Thermodynamics of Carrier-Mediated Magnetism in Semiconductors

We propose a model of carrier-mediated ferromagnetism in semiconductors that accounts for the temperature dependence of the carriers. The model permits analysis of the thermodynamic stability of competing magnetic states, opening the door to the construction of magnetic phase diagrams. As an example, we analyze the stability of a possible reentrant ferromagnetic semiconductor, in which increasing temperature leads to an increased carrier density such that the enhanced exchange coupling between magnetic impurities results in the onset of ferromagnetism as temperature is raised.

Figure 1

Schematic energy-band structure and temperature evolution of free energy. (a) In the absence of magnetization ($M=0$) both the conduction band and the donor level (at $-{\epsilon }_d$ relative to the band edge) are unsplit. (b) The onset of magnetic order ($M\ne 0$) leads to spin splitting $\Delta$ of the conduction band and ${\Delta }_d$ of the donor level . Thick arrows represent two spin projections. (c) Free energy $\mathcal{F}$ vs average electron spin $s$. Materials parameters were chosen based on Gd-doped EuO [21]: ${\Gamma }_{\mathrm{ex}}=40\mathrm{eV}·{\AA }^3$, $|{\epsilon }_d|=20\mathrm{meV}$, $N_d=3.5\times {10}^{19}{\mathrm{cm}}^{-3}$, $m^{*}=2$, lattice constant $a_0=5.15\AA$, $N_i=4/a_0^3$, $\gamma =1$, $a_B=8\AA$, and $J=7/2$. Behavior at 40 and 150 K reveals ferromagnetic states.

Figure 2

Temperature dependence of order parameters $s$ and $m$, showing reentrant behavior at $T=T_{C2}$ as temperature is raised. Curie temperatures $T_{C1,2,3}$, at which $m$ and $s$ both vanish, define phase boundaries between two ferromagnetic and two paramagnetic states. Inset: temperature dependence of the combined average spin from all electrons. Materials parameters are the same as in Fig. 1.

Figure 3

Temperature dependence of order parameters $s$ and $m$ when percolation is omitted (see text). Materials parameters are the same as in Fig. 1, except for $m^{*}=2.5$.