Shubnikov–de Haas effect

An oscillation in the conductivity of a material that occurs at low temperatures in the presence of very intense magnetic fields, the Shubnikov–de Haas effect (SdH) is a macroscopic manifestation of the inherent quantum mechanical nature of matter. It is often used to determine the effective mass of charge carriers (electrons and electron holes), allowing investigators to distinguish among majority and minority carrier populations. The effect is named after Wander Johannes de Haas and Lev Shubnikov. I m = 2 e ⋅ i h ( μ m − ∑ l ≠ m T m l μ l ) , {displaystyle I_{m}=2{frac {ecdot i}{h}}left(mu _{m}-sum _{l eq m}T_{ml}mu _{l} ight),,}     (1) D = ( 2 S + 1 ) Φ Φ 0 = 2 Φ Φ 0 . {displaystyle D=left(2S+1 ight){frac {Phi }{Phi _{0}}}=2{frac {Phi }{Phi _{0}}}.,}     (2) Δ ( 1 B ) = 1 B i + 1 − 1 B i = 2 ⋅ e n ⋅ h . {displaystyle Delta left({frac {1}{B}} ight)={frac {1}{B_{i+1}}}-{frac {1}{B_{i}}}={frac {2cdot e}{ncdot h}}.,}     (3) An oscillation in the conductivity of a material that occurs at low temperatures in the presence of very intense magnetic fields, the Shubnikov–de Haas effect (SdH) is a macroscopic manifestation of the inherent quantum mechanical nature of matter. It is often used to determine the effective mass of charge carriers (electrons and electron holes), allowing investigators to distinguish among majority and minority carrier populations. The effect is named after Wander Johannes de Haas and Lev Shubnikov. At sufficiently low temperatures and high magnetic fields, the free electrons in the conduction band of a metal, semimetal, or narrow band gap semiconductor will behave like simple harmonic oscillators. When the magnetic field strength is changed, the oscillation period of the simple harmonic oscillators changes proportionally. The resulting energy spectrum is made up of Landau levels separated by the cyclotron energy. These Landau levels are further split by the Zeeman energy. In each Landau level the cyclotron and Zeeman energies and the number of electron states (eB/h) all increase linearly with increasing magnetic field. Thus, as the magnetic field increases, the spin-split Landau levels move to higher energy. As each energy level passes through the Fermi energy, it depopulates as the electrons become free to flow as current. This causes the material's transport and thermodynamic properties to oscillate periodically, producing a measurable oscillation in the material's conductivity. Since the transition across the Fermi 'edge' spans a small range of energies, the waveform is square rather than sinusoidal, with the shape becoming ever more square as the temperature is lowered. Consider a two-dimensional quantum gas of electrons confined in a sample with given width and with edges. In the presence of a magnetic flux density B, the energy eigenvalues of this system are described by Landau levels. As shown in Fig 1, these levels are equidistant along the vertical axis. Each energy level is substantially flat inside a sample (see Fig 1). At the edges of a sample, the work function bends levels upwards. Fig 1 shows the Fermi energy EF located in between two Landau levels. Electrons become mobile as their energy levels cross the Fermi energy EF. With the Fermi energy EF in between two Landau levels, scattering of electrons will occur only at the edges of a sample where the levels are bent. The corresponding electron states are commonly referred to as edge channels. The Landauer-Büttiker approach is used to describe transport of electrons in this particular sample. The Landauer-Büttiker approach allows calculation of net currents Im flowing between a number of contacts 1 ≤ m ≤ n. In its simplified form, the net current Im of contact m with chemical potential µm reads wherein e denotes the electron charge, h denotes Planck's constant, and i stands for the number of edge channels. The matrix Tml denotes the probability of transmission of a negatively charged particle (i.e. of an electron) from a contact l ≠ m to another contact m. The net current Im in relationship (1) is made up of the currents towards contact m and of the current transmitted from the contact m to all other contacts l ≠ m . That current equals the voltage μm ⁄ e of contact m multiplied with the Hall conductivity of 2 e2 ⁄ h per edge channel.