Inelastic Scattering in Metal-H2-Metal Junctions
I. S. Kristensen, M. Paulsson, K. S. Thygesen, K. W. Jacobsen
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a y Inelastic Scattering in Metal-H -Metal Junctions I. S. Kristensen , M. Paulsson , K. S. Thygesen , and K. W. Jacobsen Center for Atomic-scale Materials Design (CAMD),Department of Physics, Technical University of Denmark, DK - 2800 Kgs. Lyngby, Denmark and Division of Physics, Department of Natural Sciences, Kalmar University, 391 82 Kalmar, Sweden (Dated: December 4, 2018)We present first-principles calculations of the dI/dV characteristics of an H molecule sandwichedbetween Au and Pt electrodes in the presence of electron-phonon interactions. The conductance isfound to decrease by a few percentage at threshold voltages corresponding to the excitation energyof longitudinal vibrations of the H molecule. In the case of Pt electrodes, the transverse vibrationscan mediate transport through otherwise non-transmitting Pt d -channels leading to an increase inthe differential conductance even though the hydrogen junction is characterized predominately bya single almost fully open transport channel. In the case of Au, the transverse modes do not affectthe dI/dV because the Au d -states are too far below the Fermi level. A simple explanation of thefirst-principles results is given using scattering theory. Finally, we compare and discuss our resultsin relation to experimental data. PACS numbers: 73.63.Rt, 72.10.Fk, 85.65.+h
In recent years it has become possible to measure theelectrical properties of single molecules captured betweenmetallic electrodes [1, 2, 3]. Such experiments providea unique opportunity to develop our understanding ofbasic quantum mechanical phenomena at the nanometerlength scale and at the same time constitute the firststeps towards molecule-based electronics. [4]Interactions between the conduction electrons and themolecule’s vibrational degrees of freedom is of particu-lar interest for the performance of molecular electron-ics devices as they determine the local temperature andstability of the device when subject to an external biasvoltage[5]. Moreover, inelastic scattering can be used toidentify the atomic structure of molecular junctions byexploiting the sensitiveness of the molecule’s vibrationalfrequencies and the electron-phonon interaction to thejunction geometry. [6, 7, 8, 9, 10, 11, 12]Perhaps the simplest molecular junction consists ofa single hydrogen molecule sandwiched between metalelectrodes, see Fig. 1. [3, 13] Shot noise measurementson Pt-D contacts show that the conductance is car-ried predominantly by a single almost fully transparentchannel[14], and density functional theory (DFT) calcu-lations have shown that this is consistent with a lin-ear bridge configuration. [3, 15, 16, 17]. An alterna-tive configuration where the H molecule is dissociatedin the contact has also been proposed, however, this junc-tion yields a conductance larger than 1 G ( G = 2 e /h is the conductance quantum) with contributions fromthree channels [18]. Inelastic point contact spectroscopyprovides information about the hydrogen molecule’s vi-brational frequencies and their variation upon stretch-ing. The data obtained from such measurements havealso been found to be consistent with the linear bridgeconfiguration.[7]The fact that the hydrogen junction supports a sin-gle, almost fully open conductance eigenchannel suggests that the inelastic scattering processeses should be par-ticularly simple to understand. Indeed, consider a junc-tion supporting a single scattering channel at the Fermienergy with a transmission probability of T = | t ( ε F ) | .At low temperatures the molecule sits in its vibrationalgroundstate and the electron looses the energy ~ Ω to themolecule during a scattering event. Assuming a bias volt-age eV = µ L − µ R > ~ Ω an electron incident on themolecule from the left with an energy just below µ L ,must end up in a left moving scattering state after in-teracting with the molecule. This follows from energyconservation and the Pauli principle. Upon inelastic scat-tering, the probability for the electron to enter the rightelectrode is thus changed from T to R = 1 − T . Conse-quently, the change in conductance due to the electron-phonon interaction should be proportional to 2 T − T < . T > . Ojunctions [22].In this paper we present DFT calculations for the dI/dV curves of Pt-H -Pt and Au-H -Au junctions inthe presence of electron-phonon interactions. For both Ptand Au electrodes, scattering on the longitudinal modeslowers the conductance by a few percentage of G in ac-cordance with the simple one-channel model discussedabove. In the case of Pt, the transverse modes can me-diate tunneling through the otherwise closed d -channelsleading to an increase in the conductance of up to 5%of G , demonstrating that the metal-H -metal junctioncannot be viewed as a simple one-channel system. ForAu, the transverse modes have no effect on the conduc-tance because only s -states are present at the Fermi leveland these do not couple via the transverse vibrations. FIG. 1: The supercell use to model the metal-H -metal junc-tion. Only the hydrogen atoms are allowed to vibrate (the”dynamic” atoms). This is a good approximation due to thelarge difference in mass between Au/Pt and H. The effect ofthe field generated by the vibrating H atoms is taken into ac-count inside the indicated inelastic region. The central region, C , is coupled to semi-infinite bulk electrodes and periodicboundary conditions are imposed in the directions perpendic-ular to the contact axis. The Hamiltonian of the system is given byˆ H = ˆ H el + ˆ H ph + ˆ H el-ph , (1)where ˆ H el is the Hamiltonian of electrons moving in the static equilibrium structure, ˆ H ph describes the vibrationsof the H molecule, and ˆ H el-ph is the interaction betweenthe electrons and the vibrating hydrogen atoms. For ˆ H el we use the Kohn-Sham Hamiltonian.Within the harmonic approximation the molecularvibrations are described by the Hamiltonian ˆ H ph = P λ ~ Ω λ ( b † λ b λ + ) where b † λ ( b λ ) creates (destroys) aphonon in mode λ . The electron-phonon interactiontakes the formˆ H el-ph = X n,m ∈ C X λ M λnm c † n c m ( b † λ + b λ ) (2)where the first sum runs over Wannier functions lo-cated in the inelastic region, see Fig. 1, and the sec-ond sum runs over vibrational modes. The electron-phonon coupling matrix, M λ , is given by M λnm = h φ n ( r ) | W λ ( r ) | φ m ( r ) i , where the displacement potential, W λ ( r ) = ∇ v s [ { R n } ]( r ) · Q λ , is the derivative of the effec-tive KS potential in the direction defined by eigenmode λ . In practice W λ is obtained as a finite difference be-tween equilibrium Hamiltonians describing the electronicsystem when the hydrogen molecule has been moved inthe positive and the negative normal direction.The current flowing into the molecule (central region C ) from lead α = L, R is calculated from the formula [28,29] I α = eh Z Tr h Σ <α ( ε ) G >C ( ε ) − Σ >α ( ε ) G
111 (2005).[28] H. Haug and A. -P. Jauho,
Quantum Kinetics in Trans-port and Optics of Semiconductors , Springer (1998).[29] Y. Meir and N. S. Wingreen, Phys. Rev. Lett. , 2512(1992).[30] Note that both the Green functions and self-energies de-pend on the bias voltage, however, for notational simplic-ity we do not show this dependence explicitly.[31] T. Frederiksen, M. Paulsson, M. Brandbyge, and A-P.Jauho Phys. Rev. B , 205413 (2007). [32] A. Gagliardi, G. C. Solomon, A. Pecchia, T. Frauenheim,A. Di Carlo, N. S. Hush, and J. R. Reimers, Phys. Rev.B , 174306 (2007)[33] J. R. Taylor, Scattering Theory , Dover, New York (2000).[34] M. Brandbyge, M. R. Sorensen, and K. W. Jacobsen,Phys. Rev. B , 14956 (1997)[35] M. Paulsson, T. Frederiksen, M. Brandbyge, J. Phys.Conf. Ser.35