Importance of the Fermi-surface topology to the superconducting state of the electron-doped pnictide Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$

We used angle-resolved photoemission spectroscopy and thermoelectric power to study the poorly explored, highly overdoped side of the phase diagram of Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ high-temperature superconductor. Our data demonstrate that several Lifshitz transitions—topological changes of the Fermi surface—occur for large $x$. The central hole barrel changes to ellipsoids that are centered at $Z$ at $x\sim 0.11$ and subsequently disappear around $x\sim 0.2$; changes in thermoelectric power occur at similar $x$ values. $T_c$ decreases and goes to zero around $x\sim 0.15$—between the two Lifshitz transitions. Beyond $x=0.2$ the central pocket becomes electron-like and superconductivity does not exist. Our observations reveal the importance of the underlying Fermiology in electron-doped iron arsenides. We speculate that a likely necessary condition for superconductivity in these materials is the presence of the central hole pockets rather than nesting between central and corner pockets.

Figure 1

Fermi maps and band dispersion around the upper zone edge $Z$ of Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ for $x=0.073$ (optimal doping), $x=0.166$ (edge of SC dome), and $x=0.42$. Upper row: Fermi mappings for the three doping levels, taken with incident photon energy $h\nu =35$ eV at temperature $T=20$ K. Red arrows show the exit slit direction of the hemispheric analyzer and the cutting direction of the band dispersion maps (lower row). The same direction is also used in Fig. 2. Note that the $X$ points marked in these figures have slightly lower $k_z$ values than the $Z$ points.

Figure 2

Band location analysis for the Lifshitz transitions. (a)–(h): Band dispersion maps along the direction shown in Fig. 1 for eight different doping levels at $T=20$ K. All data are taken with 35-eV photons. The red vertical line marks $0.195<x_{2Z}<0.27$. (i) Energy distribution curve (EDC) at $Z$ for each doping level. (j) and (k) Evolution of binding energy for the top of the hole band and the bottom of the electron band at the zone center with respect to cobalt doping. $x_{2Z}$ and $x_{2\Gamma }$ are defined as the midpoint between the two doping levels at which the electron and hole band evolves above $\mu$. Data are extracted from ARPES intensity maps taken with (j) 35-eV and (k) 49-eV photons, corresponding to $k_z$ values of $Z$ and $\Gamma$, respectively. For $x>0.195$, data points in (k) are extracted from the EDC’s at (i) by fitting with two Lorentzian functions. For $x⩽0.195$, a parabolic function is fitted to the momentum distribution curve (MDC) peak positions of the outer hole band in (a)–(d) for extracting the top of the band above $\mu$. Raw data for extracting panel (k) are not shown.

Figure 3

Pocket size analysis for the Lifshitz transition at upper zone boundary $Z$. (a) $Z$-pocket extraction for eight doping levels, done by fitting the MDC’s at the chemical potential with several Lorenzian functions. Positions of hollow circles are symmetrized from experimental data points (solid circles), proposing the band positions where ARPES intensity is suppressed by the transition matrix element. (b) Evolution of $Z$-pocket area with cobalt doping. Green shaded area indicates the boundary of the SC dome. (c) Visualization of the Lifshitz transition. Data in (a) are plotted against the cobalt doping $x$ as a third dimension. Shaded areas are approximate size and shape of the pockets. Panels (b) and (c) show a Lifshitz transition at $x_{2Z}\sim 0.2$.

Figure 4

(a) Location of the known Lifshitz transitions in the phase diagram. $T_{\mathrm{N}}$ and $T_c$ data are taken from Refs. 1 and 8. Top insets show schematic Fermi surface topology in the $a$-$b$ and $a$-$c$ plane for each region in the phase diagram. (b) Thermoelectric power vs doping for four different temperatures.