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15 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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Electronic instabilities

Electron phonon





BCS superconductor





Localization in 1D
-

CDW


Electron
-
electron

(
-
癥⁥硣桡湧攩


d
-
wave superconductor





Localization in 1D
-

SDW


Electron
-
electron

(+ve exchange)


p
-
wave superconductor





Itinerant ferromagnetism

Why is the Fermi surface so important?

Energy conservation


Momentum conservation


Exclusion principle

Interactionmediated by
virtual particles:

phonons

magnons

polarons

plasmons

Pairing or cooper
-

ative interaction

Pairing favors states with opposite momenta within
a shell of thickness
k
B
T about the Fermi surface

If
K

=
k
1

+
k
2

= 0,

then

k
1

=
-
k
2

and
k
´
1

=
-
k
´
2


q

= 2

k
1

=
2

k
2


Now,
k
1

and
k
2

can scatter
into any states in the inter
-
action shell with opposite
momenta

Superconductivity

Explains condensation into
K

= 0 pairs


The pair may have orbital momentum

name


l


spin state

s

-

睡癥w


0


獩湧汥s

p

-

睡癥w

1


瑲楰t整

d

-

睡癥w

2


獩湧汥s

Electron
-
phonon coupling favors BCS
s
-

wave

Coulomb repulsion favors
l



〠獴慴0s


桥湣攠
p
,
d

wave in strongly correlated systems

So, let’s consider the electron
-
phonon interaction first

What about low
-
dimensionality?

In the limit dimensionality


1

A single
q

(=
±
2
k
F
) can map one surface onto the other


Thus, essentially all exchange phonons have same
q

One
-
dimensional Fermi surface not essential

All that is needed are reasonable flat and parallel

sections of Fermi surface,
i.e.

an appreciable # of states
connected by the same
q


Under these circumstances, interactions are enhanced
further still, becoming singular in one
-
dimension

In BCS case, Interaction renormalizes phonon

dispersion

[w
(
q
)]
2

=
[w
o
(
q
)]

2

-

C

c(
q
)

w
o
(
q
) is the bare phonon frequency, C is a

constant related to density of states

[w
(
q
)]
2

=
[w
o
(
q
)]

2

-

C

c(
q
)


c(
q
)
diverges in 1D as T


0

Kohn anomaly

2
k
F

phonon “softens”


As T


0


v
g

and v
f



0


Result: Static lattice

distortion

2
k
F

Lattice distortion in 1D

Consequently, new zone boundary @
k

=
±


a


i.e.

at
k
F

For half filled band,
k
F

=


a,

so
q

=

/
a
,
l

㴠㉡

So, the distortion is twice the lattice spacing

Charge
-
Density
-
Wave (CDW)

Metal insulator transition

New zone

boundary

@
±
2

/
a


The opening of a gap, lowers the energy of the

system

States at zone boundary

lower in energy

If energy gain exceeds

electrostatic energy

increase associated with

lattice distortion, then

ordered state wins

Why does the system do this?

In general: deviations from half filling

Distortion has:



k
d

= 2
k
F






l
d

= 1/
k
F


New zone boundary @:

±

/
l
d

=
±
k
F


So metal insulator transition still results

However,
l
d

and
a

may not be commensurate

These lattice distortions are responsible

for BCS attraction


short lived
-

long enough to bind


As you go towards 1D, interaction becomes

stronger. However, eventually the static

distortion wins, at which point you get the

CDW. In principal this is a macroscopic

K=0 state
-

ought to superconduct. Rigid, or

incompressible
-

one impurity creates

threshold against conduction

So, e
-
ph interaction can give rise to

superconductivity. However, in the extreme

1D case, it causes locallization.


Thus, we begin to see the delicate balance

between superconductivity and insulating

behavior


What about superconductivity and magnetism?

Density waves

Kohn anomaly, lattice distortion,
etc
.. is an

example of a Peierls instability


In the phononic case, this gives rise to:




Charge Density Wave

In systems with strong Coulomb repulsion this

instability is unlikely to occur. However, a
magnetic equivalent can occur:





Spin Density Wave

Spin Density Waves

Magnons, or spin
-
wave modes soften





䅮瑩晥牲潭慧湥瑩挠潲摥d

Weak SDW

Incommensurate case

Extreme case

Metal insulator transition

Magnetic

zone

boundary

@
±
2

/
a


Small gap