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Abstract
We perform firstprinciples calculation to investigate the dynamic conductance of atomic wires of the benzenedithiol (BDT) as well as carbon chains with different length in contact with two Al(100) electrodes (AlC_{ n }Al). Our calculation is based on the combination of the nonequilibrium Green's function and the density functional theory. For ac conductance, there are two theories that ensures the current conservation: (1). the global formula which is a phenomenological theory that partitions the total displacement current into each leads so that the current is conserved.(2). the local formula which is a microscopic theory that includes Coulomb interaction explicitly so that the current is conserved automatically. In this work, we use the local formula to calculate the dynamic conductance, especially the emittance. We give a detailed comparison and analysis for the results obtained from two theories. Our numerical results show that the global formula overestimates the emittance by two orders of magnitude. We also obtain an inequality showing that the emittance from global formula is greater than that from local formula for real atomic structures. For AlC_{ n }Al structures, the oscillatory behavior as the number of carbon chain N varies from even to odd remains unchanged when local formula is used. However, the prediction of local formula gives rise to opposite response when N is odd (inductivelike) as compared with that of global formula. Therefore, one should use the local formula for an accurate description of ac transport in nanoscale structures. In addition, the ‘size effect’ of the ac emittance is analyzed and can be understood by the kinetic inductance. Since numerical calculation using the global formula can be performed in orbital space while the local formula can only be used in real space, our numerical results indicate that the calculation using the local formula is extremely computational demanding.
The authors would like to thank B. Wang, Y. Wang, Y. X. Xing, F. Xu and H. Y. Yu for helpful discussions. We gratefully acknowledge the support from Research Grant Council (HKU 705409P) and University Grant Council (Contract No. AoE/P04/08) of the Government of HKSAR.
I. INTRODUCTION
II. REVIEW OF TWO DIFFERENT FORMULAE
A. Global formula
B. Local formula
C. AC emittance
III. NUMERICAL RESULTS
A. Benchmark calculation for the AlC_{n}Al system
B. AlBDTAl system
IV. SIZE EFFECT
V. CONCLUSION
Key Topics
 Buffer layers
 15.0
 Carbon
 14.0
 Lead
 12.0
 Aluminium
 10.0
 Density functional theory
 10.0
Figures
The schematic structure of AlC_{6}Al system. An atomic wire with six carbon atoms (gray) is sandwiched between two semiinfinite atomic Al electrodes (pink). The Al electrodes extended to ±∞ along (100) direction.
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The schematic structure of AlC_{6}Al system. An atomic wire with six carbon atoms (gray) is sandwiched between two semiinfinite atomic Al electrodes (pink). The Al electrodes extended to ±∞ along (100) direction.
The LL component of emittance calculated by both the global formula and the local formula for the AlC_{n}Al system, n = 4, 5, ⋅⋅⋅, 9. The simulation box includes the carbon chain and 16 layers of buffering aluminum. In the figure, the ‘global emittance’ which means the emittance calculated by the global formula is plotted in blue, while the ‘local emittance,’ the emittance calculated by the local formula is plotted in green.
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The LL component of emittance calculated by both the global formula and the local formula for the AlC_{n}Al system, n = 4, 5, ⋅⋅⋅, 9. The simulation box includes the carbon chain and 16 layers of buffering aluminum. In the figure, the ‘global emittance’ which means the emittance calculated by the global formula is plotted in blue, while the ‘local emittance,’ the emittance calculated by the local formula is plotted in green.
The schematic structure of AlBDTAl system. The leads are two semiinfinite atomic Al electrodes (pink) along (100) direction. In between is a benzenedithiol (BDT) including two sulfur atoms (yellow), four hydrogen atoms (white) and six carbon atoms (carbon). The 2Dstructure BDT lies in the (110) plane of the aluminum.
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The schematic structure of AlBDTAl system. The leads are two semiinfinite atomic Al electrodes (pink) along (100) direction. In between is a benzenedithiol (BDT) including two sulfur atoms (yellow), four hydrogen atoms (white) and six carbon atoms (carbon). The 2Dstructure BDT lies in the (110) plane of the aluminum.
The imaginary part of the LL component of the dynamic conductance Im[G _{ LL }(ω)] calculated by either the global or the local formula (red dots), comparing with −ωE _{ LL } (blue lines) from 0 to 50THz. The linear behavior of the imaginary part of the dynamic conductance even holds up to 50THz, and the slope is none other than the negative of its corresponding emittance.
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The imaginary part of the LL component of the dynamic conductance Im[G _{ LL }(ω)] calculated by either the global or the local formula (red dots), comparing with −ωE _{ LL } (blue lines) from 0 to 50THz. The linear behavior of the imaginary part of the dynamic conductance even holds up to 50THz, and the slope is none other than the negative of its corresponding emittance.
The real part of the LL component of the dynamic conductance Re[G _{ LL }(ω)] (a.u.) calculated by either the global or the local formula. The frequency domain ranges from 0 to 50THz. Both of the curves show a binomial behavior, with a vanishing slope near zero frequency. varies faster than as the frequency increases.
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The real part of the LL component of the dynamic conductance Re[G _{ LL }(ω)] (a.u.) calculated by either the global or the local formula. The frequency domain ranges from 0 to 50THz. Both of the curves show a binomial behavior, with a vanishing slope near zero frequency. varies faster than as the frequency increases.
The LL component of emittance calculated by both the global formula (blue dots) and the local formula (green triangles) for the AlBDTAl system, against the shifted Fermi level from 1eV to 1eV. The inset is the global total density of states as a function of the shift of the Fermi level, which behaves more similarly to the global result than to the local result.
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The LL component of emittance calculated by both the global formula (blue dots) and the local formula (green triangles) for the AlBDTAl system, against the shifted Fermi level from 1eV to 1eV. The inset is the global total density of states as a function of the shift of the Fermi level, which behaves more similarly to the global result than to the local result.
The transmission function T _{ LR } for the AlBDTAl system calculated at different energy. There are five lines representing different numbers of buffer layers that are included in the scattering region, and these lines collapse to a single curve.
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The transmission function T _{ LR } for the AlBDTAl system calculated at different energy. There are five lines representing different numbers of buffer layers that are included in the scattering region, and these lines collapse to a single curve.
The LL component of emittance calculated by both the global formula (blue dots) and the local formula (green triangles) for the AlBDTAl system, against the number of buffer layers included in simulation box. It seems the one calculated by the global formula shows the generally linear behavior while the one calculated by the local formula shows the oscillatory behavior.
Click to view
The LL component of emittance calculated by both the global formula (blue dots) and the local formula (green triangles) for the AlBDTAl system, against the number of buffer layers included in simulation box. It seems the one calculated by the global formula shows the generally linear behavior while the one calculated by the local formula shows the oscillatory behavior.
1D doubledeltapotentialwell system, the LL component of the ac emittance calculated against the length of the leads out of the potential well, with both sides having equal length, by the global formula (blue) and the local formula (red) respectively. The figure shows that the emittance calculated by either method has a generally linear with periodically oscillatory property. The period is represented by the circle dots.
Click to view
1D doubledeltapotentialwell system, the LL component of the ac emittance calculated against the length of the leads out of the potential well, with both sides having equal length, by the global formula (blue) and the local formula (red) respectively. The figure shows that the emittance calculated by either method has a generally linear with periodically oscillatory property. The period is represented by the circle dots.
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Abstract
We perform firstprinciples calculation to investigate the dynamic conductance of atomic wires of the benzenedithiol (BDT) as well as carbon chains with different length in contact with two Al(100) electrodes (AlC_{ n }Al). Our calculation is based on the combination of the nonequilibrium Green's function and the density functional theory. For ac conductance, there are two theories that ensures the current conservation: (1). the global formula which is a phenomenological theory that partitions the total displacement current into each leads so that the current is conserved.(2). the local formula which is a microscopic theory that includes Coulomb interaction explicitly so that the current is conserved automatically. In this work, we use the local formula to calculate the dynamic conductance, especially the emittance. We give a detailed comparison and analysis for the results obtained from two theories. Our numerical results show that the global formula overestimates the emittance by two orders of magnitude. We also obtain an inequality showing that the emittance from global formula is greater than that from local formula for real atomic structures. For AlC_{ n }Al structures, the oscillatory behavior as the number of carbon chain N varies from even to odd remains unchanged when local formula is used. However, the prediction of local formula gives rise to opposite response when N is odd (inductivelike) as compared with that of global formula. Therefore, one should use the local formula for an accurate description of ac transport in nanoscale structures. In addition, the ‘size effect’ of the ac emittance is analyzed and can be understood by the kinetic inductance. Since numerical calculation using the global formula can be performed in orbital space while the local formula can only be used in real space, our numerical results indicate that the calculation using the local formula is extremely computational demanding.
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