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Chemical Physics 336 (2007) 127–135

A corrected NEGF + DFT approach for calculatingelectronic transport through molecular devices: Filling bound states

and patching the non-equilibrium integration

Rui Li, Jiaxing Zhang, Shimin Hou *, Zekan Qian, Ziyong Shen, Xingyu Zhao, Zengquan Xue

Key Laboratory for the Physics and Chemistry of Nanodevices, Department of Electronics, Peking University, Beijing 100871, China

Received 11 January 2007; accepted 6 June 2007Available online 15 June 2007

Abstract

We discuss two problems in the conventional approach for studying charge transport in molecular electronic devices that is based onthe non-equilibrium Green’s function formalism and density functional theory, i.e., the bound states and the numerical integration of thenon-equilibrium density matrix. A scheme of filling the bound states in the bias window and a method of patching the non-equilibriumintegration are proposed, both of which are referred to as the non-equilibrium correction. The discussion is illustrated by means of cal-culations on a model system consisting of a 4,4 bipyridine molecule connected to two semi-infinite gold monatomic chains.� 2007 Elsevier B.V. All rights reserved.

Keywords: Bound states; Molecular electronic devices; Non-equilibrium Green’s function; Density functional theory

1. Introduction

The study of electron transport through an individualmolecule connected to two metallic electrodes has attractedmore and more research efforts [1,2], since this is a crucialstep toward fabricating functional electronic devices basedon single molecules. It is a great challenge to quantitativelymodel the electrical properties of these electrode–molecule–electrode systems that are open, infinite, non-periodic andnon-equilibrium (if a non-zero bias voltage is applied). Atpresent, the most popular approach to this problem com-bines the non-equilibrium Green’s function (NEGF) for-malism with density function theory (DFT) [3–6], that is,the so-called NEGF + DFT approach [7–14]. In theNEGF formalism, the devices are characterized as a centralregion connected to electron reservoirs via non-interactingleads. Certainly, arbitrary interactions in the central regioncan be included [15]. The central region in molecular

0301-0104/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.chemphys.2007.06.011

* Corresponding author. Tel.: +86 10 62754775; fax: +86 10 62762999.E-mail address: smhou@pku.edu.cn (S. Hou).

devices always includes the molecule itself and some elec-trode atoms adjacent to it, which is called the extendedmolecule. Due to the metallic screening in the electrodes,if the extended molecule includes sufficient electrode atoms,its influence on the electrodes can be neglected. Thus, theelectronic structure of the electrodes is bulk-like, and theKohn–Sham (KS) Hamiltonian and electron density ofelectrodes can be replaced by the result of DFT calcula-tions for the corresponding periodic structure. Via this par-tition, the only unknown part is the extended molecule.The density matrix of the extended molecule can be calcu-lated from Green’s functions. The KS effective potential isthen obtained from the density matrix. Iterating these twoprocedures until self-consistence, we can get the final Ham-iltonian used for calculating transport properties.

Fig. 1 illustrates the system we studied, where electronscattering occurs at the extended molecule region. For sim-plicity, zero temperature is assumed here, in the rest ourdiscussion will be generalized to finite temperatures. Atzero bias, the whole system is at equilibrium and has a uni-form Fermi level Ef, all states below the Fermi level areoccupied, so that the energy integration of the retarded

Fig. 1. Schematic drawing of electronic states in an open systemcontaining a molecule sandwiched between two metallic electrodes thatare driven out of equilibrium by a bias voltage.

128 R. Li et al. / Chemical Physics 336 (2007) 127–135

Green’s function gives the density matrix of the extendedmolecule. However, when an external bias voltage Vb isapplied to the electrodes, the electronic system is drivenout of equilibrium and the uniform Fermi level will notexist. The electrodes are always assumed to be in local equi-librium, so that the Fermi level of the left (right) electrodecan be defined, lL = Ef + eVb/2(lR = Ef � eVb/2). In thisnon-equilibrium situation, all the states in the extendedmolecule can be divided into three categories: (1) the leftscattering states, which start deep in the left electrodeand are filled up to the left Fermi level lL; (2) the right scat-tering states, which start deep in the right electrode and arefilled up to the right Fermi level lR; (3) bound states local-ized in the extended molecule region whose occupation hasto be determined if they lie between lL and lR. To calculatethe non-equilibrium density matrix [11–14], we can split thestates in the extended molecule into two parts: (1) the equi-librium part, where the energy is below the right Fermilevel and all states are occupied; (2) the non-equilibriumpart, where the energy is within the bias window betweenthe left and right Fermi levels. It is clear that, in the biaswindow, the left scattering states are occupied and the rightscattering states are unoccupied. However, the occupationof bound states is not known if there are no additionalmechanisms for equilibration, and how to fill these boundstates has seldom been discussed in the literature [10,11,16].Although the occupied bound states do not participate inelectronic transport directly, they will contribute to theelectron density that affects the transmission of scatteringstates via the KS potential. Therefore, it is necessary toinclude the bound states in the calculation of the densitymatrix. However, since it is very difficult to determinebound states and their occupation, bound states withinthe bias window are always ignored in the conventionalNEGF + DFT implementations [12–14]. This will lead tounphysical results if bound states do exist in the extended

molecule. In this paper, we propose a novel scheme of fill-ing those bound states in the bias window, which does notdeal with them individually but their spectral integralmatrix as a whole.

In the conventional NEGF + DFT approach [11–14],the non-equilibrium density matrix is obtained from theenergy integration of the lesser Green’s function that mustbe performed along the real energy axis. Due to singulari-ties on the real axis, a very fine integration grid must beused, especially in the locations of band edges and broad-ened molecule orbitals, which makes the non-equilibriumintegration the most tremendous task in the NEGF calcu-lation. We also develop a method in which a coarse gridcan be applied and the integration error can be patchedapproximately. The scheme of filling bound states andthe method of patching the integral have the same form,and both of them are referred to as the non-equilibriumcorrection. Our corrected NEGF + DFT approach notonly includes effects of bound states, but is also more effi-cient and accurate.

The rest of the paper is organized as follows. In Section2, we discuss the theoretical formalism of our correctedNEGF + DFT approach. In Section 3, our correctedNEGF + DFT approach is applied to the gold–4,4 bipyri-dine–gold molecular junction to demonstrate the non-equi-librium correction. In Section 4, we summarize andconclude.

2. Theoretical formalism

In this section, firstly we investigate the conventionalNEGF + DFT approach in which bound states are omit-ted. Then the scheme of filling bound states and the methodof patching the integral are proposed. Finally, twoapproaches of finding bound states are introduced.

2.1. The problem of bound states in the conventional

NEGF + DFT approach

In an open system, the Green’s function matrix of theextended molecule is defined as [12]

GðzÞ ¼ ðzS � H � RLðzÞ � RRðzÞÞ�1; ð1Þ

where H and S represent the Hamiltonian matrix and theoverlap matrix of the extended molecule, respectively; theleft (right) self-energy matrix RL/R(z) incorporates the effectof the semi-infinite electrodes. From the spectralrepresentation

GðzÞ ¼X

n

unuþn

z� En; ð2Þ

the Green’s function gives the spectral function of the ex-tended molecule.

AðEÞ ¼ limg!0þ

i½GðE þ igÞ � GðE � igÞ� ¼ 2pX

n

unuþn dðE � EnÞ;

ð3Þ

min

eq

ne

Rμ Lμ

Fig. 2. The integration path in the complex energy plane used to calculatethe density matrix.

R. Li et al. / Chemical Physics 336 (2007) 127–135 129

where {un} include all the three categories of states. Usingthe retarded (advanced) Green’s function that is defined asGR/A(E) = limg!0+G(E ± ig), this relationship can be writ-ten as

AðEÞ ¼ i½GRðEÞ � GAðEÞ� ¼ �2ImGRðEÞ: ð4ÞIn order to obtain the non-equilibrium density matrix ofthe extended molecule, a further decomposition of the spec-tral function is needed. It is convenient to introduce the leftand right spectral functions [17], which are, respectively,defined as

AL ¼ iGRðRL � RþL ÞGA;

AR ¼ iGRðRR � RþRÞGA:ð5Þ

Then, we have

A ¼ AL þ AR þ 2pXbound

n

anunuþn dðE � EnÞ; ð6Þ

where the weight factor an ¼ uþn Sun does not equal unityfor a bound state that is not entirely localized in theextended molecule region, so that it is an indicator of thedegree of localization of a bound state. The rigorous proofof Eq. (6) is given in Appendix. It can be seen that the totalspectral function includes three parts: the left spectral func-tion, the right spectral function and that contributed bybound states.

In the conventional NEGF + DFT approach [11–14],the density matrix is written as

q ¼ 1

2pi

Z þ1

�1G<ðEÞdE

¼ 1

2p

Z þ1

�1½ALðEÞfLðEÞ þ ARðEÞfRðEÞ�dE; ð7Þ

where fL=RðEÞ ¼ 1=ð1þ eðE�lL=RÞ=KT Þ is the left (right) Fermifunction. It can be seen that only contributions to thedensity matrix made by scattering states are taken intoaccount, and that the left and right spectral functions arefilled according to the left and right Fermi functions,respectively. In order to evaluate the integral in Eq. (7)more efficiently, the density matrix is always divided intotwo parts: the equilibrium part qeq and the non-equilibriumpart qne:

q ¼ qeq þ qne: ð8ÞAll scattering states filled up to the right Fermi level are in-cluded in the equilibrium part,

qeq ¼ 1

2p

Z þ1

�1½ALðEÞ þ ARðEÞ�fRðEÞdE: ð9Þ

However, instead of AL(E) + AR(E), A(E) is often used inpractical implementations [12–14]. Thus, the bound stateswhose energies lie below the right Fermi level are actuallyfilled and contribute to the density matrix. Below some en-ergy level Emin, there are no states for the system consid-ered, thus we can set the lower bound of the integral to

Emin. Then the equilibrium part of the density matrix be-comes the following form:

qeq ¼ 1

2p

Z þ1

Emin

AðEÞfRðEÞdE

¼ � 1

p

Z þ1

Emin

ImGRðEÞfRðEÞdE: ð10Þ

Due to the analytic feature of the retarded Green’s func-tion, this energy integration can be performed along thecontour Ceq (as shown in Fig. 2)

qeq ¼ � 1

pIm

ZCeq

GðzÞfRðzÞdz� 2pikTX

zR

GðzRÞ !

; ð11Þ

where zR = lR + i(2n + 1)pkT are the poles of fR(z). Sincethe Green’s function behaves smoothly sufficiently awayfrom the real axis, this contour integration can be doneaccurately with a rough integration mesh which is realizedthrough the adaptive Gaussian–Lobatto quadrature [18].

The non-equilibrium part is the remaining part of thedensity matrix

qne ¼ 1

2p

Z þ1

�1ALðEÞ½fLðEÞ � fRðEÞ�dE: ð12Þ

When temperature reduces to zero, only the left scatteringstates in the bias window contribute to the non-equilibriumpart of the density matrix. There are two problems in thisconventional non-equilibrium integral: (1) the integral doesnot include those bound states whose energies are withinthe bias window. (2) Because AL is not analytic in the com-plex energy plane, the integration along the real energy axismust be performed using very fine integration grids, whichwill increase the computational cost significantly.

2.2. Filling the bound states

Integrating the total spectral function in the bias win-dow, we can get the spectral integral matrix of all the states

D ¼ 1

2p

Z þ1

�1AðEÞ½fLðEÞ � fRðEÞ�dE: ð13Þ

This integral can be evaluated efficiently and accurately byperforming the following contour integration of the

130 R. Li et al. / Chemical Physics 336 (2007) 127–135

Green’s function along the path Cne with the adaptiveGaussian–Lobatto quadrature (as shown in Fig. 2):

D ¼ � 1

pIm

ZCne

GðzÞ½fLðzÞ � fRðzÞ�dz� 2pikTX

zL

GðzLÞ �X

zR

GðzRÞ" # !

;

ð14Þwhere zR/L = lR/L + i(2n + 1)pkT are the poles of fR/L(z).The left (right) spectral integral matrix of the left (right)spectral function can also be obtained

DL=R ¼1

2p

Z þ1

�1AL=RðEÞ½fLðEÞ � fRðEÞ�dE; ð15Þ

which is realized through the simple extended Simpson rule[19].

Picking out the scattering states from the total spectralintegral matrix, we get

D� ðDL þ DRÞ ¼Z þ1

�1½fLðEÞ � fRðEÞ�

�Xbound

n

anunuþn dðE � EnÞdE

¼Xbound

n

uðEnÞanunuþn ; ð16Þ

where u(En) = fL(En) � fR(En) decays exponentially outsidethe bias window. In this way, we can obtain the spectralintegral matrix of the bound states in the bias windowwithout tremendous work of solving bound eigenstatesdirectly

Dbound ¼Xbound

n

uðEnÞanunuþn ¼ D� ðDL þ DRÞ: ð17Þ

The next problem is how to fill these bound states in thebias window. In the NEGF + DFT approach, the exact,non-Hermitian, energy-dependent, many-body self-energyoperator is approximated by the energy-independent, realDFT exchange-correlation potential [12]. Thus, all quasi-electrons are independent, and there are no interactions be-tween bound states with the continuum. Therefore, theproblem of filling bound states cannot be solved exactlywithin the framework of the NEGF + DFT approach.Since we have obtained the spectral integral matrix of allthese bound states in the bias window, instead of individu-ally determining the occupation of every bound state, wetry to devise a scheme of filling every element of this spec-tral integral matrix. We consider a system in which eachbound state is localized in either the left part or the rightpart of the extended molecule. We have assumed that theleft Fermi level goes up and the right Fermi level goes downwhen the bias voltage is applied to drive the molecular de-vice out of equilibrium adiabatically. Therefore, accordingto the adiabatic theorem in quantum mechanics [20], thosebound states entering the bias window whose positions arein the left part of the extended molecule will be occupiedaccording to the left Fermi function, while those boundstates entering the bias window whose positions are in

the right part will remain empty according to the right Fer-mi function. But the problem is how to divide the whole ex-tended molecule into two parts appropriately. Because thetransmission coefficient near the Fermi level is always muchsmaller than unity, the weight of left scattering states in theleft part is much larger than that of right scattering states.Thus, we can define a weight coefficient for each atom ofthe extended molecule that represents the filling probabilityfor all the bound states at this atomic site. The weight ofthe ith atom is given by the proportion between the leftand right spectral integrals

wi ¼Tr½DL�i

Tr½DL�i þ Tr½DR�i; ð18Þ

where [DL/R]i is the block of the left (right) spectral integralmatrix corresponding to the ith atom. These atomic fillingweights are used to obtain the non-equilibrium part of thedensity matrix contributed by bound states within the biaswindow

½qnebound�i;j ffi

ffiffiffiffiffiffiffiffiffiwiwjp ½Dbound�i;j

¼ ffiffiffiffiffiffiffiffiffiwiwjp ½D� ðDL þ DRÞ�i;j: ð19Þ

That is, each element of the spectral integral matrix ofbound states is filled according to the geometric mean ofcorresponding weights.

2.3. Our corrected NEGF + DFT approach

For the left spectral integral matrix DL obtained fromthe integration along the real axis, a numerical error DDL

is hardly avoided. If this error is known, we are able topatch this integral

D0L ¼ DL þ DDL: ð20ÞLikewise, the right spectral integral matrix DR also has anerror DDR

D0R ¼ DR þ DDR: ð21ÞIn the situation without any bound states, from Eq. (16) wecan get

DDL þ DDR ¼ D� ðDL þ DRÞ; ð22Þwhere D and DL/R are obtained as before. Thus, we get thetotal error for the left and right spectral integral matrices.

The next step is to separate DDL from the total error.Since each error should be in proportion to its own spectralintegral matrix approximately, the atomic weight in Eq.(18) can also be used to extract DDL from (DDL+DDR)

½DDL�i;j ffiffiffiffiffiffiffiffiffiffiwiwjp ½DDL þ DDR�i;j

¼ ffiffiffiffiffiffiffiffiffiwiwjp ½D� ðDL þ DRÞ�i;j: ð23Þ

Thus, the left spectral integral matrix can be patched as

½D0L�i;j ffi ½DL�i;j þffiffiffiffiffiffiffiffiffiwiwjp ½D� ðDL þ DRÞ�i;j: ð24Þ

Filling bound states and patching the non-equilibriumintegral simultaneously, we get a corrected NEGF + DFT

R. Li et al. / Chemical Physics 336 (2007) 127–135 131

approach with the non-equilibrium correction for thedensity matrix,

½qne�i;j ffi ½DL�i;j þffiffiffiffiffiffiffiffiffiwiwjp ½D� ðDL þ DRÞ�i;j: ð25Þ

By paying the cost of an extra integration of the retardedGreen’s function in the bias window that can be performedthrough the contour integral, not only are effects of boundstates included, our corrected NEGF + DFT approach isalso more efficient and accurate since the problem of usingtremendous integration grids can be circumvented.

2.4. Approaches to find the bound states

In order to demonstrate our filling scheme, bound statesmust be firstly found. Rigorously calculating bound statesof an open system is a serious challenge. Here, we just pres-ent two approximate approaches. One possible solution isto enlarge the Hamiltonian of the extended molecule byincluding more electrode cells and calculate all the eigen-states. Those states that are evanescent exponentially awayfrom the extended molecule region can be regarded asbound states. But, this pattern recognition must be donemanually at present, and the result is still coarse due toan arbitrary truncation of the infinite open system.

Another approach is to analyze the density of states(DOS) of the extended molecule to locate the energies ofbound states. The total DOS of all the states can be calcu-lated from the spectral function

DOSðEÞ ¼ 1

2pTr½AðEÞS� ð26Þ

and the DOS of scattering states can be obtained from theleft and right spectral functions

DOSscatterðEÞ ¼1

2pTr½ðAL þ ARÞS�: ð27Þ

Subtracting the DOS of scattering states from the totalDOS, we can get

DOSðEÞ �DOSscatterðEÞ ¼ TrXbound

n

anunuþn SdðE � EnÞ

" #

¼Xbound

n

a2ndðE � EnÞ; ð28Þ

which indicates that the difference between the total DOSand the DOS of scattering states will tell us the energiesof bound states. However, wave functions of bound states

Fig. 3. Schematic drawing of a 44BPD molecule connected to two semi-infiniteside is treated as the extended molecule.

are unavailable. Furthermore, to catch a d-function, veryfine grids must be applied.

3. Application

In this section, our corrected NEGF + DFT approach isapplied to a model system consisting of a single 4,4 bipyri-dine (44PBD) molecule coupled to two semi-infinite 1-dimensional gold electrodes, which is shown in Fig. 3.The extended molecule consists of the 44BPD moleculeand six gold atoms on each side. Our aim is to demonstratethe non-equilibrium correction for the non-equilibriumpart of the density matrix rather than presenting detailedanalysis of our findings, because the conductance of thegold–44BPD–gold molecular junction has been extensivelyinvestigated in the literature [21–28]. During the self-con-sistent iterations, the DFT calculation is performed withthe program package Gaussian 03 [29]. The TMSZ basisset is used for gold atoms and the CEP-31G basis set isused for C, N and H atoms [30,31], in which the effect ofthe core electrons is represented by effective core potential(ECP). For the electron exchange-correlation interaction,the BLYP functional is adopted [32,33]. The interatomicAu–Au distance in the gold monatomic chain is optimizedto be 2.73 A with periodic boundary condition (PBC) cal-culation at the theory level of BLYP/TMSZ [34].

3.1. Bound states at equilibrium

At equilibrium, the Hamiltonian and electron density ofthis open system can be calculated accurately [7]. TheFermi level is determined to be �6.44 eV by the electrodes.If some bound states appear near the Fermi level, theymaybe enter the bias window when a bias voltage isapplied. Furthermore, if these bound states are also locatednear the left electrode, they are to be filled and influence theelectron density. Hence, its necessary to investigate boundstates at equilibrium.

Firstly, we diagonalize the enlarged Hamiltonian matrixand recognize bound states manually. Within the energyrange of the Fermi level ±2 eV, i.e. between �8.44 and�4.44 eV, 12 bound states are found in the extended mol-ecule, six near the left electrode and six near the right elec-trode, and their energies are all below the Fermi level. Thebound states near the right electrode will be ignored as theynever enter the bias window. These six bound states near

gold monatomic chains. The molecule together with six gold atoms on each

132 R. Li et al. / Chemical Physics 336 (2007) 127–135

the left electrodes are all twofold degenerate, with associ-ated eigenenergies of �7.24 eV, �7.26 eV and �7.73 eV,respectively. Their charge density contours are illustratedin Fig. 4. It can be seen that they are all evanescent expo-nentially away from the interface between the extendedmolecule and the left electrode.

The analysis of DOS is also performed. At the energiesobtained from the above diagonalization, three peaks arisesharply out of the smooth background (see Fig. 5a, b),which implies the appearance of bound states. Further-more, it can be seen more clearly after subtracting theDOS of scattering states from the total DOS that thesepeaks are really bound states, as shown in Fig. 5c. In addi-tion, these three peaks only appear in the local density ofstate (LDOS) of gold atoms but not in that of the44BPD molecule (see Fig. 5d, e), indicating that all thebound states are located on gold atoms, in good agreementwith previous results obtained by diagonalizing the Hamil-tonian matrix.

3.2. Bound states out of equilibrium

Fig. 6 shows the change of eigenvalues of the boundstates near the left electrode when a bias voltage is applied.As can be seen, when the bias voltage Vb is increased, thesebound states are all shifted up with the left Fermi level.Simultaneously, the bias window is also enlarged. Whenthe bias voltage Vb exceeds a threshold, these bound statesbegin to enter the bias window. The highest bound statejust enters the bias window at Vb = 0.8 V. Soon anotherbound state is shifted into the bias window following thefurther increase of the bias voltage. After Vb is more than1.4 V, all these bound states are within the bias window.

Fig. 4. Charge density contours of the bound states near the left electrode. Tcharge density contour.

The occupation weights defined in Eq. (18) is carefullyexamined at a large bias Vb = 1.6 V. For atoms in the leftpart of the extended molecule where bound states are local-ized, the weight coefficients are all more than 0.99, indicat-ing that these bound states are uniformly filled, which is thecorrect physical picture in this situation.

Since some bound states are not completely localized inthe extended molecule region, the weight factors of thesebound states are also labeled in Fig. 6. It can be seen thatthe bias voltage also affects the details of these boundstates. Without this non-equilibrium correction, the self-consistent process with the conventional NEGF + DFTapproach cannot converge at bias voltages larger than1.4 V. Therefore, it is necessary to fill the bound states withour non-equilibrium correction for the density matrix inthe self-consistent calculation of this model system.

3.3. Effect of patching the non-equilibrium integration

The effect of our non-equilibrium correction method onpatching the non-equilibrium integration can be demon-strated directly from the comparison of the non-equilib-rium part density matrix calculated from the sameHamiltonian matrix with different number of integrationpoints. Using the density matrix obtained with an ultrafinegrid of totally 1024 integration points as the reference, wegive in Fig. 7 the maximum difference of all matrix ele-ments calculated with or without the non-equilibrium cor-rection. Obviously, our non-equilibrium correction caneither increase the integral accuracy with the same numberof integration points or reduce the number of integrationpoints at the same integral accuracy. For instance, the den-sity matrix calculated with 32 integration points using our

he two degenerate states with the same eigenenergy have almost the same

Fig. 5. DOS analysis of the bound states. (a) The total DOS of the extended molecule; (b) an enlarged image of the shadowed areas in (a); (c) the differencebetween the total DOS and the DOS of scattering states; (d) the LDOS of the 44BPD molecule; (e) the LDOS of gold atoms in the extended molecule.

Fig. 6. The change of eigenvalues of the bound states following theincrease of the bias voltage. The weight factor an is also labeled for eachbound state. The shadowed area in the figure is the bias window.

Fig. 7. The maximum difference among all matrix elements of the non-equilibrium part density matrix calculated with different number ofintegration points, the density matrix calculated with an ultrafine 1204integration points is chosen as the reference.

R. Li et al. / Chemical Physics 336 (2007) 127–135 133

non-equilibrium correction can achieve the same accuracyof 512 integration points without the correction, indicating

that our non-equilibrium correction method is very efficientand accurate.

To validate our non-equilibrium correction method inthe self-consistent process, the Hamiltonian matrix of this

Table 1The mean and maximum differences of diagonal elements between theHamiltonian matrices calculated with two different integration grids

Bias voltage (V) Mean difference (eV) Maximum difference (eV)

0.6 0.002 0.0101.0 0.003 0.0181.6 0.002 0.0152.0 0.005 0.036

134 R. Li et al. / Chemical Physics 336 (2007) 127–135

model system at different bias voltages is calculated self-consistently using two kinds of integration grids for thespectral integration along the real axis, one with a step of1 meV and the other with a step of 100 meV. The Hamilto-nian matrix obtained with the fine grid and that calculatedusing the coarse grid with the non-equilibrium correctionare compared, the mean and maximum differences of diag-onal elements between the two Hamiltonian matrices arelisted in Table 1. As we can see, when the non-equilibriumcorrection is applied, the reduction of the integration griddoes not affect the result significantly, indicating that thenon-equilibrium correction has patched the integral per-fectly. The number of the integration points has decreasedfrom 2000 to 20 under the bias voltage of 2.0 V, whichmakes the non-equilibrium integration no longer a tremen-dous work. In contrast, if the non-equilibrium correction isignored, a larger error will occur for the coarse integrationgrid when the bias is not very high (60.6 V). For example,at the bias voltage of 0.6 V, the mean and maximum differ-ences of diagonal elements between the two Hamiltonianmatrices are increased from 0.002 eV and 0.010 eV to0.008 eV and 0.106 eV, respectively. When the bias voltageis higher, the self-consistent iterations using the coarse inte-gration grid without the non-equilibrium correction evencannot reach convergence. Therefore, patching the non-

Fig. 8. The relative magnitude of average on-site energies of the atomic orb

equilibrium integration is necessary when the integrationgrid is not very fine, especially under a high bias voltage.

The average on-site energy of the atomic orbitals on anatom of the extended molecule reflects the electrostaticpotential on that atom [9]. Fig. 8 shows the relative magni-tude of the on-site energies on all atoms under different biasvoltages, using the on-site energies obtained for theextended molecule at equilibrium as reference. The majordrop of the potential (about 60%) occurs at the interfacesbetween the 44BPD molecule and the gold electrodes, thisis probably due to the weak coupling of the Au–N bond.The remaining part of the potential almost linearly dropsacross the 44BPD molecule, which is the direct result ofthe very small LDOS of the 44BPD molecule near theFermi level [17], as shown in Fig. 5d. And the potentialdrop between the two electrodes also agrees well with theapplied bias voltage, indicating that the electron densitycalculated with the non-equilibrium correction is appropri-ate for constructing the potential.

4. Summary

We have investigated two problems in the conventionalNEGF + DFT approach, that is, the bound states in thebias window and numerical difficulties of the non-equilib-rium integration of the density matrix. Two approachesof finding bound states are introduced, and a scheme of fill-ing these bound states is developed, in which a site-depen-dent filling weight is defined for every element of thespectra integral matrix of bound states and determiningthe occupation of each bound state individually is avoided.A method of patching the non-equilibrium integral is alsoproposed, which makes it possible to use a coarse integra-tion grid and save the cost of computation significantly.

itals on all atoms of the extended molecule under different bias voltages.

R. Li et al. / Chemical Physics 336 (2007) 127–135 135

Calculations on a model system consisting of a 44BPDmolecule connected to two semi-infinite gold monatomicchains demonstrate the correctness and efficiency of ourcorrected NEGF + DFT approach. It should be pointedout that our method is suitable for dealing with boundstates localized in either the left part or the right part ofthe extended molecule. When a bound state distributesover the whole extended molecule, our method may leadto unphysical results because this bound state is only par-tially filled. Therefore, further study is needed to solvethe problem of bound states completely.

Acknowledgements

This project was supported by the National Natural Sci-ence Foundation of China (Nos. 90406014, 90206048 and60671022) and the MOST of China (No. 2006CB932404).

Appendix

In this Appendix, we will derive the decomposition ofthe spectral function. Firstly, from the definition of theretarded and advanced Green’s functions, we can get

RL þ RR ¼ ðE þ igÞS � H � ðGRÞ�1

RþL þ RþR ¼ ðE � igÞS � H � ðGAÞ�1;

where g is a positive infinitesimal. Then, we have

A� ðAL þ ARÞ ¼ i½GR � GA� � iGR½RL þ RR � RþL � RþR�GA

¼ i½GR � GA� � iGR limg!0þ½ððE þ igÞS � H

� ðGRÞ�1Þ � ððE � igÞS � H

� ðGAÞ�1Þ�GA

¼ limg!0þ

2gGRSGA

¼ limg!0þ

2gX

n

unuþn

E þ ig� EnSX

n0

un0uþn0

E � ig� En0

¼ limg!0þ

Xn;n0

unuþn Sun0u

þn0

� 2gðE þ ig� EnÞðE � ig� En0 Þ

¼ limg!0þ

Xn

ðuþn SunÞunuþn

2g

ðE � EnÞ2 þ g2

¼ 2pX

n

anunuþn dðE � EnÞ;

where the weight factor is defined as an ¼ uþn Sun. Here wehave used the orthogonal property of all the eigenstates. un

in the above equation only contains the extended moleculepart of a state that has been normalized and maybe distrib-

utes over the infinite space. For scattering states, this part isonly an infinitesimal, which makes an = 0. Therefore, wehave

A� ðAL þ ARÞ ¼ 2pXbound

n

anunuþn dðE � EnÞ:

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