ISSN:0366-9033    

DOI:10.1039/DF97050FX001

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

D e190719071910191 1191219131913191319141914191519161916191719171917i9181918191819181919I91919201920A920192019211921192119211922192219231923192319231923192419241924192419241925192519261926192719271927GENERAL DISCUSSIONS OFTHE FARADAY SOCIETYSubjectosmotic prtssunHydrates in SolutionThe Constitution of WaterHigb Temporatwe WorkMagnetic Propertics of AlloysColloids and their Visc0sityThe Corrosion of iron and SteelThe Passivity of MetalsOptical Rotary PowerThe Hardening of MetalsThe Transformation of Pure IronMethods and Appliances for the Attainmat of High Temperatures in aRefractory MaterialsTraining and Work of the Chemical Engineerosmotic PressurePyrometcrs and PyrometryThe Setting of Cements and PlastersElectrical FurnacesCo-ordination of Scientific PublicationThe Occlusion of Gases by MetalsThe Present Position of the Theory of IonizationTho Examination of Materials by X-RaysThe Microscope : Its Design, Construction and ApplicationsBasic Slags : Their Production and Utilization in AgriculturePhysics and Chemistry of ColloidsElectrodeposition and ElectroplatingThe Failure of Metals under Internal and Prolonged StressPhysico-Chemical Problems Relating to the SoilCatalysis with special reference to Newer Theone of ChemiCal ActionSome Properties of Powders with special reference to Grading byThe Generation and Utilization of ColdAlloys Resistant to CorrosionThe Physical Chemistry of the Photographic ProcessThe Electronic Theory of ValencyElectrode Reactions and EquilibriaAtmospheric Corrosion.First ReportInvestigation on Oppau Ammonium Sulphate-NitrateFluxes and Slags in Metal Melting and WorkingPhysical and Physico-Chemical Problem relating to Textile FibresThe Physical Chemistry of Igneous Rock FormationBase Exchange in SoilsThe Physical Chemistry of Steel-Making ProcessesPhotochemical Reactions in Liquids and GasesExplosive Reactions in Gaseous MediaPhysical Phenomena at Interfaces, with special referencc to MolecularAtmospheric Corrosion. Second ReportThe Theory of Strong ElectrolytesCohesion and Related ProblemsLaboratorycapillarityHutriationOrientationVQlU??ETrans. 33678999101011121213131314141414151516161616171717171818191919191920202020202121222223232GENERAL DISCUSSIONS OF T H E FARADAY SOClETYDate19281929192919291930I9301931i 932193219331933I93419341935193519361936I9371937193819381939I939I940I94119411942194319441945f 9451946I946194719471947I9471948194819491 949I949I9501950I9501950195119511952195219521953195319541954SubjectHomogeneous CatalysisCrystal Structure and Chemical ConstitutionAtmospheric Corrosion of Metals.Molecular Spectra and Molecular StructureOptical Rotatory PowerColloid Science Applied to BiologyPhotochemical ProcessesThe Adsorption of Gases by SolidsThe Colloid Aspects of Textile MaterialsLiquid Crystals and Anisotropic MeitsFree RadicalsDipole MomentsColloidal ElectrolytesThe Structure of Metallic Coatings, Films and SurfacesThe Phenomena of Polymerization and CondensationDisperse Systems in Gases : Dust, Smoke and FogStructure and Molecular Forces in (a) Pure Liquids, and (b) SolutionsThe Properties and Functions of Membranes, Natural and ArtificialReaction KineticsChemical Reactions Involving SolidsLuminescenceHydrocarbon ChemistryThe Electrical Double Layer (owing to the outbreak of war the meetingThe Hydrogen BondThe Oil-Water interface"he Mechanism and Chemical Kinetics of Organic Reactions in LiquidThe Structure and Reactions of RubberModes of Drug ActionMolecular Weight and Molecular Weight Distribution in High Polymers.(Joint Meeting with the Plastics Group, Society of Chemical Industry)The Application of Infra-red Spectra to Chemical ProblemsOxidationDielectricsSwelling and ShrinkingElectrode ProcessesThird Reportwas abandoned, but the papers were printed in the Transaclions)SystemsThe Labile MoleculeSurface Chemistry.(Jointly with the Societk de Chimie Physique atColloidal Electrolytes and SolutionsThe Interaction of Water and Porous MaterialsBordeaux.) Published by Butterworths Scientific Publications, Ltd.Trans. 43Disc. 34Lipo-Proteins 6The Physical Chemistry of Process MetallurgyCrystal Growth 5Chromatographic Analysis 7Heterogeneous Catalysis 8Physico-chemical Properties and Behaviour of Nuclear Acids Trans.46Spectroscopy and Molecular Structure and Optical Methods of In-vestigating Cell Structure Disc. 9Electrical Double Layer Trans. 47Disc. 10 HydrocarbonsThe Size and Shape Factor in Colloidal Systems 11Radiation Chemistry 121314151617The Physical Chemistry of ProteinsThe Reactivity of Free RadicalsThe Equilibrium Properties of Solutions of Non-ElectrolytesThe Physical Chemistry of Dyeing and TanningThe Study of Fast ReactionsCoagulation and Flocculation 18Volume2425252526262728292930303131323233333434353535363737383940414242 A42 BDisc. 1GENERAL DISCUSSIONS OF T H E FARADAY SOCIETYDate Subject Volume19551955195619561957195819571958195919591960196019611961196219621963196319641964196519651966196619671967196819681969196919701971Microwave and Radio-Frequency SpectroscopyPhysical Chemistry of EnzymesMembrane PhenomenaPhysical Chemistry of Processes at High PressuresMolecular Mechanism of Rate Processes in Solidslnteractions in Ionic SolutionsConfigurations and lnteractions of Macromolecules and Liquid CrystalsIons of the Transition ElementsEnergy Transfer with special reference to Biological SystemsCrystal Imperfections and the Chemical Reactivity of SolidsOxidation-Reduction Reactions in Ionizing SolventsThe Physical Chemistry of AerosolsRadiation Effects in Inorganic SolidsThe Structure and Properties of Ionic MeltsInelastic Collisions of Atoms and Simple MoleculesHigh Resolution Nuclear Magnetic ResonanceThe Structure of Electronically-Excited Species in the Gas-PhaseFundamental Processes in Radiation ChemistryChemical Reactions in the AtmosphereDislocations in SolidsThe Kinetics of Proton Transfer ProcessesIntermolecular ForcesThe Role of the Adsorbed State in Heterogeneous CatalysisColloid Stability in Aqueous and Non-Aqueous MediaThe Structure and Properties of LiquidsMolecular Dynamics of the Chemical Reactions of GasesElectrode Reactions of Organic CompoundsHomogeneous Catalysis with Special Reference to Hydrogenation andBonding in Metallo-Organic CompoundsMotions in Molecular CrystalsPolymer SolutionsThe Vitreous StateOxidation1920212223242526272829303132333435363738394041424344454647484950For current availability of Discussionvolumes, see back cover

ISSN:0366-9033    

DOI:10.1039/DF970500X003

出版商:RSC    

年代:1970

数据来源: RSC

ISSN:0366-9033    

DOI:10.1039/DF97050BX006

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Conduction in Glassy and Liquid SemiconductorsBY N. F. MOTTCavendish Laboratory, CambridgeReceived 5th October, 1970It is frequently stated that amorphous semiconductors have, in contrast to crystals, a value of theconductivity which is relatively insensitive to composition. This is explained by assuming thateach atom in a glass has as many nearest neighbours as the number of bonds it can form (Ge, 4 ;As, 3 ; Te, 2), so that there are no free electrons available to carry a current. The validity of thisconcept will be examined; it is not true for some amorphous films (Mg-Bi) which are not stronglybonded. Also some glasses, when heated above the softening point, seem to change their coordina-tion numbers and become metallic.The theoretical models necessary to describe these results are outlined.In liquid metals andmost amorphous metal films, the Ziman theory should be applicable, giving a conductivity equalto SeZL/l27r3k, where S is the Fermi surface area and L the mean free path. When this is about3000n-l cm-', L is comparable with the distance between atoms and it cannot be smaller. Formaterials such as liquid Te for which the conductivity is lower, a " pseudogap " affects the conduc-tivity. The lowest possible metallic conductivity is about 200 C2-l cm-l. For materials (liquidsor non-crystalline solids) with lower conductivity, the current is due either to electrons excited tothe " mobility shoulder '' or to hopping conduction of the kind familiar in impurity conduction.A real gap (as contrasted with a pseudogap) must exist in transparent materials, and can be under-stood in terms of the tight-binding approximation.The h s t laboratory to study non-crystalline semiconductors seriously was the A.F. IoffeInstitute in Leningrad. There, Kolomiets always emphasized that in one way they arequite unlike crystalline semiconductors ; their conductivity is relatively insensitive to purityand impurities of different valency do not increase it, as they so markedly do in, for instance,crystalline germanium. The reason is clear in chemical terms ; in crystalline germaniuma pentavalent phosphorus atom forms four bonds with its neighbours leaving one electronweakly bound and free to move at room temperatures ; an amorphous material will normallytake up a structure in which each atom is surrounded by just enough atoms so that allelectrons can be taken up in bonds.Thus, a phosphorus atom will be surrounded by fiveneighbours, a tellurium atom two and so on. Recently X-ray 2 s determination of theradial distribution function in certain alloys has shown that this is indeed so for one alloyGe, lTess. Theorists working on amorphous semiconductors have simply assumed thata structure like this would give a density of states with a gap, as in fig. 1, so that a conductionand a valence band exist just as in the crystal. Only recently due to Phillips in BellLaboratories, and Klima and M~Gill,~ whose paper forms part of this Discussion, haveattempts been made to calculate the density of states though there are a number of experi-mental observations notably those of Spicer using the photoemission technique.I must emphasize that a transparent non-crystalline material like glass must have areal gap in the density of states, or a very deep minimum indeed.How this can occur isclearest if one uses the tight binding approximation. In crystalline non-metals, if one startsfrom two atomic states with energies El, E2, the gap iswhere z is the coordination number and J2, J2 overlap integrals between nearest neighboursin the two bands. In a non-crystalline material, if z is not changed, the gap cannot be lessthan this, if for J is taken the maximum value of the overlap integral due to fluctuations in8 CONDUCTION I N NON-CRYSTALLINE SEMICONDUCTORSthe interatomic distance.This would be the obvious way to calculate the gap in, say,liquid argon. In a covalent semiconductor, one could take for the states (l), (2) the bondingand antibonding states for an electron forming a bond between two atoms. One can hardlydoubt that a gap could be obtained this way, within the framework of the Hartree-Fockmodel as used for crystals. It is of interest that one can obtain at any rate a deep minimum,starting from the quite different approximation of free electrons.EC EFIG. 1 .-Showing conduction and valence bands in a non-crystalline semiconductor, and the mobilityas a function of energy E. Localized states are shaded.In this paper I have two main themes : (a) what are the wave functions like in the valenceand conduction bands, since on these depend the mobility of an electron or hole; and(b) what happens when bonds are broken?On (a) there exists a great amount of theoretical and experimental work.Two principlesare of outstanding importance; the first is that of Ioffe and Regel,7 viz., that the mean freepath of an electron cannot be less than the lattice parameter, and the second is that ofAndersonY8 which proves that, given sufficient disorder, all states in a band are localized.The second principle has been widely questioned, but it is certainly lo Applied tothe conduction band of an amorphous semiconductor one can deduce the following. Ina range AE of about 0.1-0.2 eV (Mott,ll Stern 12) at the bottom of the band (fig. 4), thestates are localized, which means that an electron can move from one to another only withthe emission or absorption of phonons ; it is an activated process, like the motion of a heavyion in a glass, giving a mobility of the formThe states are essentially shallow traps, and the motion of an electron is by the same kindof thermally-activated hopping process with which we are familiar in impurity conductionin doped semiconductors.But, surprisingly, there exists a sharp energy Ec at which Wvanishes and the states become non-localized (fig.1). Just above this the states are non-localized but the mean free path is just as short as it can e according to Ioffe and Regel.,u = (1 /6)vph(ea2/kT) exp (- WlkT). (1N. F. MOTT 9The electron moves by a sort of diffusive motion first described by Cohen l3 (fig.2a), andwith a wave function of the form shown in fig. 2b ; its mobility is(2)where vel is an electronic frequency of the order 1015 s-l. Ec is known as the “ mobilityshoulder ” (fig. 1) and it is the sudden jump from phonon-assisted hopping, withVph - 10l2 s-l,to a purely electronic process, with ~ ~ 1 - 1 0 ~ ~ s-l, that gives rise to it.p = (1/6)v,~(ca2/kT)-l- 10 cm2/V s,(a) path of electrons(b) wave functionFIG. 2.-Path of electrons moving with Brownian motion as described by Cohen, and the appropriatewave function with random phase on each atom.A “ mobility shoulder ” of this kind was first suspected because of the observation ofStuke l4 that the curves plotting In B against l/T for many amorphous semiconductorswere extremely straight, and that if one wrotethe values of C came out to about lo3 SZ-l cm-l.One can calculate C using a wave functionof the form shown in fig. 2 and it turns out to be l5 aboutThe term in brackets is about 200 0-l cm-’ ; fl is obtained from the temperature dependenceof the true activation energy Eo written asActually there is quite a variation in values of C but they fit in well with the models.I must now describe experimental evidence that this range of localized states and amobility shoulder actually exist. The most direct evidence is that of Le Comber andSpear,16 who measured the drift mobility in films of amorphous silicon 1 prn thick bygenerating excess electrons through pulsed electron beam excitation. They were able toshow that at low temperatures the mobility was of hopping type (1) with W-0.1 eV, whileat high temperatures the electrons were excited into extended states above Ec with muchhigher mobility according to formula (2).In this way they could show that AE was 0.2 eV.Now I would like to discuss what happens when certain glasses are heated through thesoftening temperature; Krebs’s l7 paper at this Discussion deals with this. For a numberof glasses, there seems to be evidence that there is a gradual transition of the conductivityfrom a semiconducting behaviour to that more like a liquid metal ; fig. 3 shows schematicallythe behaviour observed; there is some evidence that this is accompanied by an increase inthe coordination number, so that we can no longer say that all electrons are taken up inbonds.The metallic behaviour when this is so reminds one of Pauling’s l 8 description ofresonating bonds in metals ; again Krebs will speak to this. We have been trying to interpretsuch results in terms of a broadening and overlap of the valence and conduction bands asthe bonds break up as illustrated in fig. 4 and 5 ; instead of a gap, these liquids will have a“ pseudogap ” in the density of states. The question then is, will electron states in theB = Cexp (-EIkT), (3)(0.06 e2/iia) exp (j?/k). (4)Eo = E-PT. ( 5 10 CONDUCTION I N NON-CRYSTALLINE SEMICONDUCTORSb c CI\1 /TFIG. 3.-Schematic plot of In a against 1/T for a glass in the liquid range.pseudogap be localized? If so, current will be carried by electrons and holes excited tothe mobility shoulders, as in fig.5 , and the material is essentially a semiconductor. If not,it is essentially a metal. One can apply the Anderson localization theorem to find howdeep the pseudogap must be to give localization and one finds l9 that the quantity g definedbymust be +. Once the material is metallic, and so long as the mean free path is of ordera, one finds that o-g2. In other words, CF is proportional to {N(EF)}~, because it dependson the number of available electrons and the number of vacant states within -kT of the@ = {N(EF)}/{N(EF)}free- High temp. itemp.EFIG. 4.-Overlapping conduction and valence band in the glass of fig. 3N. F. MOTT 11Fermi energy EF. One envisages three regions : (a) the Zinian region (L>u) ; 0;-3000 C2-l cm-l; (6) the diffusion region (L-a) ; CJ between 200 and 3000C22-1 cm-’ ;(c) the semiconducting region ; at200 cm-l.In region (a) the Ziman weak-interactiontreatment is valid; in region (6) a is proportional to g2 ; in region (c) conductivity is due toexcited carriers at the mobility shoulders.n 2High temp.temperatureIEF EFIG. 5.-Density of states with pseudogap, with localised states shaded, and for a metal.All this finds striking confirmation in the work of Warren 2o on liquid Ga2Te3, whomeasured the Knight shift K as a function of temperature. K should be proportional to 9.Fig. 6 shows schematically what occurs. The most important conclusion is the existenceof a range of temperature for which a lies in the range 200-3000~-1 cm-’, and in whicha is proportional to K2.This gives rather direct evidence for the pseudogap model andthe diffusive motion. In this region, also, there is a striking breakdown of the Korringarelation which is discussed in Warren’s papers.regionI metaaS e miconduc t o r I+temperatureFIG. 6.-Schematic behaviour of conductivity cs and Knight shift K of liquid GazInS as a functionof T12 CONDUCTION IN NON-CRYSTALLINE SEMICONDUCTORSB. T. Kolomiets, Phys. Stat. Sol., 1964,7, 359.F. Betts, A. Bienenstock and S. R. Ovshinsky, J. Non-Cryst. Solids, 1970,4, 554.D. Adler, M. H. Cohen, E. A. Fagen and J. C. Thompson, J. Non-Cryst. Solids, 1970,3,402.J. C. Phillips, Covalent Bonding in Crystals, Molecules and Polymers (University of ChicagoPress, 1969).J. Klima and T. C. McGill, this Discussion, and J. Phys. C, 1970,3, L163.W. E. Spicer and T. M. Donovan, J. Non-Cryst. Solids, 1970,2, 66.P. W. Anderson, Phys. Rev., 1958, 109, 1492.P. W. Anderson, Comments on Solid State Phys., 1970, 2, 193.' A. F. Ioffe and A. R. Regel, Progr. Semiconductors, 1960,4, 237.'OD. Thouless, J. Phys. C, 1970, 3, 1559.I 1 N. F. Mott, Phil. Mag., 1970, 22,7.l2 F. Stern, 1970, to be published.l 3 M. H. Cohen, J . Non-Cryst. Solids, 1970, 4, 391.l4 J. Stuke, J. Non-Cryst. Solids, 1970,4, 1.E. A. Davis and N. F. Mott, Phil. Mag., 1970,22, 903.P. G. Le Comber and W. E. Spear, Phys. Reu. Letters, 1970, 25, 509.H. Krebs and P. Fischer, this Discussion.L. Pauling, The Nature of the Chemical Bond, 3rd ed. (Oxford, 1960).N. F. Mott, Phil. Mag., 1969,19, 835.2o W. W. Warren, J. Non-Cryst. Solids, 1970,4, 168; Solid State Comm., 1970, 8, 1269

ISSN:0366-9033    

DOI:10.1039/DF9705000007

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Effect of Carrier-Carrier Interactions on some TransportProperties in Disordered Semiconductors*B Y M. POLLAKtDept. of Physics and Solid State Institute, Technion, IsraelReceived 12th June, 1970Interactions between electrons are likely to play an important role in certain transport propertiesof disordered semiconductors, particularly where these properties depend on electrons in localizedstates. A working classification of these interactions into intra-site interactions, inter-site interactionsand polarization, is made. The first class is believed to affect primarily the transpo rt properties ocarriers around the mobility gap, by introducing two-electron wave functions into this region. Thesecond class can introduce an activation energy into the d.c. hopping conductivity at very low tempera-tures, affect the thermo-electric power, and increase noticeably the a.c.hopping conductivity above acertain frequency. The third class may reduce the activation energy of the d.c. conductivity; areduction to zero is theoretically possible. Instances where some of the above effects may have beenobserved are cited.It is customary to treat amorphous systems by use of one-electron theories.Even such theories become complicated, as is evidenced by the theory of the Andersonlocalization. On the other hand, one can sometimes see that correlation effects maybe important, e.g., the Mott transition in impurity conduction. It is the purpose ofthis paper to call attention to some other instances where many electron effects maybe responsible for pronounced phenomena, and thus cannot be overlooked.It is convenient here to divide the electronic states in the amorphous system intothree energy regions: the delocalized states far from the mobility edge, the regionnear the mobility edge, and the localized states within the pseudogap. The firstgroup is probably well represented by one-electron theories, as demonstrated byStuke,l and others, by experiments on the optical properties of disorderedmaterials.This paper will focus attention on the third, and, to a lesser extent, thesecond group, and will concentrate on effects caused by electron-electron interactions.In some instances experimental results from the literature will be used to give supportto the proposed effects.Since the paper is primarily concerned with localized states, it will be assumed thatoverlap-dependent many-electron energies are relatively small compared to correlationenergies.Also, as is accepted with systems of localized states, a hop of a carrierfrom one localized state to another is considered to be the basic excitation of thesystem. This means that the initial and final states of the excitation differ only inthe occupation of two localized states, say A and B, where A is occupied in the initialstate and vacant in the final state and vice versa for B. First, the effect of interactionon such a hop is considered. We deal with hops to singly occupied states, withhops to vacant states, and we discuss the possibility of the other electrons to rearrangethemselves after the hop, in order to minimize the energy.The case where this* partly sponsored by the Aerospace Research Laboratories under contract no. AF 61 (052)-825with the European Office of Aerospace Research, U.S. Air Force.t on leave from the Dept. of Physics, University of California, Riverside, U.S.A.114 CARRIER INTERACTION EFFECTS IN DISORDERED SEMICONDUCTORSrearrangement occurs simultaneously with the hop by a many-electron transition isalso discussed.DOUBLE OCCUPANCY OF A LOCALIZED STATEThe repulsion associated with this situation has been dealt with by M ~ t t , ~H ~ b b a r d , ~ and others in connection with the Mott transition. Here two-electronfunctions have an energy exceeding by Ne2/xa (a is the radius of the localized wavefunction, and IC the dielectric constant) twice the energy of a single localized electron.Using some reasonable numbers for the radius of the wavefunction, say 3 A, and thedielectric constant, say 3, one obtains for this repulsion energy a value exceeding1 eV, so that one can disregard most of these two-electron states.However, Mottvisualizes (fig. 3, ref. (5)) that the radius of the localized wavefunction increases asone approaches the mobility edge. If this is so, then there must be a piling-up oftwo electron states in the vicinity of the mobility edge. It is likely that such statesare delocalized. They are, at least, known to be delocalized in impurity conductionon the basis of experiments by D’Altroy and Fan.7 They may then determine to alarge degree the transport properties of the carriers near the mobility edges.In thisevent, the question whether the mobility changes continuously or discontinuously atthe mobility edge is academic.REPULSION BETWEEN ELECTRONS I N SINGLY OCCUPIED LOCALIZED STATESRestricting ourselves now to the energy range where the two-electron statesdiscussed above do not appear, the state of the system is specified by specifying whichof the localized states are occupied. We first concern ourselves with the effect ofinteractions on those excitations which determine such transport properties as d.c.conductivity and thermoelectric power. These are excitations where the states Aand B are relatively distant from each other. The distance between them mustexceed the average distance rc between carriers.Carrier-carrier interactions mayintroduce an activation energy in the d.c. conductivity at very low temperatures,*as well as a temperature-independent contribution to the Peltier coefficient.Since the distance between the states A and B is large, the Coulomb interactionbetween them is small, and the energies of the initial and final state can be consideredto be the energies of the n-fold occupied (neutral), and (n + 1)-fold occupied system.We wish to show that where no gap exists between the ground states of such systemsin the absence of carrier-carrier interactions, a gap can be created between them bythe interaction. In the absence of a gap, the activation energy of the conductivityvanishes as T-+O (Mott’s formula In OOCT-~),~ as does the temperature independentpart of the Peltier coefficient.In contrast, when the gap exists, the conductivity atvery low temperatures is activated and the Peltier coefficient has a temperature-independent part. The introduction of a gap by the carrier-carrier interaction isdemonstrated below for the simple example of impurity conduction at small compen-sations. For such a system it is convenient to consider the ionized majority impuritiesas the carriers. At low temperatures the energy Ei of the carrier i is well approximatedby its interaction with the nearest minority ion, Ei = - e / m i . The fact that in theabsence of carrier-carrier interaction there is no gap between the occupations by nand by n+ 1 carriers follows from the smoothness of the probability density functionp(vi) that a majority impurity is at a distance rz from the nearest minority impurity.The Fermi energy, i.e.the energy of the highest carrier, is then -e2/~rD, where@N,,r; = I , and ND is the majority concentration. Carrier-carrier interactions* I am indebted to Prof. Mott for this suggestionM. POLLAK 15leave the energy for occupation by n carriers almost unaffected, but alter considerablythe energy for occupation by n+ 1 carriers. The energy of the additional majorityion is determined by the interaction with acceptor-donor ion pairs, since no isolatedminority ion exists in the neutral region. This attraction is appreciable only in asmall volume and thus the probability of finding the energy of a majority ion decreasedby such an attraction is small.A simple approximation based on these considerationssuggests that the decrease in energy is about -0.1 e2/lcrD. Hence a gap of about0.9 e2/lcrD is created here by the interaction.For very large compensation, the gap is decreased to zero; so here Mott’s T-4dependence should be obeyed and the Peltier coefficient should not have a temperature-independent part.In addition to these effects of interactions on the d.c. conductivity and on thethermoelectric power, there may also be a pronounced effect on the a s . conductivity.While the long-range excursions of a carrier are impeded, or at best unaffected, theshort hops are enhanced by the carrier-carrier interaction.The physical reason forthis is as follows. Long-range excursions can be made only by the relatively fewcarriers whose energy is removed no more than about kT from the cheinical potential.The others cannot find a thermally accessible vacant state to hop into. Shorthops, on the other hand, can be executed by all the carriers as long as they have aclose enough state to hop into. This is so because the carrier-carrier repulsion almostguarantees that such a state will be vacant.Expressions have been developed lo for the a.c. conductivity to cover both alter-natives. These expressions area(w) = (n3/96)N2(EF)kTaor:e2 ro 9 rc (1)~ ( w ) = (n/24)NsNcawr~e2/E,,, ro 4 rcHere Ns, Nc are the densities of localized states and of carriers, respectively,N(EF) = i?Ns/aEI EF is the density of states at the Fermi level, rw is the hoppingdistance l 1 characteristic of the frequency coy and a the radius of the localized wavefunction.The correlation affects the temperature dependence of the a.c. conductivity,as can be seen when comparing eqn (1) and (2). If we take the simplified view thatfor r,brc eqn (1) applies, and for r,<rc eqn (2) applies, then we can consider thesetwo sets of states separately, and then superimpose the results for ~ ( w ) . The conduc-tivity of the set with ro>rcis, for O < Z - ~ ( ~ ~ ) , given by eqn (I), and for w>z-’(rC) is loa(w) = (n3/96)N2(EF)kTarde2/r(rC).For the set of pairs shorter than rc the conductivity for wtz-l(rc) is given by lo(3)and for co>z-l(rC) is given by eqn (2).The conductivity of the entire system isillustrated on a log log plot in fig. 1. The plot is similar to fig. 7 of Owen l2 where heplots his experimental results for As& A quantitative comparison with Owen’swork l2 is not possible because of the lack of an accurate model for the chalocogenideglasses, but an order of magnitude comparison can be done adopting the generalfeatures of the model (CFO) of Cohn, Fritzsche and 0vshin~ky.l~ This is done asfollows. From the comparison of the theoretical and experimental a(co) at lowfrequencies, the density of states at the Fermi-level can be obtained assuming somereasonable value for a. Then, using the CFO model, we can estimate Ns and Nc.From here, using eqn (1) and (2), ~ ( o ) can be predicted at high frequencies, andcompared with experiment.This is best done by evaluating the ratio of eqn (1)and (2), extrapolating in Owen’s fig. 7 a(o) from low frequency to high frequency b16 CARRIER INTERACTION EFFECTS I N DISORDERED SEMICONDUCTORSa straight line, and comparing the theoretical ratio with the ratio of Owen's highfrequency conduction and the extrapolated value. The former results in - lo3,the latter is lo2. Since the theoretical value is an upper limit, this result is reasonable.However, the proposed theory alone does not explain the observed temperatureindependence l4 of a(u) in the co2 region, even though T is not contained explicitlyin eqn (4). An implicit dependence is possible, since z(rc) may be temperaturedependent. To predict this, it is necessary to have a more detailed knowledge thanis available of phonon spectra and electron-phonon interactions in amorphousmaterials.In Q-In w/FIG.1.-A graphic representation of the theoretical behaviour of ~ ( w ) of hopping. The quantity~ ( r c ) is the relaxation time of the polarizability of pairs of size rc, the average separation betweencarriers. The dashed line is In ~ ( o ) of the set of pairs which are larger than rc, the dash-dotted lineis In ~ ( w ) of the set of pairs which are shorter than rc. The heavy line is In ~ ( w ) of both the setscombined.A comment on the CFO model l3 is in order here. It enhances the argument ofCFO that a large number of charged states exist in the chalcogenide glasses.Considera situation with two localized states as in fig. 2. Let us assume that they arise fromdisorder. The upper state is formed of conduction band functions, so it is normallyvacant; the lower is derived from the valence band, so it is normally occupied byan electron. In this situation the two states are neutral. Compare now the relativeenergies when the lower and upper state is occupied. The first situation has theenergy El. However, the energy when the upper state is occupied is not simply E2,but is E2-e2/w, where r is the distance between the centres of the states. TheCoulomb term comes from the fact that when the upper state is occupied it is negativelycharged, and the lower state is negatively charged when vacant. For large separationsthe Coulomb term vanishes.For a situation typical of chalcogenide glasses, theseparation between localized states is about lOA. With a dielectric constant of 4M. POLLAK 17the Coulomb term is about 3 eV. This energy is still increased by a factor amountingto a Madelung constant, because the pair in fig. 2 is in reality not isolated. In anycase, if (E2 -El) is less than the Coulomb term, the state at Ez will be occupied ;i.e., the pair will be ionic. One implication of this situation is that an overlapof bands like that proposed in the CFO model l3 is not needed to obtain a largenumber of charged states, or a possibility for hopping.FIG. 2.-A diagram of two localized states in the pseudogap. An aid for the text. The state atEz is derived primarily from wave functions in the conduction band, the state at El is derived pri-marily from wave functions in the valence band.POLARIZATION INTERACTIONThe transitions of carriers discussed so far do not necessarily terminate the process.A further rearrangement of electrons in response to the one hop is possible.Thiscan be due to the induction of an electric field at the sites of the other carriers bythe hop. The rearrangement is such that it lowers the energy of the system. Thetransition from the initial to the final state need not take the two steps as outlined,but can be effected in one step by a many electron transition. These two alternativesare analogous to the two modes of transport of a po1aron.15 The successive stepscorrespond to the hopping motion of the polaron, the many-body transition to themotion in a band.This analogue should not be stretched since the polaron transitionsoccur between the sites of a lattice, but the transitions here occur in a disorderedarray. The decision as to which of the two alternative processes occurs under givenconditions, depends on whether the one-carrier transition or the two-carrier (ormany-carrier) transition rate is larger. These rates have the formwhere $t and $f are the initial and final wave functions (one- or many-carrier, asthe case may be), V the interaction potential with the phonons, n(A) the populationof phonons with hv = A, A is the energy difference between the initial and finalstate of the transition, and K is some constant.The rearrangement of the localized carriers will be most important when thestates A and B are separated by a distance larger than rc.All localized carriersexperience the hop only through a relatively weak dipole field. Hence the polariza-tion interaction should affect more the d.c. than the a.c. conductivity. The effecton the d.c. conduction can be described qualitatively with the aid of an energy diagramwith states at three energy levels. The ground state, say, at energy zero ; the higheststate 23 of one electron excitations into conducting states of energy E, and in between,KI (44 I V l +I> I 2n(A)18 CARRIER INTERACTION EFFECTS I N DISORDERED SEMICONDUCTORSstate Cat E-E’, E’ being the energy gained by the polarization. The d.c.conductivityis given by nep, where n is the density of carriers in state C, and is, at a given tempera-ture, an increasing function of E’. If the polarization cannot follow the conductingcarrier, then the motion of the carrier must involve an activation to the upper level,and hence must be proportional to exp (-E’/kT). Thus, what was gained by thepresence of the state C by n is lost by p, and is unaffected by the existence of stateC. If, on the other hand, the polarization can follow the carrier with a reasonablemobility, the activation from state C to state B is not necessary, and CT is increased bythe polarization interaction.The polarization of the localized states can be described, at least approximately,by a contribution Ah: to the dielectric constant.The energy E’ is a decreasing functionof AK. An expression for Arc, suitable for disordered systems, has been derived loand is briefly outlined below. For a random distribution of polarizable centres thefollowing equations are valid :i IandHere E,,, is an applied electric field, E, the average electric field outside a polarizablecentre, V, is the volume of a centre of type i, nf is the density of centres of type i,Ei the average electric field in centres of type i, and Pr is a property of an individualcentre, defined by eqn (2) in ref. (18). It is the reciprocal ratio between the appliedfield and the average field induced in the isolated centre i by its polarization charge.From eqn (5) and (6) one obtains the relation between Eapp and E,.It is now shownthat--AK = niaiE,/Eapp,i(7)where Ah: is the dielectric increment arising from the localized states, and ai is anotherproperty of the polarizable centres i, namely the polarizability. With eqii (9, (6)and (7) one obtains the desired relationship for Arc,The interesting points to be made here are the following. (i) The localized carrierscan, through their polarizability, reduce the d.c. conductivity without actively partici-pating in the conduction process. Instances where such an effect may have beenobserved are the conductivity of random polymer aggregates, and impurityconduction. The latter may be apparent in fig. 4 of ref. (17). There the activationenergy E ~ , both experimental and theoretical, are plotted against average impurityspacing.The theoretical results do not include correlation effects. At low densitiesthe experimental and theoretical results agree ; at high densities the experimentalresults fall below the theoretical results. This may be due to the correlation effects.An estimate based on this assumption indicates that the deviation from the theoreticalvalue sets on at the right impurity separation.18If, in addition,the concentration is sufficiently large, a dielectric breakdown is predicted by eqn (S),since the activation energy of the d.c. conductivity is proportional to l / ~ . This canbe interpreted as the disappearance of such an activation energy. A similar view wasadopted by Frood l9 to account for the disappearance of the activation energy in(ii) It is possible for .the parameter /3 of eqn (8) to exceed unityM.POLLAK 19impurity conduction. We also observe * that the numbers ni in eqn (8) are notnecessarily constant but may also be activated. For example, this is the case forpolymer aggregates.20 The activation energy may again decrease with increasingIC. This creates a positive feedback on the magnitude of ni. If ni can become verylarge as a result of this feedback, AK will become very large. This may constituteanother mechanism by which the activation energy of the d.c. conductivity coulddisappear.I am grateful to Prof. Sir Nevi11 Mott F.R.S.. for many helpful discussions onthe subject of this paper. I also thank Prof. Tannhauser and his group, Dept.ofPhysics, and Solid State Institute of the Technion for their hospitality during thecourse of this work.J. Stuke, Festkorperprubleme, 1969,9,46.J. Tauc, R. Grigorovici and A. Vancu, Phys. Status Solidi, 1966, 15,627.N. F. Mott, Can. J. Phys., 1956, 34, 1356.J. Hubbard, Pruc. Roy. SUC. A, 1963,267,238.N. F. Mott, Festkorperprobleme, 1969,9, 22.M. PoUak, Int. Cot$ Phys. Semiconduction (Exeter, 1962), Inst. Phys. and Phys. SOC., p. 86.F. A. D’Altory and H. Y. Fan, Phys. Rev., 1956,104,1671.N. F. Mott, Phil. Mag., 1969, 19, 835.A. Miller and E. Abraham, Phys. Rev., 1960,120,745.lo M. Pollak, to be published.l1 M. Pollak, Phys. Rev. A, 1964,133,564.l2 A. E. Owen and J. M. Robertson, J. Non-Cryst. Suli& 1970, 2,40.l3 M. H. &hen, H. Fritzsche and S . R. Ovshinsky, Phys. Reu. Letters, 1969, 22, 1065.l4 R. F. Shaw, Ph. D. Thesis, 1969, (University of Cambridge).l5 T. Holstein, Ann. Phys., 1959, 8, 343.l6 B. Rosenberg and E. Postow, Ann. N. Y. Acad. Sci., 1969,158, 161.l7 H. Fritache and M. Cuevas, Phys. Rev., 1960,119,1238.l8 M. Poll& and M. Knotek, J. Non-Cryst. Solidr, 1970, 4, 459.l9 D. G. H. Frood, Proc. Phys. Soc., 1960,75, 184.2o H. A. Pohl and M. Knotek, private communication.* I am grateful to Mr. M. Knotek for suggesting the following point.This brings up an interesting point. It is well known that correlation associated with intra-centrerepulsion can effect a Mott transition. Here it is seen that correlation associated with inter-centrerepulsion can also, on the insulator side of the Mott transition, bring about a metal-non metaltransition

ISSN:0366-9033    

DOI:10.1039/DF9705000013

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Role of Short-Range Order in Producing An Energy Gapin Amorphous Covalent SemiconductorsBY J. KLIMA *, T. C. MCGILL -f- AND J. M. ZIMANH. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TLReceived 16th June, 1970The role of short-range order in determining the density of states of disordered covalent semi-conductors is investigated using the multiple scattering formalism. Detailed calculations of anapproximate density of states for a model consisting of clusters of carbon atoms in configurationsfound in amorphous and crystalline germanium are reported. These calculations suggest that theenergy gap in amorphous covalent semi-conductors is a result of the short-range order.1. INTRODUCTIONExperimental studies of amorphous silicon and germanium have yielded manyof the properties of the materials.The pioneering work of Ritcher and Breitlingon silicon and germanium, and the detailed studies of Girgorovici and colleagueson germanium indicate that the structures of these amorphous materials may bedescribed in the following way: long-range order is absent; however, almost allsites have four nearest neighbours which are arranged like the nearest neighboursin the perfect crystal. Measurements of the optical properties 3-8 show that thesematerials possess an optical gap which is qualitatively similar to that in the crystallinematerial.The existence of an optical gap in amorphous silicon and germanium posesan interesting theoretical question. In the standard theory of solids, heavy emphasisis placed upon the role of long range order in producing a band gap.The simul-taneous lack of long-range order and existence of an optical gap in these materialsforce one to ask whether any other factor can produce an energy gap.In this paper the authors address themselves to this question. The role of short-range order in determining the density of states is investigated using the multiplescattering f~rmalism.~~ lo The results of a detailed quantitative study of a modelconsisting of carbon atoms with short-range order like that in amorphous germaniumis reported. In 92, a short theoretical discussion of the relevant results from themultiple scattering formalism is presented; in §3, the exact details of the model andthe numerical results are presented and discussed.Conclusions are presented in$4. Throughout this calculation units in which = 1 , me = 3 and energy is measuredin rydbergs are used.2. THEORETICAL DISCUSSIONMultiple scattering formalism 9 s lo deals with the properties of an ensemble ofscattering centres. The potential about each of these centres is assumed to bespherically symmetric and not to overlap that of its neighbours (muffin tin approxi-* on leave of absence from Charles University, Prague.f NATO Postdoctoral Fellow23. KLIMA, T. C. MCGILL AND J . ha. ZIMAN 21mation). For a perfect crystal this formalism reduces to that obtained in the KKRmethod of calculating band structure. 9In this formalism the density of states may be expressed analytically in termsof the scattering properties of each centre and their spatial arrangement.Lloyd l 3has shown that the integrated density of states (the number of states with energy lessthat E ) is given byE+ 23n2 ni.2 N(E) = - - - Im Tr In (BLLIBUr - G&,(l- ll)kLI(l*, E)). (2.1)The first term is the integrated free particle density of states. The last term is theFriedel sum for the solid. In this term, f,l is the volume of the solid, L( = (1,m))refers to a real spherical harmonic, YL(Q), and 1 is the location of a scattering centre.GtL,(l- 1') is given by the expressionwhereCfiI = 1 dQYLII(~)YL(Q)YLI(Q)and K = ,/E. kL(l,E), the single site k-matrix for site 1, is related to the phase shiftfor spherical harmonic L, BL(Z,E), by the equationThe trace is taken over L and 1.To show the importance of short-range order in determining the density of states,eqn (2.1) may be rearranged so that the scattering properties of small clusters of sitesappear in place of the single-site scattering properties.The clusters of sites ofinterest here are defined in the following manner. First, a set of locations for thecentres of the clusters is selected. These locations will be labeled rj where the jrefers to thejth cluster. It is not necessary that the centre of a cluster coincide withone of the scattering centres. Secondly, each scattering centre is identified with thecluster whose centre is closest to the site. If a site belongs to cluster j , its locationwill be denoted by lj.Using the definition of a cluster given above, the last term in the integrated densityof states may be rearranged to givekL(I,E) = -tan 6,O,E)/u (2.3)E* 2N(E) = --- Im Tr 111 ( a L u 6 i j - C G,tII(ri -rj)KLIxLI(rj)) -3n2 ni.2 L I I2nQ - c j Im Tr In (8LLdljlj1 - G&r(ij - l;)kLI(l;)), (2.4)where KLrrLr(rJ), the K-matrix for cluster j , is given byKLnLI(rj) = ALIILrII(rj - l j ) ~ ~ I r ( I J .) M E I ~ ~ , ; ~ ~ v ~ ~ A L ~ ~ ~ ~ O f - ri), (2.5)L!&vwit22 ENERGY GAP IN AMORPHOUS COVALENT SEMICONDUCTORSand(see ref. (lo), (14)).The last term in eqn (2.4) contributes to the expression only when the K-matrix for thecluster or the single site k-matrix are not defined. Suitably defining the logarithmin the second term at these points allows us to discard this term.Eqn (2.4), without the last term, is the result desired since the presence of theK-matrix for the cluster makes it possible to study the influence of the presence ofshort-range order.However, even when the K-matrices for all the clusters are known,eqn (2.4) is still a difficult equation to evaluate exactly. For disordered structures,correlations in the position and orientation of nearby clusters must be known tomake an exact evaluation of the average of eqn (2.4) possible. For a perfect crystal,a complete band structure calculation is required to evaluate it exactly. To circumventthis difficulty and to show the importance of short-range order, eqn (2.4) is approxi-mated by neglecting the influence of multiple scattering between clusters.In thisapproximation, eqn (2.4) reduces towhere the sum o n j spans all the clusters in the solid. Eqn (2.9) shows that thisapproximation is equivalent to summing the Friedel sum for each of the clusters inthe solid. It has the particular advantage of not requiring that one specifies eitherthe relative angular orientations or the correlations in position of the clusters.However, this approximation is a potentially serious one. Intuitively, it would seemthat the error is related to the size of the cluster considered. Since the use of verylarge clusters in calculating the K-matrices in eqn (2.9) should lead to a density of stateswhich deviates only slightly from the exact density of states, one would expect thecorrection terms to eqn (2.9) to decrease with increasing size of the clusters used.The authors hope to give a more detailed discussion of this point in a futurepublication.3.CALCULATIONS AND RESULTSTo evaluate eqn (2.9), the geometry of the clusters and the single site k-matrixare required. In this calculation, the two types of clusters of eight carbon atomsshown in fig. 1 are used. The configuration on the left, the staggered configuration,2has exactly the same structure as the cubic cell of the diamond lattice ; the diamondlattice may be built up from these structures. The configuration on the right, theeclipsed configuration,2 is not found in the diamond lattice. However, it is thoughtto be present in amorphous gerrnaniwm2 Throughout this calculation, the inter-atomic distance is taken to have the value 6.74 ao, the crystalline value.Determination of the single site k-matrix requires a detailed knowledge of themuffin tin potential about each site.This potential is assumed to be identical foreach site in the material and its value is taken to be that appropriate for the perfectcrystal. Unfortunately, the muffin tin approximation is poor for covalently bondedsemiconductors. Consequently, finding a potential and phase shifts which generatea realistic band structure presents some difficulty. To avoid this difficulty (which iJ . KLIMA, T. C. MCGILL AND J . M. ZIMAN 23outside the range of interest of this paper). carbon was chosen as a model materialsince the large hand gap of diamond makes the existence of a gay, rather inseiisitiveto these problems.Following Mertens,' the crystal potential is approximatedsimply as the sum of the Herman Skillman potentials o f the two nearest neighboursalong their connecting line. The muffin tin zero was taken to be -2.207ry.16Integration of the radial Schriidinger equation for this potential yields the phasesshifts used in this calculation. As shown in fig. 2, the resulting p-phase shift exhibitsa weak resonance which is important in determining the character of the scatteringproperties of the cluster. The s-phase shift has little structure. Only s- andp-singlesite phase shifts were used in this calculation. When a KKR calculation was carriedout for diamond using these phase shifts, a qualitatively correct band structure wasobtained.(4 (6)FIG.1.-The staggered (a) and eclipsed (b) clusters used in the calculation of the density of states.The K-matrix for each type of cluster was evaluated numerically and the resultingK-matrix was substituted into the derivative of eqn (2.9) with respect to energy toproduce the density of states. The energy range 0-1.4 ry above the muffin tin zerowas investigated since this is the range where the KKR calculation indicated thatthe top of the valence band and the bottom of the conduction band are located.The densities of states as a function of energy obtained from these calculationsare presented in fig. 3. The values of the energy at the r $ s level, the top of the valenceband, and at the r15 level, the lowest level with r-symmetry in the conduction band,are also indicated along the energy axis in fig.3. These values were obtained froma KKR calculation with the phase shifts used in the calculation of the density ofstates.These results show three facts. First, the energy dependence of the density ofstates for a solid made up of these clusters can be divided into three ranges : from0.0 to 0.54 ry, from 0.54 to 1.05 ry, and above 1.05 ry. In the first and third ranges24 ENERGY GAP IN AMORPHOUS COVALENT SEMICONDUCTORS'T 2n9 1;iiit:-5 sa - '."4- 2-3'c4. --energy (rY)FIG. 2.-Single-site phase shifts for carbon. The broken and full lines correspond to the s- andp-phase shifts, respectively. The zero of energy is at the muffin tin zero.FIG.3.-The approximate density of states as a function of energy. The full and broken lines arethe densities of states for a solid made up of eight-atom clusters with staggered and eclipsed configura-tions, respectively. The dotted line is the density of states for a solid of clusters consisting of asingle site. The range of the forbidden gap in crystalline carbon is indicated by the arrows on theenergy axisJ . KLIMA, T. C. MCGlLL AND J . M. ZIMAN 25the density of states is approximately an order of magnitude greater than that foundin the second range.Secondly, the sharp transition region between range one and range two occurs atalmost exactly the value of the energy for the ri5 level ; the transition between rangetwo and range three occurs at approximately the value of the energy for the Tlslevel.Thus, the density of states obtained here are similar to what would be foundfor a perfect crystal. The ranges of energy giving large density of states wouldcorrespond to the valence and conduction bands, and the range of energies giving asmall density of states would correspond to the band gap.Thirdly, the similarities between the densities of states obtained for the twotypes of clusters indicate that the presence of a range of energies where the density ofstates is small bounded by ranges where the density of states is large is insensitiveto the differences in the short-range order which are found in disordered germanium.For comparison, the density of states resulting when clusters consisting of a singlesite are used is also shown in fig.3. The graph shows that the single-site clustersfail to produce a rapid increase in the density of states for energies greater that 0.5 ry.The differences between the density of states for the single-site clusters and eight-site clusters may be understood in the following way. The peak in the density ofstates for the single-site clusters results from the weak p-resonance which appears inthe single-site phase shifts. For comparison, the density of states resuIting whenclusters consisting of a single site are used is also shown in fig. 3. The graph showsthat whilst the single-site clusters produce a peak on the density of states in the firstrange, they do not produce as small a density of states in the second range as themulti-site clusters and fail to produce a peak in the third range.The differences inthis density of states and those obtained from the multi-site clusters may be explainedin the following way. In the first case the weak p-resonance produces only one peakin the density of states. However, in the multi-site cluster the s- andp-waves scatteredfrom the various sites interact to produce a number of resonances in the total scatteringproperties. These resonances fall into two distinct groups producing the double-peaked structure separated by a small density of states region shown in fig. 3. Itis the explicit inclusion of the short-range order in the multi-site cluster which givesthis result.The small-but finite-density of states in the gap region is the result of theincomplete cancellation of the free electron density of states (the first term in the eqn(2.9)) by the contribution from the multiple scattering properties (the second termin eqn (2.9)), and is probably due to the rather drastic approximation of neglectingmultiple scattering between clusters. Similar calculations are currently in progressfor silicon and germanium.While the results are not complete, it appears thatsimilar results will be obtained for these materials. The p-phase shift of both siliconand germanium have a weak p-resonance ; and preliminary calculations on siliconindicate that this weakp-resonance is split in a way analogous to that for carbon.4. CONCLUSIONThe scattering properties of clusters of eight carbon atoms have an energydependence which produces an approximate density of states with clear indicationof an energy gap.Even though this calculation is only an approximate one for adisordered material, it does suggest that the presence of an energy gap in the spectrumof amorphous covalent semi-conductors is closely related to the presence of short-range order. The qualitative similarity between this gap and that found in theperfect crystal is also explained by the relative insensitivity of the size of the gap to thedifference in short-range order found in these two materials26 ENERGY GAP IN AMORPHOUS COVALENT SEMICONDUCTORSThe authors acknowledge helpful discussions with G. J. Morgan. They areparticularly indebted to him for suggesting the possibility of a weak p-resonance inthese materials.Helpful correspondence with J. Treusch is also acknowledged.H. Richter and G. Breitling, 2. Nuturforsch., 1958, 13a, 988.R. Grigorovici and R. Manaila, J. Non-Cryst. Solids, 1969, 1, 371.J. Tauc, R. Grigorovici and A. Vancu, Phys. Status Solidi, 1966, $5, 627.A. H. Clark, Phys. Rev., 1967, 154,750.J . Wales, G. J. Lovitt and R. A. Hill, Thin Solid Films, 1967, 1, 137.T. M. Donovan, W. E. Spicer and J. M. Bennett, Phys. Reu. Letters, 1969, 22, 1058.R. Grigorovici and A. Vancu, Thin Solid Films, 1968,2, 105.D. Beaglehole and M. Zavetova, J. Non-Cryst. Solids, 1970, 4, 272.J. L. Beeby, Proc. Roy. SOC. A, 1964,279,82.lo P. Lloyd, Electrons in Metals and MuZtipZe Scattering Theory, unpublished.l1 J . Korringa, Physica, 1947, 13,392.l2 W. Kohn and N. Rostoker, Phys. Rev., 1954, 94, 1111.l3 P. Lloyd, Proc. Phys. SOC., 1967,90,207.l4 T. McGill and J. Klima, in preparation.0. Madelung and J. Treusch, Proc. 9th Int. ConJ Physics of Semiconductors, (Nauka, Lenin-grad, 1968), vol. 1, p. 38.l6 K. Mertens, Diss. (Univ. Marburg, 1967).l7 F. Herman and S. Skillman, Atomic Structure Calculations, (Prentice Hall, Inc., EnglewoodCliffs, New Jersey, 1963).l 8 Mertens l6 includes a d-phase shift in his band structure calculation. The omission of thed-phase shift in our calculation produces slight differences between the band structure obtainedby Mertens and our results

ISSN:0366-9033    

DOI:10.1039/DF9705000020

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Hall Effect, Thermoelectric Power and Electrical ConductivityMeasurements in Vitreous CdGexAs2*BY R. CALLAERTS T, M, DENAYER, F. H. HASHMI AND P. NAGELSSolid State Physics Dept., S.C.K./C.E.N., B-2400 Mol, BelgiumReceived 6th July, 1970A series of measurements of Hall effect, thermoelectric power and electrical conductivity havebeen performed on the vitreous CdGexAs2 having a Ge content of 0.2,0.3,0.4,0.6 and 1 mol respec-tively. The electrical conductivity has an exponential behaviour in the temperature range 185-500 Konly for the 0.6 and 1 mol composition. It deviates from the exponential behaviour at lower tempera-tures for the 0.2, 0.3 and 0.4 compositions. The thermoelectric power is positive for CdGeo.2Asz,negative for CdGeo.6Asz and CdGeIsOAsZ, whereas it changes sign for CdGeo.3As2 and CdGeo.4Asz.The Hall coefficient is negative for all compositions except for the CdGel.oAs2.The Hall mobilitiesdiffer widely for the different compositions and have the values of 1.5 x to 1.3 x 10-1 cm2 V-l s-lat room temperature.The compound CdGe,As, has been obtained in the glassy state for a varyingGe content (x = 0.02-1.3).1* The compound belongs to a ternary system basedon CdAs,, which is able to form amorphous substances when elements (e.g., Ge, T1,Sb, Si) which prevent crystallization, are added to it within certain concentrationimilts.The electrical and optical properties of vitreous stoichiometric CdGeAs, havebeen investigated earlier, but there still seems to be some doubt about the origin ofelectrical conduction in the compound.The band gap E,, as determined byGoryunova and Kolomiets from the absorption edge and from the spectral distri-bution of the photoconductivity, was 0.6 eV at room temperature. Vaipolin et aL4have measured the electrical conductivity of vitreous CdGeAs, in the temperaturerange 80-670 K and have observed an exponential temperature dependence above200 K. They have obtained Eg = 1.1 eV, assuming that the conduction is intrinsic.The authors, however, conclude from the large difference in E;Pt and E;' that theintrinsic conduction does not set in till 670 K. Tauc and coworkers have reportedmeasurements on the electrical conductivity and the thermoelectric power of amor-phous CdGeAsa. They have observed that the thermoelectric power is positive atlow temperatures and negative above 220 K.The authors explain it as a transitionfrom extrinsic to intrinsic conduction.The basic parameters such as mobility and carrier concentration still remain tobe determined. This is due to considerable difficulty in measuring low carriermobilities in high resistivity materials. This paper reports the temperature depen-dence of electrical conductivity, thermoelectric power and Hall coefficient of a numberof CdGe,As, glasses over a relatively wide range of temperatures. The aim of thestudy was mainly to extend the available Hall effect data in amorphous semiconductors.* Work performed under the auspices of the association R.U.C.A.-S.C.K./C.E.N.-f Rijksuniversitair Centrurn Antwerpen, Antwerpen, Belgium.1 on leave from the Pakistan Institute of Nuclear Sciences and Technology, Rawalpindi, Pakistan.228 VITREOUS CdGe,As2EXPERIMENTALSAMPLE PREPARATIONCdGe,Asz glasses of five different compositions, having 0.2, 0.3, 0.4, 0.6 and 1 mol ofGe respectively, were prepared from 5 N grade elements. The weighed amount of eachelement was put in quartz ampoules of 12 mm diam. which were then evacuated and sealed.The total amount of the materials was approximately 15 g.To avoid high pressures in theampoules, the temperature was slowly raised to 900°C (which is about 200°C higher thanthe melting point of the compound), and held there for 6h. To produce a completehomogeneity of the melt the ampoules were then rotated in the furnace for 8 h.Sincevitrification of compounds based on CdAs2 requires a fast rate of cooling, therefore, themelts with 0.2, 0.3 and 0.4mol of Ge respectively were quenched in air, whereas the oneswith 0.6 and 1 rnol were quenched by rapid immersion of the ampoules in ice water.Nevertheless, the bulk samples of the latter compositions consisted of a glassy and a crystallineportion. Whenever these two phases were present the boundary between them was clearlydistinguishable and specimens of millimeter sizes could easily be cut from the vitreousportions. Electron microscope examination of these samples after appropriate thinning inCP, showed no sign of crystallinity.TECHNIQUESThe electrical resistivity and the Hall effect were measured by the four probe method ofVan der Pauw.6 Specimens for these measurements were plane-parallel platelets of about5 x 5 mm area and 0.7 ~ll~ll thickness.Electrical contacts were made by fusing heated thinPt wires into the samples. The contacts showed good ohmic behaviour even at the lowesttemperature used in the measurement.The d.c. electrical resistivity was measured between 180 and 500K. The potentialdifference across different contacts (for the calculation of specific resistivity according toVan der Pauw's method), was measured above room temperature by a Dynamco digitalvoltmeter 2022 S and below room temperature by a Keithley electrometer model 640. ASefram verispot galvanometer was used to measure currents higher than 10-IOA and forcurrents lower than lo-'' A the method of the potential drop across a standard resistor wasemployed.The Hall effect measurements were carried out only above 300K.A d.c. magneticfield of 25 OOO gauss, the direction of which was reversed every 30 s, was used for all themeasurements. The residual voltage between the Hall probes, which in general do notlie on an equipotential line, was compensated with a Diesselhorst-type potentiometer.The Hall voltage was amplified by a photo-cell galvanometer amplifier or by a Fluke micro-volt amplifier (model 845 AB), and was recorded on a Kipp micrograph BD 2. The Hallvoltages were extremely small (sometimes as low as 15 pV) and great care was taken to avoidthe influence of other effects. These are mostly of thermal origin, e.g., change in thethermoelectric power at the Hall probes.' It was checked by the absence of any voltage atthe Hall probes for zero current and when the magnetic field was reversed.The thermoelectric power measurements were performed between 210 and 400 K.The specimens in the form of parallelepipeds (1.5 x 1.5 mm area and 5 mm long) weresoldered with In between two small copper blocks.This unit was then mounted with twosapphire discs on to the two large copper blocks which could be heated separately. Thesapphire slices give a good thermal contact and a high electrical insulation. Two calibratedchromel-constantan thermocouples were soldered close to the interface of the small copperblocks and the specimen. The assembly was then mounted in the inner chamber of acryostat which was evacuated and then filled with argon.A temperature gradient varyingbetween 2 and 10°C was maintained across the specimen. The thermocouple voltages weremeasured by a Dynamco digital voltmeter type 2006/D4 of 0.1 pV sensitivity. The Seebeckpotentials, for higher temperatures, were measured by a Dynamco digital voltameter 2022 Sand for low temperatures a Keithley electrometer model 602 was empIoyed. The measuredvalues were corrected for the absolute thermoelectric power of the thermocouple probesR . CALLAERTS, M. DENAYER, F. H . HASHMI AND P . NAGELS 29The resistance of the specimen was also measured simultaneously with the thermoelectricpower measurements by the Keithley 602 electrometer. The values of the specific resistivitythus obtained were in complete agreement with the ones obtained from Van der Pauw’smethod.RESULTSThe conductivity a of a CdGeo.2As2 sample, cut from the quenched material,was measured between room temperature and 450K.The room temperatureconductivity did not have the same value when it was again measured after thisthermal cycle. A decrease in the room-temperature a-value of approximately 20 %was found. A similar and even more marked effect was observed for the stoichio-metric CdGel .oAs2 composition. The difference in room-temperature a-value beforeand after the thermal cycle was of a factor of two. Further repeated thermal cyclingof the samples had no effect on the a value. Therefore all the measurements weremade on samples which were annealed for 24 h at 200°C.1 14 51 0 3 / ~ ( ~ - 1)0.2 (0, e) ; 0.3 (A) ; 0.4 (+) ; 0.6 (0, U) ; 1.0 (V).FIG.1 .-Temperature dependence of the electrical conductivity of CdGexAs2 glases ; Ge content :The measurements made on a number of samples cut from the same ingot showedreproducible results. This is shown in fig. 1 where log a is plotted against thereciprocal temperature for CdGe,As, samples containing respectively 0.2, 0.3, 0.4,0.6 and 1 mol of germanium. Moreover, the conductivity data obtained by Van derPauw’s method were in complete agreement with the ones obtained from the sampleswhich were used for thermoelectric power measurements. The ohmicity of th30 VITREOUS CdGe,As2I-V characteristics was checked at different temperatures by applying voltages upto 24 V cm-l across the samples. The conductivity was field independent even atthe lowest temperature employed in the experiment.The conductivity decreases with the increasing Ge content (fig.1) which is inagreement with the results reported by Hrubk et a2.l The temperature dependenceof the conductivity differs, however, for the different glasses. The curves ofCdGe, .0A~2 and CdGe,. ,As, have a linear 1 /T dependence in the whole temperaturerange investigated. The activation energy deduced from their slope in both thecases is 0.55 eV. The curves of the CdGeoe3As2 and CdGeom4As2, which overlapeach other, start to deviate from the linear behaviour below approximately 290 K.The straight-line portion of the curves yields an activation energy of 0.55 eV, i.e.,the same as for the higher contents of Ge.An even more pronounced deviationfrom the straight line is observed for the glass of the composition CdGe,,,As,.The activation energy in this case is 0.52 eV which is slightly lower than the othervalues. A difference in absolute value of the conductivity is sometimes observedbetween samples with the same composition but from different ingots. This effectis especially observed for the 0.2 and 1 mol composition and illustrated in fig. 1 for-1500-CdGeo,,As2 only.3 4103 /T(K - 1)FIG. 2.-Thermoelectric power as a function of the reciprocal absolute temperature for vitreousCdGexAsp ; Ge content ; 0.2 (0, a) ; 0.3 (A) ; 0.4 (+) ; 0.6 (0) ; 1.0 (V).The temperature dependence of the thermoelectric power is shown in fig.2.The Seebeck coefficient and the form of its dependence on temperature vary markedlyfor the different compositions of the glasses used. The thermoelectric power oR. CALLAERTS, M. DENAYER, I;. H . HASHMI AND P . NAGELS 31CdGe,.,As, is positive in the whole of the temperature range. For CdGe,.,As,and CdGe,,,As, a change of sign occurs at about 320 K, which is nearly the sametemperature where the conductivity starts to deviate from its linear behaviour. Onthe other hand, the thermoelectric power of the glasses CdGe,.,As, and CdGelaoAs2is always negative.The Hall coefficient data plotted as a function of the reciprocal temperature areshown in fig. 3. These measurements were performed between 300 and 500K.Below room temperature the Hall voltage could not be detected precisely due to thehigh resistance of the sample.One important feature, observed in these measure-ments, was the opposite sign for the stoichiometric composition. The sign of theHall coefficient is negative for glasses with Ge content x = 0.2, 0.3, 0.4 and 0.6,whereas it is positive for CdGe,.,As,.5FIG. 3.-Temperature dependence of the Hall coefficient of CdGexAsz glasses ; Ge content :0.2 (0, @I; 0.3 (A); 0.4 (-4-1; 0.6 (0, W); 1.0 67).The Hall mobilities calculated from the conductivity and the Hall coefficient dataare plotted in fig. 4. Their values vary widely from composition to composition.The Hall mobilities pH of compounds containing 0.3, 0.6 and 1.0 mol of Ge areindependent of temperature, whereas pHincreases slightly for CdGe, .4A~2 and decreasesslightly for CdGe, .,As2 with increasing temperature.Repeated measurementscarried out on other samples of the same ingot gave reproducible results within thelimits of experimental accuracy32 VITREOUS CdGe,As,FIG. 4.-Hall mobility as a function of temperature for vitreous CdGexAs2 ; Ge content : 0.2 (0, 40.3 (A); 0.4 (+I; 0.6 (0, W ; 1.0 (V).DISCUSSIONThe properties of amorphous stiochiometric CdGeAs,, in most of the previcstudies, have been compared with those of the corresponding crystalline materiCdGeAs, crystallizes in the chalcopyrite structure with a tetrahedr31 arrangemtof atoms and covalent bonds between them. Recently, however, Cervinka et ~1have investigated the structure of two CdGexAs2 glasses (x = 0.1 and 1.1) by X-Idiffraction. From the radial distribution curves of the atomic density they conchthat the position of the first maximum could best be interpreted by the distantbetween the atoms as determined for crystalline CdAs,.This compound hastetragonal lattice and is composed of tetrahedra formed by four As atoms with aatom in the centre. The individual CdAs, tetrahedra are bound together with 1common atoms of As and form networks. Then these are held together by ibon_ds between the atoms of As.Cervinka et al. have assumed that CdAs, forms an amorphous substance wherforeign atom, e.g., Ge, is added to it. The Ge atom will be bound to four As at0which are responsible for the binding of the neighbouring networks in pure CdAThey consider, therefore, that the amorphous CdGeAs, belongs to a broader systtof amorphous substances based on CdAs,.It seems thus obvious to compare tproperties of the amorphous CdGe,As, with those of crystalline CdAs,. Howevlthere exists a structural similarity in the short-range order of crystalline CdAs, aCdGeAs,, i.e., the lattices of both substances are made of the same unit of CdPtetrahedra.The electrical properties of crystalline CdAs, differ markedly with those o f R . CALLEARTS, M. DENAYER, F. H . HASMI AND P . NAGELS 33CdGeAs,. The latter compound has a similar band structure to that of the A"' BVsemiconductors and has a three-fold splitted valence band, i.e., consisting of a heavyhole, a light hole and a second heavy hole band.On the other hand, its electroneffective mass is very small (m* = 0.027 m0)* yielding mobility of the electronsmuch higher than that of the holes. On the contrary in CdAsz the effective massof the electrons is somewhat higher than that of the holes and both are stronglyanisotr~pic.~ The Hall mobility of electrons at 300 K is 100 and 400 cm2 V-1 s-'along the a- and c-axis respectively. O Hall mobility for holes has not been reportedso far. The forbidden gap as determined by the optical absorption is much higherfor CdAs2 (- 1 .O eV at 300 K) than for CdGeAs, (0.53 ev).Cervinka et al. assumed intrinsic conduction and deduced from conductivitymeasurements the band gap Eg for different CdAs, glasses with different contents offoreign elements.By extrapolating the dependence of Eg on the composition tox = 0, they obtained E, = 1.05 eV at T = 0 K for amorphous CdAs, which theyconsider to be in good agreement with the band gap of crystalline CdAs,.From these considerations, it seems reasonable to assume that all the glassesused in our study (including the one with 1 mol of Ge), have the same basic structure.This is strengthened by the fact that the linear parts of all the conductivity curveshave nearly the same activation energy.For a discussion of our experimental results, we compare first the thermoelectricpower data of the different glasses with the behaviour of their electrical conductivities.The slope of the conductivity curves of the compound with x = 0.2, 0.3 and 0.4increases with increasing temperature, but remains constant at high temperature.Since the slope is representative for the position of the Fermi level inside the band gap,it follows that the Fermi level shifts towards the middle of the gap and remains fixedat higher temperatures.The latter situation is already reached at the lowest tempera-ture of measurement for the CdGe,.,As,.The Seebeck coefficients of the 0.2, 0.3 and 0.4 compositions are positive inapproximately the same temperature range in which the conductivity deviate fromthe linear 1/T dependence. The experimental findings are consistent to a semi-conductor containing localized levels acting as acceptors and donors. ThFe levelsmay be due to lattice defects.The existence of such defects was proved by Cervinkaet al. They have measured magnetic susceptibility of CdGexAs2 and have found ahigh number of paramagnetic centres varying from 3 x 1019 ~ r n - ~ for 0.2 mol Geto 1 x 2OZ0 ~ m - ~ for 1.0 mol of Ge. The positive sign of the Seebeck coefiicientindicates that the number of holes is higher than the number of electrons or that thedonor-like states are outnumbered by the acceptor-like states. At very low tempera-ture the Fermi level is localized near the energy state Ec- EA. The shift of theFermi level as the temperature rises is generally due to the commencement of intrinsicconduction. A similar situation where the Fermi level lies also in the middle of thegap occurs in a highly compensated system.One may, therefore, consider that theintrinsic conduction occurs in the temperature range where the conductivity curvesare straight lines.Tauc et al. have performed conductivity and thermoelectric power measurementsdown to 140 K. They found a reversal of the Seebeck coefficient at approximately220 K which they ascribed to the transition from extrinsic to intrinsic conduction.For an intrinsic semiconductor the Seebeck coefficient is given byProviding the energy gap E,, is known, it is possible to obtain the value of p J p ,34 VITREOUS CdGe,As,and m:/rn; from the slope and the intercept of S plotted against l/T. In the presentexperiments this calculation can best be done by using the conductivity and thethermoelectric power curves of CdGe,.-,As,, since these are the curves which arelinear in the whole of the temperature range and where the negative sign of the Seebeckcoefficient indicates a higher electron to hole mobility.Using Eg0/2 = 0.55 eV andAn exact calculation of the mass ratio m;/rn: is impossible due to the uncertainty inthe temperature dependence of Eg and because the scattering mechanism is notknown. The observation that pn>puo in the CdGe,As, glasses is opposite to thatin the crystalline CdAs,.Care must be taken in deducing the intrinsic gap from the slope of the conductivitycurve. When applying classical band theory, the intrinsic carrier concentration isgiven byni = (NvNc)) exp [ - Eg0/2kT].Since (NvNc)* varies as T3, therefore it is preferable to construct a graph In (oT-3)against l/T.When applying this temperature correction, the intrinsic gap is 1 .O eVfor the glasses with 1.0, 0.6, 0.4 and 0.3 mol of Ge and 0.95 eV for the glassCdGeo,,As2. A further correction may also be necessary because of the temperaturedependence of the mobility, which is, however, not known in OUT case.The higher electron mobility implies a negative sign of the Hall coefficient in theintrinsic range. This has indeed been observed for the glasses with 0.2, 0.3, 0.4 and0.6 mol Ge. For the three first compositions, the Hall coefficient remains negativedown to the temperatures where the Seebeck coefficient has not yet changed in sign.This can easily be understood when applying the formula of RH for mixed conduction :RH = - ( n e d - pep,2)/(nePn + p e ~ p ) ~ *This expression, whch contains squared values of the mobilities, shows the onset ofintrinsic conduction at lower temperatures (pn > pP).The compound CdGe, .oAszexhibits a positive Hall coefficient. Thrs is an unexpected result since the behaviourof S ando seems to indicate that intrinsic conduction is dominant in the whole tempera-ture range. No satisfactory explanation can be offered for this sign anomaly.The Hall mobilities of the different glasses differ strongly. Measurements ofmagnetic susceptibility by Cervinka et al. have indicated the presence of a greatnumber of electrically charged centres in the CdGe,As, glasses. One would normallyexpect that these defects will act as scattering centres, leading to a large reduction ofmobility. This deccease will then depend on the scattering centre concentration,which according to Cervinka’s results, may vary from composition to composition.A. Hrubi and L. gtourae, Mat. Res. Bull., 1969, 4, 745.L. eervinka, A. Hrubi, M. Matyag, T. SimeEek, J. Skacha, L. Stourat, J. Tauc and V. Vorlitek,J . Non-Cryst. Solids, 1970,4,258.N. A. Goryunova and B. T. Kolomiets, Zhur. T. F., 1958,28, 1922.A. A. Vaipolin, g. 0. Osmanov and Yu. V. Rud’, Fiz. Tuerdogo Tela, 1965,7, 2266.Moscow, (Nauka, Leningrad, 1968), vol. 2, p. 1251.L. J. Van der Pauw, Phil@s Res. Reports, 1958, 13, 1 .N. A. Goryunova, F. P. Kesamanly and E. 0. Osmanov, Fiz. Tverdogo Tela, 1963, 5, 2031.M. J. Stevenson, Pliys. Rev. Letters, 1959, 3, 464.* J. Tauc, L. StouraE, V. VorlfEek and M. Zgvetova, Proc. 9th Int. Con$ Physics Semiconductors,’ H. J. Van Daal and A. J. Bosman, Phys. Rev., 1967,158,736.lo W. J. Turner, A. S. Fischler and W. E. Reese, Phys. Rev., 1961,121.759

ISSN:0366-9033    

DOI:10.1039/DF9705000027

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Electrical Conductivity of Melts and their Ability to FormGlasses in the System Ge+As+TeBY H. KREBS and P. FISCHER *Institiit fur Anorganische Chemie, Universitat Stuttgart, Stuttgart, GermanyReceived 15th June, 1970In the system Ge+As+Te, the heavy Te atom facilitates the formation of mesomericpo-bondingsystems. The glass forming regions are thus small and depend strongly on the quenching conditions.As in the systems Ge+Sb+Se and Ge+As+Se all metallically conducting melts in the systemGe+As+Te solidify to a crystalline structure even when quenched in water. However, not allsemiconducting melts belonging to this system become glassy under these conditions. As expected,the transition from melts solidifying to a glass structure to those solidifying to a crystalline structureis more gradual, as is also the transition from metallic to semiconducting melts.All the semi-conducting melts become more or less metallically conducting at temperatures between 900 and1OOO"C. The transition can be described by a parabolical or a log log dependence on temperature.In the chalcogenide systems the melting process often enforces the same bonding mechanismwith similar atomic short range order as does the application of high pressures or of strong electricfields. This is especially the case at high temperatures. The enforced mobility of the atoms, theirtighter packing and the effect of electrical conductivity often act in the same direction, enhancingstructural changes in the same direction.The melting of strongly 3-dimensionally interconnected high polymers isnecessarily accompanied by a change of structure.This structural change can con-sist in a depolymerization as, e.g., for red phosphorus which forms P,-moleculesin the melt at about 620°C. Another possibility of structural transition is a changeof the bonding system. In the melting process, crystalline Ge with diamond structurefirst assumes a strongly distorted NaC1-type short range order (Ge on the Na- andthe C1-lattice sites). At about 300°C above the melting point this distortion hasdisappeared; one only observes a wide smearing-out of the atomic positions.2 Inthe crystalline phase, the bonding system of Ge is characterized by localized valencyelectron pairs in (sp3)-hybrid quantum states, whereas in the melt it consists ofelectrons in p-quantum states which form a-bonds in two directions. The two-sidedorientation of the bonding function causes mesomerism (generally termed" resonance " in the English literature) and thus leads to metallic conductivity.Linearly polymerized products such as plastic S and glassy Se can form viscous melts.In this case the existence of only two bonds for each atom permits a higher atomicmobility.The melts are semiconducting like the glasses themselves, as in both casesthe bonding system is the same, being characterized by mainly localized electronpairs.The rule that strongly 3-dimensionally interconnected crystals-consisting ofatoms of the groups 4, 5 and 6~-form metallic melts which readily crystallize becauseof the mobility of the valency electrons and of the atoms themselves, should alsoapply to the particular case of the so-called chalcogenide systems.For the systemsGe + Sb + Se and Ge + As + Se this has already been ~erified.~* Thesis (Stuttgart, 1970)336 ELECTRICAL CONDUCTIVITY OF MELTSFor our further investigations we chose the system Ge +As + Te, first, as accordingto the literat~re,~. it shows two glass regions and secondly, because the heavyelement Te has two characteristic properties which lead us to expect a more complexbehaviour: (i) a bond containing a Te-atom dissociates more easily than bondscontaining Se or S; (ii) the transition from a more localized bond in which thep-valency electrons are hybridized to a greater extent with s-electrons, to non-localizedtwo-sided bonds in which the s-admixtures have more or less disappeared, shouldoccur more easily.6*The electrical conductivities were measured up to about 1000°C by Haisty's *inductive method, using his apparatus.In addition, differential thermal analyseswere performed.RESULTSWithin the precision of measurement, our results confirm the systems Ge + Te,9"As+Te l2 and Ge+As 13* l4 reported in the literature. Furthermore, our measure-ments of electrical conductivity agree with those given in the literature for the systemsGe + Te * and As + Te. 7* * We also found the sudden increase of the electricalconductivity of Te at the melting point and the continuous transition of the melt toa metallicallyliterature.9-2conducting state- with increasing temperature, as described in theAsFIG. 1.Electrical conductivity and glass formation in the system Ge+As+Te ; explanation in thetext.Fig. 1 shows the electrical behaviour of the samples of the ternary systemGe + As + Te in the melt and compares it with the glass-forming regions found by uswhen the samples were cooled in air (-) or quenched in water(---). However, ourresults are not in complete agreement with those of Hilton, Jones and Brau whoutilized larger samples and thus obtained lower cooling rates. Our results differmore appreciably from those of Panus and Borisova with respect to the glass-formingregions indicated. These authors do not observe the glass-forming regions rich iH . KREBS AND P.FISCHER 37Te ; the glass region rich in As, however, extends up to Ge contents of nearly 40 at. %,which appears possible in view of our electrical measurements. In the figure, opencircles (0) represent samples which were distinctly semiconducting in the melt, andwhich show glassy solidification when cooled in air. Open circles containing a dot(0) indicate semiconducting melts which only solidify in glassy form when quenchedin water. Half-filled circles (0) inside the dashed-dotted-line indicate semiconductingmelts with high conductivity which solidify mainly to a crystalline structure even whenquenched in water. Metallic or nearly metallic melts which always crystallize arerepresented by a full-filled circle (0). However, one can not always distinguishrigorously between metallically conducting and good semiconducting melts ; this iseven difficult for the element Te itself.Moreover, we had no scanning stereoelectron-microscope at our disposal for the investigations, so that it was not possibleto observe crystalline phases finely dispersed in the glass. Further, the indicationsgiven here and in the other figures pertaining to samples with more than 60 at. %As are not precise, since, because of the high vapour pressure of As, separationphenomena occurred at high temperatures. Even after quenching in water anenrichment in As was often obtained in the upper part of the samples, as could berecognized from the form of the oscillograms recorded in the electrical measurements.Fig. 2 shows the transformation points Tg and the points at which the crystalliza-tion of the various glasses sets in.The results were obtained by differential thermalTe - At'/* ASFIG. 2.-Transformation points found by D.T.A. (above) and beginning of crystallization (below),in "C.analysis. In most cases, the heating rate was 20"C/min. As far as comparison ispossible, the crystallization temperatures measured by us are situated between thoseobtained by Panus and Borisova 22 at heating rates of 7"C/min for powdered glasseson the one hand and for glassy casts on the other.Some typical examples of the curves measured are shown in fig. 3 and 4. Forthe sample Ge,,Te,, one notes the beginning of the transition to the high-temperatureresistivity behaviour characteristic of the glass forming region.The sampleGeQ5As45Te, shows the typical high-temperature metallic conductivity of thesamples which always solidify in a crystal structure.When one tries to give an analytical description of the resistivity curve in theregion represented in fig. 1 by a dash-dotted line, the formula given by Perron 21 cannot be made to fit. Busch's 2o equation can under certain circumstances be appliedif one is prepared to use five parameters.Fig. 3 and 4 show that the curves measured for the samples lying in the dash-dottedregion are parabolical at high temperatures. In some cases this relationship wa38 ELECTRICAL CONDUCTIVITY OF MELTS+5+4 -+3-+2-h> + I -c.r .-.- +a v) ." E 0-U 8-1--2--3--4tested; the resulting parameter values of the equation T2 log (p/p,) = aT+b arelisted in table 1.Even the experimental values given by Perron 21 for melts ofcomposition SezoTeso can be well fitted to this equation.~ ~ ~ 6 ~ 5 f i ~ ~ 2 ~ ~ 3 , 1 6 ~ ~ 2 2 8 ~ ~ 2 1 6 2 ~ S 1 2 8 I I 2 9 8 8 5 ?3 6) 5 O Q 21 t'c) '0.I I 4 7 F5 2 2.5 30&-4 1 1 I 1 I I I I I I 1 m 16 201 /T [103/K]FIG. 3.-Relationship between logarithm of specific resistivity and 1 /T: (a) for a typical glass formingsample (Ase5TeS5) ; (6) for a crystallizing, still semiconducting, sample (GeloTegO) ; (c) for a metal-lically conducting crystallizing sample (Ge4SAs45Te10) ; (d) for elementary Te.It is not possible to give an interpretation of this parabolical law. To explainthe linear relationship between log p and 1/T often found at higher temperatures inthe systems Ge + As + Se and Ge + Sb + Se,3 Mott 23 has proposed that the width ofthe forbidden zone varies linearly with temperature.This induced us to try tH . KREBS AND P . FISCHER 39approximate the experimental curves by an equation in which the width AE of theforbidden zone is an exponential function of temperature, as in chemical equilibria.For the evaluation, the formula was developed in the following form :AE = AEo f exp (UIkT), (2)AEof loge U A2k kT T’+-loge = B+- (3)This latter equation could often be fairly well fitted to the experimental values.Values of the constants A and B are indicated in table 1. For the sampleGe,oAs,oTe,o, the experimental values are given in fig.5 according to both equationsand to the expression of Perron.21FIG. 5.-Fitting of electrical resistivity and electrical conductivity of molten Gel OAsSOTe40 tothe functions : (a) TZ log (p/po) = aT+b ; (b) T20 as function of T; (c) log (Tlog (p/pl)) = (A/T)+B.TABLE 1composition T2 log ( p / p g ) = aT+b log [T(log (p/pi)l = (A/T)+Ba[K]-1 biK1-2 po[R cnil-1 A[KI-I B PIW cm1-IGe 1 oTe9 0 -6440 3 1 . 1 ~ 1 0 ~ 1 3500 -2.426 4 . 4 7 ~As45Te5 5 - 8160 49.6 x lo5 1 810 2.054 1 .OO xGel OAs5OTe40 - 8620 54.4 x lo5 1 1080 1.665 2.00 xGe20 Sb40 Se4o -99000 59.8~105 1 1260 1.535 3 . 1 6 ~As40Ge40Se20 5080 -2.344 5.13 xAS28.5Te7 1 * 5 -7200 39.2 x lo5 1 1515 -2.06 2.69 x 10-4Se2oTes 0 - 7780 44.4 x lo5 1 2410 -0.50 4.16 x 10-40 ELECTRICAL CONDUCTIVITY OF MELTSFor the systems Ge + As + Se and Ge + Sb + Se one often finds a linear relation-ship between log p and 1 /T up to temperatures of about 900°C in the glass region.Inthis representation, one can sometimes fit the experimental values by two straightlines.23 Often, however, the curves can not readily be represented in an analyticalform. In some samples, however, e.g., in Ge20Sb4,Se40 and Ge40As40Se,o, wefound the relationships discussed above. The corresponding parameters are alsoindicated in table 1.Te - At '10 ASFIG. 6.-Energy gap AE of glasses obtained (above); logarithm of specific conductivity at 100°C(below).Fig. 6 gives the AE-values calculated from the slopes of the conductivity diagramsof the glasses, as well as the logarithms of the specific electrical conductivities at100°C.As far as comparison is possible, the AE-values thus obtained are in generalhigher than those of Panus and B o r i s ~ v a , ~ ~ whose measurements were in the tempera-ture range from 20" to about 100°C.Ge455FIG. 7.-Logarithm of specific conductivity of melts in the system Ge+As+Te at 900-1OOO"CH. KREBS AND P. FISCHER 41Within the accuracy of measurement, the AE-values remain unchanged betweenthe T'-temperatures and the crystallization temperatures. Distinct increases of theslopes of the curves were only found in the following samples belonging to the binarysystem As + Te :AS,,Te56 : from 0.93 to apparently 1.53 eVA s ~ ~ T ~ ~ ~ : from 0.93 to apparently 1.53 eVAs4,TeS3 : from 0.93 to apparently 1.56 eV.The reason for this was not further investigated. We suspect the effect to be due toa crystallization in micro-regions.In any case, the increases are only calculatedvalues.Between 900 and 1000°C, all the samples of the system Ge+As+Te indicate atleast an approach to a metallically conducting state as the temperature coefficients ofthe electrical conductivity become small, whilst the conductivities themselves arefairly high. Fig. 7 indicates the logarithms of the specific electrical conductivitiesof the melts between 900 and 1000°C. In general, with increasing valency electronconcentration (Ge+ As-+Te) the conductivity decreases, as was to be expected.INTERPRETATION OF EXPERIMENTAL RESULTSIn the samples Telo0, AslTeg9 and As5Teg5 the increase of conductivity observedabove 453°C is similar and remains small, whereas for GeloTego, Ge12Te8,, AsloTegoup to AS20Te80, Ge,As,Teg, and Ge13As2Te8,, the conductivity increase of themelt above 400°C is more important and approaches that of the adjacent samples withhigher Ge- and As-contents.In spite of this, all these samples crystallize whenquenched in water. A crystallization of semiconducting melts had also been foundin the binary system-in the region SbloSego to Sb40Se60 when cooled in air andin the region Sb,,SeB0 to Sb4&3eS5 when quenched in water-and was explained bya catalytic influence of the Sb-at~m.~a) blFIG. 8.-PossibIe arrangement of Te-atoms in the melt: (a) tightly bound Te-Te atoms pairs lieon straight lines and interact with each other ; (b) tightly bound Te-Te atom pairs lie on straightlines together with p-electron pairs of other Te-atoms, as in the crystal. tm free p-electron pair ;to, weakly hybridized binding electron.Fig. 8a illustrates the catalytic effect of Te, which has similar origins.As aheavy atom, Te has a tendency to form a-bonds with little s- but high p-contents.Thus, in one bond direction, i.e., inside the chain, a neighbour is tightly bound,whereas the bond partner situated in the opposite direction belonging to a neighbouringchain is more weakly bound. The roles of favoured and disfavoured bond directionscan easily be changed as the hybridization can adapt to the movement of the atoms42 ELECTRICAL CONDUCTIVITY OF MELTSIn the melt, the linking of atoms to chains is thus subjected to perpetual fluctuations.This can be shown experimentally for S and Se.The dissociation energy of a bond is much lower for liquid sulphur (33.4 kcal/mol)than for organic disulphides (approximately 50 kcal/mol).During the dissociationprocess two-sided bonds can develop in the S-melt before the original bond is fullyloosened. In molten S, the lifetime of the radical state obtained by dissociation canbe measured by paramagnetic resonance experiment^.^ One obtains lifetimes of1.8 x s at 300°C.26 In liquid Se, however, the lifetimeis already so small that the e.s.r.-signal is largely smeared out and is no longer easilydi~cernible.~~ One can therefore expect that with Te the change of bonds takesplace even more quickly so that with increasing temperature the respective roles offavoured and disfavoured bond directions tend to become badly defined and thatthe chain structure is thus increasingly smeared out.According to a privatecommunication from T. Springer, the lifetime of a chemical bond of liquid Te isz ,N 6 x 10-l2 s . ~ *The properties of the heavy elements belonging to the group of semi-metals, asdiscussed for Te, should facilitate crystallization in general. Thus, the regions ofglass formation become smaller with increasing atomic weight of the constituentelements. The easy crystallization observed in the semiconducting samples containingmore than 20 at. % Ge can be interpreted in the same way.Glass formation wouldseem to be possible even in this case if the quench were performed still more rapidly,e.g., by using Duwez's method.29s at 200°C and 1 xCHANGE OF STRUCTURE AND ELECTRICAL CONDUCTIVITY INDUCED BYMELTING PROCESSES, HIGH PRESSURES AND STRONG ELECTRIC FIELDSThe present state of investigations concerning semi-metals and the chalcogenidesystems should allow some general conclusions to be made. Now, the meltingprocess brings about a change of the bonding system when strongly 3-dimensionallyinterconnected substances, such as Ge or the chalcogenide systems, have high contentsof 4- and 3-valent atoms. A transition takes place from localized (sp3)-hybridsto mesomeric pa-bonding systems which permit mobility of the atoms in the melt.This is associated with a tendency to increase the coordination number to 6 as thebond is made by electrons having preponderantly p-character.2* 3* 6* An increaseof the coordination number is associated with a more dense packing of the atoms andcan therefore also be produced by high pressures, e.g., at pressures of 120 kbar Geassumes the structure of white Sn with a distorted octahedral environment of eachatom,30 by analogy with the melt.2 The structure of the high-pressure modificationof Se and Te is not known.They are metallic conductors and at low temperaturessuperconductors.31~ 32 From this one can deduce that one of the four p-electronsis raised to an excited state, leaving three p-electrons available to bind six neighbours.In this way mesomeric bonding systems and consequently metallic conductivity candevelop.The existence of superconductivity suggests that the excited electron mayoccupy a 5s-state in the Se-atom or a 6s-state in the T e - a t ~ r n . ~ ~Fig. 8b shows an arrangement of electrons with more or less p-character realizedin a Te crystal. An orbital of a free p-electron pair (m) at a Te-atom belongingto one chain lies approximately on the same axis as the orbitals of only slightlyhybridized binding p-electrons (a) of four Te-atoms belonging to two neighbouringchains.' As free electron pairs are more weakly bound than binding electron pairs,a thermal collision will in general dissociate an electron from the first pair and raiseit to an excited state (presumably 6s).A part of the dissociation energy is thenrecovered by the formation of a mesomeric pa-bonding system connecting the fivH. KREBS AND P . FISCHER 43atoms indicated in the figure. The mesomerism is now well developed, as thereare only five valency electrons for five quantum states. If in the glass or in the melt,additional atom pairs of neighbouring chains lie in the required positions, the meso-meric chain is continued. It can wander in the melt or glass : this happens when anelectron jumps from another row of atoms with a p-electron pair to this mesomericchain, destroys this mesomeric chain and forms a new one. The mesomeric statecreated by an electron hole is mobile, as adjacent binding electron pairs of appropriateorientation are attached rapidly and as, in addition to this, the above mentionedjumping process can occur.These assumptions were corroborated in a recent investigation of Tourand andBenit 34 on the short-range order of atoms in molten Te.The smallest interatomicdistance in the melt is 2.99 A and thus 0.11 A greater than in the crystal (2.878 A),whilst the bond angle of 100" is nearly 2" smaller (101" 46'). The coordinationnumber of the first sphere is 2.7 at 660°C and increases to 3 at 930°C. At the sametime, a well-defined second maximum develops for which the coordination numberis not indicated but probably is around 3 ; its distance from the reference atom isonly 26 % greater than that of the first maximum, whereas for a van der Waalsdistance one would have to expect a difference of 60-70 %.A strong electric field favours the mesomeric chains aligned with the field, asinside these the pa-electrons can follow the influence of the field and are thus slightlydisplaced from one end towards the other end of the chain.The centre of gravityof the positively charged hole is displaced towards one end of the mesomeric chain,the interaction with p-electrons of adjacent atoms thus becomes stronger and thechain grows. Correspondingly, the current filament of a memory element onlygrows from the anode.35 In strong electric fields, extremely strong fields mustdevelop at the ends of these mesomeric chains with respect to the non-conductingsurroundings with inappropriately placed atoms. This results in a rapid growth ofthe mesomeric chains.When a sufficient number of mesomeric chains of satisfactorylength has developed, the electrical resistivity decreases until finally the mesomericchains interconnect, thus producing a low resistivity filament. This mechanism seemsto be the reason for the mobility gap postulated by M ~ t t , ~ ~ Cohen, Fritzsche andOvshinsky 37* 38 and for its annihilation in strong electric fields. In the glass and inthe melt, the atomic short-range order will be slightly changed with respect to thestate present in the absence of an applied field. This change of short-range order willbe comparable to that occurring in Te or in the melts of the system Ge+As+Tewhen the temperature of the melt is raised and a metallic conductivity is approachedor attained.Such changes of the short-range order have in the meantime also beenfound by us for the transition from glass-forming melts (GezoAs30Se50) to meltssolidifying in a crystalline structure (Ge45As30Se2s and GessA~30Se,5),39 cf. alsoref. (34).The mobility of the excited electrons is smaller 40 because these electrons can notparticipate in a mesomeric bonding system and because they do not become attachedto adjacent bonding systems as strongly as mesomeric systems with holes do. Theexcited electrons can be present to such an extent in the low resistivity filament thatthey form a band and contribute to the ond duct ion.^^ If these electrons occupy6s-states, superconductivity is to be expected.If one succeeds in cooling a current filament of a switch element to the temperatureof liquid He, one should be able to freeze in the displacement of the atoms of thefilament with respect to the normal After the current is switched off,the element should then remain in the conducting state.After de-freezing, theelement should again recover the properties of a switch element44 ELECTRICAL CONDUCTIVITY OF MELTSWe thank Dr. R. W. Haisty for his collaboration in the electrical measurements,Michael Fischer for the analytical description of the electrical measurements, andDr. H. Wesemeyer for many helpful discussions. We thank the Deutsche Forschungs-gemeinschaft, the Verband der Chemischen Indus trie (Fond der Chemischen Industrie)and the Hiittentechnische Vereinigung der De utschen Glasindustrie e.V.for theirfinancial support.W. Klemm, H. Spitzer and H. Niermann, Angew. Chem., 1960,72,985.H. Krebs, V. B. Lazarev and L. Winkler, 2. anorg. Chem., 1967,352,277.R. W. Haisty and H. Krebs, J. Non-Cryst. Soliak, 1969,1,399 and 427.A. R. Hilton, C. E. Jones and M. Brau, Phys. Chem. Glasses, 1966, 7, 105 ; Infared Phys.,1966, 6, 183. ' V. R. Panus and 2. U. Borisova, Zhur. Priki. Khim., 1966,39,987, Engl. trans., ConsultantsBureau New York, J. Appl. Chem. U.S.S.R., 1966,39,937.H. Krebs, J. Non-Cryst. Solids, 1969, 1,455. ' H. Krebs, Grundziige der anorganischen Kristallchemie, (F. Enke, Stuttgart, 1968), trans.P. H. L. Walter, (Fundamentals of Inorganic Crystal Chemistry), (McGraw-Hill, London,1968).R. W. Haisty, Rev. Sci.Instu., 1967, 38, 262; 1968, 39, 778.W. Klemm and G. Frischmuth, 2. anorg. Chem., 1934,218,249.lo J. P. McHugh and W. A. Tiller, A.S.M.E. Trans., 1960,218, 187.l1 K. Schubert and H. Fricke, 2. Naturforsch., 1951, 6a, 781.l2 J. R. Eifert and E. A. Peretti, J. Material Sci., 1968, 3, 293.l3 H. Stohr and W. Klemm, 2. anorg. Chern., 1940,244,205.l4 K. Schubert, E. Dorre and E. Giinzel, Naturwiss, 1954,41,448.l5 N. V. Kolomoets, F. Ja. Lev and L. M . Sysoeva, Fiz. Tuer. Tela., 1963, 5, 2871 ; Engl. trans.Sou. Phys. Solid State, 1963, 5,2101.l6 V. M. Glazov, A. N. Krestovnikov and N. N. Glagoleva, Dokl. Akad. Nauk S.S.S.R., 1965,162,94.l7 B. T. Kolomiets and B. V. Pavlov, Fiz. Tekh. Polyprou., 1967, 1, 426; Engl. trans. Sou. Phys.Semiconductors, 1967, 1, 350.G.P. L. Bedout, C. A. Chiroux and R. W. Heindl, French. Pat. 1465528 (CI. C. 03c H 01 l),Jan. 13, 1967, Appl. Nov. 25, 1965.l9 H. Fritzsche and A. Epstein, Phys. Reu., 1956, 93,922.2o G. Busch and Y . Tieche, Roc. 6th Znt. Con$ Physics of Semiconductors, (Exeter, 1962), p. 327.21 J. C. Perron, Adu. Phys., 1967, 16, 657.22 V. R. Panus and Z. U. Borisova, Izu. Akad. Nauk. S.S.S.R., Neorg. Mater., 1967,3,2190;23 N. F. Mott, private communication.24 V. R. Panus and Z. U. Borisova, Zhur. Prikl. Khim., 1967, 40, 998 ; Eng. trans. (Consultants25 D. M. Gardner and G. IS. Fraenkel, J. Amer. Chem. Soc., 1956,78,3279.26 H. Ksebs, 2. Naturforsch., 1957,12b, 795.27 G. K. Fraenkel, private communication.28 A. Axmann, W. Gissler, A. Kollmar and T. Springer, this Discussion.29 P. Duwez. Progr. Sol. State Chem., 1966,3,377.30 R. H. Wentorf and J. S. Kasper, Science, 1961, 139, 338.31 J. Wittig, Phys. Reu. Letters, 1965, 15, 159.32 B. T. Matthias and J. L. Olsen, Phys. Letters, 1964,13, 201.33 H. Krebs, 2. Naturforsch., 1968,23a, 332; 1969, %a, 848.34 G. Tourand and M. Benit, Compt. rend., B, 1970,270,109.35 H. Fritzsche and S. R. Ovshinsky, J. Non-Cryst. Solids, 1970,4,464.36 N. F. Mott, Adv. Phys. 1967, 16, 49 ; Phii. Mag., 1969, 19, 835; Festkiirperprobleme, Adu.37 M. H. Cohen, J. Non-Cryst. Solids, 1970,2,432; 1970,4,391.38 H. Fritzsche and S. R. Ovshinsky, J. Non-Cryst. Solids, 1970,2, 393.39 H. Krebs and F. Ackermann, presented at Third In?. Conf. Non-Cryst. Solids (Sheffield), Sept.40 F. K. Dolezalek and W. E. Spear, J. Non-Cryst. Solids, 1970,4,97.41 N. F. Mott, Can. J. Phys., 1956,34,1356.42 Compare E. A. Fagen and H. Fritzsche, J. Non-Cryst. Solids, 1970, 2, 180.Engl. trans. (Consultants Bureau New York), Inorg. Materials, 1967, 3, 1910.Bureau New York), J. Appl. Chem., U.S.S.R., 1967,40,964.Solid State Phys. (Vieweg), 1969, 9, 22.1970

ISSN:0366-9033    

DOI:10.1039/DF9705000035

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Electronic States of Simple Dielectric LiquidsBY STUART A. RICEDept. of Chemistry and The James Franck Institute, The University of Chicago,Chicago, Illinois, 60637Received 4th May, 1970This paper reviews recent studies of both the excess electron and bound electronic excited statesof simple dielectric liquids. It is argued that exciton states in liquid Ar and like substances havelong enough lifetime to be observed, and that they are aptly described using a combination of theFrenkel and Wannier exciton representations. Further, it is suggested that an excess electron inliquid Ar undergoes an extended state -+localized state transition as the density is lowered.The growth of interest in the properties of the electronic states of disorderedmaterials has been rapid during the past decade.From modest beginnings therehave developed promising theories of liquid metals and amorphous semiconductorsY2while the continued accumulation of experimental data partly confirms and partlychallenges our physical insights. Less attention has been devoted to the study ofdielectric liquids, but some key experiments have been performed and theoreticalinterpretations proposed. It is the purpose of this paper to argue that there nowexists sufficient evidence to construct a unified interpretation of both the boundelectron and excess electron states of simple liquids, e.g., Ar. The following sectionsreview the nature of the evidence and its interpretation.A SUMMARY OF IDEAS AND INTERPRETATIONSCONDUCTION ELECTRON STATES IN LIQUID ArWe examine first the nature of conducting states in the l i q ~ i d .~ Because theatoms in liquid Ar do not have a monopole field, it might be expected that theelectron-atom interaction is weak. This expectation is correct, but not at all obvious.Indeed, in liquid He the electron-atom interaction is so strongly repulsive that theminimum free energy for an electron in the liquid corresponds to localization of theelectron in a void which is large relative to the size of an He atom.4 Why does thisnot happen in liquid Ar? What is the effective electron-atom interaction in liquidAr ? In free space, at large electron-atom separation, the interaction is adequatelyapproximated by the classical charge-induced dipole potential energy. At smallelectron-atom separation the classical picture becomes inadequate and the electron-closed shell atom interaction becomes repulsive.To avoid the complications ofmany electron scattering theory, Lekner assumed as a functional form for theisolated electron-atom pair interaction at distance R the relationwhere a is the atomic polarizability and R, a parameter fitted by comparison oftheoretical and observed total scattering cross sections at zero energy. Using (1)446 ELECTRONIC STATES OF SIMPLE DIELECTRIC LIQUIDSand a constant value of R, the total momentum transfer scattering cross-section isaccurately predicted for incident electron energies up to several volts.In the liquid the effective electron-atom interaction differs from that in the gasbecause of interaction between the induced atomic dipoles.To calculate the localfield, Lekner took advantage of the fact that atomic velocities are much smallerthan those of conduction electrons, and conduction electron velocities are, in turn,much smaller than the velocities of bound core electrons. Thus, in the first orderdescription, bound atomic electrons respond instantaneously to the presence of theFIG. 1 .-The shielding function for electron-atom interaction in liquid Ar. The horizontal linemarkedfL represents the asymptotic limit. In this limit the formulation used reproduces the Lorentzformula for polarization. The dash line in the diagram gives the local field function due to a pointcharge using the Percus-Yevick pair correlation function corresponding to liquid Ar density and ahard core diameter of 3.44 A.The solid line in the diagram corresponds to the same function for acharge with the diagram of an Ar atom.conduction electron, but the nuclear motion is frozen. Clearly, the effective field,denoted (e/R2)f(R), wheref(R) is a shielding function, is the sum of the direct fieldand the indirect field arising from induced dipoles. It may be shown that (withthe neglect of fluctuations) f(R) satisfies the linear integral equation(2132sO(R,s,t) = - , ( s ~ + ~ ~ - R ~ ) ( s ~ + R ~ - ~ ~ ) + ( R ~ + 12-s2),where g(s) is the pair correlation function. The effective electron-atom interactionis, then, u:f(R). The results of a numerical solution of the integral equation definingf(R) are displayed in fig. l.5The range of the polarization field in the liquid is such that a conduction electronis in the superposed fields of many atoms, and is never in field free space.Oneconsequence of this feature of the polarization field is that the electron is bound tSTUART A . RICE 47the liquid, i.e., the energy of the excess electron in the liquid is negative relative to itsenergy in free space. Because the electron-liquid interaction is attractive, geometricrearrangements such as occur in liquid He are unfavourable. Incidentally, Kr andXe should behave as does Ar, but Ne may behave as does He.It is now assumed that the drift velocity of the electron in a given applied field islimited by scattering from fluctuations about the average field in the liquid.3a*A Boltzmann equation may be derived to describe the balance between forces arisingfrom the external electric field and forces arising from electron-atom interactions.In this analysis, care must be taken to insure that the dynamical properties of theliquid are propertly accounted for, since it is the liquid that acts as the dissipativesink.Consistency can be achieved by using the sum rules which characterize thedynamic structure function of the liquid, S ( ~ , W ) . ~ It is predicted that the electrondrift velocity is field independent only at low field strength, say, less than 100 V/cm,and saturates at high field strength, say, lo5 V/cm. In a completely disordered gasof the same number density, if such existed, there would be non-linear dependence ofthe drift velocity on field strength at fields as low as 1 V/~rn.~" A comparison oftheoretical and observed drift velocities is displayed in fig.2 ; the agreement is good.1 O2E (V cm-')FIG. 2.-The drift velocity of electrons in liquid Ar. Most of these data come from the measurementsof Spear but also represented are measurements of the Chicago group and of Swan. Curve C isderived from the Cohen-Lekner theory. Curve S is derived from the Shockley theory.We noted above that the calculated energy of the conduction electron correspondsto the liquid having an electron affinity. Using Wigner-Seitz type arguments andthe effective potential already described the conduction electron energy is calculatedto be -0.46$-0.15 eV.5 A study of the work function for injection of electrons inliquid Ar, when compared with the work function for emission into vacuum, givesfor the conduction electron energy -0.33 eV (see fig. 3),' in good agreement withtheory.TRANSITION FROM EXTENDED TO LOCALIZED STATESAs the density of liquid Ar is lowered the electronic mobility drops, as predictedThis behaviour persists for only a small range of density in the highThe data in fig.4 3d show that the drift velocity as a function ofby theory.density region48 ELECTRONIC STATES OF SIMPLE DIELECTRIC LIQUIDSFIG.-I--2.0 2.5 3.0hv (eV)3.-Photocurrent as a function of photon energy for the injection of electrons in to liquid At-.density has a maximum, and then approaches the behaviour characteristic of adilute gas as the density further decreases.Lekner has argued that there exists aliquid density such that the average s-wave scattering vanishes, or at least is verysmall, so that a kind of Ramsauer minimum in the scattering cross-section occurs.Even when the average s-wave scattering length vanishes, the mean square scattering1700-1600-1500-1400 -1300 - -- 1200 -r( IIOO-3 tooo- P 2 900-3IIv)800-700600 -500-3--*70 aim.65~ 6 00 55 - 504 ~ 3 300 -I I I I I I I I I I 1 - 6 '7 -8 -9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1Number Density (10-2A-3)FIG. 4.-Zero field mobility of excess electrons in liquid Ar as a function of density. Experimentsat different temperatures have been included by pIotting them at points appropriate to their densitiesSTUART A.RICE 49length does not. It is estimated that this mechanism, if it occurs, limits the peak driftvelocity in liquid Ar to 4ooo cm2/s V.Detailed examination of the experimental data shows that the Lekner mechanismcannot explain the trends in fig. 4. Whereas the Lekner-Cohen6 theory correctlyaccounts for the magnitude and field dependence of the drift velocity in the highdensity limit, it does not do so at the peak in fig. 4, or on the low density side of thepeak.Recent calculations by Neustadter and Co~persmith,~ and by Eggarter andCohen,l0 suggest that density fluctuations can lead to the formation of resonantscattering states (and bound states). Starting from the dilute gas side this implies arapid drop in the mobility at a critical density as the density is increased.In ourcase, fluctuations are small in the high density limit. We suggest that fluctuationsincrease in importance as the density is decreased and that, just about where theminimum in the average field scattering cross-section would occur, fluctuationsbecome large enough to generate resonant scattering states and hence lead to a sharpdecrease in mobility. It is noteworthy that the observed peak mobility is less thanthe estimated 4000cm2/sV, and that the maximum occurs near the critical pointof Ar where fluctuations in density are very large.WANNIER TYPE EXCITON STATES AND IMPURITY STATES IN SIMPLELIQUIDSWe can use information about conduction electron states to interpret the propertiesof bound excited electronic states as follows.Because the excess electron states inthe dense liquid are nearly free electron like, the wave function of an excess electronis well approximated as a plane wave. Now, studies of positive ion mobility inIiquid Ar (and also Kr, Xe, and He)ll indicate that a positive hole in the liquid isunstable relative to the formation of the diatomic positive ion X,'. All such diatomicions have large binding energy ( - 1-3 eV), and small mobility (- cm2/s V).Even in the absence of trapping we expect the hole to be less mobile than the electron.Thus, we construct a wave packet from plane wave functions to represent the electron,and introduce the coulomb interaction between the hole and the electron.12 It isin the use of plane waves as the basis functions of the wave packet that we employthe observation that the conduction electrons have large mean free path relative totheir de Broglie wavelength.Suppose, now, we neglect scattering processes.Then the stationary states of thebound electron are determined by the Schroedinger equation for the amplitudes ofthe plane waves in the electron wave packet. If the hole-electron potential were asimple coulomb interaction, this equation of motion would be the momentum repre-sentation of the hydrogenic Schroedinger equation. In this limit the manifold ofenergy levels is hydrogenic. An extension of the argument to describe shieldingof the electron-atom interaction leads to the conclusion that the hole-electron inter-action is screened, and the screening function oscillates in a manner similar to thatshown in fig.I. In this case the manifold of energy levels is non-hydrogenic ; forlarge energy (still bound) the levels are very close to hydrogenic, but for the lowestlevel there is an appreciable deviation from hydrogenic behaviour.The first experimental evidence that exciton states might be observable in asimple liquid was provided by the reflection spectrum of liquid Xe.14 Comparisonof the solid and liquid spectra showed remnant broadened levels, not of atomicparentage, between the 2Ps and 2P+ levels. These data were not completelyconvincing50 ELECTRONIC STATES OF SIMPLE DIELECTRIC LIQUIDSThe model Wannier exciton described above is easily recognized to be also amodel of impurity levels in liquid Ar.Raz and Jortner have succeeded in recordingthe spectra of liquids Ar and Kr doped with Xe (see fig. 5) and shown that thereexist new energy levels, between the 2P+ and 2Pp; levels of the impurity. These areassigned to the n = 2 Wannier hydrogenic level. Using the standard Rydbergformula, and the n = 2 identification, the energy of a conduction electron in Ar iscalculated to be - 0.45 % 0.2 eV, in good agreement with the value obtained from theelectron injection experiments and the Wigner-Seitz estimate. These observationsunambiguously demonstrate the existence of collective electronic excitations in aliquid which do not have parentage in atomic excitations.E eV10 93ppm Xe in solid Ar at 78 K1.5 I I I 1 1 11150 1200 1250 I300 1350 1400 1450 15001, AFIG.5.--Spectra of Xe in Ar showing new states without parentage in the free atom.The energy spectrum just described must be modified by scattering. Within theconfines of the model mentioned, the simplest description of scattering is based onthe assumption that the electron-atom scattering is independent of whether or not theelectron is bound or free.12 Thus, in traversing an orbit an electron is scattered bythe atoms of the fluid. If the mean free path at the particular energy is large relativeto the orbital circumference, then the exciton state should be well defined; if themean free path is much shorter than the orbital circumference, the level broadeningshould be so severe that level mixing occurs.The line width, in this simple picture,is (nz)-l where,12[ k I = k = ( 2 ~ z C ) ~ / n h , (4STUART A . RICE 51with z the mean free time, a the scattering length, G the exciton binding energy, andn the principal quantum number of the hydrogenic exciton. Note how the structureof the liquid influences the scattering. In this approximation all scattering arisesfrom disorder in the liquid, is of the Ccparticle” type, and does not include anydynamical effects (e.g., recoil, diffusion, etc.).Eqn (4) leads to the prediction that the exciton levels in pure Xe should bebroadened by about 0.1 eV, a value consistent with the broadened spectrum observed(using an optimistic interpretation of the existence of excitons as displayed by thedata).14 More impressive, Raz and Jortner show that (4) correctly predicts theratio of the line widths in liquid and solid Ar :Finally, all the available evidence suggests that the lowest excitation of the liquidis nearly atomic like.That is, the deviation from the Wannier hydrogenic pictureis so severe as to make that picture useless. It appears better, for the solid, to use aFrenkel description of this level, augmented by charge delocalization and ortho-gonality c0rrections.lFRENKEL-LIKE EXCITON STATES IN A SIMPLE LIQUIDI now turn to the case when the electronic overlap in both ground and excitedstates is very small. This limit also applies when vibrational and rotational excitonsareconsidered.One of the interesting aspects of the collective excitations is theexistence of a non-trivial dispersion relation. We have not, for Wannier excitons,worked out such a dispersion relation. However, for Frenkel excitions, it is possibleto work out the dispersion relation even including electronic excitation-translationalexcitation interaction.To study Frenkel excitons it is convenient to adopt a different approach from thatsketched thus far. Consider first a simple model, in which all the important detailscan be seen unmasked by mathematical problems. Nicolis and Rice have extendedthe Fano model of a solid to the liquid. In this model an exciton is identified witha classical polarization field. The liquid is assumed to consist of spherical atoms,with short-range repulsions of the ordinary type.Each atom has a transition-dipolemoment, and all the atoms interact via the dipole-dipole potential. The transition-dipole moment of these Drude oscillators can be chosen to match experiment. Thismodel of the solid is an accurate representation of excitons l9 ; we expect the sameto be true for the liquid state.The important point is that the dipole-dipole interaction is of long-range relativeto all other atomic interactions. This observation permits the calculation of thespectrum of polarization waves exactly to terms of the order of the ratio of the short-to long-ranged interactions, a ratio of 1/10oO in our case. It is found 2o that thecollective excitations are of transverse and longitudinal character, with dispersionrelations of the form (0 is the molecular diameter)1 wf = ~,+c~{;dR-[-~sin g - I 24n kR---- 24n 3n2cos kR - - sin kR , R (kR) (kR) kR0 1 O3 g--1 24n 24n 4nw,” = c3+c4J d R F [ -0 ( W 2 kR,sin kR + -cos kR + - sin kR .There is a gap in the spectrum at k = 0, and the spectrum is parabolic for smal52 ELECTRONIC STATES OF SIMPLE DIELECTRIC LIQUIDSk (ko< 1).The magnitude of the gap at k = 0 depends on the strength of the dipole-dipole coupling. If the theory remained valid for large k, it would predict a changein the sign of the frequency shift when n/2<ko<z. This is easily understood interms of the change in dipole-dipole coupling when near neighbour dipoles arereversed in orientation, as happens in this range.In this classical model there is no damping mechanism in the ordinary sense, butdisorder in the arrangement of atomic centres does lead to a weak decay of thepolarization waves.Nicolis and Rice used the spectrum just described to derivethe real part of the dielectric function, and then a Kramers-Kronig inversion toobtain the imaginary part of the dielectric function. It is found that the collectivepolarization waves decay into randomly phased dipolar oscillations. When ko issmall, the decay rate is small; when ko is large the decay rate is large. Since ko issmall for optically excited excitons, other dissipative mechanisms are undoubtedlymore important than this one.Fischer and Rice 21 have published a quantum mechanical analysis of Frenkelexcitons in a liquid, including the influence of interaction with collective translationalexcitations of the liquid.The key idea is to define a set of Born-Oppenheimer(B.O.) exciton states, in which the atomic positions are regarded as parameters. TheB.O. exciton states are defined for a static disordered liquid ; they are the quantummechanical analogues of the classical polarization waves described above. Asexpected, the dispersion relation for the B.O. excitons involves the liquid pair correla-tion function in a simple way. Fischer and Rice then use the nuclear kinetic energyoperator, which connects B.O. states, to introduce scattering interactions. Thenuclear kinetic energy operator has derivatives with respect to nuclear positions, andZwanzig 22 has shown how a simple relaxation time ansatz for the translationalmomentum density in a liquid leads to a good description of collective translationalexcitations.The derivatives in the kinetic energy operator are of just the formappearing in the Zwanzig theory, so the coupling between electronic excitons andcollective translational excitations is readily incorporated into the theory. In thistheory the key mathematical asumption appears under the heading ‘‘ weak coupling ”,by which it is meant that certain averages of products of exciton operators and collec-tive translational momentum density operators can be factored. It is then foundthat the line shape corresponding to a transition to an exciton state is the modifiedLorentzian :where IC is the dielectric function, o, the frequency of a collective translationalexcitation with wave vector q, z the relaxation time for the translational excitations,and eT is, basically, the canonical average of the instantaneous energy spectrum.It isshown by Fischer and Rice that cT depends on the dynamic structure function of theliquid, and on the excitation transfer matrix element. Several other models,23and formal theories,24 have been considered, in the study of scattering of excitonsin liquids. The general conclusions drawn from these studies support the resultscited above. There are not, at present, data available to test the theory of Frenkelexcitons in a liquid. The theory should be useful for molecular liquids, e.g., benzene,except for complications arising from instantaneous local deviations from sphericalsymmetrySTUART A .RICE 53DISCUSSIONThe theoretical ideas and experimental data reviewed in this paper clearly indicatethat the spectrum of excitations of a liquid is very rich. Despite inadequacies in thedata available, and the crudity of the theoretical analyses, it is possible to concludethat : (a) disorder in the liquid is, of itself, not sufficient to destroy the possibility forcollective electronic excitations of the liquid ; (6) scattering of excitation states byeither particle or collective motions of the molecules of the liquid leads to broadeningof the lowest exciton levels without excessive mixing of levels; (c) for Wannier-hydrogenic excitons a simple scattering model accounts for the magnitude of theobserved line widths in the liquid, and for the ratio of line widths in the liquid andsolid; (d) the dispersion relation for Frenkel excitons in a liquid depends upon thelocal structure.Calculations suggest that there exist both transverse and longitudinalexcitations, with a gap between them at k = 0, and a k2 dependence for small enough k.not discussed earlier, suggest that : (e) localizedexcitations, such as Raman transitions, have varying line widths which can be ration-alized in terms of a model Hamiltonian taken to be linear in the forces acting on themolecules ; (f) whenever molecule (excimer) formation is energetically favoured, itoccurs.26 Energy transfer is then dominated by mass diffusion, not excitationdiffusion. 27I conclude this paper by returning to the nature of the spectrum of liquid Ar.It has been argued that the lowest excited state is atomic like, the next hydrogenic(n = 2), and that the conduction states are very like free electron states.What of thestates in between? I believe that possibly the n = 3 and n = 4 hydrogenic excitonstates are well defined but that higher states, because of electron-atom scattering, arebroadened by more than the interlevel spacing. If true, this leads to the predictionthat the states between, say, n = 4 or n = 5 and the conduction states are mixed bythe scattering interaction. The mixed states remain bound, but the individuallevels are not resolvable, a case of inhomogeneous line broadening.It is likely,because of the scattering, that some lower energy conduction states are also mixedinto the bound states in this energy region, but such mixing will only be of importancenear to, and below, the ionization limit. In principle, if this description of the elec-tronic spectrum of liquid Ar is valid, photo-ionization of liquid Ar should occur at alower energy (ca 0.1-0.2 eV) than the ionization limit predicted from the lowestobserved (n = 2) exciton levels. Although difficult to carry out, this seems anexperiment worth attempting.Theories of localizedThe work described in this paper was carried out in the period 1966-1970 with anumber of collaborators. Without the contributions of Prof. J. Jortner, Prof.G. Nicolis, Prof.L. Meyer, Prof. M. H. Cohen and Prof. S. Fischer; and of DrJ. Lekner, Dr H. Schnyders, Dr K. Popielawski, Dr W. Greer and Dr J. Jahnke,it would not have been possible to make this report. During the entire period inwhich this research was carried out I have received support from the Directorate ofChemical Sciences, Air Force Office of Scientific Research. I also acknowledge theuse of facilities provided by Advanced Research Projects Agency for materialsresearch at the University of Chicago.see, e.g., N. H. March, Liquid Metals, (Pergamon Press, Oxford, 1968).see, e.g., M. H. Cohen, Proc. Int. Con$ Amorphous Liquid Semiconductors, (Cambridge, 1969) ;J. Noncryst. Solids, in press..Experimental data have been obtained by :(a) H. Schnyders, S.A. Rice and L. Mayer, Phys, Rev., 1966, 150, 127; Phys. Rev. Letters,1965, 15, 18754 ELECTRONIC STATES OF SIMPLE DIELECTRIC LIQUIDS(b) D. W. Swan, Proc. Phys. Soc., 1964,83,659.(c) L. S. Miller, S. Howe and W. E. Spear, Phys. Rev., 1968, 166, 861.(d) J. Jahnke, L. Meyer and S . A. Rice, Phys. Rev., 1971, A3,734.There are many papers on this subject. A recent review is J. Jortner and N. R. Kestner, Proc.Weil Symposium (Cornell, 1969).J. Lekner, Phys. Reu., 1967, 158, 130.J. Lekner and M. H. Cohen, Phys. Rev., 1967,158,305.J. Lekner, B. Halpern, S. A. Rice and R. Gomer, Phys. Rev., 1967, 156, 351.J. Lekner, Proc. Phys. Soc., 1968, 92, 1281.H. E. Neustadter and M. H. Coopersmith, Phys. Retc. Letters, 1969, 23, 585.lo T. P. Eggarter and M. H. Cohen to be published.l1 H. T. Davis, S. A. Rice and L. Meyer, J. Chem. Phys., 1962,37,947.l2 S . A. Rice and J. Jortner, J. Chem. Phys., 1966,44,4470.l3 L. Glass and S. A. Rice, unpublished calculations.l4 D. Beaglehole, Phys. Reu. Letters, 1965, 15, 551.l5 B. Raz and J. Jortner, Proc. Roy. SOC. A, in press..l6 See, for example, S.Webber, S. A. Rice and J. Jortner, J. Chem. Phys., 1964, 41, 2911.l7 G. Nicolis and S. A. Rice, J. Chem. Phys., 1967, 46,4445.'I3 U. Fano, Phys. Reu., 1960,118,451.l9 J. Hopfield, Phys. Rev., 1958, 112, 1555.2o J. Lebowitz, S. Baer and G. Stell, J. Math. Phys., 3965, 6, 1282.21 S. Fischer and S . A. Rice, Phys. Rev., 1968, 176,409. See also W. L. Greer and S. A. Rice,22 R. W. Zwanzig, Phys. Rev., 1967,156, 190.23 S. A. Rice, G. Nicolis and J. Jortner, J. Chem. Phys., 1968,48,2484.24 J. Popielawski and S. A. Rice, J. Chem. Phys., 1967,47,2292.25 W. L. Greer, S. A. Rice and G. C. Morris, J. Chem. Phys., 1970,52, 562226 L. Meyer, J. Jortner, S. A. Rice and E. G. Wilson, J. Chem. Phys., 1965,42,4250.27 L. Meyer, J. Jortner, S. A. Rice and E. G. Wilson, Phys. Rev. Letters, 1964,12,415.Adu. Chem. Phys., 1970, 17,229

ISSN:0366-9033    

DOI:10.1039/DF9705000045

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Atomic Vibrations in Vitreous SilicaBY R. J. BELL? AND P. DEAN*National Physical Laboratory, Teddington, MiddlesexReceived 26th May, 1970 -Frequency spectra and normal modes of vibration have been computed for vitreous silica. Theyhave been calculated from atomic arrangements in physical models based on the random networktheory. The positions of bands in the computed spectra agree well with observed features in theexperimental infra-red and Raman spectra of the glass. Detailed analysis of the normal modesindicates that the bands at 1050, 750 and 400 cm-l are associated with bond-stretching, bending androcking motions, respectively, of the oxygen atoms. Atomic vibrations in the glass are, on thewhole, less extended in space than the plane wave-like modes which prevail in perfect crystals.Thespatial localization tends to be greatest at high frequencies and near band edges. If non-bridgingoxygen atoms are present in the structure, the frequency spectrum exhibits an additional band ofvery intense localization, associated with bond-stretching vibrations of the non-bridging atoms.There is considerable experimental information 1-4 relating to atomic vibrationsin vitreous silica. Simon reviewed the situation up to about 1960, and discussedRaman and infra-red data from numerous sources. Since that time there has beena steady accumulation of further data, using modern experimental techniques, andwe would cite Miler and Hass as examples of recent infra-red and Raman work,and Leadbetter as providing data on the low-frequency spectrum by inelasticneutron scattering.In contrast, theoretical studies have not, until recently, provideda real understanding of atomic vibrational processes in glasses. The nature of theatomic geometry of the amorphous state has represented a real barrier to progress,and most previous workers have been restricted to studying mathematical modelsbased upon the two extreme ideas of small molecular-like units and regularly repeatingarrays of atoms. In this paper we describe a theoretical approach, one based uponusing atomic coordinate positions from realistic physical models of sample glassstructures.METHODThe basic structural unit in vitreous silica is an almost exactIy symmetrical Si04tetrahedron. Adjacent tetrahedra are connected together at a common, or bridging,oxygen atom ; thus, in fig.1, the central 0 represents an oxygen atom bridging thetwo tetrahedral units shown. The relative positions of two such adjacent units mayvary considerably because of two disordering mechanisms : first the Si-0-Siangle can take any value from about 120 to 180" ; secondly, each tetrahedral unitmay take on a range of possible positions formed by a rotation about the appropriateSi-0 line. As a first step towards calculating atomic vibrational properties, anumber of models containing up to 600 atoms were assembled by hand, using poly-styrene spheres and steel rods; these models were constructed so as to comply with7 Division of Numerical and Applied Mathematics.* Division of Quantum Metrology.556 ATOMIC VIBRATIONS IN VITREOUS SILICAthe near-neighbour restrictions above and the concept of a completely connectedrandom network structure.Details of the construction are given in a separate report.*On completing each model the atomic positions were measured and X-ray and neutronradial distribution functions calculated. The agreement with experimental datawas satisfactory. The discussion in this paper is confined to one of the modelsconstructed one containing 101 silicon atoms and 233 oxygens (of which 62 are non-bridging surface oxygen atoms).FIG. 1.-Two adjacent Si04 units, connected together by a common, or " bridging ", oxygen atom.Also shown, for the bridging oxygen atom in the figure, is the orthogonal S, B, R axis system describedin the text.Here the B, or bond-bending, axis is parallel to the bisector of the Si-O-Si angle;S , the bond-stretching axis, is perpendicular to this bisector, but still in the Si-0-Si plane ; thebond-rocking direction R (which is orthogonal to S and B axes) is normal to the Si-0-Si plane.In deriving atomic vibrationa1 properties one needs, in addition to atomic co-ordinates, a force field. We used a harmonic, two-parameter (central and non-central)force field limited to nearest neighbours. The ratio of non-central to central forceconstants was taken to be 3/17, close to the value suggested by Sak~ena.~ For fixednon-central to central force constant ratio, the absolute value of either constantenters the calculation as a scaling factor and can be adjusted to give an optimum fitto experiment; we use the value of 400 N m-' (4x lo5 dynes cm-l) proposed forthe central force constant by Saksena ; this gives reasonable agreement with experi-ment, although a value of 450 N m-' would be rather better.Calculations using two types of boundary condition were carried out.First,we used a condition in which the boundary (non-bridging) oxygen atoms at the edgeof the model were regarded as fixed in space : this we refer to as the fixed end boundarycondition. Secondly, a boundary condition in which the non-bridging oxygen atomswere allowed to vibrate freely was imposed : this we call the free end boundary condi-tion. The change of boundary condition from the fixed end to the free end conditionmade relatively little difference to the computed atomic vibrational properties and theform of the spectrum over most of the frequency range ; only in two relatively narrowregions were significant changes recorded.One of these, the appearance of a bandof frequencies at about 85Ocm-' when the free end condition was imposed, linkswell with some previous experimental work.The computational procedure has been described elsewhere. lo Briefly, theatomic co-ordinates and force field were used to generate a symmetric dynamicalmatrix of band form. (The eigenvalues of the dynamical matrix are the squared* Bell and Dean, National Physical Laboratory Mathematics Report Ma62, 1967R . J . BELL A N D P . DEAN 57frequencies of the system and the corresponding eigenvectors contain atomicvibrational amplitudes as elements.) For the model considered here, the dynamicalmatrix was of order 1002x 1002, with a half band-width of 90.The frequencyspectrum was computed using the negative eigenvalue theorem, O the repeatedapplication of which enables one to construct an eigenvalue distribution in the formof a histogram. Sample eigenvectors were computed by the method of inverseiteration,l accurate eigenvalues first having been computed by a combined bisectionand interpolation technique .VIBRATIONAL FREQUENCY SPECTRAOur computed vibrational frequency spectra are depicted in the top two diagramsof fig. 2. Here diagram (a) refers to a calculation with the fixed-end boundary condi-tion imposed, while (b) corresponds to the free-end condition.For the fixed-endnnc4c43 v00 400 8 0 0 1200 0 400 €300 1 2 0 0w (cm-’)FIG. 2.-Vibrational frequency spectra (a) and (b), assignment diagrams (c) and (d), and localizationpatterns (e) and (f) for vitreous silica. The left-hand diagrams (a), (c) and (e) refer to a calculationwith the fixed end boundary condition imposed, while the right-hand set (b), (d) and (f) correspondto the free end condition. The designations C, R, B etc. on the frequency spectra and assignmentdiagrams refer to particular types of atomic motion described in the text. In the assignment diagrams,vertical distances between adjacent curves give the proportion of energy arising from the varioustypes of atomic motions C, R, B, etc.In the localization diagrams (e) and (f) the quantity NEFF,which is defined in the text, gives an indication of the number of silicon atoms which are involvedin the vibrational motion at a particular frequency.spectrum first, the prominent features appearing at about 400, 750 and 1050cm-lin diagram (a) correspond closely with bands observed near to 500,800 and 1100 cm-lin both infra-red l* 2* and Raman 39 4* ’ measurements. The prominent shoulderbelow 350 cm-l in the calculated spectrum (a) is probably related to peaks observe58 ATOMIC VIBRATIONS IN VITREOUS SILICAin the 200 cm-1 region of the spectrum by several experimental workers.12* l3 Thecold neutron scattering results of Leadbetter * show a peak centred at around300 cm-l, but it is not clear whether this is related to the shoulder below 350 cm-lor the peak at 400 cm-1 in the calculated spectrum.The remaining feature in thefixed-end spectrum is a weak shoulder at 550cm-' (which emerges as a separatesmall peak in the free-end spectrum). This shoulder most probably corresponds tothe polarized Raman band observed near 600cm-l by Bobovich and T ~ l u b , ~Harrand and Has.' A minor feature at this frequency also occurs in the experi-mental infra-red spectra of Lippincott et aL2 and Miler.6In the free-end spectra of diagram (b) two features appear which were not evidentin the fixed end spectrum (a), viz., a small enhancement of the low-frequency shoulderat about 300 cm-l, and new band at 850 cm-l : both features are directly related tovibrations of the non-bridging oxygen atoms and the second, at 850 cm-l, has beenobserved experimentally l4 in the infra-red spectrum of neutron damaged vitreoussilica.(It has been proposed 1 4 s l5 that neutron irradiation causes intense thermalspikes which disrupt the silica network, break Si-0 bonds and thus produce non-bridging oxygen atoms in the structure.) For the spectral enhancement at 300 cm-lwhen the free-end boundary condition is imposed, no experimental counterpart hasyet been detected-probably because the feature lies in a region of the spectrum alreadydominated by strong spectral bands.The value adopted for the central force constant was 400 N m-', a value whichgives reasonably good agreement between our calculated spectra and the experimentalmeasurements.However, a choice of 450 N m-l (still with a value of 3/17 for theratio of non-central to central force constant) increases the calculated frequenciesby approximately 6 % and improves the agreement with experiment considerably.NORMAL MODE ASSIGNMENTSA detailed examination of the displacement eigenvectors at selected frequenciesthroughout the various bands reveals that the normal modes do not, in general, lendthemselves to the simple and convenient wave-number characterization availablefor normal modes in crystals; not do they fit in any obvious way into the symmetryclassifications of vibrations in simple molecules. Most of the modes in vitreoussilica are extremely complicated in character and can only usefully be described by astatistical parametrization.The description adopted here is based upon first separating the energy of eachmode into contributions from atomic motions of the bond-stretching (S), bond-bending (B) and bond-rocking (R) types.To do this, we consider separately eachoxygen atom of the structure and set up an orthogonal coordinate system which isbased on the local atomic geometry. Such a system is illustrated for a bridgingoxygen atom in fig. 1, where we defme the B (or bond-bending) direction as parallelto the bisector of the Si-0-Si angle; the S (bond-stretching) direction is perpen-dicuiar to the bisector, but still in the Si-0-Si plane, and bond-rocking R-axis(which is orthogonal to S and B axes) is normal to the Si--O-Si plane.The totalenergy of oxygen atoms in a normal mode can be divided into contributions fromatomic motions in the S, B, and R directions, by analyzing the appropriate displace-ment eigenvector. For the fixed end boundary condition, the remaining energy ofeach normal mode is that of the silicon atom, and we do not attempt to separate thisinto components. (This is because each silicon must move in response to the pullof four separate oxygen atoms and its pattern of motion is extremely difficult tocategorize in simple terms.R. J . BELL AND P. DEAN 59By such means, we have analyzed atomic motions for modes at roughly equi-distant frequencies throughout the spectrum. Results are presented in fig. 2(c) forthe fixed boundary condition. The vertical distance between adjacent curves repre-sents the proportion of energy arising from each type of motion, such as S, B, R or C(where C refers to the cation, or silicon, motion).It is clear from diagram (c) that the energy of oxygen atoms in modes at around1050 cm-l arises almost entirely from stretching motion ; we therefore label the bandat this frequency as an S (or stretching) band.Similarly, the peak at 400cm-'can be designated as an R band. Bond bending motions play an important part inthe modes in the 550 and 750 cm-1 regions. In the former case a proportion ofstretching motion is also present, so we designate the shoulder at 550cm-l by thelabel B + S ; in the latter case, the oxygen motion is accompanied by an appreciableamount of silicon motion, and we designate the peak at 750 cm-l as a B + C band.Some silicon motion is also present in all the other spectral regions, and particularlyin the low frequency shoulder, which we designate by C.In this region C, S, B andR motions contribute to the energy in approximately the proportions one wouldexpect for an overall translational movement of the network ; one can thus regard thelow-frequency shoulder as representing an acoustic region of the vitreous silicaspectrum.For the free-end boundary condition we consider, separately, the energy arisingfrom the boundary (or non-bridging) oxygen atoms. The contribution from theseatoms is separated into two categories-one from motion parallel to the single Si-0bond @e., the stretching component) which we denote by NS, and one from motiontransverse to the bond, which we denote collectively by NB (here the distinctionbetween bending and rocking disappears).Contributions to the energy are shownfor the free-end case in diagram (4 of fig. 2. For those spectral features alreadyapparent in the fixed end spectrum, the energy decomposition remains much asdescribed above, and the designations of diagram (c) can be retained for these bandsin diagram (d) also. For the two additional spectral features noted previously (theshoulder enhancement at 3OOcm-l and a band at 850cm-l) the importance of thenon-bridging oxygen contribution to the energy is evident. Thus, bending motionof the non-bridging atoms plays a large part in the motion at 300 cm-l, while modesin the 850cm-l band show a preponderance of non-bridging oxygen stretchingmotion.Accordingly, we designate these two features by NB and NS, respectively.This assignment scheme, although extremely useful for descriptive purposes,presents a somewhat simplified picture of the modes. The overall pattern of atomicmotion throughout the network is most complicated for the majority of modes, andour assignment procedure just selects out particular dominating aspects of the motionin each case.LOCALIZATION OF THE MODESOne property of the atomic vibrations which is of particular physical significanceis their degree of spatial localization. It is the spatial extent of modes which largelydetermines, e.g., the thermal conductivity of a glass (just as the spatial extension ofelectron states influences the electrical conductivity of a material).Therefore, weexamine normal modes in the various spectral bands of vitreous silica from the pointof view of their localization properties. This is done, as for the vibrational assign-ments, by analyzing representative displacement eigenvectors, chosen at roughlyequal frequency intervals throughout the spectrum. As a measure of localization,we introduce a paramter NEFF(u) which gives some indication of the number ofatoms effectively participating in a normal mode with frequency co60 ATOMIC VIBRATIONS IN VITREOUS SILICAIn order to obtain NEFF, we first define the energy moments-%(a) = [&i(m)lp,atoms iwhere c i ( a ) , the mean kinetic energy of atom i in a normal mode with frequency co,can be ascertained directly from the appropriate displacement eigenvector. Thus,e.g., Mo is identical to N, the total number of atoms in the system, while MI is justthe total mean kinetic energy of the system vibrating in a mode with frequency m.We then define NEFF, the quantity of interest byIt is easy to verify that ATEFF, which always lies between 1 and N, gives some indicationof the number of atoms participating in a normal mode of vibration.With a mon-atomic system as a simple example, one can show that for a mode in which all N atomsvibrate with equal amplitude (as in a purely translational motion of the system),NEFF = N ; if precisely P atoms are in motion with roughly equal amplitudes,N E F F N P , while for a mode involving only one atom NEFF = 1.For the typicalwave-like modes characteristic of perfect crystals, NEFF N N/2.For a system containing several types of atoms, NEFF tends to be mass dependent,even in the limiting case of a purely translational mode. This dependence can belessened, to some extent, by restricting the moment summations in (1) to one speciesof atom. In the present analysis we have restricted the summations to silicon atomsonly, of which the model contains 101 (so that AT,,, can range between 1 and 101);however, the alternative choice of the oxygen atoms would not essentially alter thegeneral result.The localization of the normal modes is shown in diagrams (e) and (f) of fig. 2,as plots of NEFF against frequency; here (e) and (f) refer to calculations with thefixed-end and free-end boundary condition respectively.The broken horizontallines represent average values of NEFF for individual spectral bands, where such bandscan conveniently be resolved.For the fixed-end case, then, from (e) the spatial extent of the modes is greatest(i.e., the localization is least) for modes in the low frequency (C and R) bands.However, even in this region, NEFF tends to be rather less for most modes than thevalue -VEFF= 50 to be expected for vibrations in crystals. Localization is greatestin the medium and higher frequency regions (i.e., for the B+S, B+C and S bands),being particularly marked near band edges where, in some cases, effective motionseems to be restricted to no more than about half a dozen molecules.In the localization pattern for the free-end case, diagram (f) shows that for mostbands there is little overall change in the spatial extent of modes as a result ofchanging the boundary condition.The one exception is the NS band, associatedwith bond-stretching vibrations of the non-bridging oxygen atoms, which appearsonly in the free-end spectrum (b). Here the modes show intense localization, theeffective motion being restricted in some instances to no more than one or two mole-cules. A full discussion of the nature of these modes is given elsewhere.'N. F. Bird and D. Hibbins-Butler wrote the computer programmes used in thiswork. The work described in this paper has been carried out at the National PhysicalLaboratory.V. A. Florinskaya and R. S. Pechenkina, Doklaa'y Akad. Nauk S.S.S. R., 1952, 85, 1265.E. R. Lippincott, A. V. Valkenberg, C. E. Weir and E. N. Bunting, J. Res. Nu?. Bur. Stand.,1958, 61, 61.Ya. S. Bobovich and T. P. Tulub, Uspekhi Fiz. Nauk S.S.S.R., 1958, 66, 3.M. Harrand, Compt. rend., 1954, 238, 784R. J . BELL AND P. DEAN 61I. Simon, Modern Aspects ofthe Vitreous State, ed. J. D. Mackenzie (Buttenvorths, London,1960), vol. 1, chap 6, p. 120.M. Miler, Czech. J. Phys. B, 1968, 18, 354. ' M. Hass, J. Phys. Chem. Solids, 1970, 31,415.* A. J. Leadbetter, J . Chern. Phys., 1969, 51, 779.B. D. Saksena, Trans. Faraday SOC., 1963, 59, 276.J. H. Wilkinson, The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, 1965).lo P. Dean and M. D. Bacon, Proc. Roy. SOC. A, 1965,283,64.12R. Hanna, J. Amer. Ceram. Soc., 1965, 48, 595.l3 R. S. Krishnan, Proc. Ind. Acad. Sci., 1953, 37, 377.141. Simon, J. Amer. Ceram. SOC., 1957, 40, 150.l5 R. J. Bell and P. Dean, Proc. Int. Con$ Localized Excitations in Solids (California, 1967), ed.R. F. Wallis (Plenum Press, New York, 1968), p. 124

ISSN:0366-9033    

DOI:10.1039/DF9705000055

出版商:RSC    

年代:1970

数据来源: RSC