Lissajous Rocking Ratchet: Realization in a Semiconductor Quantum Dot

Breaking time-reversal symmetry (TRS) in the absence of a net bias can give rise to directed steady-state nonequilibrium transport phenomena such as ratchet effects. Here we present, theoretically and experimentally, the concept of a Lissajous rocking ratchet based on breaking TRS. Our system is a semiconductor quantum dot with periodically modulated dot-lead tunnel barriers. Broken TRS gives rise to single electron tunneling current. Its direction is fully controlled by exploring frequency and phase relations between the two barrier modulations. The concept of Lissajous ratchets can be realized in a large variety of different systems, including nanoelectrical, nanoelectromechanical, or superconducting circuits. It promises applications based on a detailed on-chip comparison of radio-frequency signals.

Figure 1

(a) Current $I(V_L,V_R)$ at $V=({\mu }_R-{\mu }_L)/e=100\mu \mathrm{V}$. A double arrow indicates the charging energy $E_C^{(0)}$ at the working point (black cross). Lower inset: Scanning electron microscope image of the sample; the blue channel contains 2DES. Two $\mathrm{Ti}/\mathrm{Au}$ gates (yellow) are used to define a QD; gray gate is grounded. Upper inset: QD sketch; vertical lines are electrostatic barriers with tunnel couplings ${\mathrm{\Gamma }}_{L,R}$ controlled by $V_{L,R}$, horizontal lines QD chemical potentials ${\mu }_n$, blue areas indicate occupied states in the degenerate leads at ${\mu }_{L,R}$. (b) $V_{L,R}$ are modulated according to Eq. (1) with $f=200\mathrm{MHz}$ ($k=1$). Measured (left) versus calculated (right, $V=1\mu \mathrm{V}$) $I(A,\varphi )$. Inset: Measured $I(f)$ at $A=9\mathrm{mV}$ for $\varphi \simeq 0.3\pi$ (blue) and $\varphi \simeq 0.6\pi$ (red). (c) Vector field $\overrightarrow{a}$ (explained in text and Sec. II of Ref. [11]) versus gate voltages and typical Lissajous trajectories for $k=1,2,3$ according to Eq. (1). QD sketches indicate relative strengths of tunnel couplings.

Figure 2

Measured and computed current as a function of modulation amplitude $A$ and phase difference $\varphi$. (a) $V_L(t)$ at 50 MHz and $V_R(t)$ at 100 MHz ($k=2$) and (b) at 50 and 150 MHz ($k=3$). Solid lines indicate resonances ${\mu }_n\simeq {\mu }_{L,R}$ for $n=3,4$. Bottom: Corresponding Lissajous figures.

Figure 3

Lines depict theory, symbols are measured (offset voltages range from $-1\mu \mathrm{V}$ to $9\mu \mathrm{V}$ and are compensated by adjusting $V$); $k=1$, $f=100\mathrm{MHz}$. (a) $I(\varphi )$ for $V\simeq 0$, ${\kappa }_{\mu }A\simeq 2E_C^{(0)}$ (blue) and ${\kappa }_{\mu }A\simeq 0.5E_C^{(0)}$ (red). (b) $I(\varphi )$ for ${\kappa }_{\mu }A\simeq 2E_C^{(0)}$ including curves at finite bias. Arrows indicate the phases at which ${\mu }_n$ start to contribute to $I$. The relative shift between $n=0,-1$ and $n=1,-2$, etc. is related to our working point not being centered in the Coulomb gap, see Fig. 1.