Generalized Bloch theorem for complex periodic potentials: A powerful application to quantum transport calculations

Band-theoretic methods with periodically repeated supercells have been a powerful approach for ground-state electronic structure calculations but have not so far been adapted for quantum transport problems with open boundary conditions. Here, we introduce a generalized Bloch theorem for complex periodic potentials and use a transfer-matrix formulation to cast the transmission probability in a scattering problem with open boundary conditions in terms of the complex wave vectors of a periodic system with absorbing layers, allowing a band technique for quantum transport calculations. The accuracy and utility of the method are demonstrated by the model problems of the transmission of an electron over a square barrier and the scattering of a phonon in an inhomogeneous nanowire. Application to the resistance of a twin boundary in nanocrystalline copper yields excellent agreement with recent experimental data.

Figure 1

(Color online) Schematics of the quantum transport problem. (A) A molecule connected to metal electrodes. (B) A thin film tunnel junction.

Figure 2

(Color online) Schematics of possible approaches to compute quantum conductance. A periodic array of molecules with parts of the electrodes attached, (A) with absorbing layers inserted into the electrodes to damp out the resonances, (B) without the absorbing layers, and (C) with free surfaces.

Figure 3

(Color online) Reference system containing only the electrodes and the absorbing layers. (A) A single absorbing layer with electrodes. (B) A periodic array of supercells.

Figure 4

Transmission probability as a function of electron energy through a one-dimensional square barrier. Two curves (calculated and exact) are indistinguishable in the figure.

Figure 5

Transmission of a phonon through a nanowire with a localized defect as a function of frequency. The transverse modes are $n=0$, 1, and 2. The impurity potential is positive (negative) for the top (bottom) panel.

Figure 6

Resistance of a twin boundary in copper as a function of impurity concentration on the interface layer. The two solid lines are vacancies and Pb impurities. The dashed line is vacancies without vertex corrections (VC).