasumov et al., 2001). We will use the fact that DNA is in permanent oscillation to explain the electrical conduction through the life-molecule but the natural oscillations themselves (of DNA) are out of scope of the present work. In another work (Abdalla, 2011a,b), we have shown that localization of electrons in lowest unoccupied molecular orbits (LUMO) opposes the electrical conduction through the molecule. Here, on the contrary, we will show that the DNA permanent-oscillations enhance the electrical conduction by delocalization of some localized-electrons in the LUMO (conduction band) which makes the trunk of the present work. In fact, DNA exhibits unusual electrostatic properties that are thought to play a role not only in the fundamental biological process – packaging DNA into compact structures – but also in several medical applications (Chakraborty, 2007; Dekker and Ratner, 2001; Shoshanil et al., 2012). This is due to the quantum-mechanical motions of charges through the different bases of DNA, there is always a small but finite probability that the charges will change place, alter the charge transfer rate and give rise to mutations (Guallar et al., 1999; Jong-Chin et al., 2008). More interesting, this implies also that this transfer of charges over a distance of about one 10−8 cm in a static molecule may be of the driving forces in the evolution of living organisms. Not only in DNA molecule, but also the conformational structures of macromolecules affect their resonant ac-electric polarization (Guallar et al., 1999). Moreover, using electrical methods in genetic identification could lead to new therapeutic era (Jong-Chin et al., 2008; Wang et al., 2010). Nowadays, with advanced high-technology, static DNA strands of any predetermined sequence can be chemically prepared in vitro and the availability of such carful

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defined molecular material has led scientists to seek for novel applications of DNA outside the biological era: for example new medical therapeutic methods (Hsiao et al., 2011), DNA computers (Lipton, 1995), bio-nanomolecular devices (Yeo et al., 2011), different dielectrophoresis force applications on DNA (Cheng et al., 2010; Giraud et al., 2011; Henning et al., 2010; Siva et al., 2010) and new brain computer interface attempts using DNA molecules (Mahon et al., 2011). In fact, the transfer of charges through DNA is an urgent problem because some added, subtracted or localized, charges, to (or from) DNA molecule could lead to mutation and potential cancer (Shirahige and Katou 2006; Heckman et al., 2011). Thus, scientists, from three decades ago, are still working about the true nature of DNA electrical characteristics; but the question of whether DNA is intrinsically conducting is, till now, an unresolved problem. The published experimental data are quite different and even contradictory: is DNA an intrinsically conducting molecule? No net answer could be found and it is, till now, an unresolved problem. The published experimental data are quite different and even contradictory: While Cai et al. (2000) and Tran et al. (2000); have shown that the molecule is good conductor (an ohmic material); Ciavatta et al. (2006) have reported that DNA is a highly insulating material, but, Porath et al. (2000), Slinker et al. (2011), Heckman et al. (2011) have, on the contrary, confirmed that DNA is a semiconductor; and even a super conductor as reported by Kasumov et al. (2001). All these published papers have considered static DNA double helix and no publication account for the electrical conductivity through a moving (oscillating) DNA. Recently, ultrafast super cameras have shown that DNA molecule is in permanent motion (Phillips et al., 2011) and vibrates with certain oscillations. So, it is logical to search for the probable mechanism of electrical conduction through permanently vibrating DNA molecule and to encounter the presumed sources of experimental uncertainties (Higareda-Mendoza and Pardo-Galván, 2010; Ichimura et al., 2004; Phillips et al., 2011; Van Zandt, 1981; White et al., 2003; Zhang et al., 2011). One can categorize these uncertainties into four main categories: first, the permanent vibrations of DNA molecule affect the charge transfer through the molecule; then second the actual fine-structure and the initial conditions of DNA preparation (the molecule character): for example ropes versus single molecule, length of the DNA and the presence of external impurities initially present at DNA preparation: these impurities could lead to the creation of localized energy states in the molecule (Baba et al., 2006). Third, differences between the molecules and their environments for example influence of water and counter ions (Luan and Aksimentiev, 2010); and finally, the nature of the metallic contacts between the electrode and the DNA molecule. Some success has been achieved by Guo et al. (2008) concerning the dependence of conductivity on the nature of the contact to the electrodes as well as whether DNA is nicked or repaired. However, the same authors admit that, while some experimental parameters are rather well controlled, other important ones are not; for example how many DNA molecules are actually bridging the electrodes. However, advancements in nanotechnology can resolve most of these problems and in particular make it possible to have good ohmic metallic contacts. Storm et al. (2001) have performed extensive measurements, varying the sequence of DNA (using λ-DNA, as well as synthetic ply G-poly C DNA). In addition, they also varied the type of electrode (they have used Au and Pt) and the insulating substrate (they have used SiO2 and mica). They also have measured the zero conductances. In their experiment, Zhang et al. (2001) have stretched the DNA by a buffer flow across the gold electrodes and displayed insulating behavior at a bias potential up to 20 V. Concerning the semiconducting behavior of DNA, Porath et al. (2000) have used a single short molecule (only 30 base pairs) with homogeneous sequences poly G–poly C and have found a rather large HOMO–LUMO gap (4 eV) with the metal work functions sitting inside the gap. These authors have given evidence for the existence of coherent electronic states extended across the DNA molecule (Porath et al., 2000). The measured strong temperature dependence of the electrical conductivity (and the gap itself) is not easily explained within the coherent energy-band picture (Slinker et al., 2011).

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Moreover, Rakitin et al. (2000) have used long λ-DNA molecule with Au-contacts. The current passing through their DNA molecule is in the range of some pico-amperes. This poor value may be attributed to the presence of high contact resistances. On the other hand, short homogeneous oligomers (used by Porath et al., 2000) whose sticky ends attached the λ-DNA to the electrodes by sulfur–gold bonds may have poor electrical conduction. The bundles are laid across holes in an Au-coated foil, which served as one electrode. The conductance is measured with a metal-coated mechanical tip that touched the bundle and served as a second electrode. The experiments that confirm the Ohmic nature of DNA molecule have been carried out by Tran et al. (2000), Rakitin et al. (2000) and Yoo et al. (2001) show a small activation energy inferior than 0.2 eV at room temperature even though very different setups are used. Tran et al. (2000) have presented a unique set of experimental ac-conductivity data of λ-DNA with and without electrode contacts and with and without humidity (dry and wet λ-DNA) in buffer solution and over wide range of temperature. In their remarkable experimental data, Tran et al. (2000) have found two distinctive regimes of electrical conductivity depending on temperature: above 250 K the conductivity is activated by strong activation energy (0.33 eV); while below 200 K, the conductivity depends weakly on the temperature. They (Tran et al., 2000) have speculated that the weak temperature dependence may not be electronic in nature, but instead may be caused by ionic conduction, or else by reorientation of water dipoles. Similarly, it is pointed out that the increase in the ac conductivity with humidity might be explained by an increase of single molecule dipole relaxation losses plus collective reorientation of water clusters at strong humidity (Briman et al., 2004). Conversely, the dc-conductivity at zero bias voltage measured by Yoo et al. (2001) shows two similar temperature regimes. Concerning the increase of the ac conductance with humidity, similar experiments on frozen DNA samples with immobilized water molecules and counter ions show a similar dependence of the conductivity on the humidity (Briman et al., 2004). However, under such physical conditions reorientation of water dipoles seems unlikely. Several conduction mechanisms are presented to explain the temperature dependence of conductivity through DNA molecule: variable‐range hopping mechanism Yoo et al., 2001, small-polaron model (Triberis et al., 2005), thermally activated tunneling (Berlin et al., 2002) and thermally activated hopping (Tran et al., 2000) but all these mechanisms have considered the molecule as a rigid, homogeneous and perfectly pure body. Although this gives some idea about the conduction mechanism, neither the nature of small activation gaps nor the origin of free charge carriers is yet clear for a dynamic bio-molecule as DNA. Without considering the continuous motion of DNA and something like doping or finding states not associated with the base pair stake, it is unlikely that the small activation gap can be accounted for. Kasumov et al. (2001) have showed resistance data consistent with induced superconductivity in DNA. In their experiment, 16-μm long λ-DNA has showed super conductivity properties at very low temperature requiring true extended states. This experiment differs from all others in that a buffer with a predominantly divalent magnesium counter ion is used. At temperatures below 1 K the rhenium electrodes become superconducting and the proximity effect is observed in some samples in which a few DNA molecules are observed to span the electrodes. The existence of molecules across the electrode gap is confirmed with non-doping atomic force microscopy, and the resistance of the best samples is at the quantum wire limit of h/2e (Anderson, 1958; Thouless, 1997). This resistance value is of fundamental importance. However, since the metal‐DNA contact resistance is unknown, it is hard to determine the intrinsic resistance of a thin wire or molecule. One indication that DNA's resistance is above the maximum metallic value, the resistance quantum, stems from the experimental fact that DNA displays excess resistance, i.e., the resistance increases exponentially with the length of the wire instead of linearly, as is common for Ohmic materials. The question remains, however, what is the origin of the near-super coiled samples? Does it stem from the possible stabilization of floppy single DNA molecules by the

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bundles, or do condensed water and counter ions trapped between the DNA molecules lead to a different pathway for charge transport than though p stacked base pairs? If so, this could explain the lack of anisotropy of conductance seen in films of oriented DNA molecules (Triberis et al., 2005). There seems to be weak sequence dependence, however, arguing in favor of a one dimensional pathway through stacked base pairs. Poly G– n poly C and poly A–poly T samples showed slightly different activations gaps (Yoo et al., 2001). This, however, could also be due to the fact that poly G–poly C and poly A–poly T have different helical rises, 2.88 and 3.22 Å, respectively, which may have caused condensed water and counter ions to form different patterns. It is also interesting to note the somewhat larger conductivity for wet compared to dry DNA, but both dry and wet have the same activation gap (0.16 eV), as inferred from contactless measurements in microwave cavity (Tran et al., 2000). The contact is characterized by the work function of the electrode, as well as the second contact or do charge carriers first have to tunnel through the backbone? Unfortunately, only the work functions of metal electrodes are more or less known. In the important case of gold, it is not even clear if the Au work function is blow or above the DNA lowest unoccupied molecular orbit (LUMO). The above mentioned measurements are consistent with charge transport in a semiconductor where LUMO energy serves as the lowest conduction band energy and the highest occupied molecular orbital energy of the DNA (HOMO) serves as the highest valence band energy. Moreover, previous measurements show that the ac-conductivity is well parameterized as a power law in angular frequency ω (Almond and Bowen, 2004; Papathanassious, 2006). Such dependence can be a general hallmark of the disordered systems (Abdalla et al., 1987, 1989; Efros and Shklovskii, 1975; Fazio et al., 2011; Pistoulet et al., 1984) and led to the reasonable interpretation that intrinsic disorders can create a definite number of electronic localized states on the base pair sequences in which charge conduction could occur. However, such a pattern would lead to thermally activated conduction between localized states (by hopping) inconsistent with the very low dc-conductivity (Zhang et al., 2002). The above mentioned measurements are consistent with charge transport in a semiconductor where the lowest unoccupied molecular orbit (LUMO) energy serves as the lowest conduction band energy and the highest occupied molecular orbital energy of the DNA (HOMO) serves as the highest valence band energy. Here, a number of outstanding issues arise: are there localized states along the helix that form continuous conducting path? What is the linear density of charge carriers, in cm−1, that affects this electrical conduction? Can some sort of conduction between localized states over distances of a few bases still occur? Are there sensitive length dependencies in the DNA strand? In Abdalla (2011a), an answer to the first question has been given and has shown that there are some localized states within the four bases that form conducting channels throughout the different bases. This explores the effect of these localized states on the free charges density and will give answers to some of the above mentioned questions. The localized charges (electrons in the conduction band CB) are responsible for directing “free” positive charges (holes or even free radicals) to carry the current and conduct the electric energy. This localization of charges and the charge transfer have an ultimate goal to repair the damaged bonds and to inhibit the electrical conduction through the double helix. Let us see the biologically intelligent job of electron localization as it forms some reservoirs of electrons in potential wells inside the lowest unoccupied molecular orbit which can be used in repair processes. Several questions remain, however, without a clear response: 1 What is the nature of the DNA motion at 1 K (Kasumov et al., 2001)? 2 How does the continuous vibration of DNA affect the metal contact efficiency? Do these continuous vibrations damage the contacts or not? 3 How does the continuous vibration of DNA affect the localization of electrons in potential wells? 4 Does the model of conduction band apply at all to the highly disordered one dimensional DNA molecules?

5 What are the specific molecular effects of the DNA vibrations? 6 What are the molecular manifestations of the above mentioned terms “dislocations” and impurities in a double helix? 7 What are the sources of excess electrons in DNA molecules? We will try to give an answer to some of these questions but the answers of all questions are out of the scope of the present study; concerning the source of excess electrons in DNA: Fazio et al. (2011) have recently shown that DNA double-duplex helix containing a reduced flavin donor at the junction of two duplex with either the same or different electron acceptors in the duplex substrates can bring two electron acceptors in the duplex substrates into direct competition for injected electrons which explains how the kind of acceptor influences the transfer data (Fazio et al., 2011). Another point of view about the sources of excess electrons is that the adiabatic electron affinity would increase upon salvation and that dynamical simulations after vertical attachment indicate that the excess electron localizes around the nucleobases within a 15 femto-second time scale (Smyth and Kohanoff, 2011). Moreover, another reason for excess electron transfer, shown by Tainaka et al. (2010) is conjugated with amino-pyrene and di-phenyl-acetylene as a photosensitizing donor and an acceptor of excess electron, respectively. For the specific molecular effects of the vibrations: Chou (1984) and Merzell and Johnson (2011) have shown that the low-frequency vibrations posses some exceptional functions in transmitting biological information at the molecular level. In addition, Chen and Kiangb (1985) have reported that there is resonance at the molecular level which could play a central role in the energy transmission required during the cooperative interaction between subunits in a protein oligomer. So, one will consider these effects and in particular the electrons localization in the conduction band (LUMO) of DNA on its electric and dielectric properties. To do so, we will present a model taking into account the presence of inhomogeneous distribution of electrons in the LUMO of DNA which allows the localization of electrons in this lowest unoccupied molecular orbital (conduction band). These localized electrons inhibit the drift motion of the mobile (free) holes and resists their motion as it will be seen later. 2. Model and simulations 2.1. Electron density in the conduction band In this section, it will be demonstrating that DNA molecules are highly affected by the localization effect and that they have typical semiconducting properties which vary from semi-metal up to semi-insulating materials. This variation depends on several factors, for example the: (i) the continuous motion of the DNA molecule (Higareda-Mendoza and Pardo-Galván, 2010; Phillips et al., 2011; Van Zandt, 1981; Zhang et al., 2011) and even its collapse under ac-electric field (Zhou et al., 2011), (ii) energy difference between a base and the charges around it, (iii) energy difference between any base and the successive one, and (iv) types of impurities and dislocations that exist in the path of the electrons during their drift motion. In fact, these factors lead to the creation of localized states in the CB as it will be seen later. Now, one may ask: what is the origin of these disorder parameters (vibrations, impurities, dislocations and disorder parameters) inside the DNA molecule? It is believed that during the initial formation of DNA molecule, some undesired, and often uncontrolled, structure imperfections, dislocations, impurities, and other disorder parameters are, inevitably, present in (or between) the different bases. Moreover, as the DNA has a highly important bio-functions, its chemical bonds shouldn't be damaged, but if a bond is forced to be damaged by free radicals for example; the continuous motion will be necessary to localize the electrons in some potential wells in order to repair the potential damage (electron stores in CB to be used later). These localized electrons will resist the motion of free holes and give more time to repair the damaged bond. In addition, these disordered centers, dislocations and impurities lead to the formation of donor–acceptor pair and thus, stimulate the creation of localized states in the extended bands. In

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the same regard, a non periodic sequence will lead to disorder along the one-dimensional molecule. The charge transfer between different bases is characterized by two essential factors: The activation energy between sites and the time of transition from one site to another (charge transfer rate). This time will be considered to affect and correlate with the motion of DNA molecule itself. Also, analysis of hybrid quantum mechanics and molecular dynamics (Cheng et al., 2010; Giraud et al., 2011; Henning et al., 2010; Siva et al., 2010; Porath et al., 2000; Slinker et al., 2011; Heckman et al., 2011) will be essentially attributed to two factors: molecular vibrations of DNA and correlated motion of counter ions and water molecules surrounding the molecule. In fact, the conformational dynamics of DNA, that is, the relative motion of adjacent nucleobases, plays an important role in the electrical conduction through DNA double helix and it substantially affects the distribution of counterion charges around the molecule (Voityuk, 2008). These counter ions are solvated and their motion is correlated by their hydration shell, which partially screens their long-rang Columbic effect. For example, the energy difference between G+ and A+ in modified DNA is found to be about 0.3 eV (Voityuk et al., 2004). Moreover, as it has been demonstrated by Richardson et al. (2004), each of the DNA bases has its characteristic energy level and has its own electron density; thus the above mentioned disorder factors lead to inhomogeneous distribution of electrons all over the bases through the molecule. Because the Fermi level has a constant value at a certain temperature and electron density varies from a point to another along the helix, the lowest unoccupied molecular orbitals LUMO (CB) fluctuate around a most probable value (Abdalla et al., 1987; Pistoulet et al., 1984) and create localized states inside the LUMO throughout the bases. Moreover the continuous vibrations of DNA stimulates loosely bounded electrons to transfer from one site to another where it could be localized in potential well inside the CB (or trapped in disorder trapping center in the upper half of the energy gap). The spatial and energetic variations of the electron density makes the conduction band energy, EC fluctuate around a most probable value EC0 (Abdalla et al., 1987; Pistoulet et al., 1984). This latter energy corresponds to an ideal semiconducting compound similar to the DNA but without any disorder parameters and without localized states between the bases. It has been, also, shown (Abdalla et al., 1987; Pistoulet et al., 1984) that the fluctuations of EC around a most probable value, in a Gaussian distribution, lead to localization of electrons inside potential wells which affects drastically the electrical conduction. Here, it is considered that the electrons with energy less than EC0 (EC b EC0) are localized in potential wells, while electrons with energy greater than EC0 (EC > EC0) are considered to be free. Here, we consider that the potential wells have localization energy ΔE which is related to the disorder energy of the material itself. Now, the previous model (Abdalla et al., 1987; Pistoulet et al., 1984) will be applied to explain the mechanism by which free holes transfer from a base to another through DNA molecule. One should, first, show the role played by the adiabatic electron affinity AEA: the energy difference between the different bases through the DNA molecule is correlated with the AEA in the bases. The localized electrons affect the drift motion of the hole through DNA and thus the AEA of these localized electrons affects the hole motion. Richardson et al. (2004) have experimentally shown that the AEAs in eV for each of the DNA bases are as follows: 0.06, (A); 0.09, (G); 0.33, (C); and 0.44, (T) and have, also, found that the vertical detachment energies of dT and dC are substantial, 0.72 and 0.94 eV, and these anions should be observable; where A, C, G and T stand for DNA bases: adenine, cytosine, guanine and thymine, respectively. Moreover, Yanson et al. (1979) have experimentally found that the energy difference between A and T is about 0.56 eV. In the present work, the considered energies are the four activation energy values: 0.05 eV, 0.24 eV, 0.33 eV and 0.56 eV as they are commonly repetitive in the experimental published-data: (1) By dielectric measurements, Yakuphanoglua et al. (2003) have reported that DNA is a typical semiconductor which has moderate

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activation energy about 0.56 eV. (2) By electrical conductivity experimental data, Tran et al. (2000) have shown that the electrical conductivity of DNA molecule is thermally activated by 0.33 eV. Moreover, Povailas and Kiveris (2008) have, experimentally, reported that DNA molecule is an insulator and its electrical conductivity is thermally activated by 0.33 eV. Gutierrez et al. (2010) have obtained numerical results demonstrating that the charge transfer between G and C bases is thermally activated by 0.33 eV. (3) Similarly, Anagnostopoulou-Konsta et al. (1998) have reported, for DNA molecule, the presence of two well defined thermally stimulated depolarization current (TSDC) peaks, one of them lies at about 186.5 K and the other at about 120 K. The present authors have calculated the energy levels corresponding to these two peaks, and found that they lay at 0.33 eV and 0.238 eV, respectively. Moreover, by the same TSDC technique Pissas et al. (1992) have found that, at about 179 K, DNA molecule has a well defined peak at 178 K and the present work calculations lead to an energy at about 0.36 eV. In addition, by thermoelectric considerations on DNA molecule, Mecia (2005) has deduced the presence of activation energy of about 0.33 eV in the DNA molecule. Depending on these experimental data and on the above mentioned electron affinity data, four intrinsic thermal activation-energies, in DNA molecules at 0.05 eV, 0.24 eV, 0.33 eV and 0.56 eV will be considered in the present work, respectively; i.e. at a certain temperature, the electron transfer occurs by drift motion from a base to another which is thermally activated within the DNA molecule. This could occur by one of these four activation energies. Moreover, it is possible that certain activation energies hide the effect of the others, depending on the temperature range: for example the deep energy at 0.56 eV, could be well manifested at high temperatures and hide the effect of the shallow energy at 0.05 eV. While, on the contrary, the strong effect of the shallow level at 0.05 eV masks the effect of the deep level at lower temperatures. The application of dc electric field enhances all electrons, in the CB, to move towards the positive side. Free electrons respond directly to the field and transfer, by drift motion, to the nearest base as these free electrons have energy greater than the threshold energy EC0 and they are not affected by the disorder. On the other hand, the localized electrons in the potential wells should overcome a barrier, ΔEi to reach the nearest base; where “i” denotes any base under consideration and it may take one value between 1 and 4. The values of ΔEi are considered as: ΔE1 = 0.05 eV, ΔE2 = 0.24 eV, ΔE3 = 0.33 eV, and ΔE4 = 0.56 eV. The localized electrons are considered, spatially, distributed between the bases. Moreover, Voityuk (2008) and Voityuk et al. (2004) have found that holes can transfer from the positive guanine base G+ to the positive adonine base A + with an activation energy about 0.4 eV. Voityuk et al. (2004) have reported that the Boltzmann factor exp (ΔE/kT) is very sensitive to variations of ΔE. These authors have reported energies as: ΔE1 =0.05 eV, ΔE2 =0.24 eV, ΔE3 = 0.33 eV, and ΔE4 = 0.56 eV. Moreover, their figure number 1 (Voityuk et al., 2004) describes the hole transfer between the guanine bases and adenine ones as a function of time and they have found that the characteristic time of such relevant fluctuations is 0.3–0.4 ns. One has expanded their results using Fourier series then, using Gaussian distribution; one has estimated the most probable values for the different transition possibilities that can occur between the G+ and A+. Our calculations show that the most probable energy needed for the hole to jump from G+ to A+ is found to be 0.36 eV–0.39 eV which is in good accordance with the data in the present work. Let us, first, fix our attention on any base denoted i, within the DNA molecule. The density of the total electrons inside this base, n(ΔEi) varies from one base to another with a correlation factor αι. This factor is the conductivity-inhibitor factor and it represents the effect of DNA vibrations on the electrical conduction. It correlates localized charges in the considered base with the adjacent bases and with the adjacent

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charges depending on several factors; for example the distance between the localized electron and the nearest positive charge within its path. Grib et al. (2010) have shown that the distance-conductivity dependence is a consequence of a transition from under barrier tunneling mechanism to over-barrier propagation when the nearest neighbor hopping chosen is large enough. Their experimental evidences show that the electrical conductivity through DNA depends linearly on the number of attached bases and depends linearly on the molecule length (Grib et al., 2010). We will use this fact and apply it to a vibrating bases: the distance between bases will vary and shrink when DNA vibrates which resist the hole transfer through DNA double helix. So, one will use their work (Grib et al., 2010) in the present work and will consider that αι lies in the range: 0b αι b 1: the minimum correlation value corresponds to minimum electrical conduction and it corresponds to maximum localization of electrons (maximum vibration of DNA) and the correlation tends to vanish i.e. αι ~0 at maximum localization of electrons in the conduction band CB (LUMO). On the other hand, the free electrons have a strong correlation factor; this gives αι ~1 =100% which makes no action with the free holes. In addition, the free holes are almost completely correlated with the nearest negative charges because: (i) they can transfer immediately to the nearest charge under external electric field by drift motion and (ii) because they have very weak relaxation time as we will see in Section 2.5 i.e. they respond nearly immediately to the applied electric field. In general, localization plays a similar role as the chelate effect (Camara-Campos et al., 2009) in organic substances in which the enhanced affinity of chelating legends for a positive ion compared to the affinity of a collection of similar non-chelating legends for the same positive ion. The following analyses could be generally apply to macromolecules whether the main charge carriers are holes or electrons; in a particular case of DNA double helix the majority carriers are holes. Localization of electrons in CB directly affects the drift motion of holes and in similar way localization of holes in VB affects the motion of free electrons. So, the sum of the densities of free electrons, nF and localized ones, nL (ΔEi) could be written as: αi nðΔEi Þ ¼ nF þ αi nL ðΔEi Þ:

ð1Þ

As it is above mentioned, the free electrons are not affected with the disorder and their density in the CB, nF is given by: nF ¼ NC exp½ðEF −EC0 Þ=kT; where, EF is the Fermi energy, T is the temperature in Kelvin, NC is the density-of-states in the CB, and k is Boltzmann constant. One will consider that the free electron density is thermally activated by energy ΔEi. nF is related to the total electron density n(ΔEi) in the CB as: nF ¼

i¼4 X

nðΔEi Þαi expð−ΔEi =kTÞ:

ð2Þ

i¼1

i¼1

i ¼4 X nF ¼ α expð−ΔEi =kTÞ: nðΔEi Þ i ¼1 i

αi nðΔEi Þ ¼ NC exp½ðΔEi −EF Þ=kT:

ð3Þ

Packing of available holes, in potential hills, (electrons, in potential wells), from the highest (lowest) energy downwards (upward) to EVO (ECO) (by heating for example), lets the valance band VB (conduction band CB) to have full of free holes (electrons); the lowest (highest) level occupied by holes (electrons), in the VB (CB), is EV0 (ECO). This situation leads to thermally activated conduction, characteristic of semiconductors. Heating causes increased lattice vibrations in bases as well;

ð4Þ

The total density of electrons in the CB can be given by summing all over the possible values of ΔEi and αI as: αi n ¼ ΣnðΔEi Þ ¼ NC Σ exp½ðΔEi −EF Þ=kT:

ð5Þ

Furthermore, one will consider that in order to carry the electric current, the localized electrons must overcome the depth of the potential wells, which lay between the bases, ΔEi, then nL will be given as: αi nL ¼ NC expðEF =kTÞ½f expðΔEi =kTÞg−1:

ð6Þ

In addition, it is considered that, in the DNA molecule, at least one shallow donor trapping level with density, Nsh and another deep donor trapping one with density Nd are present. Their activation energies are Esh and Ed respectively and they are resulting from impurities, dislocations, imperfections, and disorder parameters. The shallow trapping level is considered so near to the CB that it is almost completely ionized. Consequently, inside the considered base, characterized by an energy ΔEi; the charge neutrality could be described as: −

αi nðΔEi Þ þ N

a ðΔEi Þ

þ

¼ ð1−αi ÞpðΔEi Þ þ N

d ðΔEi Þ

þ Nsh

ð7Þ

where p (ΔEi) is the hole density in the valence band, N+d (ΔEi) and N−a (ΔEi) are the densities of ionized donor level and ionized acceptor level, respectively. In the present work: the electron density in the CB is considered by far greater than the holes density in the valence band i.e. n≫p and the electron density n(ΔEi) is by far greater than the ionized acceptors which have density comparable to the holes density: N−a ≅p. Thus, the neutrality equation could be approximated as: αi nðΔEi Þ ¼ N

þ

d ðΔEi Þ

þ Nsh :

ð8Þ

Here, Nsh is temperature independent and “gdd” is the degeneracy of the level; it is considered to be 2 42. The total electron density, n(ΔEi), in the CB can be given as a function of a nearly temperatureindependent density Nsh as follows: 4 X

So, the ratio between the free electron density and the total electron density in the CB could be written as: i¼4 X

however, it simultaneously leads to higher population of charge carriers in the extended bands and hence increases in conductivity. In the same regard, packing of available holes from the highest energy downwards in a band can also end down with a completely filled band (at EV0); when T tends to infinite values (or when frequency of the applied acelectric field tends to infinite values). This leads to constant value of hole density and a corresponding limit value of conductivity, σLimit, at very high T (or very high frequencies). One will see, in Section 2.2, that this limit value corresponds to the total electron density in the CB i.e. the sum of both free and localized electrons density nlimit =nF +αinL. To convert these physical aspects into quantitative equations; one can write the density of total electrons in the CB as:

i ¼1

αi nðΔEi Þ ¼ Nsh þ

i ¼4 X

Nd : 1 þ g exp ½ ð E dd F −Ed þ ΔEi Þ=kT i ¼1

ð9Þ

Noting that; at a certain temperature, the Fermi level should be kept constant between the two sides in the neutrality equation and the equality of summations, in Eq. (9) all-over the four bases means that EF has a constant value within the different bases. It is worth mentioning that, the present model makes possible the addition of one (or several) deep or shallow trapping level to the neutrality equation (Eq. (9)) whether they are donors or acceptors, under the condition that there is at least only one exhausted shallow trapping level. By this presumed addition, the thermal behavior of n is expected to be kept the same, i.e. at high temperatures, the ionization of the deep level(s) dominates the electrical conduction mechanism

S. Abdalla, F. Marzouki / Gene 509 (2012) 24–37

29

and at lower temperatures, the exhaustion of the shallow trapping level controls the conduction as one will see in the next section.

1984) and both Eqs. (2) and (11), an expression for the dc electrical conductivity, σdc could be derived as:

2.2. Derivation of complex conductivity due to band carriers through moving DNA molecule: dc conductivity

σ dc ¼

The present work sheds light on the role played by the localized electrons in the CB which can lead to interesting range of electrical conductivity starting from insulators up to conductors. It has been previously shown that localized electrons play no role in the electrical conduction and they are stationary in the localized states; but they drastically affect the free holes and hence decrease the electrical conduction. The proposed model shows that charges can carry the electric current by overcoming potential barrier ΔE at a certain temperature T. Thus, following to the previous work (Abdalla, 2011a; Abdalla and Pistoulet, 1985; Abdalla et al., 1987, 1989; Pistoulet et al., 1984), the rate constant for hole (electron) transfer between the donor and acceptor species, ket, is given by ket =p(r) exp (−ΔΕ/kT), where p(r) is the probability for the hole (electron) transfer normalized to the number of times the molecular assembly acquires the correct configuration to pass through the intersection of the potential energy surfaces (Abdalla and Pistoulet, 1985). In the present work, ΔΕi is considered as the energy needed for the electron to transfer from one base to another base or another positive site. In addition, the electrical conductivity increases with temperature till a maximum value, σLimit, which occurs when there are no localized electrons in the CB i.e. when all electrons, in the band, participate in carrying the electrical current. Thus, one can write: σLimit =qαintotal(ΔEi)μ where q is the electronic charge in coulombs, ntotal (ΔEi) is the total electrons density in the CB in cm−3 and μ is the drift mobility of the electrons in cm2/(V·s). The application of dc electric field on the terminals of the DNA molecule is considered without any presumed electrical barriers due to the metal contacts. The electric field stimulates immediately the free electrons to transfer within the molecule with certain drift velocity while the localized electrons will be blocked against the potential well boundaries and they can't carry the electric current. Thus the DNA molecule will be considered as insulator if these free electrons have weak density i.e. the transfer of electric energy will be very weak. On the other hand, high densities of free holes (electrons) lead to rapid transfer of electric current which leads to semi metallic conduction as the energy passes easily through the molecule. To explain the electrical behavior of DNA as an insulator: at low temperatures most of the electrons in the band will be localized in potential wells and by heating the molecule, the localized electrons will attain more energy and more of them become free. Continuous heating makes more localized electrons to be converted free and thus, leads to increase the free electron density and at the end, at very high temperatures, all electrons in the CB, ntotal, will be free and will participate in the electrical conduction leading to a limit conductivity, σ Limit : σ Limit ¼

4 X

½ðσ F þ αi σ L ðΔEi Þ

ð10Þ

i ¼1

where, σF = q ⋅ nF ⋅ μF and σL = qαinL(ΔEi)μL. The summation all over the different four possibilities gives the limit conductivity at high temperatures:

σ limit ¼

i ¼4 X

q½ðμ F ⋅nF Þ þ ½μ L αi ⋅nL ðΔEi ÞÞ:

ð11Þ

4 X

σ Limit expð−ΔEi =kTÞ

ð12Þ

i ¼1

The semiconducting behavior of the DNA molecule is well manifested through the exponent term of the activation energy −ΔEi/kT in the last equation. 2.3. Derivation of complex conductivity due to band carriers: dc permittivity Investigation of dielectric properties of DNA goes back to the early 1960s and since then a reasonable number of papers have been published (Abdalla, 2011b; Basuray et al., 2010). In these studies, two relaxation modes are found, one at very low frequencies and the other at intermediate frequency. For the sake of covering vast range of frequency, it is also mentioned, here, that there are four relaxation types in the frequency range from 10 Hz up 10 G Hz corresponding to the relaxation of localized electrons between the four bases of the DNA molecule. This will be clarified as follows: let εdc(ΔEi) be the maximum value of the dielectric constant when all (free and localized) electrons in the base are polarized under dc conditions. On the other hand, under very high frequencies conditions, the minimum value of the dielectric constant establishes when only free electrons are polarized. As the free hole (electron) responds directly to the ac field, they lead to a limit dielectric constant at very high frequency, ε∞. Thus, one can write: εdc =[dielectric constant due to free electrons, ε∞]+[dielectric constant due to localized electrons, εL]= ε∞ +εL(ΔEi); which leads to: εL ðΔEi Þ ¼ εdc ðΔEi Þ−ε∞ :

ð13Þ

In fact, after the presented model, the relaxation phenomena and the dielectric behavior of the DNA molecule is attributed to the localized electrons rather than the free ones. On the other hand, the electric behavior of the DNA molecule is attributed to the free electrons rather than the localized ones. 2.4. Derivation of complex conductivity due to band carriers: ac conductivity On the contrary to the dc electric field, the localized electrons can carry the ac electric current as they follow the polarization of the electric field. In fact, the validity of the idea of electron localization is examined when applying ac electric field to the total electrons of the base under consideration: the free electrons respond immediately (we consider their delay time is, by convention, zero; as reference), while the localized ones respond with certain delay time τ due to their localization. Inside the considered base, τ depends on the energy, ΔEi and the temperature as: τðΔEi Þ ¼ τ0i expðΔEi =kTÞ

ð14Þ

where τ0i is the relaxation time of localized electrons when T tends to infinite values and it is a characteristic value for each base. The mean relaxation time of the DNA molecule (the measured relaxation time through the molecule), τ is obtained by the summation all over the possible four activation energies for the total bases: 4 4 X 1 X 1 1 ¼ ¼ : τ i ¼1 τðΔEi Þ i ¼1 τ0i expðΔEi =kTÞ

ð15Þ

i ¼1

Thus, the dc conductivity reaches the limit value at very high temperatures. So, taking into consideration the previous studies (Abdalla, 2011a; Abdalla and Pistoulet, 1985; Abdalla et al., 1987, 1989; Pistoulet et al.,

It is worth mentioning that, in the DNA molecule, the effect of all the four relaxation times is present at the same time, at the same conditions, but at a certain temperature only one relaxation time is rather manifested due to the thermal activation energy carried by the carriers and the correlation between the localized states between different bases: for example

30

S. Abdalla, F. Marzouki / Gene 509 (2012) 24–37

at high temperature, the relaxation of localized electrons at the deep energy levels is rather manifested while at low temperature, the relaxation of localized electrons at the shallow energy levels becomes dominating. Table 1 shows also these relaxation times as a function of the activation energy and the temperature. τ0i is taken as 1.96×10−11 s. The delay of the localized electrons, with respect to the free ones, leads to formation of electric dipoles (each localized electron is polarized with the nearest positive ion). The relaxation time of these dipoles gives rise to the creation of complex conductivity. σ*L (ΔEi) is considered as the conductivity due to localized electrons and σ*F is the conductivity due to the free ones. Thus, the total conductivity, σ*(ΔEi) of the base under consideration could be written as: σ  ðΔEi Þ ¼ σ  F þ σ  L ðΔEi Þ:

σ F ðΔEi Þ ¼ nF μ F qandσ L ðΔEi Þ ¼ αi nL ðΔEi Þμ L q

ð17Þ

μF and μL are the drift mobility of free and localized electrons, respectively. The above mentioned dipoles are considered as ideal dipoles of Debye type. Thus, applying the well known Debye-formula on a base of energy ΔEi, the ac complex conductivity is, thus given as the contribution of the complex conductivity due too free electrons andpffiffiffiffiffiffiffi localized n ffi ΔEi Þ−ε∞  ones: σ  ðΔEi Þ ¼ ½σ  Free þ ½εdc ð1þjωτ where j ¼ −1; this localized last equation could be rewritten as: σ  ðΔEi Þ ¼ ½σ " F þ jωε∞ Free

ω2 τ2 ½ε ðΔEi Þ−ε∞  þ σ L ðΔEi Þ þ jω dc 1 þ ω2 τ2 1 þ ω2 τ2

# ð18Þ localized

where σL(ΔEi) is the real part of the conductivity due to localized electrons and σF is the real part of the conductivity due to free electrons. The phase θ between the ac electric current due to free electrons and the ac current due to the localized electrons are considered to have ideal Debye behavior i.e. θ = 90° and as it has been mentioned earlier: θ = 0 for free electrons. Thus, average complex conductivity due to free electrons in the base is: σ  F ¼ σ F þ iωε∞

ð22Þ

σ ac ¼

4 X

σ ðΔEi Þ ¼

i ¼1

ε¼

4 X

4 X

σ F þ σ L ðΔEi Þ

i ¼1

εðΔEi Þ ¼

4 X

i ¼1

ε∞ þ

i¼1

ω2 τ2 1 þ ω2 τ2

εdc ðΔEi Þ−ε∞ : 1 þ ω2 τ2

ð23Þ

ð24Þ

The relation between σ and ε′ could be, directly known by omitting (ωτ) 2 from the relations (23) and (24) which leads to:   ′ 4 εdc −ε X σ ac σ dc ″ ¼ε ¼ σ L ðΔEi Þ ω ωðεdc −ε∞ Þ i¼1 ″

ε ¼

4 σ ac −σ F X ωτ ¼ σ L ðΔEi Þ : ω 1 þ ω2 τ2 i¼1

ð25Þ

2

ð26Þ

After the present model, the famous Cole and Cole curves can be explained in terms energy dispersion of localized electrons in the CB (Eq. (25)); i.e. the free electrons can't participate in these dispersions and one must subtract σdc from the measured ac electrical conductivity, σac before constructing the Cole and Cole curves. Furthermore, if one considers the well known power relation of the ac-electrical conductivity: σac = σdc + σ0ω s where σ0 is a fitting parameter, and compare it with Eq. (23); also, when one can combine Eqs. (15) and (23) to get the ac conductivity as a function of the frequency and the temperature as following: s

σ ac −σ dc ¼ σ 0 ω ¼

4 X i ¼1

σL

ω2 τ0i 2 expðΔEi kTÞ2 : 1 þ ω2 τoi 2 expðΔEi kTÞ2

ð27Þ

ð19Þ

where σF is the real part of σ* and ε∞ is the imaginary part. One should note that both σF and ε∞ are independent of the localization energy ΔEi. Similarly, for the localized electrons: ω2 τ2 ½ε ðΔEi Þ−ε∞  þ jω dc : 1 þ ω2 τ2 1 þ ω2 τ2

ð20Þ

Eq. (27) shows that the distribution of relaxation times over the possible four values of energies leads to interesting range of the exponent factor “s” as: 0 ≤ s ≤ 2 including the value s = 1 and this is, in fact, what one finds in the literature. Simple mathematics leads to a value of the exponent s as: ( s ¼ ln

 4  X σ L

i ¼1

Arranging Eqs. (18)–(20) as to have the real part (conductivity) of the total electrons and imaginary parts (permittivity) of the considered base: σ ðΔEi Þ ¼ σ F þ σ L ðΔEi Þ

ω2 τ2 1 þ ω2 τ2

ð21Þ

Table 1 Relaxation time of the localized electrons for different temperatures and activation energies. ΔE

τ250K

τ250K

τ300K

τ350K

eV

s

s

s

s

3.8 1.41 × 10−4 1.36 × 10−6 2.03 × 10−10

2.3 × 10−1 2.63 × 10−5 4.14 × 10−7 1.56 × 10−10

5 × 10−2 1.02 × 10−5 2.13 × 10−7 1.38 × 10−10

2.3 × 10− 1.56 × 10−6 5.63 × 10−8 1.05 × 10−10

0.56 0.34 0.24 0.05

εdc ðΔEi Þ−ε∞ : 1 þ ω2 τ2

Summation over all the four possible energies will yield the net ac conductivity and dielectric constant of the DNA molecule:

ð16Þ

Both the real parts σF (ΔEi) and σL (ΔEi) are, classically, given as:

σ L  ðΔEi Þ ¼ σ L ðΔEi Þ

εðΔEi Þ ¼ ε∞ þ

σ0

) 2 2 2 ω τ0i expðΔEi kTÞ = lnðωÞ: 1 þ ω2 τoi 2 expðΔEi kTÞ2

ð28Þ

It is easily noticed, from Eq. (28), that s is so sensitive for both frequency and temperature which is what one finds in the literature. Similarly, when combining Eqs. (15) and (24), one gets the dielectric constant as a function of ω and T: ε ¼ ε∞ þ

4 X

εdc −ε∞ : 2 2 1 þ ω τ ð expΔEi =kTÞ2 0i i−1

ð29Þ

Moreover, there is similarity between the effect of temperature on the localized electrons and the application of the ac field to these electrons, i.e. as the frequency increases, more localized electrons acquire higher energies and could be considered as free electrons which are capable of transferring the ac electric energy. As a

S. Abdalla, F. Marzouki / Gene 509 (2012) 24–37

2.5. Relaxation phenomena in DNA molecules The property of relaxation of charge carriers is used here to elucidate the role of band carriers in the electrical conduction through DNA molecule. As it is well known, the hole (electron) localized in a potential hill (well) in a certain base should exhibit a characteristic motion when an alternating electric field, with angular frequency ω, is applied. If the thermal energy, kT, of the electrons through the base remains unchanged but ω varies, one expects the characteristic motion to change accordingly, and the variation should be reflected in the dielectric relaxation spectroscopy (DRS) spectrum. At a certain temperature, the DRS should be characterized by a definite relaxation time τ. In this regard, at a certain applied frequency, f0, the localized electrons should have maximum response with their natural oscillations; i.e. they have a maximum displacement far away from their rest position. This occurs 1 when ω0τ = 1; where ω0 ¼ 2πf . After ac-conductivity experimental 0 measurements, Tomić et al. (2007) have found two types of relaxations in DNA molecules and they have claimed the presence of relaxation phenomena at two distinctive relaxation times, 3.72 × 10 −5 s and 4.28× 10−7 s, respectively. In the same regard, Long et al. (2003) have found that charge carriers relax in the range 10−2–4 × 10−2 s. In addition, Takashima et al. (1986) have found that charge carriers in the DNA molecule can relax at a relaxation time of 1.6 × 10−10 s. Thus, after these experimental evidences, one can consider the presence of four relaxation times in the DNA molecule which are at 2 × 10−2, 1.94× 10−4 s, 4.28× 10−7 s and 1.6 × 10 −10 s, respectively. These relaxation times are characterized by oscillations around a neutral localization of the charge. One should distinguish between these relaxation times (1× 10 −2 s–1.6× 10−10 s) and the rate by which a charge jumps over a potential barrier through the DNA double helix (some picoseconds (Voityuk et al., 2004)). Now, these four relaxation times will be correlated with the four activation energies stated after the work of Richardson et al. (2004) 0.06, (A); 0.09, (G); 0.33, (C); and 0.44, (T) as it will be demonstrated in Section 3.3.1 (Fig. 1). 3. Results and discussions 3.1. dc-Conductivity of moving DNA molecule After the presented model, in dc electrical conditions, the free holes are only responsible for carrying the electric current through the different bases. The DNA is considered, first, as an insulator, and has fitted the experimental values of Povailas and Kiveris (2008) with Eq. (12). The suitable fitting parameters are found to be as shown in Table 2 3.2. Electron density and mobility in DNA molecule The experimental data of electron density in the DNA molecule is rare in the literature and not easy to be performed, so it is difficult to examine the validity of Eq. (7). However, a technique is developed to estimate the electron density in the DNA from the experimental data of the dc-conductivity. First, Tran et al. (2000) have, experimentally, reported that the electrical conductivity is composed of two distinctive parts: 1 — at low temperatures (80 K) the conductivity is nearly temperature independent and 2 — the effect of shallow activation energy Nsh will mask the effect of the deep activation energy ΔEsh, thus, Eq. (9) can be approximated as:

4 X i ¼1

NC exp½ðΔEi −EF Þ=kT≈Nsh :

ð30Þ

1012

Relaxation Time of Localized charge Carriers, s

consequence, ΔE represents the depth of the potential well in which the electrons are localized.

0

31

2

4

6

8

10 12 10

108

108

104

104 Δ E = 0.56 eV

100

100

Δ E = 0.33 eV Δ E = 0.24 eV

10-4

10-4

Δ E = 0.05 eV

10-8

10-8 10-12

10-12 0

5

10

15

20

25

1000/T, K-1 Fig. 1. The relaxation time of the localized charge carriers as a function of temperature for different activation energies.

This can lead to an almost constant density of electrons in the CB. On the other hand, at high temperatures (400 K), Eq. (9) could be approximated as: 4 X

NC expðΔEi −EF Þ=kT ¼

i¼1

i¼4 X i¼1

Nd 1 þ gdd expðEF −Ed þ ΔEi Þ=kT

ð31Þ

 3 2 where NC ¼ 2πmkT is the density of states in the CB, m*is the effective h2 electrons mass; it is taken as m*=4.95 m0 (Abdalla et al., 1987) where m0 is the electron mass at rest and h is Planck's constant; thus, NC could 3 16 −3 be written as: NC ¼ 2:64  10 T2 cm . To estimate the Fermi level temperature dependence, Eq. (31) still has two unknowns: Nsh and EF. Similar to what it is done in low temperature, at high temperatures, Eq. (9) still has two unknowns: Nd and EF; where Ed =0.33 eV and gdd will be taken=2 as it has been used in Abdalla et al. (1989). To get a third equation in order to solve the neutrality equation for EF, one considers that there is a certain temperature Tc at which the electron density n is due to two equal parts: one from the shallow level Nsh and the other is due to the ionization of the deep level Nd. Thus, at Tc what it is done in low temperature, at high temperatures, Eq. (9) still has two unknowns: Nd and EF; where Ed = 0.33 eV and gdd will be taken = 2 as it has been used in references (Abdalla et al., 1987, 1989). To get a third equation in order to solve the neutrality equation for EF, one considers that there is a certain temperature Tc at which the electron density n is due to two equal parts: one from the shallow level Nsh and the other is due to the ionization of the deep level Nd. Thus, at Tc: n¼

i ¼4 X i ¼1

Nd þ Nsh 1 þ gdd exp½ðEF −Ed þ ΔEi Þ=kTc 

ð32Þ

Table 2 Variation of the correlation factor with the electron affinity through a DNA molecule. i

1 2 3 4

ΔEi, eV

0.56 0.33 0.24 0.05

[σLimit]i, Ω−1 cm−1

αi Povailas and Kiveris (2008) DNA insulator

Tran et al. (2000) DNA conductor

Povailas and Kiveris (2008) DNA insulator

Tran et al. (2000) DNA conductor

0.989 0.001 0.0001 6 × 10−10

1 1 1 1

1 1 × 10−3 1 × 10−4 6.93 × 10−10

8 × 106 4 × 105 3 × 104 8 × 10−2

32

S. Abdalla, F. Marzouki / Gene 509 (2012) 24–37

0

i¼4 X i¼1

Nd ≈Nsh : 1 þ gdd exp½ðEF −Ed þ ΔEi Þ=kTc 

ð33Þ

Now to estimate the critical temperature Tc, one extrapolates the curve of the electrical conductivity at high temperature and extrapolates the curve of the electrical conductivity at low temperature; then the intersection lies at Tc as shown in Fig. 2. Tc for the experimental data of Tran et al. (2000) is found to be about 240 K and 146 K for Povailas and Kiveris (2008). Taking into account the above mentioned values of ΔEi and NC and solving Eqs. (30) and (31); one finds that: NC = 9.82× 1019 cm−3 at 240 K, and Nsh =1.18×1016 cm−3, EF =0.11 eV; which gives Nd =1.34× 1018 cm−3 and n =2.35× 1016 cm−3. Then solving Eq. (9) for EF for different temperatures, one can find the Fermi energy as a function of temperature which is illustrated in Fig. 3 (Tran et al., 2000). From this figure, one can notice that a maximum of EF lays at about 0.1 eV at 215 K. Using Eq. (5) and the data in Fig. 3, one can calculate n as a function of temperature. Fig. 4 shows n as a function of temperature one can observe that at 240 K a steep rise of n with temperature begins with activation energy at high temperature about 0.33 eV; and n= 1.88 ×1019 cm−3 at 300 K. To calculate the drift mobility of charges through the different bases within the molecule at 300 K, one considers the above mentioned obtained value of electron density n =2.35× 1016 cm−3 with a conductivity 1 Ω−1 cm−1 which leads to an approximate mobility 257 cm2/(V·s). Moreover, the free electron density, nF, calculated after Eq. (2), is plotted as a function of temperature in Fig. 4, for Tran et al.'s (2000) experimental data. From this figure, one can compare between the free electron density, nF and the total electron density n and notice that the ratio nF/n tends to unity at high temperatures while it is about 4.64× 1015/1.31 ×1020 cm−1 =3.53×10−5 at 80 K. Because DNA has a linear structure, it is more convenient to express n in terms of cm−1 instead of cm−3; Beleznay et al. (2006) have reported that the volume of an elementary unit of DNA cell equals 1.5× 10−8 cm× 5× 10−8 cm× 10×10−8 cm; which gives an area of about 7.5×10−16 cm2; thus the linear electron density is about 1.22× 102 cm−1 (at room temperature). In addition, it is more convenient to express NC, for DNA molecule, in cm−1. The DNA unit area is about 7.5 ×10−16 cm2; thus the linear density-of-state in the CB for a DNA molecule is about 19.8 ×T1.5 cm−1 which gives (NC)300 K = 1.03× 105 cm−1 at 300 K .

102

0

5

10

15

20

Fermi Energy, EC - EF, eV

and

2

4

6

8

10

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3 DNA as Insulator

0.2

0.2

DNA as Conductor

0.1

0.1

0.0

0.0 50

100

150

200

250

300

350

Temperature, K Fig. 3. Fermi energy as a function of temperature form the data of Tran et al. (2000) (DNA conductor) dashed thick line and Povailas and Kiveris (2008) (DNA insulator) solid line.

3.3. ac-Conductivity of moving DNA molecule Georgakilas et al. (1998) have reported experimental data on mammalian DNA macro molecules at different frequencies and different temperatures. In this section, their ac complex conductivity data are analyzed in the light of the presented model. In particular, the dependence of the electrical conductivity σac and the dielectric permittivity ε′ as a function of angular frequency, ω and temperature: at 25 °C, σac starts from a constant value at 1.29 × 10 −3 Ω −1 cm −1, then increases with ω as ω 0.68 till another constant value 9.26 × 10 −3 Ω −1 cm −1. After our model: σdc = 1.29 × 10 −3 Ω −1 cm −1 and σtotal = 9.26 × 10 −3 Ω −1 cm −1. These experimental data are fitted with Eq. (23) and the best fitting parameters are given when the relaxation time of localized electrons τ = 1.74 × 10 −4 s. The experimental ac-conductivities are shown as symbols in Fig. 5; while the calculated values (after Eq. (23)) are shown as solid lines on the same figure. It is shown that there is a good accordance between the calculated and the experimental values. Fig. 6 shows the variations of σac as a function of temperature. The symbols represent experimental data and solid lines are for calculated values (after Eq. (23)). There is good agreement between lines and open circuits when taking τ=1.74×10−4 s. Subtracting the dc conductivity from the ac part; Eq. (28) gives: s~0.69 which in good accordance with their experimental value 0.6862. To calculate the dielectric

25

10-1

1E20

10-1

10-4

Calculated after Eq. 12

10-5

Experimental after Tarn et al (2000) Calculated after Eq. 12

10-7

Experimental after Povailas et al (2008)

10-9 10

-10

Electron Density, cm-3

DNA Conductor 1E19

1E18

1E17 nF Tarn et al nF Povailas et al

1E16

nTotal Tarn et al

10-13

10-13 0

5

10

15

20

25

1000/T, K-1

DNA insulator

nTotal Povailas et al

1E15 2

4

6

8

10

12

1000/T, K Fig. 2. The experimental dc-conductivities as a function of the temperature, of Tran et al. (2000) and Povailas and Kiveris (2008), are shown as symbols and the calculated values (after Eq. (12)) are shown as solid lines.

Fig. 4. The total electron density in the conduction band, ntotal and the free electron density nF as a function of temperature; for both DNA as a conductor and DNA as an insulator.

S. Abdalla, F. Marzouki / Gene 509 (2012) 24–37

100

102

104

106

108

1010

Dielectric Constant, ε'

400 K 300 K

200 K

Calculated at 200K Calculated at 300K Calculated at 400K

10

10

0

10

2

10

4

6

8

10

10

10

ac Electrical conductivity,Ω-1cm-1

12

Calculted after equation 23 for 100Hz Calculted after equation 23 for 1MHz Experimental after Povaillas for 100Hz Experimental after Povaillas for 1M Hz

0.1

0.01

frequency = 1 MHz 1E-3

1E-3

1E-4

1E-4 1E-5

frequency = 100Hz

1E-6

1E-6 4

6

8

10

4

6

8

10

12

14

12

to change accordingly, and the variation should be reflected in the dielectric relaxation spectroscopy (DRS) spectrum. This occurs when 1 ω0τ = 1; where ω0 ¼ 2πf . 0 The symbols represent experimental data and solid lines are for calculated values (after Eq. (24)). Tomić et al. (2007) have experimentally found two types of relaxations in DNA molecules and they have claimed the presence of relaxation phenomena at two distinctive relaxation times, 3.72 × 10 −5 s and 4.28 × 10 −7 s, respectively. Also, Long et al. (2003) have found that charge carriers could relax in the range 10 −2–4 × 10 −2 s. In addition, as it is above mentioned, also, Takashima et al. (1986) have found that charge carriers in the DNA molecule could relax at a relaxation time of 1.6 × 10 −10 s. Thus, one can consider the presence of four relaxation times in the DNA molecule which are at 2 × 10 −2, 1.94 × 10 −4, 4.28 × 10 −7 and 1.6 × 10 −10 s, respectively. This is well manifested in Fig. 9 where the relaxation time τ, obtained using the maxima in Fig. 8, is illustrated as a function of temperature. The electron transfer from the base G to Cvertical is accompanied by an energy difference ΔEG → A = 0.05 eV (Richardson et al., 2004); which corresponds to a relaxation of time about 1.6 × 10 −10 s. One can notice the exponential behavior of τ which verifies Eq. (15). In the same regard, one should notice that these relaxation times are correlated with the above mentioned four thermal activation energies. As a consequence, and with the consideration of Richardson et al.'s (2004) work the above mentioned activation energies are correlated with the corresponding electron transfer from one base to another.

80000

0.1

0.01

1E-5

1 MHz 1x106

2

14 1

1

Calculated at 1 M Hz

2x106

Fig. 7. The dependence of the dielectric constant, εac as a function of temperatures for different frequencies.

3.3.1. Relaxation phenomena in DNA molecule An electron localized in potential well in a certain base in DNA molecule should exhibit a characteristic motion when an alternating electric field, with angular frequency ω, is applied. If the thermal energy, kT, of the electrons through the base remains unchanged but ω varies, one expects the characteristic motion

10

Calculated at 100 Hz

1000/T, K-1

constant, ε′, Eq. (24) is used; then the experimental data of Georgakilas et al. (1998) are fitted with the calculated values. Fig. 7 shows their experimental data as symbols and the calculated values as solid lines. The accordance between the solid line and circles in both figures are good when considering the fitting parameter τ = 1.74 × 10 −4 s. Moreover, the electric dispersion ε″, is calculated, after Eq. (26), as a function of the frequency and it is found that, at 25 °C, ε″ passes by a maximum at about 910 Hz which corresponds to a relaxation time τ = 1/(2π*910) = 1.75 × 10 −4 s (Fig. 8). Moreover, the electric dispersion ε″, is calculated, after Eq. (26), as a function of the frequency and found that, at 25 °C, ε″ passes by a maximum at about 910 Hz which corresponds to a relaxation time of τ = 1/(2π*910) = 1.75 × 10 −4 s (Fig. 8). This last value coincides with the early mentioned relaxation time of τ = 1.74 × 10 −4 s. The relaxation time of the localized electrons is given by the product ε′/σac, which varies from a base to another.

8

3x10

10

Fig. 5. The experimental ac-conductivities as a function of the frequency are shown as symbols and the calculated values are shown as solid lines.

6

Experimental after Povaillas at 1M Hz 6

10-1

Frequency, Hz

4

Experimental after Povaillas at 100Hz

4x106

0

Experimental data after Povaillas et al

-1

100 Hz

5x106

100

14

Electric Dispersion ε"(σ/ω)

ac Electrical Conductivity,Ω-1cm-1

100

33

Calculated ε" 353 K Calculated ε" 328 K

60000

Calculated ε" 298 K Experimental ε" 298 K Experimental ε" 328 K

40000

Experimental ε" 353 K After Povaillas et al 2008

20000

0 102

103

104

105

Frequency, Hz

1000/T, K-1 Fig. 6. The variations of σac as a function of temperature: the open symbols represent experimental data and solid lines are for calculated values (after Eq. (23)).

Fig. 8. The electric dispersion ε″, calculated after Eq. (26), as a function of the frequency (solid lines) and experimental values (as symbols): at 25 °C, ε″ passes by a maximum at about 910 Hz which corresponds to a relaxation time of τ=1.75×10−4 s.

S. Abdalla, F. Marzouki / Gene 509 (2012) 24–37

Relaxation time τ, seconds

34

2.0x10-4

Experimental after Takashima et al Calculated after equation (14)

1.5x10-4

1.0x10-4

5.0x10-5 2.8

2.9

3.0

3.1

3.2

3.3

3.4

1000/T, K-1 Fig. 9. The relaxation time τ, calculated after Eq. (15), as a function of the temperature.

Similarly, the electron transfer from base C to base G is accompanied by a relaxation time of about 4.28 × 10 −7 s; which corresponds to an energy difference of ΔEC →G = 0.33 − 0.09 = 0.24 eV (Richardson et al., 2004), also, ΔET→A = 0.44 − 0.06 =0.38 eV and ΔET→Vertical T = 0.94− 0.44= 0.5 eV. This is summarized in Table 3. 3.4. Conductivity of a single vibrating DNA duplex with a carbon nanotube contact In this section, the presented model will be applied to fit the experimental model of Guo et al. (2008). These authors have inserted DNA strands between two single-walled carbon nano-tube (SWNT) electrodes and have measured the electrical conductivity at room temperature. The following points will be considered, when applying the present model: (1) The contact resistances between SWNT and the DNA are of major importance and they will be considered to play an important role in the conduction mechanism. (2) The configuration “Metallic contact–SWNT–DNA–SWNT–Metallic contact” is defined in this section as “the device” and is considered as a field effect transistor (FET) where the SWNTs represent the source and the drain, while DNA molecule stands for gate of the transistor (it plays the role of a p-channel as it will be proofed later in this section). (3) The polarization of the sample has been given by the authors Guo et al. (2008) as follows: positive terminal–(CH2)3–Pi–5′-AGT ACA GTC ATC GCG-3′–Pi–(CH2)3–negative terminal which makes us consider the forward bias of the DNA molecule to be positively biased. (4) In the presence of a gate potential (transfer potential), VGS, several steps will occur when applying external reverse potential VDS between the terminals of the device: (i) first, the extended bands bending in at the SWNT–DNA interface increases, eventually to the point at which holes have a high probability of jumping Table 3 The relaxation time τ, at room temperature, as a function of the activation energy and the corresponding energy transfer from a base to another. Authors Long et al. (2003) Tran et al. (2000) Tomić et al. (2007) Takashima et al. (1986)

Relaxation time τ, seconds

Hole transfer

Corresponding activation energy ΔE, eV

2 × 10−2 3.72 × 10−5 4.28 × 10−7 1.6 × 10−10

T→A C →G Tvertical →A G → Cvertical

0.56 0.33 0.24 0.05

from the localized state to the valence band (HOMO); thus they becomes free to carry the electric current, (IDS) and controlled by the gate potential VGS. The effect of higher dopant concentrations can also be realized by accumulating (or depleting) carriers from the SWNT–DNA interface (with a gate bias) for a DNA of a given dopant level. (ii) In addition to the precedent point, the energy of localized holes, ΔEh becomes an effective variable which can be changed to affect the charge transfer in the molecule. Because the localized holes in the hills (potential wells in the conduction band LUMO — for electrons) inside the VB will acquire more energy from the applied VGS and can jump to overcome these potential hills, and then become free to carry the electric energy (current). (iii) Therefore, the effect of larger localization energies can be countered by stronger accumulation (at least until quantum carrier confinement becomes significant). This allows for greater thermal operating stability without a significant loss in the drift current IDS. The accumulation of carriers around the SWNT in the presence of both VDS and VGS results in the formation of a negative Schottky field effect (Greatbanks et al., 2000). The behavior is similar to that of a p-channel metal-oxide-semiconductor FET (Eley and Spivey, 1962). The source-drain current, IDS decreases strongly with increasing gate voltage, which demonstrates that the given device operates as a field effect transistor and also that transport through the semiconducting DNA is dominated by positive carriers (holes). Neglecting the diameter‐dependence of contact resistances; for VG b 0 V, the curves describing the current as a function of the gate voltage saturates indicting that the contact resistance RC at the SWNT electrodes starts to dominate and the total resistance will be: RDS =RDNA +2RC of the device. Here RDNA denotes the gate-dependent resistance of the DNA molecule. After the experimental work of Guo et al. (2008), the saturation value of the current corresponds to R≈50 mV/16 n A=3.16 MΩ. Similar contact resistances (≈1.1 MΩ) are found for metallic SWNT (Lagerqvist et al., 2006). This leads to the conclusion that RDNA should be inferior to RDS i.e. RDNA b 3.16 MΩ. If the gate voltage (the on-site energy) is increased in reverse direction, then the drift current, IDS, will decrease until the channel will be pinched off at a critical voltage; at which IDS =0. This is quite seen in the experimental work of Guo et al., 2008, Fig. 2-a: at high negative values of the gate potential, one can observe that the maximum value of the drain current IDS is nearly constant at about 16 nA which represents the on state of the FET (maximum conductivity of the channel). At this on “state”, the potential VDS is divided into the two portions: one affects the two SWNTs, 2VC and the other potential VDNA is applied on the DNA. One can write: RDNA ¼ VISDSD −2 . Tans et al. (1998) have calculated the contact resistance between SWNT and metallic gold to be about 1.1 MΩ; thus, the last equation could give directly the DNA resistance as ≈0.96 MΩ. This gives a potential drop across the SWNT, VC =17.6 mV which leads to a potential drop across the molecule VDNA =VDS −2*VC =50 mV−2*17.6 mV=14.8 mV. One can depict the charge transfer in the device as follows: as the gold has a high work function, it can easily withdraw electrons from the SWNT leaving the conduction band (LUMO) of this latter rich in holes. Electrons from the valence band (HOMO) of this latter will be excited, and overcome the barrier of the forbidden gap energy, e.g. to compensate for the “created” holes in the conduction band and hence the valance band will be enriching with holes ready to carry the current (free holes in the VB). These free holes constitute the current IDS. Consequently, when increasing the reverse gate potential, the current in the channel, IDS starts to decrease exponentially according to the relation: IDS ¼ 2

     VC V −qVGS þ ΔEh þ Δϕ þ DNA exp nkT RC RDNA

ð34Þ

where Δϕ is the height of the potential barrier between DNA and the SWNT (which is expected to be e.g. in this case), ΔEh is the thermal activation energy (Sze et al., 1964), n is the ideality factor (Sze et al., 1964).

S. Abdalla, F. Marzouki / Gene 509 (2012) 24–37

The best fit between the experimental values of Guo et al. (2008) with the last equation (Eq. (34)) are: ΔEh =0.162 eV, n=1, Δϕ=0.791 eV. This last value is not too far from the energy gap of the SWNTs e.g. 0.8 eV (Xue and Ranter, 2003). The fit between calculated data after Eq. (34) and experimental values (Guo et al., 2008) are sown in Fig. 10. The origin of the holes is an important question to address. One possibility is that the carrier concentration is inherent to the DNA molecule itself. Another possibility is that the majority of carriers are injected at the gold–SWNT contacts; then to the molecule via the amide groups. The higher work function of gold (≈5 eV) leads to the generation of holes in the DNA by electron transfer from the DNA to the gold electrodes (Spalenka et al., 2011). Assuming that the band-bending length in their SWNT is neither very short nor very long, Tans et al. (1998) have argued that the FET operation can be explained based on this charge transfer (Spalenka et al., 2011). At VG = 0 V, the device is “on” and the Fermi energy approaches the maximum heights of the potential hills in the valence band throughout the DNA. If indeed the bandbending length is comparable to the length the SWNT, a positive gate voltage would generate an energy barrier of an appreciable fraction of eVG in the center of the channel (since the gate/DNA distance is shorter than the source/drain separation). The threshold voltage VGScr require to suppress hole conduction by depleting the DNA center would be determined by the thermal energy available for overcoming this barrier. Thus, VG should be much lower than 4 V (which is observed in their experimental data Guo et al., 2008). Therefore it is important to explore the other possibility, namely, the carriers are an inherent property of the molecule itself. In this case, it is expected that an inhomogeneous hole distribution along the DNA will affect drastically the charge transfer through the DNA. The validity of explanation can be verified by examining the charge population on the different CG and AT pairs. It can clearly be seen that the population of holes on the CG pairs is quite large, while the population on the AT one is always negligible. Thus, the hole density through the molecule varies as a function of energy and displacement vector (location) while the Fermi energy EF is kept constant value throughout the different bases. This makes the valence band energy; EV itself fluctuates around a most probable value EV0. All holes at EV having energies more than ΔEh are considered to be localized in potential hills; and similarly the conduction band energy EC fluctuates around a most probable value EC0 where all electrons having energies less than ΔEh are considered to be localized in potential wells. This can be understood as fluctuation of the highest occupied molecular orbits HOMO around a most probable value EV0 while EC0 is the most probable value due to fluctuation of the lowest unoccupied molecular orbits LUMO. This will be called potential fluctuations PFs of the extended bands (HOMO and LUMO) in DNA molecule. If one considers

Source Drain Current, ISD, nA

20

16

12

8

that each of the 15 bases mentioned in the work of Guo et al. (2008) has a partial positive charge (plausible on the hydrogen atoms throughout the molecule) and a partial negative charge (plausible on the oxygen atoms throughout the molecule), this configuration leads one to consider that any base inside DNA accumulates the charges as an capacitor with capacitance Cj. An approximate estimate of the hole density can then be obtained by writing the total charge on the 15 base DNA as:

Q ¼ VGScr

15 X

Cj

j¼1

where C ¼

15 X

Cj is the sum of all the 15 capacitances all over the mole-

j¼1

cule and is the threshold voltage necessary to completely deplete all charges from the capacitances (bases). The DNA capacitance per unit length with respect to the back gate is:

C L

0 ¼ 2πεε with r, h and L being ln2h r

the DNA radius, thickness and length respectively; ε is the relative dielectric constant. As a first approximation, one will consider that the 15 bases are equally charged and that they are linearly arranged in a straight manner. Thus, using reasonable values: L=20 nm×15 base, r=0.8 nm, h= 140 nm and ε≈88.4 (Lankhno and Fialko, 2003), the one-dimensional Q hole density, p can be evaluated as p ¼ qL ≈7 ×105 cm−1 from VG =

4 V, where q is the electronic charge. The large hole density suggests that DNA is degenerate and/or that is doped with acceptors, for example as a result of its processing (Zang et al., 2010). Assuming that the transport in DNA occurs by drift motion at room temperature, one can estimate the mobility of the holes from the transconductance of the FET as follows: in linear regime, it dI is given by dV ¼ μ h LC2 VSD . Subtracting the contact resistance G effect (transconductance of the contacts), one can obtain a DNA dI transconductance of dV =1.49×10−8 A/V at VG =50 mV, corresponding G to a hole mobility mh ≈93.9 cm2/(V·s). This value is not too far from the previous mobility (257 cm2/V·s) reported in Section 3.2 in this work. Generally, the published values of the mobility in the DNA molecule vary in a vast range: 10−10 cm2/V·sb μh b 225 cm2/V·s (Liao et al., 2010; Salieb-Beugelaar et al., 2008). However, the value reported in the present study is close to the mobility in heavily p-doped nano silicon of comparable hole density (Zang et al., 2010), but considerably smaller than 104 cm2/V·s observed in nano graphene (Castor et al., 2010). Moreover, it is difficult to accept a mobility of the order 10−10 cm2/V·s in a nano-channel. One should distinguish between the drift motion of the charge carriers alongside the molecule and the trapping rate of DNA which is insensitive to the potential (Cai et al., 2010; Kreft et al., 2008). The relatively high value of the DNA mobility is consistent with the initial assumption of drift transport and confirms that the DNA contains a large number of scatter points possibly related to defects in the DNA or structural-disorder at the DNA–gate-interface due to roughness. SWNTs are known to conform to topography of the surface so as to increase their adhesion energy. Such deformations can lead to electronic structure changes (Cai et al., 2010), which may act as scattering centers. 4. Conclusions

Experimental data after Guo et al [35]

4

35

Calculated after the present work

0 -4

-3

-2

-1

0

Gate voltage, volts Fig. 10. The figure shows the calculated values after Eq. (34) as a solid line and the experimental values of Guo et al. (2008) are shown as open squares. One can notice a good agreement between the experimental and calculated values.

In conclusion, the presented work explains the effect of natural vibrations of DNA molecule on the mechanism of charge transfer through the DNA molecule and it is able to explain the controversy of the complex conductivity as a function of frequency and temperature observed in DNA molecules in a wide region of the frequency and temperature. Moreover, the presented results suggest that the conduction mechanism through DNA is due predominantly to electronic origins rather than ionic ones.

36

S. Abdalla, F. Marzouki / Gene 509 (2012) 24–37

A strong temperature dependence of conductivity observed at high temperatures and very weak temperature dependence at weak temperatures is comprehensible in the framework of the presented model. It is worth noting that the free electron density dependences at both low and high temperatures are calculated using simple set of parameters (doesn't exceed three fitting parameters σdc, σL, and το). The correlation between different bases and the electron affinity of every base plays an essential role in the charge transport through DNA molecule. These results reveal a high possibility for the existence of four thermal activation energies corresponding to the four bases of the molecule. Further experimental data on DNA with accurate metallic contacts at high frequencies and temperatures should elucidate our proposal. It is expected that the linear drift motion of charges alongside the helix has multiple medical applications. Speak about DNA cancer charge transfer and tumor. Yet, the present results demonstrate that charge transfer through DNA molecule have sufficiently high importance in DNA reparation and in medical and technological applications.

Acknowledgments The authors would like to acknowledge with thanks the financial support of King Abdulaziz University deanship of scientific research (grant no. 1‐130-D 1432).

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