Doping-driven evolution of the superconducting state from a doped Mott insulator: Cluster dynamical mean-field theory

M. Civelli

Phys. Rev. B 79, 195113 – Published 15 May 2009

Abstract

In this paper we investigate the zero-temperature doping-driven evolution of a superconductor toward the Mott insulator in a two-dimensional electron model, relevant for high-temperature superconductivity. To this purpose we use a cluster extension of dynamical mean-field theory. Our results show that a standard $d$ -wave superconductor, realized at high doping, is driven into the Mott insulator via an intermediate superconducting state displaying unconventional physical properties. By restoring the translational invariance of the lattice, we give an interpretation of these findings in momentum space. In particular, we show that at a finite doping a strong momentum-space differentiation takes place: non-Fermi liquid and insulatinglike (pseudogap) characters rise in some regions (antinodes), while Fermi liquid quasiparticles survive in other regions (nodes) of momentum space. We describe the consequence of these happenings on the spectral properties, stressing in particular the behavior of the superconducting gap, which reveals two distinct nodal and antinodal energy scales as a function of doping, detected in photoemission and Raman spectroscopy experiments. We study and compare with experimental results the doping-dependent behavior of other physical quantities, such as for instance, the nodal quasiparticle velocity (extracted in angle-resolved photoemission) and the low-energy slopes of the local density of states and of the Raman scattering response. We then propose a description of the evolution of the electronic structure while approaching the Mott transition. We show that, within our formalism, a strong asymmetry naturally arises in the local density of states, measured in scanning tunneling spectroscopy. We investigate in detail the doping evolution of the electronic bands, focusing on the kinklike quasiparticle dispersion observed with angle-resolved photoemission in specific cuts of the momentum-energy space. We finally show the consequences of the properties presented above in the doping dependence of the Hall resistivity, measured in magnetotransport experiments.

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Received 29 August 2008

DOI:https://doi.org/10.1103/PhysRevB.79.195113

Authors & Affiliations

M. Civelli

Theory Group, Institute Laue Langevin, 6 rue Jules Horowitz, 38042 Grenoble Cedex, France

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Vol. 79, Iss. 19 — 15 May 2009

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Figure 1
(Color online) Left panel: doping $δ = 1 - ⟨ n ⟩$ versus chemical potential $μ$ . Right panel: cluster $d$ -wave order parameter $(dOP) = {⟪ c_{i ↑} c_{i + 1 ↓} ⟫}_{(τ = 0)}$ as a function of doping $δ$ .Reuse & Permissions
Figure 2
(Color online) Normal components of the cluster self-energy ${\hat{Σ}}_{c}$ vs the Matsubara frequency $ω_{n}$ as a function of doping $δ$ . The real parts are displayed in the left column; the imaginary parts are displayed in the right column. From the top row to the bottom row we show the local $Σ_{11}$ , the next-neighbor $Σ_{12}$ , and nearest-next-neighbor $Σ_{13}$ self-energies.Reuse & Permissions
Figure 3
(Color online) Eigenvalues of the normal component of the cluster self-energy ${\hat{Σ}}_{c}$ vs the Matsubara frequency $ω_{n}$ as a function of doping $δ$ . The real parts are displayed in the left column; the imaginary parts are displayed in the right column. From the top row to the bottom we show the self-energy corresponding to the points $k = (0, 0)$ , $(0, π)$ , and $(π, π)$ in the first quadrant of the Brillouin zone (see Sec. 3).Reuse & Permissions
Figure 4
(Color online) Cluster quasiparticle residua, associated with the eigenvalues of the cluster self-energy matrix, as a function of doping $δ$ .Reuse & Permissions
Figure 5
(Color online) Left panel: real part of the anomalous component of the cluster-self-energy $Σ_{ano}$ (the imaginary part is negligible) vs Matsubara frequency $ω_{n}$ for different doping $δ$ . Right panel: $Re Σ_{ano} (ω_{n} \to 0)$ versus doping $δ$ .Reuse & Permissions
Figure 6
(Color online) Local density of state $N (ω) = - \frac{1}{t π} Im {[{\hat{G}}_{c}]}_{11} (ω)$ obtained from the cluster-impurity solution [Eq. (8)] for various doping $δ$ (from top to bottom). The broadening used for display is $η = 7 \times 10^{- 3}$ . A measure of the energy scale associated with the one-particle gap (always present in the superconducting state) is given by the left (red circle) and right (green square) distances of the ED poles from the Fermi level $ω = 0$ . These are displayed as a function of doping $δ$ in the right bottom panel.Reuse & Permissions
Figure 7
(Color online) We compare the nodal and antinodal quasiparticle peaks obtained from the normal component of the superconductive ED-CDMFT solution at zero temperature and the QMC-CDMFT normal state solution at much higher temperature $T$ $(β = t ∕ T = 75)$ . The functions on the real axis with QMC are obtained with the maximum entropy method. To display the ED-CDMFT curves the slightly $ω$ -dependent broadening $η (ω) = 0.075 + ω^{2}$ was used (see discussion in Appendix ).Reuse & Permissions
Figure 8
(Color online) Top panel: reconstructed local density of states $N (ω) = - \frac{1}{t π} \sum_{k} G_{k}^{nor} (ω)$ for the normal component $(Σ_{ano} = 0)$ of Green’s function at small doping $(δ = 0.05)$ . $G_{k}^{nor} (ω)$ has been obtained by either the cumulant $M$ periodization (red point-dashed line labeled $M$ scheme) and by the $Σ$ periodization (green dashed line labeled $Σ$ scheme) and confronted with the local density of states of the cluster-impurity solution (black continuous line). Middle panel: closeup at the Fermi level. Bottom panel: the local density $N (ω)$ of states of full superconducting solution $(Σ_{ano} \neq 0)$ selects in the $k$ summation only the nodal points. A low-energy confront with the cluster-impurity solution shows a different matching of the cumulant and self-energy periodizations with the cluster-impurity result. The display on the real axis of Green’s function has been obtained by introducing a $ω$ -dependent broadening $i η (ω)$ (see Appendix for details).Reuse & Permissions
Figure 9
(Color online) Panel (a): nodal $k_{nod}$ and antinodal $k_{anod}$ $x$ -axis components of the positions of the quasiparticle peaks, in the first quadrant of the Brillouin zone, as a function of doping $δ$ . Panel (b): Nodal peaks are found always at the Fermi level and decrease with doping $δ$ in approaching the insulator. Panel (c): at high doping $(δ > δ_{c} \sim 0.08)$ antinodal peaks are also found at the Fermi level in the normal component of the system [set $Σ_{ano} = 0$ in Eq. (19)]. For $(δ < δ_{p} \sim 0.06)$ however they shifts to negative energy opening a pseudogap. Panel (d): the actual antinodal spectra [ $Σ_{ano} \neq 0$ in Eq. (19)] show to become asymmetric for $δ < δ_{c}$ when the pseudogap starts opening in the normal component [see panel (c) and bottom right panel of Fig. 6].Reuse & Permissions
Figure 10
(Color online) Quasiparticle residuum at the nodal point $Z_{nod}$ and antinodal points $Z_{anod}$ as a function of the doping $δ$ , evaluated as area of the quasiparticle peaks shown in Fig. 9 (as a cross-check, $Z_{nod}$ is also evaluated as the slope of the imaginary part of the self-energy on the Matsubara axis, green crosses). While $Z_{nod}$ monotonically decreases with doping, as in the standard picture of the Mott transition, the antinodal $Z_{anod}$ shows to stay constant upon opening of the pseudogap $(δ < δ_{c} \sim 0.08)$ . For comparison’s sake, we display the nodal $Z_{nod}$ obtained by CTQMC (Ref. 52) on the two-dimensional Hubbard model with similar parameters ( $U = 12 t$ , $t^{'} = 0.0$ , $β = 200$ ). These quasiparticle residua should be compared with their cluster counterparts in Fig. 4.Reuse & Permissions
Figure 11
(Color online) Top: Nodal velocity as a function of doping measured from the quasiparticle dispersion of different materials (figure taken from Ref. 87). To be confronted with the nodal component of the velocity $v_{nod}$ calculated in our work and displayed in the bottom. Bottom: Nodal anomalous velocity $v_{ano} = \sqrt{2} Z_{nod} Σ_{ano} (k_{nod})$ (tangent component of the nodal velocity) and nodal quasiparticle velocity $v_{nod} = Z_{nod} ∣ \nabla_{k} ξ_{k_{nod}} ∣$ (component of the nodal velocity perpendicular to the Fermi surface) as a function of doping $δ$ (units are $a_{0} t$ , where $a_{0}$ is the square lattice spacing). The trends are compared with CTQMC-CDMFT (Ref. 52) calculations on the two-dimensional Hubbard model with similar parameters ( $U = 12 t$ , $t^{'} = 0.0$ , $β = 200$ ).Reuse & Permissions
Figure 12
(Color online) The slope of the low-energy local density of states $N (ω) \sim \frac{Z_{nod}}{v_{Δ} v_{nod}} ω$ is displayed as a function of doping $δ$ . Trends are compared with CTQMC-CDMFT (Ref. 52) calculations on the two-dimensional Hubbard model with similar parameters ( $U = 12 t$ , $t^{'} = 0.0$ , $β = 200$ ).Reuse & Permissions
Figure 13
(Color online) Top: the linear coefficient of the low energy Raman response $χ_{B_{2 g}} (ω) \sim α ω$ and the low temperature superfluid stiffness (from penetration depth) $ρ_{s} (T) - ρ_{s} (0) \sim β T$ , extracted from various experiments (figure taken from Ref. 70). Quantities are normalized at the optimal doping value. Bottom: the linear coefficient of the Raman response and superfluid stiffness as extracted from our calculation $Z_{nod}^{2} ∕ (v_{Δ} v_{nod})$ and displayed as a function of doping $δ$ . Trends are compared with CTQMC-CDMFT (Ref. 52) calculations on the two-dimensional Hubbard model with similar parameters ( $U = 12 t$ , $t^{'} = 0.0$ , $β = 200$ ).Reuse & Permissions
Figure 14
(Color online) Top: Nodal $(B_{1 g})$ and antinodal $(B_{2 g})$ quasiparticle gaps extracted by different spectroscopy experiments as a function of doping (the figure is extracted from the Raman spectroscopy results in Ref. 70). Bottom: the antinodal gap $Δ_{tot}$ and the nodal $Δ_{sc}$ extracted from the spectra of Fig. 9 as a function of doping $δ$ . $Δ_{sc}$ can be directly related to the tangential component of the nodal velocity $v_{ano}$ presented in Fig. 11 (here displayed by the blue triangles). $Δ_{nor}$ is the pseudogap in the normal component extracted in panel (c) of Fig. 9.Reuse & Permissions
Figure 15
Left: the shaded region is the patch $Φ_{k}$ used in the phenomenological Boltzmann approach in Ref. 60 in the first quadrant of the Brillouin zone. Right: the patch $Φ_{k}$ used in this work [see formula (37)].Reuse & Permissions
Figure 16
(Color online) We show that the evolution of the normal-component Fermi surface (blue continuous line in the $M$ -periodization scheme; green dashed line in the $Σ$ -periodization scheme) defined has $ξ_{k} + Re Σ_{k} = 0$ in Eq. (19). Panels span from the overdoped region ( $δ = 0.13$ top-left panel) to the Mott insulator ( $δ = 0.0$ bottom-right panel). By periodizing the cumulant $M$ , the Fermi surface disappears in the insulating state and it is replaced by a line of zeroes of $G_{k}$ (red dot-dashed lines). The way this takes place is via a topological phase transition around $δ_{p} \sim 0.06 < δ < δ_{c} \sim 0.08$ from a large holelike Fermi surface to a pocketlike Fermi surface. The low doping region is marked by a phase where Fermi surface and lines of zeroes coexist. This gives origin to the pseudogap and the fragmentation of the Fermi arcs in measures of spectra of the normal state system (see, e.g., Fig. 18). By periodizing the self-energy $Σ$ , by construction, the lines of zeroes cannot appear. The pseudogap is realized only through a modulation of the spectral intensity along the Fermi surface line, which extends always on all the Brillouin zone (see also Fig. 18).Reuse & Permissions
Figure 17
(Color online) The dimension of the patch $Ψ_{k}$ in momentum space (gray shaded region) is determined in such a way that the $M$ -Fermi surface (black continuous line) is cut away and replaced by the $Σ$ -Fermi surface (green dashed line). On the red lines around the corner point ( $π, π$ ) the self-energy $Σ_{k}$ is diverging and $G_{k} = 0$ .Reuse & Permissions
Figure 18
(Color online) The spectral function $A (k, ω) = - \frac{1}{π} Im G_{k} (ω \to 0)$ in the first quadrant of the Brillouin zone obtained with the mixed periodization (top row) is confronted with one obtained from the self-energy $(Σ)$ and the cumulant $(M)$ periodizations (second and third rows from the top). The broadening $η = 0.052$ ; the color scale $x = 0.5$ (see Appendix for details).Reuse & Permissions
Figure 19
(Color online) Comparison between the $Σ$ -periodization scheme, the cumulant $M$ -periodization scheme, and the mixed-periodization scheme for a low doped system $δ = 0.05$ . On the top row (a) we show the full energy range $- 14 < ω < 14$ covering the lower Hubbard and upper Hubbard bands. The gross features are very similar. Note that the $Σ$ scheme introduces fake density of states in the Mott-Hubbard gap. In the bottom row (b) we show a closeup at low energy $- 1 < ω < 2$ , where the methods are most different. The mixed scheme is constructed to retain a $Σ$ -periodization character close to the nodal region $k \sim (\frac{π}{2}, \frac{π}{2})$ and a $M$ -periodization character in the antinodal region $k \sim (0, π)$ . Spectral functions are displayed by introducing a $ω$ dependent broadening $i η (ω)$ and the maximal color scale value is $x = 0.20 [0.25]$ for the top [bottom] panel (see Appendix ).Reuse & Permissions
Figure 20
(Color online) Local density of states $N (ω)$ as a function of the doping obtained with the mixed-periodization scheme. The V shape at $ω \to 0$ is expected from a $d$ -wave superconductor with nodal Fermi liquid quasiparticles. For doping $δ < δ_{p} \sim 0.6$ spectra develop a marked asymmetry. This result interprets the cluster result of Fig. 6 and it is in good agreement with scanning tunneling experiments (see, e.g., Ref. 69). The display on the real axis of Green’s function has been obtained by introducing a $ω$ -dependent broadening $i η (ω)$ (see Appendix for details).Reuse & Permissions
Figure 21
(Color online) Band dispersion $A (k, ω)$ along the path $(0, 0) \to (π, π) \to (0, π) \to (0, 0)$ in the first quadrant of the Brillouin zone. The Mott-insulating state $(δ = 0)$ is confronted with a small doping $(δ = 0.04)$ state. Panel (a): the full range of energies $(- 15 < ω < 15)$ covered by the one-particle spectrum is displayed. The gross band structure, displaying a LHB $(- 6 < ω < 0)$ and a UHB $(10 < ω < 15)$ separated by a Mott gap $(0 < ω < 10)$ , remains unchanged in going from the insulator into the metal. Panel (b): detail of the LHB, which rigidly shifts in going from the insulating to the metallic state. Panel (c): detail of the low energy $(- 1 < ω < 2)$ feature. In going into the metallic state, the shift of the bands is not rigid and spectral weight appears at positive energy as soon as doping is added into the system. Panel (d): detail of the UHB. Upon doping, a rigid shift in the metallic state is accompanied by a transfer of spectral weight to low energies. The color scale maximum value is always $x = 0.20$ , except for panel (d), where it is set $x = 0.25$ . We refer to Appendix for the choice of the broadening $i η (ω)$ .Reuse & Permissions
Figure 22
(Color online) Evolution of the one-particle spectrum from the overdoped (left side, $δ = 0.13$ ) to the underdoped (right side, $δ = 0.04$ ) state. The typical Fermi-liquid band, well visible in a narrow energy range $(- 1 < ω < 1)$ around the Fermi level at high doping $δ = 0.13$ , is progressively destroyed by reducing doping, while the upper and lower Hubbard bands build up acquiring spectral weight. These features are shown in detail in Figs. 23, 24. The color scale maximum value is $x = 0.20$ and we refer to Appendix for the choice of the broadening $i η (ω)$ .Reuse & Permissions
Figure 23
(Color online) Panel (a): Evolution of the LHB from the overdoped (left side, $δ = 0.13$ ) to the underdoped (right side, $δ = 0.04$ ) state. The LHB rigidly shift according to the change in chemical potential by reducing doping, acquiring structure [follow the horizontal line at $k \sim (0, π)$ ]. Panel (b): Evolution of the UHB from the overdoped (left side, $δ = 0.13$ ) to the underdoped (right side, $δ = 0.04$ ) state. The UHB rigidly shift according to the change in chemical potential by reducing doping and, as evident in the color intensity, it gains spectral weight. The color scale maximum value is $x = 0.20$ and we refer to Appendix for the choice of the broadening $i η (ω)$ .Reuse & Permissions
Figure 24
(Color online) Panel (a): we show the evolution of the low-energy one-particle spectral band from the overdoped (left side, $δ = 0.13$ ) to the underdoped (right side, $δ = 0.04$ ) state. A Fermi-liquid-like band at high doping (left side, $δ = 0.13$ ) is progressively destroyed by decreasing doping. Around the region $k = (\frac{π}{2}, \frac{π}{2})$ the dispersion presents a Fermi liquid quasiparticle crossing the Fermi level. Around the region $k = (0, π)$ , the $d$ -wave superconducting state opens a gap by removing spectral weight from the Fermi level to positive and negative energies. Panel (b): we show the contribution to the low-energy band coming from the normal component of the system (we set the anomalous self-energy $Σ_{ano} = 0$ ). In particular, we observe that in the underdoped side $(δ ⩽ 0.08)$ a gap (the pseudogap) opens in the $k = (0, π)$ region even if the superconductive term is absent. For convenience’s sake, the color scale maximum value is set $x = 0.30$ for $δ = 0.13$ and $x = 0.25$ for all the other doping values. We refer to Appendix for the choice of the broadening $i η (ω)$ .Reuse & Permissions
Figure 25
(Color online) The low-energy kink feature in the spectral intensity at the nodal point for various values of doping. For convenience’s sake, the color scale maximum values are set to $x = 0.35$ for $δ = 0.13$ and $x = 0.30$ for all the other doping values. We refer to Appendix for the choice of the broadening $i η (ω)$ .Reuse & Permissions
Figure 26
(Color online) Top panel: The low energy spectral intensity $A (k, ω)$ vs $ω$ calculated within CDMFT (left) is confronted with the experimental result in Ref. 108 in the nodal and antinodal regions. Vertical cuts are traced in $k$ space close to the node and the antinode, as traced on the figures. $k$ is expressed in units of $(π, π)$ . Bottom panel: Color-intensity plots of the spectral intensity $A (k, ω)$ in the $k − ω$ space in the nodal (left) and antinodal (right) cuts of momentum space (the same used in the top panel). The color scale is set $x = 0.30$ (see Appendix ). In the bottom row the experimental result in Ref. 108 are displayed for comparison. We refer to Appendix for the choice of the broadening $i η (ω)$ .Reuse & Permissions
Figure 27
Hall resistivity $R_{H}$ as a function of doping $δ$ . The topological phase transition of the Fermi surface is efficiently detected by a rapid change in the behavior in the Hall resistivity $R_{H}$ , which indicates a change in the carrier density. Left: our theoretical result (units are in the unitary cell volume $v_{0}$ over electron charge $e$ ). Right: the experimental result for LSCO extracted from Ref. 97.Reuse & Permissions
Figure 28
Example of the broadening $η (ω)$ used to display spectral function on the real axis. Left: the low energy range $- 1 < ω < 1$ example of Fig. 20. Right: the wide energy range example of Fig. 19. For a complete list of $η (ω)$ see table in Appendix .Reuse & Permissions
Figure 29
(Color online) Color scale code adopted in displaying the color-density plots. $x$ is the maximum value of the scale.Reuse & Permissions

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