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'''Lindhard theory'''<ref>J. Lindhard, K. Dan. Vidensk. Selsk. Mat. Fys. Medd. '''28''', 8 (1954)</ref><ref name=Ashcroft>N. W. Ashcroft and N. D. Mermin, ''Solid State Physics'' (Thomson Learning, Toronto, 1976)</ref> is a method of calculating the effects of [[electric field screening]] by electrons in a solid. It is based on quantum mechanics and the [[random phase approximation]].
[[Thomas-Fermi screening]] can be derived as a special case of the more general Lindhard formula. In particular, Thomas-Fermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the fermi wavevector, i.e. the long-distance limit.<ref name=Ashcroft/>
This article uses [[Gaussian units|cgs-Gaussian units]].
== Formula ==
Lindhard formula for the longitudinal [[dielectric function]] is given by
Here, <math>V_q</math> is <math>V_{eff}(q) - V_{ind}(q)</math> and <math>f_k</math> is the carrier distribution function which is the Fermi-Dirac distribution function(see also [[Fermi–Dirac statistics]]) for electrons in thermodynamic equilibrium.
However this Lindhard formula is valid also for nonequilibrium distribution functions.
== Analysis of the Lindhard formula ==
For understanding the Lindhard formula, let's consider some limiting cases in 3 dimensions and 2 dimensions. 1 dimension case is also considered in other way.
=== Three Dimensions ===
==== Long Wave-length Limit ====
First, consider the long wavelength limit (<math>q\to0</math>).
where we used <math>E_k = \hbar \epsilon_k</math>, <math>V_q = \frac{2 \pi e^2}{\epsilon q L^2}</math> and <math>\omega_{pl}^2(q) = \frac{2 \pi e^2 n q}{\epsilon m}</math>.
==== Static Limit ====
Second, consider the static limit (<math>\omega + i\delta \to 0</math>).
This time, let's consider some generalized case for lowering the dimension.
The lower the dimensions is, the weaker the screening effect is.
In lower dimension, some of the field lines pass through the barrier material wherein the screening has no effect.
For 1 dimensional case, we can guess that the screening effects only on the field lines which are very close to the wire axis.
==== Experiment ====
In real experiment, we should also take the 3D bulk screening effect into account even though we deal with 1D case like the single filament.
D. Davis applied the Thomas–Fermi screening to an electron gas confined to a filament and a coaxial cylinder.
For K<sub>2</sub>Pt(CN)<sub>4</sub>Cl<sub>0.32</sub>·2.6H<sub>2</sub>0, it was found that the potential within the region between the filament and cylinder varies as
<math>e^{-k_{eff}r}/r</math> and its effective screening length is about 10 times that of metallic [[platinum]].
== See also ==
* [[Electric field screening]]
== References ==
{{reflist}}
*{{cite book | author=Haug, Hartmut; W. Koch, Stephan | title=Quantum Theory of the Optical and Electronic Properties of Semiconductors (4th ed.) | publisher=World Scientific Publishing Co. Pte. Ltd. | year=2004 | isbn=981-238-609-2}}
*D. Davis ''[http://prola.aps.org/abstract/PRB/v7/i1/p129_1 Thomas-fermi screening in one dimension]'', Phys. Rev. B, 7(1), 129, (1973)
Thomas-Fermi screening can be derived as a special case of the more general Lindhard formula. In particular, Thomas-Fermi screening is the limit of the Lindhard formula when the wavevector (the reciprocal of the length-scale of interest) is much smaller than the fermi wavevector, i.e. the long-distance limit.[2]
Here, is and is the carrier distribution function which is the Fermi-Dirac distribution function(see also Fermi–Dirac statistics) for electrons in thermodynamic equilibrium.
However this Lindhard formula is valid also for nonequilibrium distribution functions.
Analysis of the Lindhard formula
For understanding the Lindhard formula, let's consider some limiting cases in 3 dimensions and 2 dimensions. 1 dimension case is also considered in other way.
Three Dimensions
Long Wave-length Limit
First, consider the long wavelength limit ().
For denominator of Lindhard formula,
,
and for numerator of Lindhard formula,
.
Inserting these to Lindhard formula and taking limit, we obtain
,
where we used , and .
(In SI units, replace the factor by .)
This result is same as the classical dielectric function.
Static Limit
Second, consider the static limit ().
The Lindhard formula becomes
.
Inserting above equalities for denominator and numerator to this, we obtain
.
Assuming a thermal equilibrium Fermi-Dirac carrier distribution, we get
here, we used and .
Therefore,
is 3D screening wave number(3D inverse screening length) defined as
.
Then, the 3D statically screened Coulomb potential is given by
The result is , 3D Debye-Hückel screening wave number.
Two Dimensions
Long Wave-length Limit
First, consider the long wavelength limit ().
For denominator of Lindhard formula,
,
and for numerator of Lindhard formula,
.
Inserting these to Lindhard formula and taking limit, we obtain
where we used , and .
Static Limit
Second, consider the static limit ().
The Lindhard formula becomes
.
Inserting above equalities for denominator and numerator to this, we obtain
.
Assuming a thermal equilibrium Fermi-Dirac carrier distribution, we get
here, we used and .
Therefore,
is 2D screening wave number(2D inverse screening length) defined as
.
Then, the 2D statically screened Coulomb potential is given by
.
It is known that the chemical potential of the 2 dimensional Fermi gas is given by
,
and .
So, the 2D screening wave number is
Note that this result is independent of n.
One Dimension
This time, let's consider some generalized case for lowering the dimension.
The lower the dimensions is, the weaker the screening effect is.
In lower dimension, some of the field lines pass through the barrier material wherein the screening has no effect.
For 1 dimensional case, we can guess that the screening effects only on the field lines which are very close to the wire axis.
Experiment
In real experiment, we should also take the 3D bulk screening effect into account even though we deal with 1D case like the single filament.
D. Davis applied the Thomas–Fermi screening to an electron gas confined to a filament and a coaxial cylinder.
For K2Pt(CN)4Cl0.32·2.6H20, it was found that the potential within the region between the filament and cylinder varies as
and its effective screening length is about 10 times that of metallic platinum.
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