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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]].
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[[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
::{|cellpadding="2" style="border:2px solid #ccccff"
| <math>\epsilon(q,\omega) = 1 - V_q \sum_k{\frac{f_{k-q}-f_k}{\hbar(\omega+i\delta)+E_{k-q}-E_k}}.</math>
|}
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>).
 
For denominator of Lindhard formula,
 
: <math>E_{k-q} - E_k = \frac{\hbar^2}{2m}(k^2-2\vec{k}\cdot\vec{q}+q^2) - \frac{\hbar^2 k^2}{2m} \simeq -\frac{\hbar^2 \vec{k}\cdot\vec{q}}{m}</math>,
 
and for numerator of Lindhard formula,
 
: <math>f_{k-q} - f_k = f_k - \vec{q}\cdot\nabla_k f_k + \cdots - f_k \simeq - \vec{q}\cdot\nabla_k f_k</math>.
 
Inserting these to Lindhard formula and taking <math>\delta \to 0</math> limit, we obtain
 
: <math>
\begin{alignat}{2}
\epsilon(0,\omega) & \simeq 1 + V_q \sum_{k,i}{ \frac{q_i \frac{\partial f_k}{\partial k_i}}{\hbar \omega_0 - \frac{\hbar^2 \vec{k}\cdot\vec{q}}{m}} }\\
& \simeq 1 + \frac{V_q}{\hbar \omega_0} \sum_{k,i}{q_i \frac{\partial f_k}{\partial k_i}}(1+\frac{\hbar \vec{k}\cdot\vec{q}}{m \omega_0})\\
& \simeq 1 + \frac{V_q}{\hbar \omega_0} \sum_{k,i}{q_i \frac{\partial f_k}{\partial k_i}}\frac{\hbar \vec{k}\cdot\vec{q}}{m \omega_0}\\
& = 1 - V_q \frac{q^2}{m \omega_0^2} \sum_k{f_k}\\
& = 1 - V_q \frac{q^2 N}{m \omega_0^2} \\
& = 1 - \frac{4 \pi e^2}{\epsilon q^2 L^3} \frac{q^2 N}{m \omega_0^2} \\
& = 1 - \frac{\omega_{pl}^2}{\omega_0^2}
\end{alignat}
</math>,
where we used <math>E_k = \hbar \omega_k</math>, <math>V_q = \frac{4 \pi e^2}{\epsilon q^2 L^3}</math> and <math>\omega_{pl}^2 = \frac{4 \pi e^2 N}{\epsilon L^3 m}</math>.
 
(In SI units, replace the factor <math>4\pi </math> by <math>1/\epsilon_{0}</math>.)
 
This result is same as the classical dielectric function.
 
==== Static Limit ====
Second, consider the static limit (<math>\omega + i\delta \to 0</math>).
The Lindhard formula becomes
: <math>\epsilon(q,0) = 1 - V_q \sum_k{\frac{f_{k-q}-f_k}{E_{k-q}-E_k}}</math>.
 
Inserting above equalities for denominator and numerator to this, we obtain
 
: <math>\epsilon(q,0) = 1 - V_q \sum_{k,i}{\frac{-q_i \frac{\partial f}{\partial k_i} }{ -\frac{\hbar^2 \vec{k}\cdot\vec{q}}{m} }}
= 1 - V_q \sum_{k,i}{\frac{q_i \frac{\partial f}{\partial k_i} }{\frac{\hbar^2 \vec{k}\cdot\vec{q}}{m} }}</math>.
Assuming a thermal equilibrium Fermi-Dirac carrier distribution, we get
: <math>\sum_{i}{ q_i \frac{\partial f_k}{\partial k_i} } = -\sum_{i}{ q_i \frac{\partial f_k}{\partial \mu} \frac{\partial \epsilon_k}{\partial k_i} } = -\sum_{i}{ q_i k_i \frac{\hbar^2}{m} \frac{\partial f_k}{\partial \mu}}
</math>
here, we used <math>\epsilon_k = \frac{\hbar^2 k^2}{2m}</math> and <math>\frac{\partial \epsilon_k}{\partial k_i} = \frac{\hbar^2 k_i}{m} </math>.
 
Therefore,
: <math>
\begin{alignat}{2}
\epsilon(q,0) & =
1 + V_q \sum_{k,i}{\frac{ q_i k_i \frac{\hbar^2}{m} \frac{\partial f_k}{\partial \mu} }{\frac{\hbar^2 \vec{k}\cdot\vec{q}}{m} }} =
1 + V_q\sum_k{\frac{\partial f_k}{\partial \mu}} =
1 + \frac{4 \pi e^2}{\epsilon q^2} \frac{\partial}{\partial \mu} \frac{1}{L^3} \sum_k{f_k} \\
& = 1 + \frac{4 \pi e^2}{\epsilon q^2} \frac{\partial}{\partial \mu} \frac{N}{L^3} =
1 + \frac{4 \pi e^2}{\epsilon q^2} \frac{\partial n}{\partial \mu} \equiv
1 + \frac{\kappa^2}{q^2}.
\end{alignat}
</math>
 
<math>\kappa</math> is 3D screening wave number(3D inverse screening length) defined as
<math>\kappa = \sqrt{ \frac{4\pi e^2}{\epsilon} \frac{\partial n}{\partial \mu} }</math>.
 
Then, the 3D statically screened Coulomb potential is given by
: <math>V_s(q,\omega=0) \equiv \frac{V_q}{\epsilon(q,\omega=0)} = \frac {\frac{4 \pi e^2}{\epsilon q^2 L^3} }{ \frac{q^2 + \kappa^2}{q^2} }  = \frac{4 \pi e^2}{\epsilon L^3} \frac{1}{q^2 + \kappa^2}</math>.
 
And Fourier transformation of this result gives
: <math>V_s(r) = \sum_q{ \frac{4\pi e^2}{\epsilon L^3 (q^2+\kappa^2)} e^{i \vec{q} \cdot \vec{r}} } = \frac{e^2}{\epsilon r} e^{-\kappa r}</math>
known as the [[Yukawa potential]].
 
[[File:Screening.png|500px|thumb|Statically screened potential(upper curved surface) and Coulomb potential(lower curved surface) in three dimensions]]
 
For a degenerating gas(T=0), [[Fermi energy]] is given by
: <math>E_f = \frac{\hbar^2}{2m}(3\pi^2 n)^{\frac{2}{3}} </math>,
So the density is
: <math>n = \frac{1}{3\pi^2}  \left(\frac{2m}{\hbar^2} E_f\right)^{\frac{3}{2}} </math>.
 
At T=0, <math>E_f \equiv \mu</math>, so <math>\frac{\partial n}{\partial \mu} = \frac{3}{2}\frac{n}{E_f}</math>.
 
Inserting this to above 3D screening wave number equation, we get
 
::{|cellpadding="2" style="border:2px solid #ccccff"
| <math>\kappa = \sqrt{ \frac{4\pi e^2}{\epsilon} \frac{\partial n}{\partial \mu} } = \sqrt{ \frac{6\pi e^2 n}{\epsilon E_f} }</math>.
|}
 
This is 3D [[Thomas–Fermi screening]] wave number.
 
For reference, [[Debye length|Debye-Hückel screening]] describes the nondegenerate limit case.
 
The result is <math>\kappa = \sqrt{ \frac{4\pi e^2 n \beta}{\epsilon} }</math>, 3D Debye-Hückel screening wave number.
 
=== Two Dimensions ===
==== Long Wave-length Limit ====
First, consider the long wavelength limit (<math>q\to0</math>).
 
For denominator of Lindhard formula,
 
: <math>E_{k-q} - E_k = \frac{\hbar^2}{2m}(k^2-2\vec{k}\cdot\vec{q}+q^2) - \frac{\hbar^2 k^2}{2m} \simeq -\frac{\hbar^2 \vec{k}\cdot\vec{q}}{m}</math>,
 
and for numerator of Lindhard formula,
 
: <math>f_{k-q} - f_k = f_k - \vec{q}\cdot\nabla_k f_k + \cdots - k_k \simeq - \vec{q}\cdot\nabla_k f_k</math>.
 
Inserting these to Lindhard formula and taking <math>\delta \to 0</math> limit, we obtain
 
: <math>
\begin{alignat}{2}
\epsilon(0,\omega) & \simeq 1 + V_q \sum_{k,i}{ \frac{q_i \frac{\partial f_k}{\partial k_i}}{\hbar \omega_0 - \frac{\hbar^2 \vec{k}\cdot\vec{q}}{m}} }\\
& \simeq 1 + \frac{V_q}{\hbar \omega_0} \sum_{k,i}{q_i \frac{\partial f_k}{\partial k_i}}(1+\frac{\hbar \vec{k}\cdot\vec{q}}{m \omega_0})\\
& \simeq 1 + \frac{V_q}{\hbar \omega_0} \sum_{k,i}{q_i \frac{\partial f_k}{\partial k_i}}\frac{\hbar \vec{k}\cdot\vec{q}}{m \omega_0}\\
& = 1 + \frac{V_q}{\hbar \omega_0} 2 \int d^2 k (\frac{L}{2 \pi})^2 \sum_{i,j}{q_i \frac{\partial f_k}{\partial k_i}}\frac{\hbar k_j q_j}{m \omega_0}\\
& = 1 + \frac{V_q L^2}{m \omega_0^2} 2 \int \frac{d^2 k}{(2 \pi)^2}  \sum_{i,j}{q_i q_j k_j \frac{\partial f_k}{\partial k_i}}\\
& = 1 + \frac{V_q L^2}{m \omega_0^2} \sum_{i,j}{ q_i q_j 2 \int \frac{d^2 k}{(2 \pi)^2} k_j \frac{\partial f_k}{\partial k_i}}\\
& = 1 - \frac{V_q L^2}{m \omega_0^2} \sum_{i,j}{ q_i q_j 2 \int \frac{d^2 k}{(2 \pi)^2} k_k \frac{\partial f_j}{\partial k_i}}\\
& = 1 - \frac{V_q L^2}{m \omega_0^2} \sum_{i,j}{ q_i q_j n \delta_{ij}}\\
& = 1 - \frac{2 \pi e^2}{\epsilon q L^2} \frac{L^2}{m \omega_0^2} q^2 n\\
& = 1 - \frac{\omega_{pl}^2(q)}{\omega_0^2},
\end{alignat}
</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>).
The Lindhard formula becomes
: <math>\epsilon(q,0) = 1 - V_q \sum_k{\frac{f_{k-q}-f_k}{E_{k-q}-E_k}}</math>.
 
Inserting above equalities for denominator and numerator to this, we obtain
 
: <math>\epsilon(q,0) = 1 - V_q \sum_{k,i}{\frac{-q_i \frac{\partial f}{\partial k_i} }{ -\frac{\hbar^2 \vec{k}\cdot\vec{q}}{m} }}
= 1 - V_q \sum_{k,i}{\frac{q_i \frac{\partial f}{\partial k_i} }{\frac{\hbar^2 \vec{k}\cdot\vec{q}}{m} }}</math>.
Assuming a thermal equilibrium Fermi-Dirac carrier distribution, we get
: <math>\sum_{i}{ q_i \frac{\partial f_k}{\partial k_i} } = -\sum_{i}{ q_i \frac{\partial f_k}{\partial \mu} \frac{\partial \epsilon_k}{\partial k_i} } = -\sum_{i}{ q_i k_i \frac{\hbar^2}{m} \frac{\partial f_k}{\partial \mu}}
</math>
here, we used <math>\epsilon_k = \frac{\hbar^2 k^2}{2m}</math> and <math>\frac{\partial \epsilon_k}{\partial k_i} = \frac{\hbar^2 k_i}{m} </math>.
 
Therefore,
: <math>
\begin{alignat}{2}
\epsilon(q,0) & =
1 + V_q \sum_{k,i}{\frac{ q_i k_i \frac{\hbar^2}{m} \frac{\partial f_k}{\partial \mu} }{\frac{\hbar^2 \vec{k}\cdot\vec{q}}{m} }} =
1 + V_q\sum_k{\frac{\partial f_k}{\partial \mu}} =
1 + \frac{2 \pi e^2}{\epsilon q L^2} \frac{\partial}{\partial \mu} \sum_k{f_k} \\
& = 1 + \frac{2 \pi e^2}{\epsilon q} \frac{\partial}{\partial \mu} \frac{N}{L^2} =
1 + \frac{2 \pi e^2}{\epsilon q} \frac{\partial n}{\partial \mu} \equiv
1 + \frac{\kappa}{q}.
\end{alignat}
</math>
 
<math>\kappa</math> is 2D screening wave number(2D inverse screening length) defined as
<math>\kappa = \frac{2\pi e^2}{\epsilon} \frac{\partial n}{\partial \mu}</math>.
 
Then, the 2D statically screened Coulomb potential is given by
: <math>V_s(q,\omega=0) \equiv \frac{V_q}{\epsilon(q,\omega=0)} = \frac{2 \pi e^2}{\epsilon q L^2} \frac{q}{q + \kappa} = \frac{2 \pi e^2}{\epsilon L^2} \frac{1}{q + \kappa}</math>.
 
It is known that the chemical potential of the 2 dimensional Fermi gas is given by
 
: <math>\mu (n,T) = \frac{1}{\beta} \ln{(e^{\hbar^2 \beta \pi n/m}-1)}</math>,
 
and <math>\frac{\partial \mu}{\partial n} = \frac{\hbar^2 \pi}{m} \frac{1}{1-e^{-\hbar^2 \beta \pi n / m}}</math>.
 
So, the 2D screening wave number is
::{|cellpadding="2" style="border:2px solid #ccccff"
| <math>\kappa = 2\pi e^2\frac{\partial n}{\partial \mu} = 2\pi e^2 \frac{m}{\hbar^2 \pi} (1-e^{-\hbar^2 \beta \pi n / m}) = \frac{2 m e^2}{ \hbar^2} f_{k=0} .</math>
|}
 
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 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)
 
[[Category:Condensed matter physics]]

Revision as of 07:10, 8 February 2014

Hi, everybody!
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