Curie temperature versus hole concentration in field-effect structures of ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$

The Curie temperature $T_{\text{C}}$ is investigated as a function of the hole concentration $p$ in thin films of ferromagnetic semiconductor (Ga,Mn)As. The magnetic properties are probed by transport measurements and $p$ is varied by the application of an external electric field in a field-effect transistor configuration. It is found that $T_{\text{C}}\sim p^{\gamma }$, where the exponent $\gamma =0.19\pm 0.02$ over a wide range of Mn compositions and channel thicknesses. The magnitude of $\gamma$ is reproduced by a $p\text{−}d$ Zener model taking into account nonuniform hole distribution along the growth direction, determined by interface states and the applied gate electric fields.

Figure 1

(Color online) Hole concentration $p$ as a function of the gate electric field $E_{\text{G}}$ measured at $T=80\text{K}$ for device B2. Arrows indicate the sweep direction of $E_{\text{G}}$. The inset shows $E_{\text{G}}$ dependence of sheet conductance $R_{\text{sheet}}^{-1}$ for three devices, A1, B2, and C5, where A1 and C5 have the lowest and highest conductivities, respectively, among all the devices investigated in the present work. Device C5 with ${\text{HfO}}_2$ gate insulator shows a larger hysteresis than A1 and B2 with ${\text{Al}}_2{\text{O}}_3$ gate insulator.

Figure 2

(Color online) Magnetic field dependence of the Hall resistance $R_{\text{Hall}}$ of device B2 measured (a) at several temperatures $T=50–80\text{K}$ at the gate electric field $E_{\text{G}}=0$ and (b) at $T=60\text{K}$ at $E_{\text{G}}=-7$, 0, and $+7\text{MV}/\text{cm}$.

Figure 3

(Color online) (a) Arrott plots for device B2 at temperatures $T=75–90\text{K}$ and at zero electric field $E_{\text{G}}=0$. (b) Temperature dependence of spontaneous Hall resistance $R_{\text{Hall}}^{\text{S}}$, obtained from the Arrott plots. (c) $T_{\text{C}}$ as function of $E_{\text{G}}$, as determined from the Arrott plot of $R_{\text{Hall}}$ (closed circles), the temperature dependence of $R_{\text{sheet}}$ (open circles) and the temperature dependence of $d{R}_{\text{sheet}}/dT$ (open triangles). The arrow indicates the sweep direction of $E_{\text{G}}$.

Figure 4

(Color online) Logarithm plots of the Curie temperature $T_{\text{C}}$ versus hole concentration $p$; device B2 (circles), device A1 with the lowest $p$ (triangles), and device C5 with the highest $p$ (squares). Solid lines represent linear fit for the data. Dashed line shows the expected relation from the $p\text{−}d$ Zener model for the three-dimensional (bulk) (Ga,Mn)As. The inset shows the exponent $\gamma$ in $T_{\text{C}}\propto p^{\gamma }$ as a function of hole concentration at $E_{\text{G}}=0$, $p_0$, for the twelve devices.

Figure 5

(Color online) (a) The hole distribution profiles $p\left(z\right)$ at 80 K obtained by solving the Poisson equation for various channel thickness $t$. (b) Curie temperature $T_{\text{C}}$ as a function of $t$ calculated by using the determined hole distribution profiles and Eq. (7) (see main text) for Mn density $x_{\text{eff}}=0.03$. The dotted line represents $T_{\text{C}}$ obtained from the $p\text{−}d$ Zener model for the three-dimensional case (Refs. 6, 30).

Figure 6

(Color online) (a) The position of the valence band in respect to the Fermi energy ${\epsilon }_{\text{V}}-{\epsilon }_{\text{F}}$ and (b) hole distribution profiles $p\left(z\right)$ for various $N_{\text{i}}$ and $N_{\text{A}}$ at 80 K and for $N_{\text{D}}=4.0\times {10}^{19}{\text{cm}}^{-3}$. Thickness of (Ga,Mn)As is $t=4.5\text{nm}$. (c) $T_{\text{C}}$ calculated by using Eq. (7) with $x_{\text{eff}}=0.03$ are shown as a function of $N_{\text{i}}$ and $N_{\text{A}}$.

Figure 7

(Color online) Double logarithmic plot showing the relation between the Curie temperature $T_{\text{C}}$ and the sheet hole concentration normalized by nominal thickness, $p_{\text{S}}/t$. Circles and stars are the values of $T_{\text{C}}$ measured for device B2 (same as those in Fig. 4) and calculated by Eq. (7), respectively. The solid lines show linear fits and the dashed line is the prediction of the $p\text{−}d$ Zener model for the three-dimensional (bulk) (Ga,Mn)As. The inset presents the computed hole distribution profiles for various gate electric fields $E_{\text{G}}$.