41 charge density wavebingweb.binghamton.edu/.../40_charge_density_wave.pdf2012/01/13 · 1 charge...
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1
Charge density wave Masatsugu Sei Suzuki and Itsuko S. Suzuki
Department of Physics, SUNY at Binghamton (Date: January 13, 2012)
Sir Rudolf Ernst Peierls, CBE (June 5, 1907, Berlin – September 19, 1995, Oxford) was a German-born British physicist. Rudolf Peierls had a major role in Britain's nuclear program, but he also had a role in many modern sciences. His impact on physics can probably be best described by his obituary in Physics Today: "Rudolph Peierls...a major player in the drama of the eruption of nuclear physics into world affairs.
http://en.wikipedia.org/wiki/Rudolf_Peierls ((John Bardeen (1941)))
Many ideas of CDWs were developed in early attemmpts to explain superconductivity. In 1941, John Bardeen suggested that "in the superconducting state there is a small periodic distortion of the lattice" that produces energy gaps, and that these gaps would lead to enhanced diamagnetism. Bardeen abandoned this idea when he realized the difficulty of obtaining an appropriate arrangement of gaps on the three-dimensional Fermi surfaces of common superconductors. ((J. Bardeen, Phys. Rev. 59, 928 (1941))) Proceedings of the American Physical Society, Minutes of Washington DC, Meeting May 1-3, 1941.
The energy discontinuities produced by the zone structure yield a decrease in the energy of the electrons at the expense of the increase in energy of the lattice resulting from the distortion. A rough estimate of the interaction between the electrons and the lattice obtained from the electrical conductivity in the normal state indicates that the superconducting state may be stable at low temperatures. The most favorable metals are those which have a high
2
density of valence electrons in a wide energy band and which have a large interaction between electrons and lattice (low conductivity). ___________________________________________________________________________ 1. One-dimensional energy band (1) Regular lattice
Suppose that the system consists of N atoms, forming a linear chain along the x direction. They are periodically arranged such that the distance between the nearest neighbor atoms is a. The size of the system is L = Na.
Fig. One dimensional chain of atoms where the nearest neighbor distance is a.
The energy gap appears at the Brillouin zone boundary )(a
k
. The energy gap is fixed at
this reciprocal lattice point. In this case there are 2N states for the first Brillouin zone
)(a
k
. The factor 2 comes from the spin of electrons. When each atom has two electrons,
there are 2N electrons in the system. Then the band is filled up to the Brillouin zone (insulator). When each atom has one electrons, there are N electrons in the system. Then the band is half-filled in the energy band (metal).
Fig. Energy band for the system where there are two electrons per atom. All states are
occupied up to the zone of the Brillouin zone (insulator)
a
k
Ek
pa
-pa
O
3
Fig. Energy band for the system with one electron per atom. All states are occupied for
|k|<kF (= /a) in the Brillouin zone (metal) (ii) Effect of lattice distortion
We still assume that there is one electron per atom in the linear chain. Now let us displace
every second atom by a small distance .
Fig. Lattice constant changes from a to 2a due the lattice distortion. This reduces the symmetry to that of a chain with spacing 2a, and the potential acquires a
Fourier component of wave number /a which in this case is equal to 2kF. This results in an
energy gap at k = kF = /2a, in accordance with the change in the periodicity from a to 2a. In this case, all states raised by the change are empty, and all states lowered are occupied, so the system becomes insulator.
k
Ek
pa
-pa
kF=p
2 a-p
2 aO
2 a
4
Fig. Energy band for the system with one electron per atom, after the lattice distortion.
The energy gap appears at k=kF = ±/2a. The system changes from metal to insulator (Peierls instability).
((Peierls, More surprise in theoretical physics, 1991))
This instability came to me as a complete surprise when I was tidying material for my book (Peierls 1955), and it took me a considerable time to convince myself that the argument was sound. It seemed of only academic significance, however, since there are no strictly one-dimensional systems in nature (and if there were, they would become disordered at any finite temperature. I therefore did not think it worth publishing the argument, beyond a brief remark in the book, which did not even mention the logarithmic behavior.
It must also be remembered that the argument relies on the adiabatic approximation, in which the atomic nuclei are assumed fixed. If their zero-point motion were taken into account, the answer might change, but this would be a difficult problem to deal with, since it involves a strongly nonlinear many-body problem. 2. Elastic energy due to the lattice distortion
Then change of the total energy of the system consists of
(i) the change of energy in electrons (Eelectronic<0) which decreases because of the appearance of energy gap.
(ii) the change of energy associated with the lattice distortion (Eelastic>0)
The total energy E is given by
latticeelectronic EEE .
k
Ek
kF=p2a-kF=-p2a O
5
If E <0, the lattice distortion occurs. This is predicted by Peierls for an ideal one dimensional conductor.
The charge density wave has a periodic function of x with the periodicity . We consider the elastic strain
)2cos()cos( xkQx F ,
where
F
F
k
k
Q 222
Note that kF is a general value and is not always equal to /2a. The spatial-average elastic energy per unit length is
44
1
)]4
cos(1[4
1
)2(cos2
1
)2(cos2
1
22
0
2
0
22
22
CC
dxxC
dxxkC
xkCE
F
Felastic
where C is the force constant of the linear metal. We next calculate Eelectronic. Suppose that the ion contribution to the lattice potential seen by a conduction electron is proportional to the deformation,
iQxiQxF eVeVxkVxU 000 )2cos(2)(
where AVUQ 0 (see Kittel, ISSP, p.422- 423).
3. Bragg diffraction; Ewald sphere (a) Typical Bragg reflections:
6
Fig. Ewald sphere: kkq ' ; k'=k. The Bragg reflection occurs when q = G. q is the
scattering vector and G is the reciprocal lattice vector (b) The case for the charge density waves;
Fig. Ewald sphere: kkq ' ; k'=k. The Bragg reflection occurs when q = Q. q is the
scattering vector and Q is equal to 2kF. Experimentally the Bragg reflection appear at
Fklaq 2*
O
G G=2kF
k k =kF k' k'=kF Origin of RL
AB
O
Q
kk'Origin of RL
AB
7
where a
a2* is the reciprocal lattice and a is the lattice constant in the absence of the
lattice distortion. The Bragg intensity is proportional the square of the order parameter (energy gap). Note that
FF k
a
kaa 22
12 *
.
If this ratio is a rational number (= p/q; p and q are integers).
q
p
a
.
There are q CDW waves in the p lattice distances. This is called the commensurate CDW. If this ratio is irrational, this is called the incommensurate CDW. 4. Calculation of the change of energy near the regions at k = kF
8
Now we consider the simplest case: mixing of only the two states: k and Gk (k≈kF),
k - G ≈ - kF, G = 2kF). The wavefunction is approximated by
GkCkC Gkk .
where only the coefficients Ck and Ck-G are dominant. The central equation (eigenvalue problems) is
9
0
0*
Gk
k
GkG
Gk
C
C
U
U
.
where
22
2k
mk
, 2
2
)(2
GkmGk
From the condition that the determinant is equal to 0, we have
0))((2 GGkk U ,
or
2
4)(22
)( GGkkGkkk
U
.
Note that the potential energy U(x) is described by
)cos(2)(00 GxUUeUeUUxU G
iGxG
iGxG ,
where we assume that UG is real:
0* VUUU GGG .
Then we have
20
2222
422
2)( })({[
16])([
4VGkk
mGkk
mk
We retain the minus sign to get minimum energy. The reduction of the lower energy is
10
20
2222
422
2
20
2222
422
22
2
)()1(
})({[16
])([4
})({[16
])([42
)(
VGkkm
Gkkm
VGkkm
Gkkm
km
kE kkelectronic
To get the reduction of the total energy per unit length, we have to integrate from -kF to kF,
multiplied by (1/2), The main contribution comes from the neighborhood of k = kF. We assume that V0 is constant. Using the relation
)(4)(2))(()( 22FFF kkkkkGGkkGkkGkk
we get
20
22)( VkE selectronic
where
m
kF2
, kkF
We note that
1)(
20
22
00
VVV
kE selectronic
As is expected, the value of 0
)(
V
kE selectronic is negative. It increases with increasing
0V
.
11
The approximations are valid only in the neighborhood of k = kF. So we need to restrict the integration to a maximum value of k, k0, such that
FkkV
00
Then we have
1
0 0
20
2220
22)1( )(1
)(2
2
dVdkVEFk
k
electronic
where the factor 2 comes from the spin degree of the electron.
01 kkF
5. Calculation of change of energy in the regions near k = -kF.
Here we show that the contribution from the regions near k = -kF is the same as that from the regions near k = kF as shown in the above. give equal contributions
0.5 1.0 1.5 2.0 2.5 3.0akV0
-1.0
-0.8
-0.6
-0.4
-0.2
0.0DEelectronickV0
12
We consider the case when k ≈-kF (<0), k+G = kF with FkG 2 . Since
2
4)( 20
2)( VGkkGkk
k
we have
20
2222
422
2
20
2222
422
22
2
)()2(
})({[16
])([4
})({[16
])([42
)(
VGkkm
Gkkm
VGkkm
Gkkm
km
kE kkelectronic
Noting that
)(4)(2))(()( 22FFF kkkkkGGkkGkkGkk
we get
13
20
22)2( )( VkE selectronic
where
m
kF2
, kkF
Then we get
10
0
20
2220
22)2( )(1
)(2
2
dVdkVEk
k
electronic
F
where the factor 2 comes from the spin degree of the electron.
01 kkF
So it is found that
)1()2(electronicelectronic EE
6. The change in the total energy due to the lattice distortion
The total energy is given by
14
]2
ln24
11[
2
)4
2(ln
ln)8
1
2
1(
ln11ln)11(
)ln()(
)(2
22
0
12
12
20
20
21
2
20
20
1
02
02
12
20
20
1
02
02
12
20
20
21
2
20
21
0
21
2201
202
122
011
0
20
22
)1()1(
1
V
VV
VV
VVVV
VVVVV
V
VVV
dV
EEE electronicelectronicelectronic
or
]2
ln24
11[
2 0
12
12
20
20
V
VVEelectronic
The important feature of this result is that it behaves for small V0 as
1
02
0
20
0
12
0
0
12
0
ln
2lnln
2ln
VV
V
V
V
V
VEelectronic
For small displacement, V0 is proportional to the displacement .
AV 0 ,
where A is constant (see Kittel, ISSP, p.422- 423). The behaviors of the reduction in electronic energy for small displacement is
15
AA
Eelectronic ln22
<0
This is interesting because there may be other effects favoring the regular spacing, = 0,
such as the repulsion between the atomic cores, but these will have an energy varying 2. Thus the electronic energy must dominate for small displacement. This suggests that the periodic chain must always be unstable. The change in the total energy E is given by
42ln
4]
2ln2
4
11[
22
1
22
2
12
12
2222
CAA
CAAA
EEE elasticelectronic
In order to find the minimum value of E, we take a derivative of E with respect to .
0)]2
ln(42[2
1)(
1
22
A
ACAd
Ed.
Then we get
)4
exp(21306.1
)4
exp()2
1exp(2
21
21
A
CA
CA
or
)4
exp(21306.1 2
21
2
mA
Ck
m
kA FF
This expression is almost the same as that derived by Kittel,
)4
exp(2
2
222
mA
Ck
m
kA FF
(Kittel, ISSP)
16
7. Electronic density )(x
At finite temperatures, there is a finite probability that a part of electrons is excited from the
lower state ( )(k )to the higher state ( )(
k ).
We consider the state given by
QkCkC Qkk
)()()(
or
xQkiQk
ikxkk eCeCxx )()()()()( )(
17
where
0
0*
Qk
k
QkQ
Qk
C
C
U
U
.
The eigenvalues are )(k and )(
k , defined by
]2
1[22
2
4)(
2
2
22
)(
Qkk
QQkkQkk
QQkkQkk
k
U
U
and
12)(2)(
Qkk CC . (Normalization).
The charge density at x is given by
k
kkkk ffN
x })()({1
)(2)()(2)()( ,
where f is the Fermi-Dirac distribution function. When Qkk ,
kQk
Q
k
Qkk
QkQkQkkk
U
U
2
2
2
)( ]2
1[2
)(
2
and
18
kQk
Q
Qk
Qkk
QkQkQkkk
U
U
2
2
2
)( ]2
1[2
)(
2
When 02QU ,
)()( )(kk ff , )()( )(
Qkk ff .
The probability is given by 2)( )(xk
,
iQxQkk
iQxQkk
iQxQkk
iQxQkkQkk
iQxQkk
xQkiQk
ikxkk
eCCeCC
eCCeCCCC
eCC
eCeCx
*)()()(*)(
*)()()(*)(2)(2)(
2)()(
2)()()(2)(
1
)(
Then the density is
]}1)[(
]1)[({1
})()({1
)(
*)()()(*)(
*)()()(*)(
2)(2)(
iQxQkk
iQxQkkQk
k
iQxQkk
iQxQkkk
kkQkkk
eCCeCCf
eCCeCCfN
ffN
x
What is the value of *)()(
Qkk CC ? We note that
0
0)(
)(
)(
*)(
Qk
k
kQkQ
Qkk
C
C
U
U
.
or
19
0)( )(*)()(
QkQkkk CUC ,
or
*
)()()( )(
Q
kkkQk
U
CC
.
Then we get
Qkk
kQ
Qkk
Q
Q
k
Q
kkk
Qkk
CU
U
U
C
U
CCC
2)(*
22)(
2)()(*)()(
)(
since
kQk
Q
kk
U
2
)(
Similarly,
0
0)(
)(
)(
*)(
Qk
k
kQkQ
Qkk
C
C
U
U
,
or
0)( )()()(
QkkQkkQ CCU .
Then we get
20
kQk
kQ
kQk
Q
Q
k
Q
kQkk
Qkk
CU
U
U
C
U
CCC
2)(
2
*
2)(
*
2)(*)(*)()(
)(
since
kQk
Q
Qkk
U
2
)( .
The electron density can be rewritten as
]}(1)[(
](1)[({1
)(
*
2)(
*
2)(
iQxQ
iQxQ
kQk
k
Qk
k
iQxQ
iQxQ
Qkk
k
k
eUeUC
f
eUeUC
fN
x
Suppose that QU is real. Thus we have
)]}cos(2
1)[(
)]cos(2
1)[({1
)(
2)(
2)(
QxUC
f
QxUC
fN
x
kQk
Qk
Qk
k Qkk
Qk
k
It is appropriate that 2
12)(2)( kk CC for 0QU .
21
kQ
Qkk
Qkk
kQkk
QxUff
N
ffN
x
)cos(])()(
[1
)]()([1
)(
We define (Q, T) as
k Qkk
kQk ff
NTQ
)()(1
),( .
Then we have
)cos(),()( 0 QxUTQx Q .
The Fourier component of )(x is given by
QQ UTQ ),( .
We note that the potential energy is given by
iQxQ
iQxQQ eUeUQxUxU
)cos(2)( .
where
realAVUUU QQQ 0* .
8. Calculation of )0,( TQ
We now calculate the susceptibility
22
kk
kk
k k Qkk
k
kQk
k
k k kQk
Qk
kQk
k
k kQk
Qkk
kQkQf
Q
m
kQQkQQf
m
ff
ffffTQN
)2
1
2
1)((
2
]2
1
2
1)[(
2
)()(
)()()()(),(
2
222
' ''
'
At T = 0 K, 1)( kf for k<kF and 0 for k>kF.
F
F
F
F
k
k
k kQk
Qkk
kQ
kQ
Q
mLQ
k
Qk
Q
mL
Qk
Qk
dkQ
mL
kQkQdk
Q
mL
ffTQN
F
F
2
2ln
2
2
2ln2
)
2
1
2
1(
2
12
2
2
2
)2
1
2
1(2
2
2
2
)()(),(
22
02
02
or
12
12
ln
2
2
2ln2
),(
2
2
F
F
FF
F
F
k
Qk
Q
k
Qnk
m
Qk
Qk
QN
mLTQ
where we take into account of the degree of spin (the factor 2). We make a plot of the function defined by
23
1
1ln
1),()(
0
x
x
x
TQxf
where x = Q/(2kF), and
20 Fnk
m
The susceptibility is found to diverge at Q = 2kF.
Fig. Scaling plot of 1
1ln
1)(
x
x
xxf with
Fk
Qx
2 .
9. The density of state at the Fermi energy for the 1D system
Q
cQ,T=0c0
2kFkFO
24
Fig. Peierls instability. Electrons with wave number k near k = kF have their energy lowered by a lattice deformation. When the length of the system is L, the allowed
value of k is k = l (2/L), where l is 0, ±1, ±2,.... In this figure (T = 0 K), the states in the lower energy band is fully occupied, while the states in the upper energy band are empty.
The lattice distortion with wavelength gives rise to an energy gap at
k . If Fkk ,
Fk
.
Then the system changes from conductor to insulator. All the states ,k (spin-up state) and
,k (spin-down state) with |k|≤kF are occupied in the lower band. Then we have
FF kL
kL
N2
)2(2
2 ,
k
Ek
kF-kF O
25
where the factor 2 comes from the spin degree, and N is the total number of electrons in the system (L). Note that N is not the number of unit cell in this case. The number density n is defined by
L
Nn .
Using the above relation, we get
Fk
n2
, or , 2
nkF
.
We note that
nnk
k
F
F
2
2222
.
This means that here are two electrons per wavelength .
The density of states is defined by
dkL
dkL
dD
2
2
22)( ,
where we take into account of (i) the spin factor (2) and (ii) the even function of k for the energy dispersion. The energy dispersion of the free electron is
22
2k
m
,
2
22
dmdk
.
l
26
Using the relation
dmLdmL
dkL
dD22
2
2
222)(
,
we have the density of states as
1
2
22)(
2 LmmL
D .
The density of states at the Fermi energy is
FFFFFF
N
k
mN
k
mLmLmLD
2
222)( 22222
.
10. The relation 0
1),2(
TkQ F
What is the temperature dependence of the susceptibility?
k kkk
kkkF
F
Fff
NTkQ
2
2 )()(1),2(
We assume that kkF
FF
FF
F
F
F
Fk
kkm
k
km
k
km
km
)2
1()2
1(2
)1(2
)(22
22
222
22
22
27
FF
FF
F
F
F
FF
Fkk
kkm
k
km
k
kkm
kkmF
)2
1()2
1(2
)1(2
)2(2
)2(2
22
222
22
22
2
where
)()(22 2
kkm
kkk
kkF
FF
F
F
F
F
This leads to
2)(2 FFkkk F.
The numerator:
1
1
1)(exp[
1)(
22
ef
Fkkkk
F
F
1
1
1)(exp[
1)(
e
fFk
k
Then we get
)2
tanh(1
1
1
1
1
1)()( 2
e
e
eeff kkk F
and
2
)2
tanh()()(
2
2
F
F
kkk
kkk ff
28
Then we have
0
0
0
0
0
0
)2
tanh(
2
)(
2
)2
tanh()(
1
2
)2
tanh()(
1
2
)2
tanh(1),2(
dxN
D
dDN
dDNN
TkQ
F
F
Fk
F
or
)13387.1ln(2
)(
1
2
)(),2(
0
0
TkN
D
N
DTkQ
B
F
FF
where
)13387.1ln()
2tanh(1 0
00
0
Tkdx
B
Note that
dkm
kd F
2
)( 0
2
0 kkm
kF
F
k0 is the lower limit of k (which was discussed above). The magnitude of 0 is not essential in our discussion. 11. The critical temperature Tc
29
The energy gap is equal to zero at T = Tc. The critical temperature Tc is derived from the energy gap equation. The derivation of Tc is almost the same as that derived for the BCS model of the superconductivity. We start with the energy gap equation with zero energy gap,
)2
ln(
])4
[ln()2
tanh()2
ln(
)(cosh
ln)]tanh([ln
)(cosh
ln)]tanh([ln
)tanh()2
tanh(1
0
00
02
)2/(0
)2/(
02
)2/(0
)2/(
000
0
0
0
00
e
Tk
TkTk
d
d
d
Tk
d
cB
cBcB
Tk
TkTk
Tk
cB
cB
cB
cB
cB
,
where = 0.577216 is the Euler's constant. Then we get
81878.0)2
ln(1 0
0
cBTk
since 81878.0])4
[ln( . This equation can be rewritten as
]1
exp[881939.0
)81878.0exp(]1
exp[2
0
0
0
cBTk
or
)1
exp(13387.10
0 cBTk .
________________________________________________________________ 12. The energy difference at finite temperatures
30
),(
])()([
[
])()(
[
)}()({
)}(][)(]{[
)}()({)}()({
2
2
22
22
)()()()(
TQNU
ffU
fUfU
ff
fU
fU
ffffE
Q
k kQk
QkkQ
k kQk
QkQ
kQk
kQ
kQkQkkk
kQk
kQk
Q
QkkkQk
Q
k
kQkQkkk
kkkkkelectronic
where
kQk
Q
kk
U
2
)( , kQk
Q
Qkk
U
2
)(
)()( )(kk ff , )()( )(
Qkk ff
and
k kQk
Qkk
k Qkk
kQk ffff
NTQ
)()()()(1
),(
. Then we have
),(2
TQNUE Qelectronic
The total energy of both electron and lattice is given by
31
}4
),({
4),(
4),(
2
2
2
2
2
A
CTQNU
UA
CTQN
CTQN
EEE
Q
Q
elasticelectronic
where AVUQ 0 . When Q → 2kF and T →0, ),( TQ shows a logarithmic divergence.
Therefore, below a characteristic temperature Tp, E becomes negative, leading to the Peierls
instability. Using the expression for ),2( TkF ,
)133887.1ln(2
)(),2( 0
TkN
DTkQ
B
FF
we get
)13387.1ln(2
)(
4),2( 0
2 TkN
D
NA
CTkQ
B
FF
or
)1
exp(13387.1])(2
exp[13387.10
020
Fc DA
CT
where 0 is a dimensionless electron-lattice interaction parameter,
C
DA F )(2 2
0
.
13. Kohn anomaly
At temperature which is sufficiently higher than the Peierls transition temperature, the
angular frequency (k = 2 kF) of the phonon dispersion curve drastically decreases, showing the softening of the phonon mode. This behavior is called the Kohn anomaly. This behavior can be explained qualitatively as follows. The electrons are influenced through the potential UQ. due to the lattice distortion. The electronic charge is newly generated in the form of
QQ UTQ ),( .
32
The force exerted on the phonon is weakened by the electron-phonon interaction. This leads to the decrease in the restoring force of the lattice distortion. The angular frequency of phonon at Q becomes softening and is reduced to zero at the critical temperature. In summary, the lattice wave with Q gives rise to the electric potential with Q, which works for the electric
charge Q, turns back to the lattice wave with Q as a positive feedback. This leads to the restoring force of the lattice.
We discuss the time dependence of the displacement Q of the phonon mode with Q and
Q . Noting that
QQ UTQ ),( , QQQ vU
we get the equation of motion for Q as
QQQ
QQQQ
QQQQQ
TQv
TQv
vdt
d
)],([
),(22
22
22
2
2
where vQ is the strength of the electron-phonon interaction, and AUQ in thermal
equilibrium. Then we have the characteristic angular frequency
),(222 TQvQQQ
Q reduces to zero when
2
2
),(Q
Q
vTQ
at Q = 2kF. Using
)13387.1ln(2
)(),2( 0
TkN
DTkQ
B
FF
we have
33
)13387.1ln(2
)( 02
2
TkN
D
v B
F
Q
Q
or
]1
exp[13387.10
0 cT ,
where
2
2
02
)(
Q
FQ
N
Dv
.
This is equal to
20
)(2
CA
D F ,
which is obtained in the previous section.
Fig. Schematic diagram of acoustic phonon dispersion relation of a one dimensional metal
at various temperatures above Tc.
Q
WQ
2kF
T>Tc
T=Tc
34
14. Order parameter
The Peierls instability occurs in a one-dimensional metal. This instability gives rise to the combination of the electronic density wave and lattice wave with the same wave number 2kF. This combined waves are called the charge density wave (CDW). The appearance of the CDW state can be experimentally found by x-ray and neutron scattering.
In the CDW state, the order parameter is the energy gap. This gap is equal to zero at T = Tc and increases with decreasing temperature. We consider the temperature dependence of the energy gap below the critical temperature. We start with
2
2
),(Q
Q
vTQ
We need to calculate the T dependence of ),( TQ .
k kk
kkF
ff
NTkQ )()(
)()( )()(1),2(
2
4)(22
)( GGkkGkkk
U
FFF
FFF
FFF
FFk
kk
kkm
k
kkkkm
kkm
)(
)(
))((2
)(2
2
2
222
FFF
FFF
FFF
FFGk
kk
kkm
k
kGkkGkm
kGkm
)(
)(
))((2
])[(2
2
2
222
35
FGkk 2 , )(2 FGkk kk
Here we assume that
m
kF2
. FkG 2 Fkk
)( kkF , QU
Then we get
22)( Fk
1)exp(
)exp(
1)exp(
1)(
22
22
22
)(
kf
1)exp(
1)(
22
)(
kf
)2
tanh(
1)exp(
1)exp()()(
22
22
22)()(
kk ff
kk
F E
E
NNTkQ
)2
tanh(
2
1)2
tanh(
2
1),2(
22
22
where
22 E
)( 00 kkF
dkd
36
0
0
0
0
22
22
)2
tanh(
2
)(
)2
tanh()(
2
1
)2
tanh(
2
1),(
E
E
dN
D
E
E
dDN
NTQ
F
k
We note that
)14.1ln(1
),2()(
2 0
0 TkTkQ
D
N
BF
F
Then we have
)14.1ln()
2tanh(1 0
00
0
Tkd
B
and
0
00
)2
tanh(1
E
E
d.
14. Calculation of order parameter
00
0
0022
22
022
22
0
)2
tanh(}
)2
tanh()2
tanh({
)2
tanh(1
Tkd
TkTkd
Tkd
BBB
B
37
Then we get
0
00
0
0
22
22
0
022
22
0
022
22
0
)]2
tanh(1
)2
tanh(1
[)2
ln(
})
2tanh()
2tanh(
{)
2tanh(
)2
tanh(1
TkTkd
e
Tk
TkTkd
Tkd
Tkd
BBB
BBB
B
Using the expansion formula
0222 )12(4
8)tanh(
n nx
xx
we get
0 0222222
20
22
2
22
22200
0
0
0
0
])12([4)
2ln(
]
)12(2
4
1
)12(2
4
1[
4)
2ln(
1
n B
BB
B
B
nBB
nTk
dTk
e
Tk
nTk
nTk
Tk
e
Tk
Here
38
3233
0222233
/
0222233
0222222
)12(4
11
])12[(
1
])12[(
1
])12([
00
nTk
xn
dx
Tk
xn
dx
TknTk
d
B
B
Tk
BB
B
Then we have
)3(8
7)
2ln(
)12(
1
4
4)
2ln(
1
222
20
03222
20
0
Tk
e
Tk
nTk
e
Tk
BB
nBB
where
)3(8
7
)12(
1
03
n n,
and (3) = 1.2206. Using the relation
)2
ln(1 0
0
e
Tk cB
,
we have
)3(8
7)
2ln()
2ln(
222
200
Tk
e
Tk
e
Tk BBcB
,
or
)3(8
7)ln(
222
2
TkT
T
Bc
.
or
39
)ln()3(7
82
22222
T
T
T
TTk c
c
cB
From this we get the expression for as,
2/1
2/12
2/12
)1(06326.3
)1(])3(7
8[
)ln(])3(7
8[)(
ccB
ccB
cccB
T
TTk
T
TTk
T
T
T
TTkT
((Mathematical note))
Suppose that
xT
Tt
c
1
where x is close to x ≈ 0 (but x>0)
xxxxx
xx
tt
tt
T
T
T
T c
c
)(03
1
2
3
)1ln()1(
)ln(
)1
ln()ln(
432
2
2
2
2
In the vicinity of T = Tc (but T<Tc), we have a good approximation,
2/1
2
)1()ln( tT
T
T
T c
c
16. Critical behavior of the order parameter (energy gap)
The energy gap is obtained as
40
2/1
2/1
2/1
2/1
)1(06326.3
)1()3(7
8
)ln()3(7
8)(
ccB
ccB
c
ccB
T
TTk
T
TTk
T
T
T
TTkT
The energy gap at T = 0 K can be evaluated from the energy gap equation at T = 0 K
]2
ln[
]ln[
1
0
0
0
20
200
02
02
0
0
d
or
)1
exp(20
00 .
Together with the relation
)1
exp(13387.10
0 cBTk ,
we get the universal relation
52774.313387.1
42 0
cBTk.
The energy gap is the order parameter. Since this order parameter continuously changes with T and reduces to zero. The phase transition is of the second order with the critical exponent = 1/2.
41
2/12/1
0
)1(73667.1)1(52774.3
06326.32)(
cccB
cB
T
T
T
T
Tk
TkT
Fig. Plot of the energy gap /0 as a function of a reduced temperature t = T/Tc. ______________________________________________________________________ REFERENCES R.E. Peierls, Ann. Phys. Leipzig 4, 121 (1930). R.E. Peierls, Suprises in Theoretical Physics (Princeton University Press, New Jersey, 1976) R.E. Peierls, More Surprises in Theoretical Physics (Princeton University, New Jersey, 1991). R.E. Peierls, Quantum theory of solids (Oxford, University Press, 1955). J. Bardeen, Phys. Rev. 59, 928 (1941). H. Frölich, Proc. R. Soc. A 223, 296 (1954). C. Kittel, Introduction to Soild State Physics, 8-th edition (John Wiley & Sons, 2005). G. Grűner, Density Waves in Solids (Addison-Wesley, New York, 1994). G. Grűner, Rev. Mod. Phys. 60, 1129 (1988). S. Kagoshima, H. Nagasawa, and T. Sambongi, One dimensional Conductors (Springer
Verlag, 1988). R.E. Thorne, Phys. Today 42, May Issue (1996). S. Kagoshima, T. Ishiguro, and H. Anzai, J. Phys. Soc. Jpn, 41, 2061 (1976). J.P. Pouget, B. Hennion, C.E.-Fioppini, and M. Sato, Phys. Rev. B 43, 8421 (1991). M. Sato, H. Fujishita, and S. Hoshino, J. Phys. C 16, L877 (1983). _______________________________________________________________________ APPENDIX Typical quasi-one dimensional metal 1. TTF-TCNQ
0.2 0.4 0.6 0.8 1.0 1.2t=TTc
0.2
0.4
0.6
0.8
1.0
DTD0
42
The tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ) is not a simple metal. Above 58 K, there is an energy gap at g = 0.14 eV and an extremely narrow conductivity
mode centered at zero frequency. Near 53 K, TTF-TCNQ undergoes a metal-insulator transition to a high-dielectric constant semiconductor in which the oscillator strength is shifted from zero frequency and pinned in the far infrared. In an earlier work.
2. Blue bronze K0.3MoO3
The quasi-one-dimensional (1D) conductor K0.3MoO3 undergoes a Peierls transition at T =183 K. Using cold-neutron scattering, Pouget have succeeded in resolving in frequency and for wave vectors parallel to the chain direction the pre-transitional dynamics and the collective excitations of the phase and of the amplitude of the charge-density-wave (CDW) modulation below T. The pre-transitional dynamics consists of the softening of a Kohn anomaly at the wave vector 2kF together with the critical growth of a central peak in the vicinity of T, . In addition we observed just above T, the beginning of a decoupling between the fluctuations of the phase and of the amplitude of the CDW.
43
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