Figure 1

(Color online) [(a) Reprinted with permission from J. Reichert, H. B. Weber, M. Mayor, and H. v. Lohneysen. Appl. Phys. Lett. 82, 4137, copyright 2003, American Institute of Physics; (b) Reprinted by permission form Macmillan Publishers Ltd: J. Park, A. N. Pasupathy, J. I. Goldsmith, C. Chang, Y. Yaish, J. R. Petta, M. Rinkoski, J. P. Sethna, H. D. Abruna, P. L. McEuen, and D. C. Ralph, Nature 417, 722, copyright (2002)]” (a), (b) Experimental and (c), (d) theoretical I-Vs for a molecular ring weakly coupled with a backbone or with conducting electrodes. Many nontrivial features such as low zero-bias conductance, sharp current onset, and a subsequent quasilinear region spanning several volts with multiple closely spaced features (a)–(d) arise from excitations in our treatment of CB. Such features, however, do not arise even qualitatively in a spin-restricted SCF (RSCF) treatment for the same parameter set[30] [inset in (c)], or from an orthodox Coulomb Blockade theory that does not capture size quantization and the physics of excitations. For asymmetric contacts, there are additional features (b), (d) including current step heights (as opposed to widths) that are asymmetric in bias, are modulated with a gate voltage,[7] and reverse polarity for gate voltages on either side of the charge degeneracy point.[16] Electron-phonon interactions smoothen out the first few low-energy plateaus, but are typically inadequate for generating the unique higher energy features.Reuse & Permissions

Figure 2

(Color online) (a) Fock space, (b) equilibrium occupancy $N-\mu$ and (c) nonequilibrium I-V (assuming equal voltage division between contacts) of a spin degenerate level $ϵ=1\mathrm{eV}$ with a single electron charging energy $U=1\mathrm{eV}$. A spin restricted SCF calculation (gray dashed line) shows fractional charge occupation and is inappropriate in the weak coupling limit. An unrestricted SCF (blue line with circles) describes integer charge transfer and matches the many-body $N-\mu$ (black solid line) plot in (b); however, it yields equal current steps in (c) corresponding to sequential removal/addition of two electrons, as opposed to a many-body calculation in which step heights are in the ratio of 2:1. Including correlations in SCF alters the current onsets and the plateau widths, but misses the essential point that consecutive removal/addition of spins need not carry equal current.[12, 22]Reuse & Permissions

Figure 3

(Color online) Comparison between SCF and Coulomb blockade type calculations. A spin unrestricted calculation (USCF) for ${\gamma }_1∕{\gamma }_2$ equal to (a) 2 and (b) 10 yields varying plateau widths along the two bias directions consistent with RSCF calculations discussed in;[6] while maintaining the original $({\gamma }_L={\gamma }_R)$ plateau heights. Contact asymmetry yields varying plateau heights as opposed to widths in a typical CB type calculation for the same parameter ratios (c) and (d).Reuse & Permissions

Figure 4

Coulomb blockade I-V features for a general molecular system, showing transitions at threshold involving (1) only ground states. Here $\mid E_F-{ϵ}_{00}^{Nr}\mid =10\mathrm{meV}$. ${ϵ}_{00}^{Nr}$ is accessed before ${ϵ}_{10}^{Nr}$ (shown in the adjacent state transition diagram). I-V characteristics [black dotted line in (c)] then has a brief intervening plateau until an excitation is accessed. (2) Threshold transition involving excited states. Current at threshold $(\mid E_F-{ϵ}_{00}^{Nr}\mid =30\mathrm{meV})$ involves a transport channel involving excited states also (say ${ϵ}_{10}^{Nr}$) i.e., ${ϵ}_{10}^{Nr}$ is accessed before ${ϵ}_{00}^{Nr}$. In this case (see text) current rise [black solid line in (c)] due to closely spaced excitations follows upon threshold.Reuse & Permissions

Figure 5

(Color online) (a) State transition diagram showing various addition and removal pathways for asymmetric contacts $({\gamma }_L⪢{\gamma }_R)$, including the possibility of populating higher excitations (b), say, via transport channel ${ϵ}_{20}^{Nr}$ at threshold. For positive bias charge removal is the rate limiting process, while for negative bias addition dominates, accounting thus for the corresponding I-V asymmetry in (c). Progressive access of higher excitations also accounts for the observed gate modulation of the current steps, as shown in (d).Reuse & Permissions