Ferromagnetism, paramagnetism, and a Curie-Weiss metal in an electron-doped Hubbard model on a triangular lattice

Motivated by the unconventional properties and rich phase diagram of ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$ we consider the electronic and magnetic properties of a two-dimensional Hubbard model on an isotropic triangular lattice doped with electrons away from half-filling. Dynamical mean-field theory (DMFT) calculations predict that for negative intersite hopping amplitudes $(t<0)$ and an on-site Coulomb repulsion, $U$, comparable to the bandwidth, the system displays properties typical of a weakly correlated metal. In contrast, for $t>0$ a large enhancement of the effective mass, itinerant ferromagnetism, and a metallic phase with a Curie-Weiss magnetic susceptibility are found in a broad electron doping range. The different behavior encountered is a consequence of the larger noninteracting density of states (DOS) at the Fermi level for $t>0$ than for $t<0$, which effectively enhances the mass and the scattering amplitude of the quasiparticles. The shape of the DOS is crucial for the occurrence of ferromagnetism as for $t>0$ the energy cost of polarizing the system is much smaller than for $t<0$. Our observation of Nagaoka ferromagnetism is consistent with the A-type antiferromagnetism (i.e., ferromagnetic layers stacked antiferromagnetically) observed in neutron scattering experiments on ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$. The transport and magnetic properties measured in ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$ are consistent with DMFT predictions of a metal close to the Mott insulator and we discuss the role of Na ordering in driving the system towards the Mott transition. We propose that the “Curie-Weiss metal” phase observed in ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$ is a consequence of the crossover from a “bad metal” with incoherent quasiparticles at temperatures $T>T^{*}$ and Fermi liquid behavior with enhanced parameters below $T^{*}$, where $T^{*}$ is a low energy coherence scale induced by strong local Coulomb electron correlations. Our analysis also shows that the one band Hubbard model on a triangular lattice is not enough to describe the unusual properties of ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$ and is used to identify the simplest relevant model that captures the essential physics in ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$. We propose a model which allows for the Na ordering phenomena observed in the system which, we propose, drives the system close to the Mott insulating phase even at large dopings.

Figure 1

Phase diagram for the electron doped Hubbard model on a triangular lattice obtained from DMFT calculations. $U∕\mid t\mid$ is the ratio of the on-site Coulomb repulsion to the absolute value of the hopping matrix element $t$ and $n$ is the average number of electrons per site. The sign of $t$ leads to qualitative changes in the phase diagram. In the $t>0$ case, a ferromagnetic metal and a metal with Curie-Weiss susceptibility are found. In contrast, for $t<0$ a paramagnetic metal appears throughout the phase diagram. To make a direct comparison with ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$ one should note that $x=n-1$.

Figure 2

Energy cost in spin polarizing the three site triangular cluster with four noninteracting electrons. For $t>0$ the ground state is degenerate and there is no energy cost in flipping one spin. In contrast, it costs $3\mid t\mid$ to polarize the $t<0$ cluster. This indicates that the tendency to ferromagnetism of the electron doped $t>0$ triangular cluster is much stronger than that of the cluster with $t<0$ even at the one-electron level.

Figure 3

(Color online) A pictorial representation of the ground states of the Hubbard model on a triangular cluster with four electrons. Doubly occupied sites are indicated by the presence of two arrows, one pointing up and the other pointing down, while unoccupied sites have no arrows. In the cases where we have only one doubly occupied site the remaining sites with only one electron can form either a singlet (indicated by a thick red line with arrows on the two sites pointing in opposite directions) or a triplet (indicated by a thick blue line with arrows on the two sites pointing in the same direction). Thus, the middle left triangle indicates the state $\frac{1}{\sqrt{2}}(\mid \upharpoonleft \downharpoonright ,\uparrow ,\downarrow ⟩-\mid \upharpoonleft \downharpoonright ,\downarrow ,\uparrow ⟩)$. The top left triangle indicates one the three states ∣↓,↓,↿⇂⟩, $\frac{1}{\sqrt{2}}(\mid \uparrow ,\downarrow ,\upharpoonleft \downharpoonright ⟩+\mid \downarrow ,\uparrow ,\upharpoonleft \downharpoonright ⟩)$ or ∣↑,↑,↿⇂⟩ (Ref. 55). To keep the figure as simple as possible we have not included normalization factors in the figure, but the correct normalizations are given in Eqs. (4, 6, 7, 8). The lower part of the figure illustrates $\mid {\Psi }_{-}⟩$, the ground state for any nonzero value of $U$ and $t<0$ [cf. Eq. (4)] (the ground state is degenerate for $t<0$ and $U=0$). The upper part of the figure illustrates all three degenerate states $\mid {\Psi }_{+}^{\downarrow }⟩$, $\mid {\Psi }_{+}^0⟩$, and $\mid {\Psi }_{+}^{\uparrow }⟩$ as these only differ by the value of the $S_z=0$ projection of the triplet. These states are the ground states for $t<0$ and any value of $U$ [cf. Eqs. (6, 7, 8)]. $\mid {\Psi }_{-}⟩$ depends on the variable $\theta$ which is a function of the ratio $U∕t$, given by Eq. (5). $\theta \to 0$ as $U\to \infty$; $\theta =\pi ∕4$ for $U=0$.

Figure 4

(Color online) The imaginary time local spin autocorrelation function for both $t>0$ and $t<0$ with $n=1.3$, $U=10\mid t\mid$ and $\beta =3∕\mid t\mid$. The larger values of $⟨S_z\left(0\right)S_z(\tau =\beta ∕2)⟩$ for $t>0$ than for $t<0$ indicates the greater degree of localization of the electrons for $t>0$.

Figure 5

(Color online) The temperature dependence of the inverse of the local spin susceptibility for different electron occupations, $n$. We present results for $U=10\mid t\mid$ and for both $t<0$ and $t>0$. The lines are guides to the eye. For the $t>0$ cases (only $n=1.3$ is shown for clarity) the coherence temperature $T^{*}⪡\mid t\mid$, whereas for the $t<0$ cases $T^{*}\sim \mid t\mid$.

Figure 6

(Color online) Curie-Weiss versus paramagnetic behavior of the uniform susceptibility, $\chi \left(T\right)$, for the Hubbard model on the triangular lattice with $U=10\mid t\mid$. The inset shows ${\chi }_0∕\chi \left(T\right)$ (where ${\chi }_0$ is the zero temperature noninteracting magnetic susceptibility). The central panel compares the strong $T$ dependence for $t>0$ with the nearly temperature independent behavior for $t<0$ (Pauli paramagnetism). For $n=1.4$ ferromagnetism appears below a critical temperature, $T_c$. The main panel displays the inverse of the uniform susceptibility, ${\chi }_0∕\chi \left(T\right)$, showing the Curie-Weiss $T$ dependence for $t>0$. The arrow indicates the coherence temperature $T^{*}$ found from fitting the local susceptibility data in Fig. 5 to Eq. (22).

Figure 7

(Color online) Electron occupation dependence of the coherence scale, $T^{*}\left(n\right)$, the Curie-Weiss scale $\theta \left(n\right)$ and the effective moment in the uniform magnetic susceptibility, $\mu$ for fixed $U=10\mid t\mid$ and $t>0$. These parameters are extracted by fitting the results of our DMFT calculations of magnetic susceptibilities to Eqs. (22, 23). For these values of $U$ and $t$ ferromagnetism is observed in the range $1.3<n<1.6$ (cf. Fig. 1). $\theta \left(n\right)>0$ for $n<1.3$ and no ferromagnetism is found at these dopings. $\mu$ decreases to zero as $n\to 2$ as the density of unpaired electrons decreases. This appears to be responsible for the absence of ferromagnetism for $n>1.6$.

Figure 8

(Color online) Comparison of the imaginary part of the self-energy for $t>0$ with the same quantities with $t<0$. Results are reported for $U=10\mid t\mid$ and various electron dopings. The linear dependence of the self-energy at low energies indicates Fermi liquid behavior. The slope near $i\omega =0$ is used to extract the quasiparticle weight $Z$, shown in Fig. 9.

Figure 9

(Color online) Dependence of the quasiparticle weight $Z$ on the Coulomb repulsion $U∕\mid t\mid$ at various electron dopings. The quasiparticles become more renormalized as the doping is decreased due to the proximity to the Mott transition. For a given value of $U$, quasiparticles on the triangular lattice with $t<0$ are less renormalized than for $t>0$. In contrast, for the half-filled case $(n=1)$, we find the same behavior of $Z$ for both $t>0$ and $t<0$ with a Mott metal-insulator transition taking place at a common value $U∕W\sim 1.66$.

Figure 10

(Color online) The variation of the effective mass, $m^{*}∕m_b=1∕Z$ with doping $n$ for both signs of $t$ for $U=10\mid t\mid$. Notice in particular the nonmonotonic variation of the effective mass with $n$ for $t>0$.

Figure 11

(Color online) The variation of the Sommerfeld-Wilson ratio, $R_W={\lim }_{T\to 0}(\chi \left(T\right)∕{\chi }_0)∕(\gamma \left(T\right)∕{\gamma }_0)$ with doping $n$ for both signs of $t$ for $U=10\mid t\mid$.

Figure 12

(Color online) Spectral densities for $U=10\mid t\mid$ and $n=1.5$ for the triangular lattice with $t<0$ and $t>0$. The noninteracting DOS is also shown (red line) for comparison and energies are relative to the Fermi energy. The majority (↑) and minority (↓) spectral densities for $n=1.5$ and $t>0$ are shown as the system is ferromagnetic.

Figure 13

(Color online) The doping dependence of the Kadowaki-Woods ratio $A∕{\gamma }^2$ for $t<0$ and $t>0$ for fixed $U=10\mid t\mid$. The lattice parameter, $a=2.84\mathrm{\AA }$ relevant to ${\mathrm{Na}}_x\mathrm{Co}{\mathrm{O}}_2$ has been used. Note the nonmonotonic doping dependence of the ratio and the fact that even at half-filling the ratios are different for the different signs of $t$.

Figure 14

(Color online) Ferromagnetism in the electron doped $t>0$ triangular lattice from DMFT. The magnitude of the spontaneous magnetic moment is shown as a function of the electron occupation $n$. As $U$ is increased above $U_c\left(n\right)$ the system magnetizes spontaneously stabilizing a ferromagnetic metal in a broad doping region. Ferromagnetism does not appear in the electron doped $t<0$ triangular lattice.

Figure 15

(Color online) Different sublattices of cobalt ions due to commensurate ordering of sodium ions. Pictorial representation of the Hamiltonians given by Eq. (38) are shown for $x=0.33$, 0.5, 0.71, and 0.75. The different letters denote the different site energies defined by Eq. (39), so that, for example a $P$ represents a site with site energy ${\epsilon }_P$. These different site energies result from the different Na order patterns, which have been observed experimentally (Ref. 83) and predicted from first principles calculations (Ref. 85). The Na ordering for the quoted filling factors are taken to be those shown in Fig. 2 of Ref. 85. Note that Roger et al. (Ref. 84) proposed different ordering patterns for $x=0.71$ and 0.75, which involved “vacancy clustering.”