NANO LETTERS
Surface-Stress-Induced Mott Transition and Nature of Associated Spatial Phase Transition in Single Crystalline VO2 Nanowires
2009 Vol. 9, No. 10 3392-3397
Jung Inn Sohn,† Heung Jin Joo,†,‡ Docheon Ahn,§ Hyun Hwi Lee,§ Alexandra E. Porter,| Kinam Kim,‡ Dae Joon Kang,†,⊥,* and Mark E. Welland†,* Nanoscience Centre, UniVersity of Cambridge, Cambridge CB3 0FF, United Kingdom, Semiconductor Research and DeVelopment Centre, Samsung Electronics Co., Yongin City, Korea, Pohang Accelerator Laboratory, Pohang UniVersity of Science and Technology, Pohang 790-784, Korea, Department of Materials, Imperial College, Exhibition Road, London SW7 2AZ, United Kindom, and BK21 Physics Research DiVision, Department of Energy Science, Institute of Basic Science, SKKU AdVanced Institute of Nanotechnology (SAINT) and Centre for Nanotubes and Nanostructured Composites (CNNC), Sungkyunkwan UniVersity, Suwon 440-746, Korea Received March 16, 2009; Revised Manuscript Received August 28, 2009
ABSTRACT We demonstrate that the Mott metal-insulator transition (MIT) in single crystalline VO2 nanowires is strongly mediated by surface stress as a consequence of the high surface area to volume ratio of individual nanowires. Further, we show that the stress-induced antiferromagnetic Mott insulating phase is critical in controlling the spatial extent and distribution of the insulating monoclinic and metallic rutile phases as well as the electrical characteristics of the Mott transition. This affords an understanding of the relationship between the structural phase transition and the Mott MIT.
One-dimensional (1D) nanomaterials continue to be of considerable interest for their potential use as building blocks of functional nanoscale devices as they exhibit significantly different properties compared with their bulk counterparts due to surface effects and unique dimensionality.1-4 In particular, the unique surface structure of 1D nanoscale building blocks, playing an important role in their physical properties, can induce spontaneous structural changes to relieve surface stress.4,5 In this regard the engineering of surface energetics through control of the surface structure and crystallographic growth directions in 1D nanoscale building blocks with high surface area is extremely important in modifying and designing 1D Mott transition systems, especially VO2 where a relatively easily accessible transition * To whom correspondence should be addressed:
[email protected] (D.J.K) and
[email protected] (M.E.W.). † Nanoscience Centre, University of Cambridge. ‡ Semiconductor Research and Development Centre, Samsung Electronics Co. § Pohang Accelerator Laboratory, Pohang University of Science and Technology. | Department of Materials, Imperial College. ⊥ BK21 Physics Research Division, DOES, SAINT, CNNC, Sungkyunkwan University. 10.1021/nl900841k CCC: $40.75 Published on Web 09/28/2009
2009 American Chemical Society
temperature at Tc,bulk ∼ 68 °C in bulk3,6 has the potential for realizing the Mott transistor. This approach provides a natural foundation for the self-organization of metallic and insulating phases in VO2 nanowires (NWs), allowing for a controlled way to observe the Mott transition. To date it has been demonstrated that bulk VO2 undergoes a metal-insulator transition (MIT), which is simultaneously accompanied by a first-order structural phase transition (SPT) from a high temperature metallic rutile (R) phase to a low temperature insulating monoclinic (M1) phase. This is effected through the heterogeneous nucleation and the propagation of domains to reduce the elastic energy, resulting in structural distortion.6-9 However, it is hard to control the spatial phase transitions involving the dynamic evolution of energetically favorable configurations in detail because of complex multidomain patterns and associated domain boundaries that arise through structural changes in bulk-scale systems.10,11 In addition, the recent discovery of an additional antiferromagnetic Mott insulating (M2) phase,12,13 which has two types of V chains that are quite different from the M1 phase and can be stabilized by doping or uniaxial high pressure in the [110] direction of an R phase in pure VO2, leads to further
complication in understanding Mott transitions. Interestingly, a Mott insulating M2 phase is a metastable state of an M1 phase, which is a superposition of two M2-type lattice distortions, although it is still unclear whether the insulating character of an M1 phase is associated with a band insulator or a Mott insulator due to the nonmagnetic and structural properties of an M1 phase with the pairing and tilting of all V atoms.8,9,12-15 Despite these features being the subject of many experimental and theoretical investigations, a comprehensive understanding of the underlying physics accounting for phase transition mechanisms and coexistence is still lacking. In particular, there is no simple strategy for the formation of Mott insulating M2 phases without introducing doping and uniaxial pressure, neither is there a clear relationship between Mott MITs and spatial phase transitions associated with the Mott insulating M2 phase. Here, we demonstrate that the Mott MIT in a single crystalline VO2 NW is strongly mediated by surface stress. This is particularly relevant for 1D NWs due to the high surface to volume ratio and the unique percolative nature of the transistion itself. Single crystalline VO2 NWs employed in this study were grown laterally on the basal c plane and out of the basal r plane of sapphire by the vapor phase transport process reported previously, and all NWs have the same crystallographic side plane and growth direction.16 To investigate the effect of surface and interface stresses on phase transition behavior, two sets of samples, epitaxially grown VO2 NWs on c-cut sapphire (E-NWs) and mechanically transferred VO2 NWs from r-cut growth substrate to c-cut substrate (T-NWs), were prepared. In order to study the electronic MITs, we carried out four-probe measurements. For this, VO2 NWs were first dispersed by sonication in isopropyl alcohol and then transferred onto a silicon dioxide substrate by dropping a suspension of VO2 NWs. Electrode patterns were defined by conventional photolithography followed by electron-beam evaporation of Ti (100 nm)/Au (50 nm) electrodes on a specific VO2 NW followed by lift-off processes. In order to investigate the detailed structural changes related to MITs in single crystalline VO2 NWs, we carried out in situ synchrotron X-ray diffraction (XRD) examinations, which are an ideal and powerful technique to observe precise lattice structural changes at the nanoscale. A high flux and high-resolution XRD experiment of VO2 NWs was performed at the 10C1 beamline at the Pohang Accelerator Laboratory in Korea. VO2 NWs were mounted on a specially designed heating stage with housing and were heated to the desired temperature in ambient conditions. At each temperature, the sample was stabilized for 300 s and then in situ XRD patterns were measured at constant temperature within (0.2 °C. Figure 1a shows high-resolution XRD data recorded during the temperature cycling from T-VO2 NWs, which do not have an epitaxial interface. Above Tc,bulk, we observed, as expected, a strong and narrow peak corresponding to the (110) plane of an R phase (JCPDS file 01-079-1655). In addition, we also observed two additional peaks indexed unambiguously to (2j01) and (201) planes of the M2 phases,17 which are consistent with the mirror planes perpendicular to the tetragonal c-axis arising from two crystallographically Nano Lett., Vol. 9, No. 10, 2009
Figure 1. (a) Temperature dependence of XRD data from T-VO2 NWs measured by in situ synchrotron X-ray during temperature cycling. (b) Convergent beam patterns taken from the middle at 60 °C (bottom), the middle and the tip at 65 °C (middle), and the tip of a T-NW 70 °C (top), respectively. In Figure 1b, white and purple in the indexing of planes and zone axes indicate M1 and M2 phases, respectively, showing that it is hard to distinguish an M2 phase from an M1 phase due to similarities in positions of diffraction patterns.
independent V sites in an M2 phase.12,13 This indicates that in contrast to bulk VO2 where an M2 phase can be stabilized by an externally applied force in the (110) plane of an R phase, a Mott insulating M2 phase in NWs can be naturally induced by the tensile surface stress due to the large surface area of NWs enclosed by (110) side planes. These are the energetically favored planes to break the symmetry of VO2 easily thus driving the structural transformation into a Mott insulating M2 phase as shown schematically in Figure 2b.13 As the temperature decreases, the intensity of the M2 peaks gradually increases and that of the R peaks continuously decreases, exhibiting the common feature of M2 and R phases and their first-order phase transitions. At 50 °C, the (011) peak of an M1 phase (JCPDS file 04-007-1466), which corresponds to the (110) plane of an R phase and (2j01) and (201) planes of an M2 phase, starts to appear and the coexistence of three distinct M1, M2, and R phases arises. When the temperature decreases, further R phases disappear and peaks of only M1 and M2 phases exist, as expected. However M2 phases, which are metastable states of an M1 phase, still remain down to room temperature. Akin to the cooling process, the progressive evolution of XRD peaks exhibiting direct structural changes is observed (right side of Figure 1a) during the heating process, but the phase transition temperature shifts upward, which probably indicates thermal hysteretic behavior of VO2 NWs as3393
evidence that a spatial phase transition occurs and rutile and monoclinic phases coexist in a VO2 NW during the transition. On further heating a convergent beam pattern corresponding to a rutile phase also appears at the tip region above 70 °C (the top of Figure 1b). This implies that the surface stress in the NWs is spatially inhomogeneously distributed along the NW length causing some degree of self-organization of the Mott insulating M2 phases. This in turn has a strong influence on the spatial phase transitions associated with the distribution of M1 and R phases and coexisting regions.
Figure 2. (a) Spatial fractional variation of metallic R (red circle and solid line, top), Mott insulating M2 (purple triangle and solid line, middle), and insulating M1 (blue square and solid line, bottom) phases as a function of temperature. Filled and open symbols are taken during the heating and cooling, respectively. Note that the Mott insulating M2 phase shows divergent behavior near the shaded region. (b) Schematic of VO2 projected into the tetragonal c-axis (left side) and along the tetragonal c-axis (right side). Large (V) and small (O) spheres indicate V and O atoms, respectively. In an M1 phase, all the V atoms on both sublattices (indicated by A and B, respectively) are paired and tilted from the position of an R phase while in an M2 phase V atoms of the A sublattice are paired, but not tilted, and V atoms of the B sublattice are tilted, but not paired.
sociated with the first-order phase transition during the temperature cycling of the sample. This implies that firstorder phase transitions between rutile and monoclinic phases, and the behavior of associated phases are closely related to the macroscopic MIT in VO2. To directly reveal the structural changes and the spatial distribution of metallic and insulating phases in VO2 NWs, in situ convergent beam experiments were performed using transmission electron microscopy. Figure 1b shows convergent beam patterns taken from a T-NW during the heating process. Below 60 °C, the same diffraction patterns are obtained from the middle and tip regions of a NW consistent with those of monoclinic phases for a [01j1] zone axis in an M1 phase (the bottom of Figure 1b), further consistent with XRD results. As the temperature increases, the convergent beam pattern for the middle region of the NW changes into the corresponding rutile phase for a [111] zone axis in an R phase at 65 °C, while that of the tip region remains in the monoclinic phase (the middle of Figure 1b). Here, it is difficult to clearly distinguish between M1 and M2 phases due to similarities in the positions of diffraction patterns for a [01j1] in M1 and corresponding [112] zone axis in the M2 phase. Further, there is a similar structural orientation between the (01j1j) and (200) planes in an M1 phase and corresponding (2j01) and (22j0) planes in an M2 phase, which is 122.6° and 122.4°, respectively.18 This provides the direct 3394
To further demonstrate the coexistence of rutile and monoclinic phases, we carried out quantitative XRD analysis to assess the fractional variation of the phases. Figure 2a shows hysteretic loops of spatial fractional variations of rutile and monoclinic domain regions obtained from XRD spectra shown in Figure 1a. Interestingly, the hysteretic behavior of structural changes of insulating M1 and metallic R phases is quite similar to electrical hysteretic loops reported in previous studies.3,6,7 During the cooling process a fraction of the rutile phase regions are transformed into Mott insulating M2 phase regions, resulting in the expansion of insulating regions through a propagation of M2 phases. The propagation rate of M2 phases is proportional to the fractional variation of remaining metallic R phases until the Mott insulating M2 phases reach the maximum amount of the fraction in M2 phases, as indicated by arrows, and then the propagation rate of insulating M1 phases is dependent on the fractional variation of the M2 phases. Contrary to the cooling process the propagation rate of M2 (R) phases on heating is directly affected by the fractional variation of M1 (M2) phases before (after) the maximum fraction of an M2 phase. Thus the system becomes increasingly metallic (R) through Mott M2 phases acting as a mediator through common phase boundaries between M1 and R phases, to transform M1 into R phases. This implies that the spatial phase transitions are dominated by the M2 phases. Specifically, surface-stressinduced Mott insulating M2 phases modify the stability of coexistence of distorted M1 and undistorted R domains through the self-organization of lattice distortions to form an energetically favorable local structure as shown in Figure 2b.19 Here, it is important to note that the expansion of insulating M1 and M2 phases tends to cause the constriction of metallic R phases and hence results in the persistence of metallic R phases far below Tc,bulk because the axial direction in VO2 NWs coincides with the a-axis in M1, the b-axis in M2, and the c-axis in R phases and a phase transition from R to M1 phases leads to the dilatation of the tetragonal c-axis to stabilize the insulating phases.12,20,21 We also note that a Mott insulating M2 phase showing divergent behavior mediates the spatial distribution of metallic and insulating phases and coexistence regions that in turn produces the hysteretic behavior of phase transitions associated with the MIT. This finding provides a clue that a highly correlated conducting state exists within the MIT regime, revealing the electronic nature of the Mott transition. In addition to demonstrating the surface-stress-induced spatial phase transition and the coexistence of metallic and insulating phases in VO2 NWs, we also performed temperNano Lett., Vol. 9, No. 10, 2009
Figure 3. Temperature dependence of XRD data from E-VO2 NWs measured during the heating. (a) Evolution of (011) and (b) (020) M1 plane-related peaks. The color of contour plots represents the peak intensity. Contour plots clearly exhibit coexisting characteristics within the temperature region of 54-64 °C and 68-80 °C, indicated by yellow dotted lines, for corresponding (011) and (020) M1 planes, respectively. These results demonstrate that a phase transition in VO2 NWs occurs differently according to each plane of VO2 due to the epitaxial interface interaction.
ature-dependent XRD measurements on E-VO2 NWs, which have an epitaxial interface, to more accurately assess how the interface stress affects the phase transition behavior compared with the surface stress in a nanoscale system. As shown in Figure 3a, the evolution of XRD spectra is clearly different from that measured on T-VO2 NWs without an epitaxial interface shown in Figure 1a. The peak corresponding to the (2j01) plane of M2 is broader than that expected at low temperature. In particular, its peak position shifts slightly upward compared to the value of an M2 phase in pure VO2 like T-VO2 NWs but is quite similar to that of an M2 phase in Al-doped VO2.22 It is clear from the shape and position of the observed peak that the domain structure of VO2 NWs grown epitaxially is strongly influenced by the stress at the interface due to the strong interaction between VO2 and the sapphire substrate and that these stresses accumulated at the interface along the NW length can also induce Mott insulating M2 phases.20-23 Figure 3b shows the evolution of XRD spectra related to the (020) planes of an M1 phase, demonstrating the effect of the interface stress on phase transition behavior due to the epitaxial relationship of (010) VO2//(0001) sapphire.16 Contour plots show unambiguously that there are shifts of SPTs to higher temperature at the (020) plane and different evolutions within the phase transition regime indicated by yellow dotted lines. A SPT in the (011) plane starts to occur from 54 °C, as shown in Figure 3a, whereas a peak of the (020) plane start to split into two peaks of (200) R and (002) M2 planes corresponding Nano Lett., Vol. 9, No. 10, 2009
to the (020) plane of an M1 phase from 68 °C, exhibiting a typical characteristic of coexistence of M2 and R phases.18 Interestingly, their peak positions are also very similar to those of Al-doped VO2,22 indicating that the interface stress has a strong influence on phase transition behavior and induces Mott insulating M2 phases. When the temperature increases further, the peak associated with an R phase shifts upward the position corresponding to the value of pure VO2 without epitaxial interface. These results clearly reveal that phase transitions in each plane of VO2 NWs occur at different temperatures due to the different stresses, resulting from different boundary conditions and interface interactions, because a (020) plane of VO2 has the direct epitaxial interface relationship with the substrate,16 which might stabilize an insulating phase to higher temperature as compared to an (011) plane. As T- and E-NWs have the same crystallographic side planes and growth directions as well as similar sizes,16 we believe that the difference in the evolution of phase transitions originates from different stresses. This finding suggests that for VO2 NWs without the presence of an epitaxial interface, surface stresses are dominantly introduced into NWs due to surface and geometric effects, while for VO2 NWs with an epitaxial interface the epitaxial interface interaction between the NW and the substrate can lead to additional stresses distributed along the interface. The understanding of the relationship between SPTs and Mott MITs is increasingly important in clarifying the mechanism of Mott transitions of VO2. In general, solidstate phase transitions occur through the heterogeneous nucleation and the percolation process of domain patterns to reduce the elastic energy at the expense of the increased domain wall energy,6-9,24 accompanying MITs in bulk VO2. In contrast, SPTs and associated electrical characteristics in VO2 NWs are quite different from those of their bulk counterparts due to the stress effect,4,10 which causes the reorganization of domains to relieve the stresses along the unconfined direction, and the different percolative nature, which controls a current path. During a SPT in VO2 NWs, spatial phase transitions are observed where domain structures of distorted M1 and undistorted R phases, as well as a Mott insulating M2 phase with an intermediate state, coexist as shown in XRD experiments. This coexistence is caused by the modification of local structural features involving the stress-induced Mott insulating M2 phase formation to minimize the energy cost and to reduce the distortion at the boundaries of competing phases as shown schematically in Figure 2b. Here, the interesting observation is that the behavior of structural changes is in excellent agreement with the temperature-dependent electrical behavior as shown in Figure 4. In particular, the behavior of the electrical MIT and the spatial fractional variations of metallic and insulating phases are qualitatively very similar. It seems that the macroscopic MIT, which is the average information of coexisting phases within the MIT regime on a macroscopic scale, is accompanied by a structural transition. Compared to the percolative nature of MITs in thin films and in the bulk, in 1D systems where the spatial extent of coexisting phases can be controlled on a nanometer scale, 3395
Figure 4. Temperature dependence of a fraction of metallic and insulating phases (top) and the resistance (bottom) obtained from a T-VO2 NW during heating. The resistance of a VO2 NW exhibits an exponential dependence on temperature in an insulating phase region (region A) with an activation energy of approximately 0.363 eV. The region B corresponds to a spatial phase transition regime where a metallic region starts to appear.
the electrical characteristics of Mott transitions and coexisting phases can be observed and distinguished within the MIT regime. Figure 4 shows the distribution of phases as a function of temperature from an ensemble of nanowires determined from XRD, alongside the resistance of a single nanowire. Here, nearly 80% of the physical quantity of VO2 (∼45% and 16% for 2D and 3D, respectively24,25) has to be in a metallic state to create a continuous current path as evidenced by the abrupt reduction in resistance. This enables us to observe temperature-dependent gradual changes of electrical features of individually competing phases involving Mott insulators distributed inhomogeneously. For the single nanowire studied we observed a new region (region C of Figure 4) where the conductivity decreases in the temperature range of a coexisting region of Mott insulating M2 and metallic R phases but where the activation energy is the almost same as that in the insulating phase regime (region A of Figure 4) before the abrupt electrical MIT, observed when a transition to the Mott insulating M2 phases occurs.26,27 This is consistent with XRD results (Figures 1a and 2a) showing that the corresponding region is the Mott insulating M2 phase dominant regime, which exhibits divergent behavior and allows electronic Mott transitions. Although we expect fluctuations between individual nanowires, the striking correlation between resistance behavior and the phase dynamics involving the M2 phase makes for compelling speculation around the combination of phase change and electrical properties within individual nanowires. A comparison of SPT and MIT behavior shows that a Mott insulating M2 phase can be viewed as the mediator to transform the insulating M1 (R) into metallic R (M1) phases. Therefore, stress-induced Mott insulating M2 phases in a pure VO2 NW play an important role in controlling the spatial 3396
distribution of coexisting phases as well as those regions closely correlated to electrical behavior through the selforganization of lattice distortions to form energetically favorable configurations. Except where an M2 phase can be stabilized by either doping or applying high pressure, these features of phase transitions related to stress-induced M2 phases have not been demonstrated in previous studies on NWs and bulk VO2 because of critical differences in their geometry associated with size and shape, crystallographic orientation, and boundary conditions. In summary, we have shown that the unique surface structure of single crystalline VO2 NWs having specific crystallographic orientation naturally induces Mott insulating M2 phases, which modify the stability of coexisting metallic and insulating phases through the self-organization of competing phases. This in turn effects control of the spatial extent of phase transitions and the electrical characteristic of the Mott transition. We also demonstrate the relationship between the SPT and Mott MIT associated with the Mott insulating M2 phase. Our observations are of great importance in understanding the fundamental physics of the Mott transition as well as the accompanying spatial phase transition and provide a route for tuning the electrical and optical properties of VO2 NWs in devices through control of residual stresses. Acknowledgment. This work is supported by Samsung Electronics, Korea, and the IRC in Nanotechnology, U.K. D.J.K. also wishes to acknowledge financial support in part by MEST through KICOS under Grant No. 2008-00656, through KRF under Grant No. KRF-2008-005-J00703, the KOSEF through CNNC at SKKU and by WCU program through the KOSEF under Grant No. R31-2008-000-10029-0. References (1) Lauhon, L. J.; Gudiksen, M. S.; Wang, D.; Lieber, C. M. Nature 2002, 420, 57–61. (2) Kuykendall, T.; Pauzauskie, P. J.; Zhang, Y.; Goldberger, J.; Sirbuly, D.; Denlinger, J.; Yang, P. Nat. Mater. 2004, 3, 524–528. (3) Wu, J.; Gu, Q.; Guiton, B. S.; Leon, N. P.; Ouyang, L.; Park, H. Nano Lett. 2006, 6, 2313–2317. (4) Diao, J.; Gall, K.; Dunn, M. L. Nat. Mater. 2003, 2, 656–660. (5) Moore, R. G.; Zhang, J.; Nascimento, V. B.; Jin, R.; Guo, J.; Wang, G. T.; Fang, Z.; Mandrus, D.; Plummer, E. W. Science 2007, 318, 615–619. (6) Morin, F. J. Phys. ReV. Lett. 1959, 3, 34–36. (7) Kim, H.-T.; Lee, Y. W.; Kim, B.-J.; Chae, B.-G.; Yun, S. J.; Kang, K.-Y.; Han, K.-J.; Yee, K.-J.; Lim, Y.-S. Phys. ReV. Lett. 2006, 97, 266401. (8) Eyert, V. Ann. Phys. (Leipzig) 2002, 11, 650–702. (9) Imada, M.; Fujimori, A.; Tokura, Y. ReV. Mod. Phys. 1998, 70, 1039– 1263. (10) Chen, C.-C.; Herhold, A. B.; Johnson, C. S.; Alivisatos, A. P. Science 1997, 276, 398–401. (11) Gu, Q.; Falk, A.; Wu, J.; Ouyang, L.; Park, H. Nano Lett. 2007, 7, 363–366. (12) Pouget, J. P.; Launois, H.; Rice, T. M.; Dernier, P.; Gossard, A.; Villeneuve, G.; Hagenmuller, P. Phys. ReV. B 1974, 10, 1801–1815. (13) Pouget, J. P.; Launois, H.; D’Haenens, J. P.; Merenda, P.; Rice, T. M. Phys. ReV. Lett. 1975, 873–875. (14) Wentzcovitch, R. M.; Schulz, W. W.; Allen, P. B. Phys. ReV. Lett. 1994, 72, 3389–3392. (15) Rice, T. M.; Launois, H.; Pouget, J. P. Phys. ReV. Lett. 1994, 73, 3042. (16) Sohn, J. I.; Joo, H. J.; Porter, A. E.; Choi, C.-J.; Kim, K.; Kang, D. J.; Welland, M. E. Nano Lett. 2007, 7, 1570–1574. (17) Chamberland, B. L. J. Solid State Chem. 1973, 7, 377–384. (18) All corresponding planes among M1, M2, and R phases could be indexed on the basis of related structures by transition matrixes given as Nano Lett., Vol. 9, No. 10, 2009
( ) ( )( ) ( ) ( )( ) a b c
a b c
M1
0 0 -2 a ) -1 0 0 b 0 1 1 c
M2
0 2 0 a ) 0 0 2 b 1 0 0 c
R
R
. (19) Ahn, K. H.; Lookman, T.; Bishop, A. R. Nature 2004, 428, 401–404. (20) Rakotoniaina, J. C.; Mokrani-Tamellin, R.; Gavarri, J. R.; Vacquier, G.; Casalot, A.; Calvarin, G. J. Solid State Chem. 1993, 103, 81–94.
Nano Lett., Vol. 9, No. 10, 2009
(21) Muraoka, Y.; Ueda, Y.; Hiroi, Z. J. Phys. Chem. Solids 2002, 63, 965–967. (22) Ghedira, M.; Vincent, H.; Marezio, M.; Launay, J. C. J. Solid State Chem. 1977, 22, 423–438. (23) Gea, L. A.; Budai, J. D.; Boatner, L. A. J. Mater. Res. 1999, 14, 2602– 2610. (24) Zallen, R. The Physics of Amorphous Solids; Wiley: New York, 1983. (25) Choi, H. S.; Ahn, J. S.; Jung, J. H.; Noh, T. W.; Kim, D. H. Phys. ReV. B 1996, 54, 4621–4628. (26) Zylbersztejn, A.; Mott, N. F. Phys. ReV. B 1975, 11, 4383–4395. (27) Chase, L. L. Phys. Lett. A 1973, 46, 215–216.
NL900841K
3397
