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concentration. However, it was not possible to reliably achievehigher concentrations due to the breakdown of the dielectric.Using dielectric constants and high dielectrics with high breakdown®elds, a more detailed investigation will be possible.

Figure 5 shows the dependence of the resistance on the magnetic®eld (perpendicular to the channel). The transition temperature isreduced, as is expected for a superconductor. The upper critical ®eldHc2 can be deduced from the resistance data, resulting in an initialslope dHc2/dT of approximately -3 T K-1 (see inset in Fig. 4). Thestandard extrapolation to zero temperature gives a value for Hc2(0)of 4.1 T, corresponding to a coherence length of approximately 80 AÊ .

We can only speculate on the microscopic pairing mechanism,but it appears that the polarizability of the molecular crystal is aprimary candidate. Assuming a conventional Bardeen±Cooper±Schrieffer mechanism with electron±phonon interaction, and thesame density of states at the Fermi level, the increase of Tc frompentacene to anthracene can be ascribed to increasing electron±phonon coupling with decreasing conjugation length (or withnumber of delocalized p-electrons), in line with theoreticalcalculations9,14. Further studies are needed, however, to shed morelight on the microscopic aspects of pairing. One approach would be

to vary the carrier concentration over a wide range, and thus sweepthe Fermi level through the density-of-states structure of the band.After this initial demonstration of superconductivity on relativelysimple molecules, it would be worthwhile to pursue further thisapproach in order to explore the many possibilities offered byorganic chemistry, using the powerful synthetic methods to engi-neer the electronic and vibrational properties of molecules. M

Received 18 April; accepted 17 July 2000.

1. Bednorz, J. G. & MuÈller, K. A. Possible high Tc superconductivity in the Ba-La-Cu-O system. Z. Phys. B

64, 189±193 (1986).

2. Glover, R. E. & Sherill, M. D. Changes in superconducting critical temperature produced by

electrostatic charging. Phys. Rev. Lett. 5, 248±250 (1960).

3. Mannhart, J., StroÈbel, J., Bednorz, J. G. & Gerber, Ch. Large electric ®eld effects in YBa2Cu3O7-d ®lms

containing weak links. Appl. Phys. Lett. 62, 630±632 (1993).

4. Konsin, P. & Sorkin, B. Electric ®eld effects in high Tc cuprates. Phys. Rev. B 58, 5795±5802 (1998).

5. Ahn, C. H. et al. Electrostatic modulation of superconductivity in ultrathin GdBa2Cu3O7-x ®lms.

Science 284, 1152±1155 (1999).

6. SchoÈn, J. H., Kloc, Ch., Haddon, R. C. & Batlogg, B. A superconducting ®eld-effect switch. Science 288,

656±658 (2000).

7. SchoÈn, J. H., Berg, S., Kloc, Ch. & Batlogg, B. Ambipolar pentacene ®eld-effect transistors and

inverters. Science 287, 1022±1023 (2000).

8. Silinsh, E. A. & Capek, V. Organic Molecular Crystals (AIP, New York, 1994).

9. Devos, A. & Lannoo, M. Electron-phonon coupling for aromatic molecular crystals: possible

consequences for their superconductivity. Phys. Rev. B 58, 8236±8239 (1998).

10. Jerome, D., Mazaud, A., Ribault, M. & Beechgard, K. Superconductivity in a synthetic organic

conductor (TMTSF)2PF6. J. Phys. Lett. 41, L95±L98 (1980).

11. Ishiguro, T., Yamaji, K. & Saito, G. Organic Superconductors (Springer, Berlin, 1998).

12. Warta, W. & Karl, N. Hot holes in naphtalene: High, electric-®eld dependent mobilities. Phys. Rev. B

32, 1172±1182 (1985).

13. Warta, W., Stehle, R. & Karl, N. Ultrapure, high mobility organic photoconductors. Appl. Phys. A 36,

163±170 (1985).

14. Kivelson, S. & Chapman, O. L. Polyacene and a new class of quasi-one dimensional conductors. Phys.

Rev. B 28, 7236±7243 (1983).

15. Mishima, A. & Kimura, M. Superconductivity of the quasi-one-dimensional semiconductor poly-

acene. Synth. Met. 11, 75±84 (1985).

16. Ginzburg, V. L. The problem of high temperature superconductivity. Contemp. Phys. 9, 355±374

(1968).

17. Kloc, Ch., Simpkins, P. G., Siegrist, T. & Laudise, R. A. Physical vapor growth of centimeter-sized

crystals of a-hexathiophene. J. Cryst. Growth 182, 416±427 (1997).

18. SchoÈn, J. H., Kloc, Ch. & Batlogg, B. Fractional quantum Hall effect in organic molecular

semiconductors. Science. 288, 2338±2340 (2000).

19. Dodabalapur, A., Torsi, L. & Katz, H. E. Organic transistors: Two-dimensional transport and

improved electrical characteristics. Science 268, 270±271 (1995).

Acknowledgements

We thank E. Bucher for use of equipment, and H. Y. Hwang, D. W. Murphy, H. StoÈrmerand C. M. Varma for discussions. J.H.S. was supported by the Deutsche Forschungsge-meinschaft.

Correspondence and requests for materials should be addressed to B.B.(e-mail: [email protected]).

Gate voltage (V)

Electrons per molecule

Normal state

Superconductingstate

Resistance

Tem

per

atur

e (K

)

1001.7

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

0.85 0.90 0.95 1.00 1.05

110 120 130

Figure 4 Channel resistance of a pentacene ®eld-effect device as a function of

temperature and gate bias. The transition to a superconducting state below 2 K and for

gate biases above 112 V is clearly observable. The electron density per molecule is

calculated assuming a homogeneous density only in the ®rst molecular layer.

Temperature (K)

Temperature (K)

PentaceneVg = 100 V

120 V

120 V

120 V

0.75 T

0.5 T

0.25 T Vg = 120 V

Nor

m. r

esis

tanc

e (a

.u.)

Hc2

(T)

1.6 1.8 2.0 2.2 2.4 2.6 2.8

0.0

0.2

0.4

0.6

0.8

1.7 1.8 1.9 20

0

1

Figure 5 Channel resistance of a pentacene ®eld-effect device as function of temperature

and magnetic ®eld (perpendicular to the channel). The inset shows the upper critical ®eld

as function of temperature (slope approximately -3 T K-1), and a coherence length of 80 AÊ

can be estimated for the superconductor.

.................................................................Room-temperature electronic phasetransitions in the continuous phasediagrams of perovskite manganitesYoung-Kook Yoo*², Fred Duewer*², Haitao Yang*, Dong Yi*, Jing-Wei Li*& X.-D. Xiang*

* Environmental Energy Technologies Division, Lawrence Berkeley NationalLaboratory, Berkeley, California 94720, USA² Department of Physics, University of California at Berkeley, Berkeley,

California 94720, USA

..............................................................................................................................................

Highly correlated electronic systemsÐsuch as transition-metaloxides that are doped Mott insulatorsÐare complex systemswhich exhibit puzzling phenomena, including high-temperaturesuperconductivity and colossal magnetoresistivity. Recent stu-dies1±3 suggest that in such systems collective electronic phenom-

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NATURE | VOL 406 | 17 AUGUST 2000 | www.nature.com 705

ena are important, arising from long-range Coulomb interactionsand magnetic effects. The qualitative behaviour of these systems isstrongly dependent on charge ®lling (the level of doping) and thelattice constant. Here we report a time-ef®cient and systematicexperimental approach for studying the phase diagrams of con-densed-matter systems. It involves the continuous mapping of thephysical properties of epitaxial thin ®lms of perovskite mangan-ites (a class of doped Mott insulator) as their composition isvaried. We discover evidence that suggests the presence of phaseboundaries of electronic origin at room temperature.

There have been numerous studies aimed at understandingthe relationship between charge dopant level n (or ionic radii)

and the physical properties of complex systemsÐparticularly low-temperature thermodynamic phase transitions involving long-range order. Two such studies were conducted in copper oxidesthat showed high-temperature superconductivity (see, for example,ref. 4 and references therein) and colossal magnetoresistive manga-nites5±7. These phase-diagram studies are very time consuming andde®cient because they usually involve growing single-crystal sam-ples of discrete (rather than continuous) compositions, and subse-quently measuring physical properties at various temperatures. Ourcontinuous phase diagram (CPD) method allows the continuousmapping of physical properties versus composition; this is muchmore time-ef®cient, thorough and systematic than the conventionalapproach.

The stability of the perovskite manganites against chemicalsubstitutions provides an opportunity to study the effect of varyingthe parameters of a highly correlated electronic/magnetic system.Here we control the charge ®lling range n of Mott insulators bycontinuous charge doping over the entire range; we also control thehopping integral t through the substrate-induced anisotropic straineffect in epitaxial ®lms and the average ionic radius of the A site.Our initial study uncovered several extremely narrow (in composi-tion) phases with very different electronic properties. These narrow,

200°

φ (degrees)

(101) planes of Nd0.7Sr0.3MnO3

0 350

(101) planes of LaAlO3

50 100 150 300250200

FWHM 0.2°

0 20 40 60 80

Inte

nsity

(arb

. uni

ts)

LAO

(100

)

NS

MO

(100

)

NS

MO

(200

)

LAO

(200

)

NS

MO

(300

)

LAO

(300

)

2θ (degrees)

Gradient depositionsof three precursors

Homogeneous mixingof amorphous precursors

Crystalline CPD

C

A

B

Automated in situ shutter system

1000°

8 target carousel

a

b c

Figure 1 Multi-layer deposition and post-annealing synthesis of the CPD, with X-ray

diffraction of a fabricated ®lm. a, High-precision in situ shutter system in a pulsed laser

deposition system. Eight-target carousel allows uninterrupted depositions without

breaking the vacuum. Vertical shutters are used to de®ne the width of each phase strip on

a substrate while precise gradient pro®les of three precursors are deposited with

horizontal shutters moving across the phase strip at constant speed. The precursor ®lms

in this study were deposited at room temperature by a pulsed laser deposition system in a

high vacuum (,10-7 torr). The forward-expanding plume in a high vacuum, coupled with

scanning laser beam across the 2 inch ´ 2 inch targets during deposition, results in a

deposition thickness uniformity of better than 1.5% over a 15 mm ´ 15 mm area,

ensuring the accuracy in stoichiometry (easily controlled by shutter). Following the

deposition, the sample was annealed at 200 8C for several days, then annealed at 400 8Cfor 30 h, followed by 2 h sintering at 1,000 8C. Low-temperature annealing is necessary to

allow homogeneous mixing of precursors into an amorphous intermediate before

crystallization at higher temperatures. b, The v/2v XRD pattern and c, the f-scan of the

(101) plane of a Nd0.7Sr0.3MnO3 (NSMO) thin ®lm made from three precursors of Nd2O3,

Mn3O4 and SrMnO3 on (001)LaAlO3.

Figure 2 Photo-re¯ection images of RE1-xAxMnO3 CPDs on NdGaO3 and SrTiO3

substrates, with the temperature-dependent study of Tm1-xCaxMnO3 CPD on SrTiO3. a,

The room-temperature CCD colour photograph (photo-re¯ection image) of 36 CPDs on

two different substrates under white light ((4.2±7.8) ´1014 Hz). The optical images were

taken using a monochrome charge-coupled device (CCD) camera and a blue ®lter.

Incident white light at 608 incidence was used to illuminate the sample and scattered light

at 308 incidence was measured as a function of temperature from 77 to 430 K. The

parentheses indicate the crystal ionic radii of the elements30. The commensurate doping

points of singular phases are denoted. The effective unit cell of (110)NdGaO3 has a9 = b9= 3.86 AÊ (a9 � b9 � Î�a2�b2 �

2). b, The room-temperature CCD colour photograph of Tm1-

xCaxMnO3 CPD on SrTiO3 under white light. Three doping regions are selected for blue

light ((6.2±7.8) ´ 1014 Hz) photo-re¯ection at three different temperatures.


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unexpected phases would be dif®cult to identify using conventionalmethods.

We determined CPDs of perovskite manganites (RE1-xAxMnO3)in epitaxial thin-®lm form; here RE is Eu, Gd, Tb, Er, Tm and Yb, Ais Ca, Sr and Ba, and x is varied continuously from 0 to 1 (Fig. 1a).Different phase diagrams were measured on six 15 mm by 15 mmsubstrates of two different single crystals, (100)SrTiO3, and(110)NdGaO3. As illustrated in Fig. 1a, gradients of RE oxides(Eu2O3, Gd2O3, Tb4O7, Er2O3, Tm2O3 and Yb2O3) are deposited asbottom-layer precursors. Each phase diagram was synthesized fromgradient depositions of three precursors, using a high precisionin situ linear shutter system (Fig. 1a) and subsequent ex situ post-annealing. Compared to the previously used composition-spreadmethod by co-sputtering of multiple targets8±10, our technique caneasily generate precisely controlled stoichiometric pro®les within asmall area. This advantage is crucial when various single-crystalsubstrates need to be used (as in this study) for high-qualityepitaxial ®lm growth. Despite the fact that the crystalline com-pounds are formed ex situ rather than in situ, we found thatappropriate post-annealing processes can generate high-qualityepitaxial thin ®lms.

Figure 1b shows a v/2v X-ray diffraction (XRD) pattern forNd0.7Sr0.3MnO3 on an LaAlO3 substrate which indicates that the®lm is (100) oriented (in order to study the crystalline quality of thesamples, we made individual samples of various compositionsselected from the phase diagrams under identical fabrication andprocessing conditions). Even when using a logarithmic scale intensity,we can hardly see any other impurity phase. Figure 1c shows f-scansof the (101) planes, indicating that the ®lm is in-plane aligned with

the substrate. We have con®rmed similar epitaxial growth in manydifferent compounds with different hole-doping levels.

To map the electronic properties of the CPDs, we choose twoprobes with very different energy scales (by 106): visible light andmicrowave. Visible colour photographsÐoptical-re¯ection imagesunder white light, (4.2±7.5) ´ 1014 HzÐof CPDs were obtained as afunction of temperature (Fig. 2). In order to map the electricalimpedance at low frequencies, we used a scanning evanescentmicrowave probe operating at 2.2 GHz (refs 11, 12). Brie¯y, theprobe design consists of a sharp metal tip protruding through theendwall of a coaxial resonant cavity. The high spatial frequencycomponents of evanescent waves generated at the tip (spatialfrequency is inversely proportional to tip radius) give rise to`super-resolution' (exceeding the Abbe barrier), currently reachingl/105 on dielectric samples. We note that l is here the wavelength inthe dielectric materials rather than the vacuum wavelength (`super-resolution' is usually not appropriate in metal samples, because thewavelengths in metals are less than 1 mm). The propagating wavesare con®ned within the cavity by a layer of thin metal shieldingcoated on the endwall. The instrument measures the compleximpedance of the probe by monitoring the changes in resonantfrequency (fr) and quality factor (Q) of the cavity. The measuredshift in fr and Q can then be used to calculate the real and imaginaryparts of the complex dielectric constant at fr, er and ei (ei = 4pj/q forconducting samples, j is the conductivity and q = 2pfr) (Fig. 3b).The details of the instrument and quantitative analysis methodswere published previously12.

We present here detailed measurements of the room-temperaturemicrowave impedance of a CPD of Er1-xCaxMnO3+d (Fig. 3a). The

Figure 3 The microwave response of the Er1-xCaxMnO3 CPD on NdGaO3, and the

calibration curve of microwave loss versus d.c. conductivity. The proximity of a sample to

the tip changes the complex impedance of the probe, changing fr and Q. In this study, the

model analysis is based on a ®rst-order approximation, which is valid when the

conductivity is low. The shift in resonant frequency is a monotonic increasing function of

the conductivity. The shift in quality factor is double-valued, approaching zero as the

sample becomes insulating or highly conducting. The shift in quality factor has been

calibrated against the measured d.c. conductivity and is consistent with the theoretical

analysis. a, The microwave response of the Er1-xCaxMnO3 CPD on NdGaO3. Microwave

loss, D(1/Q ), and frequency shift, Dfres/fres, are shown as a function of composition (x ).

The dashed lines are used to indicate phase boundaries. Inset, expanded-scale view of

the curve near the commensurate singularity. b, Microwave loss, D(1/Q ), versus d.c.

conductivity (Q-1 cm-1). Filled circles show the measured microwave loss and the

corresponding d.c. responses; the line is the theoretical ®t.


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NATURE | VOL 406 | 17 AUGUST 2000 | www.nature.com 707

room-temperature optical images of the complete sets of CPDs areprovided for comparison (Fig. 2a). First, we observed many clearboundaries in both the optical and the microwave impedanceimages (dashed lines in Fig. 3a). Microwave loss peaks at theseboundaries. X-ray data show no evidence of different structuralphases beside perovskite throughout these CPDs. The existence ofthese boundaries suggests that complex electronic orderings (withfundamentally different physical properties) occur as a function of x.

Second, a very narrow insulating (or more accurately semi-conducting) strip within a highly conducting region (appearing asa greyish-blue strip within a black band in Fig. 2a), around 7/8doping points of the CPDs of Tm1-xCaxMnO3 on NdGaO3 substrateand Yb1-xCaxMnO3 on SrTiO3 was observed at commensuratecharge-®lling point x = 7/8. The strip is so narrow in phase-widththat it would have been very dif®cult to ®nd it using the conven-tional practice of studying samples of discrete composition. Webelieve that this commensurate singularity point is related to thestatic charge (or spin) orderings observed in recent experimentaland theoretical studies in both copper oxides1±3,13±20 and manga-nites21±25. We expect to see a small structural anomaly associatedwith the singularity, owing to the strong coupling of charge orderingwith the lattice at the commensurate point.

In Fig. 2b, different doping ranges are selected for temperature-dependent comparisons of blue lightÐ(6.2±7.8) ´ 1014 HzÐre¯ection images. In strip A, as the temperature is raised, there isclear evidence of broader phase boundaries (blurring of the phase),indicating the effect of thermal ¯uctuation. Increased temperatureallows the coexistence of different phases over a broader range incomposition. It also shows the high-energy scale of the electronicphases observed. In strip B, as the temperature is lowered, thesuppression of intensity at the middle portion is clear. This tends tosharpen the phase boundary towards the right side (less doping) ofthe dark conducting phase. The conducting phase around x = 1/2weakens at low temperature. In other manganite systems such asNd1-xSrxMnO3, a similar phase near the 1/2 doping point was foundto have two or more intrinsic competing phases (phase separation)with one phase dominant as the temperature is lowered. As thesignal intensity is proportional to the amount of scattered light, thedisappearance of this dip at low temperature indicates that ahomogeneous phase dominates at 77 K. In strip C, the boundary(shown as a peak in intensity) disappears at high temperature.

The gradual change in ionic radii also induces unexpected abruptchanges in phase patterns. This effect can be observed in substitu-tions of both rare-earth and different divalent alkaline-earth ele-ments. We see relatively smooth phase patterns and the absence ofsharp transition boundaries in the Sr-doped system, and this trendis more pronounced in the Ba-doped system. The clear effect of thesmall t (or bandwidth), due to the smaller ionic size of Ca and theassociated overlap of electron waves, is observed in the rich anddiverse phase patterns in the Ca-doped system. The effect ofsubstrate-induced stress can also be observed.

The doped Mott insulators are strongly correlated electronicsystems, and the doped carriers tend to self-organize into highlyanisotropic patterns. Upon doping into the parent Mott insulatorcompounds, the charge carriers tend to separate (at a large scale)into phases of different electronic natures, that is, phase separation.The long range Coulomb interaction tends to frustrate this phaseseparation and favour an anisotropic long-range order of charge orspin (stripe phase for example)13. More speci®cally, in manganeseoxides the active orbital degree of freedom induces the highlyanisotropic electron-transfer interaction, which gives rise to com-plex spin±orbital coupling effects (including stripe phases)26,27. Apossible model for the current observation of phase boundaries isthe existence of different ground states of electronic self-organization due to corresponding orbital states mediated bylong range Coulomb interaction. With the transfer interactionheavily depending on the doping level, various orbital ordered

and disordered states may exist with concomitant spin (andoccasionally charge) orders. In La1-xCaxMnO3 and Nd1-xSrxMnO3

CPDs, we have also observed both optical and electronic manifesta-tion of such orbital states at room temperature with strongcorrelation to the low-temperature magnetic phases (ref. 28, andY.-K.Y. et al., manuscript in preparation).

However, the singular phases around x = 7/8 in the CPD ofEr1-xCaxMnO3 on NdGaO3 substrate and Yb1-xCaxMnO3 on SrTiO3

substrate are intriguing (Fig. 2a). It seems that these singularities at7/8 are due to the commensurate locking of orbital (charge)orderings to the lattice. If we accept this model, the highly con-ducting adjacent regions then suggest the existence of incommen-surate, ¯uctuating orbital orderings. Therefore, the currentobservation provides preliminary evidence of the recentlyproposed1 liquid-crystal phase (smectic phase) of orbital orderingsin this doping range, widely believed to exist in copper oxides. Thesmeared-out critical divergence in microwave loss at the boundariesobserved in this study is also consistent with this possibility. Thethermodynamic nature of the transition is clearly shown by thetemperature-dependent broadening or disappearance of the phaseboundaries at higher temperatures. However, we believe that morestudies are needed to con®rm this suggestion.

The systematic experimental data provided here should help toidentify new phenomena and elucidate the underlying physics ofthis complex system. The importance of the present work lies in thefact that we can continuously map the doping dependence ofcomplex materials, instead of extrapolating from discrete dopingpoints. For example, if doping-dependent electronic boundarieswere also present in high-Tc copper oxides such as La2-xSrxCuO4,the detailed study of the system at low temperature could beperformed, and might reveal suspected quantum critical behaviourat the critical doping point of x = 1/8 (ref. 29). M

Received 17 January; accepted 27 May 2000.

1. Kivelson, S. A., Fradkin, E. & Emery, V. J. Electronic liquid-crystal phases of a doped Mott insulator.

Nature 393, 550±553 (1998).

2. Tranquada, J. M., Sternlieb, B. J., Axe, J. D., Nakamura, Y. & Uchida, S. Evidence for stripe correlations

of spins and holes in copper oxide superconductors. Nature 375, 561±563 (1995).

3. Tranquada, J. M. et al. Coexistence of, and competition between, superconductivity and charge-stripe

order in La1.6-xNd0.4SrxCuO4. Phys. Rev. Lett. 78, 338±341 (1997).

4. Birgeneau, R. J. & Shirane, G. in Physical Properties of High Temperature Superconductors Vol. I (ed.

Ginsberg, D. M.) 152 (World Scienti®c, Singapore, 1989).

5. Schiffer, P., Ramirez, A. P., Bao, W. & Cheong, S.-W. Low temperature magnetoresistance and the

magnetic phase diagram of La1-xCaxMnO3. Phys. Rev. Lett. 75, 3336±3339 (1995).

6. Urushibara, A. et al. Insulator-metal transition and giant magnetoresistance in La1-xSrxMnO3. Phys.

Rev. B 51, 14103±14109 (1995).

7. Tomioka, Y., Asamitsu, A., Kuwahara, H., Moritomo, Y. & Tokura, Y. Magnetic-®eld-induced metal-

insulator phenomena in Pr1-xCaxMnO3 with controlled charge-ordering instability. Phys. Rev. B 53,

R1689±R1692 (1996).

8. Sawatzky, E. & Kay, E. Cation de®ciencies in r.f. sputtered gadolinium iron garnet ®lms. IBM J. Res.

Dev. 13, 696±702 (1969).

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10. van Dover, R. B., Schneemeyer, L. F. & Fleming, R. M. Discovery of a useful thin-®lm dielectric using a

composition-spread approach. Nature 392, 162±164 (1998).

11. Lu, Y. et al. Nondestructive imaging of dielectric-constant pro®les and ferroelectric domains with a

scanning-tip microwave near-®eld microscope. Science 276, 2004±2006 (1997).

12. Gao, C. & Xiang, X.-D. Quantitative microwave near-®eld microscopy of dielectric properties. Rev.

Sci. Instrum. 69, 3846±3851 (1998).

13. Emery, V. J. & Kivelson, S. A. Frustrated electronic phase separation and high-temperature super-

conductors. Physica C 209, 597±621 (1993).

14. Emery, V. J. & Kivelson, S. A. Collective charge transport in high temperature superconductors.

Physica C 235±240, 189±192 (1994).

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44±48 (1996).

16. Salkola, M. I., Emery, V. J. & Kivelson, S. A. Implications of charge ordering for single-particle

properties of high-Tc superconductors. Phys. Rev. Lett. 77, 155±158 (1996).

17. Zhang, S.-C. A uni®ed theory based on SO(5) symmetry of superconductivity and antiferromag-

netism. Science 275, 1089±1096 (1997).

18. Wells, B. O. et al. Incommensurate spin ¯uctuations in high-transition temperature superconductors.

Science 277, 1067±1071 (1997).

19. Aeppli, G., Mason, T. E., Hayden, S. M., Mook, H. A. & Kulda, J. Nearly singular magnetic ¯uctuations

in the normal state of a high-Tc cuprate superconductor. Science 278, 1432±1435 (1997).

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22. Kuwahara, H. et al. Striction-coupled magnetoresistance in perovskite-type manganese oxides. Science

272, 80±82 (1996).

23. Chen, C. H. & Cheong, S-W. Commensurate to incommensurate charge ordering and its real-space

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Acknowledgements

This work was supported by Advanced Energy Projects Division, Of®ce of Computationaland Technology Research, US Department of Energy and DARPA.

Correspondence and requests for materials should be addressed to X.-D.X.(e-mail: [email protected]).

.................................................................Stress transmission through a modelsystem of cohesionless elastic grainsMiguel Da Silva & Jean Rajchenbach

Laboratoire des Milieux DeÂsordonneÂs et HeÂteÂrogeÁnes (UMR 7603 CNRS), Case 86,

UniversiteÂ P. et M. Curie, 4 Place Jussieu, 75252 Paris Cedex 05, France

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Understanding the mechanical properties of granular materials isimportant for applications in civil and chemical engineering,geophysical sciences and the food industry1, as well as for thecontrol or prevention of avalanches and landslides2. Unlike con-tinuous media, granular materials lack cohesion, and cannotresist tensile stresses. Current descriptions of the mechanicalproperties of collections of cohesionless grains have reliedeither on elasto-plastic models classically used in civilengineering3, or on a recent model involving hyperbolicequations4,5. The former models suggest that collections of elasticgrains submitted to a compressive load will behave elastically.Here we present the results of an experiment on a two-dimen-sional model systemÐmade of discrete square cells submitted toa point loadÐin which the region in which the stress is con®ned isphotoelastically visualized as a parabola. These results, which canbe interpreted within a statistical framework, demonstrate thatthe collective response of the pile contradicts the standard elasticpredictions and supports a diffusive description of stress trans-mission. We expect that these ®ndings will be applicable toproblems in soil mechanics, such as the behaviour of cohesionlesssoils or sand piles.

The point-punch test probes the mechanical behaviour of thematerial. Provided that the partial-differential equation describingthe stress ®eld throughout the discrete medium is linear, as is thecase in all currently competing models, the response to the pointload identi®es with the Green's function for the stress equation. Weprepared the sample by cutting a 10-mm-thick elastomeric plateinto identical square grains (15 mm ´ 15 mm), whose vertices aretrimmed. The elastic grains are then packed into contact andbounded by a metal frame. The Young's modulus and Poissoncoef®cient of the polycarbonate elastomer are respectivelyY = 3.15 ´ 105 Pa and n = 0.36. The punch consists of a steel sphere3 mm in diameter and with Young's modulus Y = 2 ´ 1011 Pa. Thebidimensional packing is partially disordered: each linear array,

constituting a layer, is regular, but there is a varying shift with theunderlying layer (Fig. 1). Owing to tip effects, the real contactsexperienced by each square cell are preferentially located at itsvertices. Hence, the present tiling is convenient to model thestress propagation through different disordered contact networks,simply by varying the shift between adjacent layers. The sample isthen positioned between two polarizers at right angles, in order toperform photoelastic visualization of the strain state.

Figure 1a shows a photoelastic view of the point-loaded packing:the load P is typically 20 N. Figure 1b shows the active contactnetwork from the location of brightened contacts. In Fig. 1a, thearrow labelled 1 points to a typical compressive contact, for whichwe recognize a set of quasi-circular bright fringes, whereas the arrowlabelled 2 points to a typical shear contact, for which bright lobes areinclined at 458 relative to the interfacial plane.

When varying the packing structure we obtain different patternsfor the internal stress. It is informative to perform an ensembleaverage of such stress patterns. Figure 2 presents the average of 10digitized images corresponding to different packings. It is clear thatthe strained region is bounded by a parabola. This is a remarkableresult, because, for our model system, it refutes both the usual elasticapproach and the alternative hyperbolic proposal. Instead, it sup-ports a diffusive description of the stress transmission, accountedfor by equations of the parabolic type.

In classical engineering textbooks, static cohesionless piles madeup of elastic grains and submitted to a purely compressive externalload are considered to be elastic below the plastic threshold3,6. In theelastic theory, the stress ®eld obeys the biharmonic equation7,8,which is elliptic and implies that the internal state of stress dependson the whole set of boundary conditions. On the other hand, theMohr±Coulomb yield condition associated with the mechanicalequilibrium equations leads to a set of hyperbolic partial-differen-tial equations for the stresses in the plastic state6. In the latter case,stresses propagate as waves, and they are only sensitive to someparticular points located on the boundaries.

If the results carry over to granular systems in general, made up ofgrains of arbitrary shape and packed in random fashion, this impliesthat the elastic theory cannot account for the collective response of acompressed collection of elastic grains, in continuum modelling.Indeed, in two dimensions, the elastic response of a semi-in®nitemedium to a normal point force is as follows9. If the origin (of polar

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a b

Figure 1 Photoelastic view of a packing submitted to a point load. a, Typical photoelastic

view of a packing submitted to a point load of 20 N. Each packing is made up of regular

arrays of square grains; there is an arbitrary shift between adjacent arrays. Owing to tip

effects, the real contacts undergone by the grains are preferentially located at their

vertices. The arrow labelled 1 points to a typical compressive contact, and the arrow

labelled 2 points to a typical shear contact. b, The active contact network drawn on the

basis of the location of brightened contacts.


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