EPJ manuscript No. (will be inserted by the editor) Zero point motion effect on the electronic properties of diamond, trans-polyacetylene and polyethylene E. Cannuccia1 and A. Marini2 1 Nano-Bio Spectroscopy Group and ETSF Scientific Development Center, Dpto. F´ısica de Materiales, Universidad del Pa´ıs Vasco, Av. Tolosa 72, E-20018 San Sebasti´an, Spain 2 Istituto di Struttura della Materia (ISM), Consiglio Nazionale delle Ricerche, Via Salaria Km 29.5, CP 10, 00016 Monterotondo Stazione, Italy Received: date / Revised version: date Abstract. It has been recently shown, using ab-initio methods, that bulk diamond is characterized by a giant band–gap renormalization (∼ 0.6eV) induced by the electron–phonon interaction. This result casts doubt on the accuracy of purely electronic calculations. In this work we show that in polymers, compared to bulk materials, due to the larger amplitude of the atomic vibrations the real excitations of the system are composed by entangled electron–phonon states. We prove as the charge carriers are fragmented in a multitude of polaronic states leading, inevitably, to the failure of the electronic picture. The presented re- sults lead to a critical revision of the state–of–the–art description of carbon–based nanostructures, opening a wealth of potential implications. PACS. 71.38.-k – 63.20.dk – 79.60.Fr , 78.20.-e 1 Introduction tion effects. The natural consequence is that a solely elec- tronic theory may be inadequate leading to intrinsic errors The coupling between the electronic and atomic degrees as large as the precision of the ab-initio theories. Nev- of freedom plays a key role in several physical phenomena. ertheless, the enormous numerical difficulties connected For example it affects the temperature dependence of car- with the calculation of the electron–phonon interaction, riers mobility in organic devices [1] or the position and in- and the historical assumption that such interaction could tensity of Raman peaks [2]. The electron-phonon coupling lead only to minor corrections (order of few meV), has is also the driving force that causes excitons dissociation de-facto prevented the confirmation of the HAC predic- at the donor/acceptor interface in organic photovoltaic [3] tions. As phonons are atomic vibrations, the effect of the and the transition to a superconducting phase in molecu- electron-phonon coupling is usually associated to a tem- lar solids [4]. From the theoretical point of view the role of perature effect that vanishes as the temperature goes to the atomic vibrations has been treated in a semi-empirical zero. However this is not correct as the atoms posses an manner for long. Such approach, based on model hamil- intrinsic spatial indetermination due to their quantum na- tonians, relies on parameters that are difficult to extract ture, even at zero temperature. This is the zero–point en- from experiments. In contrast the ab-initio methods de- ergy whose effect on the electronic properties is, generally, scribe and in some cases predict in a quantitative man- neglected. ner the optical and electronic properties of electronic sys- Nowadays, the advent of more refined numerical tech- tems, without resorting to any external parameter. This niques, has made possible to ground the HAC approach goal is reached by benefiting of the predictivity and accu- in a fully ab-initio framework. This has been used to com- racy of density functional theory (DFT) [5] merged with pute the gap renormalization in carbon–nanotubes [9], many body perturbation theory (MBPT) [6]. Electronic the finite temperature optical properties of semiconduc- properties are usually described within the so-called GW tors and insulators [10], and to confirm a giant zero–point approximation [7], a purely electronic theory which allows renormalization (615 meV) of the band–gap of bulk dia- to calculate corrections to the electronic levels with a high mond [11], previously calculated by Zollner using semi– level of accuracy. empirical methods [12]. These works are opening unpre- Many years ago [8], however, some pioneering works dictable scenarios connected with the actual accuracy of of Heine, Allen and Cardona (HAC) pointed to the fact purely electronic theories and, thus, questioning decades that, even when the temperature vanishes, the effect of of results. the electron-phonon coupling can induce corrections of the electronic levels, as large as the ones induced by correla- 2 Pleasegiveashorterversionwith: \authorrunning and \titlerunning prior to \maketitle The electronic theories ground on the concept of the in the case of diamond, if the agreement between theory single particle state: whatever is the external perturba- and experiment can be somewhat fortuitous. tion or internal correlation, the electron is still assumed In the HAC approach it is possible to calculate the to be characterized by a well defined energy, width and temperature dependent energy shift of the electronic state wave–function. As a consequence the charge–carriers are | nk, with energy εnk, induced by a configuration of static assumed to be mainly concentrated on electronic levels. lattice displacements {uIs} The theoretical basis of the electronic concept is strictly qλ 2 qλ connected with the quasi–particle (QP) concept [13] which 1 | gn′nk | 1 Λnn′k assumes that the electron occupies a well–defined state, ∆εnk(T )= − × Nq εnk − εn′k′ 2 εnk − εn′k qλ n′ " # even if the electronic states are renormalized by corre- X X lation effects. Physically the electron is surrounded by × (2B(ωqλ)+1) . a correlation cloud (composed by electron–hole pairs or (1) phonons) whose effect, in the QP picture, reduces to an energy correction, a broadening of the electronic level, and The key quantities in Eq.1 are the Bose function dis- a reduction of the effective electronic charge associated to tribution B(ωqλ), the electron-phonon matrix elements the state. qλ 2 | gn′nk | and the phonon frequencies ωqλ. The last two In the past it has been shown that the electron-phonon quantities are calculated ab-initio using density functional coupling can break the quasi-particle approximation. In perturbation theory [18]. The terms in Eq.(1) correspond particular Scalapino et al. [14] predicted that in strong to Fan and Debye-Waller corrections, in order of appear- coupling superconductivity the Landau QP approxima- ance. It must be noted that when the temperature van- tion is not valid, while Eiguren et al. [15] were able to ishes the energy correction is not vanishing. reproduce the band splitting on a surface by a multiple In the many body perturbation theory scheme, the quasi particle approximation. electron-phonon self-energy is perturbatively calculated More recently we have shown [16] that the quantum at the second order in the atomic displacements [6]. The zero point motion of atoms induces strong dynamical ef- electron-phonon self-energy is then composed of two con- fects on the electronic properties in diamond and trans- tributions: the Fan self-energy polyacetylene. The amplitude of the atomic vibrations and qλ 2 the consequent electron-phonon interaction leads an un- ′ F an | gn′nk | B(ωqλ) + 1 − fn k−q expected as well as striking result: the breakdown of the Σnk (ω)= + + Nq ω − εn′k−q − ωqλ − i0 quasi-particle approximation. This result has been obtained n′qλ X by calculating the full energy-dependent spectral function B(ωqλ)+ fn′k−q (SF) of the electronic states, as reviewed in the second sec- + (2) ω − εn′ − + ω λ − i0+ tion. We interpreted the sub-gap states experimentally ob- k q q served in diamond [17] and the formation of strong struc- DW and the frequency independent Debye-Waller term, Σnk . tures in trans-polyacetylene band structures in terms of More details about the Debye-Waller term can be found, entangled electron-phonon states. In this paper we extend for example, in Ref. [12]. Subsequently the full frequency these results to another polymer: the polyethylene. We dependent Green’s function Gnk (ω) is readily defined to will show that multiple structures appear in the SFs of be the electronic states at T = 0 K and the bare electronic F an DW −1 charge is fragmented in polaronic states, coherent packets Gnk(ω)= ω − εnk − Σnk (ω) − Σnk . (3) of electrons and phonons, each corresponding to a peak in the SF. In the conclusions we will point out as these results The single particle energies are obtained as poles of Gnk(ω). represent an important step forward in the simulation of The QP approximation is based on the assumption of a nanostructures, with a wealth of possible implications in single pole in Eq. 3 due to the smooth ω-dependence of F an the development of more refined theories of the electronic ℜΣnk . As a consequence the QP energy is obtained by F an and atomic dynamics at the nano-scale. Taylor expanding ℜΣnk around the bare energy. In this way the Green’s function is characterized by a single pole with energy 2 Static and dynamical approach to the F an DW Enk ≈ εnk + ZnkℜΣnk (εnk)+ Σnk , (4) electron-phonon coupling 1 and the SF Ank (ω) ≡ π |ℑ [Gnk (ω)] | is a Lorentzian func- tion centered in Enk with renormalization factor [6] The HAC approach is a static theory of the electron- − Fan 1 phonon coupling. It assumes that the scattering between ∂ℜΣnk (ω) Znk = 1 − ∂ω . The many body for- the electrons and phonons is instantaneous. Dynamical ef- ω=εnk fects are in fact connected to the retardation in the scatter- mulation represents the dynamical extension of the HAC ing between electrons and phonons and are generated by approach, that is recovered from Eq. 2 under the condi- the time dependence of the atomic oscillations, uIs. Such tion retardation effects are normally neglected in the HAC the- |εnk − εn′k−q| ≫ ωqλ and the self-energy is static, imply- ory but it is worth wondering, as the authors of Ref.[11] do ing Znk = 1.