Magnetotransport in 2DEG

Magnetotransport in 2DEG Magnetotransport in 2DEG Contents • Classical and quantum mechanics of two-dimensional electron gas • Density of states in magnetic field • Capacitance spectroscopy • (Integer) quantum Hall effect • Shubnikov-de-Haas-oscillations (briefly) Lorentz force: Newtonian equation of motion: Perpendicular to the velocity! Magnetotransport in 2DEG Classical and quantum mechanics of 2DEG Classical motion: Cyclotron orbit Cyclotron frequency, Cyclotron radius, In classical mechanics, any size of the orbit is allowed. Magnetotransport in 2DEG Conductance becomes a tensor: Relaxation time Magnetotransport in 2DEG Conductance and resistance are tensors: For classical transport, What happens according to quantum mechanics? Equipotential lines Magnetotransport in 2DEG Bohr-Sommerfeld quantization rule: the number of wavelength along the trajectory must be integer. Only discrete values of the trajectory radius are allowed Energy spectrum: Landau levels Wave functions are smeared around classical orbits with l B is called the magnetic length Magnetotransport in 2DEG Classical picture Quantum picture Magnetotransport in 2DEG The levels are degenerate since the energy of 2DEG depends only on one variable, n . Number of states per unit area per level is Realistic picture Finite width of the levels is due to disorder Magnetotransport in 2DEG Landau quantization (reminder from QM) Magnetic field is described by the vector-potential, We will use the so-called Landau gauge, In magnetic field, Magnetotransport in 2DEG Ansatz: Cyclotron frequency Displacement Similar to harmonic oscillator [...]... Hence, the integrated density of states per Landau level is Magnetotransport in 2DEG Metal Insulator A series of metal-to-insulator transitions A way to measure – magneto-capacitance spectroscopy Magnetotransport in 2DEG Insulating spacer δ-doping The current at a phase difference π/2 to ac signal is measured by lock -in amplifier Charge injection changes the 2DEG Fermi level Magnetotransport in 2DEG “Chemical”... Vdc Magnetotransport in 2DEG The measured capacitance shows the filling of the 2DEG at Vg = 0.77 V, as well as the modulated density of states in perpendicular magnetic fields Magnetotransport in 2DEG The quantum Hall effect Ordinary Hall effect Magnetotransport in 2DEG Klaus von Klitzing, 1980 Si-MOSFET The following discussion will be oversimplified What is the origin of this fantastic phenomenon? Magnetotransport. .. in 2DEG Conductance and resistance are tensors: Therefore small corresponds to small How comes? Magnetotransport in 2DEG Equipotential lines E E Magnetotransport in 2DEG (Over)simplified explanation: Classical picture Solution in the absence of scattering cyclotron radius Drift of a guiding center + relative circular motion From that (after averaging over fast cyclotron motion): Magnetotransport in. .. in 2DEG drift velocity of the guiding center Role of edges and disorder Cyclotron motion in confined geometry Classical skipping orbits Quantum edge states Magnetotransport in 2DEG Calculated energy versus center coordinate for a 200nm-wide wire and a magnetic field intensity of 5 T The shaded regions correspond to skipping orbits associated with edge-state behavior Only possible scattering is in forward... GaAs, Magnetotransport in 2DEG An even filling factor, levels are fully occupied , means that j Landau An odd integer number of the filling factor means that one spin direction of Landau level is full, while the other is empty How one can control chemical potential of 2DEG in magnetic field? By changing either electron density (by gates), or magnetic field We illustrate that in the next slide assuming...Since kx is quantized, is also quantized, , the shift , so The values of ky are also quantized, By direct counting of states we arrive at the same expression for the density of states Magnetotransport in 2DEG Usually the so-called filling factor is introduced as For electrons, the spin degeneracy Magnetic field splits energy levels for different spins, the splitting being described by... existing theory allows to determine effective mass, as well as scattering time This is a very efficient way to find parameters of 2DEG Thus, magneto-transport studies are very popular Magnetotransport in 2DEG What has been skipped? Detailed explanation of the Integer Quantum Hall Effect Theory of the Shubnikov-de Haas effect Fractional Quantum Hall Effect (requires account of the electronelectron interaction)... through, except very close to the LL centers Magnetotransport in 2DEG Localized states in the tails cannot carry current Consequently, only extended states below the Fermi level contribute to the transport Thus is why Hall conductance is frozen and does not depend on the filling factor! Localized states in the tails serve only as reservoirs determining the Fermi level In the region close to E2 electrons can... measurement uncertainty was confirmed in November 1994 by comparison with a transfer quantum Hall standard of the International Bureau of Weights and Measures (BIPM) Magnetotransport in 2DEG Shubnikov-de-Haas oscillations In relatively weak magnetic fields quantum Hall effect is not pronounced However, density of states oscillates in magnetic field, and consequently, conductance also oscillates Mapping these... showing the suppression of backscattering for a skipping orbit in a conductor at high magnetic fields While the impurity may momentarily disrupt the forward propagation of the electron, it is ultimately restored as a consequence of the strong Lorentz force Magnetotransport in 2DEG Disorder makes the states in the tails localized! Sketch of the potential profile at different energies Lakes and mountains