Charge and Spin Density Waves

On a hot July afternoon the Mallin Washington, D.C., is over-run with sightseers. They move

earnestly in zigzag patterns carryingtheir coolers, bouncing from museumto monument to cafeteria. Most of thestreets bordering the lawns are ßat, andas many tourists stroll in one directionas in the other. Suddenly a drumroll isheard: a marching band is assembling.On the roads, displacing the confusedcrowd, are gathering serried ranks ofuniformed high school students. Soonthe band is mustered in neat rows,hardly disturbed even by a child tryingto hide between the trumpetersÕ legsfrom a pursuing parent. As the tour-ists watch, the band starts to play andthen marches forward with a clash of cymbals.

The wanderers on the Mall imitaterather closely the behavior of electronsin common metals. On cooling to tem-peratures close to absolute zero, mostmetals remain in this state; that is, theelectrons continue to wander. But insome metals the electrons organizethemselves into regular patterns likethe ranks of a marching band.

Such ordered ranks of electrons, oth-erwise known as charge-density waves,or CDWs for short, were envisaged by

the theoretical physicist Rudolf E.Peierls in the early 1930s and discov-ered in the 1970s. A related phenome-non, spin-density waves, or SDWs, werepredicted by Albert W. Overhauser in1960, while at Ford Motor Company;the waves were also Þrst seen in the1970s. At one time, CDWs were suggest-ed as being the agent of superconduc-tivity. Today we know that supercon-ductivity has a diÝerent origin, one inwhich the students dance in pairs rath-er than march; yet the many oddities ofthe marching bands themselves havekept researchers intrigued for decades.Charge-density waves may even Þndapplications one day as tunable capaci-tors in electronic circuits and as ex-tremely sensitive detectors of electro-magnetic radiation.

Hook up a battery to the ends of asolid in which a CDW exists and applya voltage across it. If the voltage issmall enough, nothing happens: theshoes of the marchers are stuck to theroad with chewing gum. (The stickingis weak, so charge-density waves havea Òdielectric constantÓ several milliontimes that of semiconductors, which al-lows them to store enormous amountsof chargeÑhence their potential use ascapacitors.) But if you increase the volt-age beyond a certain threshold, theshoes suddenly break free, and theband begins to marchÑthere is a largecurrent. The current is not proportion-al to the voltage, as in ordinary metalsobeying OhmÕs law; instead it increasesvastly with small increases in voltage.Further, a small part of the total cur-rent oscillates in time, even if only aconstant DC voltage is applied.

And charge-density waves exhibit aÒself-organizedÓ response to externallyapplied forces. In fact, the concept ofself-organized criticality grew out of ini-tial work on CDWs. This Þeld attemptsto understand the motions of complex

systems such as sandpiles or earth-quake fault networks. LetÕs say we drib-ble sand onto a surface. It piles up intoa conical shape; the cone is so steep thatoften adding a single grain will cause anavalanche. Likewise, tectonic plates per-petually poise themselves on the brinkof an earthquake. In some circumstanc-es, charge-density waves so conÞgurethemselves that any slight change in anexternally applied electric Þeld leads toa drastic change. CDWs are thus a table-top system in which we can test theo-ries of self-organization.

Why do density waves form?The underlying cause is the in-teraction between the elec-

trons in a metal. Normally the electro-static repulsion between the negativelycharged electrons is canceled out by thepresence of positive ions (atoms thathave lost one or more electrons and aretherefore positively charged), whichform the body of the metal. Then theelectrons hardly notice one another. Insuch a situation, if we picture the elec-trons as the strolling crowd describedearlier, the probability of Þnding a per-sonÑor electronÑat any one spot is thesame as at any other. So the electronsÕcharge density is uniform in space. Nowsuppose the electrons do interactÑsayby aÝecting the lattice in which the pos-itive ions are arranged. The lattice canin turn inßuence the position of a sec-ond electron, eÝectively giving rise toan interaction between the electrons.

50 SCIENTIFIC AMERICAN April 1994

Charge and Spin Density Waves

Electrons in some metals arrange into crystalline patterns that move in concert, respond peculiarly

to applied voltages and show self-organization

by Stuart Brown and George Gr�ner

STUART BROWN and GEORGE GR�NERshare an interest in the dynamics ofdriven systems such as charge-densitywaves and earthquakes. Gr�ner receivedhis Ph.D. in Budapest in 1971 and workedas a postdoctoral associate at ImperialCollege, London. He has been professorof physics at the University of California,Los Angeles, since 1981. Brown earnedhis Ph.D. in 1988 from U.C.L.A. and didpostdoctoral research at Los Alamos Na-tional Laboratory and the University ofFlorida. In 1991 he returned to U.C.L.A.as a member of the faculty.

REGULAR PATTERN of charge-densitywaves in the material tantalum disul-Þde is revealed by a scanning tunnelingmicroscope. Peaks of high charge den-sity (white ) are spaced 12 angstromsapart in a hexagonal array. Robert V.Coleman and C. Gray Slough of the Uni-versity of Virginia provided the scan.

Copyright 1994 Scientific American, Inc.


The interactions often cause the elec-trons to become paired up; the pairssubsequently repel one another. Theneach pair stays as far away as possiblefrom all other pairs, and an orderedstructure like that of the marching bandis formed; the charge density becomesbumpy. If we take into account the wavenature of the electrons, a smooth varia-tion of the charge density emerges. Thissmooth spatial variation of the chargeis called a charge-density wave.

In addition to charge, electrons car-ry around with them something calledspin. A spin is a magnetic moment as-sociated with each electron; the mo-ments can assume one of two states,labeled ÒupÓ and Òdown.Ó If electronswith the same spin orientation repelone another, then each up spin wantsto have a down spin as a neighbor. Theresult is a spin-density wave, or SDW.An SDW can be thought of as twoCDWs, one for each spin state, super-posed, with their peaks in alternate po-sitions. Note that for a charge-densitywave the charge varies in space, butnot for a spin-density wave.

In general, how the electrons inter-actÑand what kind of quantum-me-

chanical state is formedÑdepends onhow the electronsÕ motion is conÞned.In three dimensions, electrons have theability to avoid one another by simplymoving out of the way. But if they arelimited to traveling along a chain ofatoms, the electrons cannot avoid oneanother and tend to interact morestrongly. CDWs and SDWs occur mostlyin such materials, in which the atomsare lined up in chains. (Many of thesematerials were Þrst synthesized in theearly 1970s.) In some circumstances,the electron pairs attract rather thanrepel one another, forming a supercon-ducting state.

Chemists often design materials witha chainlike structure; however, they donot have much control over the natureof the electronic interactions. So wheth-er the synthesized substance developsa CDW or an SDW or becomes super-conducting cannot be predicted.

Electrons tend to pair at low tem-peratures. At absolute zero, eachelectron has its Òmate,Ó and the

structure is fully ordered. As we warmup the material, some of the pairs be-come separated; they then induce oth-

er pairs to separate. With higher andhigher temperatures, more and morepairs are divorced, until the last pairbreaks up; above this critical tempera-ture the material has only free elec-trons and is back to a metal. This pro-cess is known as a phase transition, asin the melting of an ice cube. If we re-verse the process, cooling the materialdown from high temperatures, a CDWforms when we cross the phase-transi-tion temperature. The electrons thenget stuck. Because small electric Þeldscan no longer dislodge them, and nocurrent ßows, the metal changes abrupt-ly to an insulator. This sudden changein electric conductivity in fact signalsthe formation of a CDW.

Far more direct observations of CDWshave been made using scanning tunnel-ing microscopes, which show the chargedensity even on atomic scale [see illus-

tration on preceding page ]. Further, aCDW is accompanied by distortions inthe lattice. The distortion pattern, calleda superlattice, can be seen by x-ray dif-fraction: the ions scatter x-rays ontophotographic Þlm, displaying a charac-teristic pattern that reveals their spac-ing. For example, if the superlatticewavelength is twice that of the latticewavelength, the x-ray diÝraction pat-tern will show additional spots halfwaybetween the main spots coming fromthe lattice. (The intensity of the halfwayspots relates to the size of the latticedeformation.) The Þrst experiments ofthis type were performed by RobertComes and his associates in Paris inthe 1970s.

Since a spin-density wave leads nei-ther to a charge ßuctuation nor to a lat-tice distortion, detecting it is muchharder. In principle, it could possiblybe seen through the magnetic force mi-croscope, an instrument that respondsto variations of the spin, but the devic-es are not yet sensitive enough. The Þrstdemonstration of spin-density waveswas made by scattering neutrons oÝchromium. (Neutrons, having spin andno charge, are useful for studying or-dered spin structures.) In addition, in-direct probes of the magnetic Þeld,such as magnetic resonanceÑthe sametechnique used in hospitals as a diag-nostic toolÑare now the only means ofsensing the presence of SDWs.

The eÝects of CDWs and SDWs mayalso be observed via the motions theyperform as a body. These motions canbe rather diÝerent depending on howthe wavelength of the density wave re-lates to the underlying lattice spacing.The CDW wavelength changes with thenumber of electrons in the solid: ifthere are more electrons, the wave-length becomes smaller and, in particu-


Pairing States

Charge-density waves (top ) can be thought of as electron-hole pairs aswell as electron-electron pairs. (A hole is a vacant quantum-mechanical

state; it acts much like a particle.) Looking at the array, we can think of anelectron as being paired either with a hole to its right or left or with the elec-tron opposite. (Superconductivity comes from yet another kind of electron-electron pairing.) In spin-density waves (bottom ), the opposing electron isshifted, so that the total charge density is constant but the spin density goesup and down along the array.

Which of the various pairing states occurs in a given material depends onthe strength of the various interactions between the electrons. For instance,if direct electrostatic repulsion between electrons dominates, either a spin-density wave or a spin-parallel (or “triplet”) superconducting state is favored.Electrons can also interact by distorting the lattice. The mediation of the lat-tice leads to attraction between electrons, and either a charge-density waveor a spin-antiparallel (or “singlet”) superconducting state results.

CHARGE-DENSITY WAVE

SPIN-DENSITY WAVE

SPIN-UPELECTRON

HOLE

SPIN-DOWNELECTRON


lar, may not match the original latticespacing of the ions in any neat way.Then the charge-density wave is said tobe ÒincommensurateÓ with the originallattice spacing; it ßoats around unaf-fected by the lattice until pinned downby a defect. (A defect acts like a pot-holeÑor chewing gumÑin the electricpotential surface, in which the CDWgets stuck.) But if the charge-densitywave and the original lattice spacingare ÒcommensurateÓ and Þt neatly,then, for example, every other studentstands in a depression in the road, andit is very hard to get the band moving.For this reason, incommensurate wavesare much more intriguing in the varie-ties of behavior that they display. Com-mensurate wavesÑthe ones originallyenvisioned by PeierlsÑare of largelyhistorical interest.

There are two basic motions thatcharge-density waves can indulge in asa body, called collective modes. Quan-tum mechanics allows us to think ofthese modes as particles, which are thennamed by the suÛx Òon.Ó The ßoatingof the crests back and forth and theiroccasional bunching are a kind of col-lective mode known as a phason ( it in-volves changing the ÒphaseÓ of the den-sity wave). For waves that do not Þtwell with the underlying lattice struc-ture, the ßoating takes no energy at all(unless a defect pins the wave down),but the bunching takes some. The other

way in which CDWs can change is thatthe crests can get higher. This motion,called an amplitudon, requires a lot ofenergy. The position and height of thecrests may both vary, with variationsover shorter distances having higherenergies. The energies of these motionswere Þrst calculated by Patrick A. Lee,T. Morris Rice and Philip W. Anderson,then at AT&T Bell Laboratories. TheSDWs share these collective motionswith the CDWs and in addition have apurely magnetic mode that is related tochanges in spin orientation. These exci-tations are called magnons.

The truly dramatic motions occurwhen we apply an electric Þeld to a sol-id containing a charge-density wave. Acurrent-voltage relation very diÝerentfrom OhmÕs lawÑin which conductivi-ty is constantÑwas found in 1986 ( inthe material niobium triselenide) byNai-Phuan Ong, Pierre Monceau andAlan M. Portis of the University of Cali-fornia at Berkeley. Since then, someCDW materials have displayed conduc-tivities that vary by several orders ofmagnitude when very modest electricÞelds (of less than one volt per centi-meter) are applied. We now know thatthis change in conductivity comes fromthe depinning and sudden motion ofthe entire density wave. Even more un-usual is the variation of the current astime passes, even when only a constant(DC) voltage is applied. This was Þrst

observed by Robert M. Fleming andCharles C. Grimes of AT&T. Our recentmeasurements, and those of Denis J�-rome, Silvia Tomic and others at theUniversity of Paris South and TakashiSambongiÕs group at Hokkaido Uni-versity, have shown that spin-densitywaves behave much like charge-densitywaves in the presence of electric Þelds.

The simplest model that describesthe behavior of density waves iscalled the classical particle mod-

el. It was proposed by one of us (Gr�-ner) with Alfred Zawadowski and PaulM. Chaikin, then at the University ofCalifornia at Los Angeles. The charge-density wave is represented by a singlemassive particle positioned at its cen-ter of mass. The behavior of this parti-cle reßects that of the entire array.When there are no external electricÞelds, the particle sits on a ribbed sur-face, like a marble in a cup of an eggtray. This conÞguration corresponds tothe crest of a CDW being stuck at a de-fect. If we move the CDW, the marbleclimbs over the edge of the eggcup andfalls into the next one, which meansthat the next crest of the CDW getsstuck at the same defect.

This model allows us to understandmuch of the versatile behavior of CDWs.The marble is free to move around thebottom of the eggcup and can thereforereadjust its position sensitively in re-

SCIENTIFIC AMERICAN April 1994 53

CHARGE AND SPIN DENSITIES of electrons are shown in nor-mal metals, charge-density waves and spin-density waves.The spin-up (orange ) and spin-down (purple ) electron den-

sities vary with position within the crystal. They can besummed to yield the total charge density (blue ); their diÝer-ence yields the spin density (green ).

´

ELE

CT

RO

N D

EN

SIT

YT

OT

AL

CH

AR

GE

DE

NS

ITY

SP

IND

EN

SIT

Y

POSITION

NORMAL METAL CHARGE-DENSITY WAVE SPIN-DENSITY WAVE

POSITION POSITION

0

0

0

0

SP

IN-U

PS

PIN

-DO

WN


sponse to applied electric Þelds. Be-cause the marbleÑthat is, the charge-density waveÑcarries charge, its posi-tion affects the electric Þeld within themedium. The marble usually adjusts itsposition so as to reduce the electricÞeld acting on it. Thus, materials withcharge-density waves have a large Òdi-electric constant,Ó so large that theycould be called superdielectrics. Ourmeasurements on both charge- andspin-density waves give values for thedielectric constant more than one mil-lion times larger than that of ordinarysemiconductors.

What happens if we apply a DC volt-age? The egg tray on which the marblelies will tilt. If the tiltÑthat is, the volt-ageÑis great enough, the marble canroll out of the eggcup and down the egg

tray. The marble slows down when itclimbs up an edge and speeds up whenit falls down one. Consequently, itsspeed, and the electric current, goes upand down with time. These current os-cillations, which we have mentionedearlier, are widely observed. The aver-age current is higher if the tilt in the eggtrayÑthat is, the DC voltageÑis higher.

Now suppose that instead of a DCvoltage, an AC voltage is applied, inwhich case the egg tray is rocked backand forth like a seesaw. The marble os-cillates back and forth in its cup. Thissloshing of the entire density wavescatters light of certain colors, allowingits detection in optical experiments atmicron and millimeter wavelengths, asconducted at U.C.L.A. (Conversely, theCDW can sensitively detect electromag-

netic radiation.) If we apply both a DCÞeld and an AC Þeld, the former makesthe egg tray tilt to one side, whereas thelatter makes it jiggle from side to side.Suppose the marble is rolling down theegg tray. If the time the marble takes togo from one eggcup to the next is near-ly the same as the time for which theegg tray is tilted ÒupÓ by the AC voltage,it will hop between eggcups once eachcycle of the AC Þeld. When the marbleis hopping down the egg tray with thehelp of the AC Þeld, augmenting the av-erage tilt of the egg tray by increasingthe DC voltage does not change the av-erage current. So if we plot the currentversus the DC voltage ( in the presenceof an AC voltage), we will see the cur-rent generally increasing with DC volt-age except for certain plateaus where


Density-Wave Materials

Two very different types of materials, organic and inorganic, show density waves.The inorganic materials are characterized by chains of transition-metal ions, such

as platinum. Within a crystal, each chain is well separated from its neighbors. Theelectrons move freely along the chains, but the large separation between chains im-pedes transverse motion,so that the electric conduc-tivity might be from 10 to1,000 times greater in thechain direction than across.A typical linear-chain com-pound of the inorganic va-riety, K2Pt(CN)4Br0.33H2O,or KCP for short, is shownat the left. The verticallobes (purple ) show elec-tron orbitals overlappingalong the chain. Some oth-er linear-chain compoundshaving so-called incom-mensurate charge-densitywaves are NbSe3, (TaSe4)2Iand K0.3MoO3.

The other types of CDWmaterials are grown fromflat organic molecules suchas the synthetic one, tetra-methyltetraselenafulva-lene, or TMTSF. Several of

these molecules are stacked on one another to form a crystaltogether with embedded PF6 ions, as shown at the right.Electrons are free to move up and down the stack but notfrom one stack to the next; thus, the conductivity is againhighly anisotropic.

These organic materials are of particular importance instudying electrons in solids because their properties can befine-tuned. For example, an entire family of TMTSF-basedsalts can be created by replacing the PF6 anion with ReO4, Br,SCN or AsF6, among others. Each new salt has slightly differ-ent interaction strengths between the electrons; these varia-tions can have profound effects. The crystal (TMTSF)2ClO4,if cooled slowly down to one kelvin, becomes a supercon-ductor, whereas rapid cooling gives a spin-density wave.

CN

Pt

PF6C Se

CH3


we have a Òmode lockingÓ [see bottom

illustration on this page].The model we have described, and

the equations it implies for the marbleÕsmotion, turns out to be applicable inquite diverse situations. For example, itdescribes a Josephson junction (thatbetween two superconductors), the mo-tion of ions in solids, a pendulum in agravitational Þeld and certain electron-ic circuits. Although the equations looksimple, they display a variety of solu-tions, including chaotic behavior.

Other behaviors are not so simple tounderstand. By cooling materials thatcontain spin-density waves down to al-most absolute zero, we Þnd a peculiarphenomenon that has been interpretedas the marbleÕs tunneling through to thenext eggcup instead of climbing overits edge. This purely quantum-mechan-ical eÝect has since been conÞrmed byother groups. Tunneling has been pre-dicted by Kazumi Maki of the Universi-ty of Southern California and by the lateJohn Bardeen of the University of Illi-nois, but it is too early to tell whethertheir model applies to our low-tempera-ture experimental Þndings about SDWs.

Perhaps the most bizarre behaviorof all is that of self-organization[see ÒSelf-Organized Criticality,Ó

by Per Bak and Kan Chen; SCIENTIFIC

AMERICAN, January 1991]. Susan N.Coppersmith and Peter B. Littlewood ofAT&T and Kurt A. Wiesenfeld and PerBak of Brookhaven National Laboratorywere the Þrst to deduce this phenome-non, from experiments on CDWs per-formed by researchers at AT&T and byus at U.C.L.A. Self-organization is a phe-nomenon that charge-density waveshave in common with earthquakes. Justas two tectonic plates rubbing on eachother get stuck at ragged edges andthen suddenly unstick (with catastroph-ic consequences), charge-density waves,in the presence of some electric Þelds,get stuck on defects and suddenly un-stick. But the analogy goes deeper.Earthquakes and charge-density wavestend to settle into conÞgurations inwhich a small disturbance will cause aviolent change: they organize them-selves into a critical state. The marblebalances itself exactly on the thin edgebetween two eggcups.

To study the self-organizing behav-ior, we have to reÞne our model slight-ly. Self-organization comes from self-interactions, so our model has to in-clude the push-and-pull between thediÝerent regions of the CDW. One mar-ble at the center of mass is no longerenough; we now need a series of mar-bles attached to their neighbors bysprings. This arrangement represents

the elasticity of the density wave. Sup-pose we repeatedly turn on a DC elec-tric Þeld for some time and then turn itoÝ. The marbles move some distanceduring the ÒonÓ time and roll to the bot-toms of eggcups when the Þeld isturned oÝ. We would expect them tomove farther if the ÒonÓ time is longer.But what happens is actually quite dif-ferent, as was found in simulations by

Coppersmith. Just before the Þeld isturned on, the marbles are positionedin neighboring eggcups. When the Þeldis turned oÝ, each is found to be exact-ly balanced on an edge between twoeggcupsÑno matter how long the Þeldis kept on [see illustration on next

page]. (After the Þeld is turned oÝ, themarbles roll into eggcups, sometimesto their right and sometimes to their

SCIENTIFIC AMERICAN April 1994 55

CHARGED-PARTICLE MODEL shows how current ßow in charge-density wavesdiÝers from that in normal metals. In a metal (left ) the particle rests on a ßat (elec-trical potential ) surface. If we apply a voltage, the surface tilts, and the particlestarts to move: there is a current. For a charge-density wave (right ), the surface isribbed. If the applied voltage is lowÑthat is, the tilt is smallÑthe particle changesposition only slightly, and there is no current. If the tilt is large enough to get theparticle over the barrier, the particle runs down the ribbed surface. Then the cur-rent goes up and down in time as the particle climbs over each barrier.

CURRENT VERSUS VOLTAGE is plotted for metals and charge-density waves. In ametal (blue) the current increases linearly with voltage. For a charge-density wave,there is no current until the voltage increases to a critical value; only then does thecurrent start to ßow (red ). If in addition to a direct voltage we apply an alternatingone, the curve shows plateaus (purple). The plateaus correspond to a Òmode lock-ingÓ when the ßow of the charge-density wave matches the alternating frequency.

METAL CHARGE-DENSITY WAVE

HIGHVOLTAGE

LOWVOLTAGE

NOVOLTAGE

DIRECT VOLTAGE

CU

RR

EN

T

1.0 2.00.0

1.0

2.0

3.0

CDW

CDW WITH ALTERNATINGVOLTAGE

METAL


left.) This bizarre self-regulatory be-havior is easier to study in charge-den-sity waves than in earthquakes. It hasgiven the former a particular use intesting complex dynamical theories.

In fact, the density-wave states areprobably just the simplest periodic con-Þgurations of electrons we can hope toÞnd. Several theories suggest a hierar-chy of more complex arrangements.One suggestion came from theoreticalphysicist Eugene Wigner in 1939. Wig-ner showed that if the density of elec-trons is low enoughÑsay in a collec-tion of electrons moving freely in twodimensionsÑthey would settle into acrystalline pattern. Since then, manyresearchers have searched for ÒWignercrystals.Ó In the early 1980s Grimes andGregory Adams, also at AT&T, showedthat electrons deposited on the surfaceof liquid helium form just such a crys-tal. Evidence of their presence in solid-state systems has come from groups atSaclay, France, AT&T and elsewhere.

The various properties of density-wave materials have yet to be appliedtoward enhancing our comfort. Still,plans abound. The dielectric constantsof CDW materials, besides being enor-mous, also change with the electricÞeld; they could be implemented in cir-cuits as tunable capacitors. The strongresponse of charge-density waves toelectromagnetic radiation could makethem useful as light detectors; at lowtemperatures, this sensitivity would ul-timately be limited by quantum me-chanics. Bardeen, better known for thetheory of superconductivity and the in-vention of the solid-state transistor,worked out the theory of the quantumtransport of density waves. Whetherquantum detectors such as he envis-aged can be built and put to practicaluse remains to be seen. Right now, weare happy enough just to learn moreabout the idiosyncracies of charge- andspin-density waves.


FURTHER READING

THE DIFFERENCE BETWEEN ONE-DIMEN-SIONAL AND THREE-DIMENSIONAL SEMI-CONDUCTORS. Esther M. Conwell inPhysics Today, Vol. 38, No. 6, pages 46Ð 53; June 1985.

THE DYNAMICS OF CHARGE-DENSITYWAVES. G. Gr�ner in Reviews of ModernPhysics, Vol. 60, No. 4, pages 1129Ð 1181; October 4, 1988.

CHARGE DENSITY WAVES IN SOLIDS. Edit-ed by L. P. GorÕkov and G. Gr�ner. Else-vier, 1990.

EVIDENCE ACCUMULATES, AT LAST, FORTHE WIGNER CRYSTAL. Anil Khurana inPhysics Today, Vol. 43, No. 12, pages17Ð20; December 1990.

SELF-ORGANIZED BEHAVIOR is compared for models of charge-density waves andearthquake faults. The particles attached by springs represent the position andelasticity of the charge-density wave (a). The particles rest on a ribbed surface. Ifone turns an electric Þeld on and oÝ repeatedly, the marbles are found to sit in themost unstable position possible: each on top of a hill. In the earthquake model (b)the blocks are attached to a surface moving sideways with respect to the lowersurface. The lower surface has metal strips that drag on the blocks; the blocks arealso connected by springs. After some arbitrary (unpredictable) time, the accumu-lated strain makes the blocks rearrange their positions catastrophically. But thenew positions are again unstable. The photograph shows the San Andreas fault inCarrizo Plain, east of San Luis Obispo, Calif.

CHARGE-DENSITY WAVE MODEL

EARTHQUAKE MODEL

a

b


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Charge and Spin Density Waves