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Lindhard theory

Quantum theory of interacting electron gas From Wikipedia, the free encyclopedia

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In condensed matter physics, Lindhard theory[1] is a method of calculating the effects of electric field screening by electrons in a solid. It is based on quantum mechanics (first-order perturbation theory) and the random phase approximation. It is named after Danish physicist Jens Lindhard, who first developed the theory in 1954.[2][3][4]

Thomas–Fermi screening and the plasma oscillations 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.[1] The Lorentz–Drude expression for the plasma oscillations are recovered in the dynamic case (long wavelengths, finite frequency).

This article uses cgs-Gaussian units.

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Lindhard function

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Thumb
Static Lindhard function . It has a logarithmic divergence at (dashed line).

The term

is a response function known as Lindhard function. Here is the is the Fermi–Dirac distribution function for electrons in thermodynamic equilibrium, is the kinetic energy with wave vector , is the electron mass and is a positive infinitesimal constant,.

Zero temperature functions

At zero temperature, is a Heaviside step function, where is the Fermi wave vector associated to the Fermi energy . The sum can be carried out in the continous limit using analytic continuation, resulting in , where in 3 dimensions can be written as[5]

In the static limit, when , we have that , where

is the (static) Lindhard function.[6] This function derivative diverges at . Also . Note that also appears in the calculation of the ground state energy of jellium when using Hartree–Fock method.[7]

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Formula for the dielectric function

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The electron-electron Coulomb potential , where e is the elementary charge, can be written in Fourier space as (where is the wave vector), then the effective single particle potential is

The Lindhard formula for the longitudinal dielectric function is then given by

where is Lindhard function.

This Lindhard formula is valid also for nonequilibrium distribution functions. It can be obtained by first-order perturbation theory and the random phase approximation (RPA).

To understand the Lindhard formula, consider some limiting cases in 2 and 3 dimensions. The 1-dimensional case is also considered in other ways.

Long wavelength limit

In the long wavelength limit (), Lindhard function reduces to

where is the three-dimensional plasma frequency (in SI units, replace the factor by .) For two-dimensional systems,

.

This result recovers the plasma oscillations from the classical dielectric function from Drude model and from quantum mechanical free electron model.

Derivation in 3D

For the denominator of the Lindhard formula, we get

,

and for the numerator of the Lindhard formula, we get

.

Inserting these into the Lindhard formula and taking the limit, we obtain

,

where we used and .

Derivation in 2D

First, consider the long wavelength limit ().

For the denominator of the Lindhard formula,

,

and for the numerator,

.

Inserting these into the Lindhard formula and taking the limit of , we obtain

where we used , and .

Static limit

Consider the static limit ().

The Lindhard formula becomes

.

Inserting the above equalities for the denominator and numerator, we obtain

.

Assuming a thermal equilibrium Fermi–Dirac carrier distribution, we get

here, we used and .

Therefore,

Here, is the 3D screening wave number (3D inverse screening length) defined as

.

Then, the 3D statically screened Coulomb potential is given by

.

And the inverse Fourier transformation of this result gives

known as the Yukawa potential. Note that in this Fourier transformation, which is basically a sum over all , we used the expression for small for every value of which is not correct.

Thumb
Statically screened potential(upper curved surface) and Coulomb potential(lower curved surface) in three dimensions

For a degenerated Fermi gas (T=0), the Fermi energy is given by

,

So the density is

.

At T=0, , so .

Inserting this into the above 3D screening wave number equation, we obtain

.

This result recovers the 3D wave number from Thomas–Fermi screening.

For reference, Debye–Hückel screening describes the non-degenerate limit case. The result is , known as the 3D Debye–Hückel screening wave number.

In two dimensions, the screening wave number is

Note that this result is independent of n.


Derivation in 2D

Consider the static limit (). The Lindhard formula becomes

.

Inserting the above equalities for the denominator and numerator, we obtain

.

Assuming a thermal equilibrium Fermi–Dirac carrier distribution, we get

.

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 .

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Formula for the magnetic susceptibility

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As with the dielectric function, the magnetic susceptibility for an electron gas can be calculated as

where is the Bohr magneton.[6]

In the static limit,

which corresponds to the spin susceptibility of Pauli paramagnetism.[6]

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Screening experiments on one dimensional systems

This time, consider some generalized case for lowering the dimension. The lower the dimension is, the weaker the screening effect. In lower dimension, some of the field lines pass through the barrier material wherein the screening has no effect. For the 1-dimensional case, we can guess that the screening affects only the field lines which are very close to the wire axis.

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. The Thomas–Fermi screening has been applied to an electron gas confined to a filament and a coaxial cylinder.[8] For a K2Pt(CN)4Cl0.32·2.6H20 filament, 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.[8]

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See also

References

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