Ground and Excited States of Bipolarons in Two and Three Dimensions

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1 Commun. Theor. Phys. Beijing, China pp c International Academic Pulishers Vol. 8, No. 1, July 15, 007 Ground and Excited States of Bipolarons in Two and Three Dimensions RUAN Yong-Hong and CHEN Qing-Hu Department of Physics, Zhejiang University, Hangzhou 31007, China Received August 1, 006; Revised Octoer 17, 006 Astract The properties of large ipolarons in two and three dimensions are investigated y averaging over the relative wavefunction of the two electrons and using the Lee-Low-Pines-Huyrechts variational method. The groundstate GS and excited-state energies of the Fröhlich ipolaron for the whole range of electron-phonon coupling constants can e otained. The energies of the first relaxed excited state RES and Franc Condon FC excited state of the ipolaron are also calculated. It is found that the first RES energy is lower than the FC state energy. The comparison of our GS and RES energies with those in literature is also given. PACS numers: 63.0.Kr, , Mx Key words: ipolaron, ground-state energy, excited-state energy, Lee-Low-Pines-Huyrechts variational method 1 Introduction Bipolarons have een extensively discussed oth for fundamental theoretical interest and for their importance in semiconductor materials. ] With recent advances in the farication of nanocrystals and semiconductor nanostructures, strong electron-phonon coupling is realized due to quantum confinement effects, and the properties of ipolarons in low-dimensional systems are of growing interest. [ 6] This prolem is also relevant to the proposal of the ipolaronic mechanism for electron pairing in the CuO plane in high-t c cuprates.,7] Recently, Schoenes and co-worers [8] reported that some superconductors such as YBa Cu O 8 have strong electron-phonon coupling, which also oosts the theoretical research on ipolarons. Extensive wor has already een devoted to the calculation of the ipolaron ground-state GS energy in two and three dimensions an incomplete list is given in Refs. [9] 3]. Different values of the critical electronphonon coupling constant α c aove which the ipolaron is stale and the critical ratio of dielectric constants η c elow which the ipolaron is stale are otained y different approximation approaches. Though the ipolaronic GS state has een extensively investigated in the past decade, only a few wors 17] have een devoted to the excited states so far. Actually nowledge of the excited states of the ipolaron is also important due to the relevance to electron transport, photoluminescence and photoemission. ] Smondyrev et al. ] calculated the energy spectra of the one-dimensional ipolaron in the strong-coupling limit. More recently, Sahoo 5] developed the Landau Pear variational method to get the ground and first excited states of the Fröhlich ipolaron in a multidimensional ionic crystal in the strong-coupling limit. The staility of magnetoipolarons was studied y Kandemir. 6] The system of excited terms of a ipolaron has een estalished y the method of canonical transformation of coordinate. 7] In this paper we extend the Huyrechts variational approach LLP-H 8] to the analysis of oth GS and excited states. Section presents the calculation of GS state. In Sec. 3, we calculate the energies of the first relaxed excited state and Franc Condon excited state in oth two and three dimensions. Discussions of these two types of excited states are given. Finally, comparison of our results with Sahoo s ground-state and excited-state energies is made in Sec.. Our conclusions are presented in Sec. 5. Ground-State Energy The Fröhlich Hamiltonian for the polaron in N dimensions ND has een derived y Peeters et al. 9] Accordingly, the Hamiltonian descriing a system of two electrons interacting with a longitudinal optical LO phonon field may e written as in units of m = h = ω LO = 1 H = [ p i + ] V a e i ri + h.c. i=1, + a a + U r 1 r, 1 where all vectors are N-dimensional N =, 3 and r i p i is the position momentum operator of the i-th electron i = 1,. a and a are respectively the creation and annihilation operators of the LO phonons with the wave vector. Here we should mention that the impurityphonon interactions have already een eliminated so that The project supported y the Natural Science Foundation of the Education Bureau of Zhejiang Province under Grant No and partially y National Natural Science Foundation of China under Grant No qhchen@css.zju.edu.cn

2 170 RUAN Yong-Hong and CHEN Qing-Hu Vol. 8 we assume ω = ω LO. The interaction coefficient is { Γ[N 1/] N 3/ π N 1/ α } 1/ V = i, V N N 1 where V N is the ND crystal volume, α is the dimensionless electron-phonon coupling constant, α = e 1 1, 3 ε ε 0 and ε ε 0 is the high-frequency static dielectric constant of the medium. Ur = U/r is the Coulom interaction potential etween the two electrons, where the nonscreened electron Coulom repulsion strength is given y U = e /ε, which may e rewritten as α U = 1 η, η = ε. ε 0 Since the ipolaron is a composite particle, it is convenient to introduce center-of-mass and relative coordinates and momenta, R = r 1 +r /, P = p 1 +p, r = r 1 r, p = p 1 p /, [0] in which the Hamiltonian can e rewritten as H = P + + r cos V a e i R + h.c. a a + p + U r. 5 To otain variational estimate of the ipolaron energy, we set the oscillator-type trial wavefunction as φ o r = Cr exp r, 6 where is a variational parameter. This trial wave function has een shown to e the est one in the estimate of GS in D and 3D ipolarons. 1] The next step is to average the Hamiltonian 5 over the wavefunction 6 and otain the effective Hamiltonian for the center-of-mass motion, H eff = P + B a e i R + h.c. + a a + E r. 7 Here B = V cos r/ and E r = p + U/r with denoting an averaging over the wavefunction φ o r. B and E r can e calculated as B ND = 1 V N, 8 Er 3D = π U, ED r = π + U. 9 Note that the effective Hamiltonian 7 corresponding to the center-of-mass motion is in essence equivalent to a single-polaron Hamiltonian. The differences etween the Hamiltonian 7 and the usual Hamiltonian of a single polaron are the following: i The energy is shifted y the average of the energy of the relative motion E r, and ii the electron-phonon interaction coefficient B is renormalized. We will follow the LLP-H variational method proposed y Huyrechts 8] for the single polaron prolem. First, we perform the following LLP-H transformation, U = exp ia Ra a, 10 where a is a variational parameter. In the limit a 0, the present calculation is identical to the strong-coupling regime, whereas the case a 1 corresponds to wea electron-phonon interaction. The Hamiltonian 7 can e transformed into H eff = 1 P a a a + {Ba exp[ i1 a R] + h.c.} + a a + E r. 11 For convenience we write the Hamiltonian 11 in the following more detailed form: H eff = 1 P a P a a a a a + {Ba exp[ i1 a R] + h.c.} + E r + a 1 a 1 a a 1 a. 1 1, The trial wave function in the new representation is assumed to tae the following form, = φr A, 13 where the wave function of the electron part is chosen as the following Gaussian type, φ 0 R = c exp λ Rj, 1 j with λ eing a variational parameter to e determined. A is the phonon coherent state, [ ] A = exp f a f a 0, 15 here f and f are assumed to e a function of only and to e determined variationally, 0 is the unpertured zero phonon state satisfying a 0 = 0 for all. In the next step, we average the Hamiltonian 1 over the wave function 13 and directly present the ipolaron

3 No. 1 Ground and Excited States of Bipolarons in Two and Three Dimensions 171 ground-state energy E 0,ND as E 0,ND = N λ a P f f a f + { Bf exp [ 1 a ] } + h.c. 8λ + E r + a 1 f 1 f f 1 f. 16 1, It can e noticed that the P term and the last term in E 0,ND vanish exactly if we perform the angular integration first. Therefore the neglect of these terms in the Hamiltonian 16 does not violate the variational principle. Minimizing E 0,ND with respect to f and f yields f = B exp[ 1 a /8λ] 1 + a, 17 / f = B exp[ 1 a /8λ] 1 + a. 18 / Using Eqs. 17 and 18, in Eq. 16 we otain the ground-state energy of the ipolaron in N dimensions with an aritrary electron-phonon coupling constant: E 0,ND = N λ B exp[ 1 a /λ] 1 + a +E r. 19 / Finally we calculate the ground-state energies of the ipolaron in 3D and D materials as follows: E 0,3D = 3λ α d 1 π E 0,D exp g 1 + a / + 7 = λ α 0 exp g 1 + a / π U, 0 d π U 1 with g = a λ. Thus the ground-state energy of this system can e otained y minimizing Eqs. 0 and 1 with respect to a,, and λ. Although one can further integrate the variale analytically on the right-hand side of Eqs. 0 and 1 and get a very complicated form including the complementary error function erf cx, which is of no use for practical calculations. 3 Internal Excited-State Energy Following the definition given y Devreese, ] we compute the relaxed excited state RES and Franc Condon excited state FC energies of the ipolaron. The RES is created if the electron in the polaron is excited while the lattice readapts to the new electronic configuration. One can imagine that the electron goes from a 1s- to a p-state, while the lattice polarization in the final state is adapted to the p-state of the electron. If, on the contrary, the lattice corresponds to the electron ground state, while the electron is excited, one speas of an FC state. Here we use a method of Huyrechts, 8] who computed the FC state energy [] of a single polaron. In this section, we will extend Huyrechts method to calculate the RES energy of the ipolaron y adjusting a,, λ, and f to the relaxed excited state. The wavefunction of the electron part for the first excited state is N-fold degenerate, φ 1 R = cr j exp λ j R j. 3 By Eqs. 3 and 15 the Hamiltonian 1 can e calculated as E 1,ND = λ + N λ a + + E r + a { B f exp P f f a [ 1 a 8λ f ][ 1 1 a ] } j + h.c. λ 1, 1 f 1 f f 1 f. The P term and the last term in E 1,ND also vanish exactly if performing the angular integration. For the FC state, f and f in E1,ND is the same as that in Eqs. 17 and 18. For the RES, f and f can e otained y minimizing the excited-state energy according to E 1,ND E 1,ND f = 0, = 0. 5 f Then we get the excited-state energy E 1,ND 1 + a E 1,ND = λ + Nλ + f + for the ipolaron in N dimensions, { [ Bf 1 1 a j ] exp [ 1 a λ 8λ ] } + h.c. + E r, 6

4 17 RUAN Yong-Hong and CHEN Qing-Hu Vol. 8 with for the FC state and f RES f FC = B exp[ 1 a /8λ] 1 + a / = B exp[ 1 a /8λ] [ 1 + a 1 1 a ] j / λ for the RES. The energy of the relative motion can e otained y Eqs. 8 and 9 and insert f FC Finally we otain the FC state and RES energies of the ipolaron in 3D and D, respectively, E FC,3D = 5λ π U α d 1 π exp g a 1 a / 1λ E FC,D = λ + π + U α d exp g a 1 a / 8λ E RES,3D = 5λ π U α d 1 π exp g a 1 a / 6λ E RES,D = λ + π + U α d 1 + exp g a 1 a / λ and f RES into Eq. 6. ], 9 ], a 80λ ], a 18λ ]. 3 For the FC state, the values of the parameters a,, and λ are taen from the calculation of the ground state energy E 0,ND. For the RES, the values of the parameters λ, a, and are otained y minimizing the excited-state energy E RES,ND of Eqs. 31 and 3 through numerical calculation. In Figs. 1a and 1, the ground-state energy E 0,ND, first RES energy ERES,ND, and FC state energy E FC,ND of the ipolaron are displayed as functions of the electron-phonon coupling constants for 3D and D materials. Comparing the energy of two single polarons y LLP-H method, we find the ipolaron is stale when α in three and two dimensions. We start from α = in Figs. 1a and 1 for 3D and D ipolarons. We find that in the whole range of electron-phonon coupling constants, the FC state and RES energies are negative. The difference in energy, E ND = E 1,ND E 0,ND, yields the excitation energy, which is related to optical asorption of ipolarons in semiconductor materials.,3] Fig. 1 a Ground-state energy E 0,3D, first RES energy ERES,3D, and FC state energy E FC,3D of the ipolaron are displayed as functions of the electron-phonon coupling constants for 3D materials. The ipolaron is stale for α αc 3D = 6.3 at η = 0.0. Ground-state energy E 0,D, first RES energy ERES,D, and FC state energy E FC,D of the ipolaron are displayed as functions of the electron-phonon coupling constants for D materials. The ipolaron is stale for α αc D =.9 at η = 0.0. As shown in Figs. 1a and 1, we find that the RES energy is always lower than the FC state energy. This is well nown from asorption spectrum calculations.,,5] The asorption pea due to a transition from the ground state to the first relaxed excited state corresponds to the zero-phonon pea. In contrast, an asorption transition from

5 No. 1 Ground and Excited States of Bipolarons in Two and Three Dimensions 173 the GS to the FC state is accompanied with phonon emission. If the ipolaron system is excited to the FC state, the lattice will relax towards the RES y emission of phonons. Comparison with Strong-Coupling Calculations In order to assess the heuristic value of our approach, we compare the present GS and RES energies with those otained y Sahoo, 5] who adopted the wavefunction φ o 0 r C exp r / for relative coordinates and developed a Landau Pear variational method to get the ground-state and RES energies of the Fröhlich ipolaron in the strongcoupling limit. To perform a comparison with our results, we define the relative deviation ξ = Es 0,ND E 0,ND / E0,ND s, with Es 0,ND referring to the GS energies otained y Sahoo 5] and E 0,ND given y Eqs. 0 and 1. Also, we define δ = Es RES,ND E RES,ND / Es RES,ND, Es RES,ND referring to the RES energies otained y Sahoo 5] and E RES,ND denoting our results 31 and 3. In Fig., we plot ξ and δ as functions of α for 3D and D materials at η = 0. ξ and δ are positive in the whole coupling regime, demonstrating that our GS and RES energies are smaller than those y Sahoo. It is well nown that the Landau Pear method wors well only in the strong-coupling regime, indeed ξ and δ decrease monotonously with the increase of α. Es 0,ND and E RES,ND are very close to our results at large α, which demonstrates the reliaility of our approach. It is shown that the present approach can give etter results for GS and RES energies, comparing with previous strong-coupling models. 5] Fig. Relative deviation of our ground-state energy and Sahoo s result ξ = Es 0,ND / E0,ND s as a function of α for 3D and D materials at η = 0.0. Relative deviation of our RES energy and Sahoo s result δ = Es RES,ND / Es RES,ND as a function of α for 3D and D materials at η = 0.0. E RES,ND E 0,ND 5 Conclusions We have extended the Huyrechts variational approach LLP-H to the analysis of the properties of ipolarons. By averaging over the wavefunction of the relative motion of the two electrons, the ground and first excited-state energies of the ipolaron in two and three dimensions are otained. Numerical results show that the RES energy is lower than the FC state energy. Our ground-state and RES energies are lower than the previously reported results from Landau Pear method, 5] which is due to the use of a more appropriate relative wave function and the additional parameter a in LLP-H method. Our results may e of relevance for high-t c superconductors where ipolarons are expected to play an important role. [7] References ] Y.H. Kim, A.J. Heeger, L. Acedo, G. Stucy, and F. Wudl, Phys. Rev. B ; M. Gurvitch and A.T. Fiory, Phys. Rev. Lett ; K.K. Lee, A.S. Alexandrov, and W.Y. Liang, Phys. Rev. Lett [] S. Muhopadhyay and A. Chatterjee, J. Phys.: Condens. Matter [3] E.P. Poatilov, V.M. Fomin, J.T. Devreese, et al., J. Phys.: Condens. Matter ; Phys. Rev. B [] D.V. Melniov and W.B. Fowler, Phys. Rev. B [5] R.T. Senger and A. Ercelei, Phys. Rev. B ; Eur. Phys. J. B ; Phys. Rev. B ; J. Phys.: Condens. Matter [6] Y.H. Ruan, Q.H. Chen, and Z.K. Jiao, Int. J. Mod. Phys ; J. of Zhejiang University SCI 5 00

6 17 RUAN Yong-Hong and CHEN Qing-Hu Vol ; Commun. Theor. Phys. Beijing, China [7] D. Emin, Phys. Rev. Lett ; A.S. Alexandrov and P.E. Kornilovitch, J. of Superconductivity [8] J. Schoenes, E. Kaldis, and J. Karpinsi, J. of Less- Common Metals 16 & [9] F. Bassani, M. Geddo, G. Iadonisi, and D. Ninno, Phys. Rev. B ] G. Verist, F.M. Peeters, and J.T. Devreese, Phys. Rev. B ] G. Verist, M.A. Smondyrev, F.M. Peeters, and J.T. Devreese, Phys. Rev. B ] Q.H. Chen, K.L. Wang, and S.L. Wan, Phys. Rev. B ] F. Lucza, F. Brosens, and J.T. Devreese, Phys. Rev. B ] P. Vansant, F.M. Peeters, M.A. Smondyrev, and J.T. Devreese, Phys. Rev. B ] S. Sahoo, Phys. Rev. B ] B.S. Kandemir and T. Altanhan, Eur. Phys. J. B ] V.K. Muhomorov, Phys. Stat. Sol ] W.J. Huyrechts, J. Phys. C: Sol. Stat. Phys ] F.M. Peeters, X.G. Wu, and J.T. Devreese, Phys. Rev. B [0] E.A. Kochetov, S.P. Kuleshov, V.A. Matveev, and M.A. Smondyrev, Teor. Mat. Fiz ] J.T. Devreese, Polarons in Ionic Crystals and Polar Semiconductors, North-Holland, Amsterdam 197 p. 83. [] In Huyrechts paper he taled aout RES. In fact, according to the definition in Ref. ], Huyrechts calculated the FC state energy of a single polaron ecause he used the same parameters a,, and λ in the excited state as those in the ground state. Also, the coefficient f in his calculation of the excited state has the same value as in the ground state. [3] Y.H. Kim, C.M. Foster, A.J. Heeger, et al., Phys. Rev. B [] J.T. Devreese and V.M. Fomin, Phys. Rev. B [5] J.T. Devreese, S.N. Klimin, and V.M. Fomin, Phys. Rev. B