Electron phonon interaction in the hubba

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 effectivel

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

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Bipolaron By Inter-Site Electron-Phonon Interaction

N S Mondal* 1, 2, S Nath1, S Bose1 and M Paul1

1 Department of Physics, University of Kalyani, Kalyani-741235, West Bengal, India

2

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

INTRODUCTION

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

short-range 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

FORMULATION

The 2D Hubbard Hamiltonian in the presence of

inter-site electron phonon coupling is given by,

−

=

σ σ σ

, ,

.) (

i i j

i c H c U n n c

t

+

i i

ib b

0

j

j i i

i b b n n

g

,

2 (1) 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; ω 0 is the phonon energy; g 2 is

the inter-site electron phonon interaction; b i (+) are the

phonon annihilation (creation) operators and n i is the

number of electrons on site i

A general state of the Hamiltonian H can be

written as the direct product Ψ = ∑ ⊗

p

e el ph

p e

,

, e and p label

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 operators likebi+ 1i = 1i ;

i i i

i i i

b 1 = 0 ; bi 0i = 0; 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

To show the effect of pair formation by

inter-site EP interaction in the ground state, here we have

calculated the electron-electron density correlation

functionC ( i − j ) = Ψ0 ninj Ψ0 In this paper

we have also calculated entropy per lattice site

defined as

⎟⎟⎠

⎞

⎜⎜⎝

⎛ +

=

T

H Z N

S 1 ln ; where N is the

number of lattice sites and = ∑ −

α

β E α

e

the calculations are on an 8 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]

RESULTS AND DISCUSSIONS

In this short paper we have shown the results

only for U/t = 8.0t at moderate adiabatic ratio ω0 = t

and <n> = 0.5

Figure 1 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

on-site electron phonon correlation and forms on-on-site

bipolarons (S0) There are also formations of

neighboring site (S1) bipolarons

0.0

0.1

0.2

0.3

0.4

0.5

U = 8.0t

g2 /t

d=0

d=1

d=rt2

d=2

d=rt5

d=3

FIGURE 1 Electron-electron correlation C(i-j) vs

inter-site EP interaction (g 2 ) The separation between electrons

(|i-j|=d) are given in the units of lattice constant.

In figure 2 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 1 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)

0.0 0.1 0.2 0.3 0.4 0.5

U = 8.0t

S

T/t

g2=0.0

g2=1.0t

g2=2.0t g

2 =3.0t

g2=4.0t

g2=5.0t

FIGURE 2 Entropy vs temperature for different values of

inter-site electron-phonon interaction.

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, ACKNOWLEDGMENTS

We are thankful to University of Kalyani for financial help

REFERENCES

1 A S Alexandrov et.al., Adv Condens Matter

Phys 2010, 206012 (2010)

2 L Vidmar et.al., Phys Rev Lett 103, 186401

(2009)

3 A.S Alexandrov EPL 95, 27004 (2011)

4 N S Mondal and N K Ghosh, Physica B 406,

3723-3725 (2011)

5 N S Mondal and N K Ghosh, DAE Solid State

Physics Symposium (2011), AIP Conf Proc

1447, 865-866 (2012)