Transport properties of a class of deterministic one dimensional models with mobility edges

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Transport properties of a class of deterministic one dimensional models with mobility edges

H. Cruz, S. das Sarma

To cite this version:

H. Cruz, S. das Sarma. Transport properties of a class of deterministic one dimensional mod- els with mobility edges. Journal de Physique I, EDP Sciences, 1993, 3 (7), pp.1515-1522.

�10.1051/jp1:1993197�. �jpa-00246813�

Classification Ph»sics Abstracts

03.65G 03.655 71.30 71.50

Transport properties of

_a

class of deterministic

_one

dimensional models with mobility edges

H. Cruz

(*)

and S. Das Sarma

DePartment of

Physics, University

of

Maryland, College

Park,

Maryland

20742-4111, U-S-A-

(Received 18 Marc-h J993, accepted 26 March J993)

Abstract, We calculate, using the Landauer formula, the dc resistance of _a class of

deterministic

one dimensional models with slowly spatially varying

potentials

which have recently been shown _to exhibit metal-insulator transitions. The mobility edge behavior of the model is

directly

confirmed using _a

tight-binding

_as well _{as a} Krbnig-Penny model. Effects of temperature and random disorder are included in _our calculations. The _{state at} the mobility edge is found _to have the intermediate fractal character of critical _states associated with singular continuous

spectra.

Recently,

Das

Sarma,

He and Xie studied

[1, 2]

the _one dimensional deterministic nearest-

neighbor tight-binding

model

[1-4]

(~t~ ~, + ~t~

)

₊

V~

_{~t~ =}

EN,, (1)

with

V~

₌ A _cos

(ara in [~) ⁽²⁾

being

a

slowly varying

on-site

potential

defined

by

the

strength

A

(with

_[A

~2),

the parameter a

(which

is _a real

number), and,

the exponent _v

(with

0

~ v _~ l

).

Das Sarma _et al.

showed

II, 2]

^that this

tight-binding

model has _a metal-insulator transition in _one dimension with the

mobility edges

at

E=tE~

^where

E~= [2-A[,

with all states of energy

(E( ~E~ being

extended and all states with

(E( ~E~

localized.

(The

_same model for

v m 2, and, for

~ v ~ 2 has been studied

by Griniasty

^{and Fishman}

[3]

and

by

Thouless

[4]

respectively, leading

_to the results that all _{states are} Anderson localized with _no

mobility edges

for _v

~

2, and,

that all states are

algebraically

localized with the

possible exception

of the states at E

= 0 where the

Lyapunov

exponent

diverges

_very

slowly.)

In this paper we repon a calculation of the transport

properties

of the _one dimensional model defined

by equations

(

Ii

and

(2)

for 0 _{~ v ~} l. We find, in agreement ^{with the conclusion of}

(*) Peimanent address

Departamento

de Fisica Fundamental _y

Experimental,

Universidad de

La Laguna, 38204 La

Laguna,

Tenerife,

Spain.

1516 JOURNAL DE PHYSIQUE I N° 7

references

[II

and

[2],

that the system ^{is indeed metallic and}

insulating, respectively,

^for

[E

~

[E~

^and

[E

_~

[E~ [.

^{While the conclusion of the references}

[I]

and

[2]

is based on a

calculation of the

Lyapunov exponent using

_a semiclassical WKB

technique

^{which is} exact in the

thermodynamic limit,

_we

directly compute

the system ^resistance

using

the Landauer

formula

[5].

Our

technique

enables _{us to} include effects of finite temperature ^{and random} disorder _on the calculated resistance. In addition _to

calculating

the dc resistance of the

tight- binding

^{model defined}

by equations (I)

and (2), we also

study

the Landauer resistance of the related one-dimensional continuum

Kr6nig-Penny

model defined

by

the

Schr6dinger equation (h

=

2 _m

=

1)

- (

+

( V~

⁸

(,r

_-,;~

)l~ ^(,r)

=

E~ (>.) (3)

da

n =1

with the

strength

_V~ of the n-th &-function

potential

still

given by

the

slowly varying

deterministic function defined

by equation (2).

In

equation (3),

there _are N

single~site

3- function

potentials

located _{at x}

= x~ with _n

=

1,

2,

,

N. The

tight-binding

model

(Eq. Iii

and the

Kr6nig-Penny

model

(Eq. (3)),

while

being quite

^distinct

mathematically,

will both be shown to have metal-insulator transitions in _one dimension with the calculated resistance R

(El diverging

in the

thermodynamic

limit for

[E[ ~E~.

The

slowly varying

deterministic

potential

^defined

by equation (?)

is

interesting

because it is the

only

^known

example

of _a situation in _one dimension where _a

gapless Schr6dinger spectrum

allows the existence of

mobility edges separating

localized and extended _states.

Mobility edges

do not exist in disordered

(I.e. V~ random)

one dimensional systems ^where any randomness in

V,, exponentially

^localizes all _one electron _states and the system, ^{in the}

thermodynamic

limit, is _an insulator. One-dimensional

quasiperiodic potentials,

such _as the incommensurate

Aubry

model

[6]

defined

by

_{v =} I and _{« an} irrational number in

equation (2),

also lead to either all extended _or all localized states

depending

on the

magnitude

of the

potential strength

A. While there _are one-dimensional models

exhibiting mobility edges (for example,

a model

[7]

where V~ ^is

composed

of _more than _one incommensurate

potential),

_we know of _no other _one-

dimensional model with

mobility edges

where the spectrum ^is

gapless.

Because of this

uniqueness, namely,

the

only

known

gapless

one-dimensional spectrum ^with

mobility edges~

the transport

properties

of the model studied in references

[I]

and

[2]

take _{on some}

significance.

In addition _to the theoretical _reasons outlined

above,

there is _an

experimental

motivation for

the _current calculation.

Potentially,

the _most

promising experimental technique

to observe the

predicted

[1,

2]

metal-insulator transition is _to carry out vertical

transport

measurements

[8]

along

^the

growth

direction in semiconductor

superlattices

where it

is,

in

principle, possible using computer-controlled doping techniques

to introduce _a

slowly varying

deterministic

potential

of the type

given by equation (2).

It should be

pointed

out that

spectral properties

^of

one-dimensional

quasiperiodic

systems

(e.g.

the Fibbonacci sequence and the

Aubry model)

have earlier been studied

[9]

in semiconductor

superlattices.

The transpon calculations

presented

^{in this} _paper

(which

include realistic finite temperature ^{and disorder}

effects)

show how the system resistance behaves when the Fermi energy moves

through

the

mobility edge

at the metal-insulator transition.

We

employ

the Landauer formula to calculate the system ^resistance

R(E)

R

(E

=

~

~

~

(4)

2 e~ T

where T is the transmission coefficient of the one-dimensional system.

Using

the transfer matrix

technique [10],

it is

fairly straightforward

to calculate the transmission coefficient for

the

tight-binding

and the

Kronig-Penny

model defined

by equations (I)

and

(3), respectively.

In the

following

_we discuss _our numerical results for the calculated

R(E)

based _on

equation (4).

Our transfer matrix calculation of T _uses system ^sizes of106 sites

(or

N

=

10~

_{for the}

Kronig-Penny model).

In

figure

I we show _our calculated transmission coefficient T for the

tight-binding

model for different values of the

strength

_parameter A, It is clear that

T(E)

is finite for

[E[

~

E~

₌ ² A,

and,

vanishes for

[E

~

E~, denoting

that there indeed _are

mobility edges

in the model _at E

= ±

E~.

We have

systematically

varied the parameters A, a, and _v in _our transfer matrix

calculations, finding

the result that _a and _{v are} irrelevant

parameters

^{whereas A is}

only

relevant in

determining

the

mobility edges

which _are

always

found _at E

= t

E~

= 1

[2

A

[.

The

peaked

structure of _our

T(E)

_curves arises from the finite size

(10~)

of _our system. ^{In the}

thermodynamic

limit _we expect

T(E)

to be smooth _{as a} function of energy.

t.5 5

t,o i o

o.5 o.5

0.o o-o

-2 -1 -2 -1

E E

.5 .5

.o i.o

o.5 o.5

o-o o-o

-2 -1 -2 -1

E E

Fig.

I. -Transmission coefficient _i,ersus energy ^at different potential strength values for the tight-

binding

model. We have taken _wa 0.2 and _v

=

0.7. The number of lattice sites has been taken _as N

=

10~. In the inset of each

figure

we show the

potential strength

A used in the calculation,

In

figure

² we show _our calculated R

(E)

for both the

tight-binding (Eq, (I)j

and the

Krbnig- Penny (Eq. (3)) model,

as a function of E, for _a quantum system ^with

10~

_sites and with

v =

0.7, ara = 0.2, and A

=

0.4. The existence of the _same

mobility edges

at

±E~

=

t 1.6 in both the

tight-binding

^and the

Kr0nig-Penny

model can be _seen in the results. While the

mobility edges

are at the same

energies

in the two models

(and, R(E) diverges

for

(E(

_{~ E~} in

both, indicating clearly

the existence of _a metal-insulator transition in both

models),

there is _a

striking

difference in the behavior of the _two models in the _« metallic _»

region.

In the

Kr6nig-Penny model, R(E)

decreases _{as one} goes away from the band _center

towards

E~, reaching

_a minimum _at

(E( =E~ and,

then

becoming infinitely large

for

[E[ ~E~.

One _can

easily

^{understand this behavior}

by constructing

the Poincard map

[I Ii

associated with the

Kronig-Penny

model _:

sin o

~

(2

s 9

)

4i ~~~

#~,, ~ +

4'n

+

f

~

~'~ ~~

~

1518 JOURNAL DE PHYSIQUE I N° 7

16)

4

~

/~

_

0.6 2

"w >

cN

~

~

~

~ o.4

-2 -1 O 2

E

-2 -1 o 2

E

Fig. 2. al Calculated Landauer resistance _versus energy for the

tight-binding

model. We have taken

N 10~, _ar«

= 0.2 and _v

= 0.7 b) Calculated Landauer resistance _i,eisus energy for the &function Kr6nig-Penny model. We have taken N 10~, _ar« 0.2, A 0.4 and _v 0.7.

where E

=

2 _cos 9.

Using perturbation _theory [3]

_{we can} show that for

large

N

R(E)~ ~~,

0~v<1

(6)

2e

near the band _center

(9

=

0).

For the sake of

completeness,

_we

give

here _our

perturbative

theoretic result for the band _center resistance in the

Kronig-Penny

model for _v

~ l

R

(E

) ^~

~ exp

~ ^~

$ ₍₇₎

2 _e 16 9

In the

large

N limit, the resistance and the

Lyapunov

exponent are related »ia the

equation

R(E)~ ~~exp[y(E)N] (8)

2e

where the

Lyapunov

_{exponent, y}

(E),

itself is

given by

an

integral

_over the

density

of _states

D(E)

y(E)

=

dE'D(E')

in

[E E'[ (9)

Since D

(E

for the one-dimensional

Kr6nig-Penny

model has _a square ^root

divergence,

and the

Lyapunov

exponent ^{for the cosine}

potential

model

(Eq. (2))

behaves

[1, 2]

_as

y(E)~

jE E~[

'~~, one

expects R(E)

to show _a minimum _on the extended side of E~

(I.e.,

for

E1

_~

Ec l.

To get a better idea of the

transport

^{behavior of} our model system, we consider the average resistance _p

(E)

which is defined

[12]

for _a system of N sites _as

p

(E)

₌

( ~~~' ~~

(10)

N

n =1

~

where R (E, _n is the resistance for _a system with _n sites.

Clearly

for extended states, _p goes to

a small _{constant as} N

- co, whereas for

exponentially

localized states,

p(E) diverges

exponentially

^{with N. For} critical states, which are neither localized _nor

extended,

associated with

singular

continuous spectra

(such

_as at the self-dual

point

of the

Aubry model),

_p

oscillates

[12]

with the system ^{size N. We have} evaluated _{p, as} shown in

figure 3,

for the

tight- binding

model with

v =

0.7,

_ara

= 0.2, and

= 0.4 around the

mobility edge

E

E~

₌

1.6. Below

E~,

_p ^is

practically

a constant,

indicating

the existence of extended states consistent with _our

expectation.

For

(E( ~E~, p(N)

increases

exponentially

with N,

indicating

exponentially

localized states. For E

=

E~,

we see clear oscillations in the average resistance

(similar

to that _seen

[12]

in the critical _state of the

Aubry model), indicating

that the

mobility edge

state in the

slowly varying

deterministic

potential

model defined

by equations

II and

(2j

is

actually

a fractal

« intermediate _» state which is neither localized _nor extended. The existence of these steps ^{in the} average resistance around E

=

E~

^is a reflection of the system wavefunction

being

extended in _some

regions

of space and localized in other

regions.

We note that the detailed

quantitative

behavior of _p

(E)

_{as a} function of N

depends

_on the value of the parameters v and _{ar a,} but the

qualitative

behavior is universal

(I.e. independent

of exact values of _v and ara).

In

figure

⁴ we consider the

finite-temperature

effects _{on our} calculated resistance

by obtaining

the thermal resistance Rj~ ^{of the} one-dimensional

crystal

defined

by

I T

E~

_(-

f/

%E dE

R~~(o

)

=

(I I) TE~(- _%f/%E

dE

where o is the

temperature

^and

f

is the Fermi distribution function, Note that R(o

= 0)m

R

(E

=

E~)

where

E~

is the Fermi energy or the chemical

potential

of the system. ^In

figures

4a,

b,

_we

depict

the calculated

R,~,

^for a few values of

k~

o, _{as a} function of

E~

for the

tight-

binding

and

Kronig-Penny model, respectively,

with A

= 0.4, _{ara =}

0.2,

and _v

=

0.7.

Temperature,

_{as one} would expect, ^{smoothens the behavior of the resistance} as a function of the Fermi energy.

Finally,

in

figure

5 _we show the effect of random

disorder, arising

from

impurities invariably

present ^{in real} systems, on the calculated transport

propenies.

To do this, _we add _a random _on- site

potential

_W~ to the

slowly-varying

deterministic

potential V~

in

equation (I

). ^We assume

the random

potential

to have _a uniform distribution within _± W and

impose

the restriction

[V~

+

W~[

_~ ² so that the

tight-binding

model makes _sense. For W

= A, _we the system ^has 100 fb disorder whereas for W

=

0,

the system ^{has 0 fb} disorder. In

figure

5 _we show _our calculated thermal resistance

(N

=

10~)

with

k~

o

=

0.005,

A

=

0.4,

_v

=

0.7,

and _ara

= 0.2

for _a few values of the disorder

strength

W/A

=

10

fb,

25 fb and 50 fb. As

expected,

for small disorder

(~

10 fb

),

the _states bellow the

mobility edge

remain

effectively

extended whereas for

large

disorder, all _states become Anderson localizated in the one-dimensional

crystal.

We

emphasize

that, in

principle,

all _{states are} localized in the one-dimensional model for any finite

W,

but,

clearly

for small W

(=

^{0,I A}

),

the localization

length

is

larger

^than the system ^size

(10~)

of _our calculations for

E~

up ^to

E~

₌ ^{1.6. For}

large disorder,

however, _even the _states

near the band _center get localized with the localization

length

less than the system ^size.

1520 JOURNAL DE PHYSIQUE I N° 7

la) (b)

Id)

5

je)

in

o 2000 4000 6000 8000 loooo

N

Fig. 3.

Average

resistance _versus number of lattice sites _at different energies near the mobility edge lE~ 1.6). We have taken A

= 0,4, _wa

=

0.2 and

v 0.7 a) E 1.7000 b) E 1.6005 c)

E 1.6003 _; d) E

= 1.6000 e) E

=

1.5997 _; f~ E 1.5995 _; g) ^E ₌ 1.5000.

(a)

3.o

2.5

,

q 0.30

©

^~'~

g

^~

b )

(b)

,

$

I

i I

tt

'

o-o o.5 1-o ^'

E~

o-o o.5 1-o 1.5

EF

Fig.

4. a) Thermal resistance _i,ersus Fermi energy at different temperatures (1l ) for the tight-binding model. We have taken A

=

0.4, _w« 0.2, N

=

10~ and _v

=

0.7. Solid line kB1l ₌ ^0.005. Dashed-

line kB 0.05. Dotted-dashed line kB = 0.01; b) Thermal resistance _i,ersus Fermi energy ^at

different temperatures ^{for the} &-function

Kronig-Penny

model. We have taken A =0.4, _w«

=

0.2, N

=

10~ and _v

=

0.7. Solid line kB =

0.005. Dashed-line k~ ^H = 0.05. Dotted-dashed line

kB 0.01.

' ,

,' ,

1' ,

j' _j

ii _{, j}

, ii _,

,, ii _,

ii i' _i

ii i ^' i i

ii ^{i '} i'

i

ii ^' i'

i

ii ^{i '} i' _'1

's ii ^{i '} i'

, i

~© i i i '

1~ ii

1~l ^{' j '} ,i i

% ,j ^{ii i i} ,1',

, ,

~ j _,j

<, ii ,' i i ,i 'j',

~

_~j ii 'j i' i '

i''i~j

,

j

~ , , ^j i, ii ,i i j1',

, ,

~

~,

i i

jj ((

_i~ ' i

(('i

~~

<~ii

~'/i

_' ii

'

, , ^{j i} , i i

, 'j i ' i j 'i

,

r i i ' ' j' '

j

')~'

' i'

ii

<' _j i> i /,' 'Ii 'j' ' ii

,' j, j , i

, ,i , ,,

~

j ' i , '

i '

i 'i

' 'j 'j '> ^' i

I ', _i '

j

" ' i

, i 'r ^'

,

i

o

0.8 1-O 1.2 1.4 1.6

Ey

Fig. 5. -Thermal resistance _versus Fermi energy at k~ ₌ 0.005 for the disordered

tight-binding

model. We have taken A

= 0.4, N

= 10~, _w«

= o.2 and _v

=

0.7. Solid line W/A 25 % disorder.

Dashed-line _: 50 % disorder. Dotted-dashed line _: lo % disorder, _as defined in the _text.

In summary, we have carried _{out a} transfer matrix-Landauer formula calculation of the dc resistance of _a one-dimensional

crystal

with _a

slowly varying

deterministic on-site

potential using

both the

tight-binding

and the

Kr6nig-Penny

model. Our calculated numerical results

strongly

_suppon earlier

[1, 2]

theoretical results based _{on a} semiclassical evaluation of the

Lyapunov

exponent. ^{We show that it} may be

possible

to observe the

predicted

metal-insulator

transition _as the Fermi level _moves

through

the

mobility edge

because the transition is robust in the presence of finite temperature ^{and disorder effects. Precise}

computer-controlled doping profile

in semiconductor

superlattices

should be able _to generate deterministic

potentials

of the type ^defined

by equation (2)

and _an

experimental

measurement of the system ^{resistance in the}

growth

direction _{as a} function of the Fermi energy should exhibit the

predicted

_one~

dimensional metal-insulator transition.

Acknowledgments.

This work is

supported by

the Materials

Theory Program (DMR)

of the National Science

Foundation. H-C- wishes _to

acknowledge

the financial suppon ^of a NATO research

fellowship.

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