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Lectures 8-11: Quantum transport andSingle-electron tunneling

• Low-dimensional structures: 2-D, 1-D and 0-D, density-of states

• Magnetic field induced quantization: Landau levels

• Quantum conductance, quantum point contacts

• Single-electron tunneling

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Low-dimensional structures

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High electron mobility transistor (HEMT)

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Magnetic field induced quantization

Landau levels

Conductance oscillations:

Shubnikov - de Haas effect

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Quantum Hall effect

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QHE devices - edge channels

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Quantum effects in 2D electron gases

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Ballistic transport

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Quantized conductance

Ballistic transport (no scattering) in 1-dim. wires orquantum point contacts

Conductance quantum: 2e2/h (with spin degeneracy)

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Single-electron tunneling

Single-electron boxNecessary conditions for SET:

1. Charging energy EC = e2/2C >> kBT

2. Tunneling resistance RT >> h/e2

Coulomb blockade:voltage range for fixed n

n12

e

Cg<Vg < n+

12

e

Cg

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Grey fields: Coulomb blockade regime

Single-electron transistor

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Double junction: single-electron transistor

Figure 11.3. A schematic diagram of the single electron

transistor showing two small capacitance tunnel junctions

characterized by junction resistance R and capacitance C,

and also the capacitively coupled gate.

Figure 11.4. The Helmholtz free energy of the

system as a function of Qo/e for various charge

states n at V=0.

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SET: energy diagrams

Figure 11.5. (a) Energy diagram of a SET with

symmetric junction capacitances. Coulomb blockade

exists when the tunneling process is energetically

unfavorable.

Figure 11.5. (b) When bias voltage sources

provide enough energy to overcome the charging

energy barrier, single electron tunneling occurs.

Figure 11.5. (c) At Qo=e/2, the potential of the

island is lowered by EC so that Coulomb blockade

is absent at all bias voltages.

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Coulomb staircaseSingle-electron tunnelingthrough a quantum dot

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