Disordine, localizzazione quantistica e ottimizzazione dei ...

Seminario Dipartimento 15 Maggio 2019

Disordine, localizzazione quantistica e ottimizzazione dei devices organici

S. Ciuchi Dipartimento di Scienze Fisiche e Chimiche Università dell’Aquila CNR - Istituto dei Sistemi Complessi

le referenze per questo seminario si trovano online nella mia pagina web: http://www.aquila.infn.it/ciuchi → Seminars → Seminario Dipartimento o direttamente al link: http://www.aquila.infn.it/ciuchi/seminars/DipartimentoMag2019/index.html

http://www.aquila.infn.it/ciuchi

http://www.aquila.infn.it/ciuchi/seminars/DipartimentoMag2019/index.html


My Outline

Motivation: curiosity driven research

General introduction in Statistical and Quantum mechanics

Boltzmann Anderson Parisi, dynamic and static disorder

Static disorder in statistical mechanics: an example the (spin)glass

Static disorder in quantum systems: Anderson localization

Applications to organic materials: dynamical restoring of diffusion

Group activity


Ludwig Boltzmann

Formulazione statistica dell entropia: S = kB log(W)

Equilibrazione nei sistemi a molti corpi: Equazione di Boltzmann


Philip Warren Anderson

Rottura spontanea della simmetria e sua applicazione nel caso della fisica degli stati condensati

Localizzazione quantistica dovuta al “disordine” (Anderson localization)


Giorgio Parisi

Teoria di campo medio in sistemi “disordinati” (vetri di spin)

Parametro d’ordine

Rottura della simmetria di “replica” nei vetri di spin

Lunghissimi tempi di autocorrelazione per gli spin


Il vetro quello veroPablo G. Debenedetti* & Frank H. Stillinger†‡ NATURE, 410, 259 (2001)


Il vetro quello “vero”

la localizzazione è dovuta alla mutua interazione fra le particelle che limita fortemente la loro diffusione quando la densità è elevata

to a fluid phase of clusters. In the other case, an arrestphenomenon will take place, driven by the crystallizationor—more likely because of the inherent self-generatedcluster polydispersity—by a glass transition. With thismechanism arrested disordered states with small ! canbe generated. This new route to the gel state demonstrates,in the case of weakly charged short-ranged attractivecolloids, the identity of the gel and the glass states ofmatter.

We next provide numerical evidence that the above-proposed mechanism does lead to disordered arrestedstates contiguous to a cluster phase. We perform MDsimulations of particles interacting with the potentialVSR ! VY [25], both along a constant T and along aconstant ! path. Here, we focus on the specific choiceof A " 0:05 and " " 2, referring to future publicationsfor a more extensive analysis. This choice of parameters ischaracteristic of colloidal particles in weakly polar sol-vents [4,26,27].

The cluster size distribution, shown in Fig. 3, providesa characterization of the clustering process. While at highT the cluster size is monotonically decreasing, aroundkBT=# " 0:2 the distributions develop a peak at a finitevalue, signaling the presence of stable equilibrium aggre-gates composed of 20–30 particles. We note that such aresulting cluster phase can be considered as an echo of theinterrupted spinodal decomposition, since clusteringtakes places just below the liquid-gas critical temperatureof the bare VSR potential.

For each of the studied T, the unclustered system (N "1) can be represented as a point in Fig. 2. A flow curve,starting from each of these points, can be calculatedaccording to Eqs. (3) and (4). In this way, a theoreticalprediction for the average cluster size at which crystal-lization and (MCT) vitrification are expected can be

derived. Such curves are shown in the inset togetherwith the T dependence of the average cluster size calcu-lated from the MD data. For the present choice of A, ",and !, structural arrest (either in the form of crystalli-zation or in the form of vitrification) is predicted when theaverage cluster size is * 35 particles (kBT=# # 0:15).

Before turning to the evaluation of dynamical proper-ties, we discuss the evolution of the structure with T,analyzing the static structure factor S$q%. As shown inFig. 4(a), S$q% progressively develops a peak at wavevectors q associated with distances in real space of theorder of several particle diameters, reflecting the nearestneighbor cluster-cluster distance. The amplitude of thisnovel peak changes significantly in a small temperaturerange around the T region where the flow diagram pre-dicts the crossing of the crystallization line. Therefore,the system changes from a solution of interacting par-ticles toward a system of larger and larger interactingclusters. In the cluster phase, the location of the peakchanges very weakly both with T and with !. It isinteresting to note that both the location and the ampli-tude of the cluster-cluster peak are in agreement with theexperimental data for colloidal gels reported in Ref. [3].To confirm that the aggregation process indeed generatesa disordered arrested state, we show in Fig. 4(b) thecoherent scattering function F$q; t%. It displays a (one-step) decay, with a characteristic time which increases oncooling. At the lowest studied T, the correlation functiondoes not decay to zero, signaling that the system is

FIG. 3 (color online). Cluster size distribution, on a log-logscale, for ! " 0:125. For clarity, curves for different T havebeen shifted along the vertical axis progressively by onedecade. The inset shows the T dependence of the averagecluster size. It also shows, for each T, the size at which thecrystallization and the ideal glass transition lines are crossed,according to the flow described in Fig. 2.

FIG. 4 (color online). Evolution of static and dynamic quan-tities on cooling at ! " 0:125. (a) S$q%. Note the developmentof a peak at small q, reflecting the formation of clusters in thesystem. (b) F$q; t% at q$ " 2:7. At T " 0:1, the autocorrelationfunction does not decay to zero any longer, highlighting theformation of a structurally arrested phase.

VOLUME 93, NUMBER 5 P H Y S I C A L R E V I E W L E T T E R S week ending30 JULY 2004

055701-3 055701-3

a bassa temperatura la funzione di autocorrelazione temporale della densità non decade velocemente

correlazioni a lungo range temporale

S. Mossa, R. Di Leonardo, G. Ruocco, and M. Sampoli PRE 2000

simulazione di modello classico interagente per un vetro fragile (ortoterfenile)

spanned at T!300 K the whole interesting q space in theinterval q!2"30 nm#1 !averaging on the values of q fall-ing in the same interval 2"q wide#.In Fig. 16 we show Fs(q ,t) for nearly all temperatures

investigated at q!qmax!14 nm#1; all the curves decay tozero, i.e., the length of all the simulations allows the fluctua-tions to become completely uncorrelated. We are in a‘‘good’’ thermodynamical equilibrium at every temperature,at least on the space scales corresponding to the inverse ofqmax .At temperatures lower then T!330 K the relaxation fol-

lows clearly the predicted two step pattern: on microscopictime scales the correlation is quadratic in time, this time-scale being the one on which the intramolecular vibrationshappen; on intermediate time scales we observe the forma-tion of a plateau, whose height is the nonergodicity param-eter f (q) and whose length in time is comparable to the oneof the plateau in the MSD $r2(t)%. On long-time scales weobserve the structural relaxation in the form of a stretched

exponential. At the highest temperatures no double pattern isvisible anymore and only a nearly exponential relaxation canbe recognized. A stretched exponential fit &see Eq. !29#' onthe structural time scale (( process# gives us the temperaturedependence of the three free parameters )( , *( , and f (q).The parameter )( !circles# is shown in Fig. 17; it appears

to be nearly T independent for temperature lower than T!400 K and its mean value )(!0.8 !dashed line# has to becompared with the experimental value )(!0.6. For tem-perature in the higher region it tends toward the value )(!1 !errors are clearly much more greater#; such behavior isdue to the fact that in this temperature region it is no longerpossible to sharply separate the long-time relaxation regionfrom the microscopic short-time one. The study of the tem-perature dependence of the nonergodicity parameter f (q) inthe interesting region is not possible due to our limited tem-

FIG. 16. Temperature dependence of Fs(q ,t) calculated at q!14 nm#1 for all temperatures investigated except T!410,430 K !lower temperatures on top#.

FIG. 17. Temperature dependence of the stretching parameter)( !circles# and of the nonergodicity parameter f q !squares#; thehorizontal lines indicate the mean values of )( !dashed line# and f q!dot-dashed line#.

FIG. 18. Temperature dependence of *( at q!14,19 nm#1

!circles and squares, respectively# together with the power-law fitswith T̄c

*(!296 K and +*(!2.0 !solid and dashed lines, respec-tively#; also the experimental shear viscosity ,s-*( data !full tri-angles# are reported !see &22' and reference therein# multiplied by afactor 1.5 ps/Poise. Molecular dynamics results have been shifted20 K with respect to the measured temperatures, as explained intext.

FIG. 19. Fs(q ,t) at q!14 nm#1 rescaled to t/*( ; all thecurves verify the time-temperature superposition principle.

624 PRE 62S. MOSSA, R. Di LEONARDO, G. RUOCCO, AND M. SAMPOLI


Philip Warren Anderson

Rottura spontanea della simmetria e sua applicazione nel caso della fisica degli stati condensati

Localizzazione quantistica dovuta al “disordine” (Anderson localization)


Disordine dinamico: localizzazione transiente

disordine statico disordine dinamico

il disordine dinamico ripristina uno stato diffusivo

la localizzazione quantistica permane in un transiente la cui scala di tempo è governata dalla scala di tempo della dinamica del disordine

disordine disordine

|ψ|2 |ψ|2


Semiconduttori organici cristallini

“Elettronica organica” sostituzione del silicio con materiali organici per la costruzione dispositivi come i FET

la “mobilità” elettrica (μ=eD/kBT) è una caratteristica da ottimizzate

ma la mobilità per il semiconduttori organici è una via di mezzo fra sistemi molecolari disordinati e semiconduttori cristallini “classici” come il Silicio

hopping Boltzmannlocalizzazione transiente


Semiconduttori organici cristallini

molecole organiche “aceni” cresciuti in fase cristallina

struttura stratificata

legami fra le molecole “deboli” di Van der Waals

basso disordine estrinseco (statico)

alto disordine intrinseco (dinamico) dovuto alle oscillazioni molecolari

rubrene


Semiconduttori organici cristallini: diffusività

nero: disordine statico rosso: disordine dinamico grigio: equazione di Boltzmann

S. Fratini, S. Ciuchi https://arxiv.org/abs/1903.12603

Sulla scala tloc il sistema è localizzato Sulla scala di tempo 𝜏in le fluttuazioni dinamiche dell’hopping ΔJ restaurano un regime diffusivo

https://arxiv.org/abs/1903.12603


Semiconduttori organici cristallini: ottimizzazione

modello reticolare bidimensionale “tight binding” parametri Ja,Jb,Jc

studio parametrico in funzione del rapporto Ja/Jb (Jb=Jc) parametrizzato in un “angolo” theta

S. Fratini, S. Ciuchi, D. Mayou, G. Trambly de Laissardiére and A. TroisiNat. Mater. 16, 998 (2017)


Semiconduttori organici cristallini: ottimizzazione

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 θ0 π/2 π-θ0 π

ΔJ/J

θ

rubrenepentaceneC10-DNTTC10-DNBDTTMTES-PnTESADTdiF-TESADTC8-BTBTm8-BTBTBTBTTESADTTIPS-PnTIPS-ADT

band transport

transient localization

MIR limit

�J/J

S. Fratini, S. Ciuchi https://arxiv.org/abs/1903.12603

https://arxiv.org/abs/1903.12603


Attività in corso nel dipartimentodisordine e transizione di Anderson in sistemi interagenti con vibrazioni reticolari

sistemi quantistici di spin disordinati, entanglement e correlazione quantistica in presenza di disordine

sistemi di elettroni fortemente interagenti


sistemi quantistici di spin disordinati, entanglement e correlazione quantistica in presenza di disordineKenneth D. McAlpine, Simone Paganelli, Sergio Ciuchi, Anna Sanpera, and Gabriele De Chiara"Magnetic phases of spin-1 lattice gases with random interactions"Phys. Rev. B 95, 235128 (2017)

Disordine statico

Proprietà di “Entanglement” dello stato fondamentale


Titoli di codaGrazie ai miei collaboratori qui…

Carlo Pierleoni: large polarons and their properties by Path Integral Monte Carlo

Simone Paganelli: Disorder in quantum spin chains, coherence effects in in strongly interacting mesoscopic systems

…ed all’estero

Gianluca Rastelli: (University of Konstantz DE) Polarons in polarizable medium

Domenico Di Sante: (University of Würzburg DE) Polaronic features in disordered systems oxides at low denisty, interplay with Anderson transitions and electron-phonon interaction

Simone Fratini: (CRNS Grenoble FR) a lot of things…

Vlad Dobrosavlievich: (Florida State University USA) Anderson localization and electron-phonon interaction

Didier Mayou: (CRNS Grenoble FR)Relaxation time approximation

Giorgio Sangiovanni & Alessandro Toschi: (University of Würzburg DE, TU Wien AT) The limits of perturbation expansion for multiparticle correlations in strongly interacting electronic systems


Bonus tracks


Localizzazione Dinamica in pillolel’idea è di esprimere il coefficiente di diffusione

D=L2(𝜏in)/2𝜏in

dove 𝜏in è il tempo caratteristico associato alle fluttuazioni dei parametri dell’Hamiltoniano. Questo si può derivare applicando il formalismo di Green-Kubo (risposta lineare) alla funzione di correlazione dell’anticommutatore delle velocità.

L2(𝜏in) si può calcolare nel sistema dove il disordine è statico.

Dal coefficiente di diffusione si ottiene la mobilità del sistema.

Le proprietà ottiche si ricavano nel formalismo di Green-Kubo supponendo un unico tempo caratteristico per la variazione temporale dei parametri dell’Hamiltoniano.


Transizione vetrosaPablo G. Debenedetti & Frank H. Stillinger NATURE, 410, 259 (2001)insight review articles

266 NATURE | VOL 410 | 8 MARCH 2001 | www.nature.com

molecular dimensions. One is the critical correlation length, whichgoverns density fluctuations in the absence of frustration. The second is the frustration-limited domain size. From these considerations there emerge predictions on the temperature dependence of the viscosity. Experimental data analysed according tothe theory display universality24, but at the expense of introducing anumber of fitting parameters. The improvement with respect tocompeting interpretations is a matter of controversy25.

Challenges and open questionsImportant aspects of the complex behaviour of viscous liquids closeto the glass transition can be explained qualitatively from the energylandscape perspective. Making this descriptive picture quantitativeand predictive is a major challenge. This will require investigatinghow basic landscape features such as the basin enumeration functiondepend on molecular architecture and, for a given substance or mix-ture, on density (see ref. 96 for an example of such a calculation).Equally important is the translation of qualitative pictures such asFig. 8 into precise measures of strength and fragility based on thebasin enumeration function. Uncovering the topographic nature ofthe scaling relationship between basin minima and saddle pointsholds the key to understanding the relationship between kinetics andthermodynamics in deeply supercooled liquids. All of these calculations are in principle straightforward, but computationally at the very limit of what is currently feasible. The development of theoretical models97 is therefore of paramount importance.

MCT and the landscape perspective offer complementary view-points of the same phenomena. So far, however, not enough effort hasbeen devoted to bridging the gap that separates these two approaches.Recent calculations70,74 offer the promise of establishing a clearer connection between the static landscape viewpoint and the dynamicperspective of MCT. At the least, what is required is a precise landscape-based explanation of what ‘hopping’ and ‘activatedprocesses’ really mean. Additional theoretical viewpoints of supercooling and the glass transition include the instantaneous normal-mode perspective on liquid dynamics98 and thermodynamic

treatments of the vitreous state based on invoking analogies to spin glasses99–101. Establishing a coherent theoretical perspective onsupercooled liquids and glasses is important. We believe that thelandscape formalism offers the natural technical tools for accomplishing this task. nn

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10–2 10–1 100 101 102 103 104 105

1

0.8

0.6

0.4

0.2

0.0

T = 5.0

T = 0.466

F s(q

,t)

t

Figure 9 Evolution of the self-intermediate scattering function for A-type atoms forthe same supercooled Lennard–Jones mixture as in Fig. 6, at q&AA'7.251,corresponding to the first peak of the static structure factor of species A (ref. 92). Hereq is the magnitude of the wave vector. Temperature and time are in units of %AA/kB and&AA(m/48%AA)

1/2, respectively. Temperatures from left to right are 5, 4, 3, 2, 1, 0.8,0.6, 0.55, 0.5, 0.475, and 0.466. The self-intermediate scattering function is thespace Fourier transform of the van Hove function Gs(r, t ), which is proportional to theprobability of observing a particle at r(dr at time t given that the same particle was atthe origin at t = 0. Note the two-step relaxation behaviour upon decreasing T.Molecular dynamics simulations of 1,000 atoms. (Adapted from refs 9 and 92.)

© 2001 Macmillan Magazines Ltd

funzione di van Hove

funzione intermedia di diffusione


p-spin spin glassesS. Ciuchi and A. Crisanti Eur. Phys. Lett. (2000)

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