Extended states in non-periodic superlattices

Superlattices

and Microstructures,

496

Vol. 5, No. 4, 1989

EXTENOEO

STflTES

IN

F. Laruelle*. IIILaboratoire

present

et

t Laboratoire

de Baaneux

a

Ravera

in the

envelope

function

Starting

with we

of

but not

Semiconductor provide

a

superlattices

convenient study

one-dimensional modulated

studied

extensively specie

ic

cases’ -3.

The present

beam epitaxy

high

QaAs/Oa,_.FIl.Fls

quality no

because

more of

f ield9.

allows

intentional

application Physical

an

to

the

range

property elements

OP

the

SLs

how this the

in

some

layer

the art

thicknesses,

there

conPiguration

Let

periodic basic

OP

electric

consider

us

elements

Fibonacci This

leads

to

properties6*7’9

on

we

localization

purpose iong range

In is

to order

0749-6036/89/040495

and

the

case

distinguish effects

of

our

interest

to

wauePunct ion

translational

when

disappears.

prime

invariance

approximants In the

barrier two

quasiperiodic

SLs

( X. = 0.25

stacking

ceil

of

properties

is

built

according

Pollouing

are

unit

to the

cells: built

with

rational

8 or B (28.3

the same width

diPPerant

FIB

quasiperiodic

each element

of

two

C times.

conuerlring

infinite

of

with

B+J iterated

OP the same QaAa quantum well

but

+ 04 $02.00/O

(

ePPectiue

ABWB (t=3I....SLs

OP the

C)a,_,Al.As

our

the

the

ceils

following,

Fibonacci

(namely

1.

parameters

electronic

f+f+B anb

ABA (d=2),

of

the

to a speciPic

design

A and B stacked

law

unit

is

related

of

blocks

Al concentration,

SLs whose unit

(C=l),

structure

choice

building

masses...)

these

It

Prom short

of

either

Pocus

is

different

electronic

band

OP these

choice

of

blocks)

( the specific

structure

structural’

here.

building

interactions

internal

with

both

in

exist

speciPic

We explain

be

or because

will

matrix

two basic

and with

which

made

Fibonacci

can

due to a

properties

study

deal

states

and dynamical’,*

attention

studies

transPer

been

of

external

the

are

quasiperiodic

has

invariance

structural

0P

the

the growth

random7,s...>

are related

law

in a

SLs in which

translational

(quasiperiodic*-6, the

state

use of

extended

may

can

problem

theoretically

of molecular

is

Por

which

This

non-periodic

law.

transport

potential will.

at

of

Our calculations

and

states

matrices

(SLI

system

of electronic

FRfWCE

properties

elements.

that

Those

transfer

Bagneux

approxinants

demonstrate

on the stacking

92220

approximation

periodic

C.N.R.S.

C.N.E.T.

8 Flugust 1988 1

two basic

structures.

the electronic

de Ricrostructures

the electronic

of

with

method.

non-periodic

experimental

study

built

superlattices

and B. Etienne*.

de Microelectronique

( Received

We

SUPERLRTIICES

D. Paquet’

188 Ftvenue Henri

super-lattices

WN-PERIODIC

81

SL. consists

A) and of a ( 31,l

A)

concentrations

and X, = 0.35).

To calculate

the

electronic

band structure

0 1989 Academic

Press Limited

496

Superlattices

YO use

the envelope

the

transfer

uell

the

1 inear

function

matrix

plane

enuelope at

uaues.

to

If

situated

those

this

at of

the

barrier

of

uritten

as”:

saw

X

??

cos
1”

L.

points

XtiY

-iR

iR

X-iY

reals

quan turn

separated

the

to

wave

quantum

two

two adjacent

uidth

X,Y and R are

leads

the plane

the nearest

of

the

probability

connects

thickness

T=

uhere

oP

matrix

the middle

of

interfaces

connecting

as a right

and

the

of

and

written

left

matrix

by a can be

1

-1 01 2345

numbers defined

by:

Flnure a

(.5-C-1)

sin(k,l

U)

=

INVARIANT

I

: Left: Conductionband structure as

1

function of om!er 1 for peptaitc boundary

condftl,nzs. The dashed ltne stis the energy at rhtch I=O. RI#ht: Bner#y dependence of

Sinh(K&)/2 Y

01234

FIBONACCI ORDER e

COSh(k,Ls) -

Vol. 5, No. 4. 1989

quantum

The continuity

and

matrix

coefpicients wells.

is

tuo

quantum uell

the

a 2x2 transfer

uells

of

Punction

current

in each

function

envelope

combination

propagating

approximation

technique:

and Microstructures,

sin
COSh(K&)

Invariant I.

t (2-S-1) cos(k,l

U) X(CI=2X~e-l)X(G-2)-X(t-3)

(1)

sinh(K,L,)/i! R = (Ct1-‘1

and

Sinh
ku=b-1(2au~)1

‘k,

(=,

E

Prom the uell in the uell

bottom.

hence

matrix X,

across Y

corresponding across

R

are

the unit

the diPPerant

for

Born-uon

cell

Karman

Por

The the

using

the

transfer

matrix

bg the product

The energg

boundary

T

A or B and labelled

obtained

matrices.

my

taken

The matrix

The

is

masses

can be to account

element

and

counted

Xkt’fk-R*=i.

have:

subscript.

of

given

ePPectiue

effects.

ue

and

the energy

in order

dispersion

unimodular

elements

The

dependent

non-parabolic

transfer

being

and ms in the barrier

to be energy

is

k,=b-L[2ms(V,-c)]1’e

spectrum

conditions

is

From

the energy

(A>

and

displayed

occurs

(labelled

2)

t=2

(labelled

the

width

In

the Fibonacci

the

total

the

following

case,

transfer

the half-traces

matrix

recurrence

M(t)

relation:

are

energy

1 and OP

those energy fig.2.

of SL e=l range

of

deduce

the

order

t

no

overlap

allowed

is

SL e=l of SL

increases,

C

mini-bands

becomes

mini-bands ranges

energy

1, 2, is

of

mini-bands

As order

3.

mainly

(..BAB..)

corresponding

SL 8=2

range

almost

two first

smaller.

wavefunction

of

P,=0

The electron

mini-band

3).

the

and

in the

by

CB),

C.

with

part,

and the

labelled

X(G)

t=-1

easily

order

first

the

exclusiuelg

bounded

can

evolution

mini-bands

energu

I~l~lRI

low

betueen

smaller

we

orders

in fig.1.

the

patterns or

IXlsl

(FIB)

of

any higher

at

spectrum

energy

the

bg:

G=l

spectrum

In

spectra

(..BfWB..)

of mini-bands

for

defined It

by the

follows

that

localized states

to miniband for

appear

states

1 and 3,

on in

the

2 and on in the

as shown in

Superlattices

and Microstructures,

497

Vol. 5, No. 4, 1989

EA69meV M

Ez116meV

MJ

,I”L

E=103meV

200

600

1000

2 PLllUt-e

2

Square

:

three dffferent

the flnlte

the electrontc Bottom:

range

energy State

fn

I=6 SL.

potentfal

ground

1

for

(n

aft6

ffg.?.

range labelled

energy

3

Figure

shcws

corresponding

:

Square

wevefunct tons

the

1000

600

1400

z (A,

stacklngs.

assocfated

energy

Extended state

allaed

Insert

profile.

state

labelled

the

200

of electronic

modulus

wavefunctions for enerstes of

1400

(A,

for

lsaiulus

of electronfc flnlte

rsndoa

four

Labelled fnserts sketch potentlai

proffle Bnergy corresponds to l=O.

Iifddle:

1) 21 B elements. 15 nodes in the ravefunctfon

2.

2) 14 B and 7 A elements. 16 n&es.

Top:

to I=0

31 Z?A

and 8 B elements, 17 n&es.

4) 21 A elements. 18 nodes. In

the

mini-bands overlap

high labelled

small

at a precise never

energy

order the

Cl). by the

invariant

order

remain

in

any

wauefunct ion

In fact

XtC)s

line

in

order

8

is

are

and C=2

k mini-gaps

broad.

fig.11 and

bounded which

the This

at

I

a

the recurrence

relation

I can be simply

bound to

transfer

matrices

any

[T,,TB

When

energy

the

energy

elements

and

on

order the

C. It depends internal

A and B as depicted

in fig.

of T,

elements

only

structure 1.

( Y,Rs

- VeRh 1’

12) of

the commutator

those

two

by:

J =

I is

on

basic

of

axis

In fact

As is

equal

that

commute. of

??

and is

shows

independent

with

defines

I:

I is

expressed

and T,:

Moreover

extended.

by considering

following

the spectrum,

of SL e=l

i dotted

occurs

can be understood

of

fit higher

and bands

corresponding

relation

part

4 and 5

completely.

are very

gap

energy

doa:

not

elements

the allowed depend

either

H and B are

or on the respective shown in

aboaje relat.ion

matrices

transfer

Therefore

the

to zero.

fig.3.

T, state

on

stacked

t,his

t-he way the along

proportion this

and T, 4t

e&ended

the SL

of A to B. state

is

498

Superlattices

observed

SLs made

in

and B elements. the

This

extended

state

quasiperiodic property

al.

but

not law

the building

are

similar

in a

are not

is

stacking

of

results

of random

demonstrates

to those

Energy mini-bands here 11 of

shown difPerent

SLs

wave-vectors

at which

I is

have the

same period

mini-zone, 9s.

equal

the

Since

SLs

in

fig.3

nodes

q,

and q,

SL

at

1).

per unit stacking

random respective linear

cell

(labelled

ratio

A

NI, and

Ne=l-N,,

(X,=7/21)

transfer

stacking elements

states of

matrices

satisfying

for

is

extended

Ostlund,

transport semiconductor

Schellnhuber

P.K.

Bhattacharya

B. Jusserand,

of

B.

Etienne.

by

a

Rev.

Adv.

H.J. Lett.,

in Phys.,

therein. R. Clarke.

Phys.

F.Y.

Rev.

Juang

Lett.,

55,

Joncour

and

1768 (1985). [5]

in

Herndon,

K. Bajena,

Rand,

Phys.

and references

!15)

a

and C. Tang,

D.

Siggia,

65 (19821

of

Pandit,

and E.D.

R. Berlin,

and

Kadanoff

R.

[4]

number

F. Mollot,

J.

Phys.

M.C.

(Paris)

C5.

40,

577

(1987).

[6] F. Laruelle,

V. Thierry-Mieg,

Etienne.

and B.

[7]

P./n

states

tabelled

in non-periodic

(23

measure

based

S.

L.P.

31.

elements

two elements

commute,

equation Lebesgue

be

.J.

Phys.

fl.C.

(Paris)

Joncour

C5. 40,

529

(1987). + NE q,

in fig.3

stacking

on

to

wiil

SE, 1870 (1983).

the

2

and 3 cX,,=13/211.

f!s extended from the

[2]

Lett..

the

interpolation:

as can be seen

Khomoto,

Rev.

P. ErdOs and R.C.

the

given

M.

[3]

4) has more

is

[l] Phys.

than

of SL B

B

predictions

super-lattices.

1873 (1983).

for

cells

and

energy.

difficult

designed

based

50,

of

the wave-Punction

N* q,P,/n

and its

bigger

the auerage

of

B

Brillouin

qP/n,

A

two energy

nodes is unit

of

inot

SLs A and

is

21

Therefore

the

same

qa

these

properly

experiments

the same

to be

REFERENCESRND NOTES

associate

Since

of

We hope that In

at

expected

Vol. 5, No. 4, 1989

matrices

curves

B

cell

fulfill.

commute

is

to zero.

Cl81 than the wave-function

(Labelled nodes

of

e?

our

matrices

condition

obserued

of Das Sarma”

and

unit

the local Our

dispersion

zero.

to a

the SL.

P and the

with

wave-function

of

fi

number

per

This

to

equal

wave-vector

the

waue-function

to

transfer

that

due

way since

when I is

Iof A

clearly

but

blocks

different

diagonal

stacking

and Microstructures,

on

zero.

Merlin,

of

Phys.

obviously

discrete

[l@]

Moreover

if

the

two different

require

that

all

R.

Ploog,

energies

more than states

[8]

only

set

if

Phys.

exist

the

is

SL built

FI. Chomette.

Bastard,

[9]

37. (II] Lett..

J.

E.E.

B. Deueaud, Rev.

Phys

K.

Lett.,

F. Laruelle 4816 X.C. 69.

A. Regreny

57,

Bajema,

(Paris)

Mendez.

Rev.

Lett.,

C5, 48,

J.

Nagle

2426

and K.

J.M.

Hong.

(1988).

and B. Etienne,

Phys.

Rev.

(19883. Xie

and S.

Das Sarma, Phys.

1538 (1388:~.

(3.

503 (1387).

F. Agullo-Ruedand 60.

and

1464 (19863.

Rev.

B,