Theory of charged impurities in correlated electron materials: Application to muon spectroscopy of high-$T_c$ superconductors

Single-site and cluster dynamical mean-field theory methods are used to estimate the response of a doped Mott insulator to a charged impurity. The effect of correlations on the Thomas-Fermi screening properties is determined. The charge density, the on-site and near-neighbor spin-spin correlations in the vicinity of the impurity are compared to those far from it. The theory is used to address the question of the effect of the density perturbation induced by the muon charge on the local response functions of a high-temperature superconductor. For reasonable values of the background dielectric constant and basic correlation strength, a muon is shown to lead to an observable perturbation of the local spin dynamics, raising questions about the interpretation of muon-spin-rotation experiments in metallic high-temperature superconductors.

Figure 1

(Color online) Sketch of two cases considered in paper. Left panel: charged impurity (muon) placed at center of cube of transition-metal (Cu) sites. Right panel: impurity placed in plane.

Figure 2

(Color online) Sketch of the self-consistency procedure used to calculate the charge density and hybridization functions in the vicinity of a charged impurity. Starting from an initial guess for the potential $V_i$ and the hybridization function ${\Delta }_i\left(\omega \right)$ the dynamical mean-field procedure is used to obtain a new density and hybridization function; the new density is used in Eq. (2) to obtain new potentials, and the process is iterated to self-consistency. Changes in the hybridization function are found to be sufficiently small that the DMFT loop may be solved once for the bulk material and $\partial n/\partial \mu$ obtained from this solution may then be used to update the density.

Figure 3

(Color online) Local-density change induced by locally potential $\delta {\mu }_i$. In all cases bulk density is $n=0.9$. Solid line: density change on site $i$, $\delta n_i$, induced by potential $\delta {\mu }_i$ applied on same site, computed for noninteracting electrons using tight-binding band parameters appropriate to high-temperature superconductors. Dotted line: same computation but using single-site dynamical mean-field theory with $U=13t$ and hybridization parameters taken from calculation for $U=13t$ and $\delta =0.1$. Dashed line: change in density on one site of a four-site plaquette induced by potential $\delta \mu$ applied to all four sites of plaquette computed using CDMFT dynamical mean-field theory with $U=9t$ with hybridization function corresponding to $U=9t$ and $\delta =0.1$.

Figure 4

(Color online) [(a)–(d)] Variation in conduction-band density per lattice site along line $\left(x,0,0\right)$ induced by impurities positioned at [(a) and (c)] (1/2,1/2,1/2) and [(b) and (d)] (1/2,1/2,0) calculated as described in the text using single-site DMFT with $U=13t$ and four-site DMFT with $U=9t$. [(e) and (f)] Variation in conduction-band density per lattice site along line $\left(x,0,1\right)$ induced by impurity positioned at (1/2,1/2,0) for (e) $ϵ=4$ and (f) $ϵ=15$ calculated as described in the text using single-site DMFT with $U=13t$ and four-site DMFT with $U=9t$.

Figure 5

(Color online) Impurity-model spin-spin correlation along line $\left(x,0,0\right)$ computed from single-site dynamical mean-field theory as function of Matsubara frequency at inverse temperature $\beta =20$ and dielectric constants indicated for charged impurity at position (1/2,1/2,0). Upper panels are on-site correlators for sites nearest to muon $\left(x=1\right)$, lower panels are on-site correlators for second neighbors to muon $\left(x=2\right)$, correlator for site far from muon is also included.

Figure 6

(Color online) On-site (upper panels) and first-neighbor (lower panels) impurity-model spin-spin correlation computed from four-site CDMFT cluster dynamical mean-field theory as function of Matsubara frequency at inverse temperature $\beta =20$ and dielectric constants indicated, for charged impurity at position (1/2,1/2,0). Solid line with solid squares (blue online) shows correlator computed for sites far from impurity site. Dotted line with solid circles (red online) shows correlator computed on designated site. Dashed line with open squares (blue online) shows correlated computed for bulk system with mean density equal to density on impurity site.