Mean-Field Theory of a Quasi-One-Dimensional Superconductor in a High Magnetic Field

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Mean-Field Theory of a Quasi-One-Dimensional Superconductor in a High Magnetic Field

N. Dupuis

To cite this version:

N. Dupuis. Mean-Field Theory of a Quasi-One-Dimensional Superconductor in a High Magnetic Field. Journal de Physique I, EDP Sciences, 1995, 5 (12), pp.1577-1613. �10.1051/jp1:1995219�.

�jpa-00247161�

J. Pliys. I France 5

(1995)

157î-1613 DECEMBER1995, PAGE 1577

Classification

Pliysics

Abstracts

74.20-z 74.70Kn 74.60-w

Mean-Field Theory of

_a

Quasi.One-Dimensional Superconductor

in

_a

High Magnetic Field

l~.

Dupuis

Laboratoire de

Physique

des Solides, Université Paris-Sud, 91405 Orsay, France

(Received

7

July

1995,

accepted

5

September 1995)

Abstract. In _a

quasi-lD superconductor (weakly coupled

^chains

system)

with _{an open} Fermi

surface,

_a

high magnetic

field stabilizes _a cascade of

superconducting phases

which ends _m

a strong reentrance of trie

superconducting phase.

Trie

superconducting

state evolves from _a

triangular

Abrikosov vortex lattice

m thelveak field _regime towards _a Josephson vortex lattice in

trie reentrant

phase.

We

study

trie

properties

of these

superconducting phases

from _a

microscopic

mortel in trie mean-field

approximation.

Trie critical temperature is calculated in trie quantum limit approximation

(QLA)

where only

Cooper

logarithmic

singularities

_ai-e retained while less divergent terms _are

ignored.

Trie effects of Paub _pair

breaking (PPB)

and

impurity

scattermg are

taken into account. Trie Gor'kov equations _are solved in trie _same

approximation

but

ignoring

trie PPB effect. We derive trie GL _expansion of trie free _energy and obtain trie

specific

heat

jump

at trie transition. ~ie show that _{a gap opens} at trie Fermi level _m trie

quasi-partiale

excitation spectrum. Trie

QLA clearly

shows how trie system ^{evolves from} a

quasi-2D

and BCS- like behavior in trie reentrant

phase

to~vards _a

gapless

behavior at weaker field. Trie calculation

is extended

beyond

trie

QLA

whose

validity

is discussed in detail.

1. Introduction

The

equilibrium

state of

type

^II

superconductors

_~vas first descnbed

by

Abrikosov

using

a

phenomenological Ginzburg-Landau (GL) theory iii,

which _was later

justified by

Gor'kov _in

a microscopic model [2]. ^The

Ginzburg-Landau~Abrikosov-Gor'kov (GLAG) theory

treats the

magnetic

field

semidassically

and therefore _can be

justified

in dean materials

onljr

at

high

temperature or low

magnetic

field [3]. ^In the last few _years, there has been _a lot of11>ork devoted to the theoretical

understanding

of the eflect of

magnetic

fields _on the mean-field

theory

of the

superconducting instability

from _a

completely

quantum

point

^of vie~v.

Nlost of these works bave been concerned with the eflects of Landau level

quantization

iii

superconductors

with _an isotropic

dispersion

la~v

[4-13].

On the _one

hand,

the quantum eflects of the field have been studied in the

vicinity

of the semiclassical critical field

H~2(T

=

0),

with

emphasis

_on the

precise

vortex lattice

structure,

the

quasi-partiale

excitation

spectrum,

^and the de Haas~van

Alphen

oscillations ansmg m the mixed state as a consequence of Landau level

quantization.

The first observation of these quantum

magiietic

oscillations _m the mixed state

Q

Les Editions de

Physique

1995

occurred

nearly _twenty

_{years ago} in trie

layer compound 2H-NbSe2 (14],

^{and interest has been} renewed

recently

with their observation _in several other materials

[15].

^{On the other}

hand,

it has been

proposed

that Landau level

quantization

_can lead to reentrant behavior at very

high magnetic

field ~vhen the

cyclotron

_energy becomes

larger

than the Fermi energy (uJ~ »

EF là,6].

This eflect is absent from the GLAG

theory

which

predicts

_a

complete disappearance

of the

superconducting phase

due to the orbital frustration of the order

parameter

ⁱⁿ the

magnetic

field. This reentrant behavior

originates

in the

suppression

^{of the} orbital frustration when the electrons reside _m

only

_{one or} the few lowest Landau levels.

Indeed,

when

only

_one Landau level is

occupied,

the

supercurrents

can be made to coincide with the orbital motion of the

electrons in this Landau level if the

periodicity

of the vortex lattice is

approximately equal

to the orbit radius of the lowest Landau level.

Moreover,

in this very

high

field

limit,

it has been

argued

that the destruction of

superconductivity by

the Pauli

pair breaking (PPB)

^eflect cari be avoided because the effective ID

dispersion

la~v allows _one to construct a Larkin-Ovchinnikov-

Fulde-Ferrell

(LOFF)

_[16] state which _can exist far above the Pauli limited field. The

reality

of this reentrant

superconductivity

remains however controversial

[12,13]

and there has been

no

experimental

result up ^to now.

The

quantum

eflects of the

magnetic

field _were also studied in the _case of

quasi-one-dimen-

sional

superconductors (weakly coupled

chains

systems)

with _{an open} Fermi

surface [17-21].

These eflects _are

especially pronounced

when the zero-field critical temperature T~o ^{is smaller}

(but

_not much

smaller)

than the interchain

couphng

t~ ⁱⁿ the direction

perpendicular

to the field

(we

will

only

consider this limit in this

paper). (The

chains _are

parallel

to the _{x axis.} The extemal field is

along

the _y direction and the interchain

hopping _t~

_in this direction is assumed to be much

larger

than

tz).

In this _case, the

superconductivity

is well described

semiclassically by

the

anisotropic

GL

theory.

In

particular,

there is _no

Josephson coupling

between chains

even at T

= 0. Because of the

quasi-ID

structure of the Fermi

surface,

the semidassical orbits

in the presence of the field _{are open.}

Consequently,

there is _no Landau level

quantization

but the field induces _a

3D/2D

_crossover

[19, 22, 23]:

^{the electronic motion} remains extended

along

the chains and

along

the direction

parallel

to the

field,

but becomes confined in the _z

direction ~vith _an extension

m~

ctz/uJ~ (c

_is the interchain

spacing

m the

= direction and _uJ~ is the

frequeiicy

of the semidassical

orbits).

This dimensional _crossover is at the

origin

of _{a very}

unusual

phase diagram.

^In

particular,

it leads to _a restoration of time-reversal

symmetry las

fat _as the Zeeman

splitting

is

ignored)

in very

high magnetic

field _{(uJ~ »}

tz)

which results in

a reentrant behavior of the

superconducting phase

with _a

Josephson couphng

between chains.

This

high-field-superconductivity

con survive _{even in} the presence of PPB because the

quasi-

ID Fermi surface allows _one to construct _a LOFF state

(for

_any value of the

magnetic field)

which _can exist far above the Pauli limited field

[17,18]. Although

the

origin

of the reentrant behavior is very diflerent in the

quasi-lD

_case and in the

isotropic

_case, in both _cases it appears

as a consequence of _a reduction of

dimensionahty,

^from 3D to 2D in the

quasi-lD

_case, and from 3D to lD in the

isotropic

_case. The

suppression

of the orbital frustration

originates

in this reduction of

dimensionality.

Besides the

qualitative

diflerences between isotropic ^and

quasi-lD superconductors,

there _is also _an

important quantitative

diflerence. In the

isotropic

_case, the

temperature

and

magnetic

field _ranges where quantum eflects _are

expected

to be

important

are determined

by

the Fermi eiiergy

EF.

For this _reason,

superconductivity

_is

destroyed

for intermediate

fields,

_i-e- for fields much

langer

than _m the semidassical

regime

but much smaller than in the reentrant regime.

lvloreover,

the reentrant behavior _can be observed

only

at very low

temperatures

and very

high

fields. This restricts

considerably

the

possible

candidates to the

expenmental

observation of very

high-field superconductivity

and is _one of the _reasons which

explam

the absence of

expenmental

results. In the

quasi~lD

_case, it is the

coupling

te between chains which

plays

N°12

QUASI-1D

SUPERCONDUCTOR IN A HIGH MAGNETIC FIELD 1579

the crucial role. Since tz _can be smaller than 10 K in

organic conductors,

the

temperature

and

magnetic

^field _ranges where very

high-field superconductivity

is

expected

_can be

expenmentally

accessible if trie

appropriate Ii.e., sufficiently anisotropic)

materials _are chosen

[19, 21].

^{This also}

means that

superconductivity

_can survive _even for intermediate fields between the GL and trie

very

high

field

regimes.

The interest in

quasi~lD superconductors

has been

recently

raised

by experimental

results _on the

organic compound (TMTSF)2Cl04 _(24].

Resistive measurements have shown _an anomalous behavior of the critical field

H~2 Although they

do not give a definite

answer for the existence of

high~field superconductivity

in

(TMTSF)2Cl04,

these results

might

be

interpreted

_as the

signature

of _a

high-field superconducting phase [21, 24].

The _main features of the

phase diagram

of _a

quasi-lD superconductor

_{are now} well _un- derstood

[17-21].

Between the GL

regime (where

the

superconducting

state is _a

triaiigular

Abrikosov vortex

lattice)

and the reentrant

phase (where

the

superconducting

state is a tri-

angular Josephson

vortex

lattice),

the

magnetic

field stabilizes _a cascade of

superconducting phases separated by

^{first order} transitions. In these

quantum phases,

the behavior of the sys-

tem

land

in

particular

the

periodicity

of the order

parameter)

is _not determined _{any more}

by

the

(semidassical)

GL coherence

length (z(T)

m~

lllÙ

but

by

trie _transverse

ii-e-,

_perpen-

dicular to trie

chains) magnetic length ctz/uJ~

m~

1/H (H being

trie external

magnetic field).

When

entering

the quantum

regime,

_{uJ~ m~}

T,

the transverse

magnetic length

is much

larger

than the GL coherence

length (z(T).

This results in _{an increase} of trie transverse

periodicity

of trie order

parameter

and in _a

strong

modification of trie vortex lattice

[19]:

^{in the}

quantum regime,

the

amplitude

of the order

parameter

and the current distribution show _a

symmetry

of _a laminar

type

while trie vortices still describe _a

triangular

lattice. Trie existence of this

somehow _new

superconducting

state is due to trie

symmetry

^of trie

one-partide

wave-functions which is

incompatible

with the

symmetry

^{of the Abrikosov} vortex lattice. The cascade of first order

phase

transitions

originates

in

commensurability

^eflects between trie

periodicity

of the order

parameter ii-e-,

the transverse

magnetic length)

and the

crystalhne

lattice _spaciiig.

In this paper, ~ve

study

the transition line and the

properties

of the

superconducting phases

iii the mean-field

approximation starting

from _a

microscopic

^{model. Some of} the results

presented

here _were

published

elsewhere

[20].

^We assume that the

superconductivity

is due to _an effective attractive electron-electron interaction of the BCS type. ^{We also} assume that the

quasi-ID

conductor is well descnbed above the transition line

by

the Fermi

liquid theory,

which

justifies

the _use of _a mean-field

theory.

This situation will be realized if the

system undergoes

a

single partide dimensionality

_crossover at _a

temperature

T~> >

T~o.

Below

T~i,

the

partide-partide (Cooper)

and

partide-hole (Peierls)

channels

decouple

_so that the usual mean-field

(or ladder)

approximation

is

justified provided

that the bare

parameters

^of the Hamiltonian _are

replaced by

renormahzed _{ones in} order ta take into account the eflects of ID fluctuations

[25] (see

also Refs.

[19, 21]

^for a discussion of the

validity

of the mean-field

approximation,

in

particular

for the

organic

^conductors of the

Bechgaard

salts

family).

As

pointed

out

by

Yakovenko

[13],

it is very

important

^that the electrons in the

ix, pi planes

have _a 2D behavior below

T~i.

Even if the

magnetic

^field suppresses the electron

hopping

_in the _z

direction,

it has _no eflect _on the electroii

motion in the

ix, vi planes

and the

Cooper

and Peierls channels remain

decoupled _[26].

This does not exdude the existence of

thermodynamical fluctuations,

in

particular

in the reentrant

phase

_(uJ~ »

tz)

where the

system

^becomes

eflectively quasi-2D.

In this very

high

field

hmit,

the transition from the metallic

phase

towards _a

phase

with real

superconducting long-range

order

might

be

replaced by

_a Kosterlitz-Thouless _transition

[19].

^This

aspect

^{will however} not be considered any further and _we will restrict ourselves to the mean-field

analysis.

In the next

section,

_we calculate the

eigenstates

and the Green's functious of the normal

phase

_in the _presence of _a uniform

magnetic

field

Rio, H,0).

We _use the gauge

A(Hz, 0, 0)

which

presents

the

advantage

to

yield

_{a very} clear

physical picture

^{of the} dimensional _crossover

induced

by

the

magnetic

field. In Section

3,

we denve the transition line in the quantum limit

approximation (QLA)

where

only Cooper logarithmic singularities

_are retained while less

divergent

terms are

ignored. Although

this

approximation strongly

underestimates the critical temperature, it

provides

_a dear

physical

_picture of the

pairing

mechanism

responsible

for the

superconducting instability. Moreover,

the eflects of disorder and PPB _can

easily

be

incorporated

in this

approach.

In Section

4,

we

study

the

superconducting phases

in the

QLA ignoring

the PPB eflect. We first _{construct a} vanational order

parameter using

the results of Section 3 and then solve the Gor'kov

equations.

^We derive the GL

expansion

of the free energy and obtain the

specific

heat

jump

at the transition. The

discontinuity

of the

specific

heat

jump

at the first order transitions _is related _to the

slope

of the first order transition lines. We find that each

phase

is first

paramagnetic

and then

diamagnetic

^for

increasing field, _except

the reentrant

phase

which is

always paramagnetic _[27].

We also show that _a gap opens ^at the Fermi level in the

quasi-partide

excitation spectrum. The

QLA dearly

shows how the

system

^evolves from _a

quasi-2D

and BCS-like behavior _in the reentrant

phase

towards _a

gapless

behavior at ll~eaker field. In Section

4,

we go

beyond

the

QLA.

We first obtain the transition line and coiistruct a variational order

parameter,

^thus

recovering

^{the results} obtained in reference [19]

in the gauge

A'(0, 0, -H~).

We then derive the GL

expansion

of the free _energy. We discuss the

importance

^of the

screening

of the extemal field

by

the supercurrents and also compare the results ~vith those obtained in the

QLA.

The

quasi-partide

excitation

spectrum

is obtained from the

Bogoliubov-de

Gennes

equations.

Besides gaps which _open at trie Fermi level _as

obtained in the

QLA,

_{gaps open} below and above trie Fermi level. This excitation spectrum is very reminiscent of trie _one of trie

field-induced-spin-density-wave (FISDiV) _phases

which appear ~vhen trie effective electron-electron interaction is

repulsive [28-31]. Finally,

the current

distribution _is calculated.

2. Green's Functions of the Normal Phase

In this

section,

we denve the Green's functions in the normal metallic

phase

in trie presence of

a uniform

magnetic

field

Rio, H, 0). Contrary

to ~vhat _can be found in

general

in trie hterature conceming

quasi-lD

conductors in _a

magnetic field,

_we work in trie gauge

A(Hz, 0,0).

Trie

oiie-partiale

Hamiltonian _is obtained from the Peierls substitution

7io

=

E(k

_- -iv

eA).

Trie

dispersion

law is given

by (h

=

kB

₌ 1

throughout

the _paper and trie Fermi energy

EF

is chosen _as trie

origin

of trie

energies)

Ejkj

=

ujjk~j kfi

₊

_t~ cosjlybj

+ tz

cosjkzcj, il

~vhere _{u is} trie Fermi

velocity

for trie motion

along

trie chains

ix axis)

and

t~,

tz are trie

couplings

between chains

separated by

trie distance b and _c. Trie condition

t~,

tz <

EF

_ensures that trie Fermi sui-face _{is open.}

Except

in a few _cases which will be

pointed

out when necessary, we will not

exphcitly

consider trie y direction

parallel

to trie

magnetic

field which does not

play

_any

role for _a hnearized

dispersion

law

(as long

_as

Cooper

_{pairs are} formed ~vith states of

opposite

momenta _in this

direction).

In order to take into account trie y direction. _we

just

bave to

replace

trie 2D

density

of states per spm

N(0)

=

1/7ri~c by

its 3D value

1/7ri~bc.

^It should be noted here that _no

generality

is lost at the mean-field level when

studying

_a 2D

system

^instead of _a 3D

system.

This is due to trie fact that trie kinetic energy

mainly

_comes from trie motion

along

the chains, which is not aflected

by

the field

(see below),

_so that the electron-electron interaction _can still be treated in

perturbation

for _a 2D

system.

This should be coiitrasted

~vith 2D

isotropic systems

in

high magiietic

^field where trie

perturbative

treatment

Ii.e.

trie mean-field

analysis)

of trie

superconducting instability

is

highly questionable [6,13].

N°12

QUASI-1D

SUPERCONDUCTOR IN A HIGH MAGNETIC FIELD 1581

Since the

magnetic

field does not

couple

the two sheets of the Fermi surface

[32],

we can

write _an Hamiltonian for each sheet of the Fermi surface:

7i(

~ ⁼

vi -iaô~ kF)

₊

aiiuJ~

₊ tz

cos(-icôz)

+

ah, (2)

where _a

= +

(-)

labels trie

right (left)

sheet of trie Fermi surface. ni is trie

(discrete) position operator

^{in trie} z direction and _a

= +

(-)

for

Î j) spin.

^ah ₌

apBH

is trie Zeeman energy for

a g factor

equal

to 2. We have introduced the _{energy uJ~}

= GV where G

= -eHc _{is a}

magnetic

wave vector. The

operator -iôx

commutes with the Hamiltonian _so that trie _momentum

k~

along

the _z axis is _a

good quantum

number. We therefore look for _a solution

#[~ ix, m)

=

e~~~~

ii (mi.

The Fourier transform

#[ (kz)

of

#[ (mi

is solution of trie

Schrôdinger equation

(settini

c = 1 for

simplicity)

^{~ ~}

ii°L°côL~

+ tz

Cosikziiit~ ^ikzi

"

é41~ ikz)

,

131

where ê

= e

u(ok~ kF)

ah and _{e is} the

eigenenergy

of the

eigenstate çi[ (~,m).

Trie

solution of

(3)

is ^~

jOE

jk

~_

~-ÎfilÊk~-t~s>n(k~)j

_j~~

k~ z

up ^to a normalization factor.

Going

back to real space, we obtain

~ ~~

jj~ jm)

~

/ je~k~(m-tÊ)+~ttz

S>n(k~).

là)

_~ ~r

çi[ (m)

is _{non zero}

only

if1=

aiuJ~

where1is _an

integer. Therefore,

the normalized

eigenstates

anj

_the

eigenenergies

of the Hamiltonian

7i(~

_are

given by (~vnting exphcitly

the interchain spacing

c)

4L,ll~)

"

j~~~~~Jl~m1°il' ¹⁶⁾

fÎ~,1,«

"

U(°~~ kF)

+

Olldc

+

Oh, (il

where _r

=

(~, mi

and

Ji

is the ith order Bessel functioii.

L~

is trie

length

of trie

system

ⁱⁱⁱ trie _~ direction and

1= tz/uJ~

is _a reduced interchain

coupling.

The state

#[~,~

^{is locahzed}

around the ith chain with _a

spatial

extension _in the _z direction of trie order of

ic

which

corresponds

to trie

amplitude

of trie semidassical orbits

[19].

^The

advantage

of

working

with the vector

potential A(Hz,

0,

0)

is _now clear: in this gauge, the

eigenstates

are

localized,

which

corresponds

to the real

physical

behavior of the

partides

_{as can} be _seen

by examining

the

one-particle

Green's function in real _space

[32]. Moreover,

_since the momentum

k~

_{remains a}

good quantum number,

trie motion

along

trie chains _{is in some sense} trivial. Note that trie

states

çi[

can be deduced

by

trie appropnate _gauge transformation from the locahzed states

introduclÀ _by

_Yakovenko

[33]

^{in the} gauge

Allo, 0, -H~).

In trie Hamiltonian

(2),

trie

magnetic

^field _appears

only through

trie additional term

oiuJ~.

The eflect of the

magnetic

^field is therefore similar to _an "electric field" -avH whose

sign

would

depend

_on the sheet of trie Fermi surface. This fictitious electric field iiitroduces _an

additional

"potential energy"

_{omuJ~ on} each chain _m which

competes

^{with the}

hopping

term tz

cos(-icôz)

and tends to prevent the electronic motion _in the _z direction. Trie localization in the _z direction is almost

complete

when the diflerence of

~'potential energy'

_uJ~ between two

chains _is much

larger

thaii the transfer

integral

tz ^{between chains} (uJ~ »

t=).

Trie fact that

an electric field _can localize trie _wave functions

(as

obtained here in _a

tight-binding model)

lias been known for _a

long

time

[34].

^{This eflect has}

recently

attracted _a lot of attention iii

counection with the studies of semiconductor

superlattices

submitted to _an electric field

[35].

The

quantized _spectrum

which results from this localization _is known _{as a} Wannier-Stark ladder. Trie semidassical

picture

of this eflect

yields

trie well-known Bloch oscillations of _a band electron in _an electric field.

Continuing

this

analogy

between electric field and

magnetic

field in

a

quasi-lD conductor,

_{we can}

interpret

trie semidassical

trajectories

z = zo +

c(tz _/uJ~) cos(Gx)

obtained from

7io

as Bloch oscillations of the electrons in the

magnetic

field. Most of the eflects which _are induced

by

trie

magnetic

field in _a

quasi-lD

conductor _can be understood _as trie consequence of these Bloch oscillations.

In trie next

section,

^{it will be useful} to _use Green's functions _in trie

representation

of trie states

çi[

~.

Introducing

trie creation and annihilation

operators b[~

~,

b[

~

of _a

particle

of

spin

_a

in~ihe

_state

çi[

~, we define trie Matsubara Green's

function~ô$(k~ÎÎ,'uJ) by

ilT

Gjjk~,i,

_~oj

=

dTei~jT~bL,i,«lTlbÎÎ,1,«1011

~~~ _~~~,Î,OE~ ^~

' ~~~

where _uJ e uJn =

7rT(2n

+

1)

^is a Matsubara

frequency.

3. Transition Line in the

Quantum

Limit

Approximation

The exact mean-field critical

temperature

T~ lias been calculated

numerically

and discussed in detail elsewhere

[18,19, 21].

^{In this}

section,

_we calculate T~ _in trie

QLA,

where

only Cooper logarithmic singularities

are retained while _less

divergent

terms _are

ignored. Although

this

approximation yields

a critical

temperature

several orders of

magnitude

smaller than trie exact mean-field

result,

it

provides

_a dear

physical

_picture of trie

pairing

mechanism

responsible

for trie

superconducting instability.

It will also allow _us to give a

simple description

of trie

properties

of trie ordered

phase

below T~

(see

Sec.

4). Moreover,

trie eflect of disorder _can be

easily incorporated

at this level of

approximation.

Trie total Hamiltonian is _now

lia

+

li;,,t

where the effective _attractive electron-electron Hamiltonian is described

by

the BCS model with

coupling

constant > 0:

7i;nt

=

~ / d~rt~S'~lr)t~l'~lr)t~llr)t~slr)

₁₉₎

~,a,=~,«

We _use trie notation

fd2r

=

c£~ f

dz and _zi

= -o, U ₌ -a. Trie

~lf(rl's

are fermionic

operators

for

particles moving

_on trie sheet

a of the Fermi surface. The interaction _is effective

only

between

partides

whose

energies

_are within Q of trie Fermi level.

We first note that trie

superconducting instability

_can be

qualitatively

understood from trie Wannier-Stark ladder

(7).

In

zero-field,

time-reversal

symmetry

ensures that

Ei(k)

₌

Et (-k)

so that trie

pairing

at _zero total momentum presents ^{trie usual}

(Cooper) logarithmic singulanty

_m~

In(2~iQ/7rT)

_{(~T m~} 1.781 is the

exponential

of the Euler

constant)

which results

in an

instability

of the metallic state at _a finite

temperature T~o.

^{A finite}

magnetic

field breaks down time-reversal

symmetry. Nonetheless,

_we still have

et

~~ ⁼

e(

_~ _~~

for _q~

=

iii +12)G (if

_we

ignore

the Zeeman

sphtting). Thus,

whatever

tie Îalue if tif Îeld,

some

pairing

channels

present

trie

Cooper singularity

if trie total momentum

along

trie chain _{q~ is}

a

multiple

of G. This results _in

loganthmic divergences

at low

temperature

in trie linearized gap

equation,

which destabilize trie metallic _{state at a}

temperature

T~

(0

_< T~ <

T~o) [17-19].

This

reasoning

holds also _in presence of the Zeemau

splitting

if _we consider _painng at total momentum q~ =

iii +12)G

+

2h/u

for two bars

ii,

₁₂ of trie Wannier-Stark ladder. Trie shift

N°12

QUASI-1D

SUPERCONDUCTOR IN A HIGH MAGNETIC FIELD 1583

ai a')

~

= +

~ ~ ~ ~

ùÎ iÎ

Fig.

1.

Diagrarnmatic representation

of trie ladder

approximation

for trie

two-particle

vertex function

r""'(ri,r21ri,ri).

_Trie

_zigzag

_{fine denotes trie attractive} electron-electron interaction. _o and a' _{refer to trie sheet of trie Fermi surface.}

+2h/u

^of trie total

painng

momentum, which

displaces

trie Fermi surfaces of

spin Î

and

spin j

relative to each

other, partially compensates

trie PPB eflect and

yields

to trie formation of _a

LOFF state

[17-19,21].

Besides trie most

singular

channels which present the

Cooper singularity,

there exist less

singular

channels with

singularities

m~ In

[2Q/nuJ~[ in # 0)

for T _{< uJ~ as} will be shown below.

In this

quantum

^limit _(uJ~ »

T),

_a natural

approximation

consists in

retaining only

trie most

singular

channels. This

QLA

lias been used

previously

in trie mean-field

theory

of

isotropic superconductors

in _a

high magnetic

field [6].

It is worth

pointing

out that trie _same kind of

reasoning

_can

explain

trie _appearance of trie FISDW

phases

in trie presence of _a

repulsive

electron-electron interaction

[31].

^Even if _we add to trie Hamiltonian _a second

neighbor hopping

term

t[ cos(2kzc) (assumed

to be

large enough

to suppress any SDW

instability

in _zero

magnetic field),

trie

spectrum

remains

a Wannier-Stark ladder

(Eq. (7)) although

trie

expression

^of trie

eigenstates

^{difler from}

(6).

Since

fi

~, _~ ⁼

-e(

~~ ~~ ~

for

Q~

₌

2kF

+

iii _12)G,

_some electron-hole

pairing

channels

present Îôgirithmic Àngllaiities

if

Q~ 2kF

is _a

multiple

of G. At low

temperature

^and

high enough field,

this leads to _an

instability

of trie metallic

phase

with

respect

to _a SDW

phase

at

wave vector

Q~

=

2kF

+ NG

(N integer). Thus,

trie _gauge

A(Hz, 0, 0) provides

_{a very} natural

picture

of trie

quantized nesting

mechanism

[36]

^{which is} at trie

origin

of the FISDW

phases

_in

quasi-ID

conductors

[37].

^Trie

QLA approximation

lias also been used in this context where it is known _as trie

single

_gap

approximation (SGA) _[29, 30,38].

3.1. WITHOUT DISORDER AND WITHOUT PPB We first consider the

simplest

_case where

both PPB and disorder _are

neglected.

In order to obtain trie cntical

temperature,

we consider trie

two-partide

vertex function

r°~'(ri,

r2i

r[, r[)

for _a

pair

^of

particles

_on

opposite

sides of trie Fermi surface and with

opposite spins

and Matsubara

frequencies. r""'

is evaluated in trie ladder

approximation

shown

diagrammatically

in

Figure

1. We first write trie

two-partide

vertex function _in trie

representation

of trie

eigenstates çi[

~~~'~~~'~~' _~~'~~~ ~~ ~ ~ ~ _~ÎÎ'(11,

121

ÎÎ,

_Î~)

9~ ~x,li,12 k[,1( _l~

~i~Îx,li(~i)~~~-kx,12(~2)~Î[,1[(~Î)~~~Î-k[,1[(~~)~ (1°)

Here _we bave used trie fact that trie total momentum q~ of trie

Cooper pair along

trie chains _is conserved. In trie ladder

approximation,

trie vertex

r[°'iii,12i ii, ii

is solution of trie

equation

~~Î'(Îl

₁₂₁

ÎÎ Î~)

"

(Îl,

_{ai 12,}

O(ilint

_ÎÎÎ

°'Î

_Î~

Ô)

~ _(ii,

_ai

12

£XÎ~int ÎÎÎ

_£X~'i

ÎÎ,

£X~~)

~,, p> i,,

1 2

~i~"

_{(~~ +}

(ÎÎ

₊

ÎÎ)~)~~Î'~'(ÎÎ, ÎÎ, ÎÎ, _Î~)

,

(~~)

where

Ù~" (~~ +

(ÎÎ

₊

ÎÎ)~) ) _{~j ~Î" (~Î,}

ÎÎ~'~Ù)~Î"

(~~

kÎ,

ÎÎ> ~~Ù)

(l~)

~~

w,ko

is trie

two-particle propagator

^{in the}

representation

of the states

çi[ Using

the

expression

(8)

for trie Green's function

G$(k~,1,uJ),

_we bave ^~'

À°(qx)

=

@ ^{In ~'~)l+}

il

~))

^{Re il}

)

+

j)j

,

(13)

~ ~~

where il is trie

digamma

function.

N(0)

=

1/7ruc

is the

density

of states per spm at trie Fermi level. Note that

x(q~)

=

£~ x~(q~)

is trie

pair susceptibility

at _zero

magnetic

field evaluated at the total _{momentum qx.} The first term _on the

nght-hand

side of

Ill)

is the

two-particle

matrix element of the electron-electron interaction

li;nt.

(Îi

£XÎ12

O(ilint

_{ÎÎÎ £X'ÎÎ~,}

O')ôk,~

+k2~,k(~ +k[~ ^"

= -À

/d~rçi(~~,~~(r)~ô( ^~~(r)*çi[Î ~>(r)çi$

~>

(ri

" ~~' ' ~~' ~

_àô~

~ ~, ~,

~li-12~'ll~ll

'i~+ '2~,'i~+

~~

x

/~~ e~~~i+~2~~l~~l)~Jii

-i~

(2icos z)Jij-ij (2icos ~). (14)

o ^7r

It is clear from

Ill

that

Cooper logarithmic singularities y" loi

m~

In(2~TQ/7rT)

appear when trie intermediate

two-particle

state

corresponds

to q~ + 2L"G

= 0

(which imposes

_q~ to be

a

multiple

of

G)

where L'/ is trie center of

gravity

of trie

Cooper

_pair with

respect

to trie

z direction. Intermediate states with qx + 2LllG

= nG

in # 0)

^lead to weaker

logarithmic singulanties x"(nG)

m~ In

[2Q/nuJ~[

^for uJ~ » T. Trie

QLA,

which is

expected

to be valid for uJ~ »

T,

restricts trie Hilbert _space to trie intermediate states such that _{q~ +} 2LllG

= 0. Since q~ is _a constant of motion. trie _center of

gravity

L

=

iii +12)/2

=

-q~/2G

also becomes _a constant of motion of the

Cooper pair

in trie

QLA. Thus, (11)

reduces to

rjj'~~

~

ii, il)

=

-àl~ji'

₊

~xjo) ~ l~je"rj')(~ ~jill,1'), pis)

~ ' ' ~

a,,,1,,'

^{~ '}

where 1=

ii -12

and 1'

=

ii ii

descnbe trie relative motion of the _{pair in} trie _z direction.

qx(L)

= -2LG and

~j]?'

=

a~a'~'ll,i,

with

11,1, "

(~<À(2icosx)Ji,(2icos~). ⁽¹⁶⁾

~~

7r

Note that

§jj?'

_is

mdependent

of trie center of

gravity

L of trie

Cooper

_pairs. To elimmate trie

dependence

on a and

a',

_we write

r(j'~~

~

ii,1')

₌

-Àa~a'~'r~~(L),L(1,1'). r~~~

_L~,L is solution

of trie

equation

^{~ '}

r~~~~~,~ji, i')

=

v,i,

+

àxjo) ~ v,i,,r~~~~~,~ji",1'). (ii)

i,,

The

preceding

matrix

equation

_is solved

by introducing

the

orthogonal

transformation

Ui,i,

which

diagonahzes

the matrix 11 _1,

(U~~ VU)

i i, = ôi _i,

Î

i The matrix

Ù~~~L~,L ⁼

U~~ r~~ jL),LU

is

diagonal:

Ù~ _~L)

L(1,

_i~)

=

ôi1, ~'~

(18)

~ ' l

Àx(0)111

N°12

QUASI-1D

SUPERCONDUCTOR IN A HIGH MAGNETIC FIELD 1585

0 2 4 6 8 10

H

(Toli) Fig.

2. Sohd fines: critical temperature _vs

magnetic

field for la

= 0 and la

= 1 in the

QLA.

Dashed fines: exact mean-field critical temperature

(see Fig.

1 of Ref. [19] ). ^{Tco =1-à K and} tz ₌ 20 K.

and _we obtain

rÎl'L),Lli, ^i'l

=

-àaia'~'~j ~i,i"V",i~~ lU-~h,,,i>

i,,

i

àx(ojq,,

i,,

ligj

The metallic state becomes instable when _a

pote

_appears in the

two-partide

vertex function.

Using y(0)

=

N(0) In(2~TQ/7rT),

we obtain the cntical temperature

~

7r

~~IÀN(0)Îo,i~

'

~~~~

where

Îo,io

is trie

highest eigenvalue

of the matrix V. Trie cntical temperatures are shown

in

Figure

2 for trie two

highest eigenvalues

of V. Trie

parameters

^{used in} trie numerical calculations (T~o = I.à K and tz = 20

K)

_are the _{same as} those of

Figure

and

Figures

4-10 of reference

[19]. Figure

2

clearly

indicates that there _are two hnes of

instability competing

with each other and

leading

to _a cascade of first order transitions in

agreement

with the exact

mean-field calculation of T~

[19].

The existence of two litres of

instability

results from the fact that 11,1, = 0 if1 and 1' do not bave trie _same

parity. Diagonalizing

the matrix 11,1, is then

equivalent

to

separately diagonalizing

trie matrices V21,21, and V21+1,21,+1 ^{In the}

following,

we

label these two fines

by

io _"

0,1so

that

Îo,io

# maxi

l/21+io,21+io.

^{Since 2L} =

ii +12

aiid

ii -12

have the _same

parity,

L is

integer (half-integer)

_{for io}

= 0

(io

₌

ii

and _can be writteii

as L

=

-io/2

_{+ p} with _p

integer. Correspondingly,

_we have

q~(L)

=

(io 2p)G.

It is dear that the

instability

line io

corresponds

to the

instability

hne

Q

=

ioG

which _was

previously

obtained in another

approach

where the

magnetic

Bloch _wave vector

Q Plays

the role of _a

pseudo-momentum

for the

Cooper pairs

_in the

magnetic

field

[19].

As _can be _seen from

Figure 2,

T~ calculated in the

QLA

is several orders of

magnitude

below the exact critical temperature

except

^{in the reentrant}

phase:

it has been

pointed

out

previously

that in

general

the

QLA, although qualitatively

correct,

strongly

underestimates the cntical

temperature [30].

^{In the}

QLA,

_we

neglect

intermediate _pair states with _a center of

gravity

^L"

#

L

=

-q~(L)/2G.

Since the

one-partide

states çif~ ^are

localized,

_[L"

_L[

_is

bounded

by

m~ tz

/uJ~.

This _means that the

QLA neglects

the

logarithilic divergences In(2Q/uJ~), In(Q/uJ~),

...,

ln(2Q/tz).

There _are therefore two cases where trie

QLA

becomes

quantitatively

correct. Either _{uJ~ >} tz

(which corresponds

to the _reentrant

phase)

_so that the

only loganthmic divergence

to be considered is the

Cooper singularity In(2~TQ/7rT).

Or the cutofl _energy Q is

sufficiently

low for the condition _uJ~

m~ Q to hold. In _a conventional

(isotropic) superconductor

where the attractive electron-electron interaction _is due to the

electron-phonon couphng,

Q is the

Debye frequency

and the condition _uJ~

m~ Q _{can never} be satisfied for reasonable values of

ÀÇ~

a)

^"' ',

~

>

,

~) $ $ $ j§

i i i

i i i i

>

Fig.

3.

ai Self-energy

correction in trie Born

approximation

due to impurity

scattering. b)

Vertex correction due _to

impurity

scattenng ^{for trie} pair propagator. ^Trie dashed fines with

a cross denote

impurity scattering.

trie

magnetic

field. In _a

quasi-ID superconductor,

it has been

argued

that Q

m~

T~>,

^where

Txi

is the

single partide dimensionality

_crossover

temperature

^{below which trie} system becomes 3D

[25].

^Trie reason is that trie

superconductiug instability

cannot

develop

at

energies

_e >

T~i (or equivalently

at

length

scales <

u/Txi)

where trie behavior of trie system ^{is ID. In} organic

superconductors

like trie

Bechgaard salts, T~i

_can be of trie order of10 30

K,

_so that the condition _uJ~

m~ Q could be realized in

particular

_cases

although

this remains quite

unlikely.

Moreover,

in the weak

coupling

limit T~ <

Q,

the

QLA

_{can never} be

quantitatively

correct when

entering

the quantum

regime

_{(uJc m~}

T).

From

(19),

_{one can} see that the

superconducting

condensation in trie channel

q~(L),L,io corresponds

to trie

following spatial dependence

for trie order

parameter

A~~(

_L~,

L,i~(r)

m~

~j _{o~Ui io4(}

L

+i(r)çi(

~~~_~ ^~_j

(r). (21)

x, 2 x x, 2

a,1

Noting

that trie matrices V and U have _{a range} of trie order

off ii-e- ll,i,, Ui,i,

are

important

for

iii, ii'[

<

1),

_one can see that

A~~(L),L,io

^{lias the form of} a

strip

extended in trie direction of the chains and localized in trie

perpendicular

direction _{on a}

length

of trie order of

ci.

_This

is not

surprising

since

A~~(L),L,io

^{results from}

pairing

between trie localized _states

çi[

i. Trie

expression (21)

of trie order parameter at T~ will be used in Section 4 to construct a

vaÀiational

order

parameter describing

trie ordered

phase

below

T~.

3.2. EFFECT OF DISORDER. We evaluate trie elfect of disorder _on trie critical

temperature

calculated in trie

QLA.

In presence of

impurity scattenng,

trie _pair

propagator appearing

_in trie

integral equation

^for the vertex function

r"°'

_{lias to be modified}

by self-energy

and vertex

corrections

[39]

as shown

diagrammatically

in

Figure

3. In trie Bom

approximation,

trie self- energy is

given by _[32]

~~'~~~°~

T~~~~~°~

' ~~~~

11>here

1/T

=

27rN(0)n~(2.

_n~ is the

impurity density

and

l~

is the

strength

of the electron-

impunty

interaction which is assumed to be local in rea.l space. The elastic

scattenng

^time T is not aflected

by

the

magnetic

field because the

density

of _{states per}

spin Nie, H)

=

N(0)

is

magnetic

^field

independent. Thus, self-energy

corrections _can be taken into account

by

trie usual

replacement

_{uJ - &l}

= uJ +

sgn(uJ)/2T

in trie _expression of trie

one-particle

Green's function. Because of trie _vertex corrections shown in

Figure 3,

trie _pair

propagator H[~°~ iii,12)

in the

pairing

^channel

q~(L),

L is determined

by

the matrix

equation

Hl'"~ iii,12)

=

ôn,

n~

xl'(0)

₊

uox$~ loi ~j §j°(°3H[3"~

_(13>12

,

(23)

' 1 3

0E3,13

N°12

QUASI-1D

SUPERCONDUCTOR IN A HIGH MAGNETIC FIELD 1587

where _uo

=

1/27rN(0)T.

The ilariables

(

refer _to the relative motion of trie

Cooper pair

_in trie

z direction. In

writing (23),

_we bave used trie fact that in trie

QLA,

trie

only

states which _are allowed

satisfy q~(L)

= -2LG where L is trie center of

gravity

^of trie pair with respect ^{to trie}

z direction.

y](0)

is defined

by

~p joj

₌

~ _G((k~,

_{L +}

j, ^£ô)G

i

(q~ (L) k~

^L

§ ~'

"

cÎ~

~~

~~

~ ~~

~ ~~~~

Introducing

trie matrix

Hj(ii>12)

=

~ (aia2)~~H[i~~ iii,12) (25)

oi,02

trie matrix

equation (23)

becomes

Hj(ii,12)

=

Kilo)

₊

uoxi(0) ~j lli i~Hj(13>12), (26)

i~

where

xj(0)

=

£~ y$(0).

In order to obtain the

preceding equation,

_we used the

property

that 11,1, is ^non zero

only

^if1 and 1' bave trie _same

parity.

Trie matrix

Hi

is

diagonalized by

trie transformation U:

ltjjii,12)

"

(U~~HjU)ii,i~

~

Xé(°)

(27)

~~ ~~ i

uoxalÙ)Îi,ii

In presence of

impunty scattenng,

trie

two-partide

vertex function is determined in trie

QLA by

trie

equation

r"~(/~

_~

^iii,12)

=

-À("[~~

₊ ÀT

~j ~j ~j §j°(~3H[3°4 (13,14)r°4(/~ ~(14,12 (28)

~x _, 1 2 1, 3 q~

,

" 0E3,13 0E4,14

The

dependence

_on the indices _o~ is

suppressed by wnting

r"1°2

iii,12)

=

-Ào(ia(~r(ii,12).

Using

the

property

that

Hz"'ii,1')

is _{non zero}

only

if1 and 1' have the _same

parity,

_we obtain

r~~~~~,~jii, i~)

= 1i~,i~ ^{+ àT}

~j ~j v~,i~njji~, i~)r~~~~~,~ji~, i~) j29)

w i~,i~

The

preceding

matrix

equation

^is

diagonalized by

the transformation U:

t~~~~~,~ji,i'j

=

ju-ir~~~~~,~uji,1>

"

~

_~'

1

ÀÎ iT £~ Ùj ii,1)

^~~~~

From

(27)

and

(30),

_{we see} that trie appearance of _a

pole

in trie

two-partide

vertex function

corresponds

to

1

À~IT i

i

i ^ll~io~

= 0

131)

This

equation

determines trie critical

temperature

in trie _painng channel

q~(L), L,1.

As in the pure

system,

^the

highest

T~ is obtaiued for

= io defined

by Îo,io

=

maxil1

The result

(31)

can be

expressed

_as

~/jo~

i~

j>~

-

Tis ~

i

~]IÎÎÎX~ _io~ ^Tf'~ [

_KW1°~

~ ~~~~

where

Tf~~

^{is trie critical}

temperature

in presence of disorder and T~ trie critical

temperature

of trie pure system

given by (20). Using

xwi°)

=

~)j°~ ¹³³⁾

we obtain

~~

Î~~

~ ~

Î

~

Î7rÎ)~~

^~~~~

For T~

Tf~~

<

T~,

trie

preceding equation simplifies

in

Tc TÎ'~ (35)

ç~

~

(1 Îo

_lu

~

~~j~~ '

In trie

preceding equation,

T~ is trie critical

temperature

calculated in trie

QLA

without

impu- rity scattering.

Since T~ ^is several orders of

magnitude

below trie exact mean-field

value, _Tf~~

will also be much smaller than trie exact value.

However,

a reasonable estimation of

Tf~~

can

be obtained

using

for T~ trie exact value mstead of trie

QLA

value. From

equation (35),

_one

can see that

Îo,io,

which _comes from vertex corrections in trie

pair propagator,

^tends to reduce trie diflerence T~

Tf"~ According

to

(35), impunty scattering

does not affect trie critical tem- perature ⁱⁿ the reentrant

phase

since

Îo,io

_- when T~ _-

T~o.

This is _a direct consequence of Anderson's theorem which states that trie cntical

temperature

is

independent

of _a

(weak)

disorder for _a system ^with time-reversal

symmetry [40]. Obviously, (35)

restricts the observa- tion of

high-field superconductivity

to clean

superconductorsll>ith

_a critical temperature ^not too small. As

pointed

out in reference

[19],

^{this latter condition}

advantages

materials iv.ith _a

large anisotropy.

Trie consequences of

(35)

_were discussed in detail in reference [19] ^{in trie} case

of the

Bechgaard

salts.

3.3. EFFECT _OF PPB. We evaluate trie eflect of PPB _on trie critical

temperature

calculated

in Section 3.1

(but ignonng

^{the effect of}

disorder).

Tue

equation (11)

for the

two-partide

vertex function

r(f'iii,12 _{iii iii}

_involves the

quantity X°"(q~

₊

iii'+1[/)G)

wuich is given

by (13)

with tue

replacement

_{afiq~ - ouqx} +2h.

Therefore, logarithmic singulanties

_anse

through x[" (q~

+

iii'+ il')G)

each time _we bave q~ =

-2L"G _{+ qo where} L" is the center of

gravity

of

thl _pair

_in

an intermediate state and _qo

" +2h

lu.

In trie

QLA,

_we retain

only

trie intermediate

states

corresponding

to these

loganthmic singulanties.

If

h/uJ~

is not

land

not too close

toi

an

integer,

the center of

gravity

^{L of the}

pair

becomes _a constant of motion and is related to the total momentum

by q~(L)

= -2LG + qo. The

equation

which determines the

two-partide

vertex function then reduces to

r[~l'L),Lli, i'l

=

-À(1?'

₊

~ ()?"x°"lqolr[((1),Lli~~> ^i'l

1361

,,1,,

n ,

Followmg

trie

analysis developed

in Section

3.1,

we find that _a

pote

_{appears in} trie

two-partide

vertex function when

1

ÀÎ ix(qo)

_" 0.

(37)

Using

x(qo)

_ci

~~~~

ln

'~~~

(38)

~

~~~~