Haug H.

Mads Brandbyge

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The Journal of Chemical Physics

Comment on “Rethinking first-principles electron transport theories with projection operators: The problems caused by partitioning the basis set” [J. Chem. Phys. 139, 114104 (2013)]

In a recent paper Reuter and Harrison [J. Chem. Phys.139, 114104 (2013)] question the widely used mean-field electron transport theories, which employ nonorthogonal localized basis sets. They claim these can violate an “implicit decoupling assumption,” leading to wrong results for the current, different from what would be obtained by using an orthogonal basis, and dividing surfaces defined in real-space. We argue that this assumption is not required to be fulfilled to get exact results. We show how the current/transmission calculated by the standard Greens function method is independent of whether or not the chosen basis set is nonorthogonal, and that the current for a given basis set is consistent with divisions in real space. The ambiguity known from charge population analysis for nonorthogonal bases does not carry over to calculations of charge flux

Brandbyge, Mads

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Downloaded from orbit.dtu.dk on: Dec 20, 2017Comment on "Rethinking first-principles electron transport theories with projectionoperators: The problems caused by partitioning the basis set" [J. Chem. Phys. 139,114104 (2013)]Brandbyge, MadsPublished in:Journal of Chemical PhysicsLink to article, DOI:10.1063/1.4873696Publication date:2014Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Brandbyge, M. (2014). Comment on "Rethinking first-principles electron transport theories with projectionoperators: The problems caused by partitioning the basis set" [J. Chem. Phys. 139, 114104 (2013)]. Journal ofChemical Physics, 140(17), 177103. DOI: 10.1063/1.4873696Comment on “Rethinking first-principles electron transport theories with projectionoperators: The problems caused by partitioning the basis set” [J. Chem. Phys.139,114104 (2013)]Mads Brandbyge  Citation: The Journal of Chemical Physics 140, 177103 (2014); doi: 10.1063/1.4873696 View online: http://dx.doi.org/10.1063/1.4873696 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/17?ver=pdfcov Published by the AIP Publishing  Articles you may be interested in Response to “Comment on ‘Rethinking first-principles electron transport theories with projection operators: Theproblems caused by partitioning the basis set’” [J. Chem. Phys.140, 177103 (2014)] J. Chem. Phys. 140, 177104 (2014); 10.1063/1.4873698  Rethinking first-principles electron transport theories with projection operators: The problems caused bypartitioning the basis set J. Chem. Phys. 139, 114104 (2013); 10.1063/1.4821176  Low-bias electron transport properties of germanium telluride ultrathin films J. Appl. Phys. 113, 063711 (2013); 10.1063/1.4790801  LongDistance Electron Transfer Coupled to Proton Pumping and Energy Transduction in Biological Systems: ASemiempirical FirstPrinciples Approach AIP Conf. Proc. 963, 278 (2007); 10.1063/1.2836061  Firstprinciples Theory of Inelastic Transport and Local Heating in Atomic Gold Wires AIP Conf. Proc. 893, 727 (2007); 10.1063/1.2730096    This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:192.38.67.112 On: Thu, 08 May 2014 09:02:42THE JOURNAL OF CHEMICAL PHYSICS 140, 177103 (2014)Comment on “Rethinking first-principles electron transport theories withprojection operators: The problems caused by partitioning the basis set”[J. Chem. Phys. 139, 114104 (2013)]Mads Brandbygea)Center for Nanostructured Graphene (CNG), Department of Micro- and Nanotechnology,Technical University of Denmark, DTU Build. 345 East, DK-2800 Kongens Lyngby, Denmark(Received 14 January 2014; accepted 2 March 2014; published online 7 May 2014)[http://dx.doi.org/10.1063/1.4873696]I. INTRODUCTIONReuter and Harrison (RH)1 argue that a certain “decou-pling condition” is required to be fulfilled in order to ob-tain correct results in transport calculations which employsan non-orthogonal basis (NOB) set, such as Ref. 2 and relatedin Ref. 1. The implicit assumption (decoupling condition) putforth by RH readsˆNL ˆH ˆNR = ˆ0 , (1)and violation is claimed to cause a “short circuit” problem.Here ˆNL/R are projection operators onto L/R regions definedin real-space, and ˆH is the Hamiltonian operator describingthe full system coupled via region C (see Fig. 1). RH state thattheir “short-circuit” problem resembles the ambiguity of as-signing charge to atoms or regions in charge population anal-ysis. Below we argue that this ambiguity problem does notcarry over to calculations of charge flux.We point out that Eq. (1) does not enter the NEGF deriva-tions of current.3 It is clear for infinitely separated L and R re-gions, we should be able to partition the system into scatteringstates originating from L and R, and thus Eq. (1) is, in prin-ciple, implicitly required to be asymptotically fulfilled. How-ever, RH assign the L and R regions to the explicitly defined“physically motivated” regions in actual calculations. Theyattribute the “short circuit” to the fact that HLR = 0 and SLR= 0, for Hamiltonian (H) and overlap (S) matrices for a NOBset does not secure the operator equation above,NLS−1HS−1NR = 0 . (2)Several works address the NOB in transport calcula-tions.4–9 Here we provide simple arguments why equivalentOB and NOB sets yield the same transmission when the over-lap is taken into account in the way discussed by Emberlyand Kirczenow (EK).4, 10 We demonstrate by simple calcula-tions how we get the exact results for transmission also whenEq. (2) is violated. Furthermore, we obtain the same transmis-sion across an arbitrary dividing surface in real space as usinga partitioning based on orbitals.a)Electronic mail: mads.brandbyge@nanotech.dtu.dkII. NONORTHOGONAL BASIS SETWe will argue that Eq. (2) can be violated while we ob-tain the exact transmission. First we consider a NOB chain-model with nearest neighbour hopping,7 and overlap (matrixelements t, s). It can be shown that the range of the effectivecoupling, |(S−1HS−1)ij| ∝ ts|i − j|, is infinite, although rapidlydecaying. This demonstrates how Eq. (2) is only possible asan asymptotic limit. On the other hand, the transmission canbe calculated analytically exact7 for this model by the EKmethod using a 1-site region for C thus explicitly violatingEq. (2).Next, we consider an OB and an arbitrarily big centralC region (Fig. 1). The transmission in terms of quantities de-fined inside C, reads3, 11T = Tr[ ¯L ¯G ¯R ¯G†] . (3)All matrices are given in the OB (denoted by a bar). Now werotate the basis set to introduce an overlap matrix between theorbitals inside C using an “inverse” Löwdin transformation12from the OB to the NOB,T = Tr[S 12 ¯LS 12 S− 12 ¯GS− 12 S 12 ¯RS 12 S− 12 ¯G†S− 12 ]= Tr[LGRG†] , (4)where10 G ≡ S−1/2 ¯GS−1/2, H ≡ S1/2 ¯HS1/2, and L/R≡ S1/2 ¯L/RS1/2.We now split region C into C1 and C2. The range in thetransport direction of the self-energies, , in the NOB is thatof H and S. For a big enough C and a NOB with finite range,we have zero matrix elements between L, 2 and 1, R. Thus can be written asL =((L)11 00 0)and R =(0 00 (R)22), (5)which is the typical case, and we haveT = Tr[(L)11G12(R)22G†21] . (6)We can write the GF in the NOB,G−1 =(g−111 ES12 − H12ES21 − H21 g−122), (7)where we introduce the inverse GF for region 1 without cou-pling to 2, g−111 = ES11 − H11 − 11, and likewise g22. It is0021-9606/2014/140(17)/177103/2/$30.00 © 2014 AIP Publishing LLC140, 177103-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to  IP:192.38.67.112 On: Thu, 08 May 2014 09:02:42177103-2 Mads Brandbyge J. Chem. Phys. 140, 177103 (2014)FIG. 1. The partition into left, central, and right regions used in the stan-dard transport schemes. (a) Left and right self-energies are non-zero (in thenonorthogonal basis) in the dashed regions inside C. An overlap is includedin region C. Region C is partitioned further into regions 1 and 2 to show howthe transmission is unchanged when calculated using region 1 as a “new” cen-tral region (b), and the standard procedure for including overlap in transportcalculations based on GFs.straightforward from (7) to obtainG11 =(g−111 − (ES12 − H12)g22(ES21 − H21))−1≡ (g−111 − (˜R)11)−1 (8)andG12 = −G11(ES12 − H12)g22 , (9)where we have introduced the right self-energy downfoldedonto region 1,(˜R)11 = (ES12 − H12)g22(ES21 − H21) . (10)Using (R)22 = i[(g−122 )† − (g−122 )], we rewrite Eq. (6) and geta transmission formula for region C = 1 (Fig. 1(b)),T = Tr[(L)11G11(˜R)11G†11] . (11)Thus we get exactly the same T for the NOB as for the orig-inal OB if we treat the overlap in the self-energy as Eq. (10),as is done in the “standard” EK approach.4 We note thatfor the smaller region, C = 1, we may have high values of|(S−1HS−1)LR| depending on S and H. But this is not relevantfor the derivation.In principle we can start by making a Löwdin transfor-mation of the whole space. This leads to a long ranged H, butwe may choose C big enough to ensure that ¯HLR is as small aswe want. In actual calculations we make the C region small,as shown above, and need only to consider the range of H, S.III. TRANSMISSION THROUGH A SURFACE DEFINEDIN REAL-SPACEThe partition of space in terms of the orbitals for a givenbasis set yields the same result for T compared to a real-spacedivision, when T is calculated with the same basis set. Wesketch the proof and refer to Ref. 13 for details. The right fluxnormalized scattering states (label r) at energy, E, yield thecurrent(T) through a surface, σ 12, located inside C,T = ¯mIm∑r∫σ12d σ · (∗r (r) ∇r (r)) (12)= ¯mIm∑r∫V1dr ∗r (r) ∇2r (r) . (13)We have rewritten the surface integral as an integral over re-gion 1. In region 1 r fulfils11[− ¯22m∇2 + V (r) +∫d r ′ L(r, r ′)]r (r ′) = Er (r) ,(14)where the integral is over the support of L. Note that σ 12is arbitrary as long as it is located within the freely chosenscattering region, and does not overlap with the support of .Using Eq. (13) in Eq. (14) we can obtainT = ¯mIm∑r∫V1dr (∗r (r) ∇2r (r)) (15)= 1¯∑r∫dr∫d r ′ ∗r (r)L(r, r ′)r (r ′) . (16)This is similar to the expression found by “embedding.”14The main point here is the partitioning in terms of a basisset, {φα}, which overlap in regions of real space. We writeAR(r, r ′) ≡ 1¯∑r∗r (r)r (r ′) =∑r,αβc∗rαcrβ φα(r)φβ(r ′),(17)L(r, r ′) =∑αβφα(r)(S−1LS−1)αβ φβ(r ′) ,and using the definition of AR we immediately obtain,T = Tr[LAR] = Tr[LGRG†], as we would when partition-ing using the orbital basis set instead of σ 12.1M. G. Reuter and R. J. Harrison, J. Chem. Phys. 139, 114104 (2013).2M. Brandbyge, J. L. Mozos, P. Ordejon, J. Taylor, and K. Stokbro, Phys.Rev. B 65, 165401 (2002).3H. Haug and A.-P. Jauho, Quantum Kinetics in Transport and Optics ofSemiconductors (Springer-Verlag, 1996).4E. Emberly and G. Kirczenow, Phys. Rev. Lett. 81, 5205 (1998).5J. Fransson, O. Eriksson, and I. Sandalov, Phys. Rev. B 66, 195319 (2002).6K. W. Sulston and S. G. Davison, Phys. Rev. B 67, 195326 (2003).7S. T. Rittenhouse and B. L. Johnson, Phys. Rev. B 71, 035118 (2005).8K. S. Thygesen, Phys. Rev. B 73, 035309 (2006).9J. Wang and H. Guo, Phys. Rev. B 79, 045119 (2009).10M. Brandbyge, N. Kobayashi, and M. Tsukada, Phys. Rev. B 60, 17064(1999).11S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge Univer-sity Press, 1995).12P.-O. Löwdin, J. Chem. Phys. 18, 365 (1950).13L. Zhang, B. Wang, and J. Wang, Phys. Rev. B 84, 115412 (2011).14J. E. Inglesfield, S. Crampin, and H. Ishida, Phys. Rev. B 71, 155120(2005). This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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Comment on "Rethinking first-principles electron transport theories with projection operators: The problems caused by partitioning the basis set" [J. Chem. Phys. 139, 114104 (2013)]

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