Conductance of atomic chains with electron-electron interaction: The embedding method

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1 Conductance of atomic chains with electron-electron interaction: The embedding method Jean-Louis Pichard 1,2, Rafael A. Molina 1, Peter Schmitteckert 3, Dietmar Weinmann 4, Rodolfo A. Jalabert 4, Gert-Ludwig Ingold 5 1 CEA/DSM, Service de Physique de l Etat Condensé, Gif-sur-Yvette, France 2 LPTM, Université de Cergy-Pontoise, Cergy-Pontoise Cedex, France 3 Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, Karlsruhe, Germany 4 IPCMS, UMR 7504 (CNRS-ULP), 23 rue du Loess, BP 43, Strasbourg Cedex 2, France 5 Institut für Physik, Universität Augsburg, Universitätsstraße 1, Augsburg, Germany We review a new method, the embedding method, for calculating the residual conductance of an interacting nanosystem. Using DMRG techniques, we show that a nanosystem with interacting spinless fermions embedded into a non-interacting auxiliary lead behaves as a non-interacting scatterer, but with an interaction dependent elastic transmission coefficient. The method, which requires to take the limit of infinite lead length, allows to calculate the conductance of atomic chains as a function of the number of atoms and the strength of the electron-electron interaction. The results depend on the nature (smooth or abrupt) of the two contacts between the system and the lead and on the carrier density. Length-dependent oscillations of the conductance, whose period depends on the electron density in the atomic chain, can result from electron-electron scattering. 1 The embedding method and the auxiliary lead The conductance of a nanostructure with electron-electron interactions can be obtained from the persistent current of a ring formed by the interacting nanostructure and an auxiliary noninteracting lead playing the role of a coherent reservoir of electrons. Such an embedding method, which requires to take the limit of infinite lead length keeping the mean number of particles in the interacting region fixed, has been actively pursued in the last years 1 8. We summarize its main hypotheses and some consequences for atomic chains. Without interaction, the conductance measures how electrons emitted at the Fermi energy of two incoherent reservoirs are elastically scattered by the nanosystem under study. In the reservoirs, the electron density is high, the Coulomb energy to kinetic energy ratio small, making the electron-electron interaction essentially negligible. The electrons inside the reservoirs are then well described by a Fermi distribution, and the residual conductance g (in units of e 2 /h) is given by the probability t(e F ) 2 of an electron at the Fermi energy E F to be elastically transmitted through the nanosystem. The situation becomes more complicated if electronelectron interaction is present inside the scattering region because the passage of electrons may lead to the creation of excitations. However, for temperatures smaller than the characteristic excitation energy of the nanosystem, the scattering formalism still applies 9 because all accessible states in the reservoirs with an energy lower than this excitation energy below E F are occupied. Inelastic processes are then forbidden and g is still given by t(e F ) 2. However, determining t(e F ) 2 for an interacting nanosystem remains a non-trivial task which can be achieved by the embedding method. Let us first explain why the embedding method, which can be rigorously established for a non-interacting scatterer 10, may be extended to the more difficult case of a scatterer with arbitrary electron-electron interactions. We consider a ring with a very long non-interacting lead embedding the nanosystem of interest, as depicted in the upper left figure. A flux threading the ring gives rise to a persistent current. When the lead length becomes infinitely long and

2 the electron density is kept uniform, the huge majority of electrons does not interact inside the combined setup, except the few ones inside the nanosystem. The infinite auxiliary lead plays the role of an electron reservoir with Fermi-Dirac statistics, having at zero temperature fully occupied one-particle states below the Fermi energy. This makes inelastic scattering impossible inside the embedded nanosystem, as for a conductance measurement between two incoherent reservoirs. This setup includes the main physical aspects of a transport measurement, though the ring must remain coherent for keeping a flux dependence of its ground-state energy, which contains information about electron propagation inside the interacting region and depends on the nature of the two contacts between the system and the lead. This makes a fundamental difference between the embedding method and other approaches based on the persistent current calculated for a correlated system without auxiliary lead. 2 Persistent current and transmission probability t(e F, U) 2 The relationship between the transmission probability t(e F ) 2 and the persistent current J(Φ) of a large ring of length L has only been established for non-interacting scatterers 10. J(Φ) decreases like 1/L and the leading contribution reads 4 for an odd number of particles and J(Φ) = ev F Arccos ( t(e F ) cos(φ) ) πl 1 t(ef ) 2 cos 2 (Φ) t(e F) sin(φ) (1) J(Φ) = ev F πl Arccos ( t(e F ) cos(φ π) ) t(e F ) sin(φ) (2) 1 t(ef ) 2 cos 2 (Φ) for the case of an even number of particles in the ring. This allows to obtain t(e F ) 2, and hence g, from J(Φ). t(e F ) 2 can also be obtained from the charge stiffness D = ( 1) N L 2 ( E(0) E(π) ) (3) which gives the change of the ground-state energy E from periodic (Φ = 0) to antiperiodic (Φ = π) boundary conditions. D can be obtained from a real Hamiltonian, thus avoiding the numerically more demanding DMRG algorithm required by complex Hamiltonians. One gets 4 ( ) π D t(e F ) = sin 2 D 0 where D 0 is the charge stiffness for a clean ring of length L without scatterer. To study the embedding method for an interacting scatterer, one considers a tight-binding model of N interacting spinless fermions on L sites described by the Hamiltonian H = t L L S (c i c i 1 + c i 1 c i ) + U [n i V + ] [n i 1 V + ]. (5) i=1 t = 1 is the hopping matrix element between nearest neighbors, c i (c i ) is the annihilation (creation) operator at site i, n i = c i c i, and the flux enters through the boundary condition c 0 = exp(iφ)c L. The interaction U 0 between nearest neighbors inside the scatterer (sites i = 1 to L S ), but vanishes in the lead of L L sites. To have a uniform density, we introduce a potential V + that acts as a positive background charge. For a half-filled ring ν = N/L = 1/2, V + = 1/2, a value which guarantees particle-hole symmetry even when U 0. When ν 1/2, V + i=2 (4)

3 becomes a function of U, N, L S and L L which has to be determined self-consistently. We begin with an initial guess for V +, then, by means of a DMRG run during which V + is kept constant, the number of particles contained inside the nanosystem is determined and an improved value for V + is estimated. This step is repeated within a Newton-Raphston iteration until the desired filling ν in the correlated system is reached. We have numerically investigated ground state properties of (5) by means of the DMRG method 11 in implementations for real and complex Hamiltonians. One first needs to study the length dependence of the persistent current and its extrapolation to infinite lead length. This is given in the upper right figure for three values of U and Φ. A second-order polynomial fit describes very well the deviation of the logarithm of the persistent current from its asymptotic value. In the middle left figure, one can see that the current J(Φ) of a ring containing an interacting region is quantitatively described when L by the persistent current of a ring with a non interacting scatterer only. A single parameter, the interactiondependent elastic transmission coefficient at the Fermi energy t(e F, U) 2 suffices to characterize the interacting sample, at least as far as the flux dependence of the ground state energy at zero temperature is concerned. We emphasize that the DMRG technique employed here to calculate J(Φ) does not rely on any assumption. In particular, the DMRG technique does not require that the correlated nanosystem must be a Fermi liquid. But the fact that the expressions (1) and (2) for the persistent current hold in the infinite lead length limit even in the presence of an interacting region provides strong evidence that the Fermi liquid behavior is retained in this limit, and that the embedding method can be extended to the case where there are electron-electron interactions inside the nanosystem. Let us mention that we have also used the embedding method for a superconducting nanosystem and verified 4 that the known behavior resulting from Andreev scattering at the two extremities of a superconducting nanosystem is reproduced. 3 Conductance of atomic chains with nearest neighbor repulsion 3.1 Conductance oscillations as a function of the number of atoms A recent measurement on atomic Au, Pt and Ir wires between break junction contacts 12 has revealed oscillations of the conductance as a function of the length of the chain. For an odd number of atoms in the atomic wire the conductance has maxima while it is reduced when the number of atoms is even. Using the embedding method and Hamiltonian (5), we have calculated the conductance of atomic chains. The middle right figure exhibits length-dependent oscillations of the conductance, whose period depends on the electron density in the chain, and which result from electron-electron scattering alone. Let us shortly explain the parity oscillations seen for ν = 1/2 and large U. When L S is odd, the ground state of the nanosystem has two degenerate components: one with (L S + 1)/2 particles on the odd sites, the other with (L S 1)/2 particles on the even sites. The two components are coupled through sequences of (L S +1)/2 one-particle hops. At half filling, this leads to a one-particle resonant tunneling situation and g = 1. When L S is even, the ground state is a superposition of two components having L S /2 particles, either on the even or the odd sites. These components are coupled by L S /2 one-particle hops which do not involve hops between the nanosystem and the lead and hence do not contribute to transport. The process which matters is now coherent cotunneling and one finds g 4(4t/U) L S for even L S in leading order of a t/u expansion. This is a signature of the Mott insulating behavior occurring for infinitely long chains at U > 2. Taking U = 0 in our model leads to a perfectly homogeneous ring and the conductance g = 1 for all fillings and lengths. In a non-interacting model, another scattering mechanism is necessary for reproducing the oscillating behavior of the conductance. A possibility is to suppose that the contacts between the atomic chain and the

4 0.3 5 Φ = π/4; U = 4 Φ = 3π/4; U = 4 Φ = π/2; U = 1 ln[j( Φ)/J ( Φ)] Φ = Φ π for L/2 odd Φ = Φ for L/2 even /L J(Φ)/J0(π/2) Φ U=1 U=4 1 g 0 ν = 1/2 ν = 1/3 ν =1/ N sites v=1/2 v=3/8 v=1/4 v=1/ Figure 1: Upper left: The ring used for the embedding method consists of the interacting region (grey) of length L S and a non-interacting lead (black) of length L L, threaded by an Aharonov-Bohm flux Φ. Upper right: Convergence of J( Φ) towards J ( Φ) when L = L S + L L increases with L S = 6 interacting sites and ν = 1/2. The definition of Φ is given in the figure (see Eqs. (1) and (2)). The data have been fitted by second-order 1/L polynomials to obtain the limit L. Middle left: The flux dependence of J(Φ)/J 0(Φ) with L S = 6 and ν = 1/2 for U = 1 and 4. The points represent DMRG results extrapolated to L L, J 0(Φ) being the current with U = 0. The lines represent the theoretical result (2) for a ring with a non-interacting scatterer and transmission amplitudes t = (solid line) and t = 25 (dotted line). Middle right: Conductance oscillations due to electron-electron scattering in an atomic chain of L S atoms with U = 4, connected to the lead by abrupt contacts at different fillings (ν = 1/2, ν = 1/3 and ν = 1/4). Lower left: Role of the carrier density: g is given as a function of U for different fillings ν for L S = 8 atoms in series and abrupt contacts. Lower right: Role of the contacts: g as a function of U for L eff = 12 and increasingly smooth contacts, defined by the length L A (see inset). All data correspond to even L S with L A = 0 ( ), L A = 4 ( ), and L A = 10 ( ).

5 electrodes are a source of reflections. But then the induced conductance oscillations would have a magnitude independent of the chain length, in contrast to the case where there is electronelectron scattering. 3.2 Role of the carrier density By choosing the particle number N = νl, we fix the average filling in the entire ring and thus the Fermi energy at which the conductance is evaluated. However, the interaction influences the filling in the nanosystem. This can be compensated by an appropriate choice of the potential V + in (5). In the lower left figure, the conductance of a nanosystem of length L S = 8 perfectly coupled to the lead is given as a function of U at different values of ν. Since ν is kept uniform everywhere in the ring, the curves characterize g(e F, U) at the corresponding Fermi energy E F. At ν = 1/8 (short dashed line), in average only one particle is left in the nanosystem and g 1, independently of U. For larger filling factors, the conductance g decreases with increasing interaction strength U and this decay becomes more pronounced as the filling factor is increased. The sharp drop of g occurring at half filling around U = 2 is a precursor of the Mott transition expected in the thermodynamic limit. 3.3 Smooth versus abrupt contacts The even-odd asymmetry, and the perfect transmission for the odd case, point to the importance of the contacts. In order to investigate their role, we introduce a position-dependent interaction strength U i which increases linearly from 0 to its maximum value U, inside the contacts of length L A (see inset of the lower right figure). As shown in the lower right figure for L S even, g increases when the contacts are smoother, at constant effective length L eff = L S L A of the sample. The conductance approaches the ideal situation of perfect transmission expected for a Luttinger liquid connected to leads through reflectionless contacts 13. We end by remarking that a more complete study of the embedding method and its application to atomic chains can be found in Refs. 3 5, including additional studies of the extrapolation procedure towards infinite lead length required by the method. References 1. J. Favand, F. Mila, Eur. Phys. J B 2, 293 (1998). 2. O.P. Sushkov, Phys. Rev. B 64, (2001). 3. R.A. Molina, D. Weinmann, R.A. Jalabert, G.-L. Ingold, J.-L. Pichard, Phys. Rev. B 67, (2003). 4. R.A. Molina, P. Schmitteckert, D. Weinmann, R.A. Jalabert, G.-L. Ingold, J.-L. Pichard, arxiv:cond-mat/ , to appear in Eur. Phys. J B. 5. R.A. Molina, D. Weinmann, J.-L. Pichard, arxiv:cond-mat/ V. Meden, U. Schollwöck, Phys. Rev. B 67, (2003). 7. T. Rejec, A. Ramšak, Phys. Rev. B 68, (2003). 8. T. Rejec, A. Ramšak, Phys. Rev. B 68, (2003). 9. Y. Meir, N.S. Wingreen, Phys. Rev. Lett. 68, 2512 (1992). 10. A.O. Gogolin, N.V. Prokof ev, Phys. Rev. B 50, 4921 (1994). 11. Density Matrix Renormalization A New Numerical Method in Physics, ed. by I. Peschel, X. Wang, M. Kaulke, and K. Hallberg, Lecture Notes in Physics, Vol. 528 (Springer, Berlin, 1999). 12. R.H.M. Smit, C. Untiedt, G. Rubio-Bollinger, R.C. Segers, J.M. van Ruitenbeek, Phys. Rev. Lett. 91, (2003). 13. I. Safi, H.J. Schulz, Phys. Rev. B 52, R17040 (1995).