Stacking fault analysis in layered materials

International Journal of Inorganic Materials 3 (2001) 1137–1142

Stacking fault analysis in layered materials H. Dittrich*, M. Wohlfahrt-Mehrens Center for Solar Energy and Hydrogen Research, Helmholtzstr. 8, 89081 Ulm, Germany

Abstract Ordered and disordered stacking sequences in graphite and Li-intercalated graphite were modelled. The X-ray powder patterns were simulated by the diffracted intensities from faulted xtals (DIFFaX) program. Resulting diffraction patterns show characteristic differences for hexagonal, rhombohedral and statistical intermixed stacking orders. For this reason, simulated patterns can be used for quantitative analysis of stacking faults by profile fitting, using the DIFFaX simulation parameters as fit parameters.  2001 Elsevier Science Ltd. All rights reserved. Keywords: Order–disorder transition; Stacking faults; X-ray powder diffraction; DIFFaX simulation; Graphite

1. Introduction

sequences in layered structures, may they be ordered or disordered, is of great importance.

1.1. Layered structures 1.2. Example: graphite Layered structures result from anisotropy of crystal bonding energy in different lattice directions. Mostly pronounced, this is demonstrated in structures where Van der Waals forces are involved along one principal crystallographic axe. Hexagonal symmetries are predominant in such types of structures including different stacking orders of the hexagonal layers. Because of very small differences in the lattice energy, polytypism occurs very often. Examples are widespread over inorganic material classes like sulfides (e.g., molybdenite, MoS 2 ), oxides (e.g., brucite, Mg(OH) 2 ), and silicates (e.g., montmorillonite, Na 0.3 (Al,Mg) 2 Si 4 O 10 (OH) 2 ?H 2 O). Due to the small difference in lattice energies of the different stacking orders in the polytypes, the order–disorder transition activation energy is also very low and high stacking fault concentrations are observed. Polytypes and stacking fault concentrations influence the physical properties of the layered materials. Particularly electronic properties (band gap of semiconductors) and electrochemical properties (electrode capacities and stabilities in batteries) can be strongly influenced and in the end optimized. Therefore a powerful characterization method of stacking *Corresponding author. Tel.: 149-731-9530-407; fax: 149-731-9530666. E-mail address: [email protected] (H. Dittrich).

Graphite is a very well known carbon allotrope and occurs in the 2H and 3R polytypes. Graphite-2H consists of two graphene layers at positions A and B, whereas graphite-3R consists of three graphene layers at positions A, B and C. Nearly all samples of graphite show intermixing of 2H and 3R stacking sequences along the common crystallographic c-axes. Besides many different applications, one important use for graphite is in the negative intercalation electrode for Li-ion batteries as used in the 4C market: computer, cellular phones, camcorder, and cordless tools. The energy storage and transformation in the Li-ion battery base on the redox reaction: → Li 0.4 MO 2 1 0.6LiC 6 LiMO 2 1 0.6C ← where M is a transition metal: Co, Ni, Mn.

2. The DIFFaX simulation program X-Ray diffraction methods based on kinematic scattering theory are most widely used for structural characterization of crystalline materials. We restrict ourselves to X-ray powder diffraction (XRD) methods. The simulation of diffraction patterns of materials including stacking faults dates back to Landau in the year 1937 [1] and several

1466-6049 / 01 / $ – see front matter  2001 Elsevier Science Ltd. All rights reserved. PII: S1466-6049( 01 )00143-X

1138

H. Dittrich, M. Wohlfahrt-Mehrens / International Journal of Inorganic Materials 3 (2001) 1137 – 1142

different approaches were developed [2–4]. In the year 1991, a general recursion method for calculating diffracted intensities from crystals containing coherent planar faults was published by Treacy et al. [5]. This algorithm was implemented in a Fortran computer program called diffracted intensities from faulted xtals (DIFFaX). DIFFaX exploits the recurring patterns found in randomized stacking sequences to compute the average interference wavefunction scattered from each layer type occurring in a faulted crystal. The method of working is easy and straightforward: (1) definition of a crystallographic layer unit cell for any involved layer types; (2) definition of a transition vector r ij specific for the stacking fault type; and (3) definition of a stacking probability a ij correlated to the stacking fault concentration. The result is a synthesis of randomized stacking sequences and its simulated XRD powder pattern.

3. The simulation of ordered and disordered graphite 2H and 3R To carry out DIFFaX simulations of graphite 2H and 3R intermixed stacking sequences, the graphene layer unit cell has to be defined. In Fig. 1 the unit cell and its structural data are given. To control the correctness of the DIFFaX simulations, first of all standard simulations of the ideal graphite-2H and -3R powder patterns were calculated by the PowderCell program [6]. The comparison of the 2H and 3R patterns shows that the most interesting part (because of the specific differences!) lies between 408 and 658 2u using Cu Ka wavelength for the calculations. Due to the changes from hexagonal to rhombohedral space group symmetry the set of diffraction peaks is strongly different. In a further step DIFFaX simulations of diffraction patterns of intermixed 2H and 3R stacking sequences were carried out. The stacking fault concentration was increased

from 0% 3R in 2H phase to 100% 3R in 10% increments. Fig. 2 shows the typical change in the peak profiles of the simulated patterns. Correlated to the increase of 3R stacking concentration, the following observations from the simulated patterns can be established: (1) 100-2H peak is disappearing, the full width at half maximum (FWHM) is independent; (2) 101-2H, 102-2H, and 103-2H peaks are disappearing, their FWHMs are strongly broadened; (3) 101-3R, 102-3R, 104-3R, and 105-3R peaks are appearing at 3R-concentrations .50%, the FWHMs are sharpened until the ideal 3R structure is reached (100% 3R); and (4) 004 / 006-2H / 3R peak is completely independent from stacking fault concentrations. The stacking fault concentration of measured graphite samples can be analysed by peak profile fitting of the DIFFaX simulated patterns to the measured ones. In Fig. 3 a comparison between measured and simulated profiles are given. In contrast to the graphite sample 1, which indicates a close similarity to the chosen structural model used for the DIFFaX simulation, the upper measured pattern of sample 2 is quite different. The peak profile analysis of this pattern shows a separation of 2H and 3R peaks without FWHM anisotropy. This effect of addition of the 2H and 3R patterns can only be explained by a phase segregation into 2H and 3R phase. The examples given in Fig. 3 demonstrate the possibility to introduce quality criteria for the production of graphite electrode materials and the final Li-ion battery by understanding the real structure through comparison of theoretical structure models with measured XRD powder patterns.

4. Li-intercalation in graphite The intercalation of Li into graphite proceeds according to the following equation:

Fig. 1. Definition of the graphite layer unit cell.

H. Dittrich, M. Wohlfahrt-Mehrens / International Journal of Inorganic Materials 3 (2001) 1137 – 1142

1139

Fig. 2. Series of DIFFaX simulated XRD patterns of intermixed graphite-2H and -3R stackings (bottom, 100% 2H stacking sequence; top, 0% 2H stacking sequence; increment, 10% 3R stacking fault concentration).

→ xLi 1 1 xe 2 1 C n Li x C n ← Due to electrochemical reduction of the carbon, Li-ions penetrate into the graphite and form a Li x C n intercalation compound. The reaction is reversible and the graphite host

structure remains stable. During the intercalation into graphite the stacking order of the graphite layers shifts so that the carbon hexagons in the graphene layers are positioned directly above each other ( . . . AAA . . . stacking). The interlayer distance between the graphene layers

Fig. 3. Comparison of powder patterns of measured graphite samples and a DIFFaX simulated pattern with randomized 60% 2H and 40% 3R stacking.

1140

H. Dittrich, M. Wohlfahrt-Mehrens / International Journal of Inorganic Materials 3 (2001) 1137 – 1142

Fig. 4. Structure of the LiC 6 phase.

increases moderately (about 10%) and the lithium is distributed in-plane in such manner that it avoids the occupation of the nearest neighbour sites. Therefore the ideal stoichiometry of the Li-intercalated graphite is LiC 6 . The structure of a LiC 6 layer is shown in Fig. 4.

5. The simulation of ordered and disordered LiC 6 -2H and -3R Analogous to the stacking possibilities in graphite there is also the possibility of 2H and 3R stacking sequences in the LiC 6 phase. This is caused by only partly Li-occupation of the centers of the hexagonal channels in the LiC 6 bulk. In principle three LiC 6 layer positions are possible and therefore hexagonal stacking sequences of ABABAB . . . positions and rhombohedral sequences of ABCABC . . . positions are allowed (see Fig. 5). The powder patterns of ideal LiC 6 -2H and LiC 6 -3R superstructures were simulated by the standard PowderCell program. The comparison of the resulting profiles shows distinct distinguishing features in spite of the small scattering contribution of the very lightweight Li-atom. A series of DIFFaX simulations was carried out, analogous to the graphite-2H and -3R intermixing series. The result is shown in Fig. 6. The characteristics of the observed diffraction peaks are very similar to the former graphite 2H and 3R series. Independent from stacking faults are the 002-2H, 003-3R, 110-2H, and 110-3R peaks. All other 2H or 3R peaks are strongly dependent on the stacking fault concentration with respect to their intensities and FWHMs.

6. The simulation of intermixed graphite and LiC 6 stacking sequences The LiC 6 stoichiometry represents the maximum capacity for Li-intercalation in graphite. A general feature of intercalation is the stepwise formation of superstructures at lower concentrations of lithium, called stage formation. It means the periodic and therefore ordered intermixing of graphite and LiC 6 layers. This phenomena can be easily observed in potential / composition curves for galvanostatic reduction of graphite to LiC 6 as concrete plateaus indicating a two-phase region of different superstructures. On the other side, statistical intermixing of graphite and intercalated layers are known from TEM observations. The DIFFaX simulation of both cases enables to distinguish between ordered and statistical intermixing. Therefore two structural different layer unit cells have to be defined. In total six different layers have to be considered: (1) three graphene layers at positions 1, 2, and 3, and (2) three graphene layers with lithium at positions A, B, and C. Transition vectors between these layers are well defined. The concentration of graphite or LiC 6 stacking sequences is defined by the chosen transition probability in the simulation parameter set. Three cases of statistical stacking orders of graphite and LiC 6 layers were simulated: (1) large surplus of graphite stacking interrupted by single LiC 6 layers; (2) equal accumulation of graphite and LiC 6 layers (defect clustering); and (3) Large surplus of LiC 6 layers interrupted by single graphite layers. In Table 1 the transition probabilities for the above-mentioned cases are given. Fig. 7 shows the resulting simulated XRD patterns. On a very first view drastic differences in peak positions, peak

H. Dittrich, M. Wohlfahrt-Mehrens / International Journal of Inorganic Materials 3 (2001) 1137 – 1142

1141

Fig. 5. 2H and 3R stacking in LiC 6 .

Fig. 6. Series of DIFFaX simulated XRD patterns of intermixed LiC 6 -2H and -3R stackings (bottom, 100% 2H stacking sequence; top, 0% 2H stacking sequence; increment, 10% 3R stacking fault concentration).

1142

H. Dittrich, M. Wohlfahrt-Mehrens / International Journal of Inorganic Materials 3 (2001) 1137 – 1142

Table 1 Transition probabilities for the DIFFaX simulation Transitions

Case 1

Case 2

Case 3

Graphite 2H⇔2R LiC 6 2H⇔2R Graphite⇒LiC 6 LiC 6 ⇒graphite

70% / 30% 50% / 50% 10% 90%

70% / 30% 50% / 50% 50% 50%

70% / 30% 50% / 50% 90% 10%

patterns from selected structure models. The results gives the basic know-how for the interpretation of measured patterns inclusive a possible quantitative stacking order analysis. This stacking order analysis gives a quality criterion for graphite and battery producer.

Acknowledgements intensities and peak profiles are clearly visible. For the 00l peaks a continuous shift in peak position to larger c-axis periodicities can be found correlated to the increase of LiC 6 layer concentration (LiC 6 →10% increase of layer thickness!). Within the 2u region of 400–600 dramatic changes in the peak profiles occur. It is in this region that quantitative analysis of randomized stacking orders in the graphite–LiC 6 system by peak profile fitting based on the DIFFaX simulation parameter set is possible.

7. Conclusion and outlook We have simulated XRD powder patterns for layered materials with statistical stacking orders. The graphite– LiC 6 system was chosen to demonstrate the capabilities of the DIFFaX computer program to synthesize powder

This work was supported by BMB1F Bundesminis¨ Bildung und Forschung; contract number: 13 N terium fur 7576.

References [1] Landau L. Phys Z SowjUn 1937;12:579. [2] Hendricks S, Teller E. J Chem Phys 1942;10:147. [3] Cowley JM. In: Diffraction Physics, New York and London: NorthHolland, 1981, pp. 388–400. [4] Michalski E. Acta Crystallogr A 1988;44:640. [5] Treacy MMJ, Newsam JM, Deam MW. Proc R Soc London A 1991;433:499. [6] W. Kraus, G. Nolze, PowderCell program: Federal Institute for Materials Research and Testing, download from www.bam-de

Fig. 7. DIFFaX simulated XRD patterns of intermixed graphite-2H, graphite-3R, LiC 6 -2H, and LiC 6 -3R layer stacking sequences (For layer concentrations in the cases 1–3 see transition probabilities of Table 1!) — based structure models are inserted.