Lattice exciton-polaron problem by quantum Monte Carlo simulations
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Lattice exciton-polaron problem by quantum Monte Carlo simulations
Martin Hohenadler, ∗ Peter B. Littlewood, and Holger Fehske Theory of Condensed Matter, Cavendish Laboratory, University of Cambridge, Cambridge CB3 0HE, United Kingdom Institute of Physics, Ernst-Moritz-Arndt University Greifswald, 17487 Greifswald, Germany
Exciton-polaron formation in one-dimensional lattice models with short- or long-range carrier-phonon inter-action is studied by quantum Monte Carlo simulations. Depending on the relative sign of electron and hole-phonon coupling, the exciton-polaron size increases or decreases with increasing interaction strength. Quantumphonon fluctuations determine the (exciton-) polaron size and yield translation-invariant states at all finite cou-plings.
PACS numbers: 71.35.-y, 02.70.Ss, 63.20.Ls, 71.38.-k
I. INTRODUCTION
The binding of electron-hole (E-H) excitations into exci-tons (Xs), governing the optical properties of most nonmetal-lic materials, plays a major role in, e.g., organics, nanos-tructure devices, quantum light sources, Bose-Einsteincondensation and DNA. The coupling of Xs to phonons is widely relevant, andgives rise to exciton-polaron (X-P) formation, correspond-ing to quasiparticles consisting of an E-H pair and a virtualphonon cloud. Apart from the essential role of phonons in re-laxation processes after optical excitation, lattice-coupling al-ters the X radius which determines, e.g., the oscillator strengthin optics and the overlap of X wave functions required forBose-Einstein condensation. Very recently, a direct obser-vation of an exciton-polaron in photoluminiscence spectra ofquantum dots has been reported. Examples where X-Ps of intermediate size are clearlyimplicated in current experiments include transition metaloxides, such as insulating manganites and nickelates, though the situation in cuprates is controversial. Anotherimportant class of materials is conjugated polymers (e.g.,Ref. 14). In these systems, the well-known approximationsof small Frenkel or large Wannier-Mott Xs are unjustified, re-quiring nonperturbative theories which includes relative E-Hmotion. Polaron formation is a complex, nonlinear, many-bodyproblem which cannot be completely described by renormal-ization of effective masses. In particular, the quantum natureof phonons—leading to retarded (self-)interaction—has to betaken into account. Since polaron physics is governed by lat-tice dynamics on the unit-cell scale, the discrete nature of thecrystal cannot be neglected. The resulting problem of an interacting E-H pair with cou-pling to quantum phonons represents a long-standing openquestion in condensed matter physics. Whereas some ex-act results are available without phonons, standard meth-ods such as perturbation theory, and variational or adiabaticapproximations are often of uncertain reliability.Furthermore, computational approaches are very demand-ing, and we are not aware of any exact results for quantumphonons.Here we present unbiased numerical results for the quantumlattice X-P within a simple E-H model, obtained by means of quantum Monte Carlo (QMC) simulations. This method,well established in the field of polaron physics, treats all cou-plings on the same footing and is not restricted to a specificX size or parameter region. Our model study of several dif-ferent Hamiltonians yields important results for the effects ofcarrier-phonon interaction on X properties.
II. MODEL
Extending previous work, we consider a simplemodel in one dimension (1D) defined by the Hamiltonian H = − t e X h i,j i e † i e j − t h X h i,j i h † i h j − X ij u ij ˆ n i,e ˆ n j,h (1) + ω X i (ˆ x i + ˆ p i ) − X i,j f j,i ˆ x j ( α e ˆ n i,e + α h ˆ n i,h ) with long-range Coulomb attraction u ij = ( U , i = j,U / | i − j | , i = j , (2)where U > U > (i.e., attractive interaction), and long-range carrier-phonon interaction f j,i = 1( | j − i | + 1) / . (3)Here e † i ( h † i ) creates an E (H) at site i , and ˆ x i ( ˆ p i ) denotes thedisplacement (momentum) of a harmonic oscillator at site i .The fermionic density operators are defined as ˆ n i,e = e † i e i and ˆ n i,h = h † i h i . The model parameters are the nearest-neighbor E (H) hopping integral t e ( t h ), the energy of Einsteinphonons ω ( ~ = 1 ), the E (H)-phonon couplings α e ( α h ), aswell as the local (extended) Coulomb interaction U ( U ).We consider a single E-H pair—a situation whichcan be studied experimentally —and neglect X cre-ation/recombination as well as dynamic screening of theCoulomb interaction due to other carriers or lattice polariza-tion. Spin degrees of freedom are not taken into account,and we assume a tight-binding band structure with s symme-try for both E and H, neglecting the existence of a band gap(which here only leads to a shift of energies). Of course thismodel is too simple to make a direct comparison with materi-als. Nevertheless, it does describe the physics of a Coulomb-bound, itinerant E-H pair whose constituents couple individu-ally to quantum phonons and—in the absence of coupling tothe lattice—captures the familiar crossover from a small to alarge exciton with increasing bandwidth (see Sec. IV). The exact form of the carrier-phonon coupling is subjectto X size, screening and material properties. We restrictour analysis to Holstein- and Fr¨ohlich-type interactions well-known and understood from polaron physics, and amenableto efficient numerical treatment. Important aspects arise fromthe fact that the coupling of E and H to the lattice can eitherbe of cooperative or compensating nature. The goal here is toobtain a qualitative understanding of the influence of the typeand range of the lattice coupling, as well as the nonadiabatic-ity of the lattice.Equation (1) allows for different signs of α e and α h . Thecoefficients f j,i correspond to a lattice version of the Fr¨ohlichinteraction with longitudinal optical phonons, but yield aHolstein coupling to transverse optical phonons for f j,i = δ i,j . Since E and H couple to the same phonon mode, weconsider the symmetric mass case t e = t h = t , and α h = σα e = σα with σ = ± and α > . We refer to the modelwith local respectively long-range carrier-lattice coupling asthe Holstein-X model (HXM), respectively, Fr¨ohlich-X model(FXM). These models capture the interplay of Coulomb at-traction, particle motion and coupling to the lattice.We introduce the dimensionless parameter λ =2 ε P ( P j f j, ) /W , where ε P = α / ω is the polaronbinding energy in the atomic limit and W = 4 t is thebare single-particle bandwidth. The time scales of E/H andquantum lattice dynamics are set by the adiabaticity ratio γ = ω /t . The units of energy and length are taken to be U and the lattice constant, respectively. III. METHOD
The world-line QMC method adapted here can handle long-range interactions—notoriously difficult for many other nu-merical approaches—higher dimensions, and general fermionand phonon dispersion relations.
From the partition function with discretized inverse tem-perature β = 1 / ( k B T ) and Trotter parameter ∆ τ = β/L , thefermionic trace can be evaluated using real-space basis states { r ρτ } = { r eτ , r hτ } , which define world-line configurations on a N × L space-time grid. The path integral over the phonons isdone analytically, yielding the fermionic partition function Z f = X { r ρτ } e P τ,τ ′ F ( τ − τ ′ ) P ρ,ρ ′ α ρ α ρ ′ φ ( r ρτ − r ρ ′ τ ′ ) (4) × e − ∆ τ P τ u reτ ,rhτ Y ρ Y τ I ρ ( r ρτ +1 − r ρτ ) . Here carrier-phonon coupling gives the memory function F ( τ ) = ω ∆ τ L L − X ν =0 cos[2 πτ ν/L ]1 − cos[2 πν/L ] + ( ω ∆ τ ) / (5) with φ ( r ρτ − r ρ ′ τ ′ ) = P j f j,r ρτ f j,r ρ ′ τ ′ and hopping enters via I ρ ( r ) = 1 N N − X k =0 cos(2 πkr/N ) e τt ρ cos(2 πk/N ) . (6)We calculate the X “radius” (see Ref. 17) R = *X i,j ( i − j ) ˆ n i,e ˆ n j,h + / , (7)the kinetic energy E kin = − t *X h i,j i e † i e j + h † i h j + , (8)and the binding energies E B ,U = E X ( t, U , U , λ ) − E e ( t, λ ) (9)and E B ,λ = E X ( t, U , U , λ ) − E X ( t, U , U , , (10)where E X ( E e ) denotes the X (E) energy. We further studythe E-H correlation function C eh ( r ) = X i h ˆ n i,e ˆ n i + r,h i , (11)and the E-phonon correlation function C eph ( r ) = X i h ˆ n i,e ˆ x i + r i . (12)Computer time ∼ ( β/ ∆ τ ) (Ref. 27) sets a practical lowerlimit on simulation temperatures. The Trotter error (whichcan be removed by scaling to ∆ τ = 0 , see Ref. 29) and sta-tistical errors limit the accuracy of our QMC results to typi-cally 1%, and we use periodic clusters with N = 32 . Moresophisticated QMC approaches to polaron problems, free ofTrotter errors and finite-size effects, have been devel-oped. Whereas the continuous-time method has recently beenapplied to a similar model, an extension of the diagram-matic MC method to the exciton-polaron problem is notyet available. Since all three methods are useful only for oneor two carriers coupled to phonons and hence not applicable tomore realistic systems, we have chosen the simplest approachcurrently available. IV. RESULTS
To set the stage for the following discussion of lattice ef-fects, and to demonstrate that the model defined by Eq. (1)describes the basic exciton physics, we begin with the case λ = 0 , i.e., no coupling to the lattice. Figure 1 shows exactdiagonalization (ED) and QMC results for the X size versusbandwidth. The zero-temperature ED data for N = 31 is well W / U R U / U = 0.00, β U = 15U / U = 0.00, β U = 20U / U = 0.50, β U = 15U / U = 0.75, β U = 15Frenkel X Wannier-Mott X FIG. 1: Exciton radius R as a function of bandwidth W for λ = 0 and different values of U . Dashed lines correspond to exact ground-state results ( N = 31 ). QMC error bars are smaller than the symbols. converged with respect to system size. With increasing W ,there is a crossover from a small, strongly bound Frenkel-Xwith R ≈ (i.e., E and H at the same site) to a larger Wannier-Mott-like X with R > . Note that the X is always bound in1D. Our parameters do not include the extensively studiedWannier-Mott limit, but instead cover experimentally relevantintermediate radii. The crossover point (
W/U ≈ ) sep-arates regions with opposite dependence of R on U . TheQMC results are overall in good agreement with T = 0 EDdata, with finite-temperature effects being most noticeable for U = 0 .In the sequel, we restrict ourselves to the wide-band case W/U = 3 . , highlighted in Fig. 1, for which R ( λ = 0) & .As this work is concerned with phonon effects, we focus onthe dependence on λ and γ , and only consider U /U = 0 . and βU = 15 .Discussing carrier-quantum-phonon interaction, it is crucialto distinguish between σ = + and − , as well as between theadiabatic (slow lattice, γ ≪ ) and the non-adiabatic (fastlattice, γ ≫ ) regime, taking γ = 0 . respectively γ = 4 .We begin with the HXM in the adiabatic regime and σ = + .Figure 2(a) shows R as a function of λ . With increasingcoupling, there is a gradual crossover to a small X-P due tothe increasingly strong phonon-mediated attractive interactionbetween E and H. The E- and H-polarons tend to maximizeboth the Coulomb and the lattice energy by forming a statewith small R , but compete with the kinetic energy of the sys-tem which decreases with increasing λ (Fig. 3). Similar tothe bipolaron problem with U = 0 , E- and H-polarons form a(phonon) bound state at any λ > in 1D. Most notably, thereis no discontinuity at a critical λ , a common misconceptiondue to earlier variational treatments, as quantum lattice fluc-tuations give rise to a translational invariant Bloch-like X-Pstate.The crossover is also reflected in a reduced X mobility, andin a more negative X binding energy [Fig. 4(a)]. With thepresent method, dynamic quantities such as the effective exci-ton mass cannot be accurately calculated. An alternative ob- R γ = 0.4 (+)γ = 4.0 (+)γ = 0.4 (−)γ = 4.0 (−) (a) HXM λ R γ = 0.4 (+)γ = 4.0 (+)γ = 0.4 (−)γ = 4.0 (−) (b) FXM FIG. 2: (Color online) QMC results for R as a function of λ for (a)the HXM (see text) and (b) the FXM for both the adiabatic ( γ = 0 . )and the nonadiabatic ( γ = 4 ) regime, as well as σ = ± . Here andin subsequent figures βU = 15 , W/U = 3 . , U /U = 0 . , andlines connecting data points are guides to the eye. servable which to some degree (see, e.g., Ref. 35) measuresthe mobility is the kinetic energy shown in Fig. 3. In addition,the E-H and E-phonon correlation functions in Fig. 5(a), al-ways positive for σ = + , fall off quickly with r , indicatingthat the X-P is a quasiparticle consisting of a tightly-boundE-H pair with a strongly localized surrounding lattice distor-tion. Such a state is similar to the Frenkel limit considered inRef. 36.The nonadiabatic regime γ ≫ mainly differs by a weakerdependence on λ (the important coupling parameter is ε P /ω ,see below). The results for the FXM exhibit qualitativelythe same tendencies, but the long-range interaction generallyleads to larger E- and H-polarons and a larger X-P.Turning to the case σ = − , we briefly discuss the differentpolaron ground states in the Holstein and the Fr¨ohlich model.The Holstein model in 1D exhibits a crossover from a largepolaron to a small polaron (with a predominantly onsite lat-tice distortion) with increasing λ . For γ ≪ , the latter occursnear λ ≈ , whereas for γ ≫ the condition is ε P /ω & ( λ = 2 for γ = 4 ). In contrast, the Fr¨ohlich polaron re-mains large (spatially extended lattice polarization) even forstrong coupling. While for σ = + the bipolaron effect dom-inates, these differences have a major impact for σ = − where λ - E k i n / U γ = 0.4 (+)γ = 4.0 (+)γ = 0.4 (−)γ = 4.0 (−) λ - E k i n / U HXM FXM
FIG. 3: (Color online) Kinetic energy E kin . -8-6-4-20 E B , U / U γ = 0.4 (+)γ = 4.0 (+)γ = 0.4 (−)γ = 4.0 (−) λ -10-8-6-4-20 E B , λ / U (a) HXM λ -8-6-4-20 E B , U / U γ = 0.4 (+)γ = 4.0 (+)γ = 0.4 (−)γ = 4.0 (−) λ -10-8-6-4-20 E B , λ / U (b) FXM FIG. 4: (Color online) Binding energies E B ,U and E B ,λ (inset). the Coulomb-bound E- and H-polarons remain separated with R & .In Fig. 2, strikingly different to σ = + , R initially increaseswith increasing λ , i.e., the X-P is larger for stronger coupling.In the HXM [Fig. 2(a)], R takes on a maximum at λ ≈ . andapproaches R = 1 for λ & , whereas in the FXM R increasesmonotonically and saturates at large R in the strong-couplingregime [Fig. 2(b)]. Accordingly, the kinetic energy in Fig. 3 ismuch larger compared to σ = + , but is eventually reduced forlarge λ in the HXM. The binding energy E B ,U → with in- C e h (r) γ = 0.4 (+)γ = 4.0 (+)γ = 0.4 (−)γ = 4.0 (−) r -0.200.20.40.60.8 C e ph (r) / C e ph ( ) (a) HXM r C e h (r) γ = 0.4 (+)γ = 4.0 (+)γ = 0.4 (−)γ = 4.0 (−) r -0.200.20.40.60.8 C e ph (r) / C e ph ( ) (b) FXM FIG. 5: (Color online) Electron-hole [ C eh ( r ) ] and electron-phonon[ C eph ( r ) , inset] correlation functions for λ = 1 . creasing coupling in both models (Fig. 4), whereas E B ,λ (seeinsets)—related to X-P effects—remains clearly negative.The (initial) increase of the radius with increasing λ for the(HXM) FXM (Fig. 2) is due to the fact that the X-P loseslattice energy if the compensating displacement clouds sur-rounding E and H overlap. Therefore, the E- and H-polaronsoptimize R to achieve maximum Coulomb energy and mini-mum phonon-cloud overlap. The resulting average distance R depends on the size of the individual (E and H) polarons. Forthe HXM, polarons are large for λ < , leading to large valuesof R in Fig. 2(a), and become small for λ > , causing the de-crease of R → in the strong-coupling regime. In the FXM,polarons remain large for all λ , leading to large values of R even for strong coupling [Fig. 2(b)]. The larger radius in thenon-adiabatic HXM in Fig. 2(a) as compared to γ ≪ is dueto the much larger polaron kinetic energy. A discontinuousdissociation of the X-P with increasing λ has been discussedin a continuum model with acoustic phonons and σ = − . From the σ = − results for the E-H correlation function inFig. 5 we see that the E-H separation is small in the HXM,whereas the pair is spread out in the FXM. For the HXM with γ = 0 . , we find a charge-transfer X-P with E and H mainlyon neighboring sites. Note that for σ = − , Coulomb andcarrier-phonon interaction have swapped roles as comparedto bipolaron formation where the lattice-coupling creates anattractive interaction that competes with Coulomb repulsion. Turning to the E-phonon correlation functions in Fig. 5, wehave C eph > for small r , but C eph < at larger distances asa result of the opposite distortions created by the hole. Again,the extent of the distortions is much larger for the FXM.Real materials will require more detailed modeling, but wenote that charge-transfer Xs in oxides will be better modeledby σ = − (for breathing modes), whereas the characteristiccase for a direct X in a neutral semiconductor would be σ =+ . V. CONCLUSIONS
In summary, we have studied the exciton-polaron problemwith quantum phonons by Monte Carlo simulations. Our sim-ple models encompass short- and long-range carrier-phononinteraction of either the same or opposite sign for electron andhole. There are no sharp transitions with increasing carrier-phonon coupling, and for couplings of opposite sign the ex-citon radius increases with increasing coupling as a result ofpolaron-polaron repulsion. To capture this effect (dependingon polaron size which is affected by nonadiabaticity) relative electron-hole motion and quantum phonon fluctuations mustbe taken into account. Our findings are expected to be impor-tant in materials with relatively small excitons such as organ-ics and transition metal oxides, although more realistic modelswill have to be studied for direct comparison.The present study motivates future work in a number of dif-ferent directions, including more general models with respectto band structure, phonon dispersion, spin dependence, disor-der, or dimensionality, and many-X-P as well as X-polaritonproblems. To this end, the development of more elaboratenumerical approaches is highly desirable, permitting investi-gations of spectral properties routinely studied experimentallyor even time-resolved studies of X formation.
Acknowledgments
This work was financially supported by the FWF Erwin-Schr¨odinger Grant No. J2583 and the DFG through SFB 652.We thank A. Alvermann, P. Eastham, V. Heine, and F. Laquaifor valuable discussions. ∗ Electronic address: [email protected] I. Egri, Phys. Rep. , 363 (1985). W. Barford,
Electronic and Optical Properties of ConjugatedPolymers (Oxford University Press, Oxford, 2005). F. Dubin, R. Melet, T. Barisien, R. Grousson, L. Legrand, M.Schott, and V. Voliotis, Nat. Phys. , 32 (2006). A. J. Shields, Nat. Photonics , 215 (2007). J. Kasprzak, M. Richard, S. Kundermann, A. Baas, P. Jeambrun,J. M. J. Keeling, F. M. Marchetti, M. H. Szymanska, R. Andre,J. L. Staehli, V. Savona, P. B. Littlewood, B. Deveaud, and L.Dang, Nature (London) , 409 (2006). S. G. Chou, F. Plentz, J. Jiang, R. Saito, D. Nezich, H. B. Ribeiro,A. Jorio, M. A. Pimenta, G. G. Samsonidze, A. P. Santos, M.Zheng, G. B. Onoa, E. D. Semke, G. Dresselhaus, and M. S. Dres-selhaus, Phys. Rev. Lett. , 127402 (2005). J. Frenkel, Phys. Rev. , 17 (1931). Excitons , edited by E. I. Rashba and M. D. Sturge (North-HollandPhysics Publishing, Amsterdam, 1987), Vol. II, p. 543. M. Gong, C.-F. Li, G. Chen, L. He, F. W. Sun, G.-C. Guo, Z.-C.Niu, S.-S. Huang, Y.-H. Xiong, and H.-Q. Ni, arXiv:0708.0468v2(unpublished). M. W. Kim, H. J. Lee, B. J. Yang, K. H. Kim, Y. Moritomo, J. Yu,and T. W. Noh, Phys. Rev. Lett. , 187201 (2007). E. Collart, A. Shukla, J.-P. Rueff, P. Leininger, H. Ishii, I. Jarrige,Y. Q. Cai, S.-W. Cheong, and G. Dhalenne, Phys. Rev. Lett. ,157004 (2006). J. Zaanen and P. B. Littlewood, Phys. Rev. B , 7222 (1994). K. M. Shen, F. Ronning, D. H. Lu, W. S. Lee, N. J. C. Ingle, W.Meevasana, F. Baumberger, A. Damascelli, N. P. Armitage, L. L.Miller, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, and Z.-X.Shen, Phys. Rev. Lett. , 267002 (2004). J.-F. Chang, J. Clark, N. Zhao, H. Sirringhaus, D. W. Breiby, J. W.Andreasen, M. M. Nielsen, M. Giles, M. Heeney, and I. McCul-loch, Phys. Rev. B , 115318 (2006). H. Fehske, A. Alvermann, M. Hohenadler, and G. Wellein, in
Po- larons in Bulk Materials and Systems with Reduced Dimension-ality , Proceedings of the International School of Physics “EnricoFermi”, Course CLXI, edited by G. Iadonisi, J. Ranninger, and G.De Filippis (IOS Press, Amsterdam, 2006), pp. 285–296. J. Ranninger, in
Polarons in Bulk Materials and Systems with Re-duced Dimensionality , Ref. 15, pp. 1–25. K. Ishida, H. Aoki, and T. Chikyu, Phys. Rev. B , 7594 (1993). E. A. Burovski, A. S. Mishchenko, N. V. Prokof’ev, and B. V.Svistunov, Phys. Rev. Lett. , 186402 (2001). H. Haken, J. Phys. Chem. Solids , 166 (1959). A. Suna, Phys. Rev. , A111 (1964). A. Sumi, J. Phys.: Condens. Matter , 1286 (1977). Y. Shinozuka and N. Ishida, J. Phys. Soc. Jpn. , 3007 (1995). H. Sumi and A. Sumi, J. Phys. Soc. Jpn. , 637 (1994). K. Ishida, Phys. Rev. B , 5541 (1994). K. Ishida, Phys. Rev. B , 12856 (1997). A. S. Alexandrov and P. E. Kornilovitch, Phys. Rev. Lett. , 807(1999). H. de Raedt and A. Lagendijk, Phys. Rep. , 233 (1985). M. Hohenadler and P. B. Littlewood, Phys. Rev. B , 155122(2007). P. E. Kornilovitch, J. Phys.: Condens. Matter , 10675 (1997). P. E. Kornilovitch, Phys. Rev. Lett. , 5382 (1998). P. E. Kornilovitch, Phys. Rev. B , 3237 (1999). N. V. Prokof’ev and B. V. Svistunov, Phys. Rev. Lett. , 2514(1998). J. P. Hague, P. E. Kornilovitch, J. H. Samson, and A. S. Alexan-drov, Phys. Rev. Lett. , 037002 (2007). A. Macridin, G. A. Sawatzky, and M. Jarrell, Phys. Rev. B ,245111 (2004). J. Loos, M. Hohenadler, A. Alvermann, and H. Fehske, J. Phys.:Condens. Matter , 236233 (2007). G. Wellein and H. Fehske, Phys. Rev. B , 6208 (1998). M. Hohenadler and W. von der Linden, Phys. Rev. B , 184309(2005). J. Edler, P. Hamm, and A. C. Scott, Phys. Rev. Lett.88