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DNA Spintronics: Charge and Spin Dynamics in DNA Wires Sohrab Behnia, Samira Fathizadeh, and Afshin Akhshani J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 12 Jan 2016 Downloaded from http://pubs.acs.org on January 12, 2016

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The Journal of Physical Chemistry

DNA Spintronics: Charge and Spin Dynamics in DNA Wires S. Behnia,∗ S. Fathizadeh, and A. Akhshani Department of Physics, Faculty of Science, Urmia University of Technology, Urmia, Iran E-mail: [email protected] Phone: +98 443 3554313 Abstract A combined analytical approach is proposed to investigate the spin selectivity properties of DNA nanowires considering the spin degree of freedom in the extended PeyrardBishop-Holstein model. Based on a real chain of DNA sequence, show no completely pure spin current through DNA, but no complete pure spin exists in a DNA chain, but instead one of the spin currents is dominant over another and creates a spin filtering effect. In several parameter regions, the net charge current is low; thus, a nearly pure spin current could be reported. We examined the effects of external fields, temperature, and sequence variation on spin-dependent charge transfer in DNA. The results show peaks in the DNA spin polarization in some parameter values. A DNA coder can be created according to these polarization peaks. Transporting information by using the DNA spin polarization is interested in information theory. Meanwhile, other parameter values exist, where nearly pure spin currents appear. The appearance of these islands can be confirmed and predicted using the Rényi fractal dimension approach.

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INTRODUCTION Spintronics field, which utilizes electron spin to store and process information, has attracted considerable interest over the last two decades. 1 Spintronic systems exploit the fact that electron current is composed of spin-up and spin-down carriers. These carriers transfer information encoded in their spin state and interact differently with magnetic materials. Information encoded in the spins persists when the device is switched on, can be manipulated without using magnetic fields, and can be written using low energies. 2 Recently, attempts have been gained to obtain spintronic devices that preserve and exploit quantum coherence, and fundamental investigations are shifting from metals to semiconducting 3,4 and organic nanomaterials. 5 Since the breakthrough of the giant magnetoresistance in 1988, 6 spin transport via solid-state systems has improved remarkably. Meanwhile, the ongoing trend in the miniaturization of electronic circuits and nanoelectronics is toward adopting single molecules as functional devices. 7 A set of spintronic nanodevices was proposed based on organic materials. Organic materials are utilized in applications, such as organic light-emitting diode displays and organic transistors. 8,9 The rich physics behind the organic materials and specific functions, such as switchability with light and electrical or magnetic fields, is an important motivation for using organic materials. The spin transport properties of the DNA as an organic molecule have been investigated experimentally and theoretically. Göhler et al. 10 reported the spin selectivity of photoelectron transmission through self-assembled monolayers of double-stranded DNA (dsDNA) deposited on gold substrate. Moreover, conduction through dsDNA oligomers is spin-selective when DNA is placed in a nanoparticle-dsDNAnickel complex. 11 Similar results were further verified by direct charge transport measurements of single dsDNAs connected between the two leads. 12 Various theoretical models were proposed to investigate the spin-selective properties of the DNA molecule based on single helical chain-induced Rashba spin-orbit coupling. A model based on scattering theory provides a qualitative explanation of the spin selectivity experimentally observed. 13 Gutíerrez et al. 14 proposed a minimal Hamiltonian model in order to describe the electron transmission 2


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throughout a helical electrostatic potential. The researches showed that the electric field generated by the charges accumulated along the helix leads to an unconventional Rashbalike-spin-orbit (RLSO) interaction. The more recently proposed model demonstrates that the environment-induced dephasing is also necessary in addtion to RLSO coupling, to reproduce the experimental results. 12 Guo and Sun showed that the combination of the spin-orbit coupling (SOC), the environment-induced dephasing, and helical symmetry should be considered in describing the experimentally studied system. By applying the gate voltage, Guo and Sun 15 showed that the spin polarization in dsDNA is strongly dependent on magnitude and the direction of the gate voltage. In another work, the spin-polarized current in DNA strongly depends on the direction of the wrapping and length of the helicoidal field through the DNA. 16 Notably, the strong electron-lattice interaction is one of the characteristic attributes of organic materials. This strong interaction leads to an important effect, that is, forming nonlinear excitations, such as polarons and bipolarons. 17 Polarons can move under an electric field, which is essential for organic light-emitting diodes and organic spin-valve devices. 18 Considering the spin coupling with radicals and the electron-lattice interaction is important to elucidate the property of carriers in such devices. In the current study, we attempted to investigate the electron transfer in DNA nanowires by considering the electron spin. We also considered the electron-lattice interaction through an extended PBH model. 19 In this model, a classic interaction between sites exists, which is linearly coupled with a tight-binding Hamiltonian. We also added the spin effect to the PBH Hamiltonian. The model characterizes the lattice dynamics as a classic PBH model and charge transport phenomena with the nearest neighbor tight-binding approach. The interplay of charge and lattice is considered through the Holstein-like approach with a chargelattice interaction term. Meanwhile, the spin-orbit interaction is combined with the PBH model. In this study, most of the Hamiltonians and corresponding motion equations are nonlinear

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and possess high sensitivity to initial conditions. Considering the nonlinear dynamics and chaos theory, tools can provide novel insight into the charge transport mechanism and its effects on DNA. In this paper, a method based on the multifractal approach is proposed for a better description of spin-dependent charge transfer in DNA. The results demonstrate that the Rényi dimension spectrum can be considered as a signature for the verification and prediction of spin filtering currents.

Model and methods Let us start consideration of N base-pairs spin-charge-lattice system with the Hamiltonian

H = Hlat + Hcar + Hint + Hso + Hf ields

(1)

Hlat simulates the lattice distortion, which is described by nonharmonic PBD model 20,21

Hlat =

X1 [ my˙ n2 + V (yn ) + W (yn+1 , yn )] 2 n

(2)

where V (yn ) = Dn (e−an yn − 1)2 is the Morse potential that represents the transverse displacement of the hydrogen bonds connecting complementary bases and

W (yn+1 , yn ) =

k (1 + ρe−b(yn+1 +yn ) )(yn+1 − yn )2 2

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is the stacking interaction between adjacent base-pairs. The electronic part of Hamiltonian is given by

Hcar =

XX n

σ† σ σ† σ σ [ǫn cσ† n cn − Vn,n+1 (cn cn+1 + cn+1 cn )]

(3)

σ

σ where cσ† n and cn create or destroy an excitation at the tight-binding site n with spin index

σ, respectively. ǫn is the on-site energy for base pair in n − th site. In this model, the charge hopping is restricted to nearest-neighbor base-pairs, and given by Vn+1,n . The transfer matrix is supposed to depend on the relative distance between two consecutive molecules on the chain in the following exponential fashion: 22 Vn+1,n = V0 e−βn (yn+1 −yn )

(4)

where V0 determines the constant hopping integral and the quantity β regulates how strong Vn+1,n is influenced by the distance r = yn+1 − yn . In the limit β = 0 the hopping integral reduces to V0 = constant and the Hamiltonian yields the standard PBH model. A limiting case is recovered by keeping only the linear term in an expansion of the exponential function in Eq. (4), i.e. Vn+1,n = V0 [1 − βn (yn+1 − yn )], which is justified only for small arguments βn (yn+1 − yn ). The other way to take into account the effects of charge lattice interaction is through a linear coupling of on-site energy with the lattice displacements yn , as proposed in the Holstein model. 23 This interaction is represented by a Hamiltonian term

Hint = χ

X

σ yn cσ† n cn

(5)

n

where χ is the electron-lattice coupling constant. An electron moves along a chiral molecule, is affected by an electric chiral field (Echiral ). It arises from the electrons and nuclei that comprise the chiral molecule. 10 Therefore, the

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moving electron experiences an electrostatic potential where Echiral = −∇V . In the electrons’ ~ = rest frame, the current generates a magnetic field B

~v c2

× Echiral , where ~v is the electron

velocity and c is the light velocity. For the dsDNA, the potential V is usually bigger along the radial direction rˆ than that along the helix axis (here, the helix axis is zˆ ). 24 Then, it is reasonable to consider E only in r direction. An internal field Er breaks the inversion symmetry and generate significant Rushba-spin-orbit interaction. We have considered the spin-orbit interaction as following: α Hso = − ~σ .(~r × p~) ~

(6)

~ 2 ) Er , ~σ are the Pauli matrices and p is the momentum operator. In α where α = −( 2mc

relation, m is the electron mass. However, Hso in the second quantization representation expresses by the Hamiltonian: 12,14 X

Hso =

itso c†n [σn + σn+1 ]cn+1 + H.C.

(7)

n

where tso is spin-orbit coupling constant, and σn = {σx (sin[n−1]∆φ+sin[n∆φ])−σy (cos[n− 1]∆φ + cos[n∆φ])} sin θ + 2σz cos θ and σx,y,z are Pauli matrices. Here, θ is the helix angle and φ = n∆φ is the cylindrical coordinate with φ the twist angle. To obtain the evolution equations, the introduced coupling term should be represented via creation and annihilation operators 25 which is written as

Hso =

X

↑ ↑† ↑ [2itso cos θc↑† n cn+1 − 2itso cos θcn cn−1

n ↓ ↓† ↓ − 2itso cos θc↓† n cn+1 + 2itso cos θcn cn−1 ↓ ∗ ↓† ↑ + Dn,n+1 c↑† n cn+1 − Dn,n+1 cn cn+1 ↑ ↑† ↓ ∗ c↓† + Dn−1,n n cn−1 − Dn−1,n cn cn−1 ]

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(8)

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where

Dn,n+1 = itso sin θ{sin[n∆φ] + sin[(n + 1)∆φ] + i cos[n∆φ] + i cos[(n + 1)∆φ]}

∗ that preservation of time reversal symmetry leads to Dn,n−1 = Dn−1,n .

In this regard, one could consider the spin Hall effect (SHE) driven by the spin orbit interaction, converts a charge current into a pure spin current. 26,27 In the current study, we propose the effect of external electrical and magnetic fields on charge-spin transfer in DNA. In this regard, The corresponding general Hamiltonian have the following form

(9)

Hf ields = HE + HB

where HE and HB are the Hamiltonian corresponded to the electrical and magnetic fields, respectively. 28 HE = −e

X

σ dE cos[(n − 1)∆φ]cσ† n cn

(10)

n,σ

HB =

X

↓† ↓ ↑ (−µB Bc↑† n cn + µB Bcn cn )

(11)

n

~ = E xˆ is the electrical field along the x axis and d is the radius of the It is supposed that E ~ = B zˆ is DNA or distance between sites of the strand along the helix. On the other hand, B the field along the z axis and µB =

e~ 2mc

= 5.78838 × 10− 5 eV.T −1 is the Bohr magneton.

We have attempted to examine the effect of both dc and ac fields on the DNA spin conductivity. In the dc condition, it is supposed that a constant field is applied in DNA. In the other case, a time periodic field is applied, so it provides an extra degree of freedom (frequency of the field) in addition to the field intensity to study the response of DNA to the external field. In this regard, the external field could be as follows: 29

E = E0 cos(ωt) 7


(12)


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B = B0 cos(ωt)

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(13)

where E0 and B0 are the amplitudes of electrical and magnetic fields, respectively and ω is the frequency of the fields. In this way, the evolution equations are formulated by 2an Dn −an yn −an yn kbρ −b(yn +yn−1 ) − 1) + (e e [e (yn − yn−1 )2 + e−b(yn+1 +yn ) (yn+1 − yn )2 ] m 2m k [(1 + ρe−b(yn +yn−1 ) )(yn − yn−1 ) − (1 + ρe−b(yn+1 +yn ) )(yn+1 − yn )] − m V0 βn ↑† ↑ ↑ ↑† ↓† ↓† ↑† ↑ ↑ ↓ ↓† ↓ ↓† ↓ ↓ (c c + c↑† − n cn−1 − cn cn+1 − cn+1 cn + cn−1 cn + cn cn−1 − cn cn+1 − cn+1 cn ) m n−1 n χ ↑2 (|c | + |c↓n |2 ) (14) − m n

y¨n =

To obtain the equations governing the electronic part, we used the Heisenberg approach ↑(↓)

c˙n

↑(↓)

= − ~i [cn , H], and then we have i c˙↑n = − {[ǫn + χyn − eE cos[(n − 1)∆φ]d − µB B]c↑n ~ ∗ + Wn−1,n c↑n−1 + Wn,n+1 c↑n+1 − Dn−1,n c↓n−1 + Dn,n+1 c↓n+1 }

(15)

i c˙↓n = − {[ǫn + χyn − eE cos[(n − 1)∆φ]d + µB B]c↓n ~ ∗ ∗ ∗ + Wn−1,n c↓n−1 + Wn,n+1 c↓n+1 + Dn−1,n c↑n−1 − Dn,n+1 c↑n+1 }

(16)

where Wn,n+1 = −Vn,n+1 + 2itso cos θ.

Results Electrical current for up and down spins According to the definition of electrical current operator, we could obtain the electrical current correspond to the spin-up and spin-down electrons directly from field equations. In 8


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this regard, one could calculate the steady state spin dependent current operators using the σ particle density operator in Heisenberg picture nσi (t) = eiHt nσi e−iHt , where nσi = cσ† i ci is the

charge density and I =

d(eni (t)) dt

=

−ie † [ci ci , H]. 30 ~

Therefore, we have:

d(en↑i (t)) dt −ie X ↑ ∗ ↑† ↑ = {Wn,n+1 c↑† n cn+1 + Wn−1,n cn cn−1 ~ n

I ↑ (t) =

↓ ↑† ↓ + Dn,n+1 c↑† n cn+1 − Dn−1,n cn cn−1 }.

(17)

d(en↓i (t)) I (t) = dt −ie X ↓ ∗ ↓† ↓ {Wn,n+1 c↓† = n cn+1 + Wn−1,n cn cn−1 ~ n ↓

↑ † ∗ ↓† ↑ − Dn,n+1 c↓† n cn+1 + Dn−1,n cn cn−1 }.

(18)

According to the obtained relation, the electrical current is dependent on the relative position of the base pairs and probability amplitude for the charge carriers in time. So, the electrical current shows the oscillatory behavior over time. Different agents affect on the current flowing through DNA. Using the nonlinear dynamics methods and the current operators equation, we can easily assess the effect of various factors such as magnetic and electrical fields, temperature and sequence variation. In the following, We have tried to examine the effect of such factors on the currents flowing through DNA. Now, by defining the net charge, Ic , and net spin , Is , currents, we studied the spin dependent currents in DNA:

Ic = I ↑ + I ↓ Is = I ↑ − I ↓

9


(19)


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Magnetic field effect It is worth mentioning that magnetic field and its type as uniform or variable field have considerable effects on spin dependent charge transfer in DNA. 28,31 In the current study, we examined the effect of both uniform and harmonic time variable magnetic fields on the spin currents in DNA. In the first type, it is supposed that uniform magnetic field is applied ~ = B zˆ. By considering a sequence from CH22 based sequences with along the helix axes B N = 60 bp and using the Eqs. (17 − 19), one could obtain spin-up and spin-down currents and corresponding Is and Ic . In all calculations, we used the parameter values established in Refs. 12,19,32 represented in Table. 1. On the other hand, the on-site energies and electron hopping constants are different for distinct base-pairs in DNA sequences. These values represent in Table. 2 and Table. 3 according to the Ref. 33 Table 1: Constant parameters used in the combined extended PBH model. 12,19,32 Symbol Description Units m Base-pairs mass amu aAT Width of the Morse A◦−1 potential for AT base pairs aGC Width of the Morse A◦−1 potential for GC base pairs DAT Depth of Morse eV potential for AT base pairs DGC Depth of Morse eV potential for GC base pairs potential k Coupling constant eV/A2 ρ Stiffness parameter b Damping coefficient A◦−1 βn coupling parameter A◦−1 of hopping integral χ electron-lattice coupling eV/A θ helix angle rad ∆ϕ twist angle tso spin-orbit coupling constant eV d distance between sites of A◦ the strand along the helix

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Value 300 6.9 4.2 0.05 0.075

0.04 0.5 0.35 ≃ [1 − 1.7]an [0.1 − 0.6] 0.66 π/5 0.01 5.6

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Table 2: The electron hopping constants for different base-pairs in meV. 33 base-pair sequence AA, TT AT AG, CT AC, GT TA TG, CA TC, GA GG, CC GC CG

electron hopping constant -29 0.5 3 32 2 17 -1 20 -10 -8

Table 3: The on-site energies for two possible base-pairs A-T and G-C in eV. 33 B-DNA base-pair A-T G-C

on-site energy -4.9 -4.5

By applying a constant magnetic field up to 500 mT , it is found that spin-up and spindown currents show different behaviors with respect to the field (Fig. 1). It is shown that the net spin current (Is ) is on the order of one-tenth whereas the net charge current (Is ) is much lower than it. By increasing the field intensity, one could apperceive, decreasing the Ic flowing through DNA chain. Therefore, Ic is much less than Is and could be neglected. Now, it is worth noting that Is could be used in information transport where the pure spin current is preferred. 34,35 The use of pure spin currents to process information is attractive because it potentially circumvents the constraints of capacitive time constants, resistive power dissipation, and heat buildup which accompany charge motion. It is resonable that the time dependent magnetic field has considerable effect on spin currents in DNA. 36 Then, we have applied a harmonic time variable field along the helix axes on the DNA. Afterwards, we have studied the effect of field frequency on currents. It is clear that, a 60Hz and a few mT magnetic field increases DNA strand breaks in biological cells, 37,38 but in order to study the effects of field intensities and frequencies on the spin transfer in DNA, we have increased the studied range and considered higher values of these parameters. 11



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Adding and changing the field frequency result in the creating of the currents Is and Ic which are similar to the applied currents in bioelectronics (see Fig. 2). So, one could say that recent situations would be the suitable candidate for information transmission using the spin. It is appeared in figure 2 that the net spin current is dominant over the charge current, too. We have considered the simultaneously effects of external magnetic field and its frequency on spin current in DNA, too (see Fig. 3). So that, figure 4 represents map of charge current versus magnetic field and its frequency. Regions of small charge current on the map correspond to areas of large spin current. The light green color determines the regions in which the charge current is approximately zero. It demonstrates the possibility of inducing pure spin currents in the presence external periodic magnetic field. We define the other quantity for studying the spin dependent conductivity which is the ratio between the net spin and charge currents as spin polarization, according to following:

P =

Is Ic

(20)

We have examined the effect of field frequency on the spin polarization in DNA. Spin polarization with respect to the field frequency shows the abrupt changes and characteristic peaks in the certain frequencies (see Fig. 5) as DNA shows the maximum spin polarization in these peaks. The intensity of peaks increases when the frequency is increased. Then, we could report a critical frequency ωc that spin polarization is maximum value. It may be said that these characteristic peaks predict some resonance phenomena in DNA in the presence of external time variable field. Similar phenomenon have been previously observed in DNA in the presence of external electrical field when the field frequency is equal to Bloch frequency. 25 It seems that one could produce a DNA coder according to these polarization peaks. It will be interesting in information theory to transport the information using spin polarization codes.

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Spin currents in the presence of voltage gate It is worth noting that the DNA molecule is a promising candidate for molecular electronics. Besides, gate voltage affects the spin transport of DNA molecules. 15,31 Then, we try to investigate the spin-selective tunneling of electrons through DNA in the presence of an external electric field which is perpendicular to the DNA helix axis. In this regard, the potential difference at both ends of DNA is entered through V = Ed0 in calculation, where P E is the applied electrical field on DNA chain and d0 = n d cos[(n − 1)∆φ] is the length of DNA. Here, d is the distance between the base-pairs in a chain. It is remarkable that it will be possible to distinguish between the two spin states. By comparison of currents evolution in the absence and presence of electrical field, one could see the effect of electrical field as apparent increase in Is . It is shown that the net spin current is more intuitive in recent conditions (see Fig. 6). Then, through the applying the electrical and magnetic field simultaneously, we could see Is dominates to Ic for certain values of fields and create the pure spin current (Fig. 7). Also, we have studied the I-V characteristic diagram of the net spin flowing through DNA. There are regions with negative gradient together with the regions with quasi-linear property in Fig. 8. In I-V characteristic negative slope corresponds to the negative differential resistance (NDR) behavior where quasi-linear behavior manifests the quasi-Ohmic property. NDR is a property of some devices where an increase in the current results in the decrease in the voltage. 39 This is in contrast to a simple ohmic resistor which exhibits an increase in voltage under the same conditions. This phenomenon has been previously observed experimentally in DNA. 40,41 On the other hand, quasi-Ohmic region represents the deviation from ohmic behavior and appears the nonlinear effects. 42 In this work, we reported the spin dependent negative differential resistance (SPNDR) in DNA by studying the I-V characteristic for spin current (Fig. 8). This effect is experimentally reported in a self-assembled molecular chain 43 and graphene nanoribbons, 44 previously. A SPNDR device will enrich spintronics as well. The intrinsic and switchable features of the SPNDR effect are therefore important for future spintronic device design. 45 13



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Spin polarization in different temperatures In a ferromagnetic material, magnetization is directly related to magnetic field and its temperature. 46 The temperature dependence of the transport spin polarization has been reported by previously in Refs. 47,48 On the other hand, much of the practical interest is in the value of spin polarization close to room temperature. The bulk spin polarization at elevated temperature is often estimated based on assumption P (T ) ∝ M (T ). It means that the spin polarization of the electrons close to the Fermi level is proportional to the spontaneous magnetization M (t). 47 Then, we supposed the DNA in contact with a Hoover thermostat and studied the spin polarization in different temperatures. In Hoover’s reformulation 20 of Nose’s method, the evolution equation of the thermostat is formulated by 1 X ξ˙ = [ my˙ n2 − N kB T ] M n

(21)

here, ξ is the thermodynamics friction coefficient which interacts with the particles. Also, T is the temperature maintained by heat bath and M is the constant of Nosé-Hoover thermostat that has been set to M = 1000. On the other hand, the equation (13) is modified by the term −ξ y˙ n . It is found that by considering the constant given values of external electrical and magnetic fields, the spin polarization shows high sensitivity versus temperatures (Fig. 9). The temperature variation causes to appear the extremum peaks in spin polarization. These peaks identify consecutive negative and positive polarizations. It means that the spin polarization gives the maximum values with different polarization directions at certain temperatures. In studying the spin polarization in different temperatures, 49 the polarization peaks were reported in the region about T = 300 − 350 K. We have obtained polarization peaks in this range of temperature, too. It is found that the two considerable peaks appear in the temperature interval (320 K < T < 350 K) corresponding to pre-melting region. The pre-melting regime is the region in which we observe the formation of local bubbles but DNA molecule

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still preserves its double helical structure. Therefore, it is said that the sharp change in the spin polarization versus temperature can be considered a signature of pre-melting and thermal denaturation of DNA. The pre-melting phenomenon is one of the very important questions for experimental and theoretical studies from a biological point of view. 50,51 Despite this, by studying the simultaneously effect of magnetic field and temperature, it is clear that there is a critical value for magnetic field intensity in which appears the temperature effect (Fig. 10). In some magnetic field values, there are the islands in which Ic is nearly zero. Therefore, one could report the pure spin current. But, via the enhancement of the magnetic field intensity, these islands are disappeared and there will be less probability for generating the pure spin current. Rényi Dimensions Investigating the parameter values in which the pure spin current occurs could have considerable importance in studying the spintronics devices. But, it is difficult to distinguish the pure spin regions in figure 10 due to the simultaneous roles of temperature and applied magnetic field in the formation of the spin currents and the intertwining intervals. It is generally accepted that many objects, both in real physical space and in phase space, are multifractals. This means they can be described by Self-similar probability measures. Recent studies on the DNA molecular dynamics indicate that DNA denaturation phenomenon shows a multifractal nature, 52,53 too. The generalized dimensions (Rényi dimensions) Dq 54 of attractors are important characteristics of complex dynamical systems. 55 These dimensions are related to frequencies with which typical orbits in phase space visit different regions of the attractors. Then, they provide information about dynamics of the systems. 56 If the q dependence of Dq satisfies the condition Dq > Dq´ for q < q´ then the spectra is said to exhibit multifractality. On the other hand, for homogeneous fractals Dq = Dq´. 57 The Rényi dimensions are commonly used to characterize the scaling properties of a distribution of points on an M-dimensional space. Large positive values of q then emphasize the most con-

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centrated regions of phase space, whereas large negative values emphasize the most rarely visited regions. The Rényi dimension spectrum is then given by. 57,58 PN (r) q 1 ln j=1 pj Dq = lim r→0 q − 1 ln r

(22)

where the d-dimensional phase space of the system is divided into cubes of size r. Then, N (r) is the number of nonempty cubes and pj is the probability raised to q-th power. To estimate the Rényi dimension, the generalized correlation sums for the various q´s is calculated. The generalized correlation sum is defined as 54

Cq (r) =

N 1 X 1 ( N j=1 N − 1

N X

Θ(r − |xj − xk |))q−1

(23)

(k=1,k6=j)

Then, the Rányi dimension could be defined as 1 ln Cq (r) r→0 q − 1 ln r

Dq = lim

(24)

In this paper, the variation of Dq with respect to temperature is investigated. In fact, the objective of this study is to evaluate the ability and efficiency of Dq spectrum to characterize the spin pure regions with respect to different temperatures. We have chosen the interval q ∈ [60, 60] for study the spin transfer properties in DNA. Figure 11 show that spin transport in DNA at each temperature exhibits multifractal scaling behavior as it is already mentioned that in multifractals Dq > Dq´ for q < q´. We have compared Dq spectrum for some temperature values in B = 270 mT (Fig. 11-a). It is clear that by varying the temperature in a given magnetic field, different curved obtained which each of them demonstrates the variations of fractal dimension with respect to q. Dq spectrum shows the different consequences for different temperature values. As it is shown, the most variation of Dq (the most gap between the lowest and highest Dq value) is manifested in T = 250 K in which Ic is nearly zero in Fig. 10 and one could report flowing of the pure spin current through DNA. Therefore, by 16


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definition the degree of multifractality as the difference between maximum and minimum values of Dq : ∆Dq = Dqmax − Dqmin 59,60 one could say: If ∆Dq is high, the multifractal spectrum is rich in information (Is ) and for a small ∆Dq , the resulting dimension spectrum is poor in information. This finding represents the ability of the Rényi dimension spectrum to characterize the pure spin current in different field values. On the other hand, as shown in Fig. 11-b it can be concluded that the results obtained by use of the Rényi dimension spectrum are in very good agreement with previous studies. Therefore, It could confirm the pure spin regions reported in Fig. 10. Moreover, the phase space reconstruction of the 30th base pair of the sequence at some selected temperatures is shown in Fig. 12: (a) T = 250 K in which Is is maximal value; (b) T = 300 K, and (c) T = 350 K. These figures confirm that the size of the attractor and the phase space density distribution directly depends on the temperature. Spin Hall effect The spin Hall effect driven by the spin orbit interaction converts a charge current into a pure spin current. 26 We have considered the spin Hall conductivity and the external agents effect on it. In this regard, we have defined the spin Hall conductivity as

σSH = (

~ )γSH σ 2e

(25)

where γSH is the spin Hall angle given by the ratio of spin to charge current density and written as following

γSH =

Js I↓ − I↓ = Jc πR2 σEr

(26)

in which R is the radios of DNA and Er is the radial electrical field. Figure 13 shows the variation of spin Hall conductivity with respect to the external magnetic field. The regions in which the spin Hall conductivity is positive represent the prevailing of 17



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Page 18 of 43

up spins to down spins and resulting the spin current with up polarization. The situation is reversed in the negative spin Hall conductivity and resulting in the down polarization spin current. The temperature dependence of spin Hall conductivity is obtained from Fig. 14. From the figure, it is clear that increasing the temperature up to the threshold of DNA denaturation temperature corresponds to an amplification of down polarization spin conductivity. Sequence dependent spin transport in DNA It was shown that the DNA molecule with different sequences could present different transport behaviors from conducting and semiconducting to insulating. 61,62 Also, Sequence-dependent spin-selective tunneling along double-stranded DNA is reported by Gou et al. in the Ref. 63 We have studied spin dependent currents in two genomes from CH22 sequences with N = 40 bp and N = 60 bp (see the Table. 4. It is clear that DNA shows the similar behavior in both sequences but the amplitude of net spin current increases via the increasing of chain length. Table 4: The sequences used in calculation. 63 Name CH22

CH22

The number of DNA sequence base pairs 60 bp AGGGCATCGCTAACGAGGTC GCCGTCCACAGCATCGCTAT CGAGGACACCACACCGTCCA 40 bp CAATGCAGTCTATCCACCTG ACGGACCCCGACCCGCCTTT

Conclusion We have studied the spin selective charge transfer properties of DNA sequences through a combined extended Peyrard-Bishop-Holstein model. In this approach, spin degree of freedom and hence spin-orbit coupling interaction is considered. We have studied the spin dependent 18


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conductivity in DNA molecules by considering the real chains based on CH22 sequences. The different currents correspond to up and down spins could be flowed through DNA. 28 The external effects such as electrical and magnetic field as well as temperature and sequence variation play a considerable role in spin polarization of DNA. 15,28,49,63 The obtained results represent that there is no completely pure spin current in a real chain of DNA in contradiction to what has been reported in Ref. 28 but spin polarization phenomenon is appeared. It is remarkable that in some external parameters values, the less net charge current could be ignored by the dominance of the net spin current. Therefor, one could report the nearly pure spin current. On the other hand, there are the spin polarization peaks in some parameter values. Emerging the spin polarization peaks could create the DNA gates based on zero-one codes for information transport. Using electron spin to encode information holds promise for integrating computation and storage. It is expected to it provides significantly increased immunity from environmental decoherence compared with the conventional charge based electronics. 64 Like this idea is used for controlled switching of interacting ferroelectric surface domains leads to a variety of regular and chaotic patterns and creates a binary encoding. 65 Applying the voltage gate, quasi-ohmic or SPNDR effects are detected in I-V characteristic diagram. 66 The quasi-ohmic and SPNDR regions are recognizable in the I-V diagram for the net spin current. However, DNA as a molecular wire plays a key role in exhibition of nonlinear behaviors in I-V characteristic curves. We have reported the nonlinear behavior and NDR phenomenon in characteristic curve I-V for the charge current as previously mentioned in experimental works. 39,67 It is worth mentioning that DNA sequence variation affects on DNA spin selectivity. 63 Previously, the temperature dependence of pure spin current polarization was reported. 47,48 The studying of external magnetic field and temperature, simultaneously, characterizes a critical value for the applying field in which the temperature effect becomes apparent. For field intensities less than a critical value, the islands appear that the net charge current is zero inside them. Therefore, a pure spin current flows through DNA. The variation of temperature after the critical value of the magnetic field intensity, have a destroying effect

19



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on the net spin current (Is ). Then, via the increasing of the magnetic field intensity, the probability of generating the pure spin current in different temperatures would be very low. Also, the appearing the islands with pure spin current property could be confirmed and predicted using the Rényi dimensions spectrum. The spin Hall effect is one of the phenomena to generate and detect spin currents. 68 Spin Hall effect to be used in various applications such as magnetization switching, domain wall motion, spin current detection. In this work, the spin Hall conductivity in different external magnetic fields and different temperatures show negative and positive regions, which are correspond to the up and down polarization spin current, respectively. However, by increasing the temperature, an amplified intensity is appeared in spin down current as increasing of the spin Hall conductivity reported in the previous work. 69 The effect of variation of DNA sequence type and length has examined. The results show the same behavior for both sequences, but the amplitude of spin currents increases via the increasing of the DNA length. It is clear that the selected model and boundary conditions have imposed some limitations on findings the real nature of phenomena. Therefore, whether all aspects are considered or not require that, the model, the situation of occurrence the phenomena and boundary condition to be reviewed in the future work.

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Figure captions Fig. 1 The spin currents with respect to the constant external magnetic field. Fig. 2 The currents with respect to the magnetic field frequency. Fig. 3 Variation of net spin current with respect to the simultaneously effects of external magnetic field intensities and frequencies. Fig. 4 The map of net spin current with respect to the external magnetic field intensities and frequencies. Fig. 5 Spin polarization in time variable magnetic fields with respect to the field frequency. Fig. 6 Spin dependent currents in the presence of the external electrical field. Fig. 7 The net charge current in the presence of both electrical and magnetic fields, simultaneously. Fig. 8 I-V characteristic diagram for the net spin current flowing through the DNA. Fig. 9 Spin polarization with respect to the temperature in different magnetic field intensities, (E=10 meV/A). Fig. 10 The net charge current, The effect of the magnetic field and temperature variations, simultaneously, (E=10 mV/A). Fig. 11 (a) The Rényi dimensions spectrum in different temperature values, (B=270 mT, E=10 mV/A), (b) The Rényi dimensions spectrum with respect to the temperature variation. Fig. 12 Phase space diagram for 30th base pair, (a) T=250 K, (b) T=300 K, (c) T=350 K. Fig. 13 The magnetic field dependence of the spin Hall conductivity. Fig. 14 Temperature dependence of the spin Hall conductivity. Table 1 Constant parameters used in the combined extended PBH model. Table 2 The electron hopping constants for different base-pairs in meV. Table 3 The on-site energies for two possible base-pairs A-T and G-C in eV. Table 3 The sequences used in calculation.

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0

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B (mT) Figure 1: The spin currents with respect to the constant external magnetic field.

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Is

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ω (THz) Figure 2: The currents with respect to the magnetic field frequency.

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−0.1 −0.2 −0.3 −0.4 300 250

1

200

0.8 150

0.6 100

0.4 50

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Figure 3: Variation of net spin current with respect to the simultaneously effects of external magnetic field intensities and frequencies.

31



Ic (nA) 300 0.3 250 0.2

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Figure 4: The map of net spin current with respect to the external magnetic field intensities and frequencies.

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ω (THz) Figure 5: Spin polarization in time variable magnetic fields with respect to the field frequency.

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E (mV/A) Figure 6: Spin dependent currents in the presence of the external electrical field.

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E (mV/A) Figure 7: The net charge current in the presence of both electrical and magnetic fields, simultaneously.

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0.4

Is (nA)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 36 of 43

0.2

0

SPNDR region SPNDR region

−0.2

−0.4

0

5

10

15

20

25

30

35

40

45

50

V (mV)

Figure 8: I-V characteristic diagram for the net spin current flowing through the DNA.

36


Page 37 of 43

25

B=300 B=50 B=150 B=350

20 15 10 5

P

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 −5 −10 −15 −20 −25 0

50

100

150

200

250

300

350

T (K)

Figure 9: Spin polarization with respect to the temperature in different magnetic field intensities, (E=10 meV/A).

37



Ic (nA)

0 250

−5

200

B (mT)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 43

150 −10

100 −15 50 −20 0 0

50

100

150

200

250

300

350

T (K) Figure 10: The net charge current, The effect of the magnetic field and temperature variations, simultaneously, (E=10 mV/A).

38


Page 39 of 43

1.2

(a)

T=250 T=300 T=350

1.1

1

Dq

0.9

0.8

0.7

0.6

0.5 −60

−40

−20

0

20

40

60

q 1.2

(b)

q=−60 q=−50 q=−40 q=−30 q=−20 q=−10 q=0 q=10 q=20 q=30 q=40 q=50 q=60

1.1

1

0.9

Dq

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.8

0.7

0.6

0.5 100

150

200

250

300

350

T (K) Figure 11: (a) The Rényi dimensions spectrum in different temperature values, (B=270 mT, E=10 mV/A), (b) The Rényi dimensions spectrum with respect to the temperature variation.

39



−4

x 10 1.5

(a) 1

y.30

0.5

0

−0.5

−1

−1.5 −8

−7.5

−7

−6.5

−6

−5.5

y

−5 −3

x 10

30

−4

x 10 2

(b) 1

y.30

0

−1

−2

−3

−4 −0.02

−0.015

y30

−0.01

−0.005

−4

x 10 1.5

(c) 1

0.5

0

y.30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 40 of 43

−0.5

−1

−1.5

−2

−2.5 −11

−10

−9

−8

−7

−6

y30

−5 −3

x 10

Figure 12: Phase space diagram for 30th base pair, (a) T=250 K, (b) T=300 K, (c) T=350 K.

40


Page 41 of 43

2

0

σSHE (103× Ω−1m−1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−2

−4

−6

−8

−10 0

50

100

150

200

250

300

350

400

450

500

B (mT) Figure 13: The magnetic field dependence of the spin Hall conductivity.

41



0.4

0.2

σSHE (× 103 Ω−1 m−1)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 42 of 43

0

−0.2

−0.4

−0.6

−0.8

−1 0

50

100

150

200

250

300

350

400

T (k) Figure 14: Temperature dependence of the spin Hall conductivity.

42


Page 43 of 43

4 0

Ic (n A)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


−4 −8

−12 −16 300 200 100

B (mT)

0

0

50

100

150

200

250

300

350

T (K)

Figure 15: Table of Contents (TOC) Image

43


400

DNA Spintronics - American Chemical Society

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