A Model of Electron Pairing, with Depletion of Mediating Phonons at

Fermi Surface in Cuprates

Qiang LI

Jinheng Law Firm, Beijing 100191, China

(revised and posted on Slideshare on 25 November 2010)

Abstract: We present a model of electron pairing based on nonstationary

interpretation of electron-lattice interaction. Electron-lattice system has an intrinsic

time dependent characteristic as featured by Golden Rule, by which electrons on

matched pairing states are tuned to lattice wave modes, with pairing competition

happening among multiple pairings associated with one electron state. The threshold

phonon of an electron pair having a good quality factor can become redundant and be

released from the pair to produce a binding energy. Lattice modes falling in a common

linewidth compete with one another, like modes competing in a lasing system. In

cuprates, due to near-parallel band splitting at and near Fermi Surface (EF), a great

number of electron pairs are tuned to a relatively small number of lattice wave modes,

leading to strong mode competition, transfer of real pairing-mediating phonons from

EF towards the “kink”, and depletion of these phonons at and near EF.

KEYWORDS: electron-lattice, Golden Rule, electron pairing, binding energy,

phonon transfer

1. Introduction

Electron pairing is a key factor in some models of superconductivity. In BCS

theory, the basic idea was discussed that electrons paired up to generate binding

energy.[1] After more than 20 years of the discovery of cuprates’ superconductivity, [2])

the mechanism of its electron pairing is still an open issue. In this paper, we propose a

model of electron pairing based on a nonstationary interpretation of the time

dependent behaviors of the electron-lattice system, and explain the origin of the

pairing’s binding energy. Discussions are to be made on pairing competition in the

proposed model and on the near-parallel band splitting features near Fermi Surface

1

(EF) in cuprates, to show that effective mode competition relating to the splitting

features leads to transfer of threshold phonons from EF towards the “kink” and

depletion of the phonons at and near EF.

2. Golden Rule Characteristic of Electron-Lattice Interaction

A non-quantized time-dependent Hamiltonian term of electron-lattice

interaction was presented by Huang:[3]

where Rn denotes position of the atom at the nth lattice point, V(r) is potential of one

atom, e is unit vector in the wave direction, A is magnitude of the lattice wave, ν is

frequency of the lattice wave, and q is wavevector of the lattice wave under elastic

wave approximation. The Hamiltonian term of (1) represents “periodic perturbation”.

[4] The lattice terms lead to wavevector selection rule:[3][5]

, (2)

where Kn is vector in the reciprocal lattice. The first order matrix element is:

, (3)

where . (4)

If magnitude A is constant and , then (4) will give:

, (5)

where , (5-1)

, (5-2)

, (5-3)

and . (5-4)

Formula (5) features the “Golden Rule” characteristic of one-phonon processes.[6] It can be seen from Formulas (1)-(5) that none of the wavevector relation of (2) and

2

the energy relation of depends on the specific form of the interaction term

. So phonon can be generated by all interactions with a vibrating lattice in

a general form similar to . For , transitions may happen only

between pairs of states (En, Ek) with , leading to a steady but nonstationary

(NSS) state of electrons on these states.

As a way for treating variation in magnitude A,[7] A was replaced by a switching

function f(t). As far as first quantization is presumed, A is proportional to ,

with being the phonon numbers of the lattice mode. When phonon

number fluctuates, A=f(t) can be represented by a step function, with f(t) being a

constant Aj in time segment . Then, for transition Enk~hν the

matrix element in Formula (5) becomes:

, (6)

where . According to (5), ank1 function has a width of at time t.[6]

Then, after time , where δE denotes separation of adjacent levels in crystal,

the width will become , meaning that at time tt the resolution of the energy

selection will be high enough to resolve all levels. With , there is

. If the phonon number remains constant from , then (6) will be

reduced to:

, (7)

which is a circle having a radius . Obviously, stands for

a quality factor, which goes to infinity at , where the circle of (7)

reduces to a straight line of . Formula (6) represents a curve formed by a

series of serially connected arcs in the complex plane. Such a curve typically has a

contour larger than the circle of zero phonon number.

We would argue that is a reasonable criterion for determining the

validity of Golden Rule. of (7) has side peaks at

.[8] When perturbation is turned on, the side peaks

3

advance at the “speed” 2πα toward the resonance frequency at

. As long as the displacement of ank1(t) function is a measure of the first

perturbation’s deviation from a true representation of the wavefunction, a reasonable

criterion for good approximation is that the displacement at the concerned energy be

much smaller than the width of one side peak, that is, , or

(8)

The condition of (8) is always satisfied for all in-resonance states with

(with a linewidth), so the general validity of Golden Rule for in-resonance states is

theoretically consistent. As applicability of Fermi’s golden rule for, such as,

describing adequately the electron-longitudinal-phonon interaction had been

experimentally established,[9] for in-resonance state pairs Golden Rule should be a

point of miracle where “vice is turned into a virtue,”[10] and the result of first time-

dependent perturbation can be valid for all finite time t. For an in-resonance state,

there is , so Formula (6) becomes

, (9)

which is a straight line; the “speed” of ank1’s extension along the line depends on

the current phonon number (Aj) of the mode. As to an off-resonance state (En), results

of (6) and (7) merely specify that its ank1 follow a basically circular locus. Fortunately,

off-resonance states (En) are basically not a concern in the explanation of electron

pairing of this paper. As long as a finite linewidth is associated with the lattice mode

(hν), the initial state (Ek), or/and the final state (En), all the final states (En,) that fall

within the linewidth are kept in-resonance in the Golden Rule process. Such a

linewidth of lattice mode shall exclude the contribution of the time-dependency of

electron-lattice interaction. Indeed, Heisenberg Uncertainty Principle should

not be a separate consideration in the discussion regarding the time-

dependency of phonon, for it is likely that “the energy kernel implements quantum

uncertainty”.[10] Back to Formula (5), as long as it characterizes one-boson interactions

with a vibrating lattice, the boson characterized by (5) has a time dependent

characteristic featured by Golden Rule. The lattice wave mode that stimulates

the one-boson interactions, however, does not have the same time dependent

characteristic as the bosons it generates.

4

3. Electron Pairing by Mutual Transitions

For energy states (En) in a crystal, considering the situation that the higher state

(E2) originally has been occupied at with , when t gets greater, the

lattice mode may drive the electron originally at a matching lower state E1

to transit to state E2 while the electron originally at state E2 has to transit to state E1.

Thus, a process may happen, in which the two electrons exchange their states; the

electron at the higher state E2 will emit a threshold phonon of , which is to

be absorbed by the electron at the lower state E1 for its transition to state E2; the

phonon emission/absorption do not cause any phonon exchange with the lattice mode.

Such an exchange of states between two electrons is referred to as “electron pairing

by mutual transitions”. Electron-lattice interactions at rates estimated to be as high as

at not too low temperatures and at absolute zero were cited.[11] We

argue that some of these interactions are “electron-pairing by mutual transition” and,

specifically, all the interactions at absolute zero are the electron pairing.

4. Origin of Binding Energy of Electron Pairs

Virtual particles were considered as indispensable in some physical processes.[12]

[13][14] A lattice wave mode at its ground state may “lend” a virtual threshold

phonon ( ) to an electron at a lower energy state (E1) for its transition to a

higher energy state (E2), and the electron then may return the threshold phonon to the

lattice mode in subsequent transition of . Such virtual phonon and energy

exchange is transient and could not be “observable”, as the virtual threshold phonon

cannot be “measured” to collapse the lattice mode to a state with a negative phonon

number like . The electron pairing described above does not ensure a binding

energy; but once the two electrons are in NSS state, the threshold phonon becomes

somewhat redundant because each of the two electrons could “borrow” a virtual

threshold phonon from the lattice mode (hν) for its transition of . If the

redundant threshold phonon escapes, the electron pair will be left with a binding

energy, which is typically comparable to the energy of the redundant threshold

phonon; although each electron in the pair can still go to the higher state E2 with the

5

borrowed virtual threshold phonon, its “measure” energy has to be determined with

reference to the ground state energy E1. When such an electron in such a pair is

scattered (“measured”) by a lattice mode, although it may be (transiently) at the

higher state E2 at the moment of scattering, the virtual threshold phonon will never go

with the scattered electron, the initial state of the scattering process can only be the

lower energy state E1, and the energy needed for the lattice mode to scatter the

electron will correspond to the energy difference of the final state of the scattering and

the lower energy state E1.

5. Pairing Competition Relating to One Electron State

The pairing candidates of an electron state may be determined as the

intersections of laminated curves of lattice waves and the plot of

bands,15) with the origin of the plot being placed at the state . Obviously,

each state usually has more than one pairing candidates, and the collection of

all these candidates should cover all possible one phonon lattice-electron interactions

of the electron at state . Exemplary electron pairing are shown in Fig. 1, where

points 10, 11, 12, and 13 represent states , , , and

on BB band respectively, and points 21, 22, and 23 represent states

, , and on AB band respectively; each dashed line with

double arrows indicates a pairing; that is, interband pairing is shown as between states

11 and 21 with a mediating threshold phonon of and ,

between states 12 and 22 with and , and between states 13

and 23 with and ; moreover, an intraband pairing is shown

as between states 10 and 11.

It may be relatively easy for an optical threshold phonon with to escape

from the electron-lattice mode sub-system, as it can interact with electromagnetic

wave of the same wavevector and frequency. [16] [17] But it may also be easy for it to

return to the electrons-mode sub-system. As indicated by Formula (5), a real phonon

is easily taken back from the mode by an electron at the lower state E1 for its real

transition to the higher state E2, and the cycle will repeat. And lattice modes may

couple with one another by anharmonic crystal interactions.[18]

6

As the lattice mode (hν) might drive the electron on the lower state (E1) to

perform downward transition or/and drive the electron on the upper state (E2) to

perform upward transition, NSS state would be interrupted at such a moment. After

the threshold phonon has escaped, however, such interruptions could be prevented, for

the upper electron, merely with a virtual threshold phonon, have to immediately

transit to the lower state. Therefore, strictly speaking, NSS state can only be ensured

after the threshold phonon has escaped. But a “near-NSS” state may be enough for a

threshold phonon to escape by probability, allowing both of the electrons to

“condense” to the lower state (E1). Even if a pair has released its threshold phonon, it

may still be unstable if the phonon number of its tuning lattice mode is non-zero.

When the lattice mode tunes more than one pairs, then only one of the tuned electron

pairs will lose its binding energy at a time. This effect could be significant as it

somewhat functions to multiply the effective binding energy of the electron pairs, and

might help explaining why superconductivity can exist at low c value, where Δ

is binding energy.

As far as an electron can only carry one virtual threshold phonon (one boson

process), two electrons at states and will be allowed to condense

to a common lower state (E1). But an electron already condensed to a lower state E1 is

prohibited from further condensing to an even lower state , for otherwise an

electron would have to carry two virtual threshold phonons to transit to the original

higher state . A condensed electron, either at the lower state (E1) or at the

higher state (E2), should be allowed to exchange its state with an electron from an

outside state; however, if the exchange happens at the higher state (E2) and the

incoming electron carries a real threshold phonon, the binding energy of the original

pairing will be removed.

As a candidate pairing can be characterized by its threshold phonon, whether

electron at a state (E1) is “pairing upward” or “pairing downward” will 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. Obviously, the “pairing upper” outcome is pro-

superconductivity. A threshold phonon with a greater energy seems to have an edge,

so pairing at a state would happen with respect to the greatest (threshold)

phonon energy available for the state. However, if no electron at or near EF can win in

7

their candidate pairings, the crystal would not have a superconducting phase. For

superconducting cuprates, of which the typical single band’s lineshape at and near EF

seems not in favor of “pairing upward”, the present model on pairing suggests that

interband structure be present at or near EF. A problem seems to exist with single-

layered Bi2201, Tl2201, and Hg1201, in which band splitting due to bilayer splitting

is absent. However, R. Manzke et al,[19] C. Janowitz et al,[20] and L. Dudy et al

demonstrated that the bands of Bi2201 and Pb-Bi2201 had fine splitting structures at

or near EF.[21]

6. Mode Competition and Threshold Phonon Transfer

Band splitting in bilayer cuprates were reported and discussed, [22]-[25] and more

splitting structures of YBCO were also reported.[26]) Of special interest is the near-

parallel band splitting features at and near EF;[24][25] such features allow a large number

of state pairs to be tuned to lattice modes in a relatively small region of k-

space of and in a small mode frequency range corresponding to

. So each lattice mode in these small regions of k-space and

frequency tends to tune a larger number of state pairs . Experiment results

showed the near-parallel band splitting structure of cuprates typically extended from

EF to the “kink”.[27][28][29] In Fig. 1, we may assume and ,

without losing any validity. Such an electron-lattice system is analogous to an optical

lasing (laser) system. The two systems have some differences such as: 1) phonons

cannot be emitted from a crystal, 2) the total phonon number is smaller at a

superconducting temperature (e.g. 100K), and 3) a lattice mode can never be turned

off. On the other hand, the two systems share two important features of a lasing

system: a) resonance associated with a linewidth; and b) mode competition, which

causes coupling among modes covered by a common linewidth. Population in an

electron-lattice system is provided by “good” electron pairs (pairs whose threshold

phonons may become redundant). For effective coupling among linewidths, modes in

each of the linewidths need to tune a sufficiently large population to ensure that the

outcome of mode competition be statistically well-defined.

As shown in Fig. 1, as long as states 11 and 12 are close enough to each other,

their tuning modes will be covered by a common linewidth and will compete with

each other, like optical modes competing in a laser cavity.[30] From (5)-(7), the

8

probability of transitions by electrons on states 11 and 12 is proportional to

, where Am is phonon number of the lattice mode. Thus, a Matthew Effect

will occur as “more phonons → greater competition dominance → more phonons →

….” so the real threshold phonons tuned to the modes covered by the common

linewidth would finally be concentrated onto a few dominating modes. Assuming that

mode hν2 dominates mode hν1, then mode hν1 will suffer phonon depletion.

Furthermore, while modes hν1 and hν3 might not be covered by a common linewidth

due to that the energy or/and wavevector difference between them are too great,

modes hν2 and hν3 may be covered by one common linewidth as long as hν2 and hν3

are close enough to each other, so mode hν1 will be indirectly coupled to mode hν3 via

mediation of mode hν2. Thus, modes tuning good electron pairs on a large range of

bands AB and BB will be coupled by such direct and indirect coupling, and real

threshold phonons from these pairs are transferred to a pool of good pairs tuned to a

smaller number of dominating modes. As long as these dominating modes do not

match the electron pairs at and/or near EF, phonons will be transferred away from EF

as shown by the thick arrow in Fig. 1, and the good electron pairs at and near EF will

suffer depletion of real threshold phonons.

In a single band scenario, although a state and its adjacent neighbors may be in a

common lineshape because levels and wavevectors on the band are quasi-continuous,

a sufficiently large population tuned to modes in one linewidth is not guaranteed.

Moreover, a conflict would arise that, while an increased number of pairs tuned to one

mode requires that the single band has a more straight lineshape, it is difficult for the

“upper threshold phonon” and “lower threshold phonon” of a state on such a nearly-

straight band to yield a well-defined pairing competition outcome, so pairing stability

may be in jeopardy.

Another conflict exists between pairing and mode competitions. In a single

band, pairing competition tends to force pairs to distribute in layers in E-k plot; one

layer includes two sub-layers each having a “thickness” of about hν; and the range of

phonon transfer would be limited to about hν. If phonons are effectively concentrated

in the second sub-layer from EF, the “upward pairing” outcome of electrons in the first

sub-layer might be undermined by the enhanced intensities of the lattice modes

residing in the second sub-layer. By contrast, these limitations on phonon transfer can

be completely removed in the presence of the near-parallel band splitting structure.

9

Low temperature functions to provide an increased proportion of “empty pairs”

(pairs with their threshold phonons having escaped). Effective phonon transfer can be

achieved only when the ultimate dominating modes tune a large pool of “empty pairs”

to accept a large number of threshold phonons the dominating modes seize by mode

competition. When a lattice mode tunes many “good pairs”, when the mode scatters a

good pair a phonon may be turned over by probability between the mode and the pair.

If all “good pairs” tuned to a lattice mode follow Maxwell-Boltzmann distribution,

then for a phonon energy of such as hν=20meV and T=100K, there is ,

which is the approximate average number of threshold phonon of the “good pairs”, so

threshold phonons released by all good pairs can be accommodated by only about

10% of the good pairs, which are tuned to the most dominating modes. When modes

are in competition, the set of good pairs tuned to one mode cannot be characterized as

a canonical ensemble. Rather, the good pairs of all the coupled modes should be

included in one canonical ensemble, and have one common distribution function:

with E being the total energy of the canonical ensemble and being

determined by normalization condition of: .31) So the good electron

pairs at and near EF in the near-parallel double/triple band structure may experience

depletion of real threshold phonons.

In high temperature superconducting (HTS) cuprates, the ultimate dominating

modes may not belong to the same linewidth as the modes “at” EF (modes that tune

the pairing at or near EF); there should be at least one intermediating linewidth

between the linewidth covering modes at EF and the linewidth covering the ultimate

dominating modes. But the number of such intermediating linewidths should not be

too great; otherwise, reduced population tuned to modes in each linewidth and

increased number of times of turning-over along the chain of linewidths may

undermine transfer efficiency. Moreover, if the population of any intermediating

linewidth is depleted, the chain of transfer would be broken; therefore, a population

gradient should be maintained along the chain of mediating linewidths between the

two “end” linewidths; thermal equilibrium might be a mechanism supporting such a

gradient. As a rough estimate, there should be some 1016 states in one section along

the nodal direction in k space, the energy range of the near-parallel band splitting

structure is about 1/100 of that of the entire band, so there are some 1014 states in the

near-parallel band splitting structure in one section, assuming that 1/10 of states

10

support “good pairs”, then at T=100K there would be 1013 “good pairs” maintaining a

pool of about 1012 releasable threshold phonons. Assuming some 10 intermediating

linewidths between EF and the “kink”, then each linewidth would be associated with a

pool of about 1012 good pairs and 1011 releasable threshold phonons. If a conventional

superconductor has the same mediating phonon energy (such as 20meV) as a HTS

cuprate, then a conventional superconductor having a critical temperature of ~10K

would have an average phonon number of . This indicates that for the

cuprate to reach the same level of phonon number at EF at a critical temperature of

about 100K, a threshold phonon depletion rate of 10-10 has to be reached at EF. In the

scenario of 10 intermediating linewidths, a depletion rate of 10 -10 will reduce the

average phonon number in the linewidth at EF to about 10. But such depletion is

theoretically attainable because, as the linewidth at EF is the most upstream one in the

linewidth chain, phonon number in the linewidth at EF could be modeled to be nearly

zero without affecting the effect of the intermediating linewidths as a transfer channel.

Thus, the present model would support a unified electron-pairing mechanism that is

consistently applicable to both HTS and low temperature superconductivity.

The linewidth needs not to be “small” as compared with the energy separation

δE of adjacent levels. When electrons on pairing states (E1,E2) condense to the lower

state (E1), the higher state (E2) remains an “occupied” state, with only that the

occupation is nonstationary. If the linewidth is comparable to δE, the electron at the

state adjacent to E1 may transit to E2, which leads to the state exchange with an

electron from an outside state as discussed above; and when the incoming electron

carries a real threshold phonon, the binding energy of the original pairing will be

removed. But such individual state exchange would not affect the overall statistical

distribution of real threshold phonon numbers, which numbers decide the stability of

corresponding electron pairs mediated by the respective (real or virtual) threshold

phonons.

7. Assurance of Pairing at and near EF

Interband pairing near EF is typically accompanied by an intraband pairing. As

shown in Fig. 1, state 11, which performs interband pairing with state 21, may also

perform intraband pairing with state 10. Due to the nearly-straight lineshape of band

BB, the “upper threshold phonon” between states 11 and 10 has a counterpart “lower

threshold phonon” as a nearly-identical competitor. However, the electron on state 11

11

is pinned by the double binding energies left by the two threshold phonons of

and and is thus prevented from condensing to any lower

state. Therefore, with the near-parallel band splitting structure, the “pairing upward”

attribute of the electron states on the lower band BB for interband pairing with the

electrons at and near EF on the upper band AB could be assured.

8. Conclusions

In summary, an electron-lattice system has intrinsic time dependent

characteristic as featured by Golden Rule. The primary essence of Golden Rule is

resonance, by which electrons on matched pairing states are tuned to the lattice wave

modes. If an electron pair is tuned with a sufficiently good quality factor, the

threshold phonon of the pair can become redundant and can be released by the pair to

produce a binding energy. Lattice modes falling in a common linewidth can compete

with one another, much like modes competing in a lasing system. In cuprates, due to

near-parallel band splitting features at and near EF, a great number of electron pairs

are tuned to a relatively small number of lattice wave modes, leading to strong and

effective mode competition, transfer of threshold phonons from EF towards the

“kink”, and depletion of threshold phonons at and near EF. We have also discussed

competition among multiple pairings associated with one electron state.

APPENDIX: Perturbation Treatment of electron-lattice interaction system based on

Time-Dependent Hamiltonian [Excerpt translation of Reference 3), pages 201-205.]

The first approximation of potential variation δVn of the atom at the nth lattice

point Rn caused by a lattice wave is:

, (7-86)

where V(r) is the potential of one atom, and

, (7-87)

represents displacement of the atom by the lattice wave , e is the unit vector in the

wave direction, A is the magnitude of the lattice wave , ν is frequency of the lattice

wave , and q is wavevector of the lattice wave under elastic wave approximation.

Potential variation of the entire lattice is:

12

ΔH can be treated as a perturbation. With Formula (7-88), transition from k1 to

k2 has the energy relation of:

, (7-90)

The normalized wave function is given as:

,

where N is the number of primitive cells in the crystal. The matrix element is given as

, (7-91)

and . (7-92)

Of special importance is the summation in the matrix element:

,

which yields

,

for

, (7-93)

and zero otherwise.

13

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15

Fig. 1: A schematic illustration of electron pairing and phonon depletion in near-parallel band splitting structure in cuprates. Interband pairs are formed between states 11 and 21, 12 and 22, and 13 and 23, as shown by dashed line with arrows. An intraband pair is formed between states 11 and 10.The lattice mode tuning pairing between states 11 and 21 indirectly compete with the mode between states 13 and 23 by the mediation of the mode between states 12 and 22. Phonon transfer thus happens over a wide range on bands AB and BB, as shown by the thick arrow, leading to phonon depletion at or near Fermi Surface EF. Electron on state 11 is pinned by its double pairings with electrons on states 21 and 10.

22

BB

EF

11

12

13

23

21E

q

AB

Real phonon transfer

10

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