Phonon as carrier of electromagnetic interaction between

vibrating lattice and electron and its role in electron-pairing

Qiang LI

Jinheng Law Firm 1004, Quantum Plaza, 23, Zhichun Road, Beijing 100191, China

lq@jinheng-ip.com

AbstractWith emphasis on time-dependency of electron-lattice system, we suggest the

fallacy of presumed quantization in the context of electron-lattice system and propose the definition of phonons as carriers of electromagnetic interaction between electrons and vibrating lattice. We have investigated behaviors of electron-lattice system relating to “measured” energy, identified non-stationary steady state of electrons engaging in “electron pairing by virtual stimulated transitions”, recognized some origins of binding energy of electron pairs in crystals, and 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 also studied 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 ground state, and determination of presence/absence of superconductivity by such competition.

Keywords: phonon; binding energy of electron pair; superconductivity; Heisenberg Uncertainty Principle; threshold phonon; stability of electron pair; non-stationary steady state; anharmonic crystal interactions; multiple-pairing of electrons

PACS numbers: 74.20.Mn 74.25.F-

A key factor in some models of superconductivity is electron pairing, including Cooper Pairing [1], which is the basis of BCS theory [2], 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 [3] and R. K. Shankar et al [4]. Virtual particles were often considered as playing an indispensable role in physical processes [5] [6] [7] [8] [9] [10]; Bardeen and Pines also pointed out that “in the theory of superconductivity, one need only consider virtual transitions ……” [11]. In the context of phonon-mediated electron pairing, real phonon can hardly be available as temperature T→0, but superconductivity is expected to be favored at T→0, 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, did not specify why and how mediating phonon was emitted/absorbed by the lattice and/or electrons, why and how a mediating phonon interacted with electrons, and what the physical rule/law that dictates or governs these processes was. They also did not specify the details of the state(s) at which the two electrons under pairing stayed, such as whether the two

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electrons were in one same state and how the electrons could be so without violating Pauli Exclusion Principle. Additionally, they did not explain why, in so far that an energy gap was formed at the Fermi Level (EF) under superconducting state, electron levels were “removed” from the gap, in which these levels existed in normal state.

Second quantization was often employed in developing Hamiltonians of electron-lattice systems concerned, as by Bardeen and Pines [11], and Kumar et al [4]. However, such application of second quantization seemed 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 was often believed to be the mediator of electron pairing relating to superconductivity, it had a somewhat awkward status in physics in that although it was 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” [12], what kind of interaction (electromagnetic, gravitational, or etc.) phonons carried seemed not having been well-addressed. If phonon is the carrier of electromagnetic interaction, are phonons essentially the same as photons? Typically, phonons were introduced on the basis of crystal vibrations [13]. But such an introduction could be premature, for while it was true that each oscillator generally has its energy quanta, neither the wavevector characteristics of phonons nor their identify as carriers of electromagnetic interaction could 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, both phonons’ wavevector characteristics and their 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.

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 the pairing. Electron pairing, as well as some mechanisms of generation of its binding energy, 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 EF, to identify some candidate mechanisms leading to high-temperature superconductivity (HTS) and/or lower temperatures superconductivity (LTS).

A non-quantized time-dependent Hamiltonian term of electron-lattice interaction was presented by Huang as [14]:

ΔH=ΣδVn=-(A/2)exp(-2πiνt)Σexp(2πiq•Rn)e•▽V(r-Rn)-(A/2)exp(2πiνt)Σexp(-2πiq•Rn)e•▽V(r-Rn) (1)

where Rn 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, ν is the frequency of the lattice wave, and q denotes the wavevector of the lattice wave under elastic wave approximation, and the summation

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is over all the lattice points (n).The Hamiltonian of Formula (1), which corresponds to the well-known

“periodic perturbation” [15], describes the context of crystals in that the lattice terms of Σexp(±2πiq•Rn)e•▽V(r-Rn) are included in (1), which result in the wavevector selection rule for transition from k1 to k2 [14] [16]:

k1-k2±q=-Kn (2)where Kn is vector in the reciprocal lattice.

The matrix element to the first approximation isank=δnk+ank1 (3)

ank1 =2π/(ih) ∫ΔH exp (i(En-Ek)t1/h)dt1 (4)When the lattice wave (hν) is stable, its magnitude (A) remains time-

independent soa nk1 ~(Fnk/h){exp[2πi(Enk+hν)t/h]-1}/ ( Enk +hν)-(F+

nk/h){exp[2πit(Enk-hν)t/h]-1}/ (Enk-hν) (5)with F=-(A/2)Σexp(2πiq•Rn)e•▽V(r-Rn)

F+=-(A/2)Σexp(-2πiq•Rn)e•▽V(r-Rn)Fnk=<φn| F |φk> andF+

nk=<φn| F+|φk>Regarding energy relation, however, the transitions do not necessarily

concentrate at Enk=±hν, as described by “Golden Rule” [17]. Remarkably, the transition matrix {ank} becomes time-independent at t→∞ because time t can be taken out of 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 author would request special attention to the time-dependency of the magnitude term (A) of its Hamiltonian, which was ignored in most of the existing art but which 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＋1/2)1/2, with m=0,1,2…being the number of phonons of the lattice wave mode. If the number m of phonon fluctuates, the lattice wave is no longer stable and Formula (5) is 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 nk1 ~ΣCj {Fnkj/h{exp[2πi(Enk+hν)(tj-tj-1)/h]-1}/ (Enk+hν)- F+

nkj/h{exp[2πit(Enk-hν)(tj-tj-1)/h]-1}/ (Enk-hν) } (6)where the summation is over index j; in each time segment (tj-1,tj) the number m of phonon of the lattice wave mode remains unchanged, but the number m assumes different values in different time segments (tj-1,tj), and Cj will denote a random complex number. Then, Formula (6) becomes a summation of a series of random complex numbers, so the matrix element ank including such a summation not only cannot normalize off other matrix elements but also tends to go to zero statistically. In conclusion, when the phonon number m of the lattice wave fluctuates, transition of the electron in the system can hardly converge to Enk=±hν.

(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, 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 Enk=±hν could not be established.)

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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 Enk=±hν at sufficiently low temperature (T→0) is allowable according to Formula (5).

Virtual transitions should be considered in examining such a system with respect to 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”. 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 hν=E2-E1) to an electron at a lower energy state of E1 for its transition from the lower energy state of E1 to a non-occupied higher energy state of E2, where the lower energy state of E1 and the higher energy state of E2

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 E2→E1. 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.

We now consider the situation in which the excited state E2 originally has been occupied by an electron too (that is, there is E2 <EF). In such a system with E2 <EF, according to Formula (5), as time t gets greater, in the presence of a lattice wave mode of frequency v with hν=E2-E1, the electron originally at level E1 has to transits to level E2 while the electron originally at level E2 has to transits to level E1, 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 E2 emitting a threshold phonon of hν=E2-E1, which would be absorbed by the electron originally at level E1 for its transition to level E2. 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 E1<E2<EF to enter into an NSS state than the sub-system with non-occupied E2 (>EF>E1), 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 its transition of E1→E2 and then return the borrowed virtual threshold phonon to the lattice wave in its subsequent transition of E2→E1; if the redundant threshold phonon somehow escapes, the electron pair will have an binding energy, which is typically comparable to 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

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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 [18], while in the electron-lattice system as described by the Hamiltonian of Formula (1), quantization of energy as

E2(k2)-E1(k1)=±hν (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 a deduced conclusion, then it would be a potentially fatal logical fallacy to take this element as a precondition. Contrarily, quantization should be introduced 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, which is 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 hν 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 of electron-lattice system, they are associated with non-stationary process in crystal.

An optical lattice wave can interact with incident electromagnetic wave of the same wavevector and frequency [19] [20].

“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 E2 is greater than EF and is not occupied at time t=0, that the system is isolated, and that the lattice wave (hν) is in its ground state, then after time t1, the energy of the electron originally at the level E1(<EF) can only be “measured” as having energy E1, as in conformity with the requirement of energy conservation. But this does not mean that the electron keeps staying at the eigenstate of E1 all the time, rather it just indicates that “measurement” can only “collapse” the electron to the eigenstate of E1. Conversely, according to Formula (5), the electron shall virtually transit between the eigenstates of E1 and E2 during the time period [0, t1]; in other words, the electron is in an NSS state that incorporating both energy eigenstates of E1 and E2.

With 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 (hν) is at its ground state and the system concerned is at sufficiently low temperature T so that kT<<hν; only then can its number of phonons be reliably kept constant (zero). On the other hand, a good approximation of stable lattice wave seems to be the “large quantum number limit”, where real phonons would 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 and is therefore not relevant to the discussion here.

For each state (E, k), its pairing candidate could be determined as the intersections of laminated plot of hν-q dispersion curves [21] of lattice waves and the

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plot of E-k bands of the crystal concerned, with the origin of the hν-q plot being placed at the (E, k) point (for determining pairing candidates for the excited state in the pairing, the hν-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 schematically 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 E2 at or below EF is occupied, the electron originally at the upper level E2 may emit a threshold phonon with energy hν=E2-E1, which can be absorbed by the electron originally at a matched lower level state E1, so the electrons at E1 and E2 can enter NSS state. Since a virtual phonon of energy hν=E2-E1 can be “borrowed” from the lattice wave by the electron at the ground level E1 for its transition to the excited level E2, 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 tend to destroy the dominance of matrix elements a 12

and a21 over other matrix element components, thus the NSS state tend to 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 E2. 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 it, when becomes redundant, 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 remain 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 (E1) 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 [22]. Denoting the separation of two adjacent energy levels by δE, then after time tt there could be

tt=2h/δE (11)

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that is, the resolution of the energy selection of transition will become high enough to resolve each of the energy levels. For crystals, δE is estimated as δE=Δ/(N0)1/3, where Δ is the width of conduction band and N0 is Avogadro constant. Taking Δ~1eV, we would get δE~10-8eV and tt~10-6s.

An upper limit of t<< h/δE is proposed in the art in order to “character this set of states by a density of states”. [22] We would argue, however, that this limit is too stringent, for the convergence of transitions to the frequency of the sine periodic perturbation needs not to stop at where the mathematical characterization of Golden Rule becomes invalid. It is also to be argued that in discussion concerning “Golden Rule”, the energy spread (ΔE) based on Uncertainty Principle should not be considered separately; this is because the energy spread (ΔE) 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.

On the other hand, the convergence of transitions to the frequency of the periodic perturbation is by competition among the target states, where the states closer to the frequency suppress all other states. As the ultimate state will not compete with itself, the competition is expected to stop when the range of the winning states narrows to the extent that it contains only the ultimate state. Thus, the upper time limit of the convergence of the function of transition probability, as a characterization of this Uncertainty Principle relationship, should be the time when only one state is left as a result of the competition, which could be estimated to be somewhere close to tt=2h/δE.

Electron-phonon interaction rates estimated to be as high as 1012s-1 at not too low temperatures and 5×1010s-1 at absolute zero are proposed in the art [16]. 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.

We now discuss the fate of a redundant non-optical threshold phonon. Lattice wave modes may couple with one another by anharmonic crystal interactions [23], 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 (Tc), 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

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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 E1 and an energy state E3 (>E1) does not affect stability of pairing between states E1 and E2. We now explain this. Assuming that a third energy state E3

is present in the system, with E1<E3<E2, and E3 has unstable pairing with E1 while E2

has stable pairing with E1. Then, the matrix elements A13 and A31 will oscillate between 0 and a non-zero value, and A12 and A21 will also oscillate, but this will not affect the NSS states of the electrons on levels E1 and E2; the system has three electrons and two threshold phonons, and once the threshold phonon between energy levels E1 and E2 escapes, then at sufficiently low temperature new threshold phonon will rarely enter the system, so the two electrons originally at E1 and E2 could only stay at NSS states and collapse to ground state E1 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 (E1) 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 (E1); conversely, the electrons are “staying” on a plurality of stationary states including the original excited state (E2). 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 (E1) when their pairs get stabilized.

The effect of an additional pairing between E1 and an energy level E4 (<E1) varies. Insofar as uncertainty energy spread ΔE 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 E1 and E2 (>E1) and between E1 and E4, in view of the limitation of Pauli Exclusion Principle. This can be explained that if the electron at E1 and in NSS state also pairs up with the electron at E4<E1, once it “condensate” to the state of E4 the threshold phonon corresponding to hν=E2-E1 might no longer be able to associate it with the eigenstate of E2. Thus, the two eigenstates of E4 and E1 would be co-occupied by the two electrons originally at the states of E 4 and E1, but the eigenstate of E1 must also be co-occupied by the electron originally at E2 if the latter electron is to be kept in NSS state, resulting in that the “degree of occupancy” of eigenstate of E1 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 (E1) have to compete with all its “lower neighbors” (the candidate pairings with eigenstate E1 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 E1) 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

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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 EF cannot get any win in each of their candidate pairings, the crystal concerned will never have a superconducting phase.

With the discussion above, in low temperature limit, 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 [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 EF in HTS samples, which to date include cuprates, iron-arsenic compounds and possibly MgB2. 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 [24] shows possible interband features near EF. For MgB2, in a report by H. Uchiyama et al [25] vertical interband separation close to 10-30 meV (with consideration of errors) are seen near EF.

For cuprates, interband separation near EF are not seen in many experimental reports; in a recent report by H. Anzai et al [26], however, “nodal bilayer splitting” is shown appearing in energy-momentum plots of ARPES spectra along the nodal direction of Bi2Sr2CaCu2O8+ 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

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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.

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[12] “Phonon”, http://en.wikipedia.org/wiki/Phonon [13] Kittel Charles Introduction To Solid State Physics 8Th Edition, Chapter 4.[14] “Solid State Physics”, by Prof. HUANG Kun, published (in Chinese) by People’s

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[15]Franz Schwabl, Quantum Mechanics, page 297, Fourth Edition, Springer Berlin Heidelberg, New York.

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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”.

[21] Figs, 7, 8(a), 8(b), and 11 of Chapter 4 of [13].[22] Pages 296 of [15]. [23] Pages 119-120 of [13]

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[24] H. Ding et al 2008 EPL 83 47001, doi: 10.1209/0295-5075/83/47001.[25] H. Uchiyama et al, Phys. Rev. Lett. 88, 157002 (2002)[26] H. Anzai et al, arXiv:1004.3961v1 [cond-mat.supr-con]

11

Figure 1 An exemplary scenario of the phonon-mediated electron pairing by stimulated transition

Band E1(k1)

Pairing with E2(k2)-E1(k1)<hνM

Band E2(k2)

k

Band gap

Energy Gap

Peak

Pairing with E2(k2)-E1(k1)~hνM

E= E(k)

12

EF

Figure 2 Another exemplary scenario of the phonon-mediated electron pairing by stimulated transition

Pairing

E= E(k)

k

Peak

Gap

EF width

E1(k1) E2(k2)

13