amorphousstructuresofge/sb/te alloys ...iffjones/pssaj2.pdf · amorphousstructuresofge/sb/te...

physica status solidi, 19 June 2012

Amorphous structures of Ge/Sb/Tealloys: Density functional simulations

J. Akola1,2,3, R. O. Jones*,1,4

1 Peter-Grunberg-Institut PGI-1, Forschungszentrum Julich, D-52425 Julich, Germany2 Department of Physics, Tampere University of Technology, P.O. Box 692, FI-33101 Tampere, Finland3 Centre of Excellence of Computational Nanoscience, Aalto University, FI-00076 Aalto, Finland4 German Research School for Simulation Sciences, FZ Julich, D-52425 Julich, Germany

Received XXXX, revised XXXX, accepted XXXX

Published online XXXX

Key words: Phase change materials, density functional simulations, Ge/Sb/Te alloys.

∗ Corresponding author: e-mail [email protected], Phone: +49-2461-614202, Fax: +49-2461-612850

Since their first development in the 1960’s, phase change(PC) memory materials have become essential compo-nents of optical memories (DVD-RW, Blu-ray Disc), . . . )used in countless households worldwide. They are nowpoised to play a decisive role in future non-volatile com-puter memories. PC memory materials are restricted tothose with an extremely rapid and reversible transitionbetween the amorphous and crystalline phases of an ap-propriate recording medium, and their development andoptimization has been hindered by the inherent difficul-ties in determining amorphous structures.

Alloys of germanium, antimony, and tellurium (“GST”alloys) are among the most widely used in the abovecontexts, and there has been much speculation concern-ing their amorphous structures. Theoreticians were slowto appreciate the rich array of problems awaiting theirattention, but the past few years have seen a dramaticchange in this situation. Density functional calculations,which are generally free of adjustable parameters, havenow been reported on numerous systems. We review herethe information that has resulted about the amorphousstructures of GST-alloys.

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1 Introduction The era of phase change (PC) materi-als can be viewed as beginning with the much-cited paperof Stan Ovshinsky [1]. The 1960’s had seen a range of ex-citing developments concerning amorphous materials [2],and the second author heard seminars at Cornell Universityvery soon after the publication of Stan’s work on resistiveswitching in semiconducting alloys. PC materials are basedon the extremely rapid and reversible transition betweenthe amorphous and crystalline form of nanosized “bits” ina very thin polycrystalline layer, whose states must showsufficient contrast in resistivity or optical properties to beidentified. It is clear that most materials do not satisfy thesecriteria, and great efforts across more than 20 years werenecessary to identify and develop appropriate alloys. Theoriginal paper discussed the alloy As30Ge10Si12Te48, andTe-based alloys have been well represented ever since.

Alloys of the GexSbyTe1−x−y family are common inthe PC world. GexTe1−x alloys were the first to show realpromise as PC storage media [3], and the widely stud-

ied Ge2Sb2Te5 (GST-225) is used commercially in DVD-RAM (random access memory). There has been particularfocus on the pseudobinary alloys along the tie line GeTe–Sb2Te3 in Fig. 1(a), where GST-225 and Ge8Sb2Te11 (usedin Blu-ray Disc memories) are well-known examples. Suchalloys are denoted in Fig. 1 as Group 1, and all haverock salt structures and numerous vacancies in the orderedphase [4]. Crystallization of amorphous bits in these ma-terials usually begins from small nuclei [Fig. 1(b)]. Al-loys near the Sb2Te3 end of the tie line crystallize morereadily, but have lower optical contrast than GeTe-rich al-loys. Some compromises are necessary in practice, andthese will include the crystallization temperature [5]. Sb-Tebased alloys are also well represented in commercial PCMmaterials, often with compositions that are near that of theeutectic mixture Sb70Te30, and are denoted as Group 2. Re-crystallization in these alloys proceeds from the polycrys-talline region surrounding the amorphous bit [Fig. 1(c)].


2 J. Akola, R. O. Jones: Amorphous structures of GST alloys: DF simulations

Figure 1 Phase diagram of PC materials and crystallization pat-

terns. (a) The most commonly used materials for optical record-

ing are in groups 1 and 2. (b) Nucleation dominated recrystalliza-

tion (as in GST), (c) Growth-dominated recrystallization.

Central to our understanding of the properties of thesematerials is a knowledge of the structures of the differ-ent phases, but these are difficult to determine in binaryor ternary alloys with significant numbers of vacancies. Itwas noted as late as 2005 that “the local structural order ofglasses in the ternary system GexSbyTe1−x−y is not wellestablished” [6]. Even the structure of the metastable, or-dered phase of GST is still open to question: Yamada [7]proposed that the metastable phase has a rock salt structurewith “Na” sites occupied randomly by Ge and Sb atomsand vacancies, and “Cl” sites by Te. Kolobov et al. [8],however, proposed that Ge and Sb atoms are displacedfrom their ideal positions, enabling the order-disorder tran-sition to occur as an “umbrella flip” of Ge atoms from oc-tahedral to tetrahedral positions. A model of amorphousGe1Sb2Te4 where a spinel structure (tetrahedral Ge) wascompared to the octahedral rock salt phase has also beenstudied [9], and x-ray fluorescence holography of an epi-taxial layer of GST indicated a cubic structure with tetra-hedral site symmetry about Ge atoms [10]. It is astonish-ing that PC materials could become the basis of commer-cially successful products with so much uncertainty aboutthe structures of the phases involved.

Density functional (DF) calculations are free of ad-justable parameters, and their combination with molecu-lar dynamics [11] has had a major impact on materialsscience and chemistry, including chalcogenide materials.Nevertheless, their demands on computational resourcesrestricted simulations on GST systems up to 2007 to rela-tively small unit cells (up to 56 atoms in Ref. [9], 108 atomsand 12 vacancies in Ref. [12], 168 atoms in Ref. [13]) andtime scales that are often much shorter than those measured(e.g. a total of 18 ps in Ref. [13]). In such cases, most ofthe atoms are near the “surface” of the simulation cell, andsignificant finite size effects are inevitable. Nevertheless,simulations on GST and other PC materials have the ad-

vantage that the physical time scale (several nanoseconds)is unusually short. This has encouraged us and other re-search groups to carry out extensive DF simulations on PCmaterials, particularly GST alloys, and to take advantage ofcontinuing improvements in computing hardware and soft-ware. It is now possible to simulate hundreds of atoms overtimes that approach the experimental time scale of boththe quenching and crystallization processes. We hope thatthe following article, where we restrict our focus to GST(Group 1) materials, reflects the sense of excitement thathas accompanied this work.

2 Density functional calculations The densityfunctional (DF) formalism [14] is the most widely usedmethod for determining electronic and structural proper-ties of condensed matter that does not require adjustableparameters. It is based on the electron density n(r) andreduces the problem of determining ground state proper-ties of an interacting system of electrons and ions to thesolution of single-particle Schrodinger-like equations. Anapproximation for the exchange-correlation energy Exc,one contribution to the total energy of the system, is un-avoidable, but there has been substantial progress in thisarea in recent years [15]. Crucial to the success of ap-plications to PC materials has been the coupling of DFcalculations to molecular dynamics (MD) [11], since theDF energy surfaces and forces can be used to determinethe motion of the ions at a known temperature.

Most simulations of amorphous materials are startedwith a liquid sample at high temperatures (typically above3000 K), so that all memory of the initial solid configura-tion is erased. It is important that cooling to the meltingpoint and to room temperature and below be carried out asslowly as possible, as energetically unfavourable configu-rations may otherwise be frozen in. In our calculations, wehave used total simulation times of hundreds of picosec-onds for samples of up to 630 atoms. DF calculations ofthis scale are extremely demanding of present computer re-sources, but they do allow definite statements to be madeconcerning the convergence of the simulation results as afunction of sample size.

MD methods allow us to follow the coordinates Ri andvelocities vi of all atoms throughout the simulations, andinsight into the local order can be found from the distri-butions of the bond (θijk) and dihedral angles (γijkl). Thepair distribution function (PDF) g(r) is a spherically aver-aged distribution of interatomic vectors,

g(r) =1

ρ2

⟨

∑

i

∑

i6=j

δ(ri)δ(rj − r)

⟩

, (1)

where ρ is the density. Partial PDF gαβ(r) can be calcu-lated by restricting the analysis to the elements α and β.The local structure can also be characterized by the aver-age coordination numbers, which are found by integrating


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gαβ(r) to the first minima Rmin

nαβ =

∫ Rmin

0

dr 4πr2ραβ(r)gαβ(r). (2)

The partial density ραβ = ρ√cαcβ , where cα and cβ are

the concentrations of the elements. We calculate the struc-ture factor S(Q) by Fourier transforming the gαβ(r) togive the partial structure factors:

Sαβ(Q) = δαβ +ραβ

∫ ∞

0

dr 4πr2 [gαβ(r)− 1]sin(Qr)

Qr.

(3)These quantities can then be weighted according to the

atomic fractions cα and form factors fα(Q) (for x-rays)or the Q-independent coherent scattering lengths bα (forneutrons) for direct comparison with experiment.

Figure 2 Schematic form for 2D-vacancy.

The cavity analysis is performed by introducing theconcepts illustrated in Fig. 2. A vacancy domain (I, Fig. 2)is a region where the minimum distance to a nearby atomis larger than a given cutoff (here 85% of the average Ge-Te distance, 2.8 A), and each domain is characterized bythe point where the distance to all atoms is greatest. Weinsert a test particle at the center of the largest sphere thatcan be placed inside the cavity (dashed circle in Fig. 2).This cavity centre can be used to calculate RDF, includ-ing cavity-cavity correlation functions. A vacancy cell (II,yellow in Fig. 2) is analogous to the Wigner-Seitz cell incrystals or Voronoi polyhedra in amorphous materials andare determined on an appropriate mesh with respect to thevacancy domains (I, Fig. 2). The importance of cavities andtheir relatively small number provide additional incentivesfor choosing large simulation cells.

Dynamical information includes the velocity autocor-relation function Cv:

Cv(t) =1

N

N∑

i=1

〈vi(0) · vi(t)〉〈vi(0) · vi(0)〉

, (4)

where N is the number of particles. The self-diffusion con-stant for all atoms or those of species α can be determinedfrom Cv:

Dα =1

3

∫ ∞

0

dt Cv(t) (5)

or directly from the coordinates Rα

Dα = limt→∞

⟨

|Rα(t)−Rα(0)|2⟩

6t, (6)

where the average is over all atoms of species α.Vibration frequency distributions (power spectra) can

be obtained by Fourier transforming the velocity-velocityautocorrelation function obtained from long trajectories atthe temperature in question.

3 Results and discussion The structures and theirchanges with time and temperature provide a rich sourceof information for comparison with experiment, and wedo this now for several Ge/Sb/Te compounds. We providemost details for GST-225, a prototype PC material that isoften referred to simply as GST. In these materials it is con-venient to separate the components into types A (Ge, Sb)and B (Te). More details of all calculations can be foundin the original articles.

Figure 3 System of 460 atoms and 52 vacancies in (a) c-GST and

(b) a-GST. Red: Ge, blue: Sb, yellow: Te. Cavities are shown as

light blue isosurfaces.

3.1 Ge2Sb2Te5 (GST-225) The amorphous (a-), liq-uid (ℓ-) and crystalline (c-) phases of Ge2Sb2Te5 have beenstudied using a sample of 460 atoms and 52 vacancies inthe unit cell over a total of 400 ps [16,17]. The startinggeometry [Fig. 3(a)] was a rock salt crystal structure, withthe Na sites occupied randomly by Ge and Sb atoms (20%each), and the Cl sites by Te atoms [7]. At the end of thesimulation (300 K), the amorphous structure in Fig. 3(b)was found. Cavities are shown by light blue isosurfaces,but we note that the multivacancy in Fig. 3(b) is just one ofnumerous cavities in a-GST.

The most striking (and unexpected) feature of the par-tial PDF (Fig. 4) is the medium-range order found amongTe atoms up to 10 A, with peaks at 4.16 A and 6.14 A,a minimum at 5.4 A, and additional maxima at 7.8 A and



0

1

2

3

2 3 4 5 6 7 8 9 10

Sb−Sb

Distance (Å)

0

1

2

2 3 4 5 6 7 8 9 10

Ge−Ge

1

Ge−Sb

0

1

2

2 3 4 5 6 7 8 9 10

Te−Te

Distance (Å)

0

2

4

62 3 4 5 6 7 8 9 10

Ge−Te

01234

RDFweight

Sb−Te

Figure 4 Partial PDF of a- (thick black) and ℓ-GST (red). Blue

curve and bars are for c-GST at 300 K (with different scale).

9.8 A. The weak maximum at 2.95 A and the low coordi-nation number (0.3) show that there are few Te-Te bonds,and there is no significant order at 900 K. Te atoms pre-fer coordination with Ge/Sb, with Te forming the second-neighbour shell. Thermal fluctuations in c-GST lead tobroadening of the Bragg peaks. The “homopolar” Ge-Ge,Ge-Sb, and Sb-Sb bonds in the three phases are shown inFig. 4. While these bonds are absent in c-GST, they oc-cur in the amorphous phase (“wrong bonds”). Heating to900 K results in significant changes in the Ge-Ge and Ge-Sb curves: the first peak now dominates, and the order be-tween 4 and 7 A moves to shorter range.

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Figure 5 Ring statistics for a-GST (300 K) and ℓ-GST (900 K).

ABAB refers to even-membered rings with bond alternation.

All coordination numbers for a-GST (Ge: 4.2, Sb: 3.7,Te: 2.9) are lower than in the crystal, but larger than thevalues (4, 3, 2) suggested by the oft-quoted “8 − N rule”,where N is the number of valence electrons. Liquid GST(900 K) is less ordered than the amorphous state. The sim-ilarities between ring statistics in a- and c-GST have beenproposed as the basis for the structural phase change inGST [18]. The statistics of all irreducible loops (rings) ina- and ℓ-GST have been calculated using a bond cutoff of

3.2 A and periodic boundary conditions (Fig. 5). Fourfoldrings dominate the statistics at 300 K, and the contributionof “ABAB squares” is 86 % of the total. 52 % of atomsparticipate in at least one ABAB configuration. In Fig. 6we show that alternating ABAB cubic structures also oc-cur.

Figure 6 A cubic ABAB subunit in a-GST. A Ge-Ge bond (be-

tween the cube and its neighbours) and an Sb-Sb bond are evi-

dent. Cavities occur near the cube, and a threefold coordination

is typical for Te.

Cavities play an essential role in Ge/Sb/Te alloys. Sincec-GST contains 10% vacancies, a-GST, whose density is7% lower, must have empty regions (cavities or voids, seeFig. 3). Cavities comprise 11.8% of the total volume of a-GST (slightly more than in the crystal), and this increasesto 13.8% at 900 K.

Figure 7 (a) XPS valence band spectrum of c- (thick black) and

a-GST (red) ([19], upper panel) and the calculated electronic

DOS (lower panel). (b) Theoretical DOS of a-GST projected onto

atom-centered s-, p-, and d-components and atomic types. (c)

DOS difference between c-GST and a-GST [from (a), positive

values imply a larger weight on a-GST]. (d) Electronic DOS of

ℓ-GST. The vertical dashed lines mark the Fermi energy.

The energy eigenvalues in DF calculations cannot berelated directly to the optical excitation spectra, but theelectronic density of states (DOS) and its projections oftenlead to useful insight and are shown in Fig. 7. The former



are characteristic of materials with average valence five:the two lowest bands can be assigned to s-electrons (σ-band), and the broad band between −5 eV and the Fermienergy has p-character (π-band). The calculations for c-and a-GST show band gaps of around 0.2 eV at the Fermienergy. Photoemission measurements have shown thatthere are significant differences in the valence band DOSand the core levels of a- and c-GST and other Ge/Sb/Tealloys [19]. The differences measured by hard x-ray pho-toemission spectroscopy (XPS, [19]) are compared withour results in Fig. 7. The agreement is remarkably good,particularly for the DOS difference [Fig. 7(c)].

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 50 100 150 200 250 300

Pow

erspectra

ω (cm−1)

v−v ac

Ge

Sb

Te

4−Ge

VDOS

a−Ge2Sb

2Te

5(300 K)

Figure 8 Calculated vibrational density of states for a-GST

(VDOS, lowest curve), and power spectra from simulation trajec-

tories for a-GST, for Ge, Sb and Te, and for fourfold coordinated

Ge atoms (“4-Ge”). Curves have maxima at 1.3 (a-GST) or 0.65

(projections).

The distribution of vibration frequencies in condensedmatter and molecules provides important structural infor-mation and can be calculated: (a) from the Fourier trans-form of the velocity-velocity autocorrelation function, cal-culated from trajectories of 6000 time steps of 3.025 fseach, and (b) from the dynamical matrix calculated fromthe second-order energy derivatives of a well-equilibratedstructure at 0 K. The results for a-GST (Fig. 8) show thatthe agreement between the results for the two methods isvery good. The most pronounced features are peaks near

60 cm−1 and 150 cm−1 and a tail for frequencies above180 cm−1. Projections onto the vibrations of different el-ements and structural units show that the tail is associatedwith the vibrations of the lightest element Ge, particularlyatoms that are fourfold coordinated.

The original DF calculations [16] agreed reasonablywell with measured S(Q) and g(r) for GST, [18] but sev-eral details were open to improvement. In particular, wehad used the approximation of Perdew, Burke and Ernzer-hof (PBE) [20] for Exc, and the bond lengths that resultedwere longer than those measured. A valuable collaborationbetween theory and experiment began when we discoveredthat the structure obtained in the original reverse MonteCarlo (RMC) fit to the x-ray diffraction (XRD) measure-ments [18] gave rise to a metallic density of states, i.e. noband gap at the Fermi energy. We combined the strengthsof theory and experiment with the goal of finding a struc-ture that:

–reproduces the measured S(Q) and the total pair distri-bution function g(r) for XRD and/or neutron diffrac-tion,–has bond lengths that are not artificially short and areconsistent with EXAFS measurements,–has an electronic structure with a band gap at the Fermienergy,–has a DF total energy close to the minimum.

We do this by combining data from high-energy XRD [18]and XPS measurements on a- and c-GST [19] with DF sim-ulations of a 460-atom sample over hundreds of picosec-onds. This strategy is related to the “experimentally con-strained molecular relaxation” (ECMR) of Biswas et al.[21].

We adopted the approximation of Tao et al. (TPSS)[15], which has orbital-dependent terms that increase thecomputing demands, but the resultant structures and en-ergy differences are generally better for extended sys-tems [15]. The structures of c- and a-GST [16] were re-optimized using the TPSS functional, and the PDF forc-GST agrees very well with XRD measurements (Fig. 9).Both EXAFS and XRD results indicate the presence of thedouble maximum in the PDF, and the calculated Ge-Te andSb-Te bond distributions have double maxima that werenot evident in the PBE results. The shorter Ge-Te and Sb-Te bond lengths (2.84 and 2.93 A, respectively) agree verywell with EXAFS values (2.83 and 2.91 A) [8]. The longerbonds have maxima above 3.1 A. TPSS bonds in a-GSTare 1− 2 % shorter than in the calculations using the PBEfunctional, and the coordination numbers (Ge: 3.9, Sb: 3.5,Te: 2.7) are lower than in [16] (4.2, 3.7, and 2.9). The co-hesive energy is 2.62 eV, and c-GST is 58 meV/atom morestable than a-GST.

The re-optimized a-GST geometry was taken as in-put for RMC refinement using the experimental x-ray to-tal structure factor S(Q). The experimental S(Q) can bereproduced very well with a range of input parameters,



but RMC simulations without constraints on the bond an-gles always led to a metallic DOS, even when the initialstructure was semiconducting. A semiconducting structurethat satisfies the x-ray structural information for a-GST canbe obtained in practice only by using an iterative schemewhere the bond angle distributions Te-Ge-Te, Te-Sb-Te,Ge-Te-Ge, Ge-Te-Sb, and Sb-Te-Sb are constrained to havevalues close to those found in the DF calculations. TheTPSS bond lengths are slightly shorter than PBE values,but the use of the PBE approximation instead of TPSSwould lead with the present optimization scheme to verysimilar structures.

The RMC optimization of the TPSS structure yields astructure factor S(Q) in excellent agreement with XRDresults [Fig. 9(a)], with minor differences in amplitudenear the first two peaks. The excellent description of thehigh-Q region is important, because the short-range atomiccorrelations in the XRD pair distribution function g(r)[Fig. 9(b)] is obtained by Fourier transforming Q[S(Q)−1]and is affected strongly by the high-Q portion of S(Q).The calculated g(r) follows the experimental curve almostperfectly with small deviations near the first peak, and thetotal energy is only 79 meV/atom higher than that of theDF energy minimum. The electronic DOS shows a bandgap at the Fermi level.

3.2 GeTe and Ge15Te85 Binary alloys of Ge and Tehave fascinated experimentalists and theorists for decades.They were among the first to show real promise as phasechange media [3], and there are striking changes on melt-ing, particularly near the eutectic composition Ge15Te85.We have simulated a-GeTe with 216 atoms in the unit celland compared the results with those of a-GST [22]. In bothcases there is medium-range order of Te atoms and the cru-cial structural motif is a four-membered “ABAB square”.The amorphous to crystalline phase change can then beviewed as a reorientation of such units to form an orderedlattice. The deviations from the “8−N rule” that are foundfor GST are also present in GeTe.

Simulations of the eutectic mixture Ge15Te85 [22,23]lead to partial PDF that again show signs of medium-rangeorder. Amorphous GeTe has few Te-Te bonds, while Ge-Ge bonds are rare in Ge15Te85. The average coordinationnumbers are 4 for Ge and 3 for Te, and the most prominentconfigurations in GeTe and Ge15Te85 are Ge-GeTe3 andGe-Te4, respectively. Ge atoms can be tetrahedrally and oc-tahedrally bonded, and the former dominate in Ge15Te85.Atoms in GeTe favour octahedral or cubic bond angles,although sixfold coordination is found in only 3.4 % and11.0 % of Ge atoms at 300 K and 1000 K, respectively.The GeTe network includes clusters of “ABAB squares”.Corner and edge-sharing Ge-Te4 tetrahedra are common inGe15Te85. The threefold (cubic) coordination of Te in bothGeTe and Ge15Te85 deviates from the predictions of the“8−N rule”.

As in GST, ABAB squares are essential for the phasetransition in GeTe. In a-GeTe, 75 % of the atoms are in-

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r (Å)�

g(r

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Figure 9 (a) Structure factor S(Q) in a-GST. Green: XRD [18];

black: calculated for RMC-optimized geometry. Inset: data for

high Q values. (b) Total PDF (XRD weights).

volved. Even above the melting point (1000 K), 41 % ofthe atoms participate in ABAB squares. The dominanceof Te in the composition of Ge15Te85 means that there arefewer ABAB squares, but cavities are characteristic of thisalloy, occupying more than 25 % of the total volume inboth amorphous and liquid states. The volumes and shapesvery widely, and the cavity distributions depend signifi-cantly on the temperature. There are more cavities in theliquid (680 K) and more that are smaller, whereas largercavities (70− 100 A3) are more abundant at 300 K.

3.3 Ge8Sb2Te11 (GST-8211) We have simulated630 atoms (240 Ge, 60 Sb, 330 Te) of this GeTe-richalloy in a cubic cell of dimension (27.19 A) appropriateto the liquid density (0.03134 atoms/A3). After heating to3000 K, the sample was cooled (120 ps) to the meltingpoint (950 K), where data were collected (31 ps). Dur-ing subsequent cooling to 300 K (melt-quench, 217 ps),there was a (small) adjustment of the density to that of theamorphous material. This was followed by data collection(30 ps) and cooling to 100 K (31 ps), where the structurewas optimized by simulated annealing. The total simu-



lation time of the melt-quench process was 429 ps. Theoptimized structure [Fig. 10(a)] shows a cubic column of16 atoms that demonstrates AB ordering, and Fig. 10(b)emphasizes the local coordination of Ge and Sb atoms.

Figure 10 (a) Simulation box of a-GST-8211 (27.06 A, 630

atoms). A “column” showing AB ordering is shown and Sb

atoms are highlighted with small spheres. Red: Ge, yellow: Te,

blue: Sb. (b) Green: tetrahedrally coordinated Ge atom, pink: oc-

tahedrally coordinated Sb.

The ring statistics of a-Ge8Sb2Te11, GeTe and GST-225 are compared in Fig. 11. There are many four-membered rings and few triangular configurations in allcases. As in GST-225, most of the four-membered ringsin Ge8Sb2Te11 show AB alternation, confirming that thebasic building blocks in the amorphous and crystallinephases are ABAB squares [16,24,25]. A good exampleis the ABAB column shown in Fig. 10. The odd-evenalternation is apparent in GST-8211 and GST-225, andthe ring distributions of these two materials are strikinglysimilar (Fig. 11). This may be surprising at first, sinceGST-8211 is much closer to GeTe on the pseudobinaryline, but even a small increase in Sb content appears tochange the properties of GeTe significantly.

Figure 11 Statistics of irreducible rings (300 K) in Ge2Sb2Te5,

Ge8Sb2Te11, and GeTe alloys. ABAB configurations refer to al-

ternation of atomic types A (Ge, Sb) and B (Te). The bond cutoff

distance is 3.2 A. Results for Ge2Sb2Te5 and GeTe from [16].

Cavity shapes include facets, edges, corners, and pro-trusions. They are illustrated in Fig. 12 for the optimizedstructure of a-GST-8211, where they occupy 12.4% of thetotal volume, i.e. the small increase in Sb content is accom-panied by a dramatic increase in the number of cavities toa value near that in a-GST-225. The coordination numbersaround the cavity centers are Ge: 1.3, Sb: 0.4, and Te: 3.7,and other cavities: 0.2. Cavities are then surrounded mainlyby Te atoms, and Ge is a more common neighbor than Sb.These features reflect the small Sb content and the under-lying cubic structure that resembles the crystalline phase.

3.4 As-deposited vs. melt-quenched GST-225Atomistic simulations on phase-change materials have fo-cused on melt-quenched (MQ) samples, where a liquidsample is cooled rapidly. Experimental samples are of-ten deposited on a substrate, and the structure is likely todiffer. We have developed a method to generate an “asdeposited” (AD) sample with 648 atoms at 300 K [26] andhave compared the results with those for a 460-atom MQsample, refined as described above with respect to XRDand XPS data [27].

The simulations were carried out at 300 K, begin-ning with a random template of 36 atoms in an area of27.61×27.61 A (layer thickness 1.4 A). These atoms were



Figure 12 Cavities in melt-quenched GST-8211.

fixed during deposition to minimize relaxation effects nearthe base of the system. 17 layers of 36 atoms were thendeposited on the sample and allowed to relax for 5− 10 pseach. The vertical box dimension was adjusted for eachlayer to minimize the interaction with the slab replica(vacuum region 10 A). Deposition took a total of 91 ps,and the result is shown in Fig. 13. There is limited mobil-ity at 300 K, but some atoms mix with layers depositedpreviously, and the lower coordination of Te enhances itsconcentration at the surface. The final vertical dimensionof the 648-atom sample is 15 % larger than in the bulk.

The model AD system was then prepared by releas-ing the 32 template atoms and decreasing the vertical boxsize (19 steps at 300 K) to obtain a cubic supercell (size27.61 A, density 0.0308 atoms/A3). The process lasted67 ps in steps of 3− 4 ps, and the system was equilibratedfor 34 ps before data collection (25 ps). The pressure wassmall (2.9 ± 1.8 kbar) at the final (experimental) density,and the final structure was optimized at 0 K [Figs. 13(c-d)]. The structure of the MQ sample [27] does not corre-spond to a DF energy minimum, so we performed a sim-ulation at 300 K using the same functional (PBEsol [28])and atomic density as for AD. Initialization (5 ps, 300 K)was followed by data collection (35 ps). The MQ structureis 11 meV/atom more stable than AD and 58 meV/atomless stable than the rock salt structure.

The structure factors S(Q) for the AD and MQ sam-ples are compared with experiment in Fig. 14. The curvesfor AD and MQ are strikingly similar, and the agreementwith the experiment is very good. We note only that thecalculated heights for the first two peaks (2.1 and 3.3 A−1)are somewhat low, and the shape of the third peak differsslightly. The calculated structures are from DF/MD simula-tions at 300 K, and deviations from experiment are causedin part by the approximations in the Exc functional [26].

The electronic densities of states (DOS) in AD and MQare very similar, with band gaps of 0.2 − 0.3 eV at theFermi energy. The DOS profiles are in satisfactory agree-ment with XPS data [19], the most significant differencebeing near the DOS minimum around−10 eV between twoσ-bands.

The PDF of both samples [26] show that Ge-Te andSb-Te bonds dominate (AB alternation). EXAFS measure-ments [8,29,30] have shown that Ge-Te distances shrinkupon amorphization (2.61 − 2.63 A), suggesting that Gemay have tetrahedral coordination. This differs from earlierDF simulations, which found mostly (distorted) octahedralconfigurations for Ge with Ge-Te bonds of 2.77− 2.78 A.Tetrahedrally bonded Ge atoms also occur. The discrepan-cies in the Ge-Te bond length appear to have two causes:(a) Most approximations for Exc tend to overestimate bondlengths by 2 − 3 %, but the use of the PBEsol functionalhere reduces this error (the Ge-Te PDF maximum is at 2.72A in MQ). (b) Sample preparation is crucial: AD has moretetrahedral Ge atoms, and the Ge-Te maximum shifts to2.69 A. Ge-Ge bonds (2.52 A), which correspond mainlyto tetrahedral Ge atoms here, agree well with experiment(2.47±0.03 A) [29]. Sb-Te bond lengths change little uponamorphization, and the first maximum of AD and MQ is at2.89 A.

The other PDF show clear differences between thesamples, since the number of “wrong bonds” is signifi-cantly larger for AD. This is reflected in the Ge-Te and Sb-Te PDF (AB bonds) as weaker first maxima and emergingsecond maxima at 4.2 A. The corresponding partial co-ordination numbers [26] show that Ge and Sb have twiceas many wrong bonds (1.1 on average) as Te (includingTe-Te bonds), although more than half of the atoms are Te.The total coordination numbers for AD are Ge: 4.2, Sb:3.7, and Te: 2.8. Earlier work on MQ samples [16,27,24]indicated that the total coordination numbers of Sb and Tein a-GST do not satisfy the “8 − N rule” of covalentlybonded networks, and additional support for this conclu-sion is found by calculating the bond orders (number ofchemical bonds) for each interatomic connection [26].

The AD structure of a-GST (648 atoms, over 200 ps)differs from MQ (460 atoms, [27]) in essential ways: (1)The environment of Ge atoms is predominantly tetrahe-dral in AD and disordered octahedral in MQ, and our re-sults for the Ge-Te bond lengths agree with EXAFS mea-surements [8,29,30]. This resolves the contradiction be-tween measured and bond lengths and those found in ear-lier DF calculations, (2) homopolar and Ge-Sb bonds aremore common and reduce the number of ABAB squares,which are needed for rapid crystallization. Sb and Te aremore highly coordinated than expected from the “8 − Nrule”, so that a-GST cannot be seen as a covalent networkglass. The PBEsol functional gives improved results, andboth samples tend to form fewer Te-Te bonds, and MQ is11 meV/atom more stable than AD. Their structure factors



Figure 13 AD sample of a-GST at 300 K. (a) Top view after 17 added layers (648 atoms). The uppermost atoms within 1.2 nm are

highlighted, and the red tetrahedra mark Ge atoms with tetrahedral coordination. Ge, red; Sb, blue; Te, yellow. (b) Side view of the final

surface. (c-d) Two perspectives of final sample. Cavities (cyan isosurfaces) comprise 16.3 % of the total volume.

Figure 14 (a) Structure factor S(Q) of AD (black) and MQ

(blue) samples at 300 K, and XRD (red). The calculated S(Q)are shifted by 0.4. (b) Electronic DOS of AD and MQ samples.

Vertical dashed line marks band gap separating the σ-band with

Te-5s character.

and electronic properties are similar, so that distinguishingbetween them using XRD alone remains a challenge.

4 Concluding remarks We have summarized hereresults of a range of simulations on Ge2Sb2Te5 (GST-225),GeTe, Ge8Sb2Te11, and Ge15Te85. The availability of lead-ership class computers and efficient numerical algorithmsmake density functional calculations possible for systemswith hundreds of atoms over time scales of hundreds ofpicoseconds, i.e. approaching the crystallization time ofphase change materials. This is an immensely encourag-ing development, and the calculations have enabled us toidentify the crucial pattern (“ABAB squares”) in the phasetransition in GST [16], to study the differences betweenGeTe, GST, and Ge15Te85, and to identify structural differ-

ences between as-deposited and melt-quenched structuresof GST-225.

We have presented most details of our own work onGST alloys, but other calculations are important, as areother PC materials. Concurrently with Ref. [16], Caravatiet al. [24] performed DF/MD simulations on a-GST-225(270 atoms, cooling from 2300 K to 300 K in 58 ps). Thesimulated S(Q) was very similar to that in [16], and theauthors noted the coexistence of tetrahedral and octahedralsites in a-GST: One third of Ge atoms were in a tetrahedralenvironment, while the remaining atoms showed a defec-tive octahedral environment. The same authors studied theeffects of stoichiometric defects [31] and pressure-inducedamorphization in the same material [32]. Hegedus and El-liott [25] have performed DF/MD simulations (63 atoms,1.3 ns) of the crystallization process in aGST-225. Crystal-lization is the time-limiting process in the write/erase cycleof PC materials and is the focus of much current research.

We have restricted our discussion to alloys on the tieline GeTe–Sb2Te3 in Fig. 1(a), which are group 1 materi-als where crystallization of the amorphous bits proceedsby nucleation from within. We noted above that amor-phous bits of Sb-based materials near the Sb70Te30 eutectic(group 2), also used as PC materials, crystallize from thepolycrystalline surroundings. We have simulated the amor-phous structure of Ag3.5In3.8Sb75.0Te17.7 and suggested aplausible scheme for its crystallization [33].

Some words of caution are, however, appropriate. Oneof the most remarkable results of this study was the dis-covery of medium-range order in Te atoms in a-GST andGeTe, and this would be impossible to detect in simula-tions of fewer atoms. This is also true for studies of cavi-ties, which can occur as large multivacancies in these sys-tems. Earlier simulations often considered less than 100atoms. While they certainly aided our understanding of the



processes involved, they cannot be expected to give reli-able predictions. It is also crucial that the liquid state becooled over a time scale that is physically relevant. Theuse of Born-Oppenheimer MD with an efficient predictor-corrector scheme for converging the orbital eigenfunctionshas been decisive in our own work, since the time steps aretwo orders of magnitude greater than possible with stan-dard techniques. Furthermore, high temperatures favourmetallic samples in many of these systems, and the absenceof a gap in the eigenvalue spectrum often leads to instabil-ities in the Car-Parrinello approach. Similar findings werefound in other work [24].

We have noted that the approximation used for theexchange-correlation is crucial to obtain an accurate de-scription of the structures of several PC materials. Elemen-tal Te provides an interesting material in this context, sinceexperimental and calculated structure factors and pair dis-tribution functions often disagree. The results of DF calcu-lations of structural properties of Te were found to dependsignificantly on the choice of the exchange-correlation en-ergy functional [34]. While the best agreement with exper-iment is obtained using recently developed functionals thatgo beyond the gradient of the density (TPSS [15]), this ob-servation means that it is essential to perform such checksin general.

DF simulations of hundreds of atoms over time scalesof nanoseconds can be carried out, but they remain a majorundertaking at the present time, both in terms of computa-tional resources and the effort required to analyze the re-sults. This situation is unlikely to change in the short term,and it is indeed pertinent to ask whether simulations of“hundreds of atoms” are enough to answer the problemswe face. The development of classical force fields that haveDF accuracy is important, and the recent work of Sosso etal. [35] is an interesting development. These authors used ahuge database of DF calculations on Ge/Te systems to de-velop a force field for GeTe that permits MD simulationswith much longer time scales and up to several thousandatoms.

We return at the end of this review to where we began,by acknowledging the pioneering work of Stan Ovshinskyas beginning the field of PC materials. We wish him all thebest as he begins his tenth decade.

Acknowledgements We thank all colleagues who workedwith us on these problems, particularly J. Kalikka, K. Kobayashi,S. Kohara, T. Matsunaga, and N. Yamada. J.A. thanks theAcademy of Finland for support. The calculations were car-ried out on IBM BlueGene and Intel Xeon computers in theForschungszentrum Julich with computing resources granted bythe FZ Julich and the John-von-Neumann Institute for Computing(NIC). The German Research School for Simulation Sciences is ajoint venture of the FZ Julich and the RWTH Aachen University.

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