Magnetoresistance measurements are made on the two-dimensional electron gas in GaAs-${\mathrm{Al}}_{\mathrm{x}}$${\mathrm{Ga}}_{1\mathrm{-}\mathrm{x}}$As heterostructures as a function of the sample width (W) and the potential probe spacing (L) to study the electron-electron interactions. The temperature- (T) dependent parabolic magnetoresistance, observed above 0.1 T, clearly shows the effect of electron-electron interactions. When W is large compared with $L_T$≡π(ħD/kT${)}^{1/2}$, the magnetoresistance agrees quantitatively with the two-dimensional (2D) interaction theory proposed by Altshuler and Aronov, confirming the earlier observations by Paalanen, Tsui, and Hwang. When W≊$L_T$, a 2D-to-1D crossover is observed. For W<$L_T$, the magnetoresistance agrees quantitatively with the 1D interaction theory, if boundary scattering is negligible. When L is decreased and is less than 1.8$L_T$, the zero-dimensional (0D) behavior is observed, confirming the 0D interaction theory. In the narrow channels, with W less than the elastic mean-free path ($l_e$), the orbital effects are greatly reduced by the boundary scattering, showing the precursor of the extremely 1D behavior. When the magnetic field is less than 0.1 T, a new size-dependent magnetoresistance is observed. This T-insensitive magnetoresistance is attributed to boundary scattering. In addition, when L≊$L_T$, irregular conductance fluctuations of order $e^2$/h are observed, consistent with the recent theory of Lee and Stone on universal conductance fluctuations.

• Received 6 January 1986

DOI:https://doi.org/10.1103/PhysRevB.33.8216