Density of states and magnetic susceptibilities on the octagonal tiling

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Density of states and magnetic susceptibilities on the octagonal tiling

A. Jagannathan

To cite this version:

A. Jagannathan. Density of states and magnetic susceptibilities on the octagonal tiling. Journal de

Physique I, EDP Sciences, 1994, 4 (1), pp.133-138. �10.1051/jp1:1994126�. �jpa-00246884�

Classification

Physics

Abstracts

61.40M 71.20C 71.25 75.10L

Density of states and magnetic susceptibilities

_on

the octagonal tiling

A.

Jagannathan

Laboratoire de

Physique

des

Solides(*),

Université

Paris-Sud,

91405

Orsay,

France

(Received

26

August

1993,

accepted

5

November1993)

Abstract. We

study

electronic

properties

_{as a} function of trie six types of local environments of trie

octagonal quasipenodic tiling using

_a

tight-binding

model. Trie

density

of _states bas six

characteristic

forms, although

^trie detailed structure dilfers from site ta site since _no two sites

are

equivalent

in _a

quasiperiodic tiling.

We present trie

site-dependent magnetic susceptibility

of electrons _on ibis

tiling,

which aise bas

six charactenstic

dependences

_{on energy.} We show

trie existence of _a non-local

spin susceptibility,

which

decays

with trie _square of trie distance between sites and

corresponds

to

quasiperiodic

Ruderman-Kittel oscillations. We

investigate

trie formation of local

magnetic

moments when electron-electron interactions _are included.

This

study

of _a

tight-binding

model _on the two dimensional

quasiperiodic tiling

with

eight-

fold

symmetry,

the

octagonal tiling,

is aimed at

understanding

electronic

properties

in real space. To

date,

studies _on

quasiperiodic tight-binding

models bave focused _on trie form of trie

density

of

states,

_on trie _response of energy levels _to

changes

of

boundary

condition

(which gives

_{a measure} of the

degree

of localization of trie

wavefunctions),

and _on the conductance of such

tilings (reviewed

in

iii).

These _are

site-averaged properties,

and _are characteristic of trie

tiling

as a whole.

Quasicrystals

_are however

interesting

for their local

properties

_as

well,

_since

they

allow local atomic environments dilferent from those found in

periodic

ordered structures.

One indication of

site-dependent

elfects is

given by

nuclear

magnetic

_resonance and Môssbauer

experiments

_[2] ^{which find that} in

magnetic quasicrystals,

trie value of trie local

magnetic

moment on a site appears ^to vary with trie

type

^{of site. There} are no theoretical calculations to date that

predict

trie distribution of

magnetic

moments of ions _on

quasiperiodic

structures.

This motivates _our present

study

of

local, site-dependent

electronic

properties

in

quasicrystals.

It is of _{course not} trie intent here to

produce

_a realistic band _structure for _a

quasicrystal (see [3]).

^Instead we consider

possibly

the

simplest

of

mortels,

whose

properties

we

hope

will not be

overly model-specific.

^Trie

octagonal tiling (which

is

infrequently

encountered in real

materials)

is

appealing

in that it is trie twc-dimensional cousin of trie 3d Penrose

tilings

used to model _a

(*)

Laboratoire associé _au CNRS

134 JOURNAL DE

PHYSIQUE

I N°1

0 2 4 6 8 10 12 là

Fig.

1. 239-site

tiling

with local environments labeled.

large

class of

quasicrystals,

and is closer

mathematically

to these than is trie 2d Penrose

tiling.

Thus _we

hope

that much of trie discussion that follows will be carried _{over to} three dimensions.

We

begin

^with

calculating

the local

density

of

states,

within _a

non-interacting tight-binding model,

for each of the six local environments _on the

octagonal tiling.

Next _we

present

the onsite

magnetic susceptibility

of conduction electrons _on this

tiling.

This is

proportional

to the

spin polarization density

at the site in

question

when _an externat

magnetic

^field is

applied.

Another

interesting quantity

calculated is the nonlocal

susceptibility,

which tells how _two

magnetic

moments interact with each other _{as a} function of their

positions

within the

quasicrystal.

This

changes

_{sign as a} ^{function of the}

distance,

_as in _a

crystalline medium,

the

quasicrystalline

version of Ruderman-Kittel oscillations.

Finally,

_we

present

results _on the

magnetic instability

induced

by

electron-electron interactions.

Upon adding

_an onsite

repulsion

U local moments appear on certain sites

depending

_on the

type

^{of site and} the

position

of the Fermi level.

The

approximants

of the

octagonal tiling

_are

generated by

the

projection

method described in

[4],

^which

yields

_square

pieces

of the infinite

tiling,

and allows for their

periodic repetition

in

the _x-y

plane.

The number of sites in the _square

approximants

studied _was

41,

239

(illustrated

in

Fig. l)

^and 1393. We take _a

single hopping

matrix element t = 1 between sites connected

by

a bond in

figure 1,

where trie six

types

^of site bave been labeled. This Hamiltonian possesses

a

symmetry

t _- -t and trie energy band is thus

symmetric

about _zero.

The local

density

of states

(LD OS)

is found

using

_a continued fraction

expansion

and _recur- sion method _{[5] for} the Green's functions. The continued fraction is summed to infinite

order,

m an

approximation

used with

good

results _m

crystals

and

amorphous

solids

(reviewed

in

[6]).

The continued fraction

representation

of the Green's function

Gm(z) corresponding

to a

given

site of the

approximant

can be written

Gu(z)

=

1~ (i)

where the

parameters bn

are the

offdiagonal

matrix elements m the

tridiagonalized

form of the

operator (z H)~~ (whose diagonal

elements _{are zero} due to the

symmetric

^{form of} the energy

spectrum ).

One

applies

the Hamiltonian

repeatedly

to the

original

basis vector

consisting

of _a

given

site,

generating

successive vectors of trie

tridiagonal basis,

and _one obtains thus the sequence of

parameters bn.

This

procedure

_is

stopped

_upon

reaching

the

boundary

of the

~ B c

-4 0 4 -4 0 ^{4 4 0 4}

D E F

-4 0 4 -4 0 4 -4 0 4

Fig.

^2. Local

density

of states. Two _{curves are} shown for each site type,

corresponding

to

taking

8 and 20

non-equal

b~ _m

equation (1)

and

taking

bc~ ₌ 4.48.

tiling, thereby limiting

the number of

bn

that _con be calculated. The first

twenty bn

have been calculated for the 1393 site

tiling

_m this _manner.

Knowing

that there _{are no} gaps ⁱⁿ the

spectrum [7],

one con show that if _one

replaces

the exact

bn by

_a constant value

bm,

then b~a ₌

A~/16 (where

A

=

2Em~x

is the band

width).

The band width _cari be inferred from

bm

and ~ice ~ersa, but of _course neither is known _a

priori _[8].

In

practice,

one can estimate

bm by calculating

the local

density

of states

pj(E)

₌

-limô_oIm Gm(E

₊

iô)/1r by summing

^the

infinite continued fraction and

requiring

that _p be normalized _to

unity.

The six local densities of states obtained after

averaging

_over ail sites of _a

given

coordination

are shown in

figure

2. The form of the LDOS

depends

_on the site coordination number for its overall

shape. Superposed

_on this is _a "fine structure" which varies from site to

site,

and shows the

intrinsically

multifractal

scaling property

that is associated with the spectra ^of

quasiperiodic Hamiltonians,

whether for the Id Fibonacci chain [9] or the vibrational

spectrum

on the

octogonal tiling [loi.

The two _curves

correspond

to the LDOS obtained for the 239 site

approximant,

with

eight

nontrivial

bn

in the continued

fraction,

and for 1393 sites with 20 nontrivial

bn and, clearly,

_more fine _structure. This is to be

expected

because

replacing

the exact

bn by

b~a

corresponds

to

replacing

the true

quasicrystalline

environment

by

_some

averaged

effective medium.

Retaining

20 _exact coefficients in the continued fraction

yields

the exact

quasicrystal upto

a

greater distance,

hence _{to a} titrer _energy resolution. The

tendency

is for A and B sites to have their

greatest weight

at the band

edges,

while the E and F sites _are

most active in states at the band center. The

density

of states thus found is in

good

^accord

with the _more

spiky

exact forms determmed

by

numencal

diagonalization.

A similar

study

has been carried out for sites on trie two-dimensional Penrose

tiling il il (in

that _case however trie _energy spectrum has several _gaps,

rendering

the

analysis

of the densities

of states somewhat

uncertain).

The

point

to be stressed about these LDOS is not that there

136 JOURNAL DE

PHYSIQUE

I N°1

A B C

-4 0 4 -4 0 4 4 0 4

D E F

-4 0 4 -4 0 4 -4 0 4

Fig.

3. _Xii

Plotted against

the Fermi energy. Trie continuous _curves show trie result

using

continued

fractions,

while trie points _are ^trie result of numencal

diagonalization,

for trie 239 site

tiling.

are

only

six different kinds of sites in trie

quasicrystal (which

is

false,

_{as can} be _seen from the differences between trie LDOS of sites of the _sanie

z),

but that local environments _are

helpful

in

understanding

certain

properties

in _a statistical _sense.

Also, compared

to the exact

density

of _states calculated

by diagonalizing

^the

Hamiltonian,

the smoothed _versions found

by

_our

calculation _may be

preferable

_on

physical grounds

for two _reasons: the presence of disorder in

a real

quasicrystal

will

probably

act to smoothen the

jagged edges

in trie

density

of _states of

a

perfect quasicrystal.

Finite

temperature

will also make it

impossible

to _see trie selfsimilar structure at very small energy scales.

Next trie electron

spin susceptibility

is found

using

trie relation

x~(EF)

_"

/

_dE

Im[Gij(E)Gji(E)]sgn(EF E) (2)

7r

where N is the number of

sites,

and and

j

_are site indices. This is trie standard formula for _spin

susceptibility,

written in real _space coordinates. Xv

gives

the

spin polarization

at site

induced

by

_a small

magnetic

^field at site

j.

For the onsite

susceptibility xii(EF),

_we have

used, first, equation (1)

of the Green's

function,

which

yields

_a smoothed function

similarly

to the LDOS in

figure

2.

Second,

_we have calculated it

directly by finding

the

eigenvectors

of trie

Hamiltonian, using

mixed

boundary conditions,

for

41,

239 and 1393

sites,

and

using

a discrete version of

equation

2 Trie two methods

yield

trie _same

envelope

_curves for trie local

susceptibilities

and _are shown _in

figures

3 for each site

type (averaged

_over sites of _a given

type).

The

susceptibility

is

roughly

flat for A and B

sites,

while the F sites _are "more

susceptible"

for

EF

close to zero.

Next _we examine the nonlocal

susceptibilities xij(EF)

_{as a} function of the distance of site

j

from _a

given

site1. These

oscillate, although

not with _a defined

period

_as do the RKKY oscillations in _a

crystal.

However trie number of _{zero crossings} of trie

susceptibility

increases

0.02

' ~=-3 672

'

0 01

',

, ',

0

,' ,

-0.01 ^'

/ /

a)

2 4 6 8 10 12 14

p=-0 128

' '

' '

'-

/ /

Fig.

4.

Quasi-RKKY

oscillations

along

_a

path

of 19 sites

(runmng

index

j), taking

_to be _an A site _on trie 1393 site

tiling

for two values of trie Fermi energy, _~c

a)

shows trie slower oscillations ai trie lower end of trie energy spectrum and

b)

trie rapid oscillations _near trie band center. Dashed fines

mdicate

1/r~ _decay.

_{Trie continuous} fines _are

simply

_a

guide

for trie _eye.

with the Fermi energy,

reaching

_a maximum at

EF

₌ 0.

Figures

4a and b show the oscillations for two different values of

EF,

_{over a range} of distance that,

corresponds

to half the lattice size of the 1393 site

tiling.

Also indicated is the

1/r)~ ^envelope

^{of these}

"quasi-RKKY-oscillations"

which _{are an}

amusing

demonstration of metallic character in trie

quasicrystal.

Finally,

trie eifect of _an onsite Coulomb

repulsion

U is considered in trie random

phase approximation (RPR).

Site

susceptibilities

_are calculated

using

trie

equation

xRpA =

xo(i Uxo)-~ ₍₃₎

where _{XRPA is} trie RPR

susceptibility matrix,

and _xo trie

susceptibility

matrix of

equation

2.

This

equation represents

^{trie real} _{space version} ^{of trie Stoner}

equation _[12] leading

to itinerant

ferromagnetism

_in metals. A

magnetic instability

is

signalled

when _an

eigenvalue

of this matrix

diverges.

This _occurs when U

=

ÀQ[~

=

Uc

where _we have determined

Uc(EF) numerically.

Uc

is about 0.5 _ai the band center,

increasing

to values of the order of1 _as the band

edge

is

approached.

Trie sites that _are "most

susceptible"

to

going magnetic

_are found

by studying

trie

eigenvectors

_{of xo}

corresponding

to Àm~x. These _are,

speaking statistically,

almost

exclusively

trie F sites when

EF

_is close to _zero

(about

25

~

of these

sites),

and include ail of trie A

JOURNAL DEPHYSIQUEi T 4, N' i JANUARY 1994

138 JOURNAL DE

PHYSIQUE

I N°1

and B sites when

EF

is _neai trie band

edges.

A _more detaileà look at the distribution of

susceptibilities

will be

given

elsewhere.

Acknowledgments.

I thank D.

Spanjaard

and lvf.-C.

Desjonquères

for

introducing

_me to trie continued fraction

representation,

R. Mosseri for

bringing

ref

Ill]

to my

attention,

M. Duneau for _many useful

discussions,

and H. J. Schulz for valuable comments and

suggestions.

References

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Takeo Fujiwar~ and Hirokazu Tsunetsugu,

Quasicrystals:

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R., Hippert F.,

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coupled equations

contained in trie

tight-binding eigenvalue equation H[@o)

_"

Emax[@o)

@o

everywhere positive)

is truncated to _a small set of approximate

equations

written for the

six

local environments, which _con be solved for trie energy Emax

analytically.

Trie value

(necessarily approximate)

obtained _agrees with trie result of numencal

diagonalization

_m 4.2.

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Tang C., Phgs.

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1020.

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