Theory of Curie temperature enhancement in electron-doped EuO

We present a comparative, theoretical study of the doping dependence of the critical temperature $T_C$ of the ferromagnetic insulator-metal transitions in Gd-doped and O-deficient EuO, respectively. The strong $T_C$ enhancement in ${\text{Eu}}_{1-x}{\text{Gd}}_x\text{O}$ is due to Kondo-like spin fluctuations on the Gd sites, which are absent in ${\text{EuO}}_{1-x}$. Moreover, we find that the $T_C$ saturation in ${\text{Eu}}_{1-x}{\text{Gd}}_x\text{O}$ for large $x$ is due to a reduced activation of dopant electrons into the conduction band, in agreement with experiments, rather than antiferromagnetic long-range contributions of the RKKY interaction. The results shed light on possibilities for further increasing $T_C$.

Figure 1

Conduction-band (upper panel) and Gd impurity (lower panel) spectral density across the phase transition in ${\text{Eu}}_{1-x}{\text{Gd}}_x\text{O}$ for $x=0.01$. $T_C\approx 95$ K. Solid curves represent the majority $\left(\sigma =\uparrow \right)$ and dashed curves the minority $\left(\sigma =\downarrow \right)$ spin spectral density.

Figure 2

Conduction-band (upper panel) and O vacancy (lower panel) spectral density across the phase transition in ${\text{EuO}}_{1-x}$ for $x=0.01$. The bare O-defect parameters are $E_d=0.0D_0,U=0$ (double occupancy). $T_C\approx 78$ K. Solid curves represent the majority $\left(\sigma =\uparrow \right)$ and dashed curves the minority $\left(\sigma =\downarrow \right)$ spin spectral density.

Figure 3

Curie temperature vs the doping concentration $n_I$ for Gd-doped and O-deficient EuO. Bare parameters for O vacancies: $E_d=0.0D_0,U=0$. The blue curve represents the behavior for ${\text{Eu}}_{1-x}{\text{Gd}}_x\text{O}$ including the RKKY interaction, see text.

Figure 4

Curie temperature vs on-site Coulomb repulsion $U$ in ${\text{EuO}}_{0.99}$ for various impurity level energies $E_d$. It is seen that for more tightly bound defect electrons the $T_C$ enhancement in ${\text{EuO}}_{a-x}$ is even weaker than for the parameter values of Fig. 3.

Figure 5

Longitudinal RKKY coupling $K^{\left|\right|}\left(r\right)$ for $x=n_I=0.04$ and various temperatures in ${\text{Eu}}_{1-x}{\text{Gd}}_x\text{O}$. The inset expands the AF region around two to four lattice spacings a of the FCC lattice.

Figure 6

Total magnetization $M=\langle S\rangle +\langle \sigma \rangle$ vs temperature $T$ for various doping concentrations in ${\text{Eu}}_{1-x}{\text{Gd}}_x\text{O}$ with long-range RKKY coupling (solid curves) and without RKKY coupling (dashed curves).

Figure 7

Dopant activation $n_c/n_I$ for Gd doped and O deficient EuO at the lowest temperature considered, $T=5\mathrm{K}$. The carrier concentration $n_c$ is shown in the inset.

Figure 8

Majority spectral densities for conduction electrons (upper panel) and impurity electrons (lower panel) in ${\text{Eu}}_{1-x}{\text{Gd}}_x\text{O}$ at $T=5$ K for various impurity concentrations $n_I$. The figure shows the downward shift of the impurity levels with increasing $n_I$.

Figure 9

Curie temperature vs carrier concentration $n_c$ in the conduction band for Gd-doped and O-deficient EuO. (Inset) Semilogarithmic plot of the data.

Figure 10

Diagram for the spin-dependent conduction electron polarization function ${\mathrm{\Pi }}_{ij}^{\sigma {\sigma }^{\prime }}\left(\omega \right)$, generating the RKKY interaction. The solid lines represent conduction electron propagators.