2 Electronic Structure Instability – Transition to the Antiadiabatic State
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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,