iIC/95/413

INTERNAL REPORT(Limited Distribution)

'^International Atomic Energy Agency

/ ;,-• s.-yp|(Sd..Rations Educational Scientific and Cultural Organization

•-^^frffERNATIONAL CENTRE FOR THEORETICAL PHYSICS

NATURE OF THE TRANSMISSION AND LOCALISATIONPROPERTIES IN ONE-DIMENSIONAL DISORDERED WELLS

A. Brezini1

International Centre for Theoretical Physics, Trieste, Italy,

M. SebbaniLaboratoire de Physique Electronique des Solides,

Institut dc Physique, Universite d'Oran Es Senia, 31000 Oran. Algeria,

C. Depollier and J. HardyLaboratoire d'Acoustique, Universite du Maine,

B.P. 537, 72017 Le Mans, France.

ABSTRACT

A one-dimensional electrified Kronig-Penncy model is examined to investigate thenature of the electronic states and transport properties of non-interacting electron disor-dered systems having a uniform distribution for the negative strengths of the ^-functions,i.e. the considered system is constituted by wells. It is shown that the ordered limitexhibits jumps in the transmission coefficient versus momentum and a short localisationlength occurs before the first jump. In the disordered case, the pha.se diagram in theenergy-disorder plane yields a detailed picture of the localisation properties of the eigen-atatcs ("superlocalisation", exponential localisation, power-law localisation and extendedstates}. In particular, a short range localisation is observed by means a peak occurringin the inverse participation ratio and a minimum in the localisation length. Further, thepresent data reveal two distinctive behaviours corresponding to a negative differentialresistance and resonances at particular energies, these being close to the edges of theBrillouin zones.

MIRAMARE - TRIESTE

December 1995

'Permanent address: Laboratoire des Polymeres, Institut de Chiinie, Universite d'OranEs Senia, 31000 Oran, Algeria.

1- Introduction

Over the last three decades different types of low-dimensional materials and devices

have known a growing interest including quantum wires, quantum wells, superlattices,

mesoscopic systems and polymer chains [1,2]. Nowadays, a proper understanding of their

electronic and structure properties remains an attracting goal [1-3] From a theoretical point of

view, disorder inducing localization is by now well established [4-6]. In particular, since the

scaling theory [7], al! the electronic states in one and two dimensional disordered systems are

expected to be localized in the absence of external fields [6] no matter how the amount of

disorder is.

However, recently, the subject has known a renewed challenge since the discovery of

typical models of disorder introducing the effects of correlations [3,8] as well as non-linearity

[9] exhibiting extended states at well defined energies Thus, it appears that the disorder does

not provide only destructive quantum interferences.

During the last decade, one-dimensional (1-D) disordered systems have been

extensively studied in the presence of uniform electric field which acts as a delocalization

factor of the electronic states [10-12], in order to appreciate the nature of the transition. Thus,

when the biais voltage V - FL, where F denotes the product the electric field by the electron

charge e and L the length of the chain, reaches the incoming electron energy E, the

corresponding wave functions undergo a transition from exponentially to power-law localized

For large voltages the electron gains kinetic energy from the electric field and becomes

insensitive to the random potential The relevant parameter in describing the interplay between

the effect of the electric field and the role of disorder is X = FL! E,

Up to now two models have been shown successful in examining the electronic and

transport properties in 1-D disordered systems : the Kronig-Penney model [10-13] and the

tight-binding model [11, 14, 15]. Despite its over simplified nature, the Kronig-Penney model

has been shown to be powerfull in introducing external fields in addition to its multiband

description.

Furthermore, two kinds of 1-D disordered systems have been studied : the diagonal

case [9-12], i.e. randomness in the potentials identically separated, and the off-diagonal one

[12, 13, 16] by means identical potentials separated at random.

Almost works treating the former situation have parametrized the disorder, say W, by

a uniform distribution for the strengths of the atomic potentials comprise between

- 0 7 2 and WI2, namely mixing both barriers and wells For such description, it has been

observed small jumps in the coefficient transmission T attributed to Zener type transitions [10,

12].

In recent work [17, 18], we have found that such jumps are indeed more pronounced

and regularely situated in disordered barriers or wells systems at particular energies given by

(t)

where /; is an integer and a being the lattice parameter At these lypical energies, Delyon et al

[11] also predicted the divergence of the localization length.

Moreover, we have surprisingly found for the disordered well systems before the first

jump in T a stronger localization character of the eigenstates in presence of an electric field

instead the usual delocalization observed in mixed systems The transmission coefficient in

such well systems behaves as

with / ( / • ) > ! and(2)

translating the supertocalized character of the eigenstates Indeed, such behavior has also been

reported for particular systems [19] such branched media, epoxy, resines ...

It is the purpose of the present paper to investigate these features in disordered

systems with potential wells only Firstly, we consider the jump in /' in the ordered limit lo

settle clearly their origin and for a proper understanding of the order-disorder transition In the

second step, we examine the interplay between the electric field and the disorder in localizing

and delocalizing the eigenstates In particular, the phase diagram in the (/'.", W) plane provides a

quantitative-picture and enables one to distinguish the region of the delocalization and the

stronger localization As a consequence, the negative differential resistance is also examined, as

well as other informations obtained from the participation ratio and the localization length.

2- Formalism

In the following, we consider the one-dimensional Kronig-Penney model to analyze

the nature of the electronic states

d2 L

; + V V S(x - na) - Fxdx , ,

= E (3)

where /•,' is the electron energy measured in units, a the lattice parameter which is constant2 w

and taken as the unity of length Disorder is introduced into the system by assuming that the

strengths Vn of the 5-function potential are statistical independent random variables with

uniform distribution of width W. The disordered sample extends from x =0 to x = 1. = Na, the

two ends being ohmically connected to perfect leads maintained at a constant potential

difference /•'/-. Moreover the 8-peak potentials of random strenghts Vn are uniformely

distributed following the three cases :

K e [ - #' , 0 ]

('„ E [0 , W]

V. e \ Wtl , W / 2 :

random wells

random barriers

random mixed cases

(4)

The numerical study of Eq (3) can be achieved by making use of the Poincare map

representation of the Schrodinger equation Namely such a representation relates conveniently

the wave (unction amplitude at different lattice sites. In particular, defining 4*n = KV(x = n ) ,

Bellissard et al- [20] showed in the absence of an external field that Eq.(3) can be exactly

mapped to a finite difference equation of the form

X , -%,, -K (5)

In the limit of the periodic case, i.e. alt the Vn = Va, one recovers the complete band

structure of the Kronig-Penney model. In the disordered case, Eq.(5) corresponds formally to

the so-called Anderson model

In crder to map Eq (3) to a finite second-order difference, it is convenient to

approximate the potential Fx by a step function, namely the so-called ladder approximation.

Such substitution of an electric field potential has been shown to lead to qualitatively similar

results in the periodic case [21] Such approximation is expected to effect essentially the short-

range behavior of the wave function. In the disordered case, because the localization effects are

mainly felt at long distances, significant changes in our data are not expected due to this

approximation. Once this is done, the solution of Eq.(3) in between 8-potentiaIs are plane

waves instead of Airy functions. The corresponding Poincare map can be performed and then

reads

where £,, --Jli + naF. Eq.(6) reduces to Eq.(5) in the limit F = 0 as expected This

representation of Eq.(3) appears very usefull for numerical work, since it is a recursion formula

and enables one to consider very long systems, being limited only by computer time available

To perform the iterations, we can give as initial values for T, and *¥2 t n e plane waves

M'j = e x p { - / / £ ) and ¥ 2 =exp(-2;V7?) where K is the energy of the electron before it

reaches the region where the electric field is applied.Iteration of Eq.(6) yields the wave

functions f,v+2 a n c i **V+3 which implies in turns the knowledge of the coefficient

transmission :

|exp<2/*, ) - l

where k - V E and k, ~ 4E + NaF . We have studied here the random variable in I' because it

is statistically well behaved, both in the /•' = 0 case and in the F * 0 case [6]

Furthermore we have examined the nature of the electronic eigenstates as well as the

nature of the delocalization transition by means the inverse participation (/PR) ratio and the

localization length. The inverse participation ratio defined by [22] :

/ ' - "_i

Y ITi^j I i

< i (8)

appears to be a relevant candidate in describing the nature of the eigenstates In particular, ii is

known to behave in the following way .

and :

P oc o( N ') for extended states

for localized states (10)

By now it is generally believed [10,23] that the localization transition occurs at X - — = 1

which serves as support to the study of the transmission coefficient

The localization length X is given by [24]

where the brackets - denote the average. For F = 0, the localization length can be expressed

in terms of the electron energy and the disorder parameter W, for the Kronig-Penney model, as

E

'W1

This formula applies only in the case <Vrt> = 0, i.e. mixed systems

3- Results

In this section we report the results obtained for both ordered and disordered wells

only. The interest in such situation is motivated from the fact that very few has been done in

contrast to the barrier and mixed case. In particular in the following, we focus our attention on

the jumps in the transmission coefficient T and (heir physical origin Furthermore, we dicuss

the superlocalization occuring before the first jump in T, The latter phenomenon is confirmed

by data from /PR and localization length

.?. /- Ordered Case

The jumps in T are known to originate from the ordered positions of the atoms in the

chain [12] Recently, we have shown [17, 18] that these jumps occur regularely at typical

energies corresponding to the edges of Brillouin zones.

In the limit of the ordered case, i.e. all the \\ are equal to a constant Vo, and in the

absence of external fields, the recurrence formula transforms to :

(13)

Defining the transfer matrix :

a -1I 0 U .

(14)

yields the allowed bands corresponding to :

a \ = <2 (15)

The band spectrum described in terms of the potential strength as a function of the momentum,

in it I a units, is reported in Fig. I. It is mainly observed that within the allowed bands (shaded

regions) the states are like Bloch waves while in the forbidden bands, the transmission

coefficient decreases exponentially with respect to the chain length In Fig 1, the lower edges

of the allowed bands correspond exactly to the energies defined by Eq.(l) At these edges, a

takes the values 2 or -2 The wave functions are no longer dependent on the potential strength

and the electron move freely through the system. Such feature leads to a resonance in Ihe

transmission coefficient A similar reasoning holds also in presence of an electric field where

the wave functions from the following equation

= cos(*,,, sin(/r,,)

(16)

do not depend on the potential strength V since the term sin kn , vanishes and therefore are be

not affected on the the didorder at these energies The localization length exhibits a divergence

at these points as suggested by Delyon et al [11]

As an electric field acts, the quasi-continuum is partly broken from the Brillouin zone

edges and consequently the allowed bands becomes narrower as shown in Fig 2 The states

associated to these regions are localized in the classical quantum sense leading to the

possibility in observing the so-called negative differential resistance (NDR). Indeed the

transmission coefficient decreases abruptly in these regions and discrete resonance energies

arise induced by the shift in the energy caused by the electric field This behaviour corresponds

to the well known Wannier-Stark localization effect [25].

However, it is observed from Fig.2 that this effect disappears at a critical energy, say

Et, in each band. Morever the related localization decreases in the higher bands

Basically, the electric field acts in decreasing the overlap of the wave function down

to a single site localized wave function [26] leading to the strong electro-absorption [27] In

Fig,3, we have presented calculations of the wave function over 20 sites for different fields

The results show clearly the decrease of the spatial extend of the domain of localization of the

wave functions In addition, the localization effect measured by the transmission coefficient is

also reproduced in Fig.4 as a function of the chain length for a given field and fixed energy

The transmission coefficient exhibits a strong decrease in the gap before the Brillouin zone

edge is reached while the Wannier-Stark ladder effect takes place after this edge has been

passed The latter feature is accompanied by oscillations with decreasing periods and

amplitudes.

3.2- Superlocalization

In our previous papers [17, 18], we have reported the observation for disordered

systems a strong localization character of the eigenstates, in the limit of weak electric field,

occuring before the first jump in 7. In particular the eigenstates have be found to behave as

Eq.(2) with y* 1.5

Such feature means that the current density, which is directly related to the

transmission coefficient drops with increasing the biais voltage leading to NDR In Fig 5, The

NDR is observed for disordered wells systems in contrast to the case of mixed and barriers

disordered systems for which the differential resistance is positive Capasso [1] has attributed

the origin of the NDR in terms of the non-conservation of the total energy and the lateral

momentum while the electron moves through the well in presence of an applied biais voltage.

Thus, the well stops the tunneling of electrons through the chain up to a sufficient voltage for

which the energy conservation is guaranteed.

However, this effect does not persist for all cases. Fig.6 reveals a delocalization of the

eigenstates, in presence of an electric field, in disordered systems with wells Thus, this effect

seems to be controlled by both disorder and the incoming electron energy This has prompted

us to determine the phase diagram in the (E, W) plane for these systems In Fig. 7 we have

reported this phase diagram by means 100 points where the boundaries are determined within

an uncertainty on the order of ExW = 0 5x0 5 to expected, even in the limit of high disorder,

the eigenstates are delocalized in presence of an electric field, and localized for ordered wells

exhibiting a iupei localized character for particular interplay between energy and potential

strengths. This behaviour is revealed by the decrease of Tin the gap (Fig.4) and appears clearly

in Fig 8a for E = 8 and V = -1 (in V units) but does not occur for V=-5. This feature means

that the effect of the field depends on the energy :

- in the center of the gap, it contributes to increase the transmission coefficient, i.e. the

current,

- near the band edges of the allowed bands, it decreases the transmission coefficient,

leading to a NDR.

From Eq (2), one can deduce the dominant electronic states remain confined in a

small region of the system. This confinement can be observed for disordered wells systems

from the IPR (see Fig 9). A spectacular feature in Fig 9 is the appearence of a peak in the IPR

for particular disorder, corresponding to an increase of the localization with increasing the biais

voltage This observation is confirmed by the results reported in Fig. 10 of the localization

length which exhibits a minimum describing a reduction of the spatial extend of the wave

function down to few number of atoms

However, the appearence of this minimum, which is governed by disorder, means that

beyond a critical field the states become more delocalized. The curves W = 3 and W = 4 in

Figs 9 and 10 respectively correspond to the region of delocalization in the phase diagram of

Fig 7 For such disorders, the peak in the IPR disappears translating the delocalization of the

eigenstates by the field

Morever, Figs. 9 and 10 show clearly the Wannier-Stark ladder localization after the

second jump in T as a peak in the IPR or a minimum in the localization length Such jumps in T

have been also reported in disordered barrier systems [17, 18]

4- Conclusion

We have presented extensive results describing two characteristic electronic

properties in one-dimensional ordered and disordered well systems submitted to an electric

field The former concerns the jump in the transmissio coefficient observed at the edges of the

Brillouin zones and the second is related to the "superlocalization" of the eigenstates

Indeed these two behaviors originate both from the same effect in the presence of the

electric field the decrease of the transmission coefficient leading to a NDR. In ordered well

systems, this decrease appears at both sides of the Brillouin zone edges. In the allowed bands,

the energy levels and shifted by the field yielding discrete resonances with decreasing spatial

extension of the wave functions which suffer a strong confinement towards localization at a

single site (namely the so-called Wannier-Stark ladder localization) In this regime, the wave

functions oscillate with an increase in both their periods and amplitudes. It is also observed that

above a given energy Ec within the band the states become extended.

At the Brillouin zone edges, the wave functions have been found insensitive with

respect to the potential strength and then a resonant tunneling occurs with a transmission

coefficient equal to unity.

On the other side of the Brillouin zone edges for energies within the gaps in the case

of ordered well systems, the transmission coefficient vanishes in the presence of an electric

field according to Eq.(2) while for vanishing fields it behaves exponentially decreasing. The

phase diagram in the energy-disorder plane reveals the nature of the localization following

different parameters (energy, disorder or potential strength). This aspect is also observed by

means the IPR and the localization length which exhibit a peak and a minimum respectively

This may be understood by the fact that a sufficient voltage is required leading to the

delocalization of the eigenstates In such situation the differential resistance becomes positive.

The NDR observed in such cases is expected to decrease the dimenssionality of the wave

function [10]

Finaly, the present study has opened new attracting questions which are the

purpose of a forthcoming paper

- the muttifractal character of the wave functions,

- the determination of the critical electrical field leading to the delocalization of the states

within the allowed bands,

- the behavior of the exponent y as a function of the field.

10

REFERENCES

[1] Capasso F , Springer Series in Electronics and Photonics 28, Berlin, Heidelberg, New York ,

Springer (1990).

| 2 | Fave J L., Springer Series in Solid State Sciences 107, Eds

Kuznamy H , Mehring M. and Roth S., Berlin, Heidelberg, New York, Springer p.21. (1991)

[3| Sanchez A., Macia E, and Dominguez-Adame F., Phys Rev B 49, 147 (1994)

[4) Mott,N.F.,Twose, W.D, Adv. Phys 10, 1445(1971)

| 5 | Ishii K., Prog, Theor Phys. Suppl. S3, 77 (1973)

Thouless D. J , Phys Rep C 13, 94 (1974)

|6 | Anderson P W., Thouless D J., Abrahams E and Fisher D , Phys Rev B 22, 3519 (1980).

| 7 | Abrahams E., Anderson P W , Licciardello D C and Ramakrishnan T. V , Phys. Rev Lett

42,673(1979)

[8j Phillips P. and Wu H. L , Science 252, 1805 (1991)

[9) Bourbonnais R and Maynard N , Phys Rev Lett 64, 1397 (1990)

Kivshar Y S., Gredeskul S. A., Sanchez A. and Vasquez L., Phys. Rev Lett. 64, 1693 (1990)

1101 SoukoulisC, M , Jose J. V , Economou E. N. andShengP., Phys Rev Lett SO, 764(1983),

[ l l | Deylon F , Simon B. and Souillard B , Phys. Rev. Lett. 52, 2187 (1984),

[12] Co taE , Jose J V and Azbel M Ya , Phys Rev. B 32, 6157 (1985)

[13[ Sack J. and Kramer B , Phys. Rev. B 24, 1761 (1981)

Velicky B , Masek J and Kramer B., Phys Lett. A 140, 447 (1989)

[14| PapatrianfillouC. T., Phys Rev, B 7, 5386 (1973).

115] Pichard J L , Phys. A 167, 66(1990).

116] Maschke K and Schreiber M , Phys Rev B 49, 2295 (1994).

|17| Ouasti R , Zekri N., Brezini A and Depollier C , J Phys. Condens Matter 7, 811( 1995)

[18| Brezini A., Sebbani M and Depollier C , Phys. Stat. Sol (b) 185, 359 (1994)

|19] Levy Y E and Souillard B , Europhys. Lett 4, 233 (1987)

120] Bellissard J , Formoso A., Lima R and Testard D. , Phys Rev B 26, 3024 (1982)

| 2 1 | Nagai S, and Kondo, J. Phys. Soc. (Jpn)49, 1255 ( !980)

|22| Bell R.J. and Dean P., Discuss Faraday Soc 50, 55 (1970)

(23] Mato G and Caro A„ J Phys. C 1, 901 (1989)

(24] Economou E N , Green's Functions in Quantum Physics, 2nd ed (Springer-Verlag, Berlin), 164

(1983).

[25| James H. M , Phys. Rev. 76, 1611 (1949)

WannierG H., Phys. Rev 117, 432 (I960).

[26) Mendez E E , , Springer Series in Solid State Sciences 97 Eds.

Kuchar F , Heinrichs H. and Bauer G., Berlin, Heidelberg, New York, Springer p 224 (1990)

|27| Juang C. and Chang C Y , Appl. Phys. Lett. 58, 1527 (1991).

11

FIGURE CAPTIONS

FIGURE 1 : Potential strength as a function of the momentum (in units of n/a) for ordered

systems The shaded regions correspond to the allowed bands

FIGURE 2 : Transmission coefficient vs the momentum in units of n/a for ordered systems

wells (V--1) with different values of the field : F=Q (solid curve), /''=0,005

(long-dashed curve), A'=0,01 (short-dashed curve).

FIGURE 3 : Square modulus of the wave functions of ordered wells for 20 atoms (E~10,

I-'—1) and different fields /•'=0.01 (solid curve), F=0.05 (long-dashed curve),

/•=0.1 (short-dashed curve), /"=0 5 (dotted curve) and F=\ (dash-dotted curve)

FIGURE 4 : Transmission coefficient as a function of the momentum in units of nla for

ordered systems wells. The momentum kx = -JE + FL varies by varying L from

1 to 1000 (/-;=8, l '=-l ,A=00l)

FIGURE 5 : Transmission coefficient vs the biais voltage for disordered chains at K=5,

W = Vl 2 , with barriers (£>), with wells (0) , and with mixed wells and barriers

(0

FIGURE 6 : Transmission coefficient for disordered systems with wells ( />1 , W=\) at

different values of the field : F=0 (€>), 0,005 (+), and 0.01 (0), The straight line

corresponds to the exponential decrease,

FIGURE 7 : Phase diagram in the (F, W) plane for the systems with wells. Shaded region

corresponds to the states which become superlocalized in an electric field

(Eq.2)

FIGURE 8 : Transmission coefficient of ordered wells for different fields : f=0 (6), 0 001

(+), and 0.0012 (0), a) : E=8, V=-l, b) : E=8, V=-5.

FIGURE 9 : Semi-logarithmic plot of the inverse participation ratio as a function of V/E for

disordered wells systems at A=8 and different disorders, W=\ (§), W=2 (+),

FIGURE 10 : Double logarithmic plot of the localization length as a function of V/E for

disordered systems with wells and for the same parameters as in Fig. 9.

FIGURE 1

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