- Email: [email protected]
Polaron formation in cuprates
Physica C 460–462 (2007) 263–266 www.elsevier.com/locate/physc
Polaron formation in cuprates O. Gunnarsson
b
a,*
, G. Sangiovanni a, O. Ro¨sch a, E. Koch b, C. Castellani c, M. Capone c
a Max-Planck-Institut fu¨r Festko¨rperforschung, D-70506 Stuttgart, Germany Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany c Dipartimento di Fisica, Universita` di Roma ‘‘La Sapienza’’, I-00185 Roma, Italy
Available online 20 March 2007
Abstract We calculate the electron–phonon coupling for La2CuO4, and show that it is strong enough to lead to small polarons, as seen experimentally. The calculated line shapes are in reasonable agreement with experiment. Deriving sum rules, we show that for a weakly doped system, the Coulomb interaction strongly suppresses effects of the electron–phonon interaction on the phonon but not electron selfenergy. Studying a Hubbard–Holstein model using a dynamical mean-field theory, we show that it is crucial to include antiferromagnetic correlations. The Coulomb interaction then only moderately suppresses the tendency to polaron formation, while the suppression of the phonon softening is very strong. 2007 Elsevier B.V. All rights reserved. PACS: 71.38.k; 71.27.+a; 74.72.h Keywords: Polarons; Electron–phonon coupling; Strong correlation
1. Introduction Recently there have been several experimental indications that the electron–phonon interaction (EPI) plays an important role for properties of high-Tc cuprates. For instance, there is a kink in the dispersion seen in photoemission [1], certain phonons show an anomalous softening when the system is doped [2] and there are signs of formation of small polarons in undoped cuprates [3]. This raises several questions. First of all, is the EPI strong enough to lead to polaronic behavior? Secondly, does the strong Coulomb interaction in these systems enhance or suppress the EPI? Thirdly, when answering the second question, do we have to distinguish between different properties? For instance, are effects of the EPI on electronic and phononic properties influenced the same way by the Coulomb interaction? Concerning the second question, it has been argued *
Corresponding author. E-mail address: [email protected] (O. Gunnarsson).
0921-4534/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.022
that the Coulomb interaction strongly suppresses the EPI [4], but also that the interaction with spin fluctuations enhances the EPI [5,6]. 2. Polaron formation To see if the EPI is strong enough for undoped cuprates to lead to small polaron formation, we calculate the EPI for La2CuO4 [7]. We use a shell model [8] to describe the phonon eigen vectors. From the displacements of the nuclei and shells we obtain the induced potential due to the excitation of a phonon. This potential couples to a hole created in photoemission. We also include the coupling to, in particular, breathing phonons via modulations of hopping integrals [9]. The EPI is described by 1 X H ep ¼ pffiffiffiffi gqm ð1 ni Þðbqm þ byqm ÞeiqRi ; ð1Þ N qmi where gqm gives the coupling to a phonon with frequency xqm, wave vector q and branch index m. The occupancy at site Ri is
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O. Gunnarsson et al. / Physica C 460–462 (2007) 263–266
where d is the doping. To lowest order in gq the corresponding result for noninteracting electrons is unity. A similar sum rule can be derived for the electron–phonon part of the electron self-energy Rep(k,x) for an undoped t–J model [12] Z 1 1 1 X 2 Im Rep ðk; x i0þ Þdx ¼ jgq j : ð3Þ p 1 N q
2 x HWHM [eV]
1.2 1
Ak(ω)
0.8
0.7 0.6 0.5 1.4
0.6
1.3 1.2 BE [eV]
0.4
(π/2,π/2) (0,π/2) 0.2 (π/2,π) (0,0) 0 1.4
1.2
1
0.8 0.6 BE [eV]
0.4
0.2
0
Fig. 1. ARPES spectra for an undoped t–J model with phonons at T = 0 for different k as a function of binding energies (BE) [7]. The spectra are normalized to the height of the phonon side band. The inset shows the side band width as a function of its binding energy. The width of the (0, 0) spectrum is poorly defined and not shown.
given by ni, bqm annihilates a phonon and N is the number of sites.PWe introduce the dimensionless coupling constant k qm jgqm j2 =ðDxqm N Þ, where D = 4t and t is a hopping integral. We then obtain k = 1.2. The criterion for the selftrapping crossover in the Holstein-t–J model has been found to be k > kc = 0.4 [6], and for the Hubbard–Holstein model we have obtained kc 0.55 [10]. We therefore find that La2CuO4 lies well inside the polaronic regime. To compare with experimental photoemission spectra [3], we calculate the spectrum using a method where individual phonon satellites are smeared out [7,11]. The results are shown in Fig. 1. The figure shows broad phonon side bands well below the (negligible) quasiparticles at the energy zero. As in the experimental work, we fit Gaussians to the low binding energy side of the spectra and determine the half-width (HWHM). We obtain 2 · HWHM = 0.5 eV for the top of the valence band in good agreement with the experimental estimate 0.48 eV [7]. The inset shows how the width increases with binding energy, in agreement with experiment [3]. The phonon side bands are at larger binding energies than found experimentally, suggesting that our calculated EPI is somewhat too large [7]. 3. Sum rules To address the importance of the electron–phonon interaction in a strongly correlated system, we derive sum rules [12]. We use the t–J model to describe the electronic structure, and assume that the phonons couple to the net charge on a site, which is a good approximation for, e.g., breathing phonons [9]. There is then an approximate sum rule for the imaginary part of the phonon self-energy P(q,x) [12,13] Z 1 1 X 1 jIm Pðq; xÞjdx 2dð1 dÞ; ð2Þ pN q6¼0 g2q 1
To lowest order (in g2q ), the same result is found for jIm Repj of noninteracting electrons. The sum rule in Eq. (2) shows that effects of the EPI on the phonon self-energy are strongly suppressed by the Coulomb interaction for weakly doped systems (d small). In contrast, Eq. (3) suggests that there is no such reduction for the electron self-energy. To understand the result in Eq. (2), we notice that in a system with a strong Coulomb interaction, double occupancy is suppressed. In the undoped system, with one hole per site, the system can then not respond to the perturbation of a phonon. If the system is doped, the holes doped into the system can respond, but if d is small, the response is small due to the small number of doped holes. The sum rule in Eq. (3) can be understood if we notice that the electron self-energy describes the response of the system when an additional hole is created in photoemission. The phonons respond strongly to this, even when the strong Coulomb interaction otherwise suppresses charge fluctuations. In spite of Eq. (3), the EPI could have a weak influence on Rep(k,x) for small x. The contribution to Im Rep(k,x) could be shifted to large x, and Re Rep(k,x) would then be only weakly influenced for small x. Indeed, this is exactly what happens in work finding that the EPI is strongly suppressed by the Coulomb interaction [4]. 4. Antiferromagnetic correlations Using the dynamical mean-field theory (DMFT) [14] and allowing for only a paramagnetic solution (P-DMFT), it has been found that the Coulomb interaction U strongly suppresses the EPI [4]. Work in the self-consistent Born approximation (SCBA) suggests that the effects of the EPI interaction strongly depend on the value of the quasiparticle weight Z in the absence of EPI [5]. In the P-DMFT and for a small d, Z becomes very small as U is increased. To see why, we notice that a half-filled Hubbard model is an insulator for a large U. The only way this can happen in the P-DMFT is to let Z go to zero. It is then interesting to allow for an antiferromagnetic solution, using a AFDMFT, since the unit cell is then doubled and an insulating solution can be found without Z going to zero. We have therefore studied a Hubbard–Holstein model on a Bethe lattice, using the AF-DMFT method [10]. The resulting impurity problem is solved using exact diagonalization. We use a finite number of bath sites and limit the number of phonons to obtain a finite Hilbert space. The
O. Gunnarsson et al. / Physica C 460–462 (2007) 263–266
1
0.4
n=.84 (m=.00) n=.85 (m=.20) n=.87 (m=.40) n=.92 (m=.60) n=.96 (m=.80) n=1.0 (m=.96)
0.35 0.3 0.25 Z
energies of and couplings to the bath levels are determined from a continued fraction expansion for the large U halffilled case and otherwise by a fit of the cavity Green’s function on the imaginary axis. We find that Z remains finite as U is increased [10]. For large U, we can compare with results obtained for the t–J model and find that there is good agreement with results obtained from the SCBA or exact diagonalization. Fig. 2 shows results for Z in a half-filled Hubbard–Holstein model as a function of the EPI strength k for the phonon frequency x0 = 0.025D, where D is half the band width. The figure illustrates that as k is increased Z is reduced. For some k = kc, Z becomes negligibly small, signaling polaron formation. For intermediate and large values of U, kc is somewhat larger than for U = 0. Thus the Coulomb interaction moderately suppresses the EPI. This is in strong contrast to P-DMFT calculations, where the EPI is strongly suppressed. This illustrates the importance of AF correlations. The arrow indicates the kc found in a diagrammatic Monte–Carlo calculation [6]. This result was compared with the result for a Holstein (U = 0) model with just one electron at the bottom of the band to determine the effects of the interaction with spin fluctuations [6]. To obtain the full effect of U, however, we compare with the half-filled Holstein model, since we can then increase U without changing the number of electrons. Since kc is quite different for the two cases, 0.33 (half-filled) and 1.2 (single electron), the conclusions are also different. The present method can be extended to doped cuprates, which are of particular interest. Fig. 3 shows Z as a function of k for a fixed U = 3.5D but for different dopings [10]. As the hole doping is increased (filling n reduced) the staggered magnetization m is reduced. This leads to an increase in the critical kc for polaron formation. This is the opposite trend to P-DMFT calculations [4]. Thus
265
0.2 0.15 0.1 0.05 0 0
0.2
0.4
0.6
0.8
1
λ Fig. 3. Z as a function of k for different fillings n and associated magnetic moments m for U = 3.5D and x0 = 0.025D in a Hubbard–Holstein model [10]. The figure shows how the critical kc is increased as the filling is reduced (doping is increased) due to a reduction of antiferromagnetic correlations.
the increase of kc in the AF-DMFT calculation with increased doping is due to the reduction of m, since at constant m = 0, kc decreases with doping. This further illustrates the importance of including antiferromagnetic correlations. We also observe that the variation of Z induced by a small k is reduced in the presence of U, in particularly at intermediate dopings, where the quasiparticle weight is mainly fixed by the Coulomb interaction. In Fig. 3 the AF-P transition takes place at n = 0.84, i.e., a much larger doping (0.16) than found experimentally. The main reason is that the calculation neglects AF correlations in the P state and therefore favors the AF state. A balanced treatment requires the use of a cluster DMFT method, which would introduce AF correlations also in the P state. As in the AF-DMFT calculation, these correlations would weaken as the doping is increased, and kc would presumably increase with doping in a qualitatively similar way as in Fig. 3.
U=0 0.8
5. Phonon softening U=1.7D
Z
0.6
U=3.5D
0.4
0.2
U=5.0D
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
λ Fig. 2. Z as a function of k for different U in a half-filled Hubbard– Holstein model [10]. The arrow shows kc of the t–J model for J/t = 0.3 (U/D = 3.3) [6]. The figure shows how the Coulomb interaction moderately suppresses polaron formation (Z ! 0).
We also study how the EPI influences phonons in AFDMFT calculations. Fig. 4a shows phonon spectral functions for the Hubbard–Holstein model at half-filling and small U [10]. The phonon is softened substantially as the EPI is increased. Fig. 4b and c shows that the softening is strongly reduced when U is increased. Fig. 4d shows how the softening increases as the doping is increased, due to the doped holes responding to phonons. This is in agreement with experiment [2]. The figure illustrates that the influence of the EPI on the phonon and electron selfenergies are dramatically different, in agreement with the sum rules discussed in Section 3. To summarize, we have calculated the electron–phonon interaction for La2CuO4 and found that it is strong enough
266
O. Gunnarsson et al. / Physica C 460–462 (2007) 263–266
a
λ=0.01 λ=0.05 λ=.125 λ=0.25
U=0.35D δ=0
c
ing) at half-filling, while the suppression is reduced with doping. These trends are consistent with experiment and can be understood from sum rules.
λ=.01 λ=.25 λ=.50 λ=1.0
U=3.5D δ=0
References λ=.01 λ=.25 λ=.50 λ=.80
b U=1.7D δ=0
0
0.05
0.1 ω/D
0.15
δ=.01 δ=.05 δ=.10 δ=.14 δ=.18
d U=5D λ=0.5
0.2 0
0.05
0.1 ω/D
0.15
0.2
Fig. 4. Phonon spectral function for different k in a Hubbard–Holstein model [10]. The bare phonon frequency is x0 = 0.1D and a Lorentzian broadening with the full width half maximum of 0.04 D was used. Figs. a–c show how the phonon softening at half-filling is dramatically suppressed by U and figure d that the softening increases with doping d.
to lead to small polarons. In paramagnetic DMFT calculations for the Holstein–Hubbard model, effects of the EPI on electrons (quasiparticle weight) are very strongly suppressed by the Coulomb interaction. Here we find that this suppression is only moderate when antiferromagnetic (AF) correlations are included. Increasing the doping leads to a reduction of the AF correlations, and the EPI is more suppressed. In contrast, the Coulomb interaction strongly suppresses effects of the EPI on phonons (phonon soften-
[1] A. Damascelli, Z.-X. Shen, Z. Hussein, Rev. Mod. Phys. 75 (2003) 473. [2] L. Pintschovius, Phys. Stat. Sol. (b) 242 (2005) 30. [3] K.M. Shen et al., Phys. Rev. Lett. 93 (2004) 267002. [4] G. Sangiovanni, M. Capone, C. Castellani, M. Grilli, Phys. Rev. Lett. 94 (2005) 026401; M. Capone, G. Sangiovanni, C. Castellani, C. Di Castro, M. Grilli, Phys. Rev. Lett. 92 (2004) 106401; G. Sangiovanni, M. Capone, C. Castellani, Phys. Rev. B 73 (2006) 165123. [5] A. Ramsak, P. Horsch, P. Fulde, Phys. Rev. B 46 (1992) R14305. [6] A.S. Mishchenko, N. Nagaosa, Phys. Rev. Lett. 93 (2004) 036402. [7] O. Ro¨sch, O. Gunnarsson, X.J. Zhou, T. Yoshida, T. Sasagawa, A. Fujimori, Z. Hussain, Z.-X. Shen, S. Uchida, Phys. Rev. Lett. 95 (2005) 227002. [8] S.L. Chaplot, W. Reichardt, L. Pintschovius, N. Pyka, Phys. Rev. B 52 (1995) 7230. [9] O. Ro¨sch, O. Gunnarsson, Phys. Rev. Lett. 92 (2004) 146403. [10] G. Sangiovanni, O. Gunnarsson, E. Koch, C. Castellani, M. Capone, Phys. Rev. Lett 97 (2006) 046404. [11] (a) K. Scho¨nhammer, O. Gunnarsson, Phys. Rev. B 30 (1984) 3141; (b) O. Ro¨sch, O. Gunnarsson, Eur. Phys. J. B 43 (2005) 11. [12] O. Ro¨sch, O. Gunnarsson, Phys. Rev. Lett. 93 (2004) 237001. [13] G. Khaliullin, P. Horsch, Phys. Rev. B 54 (1996) R9600. [14] A. Georges, G. Kotliar, W. Krauth, M.J. Rozenberg, Rev. Mod. Phys. 68 (1996) 13.
Polaron formation in cuprates O. Gunnarsson
b
a,*
, G. Sangiovanni a, O. Ro¨sch a, E. Koch b, C. Castellani c, M. Capone c
a Max-Planck-Institut fu¨r Festko¨rperforschung, D-70506 Stuttgart, Germany Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany c Dipartimento di Fisica, Universita` di Roma ‘‘La Sapienza’’, I-00185 Roma, Italy
Available online 20 March 2007
Abstract We calculate the electron–phonon coupling for La2CuO4, and show that it is strong enough to lead to small polarons, as seen experimentally. The calculated line shapes are in reasonable agreement with experiment. Deriving sum rules, we show that for a weakly doped system, the Coulomb interaction strongly suppresses effects of the electron–phonon interaction on the phonon but not electron selfenergy. Studying a Hubbard–Holstein model using a dynamical mean-field theory, we show that it is crucial to include antiferromagnetic correlations. The Coulomb interaction then only moderately suppresses the tendency to polaron formation, while the suppression of the phonon softening is very strong. 2007 Elsevier B.V. All rights reserved. PACS: 71.38.k; 71.27.+a; 74.72.h Keywords: Polarons; Electron–phonon coupling; Strong correlation
1. Introduction Recently there have been several experimental indications that the electron–phonon interaction (EPI) plays an important role for properties of high-Tc cuprates. For instance, there is a kink in the dispersion seen in photoemission [1], certain phonons show an anomalous softening when the system is doped [2] and there are signs of formation of small polarons in undoped cuprates [3]. This raises several questions. First of all, is the EPI strong enough to lead to polaronic behavior? Secondly, does the strong Coulomb interaction in these systems enhance or suppress the EPI? Thirdly, when answering the second question, do we have to distinguish between different properties? For instance, are effects of the EPI on electronic and phononic properties influenced the same way by the Coulomb interaction? Concerning the second question, it has been argued *
Corresponding author. E-mail address: [email protected] (O. Gunnarsson).
0921-4534/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2007.03.022
that the Coulomb interaction strongly suppresses the EPI [4], but also that the interaction with spin fluctuations enhances the EPI [5,6]. 2. Polaron formation To see if the EPI is strong enough for undoped cuprates to lead to small polaron formation, we calculate the EPI for La2CuO4 [7]. We use a shell model [8] to describe the phonon eigen vectors. From the displacements of the nuclei and shells we obtain the induced potential due to the excitation of a phonon. This potential couples to a hole created in photoemission. We also include the coupling to, in particular, breathing phonons via modulations of hopping integrals [9]. The EPI is described by 1 X H ep ¼ pffiffiffiffi gqm ð1 ni Þðbqm þ byqm ÞeiqRi ; ð1Þ N qmi where gqm gives the coupling to a phonon with frequency xqm, wave vector q and branch index m. The occupancy at site Ri is
264
O. Gunnarsson et al. / Physica C 460–462 (2007) 263–266
where d is the doping. To lowest order in gq the corresponding result for noninteracting electrons is unity. A similar sum rule can be derived for the electron–phonon part of the electron self-energy Rep(k,x) for an undoped t–J model [12] Z 1 1 1 X 2 Im Rep ðk; x i0þ Þdx ¼ jgq j : ð3Þ p 1 N q
2 x HWHM [eV]
1.2 1
Ak(ω)
0.8
0.7 0.6 0.5 1.4
0.6
1.3 1.2 BE [eV]
0.4
(π/2,π/2) (0,π/2) 0.2 (π/2,π) (0,0) 0 1.4
1.2
1
0.8 0.6 BE [eV]
0.4
0.2
0
Fig. 1. ARPES spectra for an undoped t–J model with phonons at T = 0 for different k as a function of binding energies (BE) [7]. The spectra are normalized to the height of the phonon side band. The inset shows the side band width as a function of its binding energy. The width of the (0, 0) spectrum is poorly defined and not shown.
given by ni, bqm annihilates a phonon and N is the number of sites.PWe introduce the dimensionless coupling constant k qm jgqm j2 =ðDxqm N Þ, where D = 4t and t is a hopping integral. We then obtain k = 1.2. The criterion for the selftrapping crossover in the Holstein-t–J model has been found to be k > kc = 0.4 [6], and for the Hubbard–Holstein model we have obtained kc 0.55 [10]. We therefore find that La2CuO4 lies well inside the polaronic regime. To compare with experimental photoemission spectra [3], we calculate the spectrum using a method where individual phonon satellites are smeared out [7,11]. The results are shown in Fig. 1. The figure shows broad phonon side bands well below the (negligible) quasiparticles at the energy zero. As in the experimental work, we fit Gaussians to the low binding energy side of the spectra and determine the half-width (HWHM). We obtain 2 · HWHM = 0.5 eV for the top of the valence band in good agreement with the experimental estimate 0.48 eV [7]. The inset shows how the width increases with binding energy, in agreement with experiment [3]. The phonon side bands are at larger binding energies than found experimentally, suggesting that our calculated EPI is somewhat too large [7]. 3. Sum rules To address the importance of the electron–phonon interaction in a strongly correlated system, we derive sum rules [12]. We use the t–J model to describe the electronic structure, and assume that the phonons couple to the net charge on a site, which is a good approximation for, e.g., breathing phonons [9]. There is then an approximate sum rule for the imaginary part of the phonon self-energy P(q,x) [12,13] Z 1 1 X 1 jIm Pðq; xÞjdx 2dð1 dÞ; ð2Þ pN q6¼0 g2q 1
To lowest order (in g2q ), the same result is found for jIm Repj of noninteracting electrons. The sum rule in Eq. (2) shows that effects of the EPI on the phonon self-energy are strongly suppressed by the Coulomb interaction for weakly doped systems (d small). In contrast, Eq. (3) suggests that there is no such reduction for the electron self-energy. To understand the result in Eq. (2), we notice that in a system with a strong Coulomb interaction, double occupancy is suppressed. In the undoped system, with one hole per site, the system can then not respond to the perturbation of a phonon. If the system is doped, the holes doped into the system can respond, but if d is small, the response is small due to the small number of doped holes. The sum rule in Eq. (3) can be understood if we notice that the electron self-energy describes the response of the system when an additional hole is created in photoemission. The phonons respond strongly to this, even when the strong Coulomb interaction otherwise suppresses charge fluctuations. In spite of Eq. (3), the EPI could have a weak influence on Rep(k,x) for small x. The contribution to Im Rep(k,x) could be shifted to large x, and Re Rep(k,x) would then be only weakly influenced for small x. Indeed, this is exactly what happens in work finding that the EPI is strongly suppressed by the Coulomb interaction [4]. 4. Antiferromagnetic correlations Using the dynamical mean-field theory (DMFT) [14] and allowing for only a paramagnetic solution (P-DMFT), it has been found that the Coulomb interaction U strongly suppresses the EPI [4]. Work in the self-consistent Born approximation (SCBA) suggests that the effects of the EPI interaction strongly depend on the value of the quasiparticle weight Z in the absence of EPI [5]. In the P-DMFT and for a small d, Z becomes very small as U is increased. To see why, we notice that a half-filled Hubbard model is an insulator for a large U. The only way this can happen in the P-DMFT is to let Z go to zero. It is then interesting to allow for an antiferromagnetic solution, using a AFDMFT, since the unit cell is then doubled and an insulating solution can be found without Z going to zero. We have therefore studied a Hubbard–Holstein model on a Bethe lattice, using the AF-DMFT method [10]. The resulting impurity problem is solved using exact diagonalization. We use a finite number of bath sites and limit the number of phonons to obtain a finite Hilbert space. The
O. Gunnarsson et al. / Physica C 460–462 (2007) 263–266
1
0.4
n=.84 (m=.00) n=.85 (m=.20) n=.87 (m=.40) n=.92 (m=.60) n=.96 (m=.80) n=1.0 (m=.96)
0.35 0.3 0.25 Z
energies of and couplings to the bath levels are determined from a continued fraction expansion for the large U halffilled case and otherwise by a fit of the cavity Green’s function on the imaginary axis. We find that Z remains finite as U is increased [10]. For large U, we can compare with results obtained for the t–J model and find that there is good agreement with results obtained from the SCBA or exact diagonalization. Fig. 2 shows results for Z in a half-filled Hubbard–Holstein model as a function of the EPI strength k for the phonon frequency x0 = 0.025D, where D is half the band width. The figure illustrates that as k is increased Z is reduced. For some k = kc, Z becomes negligibly small, signaling polaron formation. For intermediate and large values of U, kc is somewhat larger than for U = 0. Thus the Coulomb interaction moderately suppresses the EPI. This is in strong contrast to P-DMFT calculations, where the EPI is strongly suppressed. This illustrates the importance of AF correlations. The arrow indicates the kc found in a diagrammatic Monte–Carlo calculation [6]. This result was compared with the result for a Holstein (U = 0) model with just one electron at the bottom of the band to determine the effects of the interaction with spin fluctuations [6]. To obtain the full effect of U, however, we compare with the half-filled Holstein model, since we can then increase U without changing the number of electrons. Since kc is quite different for the two cases, 0.33 (half-filled) and 1.2 (single electron), the conclusions are also different. The present method can be extended to doped cuprates, which are of particular interest. Fig. 3 shows Z as a function of k for a fixed U = 3.5D but for different dopings [10]. As the hole doping is increased (filling n reduced) the staggered magnetization m is reduced. This leads to an increase in the critical kc for polaron formation. This is the opposite trend to P-DMFT calculations [4]. Thus
265
0.2 0.15 0.1 0.05 0 0
0.2
0.4
0.6
0.8
1
λ Fig. 3. Z as a function of k for different fillings n and associated magnetic moments m for U = 3.5D and x0 = 0.025D in a Hubbard–Holstein model [10]. The figure shows how the critical kc is increased as the filling is reduced (doping is increased) due to a reduction of antiferromagnetic correlations.
the increase of kc in the AF-DMFT calculation with increased doping is due to the reduction of m, since at constant m = 0, kc decreases with doping. This further illustrates the importance of including antiferromagnetic correlations. We also observe that the variation of Z induced by a small k is reduced in the presence of U, in particularly at intermediate dopings, where the quasiparticle weight is mainly fixed by the Coulomb interaction. In Fig. 3 the AF-P transition takes place at n = 0.84, i.e., a much larger doping (0.16) than found experimentally. The main reason is that the calculation neglects AF correlations in the P state and therefore favors the AF state. A balanced treatment requires the use of a cluster DMFT method, which would introduce AF correlations also in the P state. As in the AF-DMFT calculation, these correlations would weaken as the doping is increased, and kc would presumably increase with doping in a qualitatively similar way as in Fig. 3.
U=0 0.8
5. Phonon softening U=1.7D
Z
0.6
U=3.5D
0.4
0.2
U=5.0D
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
λ Fig. 2. Z as a function of k for different U in a half-filled Hubbard– Holstein model [10]. The arrow shows kc of the t–J model for J/t = 0.3 (U/D = 3.3) [6]. The figure shows how the Coulomb interaction moderately suppresses polaron formation (Z ! 0).
We also study how the EPI influences phonons in AFDMFT calculations. Fig. 4a shows phonon spectral functions for the Hubbard–Holstein model at half-filling and small U [10]. The phonon is softened substantially as the EPI is increased. Fig. 4b and c shows that the softening is strongly reduced when U is increased. Fig. 4d shows how the softening increases as the doping is increased, due to the doped holes responding to phonons. This is in agreement with experiment [2]. The figure illustrates that the influence of the EPI on the phonon and electron selfenergies are dramatically different, in agreement with the sum rules discussed in Section 3. To summarize, we have calculated the electron–phonon interaction for La2CuO4 and found that it is strong enough
266
O. Gunnarsson et al. / Physica C 460–462 (2007) 263–266
a
λ=0.01 λ=0.05 λ=.125 λ=0.25
U=0.35D δ=0
c
ing) at half-filling, while the suppression is reduced with doping. These trends are consistent with experiment and can be understood from sum rules.
λ=.01 λ=.25 λ=.50 λ=1.0
U=3.5D δ=0
References λ=.01 λ=.25 λ=.50 λ=.80
b U=1.7D δ=0
0
0.05
0.1 ω/D
0.15
δ=.01 δ=.05 δ=.10 δ=.14 δ=.18
d U=5D λ=0.5
0.2 0
0.05
0.1 ω/D
0.15
0.2
Fig. 4. Phonon spectral function for different k in a Hubbard–Holstein model [10]. The bare phonon frequency is x0 = 0.1D and a Lorentzian broadening with the full width half maximum of 0.04 D was used. Figs. a–c show how the phonon softening at half-filling is dramatically suppressed by U and figure d that the softening increases with doping d.
to lead to small polarons. In paramagnetic DMFT calculations for the Holstein–Hubbard model, effects of the EPI on electrons (quasiparticle weight) are very strongly suppressed by the Coulomb interaction. Here we find that this suppression is only moderate when antiferromagnetic (AF) correlations are included. Increasing the doping leads to a reduction of the AF correlations, and the EPI is more suppressed. In contrast, the Coulomb interaction strongly suppresses effects of the EPI on phonons (phonon soften-
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