Field-induced modulation of the conductance, thermoelectric power, and magnetization in ballistic coupled double quantum wires under a tilted magnetic field

The effect of a tilted magnetic field $B$ on the modulation of tunneling, the ballistic conductance, the ballistic electron-diffusion thermoelectric power, and the orbital magnetization is studied for tunnel-coupled ballistic double quantum wires. The magnetic field has a component $B_y$ along the wires and a component $B_x$ perpendicular to the plane that contains both wires. We find that $B_y$ alters the $B_x$ dependence of the electronic and transport properties drastically in the presence of interwire tunneling. The latter has been studied extensively in the literatures in the absence of $B_y$ and is known to show many interesting transport properties. The presence of $B_y$ causes the effective tunneling integral to oscillate continuously with sign changes and decay eventually for large $B_y$. The $B_y$-induced interwire tunnel coupling between different sublevels and the quenching of it under a large $B_y$ were both observed experimentally by Thomas et al. [Phys. Rev. B 59, 12252 (1999)].

Figure 1

A schematic illustration of the tunnel-coupled double quantum wires stacked in the $z$ (growth) direction. The two narrow GaAs conducting channels confined in the $x$ direction are formed by applying a negative voltage to two metallic split gates with respect to the grounded back gate (not shown). The formed quantum wires extend in the $y$ direction. There exists a thin AlGaAs barrier layer between the two wells in the $z$ direction, which allows the electron tunneling between two wires. The linear electron density in the wires can be varied by applying a negative voltage to a depletion gate (not shown) on the surface of double quantum wells. A magnetic field $\mathbf{B}=(B_x,B_y,0)$ is applied to the system within the $xy$ plane perpendicular to the growth direction. A current flows along the wires between the source and drain contacts.

Figure 2

(Color online) (a) $E_j\left(k\right)$ with $B_x=8\mathrm{T}$ as functions of $k$ for $B_y=0$, (b) $B_y=4\mathrm{T}$ and (c) $B_y=8\mathrm{T}$. Here, we only display lower-energy transverse sublevels. In (a), the tunnel-split upper (lower) branches in the absence of intersublevel tunneling between $n\ne n^{\prime }$ are denoted by red (blue) curves and the anticrossing gaps at $k=0$ between the upper and lower gap edges are indicated by black arrows for the lowest tunnel-split doublet. In (b) and (c), the higher (lower) coupled sublevels in the presence of intersublevel tunneling between $n\ne n^{\prime }$ are represented by red (blue) curves. In our calculations here, we take a large transverse-sublevel separation $\hslash {\omega }_x=1\mathrm{meV}$ for a strong confinement in the $x$ direction.

Figure 3

(Color online) $E_j\left(0\right)$ (black curves) as a function of $B_y$ for $B_x=0$. The red curve represents $E_F$ of the lower-density sample 1. The sublevel population and depopulation are indicated by three blue circles. The $E_3\left(0\right)$ is displayed by a thick black curve in the figure.

Figure 4

$E_j\left(0\right)$ as a function of $B_y$ for $B_x=0$. In (a), only the interwire tunnel coupling between $n=n^{\prime }$ sublevels $(m=0)$ is included in the calculation. In (b), the interwire tunnel coupling between the same sublevels and the nearest-neighboring sublevels $(m=0,1)$ is included in the calculation. The solid and dashed curves in (a) correspond to the sublevel edges $E_n^{(-)}\left(0\right)$ and $E_n^{(+)}\left(0\right)$ in Eq. (15), respectively, with different quantum numbers $n=0,1,2,3,\dots$.

Figure 5

(Color online) $G$ (left axis) of the lower-density sample 1 as a function of $B_y$ for $B_x=0$ at $T=0\mathrm{K}$ (solid curve), $0.3\mathrm{K}$ (dashed curve), and $3\mathrm{K}$ (dash-dot-dotted curve). For the sake of comparison, $E_F$ (red curve and right axis) is also shown.

Figure 6

(Color online) Contour plot of $G(B_x,B_y)$ in (a) and $G$ in (b) as a function of $B_x$ with a set of fixed values of $B_y$ for the higher-density sample 2. We set $T=0\mathrm{K}$ for our calculations here. For the sake of clarity, the successive curves from the bottom black curve in (b) are vertically shifted by an amount of $2(2e^2∕h)$. The curves from the bottom to the top in (b) correspond to $B_y=0,1,2,\dots ,7\mathrm{T}$, respectively.

Figure 7

(Color online) Blown-up view for $E_j\left(0\right)-E_1^z$ (black curves) close to $E_F$ in (a) as a function of $B_y$ and $S_d$ (right axis and solid curve) in (b) as a function of $B_y$ for the higher-density sample 2. For the sake of comparison, $E_F$ (red curve) in (a) as well as $G$ (dashed curve) in (b) are also shown. Here, $S_d<0$ (in unit of $-k_B∕e$) for electrons in the wires. The successive population (depopulation) of two sharp minima in (a) are indicated by blue circles. Two corresponding sharp peaks in $G$ are indicated by blue vertical arrows.

Figure 8

(Color online) Contour plot of $S_d(B_x,B_y)$ in (a) and $S_d$ in (b) as a function of $B_x$ with a set of fixed values of $B_y$ for the higher-density sample 2. We set $k_{B}T∕\hslash {\omega }_x=0.02$ for our calculations here. For the sake of clarity, the successive curves from the bottom black curve in (b) are vertically shifted by an amount of $0.2(-k_B∕e)$. The curves from the bottom to the top in (b) are associated with $B_y=0,1,2,\dots$, $7\mathrm{T}$, respectively.

Figure 9

$M_y$ (right axis and solid curve) as a function of $B_y$ for $B_x=0$ in (a) and $M_x$ as a function of $B_x$ in (b) with $B_y=0$ (solid curve) and $B_y=2\mathrm{T}$ (dashed curve) for the lower-density sample 1. For comparison, $E_F$ (left axis and dashed curve) and the average energy per electron $u_{\mathrm{av}}$ (left axis and dash-dotted curve) are also shown in (a).