First-Order Magnetic Phase Transition of Mobile Electrons in Monolayer ${\mathrm{MoS}}_2$

Evidence is presented for a first-order magnetic phase transition in a gated two-dimensional semiconductor, monolayer-${\mathrm{MoS}}_2$. The phase boundary separates a ferromagnetic phase at low electron density and a paramagnetic phase at high electron density. Abrupt changes in the optical response signal an abrupt change in the magnetism. The magnetic order is thereby controlled via the voltage applied to the gate electrode of the device. Accompanying the change in magnetism is a large change in the electron effective mass.

Figure 1

(a) Band structure and allowed optical transitions of monolayer ${\mathrm{MoS}}_2$. (b) A van der Waals heterostructure consisting of monolayer ${\mathrm{MoS}}_2$ embedded in h-BN. (c) Schematic of a trion in a two-dimensional Fermi sea. The black circles denote conduction-band electrons; the red circle a valence-band hole. The trion ($X^{-}$, Bohr radius $a_{\mathrm{tr}}$) is bound (unbound) at low (high) electron density.

Figure 2

Photoluminescence of gated monolayer-${\mathrm{MoS}}_2$ at 1.6 K and $B_z=9.00\mathrm{T}$. The excitation is either ${\sigma }^{+}$ or ${\sigma }^{-}$; the detection either ${\sigma }^{+}$ or ${\sigma }^{-}$: the top (bottom) row corresponds to ${\sigma }^{+}$ (${\sigma }^{-}$) excitation; the left (right) column to ${\sigma }^{+}$ (${\sigma }^{-}$) detection. $n_c$ denotes the critical density. At $n=n_c$ there is a jump in the PL energy: the PL process changes from a trion ($X^{-}$) to a Mahan exciton (Q). For $n>n_c$, the PL process is largely polarization conserving. Conversely, for $n<n_c$, the PL is highly polarization nonconserving: ${\sigma }^{-}$ excitation leads to strong ${\sigma }^{+}$ PL.

Figure 3

(a) The energetic separation between the PL and the absorption, $E_{\mathrm{PL}}-E_A$ (measured either with ${\sigma }^{+}$ or ${\sigma }^{-}$ polarization), versus electron density at 1.6 K and $B_z=9.00\mathrm{T}$. There are two trions, ${\mathrm{X}}_{\mathrm{LE}}^{-}$ and ${\mathrm{X}}_{\mathrm{HE}}^{-}$ (red and orange points, respectively) [6]. The $Q$ peak is plotted for both polarizations (${\sigma }^{+}$ and ${\sigma }^{-}$, open-purple and filled-purple points, respectively). (b) The PL polarization transfer [$a_2/(a_1+a_2+a_3)$ (purple) and $a_3/(a_1+a_2+a_3)$ (white), see Eq. (1)], versus electron density. These plots determine $n_c=3.0\times {10}^{12}{\mathrm{cm}}^{-2}$.

Figure 4

(a) The occupied bands in the ferromagnetic phase (left) and in the paramagnetic phase (right) along with the change in magnetization at $n=n_c$. In the ferromagnetic phase, occupation of two bands with the same spin is favored, a consequence of exchange. In the paramagnetic phase, all four bands are occupied. (b) The free energy $F$ as a function of magnetization fraction $M$ for $F=a{M}^2+b{M}^4+c|M{|}^3$. The terms $a{M}^2$ and $b{M}^4$ are the Ginzburg-Landau terms; the nonanalytic term $c|M{|}^3$ arises from corrections to Landau’s theory of the Fermi liquid. Plotted is $F$ for constant $b$ and $c$ with $c<0$ as a function of $a$. For $a<c^2/4b$ ($a>c^2/4b$), the global minimum of $F$ lies at $M=0$, a paramagnetic phase (large $M$, a ferromagnetic phase). The phase transition at $a=c^2/4b$ is first order.