The structure of crystalline membranes and graphene Doron Gazit Institute for Nuclear Theory, University of Washington. Based on DG, arXiv: 0810.1062 (PRB 79, 113411(2009)), 0903.5012, 0907.3718.
Download ReportTranscript The structure of crystalline membranes and graphene Doron Gazit Institute for Nuclear Theory, University of Washington. Based on DG, arXiv: 0810.1062 (PRB 79, 113411(2009)), 0903.5012, 0907.3718.
The structure of crystalline membranes and graphene Doron Gazit Institute for Nuclear Theory, University of Washington. Based on DG, arXiv: 0810.1062 (PRB 79, 113411(2009)), 0903.5012, 0907.3718. Introduction Crystalline membranes are solid-like structures with 2D character. Very common in our world: Cytoskeleton of red blood cells – whose structure is vital for the operation and stability of the cell – forms a triangular lattice. In soft condensed matter, one can create crystalline lattices by polymerizing liquid interfaces. In condensed matter layered materials of tens to hundreds of layers. However – the isolation of graphene, and then of other single layers, has conquered the final frontier – with many implications. Nelson, Piran, Weinberg (eds.), “statistical mechanics of membranes and surfaces”, (2004). August 2009 Doron Gazit - on crystalline membranes and graphene 2 Novoselov et al. Science, 306, 666 (2004) ; PNAS, 102, 10451 (2005). Introduction (2) Thus: Only one atom thick, graphene should represent the ultimate crystalline membrane, and can be used as a simple model, where calculations are feasible. The fact that graphene can be used to construct nanometer-sized electronic applications, has only enhanced the need of a profound understanding of its structure, which is a critical ingredient in the design and quality control of such applications. However, Experiments show that graphene possesses intrinsic ripples, of sizes 100-300 Å. Theory predicts a scale invariant cascade of corrugations. What is the origin of this difference? August 2009 Doron Gazit - on crystalline membranes and graphene 3 Outline Crystalline membranes. The structure of physical crystalline membranes within the self-consistent screening approximation. Graphene: an electronic crystalline membrane! On the correlation between charge inhomogeneities and ripples in graphene. Spontaneous buckling of hole doped graphene. Outlook. August 2009 Doron Gazit - on crystalline membranes and graphene 4 Physical Crystalline Membranes Membranes are D dimensional entities, embedded in a d dimensional world. Physical membranes: D=2, d=3. Crystalline membranes are built of a lattice with fixed connectivity. The main question: does a flat phase exist? What is its structure? Is the Mermin-Wagner theorem violated? Phase stability is a long wavelength question, thus continuum theory. August 2009 Doron Gazit - on crystalline membranes and graphene 5 The flat phase of a crystalline membrane We need to describe an almost flat phase in the continuum. We use the Monge representation: Describe a deviation from the flat phase by: r r (u ,...,u ,h ,...,h ) u ,h 1 D 1 d D 14 2 43 1 4 2 43 The metric is: in plane outof plane gij ij 2uij ; i, j 1,...,D the strain tensor: 1 r r 1 1 uij i u j j ui i h j h + i uk j uk 2 2 2 We expand around a flat surface, keeping leading orders in h and u. August 2009 Doron Gazit - on crystalline membranes and graphene 6 The flat phase of a crystalline membrane (2) In addition, bending the membrane costs energy. r r r This can be expressed using the curvature tensor: Cij n x ix j For small deformations, the energetics is a sum of bending energy and elastic energy: 2 r 1 F u,h 2 r 2 2 2 d D x T rC G T rC T rC T ru 2T ru 2 2 Note: Linear terms are not included as the membrane ij F ; M ij F uij Cij is assumed to be free. Cross terms break the symmetry between the two sides of the membrane. The Gaussian curvature energy is invariant under small deformations for D=2, thus not important for the structure. August 2009 Doron Gazit - on crystalline membranes and graphene 7 The flat phase of a crystalline membrane (3) Thus we will focus on the following free energy: 2 2 r 1 2 Dr 2 F u,h d x[ h uii 2uij ] 2 For a bulk 3-d material, small perturbations are given in terms of the elastic part only, characterized by 3d, and 3d (or equivalently the Young modulus K3d and the Poisson ratio n3d). For a physical membrane with a lateral size h: 4 h 3K3d K0 hK3d n n 3d 2 121 n 2 The height of the membrane: h 121 n 2 K0 Thus, the height of graphene is about 1 Angstrom, i.e. less than the lattice size! Graphene, in the continuum limit is truly the ultimate membrane! August 2009 Doron Gazit - on crystalline membranes and graphene 8 The flat phase of a crystalline membrane (4) The Green’s function of the h field: hqhq IdC dC With the effective bending rigidity R q kB T R qq4 r d q D kB T 2z ~ L D 4 q q 2 R q 0 The bending rigidity R q ~ q , 4 D 2z . z – the roughness exponent: h 2 L1 q 0 In principal, one can define an elasticity exponent: , ~ q u From Ward identitiesu 4 D 2 , which means that h fluctuations 0 z 1 are rdivergent. But normal-normal fluctuations… 2r 1 r L r h dq q2 2 2 2z 2 n h n qh ~ ~ L 0 2 4 r 2 L 2 R qq 1 h Thus, an asymptotically flat phase. Experimentally, August 2009 Doron Gazit - on crystalline membranes and graphene 9 The flat phase of a crystalline membrane (5) Thus we will focus on the following free energy: r 1 F u,h 2 2 2 r 2 2 d x[ h uii 2uij ] D The in-plane phonon fields enter quadratically – thus can be integrated out: The effective four-leg interaction: August 2009 Doron Gazit - on crystalline membranes and graphene 10 The flat phase of a crystalline membrane (5) What is the relevant perturbation scheme? Expand in the number of bubbles, implying that the elastic interaction is small compared to the bending energy. By power counting this is true only for q kB TY applicable for the structure issue. 2 , thus not A different scheme hides in the fact: Every interaction contributes a factor 1/dC=1/(d-D). Every h field propagator adds dC=d-D. Thus, a relevant perturbation scheme is in powers of 1/dC. Topologically, diagrams contributing to order n in perturbation theory fulfill the condition: n=NR-Lh, (NR- number of interactions, Lh-number of loops). August 2009 Doron Gazit - on crystalline membranes and graphene 11 Physical Crystalline Membranes The interaction is separable: Feynman diagrams: h propagator Interaction propagator Interaction vertex Dyson equations: August 2009 Doron Gazit - on crystalline membranes and graphene 12 Self Consistent Screening Approximation SCSA is an extension of a consistent perturbative expansion. One replaces each propagator with the dressed propagator. Cutting the expansion at a specific order. SCSA for a first order expansion: Le Doussal and Radzihovsky, PRL 69, 1209 (1992). SCSA for a second order expansion: DG, arxiv: 0907.3718 (2009). August 2009 Doron Gazit - on crystalline membranes and graphene 13 SCSA – general results The scaling relationu 4 D 2 holds to all orders. At any order the SCSA equations: and y are polynomials in z0y0-2. For D>2, the long-wavelength Poisson ratio is -1/3. The first order SCSA coincides with: First order expansion in 4-D. d=D Large dC expansion. August 2009 Doron Gazit - on crystalline membranes and graphene 14 Second order SCSA for physical crystalline membranes Second order - a naïve 2-loop expansion or a 1/dC2 expansion? No solution for a naïve 2-loop expansion! Thus, the 1/dC character of the expansion is essential. However, even though dC=1, the results are very close: z changes only by 2%. One should try and check if this still coincides with the e2 expansion, as this will give an indication of the error. August 2009 Doron Gazit - on crystalline membranes and graphene 15 Method u z References 1st order SCSA 0.821… 0.358… 0.590… PRL, 69, 1209 (1992) 2nd order SCSA 0.789… 0.421… 0.605… This work e Expansion [1] 0.96 0.08 0.52 PRL 60, 2634 (1990) Large dC [2] 2/3 2/3 2/3 EPL 5, 709 (1988) Exact RG [3] 0.849 PRE 79, 040101 (2009) Simulations [4] 0.75-0.85 0.50(1) 0.64(2) PRE 48, R651 (1993), J. Phys 6, 3521 (1996), arxiv:0903.3847 Experiment [5] - - 0.65(10) Science, 259, 952 (1993) All methods are consistent with each other. None show any unusual finite q behavior. One of the numerical simulations is for graphene – thus graphene would have been a great experimental device. The other universal parameter implies: 1 kB TK R q 3.5731 q 0 q R2 q lim Only 6% away from the 1st order SCSA prediction. August 2009 Doron Gazit - on crystalline membranes and graphene 16 Structure of Graphene Meyer et al. [Nature, 446, 60 (2007)], have characterized suspended graphene sheets: Showed stability. No defects were found even at strain > 10%. Used TEM diffraction patterns to determine that there is a characteristic ripple on the surface of 100-250 Å. Different groups have isolated suspended graphene far above SiO2 substrate. Guinea, Horovitz, and Le Doussal [Sol. St. Com. 149, 1140 (2009)], suggested a mechanism that results in ripples due to stress in the production process. It is commonly believed that these are real inherent ripples. External effects were still not ruled out, though. I suggest an inherent mechanism for ripple creation due to charge inhomogeneities, energetically favored. August 2009 Doron Gazit - on crystalline membranes and graphene 17 Atomistic simulations of graphene Los et al. [arxiv: 0903.3847] have used a carbon-carbon potential to calculate normal-normal correlation. They found a behavior consistent with the theory of physical membranes. No sign of ripples. August 2009 Doron Gazit - on crystalline membranes and graphene 18 How would ripples look in the normalnormal correlation function? Los, Fasolino, Katsnelson, Nature Materials 6, 858 [2007]. August 2009 Doron Gazit - on crystalline membranes and graphene 19 Other experiments Gerringer et al., PRL 102, 076102 (2009). The experiments in J. Folk’s group are not yet published, but show L~8nm. August 2009 Doron Gazit - on crystalline membranes and graphene 20 So… As graphene is a real membrane, this difference is rather disturbing. I suggest that the origin of this different is the additional degree of freedom: the -electrons. Considering the fact that these electrons are responsible for the specific size of the lattice (determine the resonant bond), such an effect is reasonably large. How do ripples affect the electronic structure? August 2009 Doron Gazit - on crystalline membranes and graphene 21 Dirac picture An effective theory around the Dirac points. Low momenta excitations are possible only around the Dirac point, thus: ai, e bi, e r r iK R n r r iK R n Defining: Hamiltonian: The effective H ihv f August 2009 r K i, r K i, a e b e r r iK ' R n r r iK ' R n r K' i, r K' i, a b r a K r ˆ K i,r i, K bi, r K r r K' r K† r r K'† r r * ˆ ˆ ˆ ˆ r dxdy r r r Semenoff Doron Gazit - on crystalline membranes and graphene 1984 22 An effective Dirac Action In Euclidean space: 2 S h 0 d r d xy y 2 1 1 kB T K y K' 0 i , 2 ; ; i 1 n n 3 3 gn diag1,v f ,v f August 2009 Doron Gazit - on crystalline membranes and graphene 23 Effects of corrugation on the electronic structure The Dirac picture is an effective picture, due to the tight- binding Hamiltonian. Thus, though attractive conceptually, it is not really a massless fermion in curved space (however...). Possible effects: Deformation energy. Pseudo-magnetic gauge fields. Electric gauge fields August 2009 Doron Gazit - on crystalline membranes and graphene 24 Deformation Energy In the presence of corrugations, the surface area changes: S ~ a uii 2 This changes the ion density, and thus the electron density. As a result, the electron’s chemical potential is locally changed an effective induced electric field: V r Duii ; D 20 30eV August 2009 Suzuura, Ando PRB, 65, 235412 (2002). Doron Gazit - on crystalline membranes and graphene 25 Effective Gauge Field The hopping integral changes due to the change in angles between normals and distances in the lattice. Due to corrugation and ripples: 0 u uij ij August 2009 Doron Gazit - on crystalline membranes and graphene 26 Thus: H a†kbk' i ab e r r r r r k k ' R i iaa k ' r r k ,k ' h.c i bk†bk' a†k ak' i aa e r r r r r k k ' R i iab k ' i Or… 2 S h 0 d 1 Deformation Energy + NNN effects With August 2009 r r r r r r 0 d xy 0 i r hv f i 5 Ar y 2 r r g2 2uxy Ar v f uxx uyy g2 ~ 1 3 eV NN effect, Keeping T invariance r 3a 2 2 2 r Duii g3 h 4 g3 ~ 10 eV Doron Gazit - on crystalline membranes and graphene 27 Intrinsic ripples in graphene The structure of graphene is determined by a mutual minimization of the lattice free energy and the electronic one. In the absence of electron inhomogeneities – this results in an elastic free energy. However, allowing inhomogeneities, keeping only the deformation energy: August 2009 Doron Gazit - on crystalline membranes and graphene 28 Intrinsic ripples in graphene We estimate the electron-electron interaction by: The effective screening is big, Kotov et al., PRB 78, 035119 (2008) showed that: E=3-4. But, they do it perturbatively and the series does not seem to converge, as the fine structure is of order 1. August 2009 Doron Gazit - on crystalline membranes and graphene 29 Intrinsic ripples in graphene Integrating out the in plane phonon fields and charge fluctuations: For 3D materials, negative Young modulus means instability. For 2D materials, this means inherent competition between bending and stretching. August 2009 Doron Gazit - on crystalline membranes and graphene 30 Intrinsic ripples in graphene The Dyson equations in this case: Estimating Sand in first order: Searching for maximum in the normal-normal correlation function. August 2009 Doron Gazit - on crystalline membranes and graphene 31 Size of ripples August 2009 Doron Gazit - on crystalline membranes and graphene 32 What is the meaning of all this? The best way to get some insight, is to rewrite the effective free-energy as: The Gaussian curvature: Let us assume that impurities contribute to the charge density as well: August 2009 Doron Gazit - on crystalline membranes and graphene 33 The meaning Naively, without a bending energy term, one expects a glass- phase, whose ground states are solutions to the equation: This is a basic difference from what was done previously, since they looked for a correlation in the mean curvature. August 2009 Deshpande al.graphene PRB 79, Doron Gazit - on crystalline membranesetand 205411 (2009) 34 Electron Hole puddles Ldisorder~10-30 nm Martin et al., Nature Physics, 4, 144 (2008). August 2009 Doron Gazit - on crystalline membranes and graphene 35 Results The electron-phonon coupling originating from the deformation energy competes with the electron-electron interaction. Formation of ripples correlated with electron-hole puddles is favored, both reproduce the experimental length scale. Indeed, Graphane is found to have less corrugation, consistent with no corrugation. Elias et al., Science, 323, 610 (2009). Additional work is required to specify the form of graphene in the presence of few impurities. The dynamics of the electrons was neglected. What about finite chemical potential? August 2009 Doron Gazit - on crystalline membranes and graphene 36 Doped graphene We neglect electron-electron interaction! 2 S h 0 d 1 r r r r r 0 d xy 0 i r hv f i 5 Ar y 2 Since vf>>vph we integrate out the electronic degrees of freedom. August 2009 Doron Gazit - on crystalline membranes and graphene 37 Feynman Diagrams for -Electrons i k Fermion propagator Vector pot./Fermion vertex r i 5 Electro-chem. Pot/Fermion vertex August 2009 i 0 Doron Gazit - on crystalline membranes and graphene 38 Integrating out the -electrons The resulting Lagrangian is pure gauge. 1 1 V 2 A L Aiij A j V 2 2 L S r d xL Feff 2 The structure is frozen to a very good approximation, thus polarizations can be calculated using zero frequency. August 2009 Doron Gazit - on crystalline membranes and graphene 39 “Structure” Polarization operators r k q r 1 d k k 0 0 V q0 0;q T r 2 2 2 1 f 2 k k q r 2 2 k q r 1 d k k i j ijA q0 0;q T r 5 5 2 2 2 1 f 2 k k q r 2 f n 1 k f ,k 2 2 August 2009 2 Doron Gazit - on crystalline membranes and graphene 40 Electro-chemical “Structure” Polarization Operator P hv f q hv f q 0 dxln2cosh 2 2 4 V 1 4 ln2 1 x 1 x 1 4 T 1 1 hv f q 15 1 300ÞK 100A Exact P 4 ln2 qiq j r hv f q q ij 2 4 q A ij r V q q 4hv f 4 August 2009 Doron Gazit - on crystalline membranes and graphene 41 Electrons contribution to the free energy F r dq 2 2 2 q 4hv f r 2 2 ˆ V hv A q f r r g2 2uxy Ar v f uxx uyy V Duii August 2009 Doron Gazit - on crystalline membranes and graphene 42 Elastic Free-Energy r 1 F u,h 2 r hv f q r 2 2 dq 2 2 4 q h quii 2quij A qˆ 2 4 2 2 D2q q 4hv f g22q q 8hv f The effect on the shear modulus is negligible. August 2009 Doron Gazit - on crystalline membranes and graphene 43 The effect of an external electro-chemical potential r 1 F u,h 2 r d 2q 2 2 r q h quii 2uij 2Duii q V q; Vq 2 4 F u,h D 2 2 r dq 2 2 2 V q; uii q Vq r D q 0; uii qr 0 V V 4Dln2 hv f 2 V r r d xuii x 2 r r d xuii x 2 Chemical potential leads to stress! August 2009 4Dln2 hv f 2 V Doron Gazit - on crystalline membranes and graphene 44 Buckling term in the elastic free energy Let’s assume a tensionless membrane in a negative chemical potential (hole doping). The stress is negative Buckling of graphene, as it has zero thickness! The merit of the buckling wave length: F h 1 2 r 2 r 2 d 2 x h h * F 0 q4 h q2 h 0 q h 0 q* 0 ! hv f 2 T 1/ 2 D 1/ 2 V 1/ 2 2 144A 300ÞK 20eV 100meV 4Dln2 V August 2009 Doron Gazit - on crystalline membranes and graphene 45 Physical dopings Chemical adsorption: T 1/ 2 D 1/ 2 ad 1/ 2 nad 1/ 2 150A 300ÞK 30eV 200meV 20% Gate voltage: In any case ~F1/2. August 2009 Doron Gazit - on crystalline membranes and graphene 46 Conclusions Graphene is an example of a new class of materials: electronic crystalline membranes, in which a strong interplay exists between the structure and the free electrons in the membrane. The -electrons induce: Ripples due to competition between electron-electron interaction and electron-phonon interaction. Additional stress in the presence of a chemical potential, which can lead to buckling in the case of hole doping. Outlook: Additional work is needed: incorporate electron-electron interactions without neglecting dynamics, … Phase space in the presence of impurities. Effect on transport. August 2009 Doron Gazit - on crystalline membranes and graphene 47