The Thomas-Fermi theory of screening by metallic electrons

2 The Thomas-Fermi theory of screening by metallic electrons 

It should be appreciated that in contrast to the simple free electron models used in much of our discussion of metals and semiconductors, a treatment of screening necessarily involves taking into account, on some level, the interaction between charge carriers. In the Thomas-Fermi theory this is done by combining a semiclassical approximation for the response of the electron density to an external potential with a mean field approximation on the Hartree level—assuming that each electron is moving in the mean electrostatic potential of the other electrons. 

Consider a semi-infinite metal represented by the gray area in Fig. 4.7. The homogeneous bulk metal is taken to be locally neutral, the electronic charge is compensated by the positive background and the potential is constant. Near impurities or at the surface this is not necessarily so. Suppose that the potential is given to be bs on the metal surface and t B in its interior and consider the potential 

The system is infinite in the y and z directions, so our problem is onedimensional. In the absence of potential bias we have free electrons that occupy eigenstates of the kinetic energy operator up to the Fermi energy. The density of states per unit volume is (cf. Eq. (4.59)) 

In the presence of an external potential we use a semiclassical argument as in (4.115), by which the electronic states remain the free wave eigenstates of the kinetic energy operator associated with eigenvalues E, however the corresponding electronic energies become position-dependent according to 

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