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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/>
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| This article uses [[Gaussian units|cgs-Gaussian units]].
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| == Formula ==
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| Lindhard formula for the longitudinal [[dielectric function]] is given by
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| ::{|cellpadding="2" style="border:2px solid #ccccff"
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| | <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>
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| |}
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| 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.
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| However this Lindhard formula is valid also for nonequilibrium distribution functions.
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| == Analysis of the Lindhard formula ==
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| 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.
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| === Three Dimensions ===
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| ==== Long Wave-length Limit ====
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| First, consider the long wavelength limit (<math>q\to0</math>).
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| For denominator of Lindhard formula,
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| : <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>,
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| and for numerator of Lindhard formula,
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| : <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>.
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| Inserting these to Lindhard formula and taking <math>\delta \to 0</math> limit, we obtain
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| : <math>
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| \begin{alignat}{2}
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| \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}} }\\
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| & \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})\\
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| & \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}\\
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| & = 1 - V_q \frac{q^2}{m \omega_0^2} \sum_k{f_k}\\
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| & = 1 - V_q \frac{q^2 N}{m \omega_0^2} \\
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| & = 1 - \frac{4 \pi e^2}{\epsilon q^2 L^3} \frac{q^2 N}{m \omega_0^2} \\
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| & = 1 - \frac{\omega_{pl}^2}{\omega_0^2}
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| \end{alignat}
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| </math>,
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| 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>.
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| (In SI units, replace the factor <math>4\pi </math> by <math>1/\epsilon_{0}</math>.)
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| This result is same as the classical dielectric function.
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| ==== Static Limit ====
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| Second, consider the static limit (<math>\omega + i\delta \to 0</math>).
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| The Lindhard formula becomes
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| : <math>\epsilon(q,0) = 1 - V_q \sum_k{\frac{f_{k-q}-f_k}{E_{k-q}-E_k}}</math>.
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| Inserting above equalities for denominator and numerator to this, we obtain
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| : <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} }}
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| = 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>.
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| Assuming a thermal equilibrium Fermi-Dirac carrier distribution, we get
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| : <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}}
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| </math>
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| 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>.
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| Therefore,
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| : <math>
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| \begin{alignat}{2}
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| \epsilon(q,0) & =
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| 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} }} =
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| 1 + V_q\sum_k{\frac{\partial f_k}{\partial \mu}} =
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| 1 + \frac{4 \pi e^2}{\epsilon q^2} \frac{\partial}{\partial \mu} \frac{1}{L^3} \sum_k{f_k} \\
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| & = 1 + \frac{4 \pi e^2}{\epsilon q^2} \frac{\partial}{\partial \mu} \frac{N}{L^3} =
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| 1 + \frac{4 \pi e^2}{\epsilon q^2} \frac{\partial n}{\partial \mu} \equiv
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| 1 + \frac{\kappa^2}{q^2}.
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| \end{alignat}
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| </math>
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| <math>\kappa</math> is 3D screening wave number(3D inverse screening length) defined as
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| <math>\kappa = \sqrt{ \frac{4\pi e^2}{\epsilon} \frac{\partial n}{\partial \mu} }</math>.
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| Then, the 3D statically screened Coulomb potential is given by
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| : <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>.
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| And Fourier transformation of this result gives
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| : <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>
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| known as the [[Yukawa potential]].
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| [[File:Screening.png|500px|thumb|Statically screened potential(upper curved surface) and Coulomb potential(lower curved surface) in three dimensions]]
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| For a degenerating gas(T=0), [[Fermi energy]] is given by
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| : <math>E_f = \frac{\hbar^2}{2m}(3\pi^2 n)^{\frac{2}{3}} </math>,
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| So the density is
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| : <math>n = \frac{1}{3\pi^2} \left(\frac{2m}{\hbar^2} E_f\right)^{\frac{3}{2}} </math>. | |
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| At T=0, <math>E_f \equiv \mu</math>, so <math>\frac{\partial n}{\partial \mu} = \frac{3}{2}\frac{n}{E_f}</math>.
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| Inserting this to above 3D screening wave number equation, we get
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| ::{|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>.
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| |}
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| This is 3D [[Thomas–Fermi screening]] wave number.
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| For reference, [[Debye length|Debye-Hückel screening]] describes the nondegenerate limit case.
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| The result is <math>\kappa = \sqrt{ \frac{4\pi e^2 n \beta}{\epsilon} }</math>, 3D Debye-Hückel screening wave number.
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| === Two Dimensions ===
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| ==== Long Wave-length Limit ====
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| First, consider the long wavelength limit (<math>q\to0</math>).
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| For denominator of Lindhard formula,
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| : <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>, | |
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| and for numerator of Lindhard formula,
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| : <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>. | |
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| Inserting these to Lindhard formula and taking <math>\delta \to 0</math> limit, we obtain
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| : <math> | |
| \begin{alignat}{2}
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| \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}} }\\
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| & \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})\\
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| & \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}\\
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| & = 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}\\
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| & = 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}}\\
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| & = 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}}\\
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| & = 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}}\\
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| & = 1 - \frac{V_q L^2}{m \omega_0^2} \sum_{i,j}{ q_i q_j n \delta_{ij}}\\
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| & = 1 - \frac{2 \pi e^2}{\epsilon q L^2} \frac{L^2}{m \omega_0^2} q^2 n\\
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| & = 1 - \frac{\omega_{pl}^2(q)}{\omega_0^2},
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| \end{alignat}
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| </math>
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| 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>.
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| ==== Static Limit ====
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| Second, consider the static limit (<math>\omega + i\delta \to 0</math>).
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| The Lindhard formula becomes
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| : <math>\epsilon(q,0) = 1 - V_q \sum_k{\frac{f_{k-q}-f_k}{E_{k-q}-E_k}}</math>. | |
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| Inserting above equalities for denominator and numerator to this, we obtain
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| : <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>.
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| Assuming a thermal equilibrium Fermi-Dirac carrier distribution, we get
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| : <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>
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| 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>.
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| Therefore,
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| : <math>
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| \begin{alignat}{2}
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| \epsilon(q,0) & =
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| 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} }} =
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| 1 + V_q\sum_k{\frac{\partial f_k}{\partial \mu}} =
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| 1 + \frac{2 \pi e^2}{\epsilon q L^2} \frac{\partial}{\partial \mu} \sum_k{f_k} \\
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| & = 1 + \frac{2 \pi e^2}{\epsilon q} \frac{\partial}{\partial \mu} \frac{N}{L^2} =
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| 1 + \frac{2 \pi e^2}{\epsilon q} \frac{\partial n}{\partial \mu} \equiv
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| 1 + \frac{\kappa}{q}.
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| \end{alignat}
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| </math>
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| <math>\kappa</math> is 2D screening wave number(2D inverse screening length) defined as
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| <math>\kappa = \frac{2\pi e^2}{\epsilon} \frac{\partial n}{\partial \mu}</math>.
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| Then, the 2D statically screened Coulomb potential is given by
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| : <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>.
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| It is known that the chemical potential of the 2 dimensional Fermi gas is given by
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| : <math>\mu (n,T) = \frac{1}{\beta} \ln{(e^{\hbar^2 \beta \pi n/m}-1)}</math>,
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| and <math>\frac{\partial \mu}{\partial n} = \frac{\hbar^2 \pi}{m} \frac{1}{1-e^{-\hbar^2 \beta \pi n / m}}</math>.
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| So, the 2D screening wave number is
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| ::{|cellpadding="2" style="border:2px solid #ccccff"
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| | <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>
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| |}
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| Note that this result is independent of n.
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| === One Dimension ===
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| This time, let's consider some generalized case for lowering the dimension.
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| The lower the dimensions is, the weaker the screening effect is.
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| In lower dimension, some of the field lines pass through the barrier material wherein the screening has no effect.
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| For 1 dimensional case, we can guess that the screening effects only on the field lines which are very close to the wire axis.
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| ==== Experiment ====
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| 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.
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| D. Davis applied the Thomas–Fermi screening to an electron gas confined to a filament and a coaxial cylinder.
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| 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
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| <math>e^{-k_{eff}r}/r</math> and its effective screening length is about 10 times that of metallic [[platinum]].
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| == See also ==
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| * [[Electric field screening]]
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| == References ==
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| {{reflist}}
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| *{{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}}
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| *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)
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| [[Category:Condensed matter physics]]
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