Quantum electron transport in magnetically entangled subbands

Transport properties of highly mobile two-dimensional (2D) electrons in symmetric GaAs quantum wells with two populated subbands placed in tilted magnetic fields are studied at high temperatures. Quantum positive magnetoresistance (QPMR) and magneto-intersubband resistance oscillations (MISO) are observed in quantizing magnetic fields, $B_{\perp }$, applied perpendicular to the 2D layer. QPMR displays contributions from electrons with considerably different quantum lifetimes, ${\tau }_q^{(1,2)}$, confirming the presence of two subbands in the studied system. MISO evolution with $B_{\perp }$ agrees with the obtained quantum scattering times only if an additional reduction of the MISO magnitude is applied at small magnetic fields. This indicates the presence of a yet unknown mechanism leading to MISO damping. Application of an in-plane magnetic field produces a strong decrease of both QPMR and MISO magnitude. The reduction of QPMR is explained by spin splitting of Landau levels indicating a $g$ factor, $g\approx 0.4$, which is considerably less than the $g$ factor found in GaAs quantum well with a single subband populated. In contrast to QPMR, the decrease of MISO magnitude is largely related to the in-plane magnetic field induced entanglement between quantum levels in different subbands that, in addition, increases the MISO period.

Figure 1

Dependence of the dissipative resistance $R_{xx}$ of 2D electrons on a perpendicular magnetic field taken at different angles $\alpha$ = 0, 86, 87, 88.1, and 88.6 degrees between magnetic field $B$ and the normal to the 2D layer. Curves at angles $\alpha <88.6$ degrees are shifted up for clarity. The inset shows magnetoresistance in a parallel magnetic field.

Figure 2

Magnetic field dependence of the normalized resistance without the effect of the parallel magnetic field on the resistance presented in the inset in Fig. 1. The $B_{\parallel }$ contribution is subtracted numerically from the curves shown in Fig. 1. Obtained curves are offset for clarity. The inset presents the same curves without offset. At $B_{\perp }<0.025$ T the inset demonstrates classical magnetoresistance in two subband systems that is independent of the angle $\alpha$ while at $B_{\perp }>0.025$ T progressive deviation between curves at different angles is observed and related to the quantum positive magnetoresistance.

Figure 3

Figure (a) shows nonoscillating content of magnetoresitance related to QPMR at various angles. From the top to the bottom $\alpha =0$, 80, 82.5, 85, 86, 87, 88.1, and 88.6 degrees. Figure (b) presents oscillating content of the magnetoresistance at the same set of angles as in Figs. 1 and 2.

Figure 4

Panel (a) presents the electron spectrum at $\alpha =0$ degrees in the vicinity of Fermi energy $E_F$. Vertical dashed lines indicate that intersections between all subband levels occur at the same magnetic field. Panel (b) presents the electron spectrum at $\alpha =88$ degrees in which intersections of the quantum levels are not aligned anymore.

Figure 5

In panel (a) solid lines present normalized QPMR, $R_{QPMR}/R_D=(R_{QPMR}^{*}-R_{\min })/R_D$ at different angles. From the top to the bottom $\alpha =0$, 85, 86, 87, 87.9, and 88.1 degrees. Symbols show theoretical fit to the data using Eq. (11). Panel (b) shows the dependence of the fitting parameters $A_i$ on the ratio $B/B_{\perp }$, which is compared with the dependence following from Eq. (6) at $g_1=g_2=0.43$. Panel (c) shows dependence of the quantum lifetimes in both subbands on the ratio $B/B_{\perp }\sim {\mathrm{\Delta }}_Z/\hslash {\omega }_c$.

Figure 6

Symbols present dependence of MISO period in the reciprocal magnetic field, $1/B_{\perp }$, on the parallel magnetic field taken in the vicinity of different perpendicular magnetic fields as labeled. Lines represent theoretical dependences of the MISO period obtained via numerical computations of the electron spectrum in tilted magnetic fields, using Eq. (8), for different widths $d$ of the quantum well as labeled.

Figure 7

Solid lines represent normalized swing of MISO at different angles $\alpha$ as labeled. Dashed lines represent theoretical dependencies of the MISO swing on the reciprocal perpendicular magnetic field obtained at the following fitting parameters: $A_{MISO}^{*}=A_{MISO}{A}_b=0.38cos(0.091/B_{\perp })$, $1{\tau }_q^{\left(1\right)}=1/{\tau }_q^{\left(2\right)}=125$ GHz, used for all angles and different $g$ factors shown in the inset. The dotted line represents theoretical dependence at $\alpha =88.1$ degrees, ignoring the Zeeman splitting.

Figure 8

Dependence of normalized dissipative resistance on the perpendicular component of a magnetic field at two different angles $\alpha$ between the magnetic field and normal to the sample as labeled. Open symbols represent classical magnetoresistance expected from Eq. (12) at ${\nu }_s=22.26$ GHz and ${\nu }_r=41$ GHz.