奈米碳管分子結間電子傳輸與干涉現象之第一原理研究

全文

(1)國立台灣師範大學物理學系暨物理研究所碩士班碩士論文. 奈米碳管分子結間電子傳輸與干涉現象之第一原理研究. An ab initio Study on the Interference Phenomenon in the Transmission of the CNT-molecule-CNT Junctions. 研究生:陳凱榆指導教授:陳穎叡博士. 中華民國 102 年八月.

(2) Acknowledgement 在師大的這兩年裡，碩一修課碩二忙論文，現在回頭看覺得似乎各方面都還可以再加強的；不管怎樣，總還是達到了一個小小的里程碑。雖然在師大老實說並不像在大學時期一樣認識很多人，但還是受到不少人的照顧，首先當然是指導老師陳穎叡：雖然老師自己曾說過是個很沒耐性的人，而我又不是反應很快的那一型；但每次討論時，要是我不能馬上理解，老師都很有耐心地換個方式說明，非常感謝老師不厭其煩地諄諄教誨，尤其是修改論文以及口試模擬的部分，在此之前完全沒有這方面的經驗，相信應該帶給老師不小的麻煩(笑)。另一方面，也要特別感謝中研院應科中心的關肇正老師，提供 Nanodcal 才能有這些成果；還有關老師的前任助理林俊儒，跟他學會了如何操作 Nanodcal 跟 QE；以及老師的前一個學生李欣翰學長，有一部分的內容是承繼學長當年的研究成果再做更深入的探討；論文口試當天還要特別感謝中央的唐毓慧老師，除了給了寶貴的建議外，口試前的討論也讓我在口試當下多少消除了些緊張感。研究方面以外，在這兩年多的日子裡，要特別感謝軒豪學長，有事沒事總是會一起中午吃個飯隨便亂聊；系辦鈞萍姐除了幫我們研究生處理事務外，也很常被請吃一些小點心；還有同間研究室的松勳學長跟維中學長，電腦室的雅惠學姊、世和學長跟德倫學弟；同屆同學雖然認識的不多，修課也比較少交流，還是要感謝萬暉、沐谷，有時候碰到問題會去他們 lab 找他們討論；最後是同學兼室友的祥友跟鳴桂，尤其是鳴桂，真的幫了我很多的忙，碰到問題第一個也總是想到他，真的是由衷的感謝。. i.

(3) Abstract 本篇論文以奈米碳管(Carbon Nanotube, CNT)與分子結構所形成的一維系統為題，利用第一原理(Ab initio)方法計算其傳輸性質；依據分子結構的不同，發現會有干涉現象的產生。藉由與緊束縛模型(Tight Binding model)所得的結果相互比對，透過傳輸係數(Transmission Coefficients)及態密度(Density of States, DOS) 的分析，可歸納出影響類似此種結構之奈米電子元件的電子傳輸性質為何，並且進一步地了解量子傳輸理論(Quantum Transport Theory)。. 關鍵字: 第一原理 (ab initio) 奈米碳管 (carbon nanotube) 電子傳輸 (transmission) 干涉 (interference) 分子結 (molecular junction) 密度泛函理論 (density functional theory, DFT) ii.

(4) Index Introduction……………………………………………………..1 Theory and model………………………………………………3 (I). The Density Functional Theory (DFT)………………………………………. 4. (II) (III) (IV). DFT in Nanodcal …………………………………………………………..... 4 Non-equilibrium Green’s function (NEGF)………………………………….. 5 Boundary condition as the self-consistent field criteria in Nanodcal……........ 6. (V). NEGF in Nanodcal …………………………………………………………... 7. Method………………………………………………………...11 Results and Discussion………………………………………..13 (I) (II). Transmission function……………………………………………….…….…13 Density of states…………………………………………………….…….….25. Conclusions…………………………………………………... 29 Appendix..……………………………………………………. 30 References……………………………………………………. 32. iii.

(5) Introduction. Nowadays, electronic devices whose size approximates micrometers are tried to be made smaller still to reach nano scales. In addition, more and more chemical or biological molecules are presented, or tried to be presented, as the junction in such nano-scaled electronic devices, for the property difference between the discrete molecular energy levels and the leads’ continuous density of states provides promising special functions. However, in experiments it is a crucial and difficult issue to manipulate the contacts between the electrodes and the junction molecules. On the one hand, the smaller the device is, the more the quantum effects could dominate the properties of the device. On the other hand, in such novel devices, the contact bondings between the very different natures of the leads and the molecules are usually ill-defined. Carbon-nanotubes (CNT) were first made in the laboratory in 19911,2. Since early last decade, due to its covalent bonding network which resembles the nature of the junction molecules, CNT’s or their ribbons have been considered either as leads or devices not only in nano-scaled electronic junction experiments3,4, but also in a variety of simulation models5-12, in the progress of nanotechnology. Same as with traditional electronic devices, we care for the transmission properties these novel molecular junctions possess, and especially focus on the region near the Fermi energy of the CNT leads. From the point of view of quantum transport theory, the counting of bands is the same as the counting of propagation channels. Since uncut metallic-type single walled CNT’s have bands crossing the vicinity of the Fermi energy, it is necessary to check how these channels are rebuilt after the CNT is cut and reconnected by the molecular junction. According to a previous tight-binding model study on a series of simple systems13, at E  EF a conjugate polyene molecule junction could rebuild one channel out of the original two of the CNT, provided that the number of carbon atoms on the polyene chain is odd, and the transmission T ( EF ) does not decay with the polyene length. Moreover, when multiple polyene molecules come to form the junction, interference phenomenon occur. Depending on the combination of contacts with leads, the rebuilt channel number varies from zero to two. -1-.

(6) Our current study basically presents the ab initio realization of this previous work, which used the tight binding model and focused on the interference phenomenon in the transport of a CNT-molecule-CNT junction 13. In this thesis, we consider the (8,8) armchair CNT as leads, and simple polyene molecules as the device. We basically follow the method of the non-equilibrium Green’s function (NEGF)14,15 to investigate the quantum transport properties of our system. Our ab initio work shown in this thesis presents the interference effect in transmission, and the molecular LDOS that corresponds to the transmission peaks. This study well agrees with the previous tight binding results, and the physical meanings of the slight differences are also worth discussion.. -2-.

(7) Theory and model. The system we consider has the following general scheme: 1. There are the left and the right electrodes, which are semi-infinite structures that periodically extend to   and   , respectively. Each electrode (or called lead as well) is in its own equilibrium state with a well-defined electrochemical potential,  L or  R . 2. There is also the scattering region (or called central box for our practical purpose), could be some molecular structure bridging both leads. The central box contains not only the molecular bridge, but also the edge layers from both lead. Note that the edges of the cut CNT leads are terminated with hydrogen atoms. 3. For a biased device where  L   R , the system is in a non-equilibrium state. We need to use the NEGF-DFT ab initio formulism to study the quantum transport therein. More details about DFT and NEGF methods are described in later sections. Despite that, in our current study we’re focusing on the inference cases with zero bias.. Fig.1 : A general schematic plot of two-probe device model.. -3-.

(8) (I) The Density Functional Theory (DFT) To consider a multi-electron system problem, one has in fact no idea about the exact form of the many-body wave-function. In the 1960s 16,17, Hohenberg, Kohn, and Sham established the foundation of DFT by proposing a simple argument showing that the ground state properties of a system could be uniquely determined by its electronic density n(r). Furthermore, all physical quantities of the system’s ground-state are functionals of n(r). In general, the Kohn-Sham ground-state energy functional is.   1   n ( r ) n( r ' )     E[n]  T[n]   d r n( r )[Vext (r )  Vion e (r )]   dr dr '    E xc [n] , 2 r  r'. (1). where T is the kinetic term, Vext is the external bias, Vion-e is the Coulomb potential between the ions and the electrons, and Exc is the exchange-correlation energy. Applying the variational principle to Eq.(1), one could get the Kohn-Sham equation   1 2   n( r ' )         dr '    Vext (r )  Vione (r )  Vxc (r ) i (r )   i i (r ) , r  r'  2 . (2). The exchange-correlation potential is written as  E [n] Vxc (r )  XC n(r ) Since one still has no idea about the exact form of Vxc, approximations must be used in practical computation for DFT. The most well-known approximations are local density approximation (LDA), and the generalized gradient approximations (GGA).. (II) DFT in Nanodcal For the practical computation of DFT, it is an important issue to choose a proper basis set. In Nanodcal, the choice of basis set is LCAO (Linear Combinations of Atomic Orbitals, or called zeta(ζ)-functions.) It could be classified by the angular momentum l and associated with angular momentum eigenfunctions Ylm and an atomic site RI to form an LCAO basis. The orbitals are labeled by  ’s, where -4-.

(9)  accounts for the angular momentum l and m , and the ionic site RI :      (r )  Rl ( r  RI )Yl m (r  R  ) ,. (3). I. The default for Nanodcal is the double-ζ plus polarization (DZP) orbitals. The choice of DZP is a compromise between accuracy and computation in general case.. (III) Non-equilibrium Green’s function (NEGF) 18,19 For the “left lead-scattering region-right lead” model in a general junction problem, we could write down the Schrödinger equation of the system in the following matrix form: EI  H 1  i  1   0 . . 1 EI  H  2. .    1   1   S1       2      0  ,    EI  H 2  i    2   2  S 2  0. (4).  1 (  2 ) is the coupling between the left (right) lead and the device,  1 (  2 ) is the left (right) scattering wavefunction, and S1 ( S 2 ) is the source term. Note that the Green’s function for an isolated lead is G  EI  H  i  . Using relations 1.    G    1 1. and. could. obtain. EI  H  1   2   S  G 1  . . 1.  2   2 G2 2. .  2   G2 2   . from the first and the last equations, we ,where. 1   1G1 1. . and. are self-energy terms.. The self-energy terms play an important role in the NEGF method. The case of a device connecting two contacts is an open-system problem. By treating the self-energy term as a surface effect from the lead to the device, we could deal with the problem only in the Hilbert space of the device. How should we obtain the proper Green’s functions for the leads and write down the self-energy terms for the device? Since a lead is a semi-infinite periodic structure, we could simply use the following iterative method to compute the effective surface Green’s function:  G G  VLd.  VdL   EI  H  G L     . 1.   , G  EI  H  GL   EI  H L  -5-. . . 1.

(10) In fact, for a structure that extends to infinity, G  G L  g s . We therefore obtain. g s  [  g s  ]1. To study the transmission of the junction, we basically follow the Landauer-Büttiker formalism. The time-dependent Schrödinger equation reads:   1  1   H 1  i  1 d   i   H 1   dt  2   2   0  2 .    1   1   S1       2      0  ,    H 2  i    2   2  S 2  0. . (5). . d  where I  tr      is the current, A  i G  G  is the spectral function, and  dt    is the broadening matrix. Finally we obtain   i     A. . . . . 2q I dET ( E )[ f1 ( E , 1 )  f 2 ( E ,  2 )] , where T  tr 1G2 G   h . . is called the. transmission function.. (IV) Boundary condition as the self-consistent field criteria in Nanodcal 20 After simplifying the problem from an open-system case to a closed-system case, the boundary conditions at the interface between leads and the central box play a crucial role in the practical calculation. The conditions of the Kohn-Sham effective  potential V eff  (r ) at the boundaries are as following.   Vl eff (r )  Vl eff ,bulk ( r ), z  z l    , V eff (r )   Vceff (r ), z l  z  z r V eff (r )  V eff (r ), z  z r ,bulk r  r. (6). where planes z  z l and z  z r are the left and right limits of the central box..  Vl eff / r ,bulk ( r ) is obtained from the isolated lead calculation.  Further more, we match the boundary conditions for the Hartree potential VcH  (r ) by solving the Possion equation in real space as follows   VcH (r ) zl  Vl ,Hbulk (r ) zl ,   VrH (r ) zr  VrH,bulk (r ) zr. -6-. (7).

(11)  where Vl ,Hbulk (r ). zl.  and VrH,bulk (r ). zr. are the Hartree potentials of the equivalent bulk. systems.    Since V eff  (r ) and VcH  (r ) are functionals of  ( r ) , the boundary conditions. of V eff would be satisfied as VcH converges to the bulk values.. (V) NEGF in Nanodcal 20 (i) Electrode potential First of all, we must calculate the KS potential of the electrodes (leads).. With. the LCAO basis sets, one could expand the Hamiltonian of the leads in the following matrix form:.    hz , z 1    .  hz , z hz 1, z. hz , z 1 hz 1, z 1 . hz 1, z  2 .   ,   . (8). where hz , z '   z H  z ' , and the z ’s label the unit layers in the leads’ periodic structure. With Bloch condition  k ( z )  e ikzk ( z ) , the Schrödinger equation becomes. h. z , z 1. . . . e ikz  hz , z  hz , z 1e ikz k  Ek S z , z 1e ikz  S z , z  S z , z 1e ikz k ,. where S z , z '   z |  z '. (9). represents the overlap matrix element. For a given k value,. one could diagonalize the above equation to obtain eigenvalues E k and eigenvectors.  k . Furthermore, the density matrix could also be obtained by integrating over the Brillouin zone the following:. ˆ   dk  k f Ek ,    k ,. (10). BZ. where f Ek ,   is the Fermi function with chemical potential  . The density   matrix’ projection onto the real space gives the charge distribution  (r )  ˆ (r, r) , and therefore the potential that is used in the KS equation in the next iteration. The.  KS potential of the leads Vl eff / r ,bulk ( r ) are solved when this iterative self-consistent -7-.

(12) procedure leads to convergence. (ii) Open device system Next we could complete the Hamiltonian of the entire system, including the central region and the left and right semi-infinite leads, in the following matrix form:.  hl , h  l ,l 1 H 0   0  0 . where hl ,.      0 hl ,l 1  0 0.  hl ,l hl ,l 1. hl ,l 1 hl ,l hc ,l 0 0. 0 hl ,c hc ,c hr ,c 0.    ,  hr ,r 1  hr , . 0 0 hc ,r hr ,r hr ,r 1. 0 0 0. 0   hr ,r  hl ,l 1  ,and hr ,  hr ,r 1  0 hl ,l . (11). hr ,r 1 hr ,r. 0 hr ,r !. . . 0 0  , . come from the calculation of the semi-infinite leads. Now one could introduce the self-energy terms based on the NEGF method which we described in the previous section, in order to write down the central device effective Green’s function:. g CC. hlE,l   ll ,l    hcE,l  0 . hlE,c hcE,c hrE,c. 1.   hcE,r  , hrE,r   rr ,r  0. (12). where hiE, j  E  i  hi , j , and the self-energy terms  ll ,l  hlE,l 1 g l , hlE,l 1 and.  rr ,r  hrE,r 1 g r , hrE,r 1 could be evaluated by the surface Green’s function g l / r ,  E  i  hl / r ,  , respectively. 1. (iii) Non-equilibrium condition The system is under non-equilibrium condition when it is applied a non-zero external bias. The distribution functions. f. kln. n. n. , f c , f kr.  of the left lead, the device,. and the right lead, respectively, are therefore ill-defined. The equilibrium case could be thought of as the extreme case of the non-equilibrium condition, and the Fermi distribution functions could be presentd as -8-.

(13) f. kln. ( E; l  Vl )  f. cn. ( E; c )  f. ( E;  r  Vr )  f. krn. eq. ( E; *) ,. where *   l  Vl   c   r  Vr . For a general non-equilibrium case where l  Vl   r  Vr , we use the following approximation, which is widely used in quantum transport problems:. f f. kln. ( E;  l  Vl )  f eq ( E;  l  Vl ). k rn. ( E;  r  Vr )  f eq ( E;  r  Vr ). ,. (13). In other words, it is taken as though the influence of the external bias on the system is reflected on the shifts in chemical potentials that govern the distribution functions of the leads, and that the equilibrium distribution functions still govern the statistics of the leads, under the non-equilibrium condition. Before writing down the density matrix under non-equilibrium condition, we introduce the lesser Green’s function:. G   G R   [ f kl , f kr ]G A , n. n. (14). where G R and G A are the retarded and advanced effective Green’s functions of the system (projected onto the device region only), and   [ f kl , f kr ]  2i Im( f kl  ll ,l  f kr  rr ,r ) takes twice the imaginary parts from the n. n. n. n. self-energies coming from the left and the right lead, weighted by their own distribution functions. Then the density matrix could be written as. i dEG  ( E ) ,  2 Further more, we could divide ˆ into two parts, ˆ  ˆ eq  ˆ neq , where. ˆ  . (15).  min Im  dEG R ( E )     , i  max   dEG ( E ) 2 min. (16). ˆ eq   ˆ. neq. 1. with  min  min(  r  Vr ,  r  Vr ) ,  max  max(  r  Vr ,  r  Vr ) . The reason is that for the energy range much lower than the Fermi level, the external bias could hardly change the energy states. Only the surface of the “Fermi Sea” is affected by the bias. The equilibrium part ˆ eq is related to the retarded Green’s function G R (E ) , which is analytic on the complex energy plane, while the non-equilibrium part ˆ neq could only be integrated on the real-axis of the complex energy plane, as Fig. 2. -9-.

(14) Fig.2 : A contour map on the complex energy plane E r , Ei  .. - 10 -.

(15) Method. The junction is composed of the left lead (LL), the right lead (RL) and the device. In the system we consider, both leads are made of crosswise-cut, same-species armchair carbon nanotubes (CNT’s), and the device is made of one or two polyene molecules. On the edges of the leads, except the few carbon atoms that make direct contact with the device, others are terminated by hydrogen atoms. Far from the vicinity of the device, the leads should stay as perfect CNT’s, while they should naturally distort near the edges, where the cuts and the bonding to the molecules take place. We do the followings to proceed the simulations: 1. Construct the structure of the ideal CNT. With the correct symmetry maintained, we perform the geometry relaxation via the Quantum Espresso (QE) package, where we use a plane-wave (PW) basis set and pseudopotentials for the GGA-PBE electronic density functional calculation, and the BEGF-method framework is used for the geometry relaxation. We define the extending dimension of the tube to be the z axis, and use a supercell that contains four unit cells of the (8,8) CNT. The x and y dimensions of the supercell are twice times the tube diameter, which is sufficiently large for the sake of lattice constant convergence. This relaxed structure is used for both leads, as two semi-infinite CNT’s, except the vicinity of the junction, which is treated in the next step. 2. Geometry relaxation of the device. Next we relax the central scattering region in our two-probe problem. This region consists of 3 final unit cells from each lead, and the molecule that sits between the leads. For the vicinity of each lead edge, the 2 unit cells that are closest to the perfect and semi-infinite lead remain perfect and un-relaxed, while the 1 unit cells that are closest to the molecule are relaxed along with the molecule. This central scattering region defines the supercell used in the QE relaxation. 3. NEGF-method in Nanodcal. After the structures are settled, we put the three parts of the junction together to proceed the calculation of electron transmission, again with the self-consistent field (SCF) method, based on the technique of - 11 -.

(16) non-equilibrium Green's functions (NEGF) formalism.. We apply the Nanodcal. package for the following: (a) SCF calculation for the bulk (infinite and perfect) CNT , to get the information for the left and right leads; (b) The result from (a) gives the self-energy term for the device region, and the SCF calculation is proceeded until the difference between the input and output effective potential at the lead-device boundary converges.. - 12 -.

(17) Results and Discussion (I) Transmission function In our current study, we consider the armchair (8,8) CNT as leads, and the polyene molecules as the device. In the tight-binding(TB) model, where each carbon atom (C) contributes effectively one p-orbital to form the  bond network, we assume a uniform energy t for the nearest neighbor hopping. Based on the NEGF method as described above, the effective Green’s function in the device region is 1.  E   L (E) t    t E      , G( E )        E t    t E   R ( E )  . (17). where self-energy terms come from the left and the right leads, and the hopping energy in the polyene molecules is the same as that in the CNT leads, for simplicity. By comparing the result from our ab initio calculation, and the TB analytic band structure, we find t =2.5 eV .. Fig. 3 : The values between two black lines are 0.766 t and 1.9121 eV , the hopping energy be determined as 2.5 eV .. - 13 -. t could.

(18) As shown in our band structure plot of the infinite CNT(8,8), there are two bands crossing the Fermi level (E=0), where the transmission coefficient T(E=0) is exactly two, from both the TB model and our ab initio study. The number of two is understood as the number of propagation channel, which also exactly corresponds to the number of bands crossing the energy level.. Fig. 4 : Transmission function (T(E)) of CNT (8,8) in TB model. Note that. - 14 -. t =2.5 eV ..

(19) Fig. 5 : Transmission function (T(E)) of CNT (8,8). in ab initio calculation.. We then study the transmission coefficient as the original infinite CNT is cut and reconnected by a junction made by simple polyene molecules. Note that except the contacts to the junction, the cut edges of CNT’s are terminated with hydrogen atoms. First, we consider a junction of one single polyene molecule. With the NEGF-DFT method programmed in Nanodcal, we obtain the transmission shown in Fig.6. The peak near the Fermi-level has a maximum value of one. This agrees with the result from the TB model, as the single polyene molecule restores effectively one channel for the cut CNT.. - 15 -.

(20) Fig. 6 : T(E) of single polyene junction in. ab initio calculation.. Fig. 7 : T(E) of single polyene junction in TB. - 16 -. model..

(21) Next we consider junctions made by two parallel polyene molecules. These junctions are categorized into two groups, according to their contacts along the armchair edge of the crossly cut CNT(8,8). Category 1 is named    , where both contact carbon atoms at either lead are of the same species (species named after the basis of the graphene unit cell). The rest cases are named    , whose contact carbon atoms at either lead are of different species.. Fig. 8 : The notation we used to distinguish different junctions. There are 32 carbons in a unit cell of (8,8) CNT, only 16 of them on the cutting-edge of lead. According to two different species in graphene unit cell, (4n  1)  (4m  1) is named  category will be (4n  1)  (4m) , ex:1-16.. - 17 -.   category, ex:1-17. The other   .

(22) Fig. 9 : The structure of device, after relaxation, which is 1-28 in our notation, belonged to category.. Two examples, each for one category, are shown below.. - 18 -.  .

(23) Fig. 10 : T(E) of 1-17 junction.. In cases of the    category, the central peak of the transmission does not exceed one. The interference of the two polyene molecules effectively give only one propagation channel.13. - 19 -.

(24) Fig. 11 : T(E) of 1-16 junction.. In cases of the    category, the value of the “central peak” depends. The interference between the odd and even channels varies with different contact, and can give numbers ranging from zero to two. We list all the    category results as following:. - 20 -.

(25) - 21 -.

(26) - 22 -.

(27) - 23 -.

(28) Fig. 12 : T(E) of all .   category junction.. Note that in the last two figures, obviously there are two peaks near the Fermi-level. According to TB model (Appendix), these two peaks could be understood as the “odd channel” and the “even channel”, respectively. In other words, they are the mathematical odd and even linear combinations of the originally irrelevant two polyene channels. With zero bias, the transmission at the Fermi-level is the result of the interference of the odd and the even channels. Also, note that the transmission of the polyene-molecule-reconnected CNT leads could not exceed the original T(E=0)=2 of the inifinite CNT.. 1-8. 1-12. 1-16. 1-20. 1-24. Fig. 13 : Comparisons with TB model and ab initio calculation.. - 24 -. 1-28.

(29) (II) Density of states As shown in both TB and ab initio results, the transmission of the junctions we present could be understood as the interference of two effective propagation channels. The interference of these two originally irrelevant channels is controlled by the off-diagonal matrix elements of the lead’s surface Green’s function, which relate the two contact carbon atoms. The resonance peaks in the transmission plot originate from the original molecular orbitals near the Fermi level. Let us use the TB model, and take the single polyene junction as an example. For a 7-carbon polyene, and Hamiltonian could be simply written as. 0  t 0  H  0 0  0  0. t 0 t 0 0 0 0. 0 0 0 0 0  t 0 0 0 0 0 t 0 0 0  t 0 t 0 0 , 0 t 0 t 0  0 0 t 0 t  0 0 0 t 0. (18). where t is the hopping energy, as before. This Hamiltonian could be easily diagonalized. The eigenvector corresponding to eigenvalue 0 is { -1, 0, 1, 0, -1, 0, 1 }, and is the only one that relates to the energy region of our discussion. Let us now turn to the ab initio results. The LDOS plots, projected to each carbon atoms, are shown in the right figure of Fig.14. Note that the first and last sub-figures correspond to the carbon atoms that sit on the leads and are the immediate neighbors of the polyene molecules. Their LDOS values are relatively small compared to those on the polyene.. - 25 -.

(30) Fig. 14 : LDOS of single polyene, the unit of energy (x-axis) is eV.. The DOS distribution results from our ab initio calculation shows that the transmission peaks indeed come from the central molecular levels (although one could not distinguish the positive and negative signs for the eigenvectors.) We list all the distribution of DOS (of polyene) of    category as following:. - 26 -.

(31) Fig. 15 : LDOS of 1-8, 1-12.. Fig. 16 : LDOS of 1-16, 1-20.. - 27 -.

(32) Fig. 17 : LDOS of 1-24, 1-28.. - 28 -.

(33) Conclusions. With zero bias, the transmission at the Fermi level T(E=0), could be understood as the interference between the two channels. The interference of different contact combinations comes from the symmetry of the original CNT structure. Our ab initio results, in the form of either transmission and LDOS, well agree with our TB results, which used only a few assumptions and one single parameter to show a simple picture that explains the transport properties in our molecular junctions models. It is astonishing, that with all the genuine structure relaxations that we perform in our ab initio study to mimic what would actually happen in reality, such interference phenomenon is still prominent and not to be smeared off.. - 29 -.

(34) Appendix 13. We should now illustrate the two-polyene junction problem with the TB model. We only need to look at the subspace related to the contact sites. Since there are two contact points at each side, the representation of the problem reduces to 2  2 matrices. Based on NEGF method, we could write down the embedding self-energies from left or right lead as follows:.  0  g si,i   i , j   0    g sj,i. g si. j  0   , g sj, j  0  . (19). where  is the coupling between the lead and the polyenes, g s is the surface Green’s function of the CNT-lead, and the sub-indices indicate the match of the contact sites. For the current purpose, we set EF  0 for the following discussion. For the    cases, the surface Green’s function reads:   i  i   , g s ( E  0)      i  i  on the other hand, the    cases have:.   i   , g s ( E  0)      i    where  and  are real numbers whose exact forms are not shown here.. (20). (21). Now one could easily check the matrix elements of the    cases. This matrix is actually a rank-1 matrix, although its size is two-by-two. This is also the reason that the    transmission would not exceed one. Based on the theory of Linear Algebra, we could choose a new basis sets, which are defined as “odd channel” and “even channel”. Under this new basis, the matrix of    case could be presented as:   2i 0  , g s ' ( E  0)    (22) 0   0 and there is only one effective channel that contributes to transmission. With the similar treatment, in the new basis the    cases have:. - 30 -.

(35) 0    i   , (23) g s ' ( E  0)      i     0 and these two effective channels interfere each other, the interference phenomenon is reflected in the transmission.. - 31 -.

(36) References. 1. Iijima, Sumio. Nature 354 (6348): 56–58. (1991). 2. Ebbesen, T. W.; Ajayan, P. M. Nature 358 (6383): 220–222. (1992). 3. K. Tsukagoshi1, I.Yagi, and Y. Aoyagi, Appl. Phys. Lett. 85, 1021 (2004). 4. Guo et al., Science 311 (5759) (2006). 5. C.-C. Kaun, B. Larade, H. Mehrez, J. Taylor, and H. Guo, Phys. Rev. B 65, 205416 (2002).. 6. W. Ren and J. Wang, Phys. Rev. B 69, 033306 (2004). 7. Wei Ren, Jeffrey R. Reimers, Noel S. Hush, Yu Zhu, Jian Wang, and Hong Guo, J. Phys. Chem. C. 111, 3700 (2007). 8. Nicolas A. Bruque, Rajeev R. Pandey, and Roger K. Lake, Phys. Rev. B 76, 205322 (2007). 9. San-Huang Ke, Harold U. Baranger, and Weitao Yang, Phys. Rev. Lett. 99, 146802 (2007). 10. Joachim A. Fürst, Mads Brandbyge, Antti-Pekka Jauho, and Kurt Stokbro, Phys. Rev. B 78, 195405 (2008). 11. N. A. Bruque, M. K. Ashraf, G. J. O. Beran, T. R. Helander, and R. K. Lake, Phys. Rev. B, 80, 155455 (2009). 12. M. K. Ashraf, Nicolas A. Bruque, Jeremy L. Tan, Gregory J. O. Beran, and Roger K. Lake, J. Chem. Phys. 134, 024524 (2011). 13. Yiing-Rei Chen ,Lei Zhang ,Mark S. Hybertsen Phys. Rev. B 76, 115408 (2007). 14. S. Datta, Superlattices and Microstructures, Vol. 28, No. 4 (2000). 15. Ferdows Zahid, Magnus Paulsson and Supriyo Datta , "Electrical Conduction in Molecules," chapter in Advanced Semiconductors and Organic Nano-Techniques, ed. H. Morkoc, Academic Press , ISBN: 0-12-507060-8 (2003). 16. P. Hohenberg W. Kohn. 17. W. Kohn and L. J. Sham Phys. Rev. 140, A1133–A1138 (1965). 18. S. Datta, Electronic Transport in Mesoscopic Systems (Cambridge, 1995, Paperback Edition, 1997). 19. S. Datta, Quantum Transport: Atom to Transistor, Cambridge (2005). 20. Jeremy Taylor, Hong Guo, and Jian Wang, Phys. Rev. B 63, 245407 (2001). Phys. Rev. 136, B864–B871 (1964). - 32 -.

(37)