Doping driven small-to-large Fermi surface transition and $d$-wave superconductivity in a two-dimensional Kondo lattice

We study the two-dimensional Kondo lattice model with an additional Heisenberg exchange between localized spins. In a first step, we use mean-field theory with two order parameters. The first order parameter is a complex pairing amplitude between conduction electrons and localized spins that describes condensation of Kondo (or Zhang-Rice) singlets. A nonvanishing value implies that the localized spins contribute to the Fermi surface volume. The second-order parameter describes singlet pairing between the localized spins and competes with the Kondo-pairing order parameter. Reduction of the carrier density in the conduction band reduces the energy gain due to the formation of the large Fermi surface and induces a phase transition to a state with strong singlet correlations between the localized spins and a Fermi surface that comprises only the conduction electrons. The model thus shows a doping driven change of its Fermi surface volume. At intermediate doping and low temperature, there is a phase where both order parameters coexist, which has a gapped large Fermi surface and $d$${}_{x^2-y^2}$ superconductivity. The theory thus qualitatively reproduces the phase diagram of cuprate superconductors. In the second part of this paper, we show how the two phases with different Fermi surface volume emerge in a strong-coupling theory applicable in the limit of large Kondo exchange. The large Fermi surface phase corresponds to a “vacuum” of localized Kondo singlets with uniform phase, and the quasiparticles are spin-1/2 charge fluctuations around this fully paired state. In the small Fermi surface phase, the quasiparticles correspond to propagating Kondo singlets or triplets whereby the phase of a given Kondo singlet corresponds to its momentum. In this picture, a phase transition occurs for low filling of the conduction band as well.

Figure 1

Critical temperature for the onset of the parameter ${\Delta }_{cd}$ with the true two-dimensional nearest-neighbor-hopping band structure (top) and a constant density of states in the interval $[-4t:4t]$ (bottom).

Figure 2

Development of the two order parameters with increasing $J$. The other values are $W/t=1.6$, $\delta =0.2$, and $T/t=0.001$.

Figure 3

Temperature dependence of the two order parameters for $W/t=1.6$, $J/t=0.05$, and $\delta =0.32$.

Figure 4

Self-consistent solution for ${\Delta }_{cd}$ as a function of temperature for various dopings $\delta$. The other values are $W/t=1.6$ and $J/t=0.05$.

Figure 5

Top panel: Self-consistent solution for ${\Delta }_{cd}$ as a function of temperature for $\delta =0.25$ (see Fig.  4). Bottom panel: Scans of the free energy as a function of ${\Delta }_{cd}$ whereby all other mean-field parameters have been obtained self-consistently.

Figure 6

Self-consistent solutions for ${\Delta }_{cd}$ as a function of doping for $T=0$ and different $J$. The value $W/t=2$ ($W/t=1.6$) in the upper (lower) panel of the figure.

Figure 7

Top panel: Mean-field phase diagram for the parameter values $W/t=2$ and $J/t=0.325$. Bottom panel: Difference in entropy per site between the phases 3 and 4.

Figure 8

Top panel: Band structure for phase 2 ($T=0.005$, $\delta =0.60$). The two vertical lines at $\mu$ give the Fermi momenta, i.e., the crossings of the quasiparticle band through $\mu$. Bottom panel: Band structure for phase 3 ($T=0.005$, $\Delta =0.25$). Other parameter values are $W/t=2$ and $J/t=0.325$.

Figure 9

Band structure for phase 4 ($T=0.005$, $\delta =0.45$). Entire bandwidth (top panel) and closeup of the region near $\mu$ (bottom panel). Other parameter values are $W/t=2$ and $J/t=0.325$.

Figure 10

Pairing amplitude for conduction holes ${\Delta }_{cc}$ as a function of temperature. Other parameter values are $W/t=2.0$ and $J/t=0.325$.

Figure 11

Energy gap (top panel) and spectral weight of the band below $\mu$ (bottom panel) as a function of doping for some momenta along $(1,0)$. Parameter values are $W/t=2.0$, $J/t=0.325$, and $T/t=0.05$.

Figure 12

Dispersion of the quasiparticle bands for the phase with small Fermi surface. Parameter values are $W/t=4$, $J/t=4$, and ${\chi }_{10}=-0.28$. The dashed line is the dispersion of the lowest band as obtained by second-order perturbation theory.

Figure 13

The function $\alpha \left(\delta \right)$ defined in Eq. (55) evaluated numerically in a $4\times 4$ cluster for different forms of the kinetic energy, i.e., for different values of the hopping integrals $t$, $t^{\prime }$, and $t^{\prime \prime }$. The values of the hopping integrals for the individual systems are given in Table . The line is the function (56).

Figure 14

Hole concentration ${\delta }_c$ where the phase transition from small-to-large Fermi surface occurs as a function of $W/t$. The small Fermi surface is realized for $\delta <{\delta }_c$.