Electronic transport in disordered n-alkanes: From fluid methane to amorphous polyethylene

JOURNAL OF CHEMICAL PHYSICS VOLUME 119, NUMBER 5 1 AUGUST 2003

Electronic transport in disordered n -alkanes: From fluid methaneto amorphous polyethylene

David Cubero and Nicholas Quirkea)

Department of Chemistry, Imperial College, London, SW7 2AY, United Kingdom

David F. CokerDepartment of Chemistry, Boston University, Boston, Massachusetts 02215

~Received 28 February 2003; accepted 7 May 2003!

We use a fast Fourier transform block Lanczos diagonalization algorithm to study the electronicstates of excess electrons in fluid alkanes~methane, ethane, and propane! and in a molecular modelof amorphous polyethylene~PE! relevant to studies of space charge in insulating polymers. Weobtain a new pseudopotential for electron–PE interactions by fitting to the electronic properties offluid alkanes and use this to obtain new results for electron transport in amorphous PE. From oursimulations, while the electronic states in fluid methane are extended throughout the whole sample,in amorphous PE there is a transition between localized and delocalized states slightly above thevacuum level~;10.06 eV!. The localized states in our amorphous PE model extend to20.33 eVbelow this level. Using the Kubo–Greenwood equation we compute the zero-field electron mobilityin pure amorphous PE to bem'231023 cm2/V s. Our results highlight the importance of electrontransport through extended states inamorphousregions to an understanding of electron transport inPE. © 2003 American Institute of Physics.@DOI: 10.1063/1.1587130#

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I. INTRODUCTION

Polyethylene is commonly used as an insulator for hivoltage electric cables.1 A long standing problem with poly-meric insulators is that they trap charge, to form whatusually described as a ‘‘space charge.’’ There is a vast liteture concerned with the experimental characterizationspace charge~methods2 include thermal step,3 pulsed electro-acoustic,4 thermal pulse,5 pressure pulse,6 photoconduc-tivity,7 and mirror image8!. It is believed that electricabreakdown in dielectric polymers is closely related to tcreation of space charge in the material9 linked to both itschemical and physical characteristics. Thus, a model todict the lifetime of the insulating material would naturalinclude a description of the space charge and its relatiothe chemical and physical microstructure. Several phenenological models have been developed to explainformation10 as well as the role of the space charge in eleccal aging.11 Nevertheless, a clear identification of its moleclar origin is not yet available. An understanding of spacharge accumulation and its link to dielectric breakdowould be a major scientific achievement as well as faciliting the prediction of cable lifetimes and the design of becables.

In our previous work we have identified physical achemical traps in model alkane waxes thought to appromate the local structure of amorphous polyethylene and cacterized them usingab initio methods.12,13This informationhas been used to predict the current voltage characteristimodel polyethylene with some success.14 However the ear-lier calculations of electron trapping were limited to the e

a!Electronic mail: [email protected]

2660021-9606/2003/119(5)/2669/11/$20.00

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fects of conformational defects with respect to the groustate of the all-trans long chain alkane molecule and wenot capable of capturing information on the role of locdensity changes associated with local disorder in amorphpolyethylene~PE! or the details of excess electron statesmore complete representations of crystalline polyethyleneaddition all previous calculations were for ground states.the present paper we use a fast Fourier transform block Lazos diagonalization algorithm to study the electronic staand mobility of excess electrons in amorphous polyethyle~PE!, developing a new pseudopotential for electron–PEteractions by fitting to the electronic properties of fluid akanes.

The behavior of excess electrons inn-alkane fluids hasbeen extensively investigated15–19and represents a paradigfor the study of the transport properties of excess electrondielectric media. There are numerous measurements oflow-field electron mobilitym and the conduction band energy V0 , defined as the difference of the photoelectric wofunction of a metal electrode when in contact with the marial and in vacuum. These two quantities are usuacorrelated,20 indeed for spherical molecules such as methaor neopentane the maximum inm appears at about the samdensity as the minimum inV0 . This correspondence can bexplained in terms of a deformation potential theory21 as-suming that the electronic states are extended. In chain mecules such asn-alkanes the mobility can decrease by seveorders of magnitude as the fluid density varies; this has battributed to localized states trapping electrons.22

In what follows we report results for the electronic statand zero-field mobilities of excess electrons in alkane fluand in amorphous PE using a pseudopotential technique.new data for amorphous PE allow us to clarify earlier d

9 © 2003 American Institute of Physics

IP license or copyright, see http://ojps.aip.org/jcpo/jcpcr.jsp

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2670 J. Chem. Phys., Vol. 119, No. 5, 1 August 2003 Cubero, Quirke, and Coker

cussions of space charge and its relation to electron trapby physical defects.9 We have reported some preliminary rsults for the electronic states in polyethylene and alkcrystals elsewhere.23,24 Note that in what follows we reporelectronic states corresponding to configurations taken fclassical molecular dynamics simulationunperturbed by thepresence of the excess electron. This does not affect the mairesults reported in this paper or our previous paper,25 sincewe are mainly concerned in conduction through extenstates, where the electron propagates fast enough that wneglect any perturbation of the slow molecular trajectorby the electron.

The order of magnitude of the electron mobility is detemined by the nature of the electronic wave functions~i.e.,extended versus localized!. Indeed in the modern theory othe insulating state,26 an insulator is characterized by thlocalization of the many-body ground state wave functioThere are a number of possible criteria for localization osingle electron wave function available in the literature.27–29

However, most of them involve studying the behavior of teigenstates as the system becomes larger, and thus arecult to apply in the context of periodic systems when thare computational limitations on the system size. Edwaand Thouless have proposed a method based on the senity of the energy levels to the boundary conditions.28,30Whenchanged from periodic to antiperiodic boundary conditiothe energy levels should be shifted. If the eigenstatesextended the shift varies asL22, whereL is the length of thesystem, but if localized asD the electron diffusion constanand the energy shift is exponentially small. However, tmethod is not easy to implement in the version of the Lazos diagonalization method31 that we wish to use for thisstudy. As a result we have developed a new criterionlocalization that does not involve changes in the systlength or the boundary conditions. With our approach, callations on different size systems still need to be performedecide if the properties of the small periodic systems westudy are representative of the bulk systems in which weinterested. Our strategy is to test pseudopotentials and loization methods on the fluid alkanes and then apply thisproach to the problem of the electronic structure and elecmobilities in amorphous polyethylene

The paper is organized as follows: In Sec. II we descrthe localization criterion method, the Kubo–Greenwoequation we have used to compute electronic mobilities,the simulation details. In Sec. III we describe the electroalkane pseudopotential. The results of the simulationscalculations are presented in Sec. IV. Finally, Sec. V provia short summary and conclusions.

II. METHODOLOGY

A. Localization criterion

Assume we have computed the eigenstatescE(r ) of theone-electron HamiltonianH in a box of sideL For simplic-ity, we will consider Born–von Karman boundary conditioat x56L/2. Starting from the operator formula@x,H#5 i\px /m we have


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\^cE8uxucE&1Fx~L/2!, ~1!

whereFx is a flux contribution involving a surface integraat the boundary of the box

Fx~xB!5\

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Equation~1! is usually presented without the boundary cotribution Fx , because it is assumed that the wave functgoes to zero when the boundaries are taken to infinity,this only applies for the special case of localized states. FEq. ~1! it is clear that localized eigenstates can only havanishing expected momenta~when E85E), they do notpropagate. For extended states the flux contribution is vimportant, becoming dominant when the two states cospond to energies that are very close. This can be usedsign of delocalization: when the fluxFx is comparable to thefirst term in Eq.~1! the states are extended. However, therea technical difficulty which comes from the fact that the psition operatorx and thus^cE8uxucE&, is not well definedunder periodic boundary conditions~nor with extendedstates!. Its matrix elements, as well asFx , depend on theparticular choice for the position of the origin of the systex0 ~chosen at any point inside the unit cell!, which in turndetermine the boundaries atxB5x06L/2. This is not a realproblem for extended states, because the boundary conttion becomes dominant and thus independent of the bouary position@the left-hand side of Eq.~1! is boundary posi-tion independent#, but can lead to errors for localized statesthe system size is small compared to the localization lenas discussed in the following.

Assume thatcE(r ) is localized around some point in thunit cell. Because of the boundary conditions, it will be lcalized around all equivalent points in the periodic imagesthe unit cell. If we place the boundaries in a region wherewave functioncE(r ) is small, the contributionFx will beexponentially small, indicating localization. But if thboundary is in the middle of the localized wave function,Fx

will provide an appreciable contribution, which might bwrongly interpreted as a sign of delocalization. Therefowe propose to computeFx as a function of the boundarpositionxB , and take the value that corresponds to the mmum in absolute value. If the minimum absolute value of tboundary term is comparable withcE8uxucE& or equiva-lently with the momentum matrix element, the states arelocalized. However, if this boundary term is much smalthan the volume term, the states are localized providedmodel system is large enough to contain the localized wfunction. These procedures can be applied to the other spcomponents, providing a reliable criterion for localization

We recall that because of time reversal symmetry~in theabsence of magnetic fields!, for any given eigenfunctionc(r ) with an expected momentumpx& there is anothereigenstate~the complex conjugate of the former! associatedwith same energy and propagating with momentum2^px&.As a consequence, all eigenstates can be taken to be rea~andare taken to be so in the Lanczos algorithm employed infollowing!, as shown by the transformationc15(c


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2671J. Chem. Phys., Vol. 119, No. 5, 1 August 2003 Electronic transport in n-alkanes

1c* )/&, c25(c2c* )/& i . In this case the expected momentum of all eigenstates will be zero,even if the statesrepresent delocalized states. The momentum module oforiginal eigenfunctions can be recovered by calculatingmatrix element c1upxuc2&, which is the quantity appearinin Eq. ~1!. In practice, it is known that in a finite disorderesystem the probability that an energy level is degeneratzero, being only strictly degenerate in the infinite system32

However, the minimum energy distance between twotended levels goes to zero as the size of system is increaand in fact, the matrix element considered in Eq.~1! is therelevant quantity determining the zero-frequency conducity as expressed by the Kubo equation@Eq. ~4!#.

Resta and Sorella28 have proposed a localization defintion that avoids any reference to the excitation spectruThey showed that the dimensionless complex numbez5^cuei (2p/L)xc& tends to one when the state is localized ato zero when it is delocalized in the limitL→`. This defi-nition is not restricted to a one-electron representation, talso showed that the trivial extension of the formula tomany-body system still tends to one for insulating syste~either independent electrons or correlated, either crystaor disordered! and to zero for metals. We have tried to appthis criterion to our data. However, the results are not cclusive, as the method requires the size of the system tmuch bigger than the typical localization lengthl. In thatcase it can be expanded asz;12(2p/L)2l2/2. If, for ex-ample, the size of the system is only six times the localition length, then the first correction would be 0.55, andz stillfar from unity. In contrast, the localization method we prpose depends only on the value of the wave functions atboundaries and will be valid provided the cell containslocalized wave function.

The approach we have outlined in this section for decing if states of a finite periodic system are localized or etended can be applied once we have evidence that the perties of the states of this small system are representativthe infinite system we are trying to study. The approachemploy here for this purpose involves comparing the traport properties of finite systems of different size as detain Sec. IV.

B. Kubo–Greenwood equation

The electrical conductivity in the low field limit is givenby the Kubo–Greenwood formula,33 which in the indepen-dent electrons approximation can be written as34

s52E dE s~E!] f

]E, ~3!

where f (E) is the Fermi distribution ands(E) is the con-ductivity at zero temperature,

s~E!52pe2\V

m2 u~px!abuav2 g~E!2, ~4!

whereV is volume of the system andg(E) is the density ofstates per unit energy in each spin direction, per unit voluThe suffix ‘‘av’’ denotes an average over all states havenergy within an intervalD aroundE:


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whereND is the number of states inside that energy intervThis expression together withg(E) must be averaged oveall the configurations of the atoms~considered as classicaparticles! in the ensemble. The conductivity is obtainedthe double limitV→` andD→0, which should be carriedout simultaneously.28

If the states from a range of energies are localized,expected momentum vanishes and Eq.~4! predicts no contri-bution to the conductivity. There can still be conduction dto hopping assisted by phonons, a mechanism not includethis version of the Kubo equation, but usually it involvemobilities orders of magnitude smaller than the contributdue to extended states. In our case the concentration ocess electrons is very low, and thus the electron gas is ndegenerate and the Fermi distribution becomes MaxwellTherefore, we can write the mobilitym5s/en (n being theexcess electron number density! as

m5pe\3Vb

m2

*dE K~V,E!2g~E!2e2bE

*dE g~E!e2bE , ~6!

whereb[1/kBT and

K2[u~px!abuav2 /\2. ~7!

We shall see that even at room temperaturekBT ~0.026eV! is small compared to the typical energy range ofg(E)(;0.1 eV). Consequently, the main contributions to thetegrals in the numerator and denominator in Eq.~6! comefrom a small interval aroundEc ~the mobility edgeEc sepa-rates localized and extended states! and the region close tothe minimum energyE0 , respectively.

In a perfect lattice, with a given fixed density of scatteers,K2 is a function of energy alone, and Eq.~6! gives theexpected divergence when the system volume is takeninfinity. In order to obtain a finite mobility,K2 must decay asV21 when V→`. This can be understood by meanssimple considerations due to Mott.34 Assuming that thephases of the wave function in two different regions arecorrelated if they are separated by a distance bigger thal,the coherence length~also called mean free path!, the inte-gral K can be expanded as a sum over different regionsG j ofvolumel3:

K5(j

K j ,

where

K j5EG j

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i

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Since the phases are uncorrelated in different regionsvariables K j will be uncorrelated, leading toK25( jK j

2

5(V/l3)K j2 , whereV/l3 is the number of such regions

For extended states the wave function can be writtenc(r )5f(r )/AV, wheref(r ) is a complex function with amodulus of the order of the unity. In the quasifree particpicture f(r )'exp(ik"r ), wherek is a vector of magnitude


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k5A2mE/\ and a random direction in each regionG j . Withthese approximations it is easily shown that the correcthavior in the limitV→` is satisfied:

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Note that if the size of the system is smaller than the typcoherence lengthl the integral in Eq.~9!, and thus the com-puted mobility, will give a value smaller than the actual mbility in the real ~larger! system.

C. Simulation details

We have used the molecular dynamics codeDLPOLY35 togenerate molecular configurations at finite temperature.have modeled then-alkane fluids using a united atommodel.36 The force field includes bonded~bond stretch,37 va-lence angles, and dihedrals38! and dispersion interactions. Icontrast the amorphous regions of PE atT5300 K weremodeled using anall-atom force field described in Ref. 39employing a single chain in a periodic simulation cell. Vaous PE system sizes were used~at constant density! rangingfrom 87 to 2880 CH2 units ~of three atoms!. The followingprocedure was employed: an initial low density configuratof the chain was generated using a Monte Carlo algoriththen the system was compressed in a constant pressurelecular dynamics run until the density reached the expmental value40 of amorphous PE (r50.855 g cm23) at roomtemperature. Finally, the system was brought to a local elibrium in the microcanonical ensemble. The pair correlatfunctions, dihedral angle distribution, and thermodynamproperties were independent of the procedure used to prethe system. We obtained good agreement with a systemtained by quenching, previous simulations,41 and experimen-tal data.42 Note that at 300 K simulated and experimenamorphous states of PE are above the glass transition.42

A fast Fourier transform block Lanczos diagonalizatialgorithm31,43was used to compute the adiabatic excess etronic states of the static configurations taken from the mlecular simulation. This method provides the lowest eneeigenstates for a given three-dimensional potential enegrid, and has been used successfully to describe excesstrons in simple fluids.44 We used a fixed grid spacing of 0.Å, though for the large fluid systems we used 1 Å. We hatested the convergence of our calculations with gridssmaller spacing. As an example, in amorphous PE, usgrids of 223, 323, and 643 points ~corresponding to uniformgrid spacings of 0.97, 0.67, and 0.33 Å! we obtained aground-state energy ofE0520.134 55, 20.107 91, and20.107 55 eV, respectively, which implies an energy resotion of 0.03 eV.

The polarization interaction between the electron andinduced dipoles in the system was computed sconsistently using an iterative algorithm~see Sec. III!. Theelectron–atom interactions were truncated atr c59 Å. Allreported electronic energies include a long-range correcVc based on a perturbation theory partition of the pseudotential for disordered systems:Vc522panaL /r c ,44 wherea is the isotropic polarizability of the molecules or~united!


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atoms, n the corresponding number density, andaL5@11(8/3)pna#21 the Lorentz local-field factor. For highly ordered systems the long-range correction can be compusing the formVc52A/r c , whereA is a constant obtainedby linear fitting the data at different cutoffs. As a test of tisotropic correction, using the ground-state energiesamorphous PE withr c59 and 12 Å we obtainedA53.89 Å eV, which leads to a long-range correction constent with the isotropic correction data within a margin0.007 eV.

III. PSEUDOPOTENTIALS

The potential energy between an excess electron atr anda system of~united! atoms atRi can be written as

V~r ,Ri !5(j

Vjr~ ur2Rj u!)1Vp~r ,Ri !, ~10!

whereVjr(r ) is a short-range repulsive pair potential whic

accounts for the interaction with the static charge distributas well as exchange and orthogonality of the excess elecwith the target atom. The second term represents the chainduced-dipole polarization interaction:

Vp~r ,Ri !521

2 (j

pj "Ej(0)Sj~ ur2Rj u!, ~11!

where

Ej(0)52e

Rj2r

uRj2r u3~12!

is the direct electric field due to the excess electron,

pj5a jEj ~13!

the dipole moment of the atomj , a j the corresponding polarizability tensor, andSj (r ) is a switching function whichdescribes the screening of the electric field due to the chadistribution of the atom as the electron approaches it. Tfunction goes to 1 as the electron separates from the taatom. In Eq.~13! Ej is the local electric field at the atomj ,which is the superposition of the Coulomb field of the exceelectron and the electric fields due to all the other inducdipoles

Ej5Ej(0)1(

kÞ jTjk•akEk , ~14!

with Tjk5(3r jk r jk21)/r jk3 . The set of equations~14! can be

solved self-consistently using an iterative approach. Hoever, if the atoms or molecules in our system are isotroand the macroscopic state is disordered, then we can umuch less expensive mean-field approach,44,45 where the lo-cal field is approximated bya(r )Ej

(0) . Here a(r ) is amean-field screening function solution of an integral eqtion that is a function of the density and the pair correlatifunction of the system.45 In general this function decayfrom unity to the Lorentz local factoraL in an oscillatoryfashion. With this approximation the potential energy bcomes pairwise additive, with a polarization contributioVp(r )52ae2S(r )a(r )/2r 4.


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A. e-CH4 pseudopotential

We have utilized the pseudopotential used in Ref.to describe the interaction between an excess electrona molecule of methane. It involves a repulsive compnent Vr(r )5( i 51

2 Ai exp(2Bir), with A158160 eV andA2

523590 eV, combined with the switching functionS(r )5(12e2r /r 0)6 with r 050.622 Å. The parameters in the repulsive term are summarized in Table I. These paramewere adjusted to give agreement withl 50 phase shift dataobtained fromab initio calculations.

Methane is a spherical molecule, with a polarizabilitya52.6 Å3.47 In an isotropic fluid of spherical particles meafield theory should be reliable. Figure 1 compares the grostate energies obtained with the mean-field approximawith the values from a full many-body polarization calcultion and experimental results for various methane densitieT5340 K. The data comprise averages over 10 differconfigurations of 512 methane molecules. Both compucurves agree well with the experimental values ofV0 . Simu-lations atT5180 K resulted in consistent data.

B. e-CH3 and e-CH2 pseudopotentials

In order to obtain a pseudopotential transferable to otalkanes we have considered the higher alkanes, ethanepropane. The electron–methane pseudopotential cannoused directly to describe the interaction with ethane becaof the anisotropy of the molecule.46 This anisotropy is mani-fested in both repulsive and attractive~polarization! parts ofthe pseudopotential. The polarizability tensor in ethaneanisotropic; the component along the axis of the molecul

TABLE I. Pseudopotential parameters. Energies in electron voltslengths in angstroms.

B1 B2 d

CH4 3.97 3.53CH3 3.97 3.53 0.3CH2 4.41 3.92 0.44

FIG. 1. Excess electron ground state energies in methane. Closed triaare the experimentalV0 values ~Ref. 58!, diamonds the full many-bodypolarization, and squares the mean-field results.


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larger than that in the perpendicular directions.47 In Ref. 46this anisotropy was accounted for by placing an exponenrepulsive center in the middle of the C–C bond. Althoumodeling a long-range interaction (;r 24) anisotropy with ashort-range term~exponential! is not optimal, it can be ad-justed to force agreement with the experimental data influid density range. However, when used in high density stems as polyethylene we have found that it leads to anderestimation of the conduction band level by several etron volts. Instead we have used the multicenpolarizabilities given in Ref. 48, which were obtained usiab initio calculations. In this approach the electrostatic ptential created by the alkane molecule due to the presencthe electron is reproduced by a distribution of dipoles locaat the hydrogen and carbons atoms, as well as at the centthe C–C bond. Using this set of polarizabilities we obtain tfollowing polarizability tensors~in Å 3) for the methyl andmethylene groups:

aCH35S 1.27

2.04

2.04D , ~15!

and

aCH25S 0.5

1.12

1.64D . ~16!

In Eq. ~15!, the x axis is defined along the C–C bondwhereas in Eq.~16! the x axis should be taken in the direction defined by the two nearest carbon neighbors in the mecule and they axis inside the plane of the carbon nuclei~seeFig. 2!. The set of polarizabilities is completed with the plarizability tensor of the bond C–C, which is represented a

d

les

FIG. 2. Scheme showing the axes defined in the text for the CH2 and CH3

units.


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new site centered at the middle of the C–C bond withcomponentaC–C52.103 Å3 in the bond direction, and zerin the perpendicular directions. Because these polarizabilwere calculated neglecting the intramolecular dipole–dipinteraction~up ton-butane48!, we did not compute the direcdipole–dipole interactions when the two centers were clothan a certain distance, taken asr mbmin53 Å. The resultswere insensitive to any reasonable choice of this parame

In Ref. 49 the excess electron wave function correspoing to the bottom of the conduction band of crystalline PEcalculated using density functional theory~DFT!. It can beseen in Fig. 3 of Ref. 49 that the CH2 units act as repulsivesites for the electron, displacedd;0.4 Å from the carbonsatoms in the direction of the hydrogens. This study is baon the identification of the Kohn–Sham eigenstates withcited states, which is not a rigorous procedure, and iknown that it leads to energies that underestimate bandby several electron volts.50,51 Nevertheless, the Kohn–Shaequations can be interpreted as a mean-field approximafor a single electron, and thus it may provide a qualitatdescription for electron states. Consequently, we havecluded this feature in our pseudopotential model~Table I liststhe displacementsd from the centers of the carbon atoms!. Inthe methyl group the displacement is away from the metalong the C–C bond direction, whereas in the methylegroups it is along they axis defined in Eq.~16! ~see Fig. 2!.

For the repulsive and switching functions in the pseupotential we have used the same functional form as for mane, but readjusted the parameters in order to get good ament with the experimental data for the ground state enerin ethane and propane. The parameters are listed in Tab

Figure 3 shows the calculated ground-state valuesethane and propane atT5340 K and T5426 K, respec-tively, together with the experimental data taken from Re52 and 53. The calculated values were averaged oveconfigurations of 216 alkane molecules~systems of 1728molecules produced consistent data!. As we shall see lateon, the mobility edge in these systems lies close to the aage ground-state energy, and hence to the experimental vof V0 .

FIG. 3. Excess electron energies in ethane and propane. Computed gstate values~lines!, mobility edge~closed squares!, and experimentalV0

~triangles! as a function of the number molecular density.


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In order to probe our pseudopotentials we have conered two different situations. In the first case the polarizaity centers of the methyl methylene units are kept atcenter of the carbon atoms, while the repulsive centersdisplaced as discussed before. In the second case the pizability centers are placed together with the repulsive cters. Both pseudopotentials produced energies in agreemwith the experimental data and to each other within 0.07which we consider a good estimate of the accuracy ofpseudopotentials.

C. Surface effects

The energy origin in our pseudopotential is such thatinteraction energy tends to zero when the electron separfrom the atoms. Therefore, it is tempting to identify the zeof energy with the vacuum level. However, this pseudoptential does not take into account the distortions in the etrostatic charge distribution near the surfaces, which resin a net dipole moment creating a electrostatic potendrop.54 Thus, the zero of energy is usually slightly below thvacuum level by a quantityWs , which depends on the details of the surface itself. We have performedab initio cal-culations of the crystal surfaces~001! and~110! to determineWs using the density functional theory codeCASTEP55 withinthe CGA, Perdew–Burke–Ernzerhof gradient correction56

Following the method introduced by Baldereschiet al.,57 wefirst computed the mean electrostatic stepDs across the slabsurface using the total~electronic and nuclear! charge densityobtained with the DFT code. The plane-averaged elecstatic potentialVel(z) ~taking thez axis perpendicular to thesurface! is then obtained by integrating the one-dimensioPoisson equation. We first considered the surface~110!, forwhich the PE chains are arranged in infinite layers parallethe surface. Inside the bulk these layers are separatedeach other by a distance 4.105 Å. The slab contained 6ers, with a vacuum thickness separating the slab of 17Both surfaces of the slab were kept equivalent, thereavoiding dipole interactions between the surfaces. In adtion we calculated the charge density of a simple layer isupercell of 15 Å for which surface effects are absent. Telectrostatic potential is obtained by simply adding the ctribution of each layer.Ws can be estimated as the differenbetween the dipole strengthDs in each case givingWs

50.0360.02 eV. To study the dependence on the surftype we have also computed the charge density of a~001! PEslab, where the chains are perpendicular to the surface, uthe chain fold proposed in Ref. 58. This gaveWs50.0760.03 eV, which implies an energy difference betweentwo surface types of 0.0460.03 eV, in reasonable agreemewith the value of 0.07 eV reported in Ref 58. The large erbars come from the fact that we are measuring very tdifferences~about 1% ofDs;5.5 eV). In fact, these valueare of questionable significance, as we do not expect DFbe capable of such accuracy. Nevertheless, they implycorrections due to surface effects are small and can beglected. This result agrees with simple electrostatic con

und


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erations, which predict no correction at all as long asmolecules are not drastically truncated at the surfaces salkanes are nonpolar molecules.

IV. RESULTS

A. Methane

Simple inspection of the excess electron wave functin fluid methane shows that the ground state is clearlytended throughout the whole density range shown in FigThe localization criterion presented in Sec. II cannot beplied to excess electron ground state wave functions beceven though they are extended, e.g., being Bloch functiothey are usually at the bottom of the band and thus witvanishing momentum. However, it can be applied to thecited states. Table II shows thex component of the expectemomentum of the wave functionc5(c11 ic2)/&, wherec1 andc2 are the first two excited levels, for a typical cofiguration of the fluid with number density 9.831021 cm23. This momentum can be readily evaluated uing the fast Fourier transform algorithm, and is equivalenthe matrix element in Eq.~1! except for a phase factor. Thminimum extended correctionFx is very close to the ex-pected momentum, showing the extended nature of the strum. This result is consistent with the high mobility foundexperiments.59 In order to calculate the mobility we havcomputed the density of states for a methane fluid of den9.8531021 cm23 at T5180 K. The histogram of the first 25levels, averaged over 10 configurations, can be summaras a density of states per spin~as shown in Fig. 4! which wehave fit to the formg(E)'11.9(E2E0) eV2131021 cm23

in the energy interval20.36,E,20.15 eV, whereE0

520.36 eV. The density of states was insensitive to teperature; we obtained the same results atT5340 K. Usingthe Kubo–Greenwood equation~6! we estimate m'102 cm2/V s, which is small compared to the experimenvalue,m'1000 cm2/V s. This discrepancy can be explainein terms of the deformation-potential theory,21 which whenused with the experimentalV0 values~as a function of themolecular density! provides the correct shape of the expemental mobility against the density and acceptable quantive agreement far from the critical density in methane60

This semiphenomenological theory is based on the assution of free-particle-like behavior, inferred from the high eperimental value of the mobility, and can be seen as a csequence of the mobility edge lying at the bottom of the ba

TABLE II. Localization criterion data. Energies in electron volts and lengin angstroms.

L E1 E2 px /\ Fx /\

Methane 74.55 20.343 20.342 0.0195 0.0163Ethane 53.11 0.102 0.124 0.0039 0.000

0.102 0.140 20.0055 20.00010.124 0.140 20.0088 20.00010.140 0.143 20.0276 20.0222

Amorphous PE 32.11 20.017 0.008 20.0054 0.00060.055 0.059 0.0167 0.01450.119 0.125 0.0176 0.0159


ece

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ed

-

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(Ec;E0). Following Basak and Cohen,21 at thermodynamicequilibrium only the lower levels will contribute to the conductivity. As a result, the corresponding eigenstates will haconsiderable amplitude Fourier components on the scalthe thermal de Broglie wave number, in addition to the laamplitude Fourier components on the much smaller scalthe inverse of the interatomic separation produced bylocal disorder. This involves two different coherence lengtThe shorter length is responsible for the decay of the avaged square momentum matrix elementK2 @defined in Eq.~7!# as a function of the system volume in the length scalethe simulations, which is shown in Fig. 5. The data corspond to the bottom of the band of systems with sizesL523.30, 37.28, 55.91, and 74.55 Å, and an average enintervalD50.13, 0.05, 0.02, and 0.01 eV, respectively. Hoever, as shown in the inset of Fig. 5, the largest systemis not big enough to reach a plateau ofVK2 values, becausethe thermal coherence length is still larger. Assuming anfective mass of the order of unity for the excess electron,

FIG. 4. Density of states per unit of volume, energy, and spin for methwith r59.8531021 cm23 ~crosses!, amorphous PE~diamonds! at roomtemperature, and ethane withr511.5331021 cm23 ~triangles!. Closed sym-bols are data from larger systems plotted to test consistency. The linemethane and PE are the polynomial fits used in the text, whereas for etit is a guide to the eye.

FIG. 5. Averaged momentumK2 ~7! at the mobility edge as a function othe system volume for methane.


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thermal de Broglie wavelength will be of the order of 100bigger than the largest system simulated. The corresponwave numbers are only scattered by the long-range fluctions of the potential created by local density fluctuationsproper calculation of the mobility in such a quasifree stwould involve simulating systems of at least twice the linedimension we have considered~i.e., 8 N!. Because of thegrid techniques we are using, that is beyond our presensources.

B. Ethane and propane

The ground-state energies for excess electrons in ethshown in Fig. 3 are calculated from unperturbed configutions of the ethane molecules at 340 K. It is known thathigh fluid densities the excess electron, in nonpolar and psolvents,61 may interact strongly with its environment, pusing the surrounding molecules aside and loweringenergy46,62 by creating a bubble. This is also expected frothe abrupt drop of the mobility in ethane at aboutr'1031021 cm23 ~Ref. 22! ~Fig. 6, top!. The abrupt mobilitydrop can be explained by considering the fraction of the vume occupied by the localized electron as a function offluid density. Figure 6 shows the fractionh of volume wherethe excess electron ground state has a density probablarger than 10% of its maximum plotted over a rangeunperturbed fluid ethane densities and for a fixed unit celL544.1 Å. The electron, and thus its interaction with tsolvent, is spread throughout much of the simulation boxlow fluid densities and hence will not exert a large forcenearby molecules, but the electron becomes increasiconcentrated for the higher densities. For densities ab1031021 cm23 the excess electron is restricted to a smregion of the box, where the concentrated quantum forcesable to create a bubble. This localization does not take p

FIG. 6. Fraction of the volume occupied by the excess electron as a funof the fluid density for a fixed system length ofL544.1 Å. The upper plotshows the experimental electron mobility of ethane atT5340 K ~closedcircles! andT5305.33 K~open triangles!—Ref. 22.


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when the electronic states are computed using the C4

pseudopotential centers at the ethane carbon atoms insteour proposed CH3 pseudopotentials for the samconfigurations.46 Therefore, it is a consequence of the rduced free volume seen by the electron because of theplacementd of the CH3 repulsive core from the ethane cabon atoms, which effectively breaks the paths whereexcess electron can propagate if ethane is viewed as paiCH4 centers.

The reduction of the wave function extension isgeneral property of the energy spectrum, but does not nesarily mean that all states are localized. Table II showslocalization criterion data for a typical unperturbed configration of the fluid ethane withr511.5331021 cm23. Thecomparison of the first excited levelE150.102 eV with theother levels indicates that it is localized, but levels abo0.140 eV show clear delocalization, even though the cosponding fraction of the volume occupied in this excitstate is as low ash50.03. We have repeated this analysis f10 different configurations, and obtained the average moity edgeEc50.13 eV, as shown in Figs. 3 and 4. We alpresent in Fig. 4 the corresponding density of states forfluid ethane state point, calculated averaging over 10 cfigurations of a system of 216~open triangles! and 1728~closed triangles! ethane molecules. A similar calculation fothe highest density considered,r513.1831021 cm23 in asimulation box of lengthL550.8 Å, leads to a mobilityedge located at 0.40 eV, and then only 0.05 eV aboveaveraged ground state level. Thus we see in these unturbed fluid ethane configurations that as the density iscreased the gap between localized states at low energydelocalized states at higher energies narrows to aboutkBT atthe highest fluid densities considered in these studies.

We have computed the mobility edge as a functiondensity for a system of 1728 molecules of propane followthe same procedure as for ethane. The results are showFig. 3, which shows a transition to localized states at abr57.531021 cm23.

C. Amorphous PE

The number density of carbon atoms is typically abo1.5 times that of the densest fluid alkanes discussed inprevious sections. Due to the high molecular densityamorphous PE, the lowest states are always localized inregions where the density is lowest~since PE is a solid we donot expect the electron to significantly alter the local densunlike for high density liquids!. The minimum energy electronic state obtained in the simulations is located belowvacuum level at20.32 eV. This is in good agreement witexperiments on molten long-chain alkanes, where the grostate energy was also found to be negative.63 Analysis of thefunction K2(E,V) as well as the localization criterion datlocate the mobility edgeEc at about the zero energy leveThis is confirmed in the two largest systems considered,systems containing 1215 and 2880 CH2 units provided thevalues 10.055 eV ~see Table II! and 10.072 eV, respec-tively. Figure 7 shows the ground state wave function antypical extended state above the mobility edge for a giv

on


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configuration of the material at room temperature and a stem size of 32.12 Å. The ground state is completely localizin a region of about 7 Å, whereas in the extended statewave function is spread through the entire simulation ce

We have computed the density of states of amorphPE ~averaging over 75 configurations! modeled with a singlechain of 360 CH2 units. The data can be fitted in the enerinterval 20.27,E,0.2 eV to the expressiong(E)'11.6(E2E0)2 eV2131021 cm23, where E0520.27 eV,as shown in Fig. 4. A larger system of 1215 CH2 units pro-duced consistent data~closed diamonds in Fig. 4!. In Fig. 8we present the calculated values ofK2 at the mobility edgefor fixed density PE samples (r536.731021 CH2 units percm3) with lengthsL513.38, 21.41, 32.12, and 42.80 Å, an

FIG. 7. ~Color! Volumetric representation of the excess electron denprobability for a typical configuration of amorphous PE at room tempeture. ~a! Ground state corresponding toE0520.12 eV showing localiza-tion. ~b! Typical extended state, with energy 0.12 eV. The lighter the regthe bigger the density probability.


s-de

s

an average energy intervalD50.13, 0.05, 0.02, and 0.01 eVrespectively. The inset shows convergence, indicatingthe longest coherence length of the excess electron is smthan the size of the systems considered. As a consequencare able to compute the electron mobility. In this caseEc

2E0@kBT, and thus Eq.~6! can be written as an activatemobility m5m0 exp(2(Ec2E0)/kBT), with an activation en-ergy of Ec2E0'0.33 eV. This value is in good agreemewith the apparent trap depth of 0.35 eV estimated from thmally stimulated conductivity and thermoluminescenmeasurements64,65 as well as activation energies estimatfrom electron-beam-induced conduction66 or the value 0.24eV given by Tanaka and Calderwood,67 which have alreadybeen attributed to the amorphous part of PE. Using our dwe then obtain a prefactorm0'750 cm2/V s and an excesselectron mobility of m'231023 cm2/V s. Note that thepresent energy resolution produces an uncertainty in thebility of about one order of magnitude~due to the exponential factor!. This value compares favorably with the reportvalues in PE, which is in the range 1023– 10210 cm2/V s.68

It has been suggested69 that the striking range of variation othe mobility values reported by various investigatorsmainly due to the different measurement procedures adopand in particular to the measurement time for each particexperiment. When the conduction takes place over very ltimes, carrier trapping and therefore space charge formabecome dominant, providing the lowest values of the appent mobility.

V. CONCLUSIONS

In this paper, we have presented a semiempirical pseupotential derived to be consistent with experimental datathe density dependence of the threshold of conductionfluid ethane and methane. The pseudopotential consistsrepulsive component constructed fromab initio calculationsin methane and crystalline PE and an anisotropic attracpart which accounts for the polarization interaction betwethe excess electron and the dielectric. The electronic stobtained with this pseudopotential are in good agreemwith the experimental data in alkane fluids and polyethyle

y-

,

FIG. 8. Averaged momentumK2 ~7! at the mobility edge as a function othe system volume for amorphous PE at room temperature.


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We have also introduced a new localization criterion tsuccessfully predicts the nature of the electronic statemethane and ethane. When applied to fluid propane it leus to predict a transition between an extended ground sand a localized ground state at the molecular densityr57.531021 cm23. In amorphous PE at room temperaturehas allowed us to locate the mobility edge at aboutvacuum level. This implies a trap depth energy of abouteV due to local disorder, in good agreement with the expmental data.64–67The location of the mobility edge should bcompared with the conduction band level of10.6 eV foundin crystalline PE.23 In materials with low degrees of crystalinity this energy difference would prevent the electron fropenetrating the crystalline regions and conduction woonly occur through the disordered regions.

We have been unable to compute the excess elecmobility in methane using the Kubo–Greenwood equatdue to the limited system sizes employed in the simulatiHowever the electronic coherence length is much smalleamorphous PE, allowing us to estimate the mobility at rotemperature asm'231023 cm2/V s. This value is at theupper limit of the range found in experimental estimatesreal PE, i.e., with coexisting crystalline and amorphousgions. It is consistent with the behavior reported by Koset al.,70 where an abrupt increase of the mobility is observwhen semicrystalline PE is heated beyond its melting poFurthermore, since we have ignored in this study otherportant traps such as chemical traps71 or those found atamorphous/crystalline boundaries, it is to be expectedour mobility would be higher than the values normally fouexperimentally. In particular, it is one order of magnitularger than the mobility obtained by Tanaka and Calderwousing a ‘‘run time’’ method that minimizes the processcarrier capture and hence predicts high mobilities.72

Our calculations suggest an important new mechanfor excess electron transport in polyethylene: conductthrough the excited states above the mobility edge inamorphous regions. Our results depend upon neglectinglocal conformational changes due to the presence of thexcess electrons. To support this assumption we note thacharacterization of the extended states using unpertuconfigurations can be expected to be a good approximasince the perturbation of the environment by the electronvery small when it is delocalized. In addition, preliminacalculations of the nonadiabatic trajectories in thesystems73 show that after a time the electron self-traps wan energy corresponding to the bottom of the band in FigThe energy change due to the electron distorting the enviment in this process is estimated to be small,;0.1 eV, of theorder of the error bars in the present calculations, and thfore, the activation energy for excitation to extended stashould be about 0.3 eV, which is close to the experimefindings.

ACKNOWLEDGMENTS

This work was supported by EPSRC through Grants NGR/R18222 and No. GR/M94427. D.F.C. acknowledges sport through a Schlumberger visiting Professorship of Che


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istry at the University of Cambridge. D.F.C. also acknowedges support from NSF Grant No. CHE-9978320.

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Electronic transport in disordered n-alkanes: From fluid methane to amorphous polyethylene

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Transcript of Electronic transport in disordered n-alkanes: From fluid methane to amorphous polyethylene