t Matrix Method

t Matrix Method

We find the transmission probability T(E) used in the Landauer formula, Eq.(1), by solving the Schroedinger equation directly for the scattered wavefunction of the electron. The electron is initially propagating in a Bloch wave in one of the modes of the source lead. The molecule will reflect some of this wave back into the various modes of the source lead. The molecule is represented by a discrete set of molecular orbitals (MO's) through which the electron can tunnel. Hence, some of the wave will be transmitted through the molecule and into the modes of the drain lead. By finding the scattered wavefunction it is then possible to determine how much was transmitted, yielding T(E).

We start with Schroedinger's equation, $H\vert\Psi^\alpha\rangle = E\vert\Psi^\alpha\rangle$ , where H is the Hamiltonian for the entire MW system consisting of the leads and the molecule. $\vert\Psi^\alpha\rangle$ is the wavefunction of the electron propagating initially in the $\alpha^{th}$ mode of the left lead with energy E. It is expressed in terms of the transmission and reflection coefficients, $t_{\alpha,\alpha'}$ and $r_{\alpha,\alpha'}$ and has different forms on the left lead (L), molecule (M), and right lead (R). The total wavefunction is a sum of these three, $\vert\Psi^{\alpha} \rangle = \vert\Psi^{\alpha}_{L}\rangle + \vert\Psi^{\alpha}_{M}\rangle + \vert\Psi^{\alpha}_{R}\rangle$ , where

$\displaystyle \vert\Psi^{\alpha}_{L}\rangle$	=	$\displaystyle \vert\Phi_{+}^{\alpha}\rangle + \sum_{\alpha'} r_{\alpha',\alpha} \vert\Phi_{-}^{\alpha'}\rangle$	(2)
$\displaystyle \vert\Psi^{\alpha}_{M}\rangle$	=	$\displaystyle \sum_j c_j \vert\phi_j\rangle$	(3)
$\displaystyle \vert\Psi^{\alpha}_{R}\rangle$	=	$\displaystyle \sum_{\alpha'} t_{\alpha,\alpha'} \vert\Phi_{+}^{\alpha'}\rangle$	(4)

In the above, $\vert\Phi_{\pm}^{\alpha}\rangle$ are forward/backward propagating Bloch waves in the $\alpha^{th}$ mode, and $\vert\phi_j\rangle$ is the j^th MO on the molecule.

We then consider our Hamiltonian in the tight-binding (or Hückel) approximation and use Schroedinger's equation to arrive at a system of linear equations in the unknown quantities, $r_{\alpha',\alpha}$ , c_j and $t_{\alpha',\alpha}$ . We solve numerically for the reflection and transmission coefficients for each rightward propagating mode $\alpha$ at energy E in the left lead. The total transmission is then given by

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