Coulomb drag in high Landau levels

Recent experiments on Coulomb drag in the quantum Hall regime have yielded a number of surprises. The most striking observations are that the Coulomb drag can become negative in high Landau levels and that its temperature dependence is nonmonotonous. We develop a systematic diagrammatic theory of Coulomb drag in strong magnetic fields explaining these puzzling experiments. The theory is applicable both in the diffusive and the ballistic regimes; we focus on the experimentally relevant ballistic regime (interlayer distance $a$ smaller than the cyclotron radius $R_c$). It is shown that the drag at strong magnetic fields is an interplay of two contributions arising from different sources of particle-hole asymmetry, namely the curvature of the zero-field electron dispersion and the particle-hole asymmetry associated with Landau quantization. The former contribution is positive and governs the high-temperature increase in the drag resistivity. On the other hand, the latter one, which is dominant at low $T$, has an oscillatory sign (depending on the difference in filling factors of the two layers) and gives rise to a sharp peak in the temperature dependence at $T$ of the order of the Landau level width.

Figure 1

The diagrams contributing to the drag conductivity to leading order in the interlayer interaction $U(\mathbf{q},\omega )$ (wavy lines). The full lines represent the electron Green function. The external vertices labelled by the velocity operator $v_i$ are vector (current) vertices while the internal vertices are scalar (density) vertices.

Figure 2

Diagrams defining the triangle vertex $\mathbf{\Gamma }(\mathbf{q},{\omega }_1,{\omega }_2)$.

Figure 3

Diagrammatic representation of the equation for the vertex corrections ${\gamma }_{nk,n^{\prime }{k}^{\prime }}^{\mu \nu }(ϵ+\omega ,ϵ;\mathbf{q})$ (full triangle at vertex) of the scalar (density) vertices in the SCBA. Dashed lines represent impurity scattering. We also indicate that for well-separated LL’s, the internal Green functions in the right-most diagram should be evaluated in the valence LL $N$ which can differ from the LL labels $n,n^{\prime }$ of the external Green functions.

Figure 4

The diagrams contributing to the triangle vertex in SCBA to leading order in the limits of well-separated Landau levels $\Delta ∕{\omega }_c$ and large $N$.

Figure 5

Diagram contributing to corrections of order $\Delta ∕{\omega }_c$ to the triangle vertex. Here both Green function adjacent to the vector vertex should be evaluated in Landau levels different from $N$.

Figure 6

Low-temperature drag for identical layers. Left panel: temperature dependences of the $O(\Delta ∕{\omega }_c)$ term in ${\rho }_{xx}^D\left(T\right)$ for different values of the filling factor of the highest LL, ${\nu }_N=0.5,0.2,0.8$ (from top to bottom). Right panel: dependence of the $O(\Delta ∕{\omega }_c)$ term on the filling factor ${\nu }_N$ for different values of temperature, $T∕\Delta =0.1,0.5,1$ (from top to bottom).

Figure 7

Schematic temperature dependence of drag in the ballistic regime for matched and mismatched densities. In the latter case the mismatch is chosen such that the drag is negative at low $T$ (see text). Scaling of ${\rho }_{xx}^D$ with temperature in different regions is indicated: (i) Eq. (71), (ii) Eq. (78), (iii) Eq. (85), and (iv) Eq. (87).

Figure 8

Schematic temperature dependence of low-temperature drag in different regimes: (a) diffusive, $R_c∕a⪡1$; (b) weakly ballistic, $1⪡R_c∕a⪡{\omega }_c∕\Delta$; (c) ballistic, ${\omega }_c∕\Delta ⪡R_c∕a⪡N\Delta ∕{\omega }_c$; (d) ultraballistic, $N\Delta ∕{\omega }_c⪡R_c∕a$.

Figure 9

Schematic illustration of different sources of particle-hole asymmetry: curvature of zero-$B$ spectrum $E\left(k\right)$ vs LL quantization of the density of states (DOS) $\nu \left(E\right)$. In the particle-hole $\left(p\text{−}h\right)$ symmetric case, the electronic and hole contributions to the current induced in the passive layer ($j_e$ and $j_h$, respectively) compensate each other. When the $p\text{−}h$ asymmetry is generated by a finite curvature, the velocities of electrons and holes (shown by arrows in the right panel) are different, which destroys the compensation. This is the “conventional” mechanism of the drag. When the DOS depends on energy (in the present case because of the LL quantization), an “anomalous” drag arises due to the difference in numbers of occupied electronic and hole states [also via the E dependence of the scattering time induced by $\nu \left(E\right)$, see Eq. (16)].

Figure 10

Contours for the $\omega$-integration.

Figure 11

Contours for the $ϵ$ integration.

Figure 12

Diagrams for the (scalar) vertex corrections in real space.

Figure 13

Functions $\mathcal{Q}\left(x\right)$ [Eq. (55)] and $\mathcal{H}\left(x\right)$ [Eq. (57)], and the product ${\mathcal{P}}^2\left(x\right)\mathcal{W}\left(x\right)$—Eqs. (84, D20), determining the frequency dependence of the “inelastic kernal” $I\left(\omega \right)$—Eq. (D15).