March 23, 2007

In E&M, we learned the famous formula for dielectric
materials: **D**(**r**) = ε **E**(**r**). **D**
is the electrical displacement, which is related to the electric field
caused by the free moving charge (the externally added charge).
**E** is the electric field caused by the total charge, which
includes the free moving and the bound charge (the internally induced
charge). ε is the dielectric constant, also called
permittivity, which relates the **D** to **E**. ε is
infinite for metals in the limit where the applied field is spatially
uniform. In this case, the electrons inside the metal are free to
arrange themselves until their own electric field exactly counters the
externally applied field **D**. Thus, the total field **E**
becomes 0.

However, when a spatially and temporally varying
field is applied, ε is not infinite even inside a metal.
Instead, it varies with positions inside the medium (**r**), the
frequency (ω) and the wave vector (**k**) of the field inside
the medium, temperature, and humidity, etc. Here, we will use quantum
mechanics to deduce the dielectric function of a metal in two steps. We
will first use the free electron model which treats ions as uniformly
distributed charges and assumes electrons are the only source of induced
charge. The free electron model generates the Thomas-Fermi dielectric
constant and the Lindhard dielectric constant by different
approximations. We will add ionic induced charge later, which modifies
Thomas-Fermi dielectric constant by adding an ionic term.

Fig. 1: Screening of a positive charge inside an
electron gas. |

A metal can be considered as an ionic lattice embedded inside an electron sea. The periodic ion lattice causes a periodic potential, which is very hard to calculate. Therefore, we often adopt the free electron model which treats the ions as a uniform background of positive charge even for real metals. This is called a free electron gas.

Screening is an important phenomena in a free
electron gas. When an external positive charge density
ρ^{ext}(**r**) is applied, the electrons will be
attracted to surround the positive charge (Fig. 1). The re-arrangement
of the electrons generates an induced charge distribution
ρ^{ind}(**r**). Therefore, the total charge density
ρ(**r**) =
ρ^{ext}(**r**)+ρ^{ind}(**r**) is less
positive than ρ^{ext}(**r**). Thus, The total potential
φ(**r**) is weaker than the external potential
φ^{ext}(**r**) caused by the positive charge only. The
phenomenon is called screening, which is the primary reason why the
electric placement **D** is not equal to the total field **E**,
and thus ε is not equal to one. By assuming that the applied
charge is weak enough so that the total potential and external potential
are linearly related, we get the following function in momentum space.

ε is the Fourier transform of the dielectric
constant, **k** is the wave vector of the field,
ρ^{ind}(**k**) is the Fourier transform of induced charge
density, and φ(**k**) is the Fourier transform of the total
potential. The physical consequences of one part of this formula are
easy to see. Since one part of ε is proportional to
1/k^{2}, ε is infinite in a spatially uniform applied
field where **k** is equal to zero. The other part of the formula,
ρ^{ind}(**k**)/φ(**k**), needs to be solved via
the Schrodinger equation with additional approximations. Two commonly
used thoeries are the Thomas-Fermi theory of screening and the Lindhard
theory of screening.

The Thomas-Fermi theory uses a semi-classical
approximation in which φ(**r**), the total potential, varies very
slowly with **r**. This is equivalent to k being very small. One
can then show that
ρ^{ind}(**k**)/φ(**k**)=-k_{0}^{2}/4π,
and thus

k_{0} is the Thomas-Fermi wave vector. This
formulation takes on a more obvious physical meaning after Fourier
transform back to position space, where 1/k_{0} becomes the
characteristic screening length for the exponential decay of the total
field inside the metal. For the applied potential of a point charge of
Q where φ^{ext}(**r**)=Q/r, the total screened field
becomes φ(**r**)=Q/r*exp(-k_{0}/r), which is negligible
at distance greater than 1/k_{0}. When T << T_{F}, the
Fermi temperature, which is true under everyday conditions for metals,

While the Thomas-Fermi theory holds only when k is
very small but makes no assumptions about the relationship between
ρ^{ind}(**r**) and φ(**r**), the Lindhard theory
of screening, also known as Random Phase Approximation (or RPA), takes
the complementary approach of making no assumptions about the magnitude
of k but instead assuming that ρ^{ind}(**r**) and
φ(**r**) are linearly related. Lindhard theory gives the
following expression for the dielectric constant.

This differs from the simple expression given by the
Thomas-Fermi theory by the term inside the {}. With a little math, it
can be shown that as k approaches zero, the content inside the {} goes
to 1, so that the Lindhard dielectric constant converge to the
Thomas-Fermi dielectric constant. However, as k approaches 2 kF, the
behavior of ε(**k**) is no longer analytical. In fact it
takes on very complicated behavior that’s beyond the scope of this
paper.

So far we have been dealing with a temporally (t)
invariant field where the frequency ω of the field is zero. In
general, ω could be none zero and ε(**k**) is actually
ε(**k**, ω=0). In fact, Lindhard theory can give
ε(**k**, ω) as a function of both **k** and ω.
That formula is too complex for the scope of our paper. However, in the
spatially (**r**) homogeneous limit where k=0 but ω not equal
to 0, the dielectric constant has an elegant solution:

where
ω_{pl}^{2}=4πne^{2}/m is the plasma
frequency of electrons, n is the electron density, and m is the electron
mass. From this formula, we know that when ω =
ω_{pl}, ε= 0. In this case, even when no
externally applied electric field **D** is present, the electrons can
still collectively oscillate at the plasma frequency.

The free electron model treats ions as a uniform background of charge. However, the assumption fails to notice the fact that ions can also move to screen an external charge. By taking the ions into account,

Where
Ω_{pl}^{2}=4πn_{i}(Ze)^{2}/M is
the plasma frequency of ions, ni is the density of ions, Z is the atomic
number of the ions, and M is the mass of an ion. The second term on the
right side of the formula is a function of **k**. It comes from the
screening of electrons. The third term is a function of ω. It
comes from the screening of ions. This formula reveals an interesting
point. The importance of the wave vector **k** relative to the
frequency term ω depends on the product of k and the velocity of
the charge species. Since electrons move very fast (kv>>ω) , its
contribution to ε is **k** dependent. On the other hadn,
since ions move much more slowly than electrons (kv<<ω) , its
contribution to ε is actually frequency dependent.

© 2007 Yun-Chieh Peng. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

[1]P. A. Martin and F. Rothen, *Many-Body Problems
and Quantum Field Theory*, (Springer, 2004).

[2] Ashcroft & Mermin, *Solid State Physics*
(Brooks Cole, 1976).