Shubnikov-de Haas oscillations

Shubnikov-de Haas (SDH) oscillations are observed for transport at low temperatures for high magnetic fields [100]. The example inFigure 3.9 shows the oscillations of the longitudinal resistance with increasing magnetic field.

In the following this effect is only introduced in brief in order to illustrate the information that is gained from this magnetoresistance effect. A more detailed description of underlying mechanisms and the stated formulas is found in the literature [100, 101].

The energy eigenvalues of a two dimensional system confined in z-direction are:

E_j= h²

8π²m_x^∗k_x²+ h²

8π²m_y^∗k_y²+ε_j

The confinement to two directions thus leads to a quantization of states as shown inFigure 3.10. The index j numbers different subbands.

Carriers will move in a circular motion with the cyclotron frequencyω_c=

e B m^∗, if a magnetic field is applied perpendicular to the confined plane. Only discrete kinetic energies are allowed for a quantum mechanical description of this motion, which results in the formation of Landau levels (LL) as shown

Rxx (Ω)

0 200 400 600

0 2 4 8 9

magnetic field (T) InGaAs/InP (77% indium)

n_s = 6.2 10¹¹ cm^-2 µ = 279000 cm²V^-1s^-1

Figure 3.9: Example of Rxx(B) showing Shubnikov-de Haas oscillations [103].

Figure 3.10

Energy over density of states for a two dimen-sional system with zero magnetic field and large magnetic field applied.

Separated Landau levels form for appliedB-field.

energy

in Figure 3.10. The energy of the n-th landau level for a subband j may be written as:

if Zeemann splitting is not taken into account. The number of states per LL is given by:

N_LL= e B

h g_s

with the spin degeneracy gs.

The oscillating nature of the magnetoresistance is explained by the magnetic field dependence of the LLs. The energy difference between the LLs increases with magnetic field. During this process, different LLs will pass the Fermi energy. Each time the Fermi energy is at the center of a LL there is a change in the resistivity and ρ_xx oscillates for varying magnetic field. This allows to calculate the carrier density n_s.

If the magnetic field changes, so does the number of occupied LLs. The resistivity is maximal each time the Fermi level lies at the center of a LL and the number of occupied Landau levels is a half-integer. The relation of the magnetic fields B

1 andB

2 of two successive peaks is thus:

h n_s

The number of carriers can thus be calculated from the position of two or more peaks in the resistivity. This method is more accurate than the extraction from the classical Hall resistance.

The Landau levels alone may explain the oscillating nature of ρ_xx(B) but cannot explain the amplitude and especially notρ_xx= 0. One way to illustrate this behavior is the presence of edge channels.

Every measured structure will be finite. Sample boundaries can be seen as potential barriers. At the edge of a sample, this leads to a rise of the LL energy as shown inFigure 3.11 a). The Fermi energy will thus always cross

a landau level at the edges. Electrons at the edges thus always contribute to conduction. Under a magnetic field, these edge states will exhibit a specific electron motion.

The electrons within the sample are moving in a circular motion. At the very edge, this is not possible. Here, carriers are drawn along the edge as shown inFigure 3.11. Scattering processes do not change this motion, as the electron will be drawn in the same direction as before the scattering event. For sufficient high fields, this results in zero resistance.

Each time a LL crosses the Fermi level within the sample, states between the edges states become available. Electrons from the edge channels can now be scattered into states at the opposing edge. This backscattering results in an increased resistivity.

Potential fluctuations result in a broadening of the LLs. This can be seen as the influence of scattering potentials that limit the lifetime of an electron in a certain LL, i.e. quantum state. This lifetime is called quantum lifetime τ_q and has an impact on the amplitude of the SDH oscillation [104]:

∆ρ_xx B

The implicit temperature dependence is only in χ. The effective mass m^∗ can thus be calculated by comparing the oscillations at different tempera-tures [105]. For a known effective mass, the quantum lifetime τ_q is the only unknown variable left and can be obtained by fitting the experimental data [105].

The ratio of transport lifetime τ_tr and quantum lifetime τ_q provides informa-tion about the dominant elastic scattering process. The transport lifetime is derived from the mobility which is determined from electrons moving along an applied field in one direction. Scattering processes with small angles will

contact contact

Figure 3.11: Edge channel model: a) energetic position of Landau levels over sample width with an upward band bending at the edges. b) electron motion in the sample with cyclotron motion in the center and drift at the edges due to magnetic field. c) channels at the edges carrying the current.

Figure 3.12

Schematic of electron propagating through a medium with a) large angle scattering and b) small angle scattering.τq, which accounts for every scattering event equally, is similar in a) and b).τtr increases for large angle scattering in b) compared to a).

τ_q≈τ_tr τ_q≪τ_tr a)

b) scatterer

therefore have less impact on τ_tr than large angle scattering events as is depicted inFigure 3.12. The quantum lifetime accounts for every scattering event equally. The ratio of τ_tr/τ_q is thus a measure of the scattering angle of the dominating scattering process [106, 107].