Vibronic Dephasing Model for CoherenttoIncoherent Crossover in DNA
Abstract
In this work we investigate the interplay between coherent and incoherent charge transport in cytosineguanine (GC) rich DNA molecules. Our objective is to introduce a physically grounded approach to dephasing in large molecules and to understand the length dependent charge transport characteristics and especially the crossover from coherent tunneling to incoherent hopping regime at different temperatures. Therefore, we apply the vibronic dephasing model and compare the results to the Büttiker probe model which is commonly used to describe decoherence effects in charge transport. Using the full ladder model and simplified 1D model of DNA, we consider molecular junctions with alternating and stacked GC sequences and compare our results to recent experimental measurements.
I Introduction
Scientific interest in charge transport characteristics of biological molecules, in particular doublestranded DNA, has grown over recent decades because of its fundamental importance in live sciences and potential for electronic applications. ?; ? Many experiments on charge transfer and transport have been realized showing a wide range of contradictory outcomes covering insulating ?; ?; ?; ?, semiconducting ?, and conducting ?; ?; ?; ?; ?; ? behaviour. This large variety of experimental results reflects the complexity of charge transfer processes in DNA and especially the influence of environmental effects characterized by measurement conditions as well as the structural conformation. At low temperatures the transport should be coherent, weakly influenced by interaction of electrons with conducting environment. On the other hand, the main source of dephasing at finite temperatures is elastic electronvibron interaction due to large number of soft vibrations in biological molecules.
Experimental as well as theoretical studies revealed two fundamental charge transport mechanisms in DNA. Over short distances and at low temperatures (actually, as we will see, even room temperature can be low enough) charge carriers move coherently through delocalized molecular orbitals. In the case of the Fermi level position inside the conductance gap, this coherent tunneling regime is represented by an exponentially length dependent transmission ?; ?; ?; ?. In the case of coherent resonant transport, the conductance is oscillating function of the length because of interference ?.
Contrary, in the longrange regime at high temperatures charge carriers localize on the bases and the main charge transfer occurs through incoherent hopping between them. This can be modeled by multistep hopping events between purine bases, i.e., bases with the lowest oxidation potentials and shows a linear length dependence of transmission at high temperatures.?; ?; ?; ?
The calculation of coherent charge transport is based on the LandauerBüttiker approach (also known as the scattering method), which is implemented usually with a help of the Green function technique?; ?; ?; ?.
However, due to external influences and temperature dependent vibrations a coherent theory is often not sufficient. In recent years there were some studies to expand the coherent theories to include decoherence effects.
In the framework of the Green function approach dephasing resulting from environmental effects can be accounted for using the Büttiker probe model?; ?; ?; ?; ?; ?; ?; ?; ?. It is based on the idea of virtual probes coupled to the system (see Sec. II.2). This coupling controls the strength of the dephasing and is chosen empirically.
More physical is the vibronic dephasing, the contribution of which is especially important at high temperatures. Vibrational effects can be taken into account through the corresponding selfenergy, which is dependent on the Green function at the same energy in the elastic approximation?; ?; ?; ? (see Sec. II.1).
In this paper we consider local and elastic dephasing models, which are most important at low and intermediate temperatures at low applied voltages, corresponding to typical experimental conditions.
The recent experiment? has shown intermediate coherentincoherent charge transport characteristics at room temperature in GCrich DNAs. The phenomenological explanation based on the incoherent hopping approach with addition of coherent transport was suggested?. It is important to understand transition from coherent to incoherent transport starting from unified microscopic approach. The model of such crossover was studied recently? by using Büttiker probe model of environmental caused dephasing. In this paper we will investigate decoherence effects in DNA structures based on the vibronic dephasing model proposed by some of us earlier?. Besides, we implemented both methods in the open software package DFTBXT ?.
The paper is organized as follows. In Sec. II we consider the Green function method and elastic dephasing models. In Sec. III we introduce the effective tightbinding like models of DNA and consider the particular system Hamiltonians and parameters. The results of calculations and discussion are presented in Sec. IV. And finally we give conclusions and outlook in Sec. V.
Ii Elastic dephasing
In this section we outline the nonequilibrium Green function formalism for systems with elastic dephasing. The total Hamiltonian in the case of two equilibrium electrodes can be written as
(1) 
where describes the left/right lead and represents the DNA molecule. The coupling between leads and molecule is defined by and . The corresponding retarded and advanced Green functions of the molecule are then given by
(2)  
(3) 
where is the retarded selfenergy of the left/right electrode and is the selfenergy due to dephasing. In the case of elastic dephasing, considered here, the selfenergy at some energy does not depend on other energies, but can be a function of the Green function at the same energy.
In order to determine the transport properties, we use the manybody nonequilibrium Green function formalism. The details can be found in Refs.?; ?. Within this approach the current can than be calculated via the MeirWingreen formula
(4) 
where the broadening function can be determined by:
.
The lesser/greater green functions are defined through the Keldysh equation
(5) 
where the total selfenergy contains contributions from the leads as well as from interactions. The former can be expressed by the equilibrium fermi functions of the leads via
(6)  
(7) 
By assuming that the total selfenergy contains just contributions of the leads, eq. (II) would yield the Landauer formula for coherent transport.
ii.1 Vibronic dephasing (VD) model
Incoherence effects on charge transport can be taken into account by including the electronvibron interaction. This can be realized by an additional selfenergy .
In order to obtain an expression for this selfenergy we assume that the Hamiltonian of the central device can be written as:
(8) 
The electronic structure of the molecule is given by and the electronvibron interaction can be written as:
(9) 
where describes the annihilation (creation) operators of vibrons and the same for electrons. The coupling between electrons and vibrons in mode is described by . The corresponding selfenergy reads (see details in Ref.?, actually it is enough to calculate the lesser selfenergy)
(10) 
where is the freevibron Green function.
To study the decoherence effects at low voltages (linear conductance) we use the elastic dephasing model ?; ?; ?; ? which will be refered to as Vibronic Dephasing (VD).
Within this model it is assumed that the electronvibron interaction is localized at atomic sites and is identical at all sites. Moreover, the energy of vibronic quanta is assumed to be small compared to the other energy scales in the system, so that no vibrons are excited inelastically. This approximation works good for lowfrequency acoustic type vibrations.
In the elastic approximation one can integrate (10) over assuming . The retarded and lesser selfenergies are then represented by:
(11) 
where is the strength of dephasing. It can be estimated as ?; ?
(12) 
where describes the electronvibroncoupling and is the temperature.
Since the selfenergy depends explicitly on we have to solve this problem selfconsistently.
ii.2 Büttiker probe (BP) model
Another way to include dephasing in charge transport calculations is the so called Büttiker Probe (BP) model ?; ?; ?; ?; ?; ?; ?; ?; ?. As depicted in Fig. 1, in this model virtual probes are connected to every site of the chain.
This leads to an additional selfenergy (in the simplest case siteindependent)
(13) 
where is the coupling strength of the probes to the sites. Electrons may tunnel to the probes and back instead of travelling from left to right lead directly. To describe model dephasing, in contrast to the real conducting environment, the probes have to fulfill a zero net current condition i.e. the number of electrons entering the probe has to be the same as the number of electrons leaving it again to the same site. The effective transmission between left and right leads is then given by ?
(14) 
where is the coherent transmission from left to right lead.
The second term includes the transmission corrections due to the probes
(15) 
where describes the Markov matrix:
(16) 
The reflection function is defined as:
(17) 
where describes all transmissions between probes and two leads :
(18) 
This approximation is within the LandauerBüttiker picture and we can use the coherent expression (18) for transmissions.
This model introduces decoherence effects due to reemission of electrons with random phase. This means that the coupling strength of the probes controls the dephasing strength. The transmission correction in eq. (15) is only valid for small voltages and small temperatures. Otherwise a numerical approach to adjust the chemical potential of the probes should be used to ensure zero net current.?; ?; ?
Although the vibronic dephasing model and the Büttiker probe model have different origins of dephasing, both describe elastic scattering caused by local interactions.
Iii The model of DNA
Our aim is to analyze the interplay between coherent and incoherent charge transport in DNA.
Therefore, we will investigate stacked and alternating sequences following Xiang et al.?, where A,C,G,T stand for the nucleobases adenine, cytosine, guanine and thymine respectively. These structures are doublestranded and selfcomplementary.
We follow the idea of previous authors?; ?; ?; ?; ?; ? and model these
structures by using a next neighbor tightbinding (TB) ladder models as depicted in
Fig. 2 (gray lines). Within this approach we neglect any geometrical effects and assume
that they are included in the electronic parameters. On top of that we consider lowenergy charge
transport so that just the frontier orbitals of the bases are relevant. This means that we take one
orbital per base into account, i.e., we have electronic states per molecule with base
pairs.
The corresponding Hamiltonian reads:
(19) 
where describes the onsite energy of site in strand , the interstrand hopping within a base pair and charge transfer between the base pairs (interstrand as well as intrastrand).
Furthermore we adopt the beforehand mentioned assumption that charge transport occurs only through the purine bases (T, G).
We compare the ladder models with the corresponding 1D models which just include these bases. For a visualisation see the black lines in Fig. 2.
We use the electronic parameters which were calculated on a DFT level by Senthilkumar et al.? and modify the data set to reproduce experimental results from Xiang et al. ?. A short explanation of this adjustment is given in the appendix. While most of the charge transfer integrals can be used directly we have to pay some attention to the onsite energies. In the study these values were estimated depending on the two neighboring bases and show, e.g. for guanine, a maximal variation of . Since the outer bases, i.e., A and T are not sandwiched between two bases, we follow the idea of previous studies?; ? and average the onsite energies. The numerical values of the parameter set can be found in the appendix.
We investigate a single doublestranded DNA sandwiched between two leads. The influence of the leads will be modelled in the wideband limit, i.e. we neglect any energy dependence of the coupling. Furthermore we assume that contacts are just coupled to the first and last base (thymine) of the DNA molecule. This assumption is in fair agreement with the experiment of Xiang et al. ?. The hybridization functions in Eq. (II) than reduces to
(20)  
(21) 
Therefore the coupling to the leads is controlled by the adjustable parameters .
With these parameters we can calculate the transport properties for the ladder as well as for the 1D models.
Iv Results
iv.1 Computational Details
Our calculations were performed using DFTBXT package ?, based on the DFTB ?; ? source code. The calculation of the vibrational selfenergies (11) is done selfconsistently.
The coupling to the leads is estimated to be . The vibronic dephasing parameter we estimate as from the formula (12). The voltage is taken to be to be in the linear regime. The Fermi energy of the leads are aligned with the HOMO energy , which is the standard approximation explained by the level alignment?; ?.
iv.2 Comparison of Ladder and 1D DNA models
In order to get insight into the charge transport characteristics of the two different TB models we calculate the current spectral density which is the integrand of Eq. (II). The results for the stacked and alternating DNA molecules with 8 base bairs with dephasing are shown in Fig. 3. The completely coherent current spectral densities are equivalent to the transmission functions from LandauerBüttiker theory ?; ?.
The current spectral densities in Fig. 3 of the stacked ladder and the
corresponding 1D model show a good agreement, i.e., charge transport mainly occurs through the
purine bases. Contrary the alternating models show a large discrepancy. The current spectral
density of the ladder model is shifted to lower energies and suppressed by a factor of
corresponding to the density of the 1D model.
This large difference is caused by the strong intrastrand coupling
in the alternating model which is neglected in the 1D
model. This has a stronger influence on the alternating than on the stacked DNA because it appears
just once in the latter system. Nevertheless the current spectral density of the 1D alternating DNA
model shows the same features as the ladder model, i.e. it can be used to investigate qualitative
transport properties. It should be emphasized that the 1D alternating model will lead to a higher
current and thus a lower resistance than the stacked model. Therefore we will investigate dephasing
effects utilizing the ladder model to get comparable results.
iv.3 Dephasing in the Ladder models
We apply the vibronic dephasing model and the ladder models described before to investigate the effect of dephasing. Therefore we calculate the current spectral densities for both DNA molecules for base pairs at room temperature. The results for the coherent and incoherent cases are shown in Fig. 4. The two DNA molecules show quite narrow transmission windows. For the coherent calculations (upper pannel in Fig. 4) we find gaps at the Fermi energy of the leads , which is in agreement with experimental results from Shapir et al.?; ?. The transmission functions of the 6 alternating DNA molecules in Fig. 4a are grouped in two discrete transmission windows above and below the Fermi energy of the leads. The stacked molecules (Fig. 4c) show broad distributed transmission peaks. Furthermore the molecules with an odd number of base pairs show transmission peaks very near while in the alternating molecules there are no features near .
Including decoherence effects leads to broadening of the current spectral densities of both DNA molecules. Moreover it can be seen that this broadening leads to the disappearance of the transmission gap at for alternating DNA (Fig. 4b) and the appearance of a local maximum slightly below . Additionally it can be seen that the current spectral densities of all alternating molecules have the same shape where the maxima decrease monotonously with the system length.
The incoherent current spectral densities for the stacked DNA molecules (Fig. 4d) show two different shapes. For the molecules with an odd number of base pairs the transmission gap is closed in contrast to the molecules with even number of base pairs. This gap decreases with larger system sizes. It is also noticeable that all stacked curves shows additional gaps apart . For example for (orange lines in Fig. 4d) we can find transmission gaps at eV, eV, eV and eV relative to the Fermi energy.
To compare the vibronic dephasing model to the Büttiker probe model we calculated the current spectral densities for a stacked DNA molecule with base pairs (see Fig. 5). We have chosen a dephasing strength which is smaller than in Fig. 4 to emphasize the difference between the dephasing models. The peak positions are identical for the three models but the shape differs. While the peaks of the vibronic dephasing model (grey lines in Fig. 5) are rather sharp the BP models show a more smeared shape. Furthermore the minima are not that pronounced in the BP models as in the VD model leading to the disappearance of the gap at Fermi energy. Additionally the BP model without zero net current condition (black lines) shows a much smaller current spectral density caused by leaking currents through the probes.
iv.4 Lengthdependent resistance
With the previously discussed current spectral densities we can calculate the current through the molecules according to Eq. (II) and thus the resistance. We used a vibronic dephasing strength of as in Fig. 4 and estimated to reproduce a similar behavior in the length dependence. The resistance at room temperature for the VD model (solid lines) as well as for the BP model with zero net current condition (dashed lines) is shown in Fig. 6.
The resistance of the stacked DNA molecule shows overall a slight slope and additionally an evenoddoscillation. This oscillations are slightly more pronounced by the BP model than by VD model. It is well known that these oscillations are caused by a partial delocalization of charge carriers in the stacked Gsequences. ?. This can also be seen in Fig. 4d: The current spectral densities of molecules with odd numbers show a maximum at and thus a minimum in resistance. Additionally we calculated the resistance for the stacked DNA molecules using the BP model without the zero net current condition (see Fig.7). Then, the resistance has an exponential length dependence which supresses the oscillations. This is caused by leaking currents through the Büttiker probes which number are proportional to the number of base pairs.
In comparison the resistance of the alternating molecule just shows a linear length dependence. The VD model has a slope of M per base pair similar to the experiments? ( M per base pair). In the BP model the slope is slightly steeper with M per base pair. This increase can also be understood with the current spectral density in Fig. 4b): In the incoherent case a length dependent decrease of the maxima can be seen. This leads to a lowered current and thus to a higher resistance. Lengthdependent resistance measurements of Xiang et al. ? show also a linear increase for alternating and stacked DNA and oscillations which superimpose the linear increase in the stacked molecule. However these measurements show a similar slope of the curves for stacked and alternating DNA sequences which we cannot reproduce with our model calculations.
In a previous study of Kim et al. ?, where decoherence effects caused by environmental effects were investigated using the Büttiker Probe model, a similar difference between alternating and stacked DNA molecules as in our model calculations was found. Note, that in Ref. ? the probes were considered at finite temperature, and the zerocurrent conditions for probes with a certain broadening for the derivative of the Fermi function were used.
iv.5 Temperaturedependent resistance
To get a more detailed view on the influence of dephasing we calculate the resistance of single DNA molecules dependent on the dephasing strength. The result for the stacked DNA molecule with base pairs is shown in Fig. 8.
In this calculation the dephasing strength was varied proportional to the square root of the temperature according to the formula (12). Analyzing this figure, two distinct temperature dependencies can be found. At low temperatures the resistance shows a strong variation, which is the result of level broadening in the case when the resonant transition window is shifted slightly from the Fermi energy, as is shown in Fig. 4c.
On the other hand for higher temperatures the resistance seems to be constant but if we zoom in we see a slight increase (see inset in Fig. 8). Moreover, the resistance is linear dependent on , thus linear in temperature. Such behavior is consistent with “metallic” Ohmic behavior of the resistance as a function of length. The crossover between the low temperature and high temperature regimes can be understood with the spectral current densities (see Fig. 4). As mentioned before decoherence effects cause broadening as well as decrease of spectral current density. For low voltages the current is obtained by the integration of spectral current density in the small voltage interval. In the low dephasing regime the broadening of spectral current density dominates resulting in an increasing current and thus decreasing resistance. On the contrary the metalliclike regime is mainly affected by the decrease of spectral current density since the broadening appears outside the integration window and therefore we obtain a slightly increasing resistance. Similar results were obtained also for odd number of base pairs as well as for the alternating DNA molecule.
V Conclusion
We have shown that decoherence effects decrease and broad the spectral current density. This
broadening is slightly different in the investigated dephasing models. For alternating DNA molecules
this closes the transmission gaps at Fermi energy occuring in the completely coherent case, leading
to a conducting behavior and causes a monotonous length dependent resistance. On the contrary
dephasing in stacked DNA structures just closes the transmission gaps for molecules with odd number
of base pairs and therefore molecules with even number of base pairs show
a lower conductance. This distinct transport mechanism were measured by Xiang et al ?.
Furthermore we have shown that vibronic dephasing leads to crossover from coherent tunneling regime
at low temperature to a metalliclike behavior for higher temperatures. To understand better this
crossover and low temperature transport in DNA, the conductance measurements at low temperature
would be very desired. We also suggest to investigate the interplay between vibronic and conducting
environment dephasing.
Vi Acknowledgement
We thank Bálint Aradi for valuable discussions and help with the development of the DFTBXT package. This work was supported by the Deutsche Forschungsgemeinschaft (FR2833/501, GRK 2247) and the European Graphene Flagship.
Appendix A Effective 1D models for DNA
Appendix B Charge transfer integrals and onsite energies
X  Y  

G  A  C  T  
G  0.053  0.077  0.114  0.141 
A  0.010  0.004  0.042  0.063 
C  0.009  0.002  0.022  0.055 
T  0.018  0.031  0.028  0.072 
G  A  C  T  
G  0.012  0.013  0.002  0.009 
A  0.013  0.031  0.001  0.007 
C  0.002  0.001  0.001  0.0003 
T  0.009  0.007  0.0003  0.001 
G  A  C  T  
G  0.032  0.011  0.022  0.014 
A  0.011  0.049  0.017  0.007 
C  0.022  0.017  0.010  0.004 
T  0.014  0.007  0.004  0.006 
8.178  9.722  8.631  9.464  0.055  0.047 
Appendix C Modification of CT integrals
References

In section 3 we presented the DNA ladder and the corresponding 1D models, which were compared in
section 4. The 1D alternating DNA molecule can be described by:
(22) (23)  For the sake of completeness we show the used charge transfer integrals as well as the averaged onsite energies of Senthilkumar et al. ?

As mentioned before we adjusted the DFT charge transfer integrals from Senthilkumar et
al. ? to reproduce experimental results obtained by Xiang et
al. ?. For low temperature calculations the original parameter set
reproduces the measured behavior quite well. For room temperature the calculations do not show any
significant oscillations, which were found in the experiments, for the stacked DNA molecule. This
can be explained by the fact that DFT is a zero temperature method so that the calculated parameters
cannot show all effects with growing temperature.
It is well known, that these oscillations are caused by partially delocalisation of charge carriers within the stacked Gsequences ?. That is why they can be controlled by , i.e. the intrastrand charge transfer integral between neighboring G bases. For our calculations we increased from to