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͑ a ͒ Computed current-voltage curves with ϭ 0.2, 0.4, 0.6, and 0.8 eV in Au 7 -Au 4 -Au 7 . ͑ b ͒ Broadened DOS of the terminal clusters in
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A Green's function formalism incorporating broadened density of states (DOS) is proposed for the calculation of electrical conductance. In cluster-molecule-cluster systems, broadened DOS of the clusters are defined as continuous DOS of electrodes and used to calculate Green's function of electrodes. This approach combined with density functional th...
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Context 1
... rs ͑ E ͒ ϭ ͱ 1 2 ͚ i d i e Ϫ ( E Ϫ i Ј ) 2 /2 2 C ri Ј C si Ј S rs Ј . ͑ 13 ͒ Here we dropped subscripts L and R for simplicity. The matrix element S rs Ј is the overlap integral between atomic orbitals r and s in the L ̈ wdin basis, being identical to the delta function ␦ rs . The real part of g is calculated from the Kramers-Kr ̈ nig relation: 1 ρ Im g R ͑ ͒ Re g R ͑ E ͒ ϭ P d . ͑ 14 ͒ Here P is the Cauchy principal value integral. Equations ͑ 9 ͒ – ͑ 14 ͒ allow us to calculate g from the submatrices F L Ј and F R Ј . When a bias voltage V is applied to the junctions, the electrical current is calculated from the following expression: 45 I ϭ 2 h e ͵ ρ dE Tr ͫ ⌫ L ͩ E Ϫ eV 2 ͪ G R ͑ E , V ͒ R E 2 G ͑ E , V ͒ ͓ f L ͑ E ͒ f R ͑ E ͔͒ , 15 where f L(R) is the Fermi distribution function of the left-side ͑ right-side ͒ electrode. Let us consider the electrical transmission in gold atomic wires. To determine the width parameter, let us first look at whether the quantum unit of conductance G 0 is theoretically obtained in gold atomic wires. Figure 2 shows cluster models, in which the linear Au 4 atomic wire is intercalated between two Au ͑ 111 ͒ surfaces. We first optimized the bond lengths of Au 4 in Au 7 -Au 4 -Au 7 at the B3LYP Refs. 46 and 47 ͒ level of theory with the LANL2DZ ͑ Refs. 48 and 49 ͒ basis set. Other bond lengths were fixed to be 2.88 Å, the bond length in bulk gold. The GAUSSIAN03 program package 50 was employed in the computation. Optimized lengths of the central and other two bonds in Au 4 are 2.755 and 2.629 Å, respectively. We adopted these bond lengths in other clusters shown in Fig. 2. In the computations of the current, we defined the central Au 2 atoms in these models as the ‘‘molecule ( M )’’ in Fig. 1. Computed electrical currents in Au 7 -Au 4 -Au 7 with several width parameters are given as a function of voltage in Fig. 3 ͑ a ͒ . The currents with ϭ 0.6 and 0.8 eV depend lin- early on the applied voltage, which is reasonable behavior. The broadened DOS of the terminal cluster (Au 7 ϩ Au) with ϭ 0.6 and 0.8 eV are smooth in the vicinity of the Fermi level, as shown in Fig. 3 ͑ b ͒ . 51 The computed currents with ϭ 0.2 and 0.4 eV do not show linear I - V responses in con- trast to the case with ϭ 0.6 and 0.8 eV, and the broadened DOS of the terminal cluster are not smooth in the vicinity of E F . The width parameter significantly affects computational results. The present approach is constructed with the equilib- rium Green’s function, and we thus have to explore the current-voltage responses at low voltages. By fitting I - V curves to the equation I / G 0 ϭ aV b in the range of 0–0.5 V, we calculated values ( a , b )s to be ͑ 1.047, 0.761 ͒ for ϭ 0.2 eV, ͑ 1.245, 0.948 ͒ for ϭ 0.4 eV, ͑ 1.156, 0.976 ͒ for ϭ 0.6 eV, and ͑ 1.106, 0.996 ͒ for ϭ 0.8 eV. Since a of 1.0 and b of 1.0 reproduce the best I - V response of gold atomic wires, s of 0.2 and 0.4 eV are not appropriate in the calculation of conductance. To investigate whether appropriate width parameters depend on cluster models, we considered I - V responses in other clusters shown in Fig. 2. Table I shows used width parameters and fitted ( a , b )s in the equation I / G 0 ϭ aV b . The parameters leading to reasonable I - V behavior slightly depend on the cluster models, but of 0.9 or 1.0 eV provides good results in all the models. Therefore the width parameters of 0.9–1.0 eV are a good choice for the B3LYP/LANL2DZ calculations. Let us next consider the functional dependence of the width parameter. We employed B3PW91 ͑ Refs. 46 and 52 ͒ and MPW1PW91 ͑ Ref. 53 ͒ as hybrid functionals and SVWN ͑ Refs. 54 and 55 ͒ and BPW91 ͑ Refs. 52 and 56 ͒ as pure density functionals. I - V responses were calculated in Au 7 -Au 4 -Au 7 with the LANL2DZ basis set. Fitted values ( a , b )s in the equation I / G 0 ϭ aV b are given in Table II. We found an appropriate parameter to be about 0.9 eV for the B3PW91 and MPW1PW91 functionals like in the B3LYP calculations, while we cannot obtain good parameters for the SVWN and BPW91 functionals. The reason why the pure density functionals cannot provide the quantum unit of conductance derives from the nearly degenerate MO levels shown in Fig. 4. In the SVWN calculation, the highest occu- pied MO ͑ HOMO ͒ , the lowest unoccupied MO ͑ LUMO ͒ , LUMO ϩ 1, and LUMO ϩ 2 are close lying in energy. Since the amplitudes of these MOs are localized at the terminal Au 7 parts, the eigenvalues L(R) Ј obtained from the diagonal- ization of the submatrix F L(R) Ј are also degenerate at the Fermi level. Therefore the broadening in Eqs. ͑ 9 ͒ , ͑ 10 ͒ , and ͑ 13 ͒ does not result in smooth DOS in the SVWN calcula- tions. Close-lying MOs are also seen in the BPW91 calculations ͑ not shown ͒ , and hence there are no appropriate parameters for the BPW91 as well as SVWN functionals. On the other hand, the HOMO, LUMO, and LUMO ϩ 1 in the B3LYP calculation are not degenerate in energy, leading to smooth DOS of the electrodes ͓ Fig. 3 ͑ b ͔͒ . We therefore conclude that the hybrid density functionals are suitable for the broadened DOS approach. Appropriate width parameters also depend on basis sets employed in the calculation of current. Tables III and IV show fitted values ( a , b )s obtained with the CEP ͑ Ref. 57 ͒ and SDD ͑ Ref. 58 ͒ basis sets, respectively. I - V responses were calculated in Au 7 -Au 4 -Au 7 with the hybrid functionals, B3LYP, B3PW91, and MPW1PW91. The geometry of the cluster optimized with B3LYP/CEP was used for the CEP calculations ͑ Table III ͒ and that optimized with B3LYP/SDD was used for the SDD calculations ͑ Table IV ͒ . 59 Table III shows good I - V responses in a wide range of width parameters, indicating that the CEP basis set is suitable for the broadened DOS approach. On the other hand, I - V responses obtained with the SDD basis set weakly depend on the hybrid functionals. 60 Thus, the LANL2DZ and CEP basis sets are better to employ in the present approach. We finally consider the electrical transmission in molecular junctions consisting of benzene-1,4-dithiolate ͑ BDT ͒ , 4,4 Ј -bipyridine ͑ BP ͒ , benzene-1,4-dimethanethiolate ͑ BDMT ͒ , hexane dithiolate ͑ HDT ͒ , and octane dithiolate ͑ ODT ͒ . Electrical conductances of these molecules connected with gold STM tip were measured at low bias voltages ( Ͻ ϳ 0.3 V). 15–17 We consider the on-top connections in the molecular junctions, as shown in Fig. 5. We optimized the geometries of the molecular junctions with B3LYP/ LANL2DZ and B3LYP/CEP-31G ͑ Refs. 57 and 61 ͒ under the constraint of the Au-Au bond lengths of 2.88 Å. Width parameter in the LANL2DZ calculations was set to be 1.0 eV, and that in the CEP-31G calculations was 1.4 eV. In the computations of current, we defined the Au-molecule-Au part as the ‘‘molecule ( M )’’ shown in Fig. ...
Context 2
... ) 2 /2 2 C ri Ј C si Ј S rs Ј . ͑ 13 ͒ Here we dropped subscripts L and R for simplicity. The matrix element S rs Ј is the overlap integral between atomic orbitals r and s in the L ̈ wdin basis, being identical to the delta function ␦ rs . The real part of g is calculated from the Kramers-Kr ̈ nig relation: 1 ρ Im g R ͑ ͒ Re g R ͑ E ͒ ϭ P d . ͑ 14 ͒ Here P is the Cauchy principal value integral. Equations ͑ 9 ͒ – ͑ 14 ͒ allow us to calculate g from the submatrices F L Ј and F R Ј . When a bias voltage V is applied to the junctions, the electrical current is calculated from the following expression: 45 I ϭ 2 h e ͵ ρ dE Tr ͫ ⌫ L ͩ E Ϫ eV 2 ͪ G R ͑ E , V ͒ R E 2 G ͑ E , V ͒ ͓ f L ͑ E ͒ f R ͑ E ͔͒ , 15 where f L(R) is the Fermi distribution function of the left-side ͑ right-side ͒ electrode. Let us consider the electrical transmission in gold atomic wires. To determine the width parameter, let us first look at whether the quantum unit of conductance G 0 is theoretically obtained in gold atomic wires. Figure 2 shows cluster models, in which the linear Au 4 atomic wire is intercalated between two Au ͑ 111 ͒ surfaces. We first optimized the bond lengths of Au 4 in Au 7 -Au 4 -Au 7 at the B3LYP Refs. 46 and 47 ͒ level of theory with the LANL2DZ ͑ Refs. 48 and 49 ͒ basis set. Other bond lengths were fixed to be 2.88 Å, the bond length in bulk gold. The GAUSSIAN03 program package 50 was employed in the computation. Optimized lengths of the central and other two bonds in Au 4 are 2.755 and 2.629 Å, respectively. We adopted these bond lengths in other clusters shown in Fig. 2. In the computations of the current, we defined the central Au 2 atoms in these models as the ‘‘molecule ( M )’’ in Fig. 1. Computed electrical currents in Au 7 -Au 4 -Au 7 with several width parameters are given as a function of voltage in Fig. 3 ͑ a ͒ . The currents with ϭ 0.6 and 0.8 eV depend lin- early on the applied voltage, which is reasonable behavior. The broadened DOS of the terminal cluster (Au 7 ϩ Au) with ϭ 0.6 and 0.8 eV are smooth in the vicinity of the Fermi level, as shown in Fig. 3 ͑ b ͒ . 51 The computed currents with ϭ 0.2 and 0.4 eV do not show linear I - V responses in con- trast to the case with ϭ 0.6 and 0.8 eV, and the broadened DOS of the terminal cluster are not smooth in the vicinity of E F . The width parameter significantly affects computational results. The present approach is constructed with the equilib- rium Green’s function, and we thus have to explore the current-voltage responses at low voltages. By fitting I - V curves to the equation I / G 0 ϭ aV b in the range of 0–0.5 V, we calculated values ( a , b )s to be ͑ 1.047, 0.761 ͒ for ϭ 0.2 eV, ͑ 1.245, 0.948 ͒ for ϭ 0.4 eV, ͑ 1.156, 0.976 ͒ for ϭ 0.6 eV, and ͑ 1.106, 0.996 ͒ for ϭ 0.8 eV. Since a of 1.0 and b of 1.0 reproduce the best I - V response of gold atomic wires, s of 0.2 and 0.4 eV are not appropriate in the calculation of conductance. To investigate whether appropriate width parameters depend on cluster models, we considered I - V responses in other clusters shown in Fig. 2. Table I shows used width parameters and fitted ( a , b )s in the equation I / G 0 ϭ aV b . The parameters leading to reasonable I - V behavior slightly depend on the cluster models, but of 0.9 or 1.0 eV provides good results in all the models. Therefore the width parameters of 0.9–1.0 eV are a good choice for the B3LYP/LANL2DZ calculations. Let us next consider the functional dependence of the width parameter. We employed B3PW91 ͑ Refs. 46 and 52 ͒ and MPW1PW91 ͑ Ref. 53 ͒ as hybrid functionals and SVWN ͑ Refs. 54 and 55 ͒ and BPW91 ͑ Refs. 52 and 56 ͒ as pure density functionals. I - V responses were calculated in Au 7 -Au 4 -Au 7 with the LANL2DZ basis set. Fitted values ( a , b )s in the equation I / G 0 ϭ aV b are given in Table II. We found an appropriate parameter to be about 0.9 eV for the B3PW91 and MPW1PW91 functionals like in the B3LYP calculations, while we cannot obtain good parameters for the SVWN and BPW91 functionals. The reason why the pure density functionals cannot provide the quantum unit of conductance derives from the nearly degenerate MO levels shown in Fig. 4. In the SVWN calculation, the highest occu- pied MO ͑ HOMO ͒ , the lowest unoccupied MO ͑ LUMO ͒ , LUMO ϩ 1, and LUMO ϩ 2 are close lying in energy. Since the amplitudes of these MOs are localized at the terminal Au 7 parts, the eigenvalues L(R) Ј obtained from the diagonal- ization of the submatrix F L(R) Ј are also degenerate at the Fermi level. Therefore the broadening in Eqs. ͑ 9 ͒ , ͑ 10 ͒ , and ͑ 13 ͒ does not result in smooth DOS in the SVWN calcula- tions. Close-lying MOs are also seen in the BPW91 calculations ͑ not shown ͒ , and hence there are no appropriate parameters for the BPW91 as well as SVWN functionals. On the other hand, the HOMO, LUMO, and LUMO ϩ 1 in the B3LYP calculation are not degenerate in energy, leading to smooth DOS of the electrodes ͓ Fig. 3 ͑ b ͔͒ . We therefore conclude that the hybrid density functionals are suitable for the broadened DOS approach. Appropriate width parameters also depend on basis sets employed in the calculation of current. Tables III and IV show fitted values ( a , b )s obtained with the CEP ͑ Ref. 57 ͒ and SDD ͑ Ref. 58 ͒ basis sets, respectively. I - V responses were calculated in Au 7 -Au 4 -Au 7 with the hybrid functionals, B3LYP, B3PW91, and MPW1PW91. The geometry of the cluster optimized with B3LYP/CEP was used for the CEP calculations ͑ Table III ͒ and that optimized with B3LYP/SDD was used for the SDD calculations ͑ Table IV ͒ . 59 Table III shows good I - V responses in a wide range of width parameters, indicating that the CEP basis set is suitable for the broadened DOS approach. On the other hand, I - V responses obtained with the SDD basis set weakly depend on the hybrid functionals. 60 Thus, the LANL2DZ and CEP basis sets are better to employ in the present approach. We finally consider the electrical transmission in molecular junctions consisting of benzene-1,4-dithiolate ͑ BDT ͒ , 4,4 Ј -bipyridine ͑ BP ͒ , benzene-1,4-dimethanethiolate ͑ BDMT ͒ , hexane dithiolate ͑ HDT ͒ , and octane dithiolate ͑ ODT ͒ . Electrical conductances of these molecules connected with gold STM tip were measured at low bias voltages ( Ͻ ϳ 0.3 V). 15–17 We consider the on-top connections in the molecular junctions, as shown in Fig. 5. We optimized the geometries of the molecular junctions with B3LYP/ LANL2DZ and B3LYP/CEP-31G ͑ Refs. 57 and 61 ͒ under the constraint of the Au-Au bond lengths of 2.88 Å. Width parameter in the LANL2DZ calculations was set to be 1.0 eV, and that in the CEP-31G calculations was 1.4 eV. In the computations of current, we defined the Au-molecule-Au part as the ‘‘molecule ( M )’’ shown in Fig. 1. Figure 6 shows computed transmission functions at the zero bias voltage, and Table V lists computational data. As shown in Fig. 6, conductances ͓ i.e., T ( E F ) G 0 ] computed with B3LYP/CEP-31G are comparable to the experimental data. The conductances of BDT, BP, and HDT are improved very much by using CEP-31G. A conductance of BDT computed with another algorithm using the B3PW91 functional with the LANL2DZ basis set is 0.065 G 0 . 26 This value is in good agreement with our result of BDT obtained at the B3LYP/LANL2DZ level of theory; see Table V. The ODT wire shows a smaller conductance than the HDT wire by about one order of magnitude. Since the conductance of molecular wires shows the exponential decay as a function of molecular length, 62 the decay of conductance in ODT is reasonable. On the other hand, the conductances of BDMT cal- culated with B3LYP/LANL2DZ and B3LYP/CEP-31G do not decay in comparison with those of BDT. Besides the conductances of BDMT do not correspond to the experimental result. To see why T ( E F ) of BDMT becomes larger than the experimental value, we analyzed the MOs in the vicinity of the Fermi level. The HOMO of Au 7 -BDMT-Au 7 , which is the nearest MO to the Fermi level ͑ Table V ͒ , is depicted in Fig. 6. The orbital amplitude of the HOMO is localized at the Au-S-CH 2 moieties, and small amplitude is seen on the benzene ring. We reported in previous studies that high orbital amplitude at the connecting sites with the electrodes results in the enhancement of conductance. 63,64 Accordingly, the large transmission function of BDMT ͓ T ( E F ) ϭ 0.136 with LANL2DZ and 0.018 with CEP-31G ͔ is reasonable from the theoretical viewpoint. We therefore have to elucidate what causes the low conductance in the experimental study ͑ 0.0006 G 0 of BDMT ͒ . Since the thiol group ͑ R-SH ͒ as well as the thiolate group ͑ R-S ͒ can bond to the electrodes, 65 we calculated the transmission function of benzene-1,4- dimethanethiol ͓ BDMT ͑ H ͔͒ at the B3LYP/CEP-31G level. Figure 7 shows a computed transmission function and the cluster conformation of Au 7 -BDMT ͑ H ͒ -Au 7 . The sulfur atoms are bonded to the neighboring hydrogen atoms, and thus the gold atom in the Au-SH-CH 2 -moieties can be formally regarded as a free radical ͑ i.e., Au-SH-CH 2 ͒ , leading to a diradical state of Au 7 -BDMT ͑ H ͒ -Au 7 . We performed unre- stricted B3LYP calculations and obtained the transmission functions of the ␣ and  spins independently. The T ( E ) of BDMT ͑ H ͒ depicted in Fig. 7 corresponds to ͓ T ␣ ( E ) ϩ T  ( E ) ͔ /2. 66 The computed conductance of BDMT ͑ H ͒ dra- matically decreases in comparison with that of BDMT, and shows a value comparable with the experimental result. 17 In the thiol system, delocalization of MOs over the molecule is no longer seen in the vicinity of the Fermi level. For example, the amplitude of the HOMO of the ␣ -spin part is localized at the lower site of Au 7 -BDMT ͑ H ͒ -Au 7 , as shown in Fig. 7. The orbital localization causes the decreased transmission in the vicinity of the Fermi level. Stokbro and co- workers theoretically proposed that benzene-1,4-dithiol ...
Context 3
... rs ͑ E ͒ ϭ ͱ 1 2 ͚ i d i e Ϫ ( E Ϫ i Ј ) 2 /2 2 C ri Ј C si Ј S rs Ј . ͑ 13 ͒ Here we dropped subscripts L and R for simplicity. The matrix element S rs Ј is the overlap integral between atomic orbitals r and s in the L ̈ wdin basis, being identical to the delta function ␦ rs . The real part of g is calculated from the Kramers-Kr ̈ nig relation: 1 ρ Im g R ͑ ͒ Re g R ͑ E ͒ ϭ P d . ͑ 14 ͒ Here P is the Cauchy principal value integral. Equations ͑ 9 ͒ – ͑ 14 ͒ allow us to calculate g from the submatrices F L Ј and F R Ј . When a bias voltage V is applied to the junctions, the electrical current is calculated from the following expression: 45 I ϭ 2 h e ͵ ρ dE Tr ͫ ⌫ L ͩ E Ϫ eV 2 ͪ G R ͑ E , V ͒ R E 2 G ͑ E , V ͒ ͓ f L ͑ E ͒ f R ͑ E ͔͒ , 15 where f L(R) is the Fermi distribution function of the left-side ͑ right-side ͒ electrode. Let us consider the electrical transmission in gold atomic wires. To determine the width parameter, let us first look at whether the quantum unit of conductance G 0 is theoretically obtained in gold atomic wires. Figure 2 shows cluster models, in which the linear Au 4 atomic wire is intercalated between two Au ͑ 111 ͒ surfaces. We first optimized the bond lengths of Au 4 in Au 7 -Au 4 -Au 7 at the B3LYP Refs. 46 and 47 ͒ level of theory with the LANL2DZ ͑ Refs. 48 and 49 ͒ basis set. Other bond lengths were fixed to be 2.88 Å, the bond length in bulk gold. The GAUSSIAN03 program package 50 was employed in the computation. Optimized lengths of the central and other two bonds in Au 4 are 2.755 and 2.629 Å, respectively. We adopted these bond lengths in other clusters shown in Fig. 2. In the computations of the current, we defined the central Au 2 atoms in these models as the ‘‘molecule ( M )’’ in Fig. 1. Computed electrical currents in Au 7 -Au 4 -Au 7 with several width parameters are given as a function of voltage in Fig. 3 ͑ a ͒ . The currents with ϭ 0.6 and 0.8 eV depend lin- early on the applied voltage, which is reasonable behavior. The broadened DOS of the terminal cluster (Au 7 ϩ Au) with ϭ 0.6 and 0.8 eV are smooth in the vicinity of the Fermi level, as shown in Fig. 3 ͑ b ͒ . 51 The computed currents with ϭ 0.2 and 0.4 eV do not show linear I - V responses in con- trast to the case with ϭ 0.6 and 0.8 eV, and the broadened DOS of the terminal cluster are not smooth in the vicinity of E F . The width parameter significantly affects computational results. The present approach is constructed with the equilib- rium Green’s function, and we thus have to explore the current-voltage responses at low voltages. By fitting I - V curves to the equation I / G 0 ϭ aV b in the range of 0–0.5 V, we calculated values ( a , b )s to be ͑ 1.047, 0.761 ͒ for ϭ 0.2 eV, ͑ 1.245, 0.948 ͒ for ϭ 0.4 eV, ͑ 1.156, 0.976 ͒ for ϭ 0.6 eV, and ͑ 1.106, 0.996 ͒ for ϭ 0.8 eV. Since a of 1.0 and b of 1.0 reproduce the best I - V response of gold atomic wires, s of 0.2 and 0.4 eV are not appropriate in the calculation of conductance. To investigate whether appropriate width parameters depend on cluster models, we considered I - V responses in other clusters shown in Fig. 2. Table I shows used width parameters and fitted ( a , b )s in the equation I / G 0 ϭ aV b . The parameters leading to reasonable I - V behavior slightly depend on the cluster models, but of 0.9 or 1.0 eV provides good results in all the models. Therefore the width parameters of 0.9–1.0 eV are a good choice for the B3LYP/LANL2DZ calculations. Let us next consider the functional dependence of the width parameter. We employed B3PW91 ͑ Refs. 46 and 52 ͒ and MPW1PW91 ͑ Ref. 53 ͒ as hybrid functionals and SVWN ͑ Refs. 54 and 55 ͒ and BPW91 ͑ Refs. 52 and 56 ͒ as pure density functionals. I - V responses were calculated in Au 7 -Au 4 -Au 7 with the LANL2DZ basis set. Fitted values ( a , b )s in the equation I / G 0 ϭ aV b are given in Table II. We found an appropriate parameter to be about 0.9 eV for the B3PW91 and MPW1PW91 functionals like in the B3LYP calculations, while we cannot obtain good parameters for the SVWN and BPW91 functionals. The reason why the pure density functionals cannot provide the quantum unit of conductance derives from the nearly degenerate MO levels shown in Fig. 4. In the SVWN calculation, the highest occu- pied MO ͑ HOMO ͒ , the lowest unoccupied MO ͑ LUMO ͒ , LUMO ϩ 1, and LUMO ϩ 2 are close lying in energy. Since the amplitudes of these MOs are localized at the terminal Au 7 parts, the eigenvalues L(R) Ј obtained from the diagonal- ization of the submatrix F L(R) Ј are also degenerate at the Fermi level. Therefore the broadening in Eqs. ͑ 9 ͒ , ͑ 10 ͒ , and ͑ 13 ͒ does not result in smooth DOS in the SVWN calcula- tions. Close-lying MOs are also seen in the BPW91 calculations ͑ not shown ͒ , and hence there are no appropriate parameters for the BPW91 as well as SVWN functionals. On the other hand, the HOMO, LUMO, and LUMO ϩ 1 in the B3LYP calculation are not degenerate in energy, leading to smooth DOS of the electrodes ͓ Fig. 3 ͑ b ͔͒ . We therefore conclude that the hybrid density functionals are suitable for the broadened DOS approach. Appropriate width parameters also depend on basis sets employed in the calculation of current. Tables III and IV show fitted values ( a , b )s obtained with the CEP ͑ Ref. 57 ͒ and SDD ͑ Ref. 58 ͒ basis sets, respectively. I - V responses were calculated in Au 7 -Au 4 -Au 7 with the hybrid functionals, B3LYP, B3PW91, and MPW1PW91. The geometry of the cluster optimized with B3LYP/CEP was used for the CEP calculations ͑ Table III ͒ and that optimized with B3LYP/SDD was used for the SDD calculations ͑ Table IV ͒ . 59 Table III shows good I - V responses in a wide range of width parameters, indicating that the CEP basis set is suitable for the broadened DOS approach. On the other hand, I - V responses obtained with the SDD basis set weakly depend on the hybrid functionals. 60 Thus, the LANL2DZ and CEP basis sets are better to employ in the present approach. We finally consider the electrical transmission in molecular junctions consisting of benzene-1,4-dithiolate ͑ BDT ͒ , 4,4 Ј -bipyridine ͑ BP ͒ , benzene-1,4-dimethanethiolate ͑ BDMT ͒ , hexane dithiolate ͑ HDT ͒ , and octane dithiolate ͑ ODT ͒ . Electrical conductances of these molecules connected with gold STM tip were measured at low bias voltages ( Ͻ ϳ 0.3 V). 15–17 We consider the on-top connections in the molecular junctions, as shown in Fig. 5. We optimized the geometries of the molecular junctions with B3LYP/ LANL2DZ and B3LYP/CEP-31G ͑ Refs. 57 and 61 ͒ under the constraint of the Au-Au bond lengths of 2.88 Å. Width parameter in the LANL2DZ calculations was set to be 1.0 eV, and that in the CEP-31G calculations was 1.4 eV. In the computations of current, we defined the Au-molecule-Au part as the ‘‘molecule ( M )’’ shown in Fig. 1. Figure 6 shows computed transmission functions at the zero bias voltage, and Table V lists computational data. As shown in Fig. 6, conductances ͓ i.e., T ( E F ) G 0 ] computed with B3LYP/CEP-31G are comparable to the experimental data. The conductances of ...
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... Furthermore, it has become possible to realize logic operations by using single-molecule components [30][31][32][33]. On the other hand, through the development of computational procedures to simulate the properties of single molecules, the electron conductivities of various molecules have also been estimated by theoretical calculations [34][35][36][37][38][39]. For example, our group developed a method to simulate the electron conductivity of open-shell molecules by density functional theory (DFT) and elastic scattering Green's function (ESGF) methods [40]. ...
Herein, the electron conductivities of [18]annulene and its derivatives are theoretically examined as a molecular parallel circuit model consisting of two linear polyenes. Their electron conductivities are estimated by elastic scattering Green’s function (ESGF) theory and density functional theory (DFT) methods. The calculated conductivity of the [18]annulene does not follow the classical conductivity, i.e., Ohm’s law, suggesting the importance of a quantum interference effect in single molecules. By introducing electron-withdrawing groups into the annulene framework, on the other hand, a spin-polarized electronic structure appears, and the quantum interference effect is significantly suppressed. In addition, the total current is affected by the spin polarization because of the asymmetry in the coupling constant between the molecule and electrodes. From these results, it is suggested that the electron conductivity as well as the quantum interference effect of π-conjugated molecular systems can be designed using their open-shell nature, which is chemically controlled by the substituents.
... To gain insights into the tunable ON/OFF ratios, we carried out DFT-NEGF calculations on the molecular junction models of [1 R '] n+ (n = 0, 1) with truncated bis(dimethylphosphino)methane ligands, with the pyramidal Au clusters attached to the terminal sulfur atoms (Figures 3h, i). [28][29][30] The HOMO transmission peaks of the neutral species are located close to the Fermi energy level (EF) of the electrode (Au) and are sensitive to the substituents. The HOMO transmission peaks approach EF as the electron-donating character of the substituents increases (CF3 (−0.23 eV) à H (−0.10 eV) à OMe (−0.03 eV); highlighted as the square marks in Figure 3h), which is also supported by the HOMO energies estimated from the cyclic voltammograms [CF3 (−4.93 eV) à H (−4.76 eV) à OMe (−4.61 eV)). ...
Molecular switch is one of the essential functional units of molecular electronics. Here, we report development of new molecular switches based on the electron-rich diruthenium complexes with the (2,5-di-R-substituted 1,4-diethynylbenzene)diyl linkers. The dinuclear molecular switches, {µ-p-C≡C-(2,5-R2-C6H2)-C≡C}{Ru(dppe)2(C≡C-C6H4-p-SMe)}2 1R (R= OMe, H, CF3), with various substituents (R) on the bridging phenylene rings showed two successive reversible 1e-oxidation waves, indicating stability of 1e-oxidized mixed-valence species. The solid-state structure of [1H]+ showed the charge-localized Robin-Day class II nature, while that of [1OMe]+ revealed the fully charge-delocalized class III nature. These characters were also evident from the spectroscopic data in solutions. Single-molecule conductance measurements by the scanning tunneling microscope break junction method revealed a significant dependence of the conductance on R, i.e. 1OMe turned out to be >100-times more conductive than 1H and 1CF3, whereas the substituent effect of the monocationic complexes was within a fold-change of 2. As a result, the ON/OFF ratios (the ratios of the conductance of the cationic species [1R]+ to that of the neutral species 1R) were critically dependent on R (as large as 191 when R = CF3) and even reversed (0.4 when R = OMe). Furthermore, the neutral and monocationic complexes 1H and [1H]+ fabricated into the nanogap devices showed in situ ON/OFF switching behavior. The present study demonstrates not only the rare examples of the mixed-valence complexes which were subjected to the break junction measurements but also the first examples of molecular switch, the ON/OFF ratio of which was controlled by tuning the organic linker parts.
... To gain insight into the tunable ON/OFF ratios, we carried out DFT-NEGF calculations on molecular junction models of [1 R '] n+ (n = 0, 1) with truncated bis(dimethylphosphino)methane ligands with the Au clusters attached to the terminal thiomethyl groups (Figure 5h, i). [31][32][33] The HOMO transmission peaks of the neutral species are located close to the Fermi energy level (EF) and are sensitive to the substituents. The HOMO transmission peaks approach to EF, as the substituents become more electron-donating (CF3 (-0.23 eV) à H (-0.10 eV) à OMe (-0.03 eV); highlighted as the square marks in Figure 4h) as is also supported by the HOMO energies estimated from the cyclic voltammograms (CF3 (-4.93 eV) à H (-4.76 eV) à OMe (-4.61 eV)). ...
Molecular switch is one of the essential functional units of molecular electronics. Here, we report development of new molecular switches based on the electron-rich diruthenium complexes with the (2,5-di-R-substituted 1,4-diethynylbenzene)diyl linkers. The dinuclear molecular switches, {µ-p-C≡C-(2,5-R2-C6H2)-C≡C}{Ru(dppe)2(C≡C-C6H4-p-SMe)}2 1R (R= OMe, H, CF3), with various substituents (R) on the bridging phenylene rings showed two successive reversible 1e-oxidation waves, indicating stability of 1e-oxidized mixed-valence species. The solid-state structure of [1H]+ showed the charge-localized Robin-Day class II nature, while that of [1OMe]+ revealed the fully charge-delocalized class III nature. These characters were also evident from the spectroscopic data in solutions. Single-molecule conductance measurements by the scanning tunneling microscope break junction method revealed a significant dependence of the conductance on R, i.e. 1OMe turned out to be >100-times more conductive than 1H and 1CF3, whereas the substituent effect of the monocationic complexes was within a fold-change of 2. As a result, the ON/OFF ratios (the ratios of the conductance of the cationic species [1R]+ to that of the neutral species 1R) were critically dependent on R (as large as 191 when R = CF3) and even reversed (0.4 when R = OMe). Furthermore, the neutral and monocationic complexes 1H and [1H]+ fabricated into the nanogap devices showed in situ ON/OFF switching behavior. The present study demonstrates not only the rare examples of the mixed-valence complexes which were subjected to the break junction measurements but also the first examples of molecular switch, the ON/OFF ratio of which was controlled by tuning the organic linker parts.
... To obtain further insights into the conduction mechanism, conductance calculations with the hybrid DFT and nonequilibrium Green's function (NEGF) method [29] were carried out for the molecular junction models with the S!Au coordination bonds Au 35 À n R À Au 35 (n = 1-3: R = hex, Ar; Au 35 : pyramidal Au 35 electrode). The geometries of Au 35 À n R À Au 35 were relaxed using the DFT method and used for the transmission calculations ( Figure 8). ...
In this work, the design, synthesis, and single‐molecule conductance of ethynyl‐ and butadiynyl‐ruthenium molecular wires with thioether anchor groups [RS=n‐C6H13S, p‐tert‐Bu−C6H4S), trans‐{RS−(C≡C)n}2Ru(dppe)2 (n=1 (1R), 2 (2R); dppe: 1,2‐bis(diphenylphosphino)ethane) and trans‐(n‐C6H13S−C≡C)2Ru{P(OMe)3}4 3hex] are reported. Scanning tunneling microscope break‐junction study has revealed conductance of the organometallic molecular wires with the thioacetylene backbones higher than that of the related organometallic wires having arylethynylruthenium linkages with the sulfur anchor groups, trans‐{p‐MeS−C6H4‐(C≡C)n}2Ru(phosphine)4 4ⁿ (n=1, 2) and trans‐(Th−C≡C)2Ru(phosphine)4 5 (Th=3‐thienyl). It should be noted that the molecular junctions constructed from the butadiynyl wire 2R, trans‐{Au−RS−(C≡C)2}2Ru(dppe)2 (Au: gold metal electrode), show conductance comparable to that of the covalently linked polyynyl wire with the similar molecular length, trans‐{Au−(C≡C)3}2Ru(dppe)2 6³. The DFT non‐equilibrium Green's function (NEGF) study supports the highly conducting nature of the thioacetylene molecular wires through HOMO orbitals.
... whose width parameter represents the disorder energy. This model of spectral function was used in the past for describing the impurity effects in the two-dimensional electron gas [42], but also more recently to incorporate the effect of the electrodes on the density of states in molecular nanowires [43]. ...
We consider electrons in tubular nanowires with prismatic geometry and infinite length. Such a model corresponds to a core-shell nanowire with an insulating core and a conductive shell. In a prismatic shell the lowest energy states are localized along the edges (corners) of the prism and are separated by a considerable energy gap from the states localized on the prism facets. The corner localization is robust in the presence of a magnetic field longitudinal to the wire. If the magnetic field is transversal to the wire the lowest states can be shifted to the lateral regions of the shell, relatively to the direction of the field. These localization effects should be observable in transport experiments on semiconductor core-shell nanowires, typically with hexagonal geometry. We show that the conductance of the prismatic structures considerably differs from the one of circular nanowires. The effects are observed for sufficiently thin hexagonal wires and become much more pronounced for square and triangular shells. To the best of our knowledge, the internal geometry of such nanowires is not revealed in experimental studies. We show that with properly designed nanowires these localization effects may become an important resource of interesting phenomenology.
... whose width parameter Γ represents the disorder energy. This model of spectral function was used in the past for describing the impurity effects in the two-dimensional electron gas [37], but also more recently to incorporate the effect of the electrodes on the density of states in molecular nanowires [38]. ...
We consider electrons in tubular nanowires with prismatic geometry and infinite length. Such a model corresponds to a core-shell nanowire with an insulating core and a conductive shell. In a prismatic shell the lowest energy states are localized along the edges (corners) of the prism and are separated by a considerable energy gap from the states localized on the prism facets. The corner localization is robust in the presence of a magnetic field longitudinal to the wire. If the magnetic field is transversal to the wire the lowest states can be shifted to the lateral regions of the shell, relatively to the direction of the field. These localization effects should be observable in transport experiments on semiconductor core-shell nanowires, typically with hexagonal geometry. We show that the conductance of the prismatic structures considerably differs from the one of circular nanowires. The effects are observed for sufficiently thin hexagonal wires and become much more pronounced for square and triangular shells. To the best of our knowledge the internal geometry of such nanowires is not revealed in experimental studies. We show that with properly designed nanowires these localization effects may become an important resource of interesting phenomenology.
... Along with experimental studies, theoretical calculations of molecular devices based on welldefined crystallographic structures of the nanocrystals provide important information to understand the molecular conductance. [15][16][17] Density functional theory (DFT) combined with nonequilibrium Green's function (NEGF) is the predominant calculation method in which the electron transport is dealt with through a molecular wire bridging two metal electrodes. [18][19] For alkanedithiol wires with large HOMO-LUMO gaps, predicted conductance values are very close to experimental values. ...
... [20][21][22][23][24][25] While for OPEs with conjugated electronic structure, the theoretical method generally gave larger conductance values than experimental measurements. 16,26 Two major strategies were used to overcome this problem. The first one was to take into account various possible adsorption structures, such as hollow, bridge and top adsorption structures, 27 or 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 4 various molecular geometries, such as rotation between neighboring phenyl rings in OPEs. ...
Electron transport through molecular junctions has been widely investigated experimentally and theoretically. Unfortunately, there exists discrepancy on the single molecular conductance between theoretical calculations and experimental measurements. In this paper, first-principle density functional theory combined with nonequilibrium Green's function approach is employed; we studied electronic structures, molecular lengths, and interfacial interactions of three kinds of molecular junctions, alkanedithiols, oligo(1,4-phenyleneethynylene)s, and 1,4-benzene-di(n-alkylthiol) (BDnT), embedding in nanogaps of gold electrodes. First, our approach can accurately describe the binding interaction between the thiol group and gold electrode so that the conductance of alkanedithiol in a gold junction can be well predicted. We found that a previous underestimation of HOMO-LUMO gaps in the junction system leads to the overestimated conductance for conjugate molecules with sulfur atoms binding to gold electrodes. In the study of BDnT molecular wires with a phenyl ring, our results show that the HOMO-LUMO gap reaches a constant with molecular length increasing. Moreover, a larger predicted conductance can be attributed to the overlapping between the nonbonding lone-paired orbital of sulfur atoms and the delocalized pi electrons of the phenyl ring. Finally, we found that the conductance of molecules with short length or conjugated electronic structure greatly relies on the interfacial configuration. We proposed that these findings can give a clear understanding of electron transport in junction systems and open a promising theoretical study of molecular electronics.
... In order to clarify the conducting orbital state for the metalmolecule-metal junctions, we carried out hybrid DFT calculations coupled with non-equilibrium Green's function method for molecular wires attached to two triangular pyramidal Au clusters (Au 35 ) as shown in Fig. 5. 20 The modeled molecular junctions of 1-4 are denoted as Au-1-Au etc. The calculated zero bias conductance of the Ru wires 1-3 (17.1, 10.4, 9.7 Â 10 À4 G 0 ) is higher than that of 4 (8.3 ...
The organometallic Ru molecular wires 1-3 Ru(PR3)4(C≡CC5H5N)2 [(PR3)4 = (dppe)2 (1), [P(OMe3)]4 (2), and (dmpe)2 (3)] show conductance significantly higher than that of an organic counterpart, 1,4-dipyridyl butadiyne (4). CV and UV-vis measurements and DFT calculations suggest that the high-lying HOMO orbitals of the Ru wires are the key factor for the high conductance.
... CuPc sandwiched between two semi-infinite Au electrodes has been investigated theoretically in Refs. [24,25]. The transmission coefficient, T (E), shows two peaks near the Fermi energy (E F ) which have been dissected in terms of molecular orbitals. ...
The magnetic and transport properties of the MPc and F$_{16}$MPc (M = Sc, Ti,
V, Cr, Mn, Fe, Co, Ni, Cu, Zn and Ag) families of molecules (Pc:
phthalocyanine) in contact with S-Au wires are investigated by density
functional theory within the local density approximation (LDA), including local
electronic correlations (LDA+$U$) on the central metal atom. The magnetic
moments are found to be considerably modified under fluorination. In addition,
they do not depend exclusively on the configuration of the outer electronic
shell of the central metal atom (as in isolated MPc and F$_{16}$MPc) but also
on the interaction with the leads. Good agreement between the calculated
conductance and experimental results is obtained. For M = Ag, a high spin
filter efficiency and conductance is observed, giving rise to a potentially
high sensitivity for chemical sensor applications.
... To evaluate the transport properties of the finite cylinder, we couple it to two leads via contacts, shown schematically in Fig. 1, and apply a Green's function formalism to calculate the phase-coherent conductance G in the linear-response regime [8,[20][21][22][23][24]. It follows from the spin-resolved Landauer formula ...
We model a core-shell nanowire (CSN) by a cylindrical surface of finite
length. A uniform magnetic field perpendicular to the axis of the cylinder
forms electron states along the lines of zero radial field projection, which
can classically be described as snaking states. In a strong field, these states
converge pairwise to quasidegenerate levels, which are situated at the bottom
of the energy spectrum. We calculate the conductance of the CSN by coupling it
to leads, and predict that the snaking states govern transport at low chemical
potential, forming isolated peaks, each of which may be split in two by
applying a transverse electric field. If the contacts with the leads do not
completely surround the CSN, as is usually the case in experiments, the
amplitude of the snaking peaks changes when the magnetic field is rotated,
determined by the overlap of the contacts with the snaking states.
