Atomistic modeling of dynamical quantum transport
Christian Oppenländer, Björn Korff, Thomas Frauenheim, Thomas A. Niehaus
pphysica status solidi, 31 October 2018
Atomistic modeling of dynamicalquantum transport
Christian Oppenl ¨ander , Bj ¨orn Korff , Thomas Frauenheim and Thomas A. Niehaus *,1 Department of Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany Bremen Center for Computational Materials Science, Am Fallturm 1a, 28359 Bremen, GermanyReceived XXXX, revised XXXX, accepted XXXXPublished online XXXX
Key words:
Time-dependent Density Functional Theory, TDDFT, Density Functional based Tight-Binding, DFTB, Molecular Elec-tronics ∗ Corresponding author: e-mail [email protected]
We present dynamical transport calculations based on a tight-binding approximation to adiabatic time-dependentdensity functional theory (TD-DFTB). The reduced device density matrix is propagated through the Liouville-vonNeumann equation. For the model system, 1,4-benzenediol coupled to aluminum leads, we are able to confirm theequality of the steady state current resulting from a time-dependent calculation to a static calculation in the conven-tional Landauer framework. We also investigate the response of the junction subjected to alternating bias voltageswith frequencies up to the optical regime. Here we can clearly identify capacitive behaviour of the molecular deviceand a significant resonant enhancement of the conductance. The results are interpreted using an analytical single levelmodel comparing the device transmission and admittance. In order to aid future calculations under alternating bias,we shortly review the use of Fourier transform techniques to obtain the full frequency response of the device from asingle current trace.
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The field of quantum transport at themolecular scale significantly diversified over the last years[1,2,3]. While the interest was initially to measure the con-ductance across individual molecules in an accurate andreproducible fashion, current topics involve spin transport[4], molecular transistors [5], thermoelectric effects [6,7]and device heating [8,9]. On the theoretical side muchprogress was achieved using Green’s function methods inthe energy domain [10]. Time domain methods, on thecontrary, promise easy access to dynamical properties, like ac transport, light-induced effects and higher harmonics inthe current [11]. In this contribution we report on resultsof such a method based on approximate time-dependentdensity functional theory, termed TD-DFTB [12,13]. Thescheme allows to perform dynamical transport simulationsof realistic devices taking the electronic structure of moleculeand leads into full account. Extending an earlier study ona similar topic [14], we first ask the question whether timeand energy domain methods provide the same answer forthe steady state dc current. We continue with a discussion of alternate currents and focus here especially on resonantenhancement of the admittance beyond the low frequencyregime commonly studied. In the following we present a brief descrip-tion of our simulation method. A more detailed derivationand justification of the present scheme may be found in theoriginal articles [15] and [13]. We assume a setup of themolecular electronic device as depicted in Fig. 1. The peri-odic left (L) and right (R) lead extend to infinity and are inthermal equilibrium at the chemical potential µ α = L,R with µ L = µ R at t = 0 . At t > , a time-dependent bias po-tential V ( t ) is applied that drives the central device region(D) out of equilibrium and leads to a time-dependent cur-rent. Instead of working with the full infinite system, onecan derive a Liouville equation for the device region only[15] (in atomic units): i ∂∂t σ ( t ) = [ H ( t ) , σ ( t )] − i (cid:88) α = L,R Q α ( t ) . (1) Copyright line will be provided by the publisher a r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Thomas A. Niehaus: Atomistic modeling of dynamical quantum transport
Here σ ( t ) denotes the one-particle density matrix forthe device region in a basis of localized atom-centered ba-sis functions φ µ ( r ) . The Hamiltonian H ( t ) is given in theadiabatic approximation of TDDFT [16] and depends onthe electron density ρ ( r , t ) , while Q α ( t ) incorporates alleffects due to the metallic leads, especially also dephasingand dissipation. Numerically tractable and explicit formsfor this term can be obtained from non-equilibriums Green’sfunction theory in the wide band limit (WBL). As shownby Zheng et al. [15], Q α then takes the form: Q α ( t ) = i [ Λ α , σ ( t )] + { Γ α , σ ( t ) } + K α ( t ) , (2)where Λ α describes the change of the device energy lev-els due to the presence of lead α , while Γ α renders thelifetime of the molecular levels finite. Both matrices areevaluated from first principles and depend on the device-lead interaction and the lead surface density of states. Theterm K α involves only known quantities besides the time-dependent bias potential V ( t ) and hence Eq. 1 representsa closed equation that can be numerically integrated byconventional Runge-Kutta methods. To this end, the initialdensity matrix at t = 0 may be obtained without furtherapproximations from equilibrium Green’s function the-ory in the WBL. As also shown in reference [15], knowl-edge of Q α ( t ) allows one to compute the time-dependentparticle current I α ( t ) through the left or right device-leadinterface according to I α ( t ) = − Tr [ Q α ( t )] . (3)In practical simulations the time step has to be chosenin the attosecond regime in order to resolve the electrondynamics accurately. This limits the accessible device di-mensions and total simulation time significantly. We there-fore adapted the scheme from above to the time-dependentdensity functional based tight-binding (TD-DFTB) method[17,12,13]. In essence, the TDDFT Hamiltonian matrix H µν ( t ) is replaced by H µν ( t ) = (cid:104) φ µ | H [ ρ ] | φ ν (cid:105) (4) + 12 [ δV A ( t ) + δV B ( t )] S µν , µ ∈ A, ν ∈ B. The first term on the right hand side is the DFT Hamil-tonian evaluated at a time independent reference density ρ = (cid:80) A ρ A , taken to be a sum of atomic densities ρ A foreach atom in the device region. These densities and hencealso the matrix elements can be computed beforehand. Thesecond term involves the overlap S µν of the basis functionsand takes the deviation of the electrostatic potential fromthe reference into account. The potential V A on the atomsin the device region is computed at each time step from aPoisson equation with boundary conditions determined bythe given bias potential in the leads. The charge density With respect to the article by Zheng et al. [15], the designa-tion of Γ α and Λ α is interchanged here. Square and curly brack-ets indicate a commutator and anti-commutator, respectively. L RD
Molecular Device S L S R Figure 1
Schematic setup of the molecular device showntogether with our test system. Only the atoms in the deviceregion are shown.required in this process is computed from the density ma-trix σ ( t ) [13]. Besides this adaption in the Hamiltonian,we follow the formalism of Zheng et al. without furthermodifications. We applied the TD-DFTB scheme to the junction depicted in Fig. 1. The 1,4-benzenediol molecule was optimized with passivating hy-drogens in vacuum at the DFTB level and then symmet-rically positioned inbetween Al nanowires of finite crosssections. The device region consists of the molecule and 36additional Al atoms, while the simulation cell for the leadsincluded 72 atoms. The latter is periodically replicated to + ∞ and −∞ for the right and left lead, respectively, in or-der to compute the surface Green’s function and WBL pa-rameters (see Eq. 2) at zero bias. The basis set is given byone s-type atomic orbital for H and one s-type and threep-type orbitals for the other elements. The Perdew-Burke-Ernzerhof exchange-correlation functional [18] is used inall calculations. This model structure was already used in[13] as well as in the first principles TDDFT study [14], sothat benchmark data is available for comparison. Transportthrough benzenediol is typical for conjugated molecules inmany respects. The transmission at the Fermi energy E F is rather small (T ≈ π and π ∗ frontier orbitals.In Fig. 2 we plot the time-dependent current throughthe left and right molecule-lead interface. Here and in thefollowing we integrate Eq. 1 with a time step of 2 as usinga 4-th order Runge-Kutta method. The bias voltage of 3.5V is applied to the left lead only and turned on exponen-tially with a time constant of 0.5 fs. The current initiallyovershoots, oscillates and settles into the steady state onlyafter several fs, long after the bias potential nearly reachedits maximum. Earlier we have shown [13], that the initialtransients depend on the time constant of the exponential Copyright line will be provided by the publisher ss header will be provided by the publisher 3 I L I R C u rr en t [ μ A ] B i a s [ V ] Figure 2
Absolute value of the time-dependent currentthrough the left ( I L ) and right ( I R ) interface of the molec-ular junction depicted in Fig. 1. The inset shows the biaspotential V ( t ) = V [1 − exp( − t/T )] with V = 3 . V and T = 0 . fs.turn on, but not the asymptotic value of the current. Inaddition, we could relate the decay time of the oscillationsto the imaginary part of the self energy of the device. Wellcoupled junctions reach the steady state earlier, whereasweakly coupled junctions feature persistent oscillations(see also [19]). As can also be seen in Fig. 2, the absolutevalues of the currents through the left and right interfaceequal each other asymptotically, but not in the transientphase of the simulation. Indeed, the particle current is aconserved quantity only in the dc limit. Under ac drivingthe device may become charged and one has to considerboth particle current and displacement current [20]. Bymonitoring the total device charge as a function of time,we verified that the latter indeed compensates for the dif-ference between | I L ( t ) | and | I R ( t ) | .An interesting question is now, whether the asymptoticcurrent I ∞ TD = lim t →∞ I ( t ) from the time-dependent sim-ulation equals the current obtained from a conventionalstatic calculation in the Landauer formalism. In the latterapproach the current is given by the energy integral I = G (cid:90) ∞−∞ dE [ f ( E, µ L ) − f ( E, µ R )] T ( E, V ) T ( E, V ) = Tr [ G r Γ R G a Γ L ] , (5)with f ( E, µ ) denoting Fermi distribution functions with µ L − µ R = V , the quantum of conductance G ≈ . µ S,and the bias dependent transmission function T ( E, V ) [10].The retarded ( G r ) and advanced ( G a ) device Green’s func-tions depend on the Hamiltonian and charge density n( r ).Since n( r ) depends itself on G r as well as on the appliedbias, a self-consistent determination of all quantities is re-quired. It is not a-priori evident, that the currents given by I TD∞ I NEGF C u rr en t [ μ A ] Figure 3
Asymptotic time-dependent current ( I ∞ TD ) andcurrent in the Landauer formalism ( I NEGF ) for 1,4-benzenediol as a function of applied bias. The values for I ∞ TD have been obtained from simulations with a total sim-ulation time t max of 20 fs and a bias potential V ( t ) = V [1 − exp( − t/T )] with T= 0.5 fs. The current has beenaveraged over the last 2 fs. The wide band approximationwas also employed in the Landauer calculations. The lineis a guide to the eye.Eq. 3 and Eq. 5 are identical. We have recently discussedthis question in great detail in the context of first-principlesTDDFT [14]. Here we perform similar simulations usingthe TD-DFTB method in order to show that our findingsare not restricted to a specific choice of the Hamiltonian.In Fig. 3 we compare the asymptotic currents I ∞ TD fromseveral time-dependent simulations at different bias valueswith the corresponding values from Eq. 5. Despite signif-icant formal and also algorithmic differences between thetwo approaches, one can observe nearly identical valuesover the full bias range. Like in Ref. [14], we conclude thattime-dependent simulations do in general not offer addi-tional or more accurate information when the interest is insteady state properties . As we discuss in the next section,there is however an important computational advantage for ac transport. Starting withthe work of Fu and Dudley [24], several studies addressedthe response of meso- and nanoscopic devices to an alter-nating bias potential [25,26,27]. In recent years approachesbased on energy domain Green’s functions became espe-cially popular [28,29,30,31], but also time domain tech-niques, as presented here, allow for the efficient evaluationof the admittance [32]. This statement holds for conventional local and semi-localfunctionals of the density. For non-local functionals differenceswith respect to the Landauer approach have been predicted [21,22,23].
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Thomas A. Niehaus: Atomistic modeling of dynamical quantum transport
To this end, the Fourier transform of bias and currentis numerically evaluated, e.g., V ( ω ) = (cid:90) ∞−∞ V ( t ) exp( iωt ) dt (6)to yield the complex admittance Y ( ω ) = I ( ω ) /V ( ω ) . Inelectronic circuit theory, the real and imaginary parts of Y are also often termed conductance ( G = Re ( Y ) ) and su-ceptance ( B = Im ( Y ) ), respectively. With the choice forthe sign of the Fourier transform from above (Eq. 6), ca-pacitive devices feature a negative susceptance, while in-ductive behaviour is characterized by positive values of B .We applied this approach to the 1,4-benzenediol junc-tion and experimented with different choices for the tem-poral profile of the bias potential. In principle, the form of V ( t ) is arbitrary as long as the amplitude is small enoughto remain in the linear response regime and the support ofits Fourier transform is sufficiently large. Fig. 4 depicts theabsolute value of the admittance | Y ( ω ) | for different func-tions V ( t ) . As reference, we perform simulations with aharmonic bias V ( t ) = V sin( ωt ) for different discrete val-ues of ω and determine the amplitude of I ( t ) after the ini-tial transients have died out. A sample simulation is shownin Fig. 5. Inspection of Fig. 4 reveals that an exponentialturn-on of the form V ( t ) = V [1 − exp( − t/T )] providesa reasonable estimate for the general features in the ad-mittance, but fails to convince on a quantitative level. Thereason is that the Fourier transform does not exist in thelimit ω → , unless one artificially damps V ( t ) by a factor exp( − Γ t ) to enforce convergence. For small values of Γ the admittance differs strongly from the reference, whilefor larger values the dc limit is overestimated. In responsecalculations for optical properties one often uses a Diracdelta function or Lorentzian as a perturbation. Here, theFourier transform exists for all ω and no artificial broaden-ing is required. Results for the bias potential V ( t ) = V π γ ( t − t ) + γ , (7)show excellent agreement with the reference data over nearlythe full frequency range, especially also in the dc limit. Abenefit with respect to the simulations at discrete frequen-cies is that the Fourier transform technique requires only asingle run to evaluate the full admittance. In the followingwe therefore continue with this choice.After this more technical discussion we now analyzethe admittance in more detail. Fig. 6 a) shows the con-ductance and susceptance of 1,4-benzenediol. For smallfrequencies, the negative values of the latter indicate ca-pacitive behaviour of the junction. This is in line with thesimulations shown in Fig. 5, where the current leads thevoltage signal. The negative susceptance can be rational-ized by inspection of the transmission T(E,0) (Fig. 6 c)) of Since V ( t ) = 0 for t < , this is equivalent to the Laplacetransform with imaginary argument. ReferenceLorentzianExp. with Γ = 0.1 fs -1 Exp. with Γ = 0.01 fs -1 | Y ( ω ) | [ G ] Figure 4
Absolute value of admittance | Y ( ω ) | in units of G as a function of frequency in units of [eV/ ¯ h ]. Resultsare given for the harmonic perturbation with discrete fre-quencies (Reference) and using Fourier transforms withexponential form and damping of the bias ( V = T = t max =
50 fs) as well as with Lorentzian form( V = γ = t = t max =
50 fs).
I(t)V(t) B i a s [ μ V ] −0.6−0.4−0.200.20.40.6 C u rr en t [ n A ] −0.02−0.0100.010.02 Time [fs]0 10 20 30 40 50 Figure 5 time-dependent current due to the ac bias V ( t ) = V sin( ωt ) ( V = µ V, ω = ¯ h , t max =
50 fs).the junction. For small frequencies, only the region aroundthe Fermi energy is relevant in the linear response regime.Here the transmission is low and the current effectivelyblocked, similar to a macroscopic capacitor. In a classicalRC circuit, the admittance is given by Y RC ( ω ) = − iωC + ω C R, (8)up to second order in the frequency [28]. As seen in Fig. 6a), real and imaginary part of Y ( ω ) show for small fre- Copyright line will be provided by the publisher ss header will be provided by the publisher 5 b) Re [Y]Im [Y] Y ( ω ) [ G ] −0.1−0.0500.050.10.15 Frequency [eV (2π/h)]0 1 2 3 4 5 c) T r an s m i ss i on a) Re[Y]Im[Y] Y ( ω ) [ G ] −1−0.500.511.522.5 Frequency [eV (2π/h)]0 1 2 3 4 5 Figure 6 a) Real and imaginary part of the admittance Y ( ω ) for 1,4-benzenediol. b) Analytical results for theone-level model of Fu and Dudley [24] with parameters ∆E = γ = ¯ h and 4 eV/ ¯ h with conductances that are two orders ofmagnitude larger than the dc one. This can be qualitativelyunderstood by a comparison to the analytical result of Fuand Dudley for a single-level model [24], characterized bya Breit-Wigner transmission T ( E ) = γ ( E − E ) + γ , (9)where E denotes the level energy. Here the admittance isgiven byRe { Y ( ω ) } = G γ ω (cid:20) arctan (cid:18) ∆E + ωγ (cid:19) − arctan (cid:18) ∆E − ωγ (cid:19)(cid:21) (10)andIm { Y ( ω ) } = G γ ω ln (cid:32) (cid:2) ( ∆E + ω ) + γ (cid:3) (cid:2) ( ∆E − ω ) + γ (cid:3) [ ∆E + γ ] (cid:33) , (11)with ∆E = E F − E . In Fig. 6 b) the Fu-Dudley ad-mittance is shown for ∆E = γ = ¯ hω andE - ¯ hω , which are not available in the dc limit.Admittedly, the frequency range for resonant enhance-ment is difficult to access experimentally. Current mea-surements on nanoscopic conductors hardly reach the GHzregime [33,34]. Nevertheless, appropriate gating of thedevice could move the HOMO/LUMO close to the Fermienergy, resulting in resonance enhancement at lower fre-quencies. Small gap materials like Graphene nanoribbonswould offer another route for the experimental realizationof this effect. In this study, we investigated the behaviourof a contacted 1,4-benzenediol molecule subjected to analternating bias directly and using a Fourier transform ofLorentzian and exponential voltage signals. The Lorentzianinput signal led to very good agreement with referencediscrete frequency calculations. In the admittance, capaci-tive behaviour could be identified and interpreted throughthe transmission of the junction. An analytical single levelmodel showed large qualitative similarities to our numeri-cal results. The approach we employed for these findings isa combination of the highly efficient tight-binding approx-imation to adiabatic TDDFT and a device density matrixpropagation scheme derived within the Keldysh formalismby Zheng and co-workers. Following earlier discussionson this topic, we can also confirm that a different choice ofthe self-consistent Hamiltonian does not change the equal-ity of the TD steady state with its static counterpart at theNEGF level.
Acknowledgements
Financial support by the German Sci-ence Foundation (DFG, SPP 1243 and GRK 1570) is greatly ac-knowledged.
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Yu, B. Wang, Y. Wei, Corrected article: “ac response of acarbon chain under a finite frequency bias” [j. chem. phys.[bold 127], 104701 (2007)], J. Chem. Phys. 127 (16) (2007)169901.[29] T. Yamamoto, K. Sasaoka, S. Watanabe, Universal transi-tion between inductive and capacitive admittance of metallicsingle-walled carbon nanotubes, Phys. Rev. B 82 (20) (2010)205404.[30] T. Sasaoka, K.and Yamamoto, S. Watanabe, K. Shiraishi, acresponse of quantum point contacts with a split-gate config-uration, Phys. Rev. B 84 (2011) 125403.[31] D. Hirai, T. Yamamoto, S. Watanabe, Theoretical analysisof ac transport in carbon nanotubes with a single atomic va-cancy: Sharp contrast between dc and ac responses in va-cancy position dependence, Applied Physics Express 4 (7)(2011) 075103.[32] C. Y. Yam, Y. Mo, F. Wang, X. Li, G. H. Chen, X. Zheng,Y. Matsuda, J. Tahir-Kheli, A. G. 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Frauenheim, G. Chen, T. Niehaus,An efficient method for quantum transport simulations in thetime domain, Chem. Phys. 391 (1) (2011) 69.[14] C. Y. Yam, X. Zheng, G. H. Chen, Y. Wang, T. Frauenheim,T. A. Niehaus, Time-dependent versus static quantum trans-port simulations beyond linear response, Phys. Rev. B 83(2011) 245448.[15] X. Zheng, F. Wang, C. Y. Yam, Y. Mo, G. H. Chen, Time-dependent density-functional theory for open systems, Phys.Rev. B 75 (19) (2007) 195127.[16] M. E. Casida, Recent Advances in Density Functional Meth-ods, Part I, World Scientific, 1995, Ch. Time-dependent Den-sity Functional Response Theory for Molecules, pp. 155–192.[17] T. Frauenheim, G. Seifert, M. Elstner, T. Niehaus, C. Kohler,M. Amkreutz, M. Sternberg, Z. Hajnal, A. Di Carlo,S. Suhai, Atomistic simulations of complex materi-als: ground-state and excited-state properties, Journal OfPhysics-Condensed Matter 14 (11) (2002) 3015–3047.[18] J. Perdew, K. Burke, M. 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