磁场条件下的电子输运 ( 高场量子效应 ) 0,0,,,,0,0, 1 exp Use lowest order perturbation theory to include the effect of the confining potential, 1 2,,, 1 2, No net current in the middle ;The edge states located at the two edges of the sample carry currents in opposite directions, since changes signs, 1, 1 1 1
磁场条件下的电子输运 ( 高场量子效应 ) 固体材料的电子输运 If, the states below are all filled and do not carry any net current. Net current arises from the filled states between and 0, net current I
磁场条件下的电子输运 ( 高场量子效应 ) Quantized Hall resistivity? 1 2 2, 2 1 2 2 2 2, 2 1, : integer number of edge states at the Fermi energy=number of bulk Landau levels below the Fermi energy, decrease as the field is increased
磁场条件下的电子输运 ( 高场量子效应 ) 固体材料的电子输运 0, net current I States carrying current in one direction are spatially separated from those carrying current in the opposite direction To relax momentum an electron has to be scattered from the top of the sample to the bottom of the sample, impossible since the overlap between the wavefunctions is exponentially small and there are no allowed states in the interior of the sample ( )
磁场条件下的电子输运 ( 高场量子效应 ) With this complete suppression of backscattering, electrons originating in the left contact enter the edge states carrying current to the right, similar for the other side, 0, Note that this situation arises only when the bulk chemical potentials lie between two bulk Landau levels (one single insulator point)
磁场条件下的电子输运 ( 高场量子效应 ) 固体材料的电子输运 Landau quantization and a single insulator point? Disorder in the system broadens the -functions in the DOS(E):localized states Magnetic length is small compared to the length scale of the potential fluctuations, the landau levels energies follow the potential fluctuations States inside one Landau level now have different energies and the shape and width of the peak in the density of states reflects the energy distribution of the disorder potential
磁场条件下的电子输运 ( 高场量子霍尔效应 ) insulator point: integer LLs occupied Insulator point expands to plateaus
磁场条件下的电子输运 ( 高场量子效应 ) Hall voltage plateaus: single insulator point to expanded region with localized states, which provides density of states between landau levels
磁场条件下的电子输运 ( 高场量子效应 ) With this complete suppression of backscattering, electrons originating in the left contact enter the edge states carrying current to the right, similar for the other side If half the LL is occupied and the Fermi level is on a LL, there is a continuous distribution of allowed states from one edge to the other, the backscattering gives rise to a maximum
磁场条件下的电子输运 ( 高场量子霍尔效应 ) The Hall resistance is quantized in units of /2 with an impressive accuracy in parts per million, independent of the sample geometry The accuracy arises from the near complete suppression of momentum relaxation processes, mean free paths of several millimeters have been observed Unusually long mean free paths are not from unusual purity of the samples, but because backscattering are suppressed With the impressive accuracy of the quantum Hall effect, the National Institute of Standards and Technology utilize it as a resistance standard.
磁场条件下的电子输运 ( 分数量子霍尔效应 ) 2 1 25.8128Ω 2 At very high fields, the Zeeman effect make the quantized plateaus with 1 25.8128Ω When field reaches a value with /, all the electrons are in a single LL with one spin. For a carrier density of 210 /, 8
磁场条件下的电子输运 ( 分数量子霍尔效应 ) What happens if we increase the field further? Experimentally, in very pure samples one continues to observe plateaus 1 25.8128Ω is a rational fraction like 1/3,2/5,4/7, etc. : 分数量子霍尔效应 arises from a novel many-body ground state (Laughlin, PRL 50,1395(1983))
磁场条件下的电子输运 ( 分数量子霍尔效应 ) 1 25.8128Ω is a rational fraction like 1/3,2/5,4/7 Ga(Al)As-HEMT, 100mk Willett(1987)
磁场条件下的电子输运 (why QHE happens only for 2D system) Quantum Hall effect, for three dimensions The Fermi sphere condense into cylinders of radii 2 1 2 LL evolve into Landau bands with a one-dimensional density of states, then no matter how large the magnetic field, there are always states at the Fermi energy
1. Quasi 2D electron gases two subbands in z-direction are occupied the Hall slope measures the total electron density each 2D subband causes a SdH oscillation 1 1 1 1 1 1 light indicates the sample is illuminated by light which ionizes residual neutral donors and increases the electron density in the 2DEG The magneto-oscillations are modulated in both cases: the two SdH frequencies correspond to the partial occupation of the two subbands
2. Displacement of the quantum Hall plateaux 2 1 Extrapolate the classical Hall slope into the quantum Hall regime, it should intersect the plateau at their center: implicitly assumed that the peaks of the DOS are symmetric Their shape depends on the character of scatters, not necessary symmetric all the time Their asymmetry affects the position of the quantum hall plateau
2. Displacement of the quantum Hall plateaux repulsive scatters attractive scatters
2. Displacement of the quantum Hall plateaux repulsive scatters
2. Displacement of the quantum Hall plateau repulsive scatters
3. Landau levels in graphene 4 3 3 3 2 3 2 0 0 0 0 3 2 0 0
3. Landau levels in graphene 4 3 3 0 0 0,0,,,,0,0
3. Landau levels in graphene 0 0 0 0 0,0,,,,0,0 / /,
3. Landau levels in graphene /, 2 2 1 2, Harmonic oscillator solution 1 2 1 2 2 Landau index n can be positive or negative
3. Landau levels in graphene 2 Landau index n can be positive or negative Landau levels are not equidistant as in conventional 2DEG and the largest energy separation is between the zero and the first Landau level
3. Landau levels in graphene 2 Nature 438,197 (2005)
3. Landau levels in graphene 2 Science 324,924 (2009) STM measure the DOS of graphene at strong magnetic field
4. Parallel magnetic field,0,0,, 0,,0 1 2 2 2 2 1 2,,,,,, 2 2 2 1 2,,,,
4. Parallel magnetic field,0,0,, 0,,0 1 2,, 2 2 2 1 2,,,, 2 1 2 2 2,, 2 1 2 2 2
4. Parallel magnetic field,0,0,, 0,,0 1 2,, 2 1 2 2 2,, Parabolic confinement generated by adds to the electrostatic confinement, such that the subbands shift to higher energies The effective mass in the y-direction (perpendicular to, but in the plane of the electron gas) increases, move more difficult in y-direction so the DOS increase for each subband as increases Φ
4. Parallel magnetic field Φ depletion of the 3 rd and 2 nd subbands when increasing, decreases: stronger scattering for upper band and interband scattering SdH oscillation with for different subbands
4. Parallel magnetic field Φ