Quasi-Particle Interference Probe of the Self-Energy

Quasi-particle interference (QPI) measurements have provided a powerful tool for determining the momentum dependence of the gap of unconventional superconductors. Here we examine the possibility of using such measurements to probe the frequency and momentum dependence of the electron self-energy. For illustration, we calculate the QPI response function for a cuprate-like Fermi surface with an electron self-energy from an RPA approximation. Then we try to reextract the self-energy from the dispersion of the peaks in the QPI response function using different approaches. We show that in principle it is possible to extract the self-energy from the QPI response for certain nested momentum directions. We discuss some of the limitations that one faces.


Introduction
Useful information about the interaction of electrons in metals and superconductors is contained in the quasiparticle self-energy Σ(k, ω). When the self-energy and the band structure E(k) of a metal is known, the Green's function G(k, ω) = [ω − E(k) − Σ(k, ω)] −1 provides complete information on the single particle properties of the system. Experimentally, in isotropic superconducting systems both the normal and anomalous (gap function) self-energies at the Fermi surface can be obtained from tunneling spectroscopy. [1] In anisotropic systems angular resolved photoemission spectroscopy (ARPES) can be used to obtain momentum resolved information on the self-energy. [2,3,4,5,6,7] In recent years it was shown that the momentum dependence of the gap in unconventional superconductors can be obtained from scanning tunneling microscopy (STM) employing the so-called quasi-particle interference (QPI). [8,9,10,11,12,13,14,15,16] In this technique one measures the local tunneling conductance at ω = eV around an impurity at the surface of a metal over a large grid of points. Its Fourier transform gives a wave vector power spectrum |Im Λ(q, ω)| 2 . Peaks in Im Λ(q, ω) arise from dynamic nesting processes in which quasi-particles undergo elastic backward scattering from the impuities. From these peaks and their dispersion one can obtain information on nesting properties of the Fermi surface and about the momentum dependence of the superconducting gap. [17] In the present work we want to explore, whether beyond that QPI experiments can be used to also extract information about the momentum and energy dependence of the self-energy, in particular about the effective mass and the lifetime of the quasiparticles. The main idea here is to closely investigate the dispersion and the width of the peaks in Im Λ(q, ω) as a function of energy ω and try to extract the self-energy from it. We will demonstrate that in principle this is possible and we will discuss some of the limitations that one faces.
In a metal, the dispersion and damping of quasi-particles with energy ω is described by a complex frequency dependent wave vector k(ω) = k 1 (ω)+ik 2 (ω), where k 1 (ω) determines the renormalized dispersion and k 2 (ω) the lifetime of a quasiparticle state. The band structure E(k) and the self-energy Σ(k, ω) determine k(ω) and conversely, given E(k), the structure of the self-energy is reflected in k 1 (ω) and k 2 (ω). The tunneling conductance at a particular point depends upon k(ω) and the surrounding impurity configuration. Here we will discuss how one can extract k(ω) from the structure in Im Λ(q, ω) and use it to study the real and imaginary parts of the self-energy.

Peaks in the QPI response
For weak charge impurity scattering, the wave-vector power spectrum of the local tunneling density of states depends upon the product of a static impurity structure factor and the quasi-particle interference response function. Neglecting vertex corrections, the response function can be written as [17], with ω = eV . An expression for Λ(q, ω) including vertex corrections is given in Ref. [18]. Here we consider the QPI response for a given value of ω plotted as a function of wave-vector q along certain lines in the Brillouin zone called q-cuts [11].
Examples of such q-cuts are illustrated in Fig. 1 for a 2D free electron system and in Fig. 2a for a cuprate like band structure. Along a q-cut, the QPI response function Im Λ(q, ω) peaks near wave-vectors q peak (ω) which connect equi-energy quasi-particle surfaces which have oppositely directed quasi-particle velocities. As noted, these peaks reflect a dynamic nesting which depends on the band structure E(k) as well as the quasi-particle self-energy. For the 2D free electron case the radial q-cuts all give the same information, while for the cuprate-like bandstructure the horizontal q-cut (c) shown in Fig. 2a probes the antinodal region of the Fermi surface while the diagonal q-cuts probe both the nodal (a) as well as intermediate regions (b), which depend on ω, as shown in Fig. 2b and c. For ω = 0, q = 2k F leads to nesting, while for ω = 0.1µ (dashed circle) nesting occurs for q = 2k(ω) = 2k F √ 1.1.
As we will discuss, by fitting the ω dependence of the peak structure in the QPI, one can extract information on the ω dependence of the real and imaginary parts of the self-energy. Both normal and Umklapp peaks associated with a given q-cut provide similar information and in principle the Umklapp peaks can be used to estimate the q dependent fall-off of the impurity scattering structure factor.
To begin, we consider the 2D free electron system of Fig. 1. In this case with H 0 the zeroth order Hankel function of the first kind, [17] N (0) the single particle density of states, and k(ω) is determined from the dispersion relation Here µ = k 2 F /2m. Carrying out the spatial integration in Eq. (1), the QPI response function is found to have a square-root singularity for This wave-vector connects nested equi-energy contours at k(ω) and −k(ω) along the q-cut. When impurity scattering is taken into account with the mean free path. In this case the wave vector k(ω) = k 1 (ω) + ik 2 (ω) contains information on both the real and imaginary parts of the single particle propagation.

RPA self-energy
For an interacting system, again neglecting vertex corrections with We have set Σ(k, ω) = (1 − Z(k, ω)) ω and E(k) is the bandstructure energy minus the chemical potential µ. For illustration, we will calculate Z(k, ω) = Z 1 (k, ω) + iZ 2 (k, ω) for a Hubbard model using a random phase approximation (RPA) for the spinfluctuation interaction: [19] ( Here, and t /t = −0.15, t /t = 0.075, µ/t = −0.81. The parameters of the bandstructure were taken from tight-binding fits to ARPES data appropriate for the La 2−x Sr x CuO 4 cuprate near optimum doping x = 0.15. [20] The coupling constantsŪ /t = 3 and U/t = 1.5 have been chosen such that a mass renormalization of Z 1 (ω = 0) ≈ 2 is obtained at the nodal direction and Z 1 (ω = 0) ≈ 3 at the antinodal direction. For the numerical calculations a finite broadening of δ = 0.005t has been used.
Calculating the RPA self-energy Z(k, ω) for these parameters and using it in Eq. (7), we find the QPI response shown in Fig. 3. In the following we will consider Im q x ,q y 0,Ω Π Figure 3. Plots of − 1 π Im Λ(q, ω) versus q for ω/t values separated by 0.05 for the diagonal (a) and horizontal (b) q-cuts of Fig. 2a. As ω/t increases the QPI peak disperses and broadens. − 1 π Im Λ(q, ω) as our "experimental" QPI response and explore how one can extract information about Z(k, ω) from it.

Estimation of the self-energy from the peak position
From Fig. 3, one sees that the response is characterized by a peak which disperses and broadens as ω/t increases. The peak in Fig. 3a corresponds to the nesting vector "a" in Fig. 2a, while the peak in Fig. 3b corresponds to the nesting vector "c" in Fig. 2a. There exists another peak along the diagonal q-cut at smaller values of q, which corresponds to the nesting vector "b" in Fig. 2a, but is not shown here. We do not consider the nesting vector "b" any further, as in the case discussed here it gives similar information about the self-energy for antinodal momenta as vector "c". Within a quasi-particle approximation, one finds that the peaks have similar structure to Eq. (4) with k(ω) = k 1 (ω) + ik 2 (ω) determined by and Here, v F (k 1 (ω)) = ∂E ∂k ⊥ (k 1 (ω)) is the band velocity with the derivative taken perpendicular to the surface where the q-cut crosses the ω = E (k 1 (ω)) surface. The change in sign of the wave vectors in the square-root of Eq. (12) relative to Eq. (4) arises from the change in sign of the Fermi surface curvature.
If one takes the peak value q peak (ω) as an estimate of 2k 1 (ω), neglects the imaginary part and uses a linearized dispersion E (k 1 (ω)) ≈ v F (k 1 (ω) − k F ) one finds approximately If q peak (ω) exceeds the region over which a linear approximation of the dispersion is appropriate, then one needs to use the full dispersion and In Fig. 4 the results for Z 1 (ω) obtained using q peak (ω)/2 as an estimate of k 1 (ω) for both the diagonal and horizontal q-cuts are compared with Z 1 (k 1 (ω), ω) (solid curves) obtained from Eq. (9). For the nodal direction, E(k) is well approximated by its linearized v F (k − k F ) form while for the antinodal direction a linear approximation fails due to the closeness of the Fermi level to the saddle point of the band at (0, π).
In this case it is necessary to use the full band dispersion E(k). In both cases, using q peak (ω) underestimates 2k 1 (ω) and the resulting Z 1 (ω) falls below the self-energy used in the calculation of Λ(q, ω).

Fitting of the QPI peaks
The problem with using the peak of the QPI q-cut to estimate Z 1 (ω) is that k(ω) in Eq. (12) has both real and imaginary parts. Thus a better alternative is to fit the QPI peak to the square-root form of Eq. (12) and extract a k 1 (ω) and k 2 (ω) for each ω q-cut. Figure 5 shows the results of a fit to a form  . Fitting − 1 π Im Λ(q, ω = 0.1t) along a diagonal q-cut using Eq. (17) to obtain k 1 (ω) and k 2 (ω).
Here, ω = 0.1t along a diagonal q-cut is shown. A finite q range of [0.825, 0.875]π has been used for the fitting. As can be seen in Fig. 3 the peaks for a cuprate-like bandstructure appear to have a linear "background" behind the smeared square-root singularity. For that reason we found it necessary to include a linear background B + Cq in the fitting formula Eq. (17), which improves the determination of k 1 and k 2 . The blue dot in Fig. 5 denotes the position of the extracted 2k 1 and the blue bar the width ±2k 2 for this particular energy ω. As this plot shows, the actual position of 2k 1 is slightly off from the peak position q peak to the right. The reason for this is the asymmetric line shape of the square-root singular QPI response function Eq. (12). As seen below, the fitted values of k 1 and k 2 allow a much more precise extraction of Z 1 and Z 2 than the peak position q peak /2.
To get precise values of k 1 and k 2 from the fit we found it necessary to restrict the fitting to a finite q range around the peak position to avoid the fit being spoiled by values away from the peak, where the fitting formula Eq. (17) is not valid anymore. To get a good coverage of the peak we have chosen the following q ranges: [q peak − 0.02π(1 + |ω|/t), q peak + 0.02π(1 + 2|ω|/t)] in the nodal direction and [q peak − 0.02π(1 + 4|ω|/t), q peak + 0.02π(1 + 8|ω|/t)] in the antinodal direction. These q ranges account for a minimum range of ∆q = 0.04π, they increase with increasing |ω| to account for the fact that the peaks are getting broader at higher frequencies, and the larger range in the antinodal direction accounts for the smaller Fermi velocity in this direction which leads to larger peak widths. Also note that we have chosen the q range asymmetrically around the peak position, as 2k 1 is always larger than q peak .
After k 1 (ω) and k 2 (ω) have been extracted from these fits, for the linear dispersion approximation we will compare and v F k 2 (ω) (19) with Z 1 (k 1 (ω), ω) and ωZ 2 (k 1 (ω), ω), respectively. In the following all of our estimates will be compared with Z (k 1 (ω), ω) since the self-energy does have a weak k dependence. If the dynamic range is such that the non-linearity of the dispersion is important, then the comparison will be with and In the following plots, the solid curves are the RPA self energy evaluated at k = k 1 (ω) with k 1 (ω) in this case obtained from the self-energy calculation Eq. (9). Our goal is to see how well one can extract the solid curves from the QPI response − 1 π Im Λ(q, ω). As shown in Fig. 6a and Fig. 7a for the nodal case, useful estimates of Z 1 (k 1 (ω), ω) and Z 2 (k 1 (ω), ω) can be obtained using (k 1 (ω), k 2 (ω)) and a linear approximation of the dispersion. Similarly in Fig. 6b and Fig. 7b one sees that if (k 1 (ω), k 2 (ω)) can be extracted by fitting Im Λ(q, ω) for the antinodal case one can again obtain useful estimates of the self-energy. However, in the antinodal case it is important to use the full bandstructure E(k).
Results obtained for Z 1 and Z 2 using Eqs. (22) and (23) are shown as the red triangles in Figs. 6 and 7. While this way of extracting k 1 (ω) and k 2 (ω) from the QPI response is less accurate than fitting Eq. (17), it can provide reasonable results. The B + Cq background must be removed from Im Λ(q, ω) in estimating the normalized curvature, Eq. (23).

Conclusions
While we have been able to extract from Im Λ(q, ω) the self-energy that went into the Green's functions used to calculate it, these results illustrate some of the challenges and limitations one faces. Extracting k 1 (ω) and k 2 (ω) will clearly become more difficult as ω increases and the peak broadens and decreases in amplitude [21]. As seen for the diagonal k x = k y cut in Fig. 2, there can also be multiple peaks associated with a given q-cut. For ω > 0, these peaks are well separated in q. However, as seen in Fig. 2c, for this Fermi surface there are problems for ω < 0, where the q a and q b peaks approach each other for ω < 0. In addition to k 1 (ω) and k 2 (ω) we used information on the bare bandstructure E(k) which is not a measured quantity. For the diagonal q-cut it appears that the band Fermi velocity v F would be sufficient, and in principle one might hope that at large values of ω one could estimate the bare v F . However, by these energies one is typically out of the linear region of dispersion. Thus one needs to make a reasonable estimate for E(k) based on band theory.
Finally, there is the q-dependence of the impurity scattering form factor and the effect of vertex corrections. [18] The scattering form factor will reduce the amplitude of the QPI response as q increases but should not lead to significant shifts of the k 1 (ω) and k 2 (ω) values provided its characteristic momentum is large compared with k 2 (ω). While the vertex Γ(k, q) can introduce additional structure, the continuity of the peak associated with the interference between the two propagators as ω increases along with the short range nature of the vertex corrections should generally limit its effect on k 1 (ω) and k 2 (ω).