Senouci, K

Zekri, N

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Available at: http://www.ictp.trieste.it/~pub off IC/2000/106United Nations Educational Scientic and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSCONDUCTANCE FLUCTUATIONS AND DISTRIBUTIONAT METAL-INSULATOR TRANSITION INDUCEDBY ELECTRIC FIELD IN DISORDERED CHAINKhaled Senouci1U.S.T.O., Departement de Physique, L.E.P.M., B.P. 1505 El M'Naouar, Oran, AlgeriaandThe Abdus Salam International Centre for Theoretical Physics, Trieste, ItalyandNouredine ZekriU.S.T.O., Departement de Physique, L.E.P.M., B.P. 1505 El M'Naouar, Oran, Algeria.AbstractA simple Kronig-Penney model for 1D mesoscopic systems with  peak potentials is usedto study numerically the inuence of a constant electric eld on the conductance uctuationsand distribution at the transition. We found that the conductance probability distribution hasa system-size independent form with large uctuations in good agreement with the previousworks in 2D and 3D systems.MIRAMARE { TRIESTEAugust 20001Corresponding author. E-mail address: ksenouci@meloo.com1 IntroductionDuring the last two decades several works have been devoted to understand the transportproperties in mesoscopic systems [1-8]. Experiments performed in such systems showed that theconductance g is not a self-averaged quantity [3] and then uctuates as the function of the Fermienergy, chemical potential and the sample size. In the metallic regime experimental [2-4] andtheoretical [5-8] studies indicated that the conductance uctuations are of order of e2=h and areuniversal (i.e. independent of the system size or the amount of disorder).In the localized (insulating) regime, the conductance exhibits strong uctuations which tendto diverge for large system sizes [9,10]. At critical point of the metal-insulator transition (MIT),large uctuations were predicted analytically for d = 2 + " ("  1) [11,12] but not detectednumerically by Markos and Kramer in 2D and 3D systems [13,14]. They found that the largeconductance uctuations are not the general feature of the metal-insulator transition. It wasalso found that the variance of log(g) is of order of its mean value and the standard deviationof g is also  < g > [15]Since the conductance does not obey to the central limit theorem [1] not only its meanvalue but its whole probability distribution has to be studied. Numerical results in 2D and 3Ddisordered systems showed that the conductance is gaussian distributed in the metallic regime[16] while for strongly localized systems (insulating regime) a log-normal distribution was found[16]. The correct form of the probability distribution at the transition is not well known. Insuch regime, it was proved that the conductance distribution is independent of the microscopicdetails of the model (determined by the distribution of the disorder), of the system size and of theposition of the critical point which separates the metallic and the localized regime in the spaceof external parameters (energy, disorder). This universality of the conductance distribution wasstudied and conrmed for 2D and 3D models [13-14]. The system-size invariance of P(g) at thecritical points of the MIT was conrmed for 3D and 4D orthogonal systems [17, 18]. It was alsofound that that the critical distribution depends on the dimension [15], the symmetry of thesystem [15,17] and on the boundary conditions [19].The electric eld was shown to delocalize the electronic states in 1D disordered systems(where all the states are localized) [20, 21]. In a previous work, we have shown that the Andersontransition may occur for strong elds [22].In this work, we use the Kronig-Penney model to test the size-independent probability distri-bution of the conductance at the transition in disordered mesoscopic chain under the inuenceof an electric eld.2 Model descriptionWe consider a Kronig-Penney model applied to a 1D system of equally spaced potentialV(x) with random strengths under a constant electric eld F. The corresponding Schrodinger2equation can be read:( d2dx2+Xnn(x   n)  eFx)	(x) = E	(x) (1)Here 	(x) is the single particle wavefunction at x, nthe potential strength of the n  th siteand E the single particle energy in units of h2=2m with m the electronic eective mass. Theelectronic charge e and the lattice parameter a are taken here for simplicity to be unity. Thechain length is identical to the number of scatterers (L = N). The two ends of the systemare assumed to be connected ohmically to ideal leads (where the electron moves freely) andmaintained at a constant potential dierence V = FL. The potential strength nis uniformlydistributed between  W=2 and W=2 (W being the degree of disorder).The exact solution of the equation (1) is Airy function-like. In order to reduce the com-putational time consuming we use the so-called ladder approximation which is valid only forweak elds [20, 21]. For strong elds, we use the multistep function approximation [23] whichis very accurate and use plane waves instead of the Airy functions. The second order dier-ential equation (1) can be mapped by means of the Poincare map representation in the ladderapproximation [20]	n+1=cos(kn+1) +knsin(kn+1)kn+1sin(kn)cos(kn) + nsin(kn+1)kn+1	n knsin(kn+1)kn+1sin(kn)	n 1(2)where 	nis the value of the wavefunction at site n and kn=pE + Fn is the electron wavenumber at the site n. The solution of equation (2) is carried out iteratively by taking the twoinitial wave functions at sites 1 and 2 : 	1= exp( ik) and 	2= exp( 2ik). We consider herean electron having a wave number k incident at site N + 3 from the right (by taking the chainlength L = N , i.e. N + 1 scatterers ). The transmission coecient (T ) readsT =k0kLj1  exp( 2ikL)j2j	N+2 	N+3exp( ikL)j2(3)where k0=pE and kL=pE + FN .The dimensionless 4-probe conductance (g =Ge2=h) can be obtained from the transmissioncoecient T via the Landauer formula for 1D systems [24]:g =2T1  T(4)where the factor two arises from the two possible states of the electron spin.3 Results and discussionIn this section we discuss the numerical results of the conductance uctuations and dis-tribution at the transition regime. In a previous work [22], we have shown that the real metal-insulator transition does not occur at E  V as believed from the results of Mato and Caro3[25] but this regime can be obtained for strong elds. To obtain the probability distributionof the conductance, we construct a statistical ensemble of 104samples which dier only in therealization of the disorder. The electric eld has been shown to delocalize the electronic statesin 1D disordered systems where the wave function becomes power-law decaying [20, 21], whilefor suciently large eld strengths, the eigenstates become extended [22]. In Figure 1, we showthe variance of Ln(g) as a function of the applied electric eld (for large eld strengths F > 1).The conductance uctuations has two pronounced peaks for F = 1:465 and F = 3. Theselarge uctuations are a sign of localized states even for large eld strengths. We expect thatthe metal-insulator transition may occur at these regions. To check the nature of the regime,we show in Figure 2 the probability distribution of Ln(g) (Fig.2a) and g (Fig.2b) for dierentsizes of the system (L = 500; 700; 800 and 900) and for F = 3. The conductance distribution isneither normal nor lognormal. The L-independence of the conductance distribution observed for2D and 3D systems [13-15, 18] is conrmed here. This behaviour is not observed in the metallicand insulating regime (see Fig.3) where the conductance distribution depends on the size of thesystem. The long tail of the distribution in Fig.1 is representative for large uctuations in goodagreement with the results of Shapiro [11] and Shapiro and Cohen [12] for such regime. Thedistribution of g in Fig.2b has a similar shape as found recently at the crossover region betweenthe metallic and the insulating regime in disordered quasi one-dimensional wire [26]. For anelectric eld strength corresponding to the rst peak in Fig.1, the conductance distribution hasa size dependent form (not shown). We can conclude that the MIT regime occur at F = 3 forE = 0:4 and W = 0:25. We can also observe that this regime is characterized by very largeconductances < g >' 58 (Fig.4 and Fig.2) This was not observed in the metallic regime wherethe mean conductance is small < g >' 0:4 [22]. We presented in Table I mean value of log gand its variance. We note clearly that Var(ln(g)) is not of order of ln(g) which is typical forthe localized states as found for 3D and 4D systems [15]. This shows that in our system theextended states dominates due to electric eld.We can also observe that P (g) has a hole at small g in agreement with the analytical resultin the " expansion [11, 12] and with the numerical results for 2D [27]and 3D systems [18, 28] .4 ConclusionWe have used the Kronig-Penney model in a simple1D disordered system in the presenceof an electric eld to examine the conductance uctuations and the size-independence of its dis-tribution at the metal-insulator transition. The results are in good agreement with the previousworks in 2D and 3D systems for other models [13-14] for the metal-insulator transition . It isimportant to study the universality of the conductance distribution at the transition in a systemof nite width potentials where the conductance uctuations are less important in comparisonto the present model[22] and to nd the critical points of the transition in the (energy-disorder-4electric eld) space and for dierent kinds of disorder. These problems will be the subject of aforthcoming paper.Acknowledgements One of us (K.S.) would like to acknowledge the hospitality of theAbdus Salam International Centre for Theoretical Physics, Trieste, Italy, enjoyed during theprogress of this work. He would also like to acknowledge Professor A.K. Sen for further discus-sions.References[1] B.L.Alt'shuler, A.G.Aronov and B.Z.Spivak, Sov.Phys. JETP Lett. 33, 94 (1981).[2] R.A.Webb, S.Washburn, C.P.Umbach and R.B.Laibowitz, Phys. Rev. Lett. 54, 2696 (1985).[3] S.Washburn and R.A.Webb, Adv. Phys. 35, 375 (1986).[4] J.L.Pichard and M.Sanquer, Physica A 167, 67 (1990)[5] P.A.Lee and A.D.Stone, Phys. Rev. Lett. 55, 1622 (1985); P.A.Lee, A.D.Stone andH.Fukuyama, Phys. Rev. B 35, 1039 (1987)[6] H.A.Weidenmuller, Physica A 167, 28 (1990)[7] B.L.Alt'shuler, Sov. Phys. JETP Lett. 41, 648 (1985); ibid 42, 359 (1986)[8] I.V.Lerner, Phys. Rev. Lett. A 133, 253 (1988)[9] K.S.Chase and A.McKinnon, J. Phys. C 20, 6189 (1987)[10] B.Kramer, A.Kawabata and M.Schreiber, Phil. Mag. B 65 (1992)[11] B.Shapiro, Phil. Mag. B 56, 1032 (1987); Phys. Rev. Lett. 65, 595 (1992)[12] B.Shapiro and A.Cohen, Int. J. Mod.Phys. B 6, 1243 (1992)[13] P.Markos, Europhys. Lett. 26,431 (1994); J. Phys.I France 4, 551 (1994)[14] P.Markos and B.Kramer, Phil. Mag. B 68, 357 (1993)[15] P.Markos, Phys.Rev.Lett.83, 588 (1999)[16] P.Sheng and Z.Zhang, J.Phys.Condens.Matter 3, 4257 (1991)[17] K.Slevin and T.Ohtsuki, Phys.Rev.Lett.78, 4083 (1997)5[18] K.Slevin and T.Ohtsuki, Phys.Rev.Lett.82, 669 (1999)[19] D.Braun, G.Montanmbaux and M.Pascaud, Phys.Rev.Lett.81, 1062 (1998)[20] C.M.Soukoulis, J.V.Jose, E.N.Economou and P.Sheng, Phys. Rev. Lett. 50, 764 (1983).[21] E.Cota, J.V.Jose and M.Ya.Azbel, Phys. Rev. B 32, 6157 (1985).[22] K.Senouci, N.Zekri and R.Ouasti, Physica A 234, 23 (1996).[23] Y.Ando and T.Itoh, J.Appl.Phys. 61, 1497 (1987)[24] R.Landauer, Phil. Mag. 21, 263 (1970)[25] C.Mato and A.Caro, J.Phys.Condens.Matter 1, 901 (1989)[26] P.Wole and K.A.Muttalib, Ann.Phys.(Leipzig) 8,7-9, 753 (1999)[27] X.Wang, Q.Li and C.M.Soukoulis, Phys.Rev.B 58, 3576 (1998)[28] C.M.Soukoulis, X.Wang, Q.Li and M.M.Sigalas, Phys.Rev.Lett.82, 668 (1999)6Table I Mean conductance < Ln(g) > and its variance for dierent system size L.L < Ln(g) > V ar(Ln(g))500 3.9557 0.20548600 3.9314 0.20474700 3.9596 0.21149800 3.9632 0.217177Figure CaptionsFig.1 Variance of Ln(g) versus applied electric eld for L = 500; E = 0:4 and W = 0:25:Fig.2a Probability distribution of Ln(g) for F = 3, E = 0:4 and W = 0:25 and dierent systemsizes ( L = 500; 700; 800 and 900 ) compared with a gaussian with same mean and variance(dashed curve).Fig2b Distribution of g for the same parameters as in Fig1a.Fig.3 Distribution of  Ln(g) in the insulting regime for F = 0; E = 5 andW = 2 and dierentsystem sizes (L = 500; 700 and 800) from left to right compared with a gaussian with same meanand variance (dashed curves).Fig.4 Mean conductance < Ln(g) > versus applied electric eld for the same parameters as inFig.1.8

Conductance fluctuations and distribution at metal-insulator transition induced by electric field in disordered chain

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