Regular Rather than Chaotic Origin of the Resonant Transport in Superlattices

We address the enhancement of electron transport in semiconductor superlattices that occurs in combined electric and magnetic fields when cyclotron rotation becomes resonant with Bloch oscillations. We show that the phenomenon is regular in origin, contrary to the widespread belief that it arises through chaotic diffusion. The theory verified by simulations provides an accurate description of earlier numerical results and suggests new ways of controlling resonant transport.

Figure 1

Scaled drift velocity vs the ratio between the Bloch and cyclotron frequencies: comparison of numerical calculations (5)–(7) (black thin solid line) and the asymptotic theory (10) (red thick dash-dotted line) for (a) $\theta =12°$, (b) $\theta =20°$.

Figure 2

Phase plane of the resonant Hamiltonian (12) for three characteristic values of $\delta \equiv (\omega -1)/\epsilon$: (a) $\delta =0$, (b) $0<\delta <{\delta }_{\mathrm{cr}}^{(2)}$, (c) $\delta >{\delta }_{\mathrm{cr}}^{(1)}$, where ${\delta }_{\mathrm{cr}}^{(n)}={\{(1/x)|[d{J}_1(x)/dx]|\}|}_{x=x_1^{(n)}}$. The $x_1^{(n)}$ are $n$th zeros of the Bessel function $J_1(x)$. Red circles mark the points ($+0$, $\pi /2$); the red dashed lines show outgoing trajectories [in (b),(c), the same applies to equivalent trajectories from ($+0$, $5\pi /2$)]. Dots mark saddles; separatrices are shown by solid lines. Arrows indicate directions of motion.

Figure 3

Scaled drift velocity vs the ratio between the Bloch and cyclotron frequencies: the general theory (15) and the numerical simulations for (a) the case Eq. (11) with $\theta =40°$, (b) $\tilde{\nu }=0.02$ as $\alpha \equiv \epsilon /(4\tilde{\nu })$ increases.

Figure 4

(a) Universal asymptotic dependence (S.6) of the amplitude of the resonant peak on $\alpha \equiv \epsilon /(4\tilde{\nu })$ (solid line) and its asymptotes for small and large $\alpha$ (dashed lines). The inset shows the enlarged scale for $\alpha <2.5$. (b) Comparison between (i) numerically calculated ${\tilde{v}}_d(\omega =1)$ for $\tilde{\nu }=0.02$ as a function of $\epsilon$ (blue thin solid lines) and (ii) the theory—the general theory (red dash-dotted line) i.e. Eq. (15) for $\omega =1$ while $v_d^{(res)}(\omega =1)$ reduces to the expression given in Eq. (S.6) of [28], and the small-$\alpha$ or large-$\alpha$ asymptotes (dashed lines) i.e. (15) for $\omega =1$ while $v_d^{(res)}(\omega =1)$ is approximated as $\alpha$ or $(x_1^{(1)}{)}^2/(8\alpha )$ respectively.