Stacking Textures and Singularities in Bilayer Graphene

Xingting Gong and E. J. Mele∗

Department of Physics and Astronomy University of Pennsylvania Philadelphia PA 19104

(Dated: August 24, 2018)

We study a family of globally smooth spatially varying two dimensional stacking textures inbilayer graphene. We find that the strain-minimizing stacking patterns connecting inequivalentground states with local AB and BA interlayer registries are two dimensional twisted textures ofan interlayer displacement field. We construct and display these topological stacking textures forbilayer graphene, examine their interactions and develop the composition rules that allow us torepresent other more complex stacking textures, including globally twisted graphenes and extendedone dimensional domain walls.

PACS numbers: 61.48.Gh,61.72.Bb,61.65.Pq

Bilayer graphene (BLG) features special functionalitiesthat microscopically derive from various forms of bro-ken sublattice symmetry present when graphene sheetsare stacked. These depend on relative lateral transla-tions [1–3], rotations [4, 5] and layer symmetry breakingthat can occur spontaneously [6–8] or be induced [9–20].There has been important recent progress imaging thestacking order in BLG using dark field transmission elec-tron microscopy [21, 22]. These experiments reveal richsubmicron domain structures with locally registered ABand BA regions delineated by dense irregular networks ofdomain walls, focusing attention on the inevitable com-petition between intralayer strain and interlayer stackingcommensuration energies [21, 23].

In this Letter we examine a general family of two di-mensional stacking textures in BLG and their defects.We find that the strain minimizing stacking patterns thatconnect inequivalent ground states are twisted textures ofthe interlayer displacement field. We construct and dis-play these topological stacking textures in BLG, examinetheir interactions and develop the composition rules thatallow us to represent other observed complex stackingtextures, including twisted graphenes and extended onedimensional domain walls.

A relative translation between two graphene layers isrepresented by an interlayer displacement vector ∆ =f1T1 + f2T2 where T1,2 = a exp(±iπ/3) are primi-tive translation vectors of a graphene lattice with lat-tice constant a. The lowest energy uniformly trans-lated structures (Fig. 1) align the A and B sublat-tices of the two layers at (f1, f2) = (2/3, 1/3) (ABstacking: ∆α) and its complement (1/3, 2/3) (BA stack-ing: ∆β). The interlayer potential is a periodic func-tion U(∆) = u0 +u1

∑n exp

(i(Ḡn∆ +Gn∆̄

)/2)

where

Gn = (4π/√

3a) exp(i(2n− 1)π/6) are vectors in thefirst star of reciprocal lattice vectors. For BLG u1 '2.1 meV/atom and u0 = 3u1 assigns the zero of energyto the α- and β-stacked states [23].

The configuration space for ∆ has the topology of atorus. When u1 is large the energy minima at α and β

FIG. 1: A relative lateral interlayer translation is representedby a vector ∆ = f1T1 + f2T2 (with T1(2) = ae

±iπ/3) insidea fundamental domain (shaded rhombus) with the states atα, β, γ representing AB, BA and AA stacking respectively.The lineplot gives the commensuration energy per atom ona vertical slice through connecting the three high symmetrystates as shown. The points sr, sg and sb are the saddle pointson the potential energy surface.

are deep and the shortest trajectories ∆(~r) connectingthe two inequivalent minima α and β are three saddlepoint paths crossing the points labelled sr, sg and sb inFig. 1, indexed by their winding on the two cycles ofthe torus. Each saddle point trajectory is bisected by a(straight) domain wall which runs along one of the threesymmetry-related directions as shown.

We focus on field textures ∆ that satisfy the boundaryconditions ∆(∞) = ∆α and ∆(0) = ∆β . A simple tex-ture that accomplishes this wraps a stacking domain wall[12, 13] into a loop thereby reversing the stacking orderwithin a confined region, as shown in Fig. 2 for the field∆w(z) = (1/2)(∆α + ∆β − (∆β −∆α) tanh[(|z| −R)/`])where z = x + iy is the complex coordinate in theplane. This texture connects two ground states througha transition region of width ` accumulating lattice strainin an annulus. This texture passes through the value

arX

iv:1

308.

5574

v1 [

cond

-mat

.mes

-hal

l] 2

6 A

ug 2

013

2

FIG. 2: A circular domain wall separating a disk of BLG withβ stacking from a background with α stacking is produced bya rigid translation of its interior region by ∆β − ∆α. Thistexture matches the interior and exterior textures through adomain wall (red) where every point on the wall is mappedto the state sr of Fig. 1. The density plot in the inset givesthe potential energy density and the wire-frame model of thelattice structure (background) illustrates the interlayer dis-placement field with the lattice constant greatly exaggeratedfor clarity.

(∆α + ∆β)/2 on any radial path at |z| = R and the wallevolves smoothly from tensile character to shear charac-ter as a function of arg[z] [21].

Viewed on scales much larger than ` the field patternin Fig. 2 changes sharply, and it is natural to considerother textures that are globally smooth but maintainthe same boundary conditions. Displacement fields with∂∆/∂z = 0 (antianalytic) are particularly useful sincethey are automatically divergenceless, avoiding any en-ergy penalty due to local compression or dilation, whileminimizing its other nonzero strains. The smoothest suchsingle-valued function that satisfies the boundary condi-tions is

∆> = ∆α + (∆β −∆α)z̄0/(z̄0 − z̄). (1)

The left hand panel of Figure 3 illustrates this texture,where the density plot gives the potential energy densityand the lines show the folding of the symmetry lines ofFig. 1 onto the texture. In this mapping the sharp tran-

sition in the original circular wall is collapsed to a singlepoint z0 represented by an isolated pole in ∆(z̄). Thissingularity is unavoidable for a nonconstant antianalyticfield ∆>. In this mapping ∆(0) ≡ ∆β induces perfectβ stacking at two additional points located at the samedistance from its pole.

One can regard Eqn. 1 as describing an optimallysmooth exterior elastic response to some, as yet unspec-ified, near field displacement pattern. It can also be un-derstood as the elastic response to an optimally smoothinterior stacking pattern in the region |1−z/z0| < 1 with

∆< = ∆α + (∆β −∆α)(z0 − z)/z0 (2)

which is analytic in z and matches ∆> everywhere on aboundary as shown in Fig. 3 (right panel). Since ∆< isanalytic it is automatically strain free, and since it is alinear analytic function of z it has a constant divergenceRe[−(∆β − ∆α)/z0] = 0. Physically this is required bythe boundary conditions because the interior solution hasa constant compression/dilation which integrates to zeroto match an exterior uncompressed field ∆> without anyaccumulation or depletion of material.

Since the modulus for long wavelength intralayer rota-tion is zero, the only elastic energy in the texture Eqns.(1-2) is the exterior strain energy Ue = πC|∆β −∆α|2/2where C is the two dimensional graphene shear modulus(' 130 N/m) and is independent of z0 because of the scaleinvariance of the texture. This can be compared with theelastic energy in a domain wall with line tension γ which,for the geometry of Fig. 1, scales extensively with thecircumference Uw = 2πγR. Thus Uw < Ue only for suffi-ciently small R < K|∆β −∆α|2/(3γ). For an estimatedBLG line tension ∼ 100 pN this requires R < 7 nm whichis essentially the domain wall width [21]. Thus, and asexpected, the elastic energy generically favors spatiallyvarying solutions ∆>( = ∆α +∑Ni siz̄i/(z̄i − z̄) which eliminates the

1/z̄ tail when∑Ni siz̄i = 0. An important case is

N = 3 with s1 = s2 = s3 and z2 = z1 exp(2πi/3) andz3 = z1 exp(−2πi/3) which breaks rotational symmetryby the selection of a single direction si for the triad, butnulls the monopole field by the threefold symmetry of thepositions of the defect centers. The left panel of Fig. 4illustrates the field energy density and critical lines forone such texture. The broken symmetry opens the “red”boundary curve which then links the three defects, whilethe “green” and “blue” critical lines form closed orbitsthat are confined around the individual defect centers.

One can iterate this process, uniformly distributing N

3

FIG. 3: Two globally smooth stacking textures with ∆(∞) = ∆α and ∆(0) = ∆β . The density plots give the commensuration(potential) energy densities and the lines are the mapping of the color coded symmetry lines of Fig. 1 onto the textures. Theleft hand plot represents a texture that is antianalytic everywhere and contains a pole inside its core at z = z0 (density plotis cut off at the white disk for clarity). The right hand plot represents a texture that matches an antianalytic function in theexterior region to an analytic function in the interior. The wire frame model of the lattice (background) illustrates the stackingpattern for the right hand texture (lattice constant greatly exaggerated for clarity.)

FIG. 4: (Left) A stacking texture that links three defectswith equal strength (∆β −∆α)/3 at three symmetry-relatedpositions around the origin. The density plot gives the com-mensuration (potential) energy density and the lines mapthe color coded symmetry lines of Fig. 1 onto this texture.This combination of defects screens the long range tail of thetexture and has a finite integrated commensuration energy.(Right) A stacking texture produced by clamping the originin the β stacked structure with ∆(0) = ∆β + T1 + 2T2, nu-cleating a region of twisted graphene in its interior, smoothlymapped to an incompressible and minimally strained exteriorfield.

defects with charges si on a circle at positions Reiφi

thereby cancelling its higher order multipoles. In thiscase the texture has an expansion

∆N = ∆α −∑p≥1

(R

z̄

)p N∑i

sie−ipφi ; |z| > R (3)

For si = s0 = (∆β − ∆α)/N and N → ∞ one obtains∆> = ∆α and ∆< = ∆β representing a rigid interlayer

translation for |z| < R and reproducing the annular do-main wall pattern of Fig. 1. For general N , one can re-gard ∆N with si = s0 as a family of trial solutions wherethe value of Nmin is selected to minimize the sum of theelastic and commensuration energies Ue+Uc. Ue(Uc) areincreasing(decreasing) functions of N , with a finite Nmin.The domain wall structures experimentally observed inBLG are relatively wide [21, 22] indicating that the sys-tem is indeed in the regime dominated by the elastic en-ergy, favoring smooth (smaller N) over sharp (larger N)solutions. This is expected since the potential energylandscape has quite broad minima in its low energy con-figuration space.

The textures given by Eqns. (1,2) realize identi-cal α stackings at |z| = ∞ and in the near field atz = z0 and are stable because of an additional bound-ary condition that clamps a different state at the ori-gin: ∆(0) = ∆β . However, β stacking at the ori-gin occurs for a lattice of possible clamped states, eachrelatively shifted by discrete lattice translation vectors∆l,m(0) = ∆β + lT1 +mT2. Choices of l and m give thewinding of the order parameter ∆ on the two cycles of atorus between z = 0 and z = ∞ and index topologicallydistinct solutions ∆l,m. Fig. 4 displays the field for thecase l = 1 and m = 2. Remarkably, we find that the in-terior solution represents a circular domain of uniformlyrotated (twisted) graphene continuously matched to anuntwisted and minimally strained exterior texture. Thisillustrates a plausible mechanism for the formation of theobserved complex stacking textures in BLG. Isolated do-mains likely grow with uncorrelated local rotational reg-istries forcing a complex stacking texture in a state ofminimum strain when the bilayer becomes continuous.

The textures identified here have well-studied analogsin (at least) two other physical contexts. First, they aresimilar, though not identical to, the static baby skyrmion

4

solutions of the two dimensional nonlinear sigma model[24, 25]. Normalizing our solution to its maximum valuerealized on the matching radius, ∆(z)/|∆(R)| = ~n⊥ isthe xy projection of a three dimensional unit vector n̂.With the convention that the exterior(interior) regionsmap to the upper(lower) hemisphere, our solution cov-ers the sphere with degree (1/4π)

∫d2r n̂ · ∂1n̂ × ∂2n̂ =

−1 and minimizes a projected strain energy functionalUBLG = (1/2)

∫d2r∇~n⊥ · ∇~n⊥. This breaks the full

O(3) symmetry of the nonlinear sigma model whosebaby skyrmion with the same degree instead minimizesUnlσ = (1/2)

∫d2r∇n̂(~r) · ∇n̂(~r). The BLG solution is

not obtained by a stereographic projection of the sphereonto the plane and has a slower far field relaxation ofits in plane components ~n⊥ and a faster relaxation of its(unmeasured) normal component nz. Second, our solu-tions are recognized as the classical field solutions fromordinary 2D electrostatics and magnetostatics where theyrepresent the field profile of a uniformly charged rodor current carrying wire. An interesting difference isthat in BLG the field energy density vanishes for twononzero values of the field (corresponding to degeneratevacua with α and β stacking) instead of just one state

( ~E, ~B = 0).

Our approach provides a unified treatment of stackingpoint defects, domain walls and twisted graphene andprovides a direction for further investigation of these sys-tems. For example it is possible that the complicatedsubmicron structure observed in these systems can beunderstood in terms of only a few fundamental stackingmotifs and their conformal maps onto spatially varyinggeometries imposed by pinning centers and irregulari-ties in the sample morphology. More intriguingly, it ispossible that a desired BLG stacking texture could becontrollably engineered using a combination of choice ofsubstrate, growth face, macroscopic curvature and var-ious forms of submicron templating. Finally, althoughour model is designed to study static low strain stackingconfigurations, they may also be important for nonlineartribological properties of BLG, where defects of the typestudied here are generated when an applied mechanialload exceeds a critical yield stress [26].

This work was supported by the Department of Energy,Office of Basic Energy Sciences under Grant No. DE-FG02-ER45118. We thank R. Kamien, C.L. Kane andF. Zhang for useful comments.

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mailto:mele@physics.upenn.edu

References