The physics of mesoscopic electronic systems has been explored for more than 15 years. Mesoscopic phenomena in transport processes occur when the wavelength or the coherence length of the carriers becomes comparable to, or larger than, the sample dimensions. One striking result in this domain is the quantization of electrical conduction, observed in a quasi-one-dimensional constriction formed between reservoirs of two-dimensional electron gas. The conductance of this system is determined by the number of participating quantum states or ‘channels’ within the constriction; in the ideal case, each spin-degenerate channel contributes a quantized unit of 2e2/h to the electrical conductance. It has been speculated that similar behaviour should be observable for thermal transport in mesoscopic phonon systems. But experiments attempted in this regime have so far yielded inconclusive results. Here we report the observation of a quantized limiting value for the thermal conductance, Gth, in suspended insulating nanostructures at very low temperatures. The behaviour we observe is consistent with predictions for phonon transport in a ballistic, one-dimensional channel: at low temperatures, Gth approaches a maximum value of g0 = π2k 2BT/3h, the universal quantum of thermal conductance.

Measurement of the Quantum of Thermal Conductance

In this chapter the DR in Quantum Wells of HD III–V semiconductors in the presence of magnetic field have been formulated in Sect. 1.2.1. On the basis of these fundamental equations, the DR in Nano Wires of HD III–V semiconductors in the presence of magnetic field has been investigated in Sect. 1.2.2. The Sect. 1.2.3 explores the DR in Quantum Dot of HD III–V semiconductors in the presence of magnetic field. The DR in Quantum Wells of HD III–V semiconductors in the presence of cross fields has been investigated in Sect. 1.2.4. The DR in Nano-Wires of HD III–V semiconductors in the presence of cross fields has been studied in Sect. 1.2.5. The DR in Quantum Dot of HD III–V semiconductors in the presence of cross fields has been investigated in Sect. 1.2.6. The DR in Quantum Wells of HD IV–VI semiconductors in the presence of magnetic field has been studied in Sect. 1.2.7. The DR in Nano Wires of HD IV–VI semiconductors in the presence of magnetic field has been investigated in Sect. 1.2.8. The DR in Quantum Dot of HD IV–VI semiconductors in the presence of magnetic field has been studied in Sect. 1.2.9. The DR in cylindrical Quantum Dot of III–V semiconductors in the presence of crossed electric and magnetic fields has been investigated in Sect. 1.2.10. The DR in Quantum Wells of HD III–V Semiconductors in the presence of arbitrarily oriented magnetic field has been studied in Sect. 1.2.11. The Sect. 1.4 contains 16 open research problems, which form the integral part of this chapter.

Kamakhya Prasad Ghatak

Papers

The DRs in Low Dimensional HD Systems in the Presence of Magnetic Field