5228
J. Phys. Chem. 1995, 99, 5228-5239
FEATURE ARTICLE The Metal-Nonmetal Transition: A Global Perspective P. P. Edwards The School of Chemistry, University of Birmingham, Birmingham, B I5 2TT, U.K.
T. V. Ramakrishnan Department of Physics, Indian Institute of Science, Bangalore 560 012, India
C. N. R. Rao* CSIR Centre of Excellence in Chemistry, Indian Institute of Science, Bangalore 560 012, India, and Department of Chemistry, University of Wales, Cardiff CFI 3TB, U.K. Received: October 24, 1994; In Final Form: January 26, 1995@
A wide range of condensed matter systems traverse the metal-nonmetal transition. These include doped semiconductors, metal-ammonia solutions, metal clusters, metal alloys, transition metal oxides, and superconducting cuprates. Certain simple criteria, such as those due to Herzfeld and Mott, have been highly successful in explaining the metallicity of materials. In this article, we demonstrate the amazing effectiveness of these criteria and examine them in the light of recent experimental findings. We then discuss the limitations in our understanding of the phenomenon of the metal-nonmetal transition.
1. Introduction Of all the physical properties of condensed materials, the electrical conductivity exhibits the widest range, anywhere from ohm-' cm-' in the best nonmetals to around 1O'O ohm-' cm-' in pure metals (not in the superconducting state). There are several situations where condensed phases transform from the metallic to the nonmetallic state on changing thermodynamic parameters such as temperature, pressure, and composition, with the electrical conductivity changing by factors of 103-1014 over a small range of the thermodynamic In Figure 1, we show the temperature-composition plane with the electron density varying between lo'* and 1030, with the temperature going up to 1O'O K. The normal experimental conditions where we find metals and semiconductors are indicated in the figure, as are also the conditions appropriate to stars. The elements H, Xe, Cs, and Hg shown by asterisks are at conditions close to the critical points attained by one of many routes such as shock waves, wire explosions, or MHD (magnetohydrodynamic implosion). The figure also shows the regime for a degenerate strongly coupled plasma, besides the values of the WignerSeitz radius, rs, given by
where n is the conduction electron density and a0 is the Bohr radius. The largest value of rs in metals5 is found for Cs. In all the systems in Figure 1, the possibility of a thermodynamically induced transition from the metallic to the nonmetallic regime exists. Clearly, the metal-nonmetal transition is a
* To whom correspondence should be addressed at CSIR Centre of Excellence in Chemistry, Indian Institute of Science, Bangalore 560 012, India. @Abstractpublished in Advance ACS Absrructs, March 15, 1995.
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Figure 1. A nonrelativistic window of the temperature-composition plane, showing electron density ( n ) and temperature (ZJ. Normal conditions (on earth) for semiconductors and elemental metals and conditions on the Sun, Jupiter, and the White Dwarf are shown. Experimental methods in A, B, C, and D are Tokamak, glow-discharge, laser fusion, and degenerate strongly coupled plasma, respectively. Wigner-Seitz radii, r,, are also shown (adapted from Redmefl).
problem of vital interest, being concerned with a wide range of issues from the metallization of stars to the size-induced transition in small clusters of metal^.^^^,' Between these two extrema is a myriad of examples in a variety of condensed matter The range of systems traversing the metal-nonmetal
0022-365419512099-5228$09.00/0 0 1995 American Chemical Society
Feature Article transition is continually increasing. Thus, oxides exhibiting high-temperaturesuperconductivityare close to or at the metalnonmetal and some of them actually exhibit metal to nonmetal transitions with a change in composition. In spite of the plethora of experimental findings, the status of our theoretical understanding of metal-nonmetal transitions is far from satisfactory. The difficulty is largely intrinsic to the phenomenon. Electronic states involved in charge transport(i.e., those near the Fermi energy) are spatially extended in the metal and are localized in the insulator. The localization may be due to static disorder (Anderson localization), to strong local electron-electron correlations which “freeze” the local electron number (Mott transition), or to strong electron-lattice coupling which traps the electron locally.1~2Jo~11 In all these cases, the natural modes of description of the electronic states in the different phases are diametrically opposite; it is difficult to find an approach which does both. Secondly, in many cases, more than one mechanism is operative, and one may reinforce the other. For example, in a disordered, strongly correlated oxide, Anderson localization due to disorder tends to increase the local correlation effect. In all electronic systems, the Coulomb interaction which is relatively weak and short ranged in a metal but strong and long ranged in an insulator (giving rise to bound electron-hole states) is present and can be important in promoting the insulating state. There is one mechanism for the metal-nonmetal (M-NM) transition, in a crystalline solid, that does not involve localized states. This is the transition of electrons from a fully filled band (insulator) to a partially filled band (metal) under pressure or structural change. This transition, however, appears to be uncommon. We discuss these mechanisms in some detail in section 6 (see also refs 1, 2, 10, and 11). There are certain simple criteria for the occurrence of the metal-nonmetal transition, based on powerful physical concepts which tum out to be surprisingly successful. One such criterion is the idea due to Mott that in a low carrier density metal, the screened Coulomb attraction may be strong enough to bind an electron hole pair, thus destabilizing the metal.2J2 Another useful criterion due to Mott13 is the idea of a minimum conductivity, om, that a metal can support, corresponding to the mean free path being equal to the de Broglie wavelength of the electron at the Fermi energy. Then, there is the Herzfeld metallizationcriterion of the dielectric catastrophe14which could occur for a dense collection of polarizable atoms. We discuss these criteria further in sections 3-5. We shall first present a brief overview of phenomena and systems associated with the M-NM transition.
2. Diverse Systems Exhibiting Metal-Nonmetal Transitions As mentioned earlier, a large variety of systems exhibit M-NM transitions. The systems metal-ammonia (or amine) solutions, expanded metals, doped semiconductors, metal-noble gas films, metal-metal halide melts, alloys of gold with metals such as cesium, transition metal oxides and sulfides, and other inorganic and organic solids. These systems have been adequately reviewed, and we shall briefly examine only those findings that are new or directly relevant to the later discussion. Transition metal oxides15 are especially noteworthy in that the M-NM transition in them can arise from one of many causes. Typical of the transitions found in oxide systems are the following: (i) pressure-induced transitions as in NiO, (ii) transitions as in Fe304 involving charge ordering, (iii) transitions as in Lac003 that are initially induced because of the different spin configurations of the transition metal ion, (iv)
J. Phys. Chem., Vol. 99, No. 1.5, 1995 5229
-
Cr
Ti
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0
0.04
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:
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Figure 2. Phase diagram of the metal insulator transition in V203 as a function of doping with Cr or Ti (regarded as chemical pressure) as well as actual pressure, showing a critical point (after McWhan, D. B.; et al. Phys. Rev. Lett. lW1,27,941). The three phases are paramagnetic insulator (PI), metal, and antiferromagnetic insulator (AFT). The different kinds of resistivity behavior in Figure 3 correspond to different constant pressure (or Cr, Ti content) cuts in this phase diagram.
transitions as in EuO arising from the disappearance of spin polarization band-splitting effects when the ferromagnetic Curie temperature is reached, (v) compositionally induced transitions, as in Lal-,Sr,CoO3 and LaNil-,Mn,O3, in which changes of band structure in the vicinity of the Fermi level are brought about by a change in composition or are due to disorder-induced localization, (vi) transitions as in &.3Mo03 due to chargedensity waves, and (vii) temperature-induced transitions in a large class of oxides such as TizO3, V02, and V2O3 involving more than one mechanism. The last category of M-NM transitions has attracted considerable attention. In Ti2O3, a second-order transition occurs around 410 K, accompanied by a gradual change in the rhombohedral cla ratio and a 100-fold jump in conductivity; the oxide remains paramagnetic throughout. A simple bandcrossing mechanism occurring with the change in the cla ratio can explain this transition. Accordingly, substitution of Ti by V up to 10% in Ti203 makes the system metallic; the cla ratio of this metallic solid solution and the high-temperature phase of Ti203 are similar. In VO2, a first-order transition occurs around 340 K, accompanied by a change in structure (monoclinic to tetragonal) and a 104-fold jump in conductivity; the material remains paramagnetic throughout. A crystal distortion model wherein a gap opens up in the low-temperature lowsymmetry structure adequately explains the transition. Substitution of trivalent ions such as Cr3+ and A13+ for vanadium in V02 leads to a complex phase diagram with at least two insulating phases whose properties are significantly different from those of the insulating phase of pure V02. These phases are now fairly well understood. The M-NM transition in V2O3 and its alloys has been the subject of a large number of publication^.^.^.^^-^^ Pure V203 undergoes a first-order transition (monoclinic-rhombohedral) at 150 K accompanied by a 107-foldjump in conductivity and an intiferromagnetical-paramagnetic transition. The carrier effective mass and other properties also show large changes at this transition. Application of pressure makes V203 increasingly metallic, thus suggesting that it is near a critical point. Accordingly, doping with Ti or Cr has a marked effect on the transition; the former has a positive pressure effect and the latter a negative pressure effect (Figure 2). V2O3 also shows a second-
5230 J. Phys. Chem., Vol. 99, No. 15, 1995
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Edwards et al.
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P M‘
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PM 1/T 1/T Figure 3. Schematic diagram depicting the changes of resistivity, e, with temperature, T, in the v203 system: (a) V203 doped with 1.5% Cr203, (b) V203 doped with 3% Cr203,(c) pure V203, (d) V203 doped with 1% Ti203 and nonstoichiometricV203, and (e) 5.5% Ti203 doped V203. AH,
antiferromagnetic insulator; PM (PM’), paramagnetic metal; PI, paramagnetic insulator (after Honig and Spalek’6). order transition around 400 K with a small conductivity anomaly. Mere crystal distortion or magnetic ordering cannot explain the large connectivity jump at 150 K. The current status of the V2O3 transition can be represented in terms of Figure 3. This figure also serves to indicate the complexity of the metalnonmetal transition in a relatively simple oxide system. There are many recent findings on VzO3 which are interesting. Thus, Carter et al.” have carefully measured the electrical and magnetic properties of single crystals of pure and doped Vz03 near the M-NM transition. Bao et al.I8 have examined the phase diagram of Vz-,O3 in the x, P , T space and identified an incommensurate spin density wave (SDW) in metallic VzO3 close to the transition. The optical conductivity of V2O3, a ( w ) , has been investigated by Thomas et In spite of extensive experimental and theoretical effort, a complete understanding of the transition in the V2O3 system is yet to emerge. We shall examine some of the factors responsible for this situation in section 6. Compositionally controlled M-NM transitions in oxides are worthy of special mention. We shall examine two types of compositionally controlled transitionsz0as typified by La1-,A,MO3 (A = Ca or Sr and M = V, Mn, or Co) and LaNil-,M,O3 (M = Mn or Fe). In Lal-,A,MO3, progressive substitution of trivalent La by divalent A brings about itinerant behavior of the d electrons, because every A ion creates an M4+ ion and promotes electron hopping from M3+ to M4+ ions (impurity-
band formation). This is to be contrasted with LaNil-,M,03 where LaNi03 (x = 0), which is a correlated metal, transforms to an insulator on progressive substitution of Ni by M (somewhat like a deep-impurity situation). In Figure 4 we show typical electrical resistivity data in the two types of transitions. Unlike the above two systems, A0 3Mo03 (M = K or Rb) shows a metal-nonmetal transition associated with charge-density waves.*l Many of the high-temperature superconducting cuprates show compositionally controlled M-NM transitions.8 Thus, in TlCal-,Ln,Sr2Cu2Oy (Ln = Y or rare earth), the superconducting T, shows a maximum at an optimal value of x (corresponding to the optimal value of the hole concentration). The system also shows a M-NM transition in the normal state as x is varied (Figure 5). Thus, the cuprate is metallic when x = 0.25 and insulating when x = 1.0. Rather interesting behavior occurs at x = 0.75 when the superconducting transition occurs from a seemingly semiconducting state. A compositionally controlled M-NM transition is also exhibited by Bi*Cal-,Ln,Sr2Cu20*+6 (Figure 5) which has a maximum T, at an optimal x value of 0.25. La2-,SrXCu04 shows a similar metal-nonmetal transition (x = 0 is an insulator and x = 0.3 is a metal) with change in x and the maximum T, is at x = 0.20. Laz-,Sr2CuO4 and a few other systems traverse the insulator- superconductor-metallic regimes with change in composition (increase in x from 0.0 to 0.3 in La2-,SrxCuOr). This suggests that the high-temperature
Feature Article
J. Phys. Chem., Vol. 99, No. 15, 1995 5231
-
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100 200 300 Figure 4. Compositionally controlled metal-nonmetal transitions in (a) Lal-,Sr,CoO3 and (b) L ~ N ~ I - , M ~ , O(from S Rao and GangulyZ0).
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Figure 5. Compositionally controlled metal-nonmetal transition in superconducting cuprates: (a) Bi2Sr2Cal-,NdXCuzO~and (b) TICal-,Nd&CuOa (after Rao8).
superconducting cuprates are at the boundary between metals and insulators. One of the fundamental questions that has intrigued scientists is, “how many atoms maketh a metal?”. The question is concerned with the possibility of a transition from the metallic to the nonmetallic state as the bulk metal is divided into finer particles. Recent studies of metal c l ~ s t e r s ~ *have * * ~ attempted to answer the question. Careful investigations of gold clusters have shown that the binding energy of the core levels increases markedly (relative to the bulk metal value) when the cluster size decreases below 1 nm. That the effect is not merely due to final state effects but due to the occurrence of metal to nonmetal transitions as the cluster size decreases is reinforced by valence band and Bremstraahlung isochromat spectroscopic studies. Furthermore, tunneling conductance measurements show the existence of a gap in clusters smaller than 1 nm containing around 50 atoms or less; in Hg atom clusters, the 6s-6p atomic transition gives way to a collective metallike plasmon absorption for a cluster size between 7 and 20
atoms.23 The size-dependent transition from the metallic to the nonmetallic state does not occur abruptly. The possibility of a matrix-bound insulator-metal transition in alkali metal-doped zeolites at a critical stage of loading has been suggested by Edwards et al.24 The dependence of the energetics and charge distribution of electron states on the cluster size and the dielectric constant has been examined by Rosenblit and J ~ r t n e r to ,~ shed light on the cluster size-induced metal-nonmetal transition.
3. The Herzfeld Criterion The earlier theoretical prediction of a M-NM transition is that derived from the work of GoldhammerZ5and Her~fe1d.I~ These authors considered the effect of increasing density on the atomic polarizability and suggested that there would be a divergence in the polarizability or the dielectric constant causing the release of bound electrons. The Herzfeld criterion for dielectric catastrophe is given by,
Edwards et al.
5232 J. Phys. Chem., Vol. 99, No. 15, 1995
where @ is the low-density static polarizability of the atom, N Avogadro’s number, V the molar volume, and R the molar polarizability. As a result of cooperative polarization effects, valence electrons get delocalized from the lattice sites at very high (metallic) densities26 and the Drude free electron model becomes applicable. In the metallic regime, (RIV) > 1. In Figure 6 we show how this criterion excellently delineates metals from nonmetals in the periodic table.27 Rao and Ganguly**have pointed out that the latent heat of vaporization, AHv, of elements with metallic cohesion is larger than that of elements which are insulating because of relatively weak bonding (Figure 7). There are, however, exceptions such as strongly covalently bonded solids e.g., carbon. A recent development is the realization of a link between the Herzfeld metallization view and the stress-induced transformations in solids, notably in semiconductor^.^^ Under a diamond pressure indentor, ordinary (semiconducting) silicon transforms to the much denser p-tin (metallic) structure, the critical pressure being in the range 11-12 GPa. This is consistent with the experiments which reveal a large, reversible drop in the electrical resistivity in a thin layer surrounding a microindentation in Si, as would be anticipated because of the metallic characteristics of the p-tin structure. Good correlations are found between the experimental metallization pressure and the values calculated from the Herzfeld polarization catastrophe criterion. Experimental transition pressures also correlate with Vickers hardness numbers and activation energies for dislocation motion. We show in Figure 8, a comparison of the calculated critical pressure (Herzfeld model) with the measured Vickers hardness (expressed in kilobars) for the group IV elements and Sic. Equally good correlations are obtained for all tetrahedrally bounded semiconductors and alkali and alkaline earth oxides. Fujii et al.30 have reported experimental evidence for the molecular dissociation process in Brz near 80 GPa. This transition, which is coincident with the onset of pressure-induced metallization, was first discovered in molecular/metallic iodine.31 A diatomic molecular crystal loses its molecular character in the limit when the intermolecular distance becomes equal to the intramolecular bond length. Fujii et aL30 applied the Herzfeld criterion to 12 and Br2 and estimated that the molar refractivity reaches the atomic limit around 20 GPa in I2 and 80 GPa in Br2. In both cases, the computed pressure coincides with that for molecular dissociation accompanied by metallization.
4. The Mott Criterion The Herzfeld criterion considers the M-NM transition as viewed from the nonmetallic side. Over 30 years ago, Mott12 proposed a simple model of the M-NM transition which considers how electron localization occurs as the transition is approached from the metallic side. In Figure 9, we show a schematic representation of a lattice of one-electron hydrogenic centers (P donors in Si) in the two limiting electronic regimes of high and low donor densities. At a sufficiently high density (small interparticle distance), the system would be a in this state, the system has a finite electrical conductivity at the absolute zero of temperature, Le., u(T = 0) 0. At large interparticle distances, the system must surely become nonmetallic, or insulating, having a conductivity of 0 at T = 0 K, viz., u(T = 0) 0. Mott argued that at a critical interdonor distance (dJ a first-order (discontinuous) transition from metal to nonmetal would occur (Mott transition). A discontinuous
*
-
transition at the absolute zero of temperature will always remain a tantalizing theoretical prediction. Experimentally, however, even very close to T = 0 K (down to 0.03 K), the experimental situation is e q u i ~ o c a l . ~The ~ - ~physics ~ of the problem is that, on the metallic side of the transition, the effective valence electron-cation potential in an atom is completely screened by the conduction electron gas and bound states are therefore nonexistent. However, if the conduction electron density, or equivalently the average separation between donor centers is changed, there comes a point at which bound levels (Le., nonmetallic states) appear at some critical concentration of centers such that the following condition is satisfied. (3) Here, UH* is the Bohr orbit radius of the isolated center and aC is the critical carrier density at the M-NM transition. Another way of viewing the transition is that of an electronic instability which ensues when the trapping of an electron into a localized level also removes one electron from the Fermi gas of electrons. This must clearly lead to a further reduction in the screening properties (which are themselves directly related to the conduction electron density) and a catastrophic situation then ensures the localization of electrons from the previously metallic electron gas. There appears to be little doubt that the Mott criterion given by eq 3 is an effective indicator of the critical condition at the M-NM transition itself. At the least, this simple criterion provides a numerical prediction for the metal-nonmetal transition in many situations. Figure 10 summarizes some of the experimental data.34.36 Interestingly, besides doped semiconductors, metal-ammonia and metal-noble gas systems and superconducting cuprates all follow the linear relation given by eq 3. This is truly remarkable.
5. Minimum Metallic Conductivity at the Metal-Nonmetal Transition Mott13 has argued that the M-NM transition in a perfect crystalline material at T = 0 K is discontinuous (Figure 8) and proposed that, at the transition, there exists a minimum conductivity, umin,for which the material could still be viewed as metallic, prior to the localization of electrons.* Mott’s ideas were based on arguments developed earlier by Ioffe and Rege137 for the breakdown of the theory of electronic conduction in semiconductors. The conventional Boltzmann transport theory becomes meaningless when the mean-free path, I, of the itinerant conduction electrons becomes comparable to, or less than, the interatomic spacing, d. The Ioffe-Regel mean free path, 1 1 ~ , at the minimum metallic conductivity is equal to d. Abrahams et al.38have, however, predicted a continuous M-NM transition on the basis of a scaling theory of noninteracting electrons in a disordered system,39and their results question the existence of udn in both two and three dimension^.^^^^^,^ The two possible scenarios of the transition are compared in Figure 11. 5.1. The Situation in Doped Semiconductors. There is an increasing belief amongst workers in the field that the M-NM transition is continuous, based on experimental measurements carried out at low temperatures down to 3 mK. In Figure 12, we show the experimental evidence in P-doped Si. Note that at a fixed (very low) temperature, the conductivity changes continuously with, for example, donor concentration. In addition, the extrapolated zero-temperature value of the conductivity ( ~ ( 0 varies )) continuously with impurity concentration. An example showing the variation of the extrapolated “zero-temperature” c o n d ~ c t i v i t yin~ the ~ case of B-doped Si is
J. Phys. Chem., Vol. 99, No. 15, 1995 5233
Feature Article
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I A I I A llIB I V B VB VI6 VIlB VI11 VI11 I B 116 JlIA IVA VA V I A V I I A 0 Figure 6. The occurrence of metallic vs nonmetallic character in the periodic table of the elements: the variation of R/V for naturally occumng elements of the s, p, and d blocks (adapted from Edwards and SienkoZ7). shown in Figure 13. This could be taken as strong experimental evidence for a continuous M-NM transition in doped semi~ ~ , ~ ~ questions the conductors at T = 0 K. M o b i ~ s , however, reliability of such 0 K extrapolations and suggests that these findings do not disprove the existence of finite a- at the transition. He argues that the data can be explained by a combination of a- on the metallic side and a Coulomb exp-{(Td7J1/2} on the interaction dominated a(7J ,,a insulating side, with TO 0 as the disorder decreases to the critical value. As noticed by him,35,42not all the measurements are consistent with this suggestion; furthermore, the clear observation of characteristic weak localization or precursor effects in transport and magnetotransport weakens the argument for a a,,. However, this analysis points out the fact that while the continuous conductivity transition prediction is for noninteracting disordered electrons, just on the insulating side the Coulomb interaction is by definition of long range and hence qualitatively important. It is possible that 3 mK may not be a low enough temperature since ~ B may T exceed the activation energy for conduction even at this temperature. While the presence of a high degree of disorder may wipe out the discontinuous nature of the M-NM t r a n s i t i ~ n ? ~ . ~ inclusion of strong correlation in scaling models could conversely change a continuous transition to a discontinuous one. In spite of such difficulties, however, a- continues to be a useful experimental c r i t e r i ~ n ~at~ least ~ * ~ at ~ ~the ~ “hightemperature” limit. Earlier results providing experimental evidence for a- have been reviewed by several authors. In Figure 14, we show some of the results. We note that 0scales with n,. As pointed out by Fritzsche,“6a- appears to satisfactorily represent the value of conductivity where the
-
-
activation energy for conduction disappears. This aspect is specially bome out by investigations of transition metal oxide systems. 5.2. The Situation in Transition Metal Oxides. In many of the oxide systems, especially those exhibiting compositionally controlled M-NM transitions (Figures 4 and 5), the temperature coefficient of the conductivity changes sign around a,,,,,, (-lo3 ohm-’ cm-’). Most of these oxides, including the superconducting cuprates, follow the relation shown in Figure 14. The points corresponding to these oxides fall somewhere between those of fluid alkali metals and Lal-,Sr,V03. The critical carrier concentrations in these materials from Figure 10 also give,,a values close to the observed values. Accordingly, a, is often taken to represent the separation of localized and itinerant electron regimes. Recent work of Raychaudhuri and c o - w o r k e r ~suggests ~~,~~ the need to reevaluate the status of transition metal oxides with regard to am. It appears that in these disordered oxide systems, genuine metallic states (with a(T = 0) f 0) exist even when the conductivity is activated (a -= amn).This implies that the earlier values of the critical electron density at the transition in such oxides may be overestimates. In Figure 15, we show the behavior of Na,W1-,TaYO3 where the temperature coefficient of the conductivity changes sign when (x - y ) zz 0.20, but the transition actually occurs at (x - y ) = 0.19. The curves for x = 0.35 and 0.34 compositions both show “activated conductivity” at T > 10 K (i.e., a negative temperature coefficient of the resistivity), but the a(r) saturates at a fairly high residual value in the case of the x = 0.35 sample as expected of a metal (a(T = 0) t 0). The x = 0.34 sample, however, shows a very much lower a value at low T, tending to infinity, and fits an activated
Edwards et al.
5234 J. Phys. Chem., Vol. 99, No. 15, 1995
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temperature dependence, with the behavior of a nonmetal (a(T = 0) = 0). These results show that it is difficult to differentiate metallic vs nonmetallic behavior in oxides based on hightemperature measurements. What emerges, however, is that the
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