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Proc. R. Soc. A (2011) 467, 166–173
doi:10.1098/rspa.2010.0085
Published online 30 June 2010
Charge-density waves arising from two
electron bands
BY DONALD W. BUKER*
51775 Nevin Road, Rosedale, BC, Canada V0X 1X0
A theory for a quasi-one-dimensional incommensurate charge-density wave state arising
from electron–phonon (el–phon) interaction connecting electron states in two different
bands is presented. An expression for the fundamental component of the energy gap
as a function of the effective el–phon coupling, valid for all coupling strengths, has
been found. For a single band, the expression simplifies to a reciprocal hyperbolicsine dependence on the reciprocal effective coupling. The effective coupling, although
simply related to the n assumed phonon-band frequencies, is not generally expressible
as a sum of independent functions of these frequencies. The theory is applied to
tetrathiofulvalinium–tetracyanoquinodimethane and to potassium blue bronze.
Keywords: electrons; phonons; charge-density waves
1. Introduction
Since the first description of charge-density waves (CDWs) in solids in the
1950s (Fröhlich 1954; Peierls 1955), there have been numerous advances on
the theory of their formation, including those described in the following works:
Overhauser (1971), Lee et al. (1973, 1974), Allender et al. (1974), McMillan (1975,
1976), Rice (1975, 1976), Rice et al. (1976, 1979), Wilson (1979), Mertsching &
Fischbeck (1981), Monceau (1985), Grüner & Zettl (1985), Horovitz (1986), Bjelis
(1987), Grüner (1988, 1994), Tucker et al. (1988), Degiorgi et al. (1991, 1995),
Eremko (1992), McKenzie & Wilkins (1992), McKenzie (1995), Mozos et al.
(2002), Sing et al. (2003) and Perroni et al. (2004).
There are a large number of CDW compounds. Accompanying the electron
wave is a periodic lattice displacement (PLD) of the background ions. Given the
band structure above the transition temperature, it is a fair game to predict
the ensuing CDW properties of the ground state. The goal is to make quantitative
predictions for some real materials.
The Hamiltonian is broken into electron and phonon parts, which are separately
solved for their ground-state energies, in terms of a single parameter, the electron
energy gap gc . The total energy is then minimized with respect to this parameter,
which yields a fairly simple analytic expression for the gap in terms of an effective
electron–phonon (el–phon) coupling leff .
*[email protected]
Received 10 February 2010
Accepted 1 June 2010
166
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Charge-density waves
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In a previous treatment (Buker 2009), the CDW–PLD system was solved in
the simple context of a single electron band. The present work extends the theory
to the common case where the CDW arises from two bands.
Derivation of the two-band gap equation is discussed in §2. Specialization
to the single-band case is made in §3. Possible refinements are discussed in
§4, and a sketch of how electron–electron (el–el) interactions would enter is
made in §5. The application to tetrathiofulvalinium–tetracyanoquinodimethane
(TTF–TCNQ) and K-blue bronze comprise §§6 and 7.
2. The two-band gap equation
Consider the Fröhlich Hamiltonian
†
†
†
†
H=
h̄urq bqr
ek ck c k +
bqr +
gkqr ck+q ck (bqr + b−qr ).
k
qr
(2.1)
kqr
Here the c’s are electron operators, the b’s are phonon operators and the gkqr ’s
are the el–phon coupling constants; r labels the phonon bands, running from 1
to n. The sum over k is meant to imply also a sum over electron spin. Attention
is narrowed to a state with a CDW and its accompanying PLD. It is assumed in
particular that
†
gr br + b¯r = gc .
(2.2)
r
Let kF1 and kF2 be the Fermi wavevectors of the two bands, and let Q = kF1 + kF2
be the wavevector of the CDW. Then, gr is gkqr evaluated at k = kF1 and q = Q;
gc , presently unknown, and to be determined, is the fundamental component of
the electron energy gap. br refers to bQr and b¯r refers to b−Qr . In real space, this
corresponds to a phonon wave with an amplitude y at position x of
y ∼ cos(Qx).
(2.3)
The Thomas–Fermi approximation for the gr is assumed, for which (see
Ashcroft & Mermin 1976)
ke Qe2
h̄ur
4p
2
gr =
.
(2.4)
Volume Q 2 + k02
2
Here, ke is the electric constant, 9 × 109 Nm2 C−2 , and Qe is the charge on the
electron. k0 will be called the screening wavevector.
The band structure is linearized about the Fermi level. Consider the two bands
shown in figure 1. Choose one of the bands to be band 1. States separated by the
fixed wavevector Q are paired. State |1 in band 1 at an energy e above the Fermi
level is paired with state |2 in band 2 at an energy pe below the Fermi level. The
parameter p is the ratio of the dispersion at the Fermi level of band 1 to that
of band 2. Similarly, state |3 is paired with state |4. Ec1 is the energy cutoff
for the pairing of |1 and |2; and Ec2 is the cutoff for the pairing of |3 and
|4. Summations over wavevectors in the Hamiltonian are replaced by integrals
E
D 0 c ()de, where D is the density of states at the Fermi level and Ec is the cutoff.
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168
D. W. Buker
energy
3
1
p
k
4
2
Figure 1. The two electron bands, showing states 1–4.
The part of the Hamiltonian concerning states |1, |2 is
†
†
†
†
H12 = ec1 c1 − (pe)c2 c2 + gc (c1 c2 + c2 c1 ).
(2.5)
Diagonalizing, one finds the ground-state energy E12 to be
1−p
2gc 2
1+p
2
e +
.
e−
E12 =
2
2
1+p
(2.6)
Integrating E12 over the energies e yields the total electronic energy for this part
of the band as
Eel1 = Eel1a − Eel1b ,
where
⎛
Eel1a = D1 ⎝
1−p
Ec21 −
4
and
Eel1b = D1
Proc. R. Soc. A (2011)
gc2
1+p
1+p
2
⎛
⎜
log ⎝
Ec1 +
Ec1
2
(2.7)
Ec21 +
2gc
1+p
Ec21 + (2gc /(1 + p))2
(2gc /(1 + p))
2
⎞
⎠
(2.8)
⎞
⎟
⎠.
(2.9)
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Charge-density waves
The pairings at (k → −k) of the |1, |2 pairing may be accommodated by taking
D1 to be the full density of states of band 1. Similarly, the electron energy Eel2
corresponding to pairs |3, |4 is the same expression as Eel1 , except that the
energy cutoff is now Ec2 .
Just as for the single-band case (see Buker 2009), the phonon energy is
Ephon =
D1 2
g ,
2leff c
(2.10)
where the effective coupling leff is given by
√ 2
ur
r
,
leff =
r ur
l
(2.11)
and where the dimensionless el–phon coupling l is given by
l=
D1 gr2
.
h̄ur
(2.12)
Minimization of the total energy, Eel + Ephon , with respect to gc leads now
directly to the basic two-band gap equation
⎡
⎛
1
2
⎢ ⎜1 + 1 +
=
⎣log ⎝
leff
1+p
x1
⎞
⎛
⎜1 +
⎠ + log ⎝
x12 ⎟
⎞⎤
1+
x2
x22 ⎟⎥
⎠⎦,
(2.13)
where x1 = (2gc /(1 + p))(1/Ec1 ) and x2 = (2gc /(1 + p))(1/Ec2 ).
Note that the proper coupling strength entity to use in the gap equation is
leff = fphon l, where the phonon factor fphon is given by
√ 2
ur
r
fphon = ;
r ur
(2.14)
leff is not generally expressible as a sum of independent functions of the
phonon frequencies.
The choice of which band to call band 1 leads to some variation in the
parameters used above. To clarify this, imagine that two experimenters, call them
Elliott and Finnegann, when faced with the same two-band system, happen to
choose different bands for their band 1. Then, when Elliott measures p, D1 , Ec1
and Ec2 , Finnegann measures p , D1 , Ec 1 and Ec 2 . The two sets of quantities are
related by p = 1/p, D1 = D1 /p, Ec 1 = pEc2 and Ec 2 = pEc1 .
Also, l and leff are experimenter-dependent quantities. If Elliott refers to l,
then Finnegann will be referring to l/p. To make each side of the gap equation
invariant under band interchange, one can multiply the whole equation by the
factor (1 + p).
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D. W. Buker
3. Specialization to a single band
Setting p = 1, and noting that l(1 band) = 2l(2 bands), then for a single band,
the gap equation can be simplified to
√
1 + 1 + x2
1
,
= log
leff
x
(3.1)
where x = gc /Ec . This can be solved for the gap to give
gc = Ec
1
sinh
leff
−1
.
(3.2)
This expression for the gap replaces the well-known weak-coupling result
gc = 2Ec e−1/l .
(3.3)
Considering the case of a single phonon band, so that leff = l, then the new
expression with its sinh dependence instead of exponential dependence on 1/l,
results in a 2 per cent increase in the gap at l = 0.5 and a 16 per cent increase
at l = 1.0.
The phonon factor, changing l to leff , carries vital information on the phonon
modes to the gap equation.
4. Further refinements
The present method allows the inclusion of other interactions. If there are energies
other than those so far considered, labelled as ‘Else’, then the total energy
will be
E = Ephon + Eel + Else.
(4.1)
Minimizing with respect to the gap, gc leads to the general gap equation
√
1
1 + 1 + x2
v(Else)
1
−
= log
.
x
Dgc
vgc
leff
(4.2)
The major influences so far unaccounted for are the second harmonic
interaction and the el–el interaction. Each of these interactions turns out to
yield quite a large correction with use of the above equation. However, the
additional terms depend relatively sensitively on the energies of states far away
from the Fermi surface (FS). In the present model, the bands have been linearized
about the Fermi level, so that these effects cannot here be accurately evaluated.
For transparency, the form the el–el correction takes is briefly given in the
next section.
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5. Electron–electron interactions
The el–el interaction Hamiltonian may be written as (Kittel 1987)
1
1
4pke Qe2
†
†
ck+qs ck −qs ck s cks .
Hel–el =
2
2
2
Vol
q + k0
k,k ,q,s,s
(5.1)
Here, the ck operators are plane-wave electron operators. Thus, the direct
Coulomb energy associated with the fundamental component of the CDW is
1
4pke Qe2
†
†
H direct =
ck+Q ck ck −Q ck .
(5.2)
Vol
Q 2 + k02
k
k
This time the summation over k is meant to also include spin. As an
approximation, the c operators can be taken to be electron operators for the
Bloch states of the band. Bloch-state corrections could be entertained once these
are known more accurately. The exchange energy is smaller, vanishing in the
infinite-volume limit.
6. Application to tetrathiofulvalinium–tetracyanoquinodimethane
For a comparison of theory and experiment, it is convenient to take the measured
value of the gap and use the gap equation to find the expected coupling leff . The
comparison is then between this value and fphon lbare , where in the Thomas–Fermi
approximation with k0 = Q/2, from equation (2.12)
1
1
2.19 × 106 m s−1
16
.
(6.1)
lbare =
5
a2 a3
Q2
vF1
For TTF–TCNQ, the gap determined experimentally is 2gc ≈ 40 meV,
according to Sing et al. (2003). From the density-functional band structure of
Sing et al. (2003), I find Ec1 = 0.36 eV, Ec2 = 0.3 eV and p = 1.4. This yields
x1 = 0.046 and x2 = 0.056. With these values, the basic gap equation (2.13)
predicts leff = 0.16, referring to the shallow band.
For the cross-sectional area a2 a3 per chain, Allender et al. (1974), give
114 × 10−20 m2 . From Sing et al. (2003), I find vF1 = 1.7 × 105 m s−1 , referring
to the shallow TCNQ band. With Q = 0.485 × 1010 m−1 , one has, from equation
(6.1), lbare = 0.384. The phonon factor fphon is at least unity, so that the coupling
expected from the band structure is quite a lot greater than that found using
the gap equation. However, as mentioned in §4, the second-harmonic and el–el
corrections are large, and require, for their accuracy, energy bands known more
accurately than a linear approximation at the FS.
7. Application to K-blue bronze
From a scanning tunnelling microscope experiment (Tanaka et al. 1993), the
CDW gap is 2gc = 140 ± 20 meV. From the density functional band structure of
Mozos et al. (2002), I find Ec1 = 0.23 eV, Ec2 = 0.3 eV and the ratio p of the two
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172
D. W. Buker
dispersions at the Fermi level to be p = 3.7. The gap equation (2.13) then yields
leff = 0.41, referring to the shallow band. For comparison to experiments, where
both bands are accounted for in the density of states, i.e. multiplying by the
factor ((1 + p)/p), the full el–phon coupling is leff,2 bands = 0.52.
Degiorgi et al. (1995) give the experimental coupling as lexptl = 0.5.
The apparent agreement of theoretical and experimental values for the gap is,
at this stage, somewhat illusory. For there is still the effect of el–el interactions
to consider. Also the phonon factor, fphon should be accounted for. But this is set
aside until more information on the phonon modes is available.
8. Conclusion
An equation has been found, for quasi-one-dimensional systems when two electron
bands are connected by the el–phon interaction, relating the CDW energy gap
to band-structure data above the transition temperature. The relation is valid
for all coupling strengths. When specialized to a single band, the theory yields a
simple expression for the energy gap.
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