Magnetoresistance and spin polarization in the insulating regime of a Si two-dimensional electron system

We have studied the magnetoresistance in a high-mobility Si inversion layer down to low electron concentrations at which the longitudinal resistivity ${\rho }_{xx}$ has an activated temperature dependence. The angle of the magnetic field was controlled so as to study the orbital effect proportional to the perpendicular component $B_{\perp }$ for various total strengths $B_{\mathrm{tot}}$. A dip in ${\rho }_{xx}$, which corresponds to the Landau level filling factor of $\nu =4$, survives even for high resistivity of ${\rho }_{xx}\sim {10}^8\Omega$ at $T=150\mathrm{mK}$. The linear $B_{\mathrm{tot}}$ dependence of the value of $B_{\perp }$ at the dip for low $B_{\mathrm{tot}}$ indicates that a ferromagnetic instability does not occur even in the far insulating regime.

Figure 1

Longitudinal resistivity as a function of $B_{\perp }$ at $B_{\mathrm{tot}}=9\mathrm{T}$ and $T=150\mathrm{mK}$ for various $N_s$ indicated in units of ${10}^{15}{\mathrm{m}}^{-2}$. The inset shows the data at $T=180\mathrm{mK}$ for $N_s=0.94\times {10}^{15}{\mathrm{m}}^{-2}$.

Figure 2

Data in the insulating regime for $B_{\mathrm{tot}}=6\mathrm{T}$ and $N_s=1.02\times {10}^{15}{\mathrm{m}}^{-2}$. The electron spins are expected to be fully polarized (Ref. 8). (a) $B_{\perp }$ dependence of ${\rho }_{xx}$ for various temperatures. (b) Arrhenius plots of ${\rho }_{xx}$ for $B_{\perp }=0$, 0.94, and $1.19\mathrm{T}$ (at the dip). (c) Activation energy as a function of $B_{\perp }$.

Figure 3

$B_{\perp }$ dependence of ${\rho }_{xx}$ for various $B_{\mathrm{tot}}$. (a) Data in the low resistivity region with $N_s=1.51\times {10}^{15}{\mathrm{m}}^{-2}$. (b), (c) Data in the higher resistivity region with $N_s=1.02$ and $0.94\times {10}^{15}{\mathrm{m}}^{-2}$, respectively.

Figure 4

Position of ${\rho }_{xx}$ minima $1∕{\nu }_{\min }=e{B}_{\perp }∕h{N}_s$. Data for (a) $N_s=0.94\times {10}^{15}{\mathrm{m}}^{-2}$, (b) $1.02\times {10}^{15}{\mathrm{m}}^{-2}$, (c) $1.27\times {10}^{15}{\mathrm{m}}^{-2}$, and (d) $1.51\times {10}^{15}{\mathrm{m}}^{-2}$ are shown. The arrows indicate $B_c$. The dotted lines represent ${\nu }_{\uparrow }=4$ or 6 (see text).

Figure 5

$B_{\perp }$ dependence of ${\rho }_{xx}$ at $T=150\mathrm{mK}$ and $N_s=0.86\times {10}^{15}{\mathrm{m}}^{-2}$ for various $B_{\mathrm{tot}}$. ${\rho }_{xx}$ minima at $\nu \approx 1$ and 2 are observed. The inset shows the position of the ${\rho }_{xx}$ minima.

Figure 6

(a) In-plane magnetic field dependence of ${\rho }_{xx}$ for different $N_s$ at $T=300\mathrm{mK}$. The arrows indicate $B_c$ determined from ${\rho }_{xx}$ vs $B_{\Vert }$ data obtained for various temperatures. (b) Activation energy determined from $T$ dependence of ${\rho }_{xx}$ for $N_s=0.82$ and $0.94\times {10}^{15}{\mathrm{m}}^{-2}$.

Figure 7

$B_c$ obtained from the $B_{\mathrm{tot}}$ dependence of $1∕{\nu }_{\min }$ (closed diamonds) and that from the $B_{\Vert }$ dependence of ${\rho }_{xx}$ (open circles) are shown as a ratio to the noninteracting value $B_0$. The dashed curve is an estimation from the spin susceptibility obtained by Pudalov et al.[9] in the limit of small magnetic fields. The dotted curve represents the values of $B_{\Vert }∕B_0$ for a tentative boundary in the $B_{\Vert }\text{‐}{N}_s$ plane determined from ${\rho }_{xx}$ at $T=150\mathrm{mK}$. ${\rho }_c=60\mathrm{k}\Omega$ is the critical resistivity at $N_s=N_c$ and $B=0$.