Buln.4 newOK' - Inria · Published by “Département de Recherche Fondamentale sur la Matière Condensée” DRFMC, CEA Grenoble, 17 rue des Martyrs, F-38054 Grenoble Cedex 9

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B U L L E T I NB U L L E T I NO F T H E

Published by“Département de Recherche Fondamentalesur la Matière Condensée”

DRFMC, CEA Grenoble, 17 rue des Martyrs, F-38054 Grenoble Cedex 9FRANCE

Chief EditorJean-Paul Duraud

EditorAnnie Pesnelle

Scientific CommitteeAriel Brenac, Pierre Dalmas de Réotier,Olivier Diat, Thierry Douki, Jean-Paul Duraud, Engin Molva,Jean-Paul Périn, Annie Pesnelle, Peter Seyfert, Ricardo Sousa

Layout Publicum

PhotographyTotem de Pierre, CEA

[email protected]

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The fourth edition of the Bulletin of the DRFMC (Department of Fundamental Researchon Condensed Matter) is dedicated to the theme "Theory and simulation" and contains adescription of the activities of the department in this field.

Theory and simulation are connected via statistical physics and constitute an importantsector of the research activities on condensed state physics and chemistry. Comparison withexperimental data in fact validates models which, in return, enable the behaviour of naturalphenomena of the physical world that surrounds us to be predicted or confirmed bycomputation.

At the DRFMC these activities mainly concern:- physics of quantum particles in interaction such as heavy fermions, frustration phenomena,quantum magnetism,- physics of magnetic nano-objects, semiconductors and superconductors,- physics of surfaces, interfaces and thin films,- physics of soft matter (polymers, foams), - quantum chemistry of f element complexes,- physics and chemistry at the interface with biology.

Alongside the study of fundamental subjects, simulation of the behaviour oftechnological achievements naturally takes its place on account of the department's desire toaccompany knowledge transfer to applications.

Finally all these works are backed-up and made possible by the development ofsimulation methods.

For reasons of editorial convenience, the contributions that illustrate these activities aregrouped together according to the organisation of the department, which in a first approachcorresponds to a breakdown by theme. I hope that reading this review will enable you todiscover the richness of the activities of the department in this field.

Enjoy your reading.

Jean-Paul DuraudHead of the DRFMC

F o r e w o r d

WAVELETS APPLIED TO ELECTRONIC STRUCTURE CALCULATIONS

C o n t e n t s

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QUASI-1D SPIN CHAINS: FROM THE SAWTOOTH LATTICE TO THE HEISENBERG CHAIN

OSCILLATIONS OF THE LONGITUDINAL MAGNETO-RESISTANCEIN 2D METALLIC SYSTEMS

SPATIAL STRUCTURE OF COOPER ELECTRONIC PAIRSFROM CURRENT NOISE MEASUREMENTS

THE KONDO EFFECT: FROM MACRO TO NANOSCALE

CHARGE TRANSPORT IN DNA

EPITAXY, SURFACE INSTABILITIES AND ROUGHNESS

ENHANCED MAGNETO-CALORIC EFFECT IN FRUSTRATED MAGNETS

LINKING AB INITIO AND MONTE-CARLO CALCULATIONS

A FOURFOLD COORDINATED POINT DEFECT IN SILICON

INCOMMENSURATE INTERFACES IN METALS

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SOME ASPECTS OF DNA BASES STUDIED BYQUANTUM CHEMISTRY TOOLS

BOILING CRISIS: THEORY, SIMULATION, AND EXPERIMENTS

PULSE TUBE CRYOCOOLERS

Gd3+-BASED CONTRAST AGENTS IN MAGNETIC RESONANCE IMAGING

HOW TO SEE THE LOCAL STRUCTURE OF A PROTEINWITHOUT CRYSTALLOGRAPHY

SOLIDITY OF LIQUID FOAMS

FROM DNA CHIPS TO STEALTH LIPOSOMES: BIOTECHNOLOGY-INSPIRED THEORY

CELLULAR ADHESION AND CELLULAR MOTILITY, A PLAYGROUND FOR PHYSICISTS

MULTI-TRACK READING HEADS

ELECTRON TRANSPORT PERPENDICULAR TO THE INTERFACESIN MAGNETIC MULTILAYERS

SIMULATION OF GRAZING INCIDENCE SMALL ANGLE X-RAYSCATTERING TWO-DIMENSIONAL IMAGES

QU A S I -1D S P I N C H A I N S : F R O M T H E S AW T O O T HL AT T I C E T O T H E HE I S E N B E R G C H A I N

QUASI-1D SPIN CHAINS: FROM THE SAWTOOTHLATTICE TO THE HEISENBERG CHAIN

Quasi-1D spin chains with a sawtooth shape are studied theoretically. Using an exactdiagonalization and extrapolation method, the ground state and low-energy excitations aredetermined as a function of the different magnetic couplings. Applied to YCuO2.5, the methodpredicts a dispersive spectrum of excitations, with a possibly vanishing gap.

Nous étudions théoriquement des chaînes de spin quasi-unidimensionelles en forme de dentde scie. A partir d’une diagonalisation exacte et d’une extrapolation, nous déterminons l’étatfondamental et les excitations de basse énergie en fonction des différents couplages magnétiques. Appliquée à YCuO2.5, cette méthode prédit un spectre dispersif des excitationsavec un gap pouvant devenir nul.

S. A. Blundella

and M. D. Núñez-Regueirob

a: Service de Physique Statistique,

Magnétisme et Supraconductivité

(SPSMS), DRFMC

b: Laboratoire de Physique des Solides,

Univ. Paris-Sud

There is currently great interest in thephysics of antiferromagnetic quantum spinsystems of reduced dimensionality, such asthe one-dimensional spin chains shown inFig. 1. The simplest example in 1D, shownin Fig. 1a, is the much-studied Heisenbergchain, a linear chain of spins 1/2 in whichnearest neighbours are coupled antiferro-magnetically. A more recent example is thezigzag spin ladder shown in Fig. 1b.

Here we shall consider another exam-ple, the sawtooth or ∆-chain shown in Fig.1c, which consists of spins 1/2 coupled in a1D chain of triangles. The ∆-chain has beenextensively studied for the case when allspin-spin interactions have equal strength(i.e., when the base-base and base-vertexcouplings are equal, Jbb = Jbv). The groundstate in this case may be shown to be two-

fold degenerate. It consists of pairs of spins(one belonging to the base and the other toa neighbouring vertex, as in Fig. 1d) cou-pled antiferromagnetically to form a spin-singlet state called a dimer. The degeneracyarises from the possibility for a base spin toform a dimer either with a spin located onthe vertex to its right (R-dimerization) or to itsleft (L-dimerization). There is also an ener-gy gap to the lowest-energy excitations,which are known as kinks and antikinks. Akink (antikink) consists of a domain wallseparating a region of R-dimerization on theleft (right) from a region of L-dimerizationon the right (left). Such domain walls, whichseparate the system into regions of differingtranslational symmetry, belong to the gener-al class of topological excitations introducedby Shastry and Sutherland more than twen-ty years ago.

Figure 1: Quasi-1D spin chains:(a) the Heisenberg chain, (b)the zigzag ladder, (c) the saw-tooth or ∆-chain. Each site cor-responds to a spin 1/2, andsingle lines indicate an antifer-romagnetic Heisenberg cou-pling of the form J si.s j (J > 0).(d) shows the two degenerateground states of the symmetric∆-chain (where all interactions Jare equal), with double linesindicating the formation of aspin-singlet state (or dimer) bet-ween nearest-neighbour spins.

Heisenberg chain (Jbb = 0 orJbv = 0). By studying thewavefunctions, we find thatthe gap closure is related tothe gradual disappearanceof dimerization in the quan-tum ground state.

Preliminary low-tem-perature susceptibility mea-surements on YCuO2.5 sug-gest a gap with Jbv of order80 K or more, a reasonablevalue for this system. Furtherexperiments and detailedcomparison with our numeri-cal results will permit one todetermine whether the ∆-chain model is applicable tothis compound, and if so toinfer the value of Jbb/Jbv. Thisfirst complete theoreticalstudy should assist the inter-pretation of these interestingexperiments.

Despite the large amount of theoreti-cal work on these 1D spin systems, there isa lack of clear physical realizations of anyof the various models discussed. However,experimental results that hold the promiseof displaying quantum topological excita-tions have recently become available,involving overdoped delafossites RCuO2+X

(where R=Y, La, etc.). The recent synthesis[1] of YCuO2.5 has allowed one to elucidateits detailed structure, which suggests thatthe compound forms a nice realization ofthe ∆-chain, most likely with Jbb < Jbv.

Therefore, we analyze here the ∆-chain for various ratios Jbb / Jbv of theseantiferromagnetic couplings. Now, foreither Jbb = 0 or Jbv = 0 the system is equiv-alent to the Heisenberg chain. Thus, tounderstand the YCuO2.5 compound, it isimportant to study the entire evolution of theelementary excitations from the sawtoothlattice to the Heisenberg chain as a functionof Jbb /Jbv. The transition between these twolimits is not immediately obvious: the sym-metric ∆-chain has a small dispersionlessgap with kink and antikink excitations, whilethe isotropic Heisenberg chain has no gapand pairs of so-called spinon excitationsexhibiting a strongly dispersive spectrum.

We are unable to solve analyticallyand exactly for the wavefunction and dis-persion for arbitrary Jbb / Jbv, but many ofthe important features of the spectrum canbe obtained with high numerical precisionby a careful exact diagonalization andextrapolation procedure. In this method,one extracts the eigenvalues and eigenvec-tors of the model spin Hamiltonian essen-tially exactly for finite-sized systems, in ourcase for up to 12 triangles (24 spin sites).This is possible because the Hamiltonianmatrix, while large (roughly 224 x 224), isalso highly sparse (most elements are zero),and may be diagonalized by iterative

approaches such as the Lanczos orDavidson methods. We then extrapolatethe results to the infinite-size limit. Exactdiagonalization, by now a well establishedtechnique, remains a flexible and powerfultool, capable of treating both ground andexcited states with various momenta andspin quantum numbers and of extractingexcitation gaps and dispersion spectra, andalso of studying dynamical or thermody-namical properties.

Figure 2 summarizes the behavior ofthe excitation gap for the ∆-chain. We find[2] the lowest-lying excitations to have totalspin quantum numbers Stot=0 and 1,which are degenerate within numericalerror, thus giving a four-fold degeneracyoverall. The gap is greatest when the spincouplings are equal (Jbb=Jbv), but decreas-es as Jbb/ Jbv moves away from unity; at thesame time, the spectrum, which is disper-sionless for Jbb = Jbv, acquires increasingdispersion. The gap finally closes atwavevector k = 0 for Jbb / Jbv < 1 and at k= π for Jbb / Jbv > 1. In this way, the gapped ∆-chain with equal couplings (Jbb = Jbv)goes over smoothly into the gapless

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[1] G. Van Tendeloo, O. Garlea, C. Darie, C. Bougerol-Chaillot, and P. Bordet, J. Solid State Chem. 156, 428 (2001). [2] S. A. Blundell and M. D. Núñez-Regueiro, Eur. Phys. J. B 31 (Rapid Note), 453 (2003);

J. Phys. Condens. Matter 16, S791(2004). Contact: [email protected]

Figure 2: Gap to the lowest spin-one excited states of the sawtooth lattice vs. Jbb/Jbv,after extrapolation to infinite size.

OS C I L L AT I O N S O F T H E L O N G I T U D I N A L M A G N E T O-R E S I S TA N C E I N 2D M E TA L L I C S Y S T E M S

OSCILLATIONS OF THE LONGITUDINAL MAGNETO-RESISTANCE IN 2D METALLIC SYSTEMS

The quantum theory of magneto-resistance for currents perpendicular to the layers in quasi two-dimensional metals is developed. It is found that a pseudo-gap structure in the spectralconductivity between the Landau levels is responsible for the inverse thermal activated dependenceof the resistivity maxima experimentally observed in high magnetic fields.

Nous développons la théorie quantique de la magnéto-résistance pour des courants perpendiculaires aux couches de métaux quasi bi-dimensionnels. Nous trouvons qu’une structure avec un pseudo-gap dans la conductivité spectrale entre les niveaux de Landauexplique la variation expérimentalement observée des maxima de résistivité en champ magnétique élevé. Ces derniers suivent une loi d'activation thermique inversée.

T. Champel and V. P. Mineev

Service de Physique Statistique,


(SPSMS), DRFMC

The electronic properties of metalsare essentially described in terms of a gasof electrons. The electronic states are thencharacterized by the electron momentum kand the energy ε which is a continuousfunction of the momentum: ε= h2k2/2mwhere m is the electron mass. Each statewith a given k can be occupied twice only(spin up and spin down states), which leadsat zero temperature to filling of all the stateswith energies ε below some energy εF (theso-called Fermi energy).

If we apply a magnetic field H to ametal, the continuous set of electronicstates transforms to discrete levels, calledLandau levels, which have quantized val-ues for the energyε=hω c(n+1/2)+h2 kz

2/2m, where n is aninteger, ω c=eH/mc the cyclotron pulsationand kz the momentum along the fielddirection z . These levels are stronglydegenerated and the degeneracy is pro-portional to the magnetic field. Let usneglect for a moment the dispersion alongz. Then again, at zero temperature, all fullyoccupied Landau levels lie below the Fermienergy. With increasing magnetic field,both the energy and the degeneracy of theLandau levels increase. Hence when thehighest filled Landau level intersects theFermi level the electrons are transferred tolower states. For a higher value of magnet-ic field the process is repeated with the nextLandau level, and so on. On the macro-scopic scale, this periodic process mani-fests itself through oscillations of the mag-netization (the “de Haas-van Alphen”effect) or the resistance (the “Shubnikov-deHaas” effect) with the magnetic field.

Actually, if one considers kz, the Fermienergy is crossed by many occupied Landaulevels. The non-quantized states availablefor any amplitude of magnetic field act as areservoir. The discrete nature of the Landaulevels is thus blurred. As a result, the oscilla-tions of the magnetization or of the magne-to-resistance become relatively small.

The quantitative theory of the mag-netic quantum oscillations in 3D-metalswas proposed in 1955 by Lifshitz andKosevich. It has been verified by manyexperiments and is still used to determinethe real Fermi surface of metals.

Recently, there has been a renewedinterest in the study of compounds withstrongly anisotropic electronic properties inhigh magnetic fields. For example, in lay-ered heterostructures, a 2D-electron gascan be artificially achieved. In this case,there is no Landau level broadening due todimensionality (kz=0), and the oscillationsof the thermodynamic and transport prop-erties can therefore be very strong. Also,compounds such as synthetic metals basedon organic salts are naturally low-dimen-sional electronic systems: they can usuallybe described in terms of a quasi 2D gas ofelectrons, i.e. they consist of highly con-ducting planes with a small electron hop-ping probability between the layers.

Features of the magnetic quantumoscillations are known to be noticeably dif-ferent in 2D systems compared to 3D sys-tems. For example, the 2D de Haas-vanAlphen effect is characterized by a sharpsawtooth-like shape at low temperatures,

while the oscillations of the magnetizationin 3D metals are always smooth.

In quasi 2D systems, the situation con-cerning the Landau level broadening is insome sense intermediate between the 2Dand 3D cases and depends on the distancehω c between these levels compared to theinterlayer coupling energy: the Landau levelbroadening due to dimensionality is large atrelatively small magnetic fields. At very highmagnetic fields, broadening only occursthrough impurity effects, exactly like in 2Dsystems. The (semi-phenomenological) the-ory of the de Haas-van Alphen effect inquasi 2D systems has revealed a crossoverbetween the 2D and 3D limits for thebehaviour of the magnetization oscillations.

It is with the same idea of an appar-ent dimensionality reduced by the magneticfield that we studied the magneto-resistanceoscillations in quasi 2D systems [1]. Wewere rather interested in the limit of highmagnetic fields, motivated by the experi-ment of Nam et al. [2] on the organic lay-ered conductor β''-(BEDT-TTF)2SF5CH2CF2SO3. Giant Shubnikov-deHaas oscillations of the longitudinal mag-neto-resistance ρzz have been observed athigh magnetic fields and low temperatures(see Fig. 1 taken from [2]; here z is thedirection of the magnetic field which is per-pendicular to the layers). In particular, aregime with an inverse thermally activatedbehaviour of the maxima of the magneto-resitivity ρzz has been pointed out for mag-netic fields higher than 20 T [2]. Such afeature is completely absent in the 3DShubnikov-de Haas theory.

Our analytical calculations [1], devel-oped in the framework of the quantum trans-port theory, have revealed a pseudo-gapstructure in the spectral magneto-conductivi-ty σzz(ε) between the Landau levels in highmagnetic fields. In other words, when theenergy of the electrons lies between theLandau levels, the conduction at a finite low

temperature T is determined by the thermalexcitation at the edges of the gap-like inter-val ∆ between the levels. This explains whythe maxima of ρzz have an inverse thermallyactivated temperature dependence: ρzz

max ~exp(∆/T ). At very low temperatures, quasi-particles whose energy lies in the pseudo-gap range ∆ mainly contribute to conduc-tion, then the resistivity maxima saturate.

Our numerical calculations of ρzz atdifferent finite temperatures shown in Fig. 2reproduce well the appearance of hugeoscillations under the same conditions oftemperatures and magnetic fields as inexperiment [2]. In particular we successfully

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[1] T. Champel and V. P. Mineev, Phys. Rev. B 66, 195111 (2002).[2] M. S. Nam et al., Phys. Rev. Lett. 87, 117001 (2001). Contact: [email protected]

obtain inverse thermal activa-tion of ρzz maxima for fieldsH ≥ 20 T.

Figure 2 : Theoretical oscillations of ρzz/r0. The different curves cor-respond to the same temperatures as shown in Fig. 1; ρ0 is the resis-tance at zero magnetic field and zero temperature.

Figure 1 : Oscillations of ρzz in the layered conductor β''-(BEDT-TTF)2SF5CH2CF2SO3 measured at different temperatures (from the top,0.59, 0.94, 1.48, 1.58, 1.91, 2.18, 2.68, 3.03, 3.38, 3.80, and 4.00K) (data from [2]).

SPAT I A L S T R U C T U R E O F CO O P E R E L E C T R O N I C PA I R SF R O M C U R R E N T N O I S E M E A S U R E M E N T S

SPATIAL STRUCTURE OF COOPER ELECTRONIC PAIRSFROM CURRENT NOISE MEASUREMENTS

Measuring the fluctuations of electrical current has recently proved a way to obtain new informationon mesoscopic devices. In a superconductor attached to two normal metallic leads, the cross-correlation of the currents flowing in each of them is shown to provide information on the spatialstructure of the Cooper pairs.

Il a été récemment prouvé que la mesure des fluctuations du courant électrique permet d’obtenir de nouvelles informations sur des dispositifs mésoscopiques. Nous montrons que,pour un supra-conducteur relié à deux électrodes métalliques normales, la corrélation croiséedes courants circulant dans chaque électrode fournit des informations sur la structure spatialedes paires de Cooper.

M. Houzet



(SPSMS), DRFMC

Tunnel junctions are formed with twopieces of normal metal separated by a thinoxide layer. They are named after thequantum tunnelling of single electronswhich allows electrical current to flowthrough the insulating barrier. When oneof the electrodes becomes superconduct-ing, the conduction electrons attract eachother into Cooper pairs. A gap in the ener-gy spectrum hinders single electron excita-tions. As a result, the standard tunnellingof single electrons is barred. Instead,Andreev reflection - a process involvingtwo electrons - is observed. It allows aCooper pair from the superconductor tobe transferred into two electrons on theother side of the junction. Equivalently, itcan be viewed as the reflection of an inci-dent quasi-hole from the normal electrodeinto a quasi-particle in the same electrode.This process is expected to take place on acharacteristic distance of about the size ofthe Cooper pairs.

In principle, it should be possible forthe electrons to be transmitted in two differ-ent normal terminals provided that the dis-tance between the contacts is comparablewith the size of the pairs. This process, calledthe crossed Andreev reflection (CAR), couldbe useful to probe the spatial structure of theCooper pair directly, but how can it bedetected? It was first suggested to performconductance measurements on multi-termi-nal devices like the one represented in Fig.1. Indeed, applying a voltage to lead Bwould affect the current flowing between thesuperconductor and lead A if the normalleads were close enough. However this pro-cedure has two main drawbacks. First, CARcomes along with elastic co-tunnelling (EC),the transfer of an electron from one normallead to the other one, via the superconduc-tor, see Fig. 2. Second, CAR and EC contri-butions to the average currents are domi-nated by direct Andreev reflection in eachnormal lead, i.e. by the current associated

Figure 1: Schematic representation of the three-terminal device.

with the transfer of a Cooper pair of thesuperconductor into two electrons in the samelead. Alternatively, we found that the CAR orthe EC contribution can be picked up directlyby measuring the cross-correlation of the cur-rents flowing in each normal lead [1].

As a matter of fact, due to the discretenature of the electric charges, current fluc-tuates in time around its mean value. It is characterised by the current noise,S αβ=2 ∫dtδ Iα(t)δ Iβ(0), where δ Iα(t)=Iα(t)-Iαdescribes the fluctuations of the instanta-neous current, Iα(t), around its mean value,Iα, and α, β stand for any normal lead A,B.Current noise measurement revealed apowerful probe for electronic devices. Intwo-terminal junctions, the equilibriumnoise is simply proportional to the tempera-ture T and the conductance G of the junc-tion, S=4kBTG, according to the fluctua-tion-dissipation theorem. By contrast, atvanishing temperature, the non-equilibriumnoise is proportional to the current and theabsolute electric charge q, transferredthrough the junction. In the case of rare anduncorrelated charge transfers, like in quan-tum tunnelling, S=2qI was predicted. As aresult of the Andreev reflection, whichinvolves two electrons, this noise is doubledin a hybrid superconducting/normal metal-lic tunnel junction (q=2|e|) compared to acompletely normal device (q=|e|). Suchdoubling of the noise was first observed inour laboratory [2].

The cross-correlation, SAB, in thethree-terminal hybrid structure representedin Fig. 1 also deserved some attention.Indeed, it was predicted not only to differ bya numerical factor, but also to show a signchange with respect to normal metallicstructures [3]. A simplified explanation ofthis effect is the following. If electrons in thetwo leads are emitted from one Cooper pair,a positive correlation is expected, since bothelectrons appear at the same time in each

lead. By contrast, in the normal structure,electrons arriving one by one at the barriersare transmitted either in one or the otherlead, leading to a negative correlation.

We find that CAR contributes to thecross-correlation with a positive sign, whileEC contributes with a negative sign, andthere is no contribution at all from directAndreev reflection. The CAR and EC contri-butions can be selected independently bytuning the voltages of the normal terminals.In particular, only CAR contributes when thevoltages are the same in each normal elec-trode, while only EC contributes when thevoltages are opposite. Cross-correlation intunnelling systems thus provides a direct way(i) to probe both CAR and EC and (ii) tomeasure the sign change of the correlations.

Our quantitative estimates of theseeffects show that their observation may bewithin the reach of present technology. Thecharacteristic size for the Cooper pairs givesthe important scale to be achieved in the

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[1] G. Bignon, M. Houzet, F. Pistolesi, and F. W. H. Hekking, Europhys. Lett. 67, 110 (2004).[2] F. Lefloch, C. Hoffmann, M. Sanquer, and D. Quirion, Phys. Rev. Lett. 90, 067002 (2003).[3] J. Torrès and T. Martin, Eur. Phys. J. B 12, 319 (2001).

Figure 2: Three processes for charge transfer.

Contact: [email protected]

real device: a few tens ofnanometres for standardmaterials. Then the overallamplitude of the signal to bemeasured would comparewith the resolution alreadyachieved in the experiment[2]. The main difficulty of theexperiment is to make thecontacts as close as possible,without short-circuiting them.We hope that the experimentcan be performed soon inour laboratory to comparewith our predictions.

I. THE KONDO EFFECT IN BULKSYSTEMS

The electrical resistivity of metalsmonotonously decreases when the tempera-ture is lowered. Yet it was discovered in theearly 60's that the resistivity of metals con-taining magnetic impurities exhibits a loga-rithmic increase as the temperature is low-ered. This is the Kondo effect which takesplace when an antiferromagnetic spin inter-action exists between a localized magneticimpurity and the conduction electrons. Thiseffect results from a spin-flip of the impurityafter each collision with a conduction elec-tron. The so-called Kondo temperature, TK,emerges characterizing a crossover betweenthe high temperature regime (T>TK) wherethe impurity spin is almost free, and the lowtemperature regime (T<TK) where one con-duction electron constitutes a singlet statewith the impurity spin. The enhanced resis-tivity observed at low temperature resultsfrom the increased scattering cross-sectionof the conduction electrons off this singlet.During the scattering process, the phase ofthe conduction electron wave function ischanged. This phase shift, δ, was predictedmore than three decades ago to be π/2. Ameasurement of this phase shift, out ofscope in bulk metals, has now been madepossible in QD.

II. THE KONDO EFFECT INQUANTUM DOTS

QD can be obtained in a controlledfashion thanks to recent progress in nano-lithography [1]. The QD are coupled to two

metal leads by tunnel barriers. By acting onthe gate voltage VG, the number N of elec-trons confined in the dot can be changed.Due to its small size, the electronic energyspectrum in the dot is discrete and VG permitstuning of the energy levels. This is theCoulomb blockade phenomenon where peri-odic conductance peaks as a function of VG

are separated by almost zero conductancevalleys (Fig. 1a). Two different regimes occurdepending on the parity of N. If N is even(odd), the conductance decreases (increases)when the temperature is lowered [2]. Thisasymmetry between the behaviour observedin conductance valleys of different parity, is asignature of the Kondo effect that takes placein odd valleys. Indeed, when an odd numberof electrons is trapped within the dot andwhen the energy of the topmost singly-occu-pied level in the dot is far below the Fermienergy of the leads, a virtual process takesplace (Fig. 1b) leading to a spin-flip in theQD. This is precisely the Kondo effect pre-sented above, where the impurity is substitut-ed by the QD. Note that in QD, the Kondoeffect results in an enhancement of the con-ductance and not of the resistance as in bulkmetals.

III. TRANSMISSION PHASE SHIFTS

An Aharonov-Bohm interferometer is adevice made of two conducting arms, thread-ed by a magnetic flux. It enables the quantuminterference between the wave functions ofelectrons passing in each of the arms to beobserved. By embedding a QD in one arm(see inset in Fig. 2), the phase shift δ experi-enced by the electron wave-function as it

TH E KO N D O E F F E C T: F R O M M A C R O T O N A N O S C A L ETHE KONDO EFFECT: FROM MACRO TO NANOSCALE

The Kondo effect is responsible for an anomalous low temperature behavior of resistivity in certainmetallic alloys. Recently fabricated quantum dots (QD) have led to a spectacular revival of thiseffect. We discuss here interferometry experiments with QD allowing for a direct measurement ofthe transmission phase shift, a fundamental quantity of the Kondo effect.

L’effet Kondo est responsable du comportement anormal de la résistivité à basses températuresdans certains alliages métalliques. L’élaboration récente de points quantiques (QD) a conduit à un renouveau spectaculaire de cet effet. Nous discutons ici des expériences d’interférométrie en présence de QD permettant d’accéder au déphasage de transmission, une quantité fondamentale de l’effet Kondo.

M. Lavagnaa, P. Vitushinskya, and A. Jerezb

a: Service de Physique Statistique,


(SPSMS), DRFMC

b: European Synchrotron Radiation

Facility, Grenoble

passes through the QD can be measured.The evolution of δ as a function of VG hasbeen obtained in recent experiments [3](Fig. 2). The results are surprising: a smoothincrease of δ with VG is observed, with avalue in the Kondo regime close to π whichdiffers from earlier theoretical predictions.

In order to clarify the experimentalsituation, we have theoretically taken intoaccount the difference of geometrybetween a QD and the bulk metal. Thetransmission phase shift introduced by theQD is analyzed in terms of a scatteringproblem in one dimension instead ofthree dimensions in bulk metals. We haveshown that this leads to a drastic changein the phase shift [4]. Using Landauer’sformula, we established the relationG∝ sin2(δ/2). Figure 2 compares this the-oretical prediction with experimental dataand, as can be seen, the agreement foundis very good. Moreover we have shownthat the phase shift is related to the aver-age number of particles in the localizedlevel nd according to a generalized Friedelsum rule. By performing exact Bethe-Ansatz calculations in the Andersonmodel, we can determine nd , and henceδ . In the presence of a single fittingparameter corresponding to the couplingstrength between the dot and the metalleads, the agreement between theoreticaland experimental results are very satisfactory.

Our project is to now extend ouranalysis to the presence of a magneticfield and to other situations leading to theexistence of new regimes characterized bynon-Fermi liquid behaviour reminiscent ofthe multichannel Kondo effect [5].

ACKNOWLEDGEMENTS

We thank D. Bensimon for collabora-tion on related subjects and L.Saminadayar, C. Baüerlé, P. Simon, D.Feinberg and F. Hekking with whom we arecarrying out a common project on this topicwithin the “Institut de Physique de la MatièreCondensée” of Grenoble.

11

[1] L. Kouwenhoven and L. Glazman, Physics World 14, 33 (2001).[2] W.G. van der Wiel et al., Science 289, 2105 (2000).[3] Y. Ji, M. Heiblum, and H. Shtrikman, Phys. Rev. Lett. 88, 076601 (2002).[4] P. Vitushinsky, A. Jerez and M. Lavagna, Proceedings of the Vth Rencontres de Moriond en Physique Mésoscopique,

edited by C. Glattli, M. Sanquer, and J. Trân Thanh Vân (Editions EDP Sciences, Les Ulis/France, 2004).[5] A. Jerez, M. Lavagna, and D. Bensimon, Phys. Rev. B 68, 094410 (2003); M. Lavagna, A. Jerez, and D. Bensimon

in Concepts in Electron Correlation Vol 110, edited by A.C. Hewson et V. Zlatic (Kluwer Academic Press, Amsterdam/The Netherlands, 2003) p. 219

Figure 1: (a): Experimentalconductance of a QD as afunction of the gate voltageVG which changes the num-ber of electrons, N, con-fined in the QD.Depending on the parity ofN, the conductancedecreases or increaseswhen the temperature islowered from 1K (red) to25mK (black) (from [2]).(b): Schematization of oneof the virtual tunnelling pro-cesses in the QD-leaddevice leading to a spin-flipin the QD.


Figure 2: Comparison of the theoretical (black curve) and experi-mental (red points) conductances. The experimental values (from [3])for G and δ as a function of VG, are reported as red and blue points,respectively.

CH A R G E T R A N S P O R T I N DNACHARGE TRANSPORT IN DNA

Charge transport in deoxyribonucleic acid (DNA) is investigated in view of its biologicalimplication in damage and repair, protein bonding, and integration in (bio)electronics or DNA-based chips. We discuss basics of charge transport in the quantum coherent regime, and showthat some concepts of electronic conduction need to be revisited for DNA.

Le transport de charge dans l’acide désoxyribonucléique (ADN) est étudié du fait de sonimplication dans les lésions et la réparation, les liaisons des protéines, et l'intégration dansdes composants (bio)électroniques ou basés sur l’ADN. Nous discutons les fondements du transport de charge en régime cohérent quantique et nous montrons que certains conceptsde conduction électronique doivent être réexaminés.

S. Roche



(SPSMS), DRFMC

The double-helix structure of DNA ismade from two weakly hydrogen-bondedstrands, each of which consists of asequence of four nucleotides (A-adenine, T-thymine, C-cytosine, G-guanine). Pairing ofthe bases belonging to two strands strictlyfollows the Watson-Crick complementaritylaw (i.e. interstrand pairs only appear as ATor CG). Due to the smaller ionizationpotential of the guanine, charge transportis believed to be conveyed through thehighest occupied molecular orbitals of thisbase (HOMO-G).

In recent years, two kinds of experi-ments have provided valuable informationon charge transfer in DNA. First, fluores-cence experiments have relied on the use ofintercalated metallic complexes acting asdonor and acceptor to photo-induce andfollow charge migration along small DNAfragments [1]. Such experiments havepointed towards an efficient transfer overunprecedented length scale forbiomolecules. Other experiments havefocused on contacting DNA with conduct-ing electrodes, and performing mesoscopicconductance measurements under variousphysical conditions. Presently availableresults lack consistency since all kinds oftransport regimes have been proposed,from metallic-like, superconducting, semi-conducting to strongly insulating behaviors.Transport mechanism might be purely hole-or polaron-based.

We have recently performed a theo-retical study on several kinds of both artifi-cial or genomic DNA. To discuss basictransport mechanisms, we have elaborat-

ed on a modelling of hole transportthrough the guanine bases, describedwithin an effective 1D tight bindingHamiltonian [2],

c tn (cn) is the operator corresponding to

charge creation (annihilation) at site n, tDNA

gives the amplitude of π-π couplingbetween nearest neighbour orbitals, εn thehole site energies and cos (θn,n+1) is tem-perature-dependent and describes the low-ering of coupling between base pairs dueto thermal fluctuations [3].

Η = ∑− tDNA cos(θn,n+1)(ctncn+1 + h.c.) + εnct

n cn

Landauer formal i sm

In the Landauer formalism used here, one considers anelectronic wavepacket of amplitude unity, wave vector kand energy E which is injected from a metallic contact (seefigure) inside a DNA sequence. The coefficients R and Tgive the amplitudes which are respectively reflected by theDNA sequence and transmitted to the other metallic con-tact. Current conservation implies R + T = 1. Theinverse transmission rate R/T is related to the dimension-less resistance ρ, and for weak transmission, the DNAresistivity is proportional to [(1-Tav)/Tav] SDNA/LDNA

taking Tav an average of the transmission coefficient forenergies close to the HOMO-G ionisation energy,SDNA=3.10 -18m2 the average DNA cross section,and LDNA the sequence length [3].

13

In order to evaluate the conductivityof the system, we use the Landauer formu-lation of electron transmission (see inset)and compute the energy-dependent trans-mission coefficient T(E). Our modelling issuch that the full energy dependence of Tis necessary to discuss the charge transferproperties, although the DNA chain is con-nected to metallic leads and the electrodeFermi levels are lined up to the HOMO-G.

We have considered syntheticPoly(dG)-Poly(dC) sequences consisting ofG and C bases alternated on each strandand sequences extracted from the biologi-cal λ-bacteriophage or from the HumanChromosome 22 DNA. The transmissioncoefficient was studied for finite sequencesfrom 20 nm up to 200 nm long, i.e. repre-senting 60 to 600 pairs of bases.

As expected, Poly(dG)-Poly(dC)sequences were found to follow a “semi-conducting-like” regime mainly due to atransmission gap stemming from the twospectral bands (see Fig. 1). Temperaturewas found to reduce coherent charge trans-fer, but theoretical calculation for the DNAresistivity was in good agreement with someexperimental estimates.

Dif ferent ly, λ-bacteriophage orChromosome 22-based sequences were foundto exhibit much more resistance to chargetransport [3]. Actually, each specific A,T,C,G-based sequence follows a complex orderwhich lacks periodicity, although encompass-ing long range correlations. In our modellingof hole transport through the HOMO-G, anyother A, T or C nucleotide acts as a potentialbarrier that reduces coherent tunnelling, sothat larger aperiodic chains exhibit strongbackscattering efficiency irrespective of theenergy of the injected charges [3]. This phe-nomenon is illustrated in Fig. 2 for λ-bacterio-phage-based sequences with increasing num-ber of base pairs.

The nature and contribution to chargetransfer of long range correlations that dis-play complex patterns in different genomicchains have also been the subject of a fullstudy. It was shown that if compared withrandom sequences with uncorrelated distri-bution of nucleotides, charge transfer inChromosome 22-based sequences wasfound to be much more efficient [4].

In the genomic sequences, it is notpossible to estimate any resistivity sinceresistance increases considerably with

Figure 1: Theoretical transmission coefficients for periodic Poly(dG)-Poly(dC) sequences with 60 base pairs, at twotemperatures.

[1] C. R. Treadway, M. G. Hill and J. K. Barton, Chem. Phys. 281, 409 (2002). [2] M. A. Ratner, Nature (London) 397, 480 (1999).[3] S. Roche, Phys. Rev. Lett. 91, 108101 (2003).[4] S. Roche, D. J. Bicout, E. Macià, and E. I. Kats, Phys. Rev. Lett. 91, 228101 (2003). Contact: [email protected]

sequence length. Thermalfluctuations that reduce π−πcoupling strength betweenintrastrand next neighbournucleotides also strongly dis-favour charge transport.From our calculations, itappears that strong insulat-ing behaviours are expectedfor length above ~ 10 nmin genomic sequences, andthat periodic syntheticsequences may sustain longrange transport over 20 nmat small temperatures.

To summarize, we havepresented some charge trans-port modelling and basicresults concerning artificialand biological DNA. Thecharge transfer was shown tobe critically dependent on thesequence nature and length ofbase pairs as well as thermalfluctuations.

Figure 2: Theoretical transmission coefficients for λ-bacte-riophage-based sequences with increasing number of basepairs.

EP I TA X Y, S U R FA C E I N S TA B I L I T I E S A N D R O U G H N E S SEPITAXY, SURFACE INSTABILITIES AND ROUGHNESS

Instabilities during epitaxial film growth may either be modelled by a linear equation, or follow aslow thermal activation process. It is shown here that a third model is possible in a growingcrystal, when roughness is first induced by growth and generates bumps that eventually reach acritical size.

Dans un film épitaxié, les instabilités lors de la croissance peuvent être modélisées par une équation linéaire ou suivre un processus lent d’activation thermique. Nous montrons qu’un troisième modèle est possible lorsque la rugosité est d’abord induite par la croissanceet qu’elle génère des îlots qui atteignent une taille critique.

J. Villain



(SPSMS), DRFMC

In semiconductor technology, epitaxialcrystal films are often grown in the (001)direction. Their surface is initially smooth, butbumps often appear during growth. Thesebumps can then be “buried” and used asquantum dots. A picture of such quantumdots of GaN on AlN, obtained by atomicforce microscopy, is shown in Fig. 1.

These bumps originate from an insta-bility of the smooth surface, which resultsfrom the misfit between the crystal constantsof the substrate (a0) and of the film (a = a0 +δ a). If the surface is flat, the epitaxial filmmust accept the crystal constant of the sub-strate. In contrast, if bumps are present, thisconstraint is partly relaxed at the top of abump. Thus, a surface with bumps has alower energy than a flat surface, which istherefore unstable. In the simplest cases, theheight h(t) of the bump (Fig. 2) evolves withtime t according to the linear differentialequation

ddt

and the smooth surface is unstable if K>0. Inthe case of a crystal surface, formula (1) canbe derived assuming [1,2]: (i) no dislocations(especially no misfit dislocations), (ii) a tem-perature higher than the so-called “roughen-ing transition temperature” TR of the surface,above which the surface becomes rough asan effect of thermal fluctuations. Condition (i)is verified, at least below a critical thickness,for films of Ge on Si, for InAs/GaAs, etc.Condition (ii) is probably not verified for the(001) surface of semiconductors. Also, inpractice, h in Eq. (1) is the amplitude of asinusoidal surface height modulation ofwavelength R. Because K depends on h and

R, Eq. (1) is actually a non-linear equation. Inthe remainder of this article, condition (ii) isassumed not to be verified. We therefore con-sider that film growing is at the origin of insta-bilities rather than thermal fluctuations.

The energy of a bump consists of twoterms: a capillary surface energy δFcap whichtends to reduce the bump area, and an elas-tic energy δFel which favours its creation inorder to reduce the stress in the film. A calcu-lation based on standard elasticity yields for abump of volume R2h (with h<R)δFel ≈ – Mε 0

2 Rh2 (2)where M is a combination of elastic constantsand ε0=δ a/a is the misfit between the latticeconstants of the substrate and the adsorbate.For T<TR , the surface free energy isδFcap ≈ γ s Rh (3)where γs is the free energy of a unit length

h=Kh (1)

Figure 1: Atomic force microscopy of quan-tum dots of GaN grown on a AlN substrate.The colour scale indicates the depth profile(courtesy of B. Daudin).

step. A step is the edge of a new atomic layer.γ s would vanish for T>TR. The smooth surfaceis unstable with respect to bump formation ifδFcap+ δFel<0. This implies h>h c with

Thus, the instability appears only after thebump has reached a sufficient size. Belowthat size, the energy is positive. There is anactivation energy ∆F ≈ Mε0

2 Rhc2 .

It may be shown that R should be ofthe order of hc so that ∆F ≈ Mε0

2 hc3 ≈

γ 3sM-2 ε0

-- 4. To summarize, there is an “acti-vated” instability for T<TR , while for T>TR

the instability is “linear”, i.e. described bythe linear equation (1). But this is not thewhole story. A third type of instability is pos-sible, as will be seen now [4].

In the above argument indeed,growth has been ignored. The remainderof this report is devoted to the case of agrowing crystal surface below T R. Theproblem is the following. At the tempera-tures used in epitaxial growth, the (001)surface of a semiconductor has very fewsteps at thermal equilibrium. Howeverwhen the film grows, it necessarily has anon-vanishing density of steps whichdepends on the growth rate. Tersoff [3] hassuggested that a crystal surface which issmooth at equilibrium (i.e. T<T R) becomesrough when it grows. Tersoff introducedelasticity into a work by Nozières & Galletwhich is a part of the Holy Writ of crystalgrowth. In both papers, roughness is pos-tulated rather than demonstrated. Tersoff'sconclusion is that the instability of a grow-ing epitaxial film is described by a linearequation. This can be correct in certaincases, but in many problems the effect oftime (ignored by Tersoff) should be care-fully taken into account [4].

As said above, the usual situation insemiconductor technology is that of an initial-ly smooth (001) surface. Even in the absenceof any misfit, growth generates a roughnessof the surface as a result of random nucle-ation of new atomic layers. It increases withtime. At infinite time, a growing surface is(kinematically) rough, in agreement with thestatements of Nozières & Gallet, and Tersoff.However, for an infinite surface, a steadyregime is not reached in any finite time! Themaximum lateral size ζ(t) of the bumps andholes generated by random nucleation goesto infinity with time, as does their maximumheight h1(t). Scaling arguments suggesth1(t)∝ ζα with α > 0.

A weak misfit is not expected to changethis situation for short times. After some time,however, the size of the bumps created byrandom nucleation becomes larger than thesize hc given by (4). Then an instability shouldtake place, driven by elasticity and misfitrather than nucleation randomness. At longenough times, the evolution may follow thelinear equation (1) which triggers an expo-nential increase, h(t)∝ exp(Kt). However, thisperiod is preceded by a long one where theinstability does not show up. The instabilitysets in at a fairly well-defined time, in contrast

15

[1] A. Pimpinelli, J. Villain, Physics of Crystal Growth (Cambridge University Press, Cambridge/U.K., 1998) and references therein.[2] P. Politi, G. Grenet, A. Marty, A. Ponchet, and J. Villain, Physics Reports 324, 271 (2000) and references therein.[3] J. Tersoff, Phys. Rev. Lett. 87, 156101 (2001).[4] J. Villain, C.R. Physique 4, 201 (2003) and references therein. Contact: [email protected]

with both linear and activatedinstabilities. It is a new type ofinstability [4]. The abovedescription is in qualitativeagreement with experimentfor weak misfits (δa/a<2%).The case of large misfits(which is of greater practicalinterest) is different. Nucleationoccurs suddenly at a criticalcoverage. We are currentlydeveloping a model whichtakes into account the dis-tortion at the edges of a newatomic layer, which proba-bly drives the formation ofthe bump.

hc = (4)

Figure 2: Transmission electron microscopy of a cut of GaN quan-tum dots (dark grey) in a matrix of AlN (light grey). The lateraldimension R and the height h of a dot are indicated (courtesy ofJ.-L. Rouvière).

γ s

Mε02

EN H A N C E D M A G N E T O-C A L O R I C E F F E C T I NF R U S T R AT E D M A G N E T S

ENHANCED MAGNETO-CALORIC EFFECTIN FRUSTRATED MAGNETS

The magneto-caloric effect is characterized by a temperature change in response to a variation ofan applied magnetic field; this effect is especially important in frustrated magnets. Thethermodynamics of demagnetization of such systems, from a fully polarized state at high field toa partially polarized one at low field, are theoretically discussed.

L’effet magnéto-calorique se caractérise par une variation de température en réponse à une variation du champ magnétique appliqué; cet effet est particulièrement important dans lessystèmes magnétiques frustrés. La thermodynamique de la désaimantation de tels systèmes,entre un état totalement polarisé à fort champ et un état partiellement polarisé à faiblechamp, est théoriquement discutée.

M. E. Zhitomirsky



(SPSMS), DRFMC

The term frustration generallydescribes a competition between variousforces and interactions without a definitewin. Such a competition leads to a largeresidual degeneracy of low-energy states. Itis for instance responsible for unique prop-erties of water ice, superconductingJosephson junction arrays, neural networks,and new magnetic materials [1].Geometrical frustration develops in mag-netic insulators with special types of crystallattices (see inset 1). Kagomé andpyrochlore lattices shown in Fig. 1 are twoexamples achieved, for instance, inSrCr9pGa12-9pO19 and in Gd2Ti2O7 respec-tively. Antiferromagnetic exchange interac-tion between local spins S on such lattices isunable to stabilize a unique spin configura-tion. It is always possible to rotate finitegroups of spins without any energy cost. Atlow temperatures the system fluctuatesbetween an infinite number of isoenergeticspin configurations. No conventional mag-netic ordering is possible at low tempera-tures in such frustrated magnets [2].

Macroscopic degeneracy of frustratedantiferromagnets persists in external mag-netic fields up to a so-called saturation fieldHsat [3]. This field is of the order of theexchange constant J and depends on thelattice: Hsat=6JS for Kagomé and Hsat=8JSfor pyrochlore lattices. Above the saturationfield, the energy of the magnetic momentsin the external field takes predominanceover antiferromagnetic interactions and sta-bilizes ferromagnetic alignment of all spinsparallel to the field direction. The thermo-dynamics of the reverse phase transforma-tion from a fully magnetized state at H>Hsat

into a partially magnetized state at H<Hsat

present a number of unusual features infrustrated antiferromagnets [4].

Reduction of the total magnetizationin antiferromagnets is generally controlledby flips of individual spins. Magnetic inter-actions between lattice sites transform spin-flips into collective propagating excitationswith a finite energy gap proportional to H-Hsat above the saturation field. In ordinarynon-frustrated antiferromagnets the gapvanishes at H=Hsat for a specific wave-vec-tor, which determines the ordering wave-vector of a canted antiferromagnetic struc-ture below Hsat. Frustrated antiferromagnets(Kagomé, pyrochlore, garnet etc) have, incontrast, a flat momentum-independentbranch of spin-flips with vanishing gap atH=Hsat. The presence of such a branch is aconsequence of a macroscopic degeneracyof the ground state below Hsat. Close to thesaturation field, corresponding low-energyexcitations behave as effective paramagnet-ic degrees of freedom. Such emergent exci-tation degrees experience an effective mag-netic field (H-Hsat). Therefore, as Happroaches Hsat under adiabatic conditions(see inset 2), the frustrated magnet shouldcool down in a very similar way to an adia-batic demagnetization of an ordinary para-magnetic salt. The main difference betweenthe two processes is that a frustrated mag-net tends to reach the lowest temperature atH=Hsat, while an ordinary paramagnetreaches the lowest temperature at H=0.Analytic theory of the demagnetization pro-cess of frustrated magnets requires goingbeyond a simple harmonic approximation.It is still under development and beyond the

17

scope of this article. Instead, a numericalapproach is presented here [4].

Specifically, we consider Heisenberg inter-actions between neighbouring spins. Large valuesof spins of magnetic ions in frustrated magneticmaterials: S=3/2 (Cr3+) and S=7/2 (Gd3+) allowus to approximate quantum spins as classicalmagnetic moments. Under such an approxima-tion, powerful Monte Carlo techniques can beused for simulation of large finite clusters (typical-ly N>1000) in order to derive the thermodynam-ic behaviour of macroscopic samples. The stan-dard Monte Carlo algorithm corresponds to simu-lations at a fixed temperature. We have imple-mented a modified algorithm, which makes it pos-sible to numerically simulate the adiabatic processat constant entropy.

In Fig. 2 we present results for adiabaticdemagnetization of two frustrated antiferromag-nets in comparison with an ideal paramagnet,which has a linear dependence T/H=cst (see inset2). The adiabatic temperature decrease for frus-trated magnets is much steeper than for a param-agnet in a wide interval of magnetic fields. Thetemperature of a pyrochlore antiferromagnetdrops by more than ten times as the field decreas-es from 1.5Hsat to 0.8Hsat. To achieve a similarcooling with a paramagnetic salt, an applied fieldhas to be decreased by at least a factor of ten. Thepredicted cooling effect should be easily observ-able, for instance, in Gd2Ti2O7, where Hsat=6.5 Tand JS2/kB=3.7 K. The obtained results suggestthat frustrated antiferromagnets with suitable satu-ration field values may be utilized as alternativerefrigerant materials.

Figure 2: Temperature variation during adiabatic demag-netization of pyrochlore and Kagomé antiferromagnets.The straight line corresponds to the demagnetization pro-cess of an ideal paramagnet.

[1] A. P. Ramirez in Handbook of Magnetic Materials Vol. 13, edited by K.H.J. Buschow (Elsevier, Amsterdam, 2001) p.423.[2] J. T. Chalker, P. C. W. Holdsworth, and E. F. Shender, Phys. Rev. Lett. 68, 855 (1992).[3] M. E. Zhitomirsky, A. Honecker, and O. A. Petrenko, Phys. Rev. Lett. 85, 3269 (2000).[4] M. E. Zhitomirsky, Phys. Rev. B 67, 104421 (2003).

1. Geometr i ca l f r us t ra t ion

Let us consider magnetic moments located at the vertices of equilateral triangles andcoupled antiferromagnetically (neighbouring moments have anti-parallel orienta-tions). This coupling scheme cannot be satisfied simultaneously for all these moments(see figure). This illustrates the so-called geometrical frustration.

2. Adiabat i c demagnet iza t ion

The adiabatic demagnetization technique based on the magneto-caloric effect hasbeen successfully used to reach temperatures in the sub-kelvin range. During an adi-abatic process, i.e. for a thermal isolation from environment, the total entropy of thesystem remains constant. Paramagnetic salts, which are standard refrigerant materi-als for low temperature cooling, consist of non-interacting magnetic moments (spins).The two energy scales are the Zeeman energy and temperature. Therefore, theentropy of an ideal paramagnet depends on field and temperature via H/T. Thismeans that during adiabatic demagnetization, temperature of a paramagnetic saltdecreases at best linearly with magnetic field.

Figure 1: Geometrically frustrated lattices: Kagomé (a) and pyrochlore (b).


?

LI N K I N G A B I N I T I O A N D MO N T E-CA R L OC A L C U L AT I O N S

LINKING AB INITIO AND MONTE-CARLOCALCULATIONS

Fast algorithms allow the behaviour of materials at the atomic level to be simulated on powerfulcomputers, and their macroscopic properties to be deduced. Large systems with low concentrationof alloying atoms or defects can henceforth be studied. The properties of high quality materialslike silicon and its alloys (SiGe, SiGeC) are therefore now accessible to simulation.

Des algorithmes rapides permettent de calculer le comportement des matériaux à l’échelleatomique, et d’en déduire des propriétés macroscopiques. On peut désormais étudier degrands systèmes comprenant une faible densité d’atomes d’alliage ou de défauts, donnantaccès aux propriétés de matériaux de haute qualité tels le silicium et ses alliages.

D. Caliste, Th. Deutsch, F. Lançon, and S. Goedecker

Service de Physique des Matériaux

et des Microstructures (SP2M), DRFMC

I. INTRODUCTION

In atomic simulation, tools are adapt-ed to the number of particles of the system.Methods simulating thousands of atomsgenerally rely on inter-atomic potentials.The atomic interaction is then simplified tothe interaction between couples of atoms,described by a potential energy function.The parameters are fitted on experimentaldata, or derived from calculations per-formed on reduced sets of atoms. Largenumbers of atoms can then be efficientlyaddressed, at the expense of accuracy, andprovided that the inter-atomic potentialbetween all couples of different atoms inthe compound is known…

We are developing methods to cir-cumvent these drawbacks. One is based onab initio calculations, solving (obviouslywith some degree of approximation) thebasic Schrödinger equation rather than fit-ting data. This method directly addressesthe electron orbitals, it is more accurate,and furthermore applicable to any atomiccomposition. It is also very expensive andtherefore limited to systems of a few hun-dred particles. The second method is thewidely used Monte Carlo simulation. Usinga model that gives the probability of all ele-mentary processes acting on each particle,an algorithm randomly picks one of theseprocesses according to its probability,changes the system configuration, againpicks a process, and so on… thus simulat-ing the evolution of the system. In this way,systems comprising millions of atoms can be

Figure 1: (a): Calculation of the energy barrier asso-ciated to the process exchanging a germanium atomwith a silicon neighbour (light green: Ge - darkgreen: Si - yellow: intermediate position - red: "pass"position). Both atoms move simultaneously, startingfrom the initial green position, passing through theyellow and red positions, where the system switchesover the “pass” towards the final valley. (b): The redto green curves correspond to successively calculat-ed paths. Only 3/4 of the movement is shown, fromthe initial position (left) to the "pass" position (right).The red curve represents the energy along the pathat the beginning of minimization. The green one rep-resents the final optimized path.

(a)

(b)

19

simulated. The accuracy is limited by theaccuracy in describing elementary process-es, and thus the choice of the model is ofcrucial importance. Our goal is to build amodel from accurate ab initio calculations.

With this strategy, we plan to simulateatomic movements called kinetic events, inthe SiGeC alloy, a compound of major tech-nological interest. We present here animproved method to calculate the energybarriers governing elementary processes,and the derivation of macroscopic quantitieswith Monte Carlo simulations, taking advan-tage of these ab initio calculations.

II. COMPUTING BARRIER ENERGIES

The energy needed to go from oneatomic configuration to another is called theenergy barrier: it governs kinetic events.While the stable configurations (includingsubstitutions of atoms) are generally known,the energy needed along the path inbetween is generally unknown. For simulat-ing such processes we need to explore all thebarriers involved during the movement. Forinstance, simulating the diffusion of Geatoms in SiGe requires the energy needed tomove a Ge atom from one substitutional siteto another to be known.

We use a method called "nudged elas-tic band" that finds a path from one configu-ration to another within different conditions ofslope of the energy variation, just like a walk-er could do for finding a pass from one val-ley to another. The program starts by select-ing intermediate atomic positions corre-sponding to a tentative movement. Usingaccurate ab initio calculations, it computesforces applied on atoms in each position andmoves them accordingly. It computes forcesin new positions and moves atoms again,and so on, until forces decrease below agiven criterion. The final path minimizes theenergy barrier to be overcome during themovement (see Fig. 1). Such calculationsinvolve large numerical resources (vectorial

computers or PC clusters), but are very accu-rate. The number of computable barriers islimited by the time available on computers.

III. TOWARDS HIGH SCALESIMULATIONS

We build a database of energy barriersusing the above-mentioned ab initio calcula-tions. During a simulation, the Monte Carlomodel will pick the parameters of the differ-ent elementary processes from the database.Using such an approach we simulate vacan-cy movements in pure silicon with low defectconcentration, illustrating the performance ofthe Monte Carlo algorithm.

Vacancies are expected to cluster insilicon, modifying their behaviour. However,due to their low concentration in siliconcrystals, they are difficult to track experi-mentally. Since modern Monte Carlo meth-ods manage large numbers of atoms, real-istic low concentrations of vacancy can bedealt with. We show that the diffusion ofvacancies is slowed down by a factor of sev-eral decades as soon as they meet oneanother (see Fig. 2). Simulation thereforeconfirms results that were reasonablyexpected but not yet quantified.


Figure 2: Diffusivity depending on temperature, calculated for vacan-cies and clusters with two and four vacancies in pure silicon.

IV. CONCLUSION

These two examplesillustrate the tools we havedeveloped: we manage largesystems with altogether a real-istic description of elementaryprocesses through ab initiocalculations, and an efficientsimulation of the movementsof vacancies or alloying atomsby means of powerful MonteCarlo methods. These meth-ods can now be linked for fullysimulating alloys like SiGeC,in which complex phenomenasuch as interactions betweengermanium, carbon, andinterstitial carbon atoms areexpected.

WAV E L E T S A P P L I E D T O E L E C T R O N I C S T R U C T U R EC A L C U L AT I O N S

C. Chauvin*, Th. Deutsch, and S. Goedecker.



I. INTRODUCTION

Solving the quantum mechanicalSchrödinger equation for systems of somehundreds of atoms would require hugeefforts, beyond computational capabilities.The Density Functional Theory (DFT) [1] cancircumvent the problem; in this frameworkthe Kohn-Sham equations are a powerfulapproximation, by which the equation hasto be solved for one electron. The potentialacting on this electron only depends on theelectron density of presence.

Mathematically the electron density isa function that must be developed into a setof functions called a basis. Efficiency is cru-cially determined by the choice of this set.We present here the wavelet basis thatemerged during the last decade as verypromising. However, once the basis hasbeen chosen an efficient code has to bedeveloped, especially for calculation of theelectron-electron interaction potential viathe Poisson equation.

A fast and accurate method address-ing systems with hundreds of electrons con-tained in a 3D box is needed. As computa-tions are performed in a discrete way, weevaluate potentials and electron density ona mesh grid. Rather than directly calculatingthe density on the real point grid, weexpand it on another one where computa-tion is more efficient. This new grid repre-sents the coefficients of the electron densityspanned on a suitable basis set. A goodaccuracy requires more grid points aroundnuclei where the density is larger (see Fig.

1). The complexity in solving Poisson andSchrödinger equations depends strongly onthis basis set. We now face two problems:how to express the different operations sim-ply and how to adapt the grid?

II. CHOOSING THE BASIS SET

Chemists first used localized basissets, but other sets can be considered.

(i) Since the functions describing theelectron density around atoms are gaussianand since it is large only near the nuclei, itseems natural to expand it on a localizedbasis set. In fact, operations to be per-formed on the simplified grid are tooexpensive, especially solving of the Poissonequation.

(ii) The plane waves basis describeswell quasi-free electrons in solids, but hasno physical meaning in the case ofmolecules or nanostructures. The coeffi-cients on this basis are derived through aFourier transform, optimizing solving of thePoisson equation. However the grid must beregular all over the box and it must be fineif good accuracy is required. Long calcula-tions are thus performed in almost emptyregions, leading to a high computationalcost.

(iii) The wavelet basis [2]. The densityis expressed on an adaptive grid: the cho-sen wavelets are localized at points where alarge density is expected (see Fig. 1). Thismethod emerged in the 80’s, and intuitive-ly appears as an efficient way to solve the

Simulation enables theory and experiments to be compared or experiments to be predicted. Forphenomena governed by atomic scale interactions, a key issue is solving the Schrödinger equa-tion. Electrical properties, for instance, can then be deduced therefrom. We present a way of solv-ing this equation, addressing systems like molecules or solids.

La simulation permet de comparer théorie et expériences ou de prévoir ces dernières. Pour les phénomènes régis par des interactions à l’échelle atomique, un point clef est la résolution de l’équation de Schrödinger. On peut alors en déduire, par exemple, des propriétés électriques. Nous présentons un moyen de résoudre cette équation, pour des systèmes tels que molécules et solides.

WAVELETS APPLIED TO ELECTRONIC STRUCTURECALCULATIONS

21

Schrödinger equation. One key is to geta good solver for the Poisson equation.While the wavelets cannot compete on aregular grid with plane wave methods,the fact that they are intrinsically welllocalized both in real and in frequencyspace (see Fig. 2) leads to a reasonableresult when details of the electron densi-ty are needed around atoms [3].

III. CONCLUSION

Development of the wavelet basismethod gives promising results [3]: weexpect the code to compete with tradi-tional methods. We are presentlyaddressing the design of a generalsolver for the Schrödinger equation.Next we will be implementing an adap-tive scheme, automatically generatinggrids for more general problems.

[1] W. Kohn and L. J. Sham, Phys. Rev. A 140, 1133 (1964).[2] I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conferences Series in Applied Mathematics

Vol 61 (SIAM, Philadelphia/USA, 1992).[3] S. Goedecker and C. Chauvin, J. Theor. Comput. Chem. 2, 483 (2003).

What is wavelet analysis?

Wavelet analysis was created in the 80’s for addressingvarious problems, from theory in mathematics and physicsto engineering applications. One famous success ofwavelets is the JPEG 2000 compression algorithm.

Previous compression algorithms used a discrete cosinetransform. A standard 640 by 480 pixel image was forexample divided into blocks of 8 by 8 pixels. Applying thetransform on each block of 64 numbers gives 64 othercoefficients, which are then ordered in a decreasing way.Keeping only the 8 most significant coefficients and apply-ing the inverse transform, an image with only a few coeffi-cients is restored, obviously at the expense of quality. Theresolution of the compressed image was in fact not suffi-cient, especially at the edges of blocks.

Wavelet analysis offered a decisive improvement: ratherthan dividing the image into blocks of fixed size, waveletanalysis addresses the different scales of the image.Consider the photography of a tree: the main features arethe trunk, boughs, and leaves. On a finer scale, the trunkroughness, and each leaf composing the green form can bedetailed; on a still finer scale, the leaf veins can be distin-guished. Correspondingly, wavelet analysis will decomposethe image from the coarsest scale (the tree in general) to thefinest one (each leaf). Suppressing irrelevant finest detailssuch as small variations of the sky, trunk, and leaves, theessential information is kept at a medium resolution. In thisway an efficient compression is achieved, while saving areasonable quality of the image. Wavelet analysis is there-fore well suited to data processing, but it is also very effi-cient in scientific calculation, as it consists of convolutionswith reasonably short filters.


Figure 2: Sections of a typical 3Dwavelet basis function (8-degreeDaubechies wavelet), in real space(up) and frequency space (down).

Figure 1: Example of a 3D system: a 3-atom molecule and a corres-ponding adaptive mesh grid.

* present address: [email protected]

A F O U R F O L D C O O R D I N AT E D P O I N T D E F E C TI N S I L I C O N

A FOURFOLD COORDINATED POINT DEFECTIN SILICON

Point defects result from the lack (vacancy), insertion (interstitial), or displacement (Frenkel pair)of an atom in a solid. They are considered as the basic defects in silicon. We predict a new pointdefect in silicon, fourfold-coordinated and lower in energy than the others.

Les défauts ponctuels résultent du manque (lacune), de l’insertion (interstitiel) ou du déplacement (paire de Frenkel) d’un atome dans le réseau. Ils sont considérés comme lesdéfauts de base du silicium. Nous prédisons l’existence d’un nouveau défaut ponctuel, tétra-coordonné et d’énergie de formation plus faible que les autres.

S. Goedecker, Th. Deutsch, and L. Billard



Due to their technological impor-tance, point defects in silicon are amongthe most studied physical systems. Theexperimental evidence of point defectsburied in bulk is difficult, leading experi-mentalists to use rather indirect methods.In parallel, simulations of defects havebeen performed at various levels ofsophistication. The generally acceptedconclusion of all these studies is thatvacancies and self-interstitials are thebasic point defects in silicon.

Such defects result from taking outor adding atoms to the crystal. Thereforethe fourfold coordination is destroyed,leading to rather high defect formationenergy. In addition there is another pointdefect, conserving the number of atoms,the Frenkel pair, consisting of a vacancyand an interstitial. Its formation energy isalso large since again bonds are broken(see Fig. 1).

Using state of the art plane wavedensity functional theory calculations (DFT)we found a new defect that, in contrast toother point defects, conserves the fourfoldcoordination and will therefore be calledthe fourfold coordinated defect (FFCD).Whereas the finding of classical pointdefects was based on simple symmetryconsiderations, the discovery of this newpoint defect was made possible by newalgorithmic developments. The configura-tional space was systematically exploredusing a modified basin hopping methodand an inter-atomic silicon potential. Themost promising configurations were thenrefined with a plane wave electronic struc-

ture code developed by a consortium inwhich we participate [1, 2]. Figure 2 showsthe novel defect configuration.

We have summarized in Table 1 thetotal energy of all low energy defects calcu-lated for intrinsic silicon by various authors. Inaddition we have included a low energyFrenkel pair. We did not include the tetrahe-dral interstitial and caged interstitial that werefound higher in energy and metastable. The2.4 eV formation energy of the FFCD is clear-ly lower than the formation energy of allother known point defects both in intrinsicand doped silicon. The bond length andangle do not significantly deviate from thebulk values. Whereas the bond length andangle in the bulk are 2.35 Å and 109degrees, they respectively vary from 2.25 to2.47 Å and from 97 to 116 degrees for thebonds formed by the two (red) defect atoms.In order to annihilate the defect, two bondshave to be transiently broken, correspondingto a barrier energy of 0.57 eV. The FFCDdefect is close to the so-called “bond defect”,the configuration of which was inferred fromless accurate tight binding simulations of bulksilicon [3]. We found the precise configura-tion of the stable defect and calculated its for-mation energy for the first time.

Due to the properties of this newdefect, it is hardly detectable experimentally,and as a matter of fact it was not detecteduntil now. We have explored the possible rea-

FFCD Frenkel X interstitial H interstitial Vacancy2.42 4.32 3.31 3.31 3.17

Table 1: Formation energy in eV of the low energypoint defects.

sons for this situation. Firstly, because of theperfect fourfold coordination, no unpairedelectrons exist that could be detected withElectronic Paramagnetic Resonance (EPR) andElectron Nuclear DOuble Resonance(ENDOR) experiments. Secondly the presenceof electronic levels within the gap, whichcould be evidenced through Deep LevelTransient Spectroscopy (DLTS), also seemsunlikely. To confirm this point we calculatedthe Highest Occupied Molecular Orbital(HOMO) and Lowest Unoccupied MolecularOrbital (LUMO) splitting. In fact the bandstructure is much less disturbed by the FFCDthan by the other defects. Finally the FFCDspectrum is insensitive to the doping levelwhereas the other spectra are strongly influ-enced via various Jahn-Teller distortionsbecause of their orbital degeneracy.Therefore the presence of the defect could notbe detected through its influence on dopants.Rutherford backscattering experiments onlight ion implanted silicon samples give ener-gy spectra that can be fitted on the basis ofdefects of this type, but cannot unambiguous-ly confirm a precise configuration.

In fact, due to its low formation energy,this defect is probably generated duringimplantation of light ions in silicon. The anni-hilation barrier energy leads to a relativelyshort lifetime at room temperature. However,during its lifetime, the FFCD can either pro-mote the migration of self-interstitials or

dopants, or cluster with other FFCD, leadingto more stable configurations, which could inturn play an important role in room-temper-ature amorphisation of silicon during theimplantation process. We are continuing thetrack, reviewing and calculating possibleexperimental signatures of the defect, andare also simulating more extended defectsusing the FFCD as a building block.

23

[1] CPMD Version 3.3 developed by J. Hutter, A. Alavi, T. Deutsch, M. Bernasconi, S. Goedecker, D. Marx, M. Tuckermanand M. Parrinello, Max-Planck-Institut für Festkörperforschung and IBM Zürich Research Laboratory (1995-1999).

[2] S. Goedecker, T. Deutsch, and L. Billard, Phys. Rev. B 88, 235501 (2002).[3] F. Cargoni, C. Gatti, and L. Colombo, Phys. Rev. B 57, 170 (1998).

Figure 1: Frenkel pair defect in silicon. Green spheres: Si atoms - redsphere: atom in new position - black sphere: vacancy (former atomposition) - small blue spheres: covalent bonds - large blue spheres: perturbed bonds.

Figure 2: FFCD defect in silicon. Green spheres: Si atoms- red spheres: atoms in new positions - black spheres:empty lattice sites (former positions) - small blue spheres:covalent bonds.


IN C O M M E N S U R AT E I N T E R FA C E S I N M E TA L SINCOMMENSURATE INTERFACES IN METALS

The interface between two crystals is incommensurate when the ratio of their periodicities alongthe interface is an irrational number. We have shown that such interfaces in mono-atomic metalsincluding gold and copper can have a specific behavior. Hypo-friction of the interface sliding andtriple-line reconstruction are considered here.

Une interface entre deux cristaux est dite incommensurable lorsque le rapport des périodicités le long de l'interface est un nombre irrationnel. Nous avons montré que de tellesinterfaces dans les métaux purs comme l'or ou le cuivre ont des propriétés particulières. Nous nous intéressons ici à l'hypo-friction lors du glissement de l'interface et à une reconstruction de ligne triple.

F. Lançon



Solid bodies are generally constitutedby microscopic crystalline grains separatedby interfaces called grain boundaries.While the general solid-state theory depictsthe properties of single crystals, the proper-ties of poly-crystalline solids are largelyinfluenced by the presence of these bound-aries, which affect features like mechanicalbehavior, electrical and thermal conductivi-ty, diffusion of impurities... The advent ofnanomaterials made of nanometer-sizedgrains further amplifies the role of theboundaries. Most of the studied grainboundaries deal with lattices having a com-mon period, a multiple of the periods ofboth crystals. However in most of the realcases there is no such common periodicity;the interface is then called incommensu-rate. Usually a periodic structure is foundthat describes the properties of the interfacewith rather good accuracy, but sometimes,as illustrated in this paper, the incommen-surability has specific consequences.

A simple incommensurate structurehas been found by high-resolution electronmicroscopy in gold [1] and aluminium. It isa grain boundary between two face cen-tered cubic (fcc) crystals. In order to visual-ize the grain boundary plane, a cube canbe imagined cut in two prisms as shown inFig. 1. If one half is rotated 90° with respectto the other, its face diagonal (length √2)comes along the edge of the other half(length 1). The crystalline grains have acommon axis such that atoms viewed fromthis direction form rectangles with edgesproportional to 1 and √2. The first grainhas the edge of length 1 along one side ofthe interface (x axis); the second grain is

rotated through 90° with the √2 edge com-ing along this x axis. Because √2 is an irra-tional number, the interface cannot be peri-odic along x. Close to the interface, atomsof each grain have positions perturbed bythe atoms of the other grain. As a result, theinterface has a complex incommensuratestructure.

This structure is non periodic but, byadding a virtual fourth dimension to theusual space, it can however be representedby periodic sets of lines in a 4D hyperspace(see Fig. 2). The actual atomic coordinates(0D features) are given by the intersections ofso-called “atomic lines” (1D) with the physi-cal space (3D). This theoretical tool, alreadyused in the field of quasi-crystals, condenses

Figure 1: A basic fcc cube (top) is cut into two prisms.One (bottom right) is rotated with respect to theother (bottom left), showing the incommensurabilityof the edges.

25

all the information on the atomic positions ofeach atomic layer from -∞ to +∞ inside afinite and small square unit cell.

Based on this analysis of the struc-ture, and using atomistic simulations, wehave predicted that this ideal 90° incom-mensurate interface can have the surprisingproperty of zero static friction. S. Aubryoriginally found a similar behavior in a 1Datomic chain model: he proved that a chainembedded in an external periodic potentialmay be stuck (“pinned”) or freely sliding(“unpinned”), depending on the interac-tions [2]. We have found the first example ofsuch a property in a real material by inves-tigating the case of gold. We show that thetwo possible states, pinned or unpinned,exist for the 90° grain boundary [3], andthat the dynamical property of sliding isrelated to the topology of the atomic linesdefined in the 4D framework. If the atomiclines are continuous, one grain can slidealong the other without an energy barrier:the grain boundary has the property ofhypo-friction. If the lines are discontinuousthe grains are pinned to each other, just likein standard periodic grain boundaries.

Another interesting situation ariseswhen the grain boundary interacts with thefree surface of the material. We have evi-denced a new behavior corresponding to aline reconstruction at the intersection of thisboundary with the free surface. The inter-section line is transformed into a "wire" witha cross section of a few nanometers thatwas observed by electron microscopy andexplained by our simulations [4]. Thechevron-like triangular domain shown inFig. 3 results from the balance between twoenergy terms: one is the energy of the grainboundary area that has been replaced, andthe other is the energy of the stacking faultspresent in the domain, almost transformedfrom face centred cubic to hexagonal closepacking. A detailed analysis of these termsgives the conditions for its stability [5]. Thecalculations were confirmed by the experi-mental results, for instance the reconstruc-tion is present in gold but not in aluminium.

At the fundamental level, these studieshelp our understanding of the interplaybetween crystalline surfaces. The properties ofthis type of interfaces have implications ongrain dynamics: this new type of sliding at the

[1] F. Lançon, J.-M. Pénisson, and U. Dahmen, Europhys. Lett. 49, 603 (2000).[2] S. Aubry, Solid State Science Vol. 8 (Springer-Verlag, Berlin/Germany, 1978) p. 264.[3] F. Lançon, Europhys. Lett. 57, 74 (2002).[4] T. Radetic, F. Lançon, and U. Dahmen, Phys. Rev. Lett. 89, 85502 (2002).[5] F. Lançon, T. Radetic, and U. Dahmen, Phys. Rev. B 69, 172102 (2004).

Figure 3: Interaction of the grain boundary withthe surface in copper. White rectangles: orienta-tion of the grains. At the surface, the chevrondomain replaces the grain boundary energy by amore diffuse energy, which is globally lower.


interface can increase plastici-ty by several orders of magni-tude, while the line reconstruc-tion brings a new mechanismefficiently pinning the interfaceat the surface.

ACKNOWLEDGEMENTS

The author acknowl-edges the hospitality of theNational Center of ElectronMicroscopy (NCEM) during aresearch leave at the BerkeleyNational Laboratory andthanks his collaborators U.Dahmen and T. Radetic.

Figure 2: Incommensurate interface viewed in ahyper-space. Atoms are represented by "atomiclines" when a virtual direction i is added. Theintersections of these lines with the real x-axisgive the x positions of the atoms.

SI M U L AT I O N O F G R A Z I N G I N C I D E N C E S M A L L A N G L EX-R AY S C AT T E R I N G T W O-D I M E N S I O N A L I M A G E SSIMULATION OF GRAZING INCIDENCE SMALL ANGLEX-RAY SCATTERING TWO-DIMENSIONAL IMAGES

Grazing Incidence Small Angle X-ray Scattering (GISAXS) is a powerful tool to analyze the topog-raphy and morphology of nano-objects deposited on or buried below surfaces. Extracting quan-titative information requires approximations to the general scattering theory, which are discussedon facetting of W(111) surfaces into nanopyramids.

La diffusion aux petits angles en incidence rasante (GISAXS) est une méthode puissante pour analyser la topographie et la morphologie de nano-objets déposés sur ou enterrés sous unesurface. L’accès à l’information quantitative nécessite des approximations à la théorie générale de la diffusion, discutées dans le cas du facettage de surfaces de W(111) en nanopyramides.

F. Leroy, G. Renaud, R. Lazzari*,and C. Revenant



GISAXS is becoming a common tool,at third generation synchrotron sources, tocharacterize the shape, size, and ordering ofnanoparticles deposited on, or buried below,a surface. It can even be used in situ, duringthe elaboration of nanostructures [1]. As illus-trated in Fig. 1, the incident X-ray beamimpinges on the sample at a grazing incidentangle, and the scattered intensity is recordedperpendicularly to the beam with a two-dimensional detector. Nanometer-sized non-homogeneities of the electron density ornanostructures on the surface scatter the X-rays at small angles. The figure shows how X-rays scatter on the nano-facets of small trian-gular pyramids, grown on a W(111) surfaceafter annealing of a 1ML Pt deposit [2]. In theBorn approximation (BA) single scattering ofX-rays is assumed. The GISAXS image is thusthe squared Fourier transform of the realspace topography: the smaller the lateral(resp. vertical) size of the nano-objects, thelarger the extent of the scattering parallel(resp. perpendicular) to the surface.Qualitative features can then be deducedfrom experimental data. First the three-foldsymmetry of the pattern through the rotationof the sample in its plane proves the three-fold symmetry of the nano-pyramids. Next weobserve scattering rods at an angle of 18.6°(Figs. 1 and 2a) with respect to the surface.Since a two-dimensional facet in real spaceyields a perpendicular one-dimensional rodin reciprocal space, we infer the existence of[112] facets from this angle.

Extracting quantitative informationsuch as the facet size and pyramid heightrequires a theoretical treatment. A first com-plication arises from the very small incident

grazing angle α i : not only the incident beamis present above the surface (and thus scat-tered by the objects), but the specular reflect-ed beam also is, scattered with similar ampli-tude. The additional term, multiplied by areflection coefficient (~1 for small α i ) andshifted perpendicularly to the surface by anexit angle ∆αƒ = 2αi, interferes with thedirect scattering term. This leads in Fig. 1 totwo parallel rods shifted perpendicularly tothe surface, while the single scatteringapproximation predicts only one. Thus, forsimulating the data, we use the DistortedWave Born Approximation (DWBA), whichtakes “multiple scattering” into account. Asecond complication arises from the fact thatthe islands are not identical, displaying a sta-tistical distribution of size and shape. Theonly tractable approach is then to comparewith models that take the physics of thegrowth into account.

A common approach is the LocalMonodisperse Approximation (LMA). It isassumed that neighboring islands locallyhave the same size, and that zones withislands of different sizes are separated by dis-tances larger than the coherent length of thebeam. Waves scattered by different zones arethen added incoherently, and the total inten-sity is the sum of intensities scattered by thesezones. This does not however account for fre-quent situations where islands of rather dif-ferent size are close to each other, thus scat-tering coherently.

The decoupling approximation (DA)addresses islands of different size and dis-tances in the same zone. It assumes decou-pled distributions: the size of an island does

not depend on the size or position of neigh-bors. This approximation does not take intoaccount various correlations between thesizes and/or the separation of neighboringislands. In fact the processes (growth or coa-lescence) leading to an island larger than theaverage also lead to a larger depletion areaaround it, in turn making the neighboringislands nucleate and grow farther. This yieldsa positive correlation between island sizesand separations, and generally a positivecorrelation between sizes of neighboringislands.

We have developed a software pack-age called IsGISAXS [4] to simulate, analyzeand fit GISAXS data for a large variety ofnanoparticle assemblies, either embeddedbelow or sitting on a surface. Particles mayhave the various shapes experimentallyobserved, with different lateral and verticalsize distributions. They may be located on thenode of perfect or disordered crystals withseveral variants related by symmetry, or dis-tributed in a disordered way, characterized bya specific particle-particle pair correlationfunction. Within the above two approxima-tions, the program enables the experimentalGISAXS data to be fitted and simulated inalmost all situations encountered up to now,as exemplified in Fig. 2 where LMA and DAwere used to simulate GISAXS data on small

(~10 nm) correlated nanopyramids. TheLMA describes the data better, as seen in thefigure. However electron microscopy showsthat the islands are not locally monodisperse.This points the limits of the generally adoptedapproximations; we are now developingmore sophisticated models to take diffusescattering of X-rays on correlated systems intoaccount [5].

27

[1] G. Renaud, R. Lazzari, C. Revenant, A. Barbier, M. Noblet, O. Ulrich, F. Leroy, J. Jupille, Y. Borentsztein,C. R. Henry, J.-P. Deville, F. Scheurer, J. Mane-Mane, and O. Fruchart, Science 300, 1416 (2003).

[2] J. J. Kolodziej, T. E. Madey, J. W. Keister, J. E. Rowe, Phys. Rev. B 65, 75413 (2002). [3] C. Revenant, F. Leroy, R. Lazzari, G. Renaud, and C. R. Henry, Phys. Rev. B 69, 035411 (2004). [4] R. Lazzari, J. Appl. Cryst. 35, 406 (2002).[5] F. Leroy, R. Lazzari, and G. Renaud, Acta Cryst. A (2004) in press.

Figure 1: Schematic of a GISAXS experiment, and experi-mental GISAXS image recorded on a nanofacetted W(111)surface. The horizontal is parallel to the surface. The incidentand reflected beam, as well as the specular scattering rod,are hidden by a beam stop. The GISAXS has three-fold sym-metry on rotation of the sample with respect to a vertical axis,perpendicular to the surface.

Figure 2: (a): GISAXS data of small correlated nanopyramids of the nanofacettedW(111) surface. (b) and (c): Simulations using the Distorted Wave Born Approximation(DWBA) plus two further approximations LMA (b) and DA (c).


* present address : Groupe de Physique du Solide, Univ. Paris-6

SO L I D I T Y O F L I Q U I D F O A M SSOLIDITY OF LIQUID FOAMS

Liquid foams have wide-ranging industrial applications but their flow remains largely an unsolvedphysical problem. This is because it combines elastic, plastic, and viscous responses. We havedevised an experimentation which, together with a careful theoretical analysis, shed some light onits uncommon features.

Les mousses liquides trouvent de nombreuses applications industrielles mais leur écoulementreste largement incompris en physique, car il associe des comportements à la fois élastiques,plastiques et visqueux. Nous avons construit une expérience qui, grâce à une théorie appropriée, permet de mieux comprendre ses caractéristiques peu communes.

M. Aubouy

Service des Interfaces et des Matériaux

Moléculaires et Macromoléculaires

(SI3M), UMR SPrAM 5819, DRFMC

In the XXth century, physics producedmultiple discoveries at the forefront of ourtechnological capacities in the aim to pushforward their limits. Yet everyday life is still asource of exciting issues, and amazing dis-coveries are sometimes within the reach ofwhoever simply scrutinizes our ordinary sur-rounding. The subject of liquid foamsbelongs to this family, and naive questionsdo not always have definitive answers. Inparticular, we are used to separating mate-rials into solids and liquids: what aboutfoams? Clearly this is a solid if one recallsthat shaving foam remains persistently onthe skin. However, foam flows easily in thefire-extinguisher. Soon we realize that this isan extraordinarily complicated commonmaterial. Physicists call it a “complex fluid”.

In terms of fundamental physics, weare looking for the mechanical constitutivelaw which relates the applied force to theresulting or effective deformation (or con-versely). Formulating such laws has been aquery of physicists for many centuries,which resulted in two brilliant intellectualachievements: namely the theory of elastic-ity and hydrodynamics, which respectivelydefine and theorize elastic solids and vis-cous liquids. But, in spite of their consider-able successes, these theories are unable toaccount for the flow behaviour of large cat-egories of materials: foams, micro-emul-sions, pastes, slurries, concentrated emul-sions, granular materials…

Figure 1: Instantaneous image of a two-dimensional quasi-staticflow of foam around a circular obstacle (actual size: 10 cm x 8 cm).We superimposed the local determination of the statistical straintensor represented by the solid ellipses and characterizing theeffective deformation on each bubble [1].

In this list, foams are remarkable.This is because if we squeeze a liquid foambetween two glass plates (to form what wecall a 2d-foam) we are in a position toobserve both the microscopic scale (thebubbles), and the macroscopic scale (thebulk flow). Accordingly, there is no hiddenvariable here, which makes it a model sys-tem for studying complex liquids.

As shown in Fig. 1, we devised anexperiment where a 2d liquid foam isforced to flow around a circular obstacle.But how do we estimate the effective defor-mation (strain tensor)? Presumably, the dif-ficulty is better understood if we comparethe foam to a piece of rubber. Suppose wepull an elastic band from rest: the variationof its length quantifies the applied deforma-tion (equal to the effective deformation),and the force we experience is proportionalto it, since the elastic band returns to its ini-tial state when our effort is released. In con-trast, when we squeeze a piece of foam inour hands, we soon feel that the resistancedoes not increase proportionally to theapplied deformation. For instance, thematerial relaxes to an intermediate state:not the initial state! Accordingly, we have todistinguish the effective deformation fromthe applied deformation; one strain tensordoes not enable the overall deformation ofthe material to be characterized.

As a matter of fact, this is a majorconceptual step away from both the theo-ry of elasticity and hydrodynamics, forwhich it is assumed that both deformationscoincide (the affine hypothesis). We foundan operational definition for the effectivedeformation that we called the statisticalstrain. This is a tensor that we represent byan ellipse such that the principal axes areproportional to the eigen-vectors of thelocal field of deformation. The pertinentscale for foams is the bubble size and the

overall deformation has to be discretizedat this mesoscopic scale.

We are then in a position to measurefor the first time both the stress and effectivestrain on each bubble. Since the flow isinhomogeneous, a single image displaysthe complete mechanical constitutive law.Among other results, our analysis revealsthe presence of a wake that was invisible tothe eye (see Fig. 2 and reference [2]).

Yet a lot remains to be done. Now thatwe have found the proper conceptual tools,we aim at describing the whole range ofmechanical responses: elastic, plastic, vis-cous flow. Because foams are strongly non-affine materials, these researches belong tothe emerging field of “sub-continuummechanics” (i.e. mechanics when the contin-uum medium hypothesis no longer holds).This is a field of great interest for the CEAsince foams are used as cleaning product forradioactive contamination or as a shockdamping material after ignition of a nuclearweapon (AIRIX experiment). This is also veryinteresting for the manipulation of nano-objects, for which all the classical assump-tions of elasticity fail.

In his novel The Alchemist, PauloCoelho tells the story of a boy who makes

29

[1] M. Aubouy, Y. Jiang, J. A. Glazier, and F. Graner, Granular Matter 5, 67 (2003).[2] M. Asipauskas, M. Aubouy, J. A. Glazier, F. Graner, and Y. Jiang, Granular Matter 5, 71 (2003).

Figure 2: Our analysis of effective local strain reveals the presenceof a wake around the obstacle [2].


an extraordinary journey in apursuit to find a fabuloustreasure. He eventually findsit precisely at the place wherehe was when the storybegan. Wouldn’t it be a nicestory indeed if the foams weused to play with as a childshould provide us withanswers to long-outstandingi n t e l l e c t u a l i s s u e s ?Presumably, we needed thatjourney to find this out, thenarrator would say.

ACKNOWLEDGEMENTS

I wish to thank F. Granerand C. Quilliet for their enthu-siastic collaboration.

It is too early to judge the utility of the-ory to biotechnology. However, the examplesdiscussed below support the reverse claim:biotechnology is a source of interesting prob-lems for theory. To illustrate their “flavor” wewill discuss two examples: DNA microarrays(“DNA Chips”) and stealth liposomes (SL).The first is an analytical technique of greatimportance to molecular biology andmedicine. The second is a drug delivery vehi-cle for cancer therapy.

I. DNA CHIPS

DNA chips comprise a multitude of“spots”. Each consists of numerous, termi-nally anchored, single-stranded DNAchains (ssDNA) of short length referred toas “probes”. They are used to analyze thecomposition of solutions comprising a mix-ture of short ssDNA of unknown sequenceknown as “targets”. In a typical setup fluo-rescent tags label the targets. The targetshybridize and form a double-strandedDNA on the spot carrying probes of com-plementary sequence. The composition ofthe bulk solution is deduced from the mea-sured fluorescence intensity of the differentspots. Our research [1] concerns the fac-tors determining the performance of DNAchips focusing on probe density and com-petitive hybridization. In particular, weobtain useful thermodynamics bounds forthe performance of DNA chips from theirhybridization isotherms. These relate thebulk composition and the hybridizationfraction at a given spot in thermodynamicequilibrium. Since both the probes and tar-gets are electrically charged, the hybridiza-tion isotherms emerge as Langmuir

adsorption isotherms modified to allow forelectrostatic interactions in the probe layer:each hybridization event increases thecharge of the probe layer thus raising thepenalty for the next hybridization event.Two types of competitive hybridization areinvolved (see Fig. 1): (i) Competitive sur-face hybridization occurs when two differ-ent targets can hybridize with the sameprobe; (ii) Competitive bulk hybridizationarises when the solution contains comple-mentary sequences that hybridize with thetargets but not with the probes. Thehybridization isotherms allow for the role ofcompetitive hybridization and quantify theresulting reduction in the performance ofthe chips.

Two examples of theories motivated by biomedical applications illustrate the role of theory inclarifying design parameters for biotechnology. One concerns the performance of DNA chips. Thesecond involves controlled release of drugs by stealth liposomes incorporating lipidsfunctionalized by neutral polymers.

Deux exemples illustrent le rôle de la théorie dans le domaine de la biotechnologie. Le premier concerne l’absorption d’ADN sur des biopuces et le deuxième concerne l’étudedu rôle d’un polymère neutre dans la protection contre une dégradation enzymatique d’un vecteur thérapeutique appelé liposome furtif.

A. Buhot a, E. B. Zhulina b

and A. Halperina

a: Service des Interfaces et des Matériaux



b: Institute of Macromolecular Compounds,

Russian Academy of Science, St. Petersburg

FR O M DNA C H I P S T O S T E A LT H L I P O S O M E S : B I O T E C H N O L O G Y- I N S P I R E D T H E O RY

FROM DNA CHIPS TO STEALTH LIPOSOMES: BIOTECHNOLOGY-INSPIRED THEORY

Figure 1: Schematic picture of probes, grafted ssDNA(in red) hybridizing with perfectly matched and mis-matched targets in the presence of complementaryssDNA. Competitive surface hybridization involvesreactions 1 and 2. Competitive bulk hybridizationinvolves reactions 1 and 3.

31

II. STEALTH LIPOSOME

Stealth liposomes (SL) are vesiclesformed by lipid bilayers incorporating twotypes of lipids: normal lipids and lipids func-tionalized by short chains of the neutralwater-soluble polymer Polyethylene Oxide(PEO) (see Fig. 2). SL are used to deliver anti-cancer drugs into tumors. The presence of thePEO chains increases the circulation time ofthe SL in the blood by slowing down theirelimination in the liver. The enhanced circula-tion time is attributed to repression of proteinadsorption on the liposomes, an event thattriggers their elimination. The repressionmechanism of protein adsorption by termi-nally anchored PEO chains is of interest for abetter understanding of SL function. A simpleexplanation for much of the existing dataattributes this function to the osmotic penaltyincurred by a protein on entering the PEOlayer [2]. For small proteins the penalty is ofthe form (osmotic pressure in the PEO layer)× (protein volume). It thus allows for the den-sity of the chains and the dimensions of theprotein. A new twist to the SL story recentlyemerged with the discovery that SL are moresusceptible than bare liposomes to degrada-tion by interfacial enzymes, secretory phos-pholipase A2 (sPLA2), that hydrolyzes lipidsincorporated into membranes. This discoveryis of interest because sPLA2 are more abun-dant in cancerous tissue thus affording a

mechanism for controlled release of anti-cancer medication within the tumor. Ourwork addresses two questions: Why doesn’tthe PEO repress the activity of the sPLA2?Why does it enhance their function? Theanswer to the first question is simple. ThesPLA2 are small proteins and the osmoticpenalty incurred by approaching the mem-brane is small compared to the thermal ener-gy and thus negligible. The answer to the sec-ond question emerges upon realizing that thehead group of the lipid must protrude outfrom the membrane in order to reach theactive site of the enzyme [3]. Such protrusions(see Fig. 2) are fluctuations when the lipidsare not functionalized. However, protrusionsof lipids attached to PEO chains are equilibri-um features due to the entropic penalty, aris-ing because the membrane is impenetrableto the PEO chain, which balances the price ofexposing the lipid hydrophobic tail.

III. PERSPECTIVE

The two examples discussed aboveinvolve polymers whose monomers undergodynamic equilibrium between two intercon-verting states (“two-state” polymers). DNAmonomers can exist either in a helical or acoil-like state. PEO monomers are thought tointerconvert between hydrophilic andhydrophobic states. The “two-state” feature is

[1] A. Halperin, A. Buhot and E. B. Zhulina , Biophys. J. 86, 718 (2004).[2] A. Halperin and D. E. Leckband, C.R. Acad. Sci. 1 IV, 1171 (2000).[3] A. Halperin and O. Mouritsen, submitted to Eur. Biophys. J. [4] A. Buhot and A. Halperin, Europhys. Lett. 50, 756 (2000).

Figure 2: Lipid monolayer incorporating PEO-functionalizedlipids. The inset depicts a protruding lipid. The PEO-stabilizedprotrusions enable the function of interfacial enzymes (in red) tohydrolyze the functionalized lipids.


qualitatively important in thefirst example and useful forthe quantitative analysis of thesecond. It is present in otherwater-soluble polymers andcan lead to novel configura-tional and phase transitions ofrelevance to the design ofbiomedical devices [4].

Investigation of the physical mechanisms of cell adhesion and spreading is just emerging withobservations of living cells becoming more and more accurate and rich in information. In thefollowing, we will focus on a working model when a cell adheres and crawls onto a solid substrate.

Les observations et techniques utilisées devenant de plus en plus précises et performantes, la recherche et la compréhension des mécanismes physiques de l’adhésion d’une cellule sont en plein essor. Dans l’article qui suit nous présentons un modèle de travail théorique caractérisant le processus d’adhésion et de décollement d’une cellule vivante sur un support solide.

B. Fourcadea and F. Bruckertb

a: Service des Interfaces et des Matériaux



b: Laboratoire Biochimie et Biophysique

des Systèmes Intégrés, DRDC,

CEA/Grenoble

In the last fifteen years, new experi-mental and conceptual approaches haveincreased our knowledge about cellularmachinery. Living systems having low con-trast from the optical point of view are nowcommonly observed in vivo using interferen-cial microscopy techniques and fluorescentprobes. Moreover, new micro-mechanicaldevices for single molecule experiments, arti-ficial nano-machines and bio-mimetic sys-tems are constantly invented to probe andmodel living matter. Cellular machinery hasnow also become a playground for theory.

We want to understand how and whymechanical and chemical forces work in avery coordinate way at a scale (1µm) whereBrownian motion should a priori dominate.Cellular biology tells us that the cell is a

chemical automaton which responds to anexternal stress via a cascade of internalchemical and physical reactions. There arefundamental differences between usual self-assembled systems, such as polymers oramphiphiles, and living cells: (i) living matteris out-of-equilibrium; (ii) it is a driven assem-bly system in the sense that their motion isdue to a constant reshaping of the cell struc-ture. The chemistry of this reconstruction is,however, stress-dependent. Thus under-standing how forces are distributed and howenergy is dissipated may help to understandhow the many different biochemical specieswork together to produce motion.

The simplest method to probe the cellmotion properties is to apply a mechanicalforce on a cell adhering to a solid substrate.

CE L L U L A R A D H E S I O N A N D C E L L U L A R M O T I L I T Y* , A P L AY G R O U N D F O R P H Y S I C I S T S

CELLULAR ADHESION AND CELLULAR MOTILITY*, A PLAYGROUND FOR PHYSICISTS

* motility : mobility and motricity

Figure 1: A living cell adhering on a solid substrate is submitted to a steady flow stress.Imaging the cell allows the velocity V of the centre of mass to be measured. Under the actionof the external force, the cell crawls and the adhesive bridges are stretched and detached.

This can be obtained in a hydrodynamic fluxchamber in which water flows parallel to thesubstrate [1,2]. The cells at the surfaceseem to move downstream like a passivebody. In fact, the backside exposed to thehydrodynamic flux is peeled off the sub-strate with a constant velocity while thefront-side at the opposite face advances byextending protrusions at regular time inter-vals: actually, the cells are active systems.The external hydrodynamic force is not itselfthe motor of the movement but activates it.

Our schematic model points out therole of the extreme margin of the adhesivezone where the cell is in contact with thesubstrate (see Fig. 1). This is where the cell"feels" if it has to spread or to peel off.Unlike traditional soft materials for whichadhesion occurs via short-range potentials,cellular adhesion occurs because of adhe-sive bridges. They connect the cell body tothe substrate via a complex set of proteinswhich pierce the membrane and which areanchored into the cell body. This is howadhesive bridges are able to initiate multi-ple cascades of signals.

Roughly, an adhesive bridge made ofproteins has a length of a few tenths ofnanometers. The forces of a few pico-new-tons which come into play are too small tochange the protein structure. From themechanical point of view, we consider thatthese proteins are springs with one endanchored on the substrate with a weak inter-action such as a hydrogen bond. Thus, thenature of these weak anchoring points is onlystatistic and thermal fluctuations play animportant role. In our picture, peeling the celloff the substrate amounts to disconnectingthe adhesive bridges from the substrate.However, as shown in recent single moleculespectroscopy experiments, breaking a non-covalent bond depends on the loading rateand this why the mechanical response of theadhesive belt has remarkable properties.

Because of this very dissipative mech-anism, we have shown that the connectingtime of the adhesive bridges located at theextreme margin of the cell (the adhesivebelt) is the characteristic time for peeling thecell off the substrate. Then the basic task ofstatistical physics is to model the relation-ship between the mechanical work appliedto the system and the mechanical work nec-essary to break off the microscopic bonds:to correlate the microscopic to the macro-scopic. Both corresponding energies differdrastically, the dissipation being very large.Our model enables the characteristic cellvelocity curve versus the applied flow forceto be computed.

Figure 2 shows the non-linear increaseof the peeling velocity of the cell as a functionof the applied force and points correspond todifferent sets of experimental data. The quasi-exponential behaviour of this curve is the sig-nature of breaking of the adhesive bridges ina thermo-activated regime.

Now our work is geared to under-standing how mechanical and chemicalforces work together to produce motionand this is where non-equilibrium statisticalmechanics comes into play. In ourapproach, the polymeric network of poly-

33

[1] E. Decave, D. Garrivier, Y. Brechet, F. Bruckert, and B. Fourcade, Phys. Rev. Lett. 89, 108101 (2002).[2] E. Decave, D. Rieu, J. Dalous, S. Fache, Y. Brechet, B. Fourcade, M. Satre, and F. Bruckert,

J. Cell. Sci. 116, 4331 (2003).

Figure 2: Peeling velocity versus applied stress. The solid curve corresponds tothe data adjustment with a theoretical model describing the peeling process.


merised actin molecules(actin cortex) at the cell bodyperiphery plays a fundamen-tal role. This network is con-stantly remodelling itselfunder the action of theadhesive bridges and thisreshaping is the basic pro-cess at the origin of the cellmovement. The fascinatingproperties of cell motions isjust beginning to be under-stood from both the bio-chemical and mechanicalpoints of view. A cell is neverat rest. It constantly exploresits environment and the roleof physics is to help under-stand the extreme sensitivityof the cellular machine toexternal signals.

GD3+-BASED CONTRAST AGENTS IN MAGNETIC RESONANCE IMAGINGGd3+-BASED CONTRAST AGENTS IN MAGNETICRESONANCE IMAGING

Magnetic Resonance Imaging (MRI) often involves the use of contrast agents. An in-depthcharacterization at the molecular level of the factors affecting the efficiency of these powerfulimaging probes allows their rational design and set-up.

L'Imagerie par Résonance Magnétique (IRM) fait fréquemment appel à l'utilisation d'agents de contraste. La caractérisation approfondie à l'échelle moléculaire des facteurs gouvernant l'efficacité de ces puissantes sondes d'imagerie permet d'optimiser leur conception et leur mise en place.

P. H. Fries

Service de Chimie Inorganique et

Biologique (SCIB), DRFMC

I. INTRODUCTION

MRI is a powerful and safe tech-nique in diagnostic clinical medicine andbiomedical research [1]. The expansion ofMRI has prompted the development ofnew drugs, called contrast agents (CAs).These compounds are administrated topatients in order to target specific parts ofthe body and locally increase the imagecontrast (see Fig. 1). Physicians can thusdifferentiate either normal and diseased tis-sues or different parts of the body, so as toshow organ function or blood flow. About40% of MRI examinations use CAs and thispercentage is increasing with the develop-ment of new agents and applications.

MRI is based on nuclear magneticresonance: In a magnetic field B0 , anoscillating electromagnetic signal emittedby the nuclear spins I (magnetic moments)of the hydrogen nuclei of water moleculescan be measured. The resonance frequen-cy of the signal oscillations is proportionalto B0 . In order to construct an image, aspatial gradient of magnetic field isapplied to the living system, leading to acorrespondence between the resonancefrequency and the spatial position of avoxel (three-dimensional pixel). Computerprocessing of the total signal emitted bythe nuclei of all the voxels makes theimage as the spatial juxtaposition of thesignals from the various voxels. The signalintensity is proportional to the density ofthe water molecules in the voxel.Unfortunately, this density parameter can-

not give a suitable image contrast in med-ical MRI since the water densities of thesoft tissues in living organisms often havetoo similar values.

II. LONGITUDINAL RELAXATIONCONTRAST AGENTS

To overcome this difficulty, anotherparameter is used: the longitudinal relax-ation rate R1(r) that represents the speed ins-1 at which the components of the nuclearspins I along the field B0 return to equilibri-um after a temporary magnetic perturba-tion. R1(r) depends on the physico-chemicalnature of the tissue at the spatial position r.Therefore, R1(r) varies from one voxel toanother and gives rise to a position-depen-dent brightness or contrast. Contrast agentsare used for their ability to locally enhanceR1(r). Among the possible paramagneticmetal ions, Gd3+ is particularly suitable.Indeed, because of its high electronic spinvalue S=7/2, it has a strong dipolar mag-netic interaction Hdip with the hydrogennuclear spins. In addition, Gd3+ exhibits aslow electronic relaxation. It can form verystable biocompatible complexes, which areeliminated from the body before release oftoxic free Gd3+ ions.

The efficiency of a CA depends on thestrength and time fluctuations of Hdip.These fluctuations are caused by the relativespatial motion of the water hydrogen atomswith respect to Gd3+ and by the relaxationof the electronic spin of this ion. We devel-oped various aspects of the molecular mod-

35

eling needed to predict these fluctuationswhich can be classified according to thefates of representative water molecules asshown in Fig. 2. At an arbitrary time origint=0, three situations occur. First, watermolecules such as wM (1st hydration shell)are directly bound to the metal ion M (greysphere). Second, other water moleculessuch as w2S (2nd hydration shell) have aweak H-bond with one of the associatingsites (pink) of the CA. Third, the rest of thewater molecules such as w1 and w2 (outer-shell situation) undergo a free translationaldiffusion.

All these molecules are progressivelyreplaced by other water molecules (that donot appear in the figure) initially located inother positions. After 10 ps, the outer-sphere water molecules have alreadyundergone notable displacements while theCA has only rotated by a tiny angle and the1st and 2nd sphere molecules are stilllinked to the CA. After 40 ps, the outer-sphere w1 has exchanged with the 2ndsphere w2S. A much longer duration of 1 nswas necessary to observe a 180° rotation ofthe whole CA. During this period all thewater molecules, but wM directly coordinat-ed to Gd3+, have also gone away from theCA. Eventually, after 1 µs, wM hasexchanged with a free neighbouring watermolecule. All the four previous dynamicalmechanisms affect R1(r), since they occurwithin periods shorter than the characteris-tic relaxation time 1/R1(r) which is typicallyin the order of 100 ms to 1 s.

III. CONCLUSION

Our contribution to the field of con-trast agents [2,3] allows the development ofnew accurate models and experimentalmethods that bring insight into the underly-ing physics. Such studies, which support therational design of of Gd3+ imaging probes,will be pursued in the Network of ExcellenceEMIL (European Molecular ImagingLaboratories).

ACKNOWLEDGMENTS

This research was car-ried out in the framework ofthe EC COST Action D-18"Lanthanide Chemistry forDiagnosis and Therapy".

[1] P. Caravan, J. J. Ellison, T. J. McMurry, and R. B. Lauffer, Chem. Rev. 99, 2293 (1999). [2] P. H. Fries, G. Ferrante, E. Belorizky, and S. Rast, J. Chem. Phys. 119, 8636 (2003). [3] E. Belorizky and P. H. Fries, Phys. Chem. Chem. Phys. 6, 2341 (2004).

Figure 1: MRI image of a mouse liver. (a): withoutcontrast agent. (b): one minute after injection of aGd3+-DTPA derivative (courtesy of S. Boutry, RMNLaboratory of Prof. R. N. Müller, University of Mons,Belgium, EMIL partner).


Figure 2: Theoretical description of the fates of typical water mole-cules which are near a contrast agent at an arbitrary time origint=0. The ligand (yellow) complexing the Gd3+ ion (grey sphere) isbound to a large biocompatible macromolecule (green).

Deleterious effects of ionizing radia-tion to cells result at least partly from theinduction of DNA damage. The largemajority of the underlying chemical reac-tions is triggered by radical species pro-duced upon radiolysis of the watermolecules surrounding DNA. Amongthem, hydroxyl radicals (°OH) and hydro-gen atoms (°H) are the most reactive.Elucidation of the pathways leading fromthe initial water radicals to the final muta-genic and lethal DNA lesions first requiresidentification of the DNA radicals result-ing from the reaction of bases and 2-deoxyribose units with °OH and °H.Subsequently, a more precise descriptionof the sequence of events leading to for-mation of the latter radical species maybe attempted. For both topics, theoreticalapproaches, based on quantum chemistrytools, provide useful information.

Characterization of DNA base radi-cals is experimentally mostly achieved byelectronic paramagnetic resonance (EPR).Emphasis has been placed on the radicalarising from thymine, which is the mostreactive base towards °H. The methyl-cen-tered radical (see Fig1) was the first to beidentified within irradiated DNA. The twomain products of the reaction of thyminewith °H, the 5,6-dihydro-6-yl-thymine(hereafter referred to as 6-yl) and 5,6-dihy-dro-5-yl-thymine (hereafter referred to as5-yl) radicals were the subject of structuralinvestigations during the past decades.However, this task is not simple because therelationships between EPR spectroscopicparameters and structural features are quite

indirect. This led to rather large discrepan-cies on the results of EPR studies on thymine5-yl and 6-yl radicals.

In such circumstances, computationalapproaches are particularly attractive and pro-vide additional information that partly explainsthese conflicting experimental data. As a mat-ter of fact, the EPR features of the different iso-mers of the thymine radicals can be calculatedprecisely [1]. Calculations show that in equilib-rium geometries of both 5-yl and 6-yl, thethymine ring is approximately planar with theexception of the carbon atom where hydrogenwas added. The latter carbon may be on eitherside of the average plane of the radical, in so-called half-chair configurations. Inversion ofthe half-chair is possible for the two isomers.However, energy calculation shows that oneform is more stable than the other for 6-ylwhile both are equally possible for 5-yl (seeFig. 2). In addition, determination of the vibra-tional wave functions shows that the form with

SO M E A S P E C T S O F DNA B A S E S S T U D I E D B YQ U A N T U M C H E M I S T RY T O O L S

SOME ASPECTS OF DNA BASES STUDIED BYQUANTUM CHEMISTRY TOOLS

We have applied a computational protocol to a comprehensive study of the spectroscopic propertiesof radicals arising from DNA bases. This approach also allowed the description of the pathwaysinvolved in their formation as well as calculation of thermodynamic parameters of the reactions.

Nous avons appliqué une démarche théorique pour réaliser une étude approfondie de propriétés spectroscopiques de radicaux issus des bases de l'ADN. Cette approche a aussipermis la description de chemins réactionnels impliqués dans leur formation, ainsi que le calcul de paramètres thermodynamiques de la réaction.

A. Grand and F. Jolibois

Service de Chimie Inorganique

et Biologique (SCIB), DRFMC

Figure 1: Chemical structure of the radicals arisingfrom the reaction of H° with thymine.

37

the methyl group in axial position is favored atlow temperature for 6-yl. In contrast, constantinter-conversion of one conformation into theother is expected for 5-yl. These calculationsalso show that, when increasing the tempera-ture and thus favoring higher vibrational levels,variations in the EPR spectra were expected,which actually accounted for some of the scat-tering in experimental data.

Identification of the geometry of baseradicals then enables the sequence of eventsinvolved in the addition of °H to the thyminering to be determined [2]. This description ofa chemical reaction cannot be experimental-ly observed, because intermediates are veryshort-lived species. Computation thus pointsout some important features such as thegeometry of pre-reactive and transitionstates. Thermodynamic parameters, includ-ing energy and activation barriers, can alsobe calculated. The main steps of the additionof H° to thymine are displayed in Fig. 3 whereeach point corresponds to an optimized ori-entation of the two reagents. The validity ofthe results is confirmed by the calculated ther-modynamic features of the reaction. First, alow value is found for the energy barrier (fewkcal/mole) for the formation of both 5-yl and6-yl radicals, in agreement with the efficiencyof addition of H° to thymine. In addition, theenthalpy difference is larger for formation of5-yl than 6-yl. Moreover, the activation ener-gy of the reaction leading to 5-yl is smallerthan for 6-yl. Altogether, these thermodynam-ic features are in complete agreement withthe experimental evidence showing that H°preferentially adds to the C6 position.

The results presented above mainlyfocused on the addition of °H to thymine.However, the reactivity of hydroxyl radicals°OH, the most reactive product arising fromwater radiolysis, has also been extensivelystudied using the same computationalapproach. Work is also in progress for theother DNA bases [3]. Future investigations

will deal with more complicated reactionssuch as the radical addition between twoadjacent DNA bases. The resulting informa-tion, combined with the quantification of thefinal products arising from the initial radicalspecies, will provide insights into the interac-tion between ionizing radiation and genomicmaterial.

[1] F. Jolibois, J. Cadet, A. Grand, R. Subra, N. Rega, and V. Barone, J. Am. Chem. Soc. 120, 1864 (1998).[2] F. Jolibois, A. Grand, J. Cadet, C. Adamo, and V. Barone, Chem. Phys. Lett. 301, 255 (1999).[3] C. Adamo, M. Heitzmann, F. Meilleur, A. Grand, J. Cadet, and V. Barone, J. Am. Chem. Soc. 123, 7113 (2001).

Figure 2: Potential energy (—) and first vibrational wavefunction ( ) for thymine 6-yl and 5-yl radicals on motion of C5 and C6 atoms, respectively, out of theplane of the molecule. Half-chair structure shown corresponds to conformation in thepotential wells.

Figure 3: Variation of energy on addition of H° to thy-mine. The activation energy (1) and enthalpy variation(2) are shown for formation of the 6-yl isomer.


By combining quantum-chemical calculations with electron paramagnetic resonance (EPR)measurements, we have built a model linking the local structure of the active sites of [2Fe-2S]proteins to their EPR features. This is a useful tool to characterize these proteins, on the basis of asimple EPR experiment when crystals are not available.

En combinant des calculs de chimie quantique avec des mesures de résonance paramagnétique électronique (RPE), nous avons construit un modèle reliant la structure locale du site actif de protéines [2Fe-2S] aux données de RPE. C'est un outil utile pour caractériser ces protéines, par une simple expérience de RPE, quand des cristaux ne sont pas disponibles.

J. M. Mouesca and S. Gambarelli

Service de Chimie Inorganique

et Biologique (SCIB), DRFMC

I. INTRODUCTION

Iron-Sulfur proteins are ubiquitous inthe living realm and play fundamental rolesin such processes as cellular respiration,photosynthesis, etc. Their active sites con-tain iron ions linked to sulfur containinggroups, for instance in the form of [2Fe-2S]clusters (see Inset1) (see Fig. 1a). Thegeometry of the active site appears to be akey parameter in the reactivity of the pro-tein. X-ray crystallography is the choicemethod to obtain the 3D structures.Unfortunately, in the case of [2Fe-2S] pro-teins, only a handful of proteins have beencrystallized and characterized among thehundreds isolated. In contrast, these sys-tems are easily characterized by electronparamagnetic resonance spectroscopy(EPR) (see Inset2) which requires very smallamounts of biological products. However,the basic quantity derived from EPR, the g-tensor (Fig. 1b), is difficult to correlate to thecluster electronic structures. A phenomeno-logical model of the [2Fe-2S] g-tensor pub-lished in the late 1970s (B-G model) [1]was based on a wrong geometry becausestructural data were not available at thattime.

Our strategy [2] to link EPR data togeometric features consisted firstly in find-ing which structural feature is responsiblefor the variation of the g tensors, and sec-ondly in constructing a subsequent modelwhich allows for computation of the g-ten-sors. We finally confronted our predictionswith known experimental data.

HO W T O S E E T H E L O C A L S T R U C T U R E O F A P R O T E I NW I T H O U T C RY S TA L L O G R A P H Y

HOW TO SEE THE LOCAL STRUCTURE OF A PROTEINWITHOUT CRYSTALLOGRAPHY

1. [2Fe-2S] fer redoxins

[2Fe-2S] ferredoxins are small electron-transfer pro-teins found in bacteria, plants and animals. These proteinsare folded chains of amino-acids embedding a [2Fe-2S](Cysteine)4 cluster, close to the protein sur-face, whose iron atoms are linked to the main proteicskeleton via two cysteines each. The paramagnetic EPR-active state is made of one ferrous and one ferric ion.

2. EPR & g- tensors

Electronic Paramagnetic Resonance (EPR) is a spectro-scopic technique analogous to Nuclear MagneticResonance (NMR). In both methods, spins are submittedto a static magnetic field H and interact with electromag-netic radiations of energy hv. Each spin system resonatesat a specific field value, and is thus characterized by aquantity called the g-factor for electronic spins (EPR, viahv = gµBH, where µB is the Bohr magneton)or the chemical shift for nuclear spins (NMR). In com-plex systems, because of the anisotropy, the g-factor ischanged into a g-tensor, characterized by three eigenval-ues g1, g2 and g3, which depend on the nature of themolecular orbitals involved and on their relative energies.

3. Dens i ty Func t iona l Theor y

The Density Functional Theory (DFT) is an ab initioquantum chemistry method where the fundamental quan-tity is the electron density. The DFT is particularly well suit-ed for computation of the electronic structure of poly-metallic clusters, in which the localized metal spins aremagnetically coupled via the exchange interaction.

II. THEORETICAL MODEL

We suspected that the main parameterinfluencing the g-tensor was the orientation ofthe sulfur lone pairs of the cysteine amino-acidligands, acting on the energies of the iron dorbitals. Consequently, we first listed the avail-able 3D structures of the known [2Fe-2S] pro-teins. We looked at the orientations of the cys-teines relative to the cluster, determining theposition of the sulfur lone pairs, and selectedthe relevant structural feature: the Fe-Fe-S-Cysteine dihedral angles.

In each protein, we observed that thetwo dihedral angles on the ferrous (Fe2+)site were nearly identical, the average angleΩ varying between 120° and 150° (see Fig.2). We quantum-mechanically computed theg-tensors, by DFT (see Inset3), for a simpli-fied molecular model: [2Fe-2S](SH)4,replacing the four cysteines by SH ligands,and varying Ω from 0° to 180° (see Fig. 3a).

We finally confronted our theoreticalresults with experimental data by plotting theg-tensors as a function of Ω for the few sys-tems for which both EPR and X-ray data areavailable (see Fig. 3b). The agreement isqualitatively (rhombic g-tensors forΩ=120°, axial ones for Ω=150°) andquantitatively good.

III. CONCLUSION

For the first time, it has been possibleto link the variation of the [2Fe-2S] g-ten-sors to a specific structural parameter. Withsuch a tool, the biologists’ community cannow appreciate through a simple EPRexperiment the local geometrical environ-ment of the [2Fe-2S] active sites, which alsoaffects the redox potential, the electrontransfer rate, the network of hydrogenbonds, etc. This has been a challenge froma computational point of view, but our newmodel enables the spectroscopists’ commu-nity to interpret a whole set of spectroscop-ic observables (EPR, Mössbauer) in a coher-ent fashion.

39

[1] P. Bertrand and J.-P. Gayda, Biochim. Biophys. Acta 579, 107 (1979). [2] S. Gambarelli and J.-M. Mouesca, Inorg. Chem. 43, 1441 (2004).[3] See for instance "Iron-Sulfur proteins" in Adv. Inorganic Chemistry Vol. 47,

edited by A. G. Sykes and R. Cammack (Academic Press, London, 1999).

Figure 1: (a): Schematic representation of a [2Fe-2S] cluster. (b):Typical rhombic (top) and axial (bottom) [2Fe-2S] cluster EPR spectra.The three g-values are linked to the field through the relation g = hv / (µBH) = 6883 / H(G).

Figure 2: Visualization of the planes defining a Fe-Fe-S-Cysteine-dihedral angle in a [2Fe-2S] cluster.

Figure 3: (a): Theoretical (DFT) variations of the three g values as a function of Ω.The grey zone corresponds to the Ω range observed in actual proteins. (b): Plot ofthe three experimental g values as a function of Ω for known proteins. In the latterscheme, trends in the variation of the g values are outlined by regression lines.

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We finally open the wayfor correlation between “sim-ple” spectroscopic data andprecise geometrical features inthe world of Iron-Sulfur pro-teins, of which interesting func-tions are continuously beingdiscovered [3] (various hydro-genases, nitrogenases…).

ACKNOWLEDGMENTS

We thank J. Meyer andM. Fontecave (DRDC,CEA/Grenoble), and J. C.Fontecilla (IBS/Grenoble) fortheir interest and helpful discussions.

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MU LT I -T R A C K R E A D I N G H E A D SMULTI-TRACK READING HEADS

Nowadays, the size of magneto-resistive reading heads has reached the micron range. Themagnetic behavior of such confined systems can no longer be described by macroscopic models.So, we have developed an original approach, able to deal simultaneously with magnetic systemsof very different sizes, one macroscopic and the other microscopic.

Aujourd'hui, les dimensions des têtes de lecture magnéto-résistives ont atteint le domaine du micron. Leur réponse magnétique ne peut plus être décrite par des modèles macroscopiques. Aussi, avons nous développé une approche originale, capable de traiter le couplage entre des systèmes magnétiques de tailles très différentes, l’un macroscopique et l’autre microscopique.

L. D. Buda, J. Ch. Toussaint, I. Firastrau, and J.-P. Nozières

Service de Spintronique et Technologie

des Composants (SPINTEC), URA 2512,

DRFMC

The design of advanced magnetore-sistive (MR) reading heads requires a deepunderstanding of the fine interactionsbetween the sensor micromagnetics, themedia magnetic pattern and the magneticbehavior of macroscopic elements such asthe magnetic shields (see Fig. 1). In a read-ing head the active element (the sensor) isinserted between two magnetic shieldswhich serve the purpose of concentratingthe field produced by the media. There is alarge scale difference between all these ele-ments. The shields and media are of micronsize or larger, while the sensor has dimen-sions comparable with the characteristiclengths of magnetic materials (exchangelength and magnetic domain wall width)which are in the order of the nanometer.Modeling of such systems therefore involvesa complex computational issue since someelements of the head have to be treated

macroscopically while the active elementhas to be analyzed at nanometric scale bya micromagnetic approach [1].

Up to now, the magnetic field gener-ated by the media is usually calculated bybi-dimensional models limited to idealshields having infinite permeability (withoutstray field). For three-dimensional headswith arbitrary permeability, large scaleFinite Element Method (FEM) computationcould be used. Because the MR readingheads for disk/tape drives discussed hereare systems with very different characteris-tic length scales, we preferred to use theBoundary Element Method (BEM) since, forcontinuous and linear media, only the sur-faces/interfaces are discretized and not thewhole volume as in a common FEMapproach: this therefore saves a lot ofcomputation time.

Figure 1: A shielded multi-track magnetic reading head (sensor and shields) flyingabove a longitudinal media. During the reading process, the head moves along thedowntrack direction and reads information stored along the crosstrack direction.

In the case of longitudinal magneticrecording, the magnetization lies in theplane of the thin media, and the informa-tion is stored by the presence (bit 1) orabsence (bit 0) of a magnetic transition(see Fig. 1). The thin active element of thereading head picks up the magnetic fieldgenerated by magnetic transition writtenon the media. To evaluate this field in thepresence of macroscopic shields character-ized by a finite magnetic permeability µr,the Poisson’s equation for the scalar mag-netic potential is solved. In addition, formulti-track data patterns as encountered intape drives, the magnetic scalar potentialmay be estimated with a good approxima-tion by considering a periodic system alongthe crosstrack direction (Oz). By applyingthe Fourier series expansion for the mag-netic charge distribution along thecrosstrack direction, the initial BEM three-dimensional problem can be treated as abi-dimensional problem. In fact, the mag-netic scalar potential is now the solution ofa modified Helmholtz type equation. Themethod is conceptually straightforward, butsome care had to be taken in order toensure numerical accuracy.

The spatial distribution of the magne-tostatic field evaluated by BEM serves as aninput to a classical micromagnetic descrip-tion of the sensor response. Briefly, themagnetic state of the sensor is the result ofcompetition between several interactions:the exchange interaction, the magnetocrys-talline anisotropy, the magnetostatic inter-action and Zeemann coupling. Each possi-ble magnetic stable state corresponds to alocal minimum of the total free energy.These equilibrium states may be reached byintegrating the Landau-Lifshitz-Gilbertequation [2] which governs the dynamics ofmagnetization. For a crosstrack data pat-

tern consisting of alternating transitions andguard-bands schematically drawn in Fig. 1,the magnetization distribution of the MRsensor, as predicted by micromagnetic sim-ulation, is depicted in Fig. 2a. The distribu-tion of the magnetization is directly linked tothe electrical response of the sensor.Supposing that the head moves along thedowntrack direction, the read transfer curvemay be reconstructed (see Fig. 2b) and thencompared with experimental data.

Thus by coupling the BoundaryElement Method and the micromagneticapproach, we fully describe the behaviorof the magnetic sensor in its environment.Problems such as cross-talks betweenadjacent tracks could be addressed. Thisnumerical approach is a useful tool todesign optimized multi-track magnetore-sistive reading heads.

41

[1] I. Firastrau, L. D. Buda, J.-Ch. Toussaint, and J.-P. Nozières, J. Magn. Mater. 272-276, 738 (2004). [2] O. Fruchard, J.-P. Nozières, B. Kevorkian, J.-C. Toussaint, D. Givord, F. Rousseaux, D. Decanini, and F. Carcenac,

Phys. Rev. B 57, 2596 (1998).

Figure 2: (a): Magnetic stable state of a magnetoresistive head. The head is centeredexactly above the magnetic transitions and the magnetization distribution is plotted inthe yOz plane. The magnetic volume charges of the transitions (t = s = 1µm) aredrawn below along the crosstrack direction Oz. (b): Simulation of the read transfercurve of the head for shields with various magnetic permeabilities.


The discovery of giant magneto-resis-tance in (Fe/Cr) multilayers in 1988launched very intense activity on spin-dependent transport in magnetic multilay-ers. The invention of spin-valves [1] whichexhibit giant magneto-resistance at lowfields had a tremendous impact on mag-netic recording technology. Indeed thesematerials are nowadays used as sensingelement in read heads of computer diskdrives. A metallic magnetic multilayer con-sists of a stack of magnetic layers separat-ed by a non magnetic spacer layer. Thematerials are chosen so that the relative ori-entation of the magnetization in the succes-sive layers can be varied by application ofan external field. As the relative orientationchanges from parallel to anti-parallel, themultilayer varies from low to high resis-tance. Information is retrieved from record-ed bits by monitoring the multilayer resis-tance as the read head goes through themagnetic fields they create.

In metallic magnetic materials, theelectron transport properties depend on therelative orientation of the electron spin withrespect to the local magnetization. Forinstance in Co, the conductivity of the elec-trons with spin parallel to the local magne-tization is six times larger than for electronswith antiparallel spin. As a result, the netcurrent is spin polarized parallel to themagnetization. Let us then consider aninterface separating two magnetic layers ofopposite magnetization (see Fig.1) andassume that a current is flowing from rightto left i.e. electrons are drifting from left toright. In the left ferromagnetic layer, farfrom the interface, the current is mainly car-

ried by spin ↑ electrons. The spin ↑ currentis given by

where J is the total current density and βis a characteristic scattering asymmetryparameter (about 0.7 in Co). The spin ↓electrons carry a smaller current

EL E C T R O N T R A N S P O R T P E R P E N D I C U L A RT O T H E I N T E R FA C E S I N M A G N E T I C M U LT I L AY E R S .

ELECTRON TRANSPORT PERPENDICULARTO THE INTERFACES IN MAGNETIC MULTILAYERS

When an electrical field is applied perpendicular to the interfaces of a magnetic metallic multi-layer, the drift of the conduction electrons is spin-dependent and varies from layer to layer. Thesevariations result in local spin accumulations which are counterbalanced by spin relaxation effects.

Quand un champ électrique est appliqué perpendiculairement aux interfaces d’une multicouche métallique magnétique, le mouvement de dérive des électrons dépend du spin et varie d’une couche à l’autre. Ces variations conduisent à des accumulationslocales de spin contrebalancées par des effets de relaxation de spin.

B. Dieny a, A. Vedyaev b

and N. Strelkov b

a: Service de Spintronique et Technologie

des Composants (SPINTEC),

URA 2512, DRFMC

b: Lomonosov State University, Moscow

Figure 1: Illustration of spin accumulation at a singleinterface separating two ferromagnetic layers of oppo-site magnetization. The large arrows represent mag-netization orientation. (a): Spatial variation of the dif-ference in chemical potential ∆µ= µ↑− µ↓ between thetwo species of electrons: ∆µ > 0 means an accumu-lation of spin ↑ electrons at the interface. (b): Spatialvariation in current density for each spin channel(adapted from [2]).

Jj ↑ = (1+β)2

Jj ↓ = (1-β).2

On the right-hand side, the magneti-zation is down and the roles of spin ↑ andspin ↓ electrons are inverted. More spin ↓electrons flow away from the interface thanspin ↑ electrons. Within each spin channel,there is therefore an unbalance between thenumber of electrons moving towards andaway from the interface per unit time. Thisresults in a spin accumulation which con-sists of a local excess of spin ↑ electronsaround the interface and a correlated localdeficit in spin ↓ electrons. In steady state,this spin accumulation is counterbalancedby spin flip processes (spin orbit interactionand magnon scattering). The characteristiclength scale over which the electrons main-tain their spin orientation in a given materi-al is the spin diffusion length lsf. The bal-ance between the spin accumulation andspin relaxation leads to local variations inthe difference ∆µ of electrochemical poten-tial of spin ↑ and spin ↓ electrons as illus-trated in Fig.1. The result is a small nonequilibrium local magnetization near theinterface, proportional to the current.

The Valet and Fert theory [2]addresses the effect of spin accumulation,using a macroscopic model based on twobasic equations. A first equation describesspin relaxation effects whereas the secondis a spin-dependent generalized form ofOhm’s law. By solving these equationswithin each layer and connecting themfrom layer to layer by a transfer matrixtechnique, we have developed a numericalcode which enables the resistance andmagneto-resistance of any magnetic multi-layered stack to be calculated, taking spin-flip processes into account [3]. This code iscurrently being used by head manufactur-ers for optimization of spin-valve stackswith current flowing perpendicularly to theinterfaces. This approach allows a muchfaster optimization of the stack composi-

tion. We have studied sandwich structuresof the form NiFe t F /Cu 3nm/NiFe 5nmand Co t F /Cu 3nm/Co 5nm versus thick-ness of one of the ferromagnetic layers.Figure 2 shows a comparison of the varia-tion of the magneto-resistance expressedboth in terms of absolute change in resis-tance times area (∆R×A) and relativechange of resistance between parallel andantiparallel magnetic configurations. Themain differences between NiFe and Co aretheir resistivities (25µΩ.cm for NiFe and15µΩ.cm for Co) and their spin diffusionlength ( l = 4.5 nm and l =15 nm). Thelarger NiFe resistivity is at the origin of thelarger magnetoresistance amplitudeobserved in the NiFe based structures. Italso emphasizes the role of bulk scatteringwith respect to interfacial scattering leadingto the presence of a maximum in ∆R/R notobserved with Co. In head applications, ∆Ris related to the amplitude of the signalwhereas ∆R/R relates to the signal to noiseratio. According to Fig. 2, NiFe layers withthickness about 4nm would optimize thetransport properties of the device.

43

[1] B. Dieny, V. S. Speriosu, S. Metin, S. S. P. Parkin, B. A. Gurney, P. Baumgart, and D. Wilhoit, Appl. Phys. 69, 4774 (1991).[2] T. Valet and A. Fert, Phys. Rev. B 48, 7099 (1993).[3] N. Strelkov and A.Vedyaev, Journ. Appl. Phys. 94, 3278 (2003).

Figure 2 : Example of calculation of magneto-resistance of NiFe tF/Cu3nm/NiFe 5nm and Co tF/Cu 3nm/Co 5nm sandwiches versus thickness ofone of the ferromagnetic layers. Absolute magneto-resistance × area productand relative MR ratio.


The semi-classical the-ory of spin-dependent trans-port perpendicular to theinterfaces in magnetic metal-lic multilayers was general-ized to calculate the resis-tance and magneto-resis-tance from the knowledge ofthe transport properties ofthe material constituting eachlayer. This approach is usedfor optimization of structuresimplemented in magneto-resistive heads. NiFe

sfCosf

BOILING CRISIS: THEORY, SIMULATION, AND EXPERIMENTSBOILING CRISIS: THEORY, SIMULATION, AND EXPERIMENTS

Boiling is a very efficient way to transfer heat from a heater to the liquid heat carrier. We discuss theboiling crisis, a sharp decrease in the heat transfer rate, which can cause a major accident in industrialheat exchangers. Numerical simulation of the growth of vapor bubbles has been found to be in verygood agreement with experimentally observed results.

L'ébullition est un moyen très efficace de transférer la chaleur d'un élément chauffant à un liquidecaloporteur. Nous discutons la crise d'ébullition, une chute brutale du transfert de chaleur, susceptible de causer un accident grave dans des échangeurs de chaleur industriels. La simulation numérique de la croissance des bulles de vapeur s'est révélée en excellent accord avec des résultats observés expérimentalement.

V. Nikolayev and D. Beysens

Service des Basses Températures

(SBT), DRFMC

Boiling is observed commonly ineveryday life. This is why it seems well under-stood. It is true that boiling has been exten-sively studied from an empirical point of viewfor the most common fluids and regimes, forinstance for water at atmospheric pressureand moderate heat flux supplied to fluid.However, the basic theory of boiling remainsterra incognita. These difficulties originatefrom the violence of the fluid motion that onthe one hand conceals the mechanisms ofbubble growth from detailed observation,and on the other hand hugely complicatesdirect numerical simulations. Most still unan-swered questions concern the close vicinity ofthe heating surface, down to the scale ofbubbles growing at the surface of the heaterespecially during boiling at high heat fluxescommon for industrial heat exchangers, e.g.nuclear power plant steam generators.

Two main boiling regimes can be dis-tinguished: nucleate boiling and film boil-ing. The former can be commonlyobserved in a sauce pan. It features sepa-rate vapor bubbles that nucleate (i.e. form)and grow at the heater. This regime pro-vides a very efficient heater-liquid heatexchange due to their direct contact. Theother regime features a vapor film thatseparates the liquid from the heater. Filmboiling can be also observed in the kitchenby sprinkling water onto a hot frying pan.In spite of the large temperature of theheater (the pan) the water drops survive fora long time precisely due to the thermallyinsulating vapor film that prevents the liq-uid from touching the pan. Obviously, theheater-liquid heat transfer is much lowerthan in the first case.

The subject of our study is the transi-tion from nucleate to film boiling which iscalled "boiling crisis". During this transition,separate bubbles at the heater are replacedrapidly by a continuous vapor film. Thisblocks the heater-liquid heat transfer andthus leads to a rapid increase of the heatertemperature. If the power is not switched offimmediately, the heater can melt downwhich can cause a major accident if anindustrial installation is involved. The boil-ing crisis occurs at a critical value of theheating flux (fortunately not attainableunder "kitchen" conditions).

In spite of its importance, the mecha-nism of the boiling crisis remains poorly stud-ied. Our purpose is to explain why, how, andwhen the vapor film begins to form. We havemade the fundamental hypothesis that theboiling crisis is triggered by the vapor recoilwhen liquid transforms into vapor.

Every fluid molecule evaporated fromthe liquid interface causes a recoil force anal-ogous to that created by the gas emitted by arocket engine. It pushes the interface towardsthe liquid side in the normal direction. Thisforce appears because the fluid necessarilyexpands while transforming from liquid togas phase. Obviously, the stronger the evap-oration rate, the larger the vapor recoil force.

Let us now consider the variation of theevaporation rate along the surface of a bub-ble growing at the heating surface. Since thefluid is hotter close to the heater than far fromit, the evaporation distributed along the bub-ble surface is strongest in the vicinity of theline of contact of the bubble surface with the

heater. The vapor recoil is also strongest at thecontact line and therefore tends to pull the con-tact line outward, thus spreading the dry areaor “dry spot” under the bubble (see Fig. 1a).

A numerical simulation [1] of such aprocess is presented in Fig. 1b. Such a sim-ulation requires solving of extremely delicatethermal and capillary problems. It can beseen that the dry spot is initially very smalland remains so during the initial growthstage. At about 180ms the dry spot beginsto grow suddenly, i.e. the bubble spreads.Such a spreading represents the beginningof formation of the vapor film characteristicfor the boiling crisis. Figure 1b also showsformation of a hot spot at the heater surfacein the middle of the dry area. This tempera-ture rise illustrates the already discussedblocking of the heat transfer by vapor.

This theoretical approach can becompared to an experiment [2] carried outwith SF6 fluid near its critical liquid-gaspoint. This experiment takes advantage ofso-called "critical slowing down" of the bub-ble growth observed near the critical point.In fact, the growing process of a singlevapor bubble could be observed during 45min thus allowing for a very detailed analy-sis. The choice of SF6 was made for practi-cal reasons: the critical point of this fluid isat 45.6°C, 38 bar and requires much lesssevere conditions for the experiment thanfor example water (374°C, 220 bar).However, near-critical bubble growth experi-

ments have an important drawback. Sincethe surface tension becomes very low nearthe critical point, gravity completely flattensthe liquid interface. Weightlessness condi-tions are thus necessary to preserve theusual convex bubble shape. Some of theresults of this experiment performed onboard the Mir space station are presentedin Fig. 2a. The sequential photos of thegrowing vapor bubble were taken throughthe transparent bases of the cylindrical cell,the lateral copper walls of which are beingheated. Spreading of the dry spot under thebubble similar to that in Fig. 1b can beseen. The bubble shapes calculated for dif-ferent values of the vapor recoil strength Nare presented for comparison in Fig. 2b.The correspondence is striking.

45

[1] V. S. Nikolayev, D. Beysens, G.-L. Lagier, and J. Hegseth, Int. J. Heat Mass Transfer 44, 3499 (2001).[2] Y. Garrabos, C. Lecoutre-Chabot, J. Hegseth, V. S. Nikolayev, D. Beysens, and J.-P. Delville,

Phys. Rev. E 64, 051602 (2001).

Figure 1. Vapor bubble spreading under the influence of vapor recoil. (a): Sketch illustrating the vapor recoil effect. The amplitude and direction of the vapor recoil force are shown by arrows. (b): Simulation results. The color indicates the local temperature with respect to the saturation temperature Tsat.

Figure 2. Snapshots of the vapor bubble (V) growing in near criticalliquid (L), both experimental (a) and calculated (b) for the given valuesof the vapor recoil strength N.


Generally speaking,the comparison between theresults presented aboveshows the complementarityof theory (basic assump-tions), experiment (test of thevalidity of a model) and sim-ulation (access to parametersnot attainable by experi-ments).

(a) (b)

(a)

(b)

PU L S E T U B E C RY O C O O L E R SPULSE TUBE CRYOCOOLERS

There is a growing demand for small cryocoolers due to the emergence of specific applicationsrequiring low-temperature operation, typically in the 80 K to 4 K range. Pulse tube refrigerators arevery promising in this regard. We report on efforts made in our laboratory to develop a calculationmodel for this device.

Il y a une demande croissante pour des petits cryoréfrigérateurs qui est due à l'apparition d'applications spécifiques fonctionnant à basse température, typiquement dans la gamme de 80 K à 4 K. Les réfrigérateurs à tube pulsé sont très prometteurs à cet égard. Nous évoquons les efforts faits dans notre laboratoire pour développer un modèle de calculpour cet appareil.

J. M. Poncet and R. Vallcorba

Service des Basses Températures

(SBT), DRFMC

The need for small cryocoolers aris-es in different fields of activity. For exam-ple, electromagnetic radiation detectors(infrared to x-ray) used in astronomicaland earth observation require cooling wellbelow room temperature for better sensi-tivity. Devices containing superconductingcomponents need refrigeration belowcharacteristic critical temperatures (typical-ly between 50 and 80 K). Basic researchon material properties is often performedat low temperatures. In all these applica-tions, the purpose of the cryocooler is tocool objects and to maintain them at aspecified operating temperature byabsorbing the heat released from theseobjects and from their immediate environ-ment (thermal conduction throughmechanical structures, thermal radiationfrom surrounding hot surfaces …).

In most types of cryocoolers, periodicgas expansion is used to produce the refrig-eration effect [1]. A working gas (helium)follows a closed thermodynamic cycle.Expansion and heat absorption take placeat a special section - the cold source.Compression associated with heat releaseis usually performed at room temperature(hot source). Pulse tube cryocoolers makeuse of pressure waves. The basic principle isshown on the left-hand side of Fig. 1: acompressor unit (pressure oscillator) gener-ates a pressure wave which propagatesthrough a set of two tubes, the regeneratorand the pulse tube, before reaching thesubsystem labelled impedance.

The regenerator tube is filled withporous material and, in stationary opera-tion, exhibits a temperature gradient span-

Figure 1: Pulse tube cryocooler. Left: schematic, colours roughly indicate temperaturedistribution, red – hot, blue – cold. Right : Practical example for medical applications. The regenerator and pulse tube are in a concentric configuration; only the regeneratoris visible. The device is used to cool a superconducting antenna.

After full validation it will besuitable for temperaturesdown to the 4 K range.

A sector model hasbeen developed for theimpedance device. It focuseson the study of the phaseshift effect created by a cap-illary and a buffer volume.Figure 2 shows a comparisonbetween calculated andexperimental results [3]which, among other things,reveal forbidden zones in theflow rate/phase shift plane.In the diagram, these zonesclearly appear outside blackboundary lines. For example,a phase shift of 40° can onlybe obtained with mass flowrates larger than 0.6 g/s.

ACKNOWLEDGEMENTS

We wish to thank ourcolleagues P. Seyfert and I . C h a r l e s f o r h e l p f u l discussions.

ning the gap between the cold source androom temperature. Its function is to cooldown the incoming helium gas on its way tothe cold source. It acts like a thermalsponge by alternately absorbing heat fromthe incoming gas and releasing heat to theoutgoing gas. The pulse tube is nothingmore than an empty thin-walled tubeentirely filled with the working gas whichbehaves somewhat like a virtual piston. Thephase shift between the oscillating gas flowand the pressure wave is the key parameterwhich governs the cooling effect at the coldend. For optimum performance, it shouldbe nearly zero at that location. The specialimpedance device (here a capillary associ-ated with a buffer volume) connected to thehot end of the pulse tube allows correctadjustment of this parameter. The phaseshift at the hot end is related to that at thecold end via the flow impedance of thepulse tube.

The major advantages over othertypes of cryocoolers (Stirling, Gifford-MacMahon) are the lack of moving parts atthe cold source and the associated reductionof vibration. The consequences are excellentreliability and simplified fabrication. A prac-tical example of a compact pulse tube cry-ocooler for medical application is shown onthe right-hand side of Fig. 1.

The two main features of cryocoolerperformance are cooling capacity and ther-modynamic efficiency. In the pulse tube cry-ocooler, these features are mainly deter-mined by the regenerator efficiency and thepressure/flow phase shift along the pulsetube. Many parameters come into play: theproperties of the regenerator material (spe-cific heat, thermal conductivity and diffusiv-ity), the size of the pulse and regeneratortubes, the characteristics of the impedancedevice and of the pressure wave generator.

Their number is too large and their influencetoo complex to allow a performance predic-tion based on simple empirical scaling laws.

This apparently simple technologicalobject is therefore an open study subject fortheory. As part of an ongoing developmentprogramme in our laboratory, long-standingefforts have been made to establish calcula-tion models for pulse tube operation as a toolfor performance prediction and, more gener-ally, for design of advanced prototypes.

Two types of models have been devel-oped: global models which concern theoperation of the entire cooler [2] and sectormodels which describe one of the relevantcomponents in detail. Our first globalmodel was based on analytic algorithmsand allows fast calculation of a large num-ber of geometrical configurations. Theassumptions and simplifications made inthis model (ideal gas properties for helium,simplified regenerator) do however limit itsapplication to cycles with cold source tem-peratures above 15 K.

A second global model was recentlycompleted. It includes a real gas stateequation for helium and uses numericalmethods for the regenerator operation.

47

[1] L. Duband and A. Ravex in Handbook of Cryogenics Engineering, edited by J. G. Weisend II (Taylor&Francis, Philadelphia/USA, 1998) p.287.

[2] A. Ravex, P. Bleuzé et al., 19th International Congress of Refrigeration, proceedings IIIb, (International Institute of Refrigeration, Paris/France, 1995) p.1209.

[3] I. Charles, J. M. Duval et al., Cryocoolers 12, Conference Proceedings edited by R. G. Ross Jr. (Kluwer Academic/PlenumPublishers, New York/USA, 2003) p.131.

Figure 2 : Phase shift vs. flow rate for different geometries of the capillary inthe impedance device. The diameter of the capillary is indicated by the colour,its length in mm is marked on the curves. (solid line – numerical results),(dashed line – experimental results).


BULLETINO F T H E

Direct ion des Sciences de la Mat ièreCEA GRENOBLE

BULLETINO F T H E

Direct ion des Sciences de la Mat ièreCEA GRENOBLE

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