Jason Sadowski- Interplay of Charge Density Waves and Superconductivity

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Interplay of Charge Density Waves and Superconductivity

Jason Sadowski

April 22, 2010

Abstract

The nature of the relationship between CDW and SC is not fully understood, and the literature is full

of conflicting experimental evidence. This paper will provide a review of the Greens function formalism

for both charge-density waves and superconductors, as well as discuss the nature of the long ranged orderinherent to each state. Experimental evidence will be given illustrating the non-intuitive nature of the

interplay between charge-density waves and superconductivity.

Contents

1 Introduction 2

2 Theory of CDWs 3

2.1 Greens Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Random Phase Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Dielectric Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.4 Diagonal Long Range Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Superconductivity 7

3.1 Results from the BCS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Anomalous Greens Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Off-Diagonal Long Range Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Evidence of Density Waves in Superconductors 10

4.1 Competition of CDW and SC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Enhancement of SC by CDW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Conclusion 11

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A Appendix 12

A.1 Summary of Diagram Rules for the Single Particle Propagator . . . . . . . . . . . . . . . . . . . . . . 12

A.2 Dielectric Response in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1 Introduction

The quantum theory of the solid state was developed in

the mid 1930s and had seen many triumphs in the pre-

diction of many material properties, however there were

two obstacles standing in the way of the solid state theo-

ries success. The first was the absolute failure of the solid

state theory in predicting certain metal-insulator transi-

tions (now referred to as Mott-Insulator metals). Accord-

ing to the theory compounds such as NiO should be a

metal since the electrons form a d band which is not com-

pletely occupied. The experimental fact however, is that

NiO is an insulator. The explanation first given by Mott

in 1949 says that this discrepancy comes from approxi-

mating the electrons as only weakly interacting. It turns

out that the Coulomb repulsion (which is not accounted

for in this approximation) is enough to cause them to be-

come localized at a particular lattice site, making them

insulators. These metals provided some of the first ex-

amples where the many-body effects from the electroninteractions simply could not be ignored.

In the mid 1930s Rudolph Peierls was writing an in-

troductory textbook on the quantum theory of the solid

state [1], and was in the process of designing a many-

body exercise when he made an accidental discovery. He

was considering what would happen if the crystal lattice

could be distorted, and effectively double the size of the

unit cell. It was well established that the periodic poten-

tial of the lattice will cause a gap to form at the edge of

the Brillouin zone (In a one dimensional lattice of length athe gap is at k = /a). Doubling the size of the unit cell,

he thought, should effectively halve the size of the Bril-

louin zone and the gap would open at the Fermi surface.

The question he was asking was how much energy would

be saved by the gap opening at the Fermi surface? To

his complete surprise [2] this distortion of the lattice was

always energetically favorable and such a metal would al-

ways undergo what is now known as a Peierls Transition.

Since the distortion of the lattice causes the electron den-

sity to distort in a periodic fashion, it is also known as a

charge-density wave transition (CDW).

The second problem faced by solid state theory was a

theoretical explanation for superconductivity. Supercon-

ductivity was discovered in 1911 by Keike Kamerlingh

Onnes while investigating the low temperature proper-

ties of metals. It was thought at the time that perhaps

the electron phonon interaction of the Peierls transition

could be the mechanism responsible for superconductiv-

ity. Bardeen and Frohlich were able to show that in a

pure 1D metal the sliding of electrons could generate a so

called ideal conductor, as well as the required energy

gap at the Fermi surface. However they quickly realized

that this theory could not explain the multitude of 3D

superconductors, or the Meissner effect. It wasnt until1957 when Bardeen, Cooper, and Schrieffer (BCS) were

able to successfully use the quantum theory to describe

the superconducting state in a true many body fashion.

Although it has been realized that the Peierls transi-

tion can not be reconciled with superconductivity, there

is a large body of evidence which in fact shows that these

two states can coexist with each other. It has been the

consensus that the CDW state (which exhibits insulating

properties) should compete with superconductivity. How-

ever there is also experimental evidence to show that thesuperconducting state can be enhanced by the electron-

hole pairing at the Fermi surface by the CDW state. At

this time the issue is not about the coexistence of CDW

and SC, but whether or not the CDW is favorable or

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destructive to superconductivity. The remainder of this

paper will discuss the many-body theories of both super-

conductivity,charge density waves, as well as the interplay

between the two sates. Section 2 and Section 3 discussesthe theoretical description of CDW and SC ground states

and examines intrinsic differences between them. Finally,

Section 4 provides examples of some recent experiments

which illustrate conflicting evidence about the interplay

of CDW and superconductivity.

2 Theory of CDWs

As already mentioned the charge density wave ground

state was serendipitously discovered by Peierls while

preparing materials for his book on solid state physics.At the time he did not consider the result anything more

than an academic curiosity, since in practice there is no

such thing as a strictly one dimensional metal. It was

later discovered that some materials do in fact exhibit

CDW properties at low temperatures, in agreement with

his earlier results. In his book More Surprises in Theo-

retical Physics he says,

In this case the first surprise was in the

mathematical result for the one-dimensional

case; a second surprise was that this hadsome connection with reality.

In this section the many-body formulation of the CDW

state, particularly the properties of the Greens function,

will be discussed.

2.1 Greens Functions

Recall that the single particle Greens function for the

many-body system is given by:

G(k2, t2; k1, t1) = iD0 T{ck2(t2)ck1(t1)} 0E (1)where T{} is the usual time ordering operator. Physicallythe Greens function represents the amplitude for finding

the system in its ground state with an added particle

of momentum k2 at time t2, if we were to inject a par-

ticle of momentum k1 into the ground state of the full

interacting system at time t1. Thus, knowledge of the

Greens function provides information about the natureof the many body interactions, as it represents the prop-

agation of a particle in the full many body system. Since

it is usually impossible to determine the true many body

Greens function, it is expanded in a perturbation series

to try to incorporate the most dominant contributions.

A summary of the diagram rules for the single particle

propagator perturbation expansion is given in Appendix

A.1. The expansion of the propagator using these rules is

given by [3]:

= + + +

+ + + +

However the third and fourth terms in the sum are said

to be composed of repeated simple parts since they are

composed of elements similar to the first and second.

These terms which can not be further broken down are

referred to as the irreducible self energy contribution to

the Greens function. It is possible to sum over diagramswhich are irreducible using partial summation. For exam-

ple the usual Hartree-Fock approximation to the Greens

function is given by the lowest order irreducible self ener-

gies due to exchange and Coulomb interactions. Summing

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over all of the repeated self-energies gives:

11

+

where the term represents the sum over irreducible self

energies.

= + + + (2)

Using the rules in Appendix A.1 Equation 2 becomes:

G(k, ) =1

k (k, ) + ik(3)

Equation 3 is referred to as Dysons Equation and is the

starting point in most many-body calculations. Although

the partial summation which was performed is exact, it

is in general not possible to determine (k, ) and some

approximations have to be made. For a given system,

one must determine which contributions to will be the

most dominant. For example in the low density limit the

only important contributions to the sum are the so called

ladder graphs, which are graphs containing only one hole

line. In the next section we will look at the contribution

to the irreducible self energy in the high density limit,

also known as the random phase approximation.

2.2 Random Phase Approximation

In the random phase approximation only diagrams of the

form:

DRPA =

q

k q p

q

p + q

are considered in the contributions to the irreducible self

energy. This approximation was first introduced by Gell-

Mann and Brueckner who attempted to model the screen-

ing effects in the electron gas. In order to see why thesegraphs are the dominant contribution we consider the ex-

pression for the diagram given by,

DRPA =

(1)Xq,p

ZiG0(k q, )iG0(q + p, + )iG0(p,) (4e

2)2

q4dd

(2)2

It is clear that the integral diverges at q = 0 due to the

q4 in the denominator, which is due to the long range na-

ture of the Coulomb potential. Similar diagrams of higher

order will have the same behavior, and will also diverge.

Therefore the most divergent diagrams are ones whichhave the same momentum q transferred, and we pick up

a factor of 1/q2 in the integral. Evidently the most dom-

inant contribution to the self energy are those due to the

ring diagrams.

= + + +

After performing a partial summation over the repeated

graphs this sum over infinite divergences manages to give

a finite result! It is found that the effect of the random

phase approximation is to renormalize the Coulomb po-

tential lines leaving an effective interaction. It can be

shown that this effective interaction is given by:

iVeff(RPA) =RPA

=

1 (4)

Looking at the diagram for the effective interaction wecan see that the Coulomb interaction is renormalized as a

consequence of the sum over the fermion pair loops. If a

horizontal line were drawn across the diagram represent-

ing a particular instant in time, the primary contribution

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is due to an electron-hole interaction. In a sense the sys-

tem has become virtually polarized, and the sum is occa-

sionally referred to as a sum over polarization diagrams.

Using the diagram rules Equation 4 can be written as:

Veff(RPA) =Vq

1 + Vq0(q, ) Vq

RPA(q, )(5)

and 0(q, ) is the interaction of an electron and hole pair.

The generalized dielectric constant, RPA(q, ), is a result

of the polarization of the medium due to an applied field.

Is is the properties of this dielectric response that are of

importance in understanding the Charge Density Wave

state.

2.3 Dielectric Response

Equation 5 provides the definition for the generalized di-

electric constant for a system of interacting electrons.

This result is important because the dielectric response

RPA(q, ) characterizes how the electron gas will respond

to an applied potential. It has been shown by other means

that the response is given by:

RPA(q, ) = 1 e2

0q2

Zfo(k + q) fo(k)Ek+q Ek

d3k

(2)3(6)

Where fo(Ek) is the Fermi-Dirac distribution function for

energy Ek, and and q are the Fourier components of theperturbation potential. The interesting characteristic of

the dielectric response is that in 1D it diverges. By set-

ting = 0 for a time independent potential the integral

can be performed explicitly to give:

1DRPA(q, ) = 1 + e2g(Ef) ln

q + 2kfq 2kf

Where g(Ef) is the density of states at the Fermi level.

This result is divergent at q = 2kf which implies that

the system of electrons is unstable. Where does this di-

vergence come from? Taking a closer look at Equation 6shows that the most dominant contributions to the inte-

gral occur when Ek+q Ek = 0. That is, when two statesspanned by the same wave-vector q have the same energy,

the integral diverges, and the system becomes unstable.

The reason the dielectric response leads to a Charge Den-

sity Wave state is because the screened interaction of the

RPA causes the electrons to form an induced charge den-

sity, and modulation of the surrounding lattice ions. Itcan be shown that when the dielectric response diverges,

the charge density for large distances takes the form:

cos(2kfr + )r3

(7)

where is an associated phase [4]. This is the origin of

the term charge density wave. Consequently, the mod-

ulation of the lattice has the same period of oscillation.

It is interesting to note that the screened interaction is

the origin of the so called Kohn Anomaly. The physi-

cal reason for this is given by the effective interaction in

Equation 4. All of the interactions are replaced by the

screened interactions, which depend on the dielectric re-

sponse RPA(q, ). As a result electrons and phonons

interact via this screened field. The singularity which

develops in the dielectric response manifests itself as dra-

matic dip in the phonon frequency spectrum at q = 2kf,

resulting in the Kohn Anomaly.

Nesting

In one dimension the Fermi Surface is simply 2 parallel

lines at k = kf. An electron at k = kf and a holeat k = +kf have the same energy and are separated by

q = 2kf. The wavevectors q which connect the diver-

gent points are referred to as nesting vectors. Since there

are an infinite number of these points in 1D the response

is strongly divergent. In higher dimensions however the

number of these points is reduced as shown in Figure 2.3

Figure 1: Nesting in 1,2 and 3 Dimensions. The num-

ber of points which lead to divergent terms in the

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integral is reduced in higher dimensions

It is apparent that the strength of the divergence is

strongly Dependant on the particular topology of the

Fermi Surface. Likewise the integral in Equation 6 can

be carried out in 3D to give the result first derived by

Lindhard 1

3DRPA(q, 0) = 1 +e2

0q2g(Ef)

1 +

1 22

ln

|1 + ||1 |

(8)

This expression is no longer divergent at q = 2kf,

and as a result CDWs in three dimensional materials are

quite rare. The calculation is performed for zero tem-

perature, but since Equation 6 contains the Fermi-Diracdistribution function the divergence of the dielectric re-

sponse, and thus the CDW state, is also a strong function

of temperature.

2.4 Diagonal Long Range Order

The CDW state is characterized by the interaction be-

tween the electrons and phonons. At temperatures be-

low a certain threshold the system transforms into one

with modulated lattice ions and charge density. As it was

stated elsewhere [5], the CDW exhibits amacroscopically

occupied phonon mode that characterizes an ordered state

with an associated transition temperature. In an ordered

state the value of some property of the system at a given

point is correlated to the value at some other point, po-

tentially a long distance away. The one particle density

matrix is given by:

(r r) =D

(r)(r)E

(9)

and is clearly proportional to the one particle Greens

function. Expanded in its Fourier coefficients:

(r r) = 1V

Xk,k

eik(rr)

Dakak

E(10)

which is evidently the Fourier transform of the momen-

tum distribution nk. Now recall that a macroscopically

occupied mode is a coherent quantum many particle state.

In the coherent state only one momentum states is macro-scopically occupied and thus,

nk |k|2

for the coherent state

|k1 , k2 , k3 ,

Hence the one particle density matrix is:

(r r) = 1V

Xk,k

eik(rr) |k|2

If only one of these states has a finite occupation, say No

at ko then the momentum distribution has the form:

nk Nok,ko + f(k) (11)where f(k) is assumed to be some sufficiently smooth

function. The corresponding density matrix (which is

proportional to the single particle Greens function) by

Fourier transform has the form:

(r r

) =

No

V +

2

(2)3Z

e

ik(rr)

f(k)d

3

k (12)

where V is the volume of the system. In a normal (dis-

ordered) system the one particle density matrix should

decrease with the separation distance r r, usually ex-ponentially. However, since the density decays to a con-

stant value when r r in Equation 12 the systemis said to exhibit long range order. Additionally, since

the density matrix goes to a constant value for large r, it

means that the two points in Equation 9 are statistically

independent. This means that the thermal average of the

operators can be computed independently. Hence,

(r r) =D

(r)(r)E

D

(r)E

(r)

1According to many textbooks, the Lindhard expression is the result of a trivial integral. As in most cases however, the

result is far from trivial, and is derived in Appendix A.2.

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Systems which exhibit this property are said to have diag-

onal long range order because the density matrix corre-

sponds to the normal Greens Functions defined pre-

viously. In the following sections, we will discuss theanomalous Greens functions and the off-diagonal long

ranged order in superconductors.

3 Superconductivity

The CDW state has many features in common with the

superconducting state, namely they both are due to in-

teraction of electrons with the lattice, and both have an

energy gap opening at the Fermi surface. These two states

have many features in common, but have physical prop-

erties which are worlds apart. The CDW state tends tohave very high effective dielectric constant, and thus ex-

hibits insulating behavior, while the SC state exhibits zero

resistivity, as well as expelling a magnetic field from the

interior of the sample. As was mentioned in the Introduc-

tion, the CDW and SC state were quickly realized to be

incompatible with each other and were assumed to com-

pete. Strangely, there is experimental evidence to show

that these two states can in fact coexist and do not always

compete with each other. In this section we will briefly

review the results from the usual BCS theory as well as

discuss the Greens function approach to the supercon-

ducting ground state. In this way the primary differences

between the CDW and SC state can be illustrated.

3.1 Results from the BCS Theory

Before continuing into the Greens function approach to

SC it will be beneficial to briefly review some of the re-

sults from the BCS theory. The major discovery made

by Cooper was that there existed certain scenarios where

the interaction between two electrons could beattractive

via the electron phonon interaction. He later came to

realize that if two electrons of opposite momentum were

interacting above a filled Fermi sea, then these electrons

would always form a bound state now known as a Cooper

pair. Bardeen, Cooper and Schrieffer then tried to solve

the reduced Hamiltonian,

Hred = 2Xk

kbkbk

Xk,k Vk,kbkbk (13)

where the bks are the Cooper pair creation and anhilation

operators bk = ckck. This Hamiltonian describes a gas

of Cooper pairs interacting above a filled Fermi sea. Al-

though the Hamiltonian is reduced, it is still not possible

to determine the exact form of the wavefunction. The real

breakthrough came when Schrieffer suggested a form of

coherent state wavefunction using a variational approach.

He suggested that the ground state wavefunction could

be written as:

|0 =Yk

uk + vkb

k

|0 (14)

where uk and vk are the hole and particle amplitudes for

a Cooper pair state respectively. The amplitudes uk and

vk were treated as variational parameters, and are cho-

sen such that it makes the ground state energy of the

reduced Hamiltonian minimum. After performing such a

procedure one finds:

v2k =1

21

k

Ek (15)

and u2k+v2k = 1. In addition we find the energy dispersion

law:

Ek =q

2k + 2k (16)

where k represents a gap at the Fermi surface, given by

the self consistent equation:

k = Xk

Vkkk

2Ek(17)

In contrast to the Bose-Einstein condensates, the BCS

wavefunction represents a coherent state of Cooper pairs,which are fermions. It is this fact, in addition to the Pauli

exclusion principle, which lead to the interesting char-

acteristics of the superconductor. Excited states of the

SC are determined using the reduced Hamiltonian, and

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can be obtained through the Bogoliubov-Valatin canoni-

cal transformations. These transformations are given by

[6]

p = upcp vpcp

p = upcp + vpcp

as well as their Hermitian conjugates. When these oper-

ators act on the superconducting ground state |0 onefinds the relations,

p |0 = |pp |0 = |p

p |0 = 0p |0 = 0

Evidently the operators p and p create and de-

stroy excitations in the superconducting state, where |0is the vacuum state for the superconductor. These ex-

citations are sometimes referred to as Bogoliubov quasi

particles or Bogolons.

3.2 Anomalous Greens Function

Although the Greens function approach has been tremen-

dously successful for describing many metals in the nor-

mal state, the method breaks down when applied to su-

perconductivity, and a new approach must be found. A

quick calculations shows why the usual perturbation ex-

pansion is no longer valid. It is well established that the

poles of the Greens function correspond to the energies

of the system, and from Equation 16 we would expect

something of the form:

G(k, ) 1 p2k + 2k + i

The expansion of the propagator will be the same as usual,

however we should replace the Coulomb interaction with

the BCS interaction given by Cooper. The graphs which

make the dominant contribution to the irreducible self en-ergy should be those in which two particles interact with

no net momentum transfer q = 0. These graphs are the so

called ladder graphs and the sum of them define the t-

matrix, which also obeys a Dyson like integral equation.

t

= + +

,

+

For the attractive interaction in the BCS problem the

t-matrix has the form:

t(q = 0, ) 1 + i

+1

i

Taking the Fourier transform of this equation one can see

that the pole in the right half plane will cause the result

to diverge, and the solution is unstable when time t .Although the usual perturbation theory can not be per-

formed for the SC state, it is possible to define a new typeof Greens function where the usual techniques will apply.

The technique was first introduced by Nambu and

Gorkov in an attempt to save the nice Dyson equa-

tions for the SC state. The major obstacle in defining

a perturbative expansion of the Greens function comes

from the form of the reduced Hamiltonian, where one has

to deal with terms of the form:

Hint XkVk

ckck + H.c

(18)

In this case one does not have the usual type of one parti-

cle interaction terms. Nambu discovered that it is possible

to eliminate this problem if we define the two component

field operator:

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k =

ck

ck

!k =

ck

ck

!(19)

Then the commutation relations are given bynk,

k

o= kk1 and {k, k} = 0, with these def-

initions the complication of Equation 18 are removed [7],

and one can perform the perturbation expansion in the

usual way. The Greens function becomes,

G(p,t) = i 0|T{p(t)p(0)} |0 (20)and the propagator written explicitly in matrix form is

G =

G(p , t) F(p,t)F(p,t) G(p , t)

!(21)

From the definition of the two component spinors (19)

and the Greens function it is clear that the anomalous

Greens function F is given by:

F(p,t) = i 0|T{ckck} |0 (22)F(p,t) = i

D0|T

nckc

k

o|0E

(23)

and now the perturbation can be written down by using

the matrix propagator G and the usual diagrams for G.

G therefore obeys the Dyson equation with all the inter-

actions now evaluated in the Nambu formalism:

1

1

+

(24)

and the Greens function is:

G(k, ) =1

1 k3 (k, ) (25)where t3 is the Pauli matrice. When the reduced Hamil-

tonian is solved using G the BCS results are easily repro-

duced, and give for G11 ofG:

G11 = u

2

k p2k + 2k + i + v

2

k +

p2k +

2k i (26)

which is precisely what was expected based on physical

grounds. For a detailed account of how to arrive at this

result see Schrieffer (pg 178).

3.3 Off-Diagonal Long Range Order

It was stated in the last section that the BCS wavefunc-

tion is a coherent state of Cooper pairs. This means thatbelow the critical temperature the density of Cooper pairs

picks up a finite occupation number and becomes macro-

scopically occupied state. The density matrix of Cooper

pairs is

(r r) =D

(r)(r)E

and is now the wavefunction for a Cooper pair. Simi-

larly to Section 2.4 the Fourier transform is

(r r) = 1V

Xk,k

eik(rr

)D

ckckckck

E(27)

which is clearly proportional to the anomalous Greens

functions. All of the arguments from Section 2.4 still

hold, however now it is the anomalous Greens function

which become correlated as r r . In this case theSC state is said to exhibit off-diagonal long range order

since F(p,t) comprises the off-diagonal components of the

Greens function G.

The off-diagonal long ranged order in the anoma-

lous Greens functions is the primary difference from the

charge density wave state. It is generally believed that

ODLRO is a sufficient condition (although not necessary)

for the appearance of super-phenomena, and was proved

to be capable of maintaining both the Meissner effect and

flux quantization. The CDW state exhibits DLRO, and

is responsible for the coherence properties such as slid-

ing and the locking of phase. Although they are differ-

ent, it is possible for the two types of order to exist in

the same material. The next section will give examplesof materials demonstrating coexistence of superconduc-

tivity and charge-density waves, as well as illustrate the

non-intuitive nature of the interplay between these two

states.

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4 Evidence of Density Waves in Superconduc-

tors

The goal of this section is to illustrate that the CDWstate and SC can in fact coexist with each other, and

that often the results are very non intuitive. One would

expect that due to the insulating properties of the charge

density wave state that CDW should compete with super-

conductivity. In many experiments this is in fact the case,

however there are examples where CDW can enhance SC.

This section will describe a pair of examples which show

conflicting evidence regarding the nature of the interplay

of CDW and SC.

4.1 Competition of CDW and SC

Recently (2009) H. Fu et al at the Lawrence Berkeley Na-

tional Laboratory were studying the effects of charge den-

sity wave in samples of N a0.3CoO2 1.3H2O. These sam-ples exhibit superconductivity at 4.5K, and the strength

of the superconducting state depends on the age of the

sample. As the sample ages oxygen vacancies begin form-

ing causing pair breaking to occur, and eventually super-

conductivity is killed. The group performed specific heat

measurements for samples aged at three, five, and forty

days. Superconductivity is present in the three day sam-

ple and shows measurable pair breaking by 5 days. By

40 days the superconductivity in the sample is completely

destroyed. In the 40 day sample however the specific heat

measurements still show a second order phase transition,

but it is moved from 4K to 7K. It was concluded that

the phase transition in this sample is due to a CDW state

since the transition is unaffected by a magnetic field. Fig-

ure 2 show the specific heat measurements for the 3 and40 day samples.

Figure 2: The specific heat measurements for the 3 and

40 day samples. The three day samples show the SC phasetransition at TC = 4K, and demonstrates the Meissner

Effect. The 40 sample has TCDW = 7K and is not ef-

fected by a magnetic field [8]

What is interesting about this experiment is that the

charge density wave transition was anticipated to occur at

7K by a theoretical calculation. Intriguingly however the

CDW state only occurs in the 40 day sample, and since

TCDW > TC it should occur in all of the samples. Only

once superconductivity has been destroyed does the CDW

state appear. The conclusion is that the superconduct-

ing state competes with CDW, and pair breaking frees

portions of the Fermi surface allowing the CDW state to

form.

4.2 Enhancement of SC by CDW

In 2007 Kiss et al performed angle resolved photo emis-

sion (ARPES) measurements of NbSe2 and were able to

map the Fermi surface in both the CDW and SC state. It

was shown that regions of the Fermi surface where CDW

occurs correspond to regions where the SC gap is max-

imum. The Fermi surface mapping is shown in Figure

3.

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Figure 3: Mapping of k-points corresponding to CDW

and k-dependant SC gap [9]. The upper portion cor-responds to measurements in the CDW state while the

lower portion is in the SC state. Red and blue circles

identify primary and secondary CDW nesting vectors, re-

spectively, and colored dots give the magnitude of the

superconducting gap across the Fermi surface.

The upper portion corresponds to the CDW measure-

ments and show the location of the CDW nesting vectors.

Primary vectors (red circles) form a triangle on the K2

sheet and blue circles lying on the kogome lattice labeled

2 correspond to secondary CDW vectors. The sample

was cooled into the SC state, and the magnitude of the

gap was measured at different regions of the Fermi sur-

faces as shown in the lower portion of Figure 3. It can

be seen that the points where the SC gap is maximum

correspond precisely to the location of the CDW nest-

ing vectors. These results indicate that a charge density

wave may enhance superconductivity in certain regions of

momentum space.

These results are completely contrary to those given

in the previous example. This problem is not restrictedto these particular results, and in general it possible to

find many examples of such conflict in the literature. It

is clear that there is still much to learn in the field of

many-body physics, and everyday new developments are

being made both experimentally and theoretically.

5 Conclusion

In this review it has been shown that in many cases the

CDW and SC state are capable of coexisting with each

other. At this time there is no consensus as to whether

charge density waves compete with, or enhance super-

conductivity. There seems to be numerous experimental

evidence to support both arguments, and it would not be

possible to list all of them here. For a review of the cur-

rent state of experimental studies regarding the CDW and

SC one should look at [10]. Although CDW and SC both

stem from the electron phonon interaction and have very

similar characteristics, their physical properties are verydifferent. It was shown that the two states show different

types of long ranged order ,diagonal long range order in

the charge density waves, and off-diagonal for supercon-

ductivity. It is believed that the fundamental difference

between the two types of long ranged order are what leads

to the interesting properties of these two states. Despite

the large volume of experimental evidence and theoretical

might of the Greens function approach, the true nature of

the interplay between charge density waves and supercon-

ductivity is still elusive, and remains a scientific challenge

waiting to be solved.

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A Appendix

A.1 Summary of Diagram Rules for the Single Particle Propagator

Diagram Element Factor

k or iG(k, )

k or iG0(k, ) =i

k+i

q

m

k

n

l

iVklmn or iVq (Vklmn() for time dependent interactions)

Fermion loops: (1)

Intermediate Energy R

d2

Intermediate momentum kP

k orR

d3k(2)3

(including sum over spins for unit volume)

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A.2 Dielectric Response in 3D

To find the dielectric response in three dimensions we have

to evaluate an integral of the form:

RPA(q, ) = 1 e2

0q2

Zf0(k + q) f0(k)

E(k + q) E(k) d3k

(2)3

(A.1)

Where f0(k) is the Fermi-Dirac distribution function

for energy E(k). In three dimensions the volume element

in momentum space is d3k k2 sin dddk. For the freeelectron gas the energy-momentum dispersion relation is:

Ek =2k2

2mFor simplicity we will only consider the zero frequency

response at zero temperature. We are considering the

scattering between states of momentum k and k and are

connected by wavevector q = k k.

Figure 4: Scattering of electrons from statek

tok

.At zero temperature all states less k < kf are occu-

pied and states k > kf are unoccupied

Then at zero temperature for all states k below the

Fermi-level we have:

f0(k q) = 0 (k < kf)

And the integral can be split into two parts for q

3D

RPA(q, 0) = 1 e2

0q22m

(2)32 I (A.2)

I = Z|kfkf |

1

(k + q)2 k2 +1

(k q)2 k2 d3k (A.3)

Since the integral is independent of we pick up a

factor of 2, and after a little manipulation:

I =2

q2

Z11

d(cos)

Zkf0

2

4k2

1 + 2kqq2

+k2

1 2kqq2

3

5dk

=2

q2

Zkf0

q

2kln(1 +

2k cos

q) q

2kln(1 2k cos

q)

11

=2

q

Zkf0

k ln

1 + 2kq

1 2kq

!dk

Now perform the change of variable x = 2kq and

= q2kf then the integral becomes:

I =q

2

Z1/0

x ln

1 + x

1 x

dx (A.4)

Performing the integration by parts we get:

I =q

2

x2

2ln

1 + x

1 x1/

0

12

Z1/0

x2

1 + x+

x2

1 x dx!

Now we use the result:

Zx2

ax + bdx =

b2

a3ln (|ax + b|) + ax

2 2bx2a2

+ C (A.5)

to arrive at:

I =q

2

1 +

1 22

ln

|1 + ||1 |

(A.6)

The zero frequency dielectric response in three dimen-

sions is then:

3DRPA(q, 0) = 1 +2me2kf

(2)320q2

1 +

1 22

ln

|1 + ||1 |

(A.7)

Recall that for a free electron gas the density of states

at the Fermi level is given by:

g(Ef) =1

22

2m

2

3/22m

kf (A.8)

Substituting this into the expression for (q, 0) and

accounting for both spin directions gives the Lindhard

result for the response in 3D

3DRPA(q, 0) = 1 +e2

0q2g(Ef)

1 +

1 22

ln

|1 + ||1 |

(A.9)

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References

[1] Rudolph Peierls. Quantum Theory of Solids. Clarendon Press, Oxford, 1955.

[2] Rudolph Peierls. More Surprises in Theoretical Physics. Princeton University Press, 1991.

[3] R.D. Mattuck. A Guide to Feynman Diagrams in the Many Body Problem, page 145. McGraw-Hill, 1967.

[4] T.D. Schultz. Quantum Field Theory and the Many-Body Problem, page 99. Gordan and Breach, Science

Publishers, 1964.

[5] Jason Sadowski. Interplay of charge density waves and superconductivity. Presentation for PHYS 894, March

2010.

[6] J. R. Schrieffer. Theory of Superconductivity, pages 4748. W.A. Benjamin, Inc., Publishers, 1964.

[7] J. R. Schrieffer. Theory of Superconductivity, pages 173176. W.A. Benjamin, Inc., Publishers, 1964.

[8] H.Fu and N. Oeschler et al. Competition between superconductivity and charge-density wave in

Na0.3CoO21.3H2O. Journal of Superconductivity and Novel Magnetism, 22(3):295 298, 2009.[9] T. Kiss, T. Yokoya, A. Chainani, S. Shin, T. Hanaguri, M. Nohara, and H. Takagi. Charge-order-maximized

momentum-dependent superconductivity. Nature Physics, 3:720725, October 2007.

[10] A. M. Gabovich, A. I. Voitenko, and M. Ausloos. Charge- and spin-density waves in existing superconductors:

competition between cooper pairing and peierls or excitonic instabilities. Physics Reports, 367(6):583 709,

2002.

[11] Goerge Gruner. Density Waves in Solids. Perseus Publishing, Massachusetts, 1996.

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Jason Sadowski- Interplay of Charge Density Waves and Superconductivity