奈米碳管電極之間分子結的電子傳輸研究

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(1)國立臺灣師範大學物理研究所碩士論文. 奈米碳管電極之間分子結的電子傳輸研究. Electron Transport Study of Molecular Junctions in between Carbon Nanotube Electrodes. 研究生：林明寬指導教授：陳穎叡博士. 中華民國 106 年 7 月.

(2) 致謝首先要感謝我的指導教授陳穎叡老師的指導，協助我解決物理和程式方面的問題，並且在修改論文和口試模擬的部分給了寶貴的建議。還要感謝中研院應科中心的關肇正老師和他的助理竇坤鵬協助我解決在操作 SIESTA 和 Nanodcal 軟體時遇到的問題。在口試當天，感謝陳穎叡老師、關肇正老師和胡淑芬老師對本篇論文的指導與建議。另外，感謝俊凱和晴羽在口試當天的幫忙。最後要感謝我的家人給我的幫助與支持，讓我順利完成碩士學位。. i.

(3) 摘要本篇論文以斜切的 armchair 奈米碳管(carbon nanotube)作為分子結(molecular junction)中的電極。使用緊密束縛模型(tight-binding model)計算斜切的 armchair 奈米碳管、直切的 armchair 奈米碳管和直切的 zigzag 奈米碳管從表面到內部的局域態密度(local density of states)。直切的 armchair 奈米碳管和直切的 zigzag 奈米碳管的每一層局域態密度分別顯示三層循環的週期性振盪和局域的邊緣態 (edge state)。斜切的 armchair 奈米碳管不只具有週期性振盪，也具有局域的邊緣態。在局域態密度的研究之後，我們把一條或兩條多烯(polyene)接在兩個斜切的 armchair 奈米碳管之間作為分子結。使用緊密束縛模型和第一原理(ab initio)方法研究分子結的電子傳輸性質。One-polyene 分子結在費米能量(Fermi energy)的傳輸(transmission)數值接近 1，所以它恢復了一條電子傳輸通道。Two-polyene 分子結在費米能量的傳輸數值在 0 和 2 之間變化，所以它顯示了干涉效應。儘管緊密束縛模型和第一原理的結果大致相同，但是從這兩種方法得到的結果還是有不一致之處。藉由調整緊密束縛模型中參數的大小，研究分子結的傳輸性質如何變化。我們發現分子結的傳輸性質會受到來自於分子內的鍵結(intra-molecular bonding) 強度、耦合(coupling)強度和 on-site energy 的影響。. 關鍵字：奈米碳管、分子結、緊密束縛模型、第一原理 ii.

(4) Abstract In the thesis, we study the angled-cut armchair carbon nanotubes (CNTs) taken as the electrodes in a molecular junction. Using a one-parameter tight-binding model, we investigate the local density of states (LDOS) from the edge to the interior of the angled-cut armchair CNTs, and similarly for the cross-cut armchair CNTs and the cross-cut zigzag CNTs. We find the periodic oscillation of a 3-layer-cycle and the localized edge states from the layer-by-layer LDOS of the cross-cut armchair CNTs and the cross-cut zigzag CNTs, respectively. The angled-cut armchair CNTs possess not only the periodic oscillation but also the localized edge states. Following the LDOS study, we consider one or two polyenes bridging two angled-cut armchair CNTs as our molecular junction system. We use both the tight-binding model and the ab initio approach to study the electronic transport properties of these molecular junctions. For the one-polyene cases where the polyene has odd number of carbon atoms, the value of the transmission reaches one at the Fermi energy, meaning that one electronic channel is restored by the polyene. For the two-polyene cases, the value of the transmission varies between zero and two at the Fermi energy, which shows the interference effect. Despite the qualitative agreement between the tight-binding and the ab initio results, we look into the discrepancies between the transmissions obtained from the two methods. Tuning the magnitude of the parameters in the tight-binding model, we investigate the corresponding responds in the transmission. We find that the transmission of the molecular junction suffers the influences from the intra-molecular bonding strength, the coupling strength, and the on-site energy.. Keywords: carbon nanotube, molecular junction, tight-binding model, ab initio iii.

(5) Contents 致謝............................................................................................... i 摘要.............................................................................................. ii Abstract ...................................................................................... iii Chapter 1. Introduction .......................................................... 1. Chapter 2. Theory .................................................................... 3. 2.1 Quantum transport ...................................................................3 2.1.1 Green’s function ............................................................................... 3 2.1.2 Surface Green’s function ................................................................. 4 2.1.3 Density matrix .................................................................................. 5 2.1.4 Transmission function ..................................................................... 6. 2.2 Density functional theory (DFT) .............................................7 2.3 Quantum transport calculations using Nanodcal ..................9 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5. Two-probe device model .................................................................. 9 The procedure of the lead calculation in Nanodcal ...................... 9 Boundary condition ....................................................................... 10 Green’s function of the scattering region..................................... 11 Density matrix of the scattering region ........................................ 12. Chapter 3. Method ................................................................. 14. 3.1. The layer-by-layer LDOS of the semi-infinite carbon nanotube..................................................................................14 3.1.1 Introduction to the system............................................................. 14 3.1.2 Calculation using the tight-binding model .................................. 16. 3.2 The transmission of the polyene junctions in between angled-cut armchair carbon nanotube leads.......................18 3.2.1 Introduction to the system............................................................. 18 3.2.2 Tight-binding model calculations ................................................. 21 3.2.3 Ab initio calculations ..................................................................... 22 3.2.4 The contributions from the even and odd channels in the two-polyene junctions .................................................................... 24 iv.

(6) Chapter 4. Result and Discussion......................................... 26. 4.1 The semi-infinite carbon nanotube .......................................26 4.1.1 The LDOS of the outermost layer ................................................ 26 4.1.2 The layer-by-layer LDOS at the Fermi energy ........................... 33. 4.2 The polyene junctions in between angled-cut armchair carbon nanotube leads ...........................................................38 4.2.1 The band structure and the transmission of the armchair carbon nanotube bulk .................................................................... 38 4.2.2. 4.3. The transmission of the polyene junctions in between angled-cut armchair carbon nanotube leads ............................... 40. The discrepancy of the transmission between the tight-binding model and the ab initio...................................44 4.3.1 4.3.2 4.3.3 4.3.4. The intra-molecular hopping energy ........................................... 45 The contact coupling ...................................................................... 46 The unbalanced contact couplings ............................................... 47 The on-site energy .......................................................................... 48. Chapter 5. Conclusion ........................................................... 53. References ................................................................................. 54. v.

(7) Chapter 1. Introduction. The carbon nanotubes were first discovered by Iijima in 1991 [1]. Then, the single-wall carbon nanotubes (CNTs) were synthesized in 1993 [2,3]. Depending on the direction of the chiral vector, the CNT can be categorized into three types: armchair, zigzag, and chiral [4]. The electronic properties of the CNT depend on the diameter and the helicity of the tube [5]. When defects appear, the electronic properties of the CNT can be modified [6,7]. The capped CNT is an example of defects, and exhibits localized states at the end of the tube near the Fermi energy (𝐸𝐹 ) [8-10]. Moreover, the graphene ribbons with a zigzag-shaped edge also have localized states along the edge at the vicinity of the 𝐸𝐹 [11-14]. Therefore, the frontier electronic states of these carbon materials drastically change because of the presence of the edge. When a CNT bulk is cut to become semi-infinite, an edge is formed at the incision. In our study, we consider the angled-cut armchair CNT and take it as the electrode in a single- (or multi-) molecule junction. An angled-cut armchair CNT is a semi-infinite CNT, whose tube edge exhibits a zigzag rim. To understand how the formation of the edge affects the electronic states of the angled-cut armchair CNT, we investigate the local density of states (LDOS) from the edge to the interior of the angled-cut armchair CNT. Then, we also calculate the layer-by-layer LDOS for the cross-cut armchair CNT and the cross-cut zigzag CNT to compare with the result of the angled-cut armchair CNT. We then consider a single molecule that bridges the gap between two CNT leads, and study the transmission of such an electronic device. Experimentally, the single-wall CNT is treated by ion etching to give a slot [15-21]. Then, the junction between the CNT lead and the molecule has been fabricated through the formation of the amide linkage [17]. Moreover, the electronic transport properties of the CNT-molecule-CNT junction have been simulated theoretically [22-29]. Our study is the extension of the previous study which focuses on the electronic transport properties of the polyene junctions bridged between two cross-cut armchair CNT leads [30,31]. In this study, we investigate the electronic transport properties of the polyene junctions bridged between two angled-cut armchair CNT leads. From the band structure of the armchair CNT bulk, we know that it has two bands serving as electronic channels at the vicinity of the 𝐸𝐹 . When the armchair CNT bulk is etched to become two angled-cut armchair CNT leads, its electronic channels are broken. As the gap between the two angled-cut armchair CNT leads is bridged by the molecule, in a sense the electronic channels are restored to some extent. To form the good 1.

(8) linkage between the molecule and the CNT lead in the molecular junction, we choose the molecule which is similar to the bonding within the CNT lead. According to the previous study, the one-polyene junctions can restore an electronic channel near the 𝐸𝐹 , and the two-polyene junctions show the interference effect [30,31]. In this study, we will also consider the polyene molecule(s) for the molecular junction. We use both the tight-binding model and the ab initio calculations to investigate the electronic transport properties of the polyene junctions bridging two angled-cut armchair CNT leads. With the tight-binding model, we first construct the Hamiltonian, and then obtain the surface Green’s function of the lead via the iterative method, and finally calculate the transmission of the molecular junction by the non-equilibrium Green’s function method [32,33]. In the ab initio calculation, we first optimize the geometry structure of the molecular junction with the SIESTA package [34], and then perform the transport calculation of the molecular junction with the Nanodcal package [35]. Finally, the ab initio result is compared with the tight-binding model result. In the ab initio calculation, the geometry relaxation pursues the actual bonding and orbital hybridization, and reveals the distortion at the CNT cut. Therefore, the discrepancies between the transmissions obtained from both methods bring about more interesting discussions. To get an insight from these discrepancies, we tune the magnitude of the parameters in the tight-binding model to understand how the transmission of the molecular junction is affected by these parameters, and therefore the reasons that cause the discrepancies.. 2.

(9) Chapter 2. Theory. 2.1 Quantum transport [32,33] 2.1.1 Green’s function. Fig. 2.1: A schematic plot of a device connecting two leads. The system we consider is a device connecting two leads. The device has a finite structure, while the leads are semi-infinitely periodic. The Schrödinger equation of the two isolated leads can be written as (1) [ EI  H L  i ]{ L }  {S L } , (2) [ EI  H R  i ]{ R }  {S R } , where 𝐻𝐿(𝑅) is the Hamiltonian of the left (right) lead, Φ𝐿(𝑅) is the wavefunction of the left (right) lead, and 𝑆𝐿(𝑅) is the source from the left (right) lead. We then write down the Schrödinger equation of the full system in matrix form:.  EI  H L  i   L   0 .   L EI  H D .  R.   L   L   S L       R    0 ,    EI  H R  i   R   R  S R  0. (3). where 𝐻𝐷 is the Hamiltonian of the device, 𝜓 is the wavefunction of the device, 𝜒𝐿(𝑅) is the scattered wave in the left (right) lead, and 𝜏𝐿(𝑅) is the coupling between the left (right) lead and the device. From Eq. (3), we obtain { L }  GL L { } , (4)  { R }  GR R { } , (5) (6) [ EI  H D   L   R ]{ }  {S}. In Eq. (4) and Eq. (5), 𝐺𝐿(𝑅) is the Green’s function of the left (right) lead, and it can be represented as. GL ( R )  [ EI  H L ( R )  i ]1 . 3. (7).

(10) In Eq. (6), 𝛴𝐿(𝑅) is the self-energy from the left (right) lead, and it can be written as.  L ( R )   L ( R )GL ( R ) L( R ) .. (8). Then, we represent the broadening via the self-energy:. L ( R )  i[ L ( R )   L ( R ) ] .. (9). Finally, we define the Green’s function of the device from Eq. (6): G  [ EI  H D   L   R ]1 .. (10). 2.1.2 Surface Green’s function. Fig. 2.2: The semi-infinitely periodic structure is cut into the layers. We cut the semi-infinitely periodic structure into the layers of identical supercells. Layer 0 is recognized as the A-subspace, and the remaining layers are taken as the B-subspace. The Green’s function of the system can be represented as GAB   EI  H A  i G G   AA    VAB GBA GBB  .  VAB. 1.  , EI  H B  i . (11). where the 𝑉𝐴𝐵 matrix describes the coupling between the A-subspace and the B-subspace. From Eq. (11), we can obtain the projection of 𝐺 onto the A-subspace:  1 G AA  [(G A ) 1  VAB GBVAB ] , (12) where 𝐺𝐴(𝐵) is defined as GA( B )  [ EI  H A( B )  i ]1 .. (13). Because layer 0 only connects with layer 1 in the B-subspace, the non-zero matrix elements of 𝑉𝐴𝐵 only occur in the specific region. By separating layer 1 (denoted by b) from the B-subspace, Eq. (12) can be reduced to. GAA  [(GA ) 1  VAbGbVAb ]1 , 4. (14).

(11) where 𝐺𝑏 is the projection of 𝐺𝐵 onto layer 1, and 𝑉𝐴𝑏 is the coupling between layer 0 and layer 1. However, due to the symmetry of the semi-infinite periodicity, 𝐺𝐴𝐴 = 𝐺𝑏 , and therefore we rewrite Eq. (14) as. g s  [  g s   ]1 ,. (15). where g𝑠 is the surface Green’s function. In the system of a device sandwiched between two leads, the device only connects with the surface of the two leads. We now apply the concept of the surface Green’s function to Eq. (8) and rewrite it as.  L ( R )   L ( R ) g sL( R ) L( R ) ,. (16). where g𝑠𝐿(𝑅) is the surface Green’s function of the left (right) lead.. 2.1.3 Density matrix The system which we mention above is the non-equilibrium problem, so we next discuss the non-equilibrium density matrix. Before starting the main topic, we first introduce two notations, namely, the correlation function −𝑖𝐺 < and the in-scattering −𝑖𝛴 <:. where 𝑓𝐿(𝑅).  iG   G n  G inG  ,  i    in  L f L  R f R , is the Fermi function of the left (right) lead: 1 f L ( R ) ( E;  L ( R ) )  . exp[( E   L ( R ) ) / k BTe ]  1. (17) (18). (19). The non-equilibrium density matrix is defined as. ˆ . 1 2. . . . dEG n ( E ) .. (20). Then, we define the spectral function of the left (right) lead as. AL ( R )  i[GL ( R )  GL( R ) ]  GL ( R )G  . If we substitute Eq. (18) into Eq. (17), we can obtain G n  AL f L  AR f R .. (21). (22). In this thesis, we consider infinitely small bias, i.e., almost identical 𝑓𝐿 and 𝑓𝑅 , and 𝑓𝐿 = 𝑓𝑅 = 𝑓(𝐸; 𝜇). Therefore, in our case, Eq. (22) is rewritten as. G n ( E )  A( E ) f ( E;  ) , where 𝐴 is the total spectral function: A  i[G  G  ]  AL  AR . 5. (23). (24).

(12) In the case of zero bias, the equilibrium density matrix can be written as. ˆ . 1 2. . . . dEA( E ) f ( E;  ) .. (25). Finally, we represent the density of states (DOS) with the total spectral function: DOS ( E ) . 1 1 Tr[ A( E )]   Tr{Im[G ( E )]} , 2 . (26). and the individual diagonal elements of the spectral function give the local density of states (LDOS).. 2.1.4 Transmission function In the non-equilibrium problem, the device is attached to two leads with two different Fermi functions 𝑓𝐿 and 𝑓𝑅 . The difference in the chemical potential between the two leads causes a current flow. To obtain the form of the current, we write down the time-dependent Schrödinger equation in the matrix form:.   L   L   H L  i d    i      L dt    R   R   0.  L HD. .  R.   L   L   S L       R      0  . H R  i   R   R  S R  0. (27). Using the following equation: I. d   . dt. (28). With the previous derivations, the current can be written as I. 2q  dET ( E )[ f L  f R ] , h . where 𝑇(𝐸) is the transmission function defined by T ( E )  Tr (L GR G  ) .. 6. (29). (30).

(13) 2.2 Density functional theory (DFT) [36] In 1964 [37], Hohenberg and Kohn’s paper proposed the Hohenberg-Kohn theorem. In 1965 [38], Kohn and Sham’s paper derived the Kohn-Sham equation. The heart of DFT consists of the Hohenberg-Kohn theorem and the Kohn-Sham equation. In DFT, the ground state energy of a system can be represented as a functional of the electron density 𝜌(𝑟). Moreover, other ground state physical quantities of the system are also functionals of the electron density 𝜌(𝑟). In a many-electron problem, the energy functional can be written as E[  ]  T [  ]  U [  ]  V [  ] , (31) where the right-hand-side terms correspond to the kinetic energy of interacting electrons, the electron-electron interaction energy, and the potential energy, respectively. The kinetic energy 𝑇 can be divided into two parts:. T [  ]  Ts [  ]  Ec [  ] ,. (32). where 𝑇𝑠 is the kinetic energy of noninteracting electrons, and 𝐸𝑐 is the correlation energy, namely, the leftover part not included in 𝑇𝑠 . Similarly, the interaction energy 𝑈 is also divided into two parts. Then, we can obtain. U [  ]  U H [  ]  Ex [  ] ,. (33). where 𝑈𝐻 is the Hartree energy that includes only the Coulombic repulsion, and 𝐸𝑥 is the exchange energy. We define the exchange-correlation energy 𝐸𝑥𝑐 as the sum of the exchange energy 𝐸𝑥 and the correlation energy 𝐸𝑐 , i.e.,. E xc [  ]  E x [  ]  Ec [  ] ,. (34). and the energy functional now becomes. E[  ]  Ts [  ]  U H [  ]  V [  ]  Exc [  ] .. (35). We use Eq. (35) to acquire the Kohn-Sham energy functional:   1    (r )  (r ' )     E[  ]  Ts [  ]   dr dr '     dr  (r )[vext (r )  vione (r )]  E xc [  ] , (36) 2 r  r' where the right-hand-side terms correspond to the kinetic energy of noninteracting electrons, the Hartree energy, the external potential energy, the interaction potential energy between ion cores and electrons, and the exchange-correlation energy, respectively. Then, the relation between the wavefunction 𝜓 and the electron density 𝜌(𝑟) can be represented as    (r )   [  (r )] . (37) 7.

(14) When we apply the variational principle to Eq. (36), we can derive the Kohn-Sham equation:   1 2   (r ' )          dr '    vext (r )  vione (r )  vxc (r ) i (r )   i i (r ) , r  r'  2 . (38). where 𝑣𝑒𝑥𝑡 is the external potential, 𝑣𝑖𝑜𝑛 −𝑒 is the interaction potential between ion cores and electrons, and 𝑣𝑥𝑐 is the exchange-correlation potential. The exchange-correlation potential 𝑣𝑥𝑐 can be written as  E [  ] (39) vxc (r )  xc  .  (r ) If the form of the exchange-correlation energy 𝐸𝑥𝑐 of the homogenous free electron gas is adapted the local density approximation (LDA):.   Exc [  ]    xc [  ] (r )dr .. (40). To solve the Kohn-Sham equation, we apply the self-consistent calculation. The calculation first guesses an initial electron density 𝜌(𝑟). After the Kohn-Sham Hamiltonian is constructed by this initial electron density 𝜌(𝑟), the Kohn-Sham equation is solved to obtain a set of the wavefunctions 𝜓𝑖 . A new electron density is calculated by   2  ' (r )   f ( Ei ;  ) i (r ) , (41) i. where 𝑓 is the Fermi function, and 𝜇 is the chemical potential. Then, the new electron density 𝜌′(𝑟) is compared with the old electron density 𝜌(𝑟), and the next iteration starts if their difference exceed a set-up threshold. The iterative process continues until the convergence criteria are met. Finally, the ground state energy is obtained, and other physical quantities are calculated from the converged electron density.. 8.

(15) 2.3 Quantum transport calculations using Nanodcal [35] 2.3.1 Two-probe device model. Fig. 2.3: A schematic plot of the two-probe device model. In Nanodcal, a device can be connected to two or more leads. The system we now consider is a molecule bridging two semi-infinite leads and can be divided into three parts: the left lead, the scattering region, and the right lead. The left lead and the right lead extend to −∞ and +∞ in the z direction, respectively. The scattering region contains not only the molecule but also several layers of the lead atoms. When a bias voltage is applied to the leads, the system is at a non-equilibrium state. DFT has long been used to look at electron structures of finite and periodic systems. In order to treat problems of open and non-equilibrium system like ours, the DFT self-consistent field theory is combined with the non-equilibrium Green’s function (NEGF). Then, we use this approach which is called NEGF-DFT to study the quantum transport properties of our open system.. 2.3.2 The procedure of the lead calculation in Nanodcal In Nanodcal, the basis set of the linear combination of atomic orbital (LCAO) is adopted:      (r )  Rl (| r  RI  |)Yl m (rRI ) , (42) . where 𝑙𝜇 ,𝑚𝜇 are the quantum numbers, and 𝑅𝐼𝜇 is the atomic site of the atom 𝐼. With this basis set, the Hamiltonian of the lead is represented in the matrix form:. 9.

(16)       hz , z 1 hz , z hz , z 1  ,   hz , z 1 hz , z hz , z 1       where the sub-Hamiltonian matrix hz , z '   z H  z '. (43). represents the coupling between. unit cells with labels 𝑧 and 𝑧′ along the transport axis. Since the bulk lead is periodic along the z direction, the wavefunction can be described as the Bloch wave:.  k ( z )  eikzk ( z ) .. (44). Then, the Schrödinger equation can be written as. (hz , z1eik  hz , z  hz ,z 1eik )k  Ek (sz , z1eik  sz , z  sz , z1eik )k , where sz , z '   z  z '. (45). is the overlap matrix. After we choose a given k value to. diagonalize Eq. (45), we can obtain the eigenvalues 𝐸𝑘 and the eigenfunctions 𝜙𝑘 . Then, we can calculate the density matrix of the lead:. ˆ   dk  k f ( Ek ;  )  k .. (46). When the self-consistent calculation is reached, we can obtain the effective potential of the lead. Therefore, we can apply the boundary conditions Eq. (47) and Eq. (48).. 2.3.3 Boundary condition As mentioned previously, the two-probe device is divided into three parts. In the scattering region, the molecule is sandwiched between two clusters of atoms from the lead edges. The number of layers from the lead edges must be adequate so that the effective potential 𝑉 𝑒𝑓𝑓 (𝑟) at the boundary of the scattering region matches the effective potential deep into the bulk lead, namely, along the transport axis z:.   VLeff (r )  VLeff,bulk (r ), z  z L    V eff (r )   VCeff (r ), zL  z  zR ,   eff eff V (r )  V z  zR R ,bulk ( r ),  R. (47). where 𝑧 = 𝑧𝐿(𝑅) is the plane between the left (right) lead and the scattering region. Then, the above boundary condition is applied to the Hartree potential:.   VCH (r ) | zL  VLH,bulk (r ) | zL .   VCH (r ) | zR  VRH,bulk (r ) | zR . 10. (48).

(17) Note that the Hartree potential can be obtained by solving the Poisson equation, and convergence to the boundary condition can be achieved iteratively. When Eq. (48) is established, the charge densities from different zones agree at 𝑧 = 𝑧𝐿(𝑅) : . .  C (r ) | z   L ,bulk (r ) | z ,    C (r ) | z   R ,bulk (r ) | z , L. L. R. R. (49). and the effective potentials also agree at 𝑧 = 𝑧𝐿(𝑅) . Practically, the effective potential and the charge density in the scattering region are calculated iteratively to meet the boundary condition obtained from the lead calculation.. 2.3.4 Green’s function of the scattering region The Hamiltonian of the two-probe device can be expanded in terms of the LCAO basis sets:. H total.  hL , h  L ,L1  0   0  0 . hL ,L1. 0. 0. hL ,L. hL ,C. 0. hC ,L. hC ,C. hC ,R. 0. hR ,C. hR ,R. 0. 0. hR ,R1. 0  0  0 ,  hR ,R1  hR , . (50). where ℎ𝐿,∞ and ℎ𝑅,∞ are the Hamiltonians of the semi-infinite leads, and they are defined as. hL,.    0      0 hL ,L1 hL,L hL ,L1  ,  0 0 hL ,L1 hL ,L . (51). hR ,.  hR ,R  hR ,R1  0. (52). hR ,R1 hR ,R. 0 hR ,R1. . . 0 0  . . Then, we can write down the Green’s function of the two-probe device:. Gtotal.  hLE,  E hL ,L1  0   0  0 . hLE,L1. 0. 0. hLE,L hCE,L. hLE,C hCE,C. 0. 0. E R ,C. h. 0. 0. where the matrix element is defined as 11. hCE,R E R ,R E R , R 1. h h. 1. 0   0  0  ,  hRE,R1  hRE, . (53).

(18) hiE, j  ( E  i ) I  hi , j .. (54). When we use the concept of the self-energy to fold-in the contribution of the two leads into the Hamiltonian of the scattering region, we can obtain the Green’s function of the scattering region: hLE,L   LL ,L  GC   hCE,L  0 . hLE,C hCE,C E R ,C. h. 1.   hCE,R  , hRE,R   RR ,R  0. (55). 𝐿 𝑅 where 𝛴𝐿,𝐿 and 𝛴𝑅,𝑅 are the self-energies from the left and right leads, respectively.. They are defined as.  LL ,L  hLE,L1 g L , hLE,L1 ,  RR ,R  hRE,R1 g R , hRE,R1 ,. (56). where the surface Green’s functions g𝐿,∞ and g𝑅,∞ can be written as. g L ,  [( E  i ) I  hL , ]1. g R ,  [( E  i ) I  hR , ]1.. (57). Once the Green’s function of the scattering region is known, we can use the NEGF method to calculate the density matrix of the scattering region.. 2.3.5 Density matrix of the scattering region When the bias voltage ∆𝑉𝐿(𝑅) is applied to the left (right) lead, the electrochemical potentials of the left lead and the right lead are 𝜇𝐿 + ∆𝑉𝐿 and 𝜇𝑅 + ∆𝑉𝑅 , respectively, where 𝜇𝐿(𝑅) is the chemical potential of the left (right) lead. The difference between 𝜇𝐿 + ∆𝑉𝐿 and 𝜇𝑅 + ∆𝑉𝑅 across the two-probe device makes a non-equilibrium problem, and calls for the non-equilibrium density matrix within the NEGF method. When the Green’s function is obtained, we can begin to calculate the correlation function G n  G inG  ,. (58).  in  L f L  R f R .. (59). where Before calculating the non-equilibrium density matrix, we first discuss a special example. At the absolute zero temperature 𝑇𝑒 = 0, the Fermi functions of two leads are reduced to (60) f L ( E; L  VL )  f R ( E; R  VR )  1 , 12.

(19) and Eq. (58) can be rewritten as. G n  G[L  R ]G   AL  AR  A  i[G  G  ]  2 Im[G] .. (61). The non-equilibrium density matrix is defined as. ˆ . 1 2. . . . dEG n ( E ) .. (62). We then divide Eq. (62) into two parts. One is the equilibrium charge contribution. ˆ eq which satisfies Eq. (60), and the other is the non-equilibrium charge contribution ˆ neq which doesn’t satisfy Eq. (60). Therefore, the non-equilibrium density matrix is rewritten as. ˆ  ˆ eq  ˆ neq  . min 1 Im   dEG( E )     2   . 1. max. . dEG n ( E ) ,. (63). min. where.  min  min(  L  VL ,  R  VR ).  max  max(  L  VL ,  R  VR ).. (64). As the NEGF calculation reaches self-consistency, we may continue for the study of other physical quantities.. 13.

(20) Chapter 3. Method. 3.1 The layer-by-layer LDOS of the semi-infinite carbon nanotube 3.1.1 Introduction to the system A single-wall carbon nanotube (CNT) is a hollow cylinder formed by a roll-up graphene sheet. Depending on the direction of the chiral vector 𝑪𝒉 = 𝑛𝒂𝟏 + 𝑚𝒂𝟐 ≡ (𝑛, 𝑚), the CNT can be categorized into three types: armchair (𝑛, 𝑛), zigzag (𝑛, 0) and chiral (𝑛, 𝑚), where 𝒂𝟏 and 𝒂𝟐 are the lattice vectors of the graphene [4]. For our current study, we focus on the armchair and zigzag categories. The structures of the two cases are illustrated in Fig. 3.1.. Fig. 3.1: (a) Armchair CNT bulk. (b) Zigzag CNT bulk. The rectangle frame denotes a unit cell in the CNT bulk. We first consider the electrodes made of cross-cut armchair CNT and cross-cut zigzag CNT, respectively. The cross-cut armchair CNT exhibits an armchair-shaped edge, and the cross-cut zigzag CNT exhibits a zigzag-shaped edge, exactly the way they are named. A zigzag rim is also available from an angled-cut armchair CNT which is what we consider next. Then, in all cases we consider, the tube edges of these semi-infinite CNTs are terminated by hydrogen atoms, so the dangling bonds are saturated. These CNT electrodes are illustrated in Fig. 3.2.. 14.

(21) Fig. 3.2: (a) Cross-cut armchair CNT. (b) Cross-cut zigzag CNT. (c) Angled-cut armchair CNT.. 15.

(22) 3.1.2 Calculation using the tight-binding model In the tight-binding model, each carbon atom in the semi-infinite CNT contributes one 𝑝 orbital to the 𝜋 system. We only consider the nearest neighbor hopping and assume the same hopping energy 𝑡 throughout the semi-infinite CNT. The on-site energy of each carbon atom is equal to the Fermi energy (𝐸𝐹 ) of the semi-infinite CNT, where 𝐸𝐹 ≡ 0. To investigate the layer-by-layer LDOS of the semi-infinite CNT, we cut the semi-infinite CNT into the layers of identical supercells, as shown in Fig. 3.3.. Fig. 3.3: (a) Cross-cut armchair CNT. (b) Cross-cut zigzag CNT. (c) Angled-cut armchair CNT.. 16.

(23) As illustrated in Fig. 3.3, the semi-infinite CNT is divided into two parts. The layers in the range from the edge to some depth in the interior of the semi-infinite CNT are taken as the device, and the remaining layers, extending to infinity, are considered as the lead. The Hamiltonian of the semi-infinite CNT can be represented as. H total. H   lead  .  . . HD . First we calculate the surface Green’s function of the lead. We apply the following equation to do the iterative calculation:. gs  [( E  i ) I  Hlead  gs   ]1 . In the iterative calculation, we choose a small value of 𝜂 (10−7 ) and the criterion on the norm of the recursion error (10−10 ) to achieve the desired accuracy of the calculation. Once the surface Green’s function is obtained, the self-energy from the lead is calculated by   g s  .. Then, we can obtain the Green’s function of the device from G  [( E  i ) I  H D  ]1 . We use Eq. (26) to calculate the DOS in the device. According to the definition of a layer, we sum up the LDOS of all carbon atoms in the layer, and thus obtain the layer-by-layer LDOS of the semi-infinite CNT.. 17.

(24) 3.2 The transmission of the polyene junctions in between angled-cut armchair carbon nanotube leads 3.2.1 Introduction to the system We etch a (𝑛, 𝑛) CNT bulk to become two angled-cut (𝑛, 𝑛) CNT leads. The tube edges of the two leads are terminated by the hydrogen atoms. There are 2𝑛 contact sites along each lead edge. These contact sites are labeled by 2𝑙 + 1 and 2(𝑛 + 𝑙 + 1), where 𝑙 = 0,1, ⋯ , 𝑛 − 1, as shown in Fig. 3.4. The gap between two leads is then bridged by the polyene molecule(s). In our study, we consider one- and two-polyene cases for the molecular junction, and for the convenience of the ab initio calculation, we mainly consider 𝑛 = 8 for the CNT leads.. Fig. 3.4: The labels of the sites on the rim of an angled-cut (8,8) CNT lead. The contact sites are emphasized by solid circles. We first consider the one-polyene junctions. We use one polyene which possesses 17 carbon atoms to bridge the gap between two leads. Because of the symmetry of the leads, there are effectively 8 cases. The contact combination can be represented as 2𝑙 + 1, where 𝑙 = 0,1, ⋯ ,7. These cases are illustrated in Fig. 3.5.. 18.

(25) Fig. 3.5: One-polyene junctions. We use the labels of the contact sites to represent each case.. 19.

(26) As for the two-polyene junctions, we use two identical polyenes, each with 17 carbon atoms, to bridge the gap between the two leads. In our study, the two bridging polyenes are parallel to the z direction of the leads. With the contact site labels just introduced, each combination of the two contact sites is labeled by a pair of numbers. All independent combinations are illustrated in Fig. 3.6. Note that we have excluded the cases of combinations (15,18) and (1,32), where the two bridging polyenes are too close and cannot be realized in the ab initio calculations.. Fig. 3.6: Two-polyene junctions. We use the labels of the contact sites to represent each case.. 20.

(27) 3.2.2 Tight-binding model calculations In the simplest tight-binding model, we use the same hopping energy 𝑡 for both the angled-cut (8,8) CNT leads and the polyenes, and assume the same 𝑡 for the coupling 𝜏 between the lead and the molecule. The on-site energy of each carbon atom, in the angled-cut (8,8) CNT leads and in the polyenes, is equal to the Fermi energy (𝐸𝐹 ) of the CNT leads, and 𝐸𝐹 ≡ 0.. Fig. 3.7: (a) One-polyene junction. (b) Two-polyene junction. As illustrated in Fig. 3.7, each junction can be divided into three parts: (1) the device including one or two polyenes, (2) the left and (3) the right angled-cut (8,8) CNT leads. Therefore, the Hamiltonian of the junction can be represented as. H total. H L    L  0 .  L HD.  R. 0   R  . H R . We apply the following equation iteratively to obtain the surface Green’s function of each lead: g sL( R )  [( E  i ) I  H L ( R )  g sL( R )   ]1 .. In the iterative calculation, we choose a small value of 𝜂 (10−7 ) and the criterion on 21.

(28) the norm of the recursion error (10−10 ) to achieve the desired accuracy of the calculation. Once the surface Green’s function of each lead is obtained, the self-energies from two leads are calculated by Eq. (16). Then, we can obtain the Green’s function of the device from G  [( E  i ) I  H D   L   R ]1 . We use Eq. (9) to calculate the broadening. Finally, we use Eq. (30) to obtain the transmission of the polyene junctions in between angled-cut armchair CNT leads.. 3.2.3 Ab initio calculations. Fig. 3.8: (a) One-polyene junction. (b) Two-polyene junction. We first optimize the geometry structure of the (8,8) CNT bulk with the SIESTA package. Each lead consists of four unit cells of the (8,8) CNT. We choose an energy cutoff of 200 Ryd, the Double-  plus polarization (DZP) basis, and the Ceperley-Alder local density approximation (LDA) for the exchange and correlation functional to implement this relaxation calculation. A rectangular supercell must be large enough to pass the convergence test of the bond length, so we use a rectangular supercell of 20Å × 20Å × 𝑎, where 𝑎 is the length of the lead in the z direction. The geometry relaxation process continues until the force acting on each atom of the (8,8) CNT lead is less than 0.05 𝑒𝑉/Å . Then, the relaxed structure is used for the self-consistent field (SCF) calculation of the lead in Nanodcal. We optimize the geometry structure of the scattering region in the similar way. In the scattering region, 3 and 3.5 unit cells from each lead are contained at the left and 22.

(29) the right sides of the polyenes, respectively. We use a rectangular supercell of 20Å × 20Å × 𝐷, where 𝐷 is the length of the scattering region in the z direction. In the geometry relaxation process, we fix the atomic coordinates of the unit cell that is directly connected to either lead so that they can maintain the original bulk structure, while the coordinates of the remaining atoms in the scattering region are relaxed with no geometry constraint. Then, the relaxed structure is used for the SCF calculation of the two-probe system in Nanodcal. After the relaxed structure is obtained, we can perform the transport calculation with the Nanodcal package. We use the DZP basis and the LDA_PZ81 for the exchange and correlation functional in Nanodcal calculation. To make the vacuum area sufficiently large, 30Å × 30Å × 𝑎 and 30Å × 30Å × 𝐷 are taken as the supercells of the lead and the scattering region, respectively. We first carry out the SCF calculation of the lead. Once the calculation result of the lead is obtained, the SCF calculation of the two-probe system can be performed. When the SCF calculation is complete, we can calculate the transmission coefficients by the transmission calculation.. 23.

(30) 3.2.4. The contributions from the even and odd channels in the two-polyene junctions. Fig. 3.9: The contact sites on the rim of the left (right) CNT lead are labeled by 𝑎 and 𝑏 (𝑐 and 𝑑), and the carbon atoms in two polyenes are labeled by 1~34. In the previous study, it has been shown that the transmission of the two-polyene junctions can be separated into the contributions from the even and odd channels. [30,31] We next calculate the transmission of the even and odd channels in the cases of two-polyene junctions. For the two-polyene junctions, the transmission only depends on the two contact sites on the rim of the CNT lead. Therefore, the surface Green’s function of the CNT lead can be reduced to a 2 × 2 matrix effectively:.  g sa,a g s0    g sb,a. g sa,b  , g sb,b . where 𝑎 and 𝑏 denote the two contact sites on the rim of the CNT lead. As shown in Fig. 3.9, atom 𝑎 and atom 𝑏 are considered as the lead. First we apply the following equation to obtain the surface Green’s function of a system that contains atom 1 and atom 18:. g1s  [( E  i ) I  H D  g s0  ]1 . Next, we apply the same strategy to obtain the surface Green’s function of a system that includes two atoms, namely, atom 2 and atom 19:. g s2  [( E  i ) I  H D  g1s  ]1 . 24.

(31) The procedure is repeated until we reach the pair of atom 16 and atom 33, as shown in the following equations: g s3  [( E  i ) I  H D  g s2  ]1.  15  1 g 16 s  [( E  i ) I  H D  g s  ] .. Finally, the last pair of atom 17 and atom 34 is considered as the device, whose Green’s function reads G  [( E  i ) I  H D   L   R ]1 . Note that the self-energies from the left and right leads are   L   L g16 s L ..  R   R g s0 R .. We use Eq. (9) to calculate the broadening. Then, we obtain the transmission matrix from T  LGR G  . In the above procedure, we only need to deal with a two-dimensional space, either for the surface or for the device. Due to the symmetry of the Hamiltonian of the surface and the device, an orthogonal matrix 𝑈 can be used to rotate and therefore diagonalize the space: T '  U TU , where this matrix 𝑈 is exactly constructed by the two vectors representing the even and odd channels.. 25.

(32) Chapter 4. Result and Discussion. 4.1 The semi-infinite carbon nanotube 4.1.1 The LDOS of the outermost layer To show the difference between the semi-infinite CNT and the CNT bulk, we make the comparison between the LDOS of the outermost layer of the semi-infinite CNT (as shown in Fig. 3.3) and the unit-cell-DOS of the CNT bulk (as shown in Fig. 3.1). For our current study, we focus on the cross-cut (𝑛, 𝑛) CNTs, the cross-cut (𝑛, 0) CNTs and the angled-cut (𝑛, 𝑛) CNTs, where we consider cases of 𝑛 = 8,9,10. These comparisons are illustrated in Fig. 4.1, Fig. 4.2 and Fig. 4.3, respectively. We first consider the cross-cut (𝑛, 𝑛) CNT. In Fig. 4.1, the comparison shows the agreement in the energy range (−𝑡, 𝑡). Beyond this range, the sharp peaks in the bulk DOS are smoothened. These sharp peaks in the DOS are called the van Hove singularities (VHSs). As illustrated in Fig. 4.4, the position of the VHS in the bulk DOS corresponds to the band extrema. For the LDOS of the outermost layer, the VHSs which correspond to the band extrema occurring at 𝑘 = 0 are suppressed, while the VHSs which correspond to the band extrema occurring at 𝑘 ≠ 0 are still singular. This is true for the cross-cut (𝑛, 0) CNT as well, as shown in Fig. 4.5, and in addition, an infinite peak appears at the Fermi energy (𝐸𝐹 ) in Fig. 4.2. Finally, we consider the angled-cut (𝑛, 𝑛) CNT. As shown in Fig. 4.3, all sharp peaks in the bulk DOS are smoothened, and a finite peak appears in the vicinity of the 𝐸𝐹 . From the illustration of Fig. 4.6, we note that the VHSs which correspond to the band extrema at either 𝑘 = 0 or 𝑘 ≠ 0 are all suppressed. From the above comparisons, we know that the cross-cut (𝑛, 0) CNT and the angled-cut (𝑛, 𝑛) CNT, both exhibiting a zigzag rim at the incision of the CNT, have the drastic change in the LDOS at the 𝐸𝐹 , while the cross-cut (𝑛, 𝑛) CNT does not.. 26.

(33) Fig. 4.1: The LDOS of the outermost layer of the cross-cut (𝑛, 𝑛) CNT (blue solid line) and the unit-cell-DOS of the (𝑛, 𝑛) CNT bulk (red dashed line): (a) (8,8), (b) (9,9) and (c) (10,10).. 27.

(34) Fig. 4.2: The LDOS of the outermost layer of the cross-cut (𝑛, 0) CNT (blue solid line) and the unit-cell-DOS of the (𝑛, 0) CNT bulk (red dashed line): (a) (8,0), (b) (9,0) and (c) (10,0).. 28.

(35) Fig. 4.3: The LDOS of the outermost layer of the angled-cut (𝑛, 𝑛) CNT (blue solid line) and the unit-cell-DOS of the (𝑛, 𝑛) CNT bulk (red dashed line): (a) (8,8), (b) (9,9) and (c) (10,10).. 29.

(36) Fig. 4.4: (a) The band structure of the (8,8) CNT bulk. (b) The unit-cell-DOS of the (8,8) CNT bulk. (c) The LDOS of the outermost layer of the cross-cut (8,8) CNT. The red dashed line denotes the postitions of the van Hove singularities.. 30.

(37) Fig. 4.5: (a) The band structure of the (9,0) CNT bulk. (b) The unit-cell-DOS of the (9,0) CNT bulk. (c) The LDOS of the outermost layer of the cross-cut (9,0) CNT. The red dashed line denotes the positions of the van Hove singularities.. 31.

(38) Fig. 4.6: (a) The band structure of the (8,8) CNT bulk. (b) The unit-cell-DOS of the (8,8) CNT bulk. (c) The LDOS of the outermost layer of the angled-cut (8,8) CNT. The red dashed line denotes the postitions of the van Hove singularities.. 32.

(39) 4.1.2 The layer-by-layer LDOS at the Fermi energy As demonstraed in the previous section, the considerable amount of states appear on the zigzag-shaped edge at the 𝐸𝐹 , and no such states survive on the armchair-shaped edge. This result suggests that the LDOS of the outermost layer at the 𝐸𝐹 depends on the shape of the edge. Next, we investigate how the LDOS from the edge to the interior at the 𝐸𝐹 is affected by the different incision of the CNT. Similarly, we use the cross-cut (𝑛, 𝑛) CNT, the cross-cut (𝑛, 0) CNT and the angled-cut (𝑛, 𝑛) CNT, where we consider cases of 𝑛 = 8,9,10. These results are illustrated in Fig. 4.7, Fig. 4.8 and Fig. 4.9, respectively. The LDOS per layer is uniform at the 𝐸𝐹 for the CNT bulk. However, when the CNT bulk is cut to become semi-infinite, the distribution of the LDOS changes at the 𝐸𝐹 . We first consider the cross-cut (𝑛, 𝑛) CNT. In Fig. 4.7, the layer-by-layer LDOS shows the periodic oscillation of a 3-layer-cycle at the 𝐸𝐹 . In addition, the LDOS of layer 1 is twice as large as layer 2 and layer 3 in a 3-layer-cycle. We then consider the cross-cut (𝑛, 0) CNT. With the increase of the layer number, the LDOS gradually decays until the LDOS of the cross-cut (𝑛, 0) CNT approximates the LDOS of the (𝑛, 0) CNT bulk, as shown in Fig. 4.8. The LDOS of the cross-cut (𝑛, 0) CNT at the 𝐸𝐹 clearly shows that the considerable amount of states that pop out are localized at the edge. Finally, we consider the angled-cut (𝑛, 𝑛) CNT. As illustrated in Fig. 4.9 and Fig. 4.10, an angled-cut (𝑛, 𝑛) CNT also possesses localized edge states. Peculiarly, when the LDOS at the 𝐸𝐹 decays into the bulk, the extra behavior occurs and depends on the 𝑛 number. For all 𝑛's not a multiple of 3, the periodic oscillation of a 3-layer-cycle occurs on top of the decay. For 𝑛's of multiples of 3, although some oscillations occur on top of the decay in the beginning, the LDOS at the 𝐸𝐹 finally converges to the bulk DOS value. From the above discussion, we know that the cross-cut (𝑛, 0) CNT and the angled-cut (𝑛, 𝑛) CNT have localized edge states at the 𝐸𝐹 , while the cross-cut (𝑛, 𝑛) CNT has no such states. In addition, the periodic oscillation of a 3-layer-cycle only appears for the cross-cut (𝑛, 𝑛) CNT and the interior of the angled-cut (𝑛, 𝑛) CNT which has 𝑛 ≠ 3𝑁, where 𝑁 is an integer.. 33.

(40) Fig. 4.7: The layer-by-layer LDOS at the 𝐸𝐹 for the cross-cut (𝑛, 𝑛) CNT (blue solid line) and the (𝑛, 𝑛) CNT bulk (red solid line): (a) (8,8), (b) (9,9) and (c) (10,10).. 34.

(41) Fig. 4.8: The layer-by-layer LDOS at the 𝐸𝐹 for the cross-cut (𝑛, 0) CNT (blue solid line) and the (𝑛, 0) CNT bulk (red solid line): (a) (8,0), (b) (9,0) and (c) (10,0). The inset is the magnified illustration of the original figure.. 35.

(42) Fig. 4.9: The layer-by-layer LDOS at the 𝐸𝐹 for the angled-cut (𝑛, 𝑛) CNT (blue solid line) and the (𝑛, 𝑛) CNT bulk (red solid line): (a) (8,8), (b) (9,9) and (c) (10,10). The inset is the magnified illustration of the original figure.. 36.

(43) Fig. 4.10: The layer-by-layer LDOS at the 𝐸𝐹 for the angled-cut (𝑛, 𝑛) CNT (blue solid line) and the (𝑛, 𝑛) CNT bulk (red solid line): (a) (11,11), (b) (12,12), (c) (13,13), (d) (14,14), (e) (15,15) and (f) (16,16). The inset is the magnified illustration of the original figure.. 37.

(44) 4.2 The polyene junctions in between angled-cut armchair carbon nanotube leads 4.2.1. The band structure and the transmission of the armchair carbon nanotube bulk. The hopping energy 𝑡 and 𝑒𝑉 are the energy units of the tight-binding model calculation and the ab initio calculation, respectively. To make the comparison between the tight-binding model and the ab initio, we use the band structure of the (8,8) CNT bulk to obtain the relation between the hopping energy 𝑡 and 𝑒𝑉. As illustrated in Fig. 4.11, we pick the energy range between two red dashed lines to perform the comparison of the energy unit. As a result, we can obtain the hopping energy 𝑡 = 2.458 𝑒𝑉 . Then, we use this parameter to present transmission data of junctions with (8,8) CNT leads obtained from both tight-binding model and the ab initio calculations. In Fig. 4.12, we show the transmission results of the (8,8) CNT bulk for tight-binding model and the ab initio. The comparison shows the agreement in. all the the the. energy range from – 0.9 𝑡 to 0.6 𝑡. Because the transmission is exactly two in the vicinity of the 𝐸𝐹 , this means that the (8,8) CNT bulk has two electronic channels near the 𝐸𝐹 .. 38.

(45) Fig. 4.11: The band structure of the (8,8) CNT bulk: (a) the tight-binding model result and (b) the ab initio result.. Transmission T(E). 15. 10. 5. 0 -1. -0.5. 0 E(t). 0.5. 1. Fig. 4.12: The transmission of the (8,8) CNT bulk for the tight-binding model (blue solid line) and the ab initio (red dashed line).. 39.

(46) 4.2.2. The transmission of the polyene junctions in between angled-cut armchair carbon nanotube leads. In the previous section, we mention that there are two electronic channels near the 𝐸𝐹 for the (8,8) CNT bulk. Now, when we etch the (8,8) CNT bulk to become two angle-cut (8,8) CNT leads, two electronic channels near the 𝐸𝐹 are broken. To restore the electronic channels, we use one or two polyenes to bridge the gap between two angled-cut (8,8) CNT leads. These results are illustrated in Fig. 4.13 and Fig. 4.14, respectively. From Fig. 4.13 and Fig. 4.14, we find that the ab initio result is similar to the tight-binding model result qualitatively. However, there are still discrepancies between transmissions obtained from the two methods. Compared with the tight-binding model result, the ab initio result presents an overall slight left shift in general. We will discuss the discrepancies between the two methods in the next section. Finally, we discuss whether or not the electronic channels near the 𝐸𝐹 can be restored by bridging two CNT leads with the polyene molecule(s). We first consider the one-polyene junctions. As illustrated in Fig. 4.13, there is a transmission peak of height one near the 𝐸𝐹 in each case, which stands for one electronic channel restored by a bridging polyene. Then, we consider the two-polyene junctions. From Fig. 4.14, the value of the central peak in the transmission varies between zero and two, which shows the interference effect when bridging the CNT leads with two polyenes. According to the previous study, the transmission of the two-polyene junctions can be divided into the contributions from the even and odd channels. [30,31] In Fig. 4.15, we show the transmission results of the even and odd channels in two polyene junctions. We find that the maximum of the transmission can’t exceed one for either channel. Then, the sum of the contributions from the two channels is exactly the transmission of the two-polyene junction, since the effective two-dimensional space is diagonalized by the basis of the even and odd channels. Moreover, when two peaks near the 𝐸𝐹 approach (recede from) the 𝐸𝐹 , the sum of two peaks at the 𝐸𝐹 approximates two (zero).. 40.

(47) Fig. 4.13: The transmission of the one-polyene junctions for the tight-binding model (blue solid line) and the ab initio (red dashed line): (a) (15), (b) (13), (c) (11), (d) (9), (e) (7), (f) (5), (g) (3) and (h) (1). These labels are defined in Fig. 3.5. 41.

(48) Fig. 4.14: The transmission of the two-polyene junctions for the tight-binding model (blue solid line) and the ab initio (red dashed line): (a) (13,20), (b) (11,22), (c) (9,24), (d) (7,26), (e) (5,28) and (f) (3,30). These labels are defined in Fig. 3.6.. 42.

(49) Fig. 4.15: The transmission for the even channel (blue solid line) and the odd channel (red solid line) in the two-polyene junctions: (a) (13,20), (b) (11,22), (c) (9,24), (d) (7,26), (e) (5,28) and (f) (3,30). These labels are defined in Fig. 3.6.. 43.

(50) 4.3. The discrepancy of the transmission between the tight-binding model and the ab initio. In the tight binding model, we use the single and uniform hopping energy assumption to simplify the calculation. However, in the ab initio calculation, we can obtain the actual bonding and orbital hybridization of the junction after the structure of the junction is optimized by the SIESTA package, and the geometry relaxation brings out features overlooked by the tight-binding model. Next we discuss how the geometry relaxation causes the discrepancy in the transmission between the tight-binding model and the ab initio results by tuning the magnitude of the parameter in the tight-binding model and investigating the corresponding variation in the transmission. In this discussion, we use the two-polyene junction of the contact combination (11,22) as an example.. 44.

(51) 4.3.1 The intra-molecular hopping energy We consider the possible variation of the intra-molecular hopping energy in the polyene. The C-C bond lengths of a polyene in the one-polyene junctions are listed in Table 1, and the C-C bond lengths of two polyenes in the two-polyene junctions are listed in Table 2.1 and Table 2.2. Then, these C-C bond lengths are compared with the two C-C bond lengths of the (8,8) CNT which are 1.420 Å and 1.417 Å . Note that all C-C bond lengths of the polyenes in the junction are shorter than the C-C bond lengths of the (8,8) CNT, which implies that the intra-molecular hopping energies in the polyene is stronger than that in the (8,8) CNT. We next tune up the intra-molecular hopping energy of the polyene within the tight-binding model to understand how this parameter affects the transmission. As shown in Fig. 4.16, the overall spectrum is not shifted, but the molecular resonance peaks are pulled apart as we tune up the intra-molecular hopping energy.. Transmission T(E). 2. 1.5. 1. 0.5. 0 -0.6. -0.4. -0.2. 0 E(t). 0.2. 0.4. 0.6. Fig. 4.16: The transmission of the two-polyene junction (11,22) for the tight-binding model, where the intra-molecular hopping energies of the polyene equal 1.4 𝑡 (red line), 1.2 𝑡 (blue line) and 𝑡 (black line), respectively. The black line is the tight-binding model result in Fig. 4.14(b).. 45.

(52) 4.3.2 The contact coupling We consider the coupling 𝜏 between the polyene and the CNT leads. The 𝑝𝑝𝜋 planes of the polyene and the (8,8) CNT are not aligned after the ab initio geometry relaxation, as shown in Fig. 4.22 and Fig 4.23. The cosine value of the pairwise consecutive 𝑝𝑝𝜋 plane normals at the joint of the one-polyene junction is listed in Table 3, and the cosine value of the pairwise consecutive 𝑝𝑝𝜋 plane normals at the joint of the two-polyene junction is listed in Table 4. The not aligned and twisted 𝑝𝑝𝜋 planes lead to the weakened coupling between the polyene and the (8,8) CNT. We next tune down the contact coupling in the tight-binding model to understand how this parameter affects the transmission. As shown in Fig. 4.17, the central peak is relatively unaltered, and the first pair of side peaks is brought closer to the center.. Transmission T(E). 2. 1.5. 1. 0.5. 0 -0.6. -0.4. -0.2. 0 E(t). 0.2. 0.4. 0.6. Fig. 4.17: The transmission of the two-polyene junction (11,22) for the tight-binding model, where the couplings between the polyene and the (8,8) CNT equal (𝑡, 𝑡) (black line), (0.95 𝑡, 0.95 𝑡) (blue line) and (0.9 𝑡, 0.9 𝑡) (red line), respectively. The black line is the tight-binding model result in Fig. 4.14(b).. 46.

(53) 4.3.3 The unbalanced contact couplings As shown in Table 4, although all cosine values are less than one, the cosine values of the polyene 1 and the polyene 2 are slightly different. To simulate this circumstance, we assume two different couplings in the two-polyene junctions. As shown in Fig. 4.18, we find that the shift of the transmission is similar to the illustration in Fig. 4.17, but the height of the central peak in Fig. 4.18 suffers the influence. Because of two different couplings in the two-polyene junctions, the central peaks between the tight-binding model and the ab initio have the different height in some cases of Fig. 4.14.. Transmission T(E). 2. 1.5. 1. 0.5. 0 -0.6. -0.4. -0.2. 0 E(t). 0.2. 0.4. 0.6. Fig. 4.18: The transmission of the two-polyene junction (11,22) for the tight-binding model, where the couplings between the polyene and the (8,8) CNT equal (𝑡, 𝑡) (black line) and (0.95 𝑡, 0.75 𝑡) (blue line), respectively. The black line is the tight-binding model result in Fig. 4.14(b).. 47.

(54) 4.3.4 The on-site energy We consider the on-site energy of the polyene. In Fig. 4.19, we show the transmission in the different on-site energies of the polyene. When the on-site energy of the polyene is smaller (larger) than the one of the (8,8) CNT, the transmission can shift left (right). Because of the left shift of the peak at the 𝐸𝐹 in Fig. 4.13 and Fig. 4.14, we can know that the on-site energy of the polyene is actually lower than the one of the (8,8) CNT.. Transmission T(E). 2. 1.5. 1. 0.5. 0 -0.6. -0.4. -0.2. 0 E(t). 0.2. 0.4. 0.6. Fig. 4.19: The transmission of the two-polyene junction (11,22) for the tight-binding model, where the on-site energies of the polyene equal −0.02 𝑡 (blue line), 0 𝑡 (black line) and 0.02 𝑡 (red line), respectively. The black line is the tight-binding model result in Fig. 4.14(b).. 48.

(55) Table 1: The C-C bond lengths (Å ) of a polyene C17H19 in the one-polyene junctions. (15) (13) (11) (9) (7) (5) (3) (1) C1-C2. 1.411. 1.402. 1.405. 1.405. 1.401. 1.396. 1.380. 1.365. C2-C3. 1.393. 1.390. 1.389. 1.386. 1.387. 1.391. 1.394. 1.393. C3-C4. 1.403. 1.397. 1.399. 1.398. 1.396. 1.393. 1.382. 1.373. C4-C5. 1.394. 1.391. 1.390. 1.388. 1.388. 1.390. 1.389. 1.389. C5-C6. 1.400. 1.394. 1.396. 1.395. 1.394. 1.392. 1.384. 1.377. C6-C7. 1.396. 1.392. 1.391. 1.390. 1.389. 1.390. 1.387. 1.386. C7-C8. 1.398. 1.394. 1.394. 1.394. 1.392. 1.392. 1.384. 1.379. C8-C9. 1.398. 1.393. 1.392. 1.392. 1.390. 1.391. 1.386. 1.382. C9-C10. 1.397. 1.393. 1.393. 1.392. 1.391. 1.391. 1.386. 1.382. C10-C11. 1.397. 1.394. 1.394. 1.393. 1.392. 1.391. 1.384. 1.379. C11-C12. 1.396. 1.393. 1.391. 1.390. 1.390. 1.390. 1.387. 1.386. C12-C13. 1.399. 1.394. 1.395. 1.395. 1.393. 1.392. 1.384. 1.377. C13-C14. 1.395. 1.391. 1.389. 1.388. 1.388. 1.390. 1.389. 1.389. C14-C15. 1.402. 1.396. 1.398. 1.398. 1.396. 1.393. 1.382. 1.373. C15-C16. 1.393. 1.391. 1.387. 1.387. 1.388. 1.391. 1.393. 1.394. C16-C17. 1.410. 1.402. 1.404. 1.404. 1.400. 1.396. 1.380. 1.365. Fig. 4.20: A schematic plot of the C-C bond lengths for a polyene C17H19 in the one-polyene junctions.. 49.

(56) Table 2.1: The C-C bond lengths (Å ) of two polyenes C17H19 in the two-polyene junctions. (13,20). (11,22). (9,24). Polyene 1. Polyene 2. Polyene 1. Polyene 2. Polyene 1. Polyene 2. C1-C2. 1.405. 1.405. 1.407. 1.408. 1.408. 1.408. C2-C3. 1.390. 1.390. 1.389. 1.389. 1.386. 1.386. C3-C4. 1.398. 1.398. 1.400. 1.400. 1.400. 1.400. C4-C5. 1.390. 1.391. 1.391. 1.391. 1.387. 1.388. C5-C6. 1.396. 1.395. 1.396. 1.396. 1.396. 1.396. C6-C7. 1.393. 1.392. 1.392. 1.391. 1.390. 1.390. C7-C8. 1.394. 1.394. 1.395. 1.395. 1.395. 1.395. C8-C9. 1.393. 1.393. 1.393. 1.393. 1.392. 1.392. C9-C10. 1.393. 1.393. 1.393. 1.393. 1.393. 1.393. C10-C11. 1.394. 1.394. 1.394. 1.394. 1.394. 1.394. C11-C12. 1.392. 1.392. 1.391. 1.391. 1.390. 1.390. C12-C13. 1.395. 1.395. 1.395. 1.395. 1.396. 1.396. C13-C14. 1.391. 1.391. 1.389. 1.390. 1.388. 1.388. C14-C15. 1.397. 1.397. 1.399. 1.399. 1.399. 1.399. C15-C16. 1.390. 1.390. 1.388. 1.388. 1.386. 1.386. C16-C17. 1.405. 1.404. 1.406. 1.406. 1.406. 1.406. Fig. 4.21: A schematic plot of the C-C bond lengths for two polyenes C17H19 in the two-polyene junctions.. 50.

(57) Table 2.2: The C-C bond lengths (Å ) of two polyenes C17H19 in the two-polyene junctions. (7,26). (5,28). (3,30). Polyene 1. Polyene 2. Polyene 1. Polyene 2. Polyene 1. Polyene 2. C1-C2. 1.400. 1.400. 1.399. 1.399. 1.398. 1.399. C2-C3. 1.387. 1.387. 1.389. 1.389. 1.389. 1.389. C3-C4. 1.396. 1.396. 1.394. 1.394. 1.395. 1.395. C4-C5. 1.387. 1.387. 1.389. 1.388. 1.390. 1.390. C5-C6. 1.394. 1.394. 1.393. 1.393. 1.395. 1.394. C6-C7. 1.389. 1.389. 1.389. 1.389. 1.389. 1.390. C7-C8. 1.392. 1.391. 1.393. 1.392. 1.393. 1.392. C8-C9. 1.390. 1.390. 1.390. 1.390. 1.392. 1.391. C9-C10. 1.390. 1.390. 1.390. 1.390. 1.391. 1.391. C10-C11. 1.391. 1.391. 1.392. 1.392. 1.394. 1.393. C11-C12. 1.389. 1.389. 1.389. 1.389. 1.389. 1.389. C12-C13. 1.393. 1.393. 1.393. 1.393. 1.395. 1.394. C13-C14. 1.388. 1.387. 1.388. 1.388. 1.389. 1.389. C14-C15. 1.396. 1.396. 1.394. 1.394. 1.395. 1.395. C15-C16. 1.387. 1.388. 1.389. 1.389. 1.389. 1.389. C16-C17. 1.401. 1.401. 1.398. 1.398. 1.399. 1.399. 51.

(58) Table 3: The cosine value of the torsion angle for the one-polyene junctions. 𝑐𝑜𝑠𝜃. (15). (13). (11). (9). (7). (5). (3). (1). 0.947. 0.864. 0.869. 0.863. 0.860. 0.858. 0.872. 0.998. Fig. 4.22: A schematic plot of the torsion angle for the one-polyene junctions. Table 4: The cosine value of the torsion angle for the two-polyene junctions. (13,20) 𝑐𝑜𝑠𝜃. (11,22). Polyene 1. Polyene 2. Polyene 1. Polyene 2. Polyene 1. Polyene 2. 0.872. 0.868. 0.865. 0.869. 0.869. 0.875. (7,26) 𝑐𝑜𝑠𝜃. (9,24). (5,28). (3,30). Polyene 1. Polyene 2. Polyene 1. Polyene 2. Polyene 1. Polyene 2. 0.865. 0.863. 0.868. 0.867. 0.959. 0.959. Fig. 4.23: A schematic plot of the torsion angles for the two-polyene junctions.. 52.

(59) Chapter 5. Conclusion. When the CNT bulk is cut to become semi-infinite, the incision of the semi-infinite CNT has two prototype edge shapes, namely, the armchair-shaped edge and the zigzag-shaped edge. In our study, we focus on the cross-cut armchair CNT, the cross-cut zigzag CNT, and the angled-cut armchair CNT. The cross-cut armchair CNT exhibits an armchair-shaped edge, while the cross-cut zigzag CNT and the angled-cut armchair CNT exhibit a zigzag-shaped edge. From the LDOS of the outermost layer, we note that the considerable amount of states appears on the zigzag-shaped edge at the 𝐸𝐹 , and no such states survive on the armchair-shaped edge. Because the LDOS of the outermost layer drastically change at the 𝐸𝐹 , we next investigate the layer-by-layer LDOS at the 𝐸𝐹 . We find that the cross-cut zigzag CNT and the angled-cut armchair CNT have localized edge states at the 𝐸𝐹 , while the cross-cut armchair CNT has no such states. Moreover, the periodic oscillation of a 3-layer-cycle only appears for the cross-cut armchair CNT and the interior of the angled-cut armchair CNT which has 𝑛 ≠ 3𝑁. Next, we use both the tight-binding model and the ab initio method to study the electronic transport properties of the molecular junctions constructed by the polyene molecules bridging two angled-cut armchair CNT leads. For the one-polyene junctions, because the value of the central peak in the transmission reaches one in each case, which suggests that one electronic channel is restored by a bridging polyene. For the two-polyene junctions, depending on the contact site combination at the CNT lead, the value of the transmission at the 𝐸𝐹 varies between zero and two, which shows the interference effect. In the ab initio calculation, the geometry relaxation brings more features that are overlooked by the tight-binding model, and causes the discrepancies between the transmissions obtained from the two methods. To analyze the discrepancies between the two methods, we tune the magnitude of the parameters in the tight-binding model. Then, we find that the transmission is sensitive to the intra-molecular bonding strength, the coupling strength, and the on-site energy. Therefore, the discrepancies between the two methods originate from the influences of these parameters.. 53.

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