Top page (correct Bohr model including helium. )

Spintronics of condensed matter is an illusion.

Mechanism of useless quantum chemistry

Condensed matter physics is ancient ruins.

*(Fig.1) Superconductor.*

Superconductivity is a phenomenon of zero electrical resistance and and expulsion of external magnetic fields in some materials when they are cooled below their **critical temperature** (= T_{c} < 30 K ← almost absolute zero ).

As shown on this page, this ejection of magnetic field is called "Meissner effect", which can be explained by **classical** electromagnetism.

*(Fig.2) "Phonon" can attract two electrons ? *

According to the standard theory (= **BCS** theory ), electrons are held together in Cooper pairs (= Bose particle ).

Surprisingry, though both two electrons have **negative** charges, these electrons **attract** each other, they insist.

This attraction is caused by quasiparticle "**phonons**".

They insist "phonon" is also a fermion, like electrons. Of course, this phonon does **NOT** exist.

They say this unreal particle, "phonon" is some lattice vibration of nuclei and electrons.

But they **NEVER** try to clarify what this phonon really is, which attitudes (= Shut up and calculate ! ) clearly **obstructs** the development of condensed matter physics, forever.

*(Fig.3) Giving "gap energy" (= Δ ) to superconductor → break. *

According to BCS theory, there is some "**energy gap**" (= Δ ) between superconducting and normal states.

So if you give some gap energy to superconductor, it generates **excited** phonons, which can be detected, they insist.

Of course, unreal particle "phonons" themselves **cannot** be detected, because they are unreal.

They just detect some electric currents of electrons.

In spite of this fact, why they try to **blindly** believe this **vague** existence of "phonon" ? This is strange.

*(Fig.4) Only electrons ( NOT including nuclei ! ) can attract each other ?? ← Impossible.*

They never try to clarify mysterious force of Pauli exclusion principle.

So all we can use is very **abstract** mathematical creation (= c^{†} ) and annihilation (= c ) **operators** to express some particles.

As you see in Fig.4, it is almost **impossible** to describe various **complicated** actual phenomena by using **only** these abstract **symbols**.

The important point is that potential energy (= V ) in BCS theory does **NOT** contain positive **nuclei**.

There are only **negative** electrons (= c^{†}, c ) in this Hamiltonian (= total energy ).

How can they attract each other **without** positive nuclei ? Impossible.

So this potential energy V is **unrealistic** and **cannot** be measured in the actual experiments.

As a result, this unrealistic assumption makes it **difficult** to develop the current theory into more complicated high temperature superconductivity, in which BCS theory breaks down.

*(Fig.5) Artificial mathematical definition = Quasiparticle. *

In fact, the phonon is **NOT** a natural concept at all.

It was introduced by using very **artificial** definitions.

As shown in Fig.5, besides this unreal phonon, BCS theory needs other quasiparticles (= Bogoliubov fermion ).

It is strange that the creation operator (= γ^{†} ) is the sum of creation (= c^{†} ) and annihilation (= c ) of electrons.

So this "quasiparticle" **doesn't** exist, which was introduced **only** for getting some values close to experimental ones.

*(Fig.6) ↓ Energy gap ? *

They insist the conventional BCS theory predicts various experimental values **except for** strongly-correlated and high temperature superconductors.

The most important concept which BCS can predict is **"energy gap" = Δ**.

But as shown in Fig.6, this energy gap is very **unnatural**.

Because it consists only of **two annihilation** operators, which is completely different from usual quantum field theory.

( Of course, there are **NO** clear reasons why they choose this form as energy gap. )

Again, this artificial form of Δ was introduced only for getting values close to experimental ones.

Furthermore, they use two different **contradictory** forms as this energy gap, as I explain later.

So BCS theory is completely **wrong**, not only physically but also **mathematically**.

*(Fig.7) Temperature dependence (= curve ) of energy gap shows BCS theory is right ?*

They insist BCS theory can predict correctly temperature **dependence** of energy gap, as shown in Fig.7.

Energy gap Δ(T) decreases as temperature (= T ) becomes higher.

And at critical temperature (= T_{c} ), this energy gap becomes **zero**.

This energy gap is specific to superconductor, so, energy gap remains **zero** in normal states above critical temperature, according to BCS theory.

As I explain later, this curve of Fig.7 is introduced "intentionally" ( **NOT** natural result of BCS theory ).

*(Fig.8) Pseudogap energy also in normal state → BCS theory breaks down.*

Unfortunately, they have found a lot of superconducntors in which convencional BCS theory **doesn't** apply.

One of them is **pseudogap** energy in normal state.

As I said in Fig.9, energy gap means some **barrier** between superconducting and normal states.

So in normal state, this energy gap must **NOT** exist, if the standard theory is right.

But in various materials, they found some energy gap exists also in normal conducting states.

Furthermore, in these strange superconductors, the relation between gap and critical temperature like Fig.7 is **NOT** satisfied. ( See red circles of Fig.8. )

So, BCS theory ( depending on **unreal** "phonon" ) has already **failed**.

*(Fig.9) Energy gap (= Δ ) varies in different directions.*

As shown in Fig.7, BCS theory insists energy gap depends on critical temperature universally in all materials.

But in various superconductors, this energy gap **changes** in **different** directions at the same temperature.

This is called "**anisotropic**" energy gap.

In these materials, there are some points where energy gap becomes **zero**, though it's not zero at different points.

This is called "node".

They try to introduce new concepts such as "d wave" or "p wave" superconductivity.

But this anisotropy clearly proved the conventional BCS theory already breaks down.

*(Fig.10) Various materials do NOT obey BCS theory.*

The most important and famous relation, which BCS theory predicts is the relation between energy gap (= Δ(0) ) at T = 0 and the critical temperature (= T_{c} ).

The ratio of "**3.52**" in Fig.10 is BCS's prediction.

But BCS theory relies on various **rough** approximations, as I explain later.

So these predictions by BCS are **not** so correct also in conventional superconductors.

As shown in various superconductors such as Pb, Hg, and Mgcl2 ..., these values becomes **bigger** than BCS's predictions.

*(Fig.11) The relation of specific heat does NOT satisfy BCS theory.*

The ratio of specific heats in normal (= C_{n} ) and superconducting (= C_{s} ) states becomes "**1.43**" according to BCS theory.

But also in this ratio, various superconductors give **different** values such as 2.37, 2.8, 0.80 ..

So BCS theory based on unreal "phonon" **doesn't** apply in various materials.

*(Eq.1) Intentional mistake in calculation → BCS theory is wrong.*

As I said, the concept of energy gap (= Δ ) is the most important value which BCS can predict.

So whether BCS theory is fake or not completely depends on the **definitions** of this energy gap.

In fact, they make a intentional **mistake** in calculation, when they determine the form of this energy gap.

They need to use **two** different forms of energy gap to get the ratio of Fig.10.

( Of course, these forms of Δ themselves are **NOT** natural ones. )

The transformation from the middle to lowest equations are **wrong**, as I say in Trick-1.

This is the **essence** of BCS's trick.

*(Trick-1) Trick = Deleting some terms unneeded for BCS theory.*

Like Fig.5, we create **virtual** quasiparticle (= γ ) from electrons (= c ).

As a result, energy gap Δ becomes Trick-1 middle.

Here they suddenly **delete** the last two terms (= 3 ). ← Trick !

As far as I see some websites and textbooks, there are **NO** reasonable reasons about this **artificial** manipulation.

( Most textbooks seem to **hide** this important trick ... )

I explain these things in the latter sections in detail.

*(Trick-2) Trick = Replacing only specific "c" terms by γ (= quasiparticle ).*

In Trick-2, they replace only "c" and "c^{†}" terms of **red** line with quasiparticle (= γ ).

On the other hand, "c^{†}" of the last term remains the original (= *blue* line = "b^{*}" ), and is **NOT** replaced by quasiparticle.

This new operator "b^{*}" is clearly introduced for **hiding** this artificial trick, I think.

( So they try to **ignore** ONLY the last term of Trick-2. This is strange. )

This means they intentionally **choose** some **convenient** terms for BCS theory.

**Artifiicial** choice of some terms is clearly one of important **tricks**, and shows BCS theory is **unnatural** and wrong.

*(Eq.2) *

First they define the energy gap at absolute zero ( T = 0 ), like Eq.2.

"**V**" is the constant value, which means the magnitude of **interaction energy** among electrons.

"u" and "v" are some real numbers.

L is the length of one side.

*(Eq.3) *

They define "u" and "v" like Eq.3 and Eq.4.

*(Eq.4) *

where

*(Eq.5) *

Of course, the definions of Eq.2 and Eq.3 are **artificial** things.

There are **NO** reason why the form of interacting energy becomes like Eq.2.

These are **only** for getting some values close to experimental ones.

In Eq.5, Δ is "energy gap", and ε_{k} is

*(Eq.6) *

ε_{k} means the **difference** between true energy and Fermi energy (= E_{F} ).

So when this ε_{k} is zero, it is equal to Fermi energy.

*(Fig.12) Fermi energy is very vague concept. *

Condensed matter physics often uses the concept of **Fermi energy** (= E_{F} ).

As shown on this page, the current quantum mechanics **cannot** express **concrete** electron's states.

They can **only** show mathematics such as matrices.

So they need to depend on some roughly **approximate** models.

Orbitals become occupied in order from the lowest energy levels.

Each orbit contains two electrons due to Pauli exclusion principle.

The highest energy level of outer orbital is called "Fermi level ( energy )".

So these electrons in Fermi level **cannot** enter inner orbitals, which are already filled with other electrons.

Unfortunately there are **NO** clear images in these concepts. Very **abstract** and **useless**.

*(Eq.7) *

Substituting Eq.4 into Eq.2, we get Eq.7. ( Eq.5 is used. )

Dividing both sides of Eq.7 by Δ, we get

*(Eq.8) *

Here we change discrete wave number "k" (= momentum, energy ) into continuum,

*(Eq.9) *

In Eq.9, N(0) means electron density, and ħω is the maximum energy of quasiparticles.

Using Eq.9, Eq.8 becomes

*(Eq.10) *

Here we use the formula of

*(Eq.11) *

where

*(Eq.12) *

Supposing N(0)V is very small, e^{-x} of Eq.12 becomes zero, so we get

*(Eq.13) *

where

*(Eq.14) *

As a result, we obtain the first important relation,

*(Eq.15) *

As I said, the relation of Eq.15 is that at absolute zero ( Δ = Δ(0) ).

And the **artificial** definitions of Eq.2 and Eq.3 lead to the final result of Eq.15.

*(Eq.16) *

They define BCS Hamiltonian like Eq.16.

" ↑ ↓ " mean spin direction of each electron.

Total momentum of a pair of two electrons is supposed to be zero.

So when one electron's momentum is "k", another is "-k", they insist.

"V" means transition (= interactive ) potential energy between momentums k and k' in Cooper pair.

Surprisingly, these are **NO** positive nuclei in this potential energy. (= **Only** electrons. )

But they insist, these negative electrons can **attract** each other. This is strange.

*(Eq.17) *

Here we define operator "b" like Eq.17.

"†" denotes a complex conjugate.

*(Eq.18) *

Using Eq.17 and Eq.18 in Eq.16, we get Hamiltonian of

*(Eq.19) *

In the process of getting Eq.19, they use some **trick**.

They can **arbitrarily** choose " c c^{†} " or " b b^{*} " as operators.

*(Eq.19') Trick ↓ Which do we choose " c c ^{†} " or " b b^{*} " ?? *

Here we define energy gap Δ like

*(Eq.20) *

Strange to say, this energy consists **only** of annihilation and annihilation operators (= c c ).

This means the gap energy always becomes **zero** ? If so, BCS prediction is wrong.

Using Eq.20 in Eq.19, we get

*(Eq.21) *

As I said in Trick-2 and Eq.19', the last term (= Δ_{k}b^{*} ) of Eq.21 also consists of "c^{†}" operators.

But they artificially **ignore** only this last term.

This is one of important **tricks** in BCS theory.

*(Eq.22) c = electron, γ = quasifermion ?*

In Eq.22, γ_{k0} and γ_{k1} denote virtual **quasifermions** "1" and "2", respectively.

We can also express the relation of Eq.22 like

*(Eq.23) *

If you substitute Eq.23 into Eq.22, you can find Eq.22 and Eq.23 are the **same** relations.

Here we suppose electrons (= "c" ) satisfy **anticommutative** relation,

*(Eq.24)*

If Eq.24 holds, quasiparticles of γ also satisfy

*(Eq.25)*

So they insist virtual quasiparticle γ is also "**fermion**".

But this quasifermion is just virtual and **unreal**, because

*(Eq.26) Annihilation of quasifermion from vacuum → NOT zero !?*

As shown in Eq.26, even when we apply some annihilation operator of quasiparticle to the vacuum, this vacuum is **NOT** zero.

( Because γ contains creation operator of c^{†}, too. )

So this phonon is completely **different** from ordinary fermions such as electrons.

This is the reason why they are called "**quasiparticles**".

Substituting Eq.22 into "c" and "c^{†}" in the first term of Eq.21, we get

*(Eq.27) *

In the same way, substituting Eq.22 into "c" and "c^{†}" in the remaining terms ( **except** for the last term "b" ! ← **trick** ) , and summing all terms, Hamiltonian of Eq,21 becomes

*(Eq.28) *

As I said in Trick-2, "c^{†}" operators are still **left** only inside the last term "b^{*}".

But they **don't** try to replace only these "c^{†}" inside the last b^{*} with γ using Eq.22.

This is clearly **unreasonable** calculation and trick.

They suppose Eq.29 is satisfied in Eq.28.

*(Eq.29) *

Furthermore, they **artificially** define

*(Eq.30) *

Unfortunately, the definition of Eq.30 is completely **wrong**.

Because Δ is **NOT** equal to Δ^{*}.

*(Eq.31) *

Surprisingly, they try to **ignore** the last two terms (= **red** line ) !

This is the most important trick (= Trick-1 ).

Even if you try to look for some reasonable reason in some websites and textbooks, you **wouldn't** find it.

Because, Eq.30 is completely wrong. ( see also appendix )

Using the definition of ( see appendix )

*(Eq.32) *

and

*(Eq.3) *

*(Eq.4) *

*(Eq.5) *

Hamiltonian of Eq.28 becomes ( using Eq.29 and Eq.30 )

*(Eq.33) *

So they insist the **red** line of Eq.33 means quasifermion's energy (= E ) and number operators.

And the remaining term is treated as "unneeded" constant.

But as you see in Eq.17, this "*b ^{*}*" term still contains "c

As a result, by using the replacement of Eq.22, you will find this "b^{*}" term becomes **another** quasifermion's energy, which **changes** the BCS results.

But they intentionally ignore **only** this term.

This is one of important **tricks** (= Trick 2 ), and shows the current BCS theory is wrong.

Energy gap Δ can be expressed as ( see Eq.20 )

*(Eq.34) *

Substituting Eq.22 into Eq.34, we have

*(Eq.35) *

Suprisingly, they suddenly and **secretly** erase only the last two terms (= "3" ) in Eq.35 !

(This important fact is hidden from most textbooks and websites. )

This manipulation is very **unreasonable**, of course, a mistake in calculation.

There is **NO** reason why we can erase only "3" terms in Eq.35.

( see also appendix )

And we insert the following Fermi's distribution into number operator (= γ^{†} γ ),

*(Eq.36) *

As a result,

*(Eq.37) *

where

Of course, this **virtual** fermion **doesn't** really exist. ( See Eq.26 )

So there is **NO** reality in BCS theory.

And we have to leave only terms except "3" for getting proper results.

First, this Δ itself consists only of **annihilation** operators (= "cc" ), as shown in the first equation of Eq.37.

So this energy is **NOT** conserved and vanishes into zero from the beginning. This is strange.

And it is very **artificial** trick (= Trick 1 ) to delete only "3" terms.

*(Eq.38) *

They determine interactive potential V as some constant (= -V ).

( So BCS theory is a very **rough** approximation. )

Inserting -V into Eq.37, and dividing both sides by Δ, we get Eq.38.

Here we change discrete momentum into continuum (= integral ) like

*(Eq.39) *

Using Eq.39 in Eq.38, we have

*(Eq.40) *

where

*(Eq.41) *

Eq.40 is the direct result of BCS theory.

The important point is that the original Eq.40 does **NOT** show the relation of

*(Eq.42) *

To be close to experimental result, energy gap Δ needs to be zero at the critical temperature (= T_{c} ).

They have to **artificially** apply new relation of Eq.42 in Eq.40, so partially integrating it,

*(Eq.43) *

In Eq.43, they treat ħω as infinity **only** in "1" and integral parts.

As a result, we get approximate result of Eq.43 at critical temperature (= T_{c}, Δ = 0 ).

As shown in Eq.40, the original result does **NOT** say anything about Δ value at T_{c}

*(Eq.44) *

So, Eq.43 is **NOT** a natural result.

Ee.43 becomes

*(Eq.45) *

Here we use the result at absolute zero of Eq.15

*(Eq.15) *

From Eq.45 and Eq.15, we have

*(Eq.46) Prediction.*

Eq.46 is the famous BCS prediction between energy gap ( Δ at T=0 ) and the critical temperature (= T_{c} ).

But as I said in Fig.10, there are many materials that do **NOT** obey Eq.46.

And to get Eq.46, they need to make intentional **mistakes** in calculation (= Trick 1, 2 ).

So the result of Eq.46 is **unreasonable**.

Here we use Boltzmann constant β and add two terms in right side of Eq.40

*(Eq.47) *

Here we use approximation of Eq.43 in left side and **only** the last term of right side.

*(Eq.48) *

The important point is that the temperature "T" inside β in the last term of Eq.47 ( Eq.48 ) is **NOT** critical temperature.

But they use the approximation (of critical temperature ) also in this last term.

This is also one of **tricks** to get the result in which Δ becomes zero at the critical temperature.

Eq.48 becomes

*(Eq.49) *

Again we use another approximation of

*(Eq.50) *

In Eq.49, we pick up only ( n=1 ) term, **supposing** Δ is very small.

Using also Eq.50, Eq.49 becomes

*(Eq.51) *

Dividing Eq.51 by Eq.46, we have

*(Eq.52) *

So, they insist BCS theory can predict curve like Fig.13.

But as you see several **approximations** of Eq.48, Eq.50 and Eq.51, they just intentionally try to obtain Fig.13 curve.

The direct (= original ) BCS calculating results do **NOT** show the curve of Fig.13 at all.

*(Eq.53) *

The critical magnetic field is the experimental value, we can most exactly measure.

According to these results, the relation between specific heat (= C_{s} ) and the critical temperature (= T_{c} ) becomes

*(Eq.54) *

Eq.54 is **different** from BCS prediction (= exponential ) of

*(Eq.55) *

The ratio of specific heat is gotten from the BCS theory of Eq.37.

But as I said, these calculations are just **mistakes**, which means BCS theory **cannot** predict anything.

*(Fig.14) The relation of specific heat doesn't satisfy BCS theory.*

Actually this relation of Fig.14 is **violated** in various superconductors.

*(Fig.15) Detecting infinitesimal tunnel current ?*

In fact, it is very difficult to detect energy gap Δ in superconductors.

They apply some voltage (= energy Δ ) or light to superconductor, exciting "phonons".

This excited phonons are detected as small tunnel currents, they insist.

( Though they call it "tunnel", each particle **cannot** penetrate barriers more than "nanometers", so it just indicates some de Broglie wave's effects. )

But of course, there is some detection **threshold** in amplifier and detector of this current.

If we change the setting of this amplifier, the critical voltage **changes**, too.

So this is very **similar** to a single photon (= "fiction" ) detection.

*(Fig.16) It is impossible to make tunnel current completely zero.*

Theoretically, the energy of phonon (= ε ) becomes completly zero at absolute zero ( T = 0 K, Fig.16 left ).

But it is **impossible** to get complete absolute zero temperature.

So small tunnel currents are **always** detected, even when NO voltage is applied (= Fig.16 right ).

This is also a cause of some **errors**.

Using the definitions of Aq.1 and Aq.2,

*(Ap.1) *

*(Ap.2) *

we get

*(Ap.3) *

As a result, we obtain the result of Eq.32.

*(Ap.4) *

We defined "quasifermion" like Eq.23 (= Ap.5 ),

*(Ap.5) *

Using Ap.5, we prove this quasifermion is completely different from real particles.

Because even if annihilation operator (= γ ) is applied to the vacuum (= |0> ), this vacuum does **NOT** become zero, as shown in Ap.6.

So they define new **unrealistic** quasiparticle's vacuum like

*(Ap.7) Quasiparticle (= BCS ) vacuum ??*

Using Ap.5 and Ap.7, we can prove

*(Ap.8) *

where we use

*(Ap.9) *

So we can say Ap.7 is quasiparticle's vacuum.

Of course, this vacuum is not real, because electron's annihilation operator (= c ) **cannot** make this vacuum zero.

*(Ap.10) *

By the way, what happen when the following "b" operators are put between this quasiparticle's vacuum ?

*(Ap.11) *

Using Ap.7, Ap11 and anticommutative relation, we get

*(Ap.12) *

In the same way,

*(Ap.13) *

This is the reason why they insist Δ = Δ^{*} in Eq.30, as follows,

*(Ap.14) *

But in fact, if Ap.14 is satisfied, we **cannot** get the BCS predictions of Fig.10 and Fig.11.

*(Ap.15) *

They try to delete **only** "3" terms in Ap.15, and leave "1" and "2" terms.

This is one of important tricks in BCS theory.

As you notice, this **artificial** choice is unreasonable and completely **wrong** in both usual and quasiparticle's vacuums.

If BCS vacuum contains one quasifermion, only "2" term is left ( "1" and "3" vanish ).

And if BCS vacuum contains two quasifermions, only "3" term is left.

So the assumption that "1" and "2" terms are left has **NO** ground at all.

2013/12/12 updated. Feel free to link to this site.