Localized polarons and doorway vibrons in finite quantum structures

Localized polarons and doorway vibrons in finite quantum structures


We consider transport through finite quantum systems such as quantum barriers, wells, dots or junctions, coupled to local vibrational modes in the quantal regime. As a generic model we study the Holstein-Hubbard Hamiltonian with site-dependent potentials and interactions. Depending on the barrier height to electron-phonon coupling strength ratio and the phonon frequency we find distinct opposed behaviors: Vibration-mediated tunneling or intrinsic localization of (bi)polarons. These regimes are strongly manifested in the density correlations, mobility, and optical response calculated by exact numerical techniques.


I Introduction

Recent progress in nanotechnology has triggered a systematic study of electronic transport in microscopic systems weakly coupled to external electrodes (1). In such devices the active element can be a single organic molecule, but also a suspended Carbon nanotube, and may be thought of as a quantum dot contacted to metallic leads that act as macroscopic charge reservoirs. In small quantum dots energy level quantization becomes as important as electron correlations. Additionally vibrational modes play a central role in the electron transfer through quantum dots or molecular junctions (see, e.g., the topical review Ref. (2)).

The electron-phonon (EP) interaction is found to particularly affect the dot-lead coupling. Here electronic and vibrational energies can become of the same order of magnitude, e.g. when Coulomb charging is reduced by screening due to the electrodes (3). The same circumstance holds in the polaron crossover regime, where the electrons are dressed by a phonon cloud, implying that phonon features for the current through the quantum device are of major importance (4). Phonon and polaron effects in nanoscale devices have been extensively discussed e.g. for (magnetic) molecular transistors (5); (6); (3), quantum dots (7), tunneling diodes and Aharonov-Bohm rings (8), metal/organic/metal structures (9), or Carbon nanotubes (10).

In this paper we study the electronic properties of various EP coupled quantum systems. We consider one-dimensional structures, where the “quantum device” is sandwiched between two metallic wires characterized by (tight-binding) electron hopping amplitude , local Coulomb interaction , and EP coupling (cf. Fig. 1). Such systems may be described by a generalized Holstein-Hubbard Hamiltonian. The Holstein-Hubbard model (11); (12); (13); (14) is not completely realistic, of course, as it only includes local electron-phonon and electron-electron interactions as well as a coupling to (dispersionless) optical phonons. However, we are interested in fundamental phenomena arising from the combination of electron-phonon interaction and “confinement” in discrete quantum structures. Besides many aspects of finite (EP coupled) quantum systems may be understood using such simplified effective models (5); (6); (15).

Figure 1: (Color online) Schematic representation of model devices described by the Hamiltonian (1).

Ii Model

Allowing for site-dependent potentials and electron-phonon/electron interactions, the tight-binding Holstein-Hubbard Hamiltonian takes the form


Here , where the potentials, on site , can describe a tunnel barrier, disorder, or a voltage basis. Since we will treat left and right leads in equilibrium, we choose throughout the sample, neglecting a bias between the metal leads and, in order to avoid spurious multi-scattering from the boundaries in a finite system, we take periodic boundary conditions. The parameter (), can be viewed as additional Hubbard interaction or charging energy of, e.g., a quantum dot molecule. The parameter describes the local coupling of an electron on site to an internal optical vibrational mode at the same site (16). Here denotes the corresponding polaron binding energy, and is the frequency of the optical phonon (17). In this way the model, e.g., mimics tunneling through (single or double) barriers (), trapping of electrons, polarons, or bipolarons at single-impurity or double-well sites (), or transport through quantum dots with soft dot-lead links.

On a translational invariant lattice (, , ) the Holstein-Hubbard model can be numerically solved by variational diagonalization in the one- and two-particle sectors of interest here. This holds in the thermodynamic limit, for the whole range of parameters and any dimension (for a recent review of the Holstein (bi)polaron problem see Ref. (18)). The main result is a continuous cross over with increasing EP coupling strength, from electronic quasiparticles weakly renormalized by phonons to (small) polarons or bipolarons (19). Depending on the value of the adiabaticity ratio , in one-dimensional systems, the large-to-small polaron cross over is determined by the more restrictive of the two conditions (relevant for , adiabatic regime) or (for , anti-adiabatic regime) (20).

Here we address the problems of polaron/bipolaron formation and phonon-assisted transport for the more complicated inhomogeneous barrier structures and interactions described by the above Hamiltonian.

Iii Numerical Results and Discussion

In our numerical work we combine exact diagonalization (ED) and kernel polynomial methods (21); (22) to determine the ground-state and spectral properties. All energies will be measured in units of .

iii.1 Single-electron case

We first consider a single electron that tunnels through a single quantum barrier. The barrier height is assumed to considerably exceed the electron half-bandwidth. Outside the barrier the electron is subjected to a rather moderate EP coupling, . The chosen phonon frequency reflects an adiabatic situation.

Figure 2 shows the behavior of the system’s kinetic energy


as the EP coupling strength is increased at the barrier site. Recall that both coherent and incoherent transport processes contribute to . Without loss of generality we assume the barrier to be located at site . For the barrier is almost impermeable, consequently the local electron density is near zero at site . An additional local EP interaction renormalizes the on-site adiabatic potential, i.e., it leads to a local polaronic level shift that softens the barrier. Note that the kinetic energy stays almost constant until exceeds a certain critical value, . At , the mobility of the electron is arrested, and the charge carrier becomes quasi localized at the barrier site (23). The large number of bound vibrational states () give rise to a displaced oscillator state at site , i.e. a new equilibrium state of the lattice results, which lowers the energy. The jump-like transition is in striking contrast to what is observed if we increase only the EP coupling locally (without having a barrier), or if we form a quantum well () without additional EP interaction (see inset of Fig. 2). In these cases we found a gradual transition from a nearly free electron to a rather immobile particle.

Figure 2: (Color online) Kinetic energy of a single electron on a site ring with potential barrier at site . The main panel gives (squares), the electron density (circles), and the mean phonon number (diamonds) at the barrier site as functions of an additional EP coupling . The inset shows the variation of if the potential is lowered keeping (triangles up), or if is raised with (triangles down).

The extremely sharp polaron transition is accompanied by a drastic change in the optical response. The regular part of the optical conductivity is given by


where is the current operator and label the eigenstates of with excitation energy .

Figure 3 shows and the integrated spectral weight


in the vicinity of the transition, where a tiny increase of (of about , from top to bottom) substantially changes the optical spectra. While the upper panel resembles the optical spectra of a large polaron with an absorption maximum at small frequencies and a rather asymmetric line shape, we found a bimodal signature near the transition point (middle panel) and finally the typical (almost symmetric) small polaron absorption just above (lower panel). In this manner the system acts as an optical switch.

Corresponding behavior is found if we increase the barrier (voltage bias) keeping fixed (see Fig. 4). Again the transition is “discontinuous” for small phonon frequencies, where the concept of an adiabatic energy surface holds to a good approximation. At larger phonon frequencies non-adiabatic effects become increasingly important. Here the EP coupling does not work against the (static) barrier directly and the transition softens as in normal polaronic systems. Furthermore, for , the EP coupling constant is reduced (i.e., although is fixed we leave the strong coupling regime).

Figure 3: (Color online) Optical response for the single-barrier system for various . Dashed lines give the integrated weight .

In Fig. 4 we have included the results obtained by a simple approximative analytical approach to the single-barrier problem. Assume that


with is the probability for finding the particle at site away from the barrier site . Then, for the infinite system, the ground-state energy of a polaron with radius , where corresponds to the free electron while describes a small polaron localized at the impurity site, is given as




Of course, has to be minimized with respect to . Although the kinetic energy calculated in this way neglects important contributions from multi-phonon processes (24), we see that gives a reasonable estimate for the critical value of , at least in the adiabatic regime. In the anti-adiabatic region, fails to describe the observed continuous cross over. This is a well-known shortcoming of such a kind variational approaches, which normally yield an abrupt polaron transition in the whole frequency range (11).

Figure 4: (Color online) Kinetic energy upon increasing the barrier height at constant EP coupling. Filled (open) symbols denote ED (analytical) results for ().

iii.2 Two-electron case

Next we investigate two electrons in a single-barrier structure. Now, increasing the EP coupling on a barrier site with strong Coulomb repulsion, we found two successive transitions, see Fig. 5. In the first step one electron becomes localized at the barrier site blocking, because of the large , the second one. Raising further, both particles will be trapped, forming an on-site bipolaron. This can be seen most clearly by monitoring the density correlation


as a function of the EP coupling. and clearly also show this two-step transition, being related to significant changes of the ground-state phonon distribution (19); (22), , see insets. The comparison of data for , 10 shows that there is almost no finite-size dependence of the results.

Figure 5: (Color online) Kinetic energy (squares) and mean electron/phonon numbers (circles/diamonds) for the case of two electrons in a singlet state. We assumed strong Hubbard interaction at the barrier (, ). Otherwise , , and . The insets display the weight of the -phonon state in the ground state, , for several characteristic .

Finally let us consider the double-barrier quantum dot structure sketched in Fig. 1, with two electrons in the system.

Figure 6: (Color online) Two electrons in a quantum double-barrier system (), where the potential of the central site is lowered with respect to the leads (cf. Fig. 1). Shown are the kinetic energy (squares) and the on-site density correlations (triangles) for , and , in the upper panel, while , in the lower panel.

We plot in Fig. 6 the kinetic energy and the particle occupation of the barrier and embedded dot sites as functions of the depth of the quantum well (). The upper panel describes the regime of moderate Coulomb interaction at the dot, with otherwise. Here the dot is unoccupied until its potential is lowered below a critical value. Then the particles initially located together at one of the dot-lead sites are transferred onto the dot. In this process they change their nature from a bipolaronic quasiparticle to two electrons solely (linearly dependent) bound by the potential well (impurity). Thus the ground state is a multi-phonon (few-phonon) state for (). If the system has a large Coulomb interaction everywhere, double occupancy is prohibited (lower panel). Then we find initially one polaron per barrier (lead-dot) site and only one particle tunnels to the dot at , thereby stripping its phonons away. Note that the mobility is enhanced in the transition region.

Figure 7: (Color online) Quantum-well configuration with two electrons. Parameters as in Fig. 6 (upper panel) but now . For comparison we show for a reference system without any EP coupling at the links.
Figure 8: (Color online) Optical conductivity for a “soft-linked” quantum dot system with two electrons (parameters as in Fig. 7). The insets give the corresponding phonon distribution functions.

This effect is even more pronounced if we suppose that the EP coupling acts on the dot-lead link sites only. As can be seen from Fig. 7 there is large jump-like increase of the particle’s kinetic energy if the quantum well reaches . At the bipolaron, located at one of the dot-lead link sites, dissolves and the electrons can pass over to the dot. Clearly decreases if we lower the potential of the quantum dot further but note that for the kinetic energy is still larger than for a reference system without EP coupled dot-lead link sites. In this way the local coupling to vibrational degrees of freedom of the barrier opens the gate for particle transmission, i.e., vibronic excitations play the role of “doorway states”.

To corroborate the importance of these quantum lattice fluctuation effects we determined the optical spectra below, near, and above the threshold . The data presented in Fig. 8 give clear evidence for (bi)polaron hopping transport for a shallow quantum well, with dominant phonon emission and absorption processes, but resonant vibration-mediated tunneling takes place for a deeper well.

We emphasize that the increase of in passing below is accompanied by a decrease of the total integrated weight of the regular (incoherent) part of (compare dashed lines in Fig. 8 from top to bottom). Thus, exploiting the f-sum rule,


we can conclude that the coherent contribution (Drude part ) to is amplified. The insets substantiate this interpretation. Starting from a Poisson-like distribution of , a second maximum develops at for , and finally, for , the ground state contains only zero-, one- and two-phonon states with substantial weight.

Iv Summary

To conclude, investigating finite quantum structures coupled to vibronic degrees of freedom in the framework of a generalized Holstein-Hubbard Hamiltonian, we have demonstrated that interesting new physics, such as intrinsic (bi)polaron localization or phonon-assisted transmission, emerges when the energy scales set by external potentials, Coulomb and electron-phonon interactions become comparable. In this regime the interplay between the linear effects resulting from the barriers/cavities and the nonlinearity inherent in a discrete interacting electron-phonon system is of major importance. A general understanding of vibrational effects in (molecular) quantum transport, however, is still far off. Our objects in view will be to study (i) how polaronic quasiparticles time evolve when passing through phonon-coupled nanoscale structures and (ii) how finite temperature (heating) affects the balance between coherent and incoherent transport mechanisms.


The authors would like to thank A. Alvermann, G. Schubert, and S. A. Trugman for useful discussions. This work was supported by DFG through SFB 512 and Grant No. 436 TSE 113/33, KONWIHR Bavaria, Academy of Sciences of the Czech Republic, and U.S. DOE. Numerical calculations were performed at LRZ Munich. H.F. and G.W. acknowledge hospitality at the Los Alamos National Laboratory and the Institute of Physics AS CR.


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