---
abstract: 'We present a general framework to describe the simultaneous para-to-ferromagnetic and semiconductor-to-metal transition in electron-doped EuO. The theory correctly describes detailed experimental features of the conductivity and of the magnetization, in particular the doping dependence of the Curie temperature. The existence of correlation-induced local moments on the impurity sites is essential for this description.'
author:
- Michael Arnold and Johann Kroha
title: ' Simultaneous ferromagnetic metal-semiconductor transition in electron-doped EuO'
---

At room temperature stoichiometric europiumoxide (EuO) is a paramagnetic semiconductor which undergoes a ferromagnetic (FM) transition at the Curie temperature of $T_C=69~{\rm K}$. Upon electron doping, either by O defects or by Gd impurities, this phase transition turns into a simultaneous ferromagnetic and semiconductor-metal (SM) transition with nearly 100 % of the itinerant charge carriers polarized and a sharp resistivity drop of 8 to 13 orders of magnitude, depending on sample quality [@oliver1; @oliver2; @penney; @steeneken]. Concomitant with this transition is a huge colossal magnetoresistance (CMR) effect [@shapira], much larger than in the intensely studied manganates [@tokura]. These extreme properties make electron-doped EuO interesting for spintronics applications. Known since the 1970s, these features have therefore recently stimulated more systematic experimental studies with modern techniques and improved sample quality [@steeneken; @ott; @schmehl] as well as theoretical calculations [@schiller; @sinjukow].

In pure EuO the FM ordering is driven by the Heisenberg exchange coupling between the localized Eu 4$f$ moments with spin $S_f=7/2$ [@lee]. Upon electron doping, above $T_C$, the extra electrons are bound in defect levels situated in the semiconducting gap, and the transition to a FM metal occurs when the majority states of the spin-split conduction band shift downward to overlap with the defect levels. Although this scenario is widely accepted, several questions of fundamental as well as applicational relevance have remained poorly understood. (1) Why does the magnetic ordering of the Eu 4$f$ system occur simultaneously [@steeneken] with the SM transition of the conduction electron system? (2) What is the order of the transition? While the magnetic ordering of the 4$f$ system should clearly be of 2nd order, the metallic transition requires a [*finite*]{} shift of the conduction band and, hence, seems to favor a 1st order transition. (3) How can the critical temperature $T_C$ be enhanced by doping for spintronics applications? While in the Eu-rich compound EuO$_{1-x}$ a systematic $T_C$ increase due to the O defects (i.e. missing O atoms) is not observed experimentally [@oliver1; @oliver2], a minute Gd doping concentration significantly enhances $T_C$ [@matsumoto; @ott]. An O defect in EuO$_{1-x}$ essentially binds the two excess electrons from the extra Eu 6s orbital and, therefore, should not carry a magnetic moment. As shown theoretically in Ref. [@sinjukow], the presence of O defects with two-fold electron occupancy does not enhance $T_C$, in agreement with experiments [@oliver1; @oliver2]. In the present work we focus on the Gd-doped system Eu$_{1-y}$Gd$_y$ and calculate the temperature and doping dependent magnetization and resistivity from a microscopic model. We find that the key feature for obtaining a $T_C$ enhancement is that the impurities not only donate electrons but also carry a local magnetic moment in the paramagnetic phase.

[*The model.*]{} — A Gd atom substituted for Eu does not alter the $S_f=7/2$ local moment in the Eu Heisenberg lattice but donates one dopant electron, which in the insulating high-temperature phase is bound in the Gd 5d level located in the gap. Therefore, the Gd impurities are Anderson impurities with a local level $E_d$ below the chemical potential $\mu$ and a [*strong*]{} on-site Coulomb repulsion $U>\mu - E_d$ which restricts their electron occupation essentially to one. The hybridization $V$ with the conduction band is taken to be site-diagonal because of the localized Gd 5d orbitals. The Hamiltonian for the Eu$_{1-y}$Gd$_y$O system then reads, $$\begin{aligned}
\label{hamiltonian}
H&=&\sum_{{\bf k}\sigma}\varepsilon_{{\bf k}} 
c_{{\bf k}\sigma}^{\dagger}c_{{\bf k}\sigma}^{\phantom{\dagger}}+H_{cd}+H_{cf}\\
\label{Hcd}
H_{cd}&=&E_{d} \sum_{i=1 \dots N_I,\sigma}
d_{i\sigma}^{\dagger}d_{i\sigma}^{\phantom{\dagger}} 
+  V \sum_{i=1 \dots N_I,\sigma}
(c_{i\sigma}^{\dagger} d_{i\sigma}^{\phantom{\dagger}} 
+ H.c.)\nonumber\\
&+& U \sum_{i=1 \dots N_I} d_{i\uparrow}^{\dagger} d_{i\uparrow}^{\phantom{\dagger}} 
    d_{i\downarrow}^{\dagger} d_{i\downarrow}^{\phantom{\dagger}} \\ 
\label{Hcf}
H_{cf}&=&-  \sum_{i,j} J_{ij} \vec S_{i}\cdot\vec S_{j} 
        - J_{cf}\sum_{i}\vec \sigma_{i}\cdot\vec S_{i} \ ,\end{aligned}$$ where the first term in Eq. (\[hamiltonian\]) denotes conduction electrons with spin $\sigma$. The Eu 4$f$ moments $\vec S_i$ on the lattice sites $i=1,\dots, N$ are described in terms of a Heisenberg model $H_{cf}$ with FM nearest and next-nearest neighbor couplings $J_{ij}$ and an exchange coupling $J_{cf}$ to the conduction electron spin operators at site $i$, $\vec\sigma_{i}=(1/2)\sum_{\sigma\sigma'}
c_{i\sigma}^{\dagger}\vec\tau_{\sigma\sigma'}c_{i\sigma'}^{\phantom{\dagger}}$, with $c_{i\sigma}=\sum_{\bf k} \exp(i{\bf k x_i})\,c_{{\bf k}\sigma}$ and $\vec \tau_{\sigma\sigma'}$ the vector of Pauli matrices. The Gd impurities at the random positions $i=1, ..., N_I$ are described by $H_{cd}$. For the numerical evaluations we take $U\to\infty$ for simplicity.

For the present purpose of understanding the general form of the magnetization $m(T)$ and the systematic doping dependence of $T_C$ it is sufficient to treat the 4$f$ Heisenberg lattice, $H_{cf}$, on mean field level, although recent studies have shown that Coulomb correlations in the conduction band can soften the spin wave spectrum in similar systems [@golosov; @perakis]. The effect of the latter on $m(T)$ can be absorbed in the effective mean field coupling of the 4$f$ system, $J_{4f} \equiv \sum_{j}J_{ij}$. We therefore choose $J_{4f}$ such that for pure EuO it yields the experimental value of $T_C=69~{\rm K}$ [@oliver1; @oliver2; @shapira; @steeneken]. For simplicity, we don’t consider a direct coupling $J_{df}$ between the 4$f$ and the impurity spins, since this would essentially renormalize $J_{cf}$ only. The indirect RKKY coupling will also be neglected, since for the small conduction band fillings relevant here it is FM, like $J_{ij}$, but much smaller than $J_{ij}$.

In the evaluations we use a semi-elliptical bare conduction band density of states (DOS) with a half width $D_0=8\, {\rm eV}$, (consistent with experiment [@steeneken]), centered around $\Delta _0\approx 1.05\, D_0$ above the (bare) defect level $E_d$. The other parameters are taken as $J_{4f} \equiv \sum_{j}J_{ij} =  7\cdot 10^{-5} D_{0}$, $J_{cf}=0.05 D_{0}$, $E_{d}=-0.4 D_{0}$, and $\Gamma=\pi V^{2}=0.05 D_{0}^{2}$, where $J_{cf}\gg J_{4f}$ because $J_{4f}$ involves a non-local matrix element.

[*Selfconsistent theory.*]{} — The averaging over the random defect positions is done within the single-site $T$-matrix approximation, sufficient for dilute impurities. This yields for the retarded conduction electron Green‘s function $G_{c\sigma}({\bf k},\omega)$ in terms of its selfenergy $\Sigma _{c\sigma}(\omega)$, $$\begin{aligned}
&&G_{c\sigma}({\bf k},\omega)=\left[\omega+\mu-\varepsilon_{\bf k}-\Sigma_{c\sigma}(\omega)\right]^{-1} \label{gc}\\
&&\Sigma_{c\sigma}(\omega)=n_{I} |V|^{2}G_{d\sigma}(\omega) -J_{cf}\langle S \rangle \sigma \label{se}\end{aligned}$$ where $G_{d\sigma}(\omega)$ is the defect electron propagator and $\langle S \rangle$ the average 4$f$–moment per site. In mean field theory it is obtained, together with the conduction electron magnetization $m$, as $$\begin{aligned}
&&\langle S \rangle = \frac{\sum_{S} S e^{-\beta(2J_{4f}\langle S \rangle + J_{cf}m)S}}{\sum_{S}e^{-\beta(2J_{4f}\langle S \rangle + J_{cf}m)S}}\\
&&m=\frac{1}{2}\int d \omega  f(\omega) [A_{c\uparrow}(\omega) - 
A_{c\downarrow}(\omega)]\label{magn}\end{aligned}$$ where $f(\omega)$ is the Fermi distribution function and $A_{c\sigma}(\omega)=- \sum_{{\bf k}} 
{\rm{Im}} G_{c\sigma}(k,\omega)/\pi$ the conduction electron DOS of the interacting system.

\

In order to treat the strongly correlated spin and charge dynamics of the Anderson impurities without double occupancy beyond the static approximation, we use a slave particle representation and employ the non-crossing approximation (NCA) [@grewe]. For EuO the DOS at the Fermi level is so low or even vanishing that the Kondo temperature is well below $T_C$ and Kondo physics plays no role. In this high-energy regime the NCA has been shown to give quantitatively reliable results [@costi]. This remains true even for a finite magnetization, where the NCA would develop spurious potential scattering singularities near $T_K$ only [@kirchner]. One obtains the following set of equations for $G_{d\sigma}(\omega)$ in terms of the auxiliary fermion and boson propagators $G_{f\sigma}$, $G_{b}$, their spectral functions $A_{f\sigma}$, $A_{b}$ and their selfenergies $\Sigma_{f\sigma}, \Sigma_{b}$, $$\begin{aligned}
\Sigma_{f\sigma}(\omega)&=&\Gamma \int {d\varepsilon}\left[1-f(\varepsilon)\right] A_{c\sigma}(\varepsilon)G_{b}(\omega-\varepsilon )\label{sigmaf}\\
\Sigma_{b}(\omega)&=&\Gamma \sum_{\sigma}\int {d\varepsilon} f(\varepsilon) A_{c\sigma}(\varepsilon)G_{f\sigma}(\omega+\varepsilon )\label{sigmab}\\
\nonumber
G_{d\sigma}(\omega)&=&\int \frac{d\varepsilon} {e^{\beta \varepsilon}} \left[ G_{f\sigma}(\omega+\varepsilon )A_{b}(\varepsilon)-A_{f\sigma}(\varepsilon)G^{*}_{b}(\varepsilon-\omega)\right] \\
\label{Gd}\end{aligned}$$ Note that in Eqs. (\[sigmaf\], \[sigmab\]) $A_{c\sigma}(\varepsilon)$ is the interacting DOS, renormalized by the dilute concentration of Anderson impurities and the 4$f$–spins according to Eq. (\[gc\]). For details of the NCA and its evaluation see [@costi]. The equations (\[gc\]-\[Gd\]) form a closed set of selfconsistent integral equations. They are solved iteratively, fixing the total electron number per lattice site in the system, $$\begin{aligned}
n= \sum_{\sigma}\int \!d\omega  f(\omega)\, 
\left[A_{c\sigma}(\omega)+n_I\,A_{d\sigma}(\omega)\right]=n_I
\label{pnumber}\end{aligned}$$ by the chemical potential $\mu$ in each step.

\

[*Electrical conductivity.*]{} — The current operator $\hat{\bf j}$ can be derived from the continuity equation, $\partial\hat\rho_i/\partial t + \nabla\ \cdot \hat{\bf j} =0$, and the Heisenberg equation of motion for the total local charge operator $\hat\rho_i$ at site $i$. Because the impurity Hamiltonians $H_{cf}$, $H_{df}$ conserve $\hat\rho_i$, only $c$–electrons contribute to the current, and one obtains [@schweitzer], $
\hat{\bf j}=({e}/{\hbar}) \sum_{{\bf k}\sigma}{\partial \varepsilon_{\bf k}}/
  {\partial {\bf k}} \
c_{{\bf k}\sigma}^{\dagger} c_{{\bf k}\sigma}^{\phantom{\dagger}}
$. The linear response conductivity then reads for a local selfenergy [@schweitzer], $$\sigma=\frac{\pi e^{2}}{3 \hbar V} \sum_{{\bf k}\sigma} \int d\omega \left(
  -\frac{\partial f}{\partial \omega} \right)  A_{c\sigma}^{2}({\bf k},\omega)
\left( \frac{\partial \varepsilon_{\bf k}}{\partial {\bf k}} \right)^{2} \ .
\label{cond1}$$

[*Results and discussion.*]{} — The results of the selfconsistent theory, Eqs. (\[gc\]–\[pnumber\]), and for the conductivity, Eq. (\[cond1\]), are presented in Figs. \[fig1\]–\[fig3\]. They allow to draw a complete picture of the FM semiconductor-metal transition in Gd-doped EuO. The spectral densities per lattice site above and below the transition are shown in Fig. \[fig1\]. In the paramagnetic, insulating phase the hybridization between $d$– and $c$–electrons necessarily implies the appearance of a conduction electron sideband (Fig. \[fig1\], inset), situated below $\mu$ and at the same energies inside the semiconducting gap as the impurity $d$–band. The $d$-band (not shown) has a similar width and shape as the $c$-sideband. The combined weight of the $c$–sideband and the $d$-band adjusts itself selfconsistently such that it just accommodates the total electron number, $n=n_I$. Note that the weight of the $d$–band per impurity and spin is $\lesssim 1/2$, because the doubly occupied weight is shifted to $U\to \infty$ [@costi].

\

The $c$–4$f$ exchange coupling $J_{cf}$ induces an effective FM coupling between the electrons of the $c$–$d$ system. Hence, either the 4$f$– or the $c$–$d$–electron system can drive a FM transition, depending on which of the (coupled) subsystems has the higher $T_C$. We have chosen $J_{cf}$ (see above) large enough that the transition is driven by the $c$–$d$–electrons, because this will yield detailed agreement with the experiments [@steeneken; @ott; @matsumoto]. In this case, $T_C$ is naturally expected to increase with the impurity density $n_I$. The results for the $T$-dependent conduction electron magnetization $m(T)$, Eq. (\[magn\]), and for the doping dependence of $T_C$ are shown in Fig. \[fig2\], lower panel, and in Fig. \[fig3\], right panel, respectively. It is seen that not only $T_C$ increases with the impurity concentration, in agreement with recent measurements on Eu$_{1-y}$Gd$_{y}$O$_{1-x}$ [@matsumoto; @ott], but also that $m(T)$ has a dome-like tail near $T_C$, before it increases to large values deep inside the FM phase. From our theory this feature is traced back to the mean-field-like 2nd order FM transition of the electron system, while the large dome in the magnetization further below $T_C$ is induced by the FM ordering of the 4$f$ system, whose magnetization is controlled by $J_{4f}$ and sets in at lower $T$. This distinct feature is again in agreement with the experimental findings [@matsumoto; @ott] and lends significant support for the present model for Eu$_{1-y}$Gd$_{y}$O. We note that the Eu-rich EuO$_{1-x}$ samples of Ref. [@matsumoto] also show a magnetization tail and a $T_C$ enhancement, suggesting (small) magnetic moments on the O defects. However, the nature of the O defects requires further experimental and theoretical studies. The conduction electron polarization $P(T)=m(T)/n_c(T)$ does not show this double-dome structure and below $T_C$ increases steeply to $P=1$ (not shown in Fig. \[fig2\]). The FM phase is connected with a spin splitting of the $c$– as well as the $d$–densities of states, as shown in Fig. \[fig1\]. The narrow $d$-band induces a Fano dip structure in the $c$ majority band and a small sideband in the $c$ minority band. Note that for the present scenario the existence of preformed local moments on the impurities, induced by strong Coulomb repulsion $U$, is essential. Without these moments the transition of the electron system would be purely Stoner-like, and, because of the extremely low conduction electron DOS at the Fermi level, its $T_C$ would be far below the Curie temperature of the 4$f$ system, so that no doping dependence would be expected [@sinjukow].

\

We now discuss the conductivity and the simultaneity of the FM and the SM transitions. In the paramagnetic phase, the system is weakly semiconducting, because $\mu$ lies in the gap (Fig. \[fig1\], inset). When the FM transition occurs, the impurity d-band must acquire a spin splitting in such a way that at least part of the minority $d$–spectral weight lies above the chemical potential $\mu$, in order to provide a finite magnetization. Since near the transition the spin splitting is small, the majority $d$–band must, therefore, also be shifted to have overlap with $\mu$ (Fig. \[fig1\]), and so must the hybridization-induced $c$-electron sideband (which eventually merges with the main conduction band for $T$ sufficiently below $T_C$). This immediately implies a transition to a metallic state, simultaneous with the FM transition, as seen in Fig. \[fig2\]. Because of the small, but finite thermal occupation of the states around $\mu$, we find that this shifting of spectral weight occurs continuously, which implies the FM semiconductor-metal transition to be of 2nd order (see Fig. \[fig2\]). The doping $n_I$ dependence of the conductivity is shown in Fig, \[fig3\], left panel. It is seen that the metallic transition can be driven by increasing $n_I$, if $T>T_C(n_I=0)$.

As an alternative to Gd-doping the charge carrier concentration $n$ can be controlled independently of the impurity concentration $n_I$ by varying the chemical potential $\mu$, e.g. by applying a gate voltage to an EuO thin film. The conductivity $\sigma$ and magnetization $m$ as a function of $\mu$ are shown in Fig. \[fig4\] for two temperatures. To both sides of the ungated system ($n=n_I$) $\sigma$ increases exponentially upon changing $\mu$, characteristic for semiconducting behavior. By increasing $\mu$, the FM-metallic transition is finally reached. I.e. the magnetization can be switched, in principle, by a gate voltage. The non-monotonic behavior of $\sigma$ towards more negative $\mu$ reflects the energy dependence of the $c$ sideband. A more detailed study will be presented elsewhere.

To conclude, our theory indicates that in Gd-doped EuO the existence of preformed local moments on the impurity levels inside the semicondicting gap is essential for understanding the distinct shape of the magnetization $m(T)$ near the ferromagnetic semiconductor-metal transition. The FM ordering is driven by these impurity moments which are superexchange coupled via the 4$f$ moments of the underlying Eu lattice. This scenario immediately implies an increase of the Curie temperature with the impurity concentration, in agreement with experiments. The double-dome shape of $m(T)$ arises because of the successive ordering of the dilute impurity and of the dense Eu 4$f$ systems, as $T$ is lowered. The dynamical accumulation of conduction spectral weight at the chemical potential, induced by the hybridization $V$ and the constraint of an emerging magnetization at the FM transition, implies the FM and the SM transition to be simultaneous and of 2nd order. The magnetization can be switched by applying a gate voltage. This might be relevant for spintronics applications.

We wish to thank T. Haupricht, H. Ott, and H. Tjeng for useful discussions. J.K. is grateful to the Aspen Center for Physics where this work was completed. This work is supported by DFG through SFB 608.

[10]{}

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