LI, Qiang (501 Flat 15, No.37 Xueyuan Road, Beijing 3, 100083, CN)
| Claims 1. A method of manufacturing superconducting material, comprising: forming a band system in said superconducting material, said band system having an upper band (E2(k)) and a lower band (Ei(k)) below said upper band, said upper band having an energy level on Fermi level (EF); allowing said energy level on Fermi level (EF) of said upper band to have a vertical separation of hvM to said lower band (Ei(k)), wherein VM is frequency of optical phonon in said superconducting material; allowing the tangent lines of said upper band (E2(k)) and said lower band (Ei(k)) at and/or near Fermi level to have a non-zero angle. 2. The method of claim 1, wherein VM is the frequency of longitudinal mode (LO) optical phonon in said superconducting material. 3. The method of Claim 2, wherein the vertical separation between said upper band and said lower band is made to increase with increasing energy. 4. The method of Claim 2, wherein the vertical separation between said upper band and said lower band is made to decrease with increasing energy. 5. The superconducting material prepared with the method as defined by anyone of claims 1-4. |
METHOD THEREOF
Field of the Invention
The present invention relates to superconducting material and its manufacturing method according to electron-pairing based on definition of phonon as carrier of electromagnetic interaction in electron-lattice system feature by unquantized Hamiltonian.
Background of the invention
A key factor in some models of superconductivity is electron pairing, including Cooper Pairing ( Leon N. Cooper, "Bound Electron Pairs in a Degenerate Fermi Gas," Phys. Rev. 104. 1189 (1956)1 which is the basis of BCS theory (J. Bardeen, L. N. Cooper, and J. R. Schrieffer, "Microscopic Theory of Superconductivity," Phys. Rev. 106, 162 (1957)), and a model of "interband electron pairing induced by the virtual exchange of quanta of two boson fields (photons and phonons)" proposed by Kumar et al ("Possibility of Photoinduced Superconductivity", N. Kumar and K. P. Sinha, Phys. Rev. 174, 482 (1968)) and R. K. Shankar et al ("Photon-Induced Electron Pairing", R. K. Shankar and K. P. Sinha, Phys. Rev. B 7, 4291 (1973)).
Virtual particles are often considered as playing an indispensable role in physical processes (Woo-Joong Kim, James Hayden Brownell, and Roberto Onofrio, Phys. Rev. Lett. 96, 200402 and A. Feigel, Phvs. Rev. Lett. 92. 0204041 Bardeen and Pines also pointed out that "in the theory of superconductivity, one need only consider virtual transitions " (Phys. Rev. 99, 1140 (1955): Bardeen and Pines,
"Electron-Phonon Interaction in Metals"). In the context of phonon-mediated electron pairing, real phonon can hardly be available as temperature T→ 0, where superconductivity is expected to be favored, so such electron pairing could only be understood as being mediated by virtual phonon.
However, many models of superconductivity based on phonon-mediated electron pairing, including BCS theory, do not specify why and how mediating phonon is emitted/absorbed by the lattice and/or electrons, why and how a mediating phonon interacts with electrons, and what the physical rule/law that dictates or governs these processes is. They also do not specify the details of the state(s) at which the two electrons under pairing stay, such as whether the two electrons are in one same state and how the electrons can be so without violating Pauli Exclusion Principle. Additionally, they do not explain why, in so far that an energy gap is formed at the Fermi Level (E F ) under superconducting state, electron levels are "removed" from the gap, in which these levels exist in normal state.
Second quantization is often employed in developing Hamiltonians of electron-lattice systems concerned (as by Bardeen and Pines , and Kumar et al ). However, such application of second quantization seems to have concealed or removed the time-dependency of the original Hamiltonians and of the electron-lattice systems under consideration. As is to be discussed in this paper, some mechanisms of electron pairing are associated with and/or based on time-dependency behaviors of electron-lattice system, which behaviors could hardly be seen in analyses starting with a fully quantized perturbation item(s) of Hamiltonians.
Moreover, use of second quantization in treatments of electron-lattice interactions might also result in premature introduction of phonon. While phonon is often believed to be the mediator of electron pairing relating to superconductivity, it has a somewhat awkward status in physics in that although it is typically defined as "a quasiparticle characterized by the quantization of lattice vibrations of periodic, elastic crystal structures of solids" and "a quantum mechanical description of a special type of vibrational motion" (see "Phonon", http://en.wikipedia.org/wiki/Phonon). what kind of interaction (electromagnetic, gravitational, or etc.) phonons carry seems not having been well-addressed so far. If phonon is the carrier of electromagnetic interaction, are phonons essentially the same as photons? Typically, phonons are introduced on the basis of crystal vibrations (Kittel Charles Introduction To Solid State Physics 8Th Edition, Chapter 4). But such an introduction could be premature, for while it is true that each oscillator generally has its energy quanta, neither the wavevector characteristics of phonons nor their identify as carriers of electromagnetic interaction can be derived from phonons' definition as quanta of their oscillators. On the other hand, if we delay the introduction of phonons as quanta of crystal vibrations after establishment of electron-lattice interaction relationship (as exemplified in Appendix A), both phonons' wavevector characteristics and identity of quantum carriers of electromagnetic interaction between lattice waves and electrons would be naturally derived as results of such establishment, as is to be discussed later in this paper.
Professor HUANG Kun provides an expression of unquantized Hamiltonian ("Solid State Physics", by Prof. HUANG Kun, published (in Chinese) by People's Education Publication House, with a Unified Book Number of 13012.0220, a publication date of June 1966, and a date of first print of January 1979, page 201-205). Professor HUANG describes:
"The first approximation of potential variation 5V n of the atom at the nth lattice point R n caused by a lattice wave is
6V„^„-VV(r-R n ) (7-86)
where V(r) is the potential of one atom, and
represents displacement of the atom by the lattice wave , with e being the unit vector in the wave direction, A being the magnitude of the lattice wave , v being the frequency of the lattice wave , and q being the wavevector of the lattice wave under elastic wave approximation. Then, the potential variation of the entire lattice is
AH=∑ V n =-(A/2)exp(-2mvt)∑exp(2mq'R n )e'VV(r-R n )
-(^/2)exp(2 ivt)∑exp(-2 i^'R„)e » V V(r-R n ) (7-88) where the summation is over all the lattice points (n).
AH can be treated as a perturbation. With Formula (7-88), transition from kj to k 2 has the energy relation of
The normalized wave function can be written as
Ψ*( Γ )=1/(Ν) 1 2 ε χ ρ(-2πι/^) Γ )
where N is the number of primitive cells in the limited crystal concerned. The matrix element can be written as
(A/2)(e-r /**>) [( 1 /N)∑exp { 2 i(k l -k 2 ±q)'Rn} ] (7-91)
where the summation is over all the lattice points (n), and ^^ζ -1π Μ ι )-ξ}μ\{ξ)μ ΐ! (ξ)^ν(ξ)άξ (7-92)
Of special importance is the summation in the matrix element:
( 1 /N)∑exp { 2πί(Λι-Α: 2 ±¾τ)· ? η }
which yields
and zero otherwise. "
Summary of the Invention
In this paper, without presumption of electron pairing beforehand, nonstationary behaviors of electron-lattice system is to be theoretically examined to show some candidate associations of such behaviors with electron pairing and binding energy of electrons engaging in pairing. Electron pairing, as well as some mechanisms of generation of binding energy associated with it, is to be proposed as results of application of nonstationary perturbation to the electron-lattice system. Further examination is to be made on the behaviors of such electron pairing in the context of different band topologies at or near Fermi level E F , to identify some candidate mechanisms leading to high-temperature superconductivity (HTS) and/or lower temperatures superconductivity (LTS).
In one aspect of the present invention, a method of manufacturing superconducting material is provided, comprising:
forming a band system in said superconducting material, said band system having an upper band and a lower band below said upper band, said upper band having an energy level on Fermi level;
allowing said energy level on Fermi level of said upper band to have a vertical separation of hvM to said lower band, wherein VM is frequency of optical phonon in said superconducting material;
allowing the tangent lines of said upper band and said lower band at and/or near Fermi level to have a non-zero angle.
In one further aspect of the present invention, a superconducting material prepared with the method as defined above is provided.
Brief Description of the Drawings
Fig. 1 shows an exemplary scenario of the phonon-mediated electron pairing by stimulated transition.
Fig. 2 shows another exemplary scenario of the phonon-mediated electron pairing by stimulated transition.
Fig. 3 schematically shows a scenario of pairing with virtual transition to states higher than Fermi level based on the electron pairing model proposed in the present invention.
Fig. 4 shows an embodiment of the present invention, where the bands have different topology relationship than that of Fig. 3.
Fig. 5 shows another embodiment of the present invention, where the bands have different topology relationship than that of Fig. 3.
Detailed Description of the Invention
As expressed in the above Formular (7-88), an unquantized time-dependent Hamiltonian of electron-lattice interaction presented by Huang is :
AH=∑5V n =-(A/2)exp(-2 ivt)∑exp(2 iq'R n )e » VV(r-R n )
-(A/2)exp(2ravt)∑exp(-2raq » R„)e'VV(r-R n ) (1) where R n denotes the position of the atom at the nth lattice point, V(r) is the potential of one atom, e is the unit vector in the wave direction, A is the magnitude of the lattice wave concerned, v is the frequency of the lattice wave, and q denotes the wavevector of the lattice wave under elastic wave approximation, and the summation is over all the lattice points (n).
The Hamiltonian of Formula (1), which corresponds to the well-known
"periodic perturbation" (Franz Schwabl, Quantum Mechanics, page 297, Fourth Edition, Springer Berlin Heidelberg, New York), describes the context of crystals in that the lattice terms of∑exp(±2 iq » R n )e » VV(r-R n ) are included in (1), which result in the wavevector selection rule for transition from ki to k 2 (Appendix J of Kittel):
k ! -k 2 ±q=-K n (2)
The matrix element to the first approximation is
When the lattice wave (hv) is stable, its magnitude (A) remains time-independent so
~(F nk /h){exp[2ra(E nk +hv)t/h]-l }/ ( E„k +hv)
-(F + nk /h){exp[2rat(E nk -hv)t/h]-l }/ (Enk-hv) (5) with F=-(A/2)∑exp(2raq'R n )e » V V(r-R n )
F + =-(A/2)∑exp(-2raq'R n )e » VV(r-R n )
Regardin relation, however, the transitions do not necessarily concentrate at as described by "Golden Rule" (Pages 296-297 of Franz). Remarkably, the transition matrix { ank} becomes time-independent at t→∞ because time t can be taken away from the matrix; thus, the system becomes steady but is not at any eigenstate of energy.
While such periodic perturbation (as an extension of "Golden Rule") is well-known in quantum mechanics, the present inventor would request special attention to the time-dependency of the magnitude term (A) of its Hamiltonian, which has been ignored in most of the existing art but is a critically important factor to stability of electron pairs in crystals, as is to be explained below.
Insomuch that first quantization is presumed, the magnitude (A) of the lattice wave is proportional to (m+ l/2) 1 2 , with m=0, 1,2...being the number of phonons of the lattice wave. If the number m of phonon fluctuates, the lattice wave is no longer stable and Formula (5) no longer valid; conversely, A in Formula (1) becomes a step function of time t, the integral of Formula (4) becomes segmented, and its result is no longer a single term uniform over the entire range of integration (0,t) but becomes a summation like
a„ki ~∑C j {F nkj /h{exp[2ra(E nk +hv)(t j -t j .i)/h]-l }/ (E nk +hv)
- F + nkj /h{exp[2rat(E nk -hv)(t J -t J- i)/h]-l }/ (E^-hv) } (6) where the summation is over index j; in each time segment (t j -i,t j ) the number m of phonon of the lattice wave remains unchanged, but the number m assumes different values in different time segments (t j -i,t j ), and C j will denote a random complex number. Then, Formula (6) becomes a summation of a series of random complex numbers, so the matrix element a„ k including such a summation not only cannot normalize off other matrix elements but also goes to zero statistically. In conclusion, when the phonon number m of the lattice wave fluctuates, transition of the electron in the system cannot converges to E nk =±hv.
(The above discussion has a seemingly logical defect, that as we are to postpone first and/or second quantization, discussion here with phonons seems a vicious circle. However, this is not actually so, as we actually do not need to make any quantization at this point but only need to emphases here the precondition that the magnitude of the lattice wave has to be kept constant or convergence to E nk =±hv could not be established.)
Let us now consider the energy relation of the electron-lattice system as indicated by Formula (1). Some interpretation says that at the limit of t→∞ Formula (5) indicates that the electron is exchanging phonon (boson) with the lattice wave or the outside. But such an interpretation is problematic. First, Formula (5) does not indicate the requirement of real phonon emission/absorption. Second, at finite time t transitions other than E nk =±hv at sufficiently low temperature (T→0) is allowable according to Formula (5).
The present inventor would propose that virtual transitions be considered in examining such a system, with respect to: 1) the Heisenberg uncertainty principle, and 2) the significance of "measurement". "Measurement" is usually interpreted as an intervention to the system to be measured, which makes the system "collapse" to an eigenstate of which the eigenvalue is the result of the "measurement". Apparently, virtual transitions seem not in conformity with the requirement of energy conservation. But while all energy terms we observe are "observable", some "non-observable" energy terms can get involved in a virtual transition, and the relationship of energy conservation shall cover both "observable" and "non-observable" terms of energy involved in the virtual transition concerned. For example, the lattice wave at its ground state (with m=0) still might "lend" a threshold phonon (of energy hv=E 2 -Ei) to an electron at a lower energy state of Ei for its transition from the lower energy state of Ei to a non-occupied higher energy state of E 2 , where the lower energy state of Ei and the higher energy state of E 2 has a wavevecter match as indicated by Formular (2), and the electron then would return the threshold phonon to the lattice wave in the subsequent transition of E 2 →E 1 . As such a "lending/returning" process is transient, the phonon/energy exchanges could be "non-observable" because the system cannot collapse to a state in which the lattice wave has a negative number of phonon like m= -1.
On the other hand, according to Heisenberg Uncertainty Principle, the energy of an electron in an electron-lattice system may have a spread ΔΕ, which is related to a lifetime Δΐ, by which the electron stays in a (stationary) energy state, as
ΔΕΔΐ^η/(4π) (7)
The energy spread ΔΕ is associated with the energy of the electron, not with an energy level of a stationary state, and that it is likely that the electron concerned is not "at" any stationary state at all.
Therefore, in the system as indicated by Formula (1), as long as the electron transits between the energy states of Ei and E 2 at a transition frequency of VE with
ν Ε ^4πν (8)
there will be
ΔΕ^2 (Ε 2 -Ει) (9)
which means the energy spread ΔΕ of an electron, which is "originally" at stationary state of Ei, is broad enough to cover the excited energy level E 2 , so the electron may transit to E 2 without actually absorbing any photon/phonon. In this paper, the term "non-stationary steady (NSS) state" is used to denote the stabilized non- stationary state, at which an electron has an energy spread (ΔΕ) covering at least two stationary energy levels (as Ei and E 2 discussed above) but corresponds to a time-independent transition matrix {a nk} -
Let us now consider the situation in which the excited state E 2 originally has been occupied by an electron too (that is, there is E 2 <E F ). In such a system with E 2 <E F , according to Formula (5), as time t gets greater, in the presence of a lattice wave of frequency v with hv=E 2 -Ei, the electron originally at level Ei has to transits to level E 2 while the electron originally at level E 2 has to transits to level Ei, thus, a process would happen, in which the two electrons exchange their states with each other, with the electron originally at the higher energy level E 2 emitting a threshold phonon of hv=E 2 -Ei, which would be absorbed by the electron originally at level Ei for its transition to level E 2 . The phonon emissions/absorptions are virtual in that the phonon does not result in any phonon exchange with the lattice wave which induces the transitions and in that the phonon is confined between the two electrons concerned. In this paper, such an exchange of states between two electrons by virtual phonon emission/absorption is referred to as "electron pairing by virtual stimulated transitions".
It should be easier for the two-electron sub-system with E 2 <E F to enter into an NSS state than the sub-system with non-occupied E 2 (>E F ), as the threshold phonon will balance off the energy deficit as apparent in the latter sub-system. Here we can see that electron pairing itself does not ensure a binding energy in the current model of electron pairing. However, once the two electrons are in NSS state, the threshold phonon becomes redundant as far as the NSS state is maintained, because, as discussed above, each of the two electrons could "borrow" a virtual threshold phonon from the lattice wave for the electron's transition of Ei→E 2 and then return the borrowed virtual threshold phonon to the lattice wave in the subsequent transition of E 2 →E 1 ; if the redundant threshold phonon somehow escapes, the electron pair will have an binding energy no smaller than the energy of the redundant threshold phonon. We are to explain shortly later that in the context of crystals the fate of such a threshold phonon is critical in originating the binding energy and in determining the stability of electron pairs in crystals.
Quantization of energy for stationary harmonic oscillator is seen in the solution by the standard method of analysis for the solution of the time-independent
Schrodinger equation (see Equation (3.2) of Franz), while in the electron-lattice system as described by the Hamiltonian of Formula (1), quantization of energy as
E 2 (k 2 )-E 1 (k 1 )=±hv (10)
is seen as the limit of a time process in the solution of time-dependent first
perturbation. Logically, if an element (such as quantization) is included in one way or another in the solution, then it would be a potentially fatal logical fallacy to treat this element (quantization) as a precondition. Contrarily, quantization should be introduce and interpreted on the basis of the results of stationary and/or time-dependent solutions of Schrodinger equation and/or its equivalents. Specifically, phonon should be defined on the basis of solutions (as expressed by Formulas (2) and (10)) of time-dependent first perturbation of electron-lattice system, featured by Hamiltonian like that of Formula (1), as the quantum carriers of electromagnetic interactions between vibrating ions of the lattice and electrons, with each of the carriers carrying an energy converging at hv (as time t gets sufficiently great) and a wavevector (q) endowed by its lattice wave. As such, since phonons relate to the time-dependent Hamiltonian term(s) of electron-lattice system, they are associated with
non- stationary process in crystals.
An optical lattice wave can interact with incident electromagnetic wave of the same wavevector and frequency (T. Pham and H. D. Drew, Phys. Rev. B 41, 11681-11684 (1990), "Infrared absorptivity of YBa2Cu307-x crystals"; and, A BHARATHI, Y HARIHARAN, JEMIMA BALASELVI and C S SUNDAR, Sadhana Vol. 28, Parts 1 & 2, February/ April 2003, pp. 263-272. © Printed in India, "Superconductivity in MgB2: Phonon modes and influence of carbon doping").
"Measurement" of electrons should involve not only intervention by interaction between incident photons and the electrons (as in ARPES or the like) but also electron-phonon interactions in all "real transitions", particularly the transitions in the process of electric resistance (for this reason, "real transitions" are also to be referred to as "measurement" below). If an energy process cannot be realized by human-performed "measurement", it also cannot be realized by the electron-phonon process in electric resistance mechanism. For the system as described above and as featured by Formula (1), assuming that E 2 is greater than E F and is not occupied at time t=0, that the system is isolated, and that the lattice wave (hv) is in its ground state, then after time ti, the energy of the electron originally at the level Ei(<E F ) can only be "measured" as having energy Ei, as in conformity with the requirement of energy conservation. But this does not mean that the electron keeps staying at the eigenstate of Ei all the time, rather it just indicates that "measurement" can only "collapse" the electron to the eigenstate of Ei. Conversely, according to Formula (5), the electron shall virtually transit between the eigenstates of Ei and E 2 during the time period [0, ti]; in other words, the electron is in an NSS state that incorporating both energy eigenstates of Ei and E 2 .
With (first) quantization of lattice waves, the condition "the lattice wave is stable" means "the number of phonons of the lattice wave remains unchanged". But this can hardly be ensured unless the lattice wave (hv) is at its ground state and the system concerned is at sufficiently low temperature T so that kT«hv; only then can its number of phonons be reliably kept constant (zero). The higher the frequency of the lattice wave is, and/or the lower the temperature is, the more likely and reliable that its phonon number is kept at constant zero. (A good approximation of stable lattice wave seems to be the "large quantum number limit", where real phonons seem dominant so processes relating to virtual phonons might tend to be negligible. But this corresponds to the high-temperature limit and does not relate to superconductivity; therefore we are not going to discuss it.)
For each state (E, k), its pairing candidate could be determined as the intersections of laminated plot of hv-q dispersion curves (see Figs, 7, 8(a), 8(b), and 11 of Chapter 4 of Kittel) of lattice waves and the plot of E-k bands of the crystal concerned, with the origin of the hv-q plot being placed at the (E, k) point (for determining pairing candidates for the excited state in the pairing, the hv-q plot should be placed upside-down.) Obviously, each electron usually has more than one matches of phonon-mediated electron pairing. The collection of all these matches covers all possible (one phonon)-electron interactions of the subject electron at state (E, k). If all these phonon-mediated electron pairs can "normally" become superconducting carriers, HTS would be ubiquitous, which is definitely not in conformity with the rarity of HTS in reality.
Some exemplary scenarios of the phonon-mediated electron pairing by stimulated transition are shown in Figs. 1 and 2, where exemplary pairings are indicated by dotted or dashed lines with double arrows. As shown in Figs. 1 and 2, electron pairing typically occurs between slantingly located electron states due to the dispersion of the mediating phonon. But a few of the pairs are between nearly vertically separated electron states; these correspond to "optical phonon-mediated" (OPM) pairs, which are indicated by thick dashed lines in Figs. 1 and 2.
We now discuss the fate of the threshold phonon and its effect on binding energy. As mentioned above, when an upper level E 2 at or below E F is occupied, the electron originally at the upper level E 2 may emit a threshold phonon with energy hv=E 2 -Ei, which can be absorbed by the electron originally at a matched lower level state Ei, so the electrons at Ei and E 2 can enter NSS state. Since a virtual phonon of energy hv=E 2 -Ei can be "borrowed" from the lattice wave by the electron at the ground level Ei for its transition to the excited level E 2 , the original real threshold phonon may become redundant and can be absorbed by the lattice wave. But the real threshold phonon can hardly be emitted to the outside of the crystal, unless it is an optical phonon. As the phonon is absorbed by the lattice wave, due to the stimulation as indicated by Formula (5), the phonon is easily taken back by the electron at the lower level state for real transition to the matched higher state, and the cycle restarts as the system begins to re-establish NSS state. As explained above with reference to Formula (6), such frequent exchanges of the threshold phonon between the lattice wave and the pair of electrons destroy the dominance of matrix elements ai 2 and a 2 i over other matrix element components, thus the NSS state will collapse into the stationary energy eigenstate(s), at which the real threshold phonon has to be retrieved by the electron collapsing to the excited state of E 2 . As such, a phonon-mediated electron pair would not be stable and could hardly become superconducting carriers.
But an optical phonon-mediated (OPM) pair is different in that the redundant optical threshold phonon has a definite and substantial probability of escape by radiation, although it can also be absorbed with certain probability by the lattice wave. When the optical lattice wave is at ground state and the temperature is sufficiently low, once the threshold phonon escapes, the lattice wave would be kept constant/stable (until the incidence of an outside threshold phonon) and, most notably, each of the two electrons can only collapse to the ground state (Ei) so a binding energy occurs. The NSS state will not be affected by the escape of the threshold phonon and will be maintained until the lattice wave receives a new threshold phonon, whence the new threshold phonon will allow one of two electrons to collapse to the excited state (E 2 ). Therefore, the escape of the threshold phonon is self-consistent.
We now further examine the time process concerning periodic perturbation, which is an extension of the Golden Rule. For every finite t, the function of transition probability corresponding to Formula (5) has a width of 2h/t (see pages 296 of Franz). Denoting the separation of two adjacent energy levels by δΕ, then after time t t there could be
t t =2h/5E (11)
that is, the resolution of the energy selection of transition will become high enough to
1/3 resolve each of the energy levels. For crystals, δΕ is estimated as δΕ= Δ /(No) , where Δ is the width of conduction band and No is Avogadro constant. Taking Δ-leV, we would get 5E~10 "8 eV and t t ~10 "6 s.
An upper limit of the time t is proposed in the art (see pages 296 of Franz). We argue that this limit is not necessary. First, in discussion concerning "Golden Rule", the energy spread (ΔΕ) based on Uncertainty Principle should not be considered separately. This is because the energy spread (ΔΕ) of an electron-lattice system would come from the energy spread of the phonon provided by the lattice wave when no electron is created or annihilated, which spread should be what is characterized by the width (2h/t) of the function of transition probability. As explained with Formula (6), t (as in width 2h/t) is the time during which no real phonon exchange with the lattice wave happens, so t is a measure of the age of the virtual phonon that mediates the electron-lattice interaction. Thus, the narrowing of the function of transition probability, as a characteristic of this Uncertainty Principle relationship, should not be limited.
Moreover, electron-phonon interaction rates estimated to be as high as 10 12 s _1 at not too low temperatures and 5x l0 10 s _1 at absolute zero are proposed in the art (see Appendix J of Kittel). We would argue that some of these electron-phonon interactions should be those of "electron-pairing by virtual stimulated transition" discussed above; specifically, at absolute zero all the interactions are those of the electron pairing. Furthermore, the validity of these estimations of electron-phonon interaction rates is questionable, for they seem to be based on treatment of elements of transition matrix as measures of absolute transition rates, which is not justified. Also, the process leading to NSS state is not influenced by electron-phonon interactions of the electron(s) concerned, as is clear from the above discussion, because the process to enter NSS state is influenced only by variation of the magnitude of the lattice wave concerned.
Let us now consider the fate of a redundant non-optical threshold phonon. Lattice waves may couple with one another by anharmonic crystal interactions (see pages 119-120 of Kittel), by which a redundant threshold phonon may be taken away from its electron pair so that each of the two electrons in the pair can only collapse to the ground state. However, the probability with which the threshold phonon is taken away by anharmonic crystal interactions must have to compete with the probability of occurrence of thermal noise threshold phonon of the lattice wave. Thus, even if escape of the threshold phonon by anharmonic crystal interactions can win over occurrence of thermal noise threshold phonon, non-optical phonon-mediated (NOPM) electron pairs should be stabilized only at temperatures much lower than those for OPM pairs. This indicates that binding energy along may not decide superconducting temperature (T c ), as the stability of electron pairs is subject to the effects of a plurality of factors, including the strength of anharmonic crystal interactions, the presence/absence of optical phonon-mediated pairing matches, and so on. Of course, if escape of the threshold phonon by anharmonic crystal interactions could not win over occurrence of thermal noise threshold phonon, the crystal would not have a superconducting phase.
While each electron usually has more than one matches of phonon-mediated electron pairing as explained above so "multiple pairing" is common, additional pairing between Ei and an energy state E 3 (>Ei) does not affect stability of pairing between states Ei and E 2 . We now explain this. Assuming that a third energy state E 3 is present in the system, with Ei<E 3 <E 2 , and E 3 has unstable pairing with Ei while E 2 has stable pairing with Ei. Then, the matrix elements Ai 3 and A 3 i will oscillate between 0 and a non-zero value, and A 12 and A 21 will also oscillate, but this will not affect the NSS states of the electrons on levels Ei and E 2 ; the system has three electrons and two threshold phonons, and once the threshold phonon between energy levels Ei and E 2 escapes, then at sufficiently low temperature new threshold phonon will rarely enter the system, so the two electrons originally at Ei and E 2 could only stay at NSS states and collapse to ground state Ei upon "being measured", no matter what state the third electron is in.
That both electrons in a stabilized pair can only collapse to the ground state (Ei) upon "being measured" might be interpreted as that both electrons "condensate" to the ground state; such a condensation is in a non-stationary steady (NSS) state and is a "measured" state, and represents a "measurement" effect; it does not indicate that the electrons are co-staying on the stationary ground state (Ei); conversely, the electrons are "staying" on a plurality of stationary states including the original excited state (E 2 ). Moreover, insofar that the ground state may be the common lower state of a plurality of pairings as discussed above, all electrons in these pairings will "condensate" to the common ground state (Ei) when their pairs get stabilized.
The effect of an additional pairing between Ei and an energy level E 4 (<Ei) varies. Insofar as uncertainty energy spread ΔΕ of an electron-lattice wave system approximately corresponds to the energy of the (virtual) threshold phonon associated with it, stable parings cannot be realized between Ei and E 2 (>E1) and between Ei and E , in view of the limitation of Pauli Exclusion Principle. This can be explained that if the electron at Ei and in NSS state also pairs up with the electron at E <Ei, once it "condensate" to the state of E the threshold phonon corresponding to hv=E 2 -Ei might no longer be able to associate it with the eigenstate of E 2 . Thus, the two eigenstates of E and Ei would be co-occupied by the two electrons originally at the states of E and Ei, but the eigenstate of Ei must also be co-occupied by the electron originally at E 2 if the latter electron is to be kept in NSS state, resulting in that the "degree of occupancy" of eigenstate of Ei would exceed one, which is not in conformity with the requirement of Pauli Exclusion Principle.
In view of this, a candidate pairing having the ground state (Ei) have to compete with all its "lower neighbors" (the candidate pairings with eigenstate Ei being their excited state) in order to realized itself. As each candidate pairing can be characterized by its threshold phonon, whether the electron at a level (such as Ei) is "pairing upward" or "pairing downward" can be said to depend on the competition between its "upper threshold phonon(s)" and "lower threshold phonon(s)", with the rule that if one of the "upper threshold phonons" wins then all the "upper threshold phonons" win (and vise versa).
Obviously, the "upper threshold phonons win" outcome is pro-superconductivity. It seems that the threshold phonon with greater energy (binding energy) would have an edge, but magnitude of a matrix element depends on, among other things, degree of coupling between the two states concerned, and anharmonic crystal interactions may play an important role. The question may be that whether anyone of the upper threshold phonons can eventually dissolve itself into the lower threshold phonon and something else (as T→0). If it can, the outcome could be "upper threshold phonons win"; but if it cannot, the situation could be more complicated, and superconducting phase (if one exists) could possibly be unstable and/or uncertain. So no general answer to this question can be given, except that an optical threshold phonon of LO wave, which should correspond to HTS, would definitely win. On the other hand, if all electrons at or near E F cannot get any win in each of their candidate pairings, the crystal concerned will never have a superconducting phase.
With the discussion above, virtual transition could be understood as the "normal" or "general" form of transition while real transitions could be understood as "abnormal" or "special". In virtual transitions, a process of virtual borrowing-returning of phonon happens in a continuous way while the system is in a non- stationary steady state; when a real transition occurs, the "normal" process of lending/returning of virtual phonon is interrupted and the system is reset and "temporarily" collapses to an eigenstate; then virtual transitions take place again and the system begins to re-establish its non- stationary steady state. So an eigenstate corresponds to a transient process triggered by a real transition in a time-dependent system. In other words, like in a time-dependent system at low temperature, real transitions and associated collapses to eigenstates are occasional events happening on a continuous background of virtual transitions and non- stationary steady state. Virtual processes may have real physical consequences, especially when virtual processes are (partly or entirely) interrupted or destructed, as exemplified by Casimir effect, which is well explained as what happens because virtual photons with certain wavelengths cannot exist between two parallel metal plates arranged closely [7]. By contrast, virtual transitions of electrons in pairing produce the real consequence of establishing a binding energy of the electrons, without affecting the virtual transitions and NSS states of the electrons.
The exemplary pictures of electron pairing shown in Figs. 1 and 2 are interband pairings. Intraband pairing, in which electrons from two states in one band pair up by a mediating threshold phonon, is also supported in the present model of "electron pairing by virtual stimulated transitions", as explicit from the discussion above. On the other hand, the present model suggests the existence of interband structure at or near Fermi level E F in HTS samples, which to date include cuprates, iron-arsenic compounds and possibly MgB 2 . More specifically, if the traditional relationship of kTc~3.0-4.0 is true, then the interband structure should be expected to show a vertical separation of about 10-30 meV or so. For iron-arsenic compounds, angle-resolved photoelectron spectroscopy (ARPES) results by H. Ding et al (see H. Ding et al 2008 EPL 83 47001, doi: 10.1209/0295-5075/83/47001) shows possible interband features near E F . For MgB 2 , in a report by H. Uchiyama et al (see H. Uchiyama et al, Phys. Rev. Lett. 88, 157002 (2002)) vertical interband separation close to 10-30 meV (with consideration of errors) are seen near E F .
For cuprates, interband separation near E F are not seen in many experimental reports; in a recent report by H. Anzai et al (see H. Anzai et al, arXiv: 1004.3961vl [cond-mat.supr-con]), however, "nodal bilayer splitting" is shown appearing in energy-momentum plots of ARPES spectra along the nodal direction of Bi2Sr2CaCu208+ for UD66 (underdoped, Tc = 66 K), OP91 (optimally-doped, Tc = 91 K), and OD80 (overdoped, Tc = 80 K), with a vertical interband separation of about 20 meV for each of OP91 and OD80.
In summary, with emphasis on time-dependency of electron-lattice system, we have suggested the fallacy of presumed quantization in the context of electron-lattice systems, proposed the definition of phonons as carriers of electromagnetic interaction between electrons and lattice waves, and investigated behaviors (particularly those relating to "measured" energy) of electron-lattice system. Specifically, we have identified non- stationary steady state of electrons engaging in "electron pairing by virtual stimulated transitions", have recognized some origins of binding energy of electron pairs in crystals (for HTS and/or LTS), and have explained the state of electrons under pairing. Moreover, we have recognized the behavior and role of threshold phonon, which exists in electron pairing and is released by the electron from excited state, and have recognized the redundancy of the threshold phonon when the electrons under pairing have entered non- stationary steady state. We have discussed the effect of the stability of lattice wave on the evolution of the function of transition probability and on the stability of phonon-mediated electron pairs, the competition among multiple pairings associated with one same electron state, and determination of presence/absence of superconductivity of the crystal concerned by such competition.
Fig. 3 shows a scenario of the electron pairing model proposed in the present invention, which is consistent to the nodal bilayer splitting of Bi2Sr2CaCu208+ as mentioned above, where a pair of splitting upper band E 2 (k) and lower band Ei(k) run substantially parallel to each other so the vertical separation E 2 (k)-E 2 (k) remains substantially constant (at least near the Fermi level E F ). It seems that when the vertical separation E 2 (k)-E 2 (k) equals to the energy of longitudinal mode (LO) optical phonon, increase of binding energy as well as of superconducting critical temperature is favored. But a problem is that pairing involving virtual transitions would lower binding energy of electron pairs near Fermi level E F when the two bands extend parallel near E F . We will explain this.
As shown in Fig. 3, assuming that the state Al of the upper band E 2 (k) on the Fermi level has a relation of LO optical threshold phonon mediation with a state B 1 on the lower band Ei(k), then stable pairing is easy to be established between the two states. Principally, states on upper band E 2 (k) higher than Al could not pair up with their corresponding energy states on the lower band (such as B2, B3 and etc.), even if they have the favored pairing relation of LO optical threshold phonon mediation. But some states on the lower band Ei(k) on and below Fermi level E F (such as those shown as from B4 to B3 in Fig. 3) may transit to those states on the upper band E 2 (k) slightly higher than Al to realize pairing with corresponding states (such as B2) on the lower band, thereby "condensating" to these states on the lower band. Since the energy of state B2 is obviously lower than that of states B4 and B3, such a (virtual transition + pairing) process is favored, particularly in the presence of LO optical phonon mediated pairing. Thus, while originally an electron "condensating" from the state Al to the state B l could obtain a binding energy of hv M (where VM is the frequency of LO optical threshold phonon), now as states down to B3 become empty, the binding energy of the electron "condensating" to B l is reduced to hv M -(E F -E B 3), where E B3 is the energy of the B3 state. Approximately, the binding energy could be reduced to I M/2.
A approach to avoid such defect is to make the upper and lower bands E 2 (k) and Ei(k) unparallel near the Fermi level, that is, to make the angle between the tangent lines of the upper and lower bands at and/or near Fermi level non-zero, while the state (Al) of the upper band at the Fermi level is made to have LO optical phonon mediated pairing relationship with the lower band. As virtual transition + pairing is relatively difficult, it cannot be realized at the critical temperature of high-temperature superconductivity, which corresponds to LO optical phonon mediated pairing. The option of making the vertical separation between the upper and lower bands increase with increasing energy is preferred, that is, the states (such as A2 and A3) slightly higher than Fermi level have vertical separations greater than hv M with the lower band. As shown in Fig. 5, since no optical phonon medicated pairing is allowed for any state higher than Fermi level and the number of possible non-optical phonon mediated pairings is small, electrons on the states on the upper band at and near the Fermi level can enjoy a binding energy equal to or near hv M .
Of course, the option of making the vertical separation between the upper and lower bands E 2 (k) and Ei(k) decrease with the increase of energy is also feasible, as is shown in Fig. 4, where the states (such as A2 and A3) on the upper band slightly higher than Fermi level have vertical separation smaller than hv M to the lower band, while the state (Al) of the upper band at the Fermi level is made to have LO optical phonon mediated pairing relationship with the lower band. Such an arrangement also can destroy the optical phonon mediated pairing relationship between the states slightly higher then Fermi level and the lower band, thus avoiding decrease of binding energy of electrons near or at Fermi level at the critical temperature of HTS.
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