Abstract
This chapter introduces some popular quantum transport formalisms such as the scattering matrix formalism, the Landau–Vlasov equation (which can be viewed as a quantum-mechanical equivalent of the collisionless BTE), the nonequilibrium Green’s function approach, the Wigner distribution function, the Tsu–Esaki formalism, and the Landauer–Büttiker approach for linear response transport. Applications of these formalisms are presented in Chap. 9.
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Notes
- 1.
Sometimes the Boltzmann transport model is called “semi-classical” instead of “classical”. This is only because the scattering rate \(S(\vec{k},\vec{k}^{\prime})\) is calculated using Fermi’s Golden Rule or an equivalent quantum-mechanical prescription. Other than that, the Boltzmann model is entirely classical.
- 2.
It is very important to understand that “mesoscopic” does not mean that the device is smaller than a particular dimension (like 10 or 100 nm), but rather it is small enough to exhibit quantum-mechanical effects.
- 3.
See, for example, Supriyo Datta, “Nanoscale device modeling: The Green’s function method”, Superlat. Microstruct., Vol. 28, 253 (2000).
- 4.
We deal with transmission probabilities rather than complex transmission amplitudes since electrons are phase-incoherent in the leads. Since different modes are mutually orthogonal in the leads, we should add the transmission probabilities and understand that there is no interference between the transmission amplitudes of different modes within the leads.
- 5.
The region is, however, long enough that the probability of an evanescent mode tunneling through it is virtually zero.
- 6.
Whenever we mention 4-probe Landauer resistance, we tacitly imply that the probes are weakly coupled to the device so that the 4-probe formula is valid.
- 7.
The quantity ε p, q is of course found from the geometry and transverse dimensions of the leads. For example, if the cross-section of the leads is a rectangle of dimensions W y and W z , then ε p, q = (ℏ 2 ∕ 2m l )[(pπ ∕ W y )2 + (qπ ∕ W z )2].
- 8.
The matrix I is the N d ×N d identity matrix, where N d is the number of modes in the device, and {0} is a null column vector of size N d .
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Bandyopadhyay, S. (2012). Quantum Transport Formalisms. In: Physics of Nanostructured Solid State Devices. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-1141-3_8
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