Electronic conduction in highly anisotropic systems

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Electronic conduction in highly anisotropic systems

A. Jayannavar

To cite this version:

A. Jayannavar. Electronic conduction in highly anisotropic systems. Journal de Physique I, EDP

Sciences, 1993, 3 (9), pp.1969-1972. �10.1051/jp1:1993224�. �jpa-00246844�

Classification

Physic-s

Abstiacis

72.10 72,15E

Electronic conduction in highly anisotropic systems

A. M.

Jayannavar

Institute

ofPhysics, Sachivalaya

Marg, Bhubaneswar 751005, India (Receii,ed 23 Marc-h 1993. ret,ised J8 May J993, accepted J9 May J993)

Abstract. we have calculated the ratio of conductivities

along

the chains and

perpendicular

to the chains in

highly anisotropic

quasi-one-dimensional _systems. The electronic motion

along

the

transverse direction is viewed _{as a} coherent inter-chain tunnelling between neighbouring chains

blocked by repeated intra-chain incoherent

scatterings.

Our results predict

intrinsically

the _same temperature

dependence

for both conductivities. It also predicts _a linear dependence of the

anisotropy

ratio _on the carrier concentration, which is absent in the earlier _treatments of the _same problem.

The conduction in

highly anisotropic

systems ^exhibit some unusual

striking

features. In the

recently

discovered

high

_T~

layered

oxides the transport

properties

_are different in different directions metallic transport ^{in the ah}

plane

and _non metallic

(or

metallic) transport

along

the c-axis

[1-31. Especially

the c-axis

resistivity

of

YBajCU~C~

5 is known to be metallic [41 ^with

(dp,/dT)

_~ 0. This is _some what unusual

given

the fact that the resistivities of the

samples

are

much

larger

than the Mott's maximum metallic resistance

(or

even

Mooij

value

[51)

and yet

they

show the metallic behaviour. The observed value of the c-axis

resistivity

is much

larger

than

(10

to loo

times)

the

Ioffe-Regel-Mott

limit which

provides

a criterion for metals in _a conventional Boltzmann type ^{of band} transport.

Generally

in disordered systems ^{it is well} established that, when the value of the resistance is

larger

than the Mott's maximum resistance

(~100

~Lfl cm

),

the

incipient

localization _sets in. In this

regime

_one expects ^the

negative

temperature coefficient for resistance

(TCR) arising

due to

suppression

of the quantum interference

by

the inelastic

phonon scattering.

The

negative

TCR may also arise due _to the breakdown of the adiabatic

approximation

ⁱⁿ the metallic

phase leading

to

phonon-assisted tunnelling

[51.

However,

the

high-T~ compound

mentioned above shows

completely

the

opposite

trend for TCR

compared

to all other disordered systems. ^{In such} pure

anisotropic

systems ^{effect of} localization _seems to be less

important.

It should be noted that the

localization

being purely

quantum ^{mechanical coherent back}

scattering phenomena

that

requires

electron to retain its local coherence in all directions. Thus it is not

possible

to have

states localized in one-direction

only.

Similar

problem

_seems to arise in

quasi-one-dimensional

systems

[61.

For

instance,

in

tetrathiafulvalene-tetracyanoquinodimethane (TTF-TCNQ)

crystals.

In these systems ^{band width}

anisotropy

between _a direction

(between

the

chains)

and

1970 JOURNAL DE PHYSIQUE I N° 9

b direction

(along

the

chains)

is

roughly

of the order of

t~/t~~0.01.

In these systems

conduction is metallic in both directions. The

resistivity

is _a few _m fl _cm

along

the chains and

several fl _cm between the chains _{at room} temperature. ^Even

though

the _mean free

path

perpendicular

to the chains is much shorter than the lattice constant,

dp~/dT~0.

^This

behaviour calls for _{a new} mechanism for transport ^{other than the}

simple

Boltzmann type ^band transport ^for

highly anisotropic

systems. ^Soda et al. [61 have

explained

this in _a

simple picture, supposing

coherent on-chain and incoherent inter-chain

couplings.

In the

following

based _on the _same

picture

_we derive _an

expression

for ratio of the conductivities

(resistivities) along longitudinal

and _transverse direction. However, _our final result differs from the earlier _one with _an additional factor

arising

due _to the effective number of carriers involved in the motion

along

the _transverse directions.

We

proceed

here

along

the _same lines _{as our} recent treatment _on the c-axis

resistivity

in

layered compounds [71.

We

generalize

_our treatment to

quasi-one-dimensional anisotropic

systems. ^For

simplicity

_we consider first the two-dimensional

anisotropic

conductor. In this

case we have one-dimensional chains

along

the b directions

(or

^I

direction)

and

they

_are

weakly coupled along

the _a direction

(or

i

directions).

In _our treatment intra-chain electron transport ^is characterized

by

a Boltzmann-like _mean free

path.

The intrachain

resistivity

comes

entirely

from the _mean free lifetime _r, and is

given by

the Drude relation pjj =

mjj*/ne~

r, where

mjj* is the in-chain electron effective _mass. The intra-chain inelastic

scatterings

characterized

by

_r

interrupt

and

therefore,

block the coherent

tunnelling

of the electrons _to the

neighbouring

chains-under the influence of

tunnelling

matrix element ti. ^This

blocking

of coherent motion

modifies ti to

t(

_r/h for ti ^r/h « I. The latter

inequality simply

_assumes that the

large

number of intra-chain

scattering

takes

place

before _an inter-chain

tunnelling

_occurs. This is

equivalent

to

having large anisotropy,

_pi

/pjj

» I, which will be assumed

throughout.

With these conditions successive inter-chain

tunnelling

events are incoherent

(or

uncorrelated) and therefore it is sufficient _to consider

tunnelling

between

just

two

neighbouring chains,

labeled I and 2 say.

The Hamiltonian in _our model is

given

in obvious

notation,

H

=

Ho

₊ H', with

fi0

~

i

_~K ~

/,

_Km ^~ _{l. Km} +

I

₉

Cl

Km

C2.

Ku

K « K. «

+ ti

I (C(

~~ C

~~ + hc

),

11

where H' is the

unspecified

intra-chain inelastic

scattering

term

leading

to the finite lifetime

r for the transport

along

the chains. The energy

dispersion

relation

along

the I d chains is

given by

_eK

~

2 _{tjj cos} Kb, where b is the lattice

spacing along

the chains. In absence of

H',

if _we consider _a situation where _at time t ₌ 0 the electron with _a

quasi-momentum

K is _on the chain I, then _at later time t the

probability

of electron _to

persist

_on the _same

plane Pi

is

given by

P

ji ^~

cos~ (ti

tiff ). With the introduction of

Hi

this coherence between the chains is lost.

Consequently

the survival

probability

P for the electron _to

persist

in the chain I is

given by

2

t(

_r

~'~ ~~~

fi2

~

~~~

For the details of the evaluation of P _see, references

[7, 81.

^Here _r ^is the _mean free lifetime of the electron in the chain. From the

equation (2)

_one can

easily

get ^{the inter chain transition}

rate W

(or

the

hopping probability)

_as W

=

(2

t

( r/h~).

Soda et al.

[61

have also obtained the

same

expression

for the

hopping probability

between the chains.

However,

_to calculate the

transverse

conductivity they

assume Einstein relation between the

conductivity

and diffusion coefficient

ID).

In their calculation transverse diffusion constant D is

given

at once

by

D

=

a~

_{W, where}

a is _a lattice constant

along

the _transverse direction. It should be noted that this

procedure

subsumes that the number of electrons involved in transport

along

the

longitudinal

^direction are same as those involved in the _transverse

direction,

which is _{not true.}

It has

recently

been shown [91

explicitly using

the

anisotropic tight binding

Hamiltonian that the

density

of mobile carriers

along

the _transverse direction is much less than that

along

the

longitudinal

direction. In _an

anisotropic

system when the

density

of electron increases, the electrons _{can no}

longer

be described

by plane

_waves in transverse direction due _to the

Bragg

reflection. This is because the Fermi surface will touch the Brillouin _zone

boundary along Ki

= ± w first. Hence

only

the fraction of electron of the order of

(ti/tjj )

_n can contribute _to the electrical

conductivity

ⁱⁿ the transverse direction. To calculate the transverse

conductivity

we

proceed along

the line _now well known in the

theory

^of quantum transport ⁱⁿ

mesoscopic

systems. Let _an electric field E be

applied perpendicular

to the

pair

of chains

separated by

^the

distance _a. This will generate a chemical

potential

^difference AR

= eaE between the two

chains.

Naturally

this will expose number of

AR

gj _~ of

unoccupied

states per unit

length

into which the electrons _are free _to make the inter-chain transitions. Here _g

j ~ is the I d

density

of states per unit

length.

Now the inter-chain current

density

will be

given by j

₌ e

(AR

_gj

~)

^W.

Thus the transverse

resistivity

is

given by

_pi

m

(E/j), substituting

the value for W _we get pi =

(h~/2

e~ _ag

~ t

(

r)

(3)

This is _our final result for the _transverse

resistivity

for the two-dimensional

anisotropic quasi-

one-dimensional system. ^{As mentioned} earlier the

longitudinal resistivity

is

given by

^the

Drude relation pjj =

mjj*/ne~

r. One _can

easily generalize

the relation

(3)

to 3-dimensional

anisotropic quasi-one-dimensional conductors,

where the

pi(d=3) ^is given by

pi

(d

= 3

)

=

(h~/2

^c~ _gj

~ t

( _r).

_{For the} further calculation of ratio of resistivities _we make some

simple approximations.

First the energy

dispersion

relation _e

(K)

= 2 _{tjj cos} Kb

gives

_a

value for the effective _mass mjj* to be

mjj*= (h~/2tjj ^b~).

^Since the band width of

I -d chain is

given by 4tjj

_we ^tike the value for gj~ to be _an

averaged

value

gj _~ =

l/2 _tjj

b, including

the

spin.

With these values _{one can}

immediately

_get the

anisotropic

ratio of

resistivities, namely

pi /p11 =

2

(bla)~ _(tjj/ti

_)~

(4)

This relation is

independent

^of the

dimensionality

^{and is the number of} transport

charge

carriers per site in the chain

given by (nab )

and

(na~

b for _two- and three-dimensional

quasi-

one-dimensional conductors

respectively.

Our result

(4)

differs from that of Soda et al.

[61 by

an extra factor of &. For the three- dimensional

anisotropic layered compounds

[71 factor 4 appears instead of 2 in

right

^{hand side}

of

equation (4). Interestingly

our result

predicts

_a linear

dependence

^{of the}

anisotropic

ratio _on

the carrier concentration &, which should be

experimentally

a verifiable feature. The

temperature

dependence

of pi is similar _to that of _{p ii,} as the factor _{r enters} in the _same way in the

expressions

for both cases. The additional temperature

dependence

ⁱⁿ _{p~ may} arise due to

the renormalization of the

tunnelling

matrix element ti ^due to adiabatic

phonons.

For the

discussion _on this

phonon

renormalization of ti ^{and the role of disorder} on pjj and

pi we refer _{to our} earlier work

[7]

for the details. At present we are

exploring

to extend _our work to other transport

properties

of

highly anisotropic

_systems such _as Hall coefficient and

electronic thermal conduction.

1972 JOURNAL DE PHYSIQUE ^I N° 9

References

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(1988) 2194.

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Tamegai

T., Sakakibara T., Goto T., Miura N., Takeya H. and Takei H., Physic-a C 153-155 (1988) 26.

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Giapintzakis

J., Rice J. P. and

Ginsberg

D. M., Phys. ^Rev. B 42

(1990) 6217.

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Cooper

J. R., Ayache C, and Henry J. Y., Pfiys. Ret,. B 46 j1992) 6626.

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