Bipolaron by inter-site electron-phonon interaction N S Mondal, S Nath, S Bose, and M Paul Citation: AIP Conf Proc 1512, 810 (2013); doi: 10.1063/1.4791285 View online: http://dx.doi.org/10.1063/1.4791285 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1512&Issue=1 Published by the American Institute of Physics Related Articles Polaron formation: Ehrenfest dynamics vs exact results J Chem Phys 138, 044112 (2013) Effective mass of electron in monolayer graphene: Electron-phonon interaction J Appl Phys 113, 043708 (2013) Influence of phonons on the temperature dependence of the band gap of AlN and AlxGa1−xN alloys with high AlN mole fraction J Appl Phys 113, 043501 (2013) Non-Markovian stochastic Schrödinger equation at finite temperatures for charge carrier dynamics in organic crystals J Chem Phys 138, 014111 (2013) An absence of the Anderson transition in high-resistance alloys with a high electron density Low Temp Phys 39, (2013) Additional information on AIP Conf Proc Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors Downloaded 08 Feb 2013 to 223.231.8.18 Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions Bipolaron By Inter-Site Electron-Phonon Interaction N S Mondal* 1, 2, S Nath1, S Bose1 and M Paul1 Department of Physics, University of Kalyani, Kalyani-741235, West Bengal, India Purulia Polytechnic, Purulia-723147, Government of West Bengal, India * E-mail: nil16mon@gmail.com Abstract An exact diagonalization calculation of the Hubbard model with inter-site electron-phonon interaction on a 2D square cluster shows that inter-site electron-phonon interaction effectively creates on-site (S0) bipolaron There is also formation of neighboring site (S1) bipolaron Entropy calculation shows that system goes into more ordered state with the formation of these self-trapped bipolaron Keywords: Hubbard model; electron phonon interaction; bipolaron PACS: 71.38.-k, 71.38.Mx H = −t INTRODUCTION ∑σ (c σ c σ + H c.) + U ∑ n + i j i, j , i↑ ni ↓ i + ω ∑ bi+ bi − g ∑ ni (bi+ + bi )n j There is growing consensus that high-Tc superconductivity is a phenomenon that can be explained with proper combination of Coulomb repulsion and electron-phonon (EP) interaction [1] The theory of strongly correlated electrons and phonons with on-site Coulomb repulsion and shortrange EP interaction has been mainly studied in the framework of the Holstein-Hubbard and Holstein t-J models [2] In [3], it has been shown that a peculiar cancellation of the long range Coulomb repulsion by the long range Frohlich EP interaction can also produce high-Tc superconductivity in doped polar insulators like cuprates In this paper, we have studied the Hubbard model with EP interaction using an exact diagonalization technique In addition to the on-site Coulomb repulsion U, here we emphasize only on a phonon mediated interaction [4] term betweens electrons at nearest neighbor sites i (1) i, j The summation < i, j> extends over all pairs of nearest- neighbors (NN) sites on a 2D square lattice; t is the NN hopping amplitude, U is the onsite Coulomb interaction; ω0 is the phonon energy; g2 is the inter-site electron phonon interaction; bi(+) are the phonon annihilation (creation) operators and ni is the number of electrons on site i A general state of the Hamiltonian H can be written as the direct product Ψ = ∑e el ⊗ p ph , e and p label e, p electronic and bosonic basic states respectively Here we have truncated the infinite dimensional bosonic part of the Hilbert space by considering only one phonon per doubly occupied site Here we argue that creation of an extra phonon at a site, where a phonon is already present does not change the scenario of pairing interaction at least qualitatively We consider the case where two electrons with opposite spins coupled to dispersionless optical phonons To restrict us to one phonon per site we operate phonon FORMULATION The 2D Hubbard Hamiltonian in the presence of inter-site electron phonon coupling is given by, operators + like bi 1i = 1i ; bi+ i = 1i ; bi 1i = i ; bi i = ; where 1(0) denotes presence(absence) of phonons at site i [4] SOLID STATE PHYSICS: Proceedings of the 57th DAE Solid State Physics Symposium 2012 AIP Conf Proc 1512, 810-811 (2013); doi: 10.1063/1.4791285 © 2013 American Institute of Physics 978-0-7354-1133-3/$30.00 810 Downloaded 08 Feb 2013 to 223.231.8.18 Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions To show the effect of pair formation by intersite EP interaction in the ground state, here we have calculated the electron-electron density correlation In figure we have plotted entropy with temperature for different g2 This figure says that at large interaction, system goes to more ordered state A closer look clearly shows that region of transition to ordered state in conformity with figure Hence, we can expect that at larger values of g2 (>2.5t) the inter-site EP interaction overcomes on-site Coulomb repulsion and, as a result, the two electrons coalesce on a single site (S0 bipolaron) or on neighboring sites (S1 bipolaron) function C (i − j ) = Ψ0 ni n j Ψ0 In this paper we have also calculated entropy per lattice site H ⎞ 1⎛ ⎜ ln Z + ⎟ ; where N is the N ⎜⎝ T ⎟⎠ − βE number of lattice sites and Z = ∑ e α [5] All defined as S = α U = 8.0t the calculations are on an site tilted square cluster using exact diagonalization methods as used before [4, 5] Here the interaction considered is up to NN sites and the quasi-particles formed are either localized to a site or extended up to NN sites, so present calculation on a small 8-site tilted square cluster might be good enough [4] g2=0.0 g2=1.0t g2=2.0t g2=3.0t g2=4.0t g2=5.0t 0.5 0.4 0.3 S 0.2 0.1 RESULTS AND DISCUSSIONS 0.0 In this short paper we have shown the results only for U/t = 8.0t at moderate adiabatic ratio ω0 = t and = 0.5 Figure shows variation of electron-electron density correlation function C (i-j) with inter-site EP interaction Figure clearly shows a sharp transition in electron-electron correlation, we can see that at g2>2.5t only on-site and NN site correlation are found, with dominating on-site correlation So, we can conclude that initially inter-site EP interaction was suppressed by on-site coulomb repulsion After that inter-site EP interaction effectively increases onsite electron phonon correlation and forms on-site bipolarons (S0) There are also formations of neighboring site (S1) bipolarons ACKNOWLEDGMENTS We are thankful to University of Kalyani for financial help C(i-j) REFERENCES 0.2 0.1 0.0 Due to the formation of these localized bipolarons we observe the corresponding decrease in entropy Initially, onsite Coulomb repulsion suppresses phonon mediated interaction between electrons, but, at a certain stronger repulsion it strengthen the onsite EP interaction, d=0 d=1 d=rt2 d=2 d=rt5 d=3 FIGURE Entropy vs temperature for different values of inter-site electron-phonon interaction 0.5 0.3 T/t U = 8.0t 0.4 5 g2 /t A S Alexandrov et.al., Adv Condens Matter Phys 2010, 206012 (2010) L Vidmar et.al., Phys Rev Lett 103, 186401 (2009) A.S Alexandrov EPL 95, 27004 (2011) N S Mondal and N K Ghosh, Physica B 406, 3723-3725 (2011) N S Mondal and N K Ghosh, DAE Solid State Physics Symposium (2011), AIP Conf Proc 1447, 865-866 (2012) FIGURE Electron-electron correlation C(i-j) vs intersite EP interaction (g2) The separation between electrons (|i-j|=d) are given in the units of lattice constant 811 Downloaded 08 Feb 2013 to 223.231.8.18 Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/about/rights_permissions