Extracting the electron-boson spectral function ${\alpha }^{2}F\left(\omega \right)$ from infrared and photoemission data using inverse theory

We present a method for extracting the electron-boson spectral function ${\alpha }^{2}F\left(\omega \right)$ from infrared and photoemission data. This procedure is based on inverse theory and will be shown to be superior to previous techniques. Numerical implementation of the algorithm is presented in detail and then used to accurately determine the doping and temperature dependence of the spectral function in several families of high-$T_c$ superconductors. Principal limitations of extracting ${\alpha }^{2}F\left(\omega \right)$ from experimental data will be pointed out. We directly compare the IR and angular-resolved photoemission spectroscopy ${\alpha }^{2}F\left(\omega \right)$ and discuss the resonance structure in the spectra in terms of existing theoretical models.

Figure 1

Spectral function ${\alpha }^{2}F\left(\omega \right)$ for underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.6}$ with $T_c=59\mathrm{K}$. The left panels show ${\alpha }^{2}F\left(\omega \right)$ and the right panels show the experimental $1∕\tau \left(\omega \right)$ along with $1∕{\tau }_{\mathit{cal}}\left(\omega \right)$ calculated from the corresponding spectral function [Eq. (4)]. The top panels show a previously published spectral function (Ref. 8) obtained from the scattering rate smoothed by hand. The other five pairs of panels are the data obtained using inverse theory. Different numbers of singular values are kept in the calculations, which results in different levels of smoothing. Note that the vertical scale in panels B1 and C1 is different.

Figure 2

Spectral function ${\alpha }^{2}F\left(\omega \right)$ for underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.6}$ with $T_c=59\mathrm{K}$ obtained by smoothing the experimental $1∕\tau \left(\omega \right)$ data by hand (Ref. 8) and with SVD with 11, 12, and 13 svs.

Figure 3

Model calculations of a spectral function with two Lorentzians [Eq. (10)]. The exact solution, with all 300 svs, does not agree well with the model because small singular values produce numerical instabilities in the solution. On the other hand, if too few svs are kept unphysical negative regions appear. The model spectral function is recovered with 50–100 svs.

Figure 4

Singular values at different temperatures. Top panel: first (biggest) 200 svs from IR; bottom panel: all 100 svs from ARPES. When analyzing temperature dependence of the spectral function “horizontal cuts” are more appropriate, assuming all data sets have the same signal-to-noise ratio. On the other hand, for doping dependence studies (at the same temperature) “vertical cuts,” i.e., the same number of svs, should produce similar levels of smoothing.

Figure 5

Model calculations of spectral function from the BCS scattering rate. The left panels display the calculated spectral function ${\alpha }^{2}F\left(\omega \right)$ and the right panels display the BCS scattering rate (also shown with dotted lines in the left panels) and $1∕{\tau }_{\mathit{cal}}\left(\omega \right)$ calculated from the spectral function on the left. A gap in the density of states produces similar structure in ${\alpha }^{2}F\left(\omega \right)$ as does the coupling to a Bosonic mode.

Figure 6

Spectral function ${\alpha }^{2}F\left(\omega \right)$ for underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.6}$ with $T_c=59\mathrm{K}$. A deterministic constraint ${\alpha }^{2}F\left(\omega \right)⩾0$ is applied iteratively [Eq. (12)]. The top two panels show the initial solution with 20 svs. The other four sets of panels display intermediate solutions for several different levels of iterations.

Figure 7

Doping dependence of spectral function ${\alpha }^{2}F\left(\omega \right)$ for (A) ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.60}$ with $T_c=57\mathrm{K}$ (Ref. 36), (B) ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.60}$ with $T_c=59\mathrm{K}$ (Ref. 8), and (C) ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$ with $T_c=91\mathrm{K}$ (Ref. 36). All curves are for the lowest measured temperature $T\approx 10\mathrm{K}$. Vertical cuts, i.e., the same number of singular values (15), were made for all three data sets. The right panels display $1∕\tau \left(\omega \right)$ data along with $1∕{\tau }_{\mathit{cal}}\left(\omega \right)$. Dashed lines are the results of iterative calculations. Relatively good fits without negative values in ${\alpha }^{2}F\left(\omega \right)$ can be obtained for both ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.60}$ samples, but not for ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$.

Figure 8

Spectral function ${\alpha }^{2}F\left(\omega \right)$ for optimally doped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$ calculated from Eq. (13) for the scattering rate in the superconducting state. Different values of the gap $\Delta =0\text{–}200{\mathrm{cm}}^{-1}$ were used in the calculations. For $\Delta =0$ there is a pronounced dip following the main peak. However, when finite values of the gap are introduced the negative dip gradually disappears and the main peak shifts to lower energies.

Figure 9

Model spectral function ${\alpha }^{2}F\left(\omega \right)$ (thin line) is used to calculate the scattering rate $1∕{\tau }_{\mathit{cal}}\left(\omega \right)$ from Eq. (13). For $\Delta =0$ the calculated scattering rate resembles $1∕\tau \left(\omega \right)$ of underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.60}$ (Fig. 7). However, for finite values of the gap the calculated scattering rate resembles $1∕\tau \left(\omega \right)$ of optimally doped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.95}$: there is an overshoot following the suppressed region (Fig. 7).

Figure 10

Temperature dependence of the spectral function ${\alpha }^{2}F\left(\omega \right)$ for optimally doped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8-\delta }$ with $T_c=91\mathrm{K}$. As temperature increases the main peak shifts to lower energies and loses intensity, but seems to persist even above $T_c$.

Figure 11

Spectral function ${\alpha }^{2}F\left(\omega \right)$ of Mo(110) surface at 70 K extracted from ARPES data (Ref. 44). Three different levels of smoothing are shown with 10, 15, and 20 svs. Dotted lines in the left panels represent the theoretical spectral function (Ref. 45).

Figure 12

Temperature dependence of spectral function ${\alpha }^{2}F\left(\omega \right)$ of optimally doped Bi2212 extracted from ARPES data (along nodal direction) using inverse theory. Left panels show ${\alpha }^{2}F\left(\omega \right)$ spectra calculated from Eq. (14), and right panels show ARPES quasiparticle dispersion $E_k$ (gray symbols) and calculated dispersion $E_{k,\mathit{cal}}$ (full lines) using the corresponding spectral function on the left. Also shown with dashed lines are bare quasiparticle dispersions ${ϵ}_k$ used to calculate ${\Sigma }_1\left(\omega \right)$ [Eq. (15)].

Figure 13

Comparison of spectral functions ${\alpha }^{2}F\left(\omega \right)$ extracted from IR and ARPES data for optimally doped Bi2212 with $T_c=91\mathrm{K}$. Note that ${\alpha }^{2}F\left(\omega \right)$ from ARPES is multiplied by a factor of 3. The dashed line is an ARPES calculation with a different bare dispersion (with renormalization effects up to 0.5 eV).

Figure 14

Spectral function ${\alpha }^{2}F\left(\omega \right)$ extracted from IR data for underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.6}$ with $T_c=59\mathrm{K}$ and calculated up to $\approx 0.85\mathrm{eV}$. The top panels display calculations with negative values (Sec. 2). The middle panels are iterative calculations with 2000 iterations (Sec. 5). In both cases a significant contribution to ${\alpha }^{2}F\left(\omega \right)$ persists up to very high frequencies. The bottom panels display the results of calculations of $1∕{\tau }_{\mathit{cal}}\left(\omega \right)$ with high-frequency contribution to ${\alpha }^{2}F\left(\omega \right)$ cutoff (above $1000{\mathrm{cm}}^{-1}$). In this case the calculated scattering rate tends to saturate at higher energies.