Validity of the Free Electron Model for Two-Dimensional Electrodes
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Validity of the Free Electron Model for Two-Dimensional Electrodes Kenji Kondo Laboratory of Nanostructure Physics, Research Institute for Electronic Science, Hokkaido University, Sapporo 001-0020, Japan. ABSTRACT Generally, the electrodes are regarded as free electron gases when we calculate the transport characteristics of nanostructure materials or devices. In three dimensional electrodes, there are little electron correlation. However, in low-dimensional electrodes, electron correlation becomes much larger than that in three dimensional ones. Recently, nanotechnology has made much progress to fabricate two-dimensional (2D) electrodes easily and precisely. As a result, we must consider whether two-dimensional electrodes can be regarded as free electron gases. In this study, we investigate the electron energy spectrum of 2D electrodes, taking into consideration the electron correlation. These results suggest that the free electron model is justified only at the Fermi momentum and that we should not regard 2D electrodes as free electron gases without careful consideration under high electric field and/or high temperature. INTRODUCTION Recently, nanotechnology has made much progress to make low-dimensional electrodes easily and precisely. We can control the thickness of films deposited on the substrate on an atomic scale using molecular epitaxy and/or atomic layer epitaxy techniques. Many researchers have reported various novel nano-devices for candidates of beyond CMOS devices and of new solar cells [1, 2]. As reported in them, nano-devices can have ultrathin electrodes made of metal thin films or semiconductor thin films. Therefore, these electrodes are considered to be twodimensional (2D) ones and we have to consider whether 2D electrodes can be regarded as free electron gases in order to investigate the transport characteristics of these devices. This is because electron correlation in low-dimensional electrodes becomes much larger than that in three dimensional ones. In this study, we investigate the electron energy spectrum of 2D electrodes, taking into consideration the electron correlation. This result will be also very useful for consideration of electron confinement in quantum wells and heterojunctions made of semiconductors. THEORY AND CALCULATION RESULTS We treat the Coulomb interaction among electrons in 2D electrodes as a perturbation. According to Hedin’s theory [3], a formal way of calculating the self-energy is given by a set of five coupled equations in the space-time domain as follows: (1) G (1, 2) G0 (1, 2) d (34)G0 (1, 3)(3, 4)G (4, 2), W (1, 2) v(1, 2) d (34)v (1,3) (3, 4)W (4, 2),
(2)
(1, 2) i d (34)G (1,3)(3, 2, 4)W (4,1),
(3)
(1, 2) i d (34)G (1,3)(3, 2, 4)G (4,1),
(4)
(1, 2) G (4, 6)G (7,5)(6, 7,3), (5) G (4,5) where (i) denotes (ri , ti ) , G(1, 2) is the dressed Green’s function, G0 (1, 2) is the free Green’s function, (1, 2) is the self-energy, v(1, 2) v(r1 , r2 ) (t1 t2 ) is the bare Coulomb interaction, W (1, 2) is the screened Coulomb inter
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