Many-electron correlated scattering (MECS) is a quantum transport method that relies on use of the Wigner distribution function to apply single-particle scattering boundary conditions to a many-electron wave function [1, 2]. The method has been applied to describe tunnelling in single molecule junctions consisting of benzene dithiol (BDT) [1], alkane dithiols [3], alkane diamines [4], silane dithiols and silane diamines [5]. In each case, the predicted current-voltage characteristics are in good agreement with available experimental data [6, 7, 8, 9, 10]. For these single
molecule tunnel junctions, a range of experimentally determined conductances and several conductance peaks can appear; the explicit atomic configurations
corresponding to different conductance values has not been fully unravelled. Hence direct comparison to experiment can be difficult, although it should be mentioned that there is an improvement in the agreement between measured conductances in, for example, the case of BDT [6, 7, 8], and for the cases where amine linker
molecules are used to bond the molecule to the metal electrodes for which
well-defined conductance peaks result [9, 10]. However, comparison of the MECS calculations to date with experiment have been for molecular tunnel junctions in a weak coupling regime, as the molecules studied are bonded to electrodes via a linker group (-thiol, -amine) resulting in large contact resistances [3, 4].
Theoretically, the single-particle limit of the MECS method for a single ideal conducting channel with unity transmission, or in the strong coupling limit, has been studied for a simple analytical model yielding the well-known result of the conductance quantum [11, 12]. Here we consider a single conducting channel by considering an explicit atomic model of a gold atomic point contact. Gold point contacts have been well characterised experimentally with observation of
conductance quantisation in several measurements [13, 14, 15, 16, 17, 18, 19]. Comparing to these systems where the conductance value is well established theoretically and experimentally allows exploration of MECS in a strong coupling limit and permits an assessment of the explicit finite electrode models used in the calculations and errors associated with the numerical implementation of the theory. Relatively little work has been performed studying the effect of atomic orbital expansions on prediction of electron currents in single-particle models [20, 21], and even less is known on the role of many-electron expansions about the single-particle model [22]. In this study, we consider the problem from the standpoint of the many-electron expansion in spin-coupled Slater determinants. Hartree-Fock
2. Many-electron scattering applied
to atomic point contacts 2.2 Introduction
orbitals are used as single-particle basis states. As a voltage is applied across the junction, the junction polarises. A new set of self-consistent orbitals can be recalculated for each new voltage bias point, or equivalently a single configuration interaction (CI) expansion about the zero electric field self-consistent field solution
|Ψ >= N Y i=1 M Y m=N +1 (1 + Cm i a † mai)|Φ0 > (2.1)
can be performed. Here |Φ0 >is the zero voltage Hartree-Fock determinant, i is
the index for occupied single-electron states in |Φ0 >, N is the number of electrons,
m is the index for unoccupied single-electron states, M is the number of
single-electron states included in a finite expansion, a†
, a are the electron electron
creation and annihilation operators, and the Cm
i are the CI expansion coefficients
for singly-excited determinants. Thouless’ theorem states that any |Ψ > of the form eq. 2.1 is itself a single determinant (strictly, for M → ∞). Optimising the CI coefficients Cm
i for an expansion including only singly-excited determinants allows
us to explore the single-particle limit of the MECS method [3]. Adding higher order excitations into a CI expansion allows us to estimate the role of increasing electron correlations beyond a mean field solution.
Atomic point contacts are somewhat trivial examples for Green’s functions or single-electron scattering approaches to quantum transport in that it is only necessary within these methods to calculate the transmission of an electron impinging on the junction at the Fermi energy. For a single atomic state strongly coupled to the two leads, it is inferred from a unity or near unity transmission that the conductance from Landauer’s formula is 2e2/h, where e is the electron charge
and h is Planck’s constant. In the case of many-electron scattering, the
one-electron reduced density matrix is obtained from a full many-electron density matrix and the current density is obtained from the one-electron reduced density
matrix in the direction of current flow as
Jz(r) = e~
2mi[∂z − ∂z0]ρ(r, r0)|r,r0, (2.2)
where ρ is the density matrix, z is a Cartesian coordinate along the direction of current flow, r, r0 are position vectors, e and m are the electron charge and mass,
respectively, i =√−1 and ~ is Planck’s constant divided by 2π. Hence to accurately determine the conductance, an accurate determination of the
one-electron reduced density matrix and the applied voltage across the junction is required. Thus atomic point contacts pose a stringent test case for a MECS determination of the conductance. In the following, the junction electrostatics, sensitivity of the boundary conditions to the selection of the explicit electrode model, and the effect of the many-electron expansion on the electron current are studied. It will be demonstrated that the conductance quantum can be
approximately determined using MECS in an explicit atomistic junction model, thus allowing an evaluation of the accuracy of the method with respect to computational approximations.