4.8: Spatial variation of the electrochemical potential†

Next, we try to answer the question: Where is the voltage dropped?

Once again, let‟s consider a quantum wire at T = 0K. The wire has a single scatterer with transmission probability \(\Im\). Uncompensated electrons emitted by the left contact are partly transmitted and partly reflected by the scatterer. Thus, to the right of the scatterer, only a fraction, \(\Im\), of the +k states in the energy range \(\mu_{D} < E < \mu_{S}\) are filled. To the left of the scatterer, the fraction (\(1-\Im\)) of the -k states in the energy range \(\mu_{D} < E < \mu_{S}\) are filled; see Figure 4.8.1(a).

After scattering the +k states are no longer in equilibrium and the distribution of electrons in the +k states can no longer be described by a quasi Fermi level. These electrons are said to be hot, and may travel some distance before they equilibrate. Similarly electrons in the -k states are not in equilibrium to the left of the scatterer.

In Fig. 4.8.1(b) we plot the average quasi Fermi level of both +k and -k states. The change in the average quasi Fermi levels can be interpreted as a potential change in the vicinity of the scatterer of \((1-\Im)(\mu_{S}-\mu_{D})\).

But where is the heat dissipated?

It depends where the electrons relax into equilibrium. If the relaxation occurs within the contact, then once again all the heat is dissipated in the drain. Thus, although the average potential changes at the scatterer, heat is only dissipated where the electrons relax.

\(^{†}\)This section is adapted from S. Datta, "Electronic Transport in Mesoscopic Systems" Cambridge (1995).

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