﻿ Spin Hall effect (SHE), intrinsic Rashba model and extrinsic scattering

# Spin Hall effect doesn't mean "Spin".

## "Spin - orbit" coupling is illusion.

(Fig.1) Spin - orbit coupling doesn't really exist. It is said that spin - orbit interaction is related to spin Hall effect (= SHE, see Wiki ) and magnetoresistance.
But unfortunately, these phenomena have nothing to do with unrealistic spin.

## "Spin-orbit coupling" is too weak to explain actual phenomena.

(Fig.2) Spin-orbit coupling based on relativistic effect is too weak. ← illusion ! They insist spin orbit coupling is caused by relativistic effect (= see Wiki ).
But if so, the magnitude of spin orbit coupling becomes much smaller than the experimental values ( this site p.6 and this site ).

On this site ( p.2, left ), it is written
-----------------------------------
"For example, for GaAs one finds λ = 5.3 Å2.
Thus, the SO (= spin-orbit ) coupling in n-GaAs is by six orders of magnitude stronger than in vacuum ( λ = -3.7 × 10-6 Å2 ) and has the opposite sign.
This enhancement of SO coupling is critical for developing large extrinsic spin currents.
------------------------------------

This wide discrepancy in the λ shows the concept of spin-orbit coupling is completely false.
In spite of this obvious fact, they like to use the word of "spin-orbit coupling" in various (top) journals.
Again, this wrong usage is one of mind-controls.

## Spin-orbit coupling = relativistic effect ? ← too weak.

(Fig.3) Electric field (= E ) changes into magnetic field (= B ) under Lorentz transformation ? First we explain why spin-orbit coupling means relativistic effects.
In Fig.3, an electron with spin is moving in x direction at the speed of "v" under the electric field Ey (= S frame ).

In the rest frame of this electron ( v = 0, S' frame ) , this electric field transforms into magnetic field B according to the relativistic electromagnetism.

(Eq.1) Lorentz transformation. As shown on this site, Lorentz transformation of electric field becomes like Eq.1.
B' is magnetic field from the viewpoint of moving electron (= S', rest frame ).

Here, the magnetic field in S frame is supposed to be zero ( B = 0 ).
( Only electric field Ey exists in S frame. )
So approximately, Eq.1 becomes Eq.2.

(Eq.2) The magnetic energy is magnetic moment (= μ ) × magnetic field (= B ).
The μ of an electron's spin is minus Bohr magneton.

(Eq.3) In the rest frame of an electron (= S' ), the magnetic field (= Bz ) is caused in the minus z direction.
If this electron has "spin", the magnetic interactive energies are split into two depending on spin "up" or "down" directions.

(Eq.4) σ is Pauli matrices.
So, s = 1/2 σ.

(Eq.5) The electron's velocity (= v ) can be expressed by wavenumber (= k ) as shown in this Wiki.
And Thomas factor (= 1/2 ) is added.

Substituting Eq.4 and Eq.5 into Eq.3, the magnetic energy (= - μB ) based on relativistic effect becomes

(Eq.6) As a result,
(Eq.7) ## Experimental values are completely different from relativistic effects !

This spin orbit coupling constant (= λ ) by relativity becomes

(Eq.8) Relativistic effect ? But the experimental values of spin-orbit coupling constant in solids are much larger than Eq.8, as follows,

(Eq.9) Actual solids. where
(Eq.10) If you compare Eq.8 (= relativsitic effect ) and Eq.10, you easily find the concept of spin-orbit coupling is illusion.
Originally spin itself has NO reality at all.

In spite of this fact, why the current physicists continue believing illusory spin-orbit coupling ?
There were not alternative theories of quantum mechanics in 1920's due to lack of computers.
This is the reason why strange spin has survied even now, though they try not to ask what "spin" really is.

## Spin-orbit coupling is too weak to explain actual phenomena.

(Fig.4) Spin-orbit coupling based on relativistic effect is "illusion". The serious problem is, if we continue believing this false fact, much more people all over the world would waste their lives in wrong theories.
And we cannot develop the science due to the troublesome "Shut up and calculate!" attitude in the current quantum mechanics.

As shown on this page and this page, Alkali (= isolated sodium ) doublets cannot be explained by spin-orbit interactions, because spin orbit coupling is too weak to explain actual energy splitting.

The important point is that usual textbooks are hiding this serious defect about spin -orbit, and insist "spin" has factual grounding.
This is a kind of frauds, if they know the above results.

## Band theory is completely useless.

### [ Band theory, Brillouin zone are "obsolete" concepts. ]

(Fig.5) Band theory = ancient documents, it's completely useless. Band structure consists only of wave number (= k ) and energies, NOT including space coodinate.
( Lacking x,y,z coodinate means this band theory is NOT applicable to the actual sicence. )
In spite of this old concept, it is used even now in condensed matter physics. This is disaster.

To treat many-body complicated wavefunctions, they needed to expand wavefunction in the form of plane wave solutions (= eikx ) in 1920 - 1930s.
Considering abstract periodicity of solids, they paid attention to first ( small ) wave number zone.
This is called Brillouin zone ( see this site ).

## "Band theory" and "Brillouin zone" are meaningless.

(Fig.6) "Band theory" and "Brillouin zone" are useful ? The concept of space coodinate (= x, y, z ) is indispensable for applied physics.
We often save different kinds of informations in different places inside the computer memory.
In these cases, the position (= x, y, z ) inside memory is very important.

So the band theory based ONLY on wavenumber and Brillouin zone is completely useless.
These old concepts become like ancient documents and survive only for papers in various journals (= NOT real world at all ).

## Lack of computer caused "virtual" band theory.

(Fig.7) Time gap without computers produced band theories. Solving many-body Schrodinger equations + complicated Slater determinants is impossible.
And of course, there were NO high efficient computers in 1920s - 1960s.
In spite of these conditions, they needed to treat complicated solid states consisting of infinite atoms.

Within pen and paper theories, all they could do is rely on simple plane wave solutions (= eikr ).
The problem is, even after high efficient computers appeared, these dead concepts such as band theory and Brillouin zone are left like ancient documetns, as shown on this page

As far as we stick to these old concepts and continue "Shut up and calculate!" quantum mechanics and spin, the science would never advance.

## Ab-initio band (= DFT ) calculation is NOT ab-initio.

(Fig.8) Exchange correlation functions of DFT are "free" paremeters. Band theory for solids completely depends on density functional theory (= DFT ).
Most textbooks and academic papers insist these DFT are first-principle (= ab-initio ).
But unfortunately, this DFT is NOT ab-initio at all.

DFT was introduced to treat complicated many-body solid states in 1960s.
For this calculation, they thought up many kinds of exchange correlation functionals (= Exc ).
The important point is that these exchange-correlation functionals are like free parameters we can freely choose.

## Exchange correlation functions of DFT = "free" parameters.

On this site p.9, "the central problem in DFT is that there is NO unique prescription for how to find the exchange and correlation functionals".
So the word of "ab-initio DFT" gives ordinary people wrong impressions.
This is dangerous.

(Fig.9) Exchange energy = Pauli exclusion principle ? For example, in lithium atom, the third electron cannot enter 1s orbitals.
This strong repulsive effect needs to be expressed using Slater determinants.

As you see Fig.9, these complicated determinants are one of the main reasons why the current quantum mechanics cannot be applicable to actual larger molecules.

## Exchange = Pauli, correlation = repulsive two electrons ?

To do with this problem, all they could do was rely on some approximations.
Exchange functionals (= Ex ) were approximate forms to describe this Pauli exclusion principle.
Various forms of this functionals have been produced until now, and we can freely choose one of them.

(Fig.10) Correlation energy = electron's wavefunctions repel each other ? In multi-electrons' atoms, each electron is avoiding each other.
When one of their wavefunctions is determined, other wavefunctions need to be changed to avoid it.
This means there are NO constant probability densities in multi-electron atoms, different from hydrogen.

For example, as shown on this page, they prepared five kinds of charge parameters (= Z ) inside one helium atom's wavefunction to describe repulsive motions of two electrons.

## Correlation functions can be changed freely !

In solid states, of course, they had to use some approximations (= correlation functional, Ec ).
The form and parameters in these correlation functionals are completely arbitrary.
Again, DFT is NOT ab-initio at all.

## Basic form of DFT.

(Eq.11) Hamiltonian of multi-electron atoms. Eq.11 is Hamiltonian of multi-electron atoms.
H0 is the sum of each kinetic energy and potential energy between nucleus and electron.
The second term of right side in Eq.11 represents repulsive forces of electrons.

(Eq.12) Slater determinants = Pauli exclusion principle ? Generally, determinants become zero, when two lines ( or columns ) of them are equal.
They take advantage of this property in determinants to express Pauli exclusion principle.
Because according to Pauli exclusion, two electrons cannot be in the same states (= lines ).

As I said many times, the current quantum mechanics NEVER try to investigate the origin of this Pauli exclusion principle.
As shown on this page, electron spin is too weak to explain this strong repulsive force.

(Eq.13) Using Eq.11 and Eq.12, the total energy becomes like Eq.13.
Jij is the repulsive Coulomb energy between i and j electrons.
Kij represents exchange energy based on Slater determinants, as follows.

(Eq.14) ## DFT is NOT ab-initio, it can use arbitrary functions.

In DFT, this complicated exchange part and correlations part are treated as ungrounded approximations.
The sum of them is called exchange correlation energy (= Exc ).

(Eq.15) Density functional theory is NOT ab-initio. As I said, this exchange and correlation functionals can be freely changed as we like.
So DFT is NOT first principle at all, though they always call it "ab-initio DFT" in solid physics.
This misuse is very dangerous.

## Even local density approximation (= LDA ) is very complicated.

(Eq.16) Exchange energy in LDA. On this site (p.4), "the problem is that functional Exc is NOT known.
The simplist approximation is local density approximation (LDA) using uniform electron gas.

Of course, electron's distributions in the actual solids are far from uniform.
So LDA fail in various materials.
Eq.16 is exchange term of LDA.

## LDA correlation functionals are NOT unique.

(Eq.17) LDA - VWN correlation function. On this site p.4, there are several expressions for LDA correlation energy which have been obtained by fitting to the results of accurate QMC calculations.
Most popular choice in LDA correlation functions is VWN-LDA as shown in Eq.17.

ε of Eq.17 is on this site (p.9).
This LDA correlation function contains as many as 12 fitting parameters.
So it is NOT first-principle at all.

## Correlation functionals contain "free" parameters.

(Eq.18) LDA - PW correlation functional One of other LDA correlation functionals is PW-LDA like Eq.18.
This functional contains 6 fitting parameters.

Of course, there are other correlation functionals in LDA.
So, LDA is NOT a single pattern.
We can freely change correlation functionals depending on circumstances.

Quantum Monte Carlo simulation (= QMC ) can give random and wrong results.
The results of this QMC changes depending on electron's density (= ρ ).
NO convenient LDA functionals have been found to satisfy all conditions.

## Hybrid functionals are NOT ab-initio, either.

(Eq.19) Hybrid functional. Eq.19 is B3LYP hybrid functionals, which mix Hartree Fock, GGA and LDA functionals, as shown in this site (p.8)
This B3LYP functionals have been used most often in qunatum chemistry.
The important point is that this B3LYP contains many parameters, which are based on average empirical values of many kinds of atoms and small molecules.

So, again, this B3LYP is NOT ab-initio at all.

(Eq.20) B3LYP contains B88 exchange function.
B88 is one of GGA functional, which adds some gradient of electron's density to uniform LDA.
ζ is semi-empirical parameter.

There are many kinds of GGA functionals.
Depending on the situation, one of these functionals is chosen.

## Various forms of functionals.

(Eq.21) Eq.19 contains LYP exchange functional, which is one of GGA.
There are five semi-empirical parameters in LYP.

In the current DFT, they are spending much time in finding better exchange correlation functionals.
Because the form of these functionals are completely free.
If DFT is NOT first principle, these efforts are just waste of time.

## Spin Hall effect = mathematical trick.

(Fig.11) Spin Hall effect ? In spin Hall effect, even under NO external magnetic field, "up" ( down ) spin electron turns toward left ( right ), when some electric field (= E ) is applied ( see this site or this site ).
We have NOT yet arrived at the definite conclusion about the true mechanism of this spin Hall effect.

In extrinsic mechanism, spin-polarized electrons are scattered by some impurities through spin-orbit interaction.
( This is called "skew scattering" or "side jump". )

In intrinsic mehacanism, they insist some band structure ( Berry phase ) or spin precession cause spin Hall effect.
But these intrinsic theoirs are very artificial and have some defects, as I say later.

## Spin Hall effect does NOT see "spin" itself.

(Fig.12) Spin Hall effect = Spin ?   illusion ? In spin Hall effect experiments, they did NOT detect "spin" itself ( see this site and this site ).
They just measure the slight change of reflected light polarization.

Signal is very weak. This experiment is very difficult, so they don't know the directions of "up" or "down" spins.
Only asymmetry of light polarization can be detected.
They use Kerr effect to know "spin".
But this Kerr effect is purely classical mechanical ones like Faraday effect, as shown in Wiki.

Basically it is natural each electron inside semiconductor is moving, rotating around some positive nucleus.
So we can say spin Hall effect is caused by some circular orbit (= classical rotation ) of electrons.

## Electron "Spin" is actual spinning ? → faster-than-light !

(Fig.13) Spin Hall = Magnus effect ?   Faster-than-light spinning ? Some physicists say spin Hall effect is due to Magnus effect ( see Wiki Magnus and spin Hall ).
In Magnus effect, spinning ball curves away from its principal path.
Because the directions of spin and air flow are parallel or antiparallel in each side of the spinning balls.

The problem is spinning speed of point-like electron must reach more than 100 times speed of light ( > 100c ), if "spin" is actual spinning, as shown on this page.
So electron spin cannot use this Magnus effect as the cause of spin Hall effect.

On the other hand, if we consider spin Hall effect is caused by circular orbit of electron, Magnus effect can be applied naturally. (= each de Broglie wave flow's effects )

## Classical rotation can explain spin Hall effect naturally.

(Fig.14) Directions of electric field (= E ) become opposite → "spin" Hall effect ? Besides Magnus effect, we can consider the influence of external electric field.

In Fig.11 and Fig.14, one side of this electron's rotation is parallel to the external electric field.
Another side is antiparallel.
This difference in rotational direction may be the cause of "spin" Hall effect, though it does NOT depend on strange "spin".

On the other hand, spinning electron is point-like particle.
So it causes NO difference with respect to directions of external electric field on the surface of tiny electrons.

## Spin Hall effect is due to classical rotation of electrons

(Fig.15) Spin Hall effect = Rotation. In spin Hall experiments, they detected the directions of polarized lights ( NOT spin ! ).
In normal Zeeman effect, the change of orbital angular momentum generates ( left or right ) circular polarized lights depending on the change of ± m ( see this site ).

And Kerr and Faraday effects are originally classical effects.
Considering spinning speed of electron becomes much faster than light ( see also this site ), it is quite natural that we think spin Hall effect is due to circular orbit of electrons.

## Spin scattering is real ?

(Fig.16) Point-like electron → Only random scattering is left. As I said in Fig.13 and Fig.14, point-like electrons cannot rely on classical Magnus effect and the difference in the external electric field direction as the cause of spin Hall.
So there is ONLY random scattering left to explain spin Hall effect, if you believe "spin".

But as I said in Fig.2, spin-orbit interaction by realtivity is too weak to explain this scattering.
And random scattering means electrons turn toward left or right randomly.
So it generates NO spin currents, because the average is almost zero.
This is strange.

## Intrinsic Spin Hall effect is very artificial.

### [ Intrinsic spin Hall effect may vanish due to some impurities. ]

First, we explain intrinsic effect based on Rashba model, as shown in this site and this paper.
As far as I read this paper, they seem to use some mathematical tricks to get universal values.
And it is known this intrinsic effect vanishes due to some impurities as shown in this site and this paper (p.6, 7).

This mechanism is very complicated, and based on some tricks.

## Rashba model uses spin precession as Hall effect ?

(Fig.17) Rashba intrinsic electric field. Some semiconductors originally have intrinsic electric field (= Erash ) in the z direction.
In Fig.17, an electron is moving in y direction ( ky is wavenumber ).
According to relativity, magnetic field (= Bras is in - x direction ) is generated in the rest frame of this electron.
As an electron has negative charge, its spin turns in x direction.

Δ1 is opposite to "B" direction ( in this case, +x direction ).
The hat z means the direction of - eE (= - z direction ).

## Various tricks.

(Eq.22) Hamiltonian. Considering spin-orbit coupling energy, Hamiltonian becomes like Eq.22.
λ is the Rashba coupling constant ( λ > 0 ).
σ means Pauli matrices ( s = 1/2σ ).

(Eq.23) Larmor precession. Eq.23 is Larmor precession ( see Wiki ).
"S" is electron's spin direction. "B" is magnetic field.
ge is spin g factor (= 2 ).

(Eq.24) An electron has negative charge, so negative sign is added in Eq.23 and Eq.25.

(Eq.25) "n" and "S" (= spin ) are in the same direction.
Δ(t) is also in the spin direction, which is opposite to magnetic field (= B ).
( Unit of this Δ(t) is the same as energy (= J ). )

Using Eq.25, Eq.23 becomes

(Eq.26) Δ1 is the direction in which spin first turns.
So in Fig.17, Δ1 is in x direction.

The direction of Δ2 is defined like Eq.27.
( If Δ1 is in x direction, Δ2 is in y direction ).

## Very complicated math   ← Physics ?

(Eq.27) From Eq.26, we obtain

(Eq.28) First spin direction is +x, so Δ3 (= z component ) is zero in Eq.28.
And,

(Eq.29) Δ2 component (= perpendicular to Δ1 ) of spin is very small (= spin precession ).
Spin component n2 is proportional to Δ2, so

(Eq.30) From Eq.28 and Eq.30, we have

(Eq.31) From Eq.22 and Eq.25, Δ (= - B direction ) can be expressed as

(Eq.32) The wavenumber k is p/ħ.
First, this k is supposed to turn in y direction ( k = ky ).
( And hat z is - z direction ).
So Eq.32 becomes

(Eq.33) Here the external electric field Ex is applied like Fig.18.
After that, this electron drifts toward - x direction.
And due to relativistic effect, new effective magnetic field (= Bnew ) is generated in the z direction.

As a result, this spin turns in -z direction (= down ) due to its negative charge.
But some tricks need to be used in this calculation.

## Rashba model cannot generate spin Hall effects.

(Fig.18) Electric field Ex is applied. → spin down ? From Eq.27, Δ2 is in the ky (= k ) direction in this case.
The time derivative of this "k" component (= Δ2 ) of Eq.32 becomes

(Eq.34) In Eq.34, due to new electric field Ex, this electron is accelerated in - x direction.
So we use

(Eq.35) Momentum p is equal to ħk ( k is wavenumber ).
The time derivative of momentum is electric force (= -eE ) in Eq.35.
But Eq.34 is a little strange.
Negative sign (= -e ) in inner product should not be left, I think.

Substituting Eq.33 and Eq.34 into Eq.31, and add 1/2 (= spin ), we have

(Eq.36) As you see Eq.36, depending on ky (= moving ) direction, the z component of spin becomes reversed.
So they insist this is spin Hall effect.
But it is difficult to cause spin accumulation by this theory, as I explain later.

(Eq.37) Electron's velocity is expressed as momentum (= ħk ) / m.
Spin current ( j ) is equal to spin angular moementum (= ħS ) × velocity.

## Artificial definitions.

(Eq.38) This is two-dimensional system.
So the integral of wave number becomes like Eq.38.

kF,+ and kF,- are maximum and minimum wave numbers in this system.
These maximum and minimum wave numbers are arbitrary ( ← Trick ).

(Eq.39) Substituting Eq.36 and Eq.38 into Eq.37, and integrating it, we get Eq.39.
Here they suddenly define like Eq.40.

(Eq.40) Using Eq.40, Eq.39 becomes

(Eq.41) So they insist universal electric conductivity (= -e/8π ) can be gotten.
But as you see Eq.40, this definition can be changed as we like.
The choice of Eq.40 value leads to the result of Eq.41

(Eq.42) Comparing Eq.8 and Eq.40, the difference between kF,+ and kF,- become like Eq.42.
I cannot find proper reason why the condition of Eq.42 is chosen.

## Total spin is always "zero". ← NOT spin Hall.

(Fig.19) Total (= average ) spin is always zero in this theory. According to this theory, when electron is moving in y direction, its spin becomes "down".
When electron is moving in - y direction, spin becomes "up".
The external electric field is only in x direction.

So the average velocity in y direction is always zero.
This means the average spin is always zero in any parts of this system. This result is inconsistent with the experiment of Fig.12 (= spin accumulation in each sides ).

## Intrinsic Spin Hall effect based on band structure is mathematical trick.

(Fig.20) GaAs, heavy- (HH) and light- (LH) hole bands Another main intrinsic mechanism of spin Hall effect is based on band structure. ( Science 301 1348 2003, see this paper ).

According to Luttinger Hamiltonian, the band energies are split into heavy-hole (= HH ) and light hole (= LH ) bands.
HH has total angular momentum j = ± 3/2, LH has j = ± 1/2.

(Fig.21) Spin Hall effect ? According to this theory, particles having +3/2 or +1/2 turn in the direction opposite to particles having -3/2 or -1/2.
Each particle is supposed to be moving inside either of HH or LH bands.
But this explanation is inconsistent with experimental results, I think.

(Fig.22) Spin accumulation becomes zero in each band ? As I said, each band contains ± angular momentum, because these directions are random.
If so, like Fig.19, total spin in each band becomes always zero.
Because particles with plus and minus angular momentum are circulating, turning in the opposite directions.
Again, this theory is inconsistent with experimental results which showed spin accumulation.

## ↓ This Hamiltonian is artificially introduced.

(Eq.43) Luttinger Hamiltonian with external field. In this paper, they use the approximation of Luttinger Hamiltonian.
γ1 and γ2 are semiempirical parameters, which are different in each material.

"k" is particle's wave number, and "S" denotes total angular momentum.
"-eEx" is electric energy.

(Eq.44) Eq.44 is wave number in the spherical coordinate.
Here they use the following unitary transformation.

(Eq.45) "kS" term of Eq.43 transforms like Eq.46.
Under the unitary transformation of Eq.45, spin S turns toward z direction.
After this manipulation, each partilce is moving with its spin S in the z direction, they insist.
( Of course, this manipulation is very unnatural. )

(Eq.46) "x" space variable of Eq.43 can be changed into the derivative of wave number (= k ) like Eq.47.

## Fake vector potential "A".

(Eq.47) Using Eq.47, Hamiltonian of Eq.43 changes into Eq.48 under transformation of Eq.45.

(Eq.48) Here they suddenly change the definition of "x" like Eq.49.
The last term of Eq.48 is added to the new x variable.
This is clearly artificial definition.

(Eq.49) Mathematical Trick. Magnetic field (= B ) can be expressed as ∇ × A.
( ∇ is space ( NOT wave number ! ) derivative, and A is vector potential ).

Wave number (= k ) is clearly different from space variable "x".
But they use the same form of true magnetic field in pseudo magnetic monopole like Eq.50.

## Fake magnetic monopole. ← spin Hall effect is false.

(Eq.50) "Fake" magnetic monopole ? λ of Eq.50 is total angular momentum.

(Eq.51) Using the relation of Eq.50 in Hamiltonian of Eq.43, Haisenberg equation of motion becomes

(Eq.52) Using Eq.50, Eq.52 becomes

(Eq.53) In Eq.53, depending on the sign of λ, the direction of velocity y changes.

But the new force in y direction in Eq.53 is strange.
There are NO forces perpendicular to both spin magnetic moment and electric field in the actual world.
( This strange force is neither relativistic effect nor Lorentz force. )

And each band contains ±3/2 ( or ±1/2 ) magnetic moments.
So magnetic components of Eq.53 are cancelled out. ( +3/2 - 3/2 = 0. )

(Fig.23) Spin Hall effect ? But as I said, this explanation is inconsistent with experimental results.

(Fig.24) Spin accumulation becomes zero in each band ? The opposite sign of angular momentum ( ex. ±1/2 ) inside each band cancel each other.
Because these particles are circulating in y direction.
And kz direction of Eq.53 is random.
As a result, total spin accumulation in each site always becomes zero.

And the definition of antisymmetric tensor (= Fij ) is NOT true magnetic tensor.
I couldn't find reasonable reason of the replacement of Eq.49.

## Hall effect of Four space dimensions is real ?

### [ We should stop relying on another dimension to describe actual phenomena. ]

(Fig.25) Quanum Hall effect = Four space dimensions ? = String theory ? Surprisingly, according to top journal ( Science 294 823 2001, see arxiv ), they insist four space ( NOT time ) dimensions exist. See also this news

As you know, another dimension is completely fantasy.
You may be surprized at the fact that even top journals believe extra dimensions such as string theory. 2014/2/14 updated. Feel free to link to this site.