Two- to three-dimensional crossover in a dense electron liquid in silicon

Doping of silicon via phosphine exposures alternating with molecular beam epitaxy overgrowth is a path to Si:P substrates for conventional microelectronics and quantum information technologies. The technique also provides a well-controlled material for systematic studies of two-dimensional lattices with a half-filled band. We show here that for a dense ($n_s=2.8\times {10}^{14} {\mathrm{cm}}^{-2}$) disordered two-dimensional array of P atoms, the full field magnitude and angle-dependent magnetotransport is remarkably well described by classic weak localization theory with no corrections due to interaction. The two- to three-dimensional crossover seen upon warming can also be interpreted using scaling concepts developed for anistropic three-dimensional materials, which work remarkably except when the applied fields are nearly parallel to the conducting planes.

Figure 1

The Hall bar and the principal directions, and a schematic of the device cross section showing the arsenic-doped handle (light grey), the deposited phosphorus $\delta$-layer (red), the overgrown silicon (cyan) and the aluminum contacts.

Figure 2

MT results. (a) MT curves at 22 mK and their fits. MR with magnetic field out-of-plane in blue squares, in plane in blue triangles, Hall resistance in red circles. (b) Temperature dependence of transport parameters. The blue circles are the phase coherence length. The unsaturated part which still has $L_{\phi }\gg L$ shows a power-law dependence. The green squares are the mean free path, which is temperature-independent in this range. The red circles are conductivity at zero field, showing logarithmic WL behavior. The error bars are 95% confidence intervals. (c) A temperature dependence of the experimental phase-breaking time ${\tau }_B$ vs $B_{\parallel }^2$. The black line is MOSFET data from Mensz and Wheeler [14].

Figure 3

$p$-mean fits: (a) Angular MC at 2 T, 22 mK. Blue line: measurement. Symbols: $p$-means of the perpendicular and parallel field $\mathrm{\Delta }\sigma$; green squares: $p=1$ (algebraic sum); red circles: $p=4.18\pm 0.10$. (b) Angular MC with $p$-mean fits at 2 T field magnitude, for 22 mK (outermost), 200 mK, 400 mK, 800 mK, 2 K, 4 K, 8 K, 16 K, 30 K (innermost). (c) Normalized deviation of the $p$-mean fit from the measured $\mathrm{\Delta }\sigma$ [defined as $\mathrm{\Delta }\left(\mathrm{\Delta }\sigma \left(\theta \right)\right)\equiv \frac{\mathrm{\Delta }\sigma \left(\theta \right)-\mathrm{\Delta }{\sigma }_{p-\mathrm{mean}}}{\mathrm{\Delta }\sigma \left(\theta \right)}$] vs field angle at 22 mK, showing that the most significant deviations are for small angles (i.e., nearly in-plane fields). The red symbols and the blue line are experimental data and calculation based on Ref. [14], respectively. (d) $\mathrm{\Delta }\left(\mathrm{\Delta }\sigma \left({\theta }_i\right)\right)$ vs temperature. (e) Angular MC with $p$-mean fits at 22 mK, for 2 T (outermost), 1.5 T, 1.0 T, 0.5 T, 0.2 T, 0.1 T (innermost).

Figure 4

$p$ vs magnetic flux quanta through a circular closed trajectory with circumference $L_{\phi }$. Values from temperature and field dependences are in red (empty circles) and blue (full squares), respectively. The symbols are the fitted values of $p$ to measured results and the lines are calculated at ${\theta }_i$ [see Eq. (7)], with ${\theta }_i$ derived from the WL MC in Refs. [13] and [20]. The $X$-axis errors originate in the $L_{\phi }$ errors in Fig. 2. Top and bottom insets are the temperature and field dependences vs temperature and field magnitudes, respectively.

Figure 5

Anisotropic field scaling: (a) shows $\mathrm{\Delta }\sigma$ vs field magnitude at different field angles, 22 mK. (b) shows same data as (a) but with the magnetic field scaled according to Eq. (6) with ${\xi }_0=8.92$. The full red circles and violet squares are the $\mathrm{\Delta }{\sigma }_{\perp }$ and $\mathrm{\Delta }{\sigma }_{\parallel }$ values from Fig. 2, respectively. Inset in (b): ${\xi }_0$ values at different temperatures, showing a stronger anisotropy at low temperatures. The red circles are best fits to the MC data, the blue squares and green triangles are from Eqs. (9) and (12), respectively.