Heterodyne Hall effect in a two-dimensional electron gas

We study the hitherto unaddressed phenomenon of the quantum Hall effect with a magnetic and electric field oscillating in time with resonant frequencies. This phenomenon highlights an example of a heterodyne device with the magnetic field acting as a driving force, and it is analyzed in detail in its classical and quantum versions using Floquet theory. A bulk current flowing perpendicularly to the applied electric field is found, with a frequency shifted by integer multiples of the driving frequency. When the ratio of the cyclotron and driving frequency takes special values, the electron's classical trajectory forms a loop and the effective mass diverges, while in the quantum case we find an analog of the Landau quantization. A possible realization using metamaterial plasmonics is discussed.

Figure 1

(a) A heterodyne mixes the frequency $\omega$ of the input signal with its driving frequency $\mathrm{\Omega }$. The output is a superposition of signals with frequencies $\omega +l\mathrm{\Omega },l\in \mathbb{Z}$. (b) Model under study: an input electric field directed along $y$ and a magnetic field oscillating in time along $z$, acting on a two-dimensional electron gas (2DEG): this yields an electric current along $x$, with a frequency different from that of the input electric field. In this example, the input is a single mode with an envelope function, and the output is the zero-frequency envelope, which are related by the coefficient ${\sigma }_{xy}^{-1}\left(\mathrm{\Omega }\right)$.

Figure 2

(a) Static and heterodyne conductivities of a classical particle in an oscillating magnetic field. The dashed line represents the classical result for the resistivity in a static magnetic field. The parameter $\eta$ is 0.05, $n_e=m_e=\left|e\right|=1$. $r=r_{\alpha }^{\mathrm{cl}}$ are identified as the zeros of the Bessel function $J_0\left(r\right)$, explicitly $r^{\mathrm{cl}}=\{2.40,5.52,8.67,\cdots \}$. (b) Trajectories of the charged particle for different values of $r$ in zero electric field. (0) For general values, the particle makes open detours. (i)–(iii) For $r_1^{\mathrm{cl}}=2.40$ (i), $r_2^{\mathrm{cl}}=5.52$ (ii), and $r_3^{\mathrm{cl}}=8.67$ (iii), the trajectory forms a closed periodic orbit, whose winding number per half cycle $T/2=\pi /\mathrm{\Omega }$ is an integer $\alpha =1,2,3$, respectively.

Figure 3

(a) Plot of $\xi \left(\tau \right)$ defined in (18) for $r=1.5,2,3$. (b) The intricate Floquet spectrum (solid lines) reduces, for large enough $\mathrm{\Omega }$, to equispaced energy levels of a quantum HO (dashed lines) with frequency ${\omega }_{\mathrm{eff}}$.

Figure 4

The inverse effective mass $\frac{m_e}{m_e^{*}}$ as a function of $r$. The effective mass diverges at $r=r_{\alpha }^{\mathrm{q}}$, resulting in the Landau quantization.

Figure 5

(a) Schematic representation of the particle's dynamics. The wave function is a plane wave with momentum $k_y$ along the $y$ direction (represented as a compactified dimension, assuming $L_y$ is finite) and a localized wave packet oscillating along the $x$ direction. (b) When the Landau quantization condition $r=r_{\alpha }^{\mathrm{q}}$ is met, the Floquet quasienergy becomes flat (linearly tilted) in the absence (presence) of $E_x\left(t\right)=E_x^{1}cos\left(\mathrm{\Omega }t\right)$. The many-particle state is achieved by filling the states with electrons with a spin, denoted as circles with an arrow, respecting the Pauli principle. (c) The motion of the wave-packet center for the many-particle state in (b). The initial position $X\left(0\right)$ depends on $k_y$; wave packets with different $k_y$ evolve independently and oscillate around $x=0$ according to (18).