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2 Electronic Structure Instability – Transition to the Antiadiabatic State

2 Electronic Structure Instability – Transition to the Antiadiabatic State

Tải bản đầy đủ - 0trang

490



P. Baˇnack´y



of atomic Dirac-Fock calculations [57], the INDO version used in the SOLID

package is parameterized for nearly all elements of the Periodic table of the

elements. Incorporating the INDO Hamiltonian into the cyclic cluster method (with

Born-von Karman boundary conditions) for electronic band structure calculations

has many advantages and some drawbacks as well. The method is inconvenient for

strong ionic crystals but it yields good results for intermediate ionic and covalent

systems. The main disadvantage is overestimation of the total bandwidth. On the

other hand, it yields satisfactory results for properties related to electrons at the

Fermi level (frontier-orbital properties) and for equilibrium geometries [58–60].

In practical calculations, the basic cluster of the dimension (Na × Nb × Nc ),

is generated by corresponding translations of the unit cell in the directions of

crystallographic axes, a (Na ), b (Nb ), c (Nc ). On the basic cluster, the Born-von

Karman boundary conditions are imposed and an “infinite” – 3D-periodic cyclic

cluster structure is generated in calculation of matrix elements. In particular, the

band structure calculations have been performed for the basic clusters 11 × 11 × 7

for MgB2 , 9 × 9 × 9 for YB6 , 5 × 5 × 5 for YBa2 Cu3 O7 and 11 × 11 × 11 for Nb3 Ge.

The scaling parameter 1.2 (1.0 for Nb3 Ge) has been used in calculations of the oneelectron off-diagonal two-center matrix elements of the Hamiltonian (β-“hopping”

integrals). The basic cluster of a given size generates a grid of (Na × Nb × Nc ) points

in k-space. The HF-SCF procedure is performed for each k-point of the grid with the

INDO Hamiltonian matrix elements that obey the boundary conditions of the cyclic

cluster [55]. The Pyykko-Lohr quasi-relativistic basis set of the valence electron

atomic orbitals (s,p-AO for Mg, B, Ba, O, Ge and s, p, d-AO for Cu, Y, Nb) has been

used. The number of STO-type functions is unambiguously determined by that of

valence AOs in atoms comprising the basic cluster. In general, the precision of the

results of band structure calculation increases with increasing dimension of the basic

cluster. It has been shown [55, 58–60], however, that there is an effect of saturation,

a bulk limit beyond which the effect of increasing dimension on e.g. total electronic

energy, orbital energies, HOMO-LUMO difference. . . , is negligible. In practice,

the dimension of the basic cluster and parameter selection (e.g. for calculation

of β integrals) is a matter of compromise between computational efficiency and

convergence of calculated electronic properties and equilibrium geometry to some

reference or experimental data. Note, however, that the basic efficiency and accuracy

are restricted by the INDO parameterization.



27.2.2 Band Structures

In Fig. 27.2a, c, e, g(figures: left), are band structures (BS) of the studied compounds

at equilibrium geometries. All the band structures are of adiabatic metal-like

character with a relatively low density of states at the FL (indicated by a dashed

line). Coupling to the respective phonon mode(s) in particular compounds seemingly does not change the metal-like character of BS. In all cases, however,

e-p coupling induces BS fluctuation (see the pictures on the right), which is



27 Anti-adiabatic State – Ground Electronic State of Superconductors



a



b



E

eV

10



E

eV

10



5



5



0



0



−5



−5



−10



−10



−15



Γ



L



A



M



Γ



K



−15

M



c



1



0

−5



−9



−9



−14



−14

Γ



X



R



−18



Γ



M



R



M



X



f



E

eV



L



M



Γ



K



M



Γ



X



R



Γ



M



R



M



X



E

eV



1



1



−1



−1



−3



−3



−4



−4



−6



−6

−8



−8



g



A



E

eV

5



−4



e



Γ



d



E

eV

6



−19



491



Z



G



X



S



Y



G



S



E

eV



h



Z



G



X



S



Y



G



S



E

eV

−3

−4



−4

−5



−5



−6



−6



−7



−7



−8



−8



−9



−9

G



X



R



M



G



R



M



X



G



X



R



M



G



R



M



X



Fig. 27.2 Band structures of MgB2 (a, b), YB6 (c, d), YBa2 Cu3 O7 (e, f), Nb3 Ge (g, h). Pictures

on the left (a, c, e, g,) correspond to equilibrium high-symmetry structures. On the right: band

structures (b, d, f, h) at distorted geometry with atom displacements in the respective phonon

modes



characteristic by fluctuation of the analytic critical point (ACP) of some band across

FL (cf. a-b, c-d, e-f, g-h).

In particular, for MgB2 coupling to the E2g phonon mode (in-plane stretching

vibration of B-B) results in splitting of σ bands (px , py electrons of B atoms in

a-b plane) in Γ point of the first Brillouin zone (BZ) – Fig. 27.2b. Related to band

topology, the analytic critical point (ACP, maximum) of σ bands islocated at the



492



P. Baˇnack´y



Γ point and, for a displacement ≈0.016 A◦ /B-atom out of equilibrium position,

the ACP crosses FL. This means periodic fluctuation of the BS between topologies

2 a ↔ 2 b in coupling to vibration in the E2g mode.

The situation is similar for YB6 . In this case, BS fluctuation is related to the T2g

mode (valence vibration of B atoms in the basal a-b plane of B-octahedron). At the

displacement ≈0.017 A◦ /B-atom out of equilibrium position, the ACP (saddle

point) of the band with dominance of B-p and Y-d electrons crosses FL in the M

point and the BS fluctuates between topologies 2 c ↔ 2 d.

In the case of YBa2 Cu3 O7 , the BS fluctuation is associated with coupling to

three modes, Ag , B2g , B3g , with the apical O(4) and CuO-plane O(2), O(3) atom

displacements. At displacements ≈0.031 A◦ of apical O(4) in the Ag mode and

≈0.022 A◦ of O(2) and O(3) in the B2g , B3g modes, the ACP (saddle point) of

one of the Cu-O plane (d-pσ) band in Y point crosses FL and undergoes periodic

fluctuation between topologies 2 e ↔ 2 f.

The situation for Nb3 Ge is presented in Fig. 27.2g, h. Coupling to Γ12 phonon

mode (out-of phase vibration of Nb atoms in two perpendicular chains – see

Fig. 27.1d, displacement ≈0.025 A◦ /Nb atom) induces fluctuation of ACP (maximum) of Nb(dx2 −y2 , dz2 )-bands at the R point across the FL, cf. the topology

2 g ↔ 2 h.

In all cases presented in Fig. 27.2a–h, showing superconducting compounds, the

band ACP crosses FL at a displacement which is less than the root-mean square

(rms) displacement for zero-point vibration energy in respective phonon mode. This

means, however, that in vibrations where the ACP approaches FL at a distance less

than ±ω, the Fermi energy EF (chemical potential μ ) of the electrons in the band

close to the point where ACP crosses FL is less than the vibration energy of the

corresponding phonon mode, EF < ω. In these circumstances the adiabatic BOA is

not valid and standard adiabatic theories cannot be applied. Moreover, shift of the

ACP much increases the density of states (DOS) at FL, nσ (EF ) = (∂ εσ0 /∂ k)−1

EF , and

induces a corresponding decrease in the effective electron velocity (∂ εσ0 /∂ k)EF of

the fluctuating band in this region of k-space. Under these circumstances, the system

is in the intrinsic nonadiabatic state, or even in the antiadiabatic state, EF

ω, and

electronic motion depends on nuclear coordinates Q and is influenced by nuclear

dynamics – momenta P.

Instability of the electronic structure at e-p coupling is absent in respective nonsuperconductive analogues, such as XB2 (X ≡ Al, Sc, Y, Ti, Zr, Hf, V, Nb, Ta,

Cr, Mo, W, Mn,..), CaB6 , YBa2 Cu3 O6 and Nb3 Sb. As an illustration, in Fig. 27.3

are band structures of these compounds at equilibrium high-symmetry structure

(Fig. 27.3a, c, e, g) and at distorted geometry (Fig. 27.3b, d, f, h) with the same

displacements in respective phonon modes as those in the case of corresponding

superconductors at the transition in the antiadiabatic state.

In spite that for XB2 compounds; coupling to the E2g mode induces splitting of

σ bands in the Γ point, the systems remain stable in the adiabatic state. For these

systems, the ACP of the σ band does not fluctuate across FL. In Fig. 27.3 are band

structures of AlB2 at equilibrium high-symmetry structure (3a) and at distorted



27 Anti-adiabatic State – Ground Electronic State of Superconductors



a



b



E

eV

5



E

eV

5



1



1



−2



−2



−6



−6



−9



−9



−13



−13



Γ



c



L



A



M



Γ



K



M



d



E

eV

5

1



E

eV

5



−3



−3



Γ



A



L



M



Γ



K



M



1



−7



−7



−11



−11



−15



−15



−19



e



Γ



X



R



−19



Γ



M



R



M



X



f



Γ



X



R



Γ



M



R



M



X



E

eV



E

eV

2



2



−1



−1



−4



−4



−7



−7



−10



−10

−13



−13

Z



g



G



X



S



Y



G



S



h



E

eV

−3



E

eV

−3



−5



−5



−7



−7



−9

−11



−11



−13



−13



−15



493



Z



G



X



S



Y



G



S



−9



Γ



X



R



M



Γ



−15

R



M



X



Γ



X



R



M



Γ



R



M



X



Fig. 27.3 Band structure of AlB2 , CaB6 , YBa2 Cu3 O6, and Nb3 Sb at equilibrium geometry (a, c,

e, g) and at distorted geometry (b, d, f, h)



geometry (3b) with the same B-atom displacements in the E2g phonon mode as in the

case of MgB2 in transition to a superconducting state. In spite of σ bands splitting

and nearly the same value of the e-p interaction strength (the calculated mean value

is u¯ ≈ 1.01 eV/u.cell) as that of MgB2 (u¯ ≈ 0.98 eV/u.cell), AlB2 remains in e-p

coupling in the adiabatic state as a non-superconductive compound. In this case, BS

fluctuation (bands splitting in e-p coupling) does not decrease chemical potential.

It remains in e-p coupling still larger than the vibration energy (μad > h¯ ω ) and,

consequently, there is no driving force for transition into the antiadiabatic state.



494



P. Baˇnack´y



In the case of deoxygenated YBCO that is without the chain oxygen

[61] – YBa2 Cu3 O6 (in contrast to the superconducting YBa2 Cu3 O7 ) a combination

of electron coupling to the Ag , B2g and B3g phonon modes leaves band structure

without substantial change (Fig. 27.3e–f). In the case of YBa2 Cu3 O7 , the ACP

(SP-saddle point) at Y point fluctuates across FL (see Fig. 27.2e–f), which yields

substantial reduction of chemical potential → μantiad < h¯ ω . For YBa2 Cu3 O6 the SP

does not fluctuate across FL and chemical potential remains larger than the phonon

energy spectrum, μad > h¯ ω , and the system remains in the adiabatic state.

The CaB6 is an insulator and coupling to the T2g mode does not change this

property – BS topology remains at e-p coupling without change (Fig. 27.3c–d).

The alloy Nb3 Sb of A15 class is non superconductive [62]. Metal-like character

of Nb3 Sb and topology of BS remains without significant change at e-p coupling to

Γ12 phonon mode of Nb atoms vibration –Fig. 27.3g, h.



27.2.3 Nonadiabatic Effects Induced by Transition

into Antiadiabatic State

27.2.3.1 Formation of Antiadiabatic Ground State and Gap Opening

The main part of the effect of nuclear kinetic energy on electronic motion can be

derived as diagonal correction by sequential Q, P-dependent base transformations

[52] (or quasi-particle transformation [53]). This is a generalization of adiabatic

Q-dependent transformation which yields the well-known adiabatic diagonal BO

correction (DBOC) [63, 64]. Due to diagonal approximation with factorized form

of total wave function, Ψ0 (r, Q, P) = Φ0 (r, Q, P).X0 (Q, P), the standard clamped

nuclear Hamiltonian treatment can be used and the Q, P-effect is calculated in

the form of corrections to the electronic ground-state energy (zero-particle term

correction), corrections to orbital energies (one-particle term corrections) and twoparticle term corrections (correction to the electron correlation energy).

The correction to the electronic ground-state energy in the k-space representation

due to interaction of pair of states mediated by the phonon mode r can be written

as [52–54],

εk ,max

0

ΔE(na)



= 2∑ ∑

ϕRk ϕSk



nεk (1 − fε 0 k ) d εk0

0



εk max



fε 0 k urk−k

εk,min



2



nεk



h¯ ωr

d εk0 , ϕRk

2

0

0

εk − εk − (¯hωr )2



= ϕSk



(27.1)



In general, all bands of 1st BZ of multiband system are covered. Coupling is of

inter-band character, while εk0 < εF ; εk0 > εF .



Binding energy - (E-EF), [eV]



a



495



b



8



0,00

−0,05



6



−0,10

4

−0,15

2



−0,20

−0,25

0,36 0,38 0,40 0,42 0,44 0,46 0,48 0,50



−0,08



−0,06



−0,04



−0,02



0

0,00



nsp - density of states of d-p band at Y point



27 Anti-adiabatic State – Ground Electronic State of Superconductors



k - along the line Γ-Y (0,0,0 - 0,1/2,0), kF=1/2 Energy distance of saddle point from Fermi level [eV]



Fig. 27.4 Calculated dispersion of d-pσ band in ΓY direction with the kink formation indicated

by the arrow (a) and increase of the d-pσ band DOS when ACP approaches FL (b) for YBa2 Cu3 O7



Fermi-Dirac populations fε 0 k , fε 0 k make correction (27.1) temperaturedependent. Term urk−k stands for matrix element of e-p coupling and nε k , nε k ,

are DOS of interacting bands at εk0 and εk0 . For adiabatic systems, such as metals,

this correction is positive and negligibly small (DBOC). Only for systems in the

antiadiabatic state the correction is negative and its absolute value depends on

the magnitudes of urk−k and nε k , nε k , at displacement for FL crossing. At the

moment when ACP approach FL, the system not only undergoes transition to the

antiadiabatic state but DOS of the fluctuating band is considerably increased at FL.

0

For all the studied systems at 0 K, the Δ E(na)

(which covers the effect of

nuclear momenta) prevails in absolute value the electronic energy increase

Δ Ecr = Ed,cr − Eeq at nuclear displacements Rd,cr when ACP crosses FL as

calculated for clumped nuclear adiabatic structures. For instance in case of

YBa2 Cu3 O7 , calculated [65] increase at Rd,cr is Δ Ecr = +170 meV/unitcell,

but correction to the total energy due to e-p coupling in antiadiabatic state is

0

Δ E(na)

= −204 meV/unitcell. The net effect of symmetry lowering (distortion)

is the fermionic ground state energy stabilization. It means that due to effective

nonadiabatic e-p coupling, the distorted structure for the specified displacements is

by (−204 + 170) = −34 meV/unitcell more stable than undistorted – equilibrium

structure on the BOA level. Under these circumstances, the system is stabilized in

the antiadiabatic electronic ground state at broken symmetry with respect to the

adiabatic equilibrium high-symmetry structure.

It can be identified by ARPES as a kink formation in the momentum distribution

curve at FL, i.e. as band curvature at ACP when approaching FL – see the calculated

results for YBa2 Cu3 O7 , (Fig. 27.4).

Due to translation symmetry of the lattice, the resulting antiadiabatic electronic

ground state is degenerate when distorted, with a fluxional nuclear configuration

in a given phonon mode(s) – see, e.g., Fig. 2 in Ref [50], or Fig. 27.6 below.

The ground state energy is the same for different positions of the atoms involved



496



P. Baˇnack´y



(in phonon modes which drive the system into this state) in motion over circumferences of flux circles with radii equal to characteristic displacements Δ Rcr =

|Req − Rd,cr | for FL crossing.

In transition to the antiadiabatic state, k-dependent gap Δk (T ) in quasi-continuum

of adiabatic one-electron spectrum is opened. The gap opening is related to shift

0 , ε

0

Δ εPk of the original adiabatic orbital energies εPk

Pk = εPk + Δ εPk , and to the

k-dependent change of DOS of particular band(s) at Fermi level. Shifts of orbital

energies in band ϕP (k) has the form [52–54],







Δε (Pk ) =



uk −k1



2



1 − fε 0 k



1



Rk1 >kF

2



uk−k







|uk−k1 | (1 − fε 0 k )



fε 0 k



Sk


εk0



1



εk0



2



− εk0

1



− h¯ ωk −k1



h¯ ωk−k

2

0

− εk − (¯hωk−k )2











h¯ ωk −k



2



(27.2)



for k > kF , and

Δε (Pk) =



Rk1 >kF











Sk1


h¯ ωk−k

1



2



1



εk0 − εk0



2



1



h¯ ωk−k1



2



|uk−k1 | fε 0 k



− h¯ ωk−k

1



εk0 − εk1 10



2



− h¯ ωk−k1



2



2



(27.3)



for k ≤ kF .

Replacement of discrete summation by integration, ∑ . . . → n(εk ), introduces

DOS n(εk ) into Eqs. 27.2 and 27.3, which is of crucial importance in relation

to fluctuating band – see Fig. 27.4b. For corrected DOS n(εk ), which is the

consequence of shift Δ εk of orbital energies, the following relation can be derived;

n(εk ) = 1 + (∂ (Δ εk )/∂ εk0 )



−1 0



n (εk0 )



(27.4)



Term n0 (εk0 ) stands for uncorrected DOS of the original adiabatic states of particular

band,

−1

(27.5)

n0 (εk0 ) = ∂ εk0 /∂ k

Close to the k-point where the original band, which interacts with fluctuating band,

intersects FL, the occupied states near FL are shifted downward below FL and

unoccupied states are shifted upward – above FL. The gap is identified as the

energy between peaks in the corrected DOS above FL (half-gap) and below FL.

The formation of peaks is related to the spectral weight transfer that is observed by

ARPES or tunneling spectroscopy in cooling below Tc .



27 Anti-adiabatic State – Ground Electronic State of Superconductors



b



10



8



Density of states



12



18

16

14

12



Density of states



a



497



10

6

8

6



4



4

2

2

0



0



−0,05 −0,04 −0,03 −0,02 −0,01 0,00



0,01



0,02



0,03



0,04



−0,04



0,05



Energy distance from Fermi level [eV], Y- Γ direction



−0,02



0,00



0,02



0,04



Energy distance from Fermi level [eV], X- Γ direction



c 0,40



d



0,064



0,35

0,062



Density of state [a.u.]



Density of states [a.u.]



0,30

0,25

0,20

0,15

0,10



0,060



0,058



0,056



0,054



0,05

0,00

0,000



0,005



0,010



0,015



0,020



0,052

0,000



0,025



Energy distance above Fermi level [eV]



Density of states [arb.u]



e



0,005



0,010



0,015



0,020



0,025



Energy distance above Fermi level [eV]



8



6



4



2



0

0.000



0.002



0.004



0.006



0.008



0.010



0.012



0.014



0.016



Energy distance above Fermi level [eV]



Fig. 27.5 Calculated (0 K) DOS in antiadiabatic state and gap formation near k point where

particular bands intersect FL. The DOS are for YBCO in Y-Γ (a), X-Γ (b) direction, MgB2 (c),

YB6 (d), and Nb3 Ge (e)



For the studied compounds, the calculated corrected DOS of particular band(s)

with gap opening are in Fig. 27.5. In particular, YBa2 Cu3 O7 exhibits an asymmetric gap in two directions: O1-pσ band gap is Δb (0) ≈ 35.7 meV in the Γ − Y

direction (5a) and Δa (0) ≈ 24.2 meV is in the Γ − X direction (5b). The calculated



498



P. Baˇnack´y



asymmetry, i.e. the ratio (Δ a (0)/Δb (0))theor ≈ 0.68 is very close to the experimental

value (≈0.66) that has been recorded [66] for untwined YBa2 Cu3 O7 .

Two gaps, in σ and π band, are opened in Γ − K(M) directions of MgB2 – (5c):

Δσ (0)/2 ≈ 7.6 meV and Δπ (0)/2 ≈ 2.2 meV. The result simulates tunneling spectra

at positive bias voltage and calculated half-gaps are in a very good agreement with

experimental high-precision measurements [67, 68].

A small gap opens on pd-band in the Γ − X direction of YB6 – (4d): Δpd (0)/2 ≈

2.2 meV.

In case of Nb3 Ge, we are not familiar with tunneling or ARPES spectra.

However, recent results of point-contact spectroscopy obtained for superconductor

of the same A15 group – Nb3 Sn(Tc = 18.1 K) [69], indicate that this system could

be of two-gap character. Respective half-gaps are; Δ1 /2 ≈ 3.92 meV and much

smaller gap is Δ2 /2 ≈ 0.85 meV. Our calculation for Nb3 Ge(Tc = 23.2 K) is shown

in Fig. 27.5e. Calculated DOS is also of two-gap character. The half gaps are;

Δ1 /2 ≈ 4.15 meV and much smaller gap is Δ2 /2 ≈ 1.7 meV. The gaps are opened

in M-R direction of 1st BZ.

It should be stressed that this result is the first theoretical prediction of two-gap

character for some superconductor of A15 family.



27.2.3.2 Critical Temperature Tc of Antiadiabatic State Transition

The corrections to orbital energies (2, 3) and to the ground state energy (1) are

temperature dependent and decrease with increasing T. At a critical value Tc , the

gap in one-particle spectrum [52–54],



Δ (T ) = Δ (0)tgh[Δ (T )/4kB T ]



(27.6)



as Δ (0) at 0 K, disappears – i.e. Δ (Tc ) = 0 (continuum of states is established at FL).

0 (T )| ≤ Δ E

At these circumstances holds |Δ E(na)

c

d,cr and the system undergoes

transition from the antiadiabatic into adiabatic state, which is stable for equilibrium

high-symmetry structure above Tc . With respect to Δ (0), a simple approximate

relation follows from Eq. 27.6,

Tc = Δ (0)/4kB



(27.7)



Calculated values of critical temperature for the set of studied compounds are

presented in Table 27.1. As it can be seen, the values of Tc for transition into

antiadiabatic state are in a good agreement with corresponding experimental values

of Tc for superconducting state transition of particular compounds [3, 47, 70].

It should be noticed, however, that while there is a general consensus about

importance of the e-p coupling to the Ag , B2g , B3g and E2g phonon modes in

case of YBa2 Cu3 O7 and MgB2 in transition to superconducting state, the situation

with YB6 is rather controversial. Recent studies [71, 72] advocate importance



27 Anti-adiabatic State – Ground Electronic State of Superconductors



499



Table 27.1 Calculated values of critical

temperature for transition into antiadiabatic state (Tc -theor) and experimental

values of critical temperature for transition

into superconducting state (Tc -exp)

Compound

Tc -theor

Tc -exp

MgB2

44 K

39.5 K

YB6

12.2 K

8–10 K

103.5 K

92–94 K

YBa2 Cu3 O7

24 K

23.2 K

Nb3 Ge



of e-p coupling to low-frequency (8–10 meV) phonon modes of Y-vibration for

superconducting coupling. It is associated with overall value of dimensionless ep coupling constant λ for calculation of Tc according to McMillan formula. For

medium-strong coupling λ ∼ 1 − 1.4 and Coulomb pseudopotential μ ∗ ∼ 0.1 −

0.2, the experimental Tc ∼ 6.2 – 9 K is reproduced. In these circumstances, lowfrequency Y- vibrations contribute by 84% to the overall value of λ. Our results

show that coupling to Y-vibration does not induce the adiabatic-antiadiabatic state

transition. This transition is connected to B-vibrations, in particular to T2g mode.

The value of dimensionless constant λ related to T2g mode coupling, calculated

in our study is, λT2g ∼ 0.1. This value is in full agreement with decomposition of

Eliashberg spectral function on contributions from the particular phonon modes in

YB6 calculated by Schell et al. [73]. The overall value of λ ∼ 0.48 that accounts

for nonlocal corrections on e-p coupling in the modes where B-octahedrons move

as a whole is dominated by high-frequency (30–90 meV) B-vibrations. The authors

[73] made conclusion that in transition to superconducting state in YB6 , the Bvibration phonon modes are essential. The conclusion is based on the fact that within

McMillan formula, a small increase in overall λ ∼ 0.48 can reproduce experimental

Tc . In the present work, it is shown that in spite of the fact that coupling to T2g mode

is weak (uk−k ∼ 0.1 eV), transition in superconducting state and relatively highvalue of Tc can be reached due to enormous increase of DOS of the fluctuating band

in M point at FL, from the adiabatic value 0.06 states/eV to the value 1.09 states/eV

at transition into antiadiabatic state.



27.2.3.3 Formation of Mobile Bipolarons in Real Space

From the theory of the antiadiabatic ground state [52a, b] follows that instead

of Cooper pairs, formation of mobile bipolarons arise naturally as a result of

translation symmetry breakdown at the antiadiabatic level. Bipolarons are formed

as polarized inter-site charge density distribution, mobile on the lattice without

dissipation due to degeneracy (fluxional structure) of the antiadiabatic ground state

at distorted nuclearconfigurations. Formation of polarized inter-site charge density



500



P. Baˇnack´y



distribution at transition from adiabatic into antiadiabatic state is reflected by

corresponding change of the wave function.For spinorbital (crystal orbital – band)

ϕR(k) holds [52]a,

+

r ¯ +

+

|ϕS (x, Q, P) = a+

S (x, Q, P)|0 = a¯S − ∑ cSR Qr a¯R − ∑ c SR Pr a¯R

r



rR



rR



¯ P ) |0 = |ϕS (x, 0, 0) − ∑ crSR Q¯ r |ϕR (x, 0, 0)

+ O(Q , QP,

¯2



2



rR



− ∑ c SR P¯ r |ϕR (x, 0, 0) + ......

r



(27.8)



rR



Expansion coefficients in (27.8) are coefficients of adiabatic Q-dependent canonical

transformation crPQ and of non-adiabatic P-dependent canonical transformation cˆrPQ ,

crSR =



∂ cSR (Q) r

∂ cˆSR (P)

; cˆSR =

∂ Qr

∂ Pr



(27.8a)



Approximate solution [53] yields the following analytical forms,

crSR = urSR

cˆrSR = urSR



εS0 − εR0

(¯hωr )2 − εS0 − εR0



2



;



S=R



(27.8b)



2



;



S=R



(27.8c)



h¯ ωr

(¯hωr



)2 −



εS0 − εR0



Bear in mind that for solids in reciprocal (quasi-momentum) space, the orbital

0

.

energies are k-dependent, i.e. εS0 ≡ εS0 (k) ≡ εS,k

At transition into antiadiabatic state (|εS0 (kc ) − εF0 |Req ±Q h¯ ωr ), coefficients

crRS of Q-dependent transformation matrix (27.8b) become negligibly small and

absolutely dominant for modulation of crude-adiabatic wave function are in this

case coefficients cˆrRS of P-dependent transformation matrix (27.8c). For simplicity,

let us consider that transition into antiadiabatic state is driven by coupling to a

phonon mode r with stretching vibration of two atoms (e.g. B-B in E2g mode of

MgB2 , valence T2g mode vibration of B-B atoms in basal a-b plane of B-octahedron

in YB6 , vibration motion of O2, O3 in Cu-O planes – B2g , B3g modes of YBCO,

or Γ12 phonon mode vibration of Nb-Nb atoms in chains of Nb3 Ge in a-b, a-c

or b-c plane). Let m1 and m2 are equilibrium site positions of involved nuclei on

crude-adiabatic level and d1 and d2 are nuclear displacements at which crossing

into antiadiabatic state occurs. At these circumstances the original crude-adiabatic

wave function ϕk0 (x, 0, 0), which corresponds to fluctuating crystal orbital (band)

that crosses FL at e-p coupling, is changed in a following way,



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2 Electronic Structure Instability – Transition to the Antiadiabatic State

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