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Ballistic Electronic Transport in GaAs-AlGaAs Heterojunctions

  • M. Pepper
Part of the NATO ASI Series book series (NSSB, volume 231)

Extended Abstract

The use of the high mobility electron gas in modulation doped GaAs-AlGaAs heterojunctions allows the fabrication of a range of structures whose size is less than the mean free path for electron scattering. This particular combination results in ballistic transport where the electron traverses the sample without scattering.

A particularly convenient way of converting two dimensional transport to one dimensional is by the method of split gates.1 Here two Schottky gates are placed a distance apart which can be of order ~ −10−4 cms. Application of a negative gate voltage results in the electron gas being “electrostatically squeezed” with the establishment of a series of quantised levels, (in exactly the same manner as the establishment of two dimensional levels by surface quantisation). This further quantisation can be seen by the application of a magnetic field which successively forces the levels through the Fermi energy. The resultant depopulation produces structure in the resistance2 which can be distinguished from the 2D Shubikov-de Haas effect due to Landau levels.

The combination of one dimensional confinement, produced by quantisation in the other two dimensions, in the ballistic regime yields a range of quantum effects. The first is the quantisation of the ballistic resistance which we will now consider in more detail.

We consider the application of a voltage V across a one dimensional resistor, the electronic current diffusing down the resistor toward +V is given by
$${J_ + } = \int_0^{{E_F}} {\frac{{eN(E)}}{2}} \,v\,dE$$
(1)
and the current diffusing the opposite way is
$${J_ - } = \int_{{E_{F - eV}}}^{{E_F}} {\frac{{eN(E)}}{2}} \,v\,dE$$
(xxx)
where N(E) is the 1D density of states at energy E and v is the group velocity at that energy, the factor 2 arises as only half the electrons diffuse in a particular direction. We note that the applied voltage V essentially provides a cut off energy (eV) below which electrons cannot diffuse against the applied voltage. As the 1D density of states including spin degeneracy is \({\left( {\frac{m}{{2{\pi ^2}{\hbar ^2}E}}} \right)^{1/2}}\), which can be expressed as \(\frac{2}{{hv}}\), the conductance becomes \(\frac{{2e}}{h}.\) If there are i 1D subbands then the conductance becomes \(\frac{{2{e^{{2_i}}}}}{h}.\) Each time a subband is forced through the Fermi energy, EF, by changing the width, or EFF, directly so σ changes by \(\frac{{2{e^2}}}{h}.\) Experiments reporting the observation of the quantisation are described in references 3 and 4.

The factor of 2 in the conductance formula comes from the spin degeneracy and can be lifted by the application of a magnetic field. This is most clear when a parallel field is applied, which avoids the complication of the conversion of the ballistic quantisation to the Quantum Hall Effect. However, the transverse field can be used to investigate magnetic depopulation and the nature of the confining potential.

The above approach is simplistic in that it neglects the question of the transmission of electron waves through the sample. As pointed out by Landauer5, Imry6 and Buttiker7 the correct expression is \(\sigma = \frac{{2{e^2}}}{h}T\) where T is the transmission coefficient. The lifting of h g the spin degeneracy by a parallel magnetic field is discussed in reference 8.

Application of a high voltage across a sample such that eV≈EF will, according to the derivation of the quantised conductance, result in a saturation of the current at a value of \(\frac{{2{e^2}}}{h}{E_F}.\) Calculation suggest that T will decrease with increasing V soproducing a h decreasing current, i.e. negative differential resistance. This is discussed in detail in references 9 and 10.

Modification of the transmission coefficient by reflection at entrance and exit regions of split gate devices has proved to be of importance in investigating “electrostatic squeezing”. In the presence of a magnetic field, transport can arise from electrons confined to the edge regions so defining an effective ring geometry.11, 12 The period of Aharonov-Bohm oscillations gives the area of the channel. In this way it was found that narrowing the conducting region by depletion also produced a lengthening, due to the depletion region extended out beyond the channel which was initially defined.

Quantum box structures in which a lateral box is interposed between two ballistic channels show striking evidence of resonance effects. Here, peaks in resistance rising above the quantised values are formed by a resonance in the quantum box. Application of a magnetic field results in Aharonov-Bohm oscillations modulated by a clear transmission resonance.14,15 The well defined box potential allows both weak and strong coupling cases to be studied.16

Recently Smith et al17 have constructed and measured the properties of Fabry Pérot interferometer which consists of a one dimensional resistor situated within a resonance cavity formed by reflector gates.

The large value of scattering mean free path in GaAs-AlGaAs heterostructures allows ballistic effects to be studied in integrated quantum structures. Early work showed that the resistance of two ballistic resistors in series was roughly equal to that of the narrowest, i.e. the resistance was determined by the smallest number of 1 D subbands.18 More recent work suggests that transmission effects may alter this result by ≈30%.19 A surprising effect is displayed by two resistors in parallel.20 When they are within a phase coherence length a coupling effect is found whereby they depopulate subbands together and show jumps in conductance of \(\frac{{4{e^2}}}{h}.\) This result is not fully understood but it may arise from Coulomb effects or the readiness of one dimensional systems to hybridise wavefunctions and so reduce the total energy.

Further details of the experiments described here are provided in the references listed.

Keywords

Quantum Hall Effect Negative Differential Resistance Parallel Magnetic Field Spin Degeneracy Applied Voltage Versus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • M. Pepper
    • 1
  1. 1.Cavendish LaboratoryCambridgeUK

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