Quantum Ratchet in Graphene: One-way Electron Traffic at Atomic Scale
S.D. Ganichev1 and S.A. Tarasenko2
1Terahertz Center, University of Regensburg, Germany
2Ioffe Physical-Technical Institute, St. Petersburg, Russia
A mechanical or electronic system driven by alternating force can exhibit a directed motion facilitated by thermal or quantum fluctuations. Such a ratchet effect occurs in systems with broken spatial inversion symmetry. Canonical examples are the ratchet-and-pawl mechanisms in watches, electric current rectifying diodes and transistors in electronics, and Brownian molecular motors in biology [1,2]. The ratchet suggests one-way traffic. One can pull it back and forth, but it moves predominantly in a certain direction. Therefore, the effect has fascinating ramifications in engineering and natural sciences.
Now an international consortium consisting of research groups from Germany, Russia, Sweden, and the U.S. has demonstrated that electronic ratchets can be successfully scaled down to one-atom thick layers . Specifically, it has been shown that graphene layers support a ratchet motion of electrons when placed in a static magnetic field. The ac electric field of terahertz radiation  was applied to push the Dirac electrons back and forth, while the magnetic field acted as a valve letting the electrons move in one direction and suppressing the oppositely directed motion. The resulting magnetic quantum ratchet transforms the ac power into a dc current, extracting work from the out-of-equilibrium Dirac electrons driven by undirected periodic forces.
Graphene, a one-atom-thick layer of carbon with a honeycomb crystal lattice , is usually threated as a spatially symmetric structures, as far as its electric or optical properties are concerned. Driven by a periodic electric field, no directed electric current can be expected to flow. However, if the space inversion symmetry of the structure is broken due to the substrate or chemisorbed adatoms on the surface, an electronic ratchet motion can arise. In Ref., we and our colleagues report on the observation and experimental and theoretical study of quantum ratchet effects in single-layer graphene samples, proving and quantifying the underlying spatial asymmetry.
Figure 1: Alternating electric field drives a ratchet current in graphene.
The physics behind the magnetic quantum ratchet effect in graphene is illustrated in Fig. 1. The alternating electric field E(t) drives Dirac electrons back and forth in the graphene plane. Due to the Lorentz force, the applied static magnetic field B deforms the electron orbitals such that the right-moving electrons have their centre of gravity shifted upwards, while the left-moving electrons are shifted downwards. (In quantum mechanical consideration, the shift is caused by the magnetic-field-induced coupling between σ- and π-band states). For spatial symmetric systems the net dc current would vanish. However, in a graphene layer with spatial asymmetry, e.g., caused by top adsorbates, the electrons shifted upwards feel more disorder and exhibit a lower mobility than the electrons shifted downwards and moving in the opposite direction. This difference in the effective mobility for the right- and left-moving carriers results in a net dc current. The current scales linearly with the magnetic field, changes a sign by switching the magnetic field polarity, and proportional to the square of the amplitude of the ac electric field. The linear dependence on B comes from the Lorentz force. The electric field appears twice: on the one hand, it causes the oscillating motion of carriers in the plane, and on the other, the Lorentz force itself is proportional to the electron velocity.
The ratchet motion implies that the particle flow depends on the orientation of the ac force with respect to the direction of built-in spatial asymmetry. In the case of magnetic quantum ratchets, where the asymmetry stems from the magnetic field, the relevant parameter is the angle β between the ac electric field E(t) and the static magnetic field B. Shown in Fig. 2 is the measured dependence of the dc current on the angle β. The current reaches a maximum for the perpendicular electric and magnetic fields and remains finite for the co-linear fields. The whole angular dependence is well described by the equation
jx(β) = j1 Cos(2β) + j2
with two contributions j1 and j2, which is in agreement with the developed theory [see Ref.3]. It has also been shown that the ratchet transport can be induced by a force rotating in space. By exciting the graphene samples with a clockwise or counterclockwise rotating in-plane electric field E(t), the dc current is detected. Interestingly, the current measured along the static magnetic field turns out to be sensitive to the radiation helicity being of the opposite sign for the clockwise and counterclockwise rotating fields.
Figure 2: Dependence of the ratchet current on the orientation of ac electric field. Experimental data (dots) are obtained for an epitaxial graphene on SiC at temperature 115K, magnetic field 7T, and electric field amplitude 10 kV/cm. Solid line is a theoretical fit.
Graphene may be the ultimate electronic material, possibly replacing silicon in electronic devices in the future. It has attracted worldwide attention from physicists, chemists, and engineers. The discovery of the ratchet motion in this purest possible two-dimensional system known in nature indicates that the orbital effects may appear and be substantial in other two-dimensional crystals such as boron nitride, molybdenum dichalcogenides and related heterostructures. The measurable orbital effects in the presence of an in-plane magnetic field provide strong evidence for the existence of structure inversion asymmetry in graphene.
 R. P. Feynman, R. B. Leighton, and M. Sands, "The Feynman Lectures on Physics, Vol. 1" (Addison-Wesley, 1966).
 Peter Hänggi and Fabio Marchesoni, "Artificial Brownian motors: controlling transport on the nanoscale", Review of Modern Physics, 81, 387 (2009). Abstract.
 C. Drexler, S. Tarasenko, P. Olbrich, J. Karch, M. Hirmer, F. Müller, M. Gmitra, J. Fabian, R. Yakimova, S. Lara-Avila, S. Kubatkin, M. Wang, R. Vajtai, P. Ajayan, J. Kono, and S.D. Ganichev: "Magnetic quantum ratchet effect in graphene", Nature Nanotechnology 8, 104 (2013). Abstract.
 S.D. Ganichev and W. Prettl, "Intense Terahertz Excitation of Semiconductors" (Oxford Univ. Press, 2006).
 A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, and A.K. Geim, "The electronic properties of graphene". Review of Modern Physics, 81, 109 (2009). Abstract.