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. MarzoukiPhysics Department, Faculty of Science, King Abdulaziz University at Jeddah, Saudi Arabia

Abbreviations: A, C, G and T, stand for DNA bases: athymine, respectively; AEA, adiabatic electron affinityBPs, phenomenon of bio-photons; CB, conduction bandexp (ΔE/kT), Boltzmann factor; FET, field effect transisof; HOMO, highest occupied molecular orbits; LUMO, lowbits; SWNT, single-walled carbon nano-tube; TSDC, thetion current; VB, valance band; ΔE, localization energy;⁎ Corresponding author. Tel.: +966 582343822, +96

E-mail address: [email protected] (S. Abdalla)

0378-1119/$ – see front matter © 2012 Elsevier B.V. Alhttp://dx.doi.org/10.1016/j.gene.2012.07.082

a b s t r a c t
a r t i c l e i n f o
Article history:Accepted 30 July 2012Available online 17 August 2012

Keywords:DNA moleculeElectrical conductionLocalized electronsPotential wellsRelaxation timesElectron density

Delocalization of charges thorough DNA occurs due to the natural and continuousmovements of moleculewhichstimulates the charge transfer through the molecule. A model is presented showing that the mechanism of elec-trical conduction occurs mainly by thermally-activated drift motion of holes under control of the localized car-riers; where electrons are localized in the conduction band. These localized (stationary-trapped) electronscontrol themovements 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 acti-vation energies 0.56 eV, 0.33 eV, 0.24 eV, and 0.05 eV respectively. The ac-conductivity of some published dataare 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 at300 K. Themodel shed light on the role of transfer and/or localization of charges throughDNAwhich hasmultipleapplications inmedical, nano-technical, bio-sensing and different domains. So, repairDNAbyadjusting the chargetransport through the molecule is future challenges to newmedical applications.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Every process of consuming bio-energy cannot entirely change toanother form without some loss of heat which warms the body itself.Rattemeyer et al. (1981) and Ruth and Popp (1975) have defined thephenomenon of bio-photons (BPs) (or ultra weak photon emission ofbiological cells/systems). This says that all living systems can emit BPsin ubiquitous nature; for example, our DNA molecules can absorb lightenergy: part of this energy is stored in themolecules and the rest is emit-ted from them. The stored energy inside the molecules forces themto oscillate in a simple manner and one can call the DNA as “simpleharmonic oscillator”. In principal, this oscillator should lose its energywith time, but by the continuous absorption of light energy, from bodycells, can compensate any loss in the oscillator-energy andmake an equi-librium state between absorbed and emitted energy which makes themolecule in permanent vibration. Thus, the DNA molecule acts as apermanent-resonator with a ubiquitous nature. This may confirm thefact that DNA is an organic-superconductor (Murakami, 1992). So, DNAis a wonderful superconductor that can perform its jobs at moderate

denine, cytosine, guanine and; Alpha, α correlation factor;; DNA, deoxyribonucleic acid;tor; h/2e, quantum wire limitest unoccupied molecular or-rmally stimulated depolariza-ΔEi, potential barrier.6 562010819 (Mobile)..

l rights reserved.

temperatures (for example 37 °C for human cells). Also, super conduc-tors have the ability to store light energy (Kasumov et al., 2001). Wewill use the fact that DNA is in permanent oscillation to explain the elec-trical conduction through the life-molecule but the natural oscillationsthemselves (of DNA) are out of scope of the present work. In anotherwork (Abdalla, 2011a,b), we have shown that localization of electronsin lowest unoccupied molecular orbits (LUMO) opposes the electricalconduction through the molecule. Here, on the contrary, we will showthat the DNA permanent-oscillations enhance the electrical conductionby delocalization of some localized-electrons in the LUMO (conductionband) which makes the trunk of the present work. In fact, DNA exhibitsunusual electrostatic properties that are thought to play a role not onlyin the fundamental biological process – packaging DNA into compactstructures – but also in several medical applications (Chakraborty,2007; Dekker and Ratner, 2001; Shoshanil et al., 2012). This is due tothe quantum-mechanical motions of charges through the differentbases of DNA, there is always a small but finite probability that thecharges will change place, alter the charge transfer rate and give riseto mutations (Guallar et al., 1999; Jong-Chin et al., 2008). Moreinteresting, this implies also that this transfer of charges over a distanceof about one 10−8 cm in a staticmoleculemay be of the driving forces inthe evolution of living organisms. Not only in DNAmolecule, but also theconformational structures of macromolecules affect their resonantac-electric polarization (Guallar et al., 1999). Moreover, using electricalmethods in genetic identification could lead to new therapeutic era(Jong-Chin et al., 2008; Wang et al., 2010). Nowadays, with advancedhigh-technology, static DNA strands of any predetermined sequencecan be chemically prepared in vitro and the availability of such carful

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25S. Abdalla, F. Marzouki / Gene 509 (2012) 24–37

definedmolecularmaterial has led scientists to seek for novel applicationsof DNA outside the biological era: for example new medical therapeuticmethods (Hsiao et al., 2011), DNA computers (Lipton, 1995), bio-nano-molecular devices (Yeo et al., 2011), different dielectrophoresis force ap-plications on DNA (Cheng et al., 2010; Giraud et al., 2011; Henning et al.,2010; Siva et al., 2010) and new brain computer interface attempts usingDNA molecules (Mahon et al., 2011). In fact, the transfer of chargesthrough DNA is an urgent problem because some added, subtracted orlocalized, charges, to (or from) DNA molecule could lead to mutationand potential cancer (Shirahige and Katou 2006; Heckman et al., 2011).Thus, scientists, from three decades ago, are still working about the truenature of DNA electrical characteristics; but the question of whetherDNA is intrinsically conducting is, till now, an unresolved problem. Thepublished experimental data are quite different and even contradictory:is DNA an intrinsically conducting molecule? No net answer could befound and it is, till now, an unresolved problem. The published experi-mental data are quite different and even contradictory: While Cai et al.(2000) and Tran et al. (2000); have shown that the molecule is goodconductor (an ohmic material); Ciavatta et al. (2006) have reported thatDNA is a highly insulating material, but, Porath et al. (2000), Slinker etal. (2011), Heckman et al. (2011) have, on the contrary, confirmed thatDNA is a semiconductor; and even a super conductor as reported byKasumov et al. (2001). All these published papers have considered staticDNA double helix and no publication account for the electrical conductiv-ity through amoving (oscillating) DNA. Recently, ultrafast super camerashave shown that DNA molecule is in permanent motion (Phillips et al.,2011) and vibrates with certain oscillations. So, it is logical to search forthe probable mechanism of electrical conduction through permanentlyvibrating DNA molecule and to encounter the presumed sources of ex-perimental 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 intofour main categories: first, the permanent vibrations of DNA moleculeaffect the charge transfer through the molecule; then second the actualfine-structure and the initial conditions of DNA preparation (the mole-cule character): for example ropes versus single molecule, length of theDNA and the presence of external impurities initially present at DNApreparation: these impurities could lead to the creation of localized en-ergy states in the molecule (Baba et al., 2006). Third, differences be-tween the molecules and their environments for example influence ofwater and counter ions (Luan and Aksimentiev, 2010); and finally, thenature of themetallic contacts between the electrode and the DNAmol-ecule. Some success has been achieved by Guo et al. (2008) concerningthe dependence of conductivity on the nature of the contact to the elec-trodes aswell as whether DNA is nicked or repaired. However, the sameauthors admit that, while some experimental parameters are ratherwell controlled, other important ones are not; for example how manyDNAmolecules are actually bridging the electrodes. However, advance-ments in nanotechnology can resolve most of these problems and inparticular make it possible to have good ohmic metallic contacts.Storm et al. (2001) have performed extensive measurements, varyingthe sequence of DNA (using λ-DNA, as well as synthetic ply G-poly CDNA). In addition, they also varied the type of electrode (they haveused Au and Pt) and the insulating substrate (they have used SiO2 andmica). They also have measured the zero conductances. In their exper-iment, Zhang et al. (2001) have stretched the DNA by a buffer flowacross the gold electrodes and displayed insulating behavior at a biaspotential up to 20 V. Concerning the semiconducting behavior of DNA,Porath et al. (2000) have used a single short molecule (only 30 basepairs) with homogeneous sequences poly G–poly C and have found arather large HOMO–LUMO gap (4 eV) with the metal work functionssitting inside the gap. These authors have given evidence for the exis-tence of coherent electronic states extended across the DNA molecule(Porath et al., 2000). The measured strong temperature dependence ofthe electrical conductivity (and the gap itself) is not easily explainedwithin the coherent energy-band picture (Slinker et al., 2011).

Moreover, Rakitin et al. (2000) have used long λ-DNA molecule withAu-contacts. The current passing through their DNA molecule is in therange of some pico-amperes. This poor value may be attributed to thepresence of high contact resistances. On the other hand, shorthomogeneous oligomers (used by Porath et al., 2000) whose stickyends attached the λ-DNA to the electrodes by sulfur–gold bonds mayhave poor electrical conduction. The bundles are laid across holes in anAu-coated foil, which served as one electrode. The conductance is mea-sured with a metal-coated mechanical tip that touched the bundle andserved as a second electrode. The experiments that confirm the Ohmicnature of DNA molecule have been carried out by Tran et al. (2000),Rakitin et al. (2000) and Yoo et al. (2001) showa small activation energyinferior than 0.2 eV at room temperature even though very differentsetups are used. Tran et al. (2000) have presented a unique set of exper-imental ac-conductivity data of λ-DNA with and without electrodecontacts andwith andwithout humidity (dry andwet λ-DNA) in buffersolution and overwide range of temperature. In their remarkable exper-imental data, Tran et al. (2000) have found two distinctive regimesof electrical conductivity depending on temperature: above 250 K theconductivity is activated by strong activation energy (0.33 eV); whilebelow 200 K, the conductivity depends weakly on the temperature.They (Tran et al., 2000) have speculated that the weak temperaturedependencemay not be electronic in nature, but insteadmay be causedby ionic conduction, or else by reorientation of water dipoles. Similarly,it is pointed out that the increase in the ac conductivity with humiditymight be explained by an increase of single molecule dipole relaxationlosses plus collective reorientation of water clusters at strong humidity(Briman et al., 2004). Conversely, the dc-conductivity at zero bias volt-age measured by Yoo et al. (2001) shows two similar temperatureregimes. Concerning the increase of the ac conductance with humidity,similar experiments on frozen DNA samples with immobilized watermolecules and counter ions show a similar dependence of the conduc-tivity on the humidity (Briman et al., 2004). However, under suchphysical conditions reorientation ofwater dipoles seemsunlikely. Sever-al conduction mechanisms are presented to explain the temperaturedependence of conductivity through DNA molecule: variable‐rangehopping mechanism Yoo et al., 2001, small-polaron model (Triberis etal., 2005), thermally activated tunneling (Berlin et al., 2002) and ther-mally activated hopping (Tran et al., 2000) but all these mechanismshave considered the molecule as a rigid, homogeneous and perfectlypure body. Although this gives some idea about the conduction mecha-nism, neither the nature of small activation gaps nor the origin of freecharge carriers is yet clear for a dynamic bio-molecule as DNA. Withoutconsidering the continuousmotion of DNA and something like dopingorfinding states not associated with the base pair stake, it is unlikely thatthe small activation gap can be accounted for. Kasumov et al. (2001)have showed resistance data consistent with induced superconductivityin DNA. In their experiment, 16-μm long λ-DNA has showed super con-ductivity properties at very low temperature requiring true extendedstates. This experiment differs from all others in that a buffer with a pre-dominantly divalent magnesium counter ion is used. At temperaturesbelow 1 K the rhenium electrodes become superconducting and theproximity effect is observed in some samples in which a few DNA mol-ecules are observed to span the electrodes. The existence of moleculesacross the electrode gap is confirmed with non-doping atomic forcemicroscopy, and the resistance of the best samples is at the quantumwire limit of h/2e (Anderson, 1958; Thouless, 1997). This resistancevalue is of fundamental importance. However, since the metal‐DNAcontact resistance is unknown, it is hard to determine the intrinsic resis-tance of a thin wire or molecule. One indication that DNA's resistance isabove themaximummetallic value, the resistance quantum, stems fromthe experimental fact that DNA displays excess resistance, i.e., the resis-tance increases exponentially with the length of the wire instead oflinearly, as is common for Ohmic materials. The question remains, how-ever, what is the origin of the near-super coiled samples? Does it stemfrom the possible stabilization of floppy single DNA molecules by the

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

bundles, or do condensed water and counter ions trapped between theDNA molecules lead to a different pathway for charge transport thanthoughp stackedbase pairs? If so, this could explain the lack of anisotropyof conductance seen in films of oriented DNA molecules (Triberis et al.,2005). There seems to be weak sequence dependence, however, arguingin favor of a one dimensional pathway through stacked base pairs. PolyG– n poly C and poly A–poly T samples showed slightly different activa-tions gaps (Yoo et al., 2001). This, however, could also be due to the factthat poly G–poly C and poly A–poly T have different helical rises, 2.88and 3.22 Å, respectively, which may have caused condensed water andcounter ions to form different patterns. It is also interesting to note thesomewhat larger conductivity for wet compared to dry DNA, but bothdry and wet have the same activation gap (0.16 eV), as inferred fromcontactless measurements in microwave cavity (Tran et al., 2000). Thecontact is characterized by the work function of the electrode, as well asthe second contact or do charge carriers first have to tunnel through thebackbone? Unfortunately, only the work functions of metal electrodesare more or less known. In the important case of gold, it is not evenclear if the Auwork function is blowor above theDNA lowest unoccupiedmolecular orbit (LUMO). The abovementionedmeasurements are consis-tent with charge transport in a semiconductor where LUMO energyserves as the lowest conduction band energy and the highest occupiedmolecular orbital energy of the DNA (HOMO) serves as the highest va-lence band energy.

Moreover, previous measurements show that the ac-conductivity iswell parameterized as a power law in angular frequency ω (Almondand Bowen, 2004; Papathanassious, 2006). Such dependence can be ageneral hallmark of the disordered systems (Abdalla et al., 1987, 1989;Efros and Shklovskii, 1975; Fazio et al., 2011; Pistoulet et al., 1984) andled to the reasonable interpretation that intrinsic disorders can createa definite number of electronic localized states on the base pair sequencesin which charge conduction could occur. However, such a pattern wouldlead to thermally activated conduction between localized states (by hop-ping) inconsistent with the very low dc-conductivity (Zhang et al., 2002).The above mentioned measurements are consistent with charge trans-port in a semiconductor where the lowest unoccupied molecular orbit(LUMO) energy serves as the lowest conduction band energy and thehighest occupied molecular orbital energy of the DNA (HOMO) servesas the highest valence band energy. Here, a number of outstanding issuesarise: are there localized states along the helix that form continuousconducting path? What is the linear density of charge carriers, in cm−1,that affects this electrical conduction? Can some sort of conduction be-tween localized states over distances of a few bases still occur? Arethere sensitive length dependencies in the DNA strand? In Abdalla(2011a), an answer to the first question has been given and has shownthat there are some localized states within the four bases that formconducting channels throughout the different bases. This explores theeffect of these localized states on the free charges density and will giveanswers to some of the abovementioned 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 con-duct the electric energy. This localization of charges and the charge trans-fer have an ultimate goal to repair the damaged bonds and to inhibit theelectrical conduction through the double helix. Let us see the biologicallyintelligent job of electron localization as it forms some reservoirs of elec-trons in potential wells inside the lowest unoccupied molecular orbitwhich can be used in repair processes. Several questions remain, howev-er, without a clear response:

1 What is the nature of the DNAmotion at 1 K (Kasumov et al., 2001)?2 Howdoes the continuous vibration of DNA affect themetal contact ef-

ficiency? 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 disor-

dered 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 theanswers 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 thatDNAdouble-duplex helix containing a reducedflavin donor at the junction of two duplexwith either the same or differ-ent electron acceptors in the duplex substrates can bring two electronacceptors in the duplex substrates into direct competition for injectedelectrons which explains how the kind of acceptor influences the trans-fer data (Fazio et al., 2011). Another point of view about the sources ofexcess electrons is that the adiabatic electron affinity would increaseupon salvation and that dynamical simulations after vertical attachmentindicate that the excess electron localizes around the nucleobases withina 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 aphotosensitizing donor and an acceptor of excess electron, respectively.For the specific molecular effects of the vibrations: Chou (1984) andMerzell and Johnson (2011) have shown that the low-frequency vibra-tions posses some exceptional functions in transmitting biological infor-mation at the molecular level. In addition, Chen and Kiangb (1985) havereported that there is resonance at the molecular level which could playa central role in the energy transmission required during the cooperativeinteraction between subunits in a protein oligomer. So, one will considerthese effects and in particular the electrons localization in the conductionband (LUMO) of DNA on its electric and dielectric properties. To do so, wewill present a model taking into account the presence of inhomogeneousdistribution of electrons in the LUMO of DNA which allows the localiza-tion of electrons in this lowest unoccupied molecular orbital (conductionband). 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 DNAmolecules are highlyaffected by the localization effect and that they have typical semiconduct-ing properties which vary from semi-metal up to semi-insulating mate-rials. This variation depends on several factors, for example the: (i) thecontinuous motion of the DNA molecule (Higareda-Mendoza andPardo-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) en-ergy difference between any base and the successive one, and (iv) typesof impurities and dislocations that exist in the path of the electrons duringtheir drift motion. In fact, these factors lead to the creation of localizedstates in the CB as it will be seen later. Now, one may ask: what is the or-igin of these disorder parameters (vibrations, impurities, dislocations anddisorder parameters) inside the DNAmolecule? It is believed that duringthe initial formation of DNA molecule, some undesired, and often un-controlled, structure imperfections, dislocations, impurities, and otherdisorder parameters are, inevitably, present in (or between) the differentbases. Moreover, as the DNA has a highly important bio-functions, itschemical bonds shouldn't be damaged, but if a bond is forced to be dam-aged by free radicals for example; the continuous motion will be neces-sary to localize the electrons in some potential wells in order to repairthe potential damage (electron stores in CB to be used later). These local-ized electrons will resist the motion of free holes and give more time torepair the damaged bond. In addition, these disordered centers, disloca-tions and impurities lead to the formation of donor–acceptor pair andthus, stimulate the creation of localized states in the extended bands. In


the same regard, a non periodic sequence will lead to disorder along theone-dimensional molecule. The charge transfer between different basesis characterized by two essential factors: The activation energy betweensites and the time of transition from one site to another (charge transferrate). This timewill be considered to affect and correlate with themotionof DNA molecule itself. Also, analysis of hybrid quantum mechanics andmolecular dynamics (Cheng et al., 2010; Giraud et al., 2011; Henning etal., 2010; Siva et al., 2010; Porath et al., 2000; Slinker et al., 2011;Heckman et al., 2011)will be essentially attributed to two factors: molec-ular vibrations of DNA and correlated motion of counter ions and watermolecules surrounding the molecule. In fact, the conformational dynam-ics of DNA, that is, the relative motion of adjacent nucleobases, plays animportant role in the electrical conduction through DNA double helixand it substantially affects the distribution of counterion charges aroundthe molecule (Voityuk, 2008). These counter ions are solvated and theirmotion is correlated by their hydration shell, which partially screenstheir long-rang Columbic effect. For example, the energy difference be-tween 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 itsown electron density; thus the above mentioned disorder factors leadto inhomogeneous distribution of electrons all over the bases throughthe molecule. Because the Fermi level has a constant value at a certaintemperature and electron density varies from a point to another alongthe helix, the lowest unoccupied molecular orbitals LUMO (CB) fluctu-ate around a most probable value (Abdalla et al., 1987; Pistoulet et al.,1984) and create localized states inside the LUMO throughout thebases. Moreover the continuous vibrations of DNA stimulates looselybounded electrons to transfer from one site to another where it couldbe localized in potential well inside the CB (or trapped in disorder trap-ping center in the upper half of the energy gap). The spatial and ener-getic variations of the electron density makes the conduction bandenergy, EC fluctuate around a most probable value EC0 (Abdalla et al.,1987; Pistoulet et al., 1984). This latter energy corresponds to an idealsemiconducting compound similar to the DNA butwithout any disorderparameters andwithout localized states between the bases. It has been,also, shown (Abdalla et al., 1987; Pistoulet et al., 1984) that the fluctu-ations of EC around a most probable value, in a Gaussian distribution,lead to localization of electrons inside potential wells which affectsdrastically the electrical conduction. Here, it is considered that the elec-tronswith energy less than EC0 (ECbEC0) are localized in potentialwells,while electrons with energy greater than EC0 (EC>EC0) are consideredto be free. Here, we consider that the potential wells have localizationenergy 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 transferfrom a base to another through DNA molecule. One should, first, showthe role played by the adiabatic electron affinity AEA: the energy differ-ence between the different bases through the DNAmolecule is correlatedwith theAEA in the bases. The localized electrons affect the driftmotion ofthehole throughDNAand thus theAEAof these localized electrons affectstheholemotion. Richardson et al. (2004)have experimentally shown thatthe 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 verticaldetachment 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 forDNA bases: adenine, cytosine, guanine and thymine, respectively.

Moreover, Yanson et al. (1979) have experimentally found that theenergy difference between A and T is about 0.56 eV. In the presentwork, 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 repetitivein the experimental published-data:

(1) By dielectric measurements, Yakuphanoglua et al. (2003) havereported that DNA is a typical semiconductor which has moderate

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 isthermally activated by 0.33 eV. Moreover, Povailas and Kiveris(2008) have, experimentally, reported that DNAmolecule is an in-sulator and its electrical conductivity is thermally activated by0.33 eV. Gutierrez et al. (2010) have obtained numerical resultsdemonstrating that the charge transfer between G and C bases isthermally activated by 0.33 eV.

(3) Similarly, Anagnostopoulou-Konsta et al. (1998) have reported, forDNAmolecule, the presence of two well defined thermally stimu-lated depolarization current (TSDC) peaks, one of them lies atabout 186.5 K and the other at about 120 K. The present authorshave calculated the energy levels corresponding to these twopeaks, and found that they lay at 0.33 eV and 0.238 eV, respective-ly. Moreover, by the same TSDC technique Pissas et al. (1992) havefound that, at about 179 K, DNAmolecule has a well defined peakat 178 K and the present work calculations lead to an energy atabout 0.36 eV. In addition, by thermoelectric considerations onDNA molecule, Mecia (2005) has deduced the presence of activa-tion energy of about 0.33 eV in the DNA molecule.

Depending on these experimental data and on the above mentionedelectron affinity data, four intrinsic thermal activation-energies, in DNAmolecules at 0.05 eV, 0.24 eV, 0.33 eV and 0.56 eVwill be considered inthe present work, respectively; i.e. at a certain temperature, the electrontransfer occurs bydriftmotion froma base to anotherwhich is thermallyactivatedwithin theDNAmolecule. This could occur by one of these fouractivation energies. Moreover, it is possible that certain activation ener-gies hide the effect of the others, depending on the temperature range:for example the deep energy at 0.56 eV, could be well manifested athigh 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 eVmasks 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 tothe field and transfer, by drift motion, to the nearest base as these freeelectrons have energy greater than the threshold energy EC0 and theyare not affected by the disorder. On the other hand, the localized elec-trons in the potential wells should overcome a barrier, ΔEi to reachthe nearest base; where “i” denotes any base under considerationand it may take one value between 1 and 4. The values of ΔEi are con-sidered as: ΔE1=0.05 eV, ΔE2=0.24 eV, ΔE3=0.33 eV, and ΔE4=0.56 eV. The localized electrons are considered, spatially, distributedbetween the bases. Moreover, Voityuk (2008) and Voityuk et al.(2004) have found that holes can transfer from the positive guaninebase G+ to the positive adonine base A+ with an activation energyabout 0.4 eV. Voityuk et al. (2004) have reported that the Boltzmannfactor exp (ΔE/kT) is very sensitive to variations of ΔE. These authorshave 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 (Voityuket al., 2004) describes the hole transfer between the guanine basesand adenine ones as a function of time and they have found that thecharacteristic time of such relevant fluctuations is 0.3–0.4 ns. One hasexpanded their results using Fourier series then, using Gaussian distri-bution; one has estimated the most probable values for the differenttransition possibilities that can occur between the G+and A+. Our cal-culations show that the most probable energy needed for the hole tojump from G+ to A+ is found to be 0.36 eV–0.39 eV which is ingood accordance with the data in the present work.

Let us, first, fix our attention on any base denoted i, within the DNAmolecule. The density of the total electrons inside this base, n(ΔEi)varies from one base to another with a correlation factor αι. This factoris the conductivity-inhibitor factor and it represents the effect of DNA vi-brations on the electrical conduction. It correlates localized charges inthe considered base with the adjacent bases and with the adjacent


charges depending on several factors; for example the distance betweenthe localized electron and the nearest positive charge within its path.Grib et al. (2010) have shown that the distance-conductivity depen-dence is a consequence of a transition from under barrier tunnelingmechanism to over-barrier propagation when the nearest neighborhopping chosen is large enough. Their experimental evidences showthat the electrical conductivity through DNA depends linearly on thenumber of attached bases and depends linearly on the molecule length(Grib et al., 2010). We will use this fact and apply it to a vibratingbases: the distance between bases will vary and shrink when DNA vi-brates which resist the hole transfer through DNA double helix. So,one will use their work (Grib et al., 2010) in the present work and willconsider that αι lies in the range: 0bαιb1: the minimum correlationvalue corresponds tominimum electrical conduction and it correspondsto maximum localization of electrons (maximum vibration of DNA) andthe correlation tends to vanish i.e. αι~0 at maximum localization ofelectrons in the conduction band CB (LUMO). On the other hand, thefree electrons have a strong correlation factor; this gives αι~1=100%which makes no action with the free holes. In addition, the free holesare almost completely correlated with the nearest negative charges be-cause: (i) they can transfer immediately to the nearest charge under ex-ternal electric field by driftmotion and (ii) because they have veryweakrelaxation time as we will see in Section 2.5 i.e. they respond nearly im-mediately 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 en-hanced affinity of chelating legends for a positive ion compared to theaffinity of a collection of similar non-chelating legends for the samepositive ion.

The following analyses could be generally apply to macromoleculeswhether the main charge carriers are holes or electrons; in a particularcase of DNA double helix the majority carriers are holes. Localization ofelectrons in CB directly affects the drift motion of holes and in similarway 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:

αin ΔEið Þ ¼ nF þαinL ΔEið Þ: ð1Þ

As it is above mentioned, the free electrons are not affected withthe disorder and their density in the CB, nF is given by:

nF ¼ NC exp EF−EC0ð Þ=kT½ �;

where, EF is the Fermi energy, T is the temperature in Kelvin, NC is thedensity-of-states in the CB, and k is Boltzmann constant.

Onewill consider that the free electron density is thermally activatedby energy ΔEi. nF is related to the total electron density n(ΔEi) in the CBas:

nF ¼Xi¼4

i¼1

n ΔEið Þαi exp −ΔEi=kTð Þ: ð2Þ

So, the ratio between the free electron density and the total elec-tron density in the CB could be written as:

Xi¼4

i¼1

nF

n ΔEið Þ ¼Xi ¼4

i ¼1

αi exp −ΔEi=kTð Þ: ð3Þ

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

however, it simultaneously leads to higher population of charge carriersin the extended bands and hence increases in conductivity. In the sameregard, packing of available holes from the highest energy downwardsin 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 ac-electric field tends to infinite values). This leads to constant value ofhole density and a corresponding limit value of conductivity, σLimit, atvery high T (or very high frequencies). One will see, in Section 2.2, thatthis 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. Toconvert these physical aspects into quantitative equations; one canwrite the density of total electrons in the CB as:

αin ΔEið Þ ¼ NC exp ΔEi−EFð Þ=kT½ �: ð4Þ

The total density of electrons in the CB can be given by summingall over the possible values of ΔEi and αI as:

αin ¼ Σn ΔEið Þ ¼ NCΣ exp ΔEi−EFð Þ=kT½ �: ð5Þ

Furthermore, one will consider that in order to carry the electric cur-rent, the localized electrons must overcome the depth of the potentialwells, which lay between the bases, ΔEi, then nL will be given as:

αinL ¼ NC exp �EF=kTð Þ exp ΔEi=kTð Þf g−1½ �: ð6Þ

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

αin ΔEið Þ þ N−a ΔEið Þ ¼ 1−αið Þp ΔEið Þ þNþ

d ΔEið Þ þNsh 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 presentwork: the electron density in the CB is consid-ered by far greater than the holes density in the valence band i.e. n≫pand the electrondensity n(ΔEi) is by far greater than the ionized acceptorswhich have density comparable to the holes density: N−

a≅p. Thus, theneutrality equation could be approximated as:

αin ΔEið Þ ¼ Nþd ΔEið Þ þ Nsh: ð8Þ

Here, Nsh is temperature independent and “gdd” is the degeneracyof the level; it is considered to be 242. The total electron density,n(ΔEi), in the CB can be given as a function of a nearly temperature-independent density Nsh as follows:

X4i ¼1

αin ΔEið Þ ¼ Nsh þXi ¼4

i ¼1

Nd

1þ gdd exp EF−Ed þ ΔEið Þ=kT½ �: ð9Þ

Noting that; at a certain temperature, the Fermi level should bekept constant between the two sides in the neutrality equation andthe equality of summations, in Eq. (9) all-over the four bases meansthat EF has a constant value within the different bases.

It is worth mentioning that, the present model makes possible theaddition of one (or several) deep or shallow trapping level to the neu-trality equation (Eq. (9)) whether they are donors or acceptors, underthe condition that there is at least only one exhausted shallow trap-ping level. By this presumed addition, the thermal behavior of n isexpected to be kept the same, i.e. at high temperatures, the ionizationof the deep level(s) dominates the electrical conduction mechanism


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

2.2. Derivation of complex conductivity due to band carriers throughmovingDNA molecule: dc conductivity

Thepresentwork sheds light on the role played by the localized elec-trons in the CB which can lead to interesting range of electrical conduc-tivity starting from insulators up to conductors. It has been previouslyshown that localized electrons play no role in the electrical conductionand they are stationary in the localized states; but they drastically affectthe free holes and hence decrease the electrical conduction. The pro-posed model shows that charges can carry the electric current by over-coming potential barrier ΔE at a certain temperature T. Thus, followingto the previous work (Abdalla, 2011a; Abdalla and Pistoulet, 1985;Abdalla et al., 1987, 1989; Pistoulet et al., 1984), the rate constant forhole (electron) transfer between the donor and acceptor species, ket, isgiven by ket=p(r) exp (−ΔΕ/kT), where p(r) is the probability for thehole (electron) transfer normalized to the number of times the molec-ular assembly acquires the correct configuration to pass through theintersection of the potential energy surfaces (Abdalla and Pistoulet,1985). In the present work, ΔΕi is considered as the energy needed forthe electron to transfer from one base to another base or another positivesite. In addition, the electrical conductivity increaseswith temperature tilla maximum value, σLimit, which occurs when there are no localized elec-trons in the CB i.e. when all electrons, in the band, participate in carryingthe electrical current. Thus, one canwrite:σLimit=qαintotal(ΔEi)μwhere qis the electronic charge in coulombs, ntotal (ΔEi) is the total electronsdensity in the CB in cm−3 and μ is the drift mobility of the electrons incm2/(V·s). The application of dc electric field on the terminals of theDNA molecule is considered without any presumed electrical barriersdue to the metal contacts. The electric field stimulates immediately thefree electrons to transfer within the molecule with certain drift velocitywhile the localized electrons will be blocked against the potential wellboundaries and they can't carry the electric current. Thus the DNAmole-cule will be considered as insulator if these free electrons haveweak den-sity i.e. the transfer of electric energy will be very weak. On the otherhand, high densities of free holes (electrons) lead to rapid transfer of elec-tric current which leads to semi metallic conduction as the energy passeseasily through the molecule.

To explain the electrical behavior of DNA as an insulator: at lowtemperatures most of the electrons in the band will be localized inpotential wells and by heating the molecule, the localized electronswill attain more energy and more of them become free. Continuousheating makes more localized electrons to be converted free andthus, leads to increase the free electron density and at the end, atvery high temperatures, all electrons in the CB, ntotal, will be freeand will participate in the electrical conduction leading to a limit con-ductivity,

σLimit : σ Limit ¼X4i ¼1

σF þαiσL ΔEið Þð �½ ð10Þ

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

σ limit ¼Xi ¼4

i ¼1

q μF⋅nFð Þ þ μLαi⋅nL ΔEið Þ½ Þ½ �: ð11Þ

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

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

σdc ¼X4i ¼1

σLimitexp −ΔEi=kTð Þ ð12Þ

The semiconducting behavior of the DNAmolecule is well manifestedthrough the exponent term of the activation energy −ΔEi/kT in the lastequation.

2.3. Derivation of complex conductivity due to band carriers: dc permittivity

Investigation of dielectric properties of DNA goes back to the early1960s and since then a reasonable number of papers have been published(Abdalla, 2011b; Basuray et al., 2010). In these studies, two relaxationmodes are found, one at very low frequencies and the other at intermedi-ate frequency. For the sake of covering vast range of frequency, it is alsomentioned, here, that there are four relaxation types in the frequencyrange from10 Hz up 10 G Hz corresponding to the relaxation of localizedelectrons between the four bases of the DNA molecule. This will be clari-fied as follows: let εdc(ΔEi) be the maximum value of the dielectric con-stant when all (free and localized) electrons in the base are polarizedunder dc conditions. On the other hand, under very high frequencies con-ditions, the minimum value of the dielectric constant establishes whenonly free electrons are polarized. As the free hole (electron) respondsdirectly to the ac field, they lead to a limit dielectric constant at veryhigh frequency, ε∞. Thus, one can write: εdc=[dielectric constant due tofree electrons, ε∞]+[dielectric constant due to localized electrons, εL]=ε∞+εL(ΔEi); which leads to:

εL ΔEið Þ ¼ εdc ΔEið Þ−ε∞: ð13Þ

In fact, after the presented model, the relaxation phenomena andthe dielectric behavior of the DNA molecule is attributed to the local-ized electrons rather than the free ones. On the other hand, the elec-tric behavior of the DNA molecule is attributed to the free electronsrather 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 cancarry the ac electric current as they follow the polarization of the elec-tric field. In fact, the validity of the idea of electron localization is ex-amined when applying ac electric field to the total electrons of thebase 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 totheir localization. Inside the considered base, τ depends on the ener-gy, ΔEi and the temperature as:

τ ΔEið Þ ¼ τ0i exp ΔEi=kTð Þ ð14Þ

where τ0i is the relaxation time of localized electrons when T tends toinfinite values and it is a characteristic value for each base. The meanrelaxation time of the DNA molecule (the measured relaxation timethrough the molecule), τ is obtained by the summation all over thepossible four activation energies for the total bases:

1τ¼

X4i ¼1

1τ ΔEið Þ ¼

X4i ¼1

1τ0i exp ΔEi=kTð Þ: ð15Þ

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


at high temperature, the relaxation of localized electrons at the deep en-ergy levels is rather manifested while at low temperature, the relaxationof localized electrons at the shallow energy levels becomes dominating.Table 1 shows also these relaxation times as a function of the activationenergy and the temperature. τ0i is taken as 1.96×10−11 s. The delay ofthe localized electrons, with respect to the free ones, leads to formationof electric dipoles (each localized electron is polarized with the nearestpositive ion). The relaxation time of these dipoles gives rise to the crea-tion of complex conductivity.

σ*L (ΔEi) is considered as the conductivity due to localized electronsand σ*F is the conductivity due to the free ones. Thus, the total conduc-tivity, σ*(ΔEi) of the base under consideration could be written as:

σ � ΔEið Þ ¼ σ � F þ σ � L ΔEið Þ: ð16Þ

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

σF ΔEið Þ ¼ nFμFqandσL ΔEið Þ ¼ αinL ΔEið ÞμLq ð17Þ

μF and μL are the drift mobility of free and localized electrons, respec-tively. The above mentioned dipoles are considered as ideal dipoles ofDebye type. Thus, applying the well known Debye-formula on a baseof energyΔEi, the ac complex conductivity is, thus given as the contribu-tion of the complex conductivity due to free electrons and localizedones: σ � ΔEið Þ ¼ σ �½ �Free þ εdc ΔEið Þ−ε∞½ �

1þjωτ

n olocalized

where j ¼ffiffiffiffiffiffiffiffi−1

p; this

last equation could be rewritten as:

σ � ΔEið Þ ¼ σF þ jωε∞½ �Freeþ σL ΔEið Þ ω2τ2

1þω2τ2þ jω

εdc ΔEið Þ−ε∞½ �1þω2τ2

" #localized

ð18Þ

where σL(ΔEi) is the real part of the conductivity due to localized elec-trons andσF is the real part of the conductivity due to free electrons. Thephase θ between the ac electric current due to free electrons and the accurrent due to the localized electrons are considered to have idealDebye behavior i.e. θ=90° and as it has been mentioned earlier: θ=0 for free electrons. Thus, average complex conductivity due to freeelectrons in the base is:

σ � F ¼ σF þ iωε∞ ð19Þ

where σF is the real part of σ* and ε∞ is the imaginary part. One shouldnote that both σF and ε∞ are independent of the localization energy ΔEi.Similarly, for the localized electrons:

σL � ΔEið Þ ¼ σL ΔEið Þ ω2τ2

1þω2τ2þ jω

εdc ΔEið Þ−ε∞½ �1þω2τ2

: ð20Þ

Arranging Eqs. (18)–(20) as to have the real part (conductivity) ofthe total electrons and imaginary parts (permittivity) of the consid-ered base:

σ ΔEið Þ ¼ σF þ σL ΔEið Þ ω2τ2

1þω2τ2ð21Þ

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

ΔE τ250K τ250K τ300K τ350K

eV s s s s

0.56 3.8 2.3×10−1 5×10−2 2.3×10−

0.34 1.41×10−4 2.63×10−5 1.02×10−5 1.56×10−6

0.24 1.36×10−6 4.14×10−7 2.13×10−7 5.63×10−8

0.05 2.03×10−10 1.56×10−10 1.38×10−10 1.05×10−10

ε ΔEið Þ ¼ ε∞ þ εdc ΔEið Þ−ε∞1þω2τ2

: ð22Þ

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

σac ¼X4i ¼1

σ ΔEið Þ ¼X4i ¼1

σF þ σL ΔEið Þ ω2τ2

1þω2τ2ð23Þ

ε ¼X4i ¼1

ε ΔEið Þ ¼X4i¼1

ε∞ þ εdc ΔEið Þ−ε∞1þω2τ2

: ð24Þ

The relation between σ and ε′ could be, directly known by omit-ting (ωτ)2 from the relations (23) and (24) which leads to:

σac�σdc

ω¼ ε″ ¼

X4i¼1

σL ΔEið Þεdc−ε′

� �ω εdc−ε∞ð Þ ð25Þ

ε″ ¼ σac−σF

ω¼

X4i¼1

σL ΔEið Þ ωτ2

1þω2τ2: ð26Þ

After the present model, the famous Cole and Cole curves can beexplained in terms energy dispersion of localized electrons in the CB(Eq. (25)); i.e. the free electrons can't participate in these dispersionsand one must subtract σdc from the measured ac electrical conductiv-ity, σac before constructing the Cole and Cole curves. Furthermore, ifone considers the well known power relation of the ac-electrical con-ductivity: σac=σdc+σ0ωs where σ0 is a fitting parameter, and com-pare 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 tem-perature as following:

σac−σdc ¼ σ0ωs ¼

X4i ¼1

σLω2τ0i

2 exp ΔEikTð Þ21þω2τoi

2 exp ΔEikTð Þ2: ð27Þ

Eq. (27) shows that the distribution of relaxation times over thepossible four values of energies leads to interesting range of the expo-nent 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 valueof the exponent s as:

s ¼ lnX4i ¼1

σL

σ0

� �ω2τ0i

2 exp ΔEikTð Þ21þω2τoi

2 exp ΔEikTð Þ2( )

= ln ωð Þ: ð28Þ

It is easily noticed, from Eq. (28), that s is so sensitive for both fre-quency and temperature which is what one finds in the literature.Similarly, when combining Eqs. (15) and (24), one gets the dielectricconstant as a function of ω and T:

ε ¼ ε∞ þX4i−1

εdc−ε∞1þω2τ0i

2 expΔEi=kTð Þ2: ð29Þ

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

0 5 10 15 20 2510-12

10-8

10-4

100

104

108

1012

10-12

10-8

10-4

100

104

108

1012

Rel

axat

ion

Tim

e o

f L

oca

lized

ch

arg

e C

arri

ers,

s

1000/T, K-1

Δ E = 0.56 eV

Δ E = 0.33 eV

Δ E = 0.24 eV

Δ E = 0.05 eV

0 2 4 6 8 10

Fig. 1. The relaxation time of the localized charge carriers as a function of temperaturefor different activation energies.

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

i ΔEi,eV

αi [σLimit]i, Ω−1 cm−1

Povailas andKiveris (2008)DNA insulator

Tran et al.(2000)DNA conductor

Povailas andKiveris (2008)DNA insulator

Tran et al.(2000)DNA conductor

1 0.56 0.989 1 1 8×106

2 0.33 0.001 1 1×10−3 4×105

3 0.24 0.0001 1 1×10−4 3×104

4 0.05 6×10−10 1 6.93×10−10 8×10−2


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

2.5. Relaxation phenomena in DNA molecules

The property of relaxation of charge carriers is used here to elucidatethe role of band carriers in the electrical conduction through DNA mol-ecule. As it is well known, the hole (electron) localized in a potential hill(well) in a certain base should exhibit a characteristic motion when analternating electric field, with angular frequency ω, is applied. If thethermal energy, kT, of the electrons through the base remainsunchanged but ω varies, one expects the characteristic motion tochange accordingly, and the variation should be reflected in the dielec-tric relaxation spectroscopy (DRS) spectrum. At a certain temperature,the DRS should be characterized by a definite relaxation time τ. In thisregard, at a certain applied frequency, f0, the localized electrons shouldhave maximum response with their natural oscillations; i.e. they have amaximum displacement far away from their rest position. This occurswhen ω0τ=1; where ω0 ¼ 1

2πf0. After ac-conductivity experimental

measurements, Tomić et al. (2007) have found two types of relaxationsin DNA molecules and they have claimed the presence of relaxationphenomena at two distinctive relaxation times, 3.72×10−5 s and4.28×10−7 s, respectively. In the same regard, Long et al. (2003) havefound that charge carriers relax in the range 10−2–4×10−2 s. In addi-tion, Takashima et al. (1986) have found that charge carriers in theDNA molecule can relax at a relaxation time of 1.6×10−10 s. Thus,after these experimental evidences, one can consider the presence offour 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 re-laxation times are characterized by oscillations around a neutral locali-zation of the charge. One should distinguish between these relaxationtimes (1×10−2 s–1.6×10−10 s) and the rate by which a chargejumps over a potential barrier through the DNA double helix (some pi-coseconds (Voityuk et al., 2004)).

Now, these four relaxation times will be correlated with the fouractivation 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 demonstratedin 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 holesare only responsible for carrying the electric current through the differ-ent bases. The DNA is considered, first, as an insulator, and has fitted theexperimental values of Povailas and Kiveris (2008) with Eq. (12). Thesuitable 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 israre in the literature and not easy to be performed, so it is difficult toexamine the validity of Eq. (7). However, a technique is developed toestimate the electron density in the DNA from the experimental dataof the dc-conductivity. First, Tran et al. (2000) have, experimentally,reported that the electrical conductivity is composed of two distinc-tive parts: 1 — at low temperatures (80 K) the conductivity is nearlytemperature independent and 2 — the effect of shallow activationenergy Nsh will mask the effect of the deep activation energy ΔEsh,thus, Eq. (9) can be approximated as:

X4i ¼1

NC exp ΔEi−EFð Þ=kT≈Nsh:½ ð30Þ

This can lead to an almost constant density of electrons in the CB. Onthe other hand, at high temperatures (400 K), Eq. (9) could be approxi-mated as:

X4i¼1

NC exp ΔEi−EFð Þ=kT ¼Xi¼4

i¼1

Nd

1þ gdd exp EF−Ed þ ΔEið Þ=kT ð31Þ

where NC ¼ 2πm�kTh2

� �32 is the density of states in the CB, m*is the effective

electrons mass; it is taken as m*=4.95 m0 (Abdalla et al., 1987) wherem0 is the electron mass at rest and h is Planck's constant; thus, NC couldbe written as: NC ¼ 2:64� 1016 T

32 cm−3. 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 gddwill 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 forEF, one considers that there is a certain temperature Tc at which theelectron density n is due to two equal parts: one from the shallowlevel 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 gddwill 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 neutralityequation for EF, one considers that there is a certain temperature Tcat which the electron density n is due to two equal parts: one fromthe shallow level Nsh and the other is due to the ionization of thedeep level Nd. Thus, at Tc:

n ¼Xi ¼4

i ¼1

Nd

1þ gdd exp EF−Ed þ ΔEið Þ=kTc½ � þNsh ð32Þ

50 100 150 200 250 300 350

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Fer

mi E

ner

gy,

EC -

EF, e

V

Temperature, K

DNA as Insulator

DNA as Conductor

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

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


and

Xi¼4

i¼1

Nd

1þ gdd exp EF−Edþ ΔEið Þ=kTc½ �≈Nsh: ð33Þ

Now to estimate the critical temperature Tc, one extrapolates thecurve of the electrical conductivity at high temperature and extrapolatesthe curve of the electrical conductivity at low temperature; then the in-tersection lies at Tc as shown in Fig. 2. Tc for the experimental data ofTran et al. (2000) is found to be about 240 K and 146 K for Povailasand Kiveris (2008). Taking into account the above mentioned values ofΔEi 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. Thensolving Eq. (9) for EF for different temperatures, one can find the Fermienergy as a function of temperature which is illustrated in Fig. 3 (Tranet al., 2000). From this figure, one can notice that a maximum of EFlays at about 0.1 eV at 215 K. Using Eq. (5) and the data in Fig. 3, onecan calculate n as a function of temperature. Fig. 4 shows n as a functionof temperature one can observe that at 240 K a steep rise of n with tem-perature begins with activation energy at high temperature about0.33 eV; andn=1.88×1019 cm−3 at 300 K. To calculate the driftmobil-ity of charges through the different bases within the molecule at 300 K,one considers the above mentioned obtained value of electron densityn=2.35×1016 cm−3 with a conductivity 1 Ω−1 cm−1 which leads toan approximate mobility 257 cm2/(V·s). Moreover, the free electrondensity, nF, calculated after Eq. (2), is plotted as a function of tempera-ture 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 elec-tron density n and notice that the ratio nF/n tends to unity at high tem-peratures 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 ex-press n in terms of cm−1 instead of cm−3; Beleznay et al. (2006) havereported that the volume of an elementary unit of DNA cell equals1.5×10−8 cm×5×10−8 cm×10×10−8 cm; which gives an area ofabout 7.5×10−16 cm2; thus the linear electron density is about1.22×102 cm−1 (at room temperature). In addition, it is more conve-nient to express NC, for DNA molecule, in cm−1. The DNA unit area isabout 7.5×10−16 cm2; thus the linear density-of-state in the CB for aDNA molecule is about 19.8×T1.5 cm−1 which gives (NC)300 K=1.03×105 cm−1 at 300 K .

0 5 10 15 20 2510-13

10-10

10-7

10-4

10-1

102

1000/T, K-1

Calculated after Eq. 12

Experimental after Tarn et al (2000)

Calculated after Eq. 12

Experimental after Povailas et al(2008)

0 5 10 15 20 25

10-13

10-9

10-5

10-1

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.

3.3. ac-Conductivity of moving DNA molecule

Georgakilas et al. (1998) have reported experimental data on mam-malian DNA macro molecules at different frequencies and differenttemperatures. In this section, their ac complex conductivity data areanalyzed in the light of the presented model. In particular, the depen-dence of the electrical conductivity σac and the dielectric permittivityε′ 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 increaseswith ω as ω0.68 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 timeof 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 onthe same figure.

It is shown that there is a good accordance between the calculated andthe experimental values. Fig. 6 shows the variations of σac as a function oftemperature. The symbols represent experimental data and solid lines arefor calculated values (after Eq. (23)). There is good agreement betweenlines and open circuits when taking τ=1.74×10−4 s. Subtracting thedc conductivity from the ac part; Eq. (28) gives: s~0.69which in good ac-cordance with their experimental value 0.6862. To calculate the dielectric

2 4 6 8 10 12

1E15

1E16

1E17

1E18

1E19

1E20

1000/T, K

nF Tarn et al

nF Povailas et al

nTotal

Tarn et al

nTotal

Povailas et alDNA insulator

DNA Conductor

Ele

ctro

n D

ensi

ty, c

m-3

Fig. 4. The total electrondensity in the conductionband, ntotal and the free electrondensitynFas a function of temperature; for both DNA as a conductor and DNA as an insulator.

Calculated at 200K

Calculated at 300K

Calculated at 400K

Experimental data after Povaillas et al

Frequency, Hz

100 102 104 106 108 1010

100 102 104 106 108 1010

10-1

100

10-1

100

200 K

300 K

400 K

ac E

lect

rica

l Co

nd

uct

ivit

y,Ω

-1cm

-1

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

2 4 6 8 10 12 14

0

1x106

2x106

3x106

4x106

5x106

Experimental after Povaillas at 100Hz

Experimental after Povaillas at 1M Hz

Calculated at 100 Hz

Calculated at 1 M Hz

Die

lect

ric

Co

nst

ant,

ε'

1000/T, K-1

100 Hz

1 MHz

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


constant, ε′, Eq. (24) is used; then the experimental data of Georgakilas etal. (1998) are fitted with the calculated values. Fig. 7 shows their experi-mental data as symbols and the calculated values as solid lines.

The accordance between the solid line and circles in both figuresare good when considering the fitting parameter τ=1.74×10−4 s.Moreover, the electric dispersion ε″, is calculated, after Eq. (26), as afunction of the frequency and it is found that, at 25 °C, ε″ passes bya maximum at about 910 Hz which corresponds to a relaxation timeτ=1/(2π*910)=1.75×10−4 s (Fig. 8). Moreover, the electric disper-sion ε″, is calculated, after Eq. (26), as a function of the frequency andfound that, at 25 °C, ε″ passes by a maximum at about 910 Hz whichcorresponds 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 timeof τ=1.74×10−4 s. The relaxation time of the localized electrons isgiven by the product ε′/σac, which varies from a base to another.

3.3.1. Relaxation phenomena in DNA moleculeAn electron localized in potential well in a certain base in DNA

molecule should exhibit a characteristic motion when an alternatingelectric field, with angular frequency ω, is applied.

If the thermal energy, kT, of the electrons through the base re-mains unchanged but ω varies, one expects the characteristic motion

4 6 8 10 12 141E-6

1E-5

1E-4

1E-3

0.01

0.1

1Calculted after equation 23 for 100Hz

Calculted after equation 23 for 1MHz

Experimental after Povaillas for 100Hz

Experimental after Povaillas for 1M Hz

1000/T, K-1

4 6 8 10 12 14

1E-6

1E-5

1E-4

1E-3

0.01

0.1

1

frequency = 1 MHz

frequency = 100Hz

ac E

lect

rica

l co

nd

uct

ivit

y,Ω

-1cm

-1

Fig. 6. The variations of σac as a function of temperature: the open symbols representexperimental data and solid lines are for calculated values (after Eq. (23)).

to change accordingly, and the variation should be reflected in the di-electric relaxation spectroscopy (DRS) spectrum. This occurs whenω0τ=1; where ω0 ¼ 1

2πf0.

The symbols represent experimental data and solid lines are forcalculated values (after Eq. (24)).

Tomić et al. (2007) have experimentally found two types of relax-ations in DNAmolecules and they have claimed the presence of relax-ation phenomena at two distinctive relaxation times, 3.72×10−5 sand 4.28×10−7 s, respectively. Also, Long et al. (2003) have foundthat charge carriers could relax in the range 10−2–4×10−2 s. In addi-tion, as it is above mentioned, also, Takashima et al. (1986) havefound that charge carriers in the DNA molecule could relax at a relax-ation time of 1.6×10−10 s. Thus, one can consider the presence offour 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 wellmanifested in Fig. 9 where the relaxation time τ, obtained using themaxima in Fig. 8, is illustrated as a function of temperature. The elec-tron transfer from the base G to Cvertical is accompanied by an energydifference ΔEG→A=0.05 eV (Richardson et al., 2004); which corre-sponds to a relaxation of time about 1.6×10−10 s. One can noticethe exponential behavior of τ which verifies Eq. (15). In the same re-gard, one should notice that these relaxation times are correlatedwith the above mentioned four thermal activation energies. As a con-sequence, and with the consideration of Richardson et al.'s (2004)work the above mentioned activation energies are correlated withthe corresponding electron transfer from one base to another.

102 103 104 105

0

20000

40000

60000

80000

Calculated ε" 353 K



Experimental ε" 298 K



After Povaillas et al 2008

Ele

ctri

c D

isp

ersi

on

ε"(

σ/ω

)

Frequency, Hz

Fig. 8. The electric dispersion ε″, calculated after Eq. (26), as a functionof the frequency (solidlines) and experimental values (as symbols): at 25 °C, ε″ passes by a maximum at about910 Hz which corresponds to a relaxation time of τ=1.75×10−4 s.

2.8 2.9 3.0 3.1 3.2 3.3 3.4

5.0x10-5

1.0x10-4

1.5x10-4

2.0x10-4

Rel

axat

ion

tim

e τ,

sec

on

ds

1000/T, K-1

Experimental after Takashima et al

Calculated after equation (14)

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 accompaniedby a relaxation time of about 4.28×10−7 s; which corresponds toan energy difference of ΔEC→G=0.33−0.09=0.24 eV (Richardson etal., 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 nanotubecontact

In this section, the presented model will be applied to fit the ex-perimental model of Guo et al. (2008). These authors have insertedDNA strands between two single-walled carbon nano-tube (SWNT)electrodes and have measured the electrical conductivity at roomtemperature. The following points will be considered, when applyingthe present model:

(1) The contact resistances between SWNT and theDNA are ofmajorimportance and theywill be considered to play an important rolein the conduction mechanism.

(2) The configuration “Metallic contact–SWNT–DNA–SWNT–Metalliccontact” is defined in this section as “the device” and is consideredas a field effect transistor (FET) where the SWNTs represent thesource and the drain, while DNA molecule stands for gate of thetransistor (it plays the role of a p-channel as it will be proofedlater in this section).

(3) The polarization of the sample has been given by the authors Guoet al. (2008) as follows: positive terminal–(CH2)3–Pi–5′-AGT ACAGTC ATC GCG-3′–Pi–(CH2)3–negative terminal which makes usconsider the forward bias of the DNA molecule to be positivelybiased.

(4) In the presence of a gate potential (transfer potential), VGS, severalsteps will occur when applying external reverse potential VDS be-tween the terminals of the device: (i) first, the extended bandsbending in at the SWNT–DNA interface increases, eventually tothe point at which holes have a high probability of jumping

Table 3The relaxation time τ, at room temperature, as a function of the activation energy andthe corresponding energy transfer from a base to another.

Authors Relaxation time τ,seconds

Holetransfer

Corresponding activationenergy ΔE, eV

Long et al. (2003) 2×10−2 T→A 0.56Tran et al. (2000) 3.72×10−5 C→G 0.33Tomić et al. (2007) 4.28×10−7 Tvertical→A 0.24Takashima et al.(1986)

1.6×10−10 G→Cvertical 0.05

from the localized state to the valence band (HOMO); thus theybecomes free to carry the electric current, (IDS) and controlledby the gate potential VGS. The effect of higher dopant concentra-tions can also be realized by accumulating (or depleting) carriersfrom the SWNT–DNA interface (with a gate bias) for a DNA of agiven dopant level. (ii) In addition to the precedent point, the en-ergy of localized holes, ΔEh becomes an effective variable whichcan be changed to affect the charge transfer in the molecule. Be-cause the localized holes in the hills (potential wells in the con-duction band LUMO — for electrons) inside the VB will acquiremore energy from the applied VGS and can jump to overcomethese potential hills, and then become free to carry the electric en-ergy (current). (iii) Therefore, the effect of larger localization ener-gies can be countered by stronger accumulation (at least untilquantum carrier confinement becomes significant).

This allows for greater thermal operating stability without a signifi-cant loss in the drift current IDS. The accumulation of carriers around theSWNT in the presence of both VDS and VGS results in the formation of anegative Schottky field effect (Greatbanks et al., 2000). The behavior issimilar to that of a p-channel metal-oxide-semiconductor FET (Eley andSpivey, 1962). The source-drain current, IDS decreases strongly with in-creasing gate voltage, which demonstrates that the given device operatesas a field effect transistor and also that transport through the semicon-ducting DNA is dominated by positive carriers (holes). Neglecting thediameter‐dependence of contact resistances; for VGb0 V, the curvesdescribing the current as a function of the gate voltage saturates indictingthat the contact resistance RC at the SWNT electrodes starts to dominateand the total resistance will be: RDS=RDNA+2RC of the device. HereRDNA denotes the gate-dependent resistance of the DNA molecule. Afterthe experimental work of Guo et al. (2008), the saturation value of thecurrent corresponds to R≈50 mV/16 n A=3.16 MΩ. Similar contactresistances (≈1.1 MΩ) are found for metallic SWNT (Lagerqvist et al.,2006). This leads to the conclusion that RDNA should be inferior to RDSi.e. RDNAb3.16 MΩ. If the gate voltage (the on-site energy) is increasedin reverse direction, then the drift current, IDS, will decrease until thechannel will be pinched off at a critical voltage; at which IDS=0. This isquite seen in the experimental work of Guo et al., 2008, Fig. 2-a: at highnegative values of the gate potential, one can observe that themaximum value of the drain current IDS is nearly constant at about16 nA which represents the on state of the FET (maximum conductivityof the channel). At this on “state”, the potential VDS is divided into thetwo portions: one affects the two SWNTs, 2VC and the other potentialVDNA is applied on the DNA. One can write: RDNA ¼ VSD

ISD−2 . Tans et al.

(1998) have calculated the contact resistance between SWNT andmetal-lic gold to be about 1.1 MΩ; thus, the last equation could give directly theDNA resistance as ≈0.96 MΩ. This gives a potential drop across theSWNT, VC=17.6 mVwhich leads to a potential drop across themoleculeVDNA=VDS−2*VC=50 mV−2*17.6 mV=14.8 mV. One can depict thecharge transfer in the device as follows: as the gold has a highwork func-tion, it can easilywithdraw electrons from the SWNT leaving the conduc-tion band (LUMO) of this latter rich in holes. Electrons from the valenceband (HOMO) of this latter will be excited, and overcome the barrier ofthe forbidden gap energy, e.g. to compensate for the “created” holes inthe conduction band and hence the valance band will be enriching withholes ready to carry the current (free holes in the VB). These free holesconstitute the current IDS. Consequently, when increasing the reversegate potential, the current in the channel, IDS starts to decrease exponen-tially according to the relation:

IDS ¼ 2VC

RC

� �þ VDNA

RDNAexp

−qVGS þ ΔEh þ ΔϕnkT

� �� ð34Þ

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


The bestfit between the experimental values ofGuo et al. (2008)with thelast equation (Eq. (34)) are: ΔEh=0.162 eV, n=1, Δϕ=0.791 eV. Thislast 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 pos-sibility is that the carrier concentration is inherent to the DNAmoleculeitself. Another possibility is that the majority of carriers are injected atthe gold–SWNT contacts; then to the molecule via the amide groups.The higher work function of gold (≈5 eV) leads to the generation ofholes in the DNA by electron transfer from the DNA to the gold elec-trodes (Spalenka et al., 2011). Assuming that the band-bending lengthin their SWNT is neither very short nor very long, Tans et al. (1998)have argued that the FET operation can be explained based on thischarge transfer (Spalenka et al., 2011). At VG=0 V, the device is “on”and the Fermi energy approaches themaximumheights of the potentialhills in the valence band throughout the DNA. If indeed the band-bending length is comparable to the length the SWNT, a positive gatevoltage would generate an energy barrier of an appreciable fraction ofeVG in the center of the channel (since the gate/DNA distance is shorterthan the source/drain separation). The threshold voltage VGScr requireto suppress hole conduction by depleting the DNA center would be de-termined by the thermal energy available for overcoming this barrier.Thus, VG should bemuch lower than 4 V (which is observed in their ex-perimental data Guo et al., 2008). Therefore it is important to explorethe other possibility, namely, the carriers are an inherent property ofthe molecule itself. In this case, it is expected that an inhomogeneoushole distribution along the DNA will affect drastically the charge trans-fer through the DNA. The validity of explanation can be verified by ex-amining the charge population on the different CG and AT pairs. It canclearly be seen that the population of holes on the CG pairs is quitelarge, while the population on the AT one is always negligible. Thus,the hole density through the molecule varies as a function of energyand displacement vector (location) while the Fermi energy EF is keptconstant value throughout the different bases. This makes the valenceband energy; EV itself fluctuates around a most probable value EV0. Allholes at EV having energies more than ΔEh are considered to be local-ized in potential hills; and similarly the conduction band energy EC fluc-tuates around a most probable value EC0 where all electrons havingenergies less than ΔEh are considered to be localized in potentialwells. This can be understood asfluctuation of the highest occupiedmo-lecular orbits HOMO around a most probable value EV0 while EC0 is themost probable value due to fluctuation of the lowest unoccupiedmolec-ular orbits LUMO. This will be called potential fluctuations PFs of the ex-tended bands (HOMO and LUMO) in DNA molecule. If one considers

-4 -3 -2 -1 0

0

4

8

12

16

20

Experimental data after Guo et al [35]

Calculated after the present work

So

urc

e D

rain

Cu

rren

t, I S

D, n

A

Gate voltage, volts

Fig. 10. The figure shows the calculated values after Eq. (34) as a solid line and the experi-mental values of Guo et al. (2008) are shown as open squares. One can notice a good agree-ment between the experimental and calculated values.

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 through-out the molecule) and a partial negative charge (plausible on the oxy-gen atoms throughout the molecule), this configuration leads one toconsider that any base inside DNA accumulates the charges as an capac-itor with capacitance Cj.

An approximate estimate of the hole density can then be obtained bywriting the total charge on the 15 base DNA as:

Q ¼ VGScr

X15j¼1

Cj

where C ¼X15j¼1

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

cule and is the threshold voltage necessary to completely deplete allcharges from the capacitances (bases). The DNA capacitance per unit

length with respect to the back gate is: CL ¼ 2πεε0

ln2hr

with r, h and L being

theDNA radius, thickness and length respectively; ε is the relative dielec-tric constant. As afirst approximation, onewill consider that the 15 basesare equally charged and that they are linearly arranged in a straightman-ner. 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-dimensionalhole density, p can be evaluated as p ¼ Q

qL≈7×105 cm−1 from VG=

4 V, where q is the electronic charge. The large hole density suggeststhat DNA is degenerate and/or that is doped with acceptors, for exampleas a result of its processing (Zang et al., 2010).

Assuming that the transport in DNA occurs by drift motion atroom temperature, one can estimate the mobility of the holes fromthe transconductance of the FET as follows: in linear regime, itis given by dI

dVG¼ μh

CL2

� �VSD. Subtracting the contact resistance

effect (transconductance of the contacts), one can obtain a DNAtransconductance of dI

dVG=1.49×10−8 A/V at VG=50 mV, corresponding

to a hole mobility mh≈93.9 cm2/(V·s). This value is not too far from theprevious mobility (257 cm2/V·s) reported in Section 3.2 in this work.Generally, the published values of the mobility in the DNA moleculevary in a vast range: 10−10 cm2/V·sbμhb225 cm2/V·s (Liao et al.,2010; Salieb-Beugelaar et al., 2008). However, the value reported in thepresent study is close to the mobility in heavily p-doped nano silicon ofcomparable hole density (Zang et al., 2010), but considerably smallerthan 104 cm2/V·s observed in nano graphene (Castor et al., 2010). More-over, it is difficult to accept a mobility of the order 10−10 cm2/V·s in anano-channel. One should distinguish between the drift motion of thecharge carriers alongside the molecule and the trapping rate of DNAwhich is insensitive to the potential (Cai et al., 2010; Kreft et al., 2008).

The relatively high value of the DNA mobility is consistent withthe initial assumption of drift transport and confirms that the DNAcontains a large number of scatter points possibly related to defectsin the DNA or structural-disorder at the DNA–gate-interface due toroughness. SWNTs are known to conform to topography of the sur-face so as to increase their adhesion energy. Such deformations canlead to electronic structure changes (Cai et al., 2010), which may actas scattering centers.

4. Conclusions

In conclusion, the presented work explains the effect of natural vi-brations of DNA molecule on the mechanism of charge transferthrough the DNA molecule and it is able to explain the controversyof the complex conductivity as a function of frequency and tempera-ture observed in DNAmolecules in a wide region of the frequency andtemperature. Moreover, the presented results suggest that the con-duction mechanism through DNA is due predominantly to electronicorigins rather than ionic ones.


A strong temperature dependence of conductivity observed athigh temperatures and very weak temperature dependence at weaktemperatures is comprehensible in the framework of the presentedmodel. It is worth noting that the free electron density dependencesat both low and high temperatures are calculated using simple setof parameters (doesn't exceed three fitting parameters σdc, σL, andτο). The correlation between different bases and the electron affinityof every base plays an essential role in the charge transport throughDNAmolecule. These results reveal a high possibility for the existenceof four thermal activation energies corresponding to the four bases ofthe molecule. Further experimental data on DNAwith accurate metal-lic contacts at high frequencies and temperatures should elucidateour proposal. It is expected that the linear drift motion of chargesalongside the helix has multiple medical applications. Speak aboutDNA cancer charge transfer and tumor. Yet, the present results dem-onstrate that charge transfer through DNA molecule have sufficientlyhigh importance in DNA reparation and in medical and technologicalapplications.

Acknowledgments

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

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